A multi-objective approach to determine alternative targets in ... - UFF

Journal of the Operational Research Society (2004) 55, 1090–1101

r 2004 Operational Research Society Ltd. All rights reserved. 0160-5682/04 $30.00 www.palgrave-journals.com/jors

A multi-objective approach to determine alternative targets in data envelopment analysis MP Estellita Lins*, L Angulo-Meza and AC Moreira da Silva Operational Research, Production Engineering, COPPE/UFRJ, Brazil The choice for radial projections of classic data envelopment analysis (DEA) models, resulting in a number of projections onto the Pareto-inefficient portion of the frontier, has been seen lately as a disadvantage in DEA. The search for a non-radial projection method resulted in developments such as preference structure models. These models consider a priori preference incorporation, using weights in the search for the most preferred efficient target, although presenting some implementation difficulties. In this paper, we propose a multi-objective approach that determines the bases for a posteriori preference incorporation, through individual projections of each variable (input or output) as an objective function, thus allowing one to obtain a target at every extreme-efficient point on the frontier. This multi-objective approach is shown to be equivalent to the preference structure models, yet presenting some advantages, such as the mapping of the possible weights, assigned to partial efficiencies of an observed unit, in order to reach a specific target. Journal of the Operational Research Society (2004) 55, 1090–1101. doi:10.1057/palgrave.jors.2601788 Keywords: data envelopment analysis; multi-objective linear programming

Introduction From a managerial point of view, there could be other projections that are preferable to the radial projections of classic data envelopment analysis (DEA) models. Moreover, radial projections often lead us to Pareto-inefficient portions of the frontier. These two issues have a common source: the equi-proportional variation of input or output levels, proposed by Farrell1 as a means of measuring efficiency. Alternatives to radial projections exist in DEA literature. One of the main approaches is through models that incorporate preferences using preference structures, as proposed by Thanassoulis and Dyson2 and Zhu3 or by the specification of an ideal point.2 The appropriateness of using multi-objective linear programming (MOLP) in a DEA context is supported by the concept of Pareto efficiency, which DEA and MOLP share. Both of them search for a set of different units that are better in at least one aspect (or objective function) than the others, thus characterizing an efficient facet of the feasible solution space of a problem.4 Credit for the very first work integrating DEA and MOLP is due to Golany,5 who proposed an interactive multiobjective procedure (IMOLP—interactive MOLP) to determine efficient output levels. The algorithm consists of sequential solutions to a set of related linear programming problems in which the objective function is to maximize a weighted sum of the former objectives. *Correspondence: MP Estellita Lins, Rua Belisa´rio Ta´vora 80, ap 506 Laranjeiras, Rio de Janeiro RJ, Brazil. E-mail: [email protected]

Kornbluth6 noticed that the DEA model could be expressed as a multi-objective linear fractional programming problem. The objective function of the model has the same expression as in the CCR multipliers model, but applied to maximize efficiency of every DMU, instead of one at a time, the restrictions remaining unchanged. Also, Joro et al,7 observing the problem of characterizing efficient facets, made a structural comparison of the CCR8 and BCC9 models with the reference point approach10 to solving multi-objective problems. Another methodology based upon a multi-objective model was proposed by Halme et al,11 known as Value Efficiency Analysis. In this methodology, a multi-objective model is used to determine an existing or virtual (combination of existing) DMU preferred by the decision-maker, which is called the Most Preferred Solution (MPS). Once the MPS is identified, an efficient frontier is determined in a DEA-like manner. Joro12 made an extension of the Value Efficiency Analysis method to determine targets for inefficient DMUs. Li and Reeves13 presented a multi-objective model that considers two additional efficiency measures: the minimization of the sum of the DMU distances to the frontier (minisum) and the minimization of the largest distance (minimax), besides the maximization of the classical efficiency in DEA. The Multi-Objective Simplex approach proposed in this paper, not only allows calculation of the coordinates of all alternative extreme efficient DEA targets, but also enables putting forward a new mapping of the weight range given to each input/output improvement onto those targets. This

MP Estellita Lins et al—Data envelopment analysis 1091

promotes insights into the DM preferences, regarding variations in input or output intensity. We propose an additional reference point approach that allows a better analyst-user interaction environment, benefiting from a better visualization of the DEA targets. The content of this paper is organized in the following way: the next section comprises a revision of two models to determine targets in DEA using the preference structure, their description, and the disadvantages of this approach. In the following section, the first of DEA-MOLP models is introduced: MORO (Multiple Objective Ratio Optimization), which optimizes the ratios between observed and target inputs (or outputs) of a DMU. We include a concise value judgement-driven classification of MOLP methods, which helps understand the relationship among the models discussed here. Also, a straight comparison between DEAMOLP and preference structure models is shown, revealing their formal equivalence. In the ensuing section, we propose the use of indifference regions to map the range of weights onto the target set. Then, we present a second DEA model: MOTO (Multiple Objective Target Optimization), which directly optimizes the target values, thus promoting the use of one interactive MOLP approach: the Pareto Race. Final sections are devoted to a case study using the MORO model and conclusion.

