The purpose of this paper is to develop an approach to a resource-allocation problem that typically appears in

Transcription

1 MANAGEMENT SCIENCE Vol. 50, No. 8, August 2004, pp ssn essn nforms do /mnsc INFORMS Resource Allocaton Based on Effcency Analyss Pekka Korhonen, Mkko Syrjänen Helsnk School of Economcs, P.O. Box 1210, Helsnk, Fnland The purpose of ths paper s to develop an approach to a resource-allocaton problem that typcally appears n organzatons wth a centralzed decson-makng envronment, for example, supermarket chans, banks, and unverstes. The central unt s assumed to be nterested n maxmzng the total amount of outputs produced by the ndvdual unts by allocatng avalable resources to them. We wll develop an nteractve formal approach based on data envelopment analyss (DEA) and multple-objectve lnear programmng (MOLP) to fnd the most preferred allocaton plan. The unts are assumed to be able to modfy ther producton n the current producton possblty set wthn certan assumptons. Varous assumptons are consdered concernng returns to scale and the ablty of each unt to modfy ts producton plan. Numercal examples are used to llustrate the approach. Key words: resource allocaton; data envelopment analyss; fronter analyss; multple-objectve lnear programmng Hstory: Accepted by Thomas Leblng, mathematcal programmng and networks; receved August 22, Ths paper was wth the authors 4 months for 2 revsons. 1. Introducton The problem of resource allocaton s one of the classcal applcatons n management scence. The use of data envelopment analyss (DEA) brngs a new flavor to the problem, because t s possble to consder feasble producton plans and trade-offs between nputs/outputs based on the emprcal characterzaton of a producton possblty set. Snce Charnes et al. (1978, 1979) developed DEA, t has become a very popular method for effcency analyss. DEA s a wdely used method for analyzng techncal effcency, and based on t, a number of addtonal methods are suggested for supportng effcency mprovement plannng. The allocaton problem s currently under actve research n the DEA lterature (see, e.g., Athanassopoulos 1998). Let us consder a decson-makng envronment n whch a set of unts s operatng under a central unt wth power to control some decson parameters, such as resources of those unts. The am of a central unt s to allocate resources n such a way that the overall goals of the organzaton are satsfed as well as possble, or specfcally, the amount of the total outputs of the unts wll be maxmzed. It s natural to assume that the central unt would lke to allocate the resources n such a way that the new producton plan s nondomnated. Ths means that no other allocaton wth the same assumptons leads to as good values on all outputs and a better value on at least one output at the gven level of nput. Thus, the unts are requred to modfy ther producton plan so that the total amount of output s maxmzed. When the number of outputs and nputs s more than one, the problem s a multple-crtera one, whch usually has no unque soluton. The man lmtatons of a tradtonal DEA model n resource allocaton are that (1) t does not take nto consderaton the decson maker s (DM) preferences, and (2) t analyzes one unt at a tme n relaton to the other unts. The frst lmtaton s dscussed n many artcles n the DEA context. Golany (1988) publshed the frst paper proposng the use of preference nformaton when settng performance targets n the context of DEA. He formulated a multple-objectve lnear programmng (MOLP) model, where the outputs were objectve functons. For the same purpose, Thanassouls and Dyson (1992) combned goal programmng and DEA. The dea s to fnd the most preferred target for each unt, when one unt at a tme s consdered. Golany et al. (1993) further developed the target-settng model by suggestng a number of dfferent models for cases where dfferent knds of prce, cost, and budget nformaton are avalable. Furthermore, many authors have analyzed smlartes between DEA and MOLP. For example, see Belton (1992), Stewart (1996), Joro et al. (1998), or Bouyssou (1999). When we consder the second weakness, we can frst notce that a number of DEA models that take nto account the organzatonal settng have been developed. For example, Cook et al. (1998) suggested an effcency analyss model for a herarchcal stuaton where the assessed unts operate under dfferent groups. The groups are assumed to control 1134

2 Management Scence 50(8), pp , 2004 INFORMS 1135 a part of the resources, whch affects the possbltes, and thus the observed effcency, of the ndvdual unts. Färe and Grosskopf (2000) proposed a number of models for stuatons where the producton process s dvded nto a number of subprocesses. One of the models s an allocaton model, where a fxed nput lke farmland s allocated among alternatve uses lke varous crops wthn one unt. The prmary goal of the models s to measure the effcency of the unts. Our purpose s to consder resource-allocaton models, whch help the DM to allocate avalable future resources by takng nto account the total amount of the outputs of all unts, smultaneously. Golany et al. (1993) presented an dea to use a target-settng model for allocatng resources so that the objectves of the whole organzaton are consdered. On the bass of ths dea, Golany and Tamr (1995) developed an allocaton model and a number of extensons to t. An envelopment DEA model s the framework that s used to