Encyclopedia of Decision Making and Decision Support ... - UFF

Encyclopedia of Decision Making and Decision Support Technologies Frédéric Adam University College Cork, Ireland Patrick Humphreys London School of Economics and Political Science, UK

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709

Performance Measurement: From DEA to MOLP João Carlos Namorado Clímaco Coimbra University, Portugal INESC Coimbra, Portugal João Carlos Soares de Mello Federal Fluminense University, Brazil Lidia Angulo Meza Federal Fluminense University, Brazil

INTRODUCTION Data envelopment analysis (DEA) is a non-parametric technique to measure the efficiency of productive units as they transform inputs into outputs. A productive unit has, in DEA terms, an all-encompassing definition. It may as well refer to a factory whose products were made from raw materials and labor or to a school that, from prior knowledge and lessons time, produces more knowledge. All these units are usually named decision making units (DMU). So, DEA is a technique enabling the calculation of a single performance measure to evaluate a system. Although some DEA techniques that cater for decision makers’ preferences or specialists’ opinions do exist, they do not allow for interactivity. Inversely, interactivity is one of the strongest points of many of the multi-criteria decision aid (MCDA) approaches, among which those involved with multi-objective linear programming (MOLP) are found. It has been found for several years that those methods and DEA have several points in common. So, many works have taken advantage of those common points to gain insight from a point of view as the other is being used. The idea of using MOLP, in a DEA context, appears with the Pareto efficiency concept that both approaches share. However, owing to the limitations of computational tools, interactivity is not always fully exploited. In this article we shall show how one, the more promising model in our opinion that uses both DEA and MOLP (Li & Reeves, 1999), can be better exploited with the use of TRIMAP (Climaco & Antunes, 1987, 1989). This computational technique, owing in part to

its graphic interface, will allow the MCDEA method potentialities to be better used. MOLP and DEA share several concepts. To avoid naming confusion, the word weights will be used for the weighing coefficients of the objective functions in the multi-objective problem. For the input and output coefficients the word multiplier shall be used. Still in this context, the word efficient shall be used only in a DEA context and, for the MOLP problems, the optimal Pareto solutions will be called non-dominated solutions.

BACKGROUND Ever since DEA appeared (Charnes, Cooper, & Rhodes, 1978) many researchers have drawn attention to the similar and supplementary characteristics it bears to the MCDA. As early as 1993, Belton and Vickers (1993) commented their points of view supplement each other. This is particularly relevant for MOLP. For instance, both MOLP and DEA are methodologies that look for a set of solutions/units that are non-comparable between them, that is, are efficient/non-dominated. This contribution is focused on the synergies between MOLP and DEA. Taking into consideration the vast literature and to be able to follow the theme’s evolution articles should be classified into different categories. The first two categories are those in which DEA is used for MOLP problems and vice versa. Although the differences are often not very clear, these categories can be useful to introduce the theme.

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Performance Measurement: From DEA to MOLP

Works in which DEA is used within MOLP are not the object of this article. Some of these works are those of Liu, Huang, & Yen (2000), or Yun, Nakayama, Tanino, and Arakawa (2001). Within those articles in which MOLP is used in DEA problems a further disaggregation is possible: 1. Models that use MOLP to determine non-radial targets in DEA models. Their own nature makes it imperative that these models use the DEA envelop formulation. 2. Models that, besides the classic DEA objective, use other objectives, generally considered of lesser importance. The majority of the articles concerning this approach use the multipliers formulation. 3. Models in which optimization of more than one DMU is simultaneously attempted.

Papers in Which MOLP is Used in DEA The first article explicitly written along this line is Golany’s (1988). He starts from the assumption that not all DEA efficient solutions are effective, that is, they do not equally cater to the decision maker’s preferences. The article assumes that the inputs vector is constant and an outputs vector should be computed so the DMU is both efficient and effective. So, an interactive algorithm (MOLP), based on STEM (Benayoun, Montgolfier, Tergny, & Larichev, 1971) is proposed. This algorithm checks all possible DMU benchmarks and eliminates in succession those that do not conform to the decision maker’s interests. In fact, the author uses a multi-objective model in which every output is independently maximized, maintaining all inputs constant. Joro, Korhonen, and Wallenius (1998) produce a structural comparison between MOLP and DEA. They show that the non Archimedean output-oriented CCR model displays several similarities with the reference point MOLP model. This property is used in the “Value Efficiency Analysis” (Halme, Joro, Korhonen, Salo, & Wallenius, 2002; Joro, Korhonen, & Zionts, 2003) to assist a decision maker to find the most preferred point at the frontier. Tavares and Antunes (2001) based themselves on a minimizing Chebyshev’s distance method to put forward a DEA alternative target calculation. Lins, Angulo–Meza, and Silva (2004) developed the models MORO and MORO-D. These models are 710

