Patent Publication Number: US-7711588-B2

Title: Method and computer program for field spectrum optimization

Description:
RELATED INVENTION 
     The present invention claims priority under 35 U.S.C. §119(e) to: “Enhanced Strategic Planning and Optimization System,” U.S. Provisional Patent Application Ser. No. 60/459,925, filed 3 Apr. 2003, which is incorporated by reference herein. 
    
    
     TECHNICAL FIELD OF THE INVENTION 
     The present invention relates to the field of econometrics. More specifically, the present invention relates to enhanced processes and software for computing decisions for decision variables of a planning model characterizing an enterprise. 
     BACKGROUND OF THE INVENTION 
     Enterprise modeling is the process of characterizing a real-world enterprise, using mathematical representations, graphs, and/or pictures. An enterprise planning model often utilizes mathematical modeling techniques to analyze complex real-world scenarios, typically with the goal of improving or optimizing performance. Accordingly, an enterprise planning model can ideally provide insight into past, current, and future operating performance, enabling managers to spot trends, identify opportunities, and affect outcomes. 
     Multi-dimensional optimization entails determining a set of values that maximizes (or minimizes) a function of many decision variables. The types of mathematical relationships (for example, linear, nonlinear, or discontinuous) between the objective, the constraints, and the decision variables determine how difficult the optimization is to solve. The types of mathematical relationships also determine the solution methods or algorithms that can be used for optimization and the confidence that the solution is truly optimal. Such multi-dimensional optimization is problematic in that there is no known single multi-dimensional optimization strategy that can tackle all problems in a satisfactory way. In addition, the presence of constraints, even of simple ones, enhances this difficulty. 
     One known strategy for performing multi-dimensional optimization is an exhaustive search method in which an entire configuration space of scenarios (i.e., possible combinations of the variables) is performed to select an optimum out of all possibilities. The exhaustive search method will yield the global optimum. Unfortunately, this method is extremely computationally slow and therefore not applicable in practical situations. 
     Other strategies entail the iterative gradient search methods. These methods use information of the first and possibly second order derivatives of the criterion function to derive optimal search directions towards the optimum. Gradient search methods guarantee decreasing criterion values in successive iterations. The gradient search methods improve significantly over the exhaustive search method, but are still computationally slow and costly. A further disadvantage of these methods is that they are sensitive to the initial estimates of the unknowns if the criterion function has more than one optimum. As such, the gradient search algorithm may converge to a local optimum instead of the desired global optimum. A global optimum is one in which are no other feasible solutions with better objective function values. In contrast, a local optimum is one in which there are no other feasible solutions “in the vicinity” with better objective function values. 
     Much attention has been directed toward developing algorithms that circumvent convergence toward a local optimum. Two such algorithms are simulated annealing and genetic algorithms. The simulated annealing technique is essentially a local search, in which a move to an inferior solution is allowed with a probability that decreases as the process progresses. As such, there will always be a chance that a solution with a less good value might be retained in preference to a better solution. Thus, fine tuning of parameter settings is required. 
     Genetic algorithms are search techniques based on an abstract model of Darwinian evolution. Solutions are represented by fixed length strings over some alphabet (“gene” alphabet). Each string can be thought of as a “chromosome”. The value of the solution represents the fitness of the chromosome. Survival of the fittest principle is then applied to create a new generation with slow increase of average fitness. Accordingly, genetic algorithms also have the facility of allowing some weak members to survive in the solution pool, but typically have mechanisms for favoring fitter solutions. 
     Both simulated annealing and genetic algorithms have a fair chance of circumventing convergence toward a local optimum. In addition, both methods are much faster than the exhaustive search method. However, both methods are still computationally expensive as compared with gradient search methods. 
     Another problem with conventional optimization algorithms arises when optimizing in the presence of coupled decision variables. As used herein, the term “coupled” refers to decision variables and other objectives within an enterprise planning model that are connected causally to influence one another. This coupling further complicates optimization problems, leading to even more computational expense. 
     Thus, what is needed is a technique, within an enterprise planning model, that can efficiently and cost effectively solve complex optimization problems. 
     SUMMARY OF THE INVENTION 
     Accordingly, it is an advantage of the present invention that a method and computer program are provided for enhanced optimization calculation within a planning model. 
     It is another advantage of the present invention that a method and computer program are provided for computationally efficient optimization calculation of multi-dimensional planning models. 
     Another advantage of the present invention is that a method and computer program are provided that mitigate the computational difficulties associated with the coupling between decision variables within the planning model. 
     The above and other advantages of the present invention are carried out in one form in a planning model characterizing an enterprise, by a method of computing decisions for a set of decision variables. The method calls for generating a planning function representative of the planning model, the planning function depending upon the set of decision variables. The planning function is separated into independent planning functions, each of the independent planning functions depending upon different ones of the set of decision variables. The method further calls for independently optimizing each of the independent planning functions to obtain the decisions for the different ones of the set of decision variables, and presenting an outcome of the optimizing operation, the outcome indicating the obtained decisions. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A more complete understanding of the present invention may be derived by referring to the detailed description and claims when considered in connection with the Figures, wherein like reference numbers refer to similar items throughout the Figures, and: 
         FIG. 1  shows a block diagram of an exemplary computing environment within which the method of the present invention may be practiced; 
         FIG. 2  shows a flow chart of a decision variable optimization process in accordance with a preferred embodiment of the present invention; 
         FIG. 3  shows a table of exemplary strategic constraint factor values specified in connection with the execution of the decision variable optimization process; 
         FIG. 4  shows a flow chart of an optimization subprocess in accordance with a preferred embodiment of the present invention; 
         FIG. 5  shows a graph of an optimum pricing envelope displaying profit (primary objective) versus dollar sales (strategic objective); 
         FIG. 6  shows an exemplary table of optimized decisions for decision variables of an exemplary scenario computed through the execution of the decision variable optimization process; 
         FIG. 7  shows a graph of an optimum pricing band displaying profit (primary objective) versus dollar sales (strategic objective) and volume (strategic objective); 
         FIG. 8  shows a flow chart of an embedded constraint subprocess of the present invention; 