characterze an effcent fronter and a producton possblty set. The model resembles the target-settng DEA models (e.g., Golany 1988, 1993; Thanassouls and Dyson 1992), but all the unts are consdered at the same tme, and the basc objectve s to maxmze the sum of outputs of all the unts. Further, Athanassopoulos (1995, 1998) presented two resource-allocaton models. The frst model (Athanassopoulos 1995) s based on goal programmng and DEA. In ths so-called Go-DEA model, an effcent fronter s characterzed as n an envelopment DEA model. The second model (Athanassopoulos 1998), called TARBA, s based on a framework smlar to that of Go-DEA. The model s solved n two phases. Frst, the optmal multplers for each unt are found by means of a multpler DEA model. Then, the multplers are used to defne feasble trade-offs n allocaton. The assumptons that concern the unts ablty to change ther nput-output mx and effcency are clearly some of the key factors affectng the results of the resource allocaton. Although many valuable deas have been proposed concernng these assumptons, the decson-makng unts (DMUs ) ablty to change ther nput-output mx and effcency has not been dscussed thoroughly n the lterature. In addton, the multple-crtera nature of the resourceallocaton problem has drawn only lmted attenton. The purpose of ths paper s to develop a general multple-objectve approach to resource allocaton. We assume that a central unt smultaneously controls all the unts. Hence, the settng s smlar to the abovementoned papers that ntroduce resource-allocaton models. In the developed approach, the unts abltes to change ther producton are modeled explctly. The current nput and output values are used to characterze a producton possblty set n a manner smlar to that of Golany and Tamr (1995). To guarantee a realstc allocaton, we assume that the unts are able to modfy ther producton plan wthn the producton possblty set only accordng to certan rules. In ths aspect, we generalze and extend the deas suggested by Golany et al. (1993) and Golany and Tamr (1995). The rules represent factors (manageral, envronmental, etc.) that lmt possble changes durng the plannng perod. One of the key questons s what happens to the observed neffcency. The approach s formulated as a MOLP model and s solved by usng approaches developed for MOLP. In ths aspect, our approach clearly devates from the earler papers. Usng the MOLP model, the DM (a central unt) can search for the most preferred resource-allocaton soluton by tryng to maxmze the amount of several objectves smultaneously. The approach and ts varatons are llustrated by usng a supermarket data set that s extracted from a real applcaton. The rest of ths paper s organzed as follows. Secton 2 dscusses some theoretcal aspects of MOLP and DEA. In 3, we develop the general approach for resource allocaton and present some specfc models based on dfferent assumptons. An llustraton of an allocaton problem wth multple nputs and outputs s gven n 4. Conclusons and dscusson are presented n Prelmnary Consderatons Let us assume that we have n DMUs, each consumng m nputs and producng p outputs. Let X + m n and Y p n + be the matrces, consstng of nonnegatve elements and contanng the observed nput and output measures for the DMUs. We denote by x (the th column of X) the vector of nputs consumed by DMU, and by x j the quantty of nput j consumed by DMU. A correspondng notaton s used for outputs. Furthermore, we denote 1 = 1 1 T and refer by e to the th unt vector n n Multple-Objectve Lnear Programmng Consder the followng multple-objectve lnear program: Max y Mn x ( ) y s.t. T = x {( ) y y Y x X x y 0 x 0 (1)

3 1136 Management Scence 50(8), pp , 2004 INFORMS where = n + and A b n s a feasble set, matrx A k n s of full row rank k and vector ( b k. The set T of feasble values of vector y ) x m+p s called a feasble regon. The purpose of the model s to fnd a feasble lnear combnaton of the nput/output vectors of the exstng DMUs that smultaneously maxmzes all outputs and mnmzes all nputs. The problem, lke any multple-crtera model, has no unque soluton n general. Its reasonable solutons are called effcent solutons n the multple-crtera decson-makng (MCDM) lterature. Defnton 1. A pont ( ) y x T s effcent (nondomnated) ( ff (f and only f) there does not exst another y x) T such that y y, x x, and ( ) ( y x y ) x. Defnton 2. A pont ( ) y x T s weakly effcent (weakly nondomnated) ff there does not exst another ( y x) T such that y > y and x < x. The soluton of Problem (1) s the effcent pont that the DM prefers most. That soluton s called the most preferred soluton. Note that a subset of nputs and/or outputs may be used as objectve functons as well. The nputs and outputs not used as objectve functons may be treated as constrants. The feasble set and the effcent fronter wll be changed accordngly. In prncple, a multple-objectve optmzaton problem can be solved by usng any MOLP technque (for more nformaton, see, e.g., Steuer 1986). A possble and currently popular way to search for solutons on the effcent fronter of a MOLP problem s to use the achevement (scalarzng) functon suggested by Werzbck (1980). Usng that functon, we may project any gven (feasble or nfeasble) pont n the objectve space onto the effcent fronter. The method s called a reference pont method, and the components of the gven reference ponts are called aspraton levels. The DM can vary