A HYBRID DEA/AHP MODEL OF R&D PROJECT SELECTION CONSIDERING ORDINAL EVALUATION FACTORS

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1 A HYBRID DA/AHP MDL F R&D PRJCT SLCTIN CNSIDRING RDINAL VALUATIN FACTRS Deok-Joo Lee College of Advanced ngineeing, Kyunghee Univesity, ldj@khu.ac.k Sungsik Bae Depatment of Industial ngineeing, Seoul National Univesity ABSTRACT The distinguishing featues of e poblem of selecting R&D pojects ae as follows: Fist, quantitative analysis on e efficiencies of R&D pojects is equied to guaantee objective validity fo evaluating e pojects. Fo is eason, e meodology fo selecting R&D pojects should be based on maematical models at pefom quantitative analysis. Second, geneally, ee ae many qualitative factos like Liket-scale in e data fo evaluating R&D pojects. Pevious eseaches, howeve, couldn't suggest explicit meods incopoating ese qualitative factos into models. Thid, fo e R&D poject selection poblems wi limited esouces like budget, it is necessay to decide anking of e all pojects to have flexibility of selecting pojects accoding to budget adjustments. Accodingly, e models at poduce many anks in a tie should be impoved to e moe pecise models. This pape develops a maematical model at can be applicable to e poblems of selecting R&D pojects wi e pevious featues and analyzes e chaacteistics of e model. Fo ese puposes, in is pape, we impove e oiginal DA model fo evaluating efficiency to incopoate odinal factos and suggest e new model which can decide anking of all pojects by meging e impoved DA model and AHP meod. KYWRDS: R&D poject selection, odinal facto, DA, AHP INTRDUCTIN The distinguishing featues of e poblem of selecting R&D pojects ae as follows: Fist, quantitative analysis on e efficiencies of R&D pojects is equied to guaantee objective validity fo evaluating e pojects. Fo is eason, e meodology fo selecting R&D pojects should be based on maematical models at pefom quantitative analysis. Second, geneally,

2 ee ae many qualitative factos like Liket-scale in e data fo evaluating R&D pojects. Pevious eseaches, howeve, couldn't suggest explicit meods incopoating ese qualitative factos into models. Thid, fo e poject selection poblems wi limited esouces like budget, it is necessay to decide anking of e all pojects to have flexibility of selecting pojects accoding to budget adjustments. Accodingly, e models at poduce many anks in a tie should be impoved to e moe pecise models. The eseaches fo e meod to select e most efficient one fom many altenatives have been conducted fo long time. The Data nvelopment Analysis (DA) meod of Chanes, Coope, and Rhodes(978) is one of em. The DA model detemines e efficiency values of Decision Making Units (DMUs) by solving e linea pogamming models at maximize e efficiency of each DMU to find e optimal weights fo e input factos and e output factos of e DMUs. In ode to apply e DA model to e poblems of selecting R&D pojects, we conside each poject as a DMU and detemine e efficiencies of e pojects by e DA. Finally we select e DMU wi e highest efficiency as e optimal altenative. The oiginal DA model, howeve, has e two limitations; Fist, we can't decide e anking of all e altenatives because ee ae geneally many altenatives wi e highest efficiency. Second, we can't include odinal vaiables. Since en, many extended models of e DA fo selecting pojects has been poposed. al, Kettani, and Lang(99) suggested e model which consists of e ee phases at ae selfevaluation, coss-evaluation, and selection model. specially, ey suggested e cossefficiency model which is to evaluate e oe DMUs in e point of a DMU in e phase of coss-evaluation. The data fom e coss-efficiency model is used as e base data fo incopoating analytic hieachy pocess (AHP) model. Thompson, Singleton, Thall, and Smi(986) ealized at e weights fo e input factos and/o e output factos ae excessively close to o 0 and at because of is, e oiginal CCR model poduces many optimal DMUs. To ovecome is poblem, ey poposed e Assuance Region (AR) model which esticts e weight values fo e input and output factos. Cook, Kess, and Seifod(993) paid attention to e limitation at e oiginal DA model can't evaluate qualitative factos and suggested e model at can pefom DA if ee is one odinal input facto. In e pape, ey intepet e odinal facto as weight concept not quantitative point of view. Also, Cook, Kess, and Seifod(996) extended ei pevious model wi only one odinal input facto and pesented e CKS model at incopoates multiple odinal input and/o output factos. The CKS model was suggested wi a meod at detemines e estiction value fo e weights, which was abitay value in e pevious DA