         FIG. 9  shows a table of exemplary embedded constraint factor values specified in connection with the execution of the embedded constraint subprocess; 
         FIG. 10  shows a table depicting a first exemplary planning model derived through the execution of the decision variable computation process of  FIG. 2 ; 
         FIG. 11  shows a table depicting a second exemplary planning model derived through the execution of the decision variable computation process; 
         FIG. 12  shows a table depicting a third exemplary planning model derived through the execution of the decision variable computation process; and 
         FIG. 13  shows a table depicting a fourth exemplary planning model derived through the execution of the decision variable computation process. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The following is a glossary of terms used herein: 
     Enterprise: Any public, private, or governmental organization that provides items to be consumed by others, whether or not for profit. Presumably, an enterprise competes with other enterprises for the attention of customers and potential customers. 
     Items: Are products in the form of goods, services, or a combination of goods and services. Items are broadly defined so that the same good and/or service provided in different market segments may be considered different items within the present context. Moreover, items are considered to be consumed within the present context when physically and/or legally transferred to the customer, such as when a transaction occurs. 
     Planning Model: A user-defined characterization of a real-world enterprise. 
     Planning Function: A mathematical representation of a planning model. The planning function can include of set of related mathematical functions whose purpose is to simulate the response of the enterprise being modeled. The planning function can be linear, consisting of solely linear functions, or nonlinear, involving one or more nonlinear functions. The planning function can include primary objective functions and strategic objective functions. 
     Primary Objectives: Represent the desires/goals of the decision maker/designer for the enterprise—such as to maximize profit, minimize cost, and so forth. 
     Primary Objective Function: Is a mathematical equation that represents one of the primary objectives. The primary objective function depends upon a set of decision variables. 
     Strategic Objectives: Represent factors that the decision maker/designer seeks to consider in conjunction with the primary objectives. The strategic objectives do not represent a physical restriction on the enterprise, thus need not be met rigorously. Strategic objectives include, for example, dollar sales, revenue, price image, service level, risk, product availability, product selection, market share, and so forth. 
     Strategic Objective Function: Is a mathematical equation that represents one of the strategic objectives. The strategic objective function depends upon a subset of the decision variables. 
     Decision Variables: Are variables under the control of the decision maker that could have an impact on the solution of the problem of interest. Decision variables include, for example, price, promotion type, promotion date, promotion duration, promotional discount, purchase date, product location, purchasing quantity, shelf space, product assortment, and so forth. 
     Linear Functions: Contain terms each of which is composed of only a single, continuous variable raised to (and only to) the power of one. 
     Nonlinear Functions: Are those in which more than a single variable may appear in a single term, and the variables may be raised to any power. 
     Continuous Functions: Are those in which “small” changes in the input produce “small” changes in the output. 
     Discontinuous Functions: Are those in which “small” changes in the input can produce abrupt changes in the output. 
     Those skilled in the art will appreciate that managers of an enterprise desire to understand and achieve the objectives of the enterprise. However, managers typically do not wish merely to achieve a particular primary objective, but may additionally wish to accommodate broader considerations, i.e., the strategic objectives, in conjunction with the primary objectives. The preferred embodiments of the present invention provide a useful tool for computing decisions for a set of decision variables of a planning model. Moreover, the present invention enables the economically efficient computation of decisions, while taking into account the relationship between primary objectives and strategic objectives of an enterprise. 
     The present invention is designated “field spectrum optimization.” The term “field” refers to a Lagrange multiplier that is employed in the present invention as a strategic factor and/or a constraint factor. In other words, the Lagrange multiplier is generally analogous to a “field” as that term may be employed in physics. The term “spectrum” refers to the set of all instances of a given field, i.e., all values for a particular strategic factor or embedded factor. Thus, the field spectrum optimization methodology described herein solves, i.e., optimizes, over the “spectrum” (set of values) defined for a “field” (strategic and/or constraint factor). Moreover, the field spectrum optimization methodology described herein decouples decision variables (discussed below) to further facilitate economically efficient optimization of decisions. 
     In the following discussion relating to  FIGS. 1-12 , each Figure&#39;s reference numerals are keyed-in to its respective Figure number, i.e.,  FIG. 1  has reference numerals in the  100 &#39;s,  FIG. 2  has reference numerals in the  200 &#39;s, and so forth. 
     FIG.  1   
       FIG. 1  shows a block diagram of an exemplary computing environment  100  within which the method of the present invention may be practiced.  FIG. 1  depicts a processor section  102  in communication with an input/output section  104  and a memory  106 . Nothing prevents processor section  102 , input/output section  104  and/or memory  106  from including numerous subsections that may or may not be located near each other. Thus, computing environment  100  may be provided by any of a vast number of general or special purpose computers and/or computer networks. 
     Memory  106  represents any manner of computer-readable media, including both primary memory (e.g., semiconductor devices with higher data transfer rates) and secondary memory (e.g., semiconductor, magnetic, and/or optical storage devices with lower data transfer rates). Input/output section  104  represents any manner of input elements (i.e., a keyboard, mouse, etc.) and output elements (i.e., monitors, printers, etc.). Data and computer programs may be transferred in to or out from memory  106  through input/output section  104 . 
     Memory  106  is depicted as having a code section  108  and a data section  110 . Those skilled in the art will appreciate that any distinction between sections  108  and  110  may be due merely to different types of data and need not be due to physically different types of memory devices. Code section  108  stores any number of the types of computer programs typically found on computers and/or computer networks. In addition, code section  108  includes a planning model computer program  112  that may be partitioned into a planning function definition code segment  114  and a planning function separation code segment  116 . 
     Code section  108  also includes optimization algorithm computer programs  118 . Optimization algorithm computer programs  118  of code section  108  may include specific optimization algorithms and/or general optimization algorithms. Specific optimization algorithms include, but are not limited to, closed form analytical solution, one-dimensional maximization of continuous decision variables, one-dimensional maximization of discrete decision variables, a general multi-dimensional optimization method, or other user-defined optimization algorithms, known to those skilled in the art. General optimization algorithms include, but are not limited to, ant algorithm, genetic algorithm, tabu algorithm, simulated annealing, branch and bound, and other general optimization algorithms known to those skilled in the art. 