the aspraton levels as (s)he lkes, and any effcent soluton of a MOLP problem can be generated (Werzbck 1986). The procedure can be contnued untl the most preferred soluton s found. By parameterzng the achevement scalarzng functon, t s possble to project the whole vector nstead of a sngle pont onto the fronter, as orgnally proposed by Korhonen and Laakso (1986). When a drecton s projected onto the fronter, a curve traversng across the fronter s obtaned. That method s called a reference drecton method. Korhonen and Wallenus (1988) further developed a dynamc and vsual free-search verson of the reference drecton method. The dea s mplemented under the name Pareto Race (see, e.g., Fgure 4). In Pareto Race, a reference drecton s determned by the system on the bass of preference nformaton receved from the DM. By pressng number keys correspondng to the ordnal numbers of the objectves, the DM expresses whch objectves (s)he would lke to mprove and how strongly. In ths way (s)he mplctly specfes a reference drecton. Pareto Race enables the DM to freely search any part of the effcent fronter by controllng the speed and drecton of moton. The DM sees the objectve functon values on a dsplay n numerc form and as bar graphs, as (s)he travels along the effcent fronter. The keyboard controls nclude an accelerator, gears, brakes, and a steerng mechansm. The search on the fronter resembles drvng a car. The DM can, e.g., ncrease/decrease the speed, make a turn, and brake at any moment (s)he lkes. The DM may stop the search whenever (s)he lkes. The orgnal reference pont approach can be appled to any sze of problems, because t can be mplemented n an nteractve or nonnteractve mode. Instead, Pareto Race performs best n medumsze problems n whch the DM wants to keep the control n hs/her own hands at all tmes Basc Models n Data Envelopment Analyss In the DEA lterature, the above-defned set T s called the producton possblty set. Because n DEA all unts are feasble solutons, we have to assume that the unt vectors e, = 1 n. We are nterested n recognzng effcent DMUs, whch are defned as a subset of ponts of set T satsfyng the effcency condton defned n the same way as n Defntons 1 and 2. Thus, the methods used to search for effcent (nondomnated) solutons of the MOLP problems can be used to characterze the effcent fronter n DEA. In DEA, the producton possblty set s tradtonally defned by assumng that (a) = n + or (b) = n + and 1T = 1. The frst defnton stands for the so-called Constant Returns to Scale (CRS) assumpton (Charnes and Cooper 1978, 1979); the second one stands for the Varable Returns to Scale (VRS) assumpton (Banker et al. 1984). The standard models based on the above assumptons are also called the CCR model and the BCC model, correspondngly. The output-orented neffcency score of DMU can be defned by solvng the problem { Max ( 1 + y x ) T Instead of neffcency scores, effcency scores are usually reported n the DEA lterature. In ths case, the effcency score can be calculated from the neffcency score;.e., = 1/ Development of an Approach to Effcent Resource Allocaton By the term effcent resource allocaton, we refer to the decson problem n whch a DM (a central unt) ams to allocate addtonal resources and/or to reallocate the current resources to a set of exstng unts for achevng the maxmal output. The problem usually

4 Management Scence 50(8), pp , 2004 INFORMS 1137 has no unque soluton. Any effcent soluton s a ratonal choce. In the followng, we develop a general multpleobjectve model for effcent resource allocaton. The DM s searchng for the most preferred resourceallocaton soluton by tryng to maxmze the amount of several output varables smultaneously. The approach starts from the current practce. The current measured nput and output values are used to estmate a producton possblty set. We assume that (re)allocaton does not change the set. Furthermore, we assume that the unts are able to modfy ther producton n the defned producton possblty set only accordng to certan lmts that represent factors (manageral, envronmental, etc.) that lmt possble changes durng the plannng perod. We wll consder varous assumptons that all lead to a lnear model formulaton. The general approach and ts varatons are llustrated by usng supermarket data, orgnally from a real applcaton. For smplcty and wthout loss of generalty, we assume that the nputs are resources to be allocated and the outputs are objectves to be maxmzed. The approach can easly be extended to a case where the nput varables are mnmzed or any combnaton of nput and output varables s optmzed. The allocaton problem s a multple-crtera problem when there s more than one objectve. The general multple-objectve resource-allocaton model can be formulated as follows: Max y = Y1 = y 1 y 2 y n 1 ( ) {( ) y + y s.t. y T = y Y x X x + x x y 0 x 0 = 1 2 n = { n + and A b ( ) y + y F x + x = 1 2 n x = X1 = x 1 x 2 x n 1 r where Vectors ( y + y x + x ) m+p + = 1 2 n descrbe the nput-output mx after an allocaton. To be feasble, the new producton plan of each unt has to belong to the producton possblty set T. F s the set of producton plans that unt s assumed to be able to reach, and t s called the transformaton possblty set. In general, set F ncludes all the manageral, poltcal, envronmental, etc., factors that restrct the ablty of unt to modfy (2) ts current poston n the plannng perod. The set thus descrbes the possble