Scale Efficiency in DEA and DEA-R with Weight Restrictions

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1 Available online at Int. J. Data Envelopent Analyi (ISSN X) Vol.2, No.2, Year 2014 Article ID IJDEA-00226, 5 page Reearch Article International Journal of Data Envelopent Analyi Science and Reearch Branch (IAU) Scale Efficiency in DEA and DEA-R with Weight Retriction M.Nazari a,b, M.R.Mozaffari a,b, J.Gerai a,b. (a) Departent of Applied Matheatic, Science and Reearch Branch Azad Univerity, Far, Iran. (b) Departent of Applied Matheatic, Shiraz Branch Azad Univerity, Shiraz, Iran. Received 20 February 2014, Revied 22 May 2014, Accepted 24 June 2014 Abtract In data envelopent analyze (DEA) the cale efficiency in the input-oriented CCR odel i le than or equal to the cale efficiency in DEA baed on the fractional analyi (DEA-R). Alo, the cale efficiency in cae of ultiple input and one output and vice vera the cale efficiencie are equal in DEA and DEA-R. In thi paper, firt, DEA-R odel with weight retriction when there i relative data i recoended. Second, the cale efficiencie in a contant return to cale are copared with each other in DEA and DEA-R odel. At the end, a nuerical exaple i provided for the propoed odel with 27DMU. Keyword: weight retriction, DEA, DEA-R. 1. Introduction For the firt tie, Farrell (1957) uggeted a odel to evaluate the efficiency with one input and one output. After around two decade, Charne et al. (1978) expanded Farrell view and developed a odel to eaure the efficiency with ultiple input and ultiple output, and called it data envelopent analyze (DEA). Efficiency in DEA i defined a the weighted u of input to the weighted u of output. Data envelopent analyze baed on the fractional analyi wa preented by Depic et al. in In DEA- R the efficiency i defined a the weighted u of the ratio of the input to the output. To deterine the relative efficiency of a et, for each DMU a DEA odel i olved. The vector of Correponding author: GeraiJavad@Gail.co 397

2 398 M.Nazari,et al /IJDEA Vol.2, No.2, (2014) optial weight for variou unit i uually different and in thi cae retriction in the ultiple proble hould be applied in a way that both avoid zero weight in the optial olution and how the iportance of weight. Therefore, reearcher in the field of DEA exained the odel of weight retriction by adding the to other odel. In the preent article the DEA-R odel with weight retriction i propoed in the econd ection and a nuerical exaple with 27 DMU i provided in the third ection and finally, the concluion of the article i preented in the lat ection. 2. DEA-R odel with weight retriction Efficiency in DEA i defined a the weighted u of input to the weighted u of output. The price of v i input and u r output are deterined by olving ultiple CCR odel which i of great iportant. In thi context two proble rie: firt, zero input-output weight or poitive weight in front of zero input and output ay lead to either zero nuerator or denoinator in the following proble: r 1 i 1 u r v x i y rk ik Secondly, the ue of to avoid zero weight in the firt cae do not lead to better value of objective function in the ultiple odel. Max r=1 u r y rk. t i=1 v i x ik = 1 (1) r=1 u r y rj i=1 v i x ij 0, j = 1,, n u r ku j 0 v i kv j 0 r = 1,, 1 j = r + 1,, i = 1,, 1 j = i + 1,, It becoe ore coplicated when there are weight retriction for v i and u r weight. Therefore, the ue of fractional analyi and DEA i uggeted for olving the aforeentioned proble. In DEA- R the efficiency i defined a the weighted u of the ratio of the input to the output Overall, the odel propoed by Depic et al. [1] can detect the peudo inefficiency and alo do not ake any proble when uing. Therefore, in thi paper the ue weight control in DEA-R odel i recoended. The weight control odel in input-oriented DEA-R with the auption of contant return to cale i uggeted a follow:

3 M.Nazari,et al /IJDEA Vol.2, No.2, (2014) Max θ y i=1 r=1 ( rj x iv. t w ir x ij y rk ) θ j = 1,, n i=1 r=1 w ir = 1 (2) w ir kw it 0 w ir kw tr 0 w ir 0 i = 1,, r = 1,, 1 t = r + 1,, r = 1,, i = 1,, 1 t = i + 1,, 3. Nuerical Exaple Suppoe 27 DMU with two input and two output. Here, it hould be entioned data i taken fro the paper of Shinn Sun [2]. Conidering data in Table 1 and odel of DEA and DEA-R in weight contraint the reulting cale efficiency of odel 1 and 2 i hown in Table 2. Table1. input and output data Robot No Cot Load Capacity Velocity Repeatability ($10,000) (kg) (/)

4 400 M.Nazari,et al /IJDEA Vol.2, No.2, (2014) Through analyzing Table 2 and coparing the efficiency in odel of CCR, DEA and DEA-R without weight contraint it can be found that DMU of 1, 4, 10, 13, 14, 19, 20, 27 are efficient and cale efficiency i equal to one but for 21 DMU the cale efficiency of DEA i equal to and in DEA- R it i equal to one which how a peudo-inefficiency. Taking into account the weight contraint there i no relationhip in DEA and DEA-R. But adding weight contraint doen t violate the relationhip between the DEA DEA-R. Overall, thi idea here i a there i no relationhip between cale efficiency, it i better not to ue DEA-R odel with weight contraint. But when there i only relative data (the ratio of input to output) odel (2), it i uggeted that the reult reflect the cale efficiency. Table2.coparing the efficiency in odel of CCR, DEA and DEA-R Robot No. Without weight With weight contraint contraint DEA DEA-R DEA DEA-R

5 M.Nazari,et al /IJDEA Vol.2, No.2, (2014) Concluion In DEA when there i a proportion of input and output data and input and output data are not preciely identified, DEA-R i of great iportance. In thi paper, conidering weight contraint a new odel for calculating the cale efficiency i recoended when there i relative data. For future reearch it i recoended to calculate Malquit index in DEA-R and DEA. Reference [1] Depic, O., Depic, M. and Paradi, J.C. (2007) DEA-R: Ratio-baed coparative efficiency odel, it atheatical relation to DEA and it ue in application. Journal of Productivity Analyi. 28: [2] Sun. S, W.-M. Lu, A (2005) cro-efficiency profiling for increaing dicriination in data envelopent analyi, INFOR [3] Wei, C.K., Chen, L.C., Li, R.K. and Tai, C.H. (2011a) Uing the DEA-R odel in the hopital indutry to tudy the peudo-inefficiency proble. Expert Syte with Application. 38: [4] Wei, C.K., Chen, L.C., Li, R.K. and Tai, C.H. (2011b) Exploration of efficiency underetiation of CCR odel: Baed on edical ector with DEA-R odel. Expert Syte with Application. 38: [5] Wei, C.K., Chen, L.C., Li, R.K. and Tai, C.H. (2011c) A tudy of developing an input- oriented ratio-baed coparative efficiency odel. Expert Syte with Application. 38: