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Approximation of value efficiency in DEA with negative data.

INTRODUCTION

DEA was originally proposed by Charnes et al. [2] as an method for evaluating the technical efficiency of units essentially performing the same task. In conventional DEA models the data are assumed to be positive. In many applications, such as loss when net profit is an output variable, have been emerged negative inputs and outputs.

The aim of value efficiency is to assess the efficiency of each unit based on the DM's preferences and in relation to the indifference contour of DM's value function passing through the MPS. MPS is a virtual or existing unit on the efficient frontier with the most desirable values of inputs and outputs. However, several works have been made to calculate value efficiency of units in DEA; for example: Halme et al. [3], Korhonen et al. [5], Korhonen and Syrjanen [6], Joro et al. [4] and Zohrebandian [7]. The assess of value efficiency could be done easily, if we explicitly knew the value function. But in practice, the value function is usually unknown and we cannot characterize the indifference curve precisely and we have to estimate it. It is assumed that the DM's value function is pseudo concave, and is strictly increasing in input with a maximal value at MPS.

The purpose of this study is a modification of Halme et al.'s approach [3] for measuring value efficiency of each unit as a distance to an estimated indifference contour of a DM's value function. We present a MOLP model which its functions are input/output variables subject to the defining constraints of PPS in standard DEA models. Then, using an effective method we found weights for input and output variables. To end with, we obtain a MPS via solving a one objective linear programming. Finally, we approximate the indifference contour of the unknown value function at MPS by the supporting hyper plane at MPS and then calculate the value efficiency scores using comparing the inefficient units to units having the same value as the MPS.

The paper is organized as follows. In section 2, we explain our approach to calculate of value efficiency. In section 3 present numerical examples and finally, section 4 draws the conclusive remarks.

2. Calculating value efficiency scores in DEA with negative data:

Value efficiency is an efficiency concept, as an extension of the concept of efficiency in DEA. This method is based on the assumption that the DM compares alternatives by a value function. The value function is assumed to be pseudo concave, strictly increasing in y, strictly decreasing in x and obtain its maximum at MPS on the efficient frontier. Because value function is unknown, we have to approximate it by supporting hyper plane at MPS. We consider the PPS of BCC that is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Where the n is number of units and any unit has m inputs and s outputs. The first step in value efficiency is to locate the MPS. It means that we have to solve the following MOLP:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Input and output variables, that have negative values, are allowed in the model (1) to be without any sign, i.e. be free.

2.1. A new effective method for solving the model (1):

For solve the model (1), we propose an effective method for finding the weights of weighting method. The first, we construct a payoff table for objectives. So, optimize one of the objectives, while ignoring all others. Assume that for the q th objective, the best value (ideal solution) is [u.sub.q] corresponds to the solution, that is called ([x.sup.q], [y.sup.q], [[lambda].sup.q]), i.e. [z.sub.q] ([x.sup.q], [y.sup.q], [[lambda].sup.q]) = [u.sub.q]. In the qth row of the payoff table are computed the value of the other m + s - 1 objectives implied by ([x.sup.q], [y.sup.q], [[lambda].sup.q]). We use this payoff table to develop weights on the distance of very m+s-1 objective from their ideals, when qth objective is its ideal solution. The weights [w.sub.j] are defined as

[w.sub.q] = [[[theta].sub.q]/[m+s.summation over (j=1)][[theta].sub.j]] (2)

Where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

Then we obtain a efficient solution by solving the following single linear programming:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

2.2. Measuring value efficiency scores:

The weights [w.sub.j], which obtained by the fractional (2), are weights of input and output variables. We purpose approximation of the indifference contour of unknown value function by determination of the supporting hyper planes at MPS of PPS. Since the mentioned PPS is VRS, so in need of that we obtain the weight [w.sup.*] corresponds to the constraint 1[lambda] = 1. We define the optimal value of the model (4) as the value of [w.sup.*] and optimal solution as MPS. The normal vector of supporting hyper plane obtains by the weights [w.sub.j]'s and [w.sup.*], which this the hyper plane is tangent on PPS at MPS.

At present, suppose that the equation of the hyper plane is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where ([v.sup.*], [u.sup.*]) obtain by (2). For estimation of [theta] value such that ([x.sub.j], [y.sub.j]) + [theta]([W.sup.x], [W.sup.y]) = ([bar.x], [bar.y]), where ([bar.x], [bar.y]) is the projected point of [DMU.sub.j] on the hyper plane and ([W.sup.x], [W.sup.y]) is the direction of projection, we should have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In other words, in output oriented direction ([W.sup.x], [W.sup.y]) = (0, [y.sub.j]), we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. However, it is evident that this value is positive if and only if the [DMU.sub.j] is value inefficient.

3. Illustration with numerical examples:

Example 1. We consider 12 units with one input and two outputs, which some from DMUs have negative data in the output O2. The data recorded in Table 1.

We consider the following multi objective problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

The payoff table is as following:
Table 2: The payoff data.

            [z.sub.1]   [z.sub.2]   [z.sub.3]

[z.sub.1]   -19         4           0
[z.sub.2]   -50         58          -16
[z.sub.3]   -49         0           5


Find [[theta].sub.j] for each objective as following:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[w.sub.x] = [[[theta].sub.1]/[summation over (j)][[theta].sub.j]] = 0.0586, and also [w.sub.y] = 0.9171, [w.sub.z] = 0.0243, which these weights corresponds to input/output variables foe compute the value efficiency scores of units. Now we solve the following weighting model for finding MPS and [w.sup.*] at obtaining the value efficiency scores.

max - 0.0586x + 0.9171y + 0.0243z s.t. (Constraints of the model (5)) (6)

Then we have MPS = [DMU.sub.1] = the optimal solution of model (6) and [w.sup.*] = 49.873 = the optimal value of model (6). The computed value efficiency scores explained in table 3.