changes n outputs after resource allocaton. Vector r m stands for the total amount of allowed resource changes,.e., the budget constrant. For example, r < 0 relates to a stuaton where the total use of resources s reduced. Vector r thus descrbes the objectves related to nputs. The producton possblty set descrbes all techncally feasble producton plans whle the transformaton possblty set descrbes the unt s ablty to change ts producton wthn a plannng perod. The DM s preferences and values are taken nto consderaton n the objectve functon and correspondng restrctons on nput changes. One possble bass for the transformaton possblty set s to assume that the unt s effcency stays constant durng the plannng perod. Another possble assumpton could be to allow only proportonal scalng of the exstng producton. Or, we could set a prce for the change and thus subtract a proporton of the potental output or allocated nput relatve to the amount of change. Also, t has been suggested that one could set lmts to the changes n the factors that are not fully controllable. For example, Golany and Tamr (1995) suggested relatve constrants for partally dscretonary varables to guarantee manageral feasblty n allocaton. In the transformaton possblty set we can also fx the change so that envronmental or nondscretonary varables are not changed. Dependng on the assumptons made on the transformaton possblty sets F, = 1 2 n, and the producton possblty set T, we may generate models for varous stuatons. Tradtonally, the DEA lterature assumes that F = T, = 1 2 n, and the unt s able to reach any pont on the effcent fronter. We want to emphasze that t s mportant to consder potental lmtatons of the change n the unt s producton durng the plannng perod. To llustrate the usablty of ths defnton, we consder the stuatons based on the followng assumptons: The transformaton possblty sets F, = 1 2 n: (a) allow smultaneous proportonal scalng of all the nputs and outputs, (b) specfy that a unt s allowed to change the nput and the output values, but no mprovement n effcency s allowed. A lmtaton for the change n nputs s ncluded n F,.e., x x x x. Ths descrbes ranges for feasble resource changes of unts. Note that n the stuaton where s allowed to be equal to x, t s possble that x = x. Ths means that DMU wll be taken out of busness. Thus, the model can also be used to analyze whch busness s worth contnung.

5 1138 Management Scence 50(8), pp , 2004 INFORMS Table 1 The Data Set for Illustratve Examples Thus, the allocaton model looks as follows: Unt Input Output A 1 1 B C D E Total In both cases the producton possblty set T s based on both CRS and VRS assumptons. We would lke to emphasze that the above assumptons are set up for llustratng the possbltes of our approach. The assumptons are problem dependent and should be selected carefully n each applcaton. Models based on dfferent assumptons are frst llustrated usng a smple data set consstng of fve DMUs consumng one nput and producng one output (Table 1). Usng the smple example, the results can be llustrated wth fgures. Because ths data set leads to a sngle objectve, t s not enough to llustrate how to deal wth multple nput/multple output features, whch play an essental role n our approach. For that purpose, we wll use a more complex multpleobjectve problem n 4. The latter problem also demonstrates how the approach works n practce Transformaton Possblty Sets Based on Proportonal Change n All Inputs and Outputs In ths subsecton, we assume that the unt s ablty to change s based on proportonal scalng of exstng producton. Thus, we assume that the transformaton possblty sets are {( ) y + y F = y y { xj = mn x j x + x j = 1 2 m x x The assumpton means that a DMU can transform ts output values so that the smallest rato x j /x j provdes an upper bound for changes n outputs. Note that can be negatve as well f the amount of some nput s decreased. To guarantee that the proportonal scalng s managerally feasble, we lmt the change n nputs to be x x. The proportonal scalng s thus assumed to be vald n the neghborhood of the unts current producton. Let us now assume a convex producton possblty set based on the CRS assumpton,.e., {( y ) T= y Y x X y 0 x 0 = n + x Max y = Y1 = y 1 y 2 y n 1 ( ) {( ) y + y s.t. y T = y Y x X x + x x y 0 x 0 = 1 2 n = n + ( ) y + y F x + x = = mn { xj x j {( y + y x + x j = 1 2 m x x x = x 1 x 2 x n 1 r ) y y Based on the defnton of T, we know that ( ) y x T and 1 + ( ) y x T for 1. From the defnton of ( the transformaton possblty set F, t follows that ff y ) + y x + x F, then y + y y 1 + x + x x 1 + x x. Thus, ( ) y + y x + x F ( y ) + y x + x T for all 1. Ths means that as the maxmum ncrease n outputs s the same as the mn- mum ncrease n nputs, the transformaton possblty sets F are subsets of the CRS producton possblty set. Hence, t s suffcent to only consder proportonal changes of nput and output varables, and the problem can be wrtten as a MOLP problem n the followng way: Max y = Y1 = y 1 y 2 y n 1 s.t. y y = 1 2 n (3) x x = 1 2 n (4) x = 1 2 n where x x = 1 2 n x = x 1 x 2 x n 1 r Usng the data presented n Table 1, we may llustrate possble solutons to ths problem. Now the problem has one nput and one output and s thus reduced to an LP problem. We specfy the model parameters so that = 0 r = 0 1 n x =1 = 0 2x The results of the problem are presented n Table 2 and the changes are llustrated n Fgure 1.

6 Management Scence 50(8), pp , 2004 INFORMS 1139 Table 2 Allocaton Based on the CRS Producton PossbltySet and Proportonal Change Transformaton Table 3 Allocaton Based on the VRS Producton PossbltySet and Proportonal Change Transformaton Unt Input Output Input Output A B C D E Total Unt Input Output Input Output A B C D E Total The model allows the unts to move along (actually below) the rays startng from the orgn and passng through the correspondng unt. Thus, the unt wth the hghest productvty s frst allocated more nput untl the upper bound s reached. Because the change n the unt s nput s bounded, the extra nput s allocated to the unt wth the next hghest productvty, untl the lmt s reached. Unts A, B, and C reach the relatve (+20%) upper lmt and the budget constrant s met, whle Unt D s beng allocated some more nput. Because the restrcton on the ncrease n nput s relatve, the absolute ncrease n the nput depends on the observed sze of the unt. If more approprate, we can naturally also use an absolute lmt. The model wth the CRS producton possblty set and the proportonal change s qute smple. The addtonal resources are frst allocated to effcent unts (wth the hghest margnal productvty) untl the upper bound of the nput change s reached. If we would allow the nput to be reduced, t would frst be taken from unts wth the lowest margnal productvty. Of course, the multple-objectve case s more complex, because the trade-off between objectves has to be consdered. In any case, the CRS effcency score of the unts remans unchanged, and only the relatve sze changes. Let us next assume that the producton possbltes are based on the VRS assumpton,.e., {( ) y T = y Y x X y 0 x 0 x = { n + 1T = 1 In ths case the transformaton possblty sets are not subsets of T, and thus both constrants have to be explctly ncluded n the problem formulaton. Thus, the model can be formulated as follows: Model (4) and the followng constrants: y + y Y x + x X = 1 2 n = 1 2 n 1 T = 1 = 1 2 n 0 = 1 2 n Results based on the same data and the same values of the parameters as above are presented n Table 3 and the changes are llustrated n Fgure 2. As before, the unt wth the hghest margnal productvty s the frst to be allocated more nput. In ths case, the margnal productvty of a unt s not just dependent on ts observed productvty. Unts A and E can ncrease ther output as n the prevous case, along the ray startng from the orgn and (5) Fgure 1 Output Illustraton of the Allocaton Based on the CRS Producton PossbltySet and Proportonal Change Transformaton A B E C D Fgure 2 Output Illustraton of the Allocaton Based on the VRS Producton PossbltySet and Proportonal Change Transformaton A B E C D Input Input

7 1140 Management Scence 50(8), pp , 2004 INFORMS passng through the unt, because the producton possblty set does not restrct the changes defned by the correspondng transformaton possblty sets F. Instead, the possble changes of Unts B, C, and D are restrcted by the VRS producton possblty set, and ther margnal productvty s thus lower than n the CRS case. Thus, Unts A, B, and E reach the upper lmts for nput ncrease Transformaton Possblty Sets Based on Unchanged Effcency In ths secton, we assume that the neffcency score of the unt wth the observed nput and output values ( ) y x s not worse than the neffcency score new of the transformed unt wth nput and output values ( ) y + y x + x ;.e., new. The key dea s to allow a change n the nput-output mx whle keepng the techncal neffcency unchanged. Golany and Tamr (1995) suggested ths knd of approach, as t s lkely that unts that were neffcent n the past wll reman neffcent. However, they dd not present a formal model based on ths assumpton. Dfferent effcency measures (e.g., nput- versus output-orented or radal versus addtve) can be used. They lead to dfferent results, and therefore the way of measurng the neffcency should be selected carefully. To gve an example, we buld a model based on an output-orented relatve neffcency measure. Intally, we determne neffcency scores for each unt, = 1 2 n, usng a sutable DEA model. In the allocaton model, we lmt the effcency change by formulatng the transformaton possblty sets n the followng way: {( ) y + y ( ) F = 1 + y + y T and x + x x + x x x Each set F, = 