Target Setting for Inefficient DMUs for an Acceptable Level of Efficient Performance

Transcription

1 Global Journal of Pure and Applied Mathematic. ISSN Volume 13, Number 6 (2017), pp Reearch India Publication Target Setting for Inefficient DMU for an Acceptable Level of Efficient Performance Shinto K.G Department of Mathematic Chrit College Irinjalakuda Thriur , Kerala, India. Suhama C.M Department of Mathematic NIT Calicut, Kerala India Abtract Data Envelopment Analyi i a nonparametric technique ued to rank imilar deciion making unit by evaluating the relative efficiency. It can alo be ued to et target value for inefficient DMU o a to perform efficiently. In common et of weight DEA model, a common et of weight for input and output for DMU will be determined and thi common et of weight i ued to evaluate the relative efficiency and ranking of each DMU. The mot commonly ued method to generate common et of weight i the method baed on the goal programming. In traditional model, target etting for inefficient DMU i alo conidered o a to attain a level on the efficient frontier. Sometime forcing a DMU to function o a to attain the maximum efficiency value may reult in undeirable after effect. So it i very important to conider model which ugget target value for inefficient DMU o a attain an acceptable level of efficiency. Thi paper aim to conider uch a target etting alo. Keyword: Data Envelopment Analyi (DEA), Multi-objective Optimization, Goal programming, Common et of weight model 2010 Mathematic Subject Claification: 90C29, 90C32, 90C90

2 1894 Shinto K.G and Suhama C.M 1. INTRODUCTION Data Envelopment Analyi (DEA) i an LPP baed nonparametric technique developed by Charne et al [1] in 1978 to evaluate relative efficiency of imilar deciion making unit (DMU) that ue homogeneou multiple input to produce homogeneou multiple output. In traditional DEA model, each DMU can elect the bet input and output weight by olving an LP problem to achieve the highet efficiency. So if there are n DMU, it i required to olve n LP problem to attain the weight for each DMU [2, 3]. The drawback of thee type of DEA model are explained in [4-7]. To overcome thee hortcoming they developed common et of weight model. Thee model in the cited paper ue the olution of one LP problem rather than the olution of n LP problem to et the common et of weight. For thi it i required to olve a multiple objective problem. The mot commonly ued method to olve a multiple objective LP problem i by goal programming. Uing goal programming a multiple objective LP problem can be tranformed to an LP problem. Some of the multiobjective model and goal programming model are given in [8-13]. Suppoe there are n DMU to be evaluated where each DMUj, j = 1,2,. n conume m input x ij, i = 1,2,. m to produce output y rj, r = 1,2,... To olve goal programming problem to determine the common et of weight uing Hoeinzadeh Lofti et al [10], conider the LP problem, ubjec to n minimize S j j=1 u r y rj v i x ij + j = 0, j = 1,2, n r=1 m i=1 m u r y rj v i x ij r=1 i=1 j 0, 0, j = 1,2, n j = 1,2, n u r, v i ε r = 1,2, ; i = 1,2, m (M1) Since in the model S j 0, the econd et of contraint i redundant and can be removed from the model.

3 Target Setting for Inefficient DMU for an Acceptable Level of Efficient Performance 1895 Uing the optimal olution (u r, v i, S j ) i & r, the efficiency core for DMUj i given by θ j = r=1 u r y rj m i=1 v i x ij = 1 S j m i=1 v i x ij, j = 1,2,. n (1) Baed on thi efficiency value it i poible to rank the DMU. All DMU having θ j = 1 can be conidered a efficient or non dominated DMU. Thi model i developed to rank the DMU baed on their relative efficiency value θ j. Thi model i not applicable to ugget target value for the efficient functioning of inefficient DMU or DMU having θ j < 1. The propoed paper aim to modify the model to ugget target value for a atifactory functioning of each DMU. It i propoed to conider the atifactory level of efficiency by olving the LP problem of Hoeinzadeh Lofti et al [10] by introducing a fictitiou DMU. The ret of the paper i organized a follow. In ection 2 the LP problem to determine the atifactory efficiency and common et of weight i conidered. Target etting for atifactory functioning of inefficient DMU i given ection 3. Different target etting model are given in ection 4 and example to illutrate the propoed model i given in ection DETERMINATION OF SATISFACTORY LEVEL OF EFFICIENCY In thi ection a atifactory level of efficiency i propoed by conidering a fictitiou DMU which conume the input that are the average of repective input of DMU1 to DMUn and produce output a the average of the repective output of DMU1 to DMUn. Let u name the fictitiou DMU a DMUn+1. So by the aumption, the inputoutput factor X i,n+1 and Y r,n+1 of DMUn+1 are given by n X i,n+1 = X ij n i = 1,2,.. m j=1 n Y r,n+1 = Y rj j=1 n r = 1,2,. Include DMUn+1 into the et of DMU and evaluate the efficiency a given below. Let θ n+1 denote the efficiency of DMUn+1. The peculiarity of thi efficiency value i that it conider all input of all DMU and all output of all DMU for the efficiency