     Prior to being transferred to memory  106 , computer programs  112  or  118  may have resided on a computer-readable medium  120 . Computer-readable medium  120  represents any location or storage device from which computer programs may be accessed, including remote servers, CD ROMs, and the like. Computer programs  112  and  118 , and code segments  114  and  116  thereof, provide computer software that instructs processor section  102  how to manipulate and process a planning function  122  representative of a planning model of an enterprise, and how to store the resulting solution of planning function  122 . Planning function  122  generally includes one or more primary objective functions  124  that depend upon a set of decision variables  125 . Planning function  122  optionally includes one or more strategic objective functions  126 , each of which is coupled with a strategic constraint factor  128 , and each of which also depends upon decision variables  125 . Planning function  122  is described in greater detail in connection with  FIG. 2 . 
     FIG.  2   
       FIG. 2  shows a flow chart of a decision variable optimization process  200  in accordance with a preferred embodiment of the present invention.  FIG. 2  provides a conceptual vision of the present invention for comprehensiveness of description. However, those skilled in the art will readily recognize that task flow may vary greatly from that which is presented herein in response to actual code instructions of a computer program of the present invention. In addition, it should be understood that tasks described herein may be performed manually by a decision-maker or may be carried out, at least in part, within computing environment  100  by computer programs  112  and  118 . 
     Process  200  facilitates computationally efficient optimization calculation of decision variables  125 . In addition, process  200  mitigates the computational difficulties associated with the coupling between decision variables  125  within the planning model. For purposes of the present invention, it is assumed that one or more primary objectives and, optionally, one or more strategic objectives may be incorporated into a planning model characterizing an enterprise. 
     Decision variable optimization process  200  begins with a task  202 . At task  202 , primary objective function  124  is defined. More specifically, a primary objective of the enterprise is mathematically modeled through primary objective function  124  at task  202 . Primary objective function  124 , namely V{x}, is a function of a set of decision variables  125 , namely {x}. For simplicity of illustration, the present invention is described in connection with a single primary objective function  124 . However, the present invention can be readily expanded to include additional primary objectives of the planning model. Consequently, task  202  may further define additional primary objective functions for additional primary objectives of the planning model. 
     Following task  202 , a query task  204  determines whether the planning model includes one or more strategic objectives. When no strategic objective is defined in the planning model, process control proceeds to a task  206  (discussed below). However, when query task  204  determines that the planning model includes a strategic objective, process  200  proceeds to a task  208 . 
     At task  208 , strategic objective function  126  is defined. More specifically, a strategic objective of the enterprise is mathematically modeled through strategic objective function  124  at task  208 . The strategic objectives represent significant business decisions that may be made by the decision-maker that can affect the primary objective. Strategic objective function  126 , namely STG{x}, is also a function of decision variables  125 , namely {x}. 
     A task  210 , performed in connection with task  208 , couples strategic objective function  124  with strategic constraint factor  128 , namely λ. The constant, represented by λ, is a Lagrange multiplier employed to test the effect that the strategic constraint, modeled by strategic constraint function  126 , can have on the planning model. That is, different values for strategic constraint function  126  adjust an influence that the strategic objective will have on the planning model. 
     In response to task  210 , a task  212  specifies values for strategic constraint factor  128 . These values can be user-specified or the values may be generated automatically by computing environment  100 . The result of task  212  may be a table of values associated with particular scenario identifiers, and will be described in greater detail in connection with  FIG. 3 . 
     Following task  212 , a query task  214  determines whether the planning model includes another strategic objective. When there is another strategic objective, representing another significant business decision that may be made by the decision-maker that can affect the primary objective, process  200  loops back to task  208  to define a second strategic objective function, couple the second strategic objective function with a second strategic constraint factor, and specify values for the second strategic constraint factor. Subsequent tasks in the flowchart of  FIG. 2  shall be discussed herein below, following discussion of  FIG. 3 . 
     FIG.  3   
       FIG. 3  shows a table  300  of exemplary strategic constraint factor values  128  specified in connection with the execution of decision variable optimization process  200 . In a hypothetical situation, the planning model includes two strategic objectives. Accordingly, through the execution of process  200 , two strategic objective functions  126  are defined. Similarly, values for two strategic constraint factors  128  (i.e., two fields), namely λ 1 , and λ 2 , are specified. In table  300 , each of a number of strategic constraint scenarios  302  are given a unique strategic constraint scenario identifier  304 . Constraint factor values  306  for a first one of strategic constraint factors  128 , λ 1 , are specified as the set (i.e., spectrum) including 0, 0.2, 0.4, and 0.6. Constraint factor values  308  for a second one of strategic constraint factors  128 , λ 2 , are specified as the set (i.e., spectrum) including 0, 0.25, 0.5, 0.75, and 1. Strategic constraint scenarios  302  represent every possible combination of constraint factor values  306  and  308 . Thus, an exemplary strategic constraint scenario  302 ′, represented by strategic constraint scenario identifier  304  of “G” reveals λ 1 =0.2 and λ 2 =0.25. Optimization (discussed below) will be performed for each of strategic constraint scenarios  302 . 
     FIG.  2  Continued 
     Referring back to decision variable optimization process  200 , when query task  214  determines that there are no further strategic objectives defined within the planning model, process control proceeds to task  206 . Similarly, as described above, when query task  204  determines that the planning model does not include strategic objectives, process control proceeds to task  206 . 
     Task  206  combines primary objective function  124  and strategic objective function(s)  126 , multiplied by their unique strategic constraint factors  128 , to generate planning function  122 . The generalized planning function  122  is represented as follows: 
                   SP   =       V   ⁡     (     {   x   }     )       +       ∑     s   =   1     S     ⁢           ⁢       λ   s     ⁢       STG   s     ⁡     (     {   x   }     )                     (   1   )               
where “S” is the number of strategic objectives. Equation (1) may also be subject to a set of tactical constraints (not shown). The tactical constraints operate as decision-level constraints with a possible strategic importance. Exemplary tactical constraints include a maximum or a minimum price for an item or class of items, and a defined relationship between prices, for example, the price of item ‘n’ must be less than or equal to the price of item ‘m’. Overall ceilings or floors can also be set for tactical constraints. For example, the system can be constrained so that overall price change is less than a give percentage.
 