Non-radial models to determine targets in DEA Classic DEA models, such as constant returns to scale (CRS) or variable returns to scale (VRS) yield targets obtained as the equi-proportional reduction of input values, when inputoriented, or the equi-proportional increase in output values, in the output-oriented case (without including slacks), thus providing radial efficiency measures, according to the Farrell1 measure of efficiency. From a managerial point of view, there could be several interesting alternative targets to be explored (characterized by input and output levels), besides the radial approach. Several researchers are concerned with the determination of targets through different approaches, the most popular one being based on preference structures. These models require a priori information from the user or decision-maker about the relative importance of variables, expressed in the form of weights. Two alternative preference structure models are as follows.

Thanassoulis and Dyson model Thanassoulis and Dyson2 proposed assigning weights to changes in the variables in order to implement a preference structure. For a DMU j0, outputs and inputs are classified into two categories: variables that we want to change, set R0 for the outputs and I0 for the inputs; and unchangeable


0 and
I 0. The model is: variables: R X X Maximize wrþ fr  w i ji r2R0

i2I0

0 1 X X þ e@ s srþ A i þ i2I 0

ð1Þ

r2R0

Subject to n X

fr yrj0 

lj yrj ¼ 0; r 2 R0 ;

ð2Þ

j¼1

ji xij0 

n X

lj xij ¼ 0; i 2 I0 ;

ð3Þ

lj yrj  srþ ¼ yrj0 ; r 2 R0

ð4Þ

j¼1 n X j¼1 n X

lj xij þ s i ¼ xij0 ; i 2 I0

ð5Þ

j¼1

fr X1 8r 2 R0 ;

ð6Þ

jip1 8i 2 I0

ð7Þ

lj X0 8j

ð8Þ

þ s i ; sr X0 8i 2 I 0 e 8r 2 R0 ; e40

ð9Þ

The decision-maker’s preference structure is expressed through the weights wrþ and w i for the factors fr and ji, where fr reflects the increment in the value of output r, and ji reflects the decrease in the value of input i, in order to reach the target. Thus, the objective function consists of maximizing the weighted sum of the ratios (considering output increments and input reductions) between observed and projected data. Projections are considered for sets of inputs and outputs that can be improved, R0 and I0, through constraints (2) and (3).
0 and
I 0, we have For the non-discretionary variables R constraints (4) and (5). Bounds imposed on f (6) and j (7) prevents the levels of outputs from decreasing and those of inputs from increasing.
If fr* ¼ ji* ¼ 1, 8rAR0 , 8iAI0, and srþ * ¼ s i * ¼ 0 8rAR0,
8iAI 0. then the DMU j0 is Pareto efficient. Otherwise, the DMU j0 is inefficient and the target is given by yrj0 ¼ fr yrj0 8r 2 R0 ð10Þ yrj0 ¼ yrj0 þ srþ  8r 2 R0

ð11Þ

xij0 ¼ ji xij0 8i 2 I0

ð12Þ

8i 2 I0 xij0 ¼ xij0  s i

ð13Þ

In real-life applications, this model may be used interactively, by testing different sets of weights (wrþ and w i ) and allowing the decision-maker to search for appro-

1092 Journal of the Operational Research Society Vol. 55, No. 10

priate DMU targets. It is important to remark that this model does not indicate the efficiency score for the DMU.

will allow knowledge of weight mixes that will lead to interesting solutions from the decision-maker’s perspective.

Zhu model

Multi-objective model for ratio optimization (MORO) and with dominance (MORO-D)

3

Based on Russell’s efficiency measure, Zhu presented a model to determine targets that also incorporate a preference structure: s m X X wrþ fr  w Maximize i ji ð14Þ r¼1 i¼1 Subject to n X

lj yrj  srþ ¼ fr yrj0 ; r ¼ 1; . . . ; s

ð15Þ

lj xij þ s i ¼ ji xij0 ; i ¼ 1; . . . ; m

ð16Þ

j¼1 n X

Considering an efficient DMU, we propose here that each output to be increased (or input to be reduced) can be treated as a separate objective function within a Multiple Objective Linear Programming context. Thus, the following model considers as objective functions the independent optimization of the increment/decrease of each input/output, in order to obtain an efficient target for the DMU j0: ð20Þ max f1 ... max fs

j¼1 þ ji ; fr free 8i; r; s i ; sr X0 8i; r

where

s X r¼1

wrþ 

m X

min j1

ð17Þ

ð21Þ

...

w i ¼1

min jm

i¼1

The sum of weights condition prevents the user from inadvertently testing a redundant weight set that is equal to a previous set multiplied by a constant amount. Although the Zhu model does not specify restrictions for the nondiscretionary variables, this can be considered by assigning zero weights to respective selected inputs/outputs in the objective function. The major difference in the Zhu model is the unboundedness of ji and fr, allowing output reduction and/or input increment, that is, the observed DMU is not necessarily dominated by the target DMU, like in the Thanassoulis and Dyson model. In real-life cases, from a managerial point of view, an input increment (or output reduction) to reach the efficient frontier could be feasible and preferable. For example, in the labour field where the managers would prefer a solution that implies hiring instead of dismissing workers, which could produce a negative impact on a firm’s public image. If fr* ¼ ji* ¼ 1 and srþ * ¼ s i * ¼ 0 then the DMU j0 is Pareto efficient. Otherwise, the DMU is inefficient and the target is given by yrj0 ¼ fr yrj0 þ srþ  8r

