Patent Publication Number: US-9412109-B2

Title: Analysis of clustering solutions

Description:
BACKGROUND 
     Retailers include entities that sell merchandise. In some examples, a retailer is a business that retails general merchandise to consumers. Furthermore, in some examples, a retailer is a business that wholesales merchandise to other businesses. 
     A retailer may have many stores. Each of the stores is a physical facility at which the retailer sells merchandise. For various reasons, the retailer stocks different merchandise in different ones of its stores. For example, the retailer stocks more winter clothing in northern stores than in southern stores. In another example, the retailer stocks different types of merchandise in different stores based on the demographics of people living near the stores. 
     If the retailer has many stores, the retailer may group its stores into clusters. Each store in a cluster may stock a similar assortment of merchandise. Stores in different clusters may stock different assortments of merchandise. Grouping stores into clusters may help the retailer manage costs associated with stocking different assortments of merchandise at different stores, while at the same time improving sales by targeting particular assortments of merchandise to customers of stores belonging to the same cluster. 
     SUMMARY 
     In general, a computing system implements techniques for determining an appropriate number of clusters to use when grouping the stores of a retailer into clusters. To determine the appropriate number of clusters, the computing system determines incremental values associated with a plurality of clustering solutions. Each of the clustering solutions groups stores of a retailer into clusters in a different way. For each respective clustering solution in the plurality of clustering solutions, the incremental value associated with the respective clustering solution indicates a difference between an estimated revenue associated with the respective clustering solution and revenue associated with a baseline clustering solution. The computing system then determines, based on the incremental values associated with the plurality of clustering solutions, the appropriate number of clusters. The clustering solutions that group the stores into more or fewer clusters than the appropriate number of clusters tend to be associated with incremental values that are the same or lower than the clustering solutions that group the stores into the appropriate number of clusters. 
     In one example, this disclosure describes a method that comprises determining, by a computing system, incremental values associated with a plurality of clustering solutions. Each of the clustering solutions groups stores of a retailer into clusters in a different way. For each clustering solution in the plurality of clustering solutions, the incremental value associated with the clustering solution indicates a difference between an estimated revenue associated with the clustering solution and revenue associated with a baseline clustering solution. The method also comprises determining, by the computing system and based on the incremental values associated with the plurality of clustering solutions, an appropriate number of clusters into which to group the stores of the retailer. Clustering solutions that group the stores into more or fewer clusters than the appropriate number of clusters tend to be associated with incremental values that are the same or lower than clustering solutions that group the stores into the appropriate number of clusters. 
     In another example, this disclosure describes a computing system comprising one or more processors and one or more storage devices that store instructions that, when executed by the one or more processors, cause the computing system to determine incremental values associated with a plurality of clustering solutions. Each of the clustering solutions groups stores of a retailer into clusters in a different way. For each clustering solution in the plurality of clustering solutions, the incremental value associated with the clustering solution indicates a difference between an estimated revenue associated with the clustering solution and revenue associated with a baseline clustering solution. The instructions also cause the computing system to determine an optimal cluster count based on the incremental values associated with the plurality of clustering solutions. The optimal cluster count indicates a number of clusters into which the stores are grouped. Ones of the clustering solutions that do not have the optical cluster count have a tendency to be associated with incremental values that are the same or tower than ones of the clustering solutions that have the optical cluster count. 
     In another example, this disclosure describes a computer readable storage medium that stores instructions that, when executed by one or more processors of a computing system, cause the computing system to determine, based on incremental values associated with a plurality of clustering solutions, an appropriate number of clusters such that the incremental values associated with the clustering solutions having the appropriate number of clusters have a tendency to be greater than the incremental values associated with the clustering solutions that group the stores into more or fewer clusters than the appropriate number of clusters. Each of the clustering solutions groups stores of a retailer into clusters in a different way. For each clustering solution in the plurality of clustering solutions, the incremental value associated with the clustering solution indicates a difference between an estimated revenue associated with the clustering solution and revenue associated with a baseline clustering solution. 
     The details of one or more examples are set forth in the accompanying drawings and the description below. Other features, objects, and advantages will be apparent from the description, drawings, and claims. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a conceptual diagram that illustrates an example environment in which one or more aspects of this disclosure is implemented. 
         FIG. 2A  is a conceptual diagram that illustrates a first example clustering solution. 
         FIG. 2B  is a conceptual diagram that illustrates a second example clustering solution. 
         FIG. 3  is a block diagram that illustrates an example configuration of a computing system. 
         FIG. 4  is a conceptual diagram that illustrates an example chart on which points associated with clustering solutions are plotted. 
         FIG. 5  is a flowchart illustrating an example operation of a computing system, in accordance with one or more aspects of this disclosure. 
         FIG. 6  is a flowchart that illustrates an example operation to determine an incremental value associated with a clustering solution. 
         FIG. 7  is a flowchart that illustrates an example operation to determine incremental values for stores in a particular cluster. 
         FIG. 8  is a flowchart that illustrates an example operation to determine an actual number of swaps allowed for a brand sold in a particular cluster. 
         FIG. 9  is a flowchart that illustrates a continuation of the example operation of  FIG. 8  to determine an actual number of swaps allowed for a brand sold in a particular cluster. 
         FIG. 10  is a flowchart that illustrates an example operation to determine a maximum number of swaps allowed for a brand in a particular cluster. 
         FIG. 11  is a flowchart that illustrates an example operation to determine a distance metric for a brand in a particular cluster. 
         FIG. 12  is a conceptual diagram that illustrates an example chart on which points associated with z scores of brands are plotted. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  is a conceptual diagram that illustrates an example environment in which one or more aspects of this disclosure are implemented. As illustrated in the example of  FIG. 1 , a retailer  10  includes a plurality of stores  12 A- 12 N (collectively, “stores  12 ”). Retailer  10  is an entity that sells merchandise. In some examples, retailer  10  is a business that retails general merchandise to consumers. In other examples, retailer  10  is a business that wholesales merchandise to other businesses. 
     Stores  12  are physical facilities at which retailer  10  sells merchandise. For instance, stores  12  may include standalone buildings, retail space within shopping complexes, and so on. Each of stores  12  has a plurality of aisles. Each of the aisles may have shelf and/or rack space for displaying merchandise. In some stores, at least some of the aisles have end caps for displaying additional merchandise. Each of stores  12  includes one or more checkout lanes with cash registers at which customers may purchase merchandise. In some examples, the checkout lanes may be staffed with cashiers. 
     Retailer  10  stocks different stores  12  with different assortments of merchandise. In other words, different stores  12  have different types and quantities of items available for sale. Stocking different stores  12  with different assortments of merchandise enables retailer  10  to better meet the specific demands and/or desires of customers of different stores  12  of the retailer. For instance, retailer  10  may select different assortments of merchandise for different stores based on various data regarding the locations, layouts, and customers of the stores. 
     Ideally, retailer  10  would stock each of stores  12  with the exact assortment of merchandise the store&#39;s customers want to buy. Thus, in an ideal situation, each of stores  12  would have a different assortment of merchandise. However, it may be impractical for retailer  10  to determine with exact precision the assortment of merchandise sought by the customers of each individual store at any given time. Moreover, it can be impractical, because of the large number of stores, for retailer  10  to manage the distribution of different assortments of merchandise to each of stores  12 . 
     Accordingly, retailer  10  groups stores  12  into clusters. In general, stores  12  are not in more than one cluster at a time. In general, stores  12  in the same cluster stock the same assortment of merchandise while stores  12  in different clusters stock different assortments of merchandise. In other words, stores  12  in different clusters have different assortments of items available for sale. For example, stores  12  in a first cluster stocks additional cold weather clothing during winter while stores  12  in a second cluster stocks additional beachwear during winter. In another example, stores  12  in a first cluster stocks more items belonging to a first brand than a second brand while stores  12  in a second cluster stocks more items belonging to the second brand than the first brand. 
     Retailer  10  identifies multiple clustering solutions. Each of the clustering solutions groups stores  12  of retailer  10  into clusters in a different way. For example, retailer  10  may have eight stores, denoted “A” through “H”. In this example, a first clustering solution groups stores “A” through “D” into a first cluster and stores “E” through “H” into a second cluster. Furthermore, in this example, a second clustering solution groups stores “A” and “B” into a first cluster, stores “C,” “D,” and “E” into a second cluster, and stores “F,” “G,” and “H” into a third cluster. A third clustering solution groups each of stores “A” through “H” into separate clusters.  FIGS. 2A and 2B , described in greater detail below, are conceptual diagrams that illustrate example clustering solutions. 
     After identifying the clustering solutions, retailer  10  selects one of the clustering solutions and stock assortments of merchandise at stores  12  according to the selected clustering solution. When retailer  10  stocks assortments of merchandise at stores  12  according to the selected clustering solution, retailer  10  stocks assortments of merchandise at stores  12  such that stores  12  within clusters of the selected clustering solution have similar assortments of merchandise. 
     Retailer  10  attempts to select a clustering solution that has an appropriate number of clusters. The number of clusters in the selected clustering solution may have a significant effect on the net income of retailer  10 . If the selected clustering solution has too few clusters, some of stores  12  may not be stocking the assortment of merchandise demanded by their customers. If a store does not stock the assortment of merchandise demanded by its customers, revenue from that store may not reach its full potential. Hence, if the selected clustering solution has too few clusters, the net income of retailer  10  may not reach its full potential because the revenue of retailer  10  has not reached its full potential. On the other hand, if the selected clustering solution has too many clusters, the costs of selecting assortments of merchandise for each of the clusters and distributing the selected assortments of merchandise to stores  12  in the clusters may increase faster than the incremental revenue obtained by grouping stores  12  into additional clusters. Hence, if the selected clustering solution has too many clusters, the increased cost of doing business may exceed any additional increases in revenue. 
     In accordance with the techniques of this disclosure, a computing system  14  receives sales data for stores  12 . Computing system  14  then determines, based at least in part on the sales data for stores  12 , incremental values associated with a plurality of clustering solutions. The incremental value associated with a clustering solution indicates a difference between an estimated amount of revenue associated with the clustering solution and revenue associated with a baseline clustering solution. The baseline clustering solution is a clustering solution that retailer  10  is currently using, with which retailer  10  has experience and for which retailer  10  has past sales data. In one example, the baseline clustering solution includes all of the stores into a single cluster such that all stores receive the same assortment of products. In another example, the baseline clustering solution includes clustering stores of retailer  10  into a relatively small number of different clusters based on some general characteristic of the store like the physical size of (e.g. square feet). Example processes for determining the incremental values associated with the clustering solutions are described elsewhere in this disclosure. 
