Resource Allocation through Context-dependent data envelopment analysis

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1 Avaiabe oie at It. J. Data Eveopmet Aaysis (ISSN X) Vo.2, No.3, Year 2014 Artice ID IJDEA-00232,7 pages Research Artice Iteratioa Joura of Data Eveopmet Aaysis Sciece ad Research Brach (IAU) Resource Aocatio through Cotext-depedet data eveopmet aaysis N. Ebrahimkhai Ghazi a*, M. Ahadzadeh Nami b, (a) Departmet of Mathematics, Sciece ad Research Brach, Isamic Azad Uiversity, ehra, Ira. (b) Departmet of Mathematics, Shahr-e Qods Brach, Isamic Azad Uiversity, ehra, Ira. Abstract Received 8 March 2014, Revised 15 Jue 2014, Accepted 14 August 2014 System desigs, optimizig resource aocatio to orgaizatio uits, is sti beig cosidered as a compicated probem especiay whe there are mutipe iputs ad outputs reated to a uit. he agorithm preseted here wi divide the frotiers obtaied with DEA. I this way, we ivestigate a ew approach for resource aocatio. Keywords: DEA, cotext-depedet, budget aocatio, cost-efficiecy, efficiecy. 1. Itroductio Our iterature review idicates that to this day i the evauatio of decisio makig uits (DMUs) a DEA modes have bee used: may modes are give oy the amouts of iputs ad outputs, each DMU correspodig to the cosidered iputs ad outputs seects its best set of weights; the vaues of weights vary from oe DMU to aother. Each DMU s performace score is cacuated by the DEA modes which rage betwee zero ad oe that provides its reative degree of efficiecy. Moreover, the sources ad amouts of iefficiecy i each iput ad output for every DMU are aso idetified through the modes. he subect of debate here is the possibiity of further expoitig of avaiabe data to deveop DEA-type modes which hep DMUs desig their optima systems ad ot ust evauate their existig systems. * Correspodig author: asriebrahimi2012@gmai.com 423

2 424 N. Ebrahimkhai Ghazi,et a /IJDEA Vo.2, No.3, (2014) I fact, the purpose of this paper is to show the kid of budget aocatios to DMUs as a resut of which we have maximum reveue ad miimum cost. Budget aocatio is the distributio ad divisio of services ad faciities amog peope ad existig programs. I this process, the aocatio is doe through Cotext-depedet data eveopmet aaysis [4]. I each eve we cacuate respective ad fid the frotier which if budget B is aocated to DMUs i this eve, maximum reveue ad miimum cost wi be resuted. X Y 1. Discussio ad summery 1.1. Determiig eves Assume that DMU (=1 ) produces the outputs Y (y 1,..., y s) by cosumig X (x 1,..., x m) the iputs [3]. Ad aso suppose that J {DMU ; 1,...,} is a DMUs set. We defie J 1 J E that * E {DMU ;J 1 (, k) 1} ad * (,k) is the optima soutio vaue for the foowig iear programmig probem: * (, k) Max (, k) (1), (, k) s.t. x x F(J ) F(J ) k y (,k)y ; k 0, F(J ). Where (x k, y k) preset iput ad output vector with respect to DMU k, respectivey. DMU set. F(J ) shows that J, F(.) proposed a correspodece betwee a set of DMUs ad the aaogue commo idices Mode (1) is a output orieted CCR [5], where =1 ad defies DMUs which are i the first eve efficiecy frotier. DMUs which are i E set defie th -efficiet eve. Whe =2 the mode (1), after removig DMUs i the first eve, gives us DMUs i the secod efficiet eve. I this method, mutipe efficiet eves are recogized. hese efficiet eves wi be obtaied by the foowig agorithm: Step1: set =1, assess a the DMUs set, J, through the mode (1) DMUs which are efficiet i the first eve ca be reached.