ð18Þ

xij0 ¼ ji xij0  s 8i i

ð19Þ

As in the previous model, the decision-maker can search interactively for more appropriate targets for the DMU, testing different weight sets. There is an important drawback when determining DEA targets in both the Thanassoulis and Dyson and the Zhu models: the arbitrary choice of weights to be given to factors fr and ji. We have no guidance for establishing weights before testing the models; just the trial and error experience

Subject to fr yrj0 ¼

n X

yrj lj 8r ¼ 1; . . . ; s

ð22Þ

xij lj 8i ¼ 1; . . . ; m

ð23Þ

j¼1

ji xij0 ¼

n X j¼1

fr ; ji ; lj X0 8r; i; j This DEA-MOLP model will be referred to as MORO CRS – Multiple Objective Ratio Optimization, as it optimizes ratios between inputs (outputs) of the target DMU and the observed DMU, under constant returns to scale (CRS). To consider variable returns to scale (VRS), we add the convexity restriction: n X lj ¼ 1 ð24Þ j¼1

The model including this restriction (24) will be referred to as MORO VRS version. For both versions, the decision variables are fs, lj, jm. When the problem results in optimal solutions fr* ¼ ji* ¼ 1 8r, i, then DMU j0 lies on the Pareto efficient (non-dominated) frontier. The targets are: ð25Þ yrj0 ¼ fr yrj0 8r xij0 ¼ ji xij0 8i

ð26Þ

The final value of fr* and ji* depends on the decisionmaker’s choice among the non-dominated solutions of the problem (in multi-objective linear programming we get efficient solutions, also known as non-dominated solutions).

MP Estellita Lins et al—Data envelopment analysis 1093

Malivert,20 among others, undertook the problem of finding the efficient set in MOLP.

When we require dominance of the target over the observed DMU, we will add the following restrictions: ð27Þ fr X1 8r ¼ 1; . . . ; s ð28Þ

The resulting DEA-MOLP model will be called MORO-D (Multi-objective Ratio Optimization with Dominance), both CRS—considering expressions (20)–(23) and VRS—adding the convexity restriction (24). While desirable, dominance can be dismissed in some real-world cases as Zhu3 pointed out. Indeed, besides the labour field example given in the previous section about Zhu model, there are cases where trade offs over the frontier allow a large increase of outputs, compensating for a small increase in one input. In solving the MORO and MORO-D models, we can provide the decision-maker with a complete set of extreme efficient targets for the observed DMU, supporting the implementation of value judgements.

Classification of methods for solving MOLP problems In a MOLP problem an optimal solution, that is, one that optimizes all objectives simultaneously, is generally impossible to find. So, the process of solving MOLP problems consists of finding non-dominated solutions, that is, solutions in the feasible set of the decision variable space, which cannot be altered to improve one objective function value without deteriorating at least one other. This concept is analogous with the Pareto–Koopmans efficiency. According to Clı´ maco et al,14 there are several possible classification criteria for MOLP methods. We will adopt here the following classification, which is based on the stage at which value judgements are incorporated. Three classes resulted: (i) Methods with a priori preference aggregation. These methods usually transform the MOLP problem into a single objective one (LPP) through an equivalent objective function, whose optimal solution is a nondominated solution for the original MOLP problem. This function, known as the scalarizing function, expresses the DM’s preferences. There are three kinds of scalarizing functions, which consist of: (a) the optimisation of one objective function restricting the others; (b) the weighted sum of all objective functions; and (c) the use of a distance function or reference point approach. (ii) Methods characterized by a progressive preference articulation, where the decision-maker maintains a continuous interaction with the problem. For an extended discussion of the several classification alternatives for interactive methods, readers are referred to Steuer.15 (iii) Methods that generate efficient solutions and incorporate preferences a posteriori. Geoffrion (1968),16 Evans and Steuer,17 Zeleny,18 Ecker et al19 and Armand and

In the following three sections, we will develop MORO model equivalence to preference structures and indifference regions, using the multi-objective simplex method to solve DEA-MOLP models, a method that belongs to the third class and gives all the extreme points of the efficient set. We have run both the ADBASE software21 and the DEAMOLP simplex implemented by Frota Neto.22 After that, we will implement one interactive reference point method: the Pareto Race,18 to implement the MOTO model. It is worth remarking that, although the authors did not mention it, preference structure models could be considered as weighted sum a priori preference aggregation MOLP models.

Comparison of the multi-objective and preference structure models Let us consider the following example in order to make comparisons among the MORO and MORO-D models with the Zhu, and the Thanassoulis and Dyson model: In Table 1, we consider seven DMUs that use input X to produce output Y. We applied Linear Programming software (LINDO) to run the preference structure models, testing different weights for j and f until the complete range

Table 1

Original data set Variable

DMU

X

Y

A B C D E F G

2 4 6 2 3 1 2

12 15 6 9 15 7 4

X vs Y 16 E

14 12

B

A

10 D

Y

jip1 8i ¼ 1; . . . ; m

8 F

6

C G

4 2 0 0

1

2

3

4

X

Figure 1

CRS frontier.