     Following task  206 , process  200  proceeds to a query task  216 . Query task  216  determines whether any of decision variables  125  are coupled in primary objective function  124  of planning function  122 . That is, query task  216  determines whether any decision variables  125  in primary objective function  124  are connected causally to influence one another. Examples of coupled decision variables  125  include the effects of competing products and stores; the effects of available demand to different purchasing decisions; spatial and temporal dependencies; the effect of price of a brand to a customer&#39;s choice of a brand; and so forth. The determination of coupling between decision variables  125  in primary objective function  124  of planning function further complicates the optimization calculation of decision variables  125 . 
     When query task  216  determines that decision variables  125  are coupled, process  200  proceeds to a task  218 . Task  218  causes an embedded constraint subprocess to be performed. The embedded constraint subprocess is described in detail below in connection with  FIGS. 8-9 . The determination of coupling between decision variables  125  in primary objective function  124  of planning function further complicates the optimization calculation of decision variables  125 . As will be discussed below, the embedded constraint subprocess mitigates the computational difficulties associated with this coupling. 
     Following the execution of the embedded constraint subprocess initiated at task  218 , program control proceeds to a task  220 . Similarly, when query task  216  determines that decision variables  125  are not coupled, program control also proceeds to task  220 . At task  220 , an optimization algorithm is selected from the group of optimization algorithm computer programs  118 . As discussed above, optimization algorithm computer programs  118  of code section  108  may include specific optimization algorithms and/or general optimization algorithms. The particular optimization algorithm selected will depend upon the structure of planning function  122 . 
     A task  222  is performed in response to task  220 . At task  222 , planning function  122  is separated into independent planning functions in order to simplify the optimization calculation. In a preferred embodiment, each of the independent planning functions depends upon a different set of decision variables  125 . Once separated, these independent planning functions can thus be treated as a sum of independent planning models, one for each item, as follows: 
                   SP   =       ∑   i             ⁢           ⁢       SP   i     ⁡     (       p   i     ,   λ     )                 (   2   )               
where the independent planning functions are represented by SP i , and each independent planning function for item i (i.e., SP i  depends only upon a set of decision variables  125 , represented by p i . It should be understood that the set of decision variables  125  may include one or more of decision variables  125 , although the total number of decision variables  125  is less than the total number of decision variables for the planning model represented by planning function  122 .
 
