Grouping customers for better allocation of resources to serve correlated demands

Computers & Operations Research 26 (1999) 1041}1058

Grouping customers for better allocation of resources to serve correlated demands Rajesh Tyagi *, Chandrasekhar Das
Information Technology Laboratory, General Electric Corporate Research and Development, P.O. Box 8, Schenectady, NY 12301 USA
Department of Management, University of Northern Iowa, Cedar Falls, IA 50614-0125, USA

Abstract In this paper, we discuss a common decision-making problem arising in the allocation and decentralization of resources under uncertain demand. The total resource requirements for a given service level equals the sum of mean demands plus a safety factor multiplied by the standard deviations of demands. Since the demand means are una!ected by any customer groupings, we attempt to exploit demand correlations for developing customer groups such that the sum of the standard deviations over all groups is minimized. A concave minimization model with binary variables is developed for this purpose and a heuristic partitioning method is proposed to e$ciently solve the model. The model is appropriate for both manufacturing and service management with potential applications in salesforce allocation, grouping of machines in job shops, and allocation of plant capacities. Scope and purpose In this paper, it is shown that when demands are correlated, complete aggregation of all customers in a single-service center may require more resources than is necessary to provide a given service level. A model and a solution technique are proposed to optimally aggregate/disaggregate customers into groups such that total resource requirements are minimized. Many potential applications of the proposed technique in centralization/decentralization of resources are discussed.  1999 Elsevier Science Ltd. All rights reserved. Keywords: Resource allocation; Service operations; Heuristics; Nonlinear optimization; Decentralization of facilities; Customer grouping

* Tel.: (319) 273-6224; fax: (319) 273-2922; e-mail: [email protected]. 0305-0548/99/$ - see front matter  1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 0 5 - 0 5 4 8 ( 9 9 ) 0 0 0 2 3 - 4

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1. Introduction This paper is an exploratory study on utilizing interdependence between demands for resource allocation across manufacturing or service facilities in large organizations. Customers of large companies are usually geographically distributed and they exhibit varying preferences for the company's products and services. Demands by various customers may be interdependent due to demographic, cultural, climatic, economic, geographical, and market factors. The occurrence of such interdependence between demands introduces an additional element of complexity in managing resources. In this paper, however, we exploit the demand correlations for grouping customers such that the overall resource requirements are reduced. For deterministic demands, excellent linear programming models (Charnes and Cooper [1], Dantzig [2], and Williams [3]) are available for allocating corporate resources over various departments or activities. Stochastic extensions of some of these models (Kall [4]) are also available to deal with demand uncertainty. None of the existing models, however, include the interrelationships between demands. We "nd that when correlations between demands are explicitly considered, the objective function of a resource allocation model ceases to be linear and the optimization task requires solving a stochastic nonlinear problem. In particular, the allocation model considered in this paper involves minimizing the square root of quadratic terms in binary variables. Since the exact solution to this class of problems, called global optimization problems, is computationally as well as analytically challenging (Horst and Tuy [5]), we propose a heuristic grouping method which is approximate but e$cient and easily applied. The objective of the proposed model is to minimize the total resources subject to a service constraint that a certain percentage (called service level) of customer demands be served without delay in the long run. We achieve this objective by "nding an optimal number of service centers and an allocation of customers (called customer grouping) to each service center. From statistical theory, the level of resources required to satisfy the service constraint at any center is simply the mean demand of all customers to be served by the center plus ¸ times the standard deviation, where ¸ is the z-value corresponding to a suitable percentile of the demand distribution. The problem then reduces to one of "nding the optimal partitioning of customers into groups so that the sum total of the resources required by all groups is minimized.