Example 2. in this example we make use of the data sets employed by Emrouznejad et al. [1]. Table 4 shows data for 10 DMU with activity vector (x, y, z). Each DMU uses input x to produce outputs y, z. The output z is always positive and the output y is positive for some DMUs and negative for others.

The payoff table is as follows:
            [z.sub.1]   [z.sub.2]   [z.sub.3]

[z.sub.1]   -12         0           0
[z.sub.2]   -16         26          0
[z.sub.3]   -50         -8          27


We have [[theta].sub.1] = 0.1368, [[theta].sub.2] = 0.7027 and [[theta].sub.3] = 0.9429 also [w.sub.x] = 0.0768, [w.sub.y] = 0.3942 and [w.sub.z] = 0.5290. Then we have MPS = [DMU.sub.4] and [w.sup.*] = 13.6208. The computed value efficiency scores explained in table 5.

4. Conclusion:

The main idea of this paper is utilize from the most preferable weights for a MOLP problem which is intellectually consistent with the DEA philosophy and by solve this MOLP model to obtain an efficient reference point, which is called MPS. In fact, the obtained weights generate a tangent hyperplane on PPS at the mentioned MPS. We can use from another effective approach for solve a MOLP problem that could generate weights for objectives.

ARTICLE INFO

Article history:

Received 17 September 2013

Received in revised form 24 October 2013

Accepted 5 October 2013

Available online 14 November 2013

REFERENCES

[1] Emruoznejad, A., L. Anouze, E. Thanassoulis, 2010. A semi-oriented radial measure for measuring the efficiency of decision making units with negative data, using DEA, Euro. J. of Oper. Res., 200: 297-304.

[2] Charnes, W.W. Cooper, E. Rhoders, 1978. Measuring the efficiency of decision making units, 2: 429-444.

[3] Halme, M., T. Joro, P. Korhonen, S. Salo, J. Wallenius, 1999. A value efficiency approach to incorporating preference information in data envelopment analysis, Manage, Sci., 45(1): 103-115.

[4] Joro, T., P. Korhonen, S. Zionts, 2003. An interactive approach to improve estimates of value efficiency in data envelopment analysis, Eur. JJ. Oper. Res., 149: 688-699.

[5] Korhonen, P., A. Siljamaki, M. Soismaa, 2002. On the use of value efficiency analysis and further developments, J. Prod. Anal., 17: 49-64.

[6] Korhonen, P., M.J. Syrjanen, 2005. On the interpretation of value efficiency, J. Prod. Anal., 24: 197-201.

[7] Zohrabandan, M., 2011. Using Zionts-Wallenius method to improve estimate of value efficiency in DEA, J. Elsevier, Applied Mathematical Modeling, 35: 3769-3776.

[8] Halme, M., P. Korhonen, 1999. Restricting weights in value efficiency analysis, IIASA.

[9] Korhonen, P., A. Siljamaki, M. Soismaa, 1998. Practical aspects of value efficiency analysis, IIASA.

[10] Yaman, S., C.H. Lee, 2007. A multi-objective programming approach to compromising classification performance metrics, Machine Learning for signal processing, IEEE WORKSHOP.

[11] Joro, T., P. Korhonen, J. Wallenius, 1998. Structural comparison of data envelopment analysis and multiple objective linear programming, Manage. Sci., 44: 962-970.

(1) Majid Zohrehbandian, (2) G. Reza Jahanshahloo, (1) Hossein Abbasiyan

(1) Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.

(2) Department of Mathematics, Science & Research Branch, Islamic Azad University, Tehran, Iran.

Corresponding Author: Hossein Abbasiyan, Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.

E-mail: abbasiyan58@yahoo.com
Table 1: DMUs And their input/output variable values.

DMU    (x) input   (y)output   (z)output

DMU1      50          58          -16
DMU2      48          48          -17
DMU3      49          45          -6
DMU4      49          35           5
DMU5      48          34           4
DMU6      50          25          -12
DMU7      47          25           3
DMU8      47          25          -14
DMU9      45          16           2
DMU10     48          15          -4
DMU11     47          14           1
DMU12     35          13           1
DMU13     19           4           3
DMU14     23           4          -5

Table 3: Results of proposed method for Ex.1.

DMU       DMU1     DMU2     DMU3     DMU4

Scores   0.0000   0.2082   0.2826   0.6370
DMU       DMU8     DMU9    DMU10    DMU11
Scores   1.3299   2.5667   2.8571   3.0911

DMU       DMU5     DMU6     DMU7

Scores   0.6844   1.3327   1.2881
DMU      DMU12    DMU13     DMU14
Scores   3.3463   12.628   13.4410

Table 4: DMUs and their input and output data.

DMU    X     y    Z     DMU    X     Y    Z

DMU1   12   15    11   DMU6    50   -8    27
DMU2   35   18    6    DMU7    35   -18   27
DMU3   25   20    13   DMU8    40   -10   22
DMU4   22   12    20   DMU9    25   -7    19
DMU5   40   -10   25   DMU10   16   26    8

Table 5: Results of proposed method for Ex.2.

DMU       DMU1     DMU2     DMU3     DMU4     DMU5

Scores   0.2395   0.5881   0.0528   0.0000   0.7982

DMU       DMU6     DMU7     DMU8     DMU9     DMU10

Scores   0.5689   1.2691   1.1690   1.1313   0.0254
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Title Annotation:data envelop analysis
Author:Zohrehbandian, Majid; Jahanshahloo, G. Reza; Abbasiyan, Hossein
Publication:Advances in Environmental Biology
Article Type:Report
Date:Oct 1, 2013
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