5

6

7

1094 Journal of the Operational Research Society Vol. 55, No. 10

Table 2

MORO model (VRS)

Actual value

Objective function value

DMU

Y

X

A

12

2

B

15

4

C

6

6

D

9

2

E

15

3

F

7

1

G

4

2

Max f

Target value

Min j

0.58333 1 1.25 0.46667 0.8 1 2.5 2 1.16667 0.77778 1.33333 1.66667 0.46667 0.8 1 1 1.71429 2.14286 1.75 3 3.75

0.5 1 1.5 0.25 0.5 0.75 0.5 0.33333 0.16667 0.5 1 1.5 0.33333 0.66667 1 1 2 3 0.5 1 1.5

X vs Y

X

7 12 15 7 12 15 15 12 7 7 12 15 7 12 15 7 12 15 7 12 15

1 2 3 1 2 3 3 2 1 1 2 3 1 2 3 1 2 3 1 2 3

X vs Y

16

16

E

14 12

B

4

14

E

12

A

10

B

A

10

D

D

8

Y

Y

Y

F

6

C

4

8 F

6

G

2

C

3

4

G

2

0

0

0

1

2

3

4

5

6

7

0

Figure 2

VRS frontier for DMU D.

of basic efficient solutions was obtained. For the multiobjective models, we used ADBASE,21 which gives all efficient, basic solutions, although we can also find non-basic solutions, as will be done when applying MOTO model. The results concerning CRS, for the Zhu model as for the MORO model, show that any point on the efficient CRS frontier (Figure 1) serves as a target for every efficient or inefficient DMU, given that f and j are unbounded. In this sense, there is equivalence between solutions for both Zhu and MORO models. The same equivalence is true under VRS, given that Zhu and MORO models yield the same set of efficient solutions. Results from ADBASE for the MORO VRS model show (Table 2) that the target set is unique, no matter which DMU is being projected. This set is composed of all extreme

1

2

3

4

5

6

7

X

X

Figure 3

CRS frontier, projections for DMU G.

efficient DMUs (F, A and E), corresponding to basic nondominated solutions in the MOLP problem. In Figure 2, we present the alternative targets for DMU D. For example, if the DM plans to increase the input X of DMU D by 50%, then he should consider a required output increase of 66.7%, taking DMU E as a possible efficient target, according to Table 2. Non-extreme efficient solutions were not attained, the reason for which will become clear through the concept of indifference regions, in the respective section. The MORO-D model solved in ADBASE gave efficient basic solutions that were compared to those obtained by the Thanassoulis and Dyson model, considering CRS and VRS. For the CRS frontier (Figure 3), we obtained different sets of targets for each projected DMU, shown in Table 3.

MP Estellita Lins et al—Data envelopment analysis 1095

Table 3 Actual value

MORO-D model (CRS) Objective function value

Target value

DMU

Y

X

Max f

Min j

Y

X

A

12

2

B

15

4

C

6

6

1 0.857143 1 0.535714 1

9

2

E

15

3

F G

7 4

1 2

14 12 28 15 42 6 14 9 15 21 7 4 14

2 1.71 4 2.14 6

D

1.166667 1 1.866667 1 7 1 1.555556 1 1 1.4 1 1 3.5

Table 4 Actual value

X vs Y E

14 12

B

A

10 D 8 F

6

C G

4

4 2 0 0

1

2

3

4

5

MOLP models yielded all the non-dominated solutions in one run, while preference structure models required the specification of new weighting sets in every new run,

MORO-D model (VRS) Target value

Lambdas

Y

X

Max f

Min j

Y

X

lA

A B C

12 15 6

2 4 6

1 1 2 2.5 1.166667 1.333333 1 1 1 3 1.75

1 0.75 0.333333 0.5 0.166667 1 0.7 1 1 1 0.5

12 15 12 15 7 12 9 15 7 12 7

2 3 2 3 1 2 1.4 3 1 2 1

1

2

E F G

15 7 4

3 1 2

7

VRS frontier, projections for DMU G.

Figure 4

DMU

9

6

X

Objective function value

D

2 1.29 2.23 3 1 0.57 2

16

Y

The set of possible targets (basic feasible solutions) for DMU G (Figure 3) are the efficient points 3 and 4. It is worth noting that the targets for every DMU are proportional to input and output values for the DMU F, which defines the efficient frontier under CRS. Adding the convexity condition to the MORO-D formulation, we get the VRS frontier. Thanassoulis and Dyson program was solved for DMU G, and resulted in the targets corresponding to DMUs F and A, shown in Figure 4. We obtained an identical set of basic efficient solutions using MORO-D. The Non-Pareto efficient frontier cannot be attained at all, unlike the classic models that would yield point 4 in Figure 4 as the input oriented projection of DMU G. Table 4 shows results for the MORO-D (VRS) program: the values for f and j, the targets and the strictly positive l’s corresponding to the reference set of DMUs. Results from DEA-MOLP models, compared to those from Zhu and Thanassoulis and Dyson models, pointed out the equivalence between target sets for the DMUs. DEA-

1 0.642857 0.741935 1 1 0.285714 1

lE

lF

1 1 1 1 1 0.4

0.6 1 1

1 1

1096 Journal of the Operational Research Society Vol. 55, No. 10

sometimes failing to obtain a new target, because various weighting sets can lead to the same target solution. This can be seen with the aid of the indifference regions in the corresponding section.

