Joint Cost Allocation for Multiple Lots

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Joint Cost Allocation for Multiple Lots

Author(s): Bala V. Balachandran and Ram T. S. RamakrishnanSource: Management Science, Vol. 42, No. 2 (Feb., 1996), pp. 247-258Published by: INFORMSStable URL: http://www.jstor.org/stable/2633004

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J o i n t C o s t Allocation f o r Multiple L o t sBala V. Balachandran * Ram T. S. Ramakrishnan

NorthwesternUniversity,Evanston,Illinois60201Universityof Illinoisat Chicago,Chicago,Illinois 60607

A e consider the joint cost allocation problem that arises when several lots or resources areavailable to serve different products or divisions. We provide a two-phase model,wherein the first phase the optimal set of lots to be acquired is chosen and given the optimal

set, and the products using each acquired lot is also determined. In the second phase, a stablefull cost allocation method is developed that will not induce the divisions to form coalitions toreduce the allocated joint costs. Utilizing the optimal dual solution of the lot selection phase, weprovide a joint cost allocation mechanism based on the concept of propensity to contribute andshow that this allocation is also stable. If in the first phase there is a dual gap, then we showthat there is no cost allocation in the core. A numerical illustration is provided.(Cost Allocation;Stability;CoreSolutions;Dual Methods)

1. IntroductionThe allocation of the joint costs of common resourcesacross the divisions or products that need and usethose resources is practiced widely in industry. Thispractice is under greater scrutiny now, as firms are in-creasingly recognizing the value of decentralization.By creating autonomous units with delegated respon-sibility toward personnel, administration, sales, anddistribution, companies are able to react quickly togrowing competitive global markets for inputs andoutputs. This trend has been extensively documented,and the prototype of the future corporation is a shelltype corporate center with no unallocated costs. (SeeDumaine 1992, Business Week 1992 for experiences ofJohnson & Johnson, IBM, Asea Brown Baveri, and DuPont.)

Decentralization requires full allocation of commoncosts. If the corporatecenter does not fully allocate costs,then the operating divisions will impute lower costs tothe common factor inputs and consume the unallocatedresources excessively. (See Fuchsberg 1992 for the in-creased personnel costs arising due to decentralizationand underallocation.) Atkinson (1986) documents thatover 70%of the firms allocate such common costs. Econ-omists (Thomas 1974, Stigler 1966) maintain a view thatallocation of joint or common costs is arbitraryand ir-

relevant for economic decisions, at least in the short run.However, managers in both public and private sectorsrecognize the need for cost allocation in product costing,inventory valuation, and in the measurement of the de-centralized unit's income. Modem cost accounting in-novations have emphasized the need for full cost allo-cation when advocating activity based costing and man-agement. (See Turney 1990 and Cooper 1990 for activitybased allocation and its impact on the accuracy of prod-uct costs.) In the public sector, especially in defense re-lated industries, major cost recoveries are heavily de-pendent on the allocated joint costs and transfer prices.Thomas (1974) questions the usefulness of such alloca-tions for decision making and emphasizes that alloca-tions may only be "useful in reaching understandingswhereby individuals or organizations agree to distri-butions of resources." Following Thomas (1974), sev-eral researchers have examined different "mutually sat-isfactory allocations" based on particular economicprinciples or behavioral axioms. A few have usedmathematical programming approaches and dualitytheory for optimal allocations, as seen in Kaplan andThompson (1971) and Kaplan and Welam (1974). An-other group, (for instance, Moriarity 1975, 1976; Loud-erback 1976; and Jensen 1977) defined certain sets ofaxioms or objectives that allocation models should

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BALACHANDRAN AND RAMAKRISHNANJoint CostAllocation or MultipleLots

Table 1Incremental rocessing Costs

LotLot # j Cost 1 2 ... n1 JC1 /11 /21 I. l Inl2 JC2 /12 /22 I. 2 ... In2j JCy lj 12j IiJ ... Injm - 1 JCm_1 /1,m-1 /2,m-1 ... li,m-1 ... In,m-im 0 /1m Im ... Iim Inm