     After determining the incremental values associated with the clustering solutions, computing system  14  determines, based at least in part on the incremental values associated with the clustering solutions, an appropriate number of clusters. The appropriate number of clusters is also referred to herein as an optimal cluster count. Computing system  14  determines the appropriate number of clusters in various ways. In some examples, computing system  14  determines the appropriate number of clusters by plotting points on a graph for each of the clustering solutions. The graph has a first axis (e.g., a horizontal axis) that corresponds to a number of clusters and a second axis (e.g., a vertical axis) that corresponds to the incremental values associated with the clustering solutions. In some examples, computing system  14  determines a curve based on a polynomial regression over the points in the graph. In other examples, computing system  14  determines a best fit curve over the points in the graph. In this example, the appropriate number of clusters is equal to a number of clusters associated with a turning point (e.g., a maximum point) of the curve.  FIG. 4 , described in detail below, is a conceptual diagram that illustrates such a graph. 
       FIG. 2A  is a conceptual diagram that illustrates an example clustering solution  50 . In the example of  FIG. 2A , each of the squares corresponds to one of stores  12 . The cross-hatching in each of the squares indicates the cluster to which the corresponding store belongs. For instance, in the example of  FIG. 2A , stores that correspond to squares in the top row belong to a first cluster, stores that correspond to squares in the second row belong to a second cluster, stores that correspond to squares in the third row belong to a third cluster, and stores that correspond to squares in the fourth row belong to a third cluster. 
       FIG. 2B  is a conceptual diagram that illustrates an example clustering solution  60 . As in  FIG. 2A , each of the squares in  FIG. 2B  corresponds to one of stores  12  and the cross-hatching in each of the squares indicates the cluster to which the corresponding store belongs. However, clustering solution  60  groups the same stores in a different way than clustering solution  50 . 
       FIG. 3  is a block diagram that illustrates an example configuration of computing system  14 . For purposes of illustration,  FIG. 3  is described with reference to  FIG. 1 .  FIG. 3  illustrates only one particular example of computing system  14 , and many other example configurations of computing system  14  exist. 
     As shown in the example of  FIG. 3 , computing system  14  includes one or more processors  70 , one or more input devices  72 , one or more communication units  74 , one or more output devices  76 , one or more storage devices  78 , and one or more communication channels  80 . Computing system  14  may include many other components. For example, computing system  14  may include physical buttons, microphones, speakers, communication ports, and so on. 
     Communication channel(s)  80  interconnect each of the components  70 ,  72 ,  74 ,  76 , and  78  for inter-component communications (physically, communicatively, and/or operatively). In some examples, communication channel(s)  80  include a system bus, a network connection, an inter-process communication data structure, or any other method for communicating data. In some examples, one or more of communication channel(s)  80  are implemented using a local area network (LAN) or a wide area network, such as the Internet. 
     Storage device(s)  78  within computing system  14  store information used during operation of computing system  14 . In some examples, storage device(s)  78  have the primary purpose of being a short term and not a long-term computer-readable storage medium. In some such examples, storage device(s)  78  do not retain stored data if powered off. Examples of volatile memories include random access memories (RAM), dynamic random access memories (DRAM), static random access memories (SRAM), and other forms of volatile memories known in the art. In some examples, storage device(s)  78  are further configured for long-term storage of information as non-volatile memory and retain information after power on/off cycles. Examples of non-volatile memory configurations include magnetic hard discs, optical discs, floppy discs, flash memories, or forms of electrically programmable memories (EPROM) or electrically erasable and programmable (EEPROM) memories. 
     Computing system  14  is able to receive indications of user input from input device(s)  72 . Examples of user input include tactile, audio, and video input. Example types of input device(s)  72  include presence-sensitive screens, touch-sensitive screens, mice, keyboards, voice responsive systems, video cameras, microphones, electronic pens, or other types of devices for detecting user input. 
     Communication unit(s)  74  enable computing system  14  to send data on and receive data from a communications network, such as a local area network or the Internet. In some examples, communication unit(s)  74  include wireless transmitters and receivers that enable computing system  14  to communicate wirelessly with the communications network. 
     Output device(s)  76  generate output. Examples types of output include tactile, audio, and video output. Example types of output device(s)  76  include presence-sensitive screens, sound cards, video graphics adapter cards, speakers, cathode ray tube (CRT) monitors, liquid crystal displays (LCD), or other types of devices for generating output. 
     Storage device(s)  78  store data, such as computer-executable instructions  82 . Processor(s)  70  read instructions  82  from storage device(s)  78  and execute instructions  82 . Execution of instructions  82  by processor(s)  70  configure or cause computing system  14  to provide at least some of the functionality ascribed in this disclosure to computing system  14 . As shown in the example of  FIG. 3 , instructions  82  include an operating system  84 . Execution of instructions in operating system  84  causes computing system  14  to perform various functions to manage hardware resources of computing system  14  and to provide various common services for other computer programs. 
     Furthermore, instructions  82  include a cluster analysis module  86 . Execution of instructions in cluster analysis module  86  configures or causes computing system  14  to perform various aspects of this disclosure. For example, execution of instructions in cluster analysis module  86  causes computing system  14  to determine incremental values associated with a plurality of clustering solutions. In addition, execution of instructions in cluster analysis module  86  causes computing system  14  to determine, based on the incremental values associated with the plurality of clustering solutions, an appropriate number of clusters into which to group stores of retailer  10 . 
       FIG. 4  is a conceptual diagram that illustrates an example chart on which points associated with clustering solutions are plotted. As illustrated in the example of  FIG. 4 , the chart has a horizontal axis  150  and a vertical axis  152 . Horizontal axis  150  corresponds to the number of clusters in the clustering solutions. Vertical axis  152  corresponds to incremental values associated with clustering solutions. In the example of  FIG. 4 , small squares indicate clustering solutions. A curve  154  indicates a best fit curve through the points on the chart. A point  156  indicates a maximum point of curve  154 . As indicated by line  158 , point  156  corresponds to twelve clusters. Hence, in the example of  FIG. 4 , the appropriate number of clusters is equal to twelve. 
       FIG. 5  is a flowchart illustrating an example operation  200  of computing system  14 , in accordance with one or more aspects of this disclosure. For purposes of illustration,  FIG. 5  and the other figures of this disclosure are described with reference to  FIG. 1 . However,  FIG. 5  and the other figures of this disclosure are not limited to the example of  FIG. 1 . The example of  FIG. 5  and the examples of the other figures of this disclosure illustrate only particular examples, and many other examples exist. 
     After computing system  14  starts operation  200 , computing system  14  receives sales data for stores  12  ( 202 ). The sales data for a store include entries for brands associated with items (e.g., products) sold at the store. In some examples, a brand is a name or mark that identities a set of items. For instance, types of diapers produced by different companies may be associated with different brands. An entry for a brand identifies the brand, a number of items associated with the brand, an amount of “spend” associated with the brand, and a number of individual units associated with the brand that were sold by the store. The amount of “spend” associated with the brand is the amount of revenue received by the store by selling items associated with the brand. 
     Computing system  14  receives the sales data for stores  12  in various ways. In some examples, each of stores  12  periodically (e.g., on an hourly or daily basis) sends data regarding their sales to computing system  14 . Computing system  14  stores the sales data in a database for later retrieval. In another example, stores  12  stores sates data at computing systems physically located at stores  12 . In this example, computing system  14  may periodically request the sales data from the computing systems at stores  12 . 
     Computing system  14  determines, based on the sales data for stores  12 , incremental values associated with a plurality of clustering solutions ( 204 ). As indicated above, each of the clustering solutions group stores  12  of retailer  10  into clusters in a different way. The incremental values associated with the clustering solutions indicate differences between estimated revenues associated with the clustering solutions and revenues associated with a baseline clustering solution.  FIG. 6 , described in detail below, is a flowchart that illustrates an example operation to determine an incremental value associated with a clustering solution. 
     Computing system  14  then determines, based on the incremental values associated with the clustering solutions, an appropriate number of clusters into which to group stores  12  of retailer  10  ( 206 ). Clustering solutions that group the stores into more or fewer clusters than the appropriate number of clusters tend to be (i.e., have a tendency to be) associated with incremental values that are the same or lower than clustering solutions that group the stores into the appropriate number of clusters. 
     After determining the appropriate number of clusters, a particular clustering solution is selected based at least in part on the determined appropriate number of clusters ( 208 ). In some examples, computing system  14  selects the clustering solution. In other examples, one or more people associated with retailer  10  selects the clustering solution. Furthermore, in some examples, the selected clustering solution has the appropriate number of clusters, while, in other examples, the selected clustering solution has more or fewer clusters than the appropriate number of clusters. 
     After selecting a particular clustering solution, retailer  10  distributes, based at least in part on the selected clustering solution, merchandise to stores  12  ( 210 ). For example, the selected clustering solution may include at least a first cluster and a second cluster. In this example, retailer  10  distributes a first assortment of items to stores in the first cluster and distributes a second assortment of items to stores in the second cluster. In this example, the first assortment of items includes more of a first brand and less of a second brand than the second assortment of items. 
       FIG. 6  is a flowchart that illustrates an example operation  250  to determine an incremental value associated with a clustering solution. After computing system  14  starts operation  250 , computing system  14  determines incremental values associated with each store in each cluster in the clustering solution ( 252 ). The incremental value associated with store is an estimated change in the store&#39;s revenue if the store&#39;s current (i.e., baseline) assortment of items is swapped with another assortment of items. Computing system  14  determines the incremental value associated with a store in various ways.  FIG. 7 , described in detail below, is a flowchart that illustrates an example operation to determine the incremental value associated with a store. 
     After determining the incremental values associated with each store in each cluster in the clustering solution, computing system  14  determines the incremental value associated with each respective cluster in the clustering solution ( 254 ). The incremental value associated with a cluster is an aggregation of the incremental values associated with the stores in the cluster. For example, computing system  14  determines the incremental value associated with a cluster based on a sum of the incremental values associated with the stores in the cluster. For instance, in this example, computing system  14  determines that the incremental value associated with a cluster is equal to the sum of non-negative incremental values associated with stores in the cluster. In this example, V k,c  denotes the incremental value for a store k in cluster c. In this example, computing system  14  calculates the incremental value associated with cluster c as:
 
