DEA/AHP and its application in Full Ranking of Decision Making Units

Transcription

1 Volue 1, Iue 4, Septeber 2014 DEA/AHP and it application in Full Ranking of Deciion Making Unit Mohaad Ehanifar, PhD Departent of Indutrial Engineering Ilaic Azad Univerity, Arak, Iran Abtract The cro efficiency evaluation ha long been uggeted a an alternative ethod for ranking Deciion Making Unit (DMU) in Data Envelopent Analyi (DEA). The paper ai at ranking Deciion Making Unit (DMU) with a odel cobining Data Envelopent Analyi (DEA) and Analytical Hierarchy Proce (AHP). At firt, it olve a DEA odel for each of deciion aking unit, then the DEA odel reult are preented in uch a way that the pair-wie coparion atrix i deterined and full ranking AHP odel olution are achieved. The clai would be upported with exaple. Keyword: Analytical Hierarchy Proce (AHP); Data Envelopent analyi (DEA); odified cro efficiency evaluation. 1. INTRODUCTION After introducing AHP ethod by Thoa L. Saaty [1, 2], finding relative weight and overall weight have alway been of interet to different reearcher. When pair wie coparion atrix i conitent, weight can be found eaily. But when the atrix i inconitent, the olution would be ore coplicated. Different ethod have been propoed for finding weight. Recently, different DEA ethod have helped reearcher to find weight. One of the i DEAHP uggeted by Raanthan [3]. Thi ethod reult in unreaonable weight factor due to not including all the available data. The ain reaon for thi proble hould be earched in not conidering the relation: n 3( + ) Where n, and are nuber of DMU, nuber of input and nuber of output, repectively. 49

2 Volue 1, Iue 4, Septeber 2014 Data Envelopent Analyi (DEA) i a atheatical prograing technique that eaure the relative efficiency of Deciion Making Unit (DMU) with ultiple input and output. Charne et al [4, 5]. Firt propoed DEA a an evaluation tool to eaure and copare the relative efficiency of DMU. Their odel aued Contant Return to Scale (CRS, the CCR odel), the odel with Variable Return to Scale (VRS, the BCC odel) wa developed by Banker et al [6]. The cro evaluation ethod wa developed a a DEA extenion that can be utilized to identify bet perforing DMU and to rank DMU uing cro efficiency core that are liked to all DMU (Sexton et.al [7]). The ain idea of cro evaluation i to ue DEA in a peer evaluation intead of a elf evaluation ode. There are two principal advantage of cro evaluation: (1) It provide a unique ordering of the DMU, and (2) It eliinate unrealitic weight retriction fro application area expert (Anderon et al [8]). A range of odel have been developed for ranking deciion aking unit including claic and fuzzy ulti-criteria deciion aking odel and data envelopent analyi technique. AHP ethod i one of the ranking technique with oe application in weighting deciion criteria. It deigning ai wa ubjective etiation of a et of alternative baed on variou indexe or a hierarchical tructure. At the top level are deciion aking objective and etiation indexe and at the lower level oe option are in place which their etiation are baed on an individual index and ultiately, divere option are cobined with repect to ultiple indexe and give rie to final olution. One of the iue in relation to the AHP ethod application which create oe concern aong deciion aker i ubjective judgent in pair-wie coparion atrix. A a reult, thi paper trie to fill the gap uing the DEA ethod [9]. In thi way, pair-wie coparion are not undertaken by ubjective judgent rather, they are provided by powerful DEA technique and o the previou proble are eliinated. In effort to preent the above entioned odel, it algorith ipleentation tep are explained. The tructure of the paper i a follow: ection 2 decribe the cro efficiency evaluation and Ranking with AHP ethod to illutrate nuerical exaple i entioned in ection 3. The lat ection uarize and conclude. 2. CROSS EFFICIENCY EVALUATION AND RANKING WITH AHP METHOD Firt phae: pair-wie coparion atrix i fored uing the DEA ethod a follow: (a)the CCA Claic odel i ipleented for each of Deciion Making Unit a {1,2,,n }(odel no.1): Auing that there are DMU each with input and output, the relative efficiency k1,2,..., n of a particular DMUk i obtained by olving the following fractional prograing proble: 50

3 Volue 1, Iue 4, Septeber 2014 ax kk ubject to : rk rj r1 ik ij i1 rk rk r1 ik ik i1 1 j 1,2,..., n (1) u 0 r 1,2,..., rk v 0 i 1,2,..., ik u y v x u y v x Where i the DMU index j 1,2,..., n the output index, r 1,2,..., and i the input index i 1, 2,...,, xij the value of the th output for the th DMU, the value of the input for the u th DMU, rk v the weight given to the th output, ik the weight given to the i input. DMUk i 1. efficient if and only if kk DMUk elect weight that axiize it output to input ratio, ubject to the contraint. A relative efficiency core of 1 indicate that the DMU under conideration i efficient, wherea a core le than 1 iply that it i inefficient. Thi fractional progra can be converted into a linear prograing proble where the optial value of the objective function indicate the relative efficiency of DMUk. The reforulated linear prograing proble, alo known a the Linear CCR odel, i a follow (for ore detail about odified of cro efficiency ethod ee Yanig Ming and et.al [10]): * kk kk axurk yrk r1 ubject to : ik ik i1 1 u y v x 0 j 1,2,..., n (2) rk rk ik ik r1 i1 u 0 r 1,2,..., rk v 0 i 1,2,..., ik v x The odified cro efficiency evaluation: 51

4 Volue 1, Iue 4, Septeber 2014 kh rh rh r1 ubject to : ax u y h 1,2,..., n ih ih i1 1 u y v x 0 j 1,2,..., n (3) rh rj ih ij r1 i1 * urh yrk kk vihxiik r1 i1 0 u 0 r 1,2,..., rh v 0 i 1,2,..., ih v x With the ue of above entioned odel reult and following relation, a pair-wie a coparion atrix a well a it kh eleent i obtained: kk kh 1 akh k, h 1, 2,..., n (4) ahk (5) hh hk akh The point here i that in the AHP ethod, value 1 i included in diaeter of pair-wie coparion a atrix, and eleent kh a 1 indicate the etiation of unit j with repect to k. The relation kh indicate lower etiate of unit j with repect to k. The pair-wie coparion atrix i copleted and preented for every two unit a it wa aid for unit j and k. Second phae: Ranking with AHP ethod The AHP ethod wa propoed by a reearcher with Iraqi origin naed a Toa Al- Saaty during Like huan brain echani, the ethod i able to analyze variou event. The AHP ethod enable deciion aker to deterine concurrent and reciprocal effect of any unknown and coplex ituation. Thi proce help deciion aker to adjut oe prioritie baed on their target, knowledge and experience uch that they will be able to conider all of their eotion and judgent iultaneouly [11]. The application of thi ethod depend on four ain tep: Step 1. Modeling: In thi tep, the proble and deciion aking objective are delineated hierarchically baed on related deciion eleent. Deciion eleent include" Deciion Making Indexe" and "Deciion Option". Step 2. Preferred judgent: Multiple deciion option are copared baed on individual indexe and the iportance of deciion index i deterined uing pair-wie coparion. Step 3. Calculating relative weight: Deciion eleent' iportance and weight are delineated in relation to each other through a et of nuerical calculation. Step 4. Cobining relative weight: The tep devote to ranking deciion option. The above ethod algorith i a follow: a. Calculate the u of each colun nuber. 52

5 Volue 1, Iue 4, Septeber 2014 b. Divide each eleent on the colun u and obtain a new atrix which i naed a noralized atrix. c. Calculate the ean of eleent in each row of noralized atrix. Thi ean indicate the ranking weight of each deciion aking unit. 3. Exaple Exaple: Conider Table (1),, (5), in thi Table, we have twenty with two Input and four output. Data have been taken fro Iranian Car Market... You can ee the reult of perforance new ethod for ranking in thi nuerical exaple. ( Yr are output and Xi are input) TABLE 1. TWENTY DMUS (AUTOMOBILES) WITH TWO INPUTS AND FOUR OUTPUTS Input Output DMU X 1= Price ( Toan Currency of Iran) X 2= Coplex Oil Conuption Y 1= Speed Y 2=POWER-TO- WEIGHT RATIO Y 3= Motor Capacity Y 4= Motor Power Mercede Benz C180 KOMPERESSOR Mercede Benz CLK280 Coupe/Convertible Mercede Benz CLK350 Coupe/Convertible Mercede Benz SLK Mercede Benz SLK55 AMG Mercede Benz SL Mercede Benz SL Mercede Benz CLS 350 CGI Mercede Benz CLS Mercede Benz GLK MATIC Mercede Benz GLK MATIC Mercede Benz ML MATIC Bra bu BMW 118i BMW 120i BMW 120i Convertible BMW 125i Coupe