possess and provided models that satisfy their sets ofobjectives.Several models of allocation have been developed re-cently, based on game theoretic concepts such as thecore or Shapley value. These cost allocation schemeshave the stability property, such that the divisions shar-ing the joint or common resources have no incentive toleave the common resource acquisition structure. Suchallocation models are found in Jensen (1977), Hamlenet al., (1977, 1980) Balachandran and Ramakrishnan(1981), Roth and Verrechia (1979), Gangolly (1981), andHughes and Scheiner (1980), to name a few. All thesemodels are applicable only to the situation where a sin-gle lot or resource is to be acquired for a group of di-visions or products. Very little analysis has been carriedout for the case with multiple common resources. Manydivisions may use the whole set of resources or lots, butall the divisions need not use the same common re-source. An alternate stream of current research has fo-cused on the existence of a nontrivial role for allocationsin resolving incentive conflicts in a multiple agent set-ting (Demski et al. 1988, Zimmerman 1979, Balachan-dran et al., 1988, Rajan 1992). These papers use hiddenaction models in a principal-agent framework and showthat some allocation of fixed common costs could re-duce collusion problems. In this paper the commoncosts are not fixed and need to be solved first beforeallocation. Further in those papers incentive problemsarise due to effort aversion and private information. Inthis paper the only incentive issue considered is the mo-tivation of the unsatisfied divisions to form a separateresource acquisition pool.

Recently, a leading gasket manufacturer found it prof-itable to procure multiple lots for their gasket products,where different product lines use different raw materi-als (steel, rubber, cork, etc.). They found it worthwhileto pile up such raw material inventory lots "to take ad-vantage of economies of scale from different vendors"so that the total costs were minimized. They were facedwith allocation of joint ordering costs of such multiplelots to their end products servicing original equipmentmanufacturers such as automobiles and agriculturalequipment. The firm had to select the optimal way ofprocuring the lots and had to decide on the allocationof the ordering costs to the end products.This paper considers the two problems-the selectionof optimal lot set and allocation of common costs-si-multaneously. The first problem is solved through amathematical programming approach and several pa-pers in the management science literature have mod-elled this problem and developed efficient algorithms.Crow (1972) considered an assignment method for amulti-mission system in a defense project with priori-ties, which was later extended by Jensen (1978) for sen-sitivity of priorities via a parametric analysis of the lin-ear programming solution. Since linear programs canyield fractional solutions and optimal solutions can notvary continuously when integer values are required forvariables, these methods are deficient. Further, stabilityamong participants as defined through the game-theoretic core property may not exist in such allocations.Sets of participants may have incentives to leave andform separate cost pools to acquire resources for onlythat set.This paper is aimed to reduce the gap that exists inthe multiple resource literature-between optimizationand cost allocation-and provides an allocation schemewhich has all the desirable stability properties enunci-ated by Moriarity (1975). Secondly, we provide a modelthat eliminates the fractional outcome that can arise inthe models of Jensen (1978) or Crow (1972). Our modelformulation and primal solution assignment generatesan optimal integer solution. We use the algorithms pro-posed by Erlenkotter (1978) for the fixed charge prob-lem which uses the optimal dual solutions. Thirdly, ourmodel formulation provides the solution to both theproblem of ex-ante lot selection from the existing set offeasible scarce lots and the problem of ex-post cost al-

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location, once the optimally selected lots and the divi-sions sharing the selected resources are known. Thusthis allocation scheme does not influence the decisionprocess in the first stage. Fourth, we use the "propensityto contribute" concept developed by Balachandran andRamakrishnan (BR) (1981) and the optimal dual solu-tion of the ex-ante lot selection problem to obtain aunique cost allocation. As the dual solution has an eco-nomically meaningful interpretation, the ex-post allo-cation procedure that is developed lies in the core. Thisguarantees stability, so that the overall total profit ismaximized without sacrificing either the divisional in-centives or their independence. Finally we relate thedual gap that may arise in the lot selection phase to thesubsidy that will be needed to ensure that the cost al-location is in the core.2. Multilot Model DevelopmentFor expositional ease, we adopt the same notation as inthe papers of Balachandran and Ramakrishnan (1981),Moriarity (1975), or Louderback (1976) with one exten-sion. An index j is used as a subscript for the lotnumber,since all these authors consider only a single lot.In this paper we consider multiple resources, lots, orsystems' where each one of these lots can be purchasedfor a known cost. These lots are used by a group ofdivisions, economic entities, or subsidiaries so that fur-ther processing yields the final required products (orsatisfies a mission as in Crow's (1972) problem). Sothere are two types of costs: joint cost of the lots andfurther processing costs for each product. If a productcannot be obtained from any specific lot, we define thecorresponding processing cost as a large number (infin-ity). The firm also has the option of buying the inputfor that product directly from the market. This directpurchase cost is generally higher than the incrementalprocessing cost with some lot, before any allocation ofthe lot cost. If it is not, we eliminate such products fromconsideration since the raw material for that productcould be purchased directly. Thus, without any loss ofgenerality we assume that the direct market price is

'We will use the word "lot" in a generic sense. This model could beused for other situations as well, such as systems, resources, supportcenters, and common overhead facilities.

more than the incremental processing cost for at leastone resource. Similar to the single lot case, if a lot ispurchased for the production of a specific product, thenthe entire required amount of that product can be pro-duced by further processing the specified lot.Let the number of lots for these resources or raw ma-terials to be m - 1, and let j be a typical lot so that j = 1,... ., m - 1. For the purposes of the model, let the exter-nal market also be a lot corresponding to the decisionof "buying directly from outside or open market." Theexternal market will be labelled with index m, as the lastlot. Thus there are m lots, where the first m - 1 lotscorrespond to the resources and the last one corre-sponds to "market." The firm consists of n different di-visions or products.

