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On apportioning costs to customers in centralized continuous review inventory systems.


Many manufacturing firms use common production and warehousing resources to jointly supply customers' demands that vary significantly in their variability. For example, it is a common practice in the defense manufacturing industry, while supplying both to the department of defense and the commercial market, to routinely combine safety stocks of the same part meant for different end users, thereby achieving lower overall operating costs operating costs nplgastos mpl operacionales . Whereas it might be possible to make a strong case for benefits of centralization (provided this does not result in substantial increases in administrative costs), it is not at all clear that an individual customer will necessarily benefit from centralization under some commonly used methods of allocating inventory costs to customers. For example, if inventory costs are spread evenly over all items stocked (produced), it is possible that a customer will end up paying more (in absolute terms (Alg.) such as are known, or which do not contain the unknown quantity.

See also: Absolute
) than what it would have paid on a stand-alone basis. The purpose of this paper is to first show that it is economical to consolidate. It then investigates some apparently "reasonable" and popular methods of cost allocation to demonstrate their potential for unfairness. It also proposes a method that is fair in the sense that any customer's post-centralization share of overheads does not exceed its costs without consolidation. The findings of this study are particularly relevant for customers with large steady demands, such as the department of defense, who might inadvertently be subsidizing inventory costs of customers with smaller and more variable demands under an unfair allocation scheme.


Inventory centralization, or consolidation of several uncertain demands, is known to reduce costs through risk-pooling, or statistical economies of scale (e.g., Eppen (1979); Zinn, Levy and Bowersox (1989)). An important question that arises when several customers are served from the same centralized stock, however, is how to allocate the inventory costs among them. If that is not done rationally, some customers may end up with higher allocated costs than if they were served from a dedicated stock, despite the overall benefit of consolidation. This issue arose in a dramatic fashion in 1987 during an inquiry by the U.S. Defense Contract Audit Agency The Defense Contract Audit Agency (DCAA), under the authority, direction, and control of the United States Under Secretary of Defense (Comptroller), is responsible for performing all contract audits for the United States Department of Defense (DoD), and providing accounting and , which discovered that the Department of Defense (DOD (1) (Dial On Demand) A feature that allows a device to automatically dial a telephone number. For example, an ISDN router with dial on demand will automatically dial up the ISP when it senses IP traffic destined for the Internet. ) indirectly subsidizes commercial customers of common suppliers (Production and Inventory Control, (1987)). It was argued that the DOD's demand is considerably less variable than that of the commercial customers, a fact that the MRP II (Manufacturing Resource Planning II) An information system that integrates all manufacturing and related applications, including decision support, material requirements planning (MRP), accounting and distribution. See MRP and ERP.  systems used by most suppliers to generate the cost allocation do not recognize, and hence the implicit subsidization.

In this paper we consider this issue in the context of a continuous review order quantity-reorder point (Q,r) inventory system with complete back ordering. We use this more complex framework, rather than the simpler news vendor model used by Eppen and others to assess benefits of risk pooling, since both cycle and safety stocks, rather than only the latter, are present here. After showing that aggregation always reduces the total expected costs in this system as well, we consider four cost allocation schemes: by demand volume, by individual safety stock requirements, by incremental contribution to joint costs, and proportional to stand-alone costs.

A simple numerical example with normal demands shows that the first three methods can easily lead to higher costs for particular customers than their stand-alone costs. Only cost allocation in proportion to each customer's stand-alone costs guarantees a reduction in costs allocated to it under centralization.