1 2 n, s a subset of the producton possblty set T, f T n the neffcency evaluaton s the same as n the allocaton model (as we assume). When T s based on the CRS assumpton, we can reformulate the general allocaton model (2) as the followng MOLP model: Max y = Y1 = y 1 y 2 y n 1 s.t. 1 + y + y Y = 1 2 n x + x X = 1 2 n 0 = 1 2 n (6) x = 1 2 n where x x = 1 2 n x = x 1 x 2 x n 1 r Table 4 Ineffcences of the Unts Unt Input Output CRS neffcency VRS neffcency A B C D E Total To llustrate the behavor of the model, we use the same data and the same specfcatons as n the prevous secton. The data and the neffcences based on output-orented CRS and VRS DEA models are presented n Table 4. In the smple case (one nput, one output, and CRS), the results are the same as presented n Table 2. However, n a case of multple nputs and multple outputs, the DM has to consder the trade-offs between nputs and outputs. Hence, the DM consders the allocaton problem not only as a scalng problem, but he s also free to change the mx of nputs and outputs. Ths ncreases the flexblty of the model and provdes the DM wth more possbltes. In the next secton, we demonstrate ths feature wth an example havng two outputs and two nputs. Smlarly, we can assume that the producton possblty set s based on VRS assumpton. The correspondng model s smply: Model (6) and the followng constrant: 1 T = 1 = 1 2 n (7) Results wth ths model are presented n Table 5 and llustrated n Fgure 3. The margnal productvty of a unt now depends on the margnal productvty of the fronter and the unt s neffcency. Unt A can move along the lne AB, and t has the hghest margnal productvty. Unt B has the second-hghest margnal productvty (along the lne BC). Unt E s projected to the pont E on the fronter (lne BC). The margnal productvty at pont E s the same as that of Unt B. Hence, the margnal productvty of Unt E s 1/ tmes that of Unt B. Table 5 Results of the Allocaton wth the VRS Producton Possblty Set and Constant EffcencyTransformaton Unt Input Output Input Output A B C D E Total

8 Management Scence 50(8), pp , 2004 INFORMS 1141 Fgure 3 Output Illustraton of the Allocaton wth the VRS Producton PossbltySet and Constant EffcencyTransformaton A B E E Input C D However, t s hgher than the margnal productvty of Unt C. Thus, addtonal nput has been allocated to Unts A, B, and E up to the +20% lmt. By consderng Unt E, we may demonstrate the mpact of the choce of the orentaton n the effcency measure on the allocaton. The output-orented approach projects Unt E onto lne BC, whereas the nput-orented approach projects Unt E onto lne AB. As we can see from Fgure 3, n the latter case the margnal productvty s hgher. Smlarly, e.g., the use of an addtve DEA model would lead to a dfferent margnal productvty. 4. Illustraton of a Problem wth Multple Inputs and Outputs In ths secton, we llustrate the use of the proposed models n a multple nput and output case. The am s to demonstrate the advanced features of our approach and to llustrate ts practcal relevance. For ths purpose, we wll analyze resource allocaton n supermarkets. The data set s extracted from a reallfe case, whch conssts of 25 supermarkets that are stuated n Fnland and belong to the same chan. The orgnal problem s smplfed by takng nto consderaton only two output varables (Sales and Proft) and two nput varables (Man-hours and Sze). Manhours refers to the labor force used wthn a certan perod and Sze s the total retal floor space of the supermarket. The data set s gven n Table 6. Two dfferent models are consdered as examples. In the frst case, we assume that the producton possblty set s based on the CRS assumpton, and the unts are assumed to be able to change ther nput and output values proportonally n the neghborhood of the current nput-output mx. These assumptons lead to the use of model (4). In the second case, we assume that the producton possblty set T s based on the VRS assumpton. In addton, we assume that the unt s unable to remove or reduce ts neffcency durng the plannng perod. Table 6 Data for the Multple Input and Output Illustraton Supermarket Sales 10 6 FIM Proft 10 6 FIM Man-hours 10 3 h Sze 10 3 m 2 Ineffcency CRS output Ineffcency VRS output y 1 y 2 x 1 x Total Notes. FIM = Fnnsh Mark; 1EUR = FIM.