Let i be the index for a typical product or division sothat i1, .. ., n. LetJC1: epresent the joint cost of the resource or lot j;(j= if . .. , m - 1),Iij:be the incremental cost of further processing forproduct i (i = 1, . . ., n) using lotj (j = 1,.. ., m - 1)Iim:represent the cost of buying product i directlyfrom the external market.Since there is no cost for the lot labelled m (market),

JCmwill be set equal to zero. The matrix of both jointand incremental costs, as given in Table 1, describes thevariables.3. The Two-Phase ModelThe joint cost allocation problem for the multiple lotcase will be analyzed as a two-phase model consistingof an ex-ante decision problem and an ex-post allocationproblem, given the decisions of the first phase. The firstphase involves the production or lot selection decision,while the second phase is concerned with joint cost al-location ex-post, given the production decision of thefirst phase. The production decision is obtained by theformulation of a "fixed charge" optimization modelsimilar to a fixed charge transportation problem so thatthe total corporate costs are minimized. Then utilizingthe dual solution associated with this optimal solutionand the core theory presented in the earlier literature,22 For a full discussion of core and game-theory related definitions andillustrations see Hamlen et al. (1977, 1980), Jensen (1977), and Bala-chandran and Ramakrishnan (1981).

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an allocation scheme is developed. This scheme has allthe advantages that Moriarity (1975) proposed, plus theproperty of economic justification implied by the dual-ity theory of optimization literature3 and the fairnessand stability properties of the core theory.The matrix of costs presented in Table 1 identifies twosets of costs. One is concerned with the fixed cost JCj,which is incurred if lot j is bought to be used by any setof products. Iijrepresents the incremental cost of furtherprocessing. The problem of optimally selecting (1) thelots to be bought and (2) the products assigned to usethe chosen lot for further processing, is similar to thefixed charge transportationproblem discussed in the op-erations research literature. This model has been formu-lated for (1) "facility location" decisions and (2) further"transportation"decisions given the facilities that are tobe opened from the decision in step 1. However, Erlen-kotter(1978) has developed an efficientalgorithm (a com-puter code and performanceresults are also included inthe paper) which uniformly has achieved good compu-tational results. We will make use of this algorithm andshow how the dual solution of this model has a meaning-ful economic property for joint cost allocation.Phase I: The Optimal Resource Selection ProblemThe "Fixed Charge" Multilot Problem.The firm's problem is to optimally select lots that shouldbe bought, from the available set of lots, and to opti-mally assign the divisions or products to the selectedlots. This can be formulated as a mixed-integer pro-gramming problem called the "fixed charge" multilotproblem. We use the expanded cost matrix (includingthe market lot option as the source m) and a JCvectorof all joint costs as in column one of Table 1. Let ujbe adecision variable for the purchase or rejection of avail-able lotj: LetI1, f resource/ otj is purchased,

l 0, if resource/ lot j is not purchased,= 1, if product is made using resource ,

xii 1 otewie

3 Dual approaches were used in allocation applications by Kaplan andThompson (1971) and Kaplan and Welam (1974) for linear and non-linear programs. The dual approach for integer, fixed charge trans-portation problems is fully discussed by Erlenkotter (1978).

With the costs defined in Table 1, the firm's mixed-integer program (IP) is given by:

m n mIP: Minimize ZIp= I JCju; + I I IijXi1, (1)j=1 i=1 j=1

Subject to xij = 1, for i =1, ..., n, (2)j=l

uj - Xi}2 O0,or i =1...,n and j ,... ,(3)xij:2?O, fori=1,...,n and j=1,...,m, (4)

u1 Oorl, forj=1, ..,m. (5)The objective function (1) is the sum of the joint costs

of resources or lots as given by the first term and theincremental processing costs as given by the secondterm. The constraint set given by (2) implies that eachproduct i should be produced from some lot j or j = 1,... . m - 1 or bought from outside (j = m). Constraintsets (3) and (4) require that xijcan be zero, but if it is tobe one, then the associated uj must also be one. (A lotcannot be used for production unless it is bought.) Con-straint set (5) is obvious since we either buy or rejectlotj. However, in constraint set (4) we do not require xijtobe 0 or 1 but impose only nonnegativity. This relaxationis acceptable as the optimal solution will have xijequaltoO or 1, because product i will be produced fully fromthe open lot (uj = 1) with the least incremental cost.If, in the integer program (IP) above given by (1) to(5) we replace constraint set (5) by