Maximize

s X

wrþ fr 

r¼1

m X

w i ji

ð31Þ

i¼1

Subject to fr yrj0 ¼

n X

yrj lj 8r ¼ 1; . . . ; s

j¼1

Formalizing equivalence In the following paragraphs we are going to formalize the equivalence between the models analysed. Conceptually, this arises from a result obtained by Geoffrion,16 which states the equivalence between an LP model with multiple objective functions and an LP model whose sole objective function consists of a convex linear combination of the former objective functions. He showed that, given a multi-objective model: Max CX

ð29Þ

S.t. AXpb

n X

xij lj 8i ¼ 1; . . . ; m

j¼1

fr X1 8r ¼ 1; :::; s ji p1 8i ¼ 1; . . . ; m fr ; ji ; lj X0 8r; i; j wrþ ,

w i

If are strictly greater than zero, then there will be no slacks, and we can assume: yrj0 ¼ fr yrj0 8r 2 R0 ð32Þ xij0 ¼ ji xij0 8i 2 I0

XX0 X* is an efficient solution for this model if and only if ( y*, y*40, Syi* ¼ 1, such that X* is optimal for Maxy CX

ji xij0 ¼

ð30Þ

S.t.

If we admit any component of wrþ or w i equal to zero (the unchanged variables case in the Thanassoulis and Dyson model), then, we should use n X lj yrj Xfr yrj0 ; r ¼ 1; . . . ; s j¼1

AXpb

n X

XX0

j¼1

Kornbluth23 stated that, given a MOLP problem, for every weighting set there is a non-dominated basic solution. Cohon24 added that weights should be strictly positive in order not to generate weakly efficient solutions, but Romero25 showed that, while possible from a theoretical point of view, this is not likely to happen in practice. In fact, this happens in the presence of multiple optimal solutions and can be avoided if we substitute a small infinitesimal amount for a zero coefficient in the objective function. As we will apply this theorem to a known vector y in order to obtain an optimal solution for model (30), it is easy to see P that imposing restriction yi* ¼ 1 will not be necessary. The equivalence between models (29) and (30) implies that, for any given efficient solution X*, we can use the associated y* to form a weighted function y*CX*. This linear function is an example of a utility function for the decision-maker, where y* represents the trade-offs between objectives, with X* as an optimal solution. In the case of the Zhu model, and Thanassoulis and Dyson model, this y* vector is equivalent to the weighting vector composed of wrþ and w i , given to variables j and f. We can easily show that the MORO model, presented in expressions (20)–(23), including dominance (27) and (28) may be put in the form of a preference structure model using (30) and substituting wrþ and w i for y:

lj xij pji xij0 ; i ¼ 1; . . . ; m

and add such slacks (multiplied by e) to the objective function. As we can see, the MORO model considering dominance is equivalent to the Thanassoulis and Dyson model, in which all variables are considered to improve. The Zhu model includes slacks in constraints, in order to let ji, fr be free, but not within the objective function, because wrþ or w i is not allowed to be equal to zero.

Obtaining indifference regions Let us consider that a given projection on the frontier can be achieved through various different sets of weights. Thus, each weight can be allowed to vary inside a limited (indifference) region, while projecting a given observed unit onto the same point on the frontier. As we will show, the indifference regions can be used to map the range of weights (j and f), given to each objective function, onto all the possible targets. These weights indicate the ‘importance’ given to a variable improvement, therefore revealing the preference structure that leads to a given target. According to Kornbluth,23 a solution x* is efficient in relation to any set of weights y that satisfy the Kuhn–Tucker conditions. Each set of weights y characterizes one indifference region: the interval where the weights y can vary and maintain the efficient solution x*.

MP Estellita Lins et al—Data envelopment analysis 1097

Table 5

MORO model (VRS)

Actual value

Objective function value

Target value

DMU

Y

X

Max f

Min j

Y

X

C

6

6

2.5 2 1.166667

0.5 0.333333 0.166667

15 12 7

3 2 1

The indifference regions can be determined by imposing the optimality conditions on the coefficient of DMU j in the objective function y*Cx, under the canonical form, that is, zjcjX0. Once y*Cx is the sum of q objective functions, the coefficient for each DMU j is       y1 z1j  c1j þ y2 z2j  c2j þ þ yq zqj  cqj where the optimality condition for a given non-dominated solution is       y1 z1j  c1j þ y2 z2j  c2j þ þ yq zqj  cqj X0 P and yiX0, i ¼ 1,y,q and qi ¼ 1yi ¼ 1. We will implement these conditions to find the indifference regions of the following MORO VRS model, applied to DMU C, with data from Table 1: Max f Min j Subject to 6f  12l1  15l2  6l3  9l4  15l5  7l6  4l7 ¼ 0 6j þ 2l1 þ 4l2 þ 6l3 þ 2l4 þ 3l5 þ l6 þ 2l7 ¼ 0 l1 þ l2 þ l3 þ l4 þ l5 þ l6 þ l7 ¼ 1 f; j; lj X0 This program yielded three non-dominated solutions, shown in Table 5, corresponding to the DMUs E, A and F, in Figure 4. Imposing the optimality conditions, zjcjX0, we obtained the following inequalities system for the first non-dominated solution (DMU E in Figure 4): 0:5y1  0:1667y2 X0 0:1667y2 X0

