Expert Systems with Applications 37 (2010) 4331–4336

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Expert Systems with Applications

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Relationship between MOLP and DEA based on output-orientated CCR dual model

F. Hosseinzadeh Lotfi a, G.R. Jahanshahloo a, M. Soltanifar a,*, A. Ebrahimnejad b, S.M. Mansourzadeh c

a Department of Mathematics, Sciences and Research Branch, Islamic Azad University, Tehran, Iranb Ghaemshahr Branch, Islamic Azad University, Ghaemshahr, Iranc Semnan Branch, Islamic Azad University, Semnan, Iran

a r t i c l e i n f o

Keywords:Data envelopment analysis (DEA)Multiple objective linear programming(MOLP)Min-ordering methodInteractive multiobjective programming

0957-4174/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.eswa.2009.11.066

* Corresponding author. Tel.: +98 21 77557873.E-mail address: soltanifar@khayam.ut.ac.ir (M. Sol

a b s t r a c t

Data envelopment analysis (DEA) and multiple objective linear programming (MOLP) are tools that can beused in management control and planning. While these two types of model are similar in structure, DEAis a performance measurement tool that was initially without consideration of the decision maker (DM)’spreference structure and is directed to assessing past performances as part of management control func-tion and MOLP to planning future performance targets. This paper will establish an equivalence modelbetween DEA and MOLP and show how a DEA problem can be solved interactively by transforming it intoMOLP formulation. We will use Zionts–Wallenius (Z–W) method to reflecting the DM’s preferences in theprocess of assessing efficiency. A case study will conducted to illustrate how DEA-oriented efficiencyanalysis can be conducted using the MOLP method.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

DEA and MOLP can be used as tools in management control andplanning. The structures of these two types of model have much incommon but DEA is directed to assessing past performances as partof management control function and MOLP to planning futureperformances.

The original DEA model does not include a decision maker(DM)’s preference structure or value judgments while measuringrelative efficiency, with no or minimal input from the DM. Valuejudgments are defined by Allen, Athanassopoulous, Dyson, andThanassoulis (1997) as ‘‘logical constructs, incorporated withinan efficiency assessment study, reflecting the DM’s preferencesin the process of assessing efficiency”. To incorporate DM’s pref-erence information in DEA, various techniques have been pro-posed such as the goal and target setting models of Golany(1988), Athanassopoulis (1998) and weight restrictions modelsincluding imposing bounds on individual weights (Dyson &Thanassoulis, 1988), assurance region (Thompson, Langemeier,Lee, & Thrall, 1990), restricting composite inputs and outputs,weight ratios and proportions (Wong & Beasley, 1990), and thecone ratio concept by adjusting the observed input–output levelsor weights to capture value judgment to belong to a given closedcone (Charnes & Cooper, 1990; Charnes, Cooper, Lewin, &Seiford, 1994). Alternative approaches include (Thanassoulis &Allen, 1998) whose model adopts unobserved DMUs, derived

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tanifar).

from pareto-efficient observed DMUs, and which incorporatevalue judgments; Zhu also integrated preference informationinto a modified DEA formulation (Zhu, 1996), while Golany andRoll used hypothetical DMUs to represent preference information(Golany & Roll, 1994).

However, all the above-mentioned techniques would requireprior articulated preference knowledge from the DM, which inmost cases can be subjective and difficult to obtain. In manufac-turing or service organizations, decision making can becomemore complex and often inherently uncertain, more so due tomultiple attributes and conflicting objectives. Multiobjective pro-gramming methods such as multiple objective linear program-ming (MOLP) are techniques used to solve such multiplecriteria decision making (MCDM) problems. Many decision mak-ing problems can be formulated as multiobjective optimizationproblems (MOP). There hardly exists the solution that optimizesall objective functions in MOP, and then the concept of Paretooptimal solution (or efficient solution) is introduced (Sawaragi,Nakayama, & Tanino, 1985). Usually, there exist a number ofPareto optimal solutions, which are considered as candidates offinal decision making solution (Koopmans, 1951). It is an issuehow decision makers decide the final solution from the set ofPareto optimal solutions. Interactive multiobjective optimizationmethods have been developed to this end (Gal, Hanne, &Stewart, 1999; Sawaragi et al., 1985; Wierzbicki, Makowski, &Wessels, 2000). These methods search a decision makingsolution by processing the following two stages repeatedly: (1)solving auxiliary optimization problem to obtain a Pareto opti-mal solution closest to the given aspiration level and (2) revising