0 < uj < 1 forj = 1, ...,m, (5')and keep the other constraint sets (1) to (4) as they are,then the resultant program is a linear program denotedby (LP). This problem (LP) is called the "linear relaxa-tion" of the integer program (IP). Exploiting the specialstructure in the constraints, Erlenkotter (1978) used a"dual ascent" algorithm for the associated LP to obtaintight lower bounds for the IP. The dual of this LPis thenused to obtain an integer solution for IP. The dual so-lutions lead to a good candidate primal solutions,thereby giving an upper bound. Since Erlenkotter(1978)has discussed the solution procedure to solve Phase I,our aim here is to show that the dual method immedi-ately gives a solution to Phase II, thereby yielding a so-lution to the joint cost allocation problem.



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Phase II: The Joint Cost AllocationSuppose a solution to Phase I, solving the ex-ante de-cision problem of buying lots at minimal cost, has beenobtained, i.e., the uj's are optimally solved, for the IPgiven by (1)-(5). We can then use the one-dimensional"single lot" methods considered for joint cost allocationby Jensen (1977), Hamlen et al., (1977, 1980), and Bal-achandran and Ramakrishnan (1981) (BR) to solve thePhase II problem. We use the rule that "each lot's costmust be entirely paid for by the products using that lot."

Sets of products such that all members of a given setuse only one source (lot) can then be formed. Then thecost allocation methods of the one lot problem, i.e., theone we propose here, or the Shapley Value can be usedon a set of decomposed single lot problems. Our prop-osition here is that the solution method used in Phase Ioffers an economically justifiable allocation mechanism.We show here that the ex-post allocation mechanismbased on the optimal decisions of Phase I lies in the core.Further, the two problems have been integrated into onemodel, thus linking the two objectives simultaneously.The Dual MethodWe now present a solution method used for solving IP.The "fastest" solution technique has been by dual basedmethods. The dual for LP (1)-(4) will be formulatedwith the relaxed constraint set (5') instead of (5). Let vi,i = 1, . . ., n be the dual variables of constraint set (2),and wij,i = 1, . ., n; j = 1, . . ., m, be the dual variablesfor constraint set (3). Then the dual program for the LPis

nDP: Maximize ZDP= V,, (6)nSubject to Xwij c? JCj, forj = 1, .., m, (7)

i=l

iv'-wij c Ii>, fori=1,...,nandj=1, ..., m, (8)wij 2 O, fori =,...,nandj = 1, ...,m. (9)

Since the objective function contains only vi's and be-cause of the special structure in constraints (7) and (8),it is easy and computationally efficient to solve the dualprogram, (DP). We can reduce the wij'sto their lowestpossible (nonnegative) values without violating con-straints (8) and (9) to achieve the maximum value for

the objective function ZDP (since no wi1'sare present inZDP). Thus we have the "Condensed Dual" (CD) givenbelow, which has eliminated the m x n constraints givenin (8).

nCD: Maximize ZDP= V, (6)i=l

nSubject to I MaxIO,vi - Ii>)c JCj,i=l

j= 1,...,m. (10)Here wijhas been set equal to Max(O,vi - Iij>.Note thatthis program CD has only n variables since we elimi-nated m x n variables wij. Further, it has only m con-straints, since m x n constraints given by (8) have beendropped. Good solutions for DP are relatively easier tofind via CD, even though (10) is piece-wise linear in v,.

4. Solution to the ProblemFrom any dual optimal solution vi, we can form a pri-mal integer solution through:

nuj= O, if , MaxIO,vi - Iij < JCj, (11)i=lnuj = 1, if , maxIO,vi - Ii) = JCj. (12)

i=l

Let J*-j Iuj = 11. Then, set (13)1, if Iij= minkeJ Iij,and

l0 otherwise. (14)We can also form the following complementary slack-ness conditions connecting the dual and the primal pro-grams:

(uj- xi) maxIO,vi - Iij = 0,i = 1, ..., n; j=1,...,m, (15)

nuj1[c1 - Y maxIO,vi-Iij) =,j=1,...,m. (16)

Equation (16) will be always satisfied if the solutions of(11) to (14) are used. Equation (15) will not be satisfiedif more than one j E J*has vi > Iij,for any i. There will

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be a dual gap as ZDP < ZIp,and ?7 discusses this situ-ation. Until ?7 we will assume that there is no dual gap.Dual Program Solution MethodsThough the dual program can be solved by regularmathematical programming methods, as the problemhas a special structure, we can use a simple, easy-to-understand process to obtain the optimal solution. Er-lenkotter (1978) has used the dual ascent and dual ad-justment procedures. For full details the reader is re-ferred to Erlenkotter(1978). Here we shall describe onlythe dual ascent procedure as it helps us in the solutionto the allocation problem and also since we make useof it in the example.