4 1896 Shinto K.G and Suhama C.M evaluation. So thi efficiency ha an importance in the performance evaluation of each DMU. If the efficiency value of a particular DMU i greater than θ n+1, then it can be concluded that the performance of the DMU i uperior to the average performance. So in the propoed model etting θ n+1 a the acceptable level of efficiency i jutified. Thi level, θ a i determined by olving the following LP problem. n+1 minimize S j Subject to j=1 u r y rj v i x ij + j = 0, j = 1,2, n + 1 r=1 m i=1 j 0, j = 1,2, n + 1 u r, v i ε r = 1,2, ; i = 1,2, m (M2) Uing the optimal olution (u r, v i, S n+1 ), the acceptable efficiency θ a i given by θ a = 1 S n+1 m i=1 v i x i,n (2) 2.1 Obervation 0 < θ a 1 and θ a = 1 only when all the input and output value are ame for all the DMU 3. TARGET SETTING FOR INEFFICIENT DMU Let (u r, v i, S j ) be the optimal value of (M2). Let θ j denote the efficiency value of DMUj, j = 1,2, n evaluated by ubtituting (u r, v i, S j ) in (1). Uing θ j, j = 1,2, n it i poible to claify the DMU into two group, DMU having efficiency value le than θ a, the acceptable efficiency and DMU having efficiency value greater than or equal to θ a.

5 Target Setting for Inefficient DMU for an Acceptable Level of Efficient Performance Definition The DMU having efficiency value greater than or equal to θ a are called acceptable efficient (a-efficient) DMU and if the efficiency i unity then uch a DMU i called completely efficient DMU. If the efficiency value of a particular DMU i le than θ a, then it i called an inefficient (a-inefficient) DMU. 3.2 Ranking and Target etting Thi paper i aimed to rank the DMU baed on their performance a well a to ugget target value for inefficient DMU o a to achieve an acceptable level of efficiency. Uing the LP problem given in (M2), the optimal deviation S j, j = 1,2, n and the common et of weight u r and v i r = 1,2, ; i = 1,2, m can be evaluated. By ubtituting the optimal deviation and the common et of weight in (1) the efficiency value for each DMU for j = 1,2, n can obtained eaily. Thee efficiency value can be ued to rank the DMU. To ugget target value firt of all it i required to identify the inefficient DMU. Let DMUo be an inefficient DMU which can be identified uing (M2). By olving an LP problem, the improvement in the value of each input and output for DMUo can be determined. Let α io be the improvement required in the i th input of DMUo for i = 1,2, m and β ro be the improvement required for the r th output for r = 1,2,. Then to determine α io for i = 1,2, m and β ro for r = 1,2,, the following LP problem i olved m minimize α io + β ro i=1 r=1 ubject to θ a ( v i (x io α io )) u r (y ro + β ro ) 0 i r u r (y ro + β ro ) v i (x io α io ) 0 r i α io x io, i = 1,2, m

6 1898 Shinto K.G and Suhama C.M α io, β ro 0 i & r (M4) Where θ a i the acceptable efficiency determined uing (2) and (M2), (u r, v i ) are the common et of weight determined uing (M2). The target input-output vector for DMUo, x io and y ro i then given by x io = x io α io and y ro = y ro + β ro (3) The target x io i the maximum input value of i th input to attain a-efficiency for DMUo. Similarly y ro i the minimum output value for the r th output to attain a- efficiency for DMUo. 4. DIFFERENT MODELS FOR TARGET SETTING Thi ection of the paper i devoted to decribe variant of the propoed model. The main advantage of the propoed model i that it i poible to modify the model o a to extend it to ue for a wide variety of cae. Firtly, the model given in (M4) i extended to ugget target input/output value where certain of the input/output value are held fixed o that change i poible only in the remaining et of input/output value in order to achieve a-efficiency. Suppoe we do not want to conider a change in a particular input or output. Then we can deign uch a model in the following way. Suppoe we do not want to revie the i p th input of the inefficient DMUo. Then we conider the LP problem m minimize α io + β ro i=1 i p r=1 ubject to θ a ( v i (x io α io ) + v p x po ) u r (y ro + β ro ) 0 i i p r u r (y ro + β ro ) ( v i (x io α io ) + v p x po ) 0 r i i p α io x io, i = 1,2, m, i p