4332 F.H. Lotfi et al. / Expert Systems with Applications 37 (2010) 4331–4336

their aspiration levels by making trade-off analysis. An appealingmethod to incorporate preference information, without necessaryprior judgment or target setting, is the use of an interactive deci-sion making technique that encompasses both DEA and MOLP.Golany (1988) first proposed an interactive model combiningboth of these approaches where the DM will allocate a set of le-vel of inputs as resources and be able to select the most pre-ferred set of level of outputs from alternative points on theefficient frontier. Post and Spronk (1999) combined the use ofDEA and interactive multiple goal programming where prefer-ence information are incorporated interactively with the DM byadjusting the upper and lower feasible boundaries of the inputand output levels. Then, Joro, Korhonen, and Wallenius (1998)showed that there are synergies from both DEA and MOLP, andshowed that the DEA formulation is structurally similar to thereference point approach of the MOLP formulation. Further theconcept of value efficiency analysis (VEA) has been introducedthat effectively incorporate preference information in DEA (Hal-me, Joro, Korhonen, Salo, & Wallenius, 1999; Joro, Korhonen, &Zionts, 2003; Korhonen, Somismaa, & Siljamaki, 2002; Yang,Wong, Xu, & Stewart, 2009) investigated equivalence modelsand interactive trade-off analysis procedures in MOLP, such thatDEA-oriented performance assessment and target setting can beintegrated in a way that the decision makers’ preferences can betaken into account in an interactive fashion. In fact they investi-gated three equivalence models between the output-orienteddual DEA model and the min-max reference point formulations,namely the super-ideal point model, the ideal point model andthe shortest distance model (see also (Wong, Luque, & Yang,2009)).

This paper will establish an equivalence model between DEAand MOLP and show how a DEA problem can be solved interac-tively by transforming it into MOLP formulation. We use Zionts–Wallenius(Z–W) method (Zionts & Wallenius, 1976) to reflectingthe DM’s preferences in the process of assessing efficiency. A casestudy is conducted to illustrate how DEA-oriented efficiency anal-ysis can be conducted using the MOLP method.

The remainder of the paper is organized as follows. The nextsection provides a brief description of the DEA techniques andintroduces the basic DEA models. In the subsequent sections, theequivalence model between DEA and MOLP, and the general formu-lation for our proposed model is shown. The next section describesbriefly an interactive multiobjective programming method. Finallyan application on the performance measurement of an insurancecompany in IRAN is examined

2. DEA models

DEA has evolved tremendously over the year and emerged as abody of concepts and methodologies, which consist of a collectionof models and extensions to the original work of Charnes, Cooper,and Rhodes (1978).

In DEA, the organization under study is called a DMU (DecisionMaking Unit). The definition of DMU is rather loose to allow flexi-bility in its use over a wide range of possible applications. Generi-cally a DMU is regarded as the entity responsible for convertinginputs into outputs and whose performances are to be evaluated.

As a performance measurement and analysis technique, DEA is anonparametric frontier estimation methodology based on linearprogramming for evaluating relative efficiency of a set of compara-ble DMU that share common functional goals.

Suppose an organization has n DMUs (j = 1,2, . . . ,n), produces soutputs denoted by yrj, the rth output of DMUj for r = 1,2, . . . , sand consumes m inputs denoted by xij, the ith input of DMUj fori = 1,2, . . .,m. The following LP models are the conventional CCRmodels for efficiency analysis.

2.1. Output-orientated CCR primal model

Min ho ¼Xm

i¼1

v ixio

s:t:Xm

i¼1

v ixij �Xs

r¼1

uryrj P 0 j ¼ 1;2; . . . ;nð1Þ;

Xs

r¼1

uryro ¼ 1;

v i;ur P 0 i ¼ 1;2; . . . ;m; r ¼ 1;2; . . . ; s:

ð1Þ

In the output-oriented CCR primal model (1), the weighted out-puts are fixed to unity and the weighted inputs minimized. Theoutput weights ur and inputs v i are adjusted accordingly to gener-ate an efficiency score which is given by eo ¼ 1=ho.

While the CCR primal model can generate both an efficiencyscore and the optimal output weights ur and input weights v i,the CCR dual model can be used to generate not only the efficiencyscore but also the composite inputs and outputs that the observedDMUo should benchmark against. kj is the reference weight forDMUj (j = 1,2, . . . ,n) and kj > 0 means that DMUj is used to con-struct the composite unit for DMUo. The output-oriented CCR dualmodel is given as follows:

2.2. Output-orientated CCR dual model

Max ho ¼ ho

s:t:Xn

j¼1

kjyrj P hoyro r ¼ 1;2; . . . ; s;

Xn

j¼1

kjxij 6 xio i ¼ 1;2; . . . ;m;

kj P 0 j ¼ 1;2; . . . ;n:

ð2Þ

Note that the above models are based on constant returns toscale (CRS). This, however, disregards economies of scale. Variablereturns to scale in efficiency analysis were taken into account inanother version of DEA model developed by Banker, Charns andCooper (BCC) (Banker, Charnes, & Cooper, 1984), called BCC modelwhich is different from CCR model in that the former has additionalconvexity constraint of all kj restricted to sum to 1 in the dual case.