The Dual Ascent Procedure. As the objective func-tion contains only vi's, they must be increased to theirmaximum values subject to constraint (10). The vi's canbe initiated to the lowest value of Iij n each column ofTable 1. The left-hand side of (10) will be zero as no Iijis > vi. The vi's will then be sequentially increased tothe next higher value in each column so that constraint(10) will not be violated, or to the highest value possible.We cycle through the columns until none of the vi's canbe increased. This is the dual ascent procedure. The dualadjustment procedure further improves the solution oralters it to reduce the complementary slackness viola-tions of (15).

Erlenkotter (1978) and Morris (1977) have demon-strated that in this type of problem, almost in all casesthe LP solution itself is an integer solution. From thetheory of duality, at the optimum, ZDPof the dual pro-gram is equal4 to Z LP.But if the primal solution for LPturns out to be integers in uj,ZDP= Z I in (1). Since wehave solved the DP given by (6) and (10) we have v*,the optimal solution to the dual program. The dual so-lution v* facilitates the joint cost allocation problem asone scheme we can use is:

Rule: v* is the cost allocated to product i.The total costs will be fully allocated with this scheme.

4In this section, we will assume that the dual optimal solution leadsto a primal feasible integer solution. If this is not true, then the gametheoretic properties that we derive for the allocations may not hold.This situation is shown for some cases in ?7.

This is so, since the optimal primal objective value(which is source costs plus the incremental costs) isequal to the optimal dual objective value, 1n=,v*, by theduality theory.

The economic justification is that v* is the cost of hav-ing constraint i in (2) of LP, which is =1xij = 1. Theconstraint means that we have to make product i fully.So product i must incur that cost. All the properties enu-merated by Moriarity (1975) are satisfied. Let J*be theset of open sources. Letj(i) be such that4ij(i)= Min Iij =si. (17)jeJ

The sum of the allocated joint costs in each group usingthe same source is the cost of that source,E [Vi Si,] = JCk,

i;j(i)=k

from the dual constraint (10) by definition.The most important property of the allocation wewant to consider is the question of stability. If the allo-cation scheme offers no incentives for the divisions tobreak out and form subgroups or coalitions then ourscheme will be stable. This is precisely the game-theoretic concept of core. Let N denote the index set ofj = 1, ... , n. An allocation v is in the core f the total costallocated to any coalition S C N is less than or equal tothe costs that coalition S will incur if it goes alone. Wecan prove that our dual based allocation is in the core.

THEOREM . Costallocationsbasedon any dualoptimalsolution v* are in the core.PROOF. See the appendix.Thus we see that dual solution based allocationschemes are economically defensible, and are not sub-ject to the criticisms of sterility and arbitrariness in al-locations discussed by Thomas (1974). However, there

is a problem, as the dual solution to the problem maynot be unique. This occurs because the primal solutioncan be highly degenerate with many basic variables tak-ing zero values. There may be whole space of optimaldual solutions. To find an unique solution that avoids"arbitrariness"for the allocation problem, we can makeuse of the "propensity to contribute" factor suggestedby Balachandran and Ramakrishnan (1981).

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5. Allocation ProceduresAn alternate set of allocation methods have been pro-posed in accounting literature without using the coreconcept. These methods produce unique allocations andare based on the concept of imputations (see Shubik1991, p. 141, for a summary). The core may not exist inmany problems (see ?7 for some cases), but imputationsalways exist as they guarantee only that no individualdivision has an incentive to use another source in anyproposed allocation. But there are many imputation so-lutions, and several authors in management accountingliterature (e.g., Hamlen et al. 1977, 1980; Jensen 1977)have suggested the use of alternate minimum cost as abasis for allocation.

