Patent Publication Number: US-7899691-B1

Title: Econometric engine

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is related to co-pending and concurrently filed application Ser. No. 09/742,472, filed Dec. 20, 2000, entitled “Imputed Variable Generator”, by Suzanne Valentine, Krishna Venkatraman, Phil Delurgio, Michael Neal, and Hau Lee, which is incorporated by reference herein for all purposes. 
     This application is related to co-pending and concurrently filed application Ser. No. 09/741,958, filed Dec. 20, 2000, entitled “Price Optimization System”, by, Michael Neal, Krishna Venkatraman, Suzanne Valentine, Phil Delurgio, and Hau Lee which is incorporated by reference herein for all purposes. 
     This application is related to co-pending and concurrently filed application Ser. No. 09/741,957, filed Dec. 20, 2000, entitled “Financial Model Engine”, by Phil Delurgio, Suzanne Valentine, Michael Neal, Krishna Venkatraman, and Hau Lee, which is incorporated by reference herein for all purposes. 
     This application is related to co-pending and concurrently filed application Ser. No. 09/741,959, filed Dec. 20, 2000, entitled “Econometric Optimization Engine”, by Krishna Venkatraman, Phil Delurgio, Suzanne Valentine, Michael Neal, and Hau Lee, which is incorporated by reference herein for all purposes. 
     BACKGROUND OF THE INVENTION 
     The present invention relates to providing a price optimization system. 
     In businesses, prices of various products must be set. Such prices may be set with the goal of maximizing profit or demand or for a variety of other objectives. Profit is the difference between total revenue and costs. Total sales revenue is a function of demand and price, where demand is a function of price. Demand may also depend on the day of the week, the time of the year, the price of related products, location of a store, and various other factors. As a result, the function for forecasting demand may be very complex. Costs may be fixed or variable and may be dependent on demand. As a result, the function for forecasting costs may be very complex. For a chain of stores with tens of thousands of different products, forecasting costs and determining a function for forecasting demand are difficult. The enormous amounts of data that must be processed for such determinations are too cumbersome even when done by computer. Further, the methodologies used to forecast demand and cost require the utilization of non-obvious, highly sophisticated statistical processes. 
     It is desirable to provide an efficient process and methodology for determining the prices of individual products such that profit (or whatever alternative objective) is optimized. 
     SUMMARY OF THE INVENTION 
     To achieve the foregoing and other objects and in accordance with the purpose of the present invention for creating a sales model for a plurality of products, generally a plurality demand groups are created, where each demand group is a set of at least one product and wherein at least one of the demand groups is a set of at least two products. A sales model is created for each demand group. A market share model is created for each product in each demand group. 
     In addition, the present invention provides an econometric engine for modeling sales as a function of price. An imputed variable generator and a coefficient estimator coupled to the imputed variable generator form the econometric engine, where imputed variables generated by the variable generator are used by the coefficient estimator to create a sales model. 
     These and other features of the present invention will be described in more detail below in the detailed description of the invention and in to conjunction with the following figures. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The present invention is illustrated by way of example, and not by way of limitation, in the figures of the accompanying drawings and in which like reference numerals refer to similar elements and in which: 
         FIG. 1  is a high level schematic view of an embodiment of the invention. 
         FIG. 2  is high level flow chart of the invention. 
         FIG. 3  is a more detailed schematic view of the econometric engine. 
         FIG. 4  is a more detailed schematic view of the optimization engine and support tool. 
         FIG. 5  is a block diagram to illustrate some of the transaction costs that occur in retail businesses of a chain of stores. 
         FIG. 6  is a flow chart of the preferred embodiment of the invention for providing an initial feasible solution. 
         FIGS. 7A and 7B  illustrate a computer system, which forms part of a network and is suitable for implementing embodiments of the present invention. 
         FIG. 8  is a schematic illustration of an embodiment of the invention that functions over a network. 
         FIG. 9A  is a graph of original profit from actual sales of the store using actual prices and optimal profit from optimized sales resulting from the calculated optimized prices bounded by its probability. 
         FIG. 9B  is a graph of percentage increase in profit and the probability of obtaining at least that percentage increase in profit. 
         FIG. 10  is a flow chart depicting a process flow by which raw econometric data can be input, subject to “cleansing”, and used to create an initial dataset which can then be used to generate imputed econometric variables in accordance with a preferred embodiment of the present invention. 
         FIG. 11  is a flow chart depicting a process flow depicting a process by which partially cleansed econometric data is subject to further error detection and correction in accordance with a preferred embodiment of the present invention. 
         FIG. 12A  is a flow chart depicting a process flow by which an imputed base price variable can be generated in accordance with one embodiment of the present invention. 
         FIG. 12B  is a price time diagram which illustrates an aspect of generating an imputed base price variable in accordance with one embodiment of the present invention. 
         FIG. 12C  is a price time diagram which illustrates an aspect of generating an imputed base price step function in accordance with one embodiment of the present invention. 
         FIG. 12D  is a diagram which illustrates an error correction aspect of base price imputation in accordance with one embodiment of the present invention 
         FIG. 13  is a flow chart depicting a process flow by which an imputed relative price variable can be generated in accordance with one embodiment of the present invention. 
         FIG. 14A  is a flow chart depicting a process flow by which an imputed base unit sales volume variable can be generated in accordance with one embodiment of the present invention. 
         FIG. 14B  is a diagram used to illustrate the comparative effects of sales volume increase and price discounts. 
         FIG. 15A  is a flow chart depicting a process flow by which supplementary error detection and correction in accordance with an embodiment of the present invention. 
         FIG. 15B  is a diagram used to illustrate the comparative effects of sales volume increase and price discounts. 
         FIG. 16  is a flow chart depicting a process flow by which an imputed stockpiling variable can be generated in accordance with an embodiment of the present invention. 
         FIG. 17  is a flow chart depicting a process flow by which an imputed day-of-week variable can be generated in accordance with an embodiment of the present invention. 
         FIG. 18  is a flow chart depicting a process flow by which an imputed seasonality variable can be generated in accordance with an embodiment of the present invention. 
         FIG. 19A  is a flow chart depicting a process flow by which an imputed promotional effects variable can be generated in accordance with an embodiment of the present invention. 
         FIG. 19B  is a diagram depicting the modeling effects of a promotional effects variable in accordance with an embodiment of the present invention. 
         FIG. 20  is a flow chart depicting a process flow by which an imputed cross-elasticity variable can be generated in accordance with a preferred embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The present invention will now be described in detail with reference to a few preferred embodiments thereof as illustrated in the accompanying drawings. In the following description, numerous specific details are set forth in order to provide a thorough understanding of the present invention. It will be apparent, however, to one skilled in the art, that the present invention may be practiced without some or all of these specific details. In other instances, well known process steps and/or structures have not been described in detail in order to not unnecessarily obscure the present invention. 
     I. OVERALL SYSTEM 
     To facilitate discussion,  FIG. 1  is a schematic view of a price optimizing system  100 . The price optimizing system  100  comprises an econometric engine  104 , a financial model engine  108 , an optimization engine  112 , and a support tool  116 . The econometric engine  104  is connected to the optimization engine  112 , so that the output of the econometric engine  104  is an input of the optimization engine  112 . The financial model engine  108  is connected to the optimization engine  112 , so that the output of the financial model engine  108  is an input of the optimization engine  112 . The optimization engine  112  is connected to the support tool  116  so that output of the optimization engine  112  is provided as input to the support tool  116  and output from the support tool  116  may be provided as input to the optimization engine  112 . The econometric engine  104  may also exchange data with the financial model engine  108 . 
       FIG. 2  is a high level flow chart of a process that utilizes the price optimizing system  100 . The operation of the optimizing system  100  will be discussed in general here and in more detail further below. Data  120  is provided from the stores  124  to the econometric engine  104  (step  204 ). Generally, the data  120  provided to the econometric engine  104  may be point-of-sale information, product information, and store information. The econometric engine  104  processes the data  120  to provide demand coefficients  128  (step  208 ) for a set of algebraic equations that may be used to estimate demand (volume sold) given certain marketing conditions (i.e. a particular store in the chain), including a price point. The demand coefficients  128  are provided to the optimization engine  112 . Additional processed data from the econometric engine  104  may also be provided to the optimization engine  112 . The financial model engine  108  may receive data  132  from the stores  124  (step  216 ) and processed data from the econometric engine  104 . The data  132  received from the stores is generally cost related data, such as average store labor rates, average distribution center labor rates, cost of capital, the average time it takes a cashier to scan an item (or unit) of product, how long it takes to stock a received unit of product and fixed cost data. The financial model engine  108  may process the data to provide a variable cost and fixed cost for each unit of product in a store. The processing by the econometric engine  104  and the processing by the financial model engine  108  may be done in parallel. Cost data  136  is provided from the financial model engine  108  to the optimization engine  112  (step  224 ). The optimization engine  112  utilizes the demand coefficients  128  to create a demand equation. The optimization engine is able to forecast demand and cost for a set of prices to calculate net profit. The stores  124  may use the support tool  116  to provide optimization rules to the optimization engine  112  (step  228 ). The optimization engine  112  may use the demand equation, the variable and fixed costs, and the rules to compute an optimal set of prices that meet the rules (step  232 ). For example, if a rule specifies the maximization of profit, the optimization engine would find a set of prices that cause the largest difference between the total sales and the total cost of all products being measured. If a rule providing a promotion of one of the products by specifying a discounted price is provided, the optimization engine may provide a set of prices that allow for the promotion of the one product and the maximization of profit under that condition. In the specification and claims the phrases “optimal set of prices” or “preferred set of prices” are defined as a set of computed prices for a set of products where the prices meet all of the rules. The rules normally include an optimization, such as optimizing profit or optimizing volume of sales of a product and constraints such as a limit in the variation of prices. The optimal (or preferred) set of prices is defined as prices that define a local optimum of an econometric model which lies within constraints specified by the rules When profit is maximized, it may be maximized for a sum of all measured products. Such a maximization, may not maximize profit for each individual product, but may instead have an ultimate objective of maximizing total profit. The optimal (preferred) set of prices may be sent from the optimization engine  112  to the support tool  116  so that the stores  124  may use the user interface of the support tool  116  to obtain the optimal set of prices. Other methods may be used to provide the optimal set of prices to the stores  124 . The price of the products in the stores  124  are set to the optimal set of prices (step  236 ), so that a maximization of profit or another objective is achieved. 
     Each component of the price optimizing system  100  will be discussed separately in more detail below. 
     II. ECONOMETRIC ENGINE 
       FIG. 3  is a more detailed view of the econometric engine  104 . The econometric engine comprises an imputed variable generator  304  and a coefficient estimator  308 . The data  120  from the stores  124  is provided to the imputed variable generator  304 . The data  120  may be raw data generated from cash register data, which may be generated by scanners used at the cash registers. 
     A. Imputed Variable Generator 
     The present invention provides methods, media, and systems for generating a plurality of imputed econometric variables. Such variables are useful in that they aid businesses in determining the effectiveness of a variety of sales strategies. In particular, such variables can be used to gauge the effects of various pricing or sales volume strategies. 
       FIG. 10  illustrates a flowchart  1000  which describes steps of a method embodiment for data cleansing imputed econometric variable generation in accordance with the principles of the present invention. The process, generally described in  FIG. 10 , begins by initial dataset creation and data cleaning (Steps  1011 - 1031 ). This data set information is then used to generate imputed econometric variables (Step  1033 ) which can be output to and for other applications (Step  1035 ). 
     1. Initial Dataset Creation And Cleaning 
     The process of dataset creation and cleaning (that is to say the process of identifying incompatible data records and resolving the data incompatibility, also referred to herein as “error detection and correction”) begins by inputting raw econometric data (Step  1011 ). The raw econometric data is then subject to formatting and classifying by UPC designation (Step  1013 ). After formatting, the data is subject an initial error detection and correction step (Step  1015 ). Once the econometric data has been corrected, the store information comprising part of the raw econometric data is used in defining a store data set hierarchy (Step  1017 ). This is followed by a second error detecting and correcting step (Step  1019 ). This is followed by defining a group of products which will comprise a demand group (i.e., a group of highly substitutable products) and be used for generating attribute information (Step  1021 ). Based on the defined demand group, the attribute information is updated (Step  1023 ). The data is equivalized and the demand group is further classified in accordance with size parameters (Step  1025 ). The demand group information is subjected to a third error detection and correction step (Step  1027 ). The demand group information is then manipulated to facilitate decreased process time (Step  1029 ). The data is then subjected to a fourth error detection and correction step (Step  1031 ), which generates an initial cleansed dataset. Using this initial cleansed dataset, imputed econometric variables are generated (Step  1033 ). Optionally, these imputed econometric variables may be output to other systems for further processing and analysis (Step  1035 ). 
     The process begins by inputting raw econometric data (Step  1011 ). The raw econometric data is provided by a client. The raw econometric data includes a variety of product information. The raw econometric data must specify the store from which the data is collected, the time period over which the data is collected and include a UPC (Universal Product Code) for the product, and provide a UPC description of the product. Also, the raw econometric data must include product cost (e.g., the wholesale cost to the store), number of units sold, and either unit revenue or unit price. Ordinarily, the UPC description also identifies the product brand, UOM (Unit of Measure), and product size. Such information can be very detailed or less so. For example, brand can simply be Coca-Cola®, or more detailed e.g., Cherry Coca-Cola®. A UOM is, for example, ounces (oz.), pound (lb.), liter (ltr), or count (CT). Size reflects the number of UOM&#39;s e.g., eight (8) oz or two (2) ltr. Also, the general category of product or department identification is input. A category is defined as a set of substitutable or complementary products, for example, “Italian Foods”. Such categorization can be proscribed by the client, or defined by generally accepted product categories. Additionally, such categorization can be accomplished using look-up tables or computer generated product categories. 
     Also, a more complete product descriptor is generated using the product information described above and, for example, a UPC description of the product and/or a product description found in some other look-up table (Step  1013 ). This information is incorporated into a product format. This product format provides a more complete picture of the product, but this information is stored in a separate database which is not necessarily processed using the invention. This information provides a detailed description of the product which can be called up as needed. 
     The data is then subjected to a first error detection and correction process (Step  1015 ). Typically, this step includes the removal of all duplicate records and the removal of all records having no match in the client supplied data (typically scanner data). An example of records having no match are records that appear for products that the client does not carry or stock in its stores. These records are detected and deleted. 
     Data subsets concerning store hierarchy are defined (Step  1017 ). This means stores are identified and categorized into various useful subsets. Typical subsets include (among other categorizations) stores segregated by, for example, zip codes, cities, states, specific geographical regions, rural environs, urban environs, associations with other stores (e.g., is this store part of a mall) or categorized by specific stores. A wide variety of other subsets may also be used. These subsets can be used to provide information concerning, among other things, regional or location specific economic effects. 
     The data is then subjected to a second error detection and correction process (Step  1019 ). This step cleans out certain obviously defective records. Examples include, but are not limited to, records displaying negative prices, negative sales volume, or negative cost. Records exhibiting unusual price information are also removed. Such unusual prices can be detected using cross store price comparisons (between similarly situated stores), for example, an average price for a product in the stores of a particular geographic region can be determined by averaging the prices for all such products sold in the subject stores. The standard deviation can also be calculated. Prices that lie at greater than, for example, two (2) standard deviations from the mean price will be treated as erroneous and such records will be deleted. These tools can be applied to a variety of product parameters (e.g., price, cost, sales volume). 
     This is followed by defining groups of products and their attributes and exporting this information to a supplementary file (e.g., a text file) (Step  1021 ). This product information can then be output into a separate process which can be used to define demand groups or product attributes. For example, this supplemental file can be input into a spreadsheet program (e.g., Excel®) which can use the product information to define “demand groups” (i.e. groups of highly substitutable products). Also, further product attribute information can be acquired and added to the supplementary file. Such attributes can comprise, for example, branding information, manufacturer, size, flavor or form (e.g., cherry soda) just to name a few. Such information can be gleaned from multiple sources e.g., UPC product catalogues, the client, product look-up tables, or other sources. The advantage of such supplementary files is that they maintain complete product information (including information not required by the processes of the present invention) which can be accessed when needed. In addition, updated demand group and attribute information can then be input as received (Step  1023 ). By maintaining a supplementary file containing large amounts of data, a more streamlined (abbreviated) dataset may be used in processing. This effectively speeds up processing time by deleting non-critical information from the dataset. 
     The data is further processed by defining an “equivalizing factor” for the products of each demand group in accordance with size and UOM parameters (Step  1025 ). This equivalizing factor can be provided by the client or imputed. An example of determining an imputed equivalizing factor follows. Product size and UOM information are obtained, for example, from the product description information. Typical examples of such size and UOM information is, 20 oz. (ounce), 6 CT (count), or 1 ltr (liter). A further advantageous aspect of the present invention is that, even if such size or UOM information is incomplete or not provided, it can also be imputed. An equivalizing factor can be imputed by using, for example, the median size for each UOM. Alternatively, some commonly used arbitrary value can be assigned. Once this information is gathered, all product prices and volume can be “equivalized”. In one example, a demand group (a group of highly substitutable products) is chosen having, for example, “soft drinks” as its subject category. And by further example, the soft drink product comes in 8, 12, 16, 24, 32, and 64 ounce sizes. The median size (or for that matter, any arbitrarily determined size) can then be used as the base size to which all other sizes are to be equivalized. For example, using the 8, 12, 16, 24, 32, and 64-ounce sizes discussed above, an arbitrary base size can be determined as, for example, 24 ounces. Then the 24-ounce size is determined as the equivalizing factor. Some of the uses of the equivalizing factors are detailed in the discussions below. Chiefly, the purpose of determining an equivalizing factor is to facilitate comparisons between different size products in a demand group. For example, if 16 is determined as the equivalizing factor for the above group of soft drinks, then an 8 oz. soft drink is equivalized to one half of a 16 oz. unit. In a related vein, a 32 oz. soft drink is equivalized to two (2) 16 oz. units. 
     Additionally, size information can be used to define further product attributes. For example, if the size is in the bottom tertile of sizes for that product, it will be classified as “Small” size. Correspondingly, if the size is in the middle tertile of sizes for that product, it will be classified as “Medium” size, and if the size is in the top tertile of sizes for that product, it will be classified as “Large” size. Such categorization can define product attributes such as small (8- and 12-ounce sizes), medium (16- and 24-ounce sizes), and large (32- and 64-ounce sizes) sizes. 
     The data is then subjected to a third error detection and correction process, which detects the effects of closed stores and certain other erroneous records (Step  1027 ). Keeping in mind that one advantage of the present invention is that very little client input is required to achieve accurate results, the inventors contemplate error correction without further input (or very little input) from the client. In accord with the principles of the invention, stores that demonstrate no product movement (product sales equal to zero) over a predetermined time period are treated as closed. Those stores and their records are dropped from the process. In a preferred embodiment, the predetermined time period is three (3) months. 
     With continued reference to  FIG. 10 , Step  1027 , the third error detection and correction also includes analysis tools for detecting the presence of erroneous duplicate records. The data is examined, in particular checking records for, date, product type, store at which the product was sold (or just “store”), price, units (which refers variously to units sold or unit sales volume), and causal variables. Causal variables are those factors which influence sales volume (a variable which can cause an increase in product sales e.g., coupons, sales promotion ads, sale prices, sale price on some complementary product, enhanced sales displays, more advantageous sales location within a store, etc.). Analysis is performed to remove the discrepant records such that only one of the records is kept as part of the analyzed data and that causal information for a particular time period is recorded. 
     Using the following illustrative table: 
                                                     Record                       Causal       Number   Date   Store   Product   Units   Price   Variable                                                            1   12/5   Y   D   10   1.99   1       2   12/5   Y   D   10   1.99   1       3   12/12   Y   D   10   1.99   1       4   12/12   Y   D   15   1.89   2       5   12/19   Y   D   12   1.99   1       6   12/26   Y   D   9   1.99   1                    
For example, using record #1, the date of record is December 5, the store is store “Y”, the product is product type “D”, units sold for that date are 10 at a price of 1.99. The causal variable is usually abbreviated with a code symbol (e.g., numbers). Here, “1” is a symbol for no causal variable, i.e., normal sales conditions. Whereas, examining, for example, record #3 includes a causal variable (code symbol 2) which, for example, will represent an advertisement concerning product “D”.
 
     Discrepant records are identified and corrected. For example, if two records have the same exact values (such as record #1 and record #2), it is assumed that one such record is an erroneous duplicate and only one record is kept as part of the analyzed dataset, for example, only record #1 is retained. 
     If two records with the same date, product id, and store id have multiple records with different causals, they are combined into a single record, with the two prices maintained in separate dataset variables, units summed across the two records, and the causal variables representing something other than a normal state being represented by new dataset variables. 
     The following table shows an updated version of the above table. Record 2 was deleted because it is identical to Record 1. Records 3 and 4 were combined into a single record (i.e., combined into a single Record 3) with new causal variables defined for Advertisement and Advertisement Price. Records 5 and 6 did not change because there was no duplicate information. 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                 Record 
                   
                   
                 Pro- 
                   
                 Regular 
                 Advertise- 
                 Advertise- 
               
               
                 Number 
                 Date 
                 Store 
                 duct 
                 Units 
                 Price 
                 ment 
                 ment Price 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
            
               
                 1 
                 12/5 
                 Y 
                 D 
                 25 
                 1.99 
                 No 
                 — 
               
               
                 3 
                 12/12  
                 Y 
                 D 
                 25 
                 1.99 
                 Yes 
                 1.89 
               
               
                 5 
                 12/19  
                 Y 
                 D 
                 12 
                 1.99 
                 No 
                 — 
               
               
                 6 
                 12/26  
                 Y 
                 D 
                 9 
                 1.99 
                 No 
                 — 
               
               
                   
               
            
           
         
       
     
     A further correction can be made for records having the same date and causal value but have differing prices or differing number of units sold. First, a data discrepancy must be detected. For example, if a record on a specific date in the same store for the same product and causal state has two different values for units, this is a discrepancy. Correction can be accomplished by, first calculating the average number of units sold over all dates in the modeled time interval. The discrepant records are compared with the average value. The record having the unit value closest to the calculated average units is kept and the other is discarded. The same general process can be followed for records having discrepancies as to price (i.e., the record having the price closest to average price is kept). If both price and units are determined to have a discrepancy, the record having the price and unit values closest to the average price and average units is kept. 
     After all the duplicate records eliminated, the data is reconstructed. The data can be reviewed again to insure all duplicates are removed. Optionally, an output file including all discrepancies can be produced. In the event that it becomes necessary, this output file can be used as a follow-up record for consulting with the client to confirm the accuracy of the error detection and correction process. 
     Additionally, reduced processing times may be achieved by reformatting the data (Step  1029 ). For example, groups of related low sales volume products (frequently high priced items) can optionally be aggregated as a single product and processed together. Additionally, the data may be split into conveniently sized data subsets defined by a store or groups of stores which are then processed together to shorten the processing times. For example, all stores in the state of California can be processed together, then all the stores in Texas, etc. 
     Next the process includes determining the nature of missing data records in a fourth error detection and correction step (Step  1031 ). The missing data records are analyzed again before finally outputting a cleansed initial dataset. For example, data collected over a modeled time interval is analyzed by introducing the data into a data grid divided into a set of time periods. The time periods can be preset, computer determined, or user defined. The time periods can include, but are not limited to, months, weeks, days, or hours. One preferred embodiment uses time periods of one week. The data grid so constructed is analyzed. For the time periods (e.g., weeks) having no records a determination must be made. Is the record missing because: 
     a. there were no sales that product during that week (time period); 
     b. the product was sold out and no stock was present in the store during that time period (this situation is also referred to herein as a “stock-out’); 
     c. the absence of data is due to a processing error. 
       FIG. 11  depicts a process flow embodiment for determining the nature of missing data records in a fourth error detection and correction step in accordance with the principles of the present invention. The records are compared to a grid of time periods (Step  1101 ). The grid is reviewed for missing records with respect to a particular store and product (Step  1103 ). These missing records are then marked with a placeholder (Step  1105 ). Missing records at the “edges” of the dataset do not significantly affect the dataset and are deleted (Step  1107 ). Records for discontinued products or products recently introduced are dropped for those time periods where the product was not carried in the Store (Step  1109 ). The remaining dataset is processed to determine an average value for units (sold) and a STD for units (Step  1111 ). Each missing record is compared to the average units (Step  1113 ) and based on this comparison, a correction can be made (Step  1115 ). 
     Referring again to  FIG. 11 , in Step  1101 , the data records are matched with a grid of time periods (shown here as weeks, but which can be any chosen time period). The grid can cover an entire modeled time interval, for example, as shown below, the six weeks January 7-February 14 (shown here as weeks 1, 2, 3, 4, 5, and 6). Each product in each store (here store “Z”) is gridded this way. For example: 
     
       
         
           
               
               
               
               
               
               
             
               
                   