V c =Σ k=1   m V k,c ,∀kεc;V k,c &gt;0
 
     After determining the incremental values associated with each respective cluster in the clustering solution, computing system  14  determines an incremental value for the clustering solution ( 256 ). Computing system  14  determines the incremental value associated with the clustering solution based on the incremental values associated with each of the clusters indicated by the clustering solution. For instance, V S  denotes the incremental value associated with the clustering solution S and V c  denotes the incremental value associated with a cluster c indicated by the clustering solution S. Thus, computing system  14  calculates the incremental value associated with the clustering solution S as: 
     
       
         
           
             
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       FIG. 7  is a flowchart that illustrates an example operation  300  to determine incremental values for stores in a particular cluster. After computing system  14  starts operation  300 , computing system  14  determines whether computing system  14  has processed all brands in the particular cluster ( 302 ). Computing system  114  determines that a brand in the particular cluster has been processed if computing system  14  has performed actions  304 - 312  with regard to the brand. The brands in the particular cluster are the brands sold in stores of the particular cluster. 
     If not all brands in the particular cluster have been processed (“NO” of  302 ), computing system  14  selects one of the unprocessed brands ( 304 ). Next, computing system  14  determines an actual number of swaps executed for the selected brand in stores of the particular cluster ( 306 ). A “swap” occurs when one assortment of items in a product group is substituted for a different assortment of items in the product group. For example, a swap occurs when the shelf space devoted to diapers of brand X is increased at the expense of the shelf space devoted to diapers of brand Y. A product group is a category of similar items, such as diapers, men&#39;s shirts, toothbrushes, and so on. Hence, an increase in the number of items in a product group that are associated with a first brand and sold in the particular cluster results in a corresponding decrease in the number of items in the product group that are associated with a second brand sold in the particular cluster. Retailer  10  may define the product groups according to its own needs. Computing system  14  determines the actual number of swaps executed for the selected brand in various ways.  FIGS. 8 and 9 , described in detail below, are flowcharts that illustrate an example operation to determine the actual number of swaps executed for a brand. 
     Furthermore, computing system  14  determines the average sales represented by items associated with the selected brand ( 308 ). Where S j,k  indicates the sales of brand j at store k and I j,k  indicates the number of items available for brand j at store k, the average sales represented by items associated with brand j in store k is given as: 
     