θ1=0

θ1=1

0.16667 0.25

Sol.3 F

Sol.2 A Figure 5

Sol.1 E Indifference regions.

solutions. For the second solution, corresponding to DMU A, we obtained: 0.166667py1p0.25, and for the third solution (DMU F): 0py1p0.16667. Then, imposing the optimality conditions, we can represent the indifference regions of weights y1 (and y2 ¼ 1–y1) for the three DEA extreme-efficient solutions 1 (DMU E), 2 (DMU A) and 3 (DMU F), as shown in Figure 5. It is worth noting that an efficient, but non-extremeefficient, solution can be expressed as a convex linear combination of some extreme-efficient solutions (for example: A and E in Figure 5), corresponding to contiguous indifference regions, and can be attainable with a single value of y1 ¼ 0.25 (and y2 ¼ 0.75). In this case, a preference structure model would exhibit multiple optimal solutions, which are not shown by usual LP software. Establishing the following equivalence: w1þ y1 e w 1 y2, we have a preference structure where w1þ X0.25 and w1þ þ w 1 ¼ 1 will lead us to the extreme-efficient solution E. A preference structure where 0.16667pw1þ p0.25 and w1þ þ w 1 ¼ 1 will lead us to the solution A, and, ultimately, to a preference structure where w1þ p0.16667 and w1þ þ w 1 ¼ 1 will lead us to the solution F.

Interactive method using the MOLP model for target optimization (MOTO)

1:3333y1  0:3333y2 X0

Instead of the ratio, we can consider each objective function as the projected value (target), that is, the ratio multiplied by the observed output or input. With this simple change in the objective functions in MORO (20 and 21), we obtain the MOTO model:

1:8333y1  0:16667y2 X0

max y1j0 f1

y1 þ y2 ¼ 1

...

After solving this system, we obtained the region 0.25py1p1. Using the same procedure, we determined the indifference regions for the other two non-dominated

max ysj0 fs

1:5y1 þ 0:5y2 X0 y1  0:16667y2 X0

min x1j0 j1

ð33Þ

ð34Þ

1098 Journal of the Operational Research Society Vol. 55, No. 10

... min xmj0 jm S.t. fr yrj0 ¼

n X

yrj lj 8r ¼ 1; . . . ; s

ð35Þ

xij lj 8r ¼ 1; . . . ; m

ð36Þ

j¼1

ji xij0 ¼

n X j¼1

MOTO model presents:  The same variables and restrictions as MORO model.  The possibility of imposing restrictions on f and j, as presented in (27) and (28) for MORO-D, resulting in the MOTO-D model, under CRS or VRS.  If f*r ¼ j*i ¼ 1 8r, i, then DMU j0 is a Pareto-efficient DMU. Otherwise, the targets are: ynrj0 ¼ fr yrj0 ; r ¼ 1 . . . s xnij0 ¼ ji xij0 ; i ¼ 1; . . . ; m Visualization of the target for each variable, instead of the value of the increment or reduction (f or j) given by the MORO model, becomes more informative and understandable, improving the application of interactive solution methods. One of them was applied in this paper: the Pareto Race,26 whose metric is based on a distance to a reference or ideal point, implemented in the VIG software, where the visualization of the objective function values is made through bar charts on the computer screen. A change proposed by the expert/decision-maker in an objective function, and its consequences for other objective functions are easily visualized. With this approach, one can know directly from the screen which values (input/output) can be obtained, given the changes proposed. It is important

Figure 6

to notice that interactive methods not only show extreme efficient points, but also non-extreme efficient points, that is, interactive approaches can explore the whole non-dominated set. The number of possible efficient solutions to be assessed is, however, unlimited. This method should, therefore, be used in a complementary way to the simplex multi-objective approach. Let us look at an example using the MOTO–VRS model to analyse the efficiency of DMU C (previous example). In Figure 3, we observed three possible targets for C, the extreme efficient points (DMUs) A, E and F. The points between them, inside the segments A–E and A–F are also efficient and can be explored using this interactive approach. Figure 6 shows the interface of VIG, with an efficient solution chosen arbitrarily as a target for DMU C. This solution corresponds to a virtual DMU, obtained by convex linear combination of DMU A (lA ¼ 0.9177545) and DMU E (lE ¼ 0.0822455), which is the target for DMU C: Y ¼ 12.2467 and X ¼ 2.08225 as seen in Figure 6 (in the MORO model the solutions visualized for this same target would be f ¼ 2.04112 and j ¼ 0.347042).

The case study Dental, as well as other branches of medical assistance, is a right guaranteed by the Brazilian Constitution. Thus, the Public Health System is supposed to provide free preventive and conclusive medical care for every Brazilian citizen. Although the Public Health System performs as a unified system, the way it is managed varies according to the region. In Rio de Janeiro city, the Federal Government supports public hospitals and clinics, which are managed by the local administration. Both Federal Government and local managers are interested in the assessment of the efficiency of the local units—geographical districts that constitute the muni-

Interactive method for MOTO-VRS to DMU C.