2. Potential applications Let us use an example to illustrate the principle of grouping customers for reducing resource requirements. Consider a salesforce responsible for serving eight customers, C1, C2, 2 , C8. Assume that the annual workloads generated by the customers follow correlated distributions with their means, variances, and correlation coe$cients as shown in Fig. 1. If the customer demands were completely pooled and served by a single pool of salespersons, the total resources required to provide a 95% service level would be 210.5 salesperson years. Alternatively, if the eight customers were partitioned into three groups as shown and each group was served independently by a service center (with one or more salesperson), the total resources required by the three centers would be 183.9 salesperson years. The main source of productivity gain here is due to the reduction of demand variation by optimal pooling. To see this, de"ne three random variables A, B, and C to

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Fig. 1. E!ect of pooling on resource requirements.

denote the total demand for the three groups of customers. The standard deviations of these groups for the given data are: p(A)"1.0, p(B)"22.9, and p (C)"13.1, whereas the standard deviation of the total demand for all eight customers is p (A#B#C)"53.20. The above partitioning reduces the overall standard deviation by 30% resulting in a 13% reduction in total resources. This indicates that it is bene"cial to partition customers into independent groups such that the total of group standard deviations is minimized. Optimal grouping of interdependent demands, such as those mentioned above, has potential applications in many areas of manufacturing and service management. For example, a job shop may experience uncertain demand of several categories of jobs to be processed by a group of machines. Job categories are usually correlated due to the complementary and supplementary nature of the related products. Each job may involve a sequence of tasks on the machines so that the long-run statistical characteristics of jobs de"ne a set of interdependent demand distributions for the machine resources. By extending the analogy of salesforce allocations to this case, the machines can be divided into optimal groups based on the job characteristics. Each machine group can then be managed as an independent machine center responsible for a speci"c group of job

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categories. Optimal allocation of plant capacities in a multiplant organization can be done in a similar fashion by studying the statistical characteristics of product categories. Initially, products may be categorized based on some similarity such as product line, market segment or processing characteristics. Plant capacities can then be allocated by studying the inter-correlations between the categories and allocating a group of products to a plant. Allocation of safety stock can be similarly improved by utilizing the correlations between individual items. Several authors, e.g., Zinn et al. [6], and Tallon [7] have analyzed the stock centralization/decentralization problem under di!erent sets of assumptions. The grouping method proposed here appears to be a more de"nitive tool for this purpose. In addition, the grouping logic can be adopted to allocate static resources. Allocation of computer resources such as secondary storage to di!erent groups of users on a computer network, allocation of parking spaces, and assignment of #eets of rental cars to categories of users are potential application areas in this regard. Finally, since the grouping method reduces the group variance, it can be applied to classify demand items for developing group forecasting systems in a multiproduct company. The method is also useful to stratify a correlated population (see Cochran [8]) into subpopulations for purposes of drawing random samples to estimate the population mean with minimum variance.

3. The model We assume that the population of customers can be initially classi"ed into a "nite set of categories on the basis of customer priorities, homogeneity of service requirements, or any other relevant characteristics. Demand for service by customers of a category is de"ned as the total number of service or resource units (i.e., man-hours or machine-hours) required to serve all customers of that category. For convenience, we use the term customer or customer category to refer to all customers belonging to a category. Customer demands are assumed to be normally distributed with known parameters, and are interdependent with a known correlation matrix. Each group of customers will be served completely by a service center with enough resources to provide a prespeci"ed service level. All service costs are assumed to be linear regardless of customer to service center assignments and the capacities of service centers. We also assume that there are no constraints on the number and size of these service centers. Let n "number of customer categories "maximum number of customer groups or service centers i "subscript used to denote customer category i"1, 2, 2 , n d "a random variable representing the number of service units needed to serve the annual G demand of customer i k "E (d ) G G p "
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D "total yearly demand of all customers assigned to group (or service center) k I L " Z d GI G G R "number of resource units required for service center k to provide service level ¸ I "E (D )#¸ (
 





L L L " R " E Z d #¸ GI G I G I I



 

L L " Z E(d )#¸ GI G I G L L L " k Z #¸ G GI G I I


L Z d GI G G



L L L Z
L L L Z p#2 Z Z r p p GI HI GH G H GI G G G HG

 (1)

s.t. L Z "1 for all i GI I Using Eq. (2), the objective in Eq. (1) can be simpli"ed to



L L L L L Z p#2 Z Z r p p . Min R" k #¸ GI HI GH G H GI G G G HG G I G Since k and ¸ are known parameters, the objective may be further reduce to G Min S "Sum of standard deviations of customers group demands 2



L L L L Z p#2 Z Z r p p . " GI G GI HI GH G H G HG I G

(2)

(3)

(4)

4. Need for a heuristic Since Z is a binary variable, Z in the objective function (4) may be replaced by Z . GI GI GI Nevertheless, the model is highly nonlinear and involves a large number of binary variables. In particular, the following three characteristics of the model make it di$cult to solve by any exact optimization algorithm.