3 N.Ebrahimkhai Ghazi,et a /IJDEA Vo.2, No.3, (2014) Step2: eave out DMUs that are cosidered i step1. 1 J J E (Stop whe J 1 ) Step3: Cosider a ew subset of iefficiet DMUs through the mode (1) we gai a ew set of efficiet DMUs 1 E (he ew efficiet frotier.) Step4: et =+1. Go to step2. Stoppig rue: 1 J, the agorithm stops Measurig profit efficiecy Now we cosider profit efficiecy mode that heps to optimize DMUs system desig. et s suppose DMU (x,y) preset bechmark DMU that with iputs X (x 1, x 2,..., x m correspodig with cost C (c 1,c 2,...,c m ad output vector Y (y 1, y 2,..., y s correspodig with prices P (p 1,p 2,...,p s. Note that both iput vector x ad output vector y are ow variabes. he cacuatio of maximum attaiabe profit, withi the DEA framework, is the startig poit of a profit aaysis which ca be doe by usig the mode show i [4]: Max s m Pr yr Cixi r1 i1 s.t. xi x i,i 1,..., m; 1 yr y r, r 1,...,s 1 xi x io,i 1,..., m; 1 yr y ro, r 1,...,s 1 1; 0, 1,..., 1 (2) Where P r ad w i are, respectivey, the price of r th output ad cost of it h iput DMU, ad the rest of the otatio is as previousy defied. Mode (2) assures us to reach maximizatio profit vaue.

4 424 N. Ebrahimkhai Ghazi,et a /IJDEA Vo.2, No.3, (2014) herefore, for differet weights ( P r, w i ), differet price ad cost is possibe. So Profit efficiecy of DMU o is: s m Pr yro Cixio PE r1 i1 s m * * Pr yr Cixi r1 i1 ; 2. Budget Aocatio through Cotext-depedet DEA Oe of the mai activities of maagemet is makig a strategy that is possibe to depoy. I orgaizatios that the strategic maagemet has ot bee impemeted resource aocatio is based o poitics ad persoa factors. I strategic orieted orgaizatios resource is aocated o the basis of the prefereces which are determied through the aua purposes. Budget provides the possibiity of aocatig the imited resources based o paig prefereces. Oe of the big barriers is the usuccessfu coectio betwee admiistrative programs ad specifyig priorities regardig aocatig budget to edurig guideie programs. he mode metioed above, impicity coud make the iformatio of budget resource portfoio avaiabe for DMUs, such as a optimum budget ad budget cogestio. I this sectio the purpose is to itroduce a agorithm which heps us to fairy aocate the tota avaiabe budget B amog DMUs. For compariso, the aocatio is doe by Cotext-depedet DEA method. Supposig that the tota budget B is avaiabe we foow this agorithm: Step1: Fid the efficiet eve as metioed i sectio 2.1. et these efficiet eve s ames 1 2 E, E,..., E. Step2: I each eve, disticty cacuate (x,y) i mode (2). (For more iformatio see sectio 2.2 ad 1 2 Fig 3.1) Suppose the profit vaue for each eve; E, E,..., E, respectivey, is 1 2,,...,. Step3: For fidig the percetage aocatio of budget B i each efficiet eve this idices wi be used: B 1 ; 1,..., (3), Which B1 B2 B B(... ) B

5 N.Ebrahimkhai Ghazi,et a /IJDEA Vo.2, No.3, (2014) is the aocatio rate of costat budget B to DMUs i efficiet eve E as if costat budget B is fairy aocated to a DMUs. Y abe 1.Iputs ad outputs. Fig. 3.1.eves ad reveue efficiecy vaue X Iputs Payabe iterest Persoe No-performig oas Outputs he tota sum of four mai deposits Other deposits oas grated Received iterest Fee abe 2.Iput-data for the 20 bak braches. abe 3.Output-data for the 20 bak braches. DMU x 1 x 2 x 3 DMU y 1 y 2 y 3 y 4 y

