Profit Efficiency with Kourosh and Arash Model

Transcription

1 Applied Mathematical Sciences, Vol. 8, 2014, no. 24, HIKARI Ltd, Profit Efficiency with Kourosh and Arash Model Dariush Khezrimotlagh* 1,2 1. Department of Applied Statistics, Faculty of Economics and Administration University of Malaya, Kuala Lumpur, Malaysia 2. UTM Centre for Industrial and Applied Mathematics, Faculty of Science Universiti Teknologi Malaysia, Johor, Malaysia Copyright 2014 Dariush Khezrimotlagh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract This article illustrates how Kourosh and Arash Model (KAM) is able to measure profit-efficiency of Decision Making Units (DMUs) in Data Envelopment Analysis (DEA). The advantages of KAM in comparison with the current profit efficiency DEA models proposed by Cooper et al. (2007) and Coelli et al. (2005) are also illustrated with a clear example. Mathematics Subject Classification: 90 Keywords: EA, KAM, Profit efficiency, Arash method, Kourosh method. 1. Introduction Since the original work of Charnes et al. [1] to propose Data Envelopment Analysis (DEA), a good number of studies have dramatically been applied in many areas of Management, Economics, Industrial Engineering and Mathematics. Khezrimotlagh et al. [2] recently improved the base of this young knowledge and proposed a robust method called Kourosh and Arash Model (KAM) to increase the advantages of DEA. They also extended the method called Integer-KAM (IKAM) for measuring the efficiency of DMUs inclusive real and integer values [3]. KAM is simultaneously able to estimate the production frontier, measure the * Corresponding author

2 1166 D. Khezrimotlagh relative efficiency score, and benchmark and rank DMUs. It was proposed based on Arash Method (AM) and Kourosh Method (KM) which are able to measure cost-efficiency and revenue-efficiency of DMUs [4-5]. A review on AM can be seen in [6-7]. In this paper, after the short background on KAM in Section 2, it is illustrated how KAM can measure the profit-efficiency of DMUs, respectively. Their corresponding theorems are also proved in Sections 3 and 4. Disadvantages of the current profit efficiency DEA models proposed by Coelli et al. [8] and Cooper et al. [9] are also depicted with a simple example in Section 4 and the paper is concluded in the last section. 2. Background Let us assume that there are DMUs ( ) with nonnegative inputs ( ) and nonnegative outputs ( ) for each DMU which at least one of its inputs and one of its outputs are not zero. Assume that ( ) is evaluated, and are the user specified weights obtained through values judgment,,,, and s and s are nonnegative slacks, for and. The -KAM is given by:, Subject to ( ). Note that, the constraints and are removed from the KAM constraints, because the best technical efficient targets with degree of freedom ( -DF) are only considered in this paper [3-4]. Table 1 illustrates the profit efficiency models proposed in Coelli et al. [8] and Cooper et al. [9] which respectively denoted by PF6 and PF7. The profit score is, where and are optimums of and, and and are available input and output price for and, respectively. Moreover, is profit efficient if. 3. KAM and Profit-Efficiency (PF) When the cost information of inputs, s, and the unit price of outputs, s, are available, users can easily calculate the observed profit, that is,, for each ( ) in order to find the most profitable DMU. The ratio of observed profit over maximum profit can also be defined as the profit efficiency score for each DMU. In order to depict the

3 Profit efficiency with KAM 1167 advantages of KAM, this section illustrates the validity of KAM to measure the profit efficiency, and describes disadvantages of PF6 and PF7. The following theorem is similarly deduced from the properties of AM and KM. Name PF6 PF7 Table 1: Current profit-efficiency models. Models, S.t. ; ;., S.t. ; ;. Theorem 1: -KAM is always feasible on its constraints space and its best technical efficient target with -DF is on the Farrell frontier [3]. Theorem 2: Let and, for. The results of PF6 and PF7 can be calculated by -KAM. Proof: First, let us consider PF6 in Table 1 and define,, for and, for. The objective of PF6 yields that: ( ), that is,, which is equivalent with. These equations yield that the objectives of KAM and PF6 are equally optimized. Since the and in PF6 only depend to the linear combination of s and s and do not depend to and, respectively, the large enough value of bridges over the models and rectifies the dependences of to and to in KAM. Now, it is enough to consider for the score of KAM. Second, let us consider PF7 and define in KAM. The objective of the models and their constraints yield that:, that is equivalent with, that is,. Moreover, the 0- KAM targets are the same as PF7 targets and the proof is completed. Table 2: Examples of three DMUs. DMUs Input1 Input2 Output1 Output2 A B C Let us consider DMUs in Table 2 where the cost information and unit price are,, and. The observed profits are 20, 46 and