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1. ¸arge number of binary variables: Since the model must theoretically allow for as many centers to be created as the number of customers, the number of binary variables (Z ) required in the GI model is (n)(n) or n. This makes the problem size increase exponentially with n. 2. Product of binary variables: The objective function includes quadratic terms involving products of binary variables. Most nonlinear algorithms cannot satisfactorily deal with such terms. 3. Square-root function: The square root in the objective function makes the model a concave minimization problem which is a di$cult problem to solve, especially in the presence of binary variables. The combination of the above three model characteristics makes the optimality of a solution derived by using available optimization software highly suspect. Complete enumeration is also not a practical alternative since it requires the evaluation of nL di!erent combinations in which n customers may be assigned to n service centers. We propose a solution approach that overcomes some of the aforementioned computational di$culties. We seek to develop a heuristic method with two special features. First, we prefer a tree-like decomposition to successively partition the customer set into two groups, with the following advantages: (a) it reduces the maximum problem size by limiting the number of variables, (b) it reduces the problem size as we traverse down the tree, and (c) it limits the maximum number of iterations to n, the number of customers. Secondly, we wish to use linear versions of the original model for partitioning the customers at each node of the tree. These linear models are easily solved in a short time using existing software.

5. The heuristic In this section, we develop a pair of simpli"ed auxiliary linear models for partitioning a set of customers into two groups. Below we discuss how the number of variables and the degree of nonlinearity of the objective function S can be reduced by a series of equivalent formulations and 2 approximations using these models. 5.1. Reducing the number of variables We reduce the number of binary variables by a tree-like decomposition approach with each node of the tree representing a group of customers. The process is successively applied to all nodes till no further partitioning is possible. The nodes that cannot be partitioned are called terminal nodes and represent the collection of customers to be assigned to individual service centers. The partitioning of customers at each node is achieved by restricting the number of groups in the original model to only two. For convenience, this restricted model is referred to as the two-group model. Since the number of groups is limited to two regardless of the number of customers, the

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number of binary variables (Z ) is only 2n, as opposed to n in the original model. The two-group GI model forms the basis of the proposed heuristic, and is a specialized version of the original model as described below. 5.1.1. The two-group model Let X and > represent the two groups into which a given set of customers will be partitioned. De"ne x "1 if customer i assigned to group X; 0 otherwise, G y "1 if customer i assigned to group >; 0 otherwise, G < "Variance of demands for group X customers 6 L L L " x p#2 x x r p p , G G G H GH G H G G HG < "Variance of demands for group > customers 7 L L L " y p#2 y y r p p . G H GH G H G G G G HG Formally, the objective function of the two-group model is Min S "(< #(< 2 6 7



"



L L L L L L x p#2 x x r p p # y p#2 y y r p p G G G H GH G H G G G H GH G H G G HG G G HG

(5)

s.t. x #y "1 for all i. G G 5.2. Eliminating the products of binary variables We now propose the following transformation to develop an equivalent linear model for the two-group problem. Let a "1 if customers i and j are both assigned to group X ; 0 otherwise. GH Then the product x x in Eq. (5) may be replaced by a single binary variable a if the following G H GH additional constraints are included in the model: a )x , GH G

(6a)

a )x GH H

(6b)

x #x H!e where 0.5(e(1.0. a * G GH 2

(6c)

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The following table shows the equivalence between x x and a when the above constraints are G H GH applied to all possible combinations of x and x values. G H x G

x H

x x G H

a GH

Explanation of a value GH

0 1 0 1

0 0 1 1

0 0 0 1

0 0 0 1

Constraints (6a) and (6b) are binding Constraint (6b) is binding Constraint (6a) is binding Constraint (6c) is binding

By de"ning b to be 1 if customers i and j are both assigned to group >, and 0 otherwise, a similar GH transformation can be used to replace the product y y by a single binary variable b . The G H GH aequivalent two-group model without quadratic terms can now be written as Min S "(< #(< 7 2 6