               
               
                 Grid 
                 Date 
                 Store 
                 Product 
                 Units 
                 Price 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 1 
                 1/7 
                 Z 
                 Y 
                 10 
                 1.99 
               
               
                 2 
                 1/14 
                 Z 
                 Y 
                 12 
                 2.19 
               
               
                 3 
                   
                   
                   
                   
                   
               
               
                 4 
                 1/28 
                 Z 
                 Y 
                 8 
                 1.99 
               
               
                 5 
                 2/7 
                 Z 
                 Y 
                 10 
                 1.99 
               
               
                 6 
               
               
                   
               
            
           
         
       
     
     Review of the grid (Step  1103 ) shows that records are “missing” for dates January 21 and February 14 (i.e., grid 3 and grid 6). Placeholders are set in the records defined by grid 3 and grid 6 (Step  1105 ). For example, an easily detectable or arbitrarily large value can be put in the price column of the grid, e.g. 999. Alternatively, a simple X can be placed as a placeholder in the price column. In the present example, “X&#39;s” can be placed in the price columns of grid 3 and grid 6. 
     If the first or last grid in the dataset (here grid 1 or grid 6) has few or no observations, those records are deleted from the dataset (Step  1107 ). For purposes of the above analysis, a grid having “few” observations is defined as a grid having 50% fewer observations than is normal for the grids in the middle of the dataset. Here, for example, the record for grid 6 (the last week) is deleted because no records are present for that week. Also, using client-supplied stocking information, products which have been discontinued during the modeled time interval do not have their grids filled out for the discontinued time period (Step  1109 ). Also, products which are introduced during the modeled time interval have their time grid filled out only for those time periods occurring after the product introduction date. Thus, certain data aberrations are removed from the modeled dataset, permitting more accurate modeling. 
     The mean units (sold) and the STD for units are then calculated (Step  1111 ). For example, in dataset depicted above, the mean is 10 units. The missing record is then compared with the mean value (Step  1113 ). Here, a missing record (grid 3) is assigned an initial unit value=0. If the value of zero units lies within one (1) STD of the calculated mean, it is assumed that an actual value of zero units is feasible and that record is treated as if the record is valid (unit volume=0). However, if zero lies at greater than one STD from the mean, it is assumed that the value of zero units is due to a “stock-out”. In such case, it is assumed that had product been present in the store an average number of units would have been sold. Therefore, the zero unit value for that record is replaced by a unit value equal to the calculated mean unit value, thereby correcting for the “stock-out”. In this case, units for grid 3 will be corrected to calculated mean units (i.e., 10). 
     The product histories of the dataset can also be examined. If the subject product was introduced or discontinued as a salable item at the subject store during the modeled time interval, the grid is not filled out (with either zero or average values) for those time periods where the product was not offered for sale in the subject store. In this way missing records do not corrupt the dataset. 
     Further aspects of the fourth error detection and correction include a detection and elimination of outlying price data points (outliers). A satisfactory way of accomplishing this begins with a calculation of the mean price for each product within a given store, as determined over the modeled time interval. Once a mean price and STD are determined, all price data for the modeled time interval is examined. If it is determined that a price record lies within three (3) STD from the mean price it is deemed accurate and not an outlier. However, prices lying outside three (3) STD are treated as outliers. These outliers are assigned adjusted prices. The adjusted prices have the value of the immediately preceding time period (e.g., the previous day&#39;s or week&#39;s price for that product within the store). This adjusted price data is again checked for outliers (using the original mean and STD). Again, outliers are checked against the original mean and STD and again price adjusted if necessary. This usually removes all the remaining outliers. However, the process may optionally continue, iteratively, until there are no further outliers. 
     The net result of execution of the process Steps  1011 - 1031  disclosed hereinabove is the generation of a cleansed initial dataset which can be used for its own purpose or input into other econometric processes. One such process is the generation of imputed econometric variables. 
     2. Generation of Imputed Econometric Variables 
     The foregoing steps ( 1011 - 1031 ) concern cleansing the raw econometric data to create an error detected and error corrected (“cleansed”) initial dataset. The cleansed initial dataset created in the foregoing steps can now be used to generate a variety of useful imputed econometric variables (Step  1033 ). These imputed econometric variables are useful in their own right and may also be output for use in further processing (Step  1035 ). One particularly useful application of the imputed econometric variables is that they can be input into an optimization engine which collects data input from a variety of sources and processes the data to provide very accurate economic modeling information. 
     a. Imputed Base Price 
     One imputed econometric variable that can be determined using the initial dataset created in accordance with the forgoing, is an imputed base price variable (or base price).  FIG. 12A  is a flowchart  1200  outlining one embodiment for determining the imputed base price variable. The process begins by providing the process  1200  with a “cleansed” initial dataset (Step  1201 ), for example, the initial dataset created as described in Steps  1011 - 1031  of  FIG. 10 . The initial dataset is examined over a defined time window (Step  1203 ). Defining a time window (Step  1203 ) includes choosing an amount of time which frames a selected data point allowing one to look forward and backward in time from the selected data point which lies at the midpoint in the time window. This is done for each data point in the dataset, with the time window being defined for each selected data point. The time frame can be user selected or computer selected. The time window includes T time periods and the time period for the selected data point. One preferred set of T time periods is eight (8) weeks. It is contemplated that time windows of greater or lesser size can be selected. Referring to a preferred example, the selected (or current) data point is centered in the time window having T/2 time periods before the selected data point and T/2 time periods after the selected data point. In the present example, the time window includes the four weeks preceding the selected data point and the four weeks after the selected data point. 
     Referring to  FIG. 12B , the selected data point “X” (shown as a single week) is framed by a time period of −T/2 (shown here as 4 weeks) before the data point “X” and a time period of +T/2 (shown here as 4 weeks) after the data point “X”. The time window comprising all the time (i.e., −T/2, X, T/2) between points a and b. 
     Referring again to  FIG. 12A , once the time window is defined, an “initial base price” is determined (Step  1205 ). This can be accomplished by the following process. With reference to  FIG. 12B , two price maxima are determined (M 1 , M 2 ), one for each of the T/2 time periods before and after the current data point. The lesser value of the two maxima (here M 1 ) comprises the initial base price. The actual price (in selected data point “X”) is compared with this initial base price (here, M 1 ). If initial base price is higher than the actual price (as shown in the pictured example), then the “initial base price” is reset to reflect the price for the previous time period. In the pictured example, the lesser maxima M 1  is $1.00, the actual price during the data point “X” is less than $1.00 so the initial base price is reset to the price of the previous time period “P” (here $1.00). 
     Alternatively, the initial base price can be determined using other methods. For example, the average price of the product over the −T/2 time period (4 weeks) preceding the data point X may be used as the initial base price. Whatever method used, the initial base price is generated for each time period of the modeled time interval. One by one, each data point in the modeled time frame is examined and an initial base price is determined for each time period (e.g., “X”) in the modeled time interval. 
     The initial base price values generated above provide satisfactory values for the imputed base price variable which may be output (Step  1207 ) and used for most purposes. However, optional Steps  1209 - 1217  describe an approach for generating a more refined imputed base price variable. 
     In generating a more refined imputed base price variable, the effect of promotional (or discount) pricing is addressed (Steps  1209 - 1217 ). This may be calculated by specifying a discount criteria (Step  1209 ); defining price steps (Step  1211 ); outputting an imputed base price variable and an imputed discount variable (Step  1213 ); analyzing the base price distribution (Step  1215 ); and outputting a refined base price variable (Step  1217 ). 
     Data records are evaluated over a series of time periods (e.g., weeks) and evaluated. The point is to identify price records which are discounted below a base price. By identifying these prices and not including them in a calculation of base price, the base price calculation will be more accurate. Therefore, a discount criterion is defined and input as a variable (Step  1209 ). A preferred criterion is 2%. Therefore, records having prices which are discounted 2% below the previously determined initial base price are treated as records having “promotional prices”. These records are temporarily deleted from the dataset. The remaining records, having zero or small discounts, are treated as “non-promoted” records. So the price of each product for the “non-promoted” time periods (weeks) is averaged over all time periods (weeks) in the modeled time interval. The average non-promoted price is referred to as a base price. 
     Further analysis is used to define base price “steps” (Step  1211 ). This process can be more readily illustrated with references to  FIG. 12C  which shows a distribution of base price data points  1220 ,  1221 ,  1222  and their relationship to a projected step function  1230 ,  1231 ,  1240 ,  1241  plotted on a graph of price over time. Base price data points  1220 ,  1221 ,  1222  are evaluated. Steps  1230 ,  1231  are roughly defined such that the base price data points  1220 ,  1221  lie within a small percent of distance from the step  1230 ,  1231  to which they are associated (e.g., 2%). This can be accomplished using, for example, a simple regression analysis such as is known to those having ordinary skill in the art. By defining the steps  1230 ,  1231 , the average value for base price over the step is determined. For example, price data points  1220  are averaged to determine the base price of step  1230 . Also, price data points  1221  are averaged to determine the base price of step  1231 . Thus, the average of the base prices in a step is treated as the refined base price for that step. 
     Further refining includes an analysis of the first step  1240 . If the first step  1240  is short (along the time axis) and considerably lower than the next step  1230 , it is assumed that the first step  1240  is based on a discounted price point  1222 . As such, the value of the next step  1230  is treated as the base price for the time period of the first step  1241  (represented by the dashed line). 
     At this point, absolute discount (ΔP) and base price (BP) are used to calculate percent discount (ΔP/BP) for each store product time period. Percent discounts that are less than some value (e.g. 1%) are treated as being no discount and corrected to ΔP/BP=0. The above determined base price variable and percent discount variable are then output (Step  1213 ). 
     This base price is subjected to further analysis for accuracy using cross-store checking (Step  1215 ). This can be accomplished by analyzing the base price data for each product within a given store. A curve is generated for each product. This curve defines the price distribution for each product. The 80 th  percentile for base price is then calculated for the analyzed product (i.e., the base price point below which 80% of the analyzed product (over the modeled time interval) is priced). This is referred to as the “in store 80 th  percentile” for that product. A calculation is then made of the average 80 th  percentile for price of the analyzed product across all stores (the cross-store 80 th  percentile). Each store&#39;s prices are then merged with each other store to calculate the average 80 th  percentile for base price over all stores. 
     The stores are then analyzed product by product. If the base price for a store is greater than two (2) standard deviations from the cross-store average 80 th  percentile for base price and if the in-store 80 th  store percentile is more than 50% different from the cross-store 80 th  percentile, this store is flagged as an outlier for the analyzed product. 
     
       
         
           
               
               
               
               
               
             
               
                   
               
               
                 Store 
                 Product 
                 In Store 80 th % 
                 Cross-Store 80 th % 
                 Flagged 
               
               
                   
               
             
            
               
                 Y 
                 A 
                 1.99 
                 1.99 
                 No 
               
               
                 Y 
                 B 
                 2.09 
                 1.99 
                 No 
               
               
                 Y 
                 C 
                 0.29 
                 1.99 
                 Yes 
               
               
                 Y 
                 D 
                 1.89 
                 1.99 
                 No 
               
               
                   
               
            
           
         
       
     
     The outlier store&#39;s base price is adjusted for the analyzed product such that it lies only two (2) standard deviations away from the average cross-store 80 th  percentile for base price over all stores. This is illustrated in  FIG. 12D . The average 80 th  percentile price over all stores is shown as “Q”. If a flagged store has a base price for an analyzed product beyond two (2) STD from the mean, as shown by data point  1250 , that data point is corrected by moving the data point to the “edge” at two (2) STD (as shown by the arrow) from the mean. That point  1251  is shown having a new base price of V. 
     Thus, the forgoing process illustrates an embodiment for determining an imputed base price variable. 
     b. Imputed Relative Price Variable 
     Reference is now made to the flowchart  1300  of  FIG. 13  which illustrates an embodiment for generating relative price variables in accordance with the principles of the present invention. In the pictured embodiment, the process begins with a calculation of an “equivalent price” for each product sold for each store (Step  1301 ). The following example will use soda to illustrate an aspect of the present invention. An example dataset is shown below: 
     
       
         
           
               
               
               
               
               
               
               
             
               
                   
               
               
                   
                   
                 Equivalent 
                 Actual 
                   
                 Equivalent 
                 Equivalent 
               
               
                 Product 
                 Size 
                 Factor 
                 Price 
                 Units 
                 Units 
                 Price 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 A 
                 8 
                 16 
                 1.00 
                 500 
                 250 
                 2.00 
               
               
                 B 
                 16 
                 16 
                 2.00 
                 300 
                 300 
                 2.00 
               
               
                 C 
                 32 
                 16 
                 3.00  
                 100 
                 200 
                 1.50 
               
               
                   
               
            
           
         
       
     
     Using this data, relative price may be calculated. As disclosed earlier, an equivalizing factor is defined. For this example, let the equivalizing factor be 16. Using the equivalizing factor, an equivalent price can be calculated (Step  1301 ). 
     
       
         
           
             
               Equivalent 
               ⁢ 
               
                   
               
               ⁢ 
               Price 
             
             = 
             
               Actual 
               ⁢ 
               
                   
               
               ⁢ 
               
                 Price 
                 · 
                 
                   ( 
                   
                     Equivalizingfactor 
                     size 
                   
                   ) 
                 
               
             
           
         
       
     
     Thus for 
                     A   :     Equivalent   ⁢           ⁢   Price       =       $1   ⁢   .00   ⁢     (     16   8     )       =     $2   ⁢   .00                     B   :     $2   ⁢   .00   ⁢     (     16   16     )         =     $2   ⁢   .00                   C   :     $3   ⁢   .00   ⁢     (     16   32     )         =     $1   ⁢   .50                 
the results of these calculations are shown in the “Equivalent Price” column of the table above.
 
     Next equivalent units sold (“units”) can be calculated (Step  1303 ). 
     
       
         
           
             
               Equivalent 
               ⁢ 
               
                   
               
               ⁢ 
               Units 
             
             = 
             
               units 
               · 
               
                 ( 
                 
                   size 
                   equivalizingfactor 
                 
                 ) 
               
             
           
         
       
     
     Thus for 
     
       
         
           
             
               
                 
                   
                     A 
                     : 
                     
                       Equivalent 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       units 
                     
                   
                   = 
                   
                     
                       500 
                       ⁢ 
                       
                         ( 
                         
                           8 
                           16 
                         
                         ) 
                       
                     
                     = 
                     250 
                   
                 
               
             
             
               
                 
                   
                     B 
                     : 
                     
                       300 
                       × 
                       
                         ( 
                         
                           16 
                           16 
                         
                         ) 
                       
                     
                   
                   = 
                   300 
                 
               
             
             
               
                 
                   
                     C 
                     : 
                     
                       100 
                       × 
                       
                         ( 
                         
                           32 
                           16 
                         
                         ) 
                       
                     
                   
                   = 
                   200 
                 
               
             
           
         
       
     
     In a similar vein, equivalent base price and equivalent base units are calculated (Step  1305 ) using the imputed values for base price (for example, as determined in Steps  1201 - 1207 ) and for base units (also referred to as base volume which is determined as disclosed below). 
     For each Store, each demand group, and each date, the total equivalent units is determined (Step  1307 ). For example, using the dataset above (assuming that the data is from the same store), a total of 750 (i.e., 250+300+200) equivalent units were sold. 
     Defining A, B, and C as products in a demand group, the equivalent values for the demand group are depicted below: 
     
       
         
           
               
               
               
             
               
                   
               
               
                 Product 
                 Equivalent Units 
                 Equivalent Price 
               
               
                   
               
             
            
               
                 A 
                 250 
                 $2.00 
               
               
                 B 
                 300 
                 $2.00 
               
               
                 C 
                 200 
                 $1.50 
               
               
                   
               
            
           
         
       
     
     A weighted calculation of relative equivalent price is then made (Step  1309 ). For example, such relative price value is determined as follows: 
     Equivalent price is divided by a weighted denominator. 
     The weighted denominator is calculated by multiplying equivalent units for each product times the equivalent units sold. For each product, only the values of other products are used in the calculation. This means excluding the product being analyzed. For example, if products A, B, and C are being analyzed in turn, when product A is analyzed the value for A is excluded from the denominator. Using the above data, the relative price of A is determined as follows: 
     
       
         
           
             
               
                 
                   
                     rel 
                     A 
                   
                   = 
                   
                     
                       equiv 
                       . 
                       priceofA 
                     
                     
                       [ 
                       
                         
                           
                             
                               ( 
                               
                                 equiv 
                                 . 
                                 unitsofB 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               
                                 Equiv 
                                 . 
                                 priceofB 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 equiv 
                                 . 
                                 unitsofC 
                               
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               
                                 equiv 
                                 . 
                                 priceofC 
                               
                               ) 
                             
                           
                         
                         
                           totalequivalentunits 
                           - 
                           equivalentunitsofA 
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                   
                     2 
                     
                       [ 
                       
                         
                           
                             
                               ( 
                               300 
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               200 
                               ) 
                             
                           
                           + 
                           
                             ( 
                             200 
                             ) 
                           
                           + 
                           
                             ( 
                             1.50 
                             ) 
                           
                         
                         
                           
                             ( 
                             
                               250 
                               + 
                               300 
                               + 
                               200 
                             
                             ) 
                           
                           - 
                           250 
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                   1.111 
                 
               
             
             
               
                 
                   
                     rel 
                     B 
                   
                   = 
                   
                     2 
                     
                       [ 
                       
                         
                           
                             
                               ( 
                               250 
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               2.00 
                               ) 
                             
                           
                           + 
                           
                             
                               ( 
                               200 
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               1.50 
                               ) 
                             
                           
                         
                         
                           750 
                           - 
                           300 
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                   1.125 
                 
               
             
             
               
                 
                   
                     rel 
                     C 
                   
                   = 
                   
                     1.50 
                     
                       [ 
                       
                         
                           
                             
                               ( 
                               250 
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               2.00 
                               ) 
                             
                           
                           + 
                           
                             
                               ( 
                               300 
                               ) 
                             
                             ⁢ 
                             
                               ( 
                               2.00 
                               ) 
                             
                           
                         
                         
                           750 
                           - 
                           200 
                         
                       
                       ] 
                     
                   
                 
               
             
             
               
                 
                   = 
                   0.75 
                 
               
             
           
         
       
     
     To insure that all members of a demand group are counted at least at some minimal level, if equivalent units=0, a value of “1” is added to all units. In an example where equivalent units were A=0; B=5; C=11, the units would be revalued as A=1; B=6; C=12, and the calculations as disclosed above would be conducted. Also, where the number of products in a demand group is equal to one, the weighted average equivalent price for the single product is equal to the equivalent price for that product. If a value for equivalent price is missing, the equivalent price for the previous record is used for equivalent price. 
     Also, a weighted average equivalent base price is calculated using the method disclosed hereinabove. The only difference being that instead of using the actual equivalent price, the calculated base price values per equivalent are used (Step  1311 ). Using the previously disclosed techniques, a moving average is generated for relative actual equivalent price and relative equivalent base price (Step  1313 ). 
     This moving average is generally calculated by first defining a time period window framing each analyzed date (e.g., four weeks, two weeks prior, two weeks after). This framing time period is specified as an input. Second, for each date in the time period window, a weighted average of actual equivalent price and a weighted average of equivalent base price are calculated. For time period windows where there are insufficient days preceding the analyzed date (e.g., if the time window requires two week&#39;s worth of data but only one week is available), imputed values are provided for base price or actual price. Such imputed values are just the average value for base price or actual price, respectively. Third, once the time period window is defined, calculations are made defining average relative actual equivalent price and average relative equivalent base price over the time period window, thereby defining a moving average for both relative actual equivalent price and relative equivalent base price. This, is repeatedly done with the window being moved incrementally through the dataset thereby obtaining a moving average. 
     Thus a variety of imputed relative price variables can be generated (e.g., relative equivalent price, relative equivalent base price. etc.). 
     c. Imputed Base Volume Variable 
     A flowchart  1400  shown in  FIG. 14A  illustrates one embodiment for generating an imputed base volume variable. Base volume refers to the volume of product units sold in the absence of discount pricing or other promotional effects. Base volume is also referred to herein as simply “base units”. The determination of base volume begins by receiving the cleansed initial dataset information for each product and store (Step  1401 ). The initial dataset information is processed to determine “non-promoted dates” (Step  1403 ). For example, using the percent discount (ΔP/BP) information generated above, product records having a percent price discount that is less than some predetermined discount level (e.g., 2%) are treated as non-promoted products for the time periods where the percent discount is less than the predetermined discount level (e.g., 2%). These records are used to generate a data subset defining the dates where the products are not significantly price discounted i.e., “non-promoted dates”. This data subset is also referred to herein as the non-promoted data subset. 
     Using the non-promoted data subset, an average value for “units” and a STD is calculated (i.e., an average value for product unit sales volume for each product during the non-promoted dates is calculated) (Step  1405 ). The average units are rounded up to the nearest integer value, this value shall be referred to as the “non-promoted average units”. 
     An initial value for base units (“initial base units”) is now determined ( 1407 ). This value is determined for all dates in the dataset, not just the non-promoted dates. For those records having a percent price discount that is less than the predetermined discount level (e.g., 2%) the actual units sold are treated as “initial base units”. However, where such records (those the 2% or less discount) also have an actual value for units sold which is greater than 1.5 STD from the non-promoted average unit value (as calculated above), then the actual value for units sold is not used. Instead, it is replaced with the non-promoted average unit value in calculating “initial base units”. For the other records (those having a percent price discount that is equal to or greater than the predetermined discount level (e.g., 2%)), the previously calculated non-promoted average unit value is used for “initial base units”. 
     This principle can be more readily understood with reference to  FIG. 14B . The price behavior  1450  can be compared with sales behavior  1460 . Typically, when the price drops below a certain level, sales volume increases. This can be seen at time periods  1470 ,  1471 . This can be reflective of, for example, a 2% or greater price discount. This is to be expected, and as a result, these sales records should not affect calculations of “base volume”. In such a case, the actual units sold (more than usual) are not included in a base volume determination. Rather, those records are replaced with the average volume value for the non-promoted dates (the non-promoted average unit to value, shown with the dotted lines  1480 ,  1481 ). However, where a sales volume increases during a period of negligible discount (e.g., less than 2%), such as shown for time period  1472 , the actual units sold (actual sales volume) are used in the calculation of base volume. However, if the records show a sales volume increase  1472  which is too large (e.g., greater than 1.5 standard deviations from the non-promoted average unit value), it is assumed that some other factor besides price is influencing unit volume and the actual unit value is not used for initial base units but is replaced by the non-promoted average unit value. 
     A calculated base volume value is now determined (Step  1409 ). This is accomplished by defining a time window. One preferred window is four (4) weeks, but the time window may be larger or smaller. For each store and product, the average value of “initial base units” is calculated for each time window. This value is referred to as “average base units”. This value is calculated for a series of time windows to generate a moving average of “average base units”. This moving average of the average base units over the modeled time interval is defined as the “base volume variable”. 
     d. Supplementary Error Detection and Correction 
     Based on previously determined discount information, supplementary error detection and correction may be used to correct price outliers. A flowchart  1500  illustrated in  FIG. 15A  shows one embodiment for accomplishing such supplementary error detection and correction. Such correction begins by receiving the cleaned initial dataset information for each product and store (Step  1501 ). In addition the previously calculated discount information is also input, or alternatively, the discount information (e.g., P/BP) can be calculated as needed. The initial dataset and discount information is processed to identify discounts higher than a preselected threshold (e.g., 60% discount) (Step  1503 ). For those time periods (e.g., weeks) having price discounts higher than the preselected threshold (e.g., greater than 60%), a comparison of actual units sold to calculated base volume units (as calculated above) is made (Step  1505 ). 
     The concepts are similar to that illustrated in  FIG. 14B  and may be more easily illustrated with reference to  FIG. 15B . The principles of this aspect of the present invention are directed toward finding unexplained price aberrations. For example, referring to  FIG. 15B , price discounts are depicted at data points  1550 ,  1551 ,  1552 , and  1553 . Also, corresponding sales increases are depicted by at data points  1561 ,  1562 , and  1563 . The data point  1550  has a discount greater than the threshold  1555  (e.g., 60%). So an analysis is made of data point  1550 . 
     If the number of actual units sold (shown as  1560 ) lies within two (2) STD of the calculated base volume, then it is assumed that the actual price  1550  is actually an erroneous record and the actual value  1560  is replaced with the calculated base price. 
     However, if the number of actual units sold is greater than two (2) STD from the calculated base volume, it is assumed that the volume number is correct and the price is reset to reflect a discount of 60% and the price is recalculated based on the 60% discount. In short, the discount is capped at the chosen value (here 60%). Once the data is corrected, it can be output (step  1507 ). 
     e. Determining Imputed Variables which Correct for the Effect of Consumer Stockpiling 
     With reference to  FIG. 16 , a flowchart  1600  illustrating a method embodiment for generating stockpiling variables is depicted. The pictured embodiment  1600  begins by defining the size of a “time bucket”(m), for example, the size (m) of the bucket can be measured in days (Step  1601 ). A preferred embodiment uses a bucket of one (1) week or seven (7) days. Additionally, the number (τ) of time buckets to be used is also defined (Step  1603 ). The total amount of time “bucketed” (m×τ) is calculated (Step  1605 ). 
     “Lag” variables which define the number of product units sold (“units”) in the time leading up to the analyzed date are defined (Step  1607 ). For example: 
     Lag1 (units)=number of product units sold in one (1) time period (e.g., a day or week) before the analyzed date; 
     Lag2(units)=number of product units sold in two (2) time periods (e.g., a day or week) before the analyzed date; 
     Lagt(units)=number of product units sold in t time periods (e.g., a day or week) before the analyzed date. 
     Then the total number of product units sold is calculated for each defined time bucket (Step  1609 ). For example: 
     Bucket1=sum of units sold during the previous m days; 
     Bucket2=sum of units sold from between the previous m+1 days to 2m days; 
     Bucket3=sum of units sold from between the previous 2m+1 days to 3m days; 
     Bucket(τ)=sum of units from between the previous (τ−1)m+1 days to (τ)m days. 
     Correction can be made at the “front end” of the modeled time interval. For example, the data can be viewed as follows: 
     