3. Equivalence between DEA and MOLP

In a DEA model, an efficiency score is generated for a DMU by max-imizing outputs with limited inputs, or minimizing inputs with de-sired or fixed outputs, or simultaneously maximizing outputs andminimizing inputs. Either way, this can be regarded as a kind of mul-tiple objective optimization problem. In this section, the theoreticalconsiderations of combining MOLP and DEA are presented.

Suppose an optimization problem has s objectives reflecting thedifferent purposes and desires of the DM. Such a problem can berepresented in a general form as follows:

Max f ðkÞ ¼ ½f1ðkÞ; f2ðkÞ; . . . ; frðkÞ; . . . ; fsðkÞ�s:t: k 2 K;

ð3Þ

where K is the feasible decision space and frðkÞ (r = 1,2, . . . , s) arecontinuously differentiable objective functions.

In an MOLP problem, it is generally impossible to find a solutionthat optimizes all objectives simultaneously. Therefore, the task ofan MOLP solution process is not to find an optimal solution but, in-stead, to find nondominated solutions and to help select a most pre-ferred one. Loosely speaking, a solution, represented by a point indecision variable space, is nondominated if it is not possible tomove the point within the feasible region to improve an objective

F.H. Lotfi et al. / Expert Systems with Applications 37 (2010) 4331–4336 4333

function value without deteriorating at least one of the otherobjectives. In multiple criteria terminology, a nondominated solu-tion is also called an efficient solution. To prevent possible confu-sion with DEA’s efficiency concept, we will use the term‘‘nondominated”. For more details about multiobjective optimiza-tion, readers are referred to Masud Abu Syed and Hwang (1978)and Steuer (1986). One fact we would like to point out here is thata nondominated solution set for an MOLP problem will always con-tain, but is not limited to, the optimal solutions obtained by indi-vidually optimizing each of the objectives in the MOLP problemunder the setting of single objective linear programming (LP).

In order to reach to a special nondominated extreme point, theMOLP formulation (3) can be written in min-ordering approach asfollows:

Max Min16r6s

f rðkÞ

s:t: k 2 K:ð4Þ

The min-ordering formulation can then be written as follows byintroducing an auxiliary variable h as follows:

Max h

s:t: h 6 frðkÞ r ¼ 1;2; . . . ; s;

k 2 K:

ð5Þ

This approach does not need any interobjective or other subjec-tive preference information from the DM once the problem con-straints and objectives have been defined. Thus this approachrequires that the DM be able to accept the solution obtained fromthe method. The advantage of this route is that in the process ofobtaining the solution the DM will not be disturbed by the analyst,which is preferable from the point of view of DM. But a major dis-advantage then is the necessity for the analyst to make manyassumptions about the DM’s preference. This is difficult to do witheven the best and most knowledgeable analyst.

From the formulation (2), the output-orientated CCR dual DEAmodel can be equivalently rewritten as follows:

Max ho

s:t:Xn

j¼1

kjyrj P hoyro r ¼ 1;2; . . . ; s;

k 2 Ko ¼ k :Xn

j¼1

kjxij 6 xio; i ¼ 1;2; . . . ;m;

(

kj P 0; j ¼ 1;2; . . . ; n�: ð6Þ

In order to prove that the CCR dual formulation (6) is equivalentto the min-ordering formulation in (5), certain conditions have tobe applied. The purpose for establishing the equivalence conditionsis to use the interactive techniques in MOLP to locate the most pre-ferred solution (MPS) on the efficient frontier for target setting andresource allocation. Suppose yro > 0 for any r = 1,2, . . . , s and sup-pose in formulation (5) frðkÞ defined as follows:

frðkÞ ¼1

yro

Xn

j¼1

kjyrj r ¼ 1;2; . . . ; s and k ¼ ðk1; k2; . . . ; knÞt: ð7Þ

Suppose the feasible space K in formulation (5) and Ko in for-mulation (6) are the same. Since yro > 0 (r = 1,2, . . . , s), formulation(6) can then be written as follows:

Max ho

s:t: ho 61

yro

Xn

j¼1

kjyrj r ¼ 1;2; . . . ; s;

k 2 Ko ¼ k :Xn

j¼1

kjxij 6 xio; i ¼ 1;2; . . . ;m;

(

kj P 0; j ¼ 1;2; . . . ; n�: ð8Þ

The equivalence relationship between the output-oriented CCRdual model (6) and the min-ordering formulation (5) can be estab-lished by the following theorem.