6 Volue 1, Iue 4, Septeber 2014 BMW 125i Convertible BMW 135i Coupe BMW 135i Convertible BMW 523i TABLE 2. MODIFIED CROSS EFFICIENCY EVALUATION 54

7 Volue 1, Iue 4, Septeber 2014 TABLE 3. PAIR-WISE COMPARISONS MATRIX OF DMUS USING DEA 55

8 Volue 1, Iue 4, Septeber 2014 TABLE 4. NORMALIZED MATRIX 56

9 Volue 1, Iue 4, Septeber 2014 Ultiately, DMU ranking and weight obtained through the AHP ethod are preented a follow: TABLE 5. MODELS OUTGOING (CARS RANKING) Rank Score DMU Mercede Benz SL500 Mercede Benz SLK55 AMG Mercede Benz CLS 500 Mercede Benz SL350 BMW 118i Mercede Benz CLS 350 CGI Mercede Benz C180 KOMPERESSOR BMW 120i Convertible Mercede Benz SLK350 BMW 125i Coupe Mercede Benz CLK350 Coupe/Convertible Mercede Benz ML MATIC Bra bu Mercede Benz GLK MATIC 57

10 Volue 1, Iue 4, Septeber BMW 523i Mercede Benz GLK MATIC Mercede Benz CLK280 Coupe/Convertible BMW 135i Coupe BMW 125i Convertible BMW 120i BMW 135i Convertible 3. CONCLUSIONS In the propoed ethod unoberved DMU are introduced. By introducing thee DMu, the relation hold, in which i the nuber of DMU, i the nuber of input and i the nuber of output. The reult obtained uing thi ethod how the reality of the weight which generated. The odel ugget in thi paper i ued to rank and evaluate of DMU. The reult ee to be logical and have econoic and anagerial interpretation becaue we ue cro efficiency evaluation. 4. REFERENCE [1] Saaty, T.L., the Analytic Hierarchy Proce, McGraw-Hill Copany, New York, (1980). [2] Saaty, T.L., The Analytic Hierarchy Proce, planning, priority, Reource Allocation RWS Publication USA, (1980). [3] R. Raanathan, Data envelopent analyi for weight derivation and aggregation in the analytical hierarchy proce, Coputer Operation Reearch 33(7) (2006) [4] Charne, A., Cooper, W. W., Rhode, E. (1978).Meauring the efficiency of deciion aking unit. European Journal of Operational Reearch, 2, [5] Charne, A., Cooper, W.,Lewin,A. Seiford, L. M., 1995.Data Envelopent Analyi: Theory, Methodology, and Application.Kluwer Acadeic Publiher. [6] Banker, R. D., Charne, A., Cooper, W. W., Soe odel for etiating technical and cale efficiency in data envelopent analyi. Manageent Science 31, [7] Sexton, T. R., Silkan, R. H., Hogan, A. J. (1986). Data envelopent analyi: Critique and extenion. In R. H. Silkan (Ed.), Meauring efficiency: An aeent of data envelopent 58

11 Volue 1, Iue 4, Septeber 2014 analyi. San Francico, CA: Joey-Ba. Shang, J., Sueyohi, T. (1995). A unified fraework for the election of flexible anufacturing yte. European Journal of Operational Reearch, 85, [8] Anderon, T. R., Hollingworth, K., Inan, L. (2002). The fixed weighting nature of a croevaluation odel. Journal of Productivity Analyi, 17, [9] Belton, V., and T.Gear, 1983, In a hort-coing of Saaty' ethod of Analytic Hierarchie, Oega, pp [10] Ying Ming Wang, Kwai-Sang Chin,Gary Ka Kwai Poon, A data envelopent analyi ethod with aurance region for weight generation in the analytical hierarchy proce, Deciion Support Syte 45 (2008) [11] Sinuauny-Stern, Zilla., Mehrez, Abraha., Hadad, Yoi., 2000.An AHP/DEA Methodology for Ranking Deciion Making Unit. 59