       
         
           
               
               
               
               
               
               
               
               
             
               
                   
               
               
                   
                 Week 
                   
                   
                   
                   
                   
                   
               
               
                   
                 1 
                 Week 2 
                 Week 3 
                 Week 4 
                 Week 5 
                 Week 6 
                 Week 7 
               
               
                   
               
             
            
               
                 Bucket 
                 — 
                 Week 1 
                 Week 2 
                 Week 3 
                 Week 4 
                 Week 5 
                 Week 6 
               
               
                 1 
                   
                   
                   
                   
                   
                   
                   
               
               
                 Bucket 
                 — 
                 — 
                 Week 1 
                 Week 2 
                 Week 3 
                 Week 4 
                 Week 5 
               
               
                 2 
                   
                   
                   
                   
                   
                   
                   
               
               
                 Bucket 
                 — 
                 — 
                 — 
                 Week 1 
                 Week 2 
                 Week 3 
                 Week 4 
               
               
                 3 
                   
                   
                   
                   
                   
                   
                   
               
               
                 Bucket 
                 — 
                 — 
                 — 
                 — 
                 Week 1 
                 Week 2 
                 Week 3 
               
               
                 4 
               
               
                   
               
            
           
         
       
     
     If working near the front end of a dataset, units from previous weeks cannot always be defined and in their place an averaged value for bucket sum can be used (Step  1611 ). For example, referring to Bucket 1, there is no Bucket 1 data for Week 1. As a result, the Bucket 1 data for weeks 2-7 are averaged and that value is put into Week 1 of Bucket 1. Similarly, with reference to Bucket 2, Week 1 and Week 2 are missing a value for Bucket 2, Weeks 1-3 are missing a value for Bucket 3, and Weeks 1-4 are missing a value for Bucket 4. The average values are generated for the missing values from weeks. For example, for Bucket 2, an average value for Weeks 3-7 is generated. This average value is used to fill out the missing dates of Bucket 2 (Weeks 1-2). Similarly, for Bucket 3, the average value for Weeks 4-7 are averaged and used to fill out the missing dates (Weeks 1-3). The same principle applies to Bucket 4. These Buckets define variables which are used to model the impact of promotional activity in previous time periods. The Buckets are used as variables in models which can be used to determine if there is a relationship between sales volume between a previous time as compared with a current time. The idea is to detect and integrate the effects of consumer stockpiling on into a predictive sales model. 
     f. Day of the Week Analysis 
     With reference to  FIG. 17 , a flowchart  1700  illustrating one embodiment for determining a Day of the Week variable is shown. It is necessary to have data on a daily basis for a determination of Day of the Week effects. In accordance with the principles of the present invention the embodiment begins by assigning the days of the week numerical values (Step  1701 ). A first date in the dataset is assigned. This can be arbitrarily assigned, but typically the first date for which data is available is selected as the “first date”. This date is assigned Day of Week=“1”. The next six days are sequentially assigned Days of the Week=2, 3, 4, 5, 6, 7, respectively. This process continues with the next consecutive days data starting over again with Day of Week=“1”, continuing throughout all the days of the modeled time interval. 
     Once categorized by day of the week the product units (sold) are summed for a specified dimension or set of dimensions. Dimension as used herein means a specified input variable including, but not limited to, Product, Brand, Demand Group, Store, Region, Store Format, and other input variable which may yield useful information (Step  1703 ). For example, if Region is the specified dimension (e.g., all the stores in Los Angeles, Calif.), all of the unit volume for selected products in the Los Angeles stores is summed for each Day of Week (i.e., 1, 2, 3, 4, 5, 6, and 7). 
     For each Day of Week and each dimension specified, the average units (sold) are determined (Step  1705 ). For each date, a “relative daily volume” variable is also determined (Step  1707 ). For example, relative daily volume for a given Store is provided by (total Store daily units)/(average Store daily units). Such calculation can be accomplished for any input variable. 
     One numeric example can be shown as follows. A store sells 700 units of X over a given modeled time interval. Average daily units=700/7=100. If sales for all of the Friday&#39;s of the modeled time interval amount to 150 units, it can be shown that, for that Store, Friday&#39;s relative daily volume is 1.5, i.e., more than average. This information may prove valuable to a client merchant and can comprise an input variable for other econometric models. 
     g. Imputed Seasonality Variable Generation 
     Another useful imputed variable is an imputed seasonality variable for determining seasonal variations in sales volume. One preferred approach for generating this variable is in accordance with the method described by Robert Blattberg and Scott Neslin in their book “ Sales Promotion: Concepts., Methods, and Strategies” , at pages 237-250 (Prentice Hall, N.J., 1990). 
     Referring to  FIG. 18 , a flowchart  1800  illustrating one embodiment in accordance with the present invention for determining an imputed seasonality variable is shown. The process begins with categorizing the data into weekly data records, if necessary (Step  1801 ). Zero values and missing records are then compensated for (Step  1803 ). “Month” variables are then defined (Step  1805 ). A logarithm of base units is then taken (Step  1807 ). Linear regressions are performed on each “Month” (Step  1809 ). “Months” are averaged over a specified dimension (Step  1811 ). Indexes are averaged and converted back from log scale to original scale (Step  1813 ). The average of normalized estimates are calculated and used as Seasonality index (Step  1815 ). Individual holidays are estimated and exported as imputed seasonality variables (Step  1817 ). 
     The embodiment begins by categorizing the data into weekly data records. Chiefly, this comprises aggregating daily data into weekly groups (Step  1801 ). For missing sales records or records having zero volume values, insert average volume data (Step  1803 ). 
     A set of month variables is first defined (Step  1805 ). A series of models of base units are constructed using each defined month variable as the predictor. 
     The process of defining month variables is as follows:
     1) Define the month variable
       a. Starting with Week 1, Day 1, assign a month number to each week (Month1)   b. Assume 4 weeks per month   c. Depending on the time frame of the dataset, there may be 12 or 13 months defined   
       2) Repeat definition of month variable three more times
       a. Advance Week 1 to the second week in the dataset   b. Assign a month number to each week (Month2)   c. Advance Week 1 to the third week in the dataset   d. Assign a month number to each week (Month3)   e. Advance Week 1 to the fourth week in the dataset   f. Assign a month number to each week (Month4)   
       

     
       
         
           
               
               
               
               
               
             
               
                   
               
               
                 Week 
                 Month1 
                 Month2 
                 Month3 
                 Month4 
               
               
                   
               
             
            
               
                 1 
                 1 
                 12  
                 12  
                 12  
               
               
                 2 
                 1 
                 1 
                 12  
                 12  
               
               
                 3 
                 1 
                 1 
                 1 
                 12  
               
               
                 4 
                 1 
                 1 
                 1 
                 1 
               
               
                 5 
                 2 
                 1 
                 1 
                 1 
               
               
                 6 
                 2 
                 2 
                 1 
                 1 
               
               
                 7 
                 2 
                 2 
                 2 
                 1 
               
               
                 8 
                 2 
                 2 
                 2 
                 2 
               
               
                 . . . 
                 . . . 
                 . . . 
                 . . . 
                 . . . 
               
               
                   
               
            
           
         
       
     
     The values determined for base units are now processed. By taking the log of base units the effect of extreme variations in base units can be reduced (Step  1807 ). A linear regression is run on the log of base units for Month 1 (Step  1809 ). The regression models the log of base units as a function of Month 1 levels and Week number: (log(base units)=f(Month1, Week number)). The regression analysis is repeated using months Month2, Month3, and Month4 to determine, respectively log(base units)=f(Month2, Week number); log(base units)=f(Month3, Week number); and log(base units)=f(Month4, Week number).
     3) The average value across the 12 (or 13) levels of the Month1-Month4 estimates within the specified dimension (e.g. demand group) is calculated.   4) The estimates are indexed to the average estimate value and the indexes are converted back to original scale:
       a. Seasindx1=exp(estimate of Month1−avg. estimate of Month1)   b. Seasindx2=exp(estimate of Month2−avg. estimate of Month2)   c. Seasindx3=exp(estimate of Month3−avg. estimate of Month3)   d. Seasindx4=exp(estimate of Month4−avg. estimate of Month4)   
       5) The average of the four normalized estimates is output as the Final Seasonality index
       a. Seasindx=Avg.(Seasindx1, Seasindx2, Seasindx3, Seasindx4)   b. The values for Seasindx will be centered around 1.0, and typically range from 0.7 to 1.3.   
       6) After estimating individual holidays, combine estimates with index prior to tool export   

     h. Imputed Promotional Variable 
     Another useful variable is a variable which can predict promotional effects.  FIG. 19A  provides a flowchart illustrating an embodiment enabling the generation of imputed promotional variables in accordance with the principles of the present invention. Such a variable can be imputed using actual pricing information, actual product unit sales data, and calculated value for average base units (as calculated above). This leads to a calculation of an imputed promotional variable which takes into consideration the entire range of promotional effects. 
       FIG. 19B  provides a useful pictorial illustration depicting a relationship between product price  1950 , calculated average base units  1951 , and actual units sold 1952 and the results of a simple regression model  1953  used to predict actual sales volume. 
     Referring back to  FIG. 18A , the process begins by inputting the cleansed initial dataset and the calculated average base units information (Step  1901 ). A crude promotional variable is then determined (Step  1903 ). Such a crude promotional variable can be defined using promotion flags. These promotion flags may be set by an analysis of the unit sales for each date. If the actual unit sales ( 1952  of  FIG. 19B ) are greater than two (2) STD&#39;s from the average base units value ( 1951  of  FIG. 19B ) for the same date, then the price is examined. If the price for the same date has zero discount or a small discount (e.g., less than 1%) and no other promotional devices (other than discount) are involved (based on promotional information provided by the client), then the promotional flag is set to “1”. For all other dates, if the above-mentioned conditions are not met the promotional flag is set to “0” for those dates. This set of “0&#39;s” or “1&#39;s” over the modeled time period defines a crude promotional variable. A simple regression analysis, as is known to those having ordinary skill in the art, (e.g., a mixed effects regression) is run on sales volume to obtain a model for predicting sales volume (Step  1905 ). This analysis will be designed to estimate the impact on sales volume of: price discount; the crude promotion variable; and other client supplied promotion including, but not limited to, advertisements, displays, and couponing. Using the model a sample calculation of sales volume is performed (Step  1907 ). The results of the model  1953  are compared with the actual sales data  1952  to further refine the promotion flags (Step  1909 ). If the sales volume is underpredicted (by the model) by greater than some selected percentage (e.g., 30-50%, preferably 30%) the promotion flag is set to “1” to reflect the effects of a probable non-discount promotional effect. For example, if we refer to the region shown as  1954 , and the predicted sales volume is 60 units but the actual sales volume was 100 units, the model has underpredicted the actual sales volume by 40%, greater than the preselected level of 30%. Therefore, for that date the promotion flag is set to “1”. This will reflect the likelihood that the increase in sales volume was due to a non-discount promotional effect. Since the remaining modeled results more closely approximate actual sales behavior, the promotion flags for those results are not reset and remain at “0” (Step  1911 ). The newly defined promotion flags are incorporated into a new model for defining the imputed promotional variable. 
     i. Imputed Cross-Elasticity Variable 
     Another useful variable is a cross-elasticity variable.  FIG. 20  depicts a flowchart  2000  which illustrates the generation of cross-elasticity variables in accordance with the principles of the present invention. The generation of an imputed cross-elasticity variable allows the analysis of the effects of a demand group on other demand groups within the same category. Here, a category describes a group of related demand groups which encompass highly substitutable products and complementary products. Typical examples of categories are, among many others, Italian foods, breakfast foods, or soft drinks. 
     An embodiment for generating cross-elasticity variables in accordance with the principles of the present invention will be illustrated with reference to the following example. The subject category is an abbreviated soft drink category defined by demand groups for diet soft drinks (diet), regular cola soft drinks (reg), caffeine free soft drinks (caff-free), and root beer soft drinks (RB). 
     The initial dataset information is input into the system (Step  2001 ). For each demand group the total equivalent sales volume for each store is calculated for each time period (for purposes of this illustration the time period is a week) during the modeled time interval (Step  2003 ). For each week and each demand group, the average total equivalent sales volume for each store is calculated for each week over the modeled time interval (Step  2005 ). For each demand group the relative equivalent sales volume for each store is calculated for each week (Step  2007 ). This may be calculated for each store for each week in accordance with the formula below: 
     Relative Demand Group Equivalent Sales Volume=Total Equivalent Sales Volume For a Specific Week divided by Average Total Equivalent Sales Volume as Determined For All Weeks in The Modeled Time Interval. 
     The purpose of the cross-elasticity variable is to quantify the effects of sales of one demand group upon the sales of another demand group. Therefore, when examining a first demand group, the sales of other demand groups within the same category are treated as variables which affect the sales of the first demand group. As such, the relative demand group equivalent sales volume for the other demand groups is quantified and treated as a variable in the calculation of sales volume of the first demand group, thereby generating cross-elasticity variables (Step  2009 ). This can be illustrated more easily with reference to the partial dataset illustrated in Tables A and B. These tables reflect one week&#39;s data (week 1). 
     
       
         
           
               
               
               
               
             
               
                 TABLE A 
               
               
                   
               
               
                   
                 PRO- 
                 DEMAND 
                 RELATIVE DEMAND GROUP 
               
               
                 WEEK 
                 DUCT 
                 GROUP 
                 EQUIVALENT VOLUME 
               
               
                   
               
             
            
               
                 1 
                 A 
                 Diet 
                 
                   
                     
                       
                         
                           VolA 
                           + 
                           VolB 
                           + 
                           VolC 
                         
                         
                           avg 
                           . 
                           
                             ( 
                             
                               VolA 
                               + 
                               VolB 
                               + 
                               VolC 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 1 
                 B 
                 Diet 
                 
                   
                     
                       
                         
                           VolA 
                           + 
                           VolB 
                           + 
                           VolC 
                         
                         
                           avg 
                           . 
                           
                             ( 
                             
                               VolA 
                               + 
                               VolB 
                               + 
                               VolC 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 1 
                 C 
                 Diet 
                 
                   
                     
                       
                         
                           VolA 
                           + 
                           VolB 
                           + 
                           VolC 
                         
                         
                           avg 
                           . 
                           
                             ( 
                             
                               VolA 
                               + 
                               VolB 
                               + 
                               VolC 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 1 
                 D 
                 Regular 
                 
                   
                     
                       
                         
                           VolD 
                           + 
                           VolE 
                           + 
                           VolF 
                         
                         
                           avg 
                           . 
                           
                             ( 
                             
                               VolD 
                               + 
                               VolE 
                               + 
                               VolF 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 1 
                 E 
                 Regular 
                 
                   
                     
                       
                         
                           VolD 
                           + 
                           VolE 
                           + 
                           VolF 
                         
                         
                           avg 
                           . 
                           
                             ( 
                             
                               VolD 
                               + 
                               VolE 
                               + 
                               VolF 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 1 
                 F 
                 Regular 
                 
                   
                     
                       
                         
                           VolD 
                           + 
                           VolE 
                           + 
                           VolF 
                         
                         
                           avg 
                           . 
                           
                             ( 
                             
                               VolD 
                               + 
                               VolE 
                               + 
                               VolF 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 1 
                 G 
                 Caff-free 
                 
                   
                     
                       
                         
                           VolG 
                           + 
                           VolH 
                           + 
                           VolI 
                         
                         
                           avg 
                           . 
                           
                             ( 
                             
                               VolG 
                               + 
                               VolH 
                               + 
                               VolI 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 1 
                 H 
                 Caff-free 
                 
                   
                     
                       
                         
                           VolG 
                           + 
                           VolH 
                           + 
                           VolI 
                         
                         
                           avg 
                           . 
                           
                             ( 
                             
                               VolG 
                               + 
                               VolH 
                               + 
                               VolI 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 1 
                 I 
                 Caff-free 
                 
                   
                     
                       
                         
                           VolG 
                           + 
                           VolH 
                           + 
                           VolI 
                         
                         
                           avg 
                           . 
                           
                             ( 
                             
                               VolG 
                               + 
                               VolH 
                               + 
                               VolI 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 1 
                 J 
                 RB 
                 
                   
                     
                       
                         
                           VolJ 
                           + 
                           VolK 
                           + 
                           VolL 
                         
                         
                           avg 
                           . 
                           
                             ( 
                             
                               VolJ 
                               + 
                               VolK 
                               + 
                               VolL 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 1 
                 K 
                 RB 
                 
                   
                     
                       
                         
                           VolJ 
                           + 
                           VolK 
                           + 
                           VolL 
                         
                         
                           avg 
                           . 
                           
                             ( 
                             
                               VolJ 
                               + 
                               VolK 
                               + 
                               VolL 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 1 
                 L 
                 RB 
                 
                   
                     
                       
                         
                           VolJ 
                           + 
                           VolK 
                           + 
                           VolL 
                         
                         
                           avg 
                           . 
                           
                             ( 
                             
                               VolJ 
                               + 
                               VolK 
                               + 
                               VolL 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
               
               
               
             
               
                 TABLE B 
               
               
                   
               
               
                   
                 DEMAND 
                   
                   
                   
                   
               
               
                 PRODUCT 
                 GROUP 
                 CE Diet   
                 CE Regular   
                 CE Caff-free   
                 CE RB   
               
               
                   
               
             
            
               
                 A 
                 Diet 
                 - 
                 X 
                 X 
                 X 
               
               
                 B 
                 Diet 
                 - 
                 X 
                 X 
                 X 
               
               
                 C 
                 Diet 
                 - 
                 X 
                 X 
                 X 
               
               
                 D 
                 Regular 
                 X 
                 - 
                 X 
                 X 
               
               
                 E 
                 Regular 
                 X 
                 - 
                 X 
                 X 
               
               
                 F 
                 Regular 
                 X 
                 - 
                 X 
                 X 
               
               
                 G 
                 Caff-free 
                 X 
                 X 
                 - 
                 X 
               
               
                 H 
                 Caff-free 
                 X 
                 X 
                 - 
                 X 
               
               
                 I 
                 Caff-free 
                 X 
                 X 
                 - 
                 X 
               
               
                 J 
                 RB 
                 X 
                 X 
                 X 
                 - 
               
               
                 K 
                 RB 
                 X 
                 X 
                 X 
                 - 
               
               
                 L 
                 RB 
                 X 
                 X 
                 X 
                 - 
               
               
                   
               
            
           
         
       
     
     With reference to Table A it is shown that a calculation of Relative Demand Group Equivalent Volume for product A (a diet soda) is the total of all equivalent sales volume for the diet soda demand group for the time period (here week 1). This includes the sum of all equivalent sales volume for diet soda A, all equivalent sales volume for diet soda B, and all equivalent sales volume for diet soda C. This sum is divided by the average sum of equivalent sales volume for diet soda A, diet soda B, and diet soda C. This Relative Demand Group Equivalent Volume is a cross-elasticity coefficient (CE diet ) for products other than diet soda (here, regular soda, caffeine-free soda, and root beer). The same type of calculation is performed with respect to regular soda (reg) and, for that matter, Caffeine-Free (caff-free) and Root Beer (RB) as well. This yields four cross-elasticity coefficients (CE diet , CE reg , CE caff-free , and CE RB ). Table B illustrates the relationship between each product, demand group, and the four cross-elasticity coefficients (CE diet , CE reg , CE caff-free , and CE RB ). The cross-elasticity coefficients are used generate cross-elasticity variables for each product. In Table B the “-” means the indicated cross-elasticity coefficient is not applicable to the indicated product. An “x” means the indicated cross-elasticity coefficient is applicable to the indicated product. For example, if product D (Regular soft drink) is examined, beginning with Table A, the equation for Relative Demand Group Equivalent Volume (for product A) is shown. This equation also yields the cross-elasticity coefficient (CE reg ) for the demand group regular soda. Referring now to Table B, the row for product D is consulted. There are “x&#39;s” for the coefficients which apply to a determination of a cross-elasticity variable for product D. Thus, cross-elasticity for product D is a function of cross-elasticity coefficients CE diet , CE caff-free , and CE RB . Therefore, the cross-elasticity variable for product D includes cross-elasticity coefficients CE diet , CE caff-free , and CE RB . 
     The calculated imputed variables and data are outputted from the imputed variable generator  304  to the coefficient estimator  308 . Some of the imputed variables may also be provided to the financial model engine  108 . 
     B. Coefficient Estimator 
     The coefficient estimator  308  uses the imputed variables and data to estimate coefficients, which may be used in an equation to predict demand. In a preferred embodiment of the invention, sales for a demand group (S) is calculated and a market share (F) for a particular product is calculated, so that demand (D) for a particular product is estimated by D=S·F. A demand group is defined as a collection of highly substitutable products. In the preferred embodiments, the imputed variables and equations for sales (S) of a demand group and market share (F) are as follows: 
     1. Modeling Framework 
     The econometric modeling engine relies on a mixed-model framework, simultaneously utilizing information across all stores and products in a client category, where a category is defined as a collection of substitutable or complementary products. The mixed model methodology is also referred to as “Bayesian Shrinkage” Modeling, because by combining data from various stores and/or products, one can “shrink” individual parameter estimates towards the average estimate, dampening the extreme values that would result if traditional regression were used. A basic rationale for this approach is as follows. 
     In developing product-level volume models for each store within a chain, one may be presented with a wide range of historical data in terms of modeling sufficiency. For some stores and/or products, the history will be quite rich, with many price changes and a variety of promotion patterns. For other stores and/or products, the history will be quite sparse, with very few price changes and little promotional activity. To maximize the stability of estimated model parameters, one might consider developing a single regression model across stores and/or products. This model might have stable parameter estimates; however, it would not fully leverage the store and/or product variability, and the resulting model would likely not predict well for a particular store and/or product. On the other hand, one might consider developing individual regression models for each store and/or product, to utilize the specific information contained in the data history. While these models might fit and predict well for the stores and/or products with substantial price variability, models would not be estimable for stores and/or products without a rich data history. 
     A mixed-model framework addresses the need for both highly predictive models and the existence of an estimable model for each store and product. In a mixed-effect model, information (in the form of data history) is leveraged across all stores and products, and a single cohesive model is built. Stores and products with little or no information in their data history default to an “average” (fixed-effect) model. Likewise, stores and products with a wealth of information in their data history will end up with unique parameter estimates tailored to their response pattern, via estimation of non-zero store and/or product-specific adjustment factors (random effects) which are added to the fixed-effect portion of the model. 
     2. Terminology 
     The equivalent price of a product is defined as the price of a standardized unit of measure, which may be calculated based on the product description and the spread of sizes/counts that apply to that description. Each individual product price is divided by this standardized unit of measure to obtain the equivalent price. 
     A demand group is defined as a set of products that are substitutes or near substitutes for each other. A product can belong to only one demand group. A product category consists of one or more demand groups. For this example, attention is restricted to a single category consisting of multiple demand groups. 
     Both Models: 
     Subscript i: Demand group (primary). A demand group is a collection of highly substitutable products. 
     Subscript j: Demand group (secondary). A secondary demand group is another demand group in the same category as the primary demand group, where a category is defined as a collection of substitutable or complementary products. 
     Subscript k: Product, where products are items with common UPC numbers. 
     Subscript t: Time period, which may be days, weeks, or hours. 
     Subscript B: Baseline, which is a state of product if there was no promotion. 
     Subscript n: Number of time periods away from current time period. 
     ε: Error term for regression equations, with appropriate subscripts per context. 
     3. Stage 1 (Sales) Model 
     a. Sales Model Multipliers (Data Values, or Covariates) and Dependent Variables 
     S i,t : The equivalent sales of demand group i in period t in store s in dollars. Equivalent sales may be defined as sales of equivalent units of products being compared. 
     S Bi,t : The equivalent baseline sales of demand group i in store s in period t. 
       S   i,t : The equivalent sales of demand group i in store s averaged over periods leading up to period t. 
     R i,t : The equivalent revenue of demand group i in period t. 
     R Bi,t : The equivalent baseline revenue of demand group i in period t, which would be baseline sales times the baseline price. 
       R   i,t : The equivalent revenue of demand group i averaged over periods leading up to period t. 
     P i,t : The equivalent price of demand group i at store s in time period t, calculated as total equivalent revenue of demand group i divided by total equivalent sales in the period (S i,t /R i,t ), and where the equivalent price is the price of an equivalent unit, such as a price per ounce.
   P   i,t : The average equivalent price of demand group i in store s for time period t, calculated as average total equivalent revenue of demand group i divided by average total equivalent sales (  S   i,t /  R   i,t ).
   P   i,t : The average competitor (e.g. a competing store in the area) equivalent price of demand group i in store s for time period t, calculated as average competitor total equivalent revenue of demand group i divided by average competitor total equivalent sales.
 