4 1168 D. Khezrimotlagh 38 for DMUs A, B and C, respectively. By dividing these values on the maximum profit, that is, 46, the profit efficiency scores of DMUs A, B and C are (20/46) = , (46/46) = 1 and (38/46) = , respectively. Now, assume Variable Returns to Scale (VRS) by adding the convexity constraint, that is,, to the constraints of PF6, PF7 and -KAM. Table 3 represents the profit efficiency scores of PF6, PF7 and KAM where, and is 0, 0.001, 5, 10, 50 and 100. Table 3: The scores of PF6, PF7 and -KAM in VRS. DMU A B C As can be seen, the scores of PF6 in the second column of Table 3 are the same as what were simply calculated for DMUs, whereas PF7 and 0-KAM are not able to discriminate A, B and C. This is the disadvantage of PF7 and 0-KAM. However, KAM significantly characterizes the most profitable DMU when. The scores of KAM and PF6 are also the same when the value of epsilon is increased. On the other hand, s in PF6 may approach infinity in Constant Returns to Scale (CRS) [1] and it makes the model indeterminate in order to suggest maximum profit. For example, let us suppose CRS and apply the models. From the second and third columns of Table 4, PF7 and 0-KAM are not able to show the most profitable DMU. PF6 is also indeterminate in this case. However, KAM is easily able to find the most profitable DMU when. Table 4: The scores of PF models and -KAM in CRS. DMU A B C Table 5: The results of -KAM for DMU B. KAM and Profit Columns six to nine of Table 4 illustrate that the scores of KAM approach to zero by increasing the value of epsilon. Since Theorem 2 demonstrates that PF6 and KAM are equivalent, it is deduced that the scores of PF6 also approach to zero. In other words, the limit of is zero as epsilon approaches infinity for each DMU, that is, the denominator as well as the PF6 s objective approaches to infinity. Table 5 illustrates the values of s, s and s in KAM for DMU B where is 5, 10, 50 and 100.

5 Profit Efficiency with KAM Conclusion This paper illustrates one of the advantages of KAM to measure the profitefficiency of DMUs where the unit price and cost information are available. The paper also demonstrates the disadvantages of current profit-efficiency DEA models proposed by Coelli et al. [8] and Cooper et al. [9] with a clear counter example. The example depicts how KAM easily measures profit-efficiency by selecting different values of epsilons. The paper suggests applying KAM for assessing the performance evaluation of DMUs either the weights of data are available or unknown. References [1] A. Charnes, W.W. Cooper and E. Rhodes, Measuring the efficiency of decision making units, European Journal of Operational Research, 2 (1978) 6: [2] D. Khezrimotlagh, S. Salleh and Z. Mohsenpour, A New Method for Evaluating Decision Making Units in DEA, Journal of the Operational Research Society, In Press, Published Online (2013). [3] D. Khezrimotlagh, S. Salleh and Z. Mohsenpour, A New Robust Mixed Integer Valued Model in DEA, Applied Mathematical Modelling, 37 (2013), 24: [4] D. Khezrimotlagh, Z. Mohsenpour and S. Salleh, Cost-Efficiency by Arash Method in DEA, Applied Mathematical Sciences, 6 (2012) 104: [5] P. Mohsenpour, S. Munisamy and D. Khezrimotlagh, Profit Efficiency and Kourosh Method in DEA, Applied Mathematical Sciences, 7 (2013) 140: [6] D. Khezrimotlagh, P. Mohsenpour, S. Salleh, and Z. Mohsenpour, A Review on Arash Method in Data Envelopment Analysis. The 11th International Conference on Data Envelopment Analysis, Samsun, Turkey, June 27-30, [7] D. Khezrimotlagh, S. Salleh and Z. Mohsenpour, Airport Efficiency with Arash Method in DEA, Journal of Basic and Applied Scientific Research, 2 (2012) 12:

6 1170 D. Khezrimotlagh [8] T.J. Coelli, D.S.P. Rao, C.J. O'Donnell and G.E. Battese, An Introduction to Efficiency and Productivity Analysis, Second Edition, Springer, New York, [9] W.W. Cooper, L.M. Seiford L.M. and K. Tone, Data Envelopment Analysis, A comprehensive Text with Models, Applications, References and DEA-Solver Software, Second Edition, Springer, New York, Received: January 25, 2014