6 424 N. Ebrahimkhai Ghazi,et a /IJDEA Vo.2, No.3, (2014) Appicatio Now we wi cosider the braches of oe of the Ira s commercia baks with 3 iputs ad 5 outputs (see abe2 ad abe3) as our DMUs. he iputs are payabe iterest, persoe ad o-performig oas ad the outputs are the tota sum of four mai deposits, other deposits, oas grated, received iterest ad fee (see tabe1). hese data were coected i First based o the agorithm i sectio 2.1 we fid three eves: eve1 = {1,4,6,7,8,9,10,11,15,17,19} eve2 = {2, 3, 5, 14, 16, 18, 20} eve3 = {12, 13} Cosider the data give i abe 1 ad abe 2. et B = 1,000,000, C = (4, 2, 5) ad P = (6, 7, 5, 4, 8). he uified mode (2) is a iear program ad thus ca be soved by ay P agorithm. So we have: Profit efficiecy i first frotier is: 29,075, Profit efficiecy i secod frotier is: 2,334, Profit efficiecy i third frotier is: 5,888, As show above the first frotier has the maximum reveue ad miimum cost vaue but ogicay it is ot a good way to aocate a the budget to the first eve because DM wats to have fair udgmet i the society. Not oy the mai aim of DM is havig fair distributio of budget so that a the DMUs have a sufficiet resource but aso esures that the maximum profit wi satisfy DM. We cotribute 3 eves i aocatio i this way: 1, 000, 00029, 075, , 000, , 075, , , 075, , 334, , 888, , 299, , 000, 000 2,334, , 000, 000 2,334, , , 075, ,334, , 888, , 299, , 000, 0005, 888, , 000, 000 5, 888, , , 075, , 334, , 888, , 299, It meas that the best way to aocate budget i this exampe is givig 779,520 of budget B to the first eve ad 62,600 of budget B to the secod eve ad 157,880 of budget B to the secod eve.

7 N.Ebrahimkhai Ghazi,et a /IJDEA Vo.2, No.3, (2014) Cocusio We ca use the above approach, wherever there is a huma eed, to aocate a imited budget to a umber of teams or some of the activities so that the case which is chose from a variety of combiatios of aocatios wi have the most output vaue. Appyig the optimizatio modes aows a the various aocatio cases to be cosidered ad the most optimized of them which is based o obective fuctio to be seected. he preseted agorithm wi eabe us to aocate the imited budget to a DMUs. his was propery examied i a case study i a Iraia bak. Athough obviousy the first frotier has the argest proportio of budget, the rest of eves are fairy aocated. Refereces [1] Aderse, P., Peterse, N.C., (1993). A procedure for rakig efficiet uits i data eveopmet aaysis. Maagemet Sciece 39, [2] Chares A, Cooper WW, Rhodes E, (1978) Measurig the efficiecy of decisio makig uits. Europea Joura of Operatioa Research; 2: [3] Cooper, W. W., Seiford,. M., & oe, K. (2007). Data eveopmet aaysis: A comprehesive text with modes, appicatio, refereces ad DEA-sover software. [4] Fare, R., S. Grosskopf ad C.A. K. ove (1994), Productio Frotiers, Cambridge: Cambridge Uiversity Press. [HB241.F336] [5] Farre MJ, (1957). he measuremet of productive efficiecy. Joura of Roya Statistica Society; 120 (3): [6] Quaig Wei., sug-sheg Chag.,(2011). Optima profit-maximizig system desig data eveopmet aaysis modes. Computer & Idustria Egieerig,

Utilizing recent developments in data envelopment analysis (DEA), this paper examines

Utilizing recent developments in data envelopment analysis (DEA), this paper examines Profitability ad Marketability of the Top 55 U.S. Commercial Baks Lawrece M. Seiford Joe Zhu Departmet of Mechaical & Idustrial Egieerig, Uiversity of Massachusetts at Amherst, Amherst, Massachusetts 01003