Theorem 1. Suppose yro > 0 for all r = 1,2, . . . , s. The output-orien-tated CCR dual model (6) can be equivalently transformed to the min-ordering formulation (5) using formulations (7) and (8) and followingequations:

K ¼ Ko; ð9Þh ¼ ho: ð10Þ

Proof. Using Eqs. (8)–(10) and (7), the output-orientated CCR dualmodel can be equivalently rewritten as follows:

Max h

s:t: h 6 frðkÞ r ¼ 1;2; . . . ; s;

k 2 K:

The above model is the min-ordering model (5). h

From Theorem 1, the output CCR dual model can be equivalentlyrewritten as the min-ordering formulation (5). The output-orien-tated BCC dual model can also be transformed to an equivalentmin-ordering formulation similar to the formulation (5), by addingthe convexity constraint

Pnj¼1kj ¼ 1.

Alternatively, since the formulation (5) gives a special weakefficient point of formulation (3) (Ehrgott, 2005), then formula-tion (6) also gives a special weak efficient point of followingformulation:

Max1

y10

Xn

j¼1

kjy1j;1

y20

Xn

j¼1

kjy2j; . . . ;1

ys0

Xn

j¼1

kjysj

" #;

s:t:Xn

j¼1

kjxij 6 xio i ¼ 1;2; . . . ;m

kj P 0 j ¼ 1;2; . . . ; n

ð11Þ

that does not include DM’s preference structures or value judg-ments in measuring relative efficiency and setting target values,so the efficiency score of the oth DMU can be generated by solvingformulation (11). Hence, an interactive MOLP method can be used tosolve the DEA problem.

4. An interactive multi objective programming

The method of Zionts–Wallenius(Z–W) can be used to design aninteractive procedure for searching for most preferred solution(MPS), that maximizes the DM’s implicit utility function.

Zionts–Wallenius(Z–W) method was introduced by Zionts andWallenius in 1976 and updated in 1983. It is applicable to problemin (3) where the objective functions are concave and K is a convexset. The overall utility function is assumed to be unknown explic-itly to the DM, but is implicitly a linear function and more generallya concave function of the objective functions. The method makesuse of such an implicit function on an interactive basis.

The first step of the method is to choose an arbitrary set of po-sitive multipliers or weights and generate a composite objectivefunction or utility function using these multipliers. The compositeobjective function is then optimized to produce a nondominatedsolution to the problem. From the set of nonbasic variables, a sub-set of efficient variables is selected (an efficient variable is onewhich, when introduced into the basis, cannot increase one objec-tive without decreasing at least one other objective). For each effi-cient variable a set of trade-offs is defined by which someobjectives are increased and others reduced. A number of suchtrade-offs are presented to the DM, who is requested to statewhether the trade-offs are desirable, undesirable or neither. From

4334 F.H. Lotfi et al. / Expert Systems with Applications 37 (2010) 4331–4336

his/her answers a new set of consistent multipliers is constructedand the associated nondominated solution is found. The processis then repeated, and a new set of trade-offs is presented to theDM at the current solution, convergence to an overall optimal solu-tion with respect to the DM’s implicit utility function is assured(taken from Masud Abu Syed & Hwang, 1978).

5. A case study

We evaluate 21 branches of an insurance company in IRAN atthis section. Each branch uses four inputs in order to produce

Table 2The data for 21 branches of an insurance company.

Input 1 Input 2 Input 3 Input 4

1 94.91666667 85 4000 89300200.672 77 93 2565 850255803 76.5 87 1343 79998269.54 92.91 93 1500 911447170.85 102.66 97 3750 78966919.676 94.5 90.5 3313 1307965137 86.16 92.41 1500 79493485.178 103.41 92 1600 153439927.29 95 79 1920 100704537.310 88.25 95 2800 59769138.4211 81.33 93.66 1630 75945943.8312 89 85.41 1127 71782212.6713 114.16 93.91 1304 79763145.514 86.58 101 1340 83499803.4215 113.41 123 2191 108737313.316 79 100 2140 16284663517 87.41 93.5 1231 60940728.1718 98.41 90 1960 93199199.6719 74.83 84.5 3375 75652227.8320 99.33 97.25 1603 476709469.221 89.5 92 2930 80225938.08

Table 3DEA efficiency results.

e 1 2 3 4 5 6 7 8 9 10

1 0.76 0.27 0.09 0.422 0.79 0.183 1 14 0.70 0.36 0.155 1 16 1 17 1 18 1 19 1 110 1 111 0.93 0.1212 0.83 0.213 114 115 116 117 118 119 0.90 0.56 0.11 0.2420 121 1

Table 1The labels of inputs and outputs.