MP Estellita Lins et al—Data envelopment analysis 1099

Figure 7

Geographical regions of Rio de Janeiro municipality.

cipality, shown on the map (Figure 7). These efficiencies were assessed through the MORO-D model. The final formulation considered the number of dentists and potential target demand (which comprises both the number of children and pregnant women belonging to lowincome families) as inputs and the total number of preventive plus conclusive dental procedures as the output, according to data shown in Table 6. The following DMUs revealed a 100% efficiency using the BCC Output oriented model: (III) Rio Comprido, (IV) Botafogo, (V) Copacabana, (IX) Vila Isabel, (XIX) Santa Cruz, (XX) Ilha do Governador, (XXI) Paqueta´, (XXIII) Santa Tereza, (XXV) Pavuna and (XXVI) Guaratiba. The number of possible targets for each inefficient DMU according to MORO-D/VRS formulation varies, depending on the enveloping frontier and on the DMU input–output vector itself. The Multi-objective Simplex algorithm based on Guerreiro27 was implemented, resulting in a number of targets for the inefficient units, ranging from three (Portua´ria and Centro) to 19 (Penha). These targets, apart from any further analysis, are already a useful tool for the DecisionMakers. Table 6 also shows the number of targets for each unit. Notice that efficient DMUs still project only on themselves. Take, for instance, the targets for DMU (II) Centro shown in Table 7: These three targets belong to the efficient frontier. The DM could choose to dismiss/relocate 11 dentists and keep production constant (target 1), or relocate nearly nine dentists and increase production to 30 615 procedures (target

2). Target 3 implies reducing the target population, which is not so straightforward, but could mean arranging for local population to be assigned to clinics in another district. The three targets can also be expressed as the following convex linear combination of the inputs/ outputs from the DMUs (IV) Botafogo, (V) Copacabana and (XXI) Paqueta´. 7 25 5667  0; 2377 þ 2612  0; 2284 22477 83629 10; 21 5 þ 205  0; 5339 ¼ 2053 26884 3507 11; 767 5 25 5667  0; 3383 þ 205  0; 6617 ¼ 2053 30615 3507 83629 10; 83 5 25 5667  0; 2918 þ 205  0; 7082 ¼ 1798 26884 3507 83629 It is possible, therefore, to explore non-basic feasible solutions, as any convex linear combination of the targets 1–3. We can note that any other target aiming to increase the number of procedures (as an output oriented model would try to do) will also decrease the number of dentists (once the new target should be expressed itself as a convex linear combination of the three targets presented) in order to be

1100 Journal of the Operational Research Society Vol. 55, No. 10

Table 6 DMUs Districts

Original data and number of alternative targets Inputs Target population

Preventive plus conclusive procedures

No. of targets

6 21 17 25 7 31 12 13 38 22 19 21 50 20 25 46 44 39 46 32 5 10 6 21 8 11

5720 2053 7529 5667 2612 8892 8604 5705 7023 50 726 34 172 13 320 25 721 16 652 35 332 42 185 81 238 56 396 49 542 13 973 205 19 521 3493 6293 30 258 15 181

22 105 26 884 66 836 83 629 22 477 44 985 42 101 35 818 91 465 34 340 58 424 63 294 103 245 71 818 92 136 101 144 170 336 146 758 253 913 122 375 3507 47 231 23 416 30 488 42 655 59 033

3 3 1 1 1 14 9 9 1 18 19 14 13 9 10 13 12 12 1 1 1 4 1 14 1 1

Table 7 Variables

Targets

MORO-D Results

Number of dentists

I Portua´ria II Centro III Rio Comprido IV Botafogo V Copacabana VI Lagoa VII Sa˜o Cristo´va˜o VIII Tijuca IX Vila Isabel X Ramos XI Penha XII Inhau´ma XIII Me´ier XIV Iraja´ XV Madureira XVI Jacarepagua´ XVII Bangu XVIII Campo Grande XIX Santa Cruz XX Ilha do Governador XXI Paqueta´ XXII Anchieta XXIII Santa Tereza XXIV Barra da Tijuca XXV Pavuna XXVI Guaratiba

Observed DMU

Output

Alternative targets for DMU Centro No. of dentists

II Centro 1 2 3

Target population

No. of procedures

21

2053

26 884

10.21 11.76 10.83

2053 2053 1799

26 884 30 615 26 884

efficient. We recall, indeed, that oriented models in practice frequently yield non-oriented projections. Under the MORO model without dominance, the (basic feasible solution) targets for DMU (II) Centro would correspond to each one of the efficient DMUs. Even so, this DMU could hardly avoid dentists being relocated, unless a huge increase in the production and decrease in target population were attained. Next, we will focus on the DMU (VI) Lagoa, which has 14 targets, according to MORO-D. We will select three from among a group of six that present an unchanged target population. The first one implies reduction in the number of dentists from 31 to roughly nine, while also keeping the number of procedures constant. A second target is characterized by 18 dentists, but with an increase of 62% in the number of procedures. Yet a third target implies no reduction in dentists, but an increase of 120% in the number

of procedures. All of them behave strictly as Pareto-efficient points on the frontier. So, it is up to the DM to decide which one would be more adequate.