Suppose that each separate warehouse, as well as the central warehouse, has a fixed ordering (setup) cost A, a unit holding cost h per unit time, and a unit shortage cost [pi] independent of the duration of shortage. Ordinarily, these costs are not entirely insensitive to the demand volume which will be considerably higher for the centralized facility. Furthermore, additional administrative expenses may be required for managing large (common) warehouses. This issue is not so critical for large defense suppliers as they already have MRP (Material Requirements Planning) An information system that determines what assemblies must be built and what materials must be procured in order to build a unit of equipment by a certain date.  systems in place to manage resources to meet demands generated by a heterogenous (spelling) heterogenous - It's spelled heterogeneous.  mix of customers. These additional costs might make centralization less desirable, particularly after the common warehouse crosses a size threshold. However, inclusion of these factors would not affect the main finding of this paper which is to demonstrate the potential for unfairness in some seemingly "reasonable" schemes for allocating costs in common inventory systems. Therefore, for simplicity of exposition, we assume that the size threshold has not been crossed and that the cost parameters A, h and [pi] are constants. Additional administrative costs, although easy to model, will also not be considered. The lead time, whether fixed or random, is assumed to be the same in all separate warehouses and the central warehouse. Denote the (expected) demand per unit time, and the random lead time demand at warehouse i, i = 1,2, ... N, by [D.sub.i] and [X.sub.i] respectively. Let the density, the cumulative distribution and the mean of [X.sub.i] be [g.sub.i], [G.sub.i] and [mu.sub.i] respectively. Thus, in the decentralized system, the approximate expected relevant cost per unit time at warehouse i, that replenishes stock by an amount [Q.sub.i] each time the inventory level reaches [r.sub.i], is:

[C.sub.i]([Q.sub.i], [r.sub.i]) = A[D.sub.i]/[Q.sub.i] + h([Q.sub.i]/2 + [r.sub.i]-[mu.sub.i]) +


(1) where
  [b.sub.i][(r.sub.i)]   E[([X.sub.i]-[r.sub.i])*]   [ .sub.r.sub.i.sup.infinity

is the expected number of units short during a replenishment cycle (Hadley and Whitin (1963)). The safety stock is [r.sub.i] - [mu.sub.i]. The optimal order quantity-reorder point pair ([Q.sub.i.sup.*], [r.sub.i.sup.*]) is the solution to the following simultaneous equations:
  [Q.sub.i.sup.*] = [square root] 2[D.sub.i][A +  [pi][b.sub.i][(r.sub.i.sup.*)]
/h               (3)
  [ .sub.i][(r.sub.i.sup.*)] = h[Q.sub.i.sup.*]/[pi][D.sub.i]

where [G.sub.i] = 1-[G.sub.i].

In the centralized system In telecommunications, a centralized system is one in which most communications are routed through one or more major central hubs. Such a system allows certain functions to be concentrated in the system's hubs, freeing up resources in the peripheral units. , let G be the aggregate cumulative lead time demand distribution, its mean is [mu] = [EPSILON.sub.i = 1.sup.N][mu.sub.i], and D = [EPSILON.sub.i = 1.sup.N][D.sub.i]. Then, the approximate expected relevant cost per unit time of ordering Q units each time the inventory level reaches r, is:

C(Q,r) = AD/Q + h(Q/2+r-[mu]) + [pi]Db(r)/Q.


The optimal pair (Q*,r*) is the solution to the following simultaneous equations:
  Q* = [square root] 2d[A = [pi]b(r*)]/h
   (r*) = hQ*/[pi]D.

We now want to prove that centralization is always beneficial in this model. While this is intuitive, it is not as obvious (or easy to prove) here as in the newsvendor Noun 1. newsvendor - someone who sells newspapers
newsagent, newsdealer, newsstand operator

market keeper, shopkeeper, storekeeper, tradesman - a merchant who owns or manages a shop
 model, since here, due to the "simultaneous" ordering, the centralized system is not a simple "relaxation" of the decentralized one. So we want to prove that:

C(Q*, r*) [is less than or equal to] [EPSILON.sub.i = 1.sup.N] [C.sub.i]([Q.sub.i.sup.*]

[r.sub.i.sup.*] = [EPSILON.sub.i = 1.sup.N][(A[D.sub.i]/[Q.sub.i.sup.*]) = h([Q.sub.i.sup.*]/2 +

[r.sub.i.sup.*]-[mu.sub.i]) = [pi][D.sub.i][ .sub.i][(r.sub.i.sup.*)]/[Q.sub.i.sup.*]]. (8)

Let Q = [EPSILON.sub.i = 1.sup.N][Q.sub.i.sup.*] and r = [EPSILON.sub.i = 1.sup.N]. Since (Q, r)

is usually not optimal for the centralized system,

C(Q*, r*) [is less than or equal to] C(Q, r). So if we manage to prove the stronger result C(Q, r) [is less than or equal to] [EPSILON.sub.i = 1.sup.N][C.sub.i] (Q.sub.i.sup.*], [r.sub.i.sup.*]), it will imply the weaker result of interest.

Proposition 1:

C(Q, r) [is less than or equal to] [EPSILON.sub.i = 1.sup.N][C.sub.i]([Q.sub.i.sup.*],




[EPSILON.sub.i = 1.sup.N] [C.sub.i]([Q.sub.i.sup.*], [r.sub.i.sup.*]) - C(Q, r) = A[

[EPSILON.sub.i = 1.sup.N] ([D.sub.i]/Q.sub.i.sup.*])-[EPSILON.sub.i = 1.sup.N][D.sub.i]/

[EPSILON.sub.i] = 1.sup.N][Q.sub.i*]]


+ [pi]{[EPSILON.sub.i = 1.sup.N][[D.sub.i][b.sub.i]([r.sub.i.sup.*])/[Q.sub.i.sub.*]]-b(r)

[EPSILON.sub.i = 1.sub.N][D.sub.i]/[EPSILON.sub.i = 1.sub.N][Q.sub.i.sup.*]}.

To prove that the above is positive, it suffices to show that: (i) b(r) [is less than or equal to] [EPSILON.sub.i = 1.sup.N][b.sub.i][(r.sub.i.sup.*)]; and

[Mathematical Expression Omitted]

variables. Inequality (ii) is a standard property of positive numbers (it can be proved, for example, by induction). Hence the claim is proved.


Perhaps the simplest and most common way to allocate the inventory costs C(Q*,r*) of the centralized system is proportionally to the realized demands (or uniformly to each unit stocked). Since over the long run the average realized demands per unit time will be close to their expected values, this is approximately equivalent to allocating costs to the [] customer proportionally to [D.sub.j]/D. This method completely ignores the differentials in demands' variabilities, and is the likely.major reason for the implicit subsidization of small and "variable" customers by large and "stable" ones previously discussed. Indeed, we will show an example where the allocated costs to a large and stable customer are higher than the costs when it is served in a stand-alone manner.

Since demand variability primarity affects safety rather than cycle stocks, another method would be to allocate total costs according to according to
1. As stated or indicated by; on the authority of: according to historians.

2. In keeping with: according to instructions.

 safety stocks only. That is, costs are allocated proportionally to [(r.sub.j.sup.*]-[mu.sub.j)]/[EPSILON.sub.i = 1.sup.N] ([r.sub.i.sup.*]-[mu.sub.i].). Note that the [r.sub.j.sup.*]'s are the reorder re·or·der  
v. re·or·dered, re·or·der·ing, re·or·ders
1. To order (the same goods) again.

2. To straighten out or put in order again.

3. To rearrange.

 points in the decentralized system. We will present an example showing that with such a method small and variable customers may end up being allocated much higher costs than the costs of their standalone systems.

Another approach is to use the costs of the individual systems as a basis for allocation. That is, to allocate the total costs in proportion to [C.sub.j]([Q.sub.i.sup.*], [r.sub.j.sup.*])/ [EPSILON.sub.i = 1.sup.N] ri*). Now, since we have proved that C([Q.sub.*, [r.sup.*]) [is less than or equal to] [EPSILON.sub.i = 1.sup.N] [C.sub.i]([Q.sub.i.sup.*], [r.sub.i.sup.*], it immediately follows that:

Proposition 2:

C([Q*, r*){[C.sub.j]([Q.sub.j.sup.*], [r.sub.j.sup.*])/[EPSILON.sub.i = 1.sup.N] [C.sub.i]

([Q.sub.i.sup.*], [r.sub.i.sup.*])} [is less than or equal to] [C.sub.j]([Q.sub.j.sup.*],



So this method guarantees that the costs allocated to each customer will be lower than its standalone costs. The last method we shall mention is to allocate total costs in proportion to each customer's incremental contribution to them. Denoting the optimal costs of a centralized system without customer j by [C.sub.j]([Q.sub.j.sup.*], [r.sub.j.sup.*]), this method amounts to allocating costs according to the proportions:

{C(Q*, r*)-[C.sub.j]([Q.sup.jj*], [r.sup.j*])}/{NC(Q*, r*)-[EPSILON.sub.i = 1.sup.N][C.sup.i]

([Q.sup.i*], [r.sup.i*])}


Our examples will first show that this scheme does not reduce to the previous one when demands are independent. Furthermore, it is possible that this method will allocate to a large and stable customer a higher cost than its stand-alone costs even when demands are independent.


A numerical example illustrating the unfairness of every scheme discussed in the previous section other than allocation in proportion to stand-alone costs is presented in Table 1. In the example, the same product is demanded by three customers with independent normally distributed demands. Customer 1 has high volume steady demand, whereas customers 2 and 3 have low volume, highly variable demands. We assume a 250 working day year with average daily demands for customers 1, 2 and 3 being 320, 8 and 4 units respectively. The lead time is assumed to be constant at 50 days (same for all customers). The mean and standard deviations of lead time demand for the three customers are: (16000, 50), (400, 120) and (200, 85) respectively.


Uniformly allocating overheads to every unit sold (procedure 1) results in a higher charge being made to customer 1 (an increase of 4.35%) than would have resulted from a stand alone stock, as this customer accounts for over 95% of the total demand for the item. Allocation according to safety stocks (procedure 2) assigns a very small cost to customer 1 and virtually all of the inventory system cost is shared by customers 2 and 3. For example, the inventory costs of customer 2 increase by nearly 124%. Allocating costs proportionally to stand-alone costs (procedure 3) results in a lowering of each customer's costs. Finally, allocation by incremental contribution to costs (procedure 4) also results in an unfairly high charge to customer 1, but for reasons quite different from those of procedure 1. In this case, the high demand volume of customer 1 causes a large relative increase in inventory system costs when its demand is consolidated with that of the remaining two customers.

Interestingly, the total safety stock requirement actually goes up upon consolidation: 299 units after as opposed to 249 units before centralization. However, centralization is clearly beneficial as it reduced inventory costs by approximately 19% ($1144.37).

Examples showing unfairness of all methods except procedure 3 when demands' distributions are dependent are just as easy to find.


This paper shows that for centralized stores allocating inventory costs either according to volume of demand or safety stocks or incremental contribution to total costs may result in situations where some customers are charged more than what they would have paid when separate stocks are maintained for each customer. This manifests itself in customers with large and stable demand subsidizing the highly variable low demand customer or vice versa VICE VERSA. On the contrary; on opposite sides. . A method that allocates costs according to each customer's stand-alone costs is found to be "fair" in this respect.


Eppen, G.D. "Effects of Centralization on Expected Costs in a Multi-location Newsboy Problem." Management Science, vol. 25, 1979, 498-501. Hadley, G., and Whitin, TM. Analysis of inventory Systems. Englewood Cliffs, NJ: Prentice Hall Prentice Hall is a leading educational publisher. It is an imprint of Pearson Education, Inc., based in Upper Saddle River, New Jersey, USA. Prentice Hall publishes print and digital content for the 6-12 and higher education market. History
In 1913, law professor Dr.
, 1963. Production and Inventory Control, "Government Cost Probe May Scuttle MRP II." Production and Inventory Control, vol. 22, no. 1, 1987, 7. Zinn, W, Levy, M., and Bowersox, D.J. "Measuring the Effect of Inventory Centralization/Decentralization on Aggregate Safety Stock: The |Square Root Law' Revisited." Journal of Business Logistics, vol. 10, 1989, 1-13.

Mesbah U. Ahmed "A Volume and Material Handling Cost Based Heuristic A method of problem solving using exploration and trial and error methods. Heuristic program design provides a framework for solving the problem in contrast with a fixed set of rules (algorithmic) that cannot vary.

 for Designing Cellular Manufacturing Cellular Manufacturing is a model for workplace design, and is an integral part of lean manufacturing systems. The goal of lean manufacturing is the aggressive minimisation of waste (or, more precisely, muda) in order to achieve maximum efficiency of resources.  Cells") is Professor of Information Systems and Operations Management at the University of Toledo National recognition
In its 125-year history UT has garnered several national accolades. The University’s programs, faculty and facilities have been highlighted in the media, including
. He received his M.S. and Ph.D. in Industrial Engineering from Texas Tech University. His research interests are in manufacturing systems, information technology, expert systems and decision support systems. He has published in Decision Sciences, Naval Research Logistics Quarterly, MIS Quarterly, Industrial Engineering Transaction, Information and Management, and Journal of Systems Management.