a generalization of Golany’s (1988) model. The multiobjective method allows simultaneously for output maximization and input minimization. Quariguasi Frota Neto and Angulo-Meza (2007) have analyzed the characteristics of the MORO and MORO-D models and used them to evaluate dentists’ offices in Rio de Janeiro. The fuzzy-DEA multidimensional model (Soares de Mello, Gomes, Angulo-Meza, Biondi Neto, & Sant’Anna, 2005) used MORO-D as an intermediary step to find optimist and pessimist targets in the fuzzy DEA frontier. Korhonen, Stenfors, & Syrjanen (2003) minimize the distance to a given reference point to find alternative and non-radial targets. In an empirical way, they show that radial targets are very restrictive.

DEA Models with Additional Objective Functions The first models of this type were not recognized as multi-objective by their authors and are rarely mentioned as such. They include the two step model (Ali & Seiford, 1993) and the aggressive and benevolent cross evaluation (Doyle & Green, 1994; Sexton, Silkman, & Hogan, 1986). These models are not usually accepted as multi-objective ones. However, as they optimize in sequence two different objective functions, they can be considered as a bi-objective model solved by the lexicographic method (Clímaco, Antunes, & Alves, 2003). Kornbluth (1991) remarked that the formulation of multipliers for DEA can be expressed as multi-objective fractional programming. A similar approach by Chiang and Tzeng (2000) optimizes simultaneously the efficiencies of all DMUs in the multipliers model. An objective function corresponds to each DMU. The problem is formulated in the fractional form so as to avoid its becoming unfeasible owing to the excessive number of equality restrictions. The authors use fuzzy programming to solve this multiobjective fractional problem. Optimization is carried out in such a manner as to maximize the efficiency of the least efficient DMU. The last two models can also be classified as models in which more than one DMU are simultaneously optimized. Owing to its importance, Li and Reeves (1999) model is detailed hereafter.


The Li and Reeves Model (1999) Li and Reeves (1999) presented a multi-objective model with the aim to solve two common problems in DEA: (1) increasing discrimination among DMUs and (2) promoting a better multiplier distribution for the variables. The first problem occurs when we have a small number of DMUs or a great number of inputs and outputs (as standard models class too many DMUs as efficient). The second problem arises as a DMU becomes efficient with non nil multipliers in just a few variables. This benefits those that display a good performance and leaves aside those that do not. These two problems are intertwined. To get around them, the authors proposed a multicriteria approach for DEA in which additional objective functions are included in the classic CCR multipliers model (Charnes et al., 1978). The additional objective functions restrict the flexibility of the multipliers. In DEA, a given DMU O is efficient when hO = 1, meaning that the constraint relative to that DMU is active and, thus, its slack is nil. The basic idea is to consider this slack as an efficiency measurement instead of h. The slack symbol becomes d. As h equals one minus d, the CCR model in (1a) can be reformulated as (1b): s

Max hO = ∑ ur yrj0

i =1

i ij0

=1

r =1

i =1

ur ,vi ≥ 0, ∀r,i

Min Max d j subject to

r =1

i ij0

(1a)