L L L a p#2 a r p p # GG G GH GH G H G G HG s.t. x #y "1 for all i, G G a )x for all i, j, GH G a )x for all i, j, GH H x #x H!e for all i, j, a * G GH 2 "

b )y for all i, j, GH G b )y for all i, j, GH H y #y H!e for all i, j, b * G GH 2

L L L b p#2 b r p p GG G GH GH G H G G HG

(7) (8a) (8b) (8c) (8d) (8e) (8f) (8g)

x , y , a , b are all binary variables. G G GH GH The above transformation is a signi"cant step in reducing the overall nonlinearity of the original model. 5.3. Avoiding concavity The square root in the objective function is the source of concavity which threatens the optimality of solutions obtained by available software. We, therefore, propose an approximation

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that avoids the use of the square-root operator. Let us de"ne: < "Variance of all demands treated as a single group (i.e., complete pooling) 2 L L L " p#2 r p p GH G H G G G HG C "Sum of covariances between group X and group > customers 67 L L "2 (1!x x !y y ) r p p G H G H GH G H G HG L L "2 (1!a !b ) r p p . GH GH GH G H G HG

(9)

Note that (1!a !b )"1 only if customers i and j are assigned to separate groups and, GH GH therefore, both a and b are zero. From Fig. 2, the following equality also holds: GH GH < "< #< #C . 2 6 7 67 Let us now investigate the conditions under which partitioning of customers into two groups will be better than no partitioning. If the customers are assigned to a single group, then the objective

Fig. 2. Components of < . 2

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function in (7) reduces to (< due to Eq. (9); otherwise it is (< #(< from Eq. (5). Therefore, 2 6 7 for partitioning to be bene"cial, the necessary condition is (< #(< ((< "(< #< #C 7 2 6 7 67 6 or < #< #2 (< (< (< #< #C 7 6 7 67 6 7 6 or (10) 2 (< (< (C . 7 67 6 It appears from Eq. (10) that the best partitioning would be achieved when the di!erence C !2 (< (< is maximized. This suggests that the objective in Eq. (7) is realized if we 7 67 6 (11) Max C !2 (< (< 7 67 6 under the same constraints (Eqs. (8a)}(8g)). We now apply heuristic reasons to maximize the nonlinear di!erence in (11) by either maximizing the "rst term, C , or minimizing the second term, 67 2 (< (< . Two auxiliary models developed for this purpose are described below. 6 7 Auxiliary Model 1: Max C 67 s.t. Constraints (8a)}(8g), and L x *1, (12a) G G L y *1. (12b) G G Constraints (12a) and (12b) are necessary to ensure that the set of customers is split into two groups by requiring each group to contain at least one customer. Next, we consider the following model. Auxiliary Model 2: Min 2(< (< 6 7 under constraints (8a)}(8g), (12a), and (12b). Next, if we succeed in reducing 2(< (< 6 7 to a negligible quantity, then the original objective function in Eq. (7) can be squared and approximated by Min (< #(< , 6 7

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or Min ((< #(< ), 7 6 or Min < #< #2(< (< 7 6 7 6 or approximately, Min < #< . 6 7

(13) (14)

Note that approximation of Eq. (13) by Eq. (14) is valid as long as 2 (< (< is small. This is true 7 6 when either < or < is kept small, perhaps with the help of a constraint. In particular, we 6 7 introduce an upper bound, < , on the value of < . This problem-speci"c upper bound is de"ned

 6 as the smallest feasible value of < that can be achieved by any non-null subset or group of 6 customers. Mathematically, the model is: Min < #< 6 7 s.t. Constraints (8a)}(8g), (12a), (12b), and < )< 6

 where < "Min <

 6 s.t. L 1) x )n!1. G G The purpose of the above constraint is to determine the subset of customers that produces the minimum possible < . The auxiliary model 2 is, thus, employed in two stages. In the "rst 6 stage, the appropriate value of < is determined which is then used to constrain < in the

 6 second stage. The above two models are used to partition the customers into two groups X and >, repeatedly. 5.4. Steps of customer partitioning heuristic The strategy of the heuristic is as follows: start by assigning all customers to a single group and calculate S for this group. Use the two auxiliary models to partition this group into two 2 groups of customers if it results in a lower S . If partitioning is not bene"cial (i.e., neither of 2 the two auxiliary models reduces S ), then the corresponding customers are assigned to a service 2 center. The process is repeated for each service center until further partitioning fails to reduce S . The complete heuristic is shown in Fig. 3 and it is further illustrated with an example in 2 the next section.