M i,t : The promotion level for demand group i in store s in period t.
 
X i,t : The seasonality index for demand group i in store s in period t.
 
TS t : The total dollar sales for the entire store in period t, computed using historical data.
 
T  S   t : The total dollar sales for the region in period t, computed using historical data. A region would be a grouping of stores possibly in a certain geographical area.
 
     b. Sales Model Factors 
     Parameters to be Estimated 
     γ i : The price elasticity factor for demand group i measured with respect to the deviations of the weighted average price of the demand group from the past weighted average price of the group. It measures the sensitivity of sales of equivalized units of the demand group with respect to the group price. 
     ν i : The promotion factor for demand group i. This factor measures the sensitivity of the equivalent sales of the demand group to the general promotion level of the group. 
     ψ i : The seasonality factor for demand group i. This factor measures the sensitivity of the equivalent sales of the demand group to seasonality. 
     κ i : The seasonality-price interaction factor that measures the interaction of weighted average price deviations and seasonality for demand group i. The seasonality and the group price may interact with each other in a nonlinear way. This factor measures a degree of nonlinearity.
 
δ i,n : The time lag factor for demand group i and delay of n weeks. The time lag factor measures the level of forward buying or stockpiling activity by customers. Note that this is the only factor that is estimated at the demand group level rather than the store-demand group level.
 
φ i,j : The cross elasticity factor for demand group i and demand group j. This factor measures how sales of a demand group are affected by the sales of other demand groups in the same category.
 
η i,t : The competitive price factor for demand group i measured with respect to the difference between the weighted average price of the demand group within the store and outside competitors. This factor measures the effect of competitive activity on the sales of products in a given demand group.
 
π i : The traffic factor for demand group i. Sales may be affected by the overall traffic through a store. This factor quantifies the relationship.
 
θ i : The day-of-week (DOW) effect for demand group i. Each day of a week could have a different sales pattern. This factor quantifies the relationship.
 
K i : The constant (intercept) associated with demand group i.
 
     c. The Sales Model is 
     
       
         
           
             
               
                 
                   
                     ln 
                     ⁡ 
                     
                       ( 
                       
                         
                           
                             S 
                             ^ 
                           
                           
                             i 
                             , 
                             t 
                           
                         
                         
                           S 
                           
                             Bi 
                             , 
                             t 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         K 
                         ^ 
                       
                       i 
                     
                     + 
                     
                       
                         
                           γ 
                           ^ 
                         
                         1 
                       
                       ⁢ 
                       
                         
                           P 
                           
                             i 
                             , 
                             t 
                           
                         
                         
                           
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                             _ 
                           
                           
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                             , 
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                     + 
                     
                       
                         
                           v 
                           ^ 
                         
                         i 
                       
                       ⁢ 
                       
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                           i 
                           , 
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                     + 
                     
                       
                         
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                           ^ 
                         
                         i 
                       
                       ⁢ 
                       
                         X 
                         
                           i 
                           , 
                           t 
                         
                       
                     
                     + 
                     
                       
                         
                           κ 
                           ^ 
                         
                         i 
                       
                       ⁢ 
                       
                         X 
                         
                           i 
                           , 
                           t 
                         
                       
                       ⁢ 
                       
                         
                           P 
                           
                             i 
                             , 
                             t 
                           
                         
                         
                           
                             P 
                             _ 
                           
                           
                             i 
                             , 
                             t 
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           1 
                         
                         τ 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             δ 
                             ^ 
                           
                           
                             i 
                             , 
                             n 
                           
                         
                         ⁢ 
                         
                           
                             
                               ∑ 
                               
                                 r 
                                 = 
                                 
                                   t 
                                   - 
                                   mn 
                                 
                               
                               
                                 t 
                                 - 
                                 
                                   m 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               S 
                               
                                 i 
                                 , 
                                 r 
                               
                             
                           
                           
                             
                               ∑ 
                               
                                 r 
                                 = 
                                 
                                   t 
                                   - 
                                   mn 
                                 
                               
                               
                                 t 
                                 - 
                                 
                                   m 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 S 
                                 _ 
                               
                               
                                 i 
                                 , 
                                 r 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
             
           
         
       
     
     In the above model (Equation 1), the dependent variable, demand group equivalent sales, is indexed (divided) by baseline demand group equivalent sales to provide a type of normalization. This normalizing allows easier comparison within a demand group and between demand groups. If a reasonable approximation of baseline demand group sales cannot be imputed, the dependent variable may alternatively be indexed by demand group equivalent sales averaged over a specified number of time periods prior to the current one (  S   i,t ). 
     In the time lag term, z represents the number of “time buckets” preceding a time period that will be included in the model, and m represents the size of the “time bucket,” in number of time periods. 
     Inclusion of several covariates (day-of-week, store traffic, within-market competitive pricing) is contingent upon the time dimension and scope of available client data. Therefore, if data is reported on a weekly basis, so that there is no data according to day of the week, the day of the week parameters will not be included. 
     d. Sales Model Unbiasing Factor 
     In regression calculations, returning to the original scale after a logarithmic transformation creates a bias. To correct for this bias, the Baskersville&#39;s method is used which consists of adding an unbiasing factor to the equation. This factor is the mean square error ({circumflex over (σ)} 2 ) of the sales model divided by 2. 
     The equation for predicting demand group sales is thus: 
     
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           S 
                           ^ 
                         
                         
                           i 
                           , 
                           t 
                         
                       
                       
                         S 
                         
                           Bi 
                           , 
                           t 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     exp 
                     ( 
                     
                       
                         
                           K 
                           ^ 
                         
                         i 
                       
                       + 
                       
                         
                           
                             γ 
                             ^ 
                           
                           i 
                         
                         ⁢ 
                         
                           
                             P 
                             
                               i 
                               , 
                               t 
                             
                           
                           
                             
                               P 
                               _ 
                             
                             
                               i 
                               , 
                               t 
                             
                           
                         
                       
                       + 
                       
                         
                           
                             v 
                             ^ 
                           
                           i 
                         
                         ⁢ 
                         
                           M 
                           
                             i 
                             , 
                             t 
                           
                         
                       
                       + 
                       
                         
                           
                             ψ 
                             ^ 
                           
                           i 
                         
                         ⁢ 
                         
                           X 
                           
                             i 
                             , 
                             t 
                           
                         
                       
                       + 
                       
                         
                           
                             κ 
                             ^ 
                           
                           i 
                         
                         ⁢ 
                         
                           X 
                           
                             i 
                             , 
                             t 
                           
                         
                         ⁢ 
                         
                           
                             P 
                             
                               i 
                               , 
                               t 
                             
                           
                           
                             
                               P 
                               _ 
                             
                             
                               i 
                               , 
                               t 
                             
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           τ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               δ 
                               ^ 
                             
                             
                               i 
                               , 
                               n 
                             
                           
                           ⁢ 
                           
                             
                               
                                 ∑ 
                                 
                                   r 
                                   = 
                                   
                                     t 
                                     - 
                                     mn 
                                   
                                 
                                 
                                   t 
                                   - 
                                   
                                     m 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 S 
                                 
                                   i 
                                   , 
                                   r 
                                 
                               
                             
                             
                               
                                 ∑ 
                                 
                                   r 
                                   = 
                                   
                                     t 
                                     - 
                                     mn 
                                   
                                 
                                 
                                   t 
                                   - 
                                   
                                     m 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
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                               ⁢ 
                               
                                   
                               
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                                 S 
                                 
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                                   , 
                                   r 
                                 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             j 
                             ≠ 
                             i 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               ϕ 
                               ^ 
                             
                             
                               i 
                               , 
                               j 
                             
                           
                           ⁢ 
                           
                             
                               
                                 S 
                                 ^ 
                               
                               
                                 j 
                                 , 
                                 t 
                               
                             
                             
                               
                                 S 
                                 _ 
                               
                               
                                 j 
                                 , 
                                 t 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           
                             η 
                             ^ 
                           
                           
                             i 
                             , 
                             t 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 
                                   P 
                                   _ 
                                 
                                 
                                   i 
                                   , 
                                   t 
                                 
                               
                               - 
                               
                                 
                                   
                                     P 
                                     _ 
                                   
                                   _ 
                                 
                                 
                                   i 
                                   , 
                                   t 
                                 
                               
                             
                             
                               
                                 
                                   P 
                                   _ 
                                 
                                 _ 
                               
                               
                                 i 
                                 , 
                                 t 
                               
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           
                             π 
                             ^ 
                           
                           i 
                         
                         ⁢ 
                         
                           
                             TS 
                             t 
                           
                           
                             T 
                             ⁢ 
                             
                               
                                 S 
                                 _ 
                               
                               t 
                             
                           
                         
                       
                       + 
                       
                         
                           
                             θ 
                             ^ 
                           
                           i 
                         
                         ⁢ 
                         
                           
                             S 
                             
                               i 
                               , 
                               
                                 t 
                                 - 
                                 7 
                               
                             
                           
                           
                             
                               S 
                               _ 
                             
                             
                               i 
                               , 
                               
                                 t 
                                 - 
                                 7 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           
                             σ 
                             ^ 
                           
                           2 
                         
                         2 
                       
                     
                     ) 
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     4. Stage 2 (Share) Model 
     a. Share Model Multipliers (Data Values, or Covariates ) and Dependent Variables 
     F i,k,t : The fraction of demand group i equivalent sales comprised by product k in time period t (market share of product k). 
       F   i,•,t : The average fraction of demand group i equivalent sales with respect to time period t. To allow linear modeling of the regression equation, this data value is used to provide centering. 
     P Bi,k,t : The equivalent base price of product k in demand group i in time period t. 
       P   Bi,(k),t : The average equivalent base price of all products other than product k in demand group i for time period t. 
     P R Bi,k,t : The relative equivalent base price of product k in demand group i for time period 
               t   ⁡     (     =         P   Bi     ,   k   ,   t         P   _       Bi   ,     (   k   )     ,   t           )       .           P   R Bi,•,t : The average relative equivalent base price in demand group i for time period t.
 
M p,i,k,t : The level of promotion type p (kind of promotion) for product k in demand group i in time period t. There can be up to n p  promotion factors estimated in the model.
   M   p,i,•,t : The average level of promotion type p in demand group i for time period t.
 
     b. Share Model Factors 
     Parameters to be Estimated 
     ρ i,k : The relative base price elasticity factor for product k in demand group i. 
     σ p,i,k : The promotion factor p for product k in demand group i. There can be up to n p  promotion factors estimated in the model. 
     χ i,k,n : The time lag factor for product k in demand group i and delay of n weeks. 
     Λ i,k : The constant (intercept) associated with product k in demand group I 
     The model for predicting product share (market share) is: 
     
       
         
           
             
               
                 
                   
                     
                       F 
                       ^ 
                     
                     
                       i 
                       , 
                       k 
                       , 
                       t 
                     
                   
                   = 
                   
                     
                       exp 
                       ⁢ 
                       
                         { 
                         
                           
                             
                               Λ 
                               ^ 
                             
                             
                               i 
                               , 
                               k 
                             
                           
                           + 
                           
                             
                               
                                 ρ 
                                 ^ 
                               
                               
                                 i 
                                 , 
                                 k 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 P 
                                 
                                   Ri 
                                   , 
                                   k 
                                   , 
                                   t 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 p 
                                 = 
                                 1 
                               
                               
                                 n 
                                 p 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 
                                   σ 
                                   ^ 
                                 
                                 
                                   p 
                                   , 
                                   i 
                                   , 
                                   k 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   M 
                                   
                                     p 
                                     , 
                                     i 
                                     , 
                                     k 
                                     , 
                                     t 
                                   
                                 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 1 
                               
                               τ 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 
                                   χ 
                                   ^ 
                                 
                                 
                                   i 
                                   , 
                                   k 
                                   , 
                                   n 
                                 
                               
                               ⁢ 
                               
                                 
                                   ∑ 
                                   
                                     r 
                                     = 
                                     
                                       t 
                                       - 
                                       mn 
                                     
                                   
                                   
                                     t 
                                     - 
                                     
                                       m 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           n 
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                     
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     F 
                                     
                                       i 
                                       , 
                                       k 
                                       , 
                                       r 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         } 
                       
                     
                     
                       
                         ∑ 
                         
                           l 
                           ∈ 
                           
                             Dem 
                             i 
                           
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         exp 
                         ⁢ 
                         
                           { 
                           
                             
                               
                                 Λ 
                                 ^ 
                               
                               
                                 i 
                                 , 
                                 l 
                               
                             
                             + 
                             
                               
                                 
                                   ρ 
                                   ^ 
                                 
                                 
                                   i 
                                   , 
                                   l 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   P 
                                   
                                     Ri 
                                     , 
                                     l 
                                     , 
                                     t 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   p 
                                   = 
                                   1 
                                 
                                 
                                   n 
                                   p 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   
                                     σ 
                                     ^ 
                                   
                                   
                                     p 
                                     , 
                                     i 
                                     , 
                                     t 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     M 
                                     
                                       p 
                                       , 
                                       i 
                                       , 
                                       l 
                                       , 
                                       t 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   1 
                                 
                                 τ 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   
                                     χ 
                                     ^ 
                                   
                                   
                                     i 
                                     , 
                                     k 
                                     , 
                                     n 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     ∑ 
                                     
                                       r 
                                       = 
                                       
                                         t 
                                         - 
                                         mn 
                                       
                                     
                                     
                                       t 
                                       - 
                                       
                                         m 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             n 
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                       
                                       - 
                                       1 
                                     
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       F 
                                       
                                         i 
                                         , 
                                         l 
                                         , 
                                         r 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           } 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   3 
                 
               
             
           
         
       
     
     This model calculates demand for a product divided by demand for the demand group of the product. 
     The product intercept Λ i,k  is not individually estimated in the model. Instead, each product is characterized according to product attributes, such as brand, size group (small/medium/large), form, flavor, etc. . . . . A store-specific estimate of the effect corresponding to each attribute level effects is obtained, and each product intercept is then constructed by summing over the applicable attribute level estimates. 
     Thus, 
                   Λ   ^       i   ,   k       =       ∑     a   =   1       n   a       ⁢           ⁢       ∑     b   =   1       n     b   ⁡     (   a   )           ⁢           ⁢         ξ   ^       a   b       ·     I     k   ,     a   b                 ,         
where
 
ξ a     b    is the effect of attribute a, level b, and
 
I k,a     b    is an indicator variable for product
 
     
       
         
           
             k 
             , 
             
               = 
               
                 { 
                 
                     
                 
                 ⁢ 
                 
                   
                     
                       
                         1 
                         , 
                         
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           product 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           has 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           level 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           b 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           of 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           a 
                         
                       
                     
                   
                   
                     
                       
                         0 
                         , 
                         else 
                       
                     
                   
                 
                 } 
               
             
           
         
       
     
     The attribute values may be used to predict sales of new products with various attribute combinations. 
     5. Linearization of Multinomial Logistic Equation 
     The multinomial logistic function that defines the market share equations for each product is nonlinear but there exist standard techniques to transform it to a linear function instead, which may make modeling easier. An example of a transformation to a linear function is as follows: 
     In this section the store index is ignored. 
     For a particular time period t: 
                 Let   ⁢           ⁢     F     i   ,   k         =       exp   ⁡     (       α     i   ,   k       +       ∑     p   =   1     P     ⁢           ⁢       β   p     ·     X     pi   ,   k           +     ɛ     t   ,   k         )           ∑     j   =   1     k     ⁢           ⁢     exp   ⁡     (       α     i   ,   j       +       ∑     p   =   1     P     ⁢           ⁢       β   p     ·     X     pi   ,   j           +     ɛ     t   ,   j         )             ,         
where α i  is the intercept for demand group i, β are the covariates, and P is the number of covariates in the share model
 
                     log   ⁡     (     F     i   ,   k       )       =       α     i   ,   k       +       ∑     p   =   1     P     ⁢           ⁢       β   p     ·     X     pi   ,   k           +     ɛ     i   ,   k       -     log   ⁡     (       ∑     j   =   1     k     ⁢           ⁢     exp   ⁡     (       α     i   ,   j       +       ∑     p   =   1     P     ⁢           ⁢       β   p     ·     X     pi   ,   j           +     ɛ     i   ,   j         )         )                       Let   ⁢           ⁢     log   ⁡     (       F   ~     1     )         =       1   k     ⁢       ∑     j   =   1     k     ⁢           ⁢     log   ⁡     (     F     i   ,   k       )                       =         α   _     i     +       ∑     p   =   1     P     ⁢           ⁢       β   p     ·       X   _     pi         +       ɛ   _     i     -     log   ⁡     (       ∑     j   =   1     k     ⁢           ⁢     exp   ⁡     (       α     t   ,   l       +       ∑     p   =   1     P     ⁢           ⁢       β   p     ·     X     pi   ,   j           +     ɛ     t   ,   j         )         )                     Thus   ,       log   ⁡     (       F     i   ,   k           F   ~     t       )       =       α     i   ,   k       +       ∑     p   =   1     P     ⁢           ⁢       β   p     ·     X     pt   ,   k           +     ɛ   t     -     log   ⁡     (       ∑     j   =   1     k     ⁢           ⁢     exp   ⁡     (       α     i   ,   j       +       ∑     p   =   1     P     ⁢           ⁢       β   p     ·     X     pi   ,   j           +     ɛ     i   ,   j         )         )                         -       α   _     i       -       ∑     p   =   1     P     ⁢           ⁢       β   p     ·       X   _     pi         -       ɛ   _     i     +     log   ⁡     (       ∑     j   =   1     k     ⁢           ⁢     exp   ⁡     (       α     i   ,   j       +       ∑     p   =   1     P     ⁢           ⁢       β   p     ·     X     pi   ,   j           +     ɛ     i   ,   j         )         )                   =       (       α     i   ,   k       -       α   _     i       )     +       ∑     p   =   1     P     ⁢           ⁢       β   p     ⁡     (       X     pi   ,   k       -       X   _     pi       )         +     (       ɛ     i   ,   j       -       ɛ   _     i       )                   
To model share in a linear framework, we simply center all covariates and model log(F i,k )−log({tilde over (F)} i ), where {tilde over (F)} i  is geometric mean of F i,k :
 
     
       
         
           
             
               
                 log 
                 ⁡ 
                 
                   ( 
                   
                     F 
                     
                       i 
                       , 
                       k 
                       , 
                       t 
                     
                   
                   ) 
                 
               
               - 
               
                 log 
                 ⁡ 
                 
                   ( 
                   
                     
                       F 
                       ~ 
                     
                     
                       i 
                       , 
                       
                         · 
                         
                           , 
                           t 
                         
                       
                     
                   
                   ) 
                 
               
             
             = 
             
               
                 Λ 
                 
                   i 
                   , 
                   k 
                   , 
                   t 
                 
               
               + 
               
                 
                   ρ 
                   
                     1 
                     , 
                     k 
                   
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       P 
                       
                         
                           R 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           i 
                         
                         , 
                         k 
                         , 
                         t 
                       
                     
                     - 
                     
                       
                         P 
                         _ 
                       
                       
                         Ri 
                         , 
                         
                           · 
                           
                             , 
                             t 
                           
                         
                       
                     
                   
                   ) 
                 
               
               + 
               
                 
                   ∑ 
                   
                     p 
                     = 
                     1 
                   
                   
                     n 
                     p 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     σ 
                     
                       p 
                       , 
                       i 
                       , 
                       k 
                     
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         M 
                         
                           p 
                           , 
                           i 
                           , 
                           k 
                           , 
                           t 
                         
                       
                       - 
                       
                         
                           M 
                           _ 
                         
                         
                           p 
                           , 
                           i 
                           , 
                           
                             · 
                             
                               , 
                               t 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   ∑ 
                   
                     n 
                     = 
                     1 
                   
                   τ 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     χ 
                     
                       i 
                       , 
                       k 
                       , 
                       n 
                     
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         r 
                         = 
                         
                           t 
                           - 
                           mn 
                         
                       
                       
                         t 
                         - 
                         
                           m 
                           ⁡ 
                           
                             ( 
                             
                               n 
                               - 
                               1 
                             
                             ) 
                           
                         
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           F 
                           
                             i 
                             , 
                             k 
                             , 
                             r 
                           
                         
                         - 
                         
                           
                             F 
                             _ 
                           
                           
                             i 
                             , 
                             
                               · 
                               
                                 , 
                                 r 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     C. Combined (Product Sales) Model 
     The model for predicting sales for product k in demand group i in time period t is thus given by:
 