=1

s

m

r =1

i =1

∑ ur yrj − ∑ vi xij + d j = 0, j=1,..,n ur , vi , d j ≥ 0, ∀r, i

=1 m

r

rj

i =1

i ij

+ d j = 0, j=1,...,n

ur , vi ≥ 0, ∀ r,i,j

s   Min dO  or Max hO = ∑ ur yrj0  r =1   subject to

i ij0

j

∑u y − ∑v x

∑ ur yrj − ∑ vi xij ≤ 0, j = 1,.., n

i =1

j =1

s

m

∑v x

n

∑d

Min

i =1

s

m

Min dO

m

subject to

∑v x

– dO. So, the lesser is d0, the more efficient is the DMU. To restrict multipliers freedom of choice, the MCDEA model takes into account two other objective functions: a “general benevolence” objective, the minimization of the deviations sum, and another, an “equity” one, that is, the minimization of the maximum deviation. Thus, the multiobjective programming problem known as multi-criteria data envelopment analysis (MCDEA) is formulated as in (2):

∑v x

r =1

m

where vi e ur are respectively the input i, i=l,...,m, and output r, r = 1,...,s, multipliers; xij and yrj DMU j, j = l,...,n, inputs i and outputs r ; xio and yro are DMU O inputs i and outputs r. Besides, d0 is the deviation variable for DMU O and dj is the deviation variable for DMU j, that is, how much the DMU is away from efficiency. In this model, DMU O is efficient if, and only if dO = 0, this being the same as hO = 1. If DMU O is not efficient, its measure of efficiency is hO = 1

(1b)

(2)

The first objective function is the classical efficiency maximization, the second is the equity function and the third that of “general benevolence.” The intention to optimize the evaluation of all DMUs, as a whole, from the view point of the DMU under analysis, is bred from the “general benevolence” function. The relative efficiency of a given DMU corresponding to the second and third objectives can be defined thus: a DMU O is minimax efficient if, and only if, the dO value that corresponds to the solution that minimizes the second objective function of model (2) is zero; likewise, a DMU O is minisum efficient if, and only if, the dO value that corresponds to the solution that minimizes the third objective function of model (2) is zero. 711

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In their original work, the authors have only used the weighted sum as a solution method and then proceed to draw considerations about the behavior of either function when the other one is optimized. They have used the ADBASE software, Steuer (1986). As their model is a three objective one, TRIMAP has some advantages in terms of content and user friendliness. A brief description follows.

The Trimap The TRIMAP package (Clímaco & Antunes, 1987, 1989) is an interactive environment. It assists the decision makers to perform a progressive and selective search of the feasible polyhedron non-dominated region. The aim is helping the decision maker to eliminate the subsets of the non-dominated solutions which are of no interest to her/him. It combines three main procedures: weights space decomposition, introduction of constraints on the objective functions space and on the weights space. In each step of the human/computer dialogue the decision maker just has to give indications about the regions of the weights space where the search for new non-dominated solutions should be carried out optimizing weighted sums of the objective functions. By cognitive reasons, usually this is performed indirectly introducing progressively during the interactive process bounds on the objective functions values, that is, reservation values taking into account the present knowledge on the non-dominated solutions set. Note that the introduction of constraints on the objective functions values is automatically translated into the weights space. The TRIMAP package is intended for tri-criteria problems. This fact allows for the use of graphical means and for the particular case of the MCDEA model is not a limitation. For instance, the indifference regions on the weights space (which is a triangle for the tri-objective case) corresponding to the non-dominated bases previously calculated are displayed graphically on the triangle. Other important tools and graphical facilities not referred to in this summary are available in TRIMAP. However, the graphic decomposition of the weights space in indifference regions corresponding to each of the previously calculated non-dominated bases is especially interesting to study the MCDEA model. Besides the graphical aspect, TRIMAP supplies a text that condenses all the numerical results. Among other data, the values of the basic variables and those 712

of the objective functions corresponding to the nondominated solutions, the share of the area occupied by the indifference regions, and so forth.

THE LI AND REEVES (1999) MODEL AND THE TRIMAP In the MCDEA model, the knowledge of the weights space decomposition, as obtained through TRIMAP, allows researchers to evaluate the stability of the DEA efficient solutions. Large indifference regions mean evaluations remained unchanged when moderate changes in the formulations of the objective functions occur. The existence of optimal alternatives is a check as to whether the optimal evaluation of a given DMU depends on a unique multipliers vector. It is easy enough to check whether any DMU is minimax or minisum efficient: it suffices that indifference regions include simultaneously the triangle corners that correspond to the optimization of the classical objective function and one of the other two. As TRIMAP supplies, for each indifference region, the multiplier values, a solution can be chosen with a multipliers distribution that is acceptable to the decision maker: for instance, one without nil multipliers. Table 1 shows input and output data for the DMUs used in an example. They are all CCR efficient. Figures 1 to 5 show weights space decompositions for the example used. It is worthy of notice that just by looking at the right hand bottom corner of the triangles that only DMUs 3 and 5 have non-dominated alternative optima for the first objective function. Thus, DMUs 3 and 5 are DEA extreme efficient. The same can not be concluded on