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Fig. 3. The partitioning heuristic.

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1053

6. Example We apply the heuristic to the eight customer example (see Fig. 1). The data for this example was randomly generated from uniform distributions as follows: the means range from 5 to 25, the standard deviations range from 10 to 15, and the correlation coe$cients cover a range from !1 to 1. The heuristic begins by assigning all eight customers to a single group as represented by node 1 in Fig. 4. The standard deviation for this group is S "(< "53.3, where < is determined by 2 2 2 Eq. (9). (Note that the formula for < does not involve any decision variable and, thus, does not 2 require any mathematical model). Auxiliary model 1 is now applied resulting in an S value of 47.6 2 ("26.1#21.5) units from the following partitioning:

Customers (<

Group X

Group >

1, 2, 3, 5 26.1

4, 6, 7, 8 21.5

Next, auxiliary model 2 is applied in two stages: the "rst stage yields a < value of 1 for

 customer subset +1, 3,. When used as an upper bound in the second stage, this value of < results

 in the following partitioning of customers with S of 39.6 ("38.6#1.0) units. 2

Customers (<

Group X

Group Y

1, 3 1.0

2, 4, 5, 6, 7, 8 38.6

Since the partitioning due to model 2 has a lower objective function value, the original customer set is partitioned into two groups, +1, 3, and +2, 4, 5, 6, 7, 8,, as represented by nodes 1.1 and 1.2 in Fig. 4. The heuristic procedure is now applied to these two nodes and the partitioning process continued until further partitioning is not possible. We thus arrive at terminal nodes which represent customer groupings or service centers. In the example, the eight customers were allocated to three groups and the resulting standard deviations and resource requirements are shown in Fig. 4.

7. Test data and computational experience Four data sets were randomly generated to determine the e!ect of optimal pooling as compared to complete pooling and to test the e$ciency of the heuristic partitioning method. The means and standard deviations for these data sets are the same as those used in the above example but with di!erent patterns of correlations, as shown in Table 1. Correlation matrix R1 (same as in the

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Fig. 4. Solving the example problem with heuristic partitioning.

example) was randomly generated with uniformly distributed correlations ranging from !1 to 1. Correlation matrix R2 is derived from R1 with all negative correlations replaced by positive correlations of the same magnitudes. Matrix R3 represents correlations of equal but moderate

Table 1 Data sets and solutions of eight-customer problem k"[13 18 12 11 15 16 23 15]

R1"

 

!1 0.8 0.7 !0.1 0.3 0.3 0.4 !0.1 0.4 0.4 0.2 0.6 1 !0.2 !0.3 0.8 0.9 0.8 1 0.8 !0.3 !0.3 0.1 1 0 !0.5 1 1 !0.4 !0.2 1 0.6 1

1 !0.5 !0.5 0.5 !0.5 0.5 0.5 !0.5 1 0.5 !0.5 0.5 0.5 0.5 0.5 1 !0.5 0.5 !0.5 !0.5 0.5 1 !0.5 0.5 0.5 !0.5 R3" 1 0.5 !0.5 0.5 1 0.5 0.5 1 !0.5 1

Data Complete set pooling

Optimal pooling

Resource Customer requirements groupings 210.5

R2

237.0

R3

188.8

R4

240.3

A"+1, 3, B"+2, 4, 8, C"+5, 6, 7, A"+1, 2, 4, 6, 7, 8, B"+3, 5, A"+1, 2, 4, 5, 8, B"+3, 6, 7, A"+1, 2, 3, 4, 5, 6, 7, 8,

R4"