For each product let bibe the alternate cost that it willincur if it does not use the source assigned to it basedon the minimum independent cost. Let si denote the in-cremental cost Iijincurred by product i in the optimalsolution as defined in (17). For product i, si is the basevalue for allocated costs, and it also has to take a shareof the fixed cost of the source it is using in the finalallocation. Let the share of the source costs be ri so thatthe final allocation ai is si + ri. As enunciated in Bala-chandran and Ramakrishnan (1981) for the single lotcase, the propensity to contribute (PTC) is then definedby:

PTCi(Propensityo)contributeJ

.bi Si (18)_ Least cost by \ (Incremental\alternate methodsJ V cost

We can use the PTCs as the weighting factors withineach group. The difficulty in applying this method inthe multiproduct case is that the least cost by alternatemethods is difficult to characterize. The main purposeof this section is to propose a definition of this cost. Letj(i) be the lot assigned to product i as defined in (17)and J* be the set of open sources. Let bqbe the incre-mental processing cost using the second-best opensource for product i, i.e.,

b= Min (IIi.jEJ*,j*j(i)

b. cannot be greater thanb', as division i can always useanother open source to get the next best incremental cost.This is analogous to using I,r (the alternatesource cost) inLouderback's 1976 method, for the one-lot problem.

Also, if bi ncreased excessively for product i, it mightbe optimal for the division to consider buying an un-used source after paying the fixed cost of that source.The dual procedure can be used to check this. Source jwill be bought if the vi's change such that the slack onconstraint (10), viz.,

niC1 max(O, vi - Ii),i=l

becomes zero. We can first use the base value si's forvi's and obtain the slack on each unused source. Thus,for eachunopenedsourcej i J*,et,qij= Ij + JCj- S max(O,Sk - Ikjl. (19)k*i

Division i will suggest opening source j if the allocatedcost is increased to qij.We have to find the minimum ofall such values over all j * J (analogous to finding thesecond-best among open sources), so that14= Min(qij). (20)j0rThe maximum alternate cost bimust be the minimumof 14 nd bqso that product i will not switch to someother open source or go out and try to open a newsource. Then biQmin(I4, bq . LetH(i)be the set of prod-ucts using the source of product i, denoted by j(i). SoH(i) = (k: (k) = j(i)). Within groups we can use thePTC's derived from biusing (18) as the weighting fac-tors so that the final allocation is:

bi- sia= Hbk - S) JC1(i) Si. (21)-k C=HU) ( k SO)We can easily show5 that lkeH(i) (bk - Sk) 2 JCj(i).So from(21), ai c bi.As bi= min(I4, bq , bi c bqand biC 14 The

For any dual solution, vi, as there is no dual gap, from (15), vib?.The primal solution assignment (11) imply vi < Iij + JCj-Xk*i maxi0, Vk - Ikjl for every unopened source j. From (20),bK= Iij+ JCj- -k*-imaxi0, Sk - IkjI for some unopened source j. V1k 2 Sk, thelowest cost among open sources as otherwise Vk can be increased with-out violating any dual constraints. This implies vi _ bi and v,i bi.From (12), for the open sources the dual constraint is binding and Sk- Ikj(i)or all k i H(i). So XkEH(,) (Vk - Sk) = JCj(i) < XkEH(i) (bk - Sk).

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Table 2Incremental rocessing Costs IjLot Product

Lot # Costj JCj 1 2 3 4 5 6 71 50 60 180 100 00 100 20 1052 60 55 110 00 80 100 30 1003 70 65 co 50 100 40 40 04 0 80 160 120 140 50 25 170

allocated cost is less than the independent cost of the sec-ond-best open source for each i, and the minimum costincurred to open a new source.This PTC based allocationscheme a is an alternativeto the core allocation.To sum-marize,the essential differencesare: (1) the coreallocationmay not exist in some situations (see ?7) while a alwaysexists, (2) the core allocationmay not be unique while a isunique, and (3) core satisfies group rationalitywhile a sat-isfies only individual rationality.Illustration of the AllocationsThe following example illustrates the dual method andthe allocation procedure. Consider the data in Table 2associated with symbols of Table 1 for three lots andseven products.

Source 4 corresponds to the alternative of directlybuying from the outside market.A cost of oo mplies thatthe product concerned cannot be served by the resourcementioned in the row. Further,the underlined costs are

the lowest independent costs for each product. Table 3shows the dual solution for the first three iterations:The dual ascent procedure is complete. The current so-lution is also optimal. LetJ*= (2, 3, 4) and assign products1, 2, 4, and 7 to source 2, products 3 and 5 to source 3, andproduct 6 to source 4 (open market). Since among opensourcesJ*,only one source is availablethroughthe condi-tions vij- Iij> 0, for all i, the complementarityslacknessconditions(15) and (16) are satisfied. Hence we have a pri-mal optimalsolution.The minimum independentcosts are

si= 55 110 50 80 40 25 100 for i = 1, ..,7.Allocation of CostsThe maximum allocations that can be charged to eachdivision i based on open set (2, 3, 4) are:

V = 65 160 120 100 50 30 170 for i = 1, ...,7.Using the si's we can calculate qil, the maximum slackof the only closed source, j = 1, as:

qil = Ij + 50 - I maxI0, Sk - Ikl = bK.k*iThe maximum allocations that can be charged to eachproduct based on closed sets are:

bK 105 225 145oo145 70 150 for i = 1, ...,7.Note that it is optimal only for division 7 to go forsource 1 if source 2 is not available by offering to pay45 for the joint costs. Division 7 will be willing to paythe rest of the joint cost 5 as it benefits by going to

Table 3Slack = JCj- Dualsolution v',max 0, k -IkjiIteration k*i 1 2 3 4 5 6 7

1 50 55 110 50 80 40 20 1006070 '-' = 4552 45 60 160 100 85 50 25 1000

10 v = 5803 35 60 160 110 85 50 25 10000 ,i= 590

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source 1. bi, the maximum charge is the minimum ofb7 and b,, i.e.,bi = 65 160 120 100 50 30 150 for i = 1, ...,7.

Hence the propensity to contribute PTCi = bi - siPTCi = 10 50 20 50 for i = 1, 2, 4, 7PTCi = 70 10 fori = 3,5PTC6= 5

For source 2 we have I PTCi = 130 and r, = 10 60= 4.6 and so on, to get ri = (4.6, 23.1, 9.2, 23.1) for i= 1, 2, 4, 7. For source 3 we have L PTCi = 80 and ri= (61.3, 8.7) for i = 3, 5 and r6 = 0. So the final allo-cated costs areai = si + ri = (59.6, 133.1, 111.3, 89.2, 48.7, 25, 123.1)

for 1, . .,7.7. Dual gap and the existence ofcore solutionsIn ?4 we assumed that the optimal dual solution leadsto an integer primal solution that attains the same ob-jective value. From the theory of duality, the optimaldual objective value will equal the optimal objectivevalue of only for the unconstrained primal program(not restricted only to integer solutions). While in manyapplications the dual gap may not arise (see Tables II-VI in Erlenkotter 1978), the interesting question iswhether we can obtain stable ull cost allocation schemesif there is a dual gap. In this section, we will first illus-trate this situation with an example and derive a generalresult connecting the dual gap and the existence of coreallocation solutions. Consider the following example inTable 4.

It is easy to see that the optimal dual solution is vi= 100 for i = 1,2,3, and the optimal dual objective valueis L vi = 300. The constraints (10) are binding for j = 1,2,3, but not for] =4. The corresponding primal solutionis uj =forj = 1, 2, 3 and ai =only if Iij< oo or j = 1,2, 3. Note that the primal objective value Ej JCj1u+ Ei Ej Iijaij lso equals 300. But the optimal primal in-tegersolution is to open and use only source 4 for all theproducts. The primal objective value is 330, and there isa dual gap. As all the products are identical, the only

Table4

Lot# Lot Costi Jjc 1 2 31 100 50 50 02 100 50 0 503 100 0 50 504 210 40 40 40

equitable full cost allocation rule is to allocate 110 toeach product. But this allocation is not in thecore, as anytwo divisions can acquire an appropriate source andservice themselves for a total cost of 200 while they arecharged 220 in the joint cost allocation procedure. It isclear that there is no cost allocation that is in the corefor this example.

We will analyze the existence of core solutions in' aspecialized setting where all the products are identical.In that case, a source's cost structure depends only onits capacity expressed in the number of products that itcan serve. Let s (1 c s c n) denote the capacity andJC(s) denote the joint cost of a source that can serve amaximum of s products at an incremental cost of I(s)for each product that it serves. Without loss of gener-ality we will assume super additivity, i.e., for 1 c k c s,

iC(s) + sI(s) c [JC(k)+ kI(k)I+ [JC(s - k) + (s - k)I(s - k)]. (22)

If super additivity is not satisfied, then the source of agiven size s never needs to be considered and can bereplaced by two sources of capacity k and s - k withlower total costs. Note that the cost structure describedin Table 4 satisfies super additivity. The Iij matrix ofTable 1 can be completed as follows: for sources of sizes, the independent cost Iij is I(s) for s products andequals m for the rest of n - s products. With this coststructure we can show the following.