Nazim U. Ahmed ("A Volume and Material Handling Cost Based Heuristic for Designing Cellular Manufacturing Cells") is an Associate Professor in the Department of Management Science at Ball State University. He has published in International Journal of Production Research, Transportation Research, Journal of Academy of Marketing Science, Production and Inventory Management and in many other journals. He is also certified as a CPIM by the American Production and Inventory Control Society and serves on the editorial review board of several journals.

Lawrence D. Fredendall ("Load Smoothing by the Planning and Order Review/Release Systems: A Simulation Experiment") is Assistant Professor of Production and Operations Management at Clemson University. He received his PhD in operations management from Michigan State University Michigan State University, at East Lansing; land-grant and state supported; coeducational; chartered 1855. It opened in 1857 as Michigan Agricultural College, the first state agricultural college. . His current research interests include job shop scheduling, synchronous manufacturing, and quality control.

Yigal Gerchak ("On Apportioning Costs to Customers in Centralized Continuous Review Inventory Systems") is Associate Professor of Management Science at the University of Waterloo The University of Waterloo (also referred to as UW, UWaterloo, or Waterloo) is a medium-sized research-intensive public university in the city of Waterloo, Ontario, Canada. The school was founded in 1957. , Canada. Dr. Gerchak's operations management research has focused on the effect various sources of uncertainty have on inventory policies and costs, and on the trade-offs involved in reducing these uncertainties. This study was motivated by an article appearing in Production and Inventory Control, "IE News," 1987, that attributes the problem of unfair allocation of costs post consolidation to MRP II systems.

Diwakar Gupta ("On Apportioning Costs to Customers in Centralized Continuous Review Inventory Systems") is Assistant Professor of Production and Management Science at McMaster University, Canada. Dr. Gupta's primary research interest lies in stochastic models Stochastic models

Liability-matching models that assume that the liability payments and the asset cash flows are uncertain. Related: Deterministic models.
 of production systems for performance and investment evaluation. This study was motivated by an article appearing in Production and Inventory Control, "IE News," 1987, that attributes the problem of unfair allocation of costs post consolidation to MRP II systems.

Maliyakal Jayakumar ("Measurement of Manufacturing Flexibility: A Value-based Approach") is Assistant Professor of Management Sciences at the University of North Texas where he teaches courses in Statistics and Mathematical Programming. He graduated from the Indian Institute of Technology, Madras and received an M.S. from the University of California The University of California has a combined student body of more than 191,000 students, over 1,340,000 living alumni, and a combined systemwide and campus endowment of just over $7.3 billion (8th largest in the United States).  at Santa Barbara and a Ph.D. from Pennsylvania State University Pennsylvania State University, main campus at University Park, State College; land-grant and state supported; coeducational; chartered 1855, opened 1859 as Farmers' High School. . His research interests are in large scale mathematical programming, parallel algorithms and their applications in operations management.

Steven A. Melnyk ("Load Smoothing by the Planning and Order Review/Release Systems: A Simulation Experiment"), a graduate of the University of Western Ontario Western is one of Canada's leading universities, ranked #1 in the Globe and Mail University Report Card 2005 for overall quality of education.[2] It ranked #3 among medical-doctoral level universities according to Maclean's Magazine 2005 University Rankings.  (1981), is Professor of Operations Management at Michigan State University. Currently, Dr. Melnyk is involved in research pertaining to buyer/supplier relationships, time-based competition, tooling management and control and shop floor management. He is the author of six books and numerous articles which have appeared in national and international journals. Dr. Melnyk sits on the manufacturing process committee of CIPM CIPM Comité International des Poids et Mesures (International Committee of Weights and Measures)
CIPM Center for Integrated Pest Management
CIPM Certificate in Investment Performance Measurement
 for APICS APICS Association for Operations Management
APICS Educational Society for Resource Management (formerly American Production and Inventory Control Society)
APICS American Production & Inventory Control Society
. Finally, he is the software editor for APICS: The Performance Advantage.

Udayan Nandkeolyar "A Volume and Material Handling Cost Based Heuristic for Designing Cellular Manufacturing Cells") is an Assistant Professor of Information Systems and Operations Management at the University of Toledo. He obtained his Ph.D. in Operations management from Pennsylvania State University. His research interests are in the areas of manufacturing flexibility, manufacturing systems design and small lot size manufacturing. He has published in the International Journal of Flexible Manufacturing Systems and International Journal of Production Research.