       
         
           
             
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     Next, computing system  14  multiplies the average sales represented by items associated with the selected brand by the actual number of swaps executed for the brand ( 310 ). Computing system  14  then adds the product of this multiplication to an intermediate version of the incremental value associated with the store ( 312 ). After adding the product to the incremental value associated with the store, computing system  14  determines whether there are any additional unprocessed brands sold at the store ( 302 ). If so, computing system  14  performs actions  304 - 312  with respect to another unprocessed brand. If all brands in the particular cluster have been processed (“YES” of  302 ), computing system  14  ends operation  300 . 
     For example, R j,k  indicates the average sales represented by items associated with a brand j in store k. NI j,c  indicates the actual number of swaps executed for brand j in the cluster c. As described below with regard to  FIG. 10 , CP j,c  denotes the proportion of items to be swapped for brand j in cluster c. In this example, computing system  14  calculates the incremental value associated with store k in cluster c as: 
     
       
         
           
             
               
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     Although the example of  FIG. 7  is explained with reference to brands, the example of  FIG. 7  is generally applicable to other classification parameters. A brand is just one example of a classification parameter. Other example classification parameters include attributes of items, such as physical size of items, physical weight of items, sizes of apparel items, color silhouette of items, and so on. In this way, a plurality of values of a classification parameter is associated with items sold at the stores in the cluster. Thus, the example of  FIG. 7  may be generalized such that for each respective store in the cluster and for each value of the classification parameter sold at the respective store, computing system  14  determines an actual number of swaps executed for the value of the classification parameter. Each of the swaps is a substitution of an assortment of items associated with the value of the classification parameter with an assortment of items associated with a different value of the classification parameter. Furthermore, computing system  14  determines an average sales represented by items associated with the value of the classification parameter. Computing system  14  then multiplies the average sales represented by items associated with the value of the classification parameter by the actual number of swaps executed for the value of the classification parameter. Computing system  14  then adds a product of the multiplication to an intermediate version of the incremental value associated with the respective store. 
       FIG. 8  is a flowchart that illustrates an example operation  350  to determine an actual number of swaps allowed for a brand sold in a particular cluster. After computing system  14  starts operation  350 , computing system  14  determines a maximum number of available swaps for the brand ( 352 ). Computing system  14  determines the maximum number of swaps available for the brand in various ways. For example,  FIG. 10 , described in detail below, is a flowchart that illustrates an example operation to determine the maximum number of swaps available for a brand sold in a particular cluster. 
     In addition, computing system  14  determines a maximum allowable number of swaps for the cluster ( 354 ). The maximum number of swaps allowed for the cluster constrains the maximum number of swaps performed in the cluster. Computing system  14  uses the maximum number of swaps allowed for the cluster to constrain the maximum number of swaps performed in the cluster because the space in stores  12  allocated to a product group may be fixed. 
     The maximum allowable number of swaps for a cluster c is denoted as MAS c  and the maximum number of available swaps for a brand j sold in cluster c is denoted as MS j,c . Computing system  14  determines MAS c  as: 
               MAS   c     =       min     M   ⁢           ⁢     S     j   ,   c           ⁢     {         ∑     j   :       MS     j   ,   c       ≥   0         ⁢     MS     j   ,   c         |       ∑     j   :       MS     j   ,   c       ≤   0         ⁢     -     MS     j   ,   c             }             
That is, computing system  14  determines MAS c  as the lesser of the sum of non-negative maximum available swaps for the brands sold in cluster c or the sum n of negatives of the non-positive maximum available swaps for the brands sold in cluster c.
 
     Computing system  14  also determines whether the brand is over-performing, at stores within the cluster when compared to average sales of the brand across retailer  10 , given the space allocated for the brand at the stores within the cluster ( 356 ). Computing system  14  determines, based on a distance metric for the brand, whether the brand is over-performing, at stores within the cluster when compared to the average sales of the brand across retailer  10 , given the space allocated for the brand at the stores within the cluster. For instance, if the distance metric of the brand exceeds a threshold, computing system  14  determines that the brand is over-performing, at stores within the cluster, given the space allocated for the brand at the stores within the cluster. In this way, computing system  14  determines, based at least in part on a distance metric for the brand, the actual number of swaps executed for the brand. 
     If computing system  14  determines that the brand is not over-performing, at stores within the cluster when compared to the average sales of the brand across retailer  10 , given the space allocated for the brand at the stores within the cluster (“NO” of  356 ), computing system  14  performs the portion of operation  350  illustrated in  FIG. 9 . However, if computing system  14  determines that the brand is over-performing, at stores within the cluster, given the space allocated for the brand at the stores within the cluster (“YES” of  356 ), computing system  14  determines a value MS 1,c  ( 358 ). The value MS 1,c  is equal to the greatest of MS j,c . In other words, MS 1,c =max jεc MS j,c . 
     Next, computing system  14  determines whether MS 1,c  is greater than or equal to MAS c  ( 360 ). In other words, computing system  14  determines whether MS 1,c  is greater than or equal to the maximum allowable swaps for the cluster. If MS 1,c  is greater than or equal to MAS c  (“YES” of  360 ), computing system  14  determines that NI j,c  is equal to MAS c  ( 362 ). That is,
 
NI j,c =MAS c , if MS 1,c ≧MAS c  
 
     On the other hand, if MS 1,c  is not greater than or equal to MAS c  (“NO” of  360 ), computing system  14  determines whether the total number of available swaps for brands sold in the cluster is greater than or equal to MAS c  ( 364 ). Utile total number of available swaps for brands sold in the cluster is greater than or equal to MAS c  (“YES” of  364 ), computing system  14  determines that NI j,c  is equal to a remaining number of swaps available for the brand ( 366 ). The remaining number of swaps available for brand j in cluster c is denoted as r j,c . r c,j  is equal to MAS c  minus the sum of MS j,c  for all brands in cluster c, except the brand having the greatest number of maximum available swaps among the brands in cluster c. That is, where MS j,c  denotes the number of swaps allowed for brand j in cluster c, computing system  14  determines r c,j  as: 
     
       
         
           
             
               
                 r 
                 
                   j 
                   , 
                   c 
                 
               
               = 
               
                 
                   MAS 
                   c 
                 
                 - 
                 
                   
                     ∑ 
                     
                       j 
                       ∈ 
                       c 
                     
                     
                         
                     
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     MS 
                     
                       j 
                       , 
                       c 
                     
                   
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               ∀ 
               
                 j 
                 ≠ 
                 1 
               
             
           
         
       
     
     However, if the total number of available swaps for brands sold in the cluster is not greater than or equal to MAS c  (“NO” of  364 ), computing system  14  determines that NI j,c  is equal to MS j,c  ( 368 ). As indicated above, MS j,c  denotes the number of swaps allowed for brand j in cluster c. 
       FIG. 9  is a flowchart that illustrates a continuation of the example operation  350  of  FIG. 8 . After computing system  14  starts the continuation of operation  350  illustrated in  FIG. 9 , computing system  14  determines whether an absolute value of MS 1,c  is greater than or equal to MAS c  ( 380 ). In the context of  FIG. 9 , MS j,c  is equal to the least of MS j,c . In other words, in the context of  FIG. 9 , MS 1,c =min jεc MS j,c . If MS 1,c  is greater than MAS c  (“YES” of  380 ), computing system  14  determines that NI j,c  is equal to the negative of MAS c  ( 382 ). 
     On the other hand, if MS 1,c  is not greater than MAS c  (“NO” of  380 ), computing system  14  determines whether the absolute value of the total number of available swaps for brands sold in the cluster is greater than or equal to MAS c  ( 384 ). If the absolute value of the total number of available swaps for brands sold in the cluster is greater than or equal to MAS c  (“YES” of  384 ), computing system  14  determines that NI j,c  is equal to the negative of the remaining number of swaps available for the brand ( 386 ). In other words, computing system  14  determines that NI j,c =−r j,c . In the context of  FIG. 9 , computing system  14  determines r j,c  as: 
     