     Consequently, a generalized independent planning function can be presented as follows: 
                       SP   i     ⁡     (     x   i     )       =         max     x   i       ⁢       V   i     ⁡     (     {     x   i     }     )         +       ∑     s   =   1     S     ⁢           ⁢       λ   s     ⁢       STG   s     ⁡     (     {     x   i     }     )                     (   3   )               
where “S” is the number of strategic objectives, and the primary objective function, V i (x i ) is to be maximized for decision variable (x i ). The ability to separate planning function  122  into a number of smaller components, i.e., independent planning functions, for subsequent optimization advantageously enhances the optimization process. Significant savings are realized in terms of required computing time and computing power by reducing a multi-dimensional planning function into lower order independent planning functions.
 
     Following task  222 , a task  224  is initiated. Task  224  causes an optimization subprocess to be performed. The optimization subprocess is described in detail below in connection with  FIG. 4 . The optimization subprocess is performed to optimize a set of decision variables  125  for each of the independent planning functions, SP i , of task  222 . Subsequent tasks in the flowchart of  FIG. 2  shall be discussed herein below, following discussion of  FIG. 4 . 
     FIG.  4   
       FIG. 4  shows a flow chart of an optimization subprocess  400  in accordance with a preferred embodiment of the present invention. Optimization entails solving the independent planning function, SP i , to determine an optimum decision or set of decisions for that independent planning function, SP i , at each scenario (for example, at each of strategic constraint scenarios  302 ).  FIG. 4  provides a conceptual vision of the present invention for comprehensiveness of description. However, those skilled in the art will readily recognize that task flow may vary greatly from that which is presented herein in response to actual code instructions of a computer program of the present invention. In addition, tasks described herein may be performed manually by a decision-maker or may be carried out, at least in part, within computing environment  100 . 
     Optimization subprocess  400  begins with a task  402 . A task  402  a “next” scenario is selected. For purposes of the present invention strategic constraint scenarios  302  are the scenarios selectable at task  402 . It should also be noted that at a first iteration of task  402 , the “next” scenario  302  is a first one of the scenarios  302 , in this case represented by strategic constraint scenario identifier  304 , “A”. Thereafter, the term “next” applies. 
     A task  404  is performed in response to task  402 . At task  404 , a “next” independent planning function, SP i , is selected. Again is should be noted that at a first iteration of task  404 , the “next” independent planning function is a first one of the independent planning functions, SP i , determined at task  222  of decision variable optimization process  200 . Thereafter, the term “next” applies. 
     Following task  404 , a task  406  optimizes a set of one or more decision variables  125  for the selected independent planning function, SP i , and the selected scenario  302 . Optimization calculations are performed utilizing the optimization algorithm selected at task  220  of decision variable optimization process  200 . 
     Following optimization task  406 , a query task  408  is performed to determine whether there is another one independent planning function, SP i , for which optimization is to be performed at the selected scenario. When there is another independent planning function, SP i , program control loops back to task  404  to select the next independent planning function and perform another optimization calculation for the next independent planning function. However, when query task  408  determines that there are no further independent planning functions, SP i , for which optimization is to be performed, optimization process  200  proceeds to a task  410 . 
     Task  410  causes the results of the optimization calculations to be compiled for the selected scenario. As mentioned above, optimization task  406  solves each independent planning function, SP i , to determine an optimum decision or decisions for that independent planning function, SP i , at the selected scenario  302 . Furthermore, as discussed above, the independent planning functions, SP i , can be treated as a sum of independent planning models. Accordingly, an outcome of such an iterative approach is a set of optimum decisions, {x*}, for each of decision variables  125  at the selected scenario  302 . 
     Once the optimum decisions {x*} are determined at task  406 , metrics that are functions of the optimum decisions can be calculated. These metrics can include, for example, financial metrics corresponding to the primary objective and strategic metrics corresponding to the strategic objective(s). The financial metrics can include, for example, gross profit, net profit, and so forth. The strategic metrics can include, for example, dollar sales, revenue, price image, service level, risk, product availability, product selection, market share, and so forth. Compilation of the results of the optimization calculations may entail storing the results in database form in data section  110  in associated with one of scenario identifiers  304  associated with the selected one of scenarios  302 . 
     Following task  410 , a query task  412  is performed to determine whether there is another one of scenarios  302  for which optimization computations are to be performed. When there is another one of scenarios  302 , program control loops back to task  402  to select the next one of scenarios  302 , and perform optimization calculations for the next one of scenarios  302 . However, when query task  412  determines that there are no further scenarios  302  at which optimization is to be performed, optimization subprocess  400  exits. 
     FIG.  2  Continued 
     Referring back to decision variable optimization process  200 , following the execution of optimization subprocess  400  at task  224 , a task  228  is performed. At task  228 , an outcome of optimization subprocess  400  is presented to the user via input/output section  104 . As mentioned above, optimization subprocess  400  optimizes each independent planning function, SP i , to determine an optimum decision or set of optimum decisions for that independent planning function, SP i , at each scenario (for example, at each of strategic constraint scenarios  302 ). Furthermore, as discussed above, the optimization results, i.e., the optimum decisions, for all of the independent planning functions, are compiled at task  410 , and metrics that are functions of the optimum decisions are calculated. Task  228  presents these metrics to a decision-maker. The metrics can be presented in a number of formats. For example, a textual list of optimized decisions and/or a variety of graphical representations may be displayed. Following task  228 , decision variable optimization process exits. 
     FIG.  5   
       FIG. 5  shows an exemplary graph  500  of an optimum pricing envelope  501  displaying profit (primary objective) versus dollar sales (strategic objective). That is, graph  500  shows profit versus dollar sales, when decision variables  125  are prices, pi, for items (i.e., goods, services, or a combination of goods and services offered by the enterprise). As discussed above, the independent planning functions can be treated as a sum of independent strategic models. Accordingly, graph  500  represents the sum of the independent planning functions. Graph  500  shows that, starting from a very low dollar sales, profits may increase as the aggregate of prices (decisions for decision variables  125 ) increases. However, at some point, further increase in the aggregate of prices causes dollar sales to increase, but results in a drop in profit. 
     When the enterprise offers many different products, a vast number of combinations of different pricing scenarios may be devised. Each pricing scenario represents a different mix of prices that may be offered for a set of products being evaluated. Each scenario is represented by a point  502  in graph  500 , of which only a few of all possible pricing scenario points  502  are shown. Only those pricing scenario points  502  on envelope  501  are optimum pricing scenarios. There are two regions of interest in graph  500 . Pricing scenario points  502  depicted in a region  504 , which reside within or underneath envelope  501 , are inefficient. Whereas, pricing scenario points  502  depicted in a region  506 , which reside outside envelope  501  are unachievable. By selecting one of pricing scenario points  502  on envelope  501 , a user can be presented with the outcome, i.e., a set of optimized decisions for decision variables  125 . 
     Each of pricing scenario points  502  on envelope  501  represents a set of optimized decisions for decision variables  125  at one of strategic constraint scenarios  302 . A decision-maker may employ the outcome of the optimization calculations as a guide to determine a preferred strategy, in this case, a preferred pricing strategy, for the enterprise. For example, a decision-maker may elect to sacrifice some profit to gain some amount of dollar sales. If such is the case, the decision-maker may select one of pricing scenario points  502  toward the right side of envelope  501 , such as an exemplary pricing scenario point  502 ′. As readily illustrated in exemplary graph  500 , some reduction in profit and a gain in dollar sales may occur. The decision-maker may determine that a short term reduction in profit, with increasing dollar sales, may be strategically important to the long term success of the enterprise. 
     FIG.  6   
       FIG. 6  shows an exemplary table  600  of optimized decisions  602  for decision variables  125  of exemplary pricing scenario point  502 ′ computed through the execution of decision variable optimization process  200 . Table  600  is illustrated for only one of pricing scenario points  502  for simplicity of illustration. However, it should be understood, that depending upon which strategy (in this case, profit versus dollar sales) that a decision-maker elects, the decision-maker can select any of pricing scenario points  502  along envelope  501 . Thus, a decision-maker may highlight any of pricing scenario points  502  along envelope  501  of graph  500 , to obtain a set of optimized decisions, such as, prices for items, promotions, and so forth, related to a particular strategic constraint scenario  302 . 
     Table  600  shows a list of items  604 , uniquely identified by item identifiers  606 , each being associated with one of decisions  602 . In this case, the optimized decisions  602  are prices for each of items  604 . Table  600  may also include non-price parameters  606  in association with items  604 . Non-price parameters  608  may include timing, availability, promotion type, customer type, price thresholds, and other user-specified parameters of interest. 
     FIG.  7   
       FIG. 7  shows a graph  700  of an optimum pricing band  702  displaying profit versus dollar sales and volume. In this exemplary situation, two strategic constraints (dollar sales and value) may influence the outcome of the optimization. In such a situation, a general form for planning function  122  may be as follows:
   SP=V ({ x })+λ 1   STG   1 ({x})+λ 2   STG   2 ({ x })  (4) 
     The generalized planning function of equation (4) illustrates an interaction between primary objective function  124  and each of strategic objective functions  126 . An optimization for each of strategic constraint scenarios  302  yields a series of optimum curves, each slightly shifted from one another. The result is optimum pricing band  702  of scenario points  704 . Like graph  500 , scenario points  704  depicted in a region  706 , which reside within or underneath band  702 , are inefficient. Whereas, scenario points  704  depicted in a region  708 , which reside outside band  702  are unachievable. By selecting one of scenario points  704  within band  702 , a decision-maker can be presented with the outcome, i.e., a set of optimized decisions for decision variables  125 , when two strategic constraints have been considered, i.e., at one of strategic constraint scenarios  302 . 
     FIG.  8   
       FIG. 8  shows a flow chart of an embedded constraint subprocess  800  of the present invention.  FIG. 8  provides a conceptual vision of the present invention for comprehensiveness of description. However, those skilled in the art will readily recognize that task flow may vary greatly from that which is presented herein in response to actual code instructions of a computer program of the present invention. In addition, tasks described herein may be performed manually by a decision-maker or may be carried out, at least in part, within computing environment  100 . It should be recalled that when query task  216  of optimization process  200  determines that decision variables  125  are coupled in primary objective function  124 , a task  218  initiates embedded constraint process  800 . 
     A generalized primary objective function  124  that includes a coupling between decision variables may be represented as follows: 
                     V   ⁡     (     {   p   }     )       =       ∑   i             ⁢           ⁢           h   i     ⁡     (     p   i     )       ⁢       g   i     ⁡     (     p   i     )             ∑   k             ⁢           ⁢       g   k     ⁡     (     p   k     )                     (   5   )               
The function h i  could be any non-linear function of decision variables  125  for decision i. In this generalized primary objective function  124 , the decision variable  125  is the price of an item, and the denominator Σ k g k (p k ) couples all of decision variables  125  together. Without this coupling, it would be possible to optimize each decision variable  125  independently. However, with this coupling, the optimization becomes more complex. A standard technique for determining the set of prices {p} which would maximize (or minimize) primary objective function  124 , V({p}), is to use some general optimization method. If the function is continuous, a gradient search may be employed. If the function is discontinuous, simulated annealing, genetic algorithm, or some other general method may be employed. However, such algorithms are expensive in terms of computational time and expense. Embedded constraint subprocess  800  is performed to reduce or eliminate this coupling of decision variables  125 , thus mitigating computational difficulties and expense associated with the coupling between decision variables.
 