1 0.5 0.5 0.5 1 0.5 0.5 1 0.5 1

Heuristic solution

Resource Customer requirements groupings 183.9

232.6 185.5 240.3

A"+1, 3, B"+2, 4, 8, C"+5, 6, 7, A"+1, 6, 7, 8, B"+2, 3, 4, 5, A"+1, 2, 4, 5, 8, B"+3, 6, 7, A"+1, 2, 3, 4, 5, 6, 7, 8,

0.5 0.5 0.5 0.5 1

0.3 0.2 0.9 0.3 0.5 0.4 1

0.5 0.5 0.5 0.5 0.5 1

0.3 0.6 0.8 0.1 1 0.2 0.6 1

0.5 0.5 0.5 0.5 0.5 0.5 1





0.5 0.5 0.5 0.5 0.5 0.5 0.5 1

Savings due to optimal pooling (%)

Heuristic solution penalty (%)

Resource Resource Reduction Resource Reduction requirements requirements in std. dev. requirements in std. dev. 183.9

12.7

30.4

0

0

235.2

1.9

3.9

1.1

2.4

185.5

1.7

5.0

0

0

240.3

0

0

0

0

1055

R1

1 0.1 1 0.8 0.7 0.1 1 0.4 0.1 0.4 0.4 1 0.2 0.3 0.8 1 0.8 0.3 R2" 1 0 1

R. Tyagi, C. Das / Computers & Operations Research 26 (1999) 1041}1058

 

1 0.1 1

p"[13 11 12 10 13 10 14 12]

1056

Data Complete set pooling

Optimal pooling

Resource Customer requirements groupings 1

179.2

2

191.8

3

205.9

4

247.5

5

203.4

6

170.2

Heuristic solution

Resource Customer requirements groupings

A"+1, 2, 3, 164.3 B"+4, 5, 7, 10, C"+6, 8, 9, A"+1, 4, 7, 8, 9, 10, 165.0 B"+2, 3, 5, 6, A"+1, 2, 3, 4, 7, 8, 177.8 B"+5, 6, C"+9, 10, A"+1, 2, 6, 222.7 B"+3, 5, 7, 10, C"+4, 8, 9, A"+1, 3, 6, 193.4 B"+2, 9, 10, C"+4, 5, 7, 8, A"+1, 5, 134.4 B"+2, 3, 4, 8, C"+6, D"+7, 9, 10,

A"+1, 2, 3, B"+4, 5, 7, 10, C"+6, 8, 9, A"+1, 4, 7, 8, 9, 10, B"+2, 3, 5, 6, A"+1, 2, 3, 4, 7, 8, B"+5, 6, C"+9, 10, A"+1, 2, 6, B"+3, 5, 7, 10, C"+4, 8, 9, A"+2, 3, 6, B"+1, 4, 5, 7, 8, 9, 10, A"+1, 5, 7, B"+2, 3, 4, 8, C"+6, 9, 10,

Savings due to optimal pooling (%)

Heuristic solution penalty (%)

Resource Resource Reduction Resource Reduction requirements requirements in std. dev. requirements in std. dev. 164.3

8.3

32.3

0.0

0.0

165.0

14.0

56.1

0.0

0.0

177.8

13.7

49.8

0.0

0.0

222.7

10.0

32.0

0.0

0.0

194.4

5.0

18.9

0.5

2.4

135.2

21.1

41.0

0.6

1.6

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Table 2 Solutions of ten-customer data sets

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magnitude (0.5) with randomly generated mixed signs. Matrix R4 is a variation of R3 with negative coe$cients replaced by positive ones. The results are shown in Table 1 and are very encouraging with respect to the bene"ts of pooling for optimal utilization of resources. The average reduction in standard deviation is approximately 10%. In terms of total resource requirements, the percentage savings averaged approximately four percent for the four data sets. Note that a large portion of the total requirements is due to mean demands which remain una!ected by customer groupings (see Eqs. (3) and (4)). Therefore, savings in standard deviations may be considered a more relevant measure of pooling e!ectiveness. The savings due to optimal pooling are especially signi"cant for correlation matrix R1 with correlations of mixed signs and random magnitudes. To further investigate savings for such correlation matrix structures, we tested six additional cases with ten customers each. The means, standard deviations, and the correlation matrices were independently generated for each case except that all correlation matrices follow the pattern of R1. As shown in Table 2, the average savings due to optimal pooling as compared to complete pooling is twelve percent for overall resource requirements, whereas the average reduction in standard deviations is thirty eight percent. 7.1. Performance of the heuristic From the test results, the heuristic appears to be both accurate and e$cient. For the eight customer data sets, optimal solutions were obtained in three of the four data sets; and for the 10 customer examples, four of the six solutions are optimal. In cases where heuristic solutions are not optimal, the solution penalty is found to be small (less than 2.5% for resources required or standard deviations) and customer groupings of heuristic solutions di!er only slightly from optimal groupings. For all data sets, the exact optimal solutions are generated by complete enumeration using the Pascal programming language on a DEC Alphaserver 2000 4/200 computer system running under OpenVMS AXP V6.1 operating system. The processing time, which is proportional to nL, is about 8 hours for a ten customer problem. In comparison, the total processing time for the auxiliary models typically ranges between 3 and 10 min per data set using LINGO [9] optimization software on a Pentium (66 MHz) personal computer.