       
         
           
             
               
                 r 
                 
                   j 
                   , 
                   c 
                 
               
               = 
               
                 
                   MAS 
                   c 
                 
                 - 
                 
                    
                   
                     
                       ∑ 
                       
                         j 
                         ∈ 
                         c 
                       
                       
                           
                       
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       MS 
                       
                         j 
                         , 
                         c 
                       
                     
                   
                    
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               ∀ 
               
                 j 
                 ≠ 
                 1 
               
             
           
         
       
     
     If the absolute value of the total number of available swaps for brands sold in the cluster is not greater than or equal to MAS c  (“NO” of  384 ), computing system  14  determines that NI j,c  is equal to the negative of the number of available swaps for the brand in the cluster ( 388 ). In other words, computing system  14  determines that NI j,c =MS j,c . 
       FIG. 10  is a flowchart that illustrates an example operation  400  to determine a maximum number of swaps allowed for a brand in a particular cluster. 
     After computing system  14  starts operation  400 , computing system  14  determines an upper bound for the brand in the particular cluster ( 402 ). Computing system  14  determines the upper bound as the maximum number of items available for the brand in stores in the cluster. For example, UB j,c  denotes the upper bound for brand j in cluster c and I j,k  denotes the number of items available for brand j in store k. In this example, computing system  14  determines UB j,c  as:
 
 UB   j,c =max kεc   {I   j,k }
 
     In the example of  FIG. 10 , the maximum allowable number of swaps for a brand in a particular cluster is based on an average assortment for retailer  10 , as a whole. For example, if stores in the cluster have 15 items of the brand and the average number of items of the brand in all stores of retailer  10  is 19, the upper bound is 19. Furthermore, in this example, computing system  14  may determine, based on the brand&#39;s sales differentiation from other brands, that 7 additional items should be sold for the brand at stores of the cluster. Adding 7 additional items for the brand would result in 22 items of the brand being sold at stores of the cluster. Because 22 items is above the average assortment for the brand for retailer  10  as a whole (i.e., 19), limiting the number of items added for the brand to 19 may keep the number of items of the brand sold at stores of the cluster to a reasonable amount. Thus, in this example, the maximum allowable number of swaps is equal to 4. 
     In addition, computing system  14  determines a lower bound for the brand in the particular cluster ( 404 ). Computing system  14  determines the lower bound as the minimum number of items available for the brand in stores in the cluster. For example, LB j,c  denotes the lower bound for brand j in cluster c and I j,k  denotes the number of items available for brand j in store k. In this example, computing system  14  determines LB j,c  as:
 
 LB   j,c =min kεc   {I   j,k   }∀j,k,c  
 
The upper bound and the lower bound set the upper and lower bounds for the number of items being swapped.
 
     Computing system  14  determines the upper bound and the lower bound in order to constrain the swaps to the maximum and minimum level of assortment availability across retailer  10 . For example, stores in a cluster c may sell, on average, fifteen items associated with a brand j. However, in this example, the maximum number of items associated with brand j sold in any cluster of retailer  10  may be nineteen items. Hence, in this example, the upper bound for brand j in cluster c (i.e., UB j,c ) is equal to nineteen. In this example, the upper bound for brand j in cluster c prevents computing system  14  from performing more than four swaps. For instance, the upper bound restricts computing system  14  such that computing system  14  may not replace five items associated with other brands with five items associated with brand j, but may replace four items associated with other brands with four items associated with brand j. Similar examples are applicable to the lower bound. 
     Furthermore, computing system  14  determines a distance metric for the brand ( 406 ). The distance metric is an indicator of performance of the brand. More generally, the distance metric is an indicator of performance of items associated with a value of a classification parameter, such as a brand. Computing system  14  determines the distance metric for the brand in various ways.  FIG. 11 , described in detail below, is a flowchart that illustrates an example operation to determine the distance metric for the brand. 
     Computing system  14  also determines an average number of items for the brand in the particular cluster ( 408 ). For example, I j,c   avg  denotes the average number of items for brand j in cluster c and N C  indicates a number of stores in cluster c. Computing system  14  determine I j,c   avg  as: 
                 I     j   ,   c     avg     =           ∑     k   ∈   c               ⁢           ⁢     I     j   ,   c           N   c       ⁢     ∀   j         ,   c         
In addition, computing system  14  determines a cumulative distribution function for the z score represented by a distance metric for the brand in the particular cluster ( 410 ). For example, φ(t) denotes the cumulative distribution function for the z score represented by a distance metric D j,c  for a brand j in a cluster c.
 
     Computing system  14  determines, based on the cumulative distribution function and the distance metric for the brand, a change percentage for the brand ( 412 ). If the distance metric for the brand in the particular cluster is less than −0.5 or greater than 0.5, computing system  14  determines that a change percentage for the brand in the particular cluster is equal to φ(t)−0.5. Otherwise, if the distance metric for the brand in the particular duster is not less than −0.5 and not greater than 0.5, computing system  14  determines that the change percentage for the brand in the particular cluster is equal to 0. That is, 
     
       
         
           
             
               CP 
               
                 j 
                 , 
                 c 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         ∅ 
                         ⁡ 
                         
                           ( 
                           t 
                           ) 
                         
                       
                       - 
                       0.5 
                     
                   
                   
                     
                       
                         ∀ 
                         
                           
                             D 
                             
                               j 
                               , 
                               c 
                             
                           
                           &lt; 
                           
                             - 
                             0.5 
                           
                         
                       
                       ; 
                       
                         ∀ 
                         
                           
                             D 
                             
                               j 
                               , 
                               c 
                             
                           
                           &gt; 
                           0.5 
                         
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       O 
                       ⁢ 
                       therwise 
                     
                   
                 
               
             
           
         
       
     
     The change percentage for the brand in the particular cluster determines the proportion of items to be swapped thr the brand sold in the particular cluster based on the distance metric for the brand sold in the particular cluster. Consequently, computing system  14  performs more swaps on a brand that has high or low performance as compared to a brand whose performance is closer to the all chain average. Thus, if the distance metric is higher, the number of swaps may be higher, and vice versa, subject to the upper and lower bounds. 
     The change percentage for the brand imposes a constraint in addition to the upper bound and lower bound. For example, there may be two brands, brand  1  and brand  2 . In this example, brand  1  has an assortment of fifteen items and brand  2  has an assortment of five items. Furthermore, in this example, brand  1  and brand  2  are differentiated similarly, having distance metrics of 1.23, which represents 30% of the variation. In this example, swapping is proportional to this variation. Thus, brand  1  may swap up to 4.5 items (15*30%) and brand  2  may swap up to 1.5 items (5*30%). In this way, the swapping may be proportional. 
     After determining the change percentage for the brand in the particular cluster, computing system  14  determines the maximum number of swaps allowed for the brand ( 414 ). Using the upper bound UB j,c , lower bound LB j,c , average number of items I j,c   avg , distance metric D j,c , and change percentage CP j,c  for a brand j in a cluster c, computing system  14  determines the maximum number of swaps MS j,c  for brand j in cluster c as follows: 
     
       
         
           
             
               MS 
               
                 j 
                 , 
                 c 
               
             
             ⁢ 
             
               { 
               
                 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 CP 
                                 
                                   j 
                                   , 
                                   c 
                                 
                               
                               * 
                               
                                 I 
                                 
                                   j 
                                   , 
                                   c 
                                 
                                 avg 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 I 
                                 
                                   j 
                                   , 
                                   c 
                                 
                                 avg 
                               
                               * 
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     CP 
                                     
                                       j 
                                       , 
                                       c 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                         ≥ 
                         