     Embedded constraint subprocess  800  begins with a task  802 . At task  802 , an embedded constraint variable representing an embedded constraint is included in primary objective function  124 . Utilizing the exemplary primary objective function  124  of equation (5), the introduction of the embedded constraint variable is as follows: 
                     V   ⁡     (     {   p   }     )       =       ∑   i             ⁢           ⁢           h   i     ⁡     (     p   i     )       ⁢       g   i     ⁡     (     p   i     )         Z               (   6   )               
The embedded constraint variable “Z” is introduced as a new variable to remove the coupling. However, so that the functional form of the original primary objective function  124  remains unchanged, the variable “Z” must satisfy an embedded constraint function.
 
     A task  804 , performed in connection with task  802  defines an embedded constraint function. The embedded constraint function is defined to include the embedded constraint variable “Z” as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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     Next, a task  806  couples the embedded constraint function with a Lagrange multiplier, i.e., an embedded constraint factor, γ, and a task  808  includes the embedded constraint function coupled with the embedded constraint factor, γ, with primary objective function  124  of planning function  122 . Thus, primary objective function  124  becomes an effective objective function {tilde over (V)}, as follows: 
     
       
         
           
             
               
                 
                   
                     
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     As such, the objective function V({p},Z) can be maximized (or minimized) while satisfying its constraint by introducing embedded constraint factor, γ, and defining an effective primary objective function {tilde over (V)}. This transformation has allowed the coupling between decision variables  125  to be broken. Each optimum p* can now be determined independently by maximizing the following:
 
p i *(γ):max (h i (p i )g i (p i )+γg i (p i ))  (9)
 
     Following task  808 , a task  810  specifies values for the embedded constraint factor, γ. Embedded constraint factor, γ, can be determined in a number of ways. If the effective primary objective function is continuous, a gradient method such as Newton&#39;s Method could be employed. If the effective primary objective function is discontinuous, a bisection, grid search, or other discontinuous method may be employed. Alternatively, an estimate for the values of γ may be determined by setting: 
                         ∂       V   ~     ⁡     (       {   p   }     ,   Z     )           ∂   Z       ⁢     ❘     p   *         =   0           (   10   )               
where p* represents a set of optimized decisions (for example, prices), which leads to
 
                   γ   =         -       h   i     ⁡     (     p   i   *     )         ⁢       g   i     ⁡     (     p   i   *     )             ∑   k     ⁢       g   k     ⁡     (     p   k     )                   (   11   )               
Since the optimum decision (such as, optimum prices) are not known, estimates of the optimum decisions can also be used to provide a good estimate of the embedded constraint factor, γ. The result of task  810  may be a table of values associated with particular scenario identifiers, and will be described in greater detail in connection with  FIG. 9 .
 