Patrick R. Philipoom ("Sequencing Rules and Due Date Setting Procedures in Flow Line Cells with Family Setups") is Associate Professor of Management Science at the University of South Carolina
''This article is about the University of South Carolina in Columbia. You may be looking for a University of South Carolina satellite campus.

. He completed his Ph.D. in 1986 at Virginia Tech. He has published in Decision Sciences, IIE See Apple II.  Transactions, Journal of Operations Management, International Journal of Production Research and several others. He is a member of Decision Sciences Institute (DSI (Dynamic Systems Initiative) An umbrella term for a suite of Microsoft products that help manage the Windows environment in large enterprises. DSI was introduced in 2003. ), The Institute of Management Science (TIMS TIMS Thermal Ionization Mass Spectrometry
TIMS The Institute of Management Sciences
TIMS Thermal Infrared Multispectral Scanner
TIMS Transportation Information Management System
TIMS The International Molinological Society
TIMS Tuberculosis Information Management System
) and the American Production and Inventory Control society (APICS). His primary areas of interest include shop floor control, just-in-time manufacturing and computer simulation.

Ram Rachamadugu ("Assembly Line Design with Incompatible Task Assignments") earned his doctoral degree in Industrial Administration from Carnegie Mellon University Carnegie Mellon University, at Pittsburgh, Pa.; est. 1967 through the merger of the Carnegie Institute of Technology (founded 1900, opened 1905) and the Mellon Institute of Industrial Research (founded 1913). . His prior work experience includes academic positions at the Indian Institute of Management, Bangalore, India, and the University of Michigan (body, education) University of Michigan - A large cosmopolitan university in the Midwest USA. Over 50000 students are enrolled at the University of Michigan's three campuses. The students come from 50 states and over 100 foreign countries. , Ann Arbor. His research publications have appeared in the Journal of Operations Management, Journal of the Operational Research Society, International Journal of Production Research, Naval Research Logistics, and Management Science. Dr. Rachamadugu has research interests in the design and operational control of multiple option assembly lines, lot sizing with learning effects in setups, and study of the impact of flexibility on scheduling decisions.

Gary L. Ragatz ("Load Smoothing by the Planning and Order Review/Release Systems: A Simulation Experiment") is Associate Professor of Operations Management in the Eli Broad Graduate School of Management at Michigan State University. He holds an MBA in quantitative analysis Quantitative Analysis

A security analysis that uses financial information derived from company annual reports and income statements to evaluate an investment decision.

 from the University of Wisconsin and a PhD in operations management from Indiana University. His research on production scheduling and capacity planning has appeared in The Journal of Operations Management, Production and Inventory Management Journal, Decision Sciences, and The International Journal of Production Research. He is an active member of APICS, the Institute of Industrial Engineers, and the Decision Sciences Institute.

Ranga Ramasesh ("Measurement of Manufacturing Flexibility: A Value-based Approach") is Assistant Professor of Decision Sciences at Texas Christian University Texas Christian University, at Fort Worth; Christian Church (Disciples of Christ); coeducational; opened 1873 at Thorp Spring, chartered 1874 as Add Ran Male and Female College. It assumed its present name in 1902 and moved to Fort Worth in 1910. , where he teaches courses in operations management and management science. he received a Ph.D. in Operations Management from Pennsylvania State University, an M.B.A. in Finance from the University of Rochester The University of Rochester (UR) is a private, coeducational and nonsectarian research university located in Rochester, New York. The university is one of 62 elected members of the Association of American Universities. , and an M.E. in Mechanical Engineering from the Indian Institute of Science. His research interests include modeling and analysis of stochastic inventory systems and flexible manufacturing systems.

Gregory R. Russell ("Sequencing Rules and Due Date Setting Procedures in Flow Line Cells with Family Setups") is a Ph.D. candidate majoring in production and operations management at the University of South Carolina. He received a B.S. in Mechanical Engineering and an M.B.A. from the University of Kentucky Coordinates:  The University of Kentucky, also referred to as UK, is a public, co-educational university located in Lexington, Kentucky. . His research interests include shop floor control, manufacturing accounting systems, world class manufacturing, and time-based manufacturing. He is a student member of the American Production and Inventory Control Society, Decision Sciences Institute, and The Institute of Management Science.
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Author:Gerchak, Yigal; Gupta, Diwakar
Publication:Journal of Operations Management
Date:Oct 1, 1991
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