Table 1. Numerical example data DMU

Input 1

Input 2

Output

1

0,5

5

8

2

2

1

4

3

3

5

20

4

4

2

8

5

1

1

4


Figures 1, 2, and 3. Decomposition of the weights space for DMUs 1, 2 and 3, respectively

(1)

(2)

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(3)

Figures 4 and 5. Decomposition of the weights space for DMUs 4 and 5, respectively

(4)

this respect about DMUs 1, 2, and 4 as TRIMAP shows only non-dominated solutions. Eventual alternative optima concerning those DMUs have no representation on the triangle, because are dominated, and so are DEA efficient but worse than the non-dominated one for at least one of the other two criteria. A classical DEA analysis of the PPLs reveals that DMU 1 has alternative optimums, which is not the case for DMUs 2 and 4. These DMUs are not Pareto efficient. The analysis of the bottom left hand side and top corners shows that DMU 3 is both minisum and minimax efficient. DMU 5 is only minisum efficient. In a problem to choose one of these five efficient DMUs it would be acceptable to get rid of DMUs 1, 2, and 4.

(5)

CONCLUSION AND FUTURE RESEARCH TRIMAP can help to analyze the MCDEA model with greater detail than the one provided by Li and Reeves (1999) themselves because it is a very flexible decision support tool dedicated to tri-criteria problems. Several conclusions can reach from mere visual analysis. The possible uses of TRIMAP for the MCDEA model do not end here. Two types of development look particularly promising. The first is to modify the model’s objective functions. The minimax function can be replaced by a minimization of a weighted Chebyshev distance so as to control the restrictive character of the objective function. 713


The minisum or minimax functions can also be replaced by the maximization of the smaller multiplier. This replacement partly solves the too common DEA problem of having multipliers with a very small value. TRIMAP capability to render into the weights space constraints on the objective function values can be used in MCDEA for future interactivity processes of the decision maker. Another development could be to create an aggregate index of all analyzed properties.

Clímaco, J. C. N., Antunes, C. H., & Alves, M. J. G. (2003). Programação Linear Multiobjectivo [Multiobjective Linear Programming] Imprensa da Universidade, Coimbra. (Portuguese)

REFERENCES

Halme, M., Joro, T., Korhonen, P., Salo, S., & Wallenius, J. (1998). A value efficiency approach to incorporating preference information in data envelopment analysis. Management Science, 45(1), 103-115.

Ali, A. I., & Seiford, L. M. (1993). The mathematical programming approach to efficiency analysis. In H. O. Fried, C. A. K. Lovell, & S. S. Schmidt (Eds.), The measurement of productive efficiency (pp. 120-159). New York: Oxford University Press. Belton, V., & Vickers, T. J. (1993). Demystifying DEA—A visual interactive approach based on multiple criteria analysis. Journal of Operational Research Society, 44(9), 883-896. Benayoun, R., Montgolfier, J., Tergny, J., & Larichev, O. (1971). Linear programming and multiple objective functions: STEP method (STEM). Mathematical Programming, 1(3), 366-375. Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measurering the efficiency of decision-making units. European Journal of Operational Research, 2, 429444. Chiang, C. I., & Tzeng, G. H. (2000). A multiple objective programming approach to data envelopment analysis. New Frontiers of Decision Making for the Information Tehnology Era, 270-285. Singapore, New Jersey, London, Hong Kong: World Scientific.

Doyle, J. R., & Green, R. H. (1994). Efficiency and Ccoss-efficiency in DEA: Derivations, meanings and uses. Journal of the Operational Research Society, 45(5), 567-578. Golany, B. (1988). An interactive MOLP procedure for the extension of DEA to effectiveness analysis. Journal of the Operational Research Society, 39(8), 725-734.