Input Output

1 The number of personals The total number of insured persons2 The total number of

computersThe number of insured persons’agreements

3 The area of the branch The total number of life-pensionreceivers

4 Administrative expenses The receipt total sum (Income)

four outputs. The labels of inputs and outputs are presented inTable 1.

The data set for 21 branches of that insurance company is givenin Table 2.

The result of CCR DEA dual formulation is shown in Table 3,which is maximizing the amount by which outputs must be pro-portionally increased for observed DMU to be efficient. For exam-ple, the efficiency score for branch 1 is 0.76, implying that it isoperating as an inefficient branch whit respect to all 21 branchesin that insurance company.

However, the DEA efficiency results generated do not considerthe value judgments of the DM. Hence, interactive MOLP methodswill be used to search for the MPS along the efficient frontier forinefficient branches. We will use Zionts–Wallenius(Z–W) methodto reflecting the DM’s preferences in the process of assessingefficiency.

The full solution process of inefficient branches using the Z–Wmethod is shown in Table 4. For example, in iteration 1, the pro-posed MPS for inefficient branch 2 is a convex combination of thebranch 10 and the branch 16. In our case, the achieved output va-lue of the total number of life-pension receivers for branch 16 is2271. However, the DM is not satisfied with the current level of

Output 1 Output 2 Output 3 Output 4

58542.41667 50.33 1134.33 221.8337047.66667 23.16 8786.41 233.9133421.41667 32.58 6597.58 438.8336643.41 40.5 9428.25 234.7571423.33 16.16 8002.33 407.7538587.58 102.16 13987.16 344.7550154.08 20.75 1564.66 419.588476.5 61.83 12449.25 466.4135709.75 215.83 12624.5 250.4188752.08 48.416 714.08 235.9148139.41 21.83 10289.83 155.1644283.75 31.5 7849.16 197.5879239.83 29.41 4868.58 217.5831717.58 64.58 14734.41 366.66

105607.75 58.5 2658.5 362.0834271.33 37.91 2271 484.558043.91 62.58 10340.5 21281974.83 67.66 4780.83 285.2544548.75 30.83 619.83 317.0869758.91 94.75 13229.58 206.5862564.25 30.91 12263.41 279.5

11 12 13 14 15 16 17 18 19 20 21

0.140.33 0.33

0.48

0.21 0.31 0.260.07 0.58

11

11

11

11

Table 4Zionts–Wallenius(Z–W) interactive results.

Efficient branches

5 6 8 9 10 14 15 16 17

Iteration 11 0.63

354181230496

0.255939610939336

2 0.603367850055881

0.300668192817323

4 0.770985150556854

0.138762374535955

11 0.0624586390080379

0.652306870529137

0.021049112468039

12 0.215406386048073

0.635540034381058

19 0.0789770439666678

0.0387805668197815

0.735533789527266

3 8 9 10 14

Iteration 21 0.537625832745

4560.42106494270405

2 0.394177582651781

0.410629985019596

4 0.305246422893482

0.682034976152623

12 0.743862307072423

0.0799955760010851

5 8 9 14

Iteration 31 0.484674563606411 0.0720032854731259 0.3969906970462094 0.383529906763286 0.138350374435963 0.463223455419175

F.H. Lotfi et al. / Expert Systems with Applications 37 (2010) 4331–4336 4335

outputs as he wants to improve the total number of life-pensionreceiver. In iteration 2, the Z–W method suggested the branch 8 in-stead of the branch 16 that its value of the total number of life-pen-sion receivers is 12449.25. Here the DM is fully satisfied with theindifference trade-offs between the objectives, the interactive pro-cess will terminate and the MPS is found.

6. Conclusion

This paper establishes an equivalence model between DEA andMOLP and show how a DEA problem can be solved interactivelyby transforming it into MOLP formulation. This provides the basisto apply interactive techniques in MOLP to solve DEA problemsand further locate the MPS along the efficient frontier for each inef-ficient DMU. We use Zionts–Wallenius(Z–W) method to reflectingthe DM’s preferences in the process of assessing efficiency. Thecase study illustrated how the equivalence models and the interac-tive procedure can be implemented to support integrated effi-ciency analysis and target setting.

Acknowledgement

The authors would like to express thanks in particular to theanonymous referees and honorable editor for accepting the origi-nal paper.

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