{circumflex over (D)} i,k,t ={circumflex over (F)} i,k,t Ŝ i,t  
 
     III. FINANCIAL MODEL ENGINE 
     The financial model engine  108  receives data  132  from the stores  124  and may receive imputed variables (such as baseline sales and baseline prices) and data from the econometric engine  104  to calculate fixed and variable costs for the sale of each item. To facilitate understanding,  FIG. 5  is a block diagram to illustrate some of the transaction costs that occur in retail businesses of a chain of stores. The chain of stores may have a headquarters  504 , distribution centers  508 , and stores  512 . The headquarters  504  may place an order  516  to a manufacturer  520  for goods supplied by the manufacturer  520 , which generates an order placement cost. The manufacturer  520  may ship the goods to one of the distribution centers  508 . The receiving of the goods by the distribution center  508  generates a receiving cost  524 , a cost for stocking the goods  528 , and a cost for shipping the goods  532  to one of the stores  512 . The store  512  receives the goods from one of the distribution centers  508  or from the manufacturer  520 , which generates a receiving cost  536  and a cost for stocking the goods  540 . When a customer purchases the item, the stores  512  incur a check-out cost  544 . With the large number of retail chains, different purchasing and delivery processes may be used. Even within a single chain, different manufacturers may provide different invoicing and delivery procedures and costing system, which may be different than the processes illustrated in  FIG. 5 . 
     The financial model engine  108  should be flexible enough to provide a cost model for these different procedures. These different costs may have variable cost components where the cost of an item is a function of the amount of sales of the item and fixed cost components where the cost of an item is not a function of the amount of sales of the item. The financial model engine  108  uses these fixed and variable costs to determine C s,i,k,t , where C s,i,k,t , is a cost for a particular product (k) given a store (s), demand group (i), and a day (t). The financial model engine  108  uses industry data to provide standard estimates. For example, instead of measuring how long it takes to stock a box of an item, an industry data may be used to estimate this time. The standard estimates helps to reduce the amount of data that must be collected. In a preferred embodiment of the invention, the stores may only need to supply labor costs of the stores and distribution centers, cost of capital, size of an item, and number of items in a case to allow a cost modeling. Likewise, the preferred embodiment may infer the amount of shelf-space an item utilizes from the cubic feet of the item, the volume of sales of the item, and how often the store is replenished with the item. By using these estimations, costs may be more easily calculated on a store level, instead of being averaged over all of the stores. This is because if large amounts of data are measured by hand, measuring such data for each store would be difficult. The tailoring of costs per store allows the maximization of profits for each store. In an example of the preferred embodiment of the invention, the financial model engine  108  comprises an activity-based costing module that uses the parameters and calculations as follows: 
     A. DEFINITIONS 
     The Activity-based costing module computes variable and fixed costs for products at specific store locations. The ABC modeling component introduces the Distribution Center entity to track costing activities at retailers&#39; distribution centers and regional stores. 
     Some numbers related to properties of the product. For each product a CasePack number is provided, which is the number of products in a case. A CaseCube number provides the cubic feet of each case of the product. 
     Certain specifiers are used to determine costs related to a distribution center. A ReceivingMethod specifier allows the specification of receiving the items by pallet or by individual cases, which specify different stocking costs. A ShelfStockMethod specifier allows the specification of the type of method that may by used to stock a shelf. For example, with baby food with many individual bottles, if the case provides a tray, which allows all bottles in the case to be quickly slid onto the shelf, then the stocking costs would be less than for a case without a tray, where all bottles must be manually lifted out of the case and placed on the shelf. If an entire box or case of an item may be placed on a shelf, the shelf stocking costs would also be less. A CaseListCost specifier specifies the wholesale price for a case. The CaseAllowance specifier specifies any discount off of the wholesale price of a case. The AllowBackhaul specifier allows the specification of a deduction provided by a manufacturer when a store is near the manufacturer so that on a return trip to a distribution center the store hauls the goods from the manufacturer. The BackhaulPerCase is the amount of the back haul discount per case. The VendorDaysCredit specifier allows the specification of the number of days of credit a manufacturer may give after shipment. The InventoryInDays specifier allows the specification of the number of days that the product is held as inventory at the distribution center. 
     If a wholesaler or manufacturer ships the product directly to the store, then different cost factors may be required, while distribution center related costs may be eliminated. The DropShipMethod specifier allows the specification of whether the directly shipped product is directly shipped to the shelf, a store back room or to a retailers distribution center. 
     The AvgWklyCase specifier specifies the average weekly number of cases of product sold by a store. This would be the average number of units of product sold by the store divided by the number of units of product per case. 
     Various costs may be associated with the process of having a product shipped directly to a store. The DropShipMethod specifier is used to specify whether a manufacturer delivers a product to a store shelf, or to a store back room, or to a distribution center. 
     Various labor costs may be associated with different distribution centers and stores. The DCLaborRate is a specifier used to specify the average labor rate at each distribution center. The StoreStockLaborRate is a specifier used to specify the average labor rate of a stock clerk at a store. The StoreCheckoutLaborRate is a specifier used to specify the average labor rate of a cashier at a store. Each of these labor rate averages may be different between different stores and different distribution centers. Each average per store or per distribution center may be used in the database. 
     A ProductStorageType specifier allows the specification of dry warehouse storage for dry items, or refrigerated storage, or frozen storage, which each have different costs. A DCDryCostPFT specifier is used to specify the cost of storing dry items, per cubic foot at the distribution center. This cost may be provided by the distribution center, but an industry average may be used as a default value. A DCRefrigCostPFT specifier is used to specify the cost of storing refrigerated items, per cubic foot at the distribution center. This cost may be provided by the distribution center, but an industry average may be used as a default value. A DCFrozenCostPFT specifier is used to specify the cost of storing frozen items, per cubic foot at the distribution center. This cost may be provided by the distribution center, but an industry average may be used as a default value. A StoreDryCostPFT specifier is used to specify the cost of storing dry items, per cubic foot at the store. This cost may be provided by the store, but an industry average may be used as a default value. A StoreRefrigCostPFT specifier is used to specify the cost of storing refrigerated items, per cubic foot at the store. This cost may be provided by the store, but an industry average may be used as a default value. A StoreFrozenCostPFT specifier is used to specify the cost of storing frozen items, per cubic foot at the store. This cost may be provided by the store, but an industry average may be used as a default value. 
     A DCCubeSecs specifier specifies the number of seconds it takes to handle a cubic foot of product at the distribution center. This time may be provided by the distribution center or an industry average may be used as a default value. A DCCaseUnitizedSecs specifier specifies the average number of seconds required to handle a palletized load at the distribution center. This time may be provided by the distribution center or an industry average may be used as a default value. A DCCaseDeadPileSecs specifier specifies the average number of seconds required to handle a non-palletized load (a dead pile) at the distribution center. This time may be provided by the distribution center or an industry average may be used as a default value. 
     A StoreCubeNTSecs specifier specifies a time in seconds to handle a non-tray cubic foot of product at a store. This time may be provided by the store or an industry average may be used as a default value. A StoreCaseNTSecs specifier is used to specify a time in unitized seconds to handle a non-tray case of a product at a store. This time may be provided by the store or an industry average may be used as a default value. A StorePkgNTSecs specifier is used to specify a time in unitized seconds to handle a non-tray package of a product at a store. This time may be provided by the store or an industry average may be used as a default value. So in a case with twenty four packages, when a package is purchased one twenty fourth of a case is purchased, which may have a volume of a tenth of a cubic foot. 
     A StoreCubeTraySecs specifier specifies a time in seconds to handle a cubic foot of a tray product at a store. This time may be provided by the store or an industry average may be used as a default value. A StoreCaseTraySecs specifier is used to specify a time in unitized seconds to handle a case of a tray product at a store. This time may be provided by the store or an industry average may be used as a default value. A StorePkgTraySecs specifier is used to specify a time in unitized seconds to handle a package of a tray product at a store. This time may be provided by the store or an industry average may be used as a default value. 
     A StoreCheckoutPkgSecs specifier specifies the time it takes to check out a package of product. This time may be provided by the store or an industry average may be used as a default value. An AvgDeliveryDistance is an average distance for shipping from a distribution center to a store. A TruckCubic specifier specifies the number of cubic feet that may be loaded on a truck. A StoreLaborPct specifier specifies the percentage of store labor that is required for the different delivery methods. If a distribution center delivers an item to a store, then the StoreLaborPct is 100%, since such products must be fully processed. If a vendor delivers a product directly to a backroom of the store, then a StoreLaborPct of 75% may be used, since the store does not need to do as much work. If a vendor delivers a product directly to a store shelf, then a StoreLaborPct of 25% may be used, since the store may be allowed to do less work for such deliveries. 
     An AnnualOpptyCost specifier specifies the annual opportunity cost of the inventory in the distribution centers and stores. Annual opportunity cost may be calculated by determining the inventory in the distribution centers and the inventory in the stores to determine the dollar value of the inventory and then subtract the dollar value of inventory that is on credit and multiplying the difference by the cost of capital percentage. Generally, the inventory may be calculated from the raw data, so that the customer only needs to provide the cost of capital to determine annual opportunity cost. An InvoiceProcessing specifier specifies the cost of providing an invoice (placing an order for a product with a vendor) per case. A Transportation specifier specifies a cost of transporting an item per mile. A ShoppingBag specifier specifies the cost of a shopping bag. 
     A database would provide an identification code for each distribution center and would provide for the specification of the location of each distribution center. Similarly, each store would have an identification code and the database would also allow the specification of the location of each store. A DeliveryFrequency specifier allows the specification of the number of days between deliveries to a store. 
     B. COST CALCULATION 
     To calculate C s,i,k,t  (the cost of a product (i) in a demand group (k) in a store (s) at a time (t) fixed and variable cost components are computed from the given data structures, as follows: 
     The bag cost calculation computes total shopping bag costs for a product in a distribution center&#39;s region. 
     1. Bag Costs 
     The bag cost calculation is:
 
BagCost=(DistributionCenter.ShoppingBag/0.6)*Product.CaseCube/Product.CasePack,
 
where one bag in this example is 0.6 cubic feet, so that bag cost is the cost of a bag divided by the volume of a bag, the quantity times the volume of a case of a product divided by the number of products in a case. The distribution center to identifier used in the equation is so that if bags have different costs at different distribution centers, the different cost may be noted by the distribution center identifier.
 
     2. Checkout Labor Costs 
     The checkout labor costs is computed as cost per package from the rate at which items are checked out and the checkout clerk&#39;s pay rate. The Checkout Labor Cost Calculation is:
 
CheckoutCostPerPkg=StoreCheckoutPkgSecs*StoreCheckoutLaborRate/3600,
 
so that the Checkout Labor cost is the product of the store check out rate in packages per second and the store labor rate in dollars per hour divided by 3600 seconds per hour.
 
     3. Distribution Center Inventory Costs 
     The Distribution Center Inventory Costs computes the inventory cost per package at a given distribution center. If a product is not stored at a distribution center, then the inventory cost is zero. If a product can be backhauled, the inventory cost is reduced by the backhaul rate. The Distribution Center Inventory Cost Calculation is as follows:
 
DCInvCostPerPkg=ISNULL(ProductDistributionCenter.InventoryInDays,0)*(DistributionCenter.AnnualOpptyCost/365)*(Product.CaseListCost−Product.CaseAllowance− IIf (Product.AllowBackHaul=1,Product.BackhaulPerCase,0))/Product.CasePack.
 
     The zero in the first factor allows the cost to be zero if direct shipping is used. Otherwise, the inventory costs are the inventory costs for a distribution center times the annual opportunity cost for the distribution center divided by 365 days times the cost per case minus the allowances such as a product case allowance and back haul, with the back haul discount being zero if there is no back haul, where the product is divided by the number of products per case or package. 
     4. Distribution Center Labor Costs 
     The Distribution Center&#39;s Labor Costs per package is computed depending upon the distribution center&#39;s receiving method. If a product is not shipped from the distribution center, the cost is zero. The calculation of labor costs at the distribution center calculates the time it takes to process a product at the distribution center and multiplies it by a labor rate. An example of an algorithm for calculating distribution center labor costs is as follows: 
     Distribution Center Labor Cost Calculation 
     If Product. ReceivingMethod=1 Then
 
DCLaborCostPerPkg=((DistributionCenter.DCCubeSecs*Product.CaseCube/Product.CasePack)+(DistributionCenter.DCCaseUnitizedSecs/Product.CasePack))*(DistributionCenter.DCLaborRate/3600)
 
Else
 
DCLaborCostPerPkg=((DistributionCenter.DCCubeSecs*Product.CaseCube/Product.CasePack)+(DistributionCenter.DCCaseDeadPileSecs/Product.CasePack))*(DistributionCenter.DCLaborRate/3600)
 
     In the first branch of the algorithm for a unitized (palletized) load, first the number of seconds to process a cubic foot of product processed by the distribution center is multiplied by the number of cubic feet per case divided by the number of products per case to obtain the number of seconds to process a product at the distribution center. The number of seconds to process a unitized (palletized) case by the distribution center divided by the number of products per case yields the number of seconds for each product being processed by the distribution center. These two products are added to obtain a total number of second to process a product processed by the distribution center. This total is multiplied by the distribution center labor rate per hour divided by 3600 seconds per hour to obtain the distribution center labor cost for palletized product. 
     In the second branch of the algorithm for a dead pile (non-palletized) load, first the number of seconds to process a cubic foot of product processed by the distribution center is multiplied by the number of cubic feet per case divided by the number of products per case to obtain the number of seconds to process a product at the distribution center. The number of seconds to process a dead pile (non-palletized) case by the distribution center divided by the number of products per case yields the number of seconds for each product being processed by the distribution center. These two products are added to obtain a total number of seconds to process a product processed by the distribution center. This total is multiplied by the distribution center labor rate per hour divided by 3600 seconds per hour to obtain the distribution center labor cost for non-palletized products. 
     5. Distribution Center Calculations 
     The Distribution Center figures are aggregated by Product Type. 
     a. Sum of Average Weekly Cases 
     The SumOfAvgWklyCases specifier specifies the total of all average weekly cases of all stores for a particular product for a particular distribution center. This specifier may be calculated as follows:
 
SumOfAvgWklyCases=SUM(ProductLocDC.AvgWklyCases)
 
     b. Count of Product Location Distribution Centers 
     The CountProdLocDC specifier specifies the number of stores per distribution center that sell the product. 
     c Average Distribution Center Inventoried Days 
     The AvgOfDCInvDays specifier specifies the average number of days of inventory of a particular product at a distribution center. The distribution center may provide this number or an industry standard may be used as a default value. 
     d. Average Product Case Cubes 
     The AvgOfCaseCube specifier specifies an average cubic foot volume of a case of a product. This average may use an industrial standard as a default if a volume of a case of a product is not provided. 
     e. Distribution Center Space 
     The DCSpace specifier is a calculation of the volume of space of a product in the inventory of a distribution center. The calculation for DCSpace is as follows:
 
DCSpace=(AvgOfDCInvDays/7)*SumOfAvgWklyCase*AvgOfCaseCube
 
     The DCSpace may be the product of the Average Distribution Center Inventoried Days divided by seven (to convert days into weeks) times the Sum of the Average Weekly Cases of the product times the Average Product Case Cube to yield the volume. 
     f. Distribution Center Cube 
     The DC_Cube specifier specifies the volume of the distribution center allocated to the product. The distribution center may be divided into slots. In this embodiment, only whole numbers of slot spaces may be allocated to a product. In this example, the slot space is 80 cubic feet. Therefore, the volume used by inventory must be rounded up to the next 80 cubic feet increment. In such an example, an algorithm for calculating the number of slots allocated to the product is as follows: 
     If DC_Space &gt;0 Then
 
DC_Cube= Int ((DC_Space/80)+0.999)*80
 
Else
 
DC_Cube=0
 
     In the first branch, if the Distribution Center Space is greater than zero, then the Distribution Center Space is divided by 80 cubic feet and the result is added to 0.999. The integer value of the resulting sum is used, so that the non-integer part of the sum is truncated. The integer value is multiplied by 80 cubic feet to convert the number of slots that are allocated to the product to a volume. 
     In the second branch, if the Distribution Center Space is equal to zero, no slots are allocated to the product. 
     g. Distribution Center Space Cost Per Store 
     The DCSpaceCostPerStore specifier specifies distribution center space cost per store. In this embodiment, an algorithm to calculate DCSpaceCostPerStore, where dry goods are ProductStorageType 1, refrigerated goods are ProductStorageType 2 and frozen goods are ProductStorageType 3, is as follows: 
     If ProductStorageType=1 Then
 
DCSpaceCostPerStore=DC_Cube*DCDryCostPFT/(Product.CasePack*52*CountProdLocDC)
 
If ProductStorageType=2 Then
 
DCSpaceCostPerStore=DC_Cube*DCRefrigCostPFT/(Product.CasePack*52*CountProdLocDC)
 
If ProductStorageType=3 Then
 
DCSpaceCostPerStore=DC_Cube*DCFrozenCostPFT/(Product.CasePack*52*CountProdLocDC)
 
     In the first branch of the algorithm for dry goods, first the Distribution Center Cube is multiplied by the DCDryCostPFT. This product is then divided by the product of the number of product per case times 52 weeks in a year times the number of stores per distribution center to obtain the Distribution Center Space Cost Per Store. 
     In the second branch of the algorithm for refrigerated goods, first the Distribution Center Cube is multiplied by the DCRefrigCostPFT. This product is then divided by the product of the number of product per case times 52 weeks in a year times the number of stores per distribution center to obtain the Distribution Center Space Cost Per Store. 
     In the third branch of the algorithm for refrigerated goods, first the Distribution Center Cube is multiplied by the DCFrozenCostPFT. This product is then divided by the product of the number of product per case times 52 weeks in a year times the number of stores per distribution center to obtain the Distribution Center Space Cost Per Store. 
     6. Invoice Processing 
     The Invoice Processing calculation computes the Invoice processing cost per Product. To calculate Invoice Processing (HQInvPro), the cost of Invoice Processing per case is divided by the number of products per case, as follows:
 
HQInvProc=InvoiceProcessing/Product.CasePack
 
     7. Product Location Space 
     The Product Location (store) Space calculation computes a store&#39;s available space for a product. 
     a. Average Delivery Frequency Calculation 
     The AvgDeliveryFreq specifier specifies the average number of deliveries to a store per week averaged over several weeks. An equation to calculate AvgDeliveryFreq is:
 
AvgDeliveryFreq=AVG(Location.DeliveryFrequency)
 
     b. Sum of Average Weekly Cases Calculation 
     The SumOfAvgWklyCases specifier is the sum of the AvgWklyCases over several weeks.
 
SumOfAvgWklyCases=SUM(.AvgWklyCases)
 
     c. Average Case Cube Calculation 
     The AvgCaseCube specifier specifies the average volume of a case of a product. 
     d. Location (Store) Cube Calculation 
     The LocationCube specifier specifies the volume of the store (location) that the product takes up on the shelf and/or back room. An equation for calculating LocationCube may be as follows:
 
LocationCube=(0.5*Product.CaseCube)+(Product.CaseCube*AvgWklyCases/DeliveryFrequency)
 
     A volume of half a case is the first term. The second term is the volume of a case times the average number of cases delivered weekly divided by the frequency of deliveries per week, which estimates the volume of inventory delivered to the store with each delivery. So this sum estimates that the store will need a volume of about the volume for each delivery plus half a case to be used for the product. 
     e. Location Space Cost Calculation 
     After determining the volume of a product in the store, the store space cost (LocationSpaceCost) may be calculated. This cost is dependent upon whether the product is a dry item ProductStorageType=1, a refrigerated item ProductStorageType=2, or a frozen item ProductStorageType=3. 
     If ProductStorageType=1 Then
 
LocationSpaceCost=LocationCube*StoreDryCostPFT/(Product.CasePack*52)
 
If ProductStorageType=2 Then
 
LocationSpaceCost=LocationCube*StoreRefrigCostPFT/(Product.Casepack*52)
 
If ProductStorageType=3 Then
 
LocationSpaceCost=LocationCube*StoreFrozenCostPFT/(Product.Casepack*52)
 
     In the first branch of the algorithm for dry goods, first the Location (store) Cube is multiplied by the StoreDryCostPFT to obtain a store storage cost per week. This product is then divided by the product of the number of product per case times 52 weeks in a year to obtain the Location Space Cost, which is the store storage cost per week per item of product. 
     In the second branch of the algorithm for refrigerated goods, first the Location (store) Cube is multiplied by the StoreRefrigCostPFT to obtain a store storage cost per week. This product is then divided by the product of the number of product per case times 52 weeks in a year to obtain the Location Space Cost, which is the store storage cost per week per item of product. 
     In the third branch of the algorithm for refrigerated goods, first the Location (store) Cube is multiplied by the StoreFrozenCostPFT to obtain a store storage cost per week. This product is then divided by the product of the number of product per case times 52 weeks in a year to obtain the Location Space Cost, which is the store storage cost per week per item of product. 
     8. Transportation Cost Per Package 
     The Transportation Cost Per Package (TransCostPerPkg) computes the cost to truck products to stores. Transportation Cost Per Package may be calculated as follows:
 
TransCostPerPkg=Product.CaseCube/(TruckCubic*Product.CasePack)*Transportation*AvgDeliveryDistance
 
     The volume of the case is divided by the product of the volume of the truck times the number of items in each case. The result is multiplied by the cost per mile of delivery times the average delivery distance. 
     9. Product Location (Store) Inventory Cost 
     The Product Location Inventory Cost calculation computes a store&#39;s inventory cost for a product. An equation for calculating Product Location Inventory Cost may be as follows:
 
ProductLocInvCost=(((LocationCube*0.75/Product.CaseCube)/SumOfAvgWklyCases)−(VendorDaysCredit/7))*((Product.CaseListCost−Product.CaseAllowance− IIf (Product.AllowBackhaul=1,Product.BackhaulPerCase,0))/Product.CasePack)*AnnualOpptyCost/52
 
     The volume of inventory in a store LocationCube is multiplied by 0.75. The 0.75 is an estimated adjustment since items are always being sold so that the actual inventory is a bit less than the actual volume of product delivered. This product is divided by the number of items per case. This result is divided by the Sum of the Average Weekly Cases to obtain an average cost per item. Then the Vendors Days Credit is divided by seven to get Vendor weeks credit, which is subtracted from the average cost per item. This difference is multiplied by the product list cost minus any allowances or discounts and back haul discounts and divided by the number of products per case to obtain the cost per product. This product is multiplied by the annual opportunity cost divided by 52 weeks in a year. 
     10. Location Receiving Stock Cost 
     The Location Receiving Stock Cost computes the labor cost to stock shelves. The cost depends upon tray and non-tray stocking rates. Products are marked as tray or non-tray stock depending upon the product&#39;s shelf stocking method. 
     An algorithm for calculating Location Receiving Stock is as follows: 
     If Product.ShelfStockingMethod=2 Then
 
LocationRcvgStkCost=((DistributionCenter.StoreCubeNTSecs*Product.CaseCube/Product.CasePack)+(DistributionCenter.StoreCaseNTSecs/Product.CasePack)+DistributionCenter.StorePkgNTSecs)*DistributionCenter.StoreStockLaborRate/3600
 
Else
 
LocationRcvgStkCost=((DistributionCenter.StoreCubeTraySecs*Product.CaseCube/Product.CasePack)+(DistributionCenter.StoreCaseTraySecs/Product.CasePack)+DistributionCenter.StorePkgTraySecs)*DistributionCenter.StoreStockLaborRate/3600
 
     In the first branch of the algorithm for a non-tray stocked item, first the time in seconds to handle a cubic foot of a non-tray product at a store is multiplied by the cubic feet of each case of the product divided by the number of products per case to obtain the number of products per second of the non-tray product is stocked on the shelves. The time in unitized seconds to handle a non-tray case of a product at a store divided by the number of products per case yields the time per product of handling a product in stocking the shelves. These two products are added to obtain a total number of seconds per product for stocking the non-tray product on the shelves. This total is multiplied by the store stocking labor rate per hour divided by 3600 seconds per hour to obtain the cost of stocking the non-tray product on the shelf. 
     In the second branch of the algorithm for a tray stocked item, first the time in seconds to handle a cubic foot of a tray product at a store is multiplied by the cubic feet of each case of the product divided by the number of products per case to obtain the number of products per second of the tray product is stocked on the shelves. The time in unitized seconds to handle a tray case of a product at a store divided by the number of products per case yields the time per product of handling a product in stocking the shelves. These two products are added to obtain a total number of seconds per product for stocking the tray product on the shelves. This total is multiplied by the stocking labor rate per hour divided by 3600 seconds per hour to obtain the cost of stocking the tray product on the shelf. 
     11. Product Location Variable Cost Calculation 
     The ProdLocVariableCost specifier specifies variable cost for each product given a store, demand group, and day. In other embodiments of the invention the time period may be a week instead of a day. The ProdLocVariableCost specifier may be computed from the following equation:
 
ProdLocVariableCost=BagCost+ProductLocInvCost+CheckoutCostPerPkg+LocationRcvgStkCost+TransCostPerPkg+DCInvCostPerPkg+DCLaborCostPerPkg+HQInvProc
 
     12. Fixed Cost Calculation 
     ProdLocFixedCost is the Fixed cost for each product given a store, demand group, and day, may be computed from a combination of costs computed above. 
     The fixed cost of a product may be calculated by the equation:
 
ProdLocFixedCost=LocationSpaceCost+DCSpaceCostPerStore
 
     13. C s,i,k,t  Calculation 
     Where C s,i,k,t , is a cost for a particular product (k) given a store (s), demand group (i), and a day (t), C s,i,k,t , may be calculated as:
 
 C   s,i,k,t =ProdLocVariableCost+ProdLocFixedCost.
 