Joro, T., Korhonen, P., & Wallenius, J. (1998). Structural comparison of data envelopment analysis and multiple objective linear programming. Management Science, 44(7), 962-970. Joro, T., Korhonen, P., & Zionts, S. (2003). An interactive approach to improve estimates of value efficiency in data envelopment analysis. European Journal of Operational Research, 149, 688-699. Korhonen, P., Stenfors, S., & Syrjanen, M. (2003). Multiple objective approach as an alternative to radial projection in DEA. Journal of Productivity Analysis, 20(3), 305-321. Kornbluth, J. S. H. (1991). Analysing policy effectiveness using cone restricted data envelopment analysis. Journal of the Operational Research Society, 42(12), 1097-1104. Li, X.-B., & Reeves, G. R., (1999). A multiple criteria approach to data envelopment analysis. European Journal of Operational Research, 115(3), 507-517.

Clímaco, J. C. N., & Antunes, C. H. (1987). TRIMAP— An interactive tricriteria linear programming package. Foundations of Control Engineering, 12, 101-119.

Lins, M. P. E., Angulo-Meza, L., & Silva, A. C. E. (2004). A multiobjective approach to determine alternative targets in data envelopment analysis. Journal of the Operational Research Society, 55(10), 1090-1101.

Clímaco, J. C. N., & Antunes, C. H. (1989). Implementation of a user friendly software package—A guided tour of TRIMAP. Mathematical and Computer Modelling, 12, 1299-1309.

Liu, F.-H. F., Huang, C.-C., & Yen, Y.-L. (2000). Using DEA to obtain efficient solutions for multi-objective 0-1 linear programs. European Journal of Operational Research, 126, 51-68.

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Quariguasi Frota Neto, J., & Angulo-Meza, L. (2007). Alternative targets for data envelopment analysis through multi-objective linear programming: Rio De Janeiro odontological public health system case study. Journal of the Operational Research Society, 58, 865-873. Roy, B. (1987). Meaning and validity of interactive procedures as tools for decision making. European Journal of Operational Research, 31, 297-303. Sexton, T. R., Silkman, R. H., & Hogan, A. (1986). Data envelopment analysis: Critique and extensions. In Richard H. Sikman (Ed.), Measuring efficiency: An assessment of data envelopment analysis (Publication No. 32 in the series New Directions of Program). San Francisco. Soares de Mello, J. C. C. B., Gomes, E. G., AnguloMeza, L., Biondi Neto, L., & Sant’Anna, A. P. (2005). Fronteiras DEA Difusas [Fuzzy DEA Frontiers]. Investigação Operacional, 25(1), 85-103. Steuer, R. (1986). Multiple criteria optimization: Theory, computation and application. New York: Wiley. Tavares, G., & Antunes, C. H. (2001). A Tchebychev DEA model. Working Paper. Piscataway, NJ: Rutgers University. Wei, Q. L., Zhang, J., & Zhang, X. S. (2000). An inverse DEA model for inputs/outputs estimate. European Journal of Operational Research, 121(1), 151-163.

KEY TERMS Benchmark: Benchmark is an efficient DMU with management practices that are reference for some inefficient DMUs. Data Envelopment Analysis (DEA): DEA is a non-parametric approach to efficiency measurement. Decision Making Unit (DMU): DMU is a unit under evaluation in DEA. Efficient DMU: An efficient DMU is one located on the efficient frontier. Multiple Criteria Data Envelopment Analysis (MCDEA): MCDEA is a tri-objective linear model proposed by Li and Reeves (1999). Multiple Objective Linear Programming (MOLP): MOLP is a linear program with more than one objective function. Non-Dominated Solution: A feasible solution is non-dominated whether does not exist another feasible solution better than the current one in some objective function without worsening other objective function. Target: Target is a point in the efficient frontier that is used as a goal for an inefficient DMU. TRIMAP: TRIMAP is an interactive tri-objective interactive solver package.

Yun, Y. B., Nakayama, H., Tanino, T., & Arakawa, M. (2001). Generation of efficient frontiers in multiple objective optimization problems by generalized data envelopment analysis. European Journal of Operational Research, 129, 586-595.

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