7 Target Setting for Inefficient DMU for an Acceptable Level of Efficient Performance 1899 α io, β ro 0 i & r (M5) Similarly if we do not want to revie any particular output ay r q th output. Then a above we can develop the LP model to ugget target value. By extending thee model it i poible to develop the input oriented a well a the output oriented model. To develop input oriented or output oriented model for target etting, we can directly modify (M4). In cae of output oriented target etting we will keep the given output and will try to ugget correponding input. However in the cae of input oriented model, we will keep the given input fixed and will ugget modified output for efficiency. However the target etting uing an LP problem ha ome limitation. A in the example, we can ee that the target uggetion uing (M4) and (M5) may concentrate on a ingle component or ome particular component of input or output for the improvement. But in real world ituation thi may not be a correct mechanim. We can imply modify our model to accommodate the ituation. Suppoe we need the utilization of the input-output factor in a particular ratio for the improvement, then we can include it in our model a contraint. The main advantage of thi dicuion i that the propoed model can concentrate a particular input/output for the improvement or it can revie all input and output with equal importance baed on a predefined ratio. Thi model i hence giving more power for the deciion maker. The following ection explain the model with an example. 5. EXAMPLE Table 1 (Given data on 5 DMU with 2 input and 2 output) X1 X2 Y1 Y2 DMU DMU DMU DMU DMU Conider the example (given in Subhah C. Ray) of 5 DMU having 2 input and 2 output. Data i given in Table 1. The Dummy DMU of our dicuion i given in Table2.

8 1900 Shinto K.G and Suhama C.M Table 2(Data for fictitiou DMU) Dummy DMU X1 X2 Y1 Y The LP problem given in (M2) correponding to the data i given by 6 minimize S j 9u u 2 6v 1 5v 2 + S 1 = 0 10u u 2 4v 1 2v 2 + S 2 = 0 12u u 2 3v 1 7v 2 + S 3 = 0 14u u 2 6v 1 8v 2 + S 4 = 0 18u 1 + 4u 2 2v 1 10v 2 + S 5 = u u 2 4.2v 1 6.4v 2 + S 6 = 0 u r, v i ε for r = 1,2 and i = 1,2 S j 0 for j = 1,2,.5 Correponding to ε = 0.01 We get (u 1, u 2, v 1, v 2 ) = (0.0115, 0.01, , 0.01) j=1 From table 3 we get the acceptable level of efficiency θ a = 0.75, DMU 2 and DMU 5 are completely efficient, DMU 1 and DMU 4 are inefficient. Table 3(Efficiency and optimal lack) DMU Dummy DMU S j θ j

9 Target Setting for Inefficient DMU for an Acceptable Level of Efficient Performance 1901 The target value for the inefficient DMU are given in Table 4 Table 4(Target input-output value for inefficient DMU) Baic model Output oriented Input oriented x 1j x 2j y 1j y 2j x 1j x 2j y 1j y 2j x 1j x 2j y 1j y 2j DMU DMU From Table 4, it i clear that the target etting i uggeted by concentrating on a particular factor. Thi we can overcome by impoing additional contrain baed on the importance and availability of input-output factor. For example if we impoe additional contraint 3α 1j α 2j, β 1j 2β 2j, α 1j + α 2j β 1j + β 2j for j = 1, 4. Then the target value of inefficient DMU with θ a = 0.75 for the baic model i given in Table 5. Table 5(Target input-output value for inefficient DMU) x 1j x 2j y 1j y 2j DMU DMU CONCLUSION The method dicued in thi paper can be ued to claify the given DMU into two categorie efficient and inefficient. It alo can be ued to define an acceptable level of efficiency and ugget target value to attain the acceptable level of efficiency. The main advantage of the model i that it i adaptable to a wide variety of ituation like input/output weight retriction, retriction on change in certain input/output etc. o thi model help the deciion maker to make analyi on the performance of DMU in many different way. REFERENCES [1] Charne, A., Cooper, W. W. and Rhode, E.: Meauring the Efficiency of Deciion Making Unit. European Journal of Operational Reearch. 2(6), (1978). [2] Charne, A., Cooper, W. W., Lewin, A. Y. and Seiford, L. M.: Data Envelopment Analyi- Theory, Methodology and Application. Kluwer Academic Publiher, Boton (1994).

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