     Following task  810 , embedded constraint subprocess  800  exits, and program control returns to task  222  of decision variable optimization process  200 . Given the strategic constraint factor γ, each optimum decision may be readily computed, and given the optimum decision, the original objective function, V({p*(γ)}), can be calculated. Consequently, a multi-dimensional optimization problem can be reduced to a 1-dimensional optimization problem {tilde over (V)}(γ): 
     
       
         
           
             
               
                 
                   
                     
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     FIG.  9   
       FIG. 9  shows a table  900  of exemplary embedded constraint factor values  902  specified in connection with the execution of the embedded constraint subprocess  800 . Through the execution of embedded constraint subprocess  800 , an embedded constraint function, represented by equation (7) is defined, and values for an embedded constraint factor  903  (i.e., a field), namely Υ, is specified. In table  900 , each of a number of embedded constraint scenarios  904  are given a unique embedded constraint scenario identifier  906 . Embedded constraint factor values  902  for embedded constraint factor  903  are specified as the set (i.e., spectrum) shown in table  900 . Thus, embedded constraint scenarios  904  represent each of embedded constraint factor values  902 . 
     Although not shown herein, it should be understood that a table may be generated that includes both strategic constraint factors  128  and embedded constraint factor  903 . In such an instance, the resulting scenarios (not shown) would represent every possible combination of constraint factor values (such as those shown in  FIG. 3 ), and embedded constraint factor values (such as that shown in  FIG. 9 ). 
     FIGS.  10 - 13   
       FIGS. 10-13  provide tables of exemplary planning models and illustrate the enhanced optimization capabilities of the above described processes. The examples described in connection with  FIGS. 10-13  are for illustrative purposes only. The present invention may be readily utilized to solve many complex optimization problems by separating multi-dimensional optimization problems into smaller components, as discussed above. 
     The notation commonly employed in each of the exemplary planning models of  FIGS. 10-13  to represent functions, variables, and other parameters, is listed below for clarity of illustration:
         V({_})—primary objective function   STG({_})—strategic objective function   EMB({_})—embedded constraint function   SP({_})—planning function, also known as a strategic planning function when strategic objectives are included in the planning model   SP i ({_})—independent planning function   P i —price of item i   c i —cost of item i   p i *—optimized price of item i   p i   c —current price of item i   λ—Lagrange multiplier utilized as a strategic constraint factor   γ—Lagrange multiplier utilized as an embedded constraint factor   Z—embedded constraint   h(_)—non-linear function   g(_)—function   US i —unit sales of item i   q i   o —demand parameter for a demand model   β i —demand parameter for a demand model   AC i —activity cost for item i   δ(x)—is a generalized function having the value 0 except at 0. In particular, δ(x)=1 if x=0, and δ(x)=0 if x≠0.   D—demand   ms i —market share of item i       