                           
                             LB 
                             
                               j 
                               , 
                               c 
                             
                           
                           ⁢ 
                           
                             ∀ 
                             c 
                           
                         
                       
                       , 
                       
                         j 
                         ; 
                         
                           
                             D 
                             
                               j 
                               , 
                               c 
                             
                           
                           &lt; 
                           
                             - 
                             0.5 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 LB 
                                 
                                   j 
                                   , 
                                   c 
                                 
                               
                               - 
                               
                                 I 
                                 
                                   j 
                                   , 
                                   c 
                                 
                                 avg 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 I 
                                 
                                   j 
                                   , 
                                   c 
                                 
                                 avg 
                               
                               * 
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     CP 
                                     
                                       j 
                                       , 
                                       c 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                         &lt; 
                         
                           
                             LB 
                             
                               j 
                               , 
                               c 
                             
                           
                           ⁢ 
                           
                             ∀ 
                             c 
                           
                         
                       
                       , 
                       
                         j 
                         ; 
                         
                           
                             D 
                             
                               j 
                               , 
                               c 
                             
                           
                           &lt; 
                           
                             - 
                             0.5 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 CP 
                                 
                                   j 
                                   , 
                                   c 
                                 
                               
                               * 
                               
                                 I 
                                 
                                   j 
                                   , 
                                   c 
                                 
                                 avg 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 I 
                                 
                                   j 
                                   , 
                                   c 
                                 
                                 avg 
                               
                               * 
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     CP 
                                     
                                       j 
                                       , 
                                       c 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                         ≤ 
                         
                           
                             UB 
                             
                               j 
                               , 
                               c 
                             
                           
                           ⁢ 
                           
                             ∀ 
                             c 
                           
                         
                       
                       , 
                       
                         j 
                         ; 
                         
                           
                             D 
                             
                               j 
                               , 
                               c 
                             
                           
                           &gt; 
                           0.5 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             ( 
                             
                               
                                 UB 
                                 
                                   j 
                                   , 
                                   c 
                                 
                               
                               - 
                               
                                 I 
                                 
                                   j 
                                   , 
                                   c 
                                 
                                 avg 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           if 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 I 
                                 
                                   j 
                                   , 
                                   c 
                                 
                                 avg 
                               
                               * 
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     CP 
                                     
                                       j 
                                       , 
                                       c 
                                     
                                   
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                         &gt; 
                         
                           
                             UB 
                             
                               j 
                               , 
                               c 
                             
                           
                           ⁢ 
                           
                             ∀ 
                             c 
                           
                         
                       
                       , 
                       
                         j 
                         ; 
                         
                           
                             D 
                             
                               j 
                               , 
                               c 
                             
                           
                           &gt; 
                           0.5 
                         
                       
                     
                   
                 
               
             
           
         
       
     
       FIG. 11  is a flowchart that illustrates an example operation  450  to determine a distance metric for a brand in a particular cluster. After computing system  14  starts operation  450 , computing system  14  determines whether there are additional clusters in the clustering solution yet to process ( 452 ). Computing system  14  determines that a cluster has been processed if computing system  14  has performed actions  454 - 472  with regard to the cluster. If there are one or more clusters in the clustering solution yet to process (“YES” of  452 ), computing system  14  selects one of the unprocessed clusters ( 454 ). Computing system  14  then determines whether there are one or more unprocessed stores in the selected cluster ( 456 ). Computing system  14  determines that a store is an unprocessed store if computing system  14  has not yet performed actions  458 - 464  with regard to the store. If there are one or more unprocessed stores in the selected cluster (“YES” of  456 ), computing system  14  selects one of the unprocessed stores in the selected cluster ( 458 ). 
     After selecting the unprocessed store, computing system  14  determines, based at least in part on the sales data for the selected store, an amount of spend for the brand at the selected store ( 460 ). For example, S j,k  denotes the sales of a brand j at store k, I j,k  denotes the number of items available for brand j at store k, and R j,k  denotes the amount of spend for brand j at store k. In this example, computing system  14  determines the amount of spend per item for the brand at the selected store as: 
     
       
         
           
             
               R 
               
                 j 
                 , 
                 k 
               
             
             = 
             
               
                 S 
                 
                   j 
                   , 
                   k 
                 
               
               
                 I 
                 
                   j 
                   , 
                   k 
                 
               
             
           
         
       
     
     In addition, computing system  14  determines, based at least in part on the sales data for the selected store, a share of average spend for the brand at the selected store ( 462 ). For example, R j,k  denotes the amount of spend per item for brand j at store k and SS j,k  denotes the share of spend for brand j at store k. In this example, computing system  14  determines the share of spend for brand j at store k as: 
     
       
         
           
             
               SS 
               
                 j 
                 , 
                 k 
               
             
             = 
             
               
                 S 
                 
                   j 
                   , 
                   k 
                 
               
               
                 
                   ∑ 
                   
                     j 
                     , 
                     k 
                   
                   
                       
                   
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   S 
                   
                     j 
                     , 
                     k 
                   
                 
               
             
           
         
       
     
     Computing system  14  also determines, based at least in part on the sales data for the selected store, a share of items for the brand at the selected store ( 464 ). For example, I j,k  denotes the number of items available for brand j at store k and SI j,k , denotes the share of items for brand j at store k. In this example, computing system  14  determines the share of items for brand j at store k as: 
               S   ⁢           ⁢     I     j   ,   k         =       I     j   ,   k           ∑     j   ,   k       ⁢     I     j   ,   k                 
After determining the share of items for the brand in the selected store, computing system  14  determines whether there are any unprocessed stores in the selected cluster ( 456 ). If so, computing system  14  selects another unprocessed store in the selected cluster, and so on. In this way, computing system  14  determines a share of spend for the brand at each store in the selected cluster and a share of items for the brand in each store in the selected cluster.
 
     Table 1, below, illustrates example sales data and baseline metrics for store  83  in a cluster (i.e., cluster  1 ). The baseline metrics in the following table are a brand&#39;s spend and items at a store, the brand&#39;s share of spend at the store associated with the brand, and the brand&#39;s share of items at the store. 
     
       
         
           
               
               
               
               
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                   
                   
                   
                   
                   
                 Share of 
                 Share of 
               
               
                 Cluster 
                 Store 
                 Brand 
                 Spend 
                 Items 
                 spend 
                 items 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 1 
                 83 
                 1 
                 120 
                 1 
                 0.002 
                 0.02 
               
               
                 1 
                 83 
                 2 
                 17619 
                 17 
                 0.282 
                 0.29 
               
               
                 1 
                 83 
                 3 
                 1645 
                 5 
                 0.026 
                 0.09 
               
               
                 1 
                 83 
                 4 
                 18679 
                 18 
                 0.299 
                 0.31 
               
               
                 1 
                 83 
                 5 
                 24503 
                 17 
                 0.392 
                 0.29 
               
               
                   
               
            
           
         
       
     
     If there are no unprocessed stores in the selected cluster (“NO” of  456 ), computing system  14  determines an average share of spend for the brand in stores of the selected cluster ( 466 ). For example, SS′ j,c  denotes the average share of spend for brand j for stores in cluster c and N c  denotes the number of stores in cluster c. In this example, computing system  14  determines SS′ j,c  as: 
     
       
         
           
             
               S 
               ⁢ 
               
                   
               
               ⁢ 
               
                 S 
                 
                   j 
                   , 
                   c 
                 
                 ′ 
               
             
             = 
             
               
                 
                   ∑ 
                   
                     k 
                     ∈ 
                     c 
                   
                 
                 ⁢ 
                 
                   S 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     S 
                     
                       j 
                       , 
                       k 
                     
                   
                 
               
               
                 N 
                 c 
               
             
           