9 1142 Management Scence 50(8), pp , 2004 INFORMS Fgure 4 Searchng for the Most Preferred Values for the Outputs Pareto Race Goal 1 (max): Total Sales <== Goal 2 (max): Total Proft ==> Bar: Accelerator F1: Gears (B) F3: Fx num: Turn F5: Brakes F2: Gears (F) F4: Relax F10: Ext Thus, the neffcency score of a unt cannot decrease. We also relax the assumpton that a supermarket could adjust ts floor space n a short run. Labor force s assumed to be adjustable. To solve the problem based the assumptons above, we use model (7). As n theoretcal consderatons, we adopt outputorented approaches. The neffcences of all unts produced by the output-orented CRS and VRS models are presented n Table 6. In the frst CRS model, the total sum of both nputs of all the unts s allowed to ncrease by at most 1% (r = 0 01 n =1 x ). To guarantee manageral feasblty, the change n nputs of each unt s lmted to 10% decrease ( = 0 1x ) and 30% ncrease ( = 0 3x ). Note that n ths model, labor force and floor space are vared n the same proporton. The output values wll change n the same proporton as well. Based on the specfcatons presented above, we assume that the chan management s wllng to allocate resources to the supermarkets as effcently as possble. The term effcently means that the management tres to maxmze the total sales and net proft wthn the constrants specfed above. Thus, the reallocatons of resources are allowed to some extent. The MOLP model (4) s smple, and ts sze remans reasonable even for many unts, makng t possble to use an nteractve MOLP approach to solve the model. We use Pareto Race (see Fgure 4) to solve the optmzaton problem n the frst case. The current problem s qute smple, because t conssts of two objectves, Sales and Proft. The DM can search for the most preferred trade-off for these objectves on the effcent fronter. When (s)he has found the fnal soluton, (s)he has solved the allocaton problem. Relatve changes n the nput and output values of the unts based on three alternatve fnal solutons are presented n Table 7. The frst soluton corresponds to Table 7 Proportonal Changes Based on Allocatons for 25 Supermarkets Model 2 Model 1 Soluton 1 Soluton 2 Soluton 3 Soluton 1 Soluton 2 Supermarket scalng scalng scalng Sales Proft Man-hours Sales Proft Man-hours y 1 /y 1 y 2 /y 2 x 1 /x 1 y 1 /y 1 y 2 /y 2 x 1 /x

10 Management Scence 50(8), pp , 2004 INFORMS 1143 Table 8 Changes n the Total Amount of Inputs and Outputs Model 1 Model 2 Output / nputs Soluton 1 Soluton 2 Soluton 3 Soluton 1 Soluton 2 Sales 10 6 FIM (%) Proft 10 6 FIM (%) Man-hours 10 3 h (%) Sze 10 3 m 2 (%) (0) 0 (0) the strategy n whch only addtonal sales are emphaszed, the second one to a strategy wth proft focus, and the thrd one to a balanced strategy where an optmal trade-off s sought. Table 8 presents the correspondng changes n the total amount of nputs and outputs. We notce that for most of the unts the allocaton remans the same n all three solutons. The maxmal amount of addtonal resources s allocated to the effcent Supermarkets 3, 10, and 23 n all three solutons of Model 1 n Table 7, and to the fourth effcent Supermarket 25 only n Soluton 2, where Proft s emphaszed. On the other hand, no neffcent unt receves addtonal resources n all three solutons. More resources are allocated to the neffcent Supermarket 4 when Proft s emphaszed (Soluton 2), and the neffcent Supermarkets 7, 16, 22, and 24 perform better when Sales s emphaszed (Soluton 1). Because the maxmal decrease s smaller than the maxmal ncrease, the number of the unts that receve less resources s larger than the number of the unts that receve more resources. Because the model s based on a CRS producton possblty set and the optmzaton problem s lnear, the solutons tend to be extreme. In the second case, the Sze of each unt s assumed to be fxed durng the plannng perod ( 2 = 2 = 0). Thus, we may drop the varables x 2 from the model. The total sum of Man-hours of all the unts s allowed to ncrease by at most 1%. To guarantee manageral feasblty, the change n Man-hours of each unt s lmted to 10% decrease ( 1 = 0 1x 1 ) and 30% ncrease ( 1 = 0 3x 1 ). Thus, n ths case, the problem s bascally to reallocate the labor force to the unts. If the number of the unts s n, the number of the varables of the model s O n 2. Because the sze of the latter model wth a large number of unts makes t unrealstc to use an nteractve approach to that problem, we use the orgnal reference pont method (Werzbck 1980) for ths problem. Two alternatve solutons are generated n Table 7. The frst soluton corresponds to a strategy n whch addtonal sales are emphaszed, and the second one corresponds to a strategy wth proft focus. The solutons are based on aspraton levels 2,896 and for Sales and Proft n the frst case, and 2,663 and n the second case. As before, the allocaton of many unts remans the same n both solutons. Some nterestng observatons can be made. When Sales s emphaszed (Soluton 1), some addtonal labor force (5%) s allocated to the neffcent Supermarket 2. The supermarket s assumed to ncrease Sales (9%), and to decrease Proft (9%). If Proft s consdered mportant (Soluton 2), Supermarket 2 does not get any addtonal labor force, but Proft s assumed to ncrease (27%), whle