Conclusions This paper proposes the use of a multi-objective linear programming approach to determine non-radial projections for inefficient DMUs onto the Pareto efficient frontier. We established the equivalence between the efficient solutions set obtained by the multi-objective (MORO and MORO-D) models and the optimal solutions set obtained in models with preference structure (Thanassoulis and Dyson, and Zhu). We pointed out the advantages presented by the proposed multi-objective models: they allow a posteriori preference incorporation and do not require a priori

MP Estellita Lins et al—Data envelopment analysis 1101

specifications of the weights given to each objective (input or output improvement) and yield indifference regions that map weight ranges onto the target alternatives. For each set of weights, this method finds an efficient point or tangential point of a family of segments y1 f1 (x) þ y2 f2 (x) ¼ l to a polygon, in the case of two objectives. In the generalized case, parallel hyperplanes will be tangential to a convex polyhedron set at an extreme-point.19 This is why the Thanassoulis and Dyson and Zhu preference structure models will always give at least one extremeefficient point as a target. We also proposed the use of an interactive approach (Pareto race) applied to solve the MOTO model, exploring the non-extreme efficient solutions. We presented a case study where the MORO-D model was implemented, illustrating the diversity of Pareto-efficient solutions that could possibly be obtained. Finally, we consider that both approaches could rather be integrated, as the mapping of the weights onto the efficiency regions can guide the DM on his/her interactive search for the ultimate target. After obtaining extreme points and indifference regions, the DM could explore non-extreme efficient solutions through interactive methods.

References 1 Farrell MJ (1957). The measurement of productive efficiency. J Roy Stat Soc Ser A 120: 253–281. 2 Thanassoulis E and Dyson RG (1992). Estimating preferred target input–output levels using data envelopment analysis. Eur J Opl Res 56: 80–97. 3 Zhu J (1996). Data envelopment analysis with preference structure. J Opl Res Soc 47: 136–150. 4 Stewart TJ (1996). Relationships between data envelopment analysis and multicriteria decision analysis. J Opl Res Soc 47: 654–665. 5 Golany B (1988). An interactive MOLP procedure for the extension of DEA to effectiveness analysis. J Opl Re Soc 39: 725–734. 6 Kornbluth JSH (1991). Analysing policy effectiveness using cone restricted data envelopment analysis. J Opl Res Soc 42: 1097–1104. 7 Joro T, Korhonen P and Wallenius J (1998). Structural comparison of data envelopment analysis and multiple objective linear programming. Mngt Sci 44: 962–970. 8 Charnes A, Cooper WW and Rhodes E (1978). Measuring efficiency of decision making unit. Eur J Opl Res 2: 429–444.

9 Banker RD, Charnes A and Cooper WW (1984). Some models for estimating technical and scale inefficiencies in data envelopment analysis. Mngt Sci 30: 1078–1092. 10 Wierzbicki A (1980). The use of reference objectives in multiobjective optimization. In: Fandel G and Gal T (eds). Multiple Objective Decision Making, Theory and Application. SpringerVerlag, New York. 11 Halme M et al. (1998). A value efficiency approach to incorporating preference information in data envelopment analysis. Mngt Sci 45: 103–115. 12 Joro T (1998). Models for identifying target units in data envelopment analysis: comparison and extension. IIASA, Interim Report IR-98-055. 13 Li X-B and Reeves GR (1999). A multiple criteria approach to data envelopment analysis. Eur J Opl Res 115: 507–517. 14 Clı´ maco J, Antunes CH and Alves MJG (1996). Programac¸a˜o Linear Multiobjetivo: Me´todos Interactivos, ‘Software’ e Aplicac¸o˜es. Faculdade de Economia da Universidade de Coimbra e INESC. 15 Steuer RE (1986). Multiple Criteria Optimization: Theory, Computation, and Application. Krieger Publishing: Malabar, FL. 16 Geoffrion AM (1968). Proper efficiency and the theory of vector maximisation. J Math Anal Appl 22: 618. 17 Evans JP and Steuer RE (1973). A revised simplex method for multiple objective programs. Math Programming 5: 54–72. 18 Zeleny M (1973). Compromise programming. In: Cochrane JL and Zeleny M (eds). Multiple Criteria Decision Making. University of South Carolina Press, Columbia, pp 262–301. 19 Ecker JG, Hegner NS and Kouada IA (1980). Generating all maximal efficient faces for a multiple objective linear program. J Optim Theory Appl 30: 353–381. 20 Armand P and Malivert C (1991). Determination of the efficient set in multiobjective linear programming. J Optim Theory Appl 70: 467–489. 21 Steuer RE (1992). ADBASE, Department of Management Science and Information Technology, University of Georgia, Athens, GA. 22 Frota Neto (2002). Implementac¸a˜o e Ana´lise de Programac¸a˜o Multiobjetivo na Soluc¸a˜o de Problemas de Ana´lise Envolto´ria de Dados. M.Sc. dissertation, COPPE/UFRJ, Brazil. 23 Kornbluth JSH (1974). Duality, indifference and sensitivity analysis in MOLP. Opl Res Quart 25: 599–614. 24 Cohon JL (1978). Multiobjective Programming and Planning. Academic Press: New York. 25 Romero C (1993). Teorı´ a de la Decisio´n Multicriterio: Conceptos, te´cnicas y aplicaciones. Alianza Editorial: Madrid. 26 Korhonen P and Wallenius J (1988). A Pareto Race. Naval Res Logist 35: 615–623. 27 Guerreiro J et al. (1985). Programac¸a˜o Linear. McGraw-Hill: Portugal.

Received September 2002; accepted April 2004 after three revisions

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.