         
       
     
     In addition, computing system  114  determines a standard deviation for the average share of spend for the brand at stores of the selected cluster ( 468 ). For example, σ j,c   SS  denotes the standard deviation for the average share of spend for brand j in cluster c. In this example, computing system  14  determines σ j,c   SS  as: 
     
       
         
           
             
               σ 
               
                 j 
                 , 
                 c 
               
               
                 S 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 S 
               
             
             = 
             
               
                 
                   
                     ( 
                     
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           S 
                           
                             j 
                             , 
                             k 
                           
                         
                       
                       - 
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           S 
                           
                             j 
                             , 
                             k 
                           
                           ′ 
                         
                       
                     
                     ) 
                   
                   2 
                 
                 
                   
                     N 
                     c 
                   
                   - 
                   1 
                 
               
             
           
         
       
     
     Computing system  14  determines an average share of items for the brand in all stores of the selected cluster ( 470 ). For example, SI′ j,c  denotes the average share of items associated with brand j for stores in cluster c and SI j,k  denotes the share of items for brand j at store k. In this example, computing system  14  determines SI′ j,c  as: 
     
       
         
           
             
               S 
               ⁢ 
               
                   
               
               ⁢ 
               
                 I 
                 
                   j 
                   , 
                   c 
                 
                 ′ 
               
             
             = 
             
               
                 
                   ∑ 
                   
                     k 
                     ∈ 
                     c 
                   
                 
                 ⁢ 
                 
                   S 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     I 
                     
                       j 
                       , 
                       k 
                     
                   
                 
               
               
                 N 
                 c 
               
             
           
         
       
     
     Computing system  14  also determines a standard deviation for the average share of items for the brand in all stores of the selected cluster ( 472 ). For example, σ j,c   SI  of denotes the standard deviation tier the average share of items for the brand. In this example, computing system  14  determines σ j,c   SI  as: 
     
       
         
           
             
               σ 
               
                 j 
                 , 
                 c 
               
               
                 S 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 I 
               
             
             = 
             
               
                 
                   
                     ( 
                     
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           I 
                           
                             j 
                             , 
                             k 
                           
                         
                       
                       - 
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           I 
                           
                             j 
                             , 
                             k 
                           
                           ′ 
                         
                       
                     
                     ) 
                   
                   2 
                 
                 
                   
                     N 
                     
                       c 
                       ⁢ 
                       
                           
                       
                     
                   
                   - 
                   1 
                 
               
             
           
         
       
     
     Computing system  14  then determines again whether there are any unprocessed clusters in the clustering solution ( 452 ). If there are one or more unprocessed clusters, computing system  14  selects another unprocessed cluster, and performs acts  454 - 472  with another unprocessed cluster in the clustering solution. 
     Table 2, below, illustrates example cluster-level data for brands in a particular cluster. 
     
       
         
           
               
               
               
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
               
                   
                   
                 Average cluster 
                 Average cluster 
                 Standard deviation of 
                 Standard deviation of 
               
               
                 Cluster 
                 Brand 
                 share of items 
                 share of spend 
                 cluster share of items 
                 cluster share of spend 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 4 
                 1 
                 0.015198 
                 0.00436 
                 1E−04 
                 0.003 
               
               
                 4 
                 2 
                 0.030602 
                 0.02633 
                 0.0004 
                 0.0152 
               
               
                 4 
                 3 
                 0.270482 
                 0.25241 
                 0.005 
                 0.0371 
               
               
                 4 
                 4 
                 0.076506 
                 0.09498 
                 0.0009 
                 0.0442 
               
               
                 4 
                 5 
                 0.030602 
                 0.02713 
                 0.0004 
                 0.0125 
               
               
                 4 
                 6 
                 0.275421 
                 0.22299 
                 0.0034 
                 0.0334 
               
               
                 4 
                 7 
                 0.306024 
                 0.37318 
                 0.0037 
                 0.0271 
               
               
                   
               
            
           
         
       
     
     Otherwise, if there are no unprocessed clusters in the clustering solution (“NO” of  452 ), computing system  14  determines a retailer average share of spend associated with the brand ( 474 ). For example, SS″ j  denotes the retailer average share of spend associated with brand j. In this example, computing system  14  determines SS″ j  as: 
     
       
         
           
             
               S 
               ⁢ 
               
                   
               
               ⁢ 
               
                 S 
                 j 
                 ″ 
               
             
             = 
             
               
                 
                   ∑ 
                   
                     k 
                     = 
                     1 
                   
                   n 
                 
                 ⁢ 
                 
                   S 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     S 
                     
                       j 
                       , 
                       k 
                     
                   
                 
               
               m 
             
           
         
       
     
     In addition, computing system  14  determines a retailer variance for the share of spend associated with the brand ( 476 ). For example, σ SS  denotes the retailer variance for the share of spend associated with the brand. In this example, computing system  14  determines σ SS  as: 
     
       
         
           
             
               σ 
               
                 S 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 S 
               
             
             = 
             
               
                 
                   
                     ( 
                     
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           S 
                           
                             j 
                             , 
                             c 
                           
                           ′ 
                         
                       
                       - 
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           S 
                           j 
                           ″ 
                         
                       
                     
                     ) 
                   
                   2 
                 
                 
                   
                     ∑ 
                     
                       c 
                       = 
                       1 
                     
                     1 
                   
                   ⁢ 
                   c 
                 
               
             
           
         
       
     
     Computing system  14  then determines a retailer average share of items associated with the brand ( 478 ). For example, SI″ j  denotes the retailer average share of items associated with brand j. In this example, computing system  14  determines SI″ j  as: 
     
       
         
           
             
               S 
               ⁢ 
               
                   
               
               ⁢ 
               
                 I 
                 j 
                 ″ 
               
             
             = 
             
               
                 
                   ∑ 
                   
                     j 
                     = 
                     1 
                   
                   n 
                 
                 ⁢ 
                 
                   S 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     I 
                     
                       j 
                       , 
                       k 
                     
                   
                 
               
               m 
             
           
         
       
     
     Computing system  14  also determines a retailer variance for the share of items associated with brands sold at retailer  10  ( 480 ). For example, σ SI  denotes the retailer variance for the share of items associated with brands sold at retailer  10 . In this example, computing system  14  determines σ SI  as: 
     
       
         
           
             
               σ 
               
                 S 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 I 
               
             
             = 
             
               
                 
                   
                     ( 
                     
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           I 
                           
                             j 
                             , 
                             c 
                           
                           ′ 
                         
                       
                       - 
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           I 
                           j 
                           ″ 
                         
                       
                     
                     ) 
                   
                   2 
                 
                 
                   
                     ∑ 
                     
                       c 
                       = 
                       1 
                     
                     p 
                   
                   ⁢ 
                   c 
                 
               
             
           
         
       
     
     Next, computing system  14  determines a z score for the brand&#39;s share of spend ( 482 ). For example, Z j,c   SS  denotes the z score for the brand&#39;s share of spend. In this example, computing system  14  determines Z j,c   SS  as: 
     
       
         
           
             
               Z 
               
                 j 
                 , 
                 c 
               
               
                 S 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 S 
               
             
             = 
             
               
                 
                   S 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     S 
                     
                       j 
                       , 
                       c 
                     
                     ′ 
                   
                 
                 - 
                 
                   S 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     S 
                     j 
                     ″ 
                   
                 
               
               
                 σ 
                 SS 
               
             
           
         
       
     
     In addition, computing system  14  determines a z score for the brand&#39;s share of items sold by retailer  10  ( 484 ). For example, Z j,c   SI  denotes the z score for the brand&#39;s share of items. In this example, computing system  14  determines Z j,c   SI  as: 
     
       
         
           
             
               Z 
               
                 j 
                 , 
                 c 
               
               
                 S 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 I 
               
             
             = 
             
               
                 
                   S 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     I 
                     
                       j 
                       , 
                       c 
                     
                     ′ 
                   
                 
                 - 
                 
                   S 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     I 
                     j 
                     ″ 
                   
                 
               
               
                 σ 
                 
                   S 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   I 
                 
               
             
           
         
       
     
     Table 3, below, illustrates example retailer-level data for brands. 
     