Sales decreases (3%). Another nterestng case s the effcent Supermarket 7. When Sales s emphaszed (Soluton 1), there are no changes n the nput and output values, but when Proft s emphaszed (Soluton 2), Proft ncreases (233%) and Sales decreases (1%). Some addtonal resources (1%) are allocated. The purpose of the examples was to demonstrate varous features of the support our approach provdes to the DM. It helps the DM to keep the problem n hs/her own hands, but on the other hand, t guarantees the soluton to be effcent. The DM has the possblty to search from a set of effcent solutons that are specfed by takng nto account the assumptons concernng the producton possblty set and the transformaton possblty sets. 5. Concluson In ths paper, we have developed a general framework for resource allocaton. We consder a decsonmakng envronment n whch a central unt controls the resources of a set of unts. We combne DEA and MOLP to obtan a tool that provdes the DM wth a possblty to ncorporate preference nformaton concernng the relatve mportance of nputs and outputs nto the analyss. DEA s used to characterze the producton possblty set, whch defnes the producton plans that are feasble n general. The set s based on the observed performance of the unts under consderaton. The producton possblty set s a well-known concept n producton economcs. Second, we defne for each unt another set called the transformaton possblty set, whch characterzes the possble shortterm changes for the unt. Each unt has ts own transformaton possblty set that specfes feasble changes for the unt wthn a certan tme frame. The

11 1144 Management Scence 50(8), pp , 2004 INFORMS transformaton possblty set s a concept created n ths paper. The resource-allocaton problem s formulated as a MOLP problem that usually has no unque soluton. It depends on the DM s preferences and the assumptons (s)he makes about the producton possblty set and the transformaton possblty sets. To demonstrate the practcal relevance of our approach, we have llustrated ts use wth a numercal example extracted from a real applcaton. In ths paper, we have consdered the MOLP models where outputs are objectves and nputs are constrants. It s also possble to use outputs as constrants and nputs as objectves, or to use outputs and nputs as objectves smultaneously. We may apply our approach to the cases where DM controls only a part of the unts. In the presented models, we have further assumed that all unts have a common producton possblty set. However, we may extend the approach to the cases where the total set of unts conssts of subsets wth dfferent producton possblty sets. All the models n ths paper are lnear, but the approach can be extended to nonlnear and nteger models as well. A drecton for further research mght be to study the problems where there are nterdependences between unts. References Athanassopoulos, A. D Goal programmng & data envelopment analyss (GoDEA) for target-based mult-level plannng: Allocatng central grants to the Greek local authortes. Eur. J. Oper. Res. 87(3) Athanassopoulos, A. D Decson support for target-based resource allocaton of publc servces n multunt and multlevel systems. Management Sc. 44(2) Banker, R. D., A. Charnes, W. W. Cooper Some models for estmatng techncal and scale neffcences n data envelopment analyss. Management Sc. 30(9) Belton, V Integratng data envelopment analyss wth multple crtera decson analyss. A. Gocoechea, L. Ducksten, S. Zonts, eds. Multple Crtera Decson Makng. Sprnger- Verlag, Berln, Germany, Bouyssou, D Usng DEA as a tool for MCDM: Some remarks. J. Oper. Res. Soc. 50(9) Charnes, A., W. W. Cooper, E. Rhodes Measurng effcency of decson makng unts. Eur. J. Oper. Res. 2(6) Charnes, A., W. W. Cooper, E. Rhodes Short communcaton: Measurng effcency of decson makng unts. Eur. J. Oper. Res. 3(4) 339. Cook, W. D., D. Cha, J. Doyle, R. Green Herarches and groups n DEA. J. Productvty Anal. 10(2) Färe, R., S. Grosskopf Network DEA. Soco-Economc Plannng Sc. 34(1) Golany, B An nteractve MOLP procedure for the extenson of DEA to effectveness analyss. J. Oper. Res. Soc. 39(8) Golany, B., E. Tamr Evaluatng effcency-effectvenessequalty trade-offs: A data envelopment analyss approach. Management Sc. 41(7) Golany, B., F. Y. Phllps, J. J. Rousseau Models for mproved effcences based on DEA effcency results. IIE Trans. 25(6) Joro, T., P. Korhonen, J. Wallenus Structural comparson of data envelopment analyss and multple objectve lnear programmng. Management Sc. 44(7) Korhonen, P., J. Laakso A vsual nteractve method for solvng the multple crtera problem. Eur. J. Oper. Res. 24(2) Korhonen, P., J. Wallenus A Pareto Race. Naval Res. Logst Steuer, R. E Multple Crtera Optmzaton: Theory, Computaton, and Applcaton. Wley, New York. Stewart, T. J Relatonshps between DEA and MCDM. J. Oper. Res. Soc. 47(5) Thanassouls, E., R. G. Dyson Estmatng preferred target nput-output levels usng data envelopment analyss. Eur. J. Oper. Res. 56(1) Werzbck, A The use of reference objectves n multobjectve optmzaton. G. Fandel, T. Gal, eds. Multple Objectve Decson Makng, Theory and Applcaton. Sprnger-Verlag, New York, Werzbck, A On the completeness and constructveness of parametrc characterzatons to vector optmzaton problems. OR Spektrum