     IV. OPTIMIZATION ENGINE AND SUPPORT TOOL 
     A. Overview 
       FIG. 4  is a more detailed schematic view of the optimization engine  112  and the support tool  116 . The optimization engine  112  comprises a rule tool  404  and a price calculator  408 . The support tool  116  comprises a rule editor  412  and an output display  416 . 
     In operation, the client (stores  124 ) may access the rule editor  412  of the support tool  116  and provides client defined rule parameters (step  228 ). If a client does not set a parameter for a particular rule, a default value is used. Some of the rule parameters set by the client may be constraints to the overall weighted price advance or decline, branding price rules, size pricing rules, unit pricing rules, line pricing rules, and cluster pricing rules. These rules will be discussed in more detail regarding the preferred embodiment of the invention. The client defined parameters for these rules are provided to the rule tool  404  of the optimization engine  112  from the rule editor  412  of the support tool  116 . Within the rule tool  404 , there may be other rules, which are not client defined, such as a group sales equation rule. The rule parameters are outputted from the rule tool  404  to the price calculator  408 . The demand coefficients  128  and cost data  136  are also inputted into the price calculator  408 . The client may also provide to the price calculator  408  through the support tool  116  a desired optimization scenario rules. Some examples of scenarios may be to optimize prices to provide the optimum profit, set one promotional price and the optimization of all remaining prices to optimize profit, or optimized prices to provide a specified volume of sales for a designated product and to optimize price. The price calculator  408  then calculates optimized prices. The price calculator  408  outputs the optimized prices to the output display  416  of the support tool  116 , which allows the stores  124  to receive the optimized pricing (step  232 ). In preferred embodiments, the rules and optimizing algorithm are as follows: 
     B. Preferred Embodiment of Optimization Module 
     The optimization engine uses the group sales equation and the market share equation previously defined in the section on the econometric engine to predict group sales and product market share, respectively. These two are then combined to predict product sales at the store level. The three equations are produced here: 
     1. First Stage Prediction 
     The predictive model for the group sales is: 
     
       
         
           
             
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                       , 
                       i 
                       , 
                       t 
                     
                   
                   
                     
                       P 
                       _ 
                     
                     
                       s 
                       , 
                       i 
                       , 
                       t 
                     
                   
                 
               
               + 
               
                 
                   ∑ 
                   
                     n 
                     = 
                     1 
                   
                   τ 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     
                       δ 
                       ^ 
                     
                     
                       i 
                       , 
                       n 
                     
                   
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           r 
                           = 
                           
                             t 
                             - 
                             mn 
                           
                         
                         
                           t 
                           - 
                           
                             m 
                             ⁡ 
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         S 
                         
                           s 
                           , 
                           i 
                           , 
                           r 
                         
                       
                     
                     
                       
                         ∑ 
                         
                           r 
                           = 
                           
                             t 
                             - 
                             mn 
                           
                         
                         
                           t 
                           - 
                           
                             m 
                             ⁡ 
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         S 
                         
                           s 
                           , 
                           i 
                           , 
                           r 
                         
                       
                     
                   
                 
               
               + 
               
                 
                   ∑ 
                   
                     j 
                     ≠ 
                     i 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     
                       ϕ 
                       ^ 
                     
                     
                       s 
                       , 
                       i 
                       , 
                       j 
                     
                   
                   ⁢ 
                   
                     
                       
                         S 
                         ^ 
                       
                       
                         s 
                         , 
                         j 
                         , 
                         t 
                       
                     
                     
                       
                         S 
                         _ 
                       
                       
                         s 
                         , 
                         j 
                         , 
                         t 
                       
                     
                   
                 
               
               + 
               
                 
                   
                     η 
                     ^ 
                   
                   
                     s 
                     , 
                     i 
                     , 
                     t 
                   
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       
                         
                           P 
                           _ 
                         
                         
                           s 
                           , 
                           i 
                           , 
                           t 
                         
                       
                       - 
                       
                         
                           
                             P 
                             _ 
                           
                           _ 
                         
                         
                           s 
                           , 
                           i 
                           , 
                           t 
                         
                       
                     
                     
                       
                         
                           P 
                           _ 
                         
                         _ 
                       
                       
                         s 
                         , 
                         i 
                         , 
                         t 
                       
                     
                   
                   ) 
                 
               
               + 
               
                 
                   
                     π 
                     ^ 
                   
                   
                     s 
                     , 
                     i 
                   
                 
                 ⁢ 
                 
                   
                     TS 
                     
                       s 
                       , 
                       t 
                     
                   
                   
                     T 
                     ⁢ 
                     
                       
                         S 
                         _ 
                       
                       
                         s 
                         , 
                         t 
                       
                     
                   
                 
               
               + 
               
                 
                   
                     θ 
                     ^ 
                   
                   
                     s 
                     , 
                     t 
                   
                 
                 ⁢ 
                 
                   
                     S 
                     
                       s 
                       , 
                       i 
                       , 
                       
                         t 
                         - 
                         7 
                       
                     
                   
                   
                     
                       S 
                       _ 
                     
                     
                       s 
                       , 
                       i 
                       , 
                       
                         t 
                         - 
                         7 
                       
                     
                   
                 
               
             
           
         
       
     
     2. Second Stage Prediction 
     The predictive model for estimating the fraction of group sales due to a product is: 
     
       
         
           
             
               
                 F 
                 ^ 
               
               
                 s 
                 , 
                 i 
                 , 
                 k 
                 , 
                 t 
               
             
             = 
             
               
                 ( 
                 
                   exp 
                   ⁢ 
                   
                     { 
                     
                       
                         
                           Λ 
                           ^ 
                         
                         
                           s 
                           , 
                           i 
                           , 
                           k 
                         
                       
                       + 
                       
                         
                           
                             ρ 
                             ^ 
                           
                           
                             s 
                             , 
                             i 
                             , 
                             k 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             P 
                             
                               s 
                               , 
                               Ri 
                               , 
                               k 
                               , 
                               t 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             p 
                             = 
                             1 
                           
                           
                             n 
                             p 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               σ 
                               ^ 
                             
                             
                               s 
                               , 
                               p 
                               , 
                               i 
                               , 
                               k 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               M 
                               
                                 s 
                                 , 
                                 p 
                                 , 
                                 i 
                                 , 
                                 k 
                                 , 
                                 t 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           τ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               χ 
                               ^ 
                             
                             
                               s 
                               , 
                               i 
                               , 
                               k 
                               , 
                               n 
                             
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 r 
                                 = 
                                 
                                   t 
                                   - 
                                   mn 
                                 
                               
                               
                                 t 
                                 - 
                                 
                                   m 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 F 
                                 
                                   s 
                                   , 
                                   i 
                                   , 
                                   k 
                                   , 
                                   r 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     } 
                   
                 
                 ) 
               
               
                 ( 
                 
                   
                     ∑ 
                     
                       1 
                       ∈ 
                       
                         Dem 
                         1 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     exp 
                     ⁢ 
                     
                       { 
                       
                         
                           
                             Λ 
                             ^ 
                           
                           
                             s 
                             , 
                             i 
                             , 
                             l 
                           
                         
                         + 
                         
                           
                             
                               ρ 
                               ^ 
                             
                             
                               s 
                               , 
                               i 
                               , 
                               l 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               P 
                               
                                 s 
                                 , 
                                 Ri 
                                 , 
                                 l 
                                 , 
                                 t 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ∑ 
                             
                               p 
                               = 
                               1 
                             
                             
                               n 
                               p 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 σ 
                                 ^ 
                               
                               
                                 s 
                                 , 
                                 p 
                                 , 
                                 i 
                                 , 
                                 l 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 M 
                                 
                                   s 
                                   , 
                                   p 
                                   , 
                                   i 
                                   , 
                                   l 
                                   , 
                                   t 
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             τ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 χ 
                                 ^ 
                               
                               
                                 s 
                                 , 
                                 i 
                                 , 
                                 k 
                                 , 
                                 n 
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   r 
                                   = 
                                   
                                     t 
                                     - 
                                     mn 
                                   
                                 
                                 
                                   t 
                                   - 
                                   
                                     m 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   F 
                                   
                                     s 
                                     , 
                                     i 
                                     , 
                                     l 
                                     , 
                                     r 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     The predictive model for demand for a given product is then given by
 
{circumflex over (D)} s,i,k,t ={circumflex over (F)} s,i,k,t Ŝ s,i,t .
 
     3. The Optimization Model 
     Regression models and the predictive equations derived from the above equations are used to construct the optimization model. 
     The objective is to maximize profit: 
                 ∑     i   ∈   G       ⁢           ⁢       ∑     k   ∈     Dem   i         ⁢           ⁢         D   ^       s   ,   i   ,   k   ,   t       ⁡     (       P     s   ,   i   ,   k   ,   t       -     C     s   ,   i   ,   k   ,   t         )           =       ∑     i   ∈   G       ⁢           ⁢       ∑     k   ∈     Dem   i         ⁢           ⁢         F   ^       s   ,   i   ,   k   ,   t       ⁢         S   ^       s   ,   i   ,   t       ⁡     (       P     s   ,   i   ,   k   ,   t       -     C     s   ,   i   ,   k   ,   t         )                   
and the constraints are:
 
     a. Obey the Regression Equations Governing 
     
       
         
           
             
               
                 
                   S 
                   ^ 
                 
                 
                   i 
                   , 
                   t 
                 
               
               ⁢ 
               
                 ∀ 
                 i 
               
             
             , 
             k 
           
         
       
       
         
           
             
               ln 
               ⁡ 
               
                 ( 
                 
                   
                     
                       S 
                       ^ 
                     
                     
                       s 
                       , 
                       i 
                       , 
                       t 
                     
                   
                   
                     S 
                     
                       s 
                       , 
                       Bi 
                       , 
                       t 
                     
                   
                 
                 ) 
               
             
             = 
             
               
                 
                   K 
                   ^ 
                 
                 
                   s 
                   , 
                   i 
                 
               
               + 
               
                 
                   
                     γ 
                     ^ 
                   
                   
                     s 
                     , 
                     i 
                   
                 
                 ⁢ 
                 
                   
                     P 
                     
                       s 
                       , 
                       i 
                       , 
                       t 
                     
                   
                   
                     
                       P 
                       _ 
                     
                     
                       s 
                       , 
                       i 
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                       t 
                     
                   
                 
               
               + 
               
                 
                   
                     v 
                     ^ 
                   
                   
                     s 
                     , 
                     i 
                   
                 
                 ⁢ 
                 
                   M 
                   
                     s 
                     , 
                     i 
                     , 
                     t 
                   
                 
               
               + 
               
                 
                   
                     Ψ 
                     ^ 
                   
                   
                     s 
                     , 
                     i 
                   
                 
                 ⁢ 
                 
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                     s 
                     , 
                     i 
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                     t 
                   
                 
               
               + 
               
                 
                   
                     κ 
                     ^ 
                   
                   
                     s 
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                     i 
                   
                 
                 ⁢ 
                 
                   X 
                   
                     s 
                     , 
                     i 
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                     t 
                   
                 
                 ⁢ 
                 
                   
                     P 
                     
                       s 
                       , 
                       i 
                       , 
                       t 
                     
                   
                   
                     
                       P 
                       _ 
                     
                     
                       s 
                       , 
                       i 
                       , 
                       t 
                     
                   
                 
               
               + 
               
                   
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   
                     
                         
                     
                     ∑ 
                   
                   
                     n 
                     = 
                     1 
                   
                   τ 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     
                       δ 
                       ^ 
                     
                     
                       i 
                       , 
                       n 
                     
                   
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           r 
                           = 
                           
                             t 
                             - 
                             mn 
                           
                         
                         
                           t 
                           - 
                           
                             m 
                             ⁡ 
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         S 
                         
                           s 
                           , 
                           i 
                           , 
                           r 
                         
                       
                     
                     
                       
                         ∑ 
                         
                           r 
                           = 
                           
                             t 
                             - 
                             mn 
                           
                         
                         
                           t 
                           - 
                           
                             m 
                             ⁡ 
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         S 
                         
                           s 
                           , 
                           i 
                           , 
                           r 
                         
                       
                     
                   
                 
               
               + 
               
                 
                   ∑ 
                   
                     j 
                     ≠ 
                     i 
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     
                       ϕ 
                       ^ 
                     
                     
                       s 
                       , 
                       i 
                       , 
                       j 
                     
                   
                   ⁢ 
                   
                     
                       
                         S 
                         ^ 
                       
                       
                         s 
                         , 
                         j 
                         , 
                         t 
                       
                     
                     
                       
                         S 
                         _ 
                       
                       
                         s 
                         , 
                         j 
                         , 
                         t 
                       
                     
                   
                 
               
               + 
               
                 
                   
                     η 
                     ^ 
                   
                   
                     s 
                     , 
                     i 
                     , 
                     t 
                   
                 
                 ( 
                 
                     
                 
                 ⁢ 
                 
                   
                       
                     
                       
                         
                           
                             P 
                             _ 
                           
                           
                             s 
                             , 
                             i 
                             , 
                             t 
                           
                         
                         - 
                         
                           
                             
                               P 
                               _ 
                             
                             _ 
                           
                           
                             s 
                             , 
                             i 
                             , 
                             t 
                           
                         
                       
                       
                         
                           
                             P 
                             _ 
                           
                           _ 
                         
                         
                           s 
                           , 
                           i 
                           , 
                           t 
                         
                       
                     
                     ) 
                   
                   + 
                   
                     
                       
                         π 
                         ^ 
                       
                       
                         s 
                         , 
                         i 
                       
                     
                     ⁢ 
                     
                       
                         TS 
                         
                           s 
                           , 
                           t 
                         
                       
                       
                         T 
                         ⁢ 
                         
                           
                             S 
                             _ 
                           
                           
                             s 
                             , 
                             t 
                           
                         
                       
                     
                   
                   + 
                   
                     
                       
                         θ 
                         ^ 
                       
                       
                         s 
                         , 
                         i 
                       
                     
                     ⁢ 
                     
                       
                         S 
                         
                           s 
                           , 
                           i 
                           , 
                           
                             t 
                             - 
                             7 
                           
                         
                       
                       
                         
                           S 
                           _ 
                         
                         
                           s 
                           , 
                           i 
                           , 
                           
                             t 
                             - 
                             7 
                           
                         
                       
                     
                   
                 
               
             
           
         
       
     
     b. Obey the Market Share Equations 
     
       
         
           
             
               
                 F 
                 ^ 
               
               
                 s 
                 , 
                 i 
                 , 
                 k 
                 , 
                 t 
               
             
             = 
             
               
                 ( 
                 
                   exp 
                   ⁢ 
                   
                     { 
                     
                       
                         
                           Λ 
                           ^ 
                         
                         
                           s 
                           , 
                           i 
                           , 
                           k 
                         
                       
                       + 
                       
                         
                           
                             ρ 
                             ^ 
                           
                           
                             s 
                             , 
                             i 
                             , 
                             k 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             P 
                             
                               s 
                               , 
                               Ri 
                               , 
                               k 
                               , 
                               t 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             p 
                             = 
                             1 
                           
                           
                             n 
                             p 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               σ 
                               ^ 
                             
                             
                               s 
                               , 
                               p 
                               , 
                               i 
                               , 
                               k 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               M 
                               
                                 s 
                                 , 
                                 p 
                                 , 
                                 i 
                                 , 
                                 k 
                                 , 
                                 t 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             1 
                           
                           τ 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             
                               χ 
                               ^ 
                             
                             
                               s 
                               , 
                               i 
                               , 
                               k 
                               , 
                               n 
                             
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 r 
                                 = 
                                 
                                   t 
                                   - 
                                   mn 
                                 
                               
                               
                                 t 
                                 - 
                                 
                                   m 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       n 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ( 
                               
                                 F 
                                 
                                   s 
                                   , 
                                   i 
                                   , 
                                   k 
                                   , 
                                   r 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     } 
                   
                 
                 ) 
               
               
                 ( 
                 
                   
                     ∑ 
                     
                       l 
                       ∈ 
                       
                         Dem 
                         i 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     exp 
                     ⁢ 
                     
                       { 
                       
                         
                           
                             Λ 
                             ^ 
                           
                           
                             s 
                             , 
                             i 
                             , 
                             l 
                           
                         
                         + 
                         
                           
                             
                               ρ 
                               ^ 
                             
                             
                               s 
                               , 
                               i 
                               , 
                               l 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               P 
                               
                                 s 
                                 , 
                                 Ri 
                                 , 
                                 l 
                                 , 
                                 t 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           
                             ∑ 
                             
                               p 
                               = 
                               1 
                             
                             
                               n 
                               p 
                             
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 σ 
                                 ^ 
                               
                               
                                 s 
                                 , 
                                 p 
                                 , 
                                 i 
                                 , 
                                 l 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 M 
                                 
                                   s 
                                   , 
                                   p 
                                   , 
                                   i 
                                   , 
                                   l 
                                   , 
                                   t 
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               1 
                             
                             τ 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 χ 
                                 ^ 
                               
                               
                                 s 
                                 , 
                                 i 
                                 , 
                                 k 
                                 , 
                                 n 
                               
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   r 
                                   = 
                                   
                                     t 
                                     - 
                                     mn 
                                   
                                 
                                 
                                   t 
                                   - 
                                   
                                     m 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         n 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                   
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   F 
                                   
                                     s 
                                     , 
                                     i 
                                     , 
                                     l 
                                     , 
                                     r 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       } 
                     
                   
                 
                 ) 
               
             
           
         
       
     
     c. Constrain Price Changes to be within a Given Range of Current Prices 
       PMIN s,i,k,t ≦P s,i,k,t ≦PMAX s,i,k,t  
 
To simplify notation, the time index may be removed from the equations. Next, all constant terms in each of the expressions may be grouped together. So the objective becomes:
 
Maximize:
 
     
       
         
           
             
               ∑ 
               
                 i 
                 ∈ 
                 G 
               
             
             ⁢ 
             
                 
             
             ⁢ 
             
               
                 ∑ 
                 
                   k 
                   ∈ 
                   
                     Dem 
                     i 
                   
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   
                     F 
                     ^ 
                   
                   
                     s 
                     , 
                     i 
                     , 
                     k 
                   
                 
                 ⁢ 
                 
                   
                     
                       S 
                       ^ 
                     
                     
                       s 
                       , 
                       i 
                     
                   
                   ⁡ 
                   
                     ( 
                     
                       
                         P 
                         
                           s 
                           , 
                           i 
                           , 
                           k 
                         
                       
                       - 
                       
                         C 
                         
                           s 
                           , 
                           i 
                           , 
                           k 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     The regression equations for Ŝ i,t ∀i,k become 
     
       
         
           
             
               log 
               ⁡ 
               
                 ( 
                 
                   
                     S 
                     ^ 
                   
                   
                     s 
                     , 
                     i 
                   
                 
                 ) 
               
             
             = 
             
               
                 a 
                 
                   s 
                   , 
                   i 
                 
               
               + 
               
                 
                   b 
                   
                     s 
                     , 
                     i 
                   
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       k 
                       ∈ 
                       
                         Dem 
                         i 
                       
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       P 
                       
                         s 
                         , 
                         i 
                         , 
                         k 
                       
                     
                     ⁢ 
                     
                       F 
                       
                         s 
                         , 
                         i 
                         , 
                         k 
                       
                     
                   
                 
               
               + 
               
                 
                   ∑ 
                   
                     
                       
                         
                           j 
                           ∈ 
                           G 
                         
                       
                     
                     
                       
                         
                           j 
                           ≠ 
                           i 
                         
                       
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     c 
                     
                       s 
                       , 
                       i 
                       , 
                       j 
                     
                   
                   ⁢ 
                   
                     S 
                     
                       s 
                       , 
                       j 
                     
                   
                 
               
               + 
               
                 const 
                 
                   s 
                   , 
                   i 
                 
               
             
           
         
       
     
     where 
     
       
         
           
             
               
                 
                   
                     
                       c 
                       
                         s 
                         , 
                         i 
                         , 
                         j 
                       
                     
                     = 
                     
                       
                         
                           
                             S 
                             _ 
                           
                           
                             s 
                             , 
                             i 
                           
                         
                         
                           
                             S 
                             _ 
                           
                           
                             s 
                             , 
                             j 
                           
                         
                       
                       ⁢ 
                       
                         
                           ϕ 
                           ^ 
                         
                         
                           s 
                           , 
                           i 
                           , 
                           j 
                         
                       
                     
                   
                   , 
                 
               
             
             
               
                 
                   
                     b 
                     
                       s 
                       , 
                       i 
                     
                   
                   = 
                   
                     
                       
                         
                           S 
                           _ 
                         
                         
                           s 
                           , 
                           i 
                         
                       
                       
                         
                           P 
                           _ 
                         
                         
                           s 
                           , 
                           t 
                         
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             γ 
                             ^ 
                           
                           
                             s 
                             , 
                             t 
                           
                         
                         + 
                         
                           
                             κ 
                             ^ 
                           
                           ⁢ 
                           
                             X 
                             
                               s 
                               , 
                               i 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     a 
                     
                       s 
                       , 
                       i 
                     
                   
                   = 
                   
                     
                       
                         
                           S 
                           _ 
                         
                         
                           s 
                           , 
                           i 
                         
                       
                       ⁢ 
                       
                         π 
                         
                           s 
                           , 
                           i 
                         
                       
                     
                     
                       
                         TS 
                         _ 
                       
                       s 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               Const 
               
                 s 
                 , 
                 i 
               
             
             = 
             
               
                 log 
                 ⁡ 
                 
                   ( 
                   
                     
                       S 
                       _ 
                     
                     
                       s 
                       , 
                       i 
                     
                   
                   ) 
                 
               
               + 
               
                 
                   ∑ 
                   
                     n 
                     = 
                     1 
                   
                   τ 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     
                       δ 
                       ^ 
                     
                     
                       s 
                       , 
                       i 
                       , 
                       n 
                     
                   
                   ⁢ 
                   
                     
                       
                         
                           ∑ 
                           
                             r 
                             = 
                             
                               t 
                               - 
                               mn 
                             
                           
                           
                             t 
                             - 
                             
                               m 
                               ⁡ 
                               
                                 ( 
                                 
                                   n 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           S 
                           
                             s 
                             , 
                             i 
                             , 
                             r 
                           
                         
                       
                       ⁢ 
                       
                           
                       
                     
                     
                       
                         ∑ 
                         
                           r 
                           = 
                           
                             t 
                             - 
                             mn 
                           
                         
                         
                           t 
                           - 
                           
                             m 
                             ⁡ 
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           S 
                           _ 
                         
                         
                           s 
                           , 
                           i 
                           , 
                           r 
                         
                       
                     
                   
                 
               
               + 
               
                 
                   
                     Ψ 
                     ^ 
                   
                   
                     s 
                     , 
                     i 
                   
                 
                 ⁢ 
                 
                   X 
                   
                     s 
                     , 
                     i 
                   
                 
               
               - 
               
                 
                   γ 
                   ^ 
                 
                 
                   s 
                   , 
                   i 
                 
               
               + 
               
                 
                   
                     η 
                     ^ 
                   
                   
                     s 
                     , 
                     i 
                   
                 
                 ⁡ 
                 
                   ( 
                   
                     
                       
                         P 
                         _ 
                       
                       
                         s 
                         , 
                         i 
                         , 
                         t 
                       
                     
                     - 
                     
                       
                         
                           P 
                           _ 
                         
                         _ 
                       
                       
                         s 
                         , 
                         i 
                         , 
                         t 
                       
                     
                   
                   ) 
                 
               
               - 
               
                 
                   
                     κ 
                     ^ 
                   
                   
                     s 
                     , 
                     i 
                   
                 
                 ⁢ 
                 
                   X 
                   
                     s 
                     , 
                     i 
                   
                 
               
               - 
               
                 
                   v 
                   ^ 
                 
                 
                   s 
                   , 
                   i 
                 
               
               + 
               
                 R 
                 
                   s 
                   , 
                   i 
                 
               
               + 
               
                 
                   
                     θ 
                     ^ 
                   
                   
                     s 
                     , 
                     i 
                   
                 
                 ⁢ 
                 
                   
                     S 
                     
                       s 
                       , 
                       i 
                       , 
                       
                         t 
                         - 
                         7 
                       
                     
                   
                   
                     
                       S 
                       _ 
                     
                     
                       s 
                       , 
                       i 
                       , 
                       
                         t 
                         - 
                         7 
                       
                     
                   
                 
               
               + 
               
                 K 
                 
                   s 
                   , 
                   i 
                 
               
             
           
         
       
     
     Similarly, the regression equations for the market share fractions become: 
     
       
         
           
             
               
                 F 
                 ^ 
               
               
                 s 
                 , 
                 i 
                 , 
                 k 
               
             
             = 
             
               
                 exp 
                 ⁢ 
                 
                   { 
                   
                     
                       
                         
                           
                             ρ 
                             ^ 
                           
                           
                             s 
                             , 
                             i 
                             , 
                             k 
                           
                         
                         
                           
                             P 
                             _ 
                           
                           
                             s 
                             , 
                             i 
                             , 
                             k 
                           
                         
                       
                       ⁢ 
                       
                         P 
                         
                           s 
                           , 
                           i 
                           , 
                           k 
                         
                       
                     
                     + 
                     
                       Const 
                       
                         s 
                         , 
                         i 
                         , 
                         k 
                       
                     
                   
                   } 
                 
               
               
                 
                   ∑ 
                   
                     l 
                     ∈ 
                     
                       Dem 
                       i 
                     
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   exp 
                   ⁢ 
                   
                     { 
                     
                       
                         
                           
                             
                               ρ 
                               ^ 
                             
                             
                               s 
                               , 
                               i 
                               , 
                               l 
                             
                           
                           
                             
                               P 
                               _ 
                             
                             
                               s 
                               , 
                               i 
                               , 
                               l 
                             
                           
                         
                         ⁢ 
                         
                           P 
                           
                             s 
                             , 
                             i 
                             , 
                             l 
                           
                         
                       
                       + 
                       
                         Const 
                         
                           s 
                           , 
                           i 
                           , 
                           l 
                         
                       
                     
                     } 
                   
                 
               
             
           
         
       
     
     where 
     
       
         
           
             
               Const 
               
                 s 
                 , 
                 i 
                 , 
                 k 
               
             
             = 
             
               
                 - 
                 
                   ρ 
                   
                     s 
                     , 
                     i 
                     , 
                     k 
                   
                 
               
               + 
               
                 
                   
                     σ 
                     ^ 
                   
                   
                     s 
                     , 
                     i 
                     , 
                     l 
                   
                 
                 ⁢ 
                 
                   R 
                   
                     s 
                     , 
                     i 
                     , 
                     k 
                   
                 
               
               + 
               
                 
                   ∑ 
                   
                     n 
                     = 
                     1 
                   
                   τ 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     
                       χ 
                       ^ 
                     
                     
                       s 
                       , 
                       i 
                       , 
                       k 
                       , 
                       n 
                     
                   
                   ⁢ 
                   
                     
                       
                         ∑ 
                         
                           r 
                           = 
                           
                             t 
                             - 
                             mn 
                           
                         
                         
                           t 
                           - 
                           
                             m 
                             ⁡ 
                             
                               ( 
                               
                                 n 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         F 
                         
                           s 
                           , 
                           i 
                           , 
                           k 
                           , 
                           r 
                         
                       
                     
                     
                       
                         ∑ 
                         
                           r 
                           = 
                           
                             t 
                             - 
                             mn 
                           
                         
                         
                           
                               
                           
                           ⁢ 
                           
                             t 
                             - 
                             
                               m 
                               ⁡ 
                               
                                 ( 
                                 
                                   n 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                             - 
                             1 
                           
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           F 
                           _ 
                         
                         
                           s 
                           , 
                           i 
                           , 
                           k 
                           , 
                           r 
                         
                       
                     
                   
                 
               
               + 
               
                 
                   Λ 
                   ^ 
                 
                 
                   s 
                   , 
                   i 
                   , 
                   k 
                 
               
             
           
         
       
     
     With this, the optimization problem may be written as: 
     4. Problem P 1   
     Maximize: 
               ∑     s   ∈   Stores       ⁢       ∑     t   ∈   G       ⁢       ∑     k   ∈     Dem   t         ⁢         exp   ⁢     {             ρ   ^       s   ,   i   ,   k           P   _       s   ,   t   ,   k         ⁢     P     s   ,   t   ,   k         +     Const     s   ,   t   ,   k         }           ∑     t   ∈     Dem   t         ⁢     exp   ⁢     {             ρ   ^       s   ,   i   ,   l           P   _       s   ,   t   ,   l         ⁢     P     s   ,   t   ,   l         +     Const     s   ,   t   ,   l         }           ⁢         S   ^       s   ,   t       ⁡     (       P     s   ,   t   ,   k       -     C     s   ,   t   ,   k         )                   
Subject to
 
     
       
         
           
             
               
                 
                   a 
                   . 
                 