8. Discussion and conclusions In this paper, we address a common decision problem of resource allocation and decentralization of facilities arising in many situations involving correlated demands. Our main contribution is to demonstrate that when demands are interdependent, a partial pooling of customers is more cost-e$cient than complete pooling if the customers' demand characteristics are suitably exploited. The suggested grouping procedure is based on a heuristic solution to a concave minimization problem with quadratic terms in binary variables. Using linearization, this method substantially reduces the problem size and is useful for solving large-sized problems. It can also be adapted to include "xed costs for facilities, upper and lower bounds on group sizes, and apriori logical relationships between group members. By using this grouping procedure, the productivity of an organization can be increased through a reduction of input resources. The method is suitable for macro-level allocation of resources, and

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its implementation requires a possible reorganization of facilities and workloads with little or no additional investment. Application of any demand management technique such as di!erential pricing can further increase the method's e!ectiveness. Unsatis"ed demand, whether backordered or lost, requires the same treatment as in the case of complete pooling. For example, backordered demand could be "lled by means of expedited shipments. For some applications such as allocation of parking spaces or secondary storage, the method requires only a logical and not a physical breakdown of the resources. Potential applications where customer groupings may be utilized for optimal resource allocation were described in an earlier section. In some of the situations, other considerations may limit the application of this procedure. For example, allocation of customers to salespersons requires close geographic proximity of the customers for each salesperson. Similarly, distribution costs from warehouses may favor geographic clustering. The requirement of geographic proximity to ensure fast response time may also limit its applicability. Other factors that may limit the method's direct use include a high "xed cost of operating a service center, and situations where goods and services are bundled for marketing or pricing purposes.

References [1] Charnes A, Cooper WW. Management models and industrial applications of linear programming. New York: Wiley, 1961. [2] Dantzig GB. Linear programming and extensions. Princeton, NJ: Princeton University Press, 1963. [3] Williams HP. Model building in mathematical programming. New York: Wiley, 1985. [4] Kall P. Stochastic linear programming. Berlin: Springer, 1976. [5] Horst R, Tuy H. Global optimization: deterministic approaches. Berlin: Springer, 1990. [6] Zinn W, Levy M, Bowersox D. Measuring the e!ect of inventory centralization/decentralization on aggregate safety stock: the &&square root law'' revisited. Journal of Business Logistics 1989;10:1}14. [7] Tallon WJ. The impact of inventory centralization on aggregate safety stock: the variable supply lead time case. Journal of Business Logistics, 1993;14:185}203. [8] Cochran WG. Sampling techniques. New York: Wiley, 1963. [9] Lindo Systems Inc., LINGO: The Modeling Language and Optimizer. Lindo Systems Inc., Chicago, IL, 1994.

Rajesh Tyagi is a supply chain scientist at the General Electric Corporate Research and Development Center in Schenectady, NY. He has published in Operations Research, ¹ransportation Research, Journal of Business ¸ogistics, and International Journal of Physical Distribution and ¸ogistics Management, among others. His current research interests include spare parts planning, and asset management in equipment leasing industries. Chandrasekhar Das is Professor of Management at the University of Northern Iowa. His research interests include inventory management, location and distribution systems, and quality control. He has published over "fty articles on these topics in Management Science, Operations Research, Decision Sciences, European Journal of Operational Research, Naval Research ¸ogistics, IIE ¹ransactions, and other journals.