     Those skilled in the art will recognize that the nomenclature can alter greatly from that which is shown herein depending upon the parameters of the particular optimization problem to be solved. 
     FIG.  10   
       FIG. 10  shows a table  1000  depicting a first exemplary planning model  1002  derived through the execution of decision variable computation process  200 . First exemplary planning model  1002  includes a primary objective  1004 , gross profit, and a strategic objective  1006 , dollar sales. Decision variables  1008  are defined as being prices of items, i. However, first exemplary planning model  1002  is not limited by tactical constraints. 
     As shown, a primary objective function  1010  is defined as being a function of the unit sales, US i , of item, i, the price, p i , of item, i, and the cost, c i , of item, i. The unit sales has an exponential dependency on price, as shown, subject to demand parameters of a demand model. A strategic objective function  1012  is revenue or dollar sales, and is defined as being a function of the price, p i , of item, i, and the unit sales, US i . Primary objective function  1010  and strategic objective function  1012 , coupled with a strategic constraint factor  1014 , are combined to yield a planning function  1016 . 
     A selected optimization algorithm  1018  is a closed form analytical algorithm for independently optimizing each of independent planning functions  1020 . More specifically, independent planning functions  1020  yield an optimization equation  1022  that is readily calculated directly to determine an optimum price, p i *, of item i. 
     FIG.  11   
       FIG. 11  shows a table  1100  depicting a second exemplary planning model  1102  derived through the execution of decision variable computation process  200 . Second exemplary planning model  1102  represents a non-linear model with continuous decision variables. Accordingly, model  1102  is more complex than model  1002 . 
     Second exemplary planning model  1102  includes a primary objective  1104 , gross profit, and a strategic objective  1106 , dollar sales. Decision variables  1108  are defined as being prices of items, i. However, second exemplary planning model  1102  is not limited by tactical constraints. As shown, a primary objective function  1110  is defined as being a function of the unit sales, US i , of item, i, the price, p i , of item, i. Moreover, the price, p i , is subject to a non-linear component, i.e., function h. In this example, second exemplary planning model  1102 , derived through the execution of process  200 , decouples the dependency of the prices. 
     A strategic objective function  1112  is revenue or dollar sales, and is defined as being a function of the price, p i , of item, i, and the unit sales, US i . Primary objective function  1110  and strategic objective function  1112 , coupled with a strategic constraint factor  1114 , are combined to yield a planning function  1116 . 
     A selected optimization algorithm  1118  is a one-dimensional optimization algorithm, such as, an exhaustive search, Newton&#39;s method, and so forth, for continuous variables for independently optimizing each of independent planning functions  1120 . Thus, independent planning functions  1120  yield an optimization equation  1122  that can be computed to determine an optimum price, p i *, of item i that maximizes the particular one of independent planning functions  1120 . 
     FIG.  12   
       FIG. 12  shows a table  1200  depicting a third exemplary planning model  1202  derived through the execution of decision variable computation process  200 . Third exemplary planning model  1202  represents a non-linear model with discrete decision variables. Accordingly model  1202  is more complex than either of models  1002  and  1102 . 
     Third exemplary planning model  1202  includes a primary objective  1204 , net profit, and a strategic objective  1206 , dollar sales. Decision variables  1208  are defined as being prices of items, i, and are discrete. Third exemplary planning model  1202  is also subject to tactical constraints  1209 . Tactical constraints  1209  require each price, p i , to be within minimum and maximum boundaries for each price. 
     As shown, a primary objective function  1210  representing net profit is generally defined as being a function of gross profit minus activity costs. In particular, primary objective function  1210  is defined as being a function of the unit sales, US i , of item, i, and the price, p i , of item, i. However, the price, p i , is subject to a non-linear component, i.e., function h. In addition, primary objective function  1210  is affected by activity cost, AC i  associated with changing the price of item i plus the cost of an employee finding the item and changing its shelf price. Through the use of the delta function, δ(x), when there is no change in the price of item i, there is no associated activity cost. However, when the price of item i is adjusted, the activity cost affects the gross profit to yield net profit. A strategic objective function  1212  is revenue or dollar sales, and is defined as being a function of the price, p i , of item, i, and the unit sales, US i . 
     Primary objective function  1210  and strategic objective function  1212 , coupled with a strategic constraint factor  1214 , are combined to yield a planning function  1216 . Accordingly, planning function  1216  takes into account the cost to implement price changes to determine the net profit. 
     A selected optimization algorithm  1218  is a one-dimensional optimization algorithm for discrete decision variables for independently optimizing each of independent planning functions  1220 . The one-dimensional optimization is performed utilizing a simple search through all acceptable price points, p a , and selecting the price, p i *, that maximizes independent planning functions  1220  as represented by an optimization equation  1222 . As mentioned above, standard routines for solving one-dimensional optimization problems, include, but are not limited to Newton&#39;s method, Brent&#39;s method, golden section search in one-dimension, and exhaustive search over acceptable boundaries (for example, price points). 
     FIG.  13   
       FIG. 13  shows a table  1300  depicting a fourth exemplary planning model  1302  derived through the execution of decision variable computation process  200 . Fourth exemplary planning model  1302  is provided to illustrate how process  200  employing embedded constraint subprocess  800  efficiently handle multinomial Logit type models. The present invention advantageously decouples the optimization in a non-linear planning model by introducing an embedded constraint, thereby decoupling the multinomial Logit form into a form, which can be solved analytically or using a one-dimensional optimization method for discontinuous planning models. Although the decoupling of decision variables is illustrated in connection with multinomial Logit type models, embedded constraint subprocess  800  can be readily applied to other types of models. 
     Fourth exemplary planning model  1302  includes a primary objective  1304 , but does not include a strategic objective for simplicity of illustration. However, a strategic objective may be readily incorporated into fourth exemplary planning model  1302 , as discussed above. Decision variables  1308  are defined as being prices of items, i. However, fourth exemplary planning model  1302  is not limited by tactical constraints. 
     As shown, a primary objective function  1310  is defined as being a function of unit sales, US i , of item, i; price, p i  of item, i; and cost, c i  of item, i. However, unit sales is a function of demand, D, and market share, ms i , of item, i. The market share is described by a multinomial Logit model that allocates the available demand to the different purchasing decisions. For instance, market share may describe the customer choice between competing products or stores. The denominator, Σ k g k , within primary objective function  1310  couples decision variables  1308 . 
     In fourth exemplary planning model  1302 , the function g i  reflects the exponential dependency of a utility function (not shown) for item i. A utility function could take into account parameters that might influence the sales of item i in a store. These parameters include, for example, whether item i is on a display, whether it has signage, how much shelf space is allocated to the item, coupons, discounts, and so forth. 
     Through the execution of embedded constraint subprocess  800 , an embedded constraint, Z, is introduced into primary objective function  1310 . The embedded constraint, Z, replaces the function Σ k g k  in primary objective function  1310 . In addition, an embedded constraint function  1312  is utilized to enforce the variable Z. Primary objective function  1310  and embedded constraint function  1312 , coupled with an embedded constraint factor, γ,  1316  are combined to yield a planning function  1318 . The inclusion of embedded constraint function  1312  with primary objective function  1310  serves to reduce the coupling between the decision variables of primary objective function  1310 . It should be apparent that since no strategic objective is defined in fourth exemplary planning model  1302 , it also follows that no strategic objective function is provided in table  1300  for subsequent inclusion in planning function  1318 . 
     In fourth exemplary planning model  1302 , a selected one-dimensional optimization algorithm  1320  may be a search algorithm for independently optimizing each of independent planning functions  1322 . The one-dimensional optimization algorithm  1320  is performed to select the price, p i *, that maximizes independent planning functions  1322  as represented by an optimization equation  1324  and satisfies the embedded constraint Z. That is, once values for each embedded constraint factor, γ, are specified at task  810  of embedded constraint subprocess  800 , a suitable value for embedded constraint factor, γ, is found that maximizes (or minimizes) primary objective function  1310  and concurrently satisfies the embedded constraint Z. 
     Through the execution of embedded constraint subprocess  800 , a coupled N-dimensional, nonlinear optimization problem can be reduced to lower order independent planning functions. A lower order optimization problem yields a significantly shorter processing time and greater stability of the optimization algorithm than prior art algorithms that handle coupled N-dimensional, nonlinear optimization problems. 
     In summary, the present invention teaches of method and computer program for enhanced optimization calculation within a planning model. The method and computer program produce computationally efficient optimization calculation by separating a multi-dimensional planning model into a set of independent planning models and determining optimal decisions for each of the independent planning models. Moreover, the method and computer program mitigate the computational difficulties associated with the coupling between decision variables within the planning model through the introduction of an embedded constraint. 
     Although the preferred embodiments of the invention have been illustrated and described in detail, it will be readily apparent to those skilled in the art that various modifications may be made therein without departing from the spirit of the invention or from the scope of the appended claims. For example, those skilled in the art will appreciate that the tasks depicted in  FIGS. 2 ,  4 , and  8  may be partitioned and sequenced in a wide variety of ways other than those specifically described here.