               
               
                 
                   
                     
                       log 
                       ⁡ 
                       
                         ( 
                         
                           
                             S 
                             ^ 
                           
                           
                             s 
                             , 
                             t 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         a 
                         
                           s 
                           , 
                           t 
                         
                       
                       + 
                       
                         
                           b 
                           
                             s 
                             , 
                             t 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               k 
                               ∈ 
                               
                                 Dem 
                                 t 
                               
                             
                           
                           ⁢ 
                           
                             
                               P 
                               
                                 s 
                                 , 
                                 t 
                                 , 
                                 k 
                               
                             
                             ⁢ 
                             
                               F 
                               
                                 s 
                                 , 
                                 t 
                                 , 
                                 k 
                               
                             
                           
                         
                       
                       + 
                       
                         
                           ∑ 
                           
                             
                               
                                 
                                   J 
                                   ∈ 
                                   G 
                                 
                               
                             
                             
                               
                                 
                                   J 
                                   ≠ 
                                   t 
                                 
                               
                             
                           
                         
                         ⁢ 
                         
                           
                             C 
                             
                               s 
                               , 
                               t 
                               , 
                               J 
                             
                           
                           ⁢ 
                           
                             S 
                             
                               s 
                               , 
                               J 
                             
                           
                         
                       
                       + 
                       
                         const 
                         
                           s 
                           , 
                           t 
                         
                       
                     
                   
                   , 
                 
               
             
             
               
                 
                   b 
                   . 
                 
               
               
                 
                   
                     
                       F 
                       ^ 
                     
                     
                       s 
                       , 
                       i 
                       , 
                       k 
                     
                   
                   = 
                   
                     
                       
                         exp 
                         ⁢ 
                         
                           { 
                           
                             
                               
                                 
                                   
                                     ρ 
                                     ^ 
                                   
                                   
                                     s 
                                     , 
                                     i 
                                     , 
                                     k 
                                   
                                 
                                 
                                   
                                     P 
                                     _ 
                                   
                                   
                                     s 
                                     , 
                                     i 
                                     , 
                                     k 
                                   
                                 
                               
                               ⁢ 
                               
                                 P 
                                 
                                   s 
                                   , 
                                   i 
                                   , 
                                   k 
                                 
                               
                             
                             + 
                             
                               Const 
                               
                                 s 
                                 , 
                                 t 
                                 , 
                                 k 
                               
                             
                           
                           } 
                         
                       
                       
                         
                           ∑ 
                           
                             l 
                             ∈ 
                             
                               Dem 
                               t 
                             
                           
                         
                         ⁢ 
                         
                           exp 
                           ⁢ 
                           
                             { 
                             
                               
                                 
                                   
                                     
                                       ρ 
                                       ^ 
                                     
                                     
                                       s 
                                       , 
                                       i 
                                       , 
                                       l 
                                     
                                   
                                   
                                     
                                       P 
                                       _ 
                                     
                                     
                                       s 
                                       , 
                                       i 
                                       , 
                                       l 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   P 
                                   
                                     s 
                                     , 
                                     t 
                                     , 
                                     l 
                                   
                                 
                               
                               + 
                               
                                 Const 
                                 
                                   s 
                                   , 
                                   i 
                                   , 
                                   l 
                                 
                               
                             
                             } 
                           
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     and 
                   
                 
               
             
             
               
                 
                   c 
                   . 
                 
               
               
                 
                   
                     P 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       MIN 
                       
                         s 
                         , 
                         t 
                         , 
                         k 
                       
                     
                   
                   ≤ 
                   
                     P 
                     
                       s 
                       , 
                       t 
                       , 
                       k 
                     
                   
                   ≤ 
                   
                     P 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       MAX 
                       
                         s 
                         , 
                         t 
                         , 
                         k 
                       
                     
                   
                 
               
             
           
         
       
     
     Now, in addition to the constraints listed above, the preferred embodiment may model several other business rules via constraints. These include limits on group price advance or decline, brand pricing rules, size pricing rules, and unit pricing rules. 
     d. Group Price Advance or Decline 
     Since demand groups are made up of like or substitutable products, managers often wish to constrain the overall weighted price advance or decline in price for them, where the weights are the market shares of the products that constitute the demand groups. The constraints are: 
     
       
         
           
             
               ∀ 
               s 
             
             , 
             i 
             , 
             
               
                 PMIN 
                 
                   s 
                   , 
                   t 
                 
               
               ≤ 
               
                 
                   ∑ 
                   k 
                 
                 ⁢ 
                 
                   
                     F 
                     
                       s 
                       , 
                       t 
                       , 
                       k 
                     
                   
                   ⁢ 
                   
                     P 
                     
                       s 
                       , 
                       i 
                       , 
                       k 
                     
                   
                 
               
               ≤ 
               
                 PMAX 
                 
                   s 
                   , 
                   i 
                 
               
             
           
         
       
     
     e. Brand Pricing Rules 
     Products are described and categorized by attributes such as brand, size and flavor. Therefore, each product may be associated with a set of attributes and values. These attributes are useful to us for several reasons. The attributes may be used in the regression modeling in order to create a parsimonious set of regression coefficients. They may also be used in setting rules. For instance, category managers might wish to offer store brands at lower prices compared to the competing national brands. They might also wish to constrain some product brands to be less expensive than others, either for strategic considerations or to meet their contractual obligations. Specifically, a manager can create a set Brand=b s,i ≡{(p s,i,l     1   , p s,i,k,     2   ):p s,i,k     1    must cost less than p s,i,k     2   }, which leads to the following constraints:
 
∀ s,i  and( p   s,i,k     1     ,p   s,i,k     2   )εBrand s,i   p   s,i,k     1     ≦p   s,i,k     2    
 
     f. Size Pricing Rules 
     Managers might also wish to create rules that relate the price of one product versus another based on their sizes. For instance, they might wish for products belonging to the same brand and sharing other attributes, but with different sizes to be priced such that the larger sized product costs less per equivalent unit of measure than a smaller one. They cSize s,i ≡{(p s,i,l     1   ,p s,i,k     2   ):p s,i,k     1    must cost less than p s,i,k     2    an create a set, which leads to the following constraints:
 
∀ s,i  and ( p   s,i,k     1     ,p   s,i,k     2   )εSize s,i   p   s,i,k     1     ≦p   s,i,k     2    
 
     g. Unit Pricing Rules 
     Continuing in the same vein, managers might wish to ensure that two products that are identical in every respect but size should be priced such that the larger product costs more than the smaller one. This rule is closely related to the size pricing rule. To implement this rule, managers can create a set Unit s,i ≡{(p s,i,l     1   ,p s,i,k     2   ):e s,i,k     1   p s,i,k     1    must cost less than e s,i,k     2   p s,i,k     2   }, where e s,i,k  is the multiplicative factor to convert equivalent units into whole product units. This leads to the following constraints:
 
∀ s,i  and( p   s,i,k     1     ,p   s,i,k     2   )εUnit s,i   e   s,i,k     1     p   s,i,k     1     ≦e   s,i,k     2     p   s,i,k     2    
 
     h. Line Pricing Rules 
     Retail customers expect common prices for certain groups of products such as, for example, cough lozenges of the same brand but different flavors. These rules may be classified as line pricing rules and implement them as follows. Product groups called line price groups may be defined as L l , l=1, . . . ∥L∥, where every product within a line group must have the same price in a store; i.e.,
 
∀ L   l   ,l= 1, . . . ∥ L∥ , and ∀ k   1   ,k   2 εL l P s,i,k     1     =P   s,i,k     2    
 
     i. Cluster Pricing 
     Retailers define geographic and other regions within which they maintain the same price for a given product. This may be translated to the notion of store clusters. A store cluster Cluster c  is a set of stores such that the price of every product is invariant within the cluster. In other words, every product has the same price in every store within the cluster although each product may have a different price. In order to implement these constraints the set of stores may be partitioned into store clusters Cluster 1 , . . . , Cluster ∥c∥  where C≡{Cluster 1 , . . . , Cluster ∥C∥ }, and define a new price variable P c,i,k  which represents the common price for product k in cluster Cluster c . This variable may be used in place of the original price variable P s,i,k  whenever sεCluster c . The entire problem P 1  can be rewritten as P 2 : 
     5. Problem P 2 : 
     Maximize: 
               ∑     c   =   1          C          ⁢       ∑     s   ∈     Cluster   c         ⁢       ∑     t   ∈   G       ⁢       ∑     k   ∈     Dem   t         ⁢         exp   ⁢     {             ρ   ^       s   ,   t   ,   k           P   _       s   ,   t   ,   k         ⁢     P     c   ,   t   ,   k         +     Const     s   ,   t   ,   k         }           ∑     l   ∈     Dem   t         ⁢     exp   ⁢     {                   ρ   ^       s   ,   i   ,   l           P   _       s   ,   t   ,   l         ⁢     P     c   ,   t   ,   l         +               Const     s   ,   t   ,   l             }           ⁢         S   ^       s   ,   t       ⁡     (       P     c   ,   i   ,   k       -     C     s   ,   t   ,   k         )                     
(Profit Maximizing Objective)
 
Subject to:
 
     
       
         
           
             
               
                 
                   a 
                   . 
                 
               
               
                 
                   
                     log 
                     ⁡ 
                     
                       ( 
                       
                         
                           S 
                           ^ 
                         
                         
                           s 
                           , 
                           t 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       a 
                       
                         s 
                         , 
                         t 
                       
                     
                     + 
                     
                       
                         b 
                         
                           s 
                           , 
                           t 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             k 
                             ∈ 
                             
                               Dem 
                               i 
                             
                           
                         
                         ⁢ 
                         
                           
                             P 
                             
                               s 
                               , 
                               t 
                               , 
                               k 
                             
                           
                           ⁢ 
                           
                             F 
                             
                               s 
                               , 
                               t 
                               , 
                               k 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           
                             
                               
                                 J 
                                 ∈ 
                                 G 
                               
                             
                           
                           
                             
                               
                                 J 
                                 ≠ 
                                 t 
                               
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           c 
                           
                             s 
                             , 
                             t 
                             , 
                             J 
                           
                         
                         ⁢ 
                         
                           S 
                           
                             s 
                             , 
                             t 
                           
                         
                       
                     
                     + 
                     
                       const 
                       
                         s 
                         , 
                         t 
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Group 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Sales 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     equation 
                   
                   ) 
                 
               
             
             
               
                 
                   b 
                   . 
                 
               
               
                 
                   
                     
                       F 
                       ^ 
                     
                     
                       s 
                       , 
                       i 
                       , 
                       k 
                     
                   
                   = 
                   
                     
                       exp 
                       ⁢ 
                       
                         { 
                         
                           
                             
                               
                                 
                                   ρ 
                                   ^ 
                                 
                                 
                                   s 
                                   , 
                                   i 
                                   , 
                                   k 
                                 
                               
                               
                                 
                                   P 
                                   _ 
                                 
                                 
                                   s 
                                   , 
                                   i 
                                   , 
                                   k 
                                 
                               
                             
                             ⁢ 
                             
                               P 
                               
                                 s 
                                 , 
                                 i 
                                 , 
                                 k 
                               
                             
                           
                           + 
                           
                             Const 
                             
                               s 
                               , 
                               i 
                               , 
                               k 
                             
                           
                         
                         } 
                       
                     
                     
                       
                         ∑ 
                         
                           l 
                           ∈ 
                           
                             Dem 
                             t 
                           
                         
                       
                       ⁢ 
                       
                         exp 
                         ⁢ 
                         
                           { 
                           
                             
                               
                                 
                                   
                                     ρ 
                                     ^ 
                                   
                                   
                                     s 
                                     , 
                                     t 
                                     , 
                                     l 
                                   
                                 
                                 
                                   
                                     P 
                                     _ 
                                   
                                   
                                     s 
                                     , 
                                     i 
                                     , 
                                     l 
                                   
                                 
                               
                               ⁢ 
                               
                                 P 
                                 
                                   s 
                                   , 
                                   i 
                                   , 
                                   l 
                                 
                               
                             
                             + 
                             
                               Const 
                               
                                 s 
                                 , 
                                 i 
                                 , 
                                 l 
                               
                             
                           
                           } 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Market 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Share 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     equation 
                   
                   ) 
                 
               
             
             
               
                 
                   c 
                   . 
                 
               
               
                 
                   
                     ∀ 
                     c 
                   
                   , 
                   
                     s 
                     ∈ 
                     
                       Cluster 
                       c 
                     
                   
                   , 
                   i 
                   , 
                   
                     
                       P 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         MIN 
                         
                           s 
                           , 
                           t 
                         
                       
                     
                     ≤ 
                     
                       
                         ∑ 
                         k 
                       
                       ⁢ 
                       
                         
                           F 
                           
                             s 
                             , 
                             t 
                             , 
                             k 
                           
                         
                         ⁢ 
                         
                           P 
                           
                             s 
                             , 
                             t 
                             , 
                             k 
                           
                         
                       
                     
                     ≤ 
                     
                       P 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         MAX 
                         
                           s 
                           , 
                           t 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Group 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     price 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     advance 
                     ⁢ 
                     
                       / 
                     
                     ⁢ 
                     decline 
                   
                   ) 
                 
               
             
             
               
                 
                   d 
                   . 
                 
               
               
                 
                   
                     
                       P 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         MIN 
                         
                           c 
                           , 
                           t 
                           , 
                           k 
                         
                       
                     
                     ≤ 
                     
                       P 
                       
                         c 
                         , 
                         t 
                         , 
                         k 
                       
                     
                     ≤ 
                     
                       P 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         MAX 
                         
                           c 
                           , 
                           t 
                           , 
                           k 
                         
                       
                     
                   
                   , 
                   
                     ∀ 
                     i 
                   
                   , 
                   k 
                 
               
               
                 
                   ( 
                   
                     Product 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     price 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     advance 
                     ⁢ 
                     
                       / 
                     
                     ⁢ 
                     decline 
                   
                   ) 
                 
               
             
             
               
                 
                   e 
                   . 
                 
               
               
                 
                   
                     ∀ 
                     c 
                   
                   , 
                   
                     
                       i 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             p 
                             
                               c 
                               , 
                               t 
                               , 
                               
                                 k 
                                 1 
                               
                             
                           
                           , 
                           
                             p 
                             
                               c 
                               , 
                               t 
                               , 
                               
                                 k 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                     
                     ∈ 
                     
                       
                         
                           Brand 
                           
                             c 
                             , 
                             t 
                           
                         
                         ⁢ 
                         
                           p 
                           
                             c 
                             , 
                             t 
                             , 
                             
                               k 
                               1 
                             
                           
                         
                       
                       ≤ 
                       
                         p 
                         
                           c 
                           , 
                           t 
                           , 
                           
                             k 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Brand 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Pricing 
                   
                   ) 
                 
               
             
             
               
                 
                   f 
                   . 
                 
               
               
                 
                   
                     ∀ 
                     c 
                   
                   , 
                   
                     
                       i 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             p 
                             
                               c 
                               , 
                               t 
                               , 
                               
                                 k 
                                 1 
                               
                             
                           
                           , 
                           
                             p 
                             
                               c 
                               , 
                               t 
                               , 
                               
                                 k 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                     
                     ∈ 
                     
                       
                         
                           Size 
                           
                             c 
                             , 
                             t 
                           
                         
                         ⁢ 
                         
                           p 
                           
                             c 
                             , 
                             t 
                             , 
                             
                               k 
                               1 
                             
                           
                         
                       
                       ≤ 
                       
                         p 
                         
                           c 
                           , 
                           t 
                           , 
                           
                             k 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Size 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Pricing 
                   
                   ) 
                 
               
             
             
               
                 
                   g 
                   . 
                 
               
               
                 
                   
                     ∀ 
                     c 
                   
                   , 
                   i 
                   , 
                   
                     
                       ( 
                       
                         
                           p 
                           
                             c 
                             , 
                             t 
                             , 
                             
                               k 
                               1 
                             
                           
                         
                         , 
                         
                           p 
                           
                             c 
                             , 
                             t 
                             , 
                             
                               k 
                               2 
                             
                           
                         
                       
                       ) 
                     
                     ∈ 
                     
                       Unit 
                       
                         c 
                         , 
                         t 
                       
                     
                   
                   , 
                   
                     s 
                     ∈ 
                     
                       Cluster 
                       c 
                     
                   
                   , 
                   
                     
                       
                         e 
                         
                           s 
                           , 
                           i 
                           , 
                           
                             k 
                             1 
                           
                         
                       
                       ⁢ 
                       
                         p 
                         
                           c 
                           , 
                           t 
                           , 
                           
                             k 
                             1 
                           
                         
                       
                     
                     ≤ 
                     
                       
                         e 
                         
                           s 
                           , 
                           t 
                           , 
                           
                             k 
                             2 
                           
                         
                       
                       ⁢ 
                       
                         p 
                         
                           c 
                           , 
                           t 
                           , 
                           
                             k 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Unit 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Pricing 
                   
                   ) 
                 
               
             
             
               
                 
                   h 
                   . 
                 
               
               
                 
                   
                     ∀ 
                     
                       L 
                       l 
                     
                   
                   , 
                   
                     l 
                     = 
                     1 
                   
                   , 
                   
                     … 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                        
                       L 
                        
                     
                   
                   , 
                   
                     ∀ 
                     
                       k 
                       1 
                     
                   
                   , 
                   
                     
                       k 
                       2 
                     
                     ∈ 
                     
                       L 
                       l 
                     
                   
                   , 
                   
                     ∀ 
                     c 
                   
                   , 
                   
                     
                       P 
                       
                         c 
                         , 
                         i 
                         , 
                         
                           k 
                           1 
                         
                       
                     
                     = 
                     
                       P 
                       
                         c 
                         , 
                         t 
                         , 
                         
                           k 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Line 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Pricing 
                   
                   ) 
                 
               
             
           
         
       
     
     The optimization problem P 2  has the following features: the objective and the group sales equations are nonlinear, while the rest of the constraints are linear. In the preferred embodiment, a heuristic approach is used to generate a good feasible starting point for the problem, if one exists. The heuristic works by repeatedly solving a related problem involving all the linear constraints 4-8, a set of linear constraints that approximate the group price advance and decline constraints (3), and an objective that involves the marginal contribution to profit of each product. 
     6. Heuristic to Generate the Feasible Solution: 
     The problem may be decomposed by store cluster and hence attention may be restricted to a single cluster Cluster c . for this explanation. 
     In the preferred embodiment, the heuristic focuses on finding good initial feasible solutions to the problem. First, a related problem is defined. 
     The market shares of all products are calculated based on a set of initial prices. Then these initial market shares are used as fixed weights to develop linear constraints that limit the advance and decline of demand group prices at each store. Specifically, let 
                   p   _         Cluster   c     ,   t       =     (           P     c   ,   t   ,   k               ⋮             P     c   ,   t   ,          Dem   t                    )       ,         p   _       Cluster   c       =     (             p   _         Cluster   c     ,   1               ⋮               p   _         Cluster   c     ,        G                  )       ,   and                     G     s   ,   t   ,   k       ⁡     (       p   _         Cluster   c     ,   t       )       ≡         exp   ⁢     {             ρ   ^       s   ,   i   ,   k           P   __       s   ,   t   ,   k         ⁢     P     c   ,   t   ,   k         +     Const     s   ,   t   ,   k         }           ∑     l   ∈     Dem   t         ⁢     exp   ⁢     {             ρ   ^       s   ,   t   ,   l           P   _       s   ,   t   ,   l         ⁢     P     c   ,   i   ,   l         +     Const     s   ,   t   ,   l         }           ⁢     ∀     s   ∈     Cluster   c             ,         
the market share of product k at store s given price vector  p   Cluster     c     ,i  at the store for demand group i. Then, if the initial price vectors are
   q   Cluster     c     ,i , ∀sεCluster c ,i, the approximate group price constraints are defined to be:
 
     
       
         
           
             
               ∀ 
               
                 s 
                 ∈ 
                 
                   Cluster 
                   c 
                 
               
             
             , 
             
               ∀ 
               i 
             
             , 
             
               
                 P 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   MIN 
                   
                     s 
                     , 
                     t 
                   
                 
               
               ≤ 
               
                 
                   ∑ 
                   k 
                 
                 ⁢ 
                 
                   
                     
                       G 
                       
                         s 
                         , 
                         t 
                         , 
                         k 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           q 
                           _ 
                         
                         
                           
                             Cluster 
                             c 
                           
                           , 
                           t 
                         
                       
                       ) 
                     
                   
                   ⁢ 
                   
                     P 
                     
                       c 
                       , 
                       t 
                       , 
                       k 
                     
                   
                 
               
               ≤ 
               
                 P 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     MAX 
                     
                       s 
                       , 
                       t 
                     
                   
                   . 
                 
               
             
           
         
       
     
     Note that the weights G s,i,k (  q   Cluster     c     ,i ) are constants and so the constraints are linear in P c,i,k . Now problem P 3  is defined as follows: 
     7. Problem P 3 (  q   Cluster     c   ) 
     Maximize: 
     
       
         
           
             
               ∑ 
               
                 s 
                 ∈ 
                 
                   Cluster 
                   c 
                 
               
             
             ⁢ 
             
               
                 ∑ 
                 
                   i 
                   ∈ 
                   G 
                 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     k 
                     ∈ 
                     
                       Dem 
                       t 
                     
                   
                 
                 ⁢ 
                 
                   
                     
                       exp 
                       ⁢ 
                       
                         { 
                         
                           
                             
                               
                                 
                                   ρ 
                                   ^ 
                                 
                                 
                                   s 
                                   , 
                                   t 
                                   , 
                                   k 
                                 
                               
                               
                                 
                                   P 
                                   _ 
                                 
                                 
                                   s 
                                   , 
                                   t 
                                   , 
                                   k 
                                 
                               
                             
                             ⁢ 
                             
                               P 
                               
                                 c 
                                 , 
                                 t 
                                 , 
                                 k 
                               
                             
                           
                           + 
                           
                             Const 
                             
                               s 
                               , 
                               t 
                               , 
                               k 
                             
                           
                         
                         } 
                       
                     
                     
                       
                         ∑ 
                         
                           l 
                           ∈ 
                           
                             Dem 
                             t 
                           
                         
                       
                       ⁢ 
                       
                         exp 
                         ⁢ 
                         
                           { 
                           
                             
                               
                                 
                                   
                                     ρ 
                                     ^ 
                                   
                                   
                                     s 
                                     , 
                                     t 
                                     , 
                                     l 
                                   
                                 
                                 
                                   
                                     P 
                                     _ 
                                   
                                   
                                     s 
                                     , 
                                     t 
                                     , 
                                     l 
                                   
                                 
                               
                               ⁢ 
                               
                                 P 
                                 
                                   c 
                                   , 
                                   t 
                                   , 
                                   l 
                                 
                               
                             
                             + 
                             
                               Const 
                               
                                 s 
                                 , 
                                 t 
                                 , 
                                 l 
                               
                             
                           
                           } 
                         
                       
                     
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         P 
                         
                           c 
                           , 
                           t 
                           , 
                           k 
                         
                       
                       - 
                       
                         C 
                         
                           s 
                           , 
                           t 
                           , 
                           k 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
     Subject to: 
     
       
         
           
             
               
                 1. 
               