       
         
           
               
               
               
               
               
             
               
                 TABLE 3 
               
               
                   
               
               
                 Overall 
                 Average all store 
                 Average all store 
                 Standard deviation of all 
                 Standard deviation of all 
               
               
                 average 
                 share of items 
                 share of spend 
                 store share of items 
                 stores share of spend 
               
               
                   
               
             
            
               
                 1 
                 0.016375 
                 0.015597 
                 0.000823 
                 0.008843 
               
               
                 2 
                 0.028389 
                 0.058533 
                 0.004959 
                 0.033206 
               
               
                 3 
                 0.291167 
                 0.234393 
                 0.014156 
                 0.038556 
               
               
                 4 
                 0.081727 
                 0.115258 
                 0.004495 
                 0.035737 
               
               
                 5 
                 0.026258 
                 0.020625 
                 0.006394 
                 0.017699 
               
               
                 6 
                 0.295289 
                 0.245428 
                 0.014024 
                 0.033116 
               
               
                 7 
                 0.297309 
                 0.363639 
                 0.009292 
                 0.047151 
               
               
                   
               
            
           
         
       
     
     After determining the z scores for the brand&#39;s share of spend and the brand&#39;s share of items sold by retailer  10 , computing system  14  plots a point for the brand on a chart ( 486 ). A first axis of the chart corresponds to z scores for share of spend and a second axis of the chart corresponds to z scores for shares of items sold by retailer  10 . After plotting the point fir the brand on the chart, computing system  14  determines the distance metric for the brand for the particular cluster as a Cartesian perpendicular distance from the point to an even productivity line ( 488 ). The even productivity line is a baseline measure of performance that represents a situation where z scores for share of spend is equal to the z scores for share of items. In some examples, D c,j  denotes the distance metric brand j in cluster c, Z j,c   SS  denotes the z score for brand j in cluster c, Z j,c   SI  denotes the z score for the share of items of brand j in cluster c, and computing system  14  determines the distance metric as: 
     
       
         
           
             
               D 
               
                 c 
                 , 
                 j 
               
             
             = 
             
               
                 
                   2 
                   ⁢ 
                   
                     
                       ( 
                       
                         
                           Z 
                           
                             j 
                             , 
                             c 
                           
                           
                             S 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             S 
                           
                         
                         - 
                         
                           Z 
                           
                             j 
                             , 
                             c 
                           
                           
                             S 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             I 
                           
                         
                       
                       ) 
                     
                     2 
                   
                 
                 2 
               
             
           
         
       
     
     Furthermore, in some examples, the even productivity line may be the set of points where Z j,c   SS  is equal to Z j,c   SI  for all brands j and cluster c. 
     In this way, computing system  14  determines the distance metric as a Cartesian perpendicular distance from an even productivity line to a point for a value of a classification parameter (e.g., a brand). A location of the point is based on a z score for a share of spend associated with the value of the classification parameter and is based on a z score for a share of items sold at the respective store that are associated with the value of the classification parameter. 
       FIG. 12  is a conceptual diagram that illustrates an example chart on which points associated with z scores of brands are plotted. As illustrated in the example of  FIG. 12 , the chart has a horizontal axis  500  and a vertical axis  502 . Horizontal axis  500  corresponds to z scores for shares of items sold by retailer  10 . Vertical axis  502  corresponds to z scores for share of spend. In the example of  FIG. 12 , squares indicate points associated with brands sold in a cluster. A line  504  indicates the even productivity line. A line  506  extends from a point  508  to line  504 . Line  506  meets line  504  at right angles. The distance metric for the brand associated with point  508  is equal to the length of line  506 . 
     The attached drawings illustrate examples. Elements indicated by reference numbers in the attached drawings correspond to elements indicated by like reference numbers in the following description. In the attached drawings, ellipses indicate the presence of one or more elements similar to those separated by the ellipses. Alphabetical suffixes on reference numbers for similar elements are not intended to indicate the presence of particular numbers of the elements. In this disclosure, elements having names that start with ordinal words (e.g., “first,” “second,” “third,” and so on) do not necessarily imply that the elements have a particular order. Rather, such ordinal words are merely used to refer to different elements of a same or similar type. 
       FIGS. 8-12  are explained with regard to brands. However, like the example of  FIG. 7 ,  FIGS. 8-12  are generally applicable to other classification parameters instead of brands. For example,  FIGS. 8-12  may be applicable to item weight. In this example, different values of the classification parameter (i.e., item weight) may be different weight ranges, such as less than 1 kg, between 1 kg and 2 kg, between 2 kg and 5 kg, and so on. 
     In one or more examples, the functions described are implemented in hardware, software, firmware, or any combination thereof. If implemented in software, the functions are stored on or transmitted over, as one or more instructions or code, a computer-readable medium and executed by a hardware-based processing unit. Computer-readable media include computer-readable storage media, which corresponds to a tangible medium such as data storage media, or communication media including any medium that facilitates transfer of a computer program from one place to another, e.g., according to a communication protocol. In this manner, computer-readable media generally correspond to (1) tangible computer-readable storage media which is non-transitory or (2) a communication medium such as a signal or carrier wave. Data storage media are any available media that can be accessed by one or more computers or one or more processors to retrieve instructions, code and/or data structures for implementation of the techniques described in this disclosure. 
     By way of example, and not limitation, such computer-readable storage media can comprise RAM, ROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage, or other magnetic storage devices, flash memory, or any other medium that can be used to store desired program code in the form of instructions or data structures and that can be accessed by a computer. Also, any connection is properly termed a computer-readable medium. For example, if instructions are transmitted from a website, server, or other remote source using a coaxial cable, fiber optic cable, twisted pair, digital subscriber line (DSL), or wireless technologies such as infrared, radio, and microwave, then the coaxial cable, fiber optic cable, twisted pair, DSL, or wireless technologies such as infrared, radio, and microwave are included in the definition of medium. It should be understood, however, that computer-readable storage media and data storage media do not include connections, carrier waves, signals, or other transient media, but are instead directed to non-transient, tangible storage media. Disk and disc, as used herein, includes compact disc (CD), laser disc, optical disc, digital versatile disc (DVD), floppy disk and Blu-ray disc, where disks usually reproduce data magnetically, while discs reproduce data optically with lasers. Combinations of the above should also be included within the scope of computer-readable media. 
     Instructions may be executed by one or more processors, such as one or more digital signal processors (DSPs), general purpose microprocessors, application specific integrated circuits (ASICs), field programmable logic arrays (FPGAs), or other equivalent integrated or discrete logic circuitry. Accordingly, the term “processor,” as used herein refers to any of the foregoing structure or any other structure suitable for implementation of the techniques described herein. In addition, in some aspects, the functionality described herein is provided within dedicated hardware and/or software modules. Also, the techniques may be fully implemented in one or more circuits or logic elements. 
     The techniques of this disclosure may be implemented in a wide variety of devices or apparatuses, including a wireless handset, an integrated circuit (IC) or a set of ICs (e.g., a chip set). Various components, modules, or units are described in this disclosure to emphasize functional aspects of devices configured to perform the disclosed techniques, but do not necessarily require realization by different hardware units. Rather, as described above, various units may be combined in a hardware unit or provided by a collection of interoperative hardware units, including one or more processors as described above, in conjunction with suitable software and/or firmware. 
     Various examples have been described. These and other examples are within the scope of the following claims.