THEOREM. For hecaseof denticalroducts, coreullcostallocation uleexists fandonly f theres no dualgap.The intuition behind the theorem can be seen by ob-

serving that, as the cost function is super additive, theprimal optimal solution is to choose the largest capacity

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source of size n. The total cost will be JC(n) + nI(n).The full cost allocation to each product is vi4 = JC(n)/n + I(n). Is this allocation a feasible solution in the dualprogram? If for sources of size s, v I(s), then con-straints (10) of the dual program will not be binding. Iffor sources of size s, vi > I(s), then constraint (10) is:

s(JC(n) + I(n) - I(s)) c JC(s).nCombining these two cases, we can see that there willbe no dualgap if and only if

JC(n) Cs)+I(n) )+ I(s) V s, 1 c s c n. (23)n sBut note that this is also the condition for a coalition ofsize s not to break away from the whole group and ac-quire a source of size s only for the coalition. The allo-cated cost to the coalition s(JC(n)In + I(n)) should beless than JC(s) + sI(s), the cost for the coalition alone.Since (23) implies that allocation v* is in the core, thetheorem is proved.

For the situation with symmetrical products, the es-sential condition is (23), which says that the productcosts computed using the full costing method must bedecreasing as the capacity increases. If this is true, thenthere will be no dual gap, and there are cost allocationsthat are in the core. If there is a dual gap and if the firmwants to preserve stability, then the firm must subsidizejoint resource acquisition. The minimum subsidyneeded is the dual gap.

Various economic rationale for allocating commoncosts (relating to pricing for usage to reduce congestioncosts, etc.,) are provided in Banker et al. (1988), Millerand Buckman (1987). Cohen and Loeb (1990) show thatfirms facing price competition will bid as if they areplanning to allocate fixed costs completely ex-ante.These papers assume that one agent, i.e., the centralheadquarters, can decide on the allocation scheme andignore the motivation of the individual divisions to ac-cept the cost allocation. The results of this section com-plement the result of those papers as we show that asubsidy (equal to the dual gap) will be required if tidyallocation methods are to be used. But once the firmstarts subsidizing common resource acquisition, then allthe advantages of decentralization, accurate product

costs, etc., outlined in the introduction, may be compro-mised.

8. ConclusionWe have provided a two-phase model which simulta-neously handles the acquisition-assignment decision ofmultiple lots (ex-ante) in Phase I and the joint cost al-location (ex-post), given the optimal selection of lots inPhase I. Utilizing the optimal dual solutions of the lotselection problem, a joint cost allocation mechanismbased on the concept of propensity to contribute is pro-vided. This allocation is shown to be in core if there isno dual gap. For the case of identical products, we alsoshow that if there is a dual gap, then there is no corefull cost allocation rule.

The existing literatureessentially has considered differ-ent cost allocationschemes for a single lot case. Here weconsider multiple lots and prove stability,a concern thatdoes not arise with one lot. Further,these cost allocationschemes are done on an ex-post basis, while here we dis-cuss ex-antedecisions as well. Severalcriteria orevaluatingcost allocationschemes are discussed in the literature,butthe emphasis has been on (1) rationalityand fairnessand(2) neutralitywith respect to decision making (Hamlen etal. 1977,p. 616). However, most of these models considerthe product/purchase decision independent of the cost al-location problem. Our model permits the simultaneousanalysis of ex-ante producdiondecision and ex-post allo-cation mechanisms. Further,our allocationssatisfy all thecriteria nunciatedearlierand those of Moriarity 1975).Wecombine the richness of duality theory, the stabilityof thecore,the incentivecompatibilityrequirementsof game the-ory, and the propensity to contributeaspects appealing tofairness.Finallya numerical llustration s provided, basedon the algorithm presentedin the paper.66 The authors would like to thank the referee and the editor for theirvaluable comments.

AppendixTHEOREM1. Allocationsbasedon any dual solution are in the core.PROOF. For any allocation v to be in the core, no coalition should

have any incentives to break away and incur lower total costs. Let Sdenote any subset of products and let c(S) be the minimum cost thatthe coalition will incur, if it acquires a separate set of resources. Sup-



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pose that when coalition S is alone it acquires the set of resourcesdenotedby J.Then,c(S) = E MinIIij+ XJCj. (24)ies Ji-I jl

Once the set of resources J is bought, each product will be producedfrom the source in J that has lowest incremental cost Iij.The first termin (24) is the total incremental processing cost and the second term isthe total source cost. Allocation v will be in the core only if

c(S)X v vi VSCN, (25)ieS

where the vi's are the allocated costs based on a dual allocation andN is the set of all products.

As the dual solutions satisfy constraints (1), we haveY, Max 0, vi -Iij} JC; for =1,. M.mieN

Adding these constraints for j E J,X X Max(O,vi - Iij jICj. (26)iGN je1eLet](i) be defined such that,

Ii,j-(i) Min(IiiJ. (27)jelThen from (26) we have,X Cjc E (Max(O, i- + E_ MaxO0, i Iij)jel ieN jel,j*j(i)

X Max(O,vii, I,()) I E Max(O, vi - I,j(j) since S C NieN ieS

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Joint Cost Allocation for Multiple Lots