               
                 
                   
                     ∀ 
                     
                       s 
                       ∈ 
                       
                         Cluster 
                         c 
                       
                     
                   
                   , 
                   
                     ∀ 
                     i 
                   
                   , 
                   
                     
                       P 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         MIN 
                         
                           s 
                           , 
                           t 
                         
                       
                     
                     ≤ 
                     
                       
                         ∑ 
                         k 
                       
                       ⁢ 
                       
                         
                           
                             G 
                             
                               s 
                               , 
                               t 
                               , 
                               k 
                             
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 q 
                                 _ 
                               
                               
                                 
                                   Cluster 
                                   c 
                                 
                                 , 
                                 t 
                               
                             
                             ) 
                           
                         
                         ⁢ 
                         
                           P 
                           
                             c 
                             , 
                             t 
                             , 
                             k 
                           
                         
                       
                     
                     ≤ 
                     
                       P 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         MAX 
                         
                           s 
                           , 
                           t 
                         
                       
                     
                   
                 
               
               
                 
                   (Group  Pricing - approximate) 
                 
               
             
             
               
                 2. 
               
               
                 
                   
                     
                       P 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         MIN 
                         
                           c 
                           , 
                           t 
                           , 
                           k 
                         
                       
                     
                     ≤ 
                     
                       P 
                       
                         c 
                         , 
                         t 
                         , 
                         k 
                       
                     
                     ≤ 
                     
                       P 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         MAX 
                         
                           c 
                           , 
                           i 
                           , 
                           k 
                         
                       
                     
                   
                   , 
                   
                     ∀ 
                     i 
                   
                   , 
                   k 
                 
               
               
                 
                   ( 
                   
                     Product 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     price 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     advance 
                     ⁢ 
                     
                       / 
                     
                     ⁢ 
                     decline 
                   
                   ) 
                 
               
             
             
               
                 3. 
               
               
                 
                   ∀ 
                   
                     
                       i 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             p 
                             
                               c 
                               , 
                               t 
                               , 
                               
                                 k 
                                 1 
                               
                             
                           
                           , 
                           
                             p 
                             
                               c 
                               , 
                               t 
                               , 
                               
                                 k 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                     
                     ∈ 
                     
                       
                         
                           Brand 
                           
                             c 
                             , 
                             t 
                           
                         
                         ⁢ 
                         
                           p 
                           
                             c 
                             , 
                             i 
                             , 
                             
                               k 
                               1 
                             
                           
                         
                       
                       ≤ 
                       
                         p 
                         
                           c 
                           , 
                           t 
                           , 
                           
                             k 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Brand 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Pricing 
                   
                   ) 
                 
               
             
             
               
                 4. 
               
               
                 
                   ∀ 
                   
                     
                       i 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             p 
                             
                               c 
                               , 
                               t 
                               , 
                               
                                 k 
                                 1 
                               
                             
                           
                           , 
                           
                             p 
                             
                               c 
                               , 
                               t 
                               , 
                               
                                 k 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                     
                     ∈ 
                     
                       
                         
                           Size 
                           
                             c 
                             , 
                             t 
                           
                         
                         ⁢ 
                         
                           p 
                           
                             c 
                             , 
                             t 
                             , 
                             
                               k 
                               1 
                             
                           
                         
                       
                       ≤ 
                       
                         p 
                         
                           c 
                           , 
                           t 
                           , 
                           
                             k 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Size 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Pricing 
                   
                   ) 
                 
               
             
             
               
                 5. 
               
               
                 
                   
                     ∀ 
                     i 
                   
                   , 
                   
                     
                       ( 
                       
                         
                           p 
                           
                             c 
                             , 
                             t 
                             , 
                             
                               k 
                               1 
                             
                           
                         
                         , 
                         
                           p 
                           
                             c 
                             , 
                             t 
                             , 
                             
                               k 
                               2 
                             
                           
                         
                       
                       ) 
                     
                     ∈ 
                     
                       Unit 
                       
                         c 
                         , 
                         t 
                       
                     
                   
                   , 
                   
                     s 
                     ∈ 
                     
                       Cluster 
                       c 
                     
                   
                   , 
                   
                     
                       
                         e 
                         
                           s 
                           , 
                           i 
                           , 
                           
                             k 
                             1 
                           
                         
                       
                       ⁢ 
                       
                         p 
                         
                           c 
                           , 
                           t 
                           , 
                           
                             k 
                             1 
                           
                         
                       
                     
                     ≤ 
                     
                       
                         e 
                         
                           s 
                           , 
                           t 
                           , 
                           
                             k 
                             2 
                           
                         
                       
                       ⁢ 
                       
                         p 
                         
                           c 
                           , 
                           t 
                           , 
                           
                             k 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Unit 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Pricing 
                   
                   ) 
                 
               
             
             
               
                 6. 
               
               
                 
                   
                     ∀ 
                     
                       L 
                       l 
                     
                   
                   , 
                   
                     l 
                     = 
                     1 
                   
                   , 
                   
                     … 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                        
                       L 
                        
                     
                   
                   , 
                   
                     ∀ 
                     
                       k 
                       1 
                     
                   
                   , 
                   
                     
                       k 
                       2 
                     
                     ∈ 
                     
                       L 
                       l 
                     
                   
                   , 
                   
                     
                       P 
                       
                         c 
                         , 
                         i 
                         , 
                         
                           k 
                           1 
                         
                       
                     
                     = 
                     
                       P 
                       
                         c 
                         , 
                         t 
                         , 
                         
                           k 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Line 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     Pricing 
                   
                   ) 
                 
               
             
           
         
       
     
     Problem P 3  has a nonlinear objective but only linear constraints. Note that the Problem P 3  does not include group sales. The objective maximizes the sum of marginal profit for all products over all the stores in the cluster. This approach may be easily extended to include a multiplier that approximates the group sales of all groups over all stores. 
     To facilitate understanding,  FIG. 6  is a flow chart of the preferred embodiment of the invention for providing a good initial feasible solution derived by solving Problem P 3  iteratively as follows:
     1. Set current incumbent prices to equal initial prices, which are prices currently used by the stores (step  604 ).   2. Set q Cluster     c    to be the set current incumbent prices for products at the stores in the cluster. Calculate the product market shares for each product at each store and generate the approximate group price constraints as a function of q Cluster     c    (Step  606 ). Solve problem P 3 (  q   Cluster     c   ) for the optimal solution  q * Cluster     c    (Step  608 ). Check to see if problem is feasible (Step  612 ). If the problem is infeasible, no feasible solution exists, terminate optimization (Step  616 ).   3. Compare the optimal solution  p * Cluster     c    to the problem P 3 (  q   Cluster     c   ) to  q   Cluster     c    (Step  620 ). If they are the same, then go to step 5. Else   4. Let  q   Cluster     c   =  p * Cluster     c    (Step  624 ) and go to step 2.   5. Redefine the bounds PMIN c,I  and PMAX c,I  as follows:   

     
       
         
           
             
               P 
               ⁢ 
               
                   
               
               ⁢ 
               
                 MIN 
                 
                   s 
                   , 
                   t 
                 
               
             
             ≡ 
             
               
                 MIN 
                 ( 
                 
                   
                     P 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       MIN 
                       
                         s 
                         , 
                         t 
                       
                     
                   
                   , 
                   
                     
                       ∑ 
                       k 
                     
                     ⁢ 
                     
                       
                         
                           G 
                           
                             s 
                             , 
                             t 
                             , 
                             k 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               p 
                               _ 
                             
                             
                               
                                 Cluster 
                                 c 
                               
                               , 
                               t 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         P 
                         
                           c 
                           , 
                           t 
                           , 
                           k 
                         
                       
                     
                   
                 
                 ) 
               
               ⁢ 
               
                   
               
               ⁢ 
               and 
             
           
         
       
     
               P   ⁢           ⁢     MAX     s   ,   t         ≡       MAX   (       P   ⁢           ⁢     MAX     s   ,   t         ,       ∑   k     ⁢         G     s   ,   t   ,   k       ⁡     (       p   _         Cluster   c     ,   t       )       ⁢     P     c   ,   i   ,   k             )     .           
This ensures that  p * Cluster     c    is feasible for P 2  (Step  628 ).
     6. Compute the group sales corresponding to  p * Cluster     c   . This is done by solving P 2  with  p   Cluster     c    fixed (Step  632 ).   7. Use resulting optimal price vector  p * Cluster     c    and group sales as a feasible starting solution to P 3  (Step  636 ).   

     In practice, convergence is declared in step 2 when the change in prices is below an acceptable tolerance threshold. 
     This method turns out to have several advantages. First, it finds an initial feasible solution by solving iteratively a nonlinear problem with linear constraints. Problem P 3  is considerably easier to solve than Problem P 1  and Problem P 2 . Second, the initial feasible solution turns out to be a good starting solution to Problem P 2 . It converges quite quickly from this starting point. Therefore these initial feasible solutions are provided as a starting point for solving Problem P 2 . 
     An example of this would have two products A and B. Each would be currently priced in the store at $1. In this example the rules only provide one constraint, which in this example is the Group price advance/decline constraint so that 
               ∀   c     ,     s   ∈     Cluster   c       ,   i   ,       P   ⁢           ⁢     MIN     s   ,   t         ≤       ∑   k     ⁢       F     s   ,   t   ,   k       ⁢     P     s   ,   t   ,   k           ≤     P   ⁢           ⁢       MAX     s   ,   t       .               
Therefore, for step  604 , the current incumbent prices are set equal to the current price in the store ($1). Using the demand and market share models, the market share for each product A and B is calculated for the given incumbent prices of $1. For example, the market share of A may be calculated to be 50% (0.5) and the market share of B may be calculated to be 50% (0.5). Therefore, the constraint
 
               P   ⁢           ⁢     MIN     s   ,   t         ≤       ∑   k     ⁢       F     s   ,   t   ,   k       ⁢     P     s   ,   t   ,   k           ≤     P   ⁢           ⁢     MAX     s   ,   t               
may be approximated by PMIN s,i ≦F A P A +F B P B ≦PMAX s,i , which yields the constraint equation PMIN s,i ≦(0.5)P A +(0.5)P B ≦PMAX s,i  (Step  606 ). The present incumbent prices of $1, the linear constraint equation, and the non-linear objective of maximizing profit are solved with a non-linear optimization problem software package to find optimized prices when the constraint equation is PMIN s,i ≦(0.5)P A +(0.5)P B ≦PMAX s,i  (Step  608 ). An example of such non-linear optimization problem software packages are MINOS™, which was invented by Bruce Murtagh and Michael Saunders and is marketed by Stanford Business Software, Inc, and CONOPT™, which was written by Arne Stolbjerg Drud and marketed by ARKI Consulting and Development A/S. Such software packages would provide a flag if the problem is not feasible (Steps  612  and  616 ).
 
     In this example the non-linear optimization software package provides an optimized price for product A to be $0.70 and an optimized price for product B to be $1.20. The current incumbent prices are compared to the optimized price. Since for product A $1≠0.70 or for product B $1≠1.20 (Step  620 ), then the incumbent prices are set equal to the optimized price (Step  624 ). From the new incumbent prices new market shares are calculated, using the model for predicting product share. For example, the new calculated market share may be 65% (0.65) for Product A and 35% (0.35) for Product B. The new market shares are placed into the constraint equation to provide a new constraint equation (Step  606 ), such as PMIN s,i ≦(0.65)P A +(0.35)P B ≦PMAX s,i . The present incumbent prices of $0.70 and $1.20, the linear constraint equation, and the non-linear objective of maximizing profit are solved with a non-linear optimization problem software package to find optimized prices when the constraint equation is
 
PMIN s,i ≦(0.65) P   A +(0.35) P   B ≦PMAX s,i (Step  608 ).
 
     In this example the non-linear optimization software package provides an optimized price for Product A to be $0.70 and an optimized price for Product B to be $1.20. Since the optimized prices are now equal to (or are within a threshold delta) the incumbent prices (Step  620 ), the group price bonds are reset so that the optimized prices for Product A ($0.70) and Product B ($1.20) are feasible for Problem P 2  (Step  628 ). The optimized prices are placed in the demand equation to generate group sales (Step  632 ). The optimized prices and group sales are then used as a feasible starting solution to Problem P 3  (Step  636 ). This means that the optimized prices, or Product A ($0.70) and Product B ($1.20), the generated group sales, the Profit Optimization Object, and the constraints of Problem P 2  are provided to a non-linear optimization problem software, such as MINOS or CONOPT. The use of the optimized prices allows for a quick convergence and an initial feasible solution for the solving of Problem P 2 . The initial feasible solution is important because in a non-linear equation there may be several local optimal locations and there are several areas where a solution is not feasible or there is no convergence or convergence is very slow. The initial feasible solution provides a feasible local optimal location where there is a faster convergence. 
     In general, optimizations with a linear objective and linear constraints are the easiest to solve. Optimizations with non-linear objectives and linear constraints may be harder to solve. Optimizations with non-linear objectives with non-linear constraints may be the hardest to solve. Therefore the heuristic uses an iterative approximation to convert an optimization with a non-linear objective and non-linear constraints to an optimization with a non-linear objective and linear constraints to provide an initial solution. Once the initial solution is solved, the initial solution is used as a starting point to optimize a non-linear objective with non-linear constraints. Such optimizations may provide a gradient around the initial solution to determine a direction in which to go. Prices are used in that direction to obtain a next point. A gradient around the new prices are then used to determine a new direction. This may be iteratively done until a boundary is reached or until the prices do not change. Other optimization heuristics and other approaches may be used in different embodiments of the invention. 
     The use of calculating demand group demand as a function of price and then using market share to calculate a product&#39;s demand from the demand group demand in an optimization scheme is a novel aspect of the invention. 
       FIGS. 7A and 7B  illustrate a computer system  900 , which forms part of the network  10  and is suitable for implementing embodiments of the present invention.  FIG. 7A  shows one possible physical form of the computer system. Of course, the computer system may have many physical forms ranging from an integrated circuit, a printed circuit board, and a small handheld device up to a huge super computer. Computer system  900  includes a monitor  902 , a display  904 , a housing  906 , a disk drive  908 , a keyboard  910 , and a mouse  912 . Disk  914  is a computer-readable medium used to transfer data to and from computer system  900 . 
       FIG. 7B  is an example of a block diagram for computer system  900 . Attached to system bus  920  are a wide variety of subsystems. Processor(s)  922  (also referred to as central processing units, or CPUs) are coupled to storage devices, including memory  924 . Memory  924  includes random access memory (RAM) and read-only memory (ROM). As is well known in the art, ROM acts to transfer data and instructions uni-directionally to the CPU and RAM is used typically to transfer data and instructions in a bi-directional manner. Both of these types of memories may include any suitable of the computer-readable media described below. A fixed disk  926  is also coupled bi-directionally to CPU  922 ; it provides additional data storage capacity and may also include any of the computer-readable media described below. Fixed disk  926  may be used to store programs, data, and the like and is typically a secondary storage medium (such as a hard disk) that is slower than primary storage. It will be appreciated that the information retained within fixed disk  926  may, in appropriate cases, be incorporated in standard fashion as virtual memory in memory  924 . Removable disk  914  may take the form of any of the computer-readable media described below. 
     CPU  922  is also coupled to a variety of input/output devices, such as display  904 , keyboard  910 , mouse  912  and speakers  930 . In general, an input/output device may be any of: video displays, track balls, mice, keyboards, microphones, touch-sensitive displays, transducer card readers, magnetic or paper tape readers, tablets, styluses, voice or handwriting recognizers, biometrics readers, or other computers. CPU  922  optionally may be coupled to another computer or telecommunications network using network interface  940 . With such a network interface, it is contemplated that the CPU might receive information from the network, or might output information to the network in the course of performing the above-described method steps. Furthermore, method embodiments of the present invention may execute solely upon CPU  922  or may execute over a network such as the Internet in conjunction with a remote CPU that shares a portion of the processing. 
     In addition, embodiments of the present invention further relate to computer storage products with a computer-readable medium that have computer code thereon for performing various computer-implemented operations. The media and computer code may be those specially designed and constructed for the purposes of the present invention, or they may be of the kind well known and available to those having skill in the computer software arts. Examples of computer-readable media include, but are not limited to: magnetic media such as hard disks, floppy disks, and magnetic tape; optical media such as CD-ROMs and holographic devices; magneto-optical media such as floptical disks; and hardware devices that are specially configured to store and execute program code, such as application-specific integrated circuits (ASICs), programmable logic devices (PLDs) and ROM and RAM devices. Examples of computer code include machine code, such as produced by a compiler, and files containing higher level code that are executed by a computer using an interpreter. 
       FIG. 8  is a schematic illustration of an embodiment of the invention that functions over a computer network  800 . The network  800  may be a local area network (LAN) or a wide area network (WAN). An example of a LAN is a private network used by a mid-sized company with a building complex. Publicly accessible WANs include the Internet, cellular telephone network, satellite systems and plain-old-telephone systems (POTS). Examples of private WANs include those used by multi-national corporations for their internal information system needs. The network  800  may also be a combination of private and/or public LANs and/or WANs. In such an embodiment the price optimizing system  100  is connected to the network  800 . The stores  124  are also connected to the network  800 . The stores  124  are able to bi-directionally communicate with the price optimizing system  100  over the network  800 . 
     V. EXAMPLES 
     In an example of the invention, using a single category for a chain of stores, the category had 430 products, 6 demand groups, and 433 stores. Table 1 shows the number of products in each demand group. Within each demand group products compete against each other, such as an 8 oz. Coke and a 24 oz. Coke. Within the category products may compete or may be complementary. An example of a category may be drinks. In this example, 12 months of scanner data was collected from the stores to provide the econometric demand model coefficients. In this example costs were not directly accounted for, but instead profit was measured by the difference between selling price and the unit cost. 
     
       
         
           
               
               
               
               
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
             
            
               
                 Demand Group 
                 A 
                 B 
                 C 
                 D 
                 E 
                 F 
               
               
                 Number of Products 
                 5 
                 44 
                 18 
                 17 
                 74 
                 21 
               
               
                   
               
            
           
         
       
     
     The econometric demand model coefficients were calculated by the use of the SAS™ program provided by SAS Institute Inc., of Cary N.C. An error analysis was performed on the demand model to determine a confidence interval for one standard deviation (68%). Table 2 shows the error for a one standard deviation confidence interval. Pct. Error is measured as 100*(forecasted value−actual value)/(forecasted value), and Abs. Pct. Error is calculated as the absolute value of Pct. Error. 
     
       
         
           
               
               
               
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
               
                   
                   
                   
                 Error 
                 Error 
                 Error 
               
               
                   
                   
                   
                 Weighted 
                 Weighted 
                 Weighted 
               
               
                   
                   
                   
                 Average 
                 Standard 
                 Average 
               
               
                 Level 
                 N 
                 Type 
                 Mean 
                 Deviation 
                 Median 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 Store  
                 22821 
                 Pct. Error 
                 −13.22% 
                 46.20% 
                 −1.97% 
               
               
                 Product 
                   
                 (Volume) 
                   
                   
                   
               
               
                 Store  
                 22821 
                 Abs. Pct. Error 
                 47.54% 
                 32.61% 
                 38.48% 
               
               
                 Product  
                   
                 (Volume) 
                   
                   
                   
               
               
                 Store  
                 22821 
                 Pct. Error  
                 −13.26% 
                 46.59% 
                 −2.01% 
               
               
                 Product  
                   
                 (Profit) 
                   
                   
                   
               
               
                 Store  
                 22821 
                 Abs. Pct. Error  
                 47.62% 
                 32.97% 
                 38.52% 
               
               
                 Product  
                   
                 (Profit) 
                   
                   
                   
               
               
                 Store  
                 2361 
                 Pct. Error  
                 −0.45% 
                 14.11% 
                 −1.06% 
               
               
                 Demand 
                   
                 (Volume) 
                   
                   
                   
               
               
                 Group 
                   
                   
                   
                   
                   
               
               
                 Store  
                 2361 
                 Abs. Pct. Error  
                 12.87% 
                 9.34% 
                 12.12% 
               
               
                 Demand 
                   
                 (Volume) 
                   
                   
                   
               
               
                 Group 
                   
                   
                   
                   
                   
               
               
                 Store  
                 2361 
                 Pct. Error 
                 −0.57% 
                 17.44% 
                 0.52% 
               
               
                 Demand 
                   
                 (Profit) 
                   
                   
                   
               
               
                 Group 
                   
                   
                   
                   
                   
               
               
                 Store  
                 2361 
                 Abs. Pct. Error  
                 14.34% 
                 12.28% 
                 12.99% 
               
               
                 Demand 
                   
                 (Profit) 
                   
                   
                   
               
               
                 Group 
                   
                   
                   
                   
                   
               
               
                 Store  
                 394 
                 Pct. Error  
                 0.15% 
                 7.27% 
                 0.32% 
               
               
                 Category 
                   
                 (Volume) 
                   
                   
                   
               
               
                 Store  
                 394 
                 Abs. Pct. Error 
                 6.27% 
                 4.77% 
                 5.86% 
               
               
                 Category 
                   
                 (Volume) 
                   
                   
                   
               
               
                 Store  
                 394 
                 Pct. Error  
                 0.87% 
                 7.59% 
                 1.52% 
               
               
                 Category 
                   
                 (Profit) 
                   
                   
                   
               
               
                 Store  
                 394 
                 Abs. Pct. Error  
                 6.80% 
                 5.02% 
                 6.41% 
               
               
                 Category 
                   
                 (Profit) 
               
               
                   
               
            
           
         
       
     
     Table 2 shows that at the Store Category level, the error weighted to standard deviation in volume of sales between predicted sales and actual sales was 7.27%. This means that there is a 68% confidence (one standard deviation) that the actual sales would fall within 7.27% of predicted sales or in other words the probability that the actual price would be more than one standard deviation away from the predicted price would be 32%. Examining the Store Demand Group level the error weighted standard deviation in volume increases to 14.11%. At the product level, the error weighted standard deviation in volume increases to 46.20%. Generally, the stores in this example carry extra inventory as safety stock, maybe as long as 4 weeks. It has been estimated that the 46.20% error would be covered by the general safety stock policies in most stores. 
     In this example group prices were allowed to vary between 85% and 105% of their current prices and the product prices were allowed to vary between 70% and 115% of their current levels. Therefore in this example 
     PMIN s,i    
     there were only two constraints, which were: 
     
         
         1. ∀sεCluster c , ∀i, 
       
    
               P   ⁢           ⁢     MIN     s   ,   i         ⁢     &lt;   _     ⁢       ∑   k     ⁢           ⁢         G     s   ,   i   ,   k       ⁡     (       q   _         Cluster   c     ,   1       )       ⁢     P     c   ,   i   ,   k           ⁢     &lt;   _     ⁢     P   ⁢           ⁢     MAX     s   ,   i               
(Group Pricing—approximate) with PMIN being 85% of current prices and PMAX being 105% of current prices.
     2. PMIN c,i,k ≦P c,i,k ≦PMAX c,i,k , ∀i,k (Product price advance/decline) with PMIN being 70% of current prices and PMAX being 115% of current prices   

     Profit was then optimized to provide new optimized prices, using the two constraints. This optimization is of profit is measured at the category level, since aggregated profit is of more interest than profit by demand group. This is because a store may be willing to have lower profits for a single demand group if the profit of an entire category is raised. 
       FIG. 9A  is a graph of original profit  964  from actual sales of the store using actual prices and optimal profit  968  from optimized sales resulting from the calculated optimized prices bounded by its probability. Along the x axis is profit in dollars. Along the y axis is the probability of achieving a particular profit value. The probability of getting at least a certain dollar profit is equal to the area under the right side of the curve. For the original profit there was a highest probability of obtaining between $1,200 and $1,600 profit. For the optimal profit there was a highest probability of obtaining between $3,200 and $4,800 profit. Thus showing a possible two fold increase in profit. 
       FIG. 9B  is a graph  972  of percentage increase in profit calculated by the optimal profit minus the original profit the difference divided by the original profit, 
                 Δ   ⁢           ⁢   %     =       (       P   optimal     -     P   original       )       P   original         ,         
and the probability of obtaining at least that percentage increase in profit. The x axis is the percentage increase in profit and the y axis is the probability of at least obtaining that percentage increase. According to this graph there is almost a 100% chance of increasing profit at least 104.3%.
 
     In the specification, examples of product are not intended to limit products covered by the claims. Products may for example include food, hardware, software, real estate, financial devices, intellectual property, raw material, and services. The products may be sold wholesale or retail, in a brick and mortar store or over the Internet, or through other sales methods. 
     While this invention has been described in terms of several preferred embodiments, there are alterations, permutations, and substitute equivalents, which fall within the scope of this invention. It should also be noted that there are many alternative ways of implementing the methods and apparatuses of the present invention. It is therefore intended that the following appended claims be interpreted as including all such alterations, permutations, and substitute equivalents as fall within the true spirit and scope of the present invention.