3 models, by using a linea pogamming meod. Finally, e DA model at can include bo quantitative and qualitative factos was pesented. Saaty and Roges(976) poposed e AHP as a multi-citeia decision making meod. In is model, we constuct a hieachy of decision-making pocess, and detemine e impotances between each hieachy, and choose e final altenative. The DA model and e AHP model ae e models fo selecting final altenatives fom many altenatives. So, ecently, eseaches to mege e two models have been conducted. Sakis and Tallui(999) empiically analyzed e poblems of selecting altenative when ee exist bo odinal factos and cadinal factos based on e CKS model. Sinuany-Sten, Mehez, and Hadad(2000) suggested e new model at ovecomes e limitations of bo of e models by incopoating e AHP model into e DA model. This model gets ove e limitation at e oiginal DA model poduces multiple DMUs wi e highest efficiency by using e AHP model and e limitation at ee may be subjectivity of e evaluatos in e AHP model by using e DA model. Lim(2000) pesented anoe hybid DA/AHP model at uses e output of e AHP model as e bounday estictions of e DA-AR model. Howeve, ee ae some difficulties in applying e existing meodologies fo selecting pojects to e poblems of selecting R&D pojects. The distinguishing featues of e poblems of selecting R&D pojects against e poblems of selecting pojects ae as follows; Fist, quantitative analysis on e efficiencies of R&D pojects is equied to guaantee objective validity fo evaluating e pojects. The DA model can handle is. Second, geneally, ee ae many qualitative factos in e data fo evaluating R&D pojects. So, we need e DA model incopoating ese qualitative factos. Thid, fo e poject selection poblems wi limited esouces like budget, it is necessay to decide anking of e all pojects to have flexibility of selecting pojects accoding to budget adjustments. The AHP model can handle is. The pupose of is pape is to develop a moe ealistic maematical model which can be applicable to e poblems of selecting R&D pojects. Fo e pupose, is pape establishes e model at incopoates e CKS model, which can handle odinal factos, into e existing model of Sinuany-Sten et al.(2000). And we mege e AHP model into is model to develop e new model at can decide anking of all e R&D pojects. A simple example follows to compae e esults of e new model wi e esults of e existing models.

4 MDL Suppose at ee ae N R&C pojects to be selected. Assume at ee ae R output factos at consist of R C cadinal output factos and R odinal output factos ( R = RC + R ), and ee ae I input factos at consist of I C cadinal input factos and I odinal input factos ( I = IC + I). Let e RC -dimensional vecto of cadinal outputs fo poject n be denoted Y u n fo n=, 2,..., N, and e I C -dimensional vecto of cadinal uu inputs fo poject n be denoted X n fo n=, 2,..., N. Assume at all e odinal input and output factos have e same numbe of ank positions, L. In egad to e odinal factos, define e L -dimensional unit vectos γ n = ( γ n,..., γnl,..., γnl) and δni = ( δni,..., δnil,..., δnil ) whee γ nl = 0 oewise if poject n is ated in l place on e odinal output δ nil = 0 oewise if poject n is ated in l place on e i odinal input Let µ be e decision vaiable o weight of cadinal output facto fo =, 2,..., RC and ν i be e decision vaiable o weight of cadinal input facto i fo i =, 2,..., I C. Using vecto notation, µ = ( µ, µ 2,..., µ RC ) and ν = ( ν, ν2,..., ν IC ). Let w l be e decision vaiable o weight of e odinal output facto associated wi being ated e l position fo =, 2,..., R ; l =, 2,..., L and u il be e decision vaiable o weight of e i odinal input facto associated wi being ated e l position fo i =, 2,..., I; l =, 2,..., L. Using vecto notation, W = ( w,..., wl,..., wl) and U = ( u,..., u,..., u ). i i il il In e fist stage, DA paiwise compaisons ae pefomed. Fo any pai of DMUs A and B, we pefom e following DA (Poblem AA) uns as if only ese two units exist. Next e coss evaluation of DMU B using e optimal weights of unit A is pefomed using Poblem BA. And symmetically, Poblems BB and AB ae solved. Then we can obtain e values of BB and AB.

5 Poblem AA R u uu = max µ Y A + W γ AA uu u µν,, W, Ui subject to: = I uu uu ν X A + Uiδ Ai = i= A uu R uu µ Y + W γ i A A = uu R uu uu I uu µ Y B + W γ νx U δ 0 B B i Bi = i= µ ε fo =, 2,..., R ν ε fo i =, 2,..., I w w ε fo =, 2,..., R ; l =, 2,..., L l, l+ w ε fo =, 2,..., R L u w ε fo i =, 2,..., I ; l =, 2,..., L il i, l+ w ε fo i =, 2,..., I il C C () Poblem BA uu R uu = max µ Y B + W γ BA u uu u µν,, W, Ui subject to: = uu νx I uu + U δ = i= B B i Bi uu R uu µ Y + W γ i B B = uu R uu uu I uu µ Y A + W γ ( νx U δ ) 0 A AA A i Ai = i= µ ε fo =, 2,..., R ν ε fo i =, 2,..., I w w ε fo =, 2,..., R ; l =, 2,..., L l, l+ w ε fo =, 2,..., R L u w ε fo i =, 2,..., I ; l =, 2,..., L il i, l+ w ε fo i =, 2,..., I il C C (2)

6 Finally, based on ese esults, we constuct e N N paiwise compaison matix A needed fo AHP fom e esults of e paied DA descibed above, so at evey pai of DMUs n and m : a nm = mm + + nm mn, a = fo n=, 2,..., N; m=, 2,..., N (3) Note at in AHP, e elements a nm of e paiwise compaison matix A eflect e evaluation of DMU n ove DMU m fo n=, 2,..., N; m=, 2,..., N. The compaison matix A can be obtained as follows Μ n n Μ n2 2n Λ Λ Ο Λ Μ n n 2n n2 (4) In e second stage, AHP anking of a single hieachical level AHP based on e paiwise compaison matix A, geneated in e fist stage, is un to calculate e maximal eigenvalue λ max and its coesponding eigenvecto α = ( α, α2,..., α N ). The n element of α eflects e elative impotance given to e n DMU fo n=, 2,..., N. We assign e ank to e DMU wi e maximal value of α n, etc., in a deceasing ode of α n. ILLUSTRATIV NUMRICAL XAMPL In e pevious section, we pesented e model at can povide objective guideline fo decision makes to select one fom many altenatives in consideation of efficiencies of e R&D pojects. In is section, we apply e oiginal DA model, CKS model, and e model fom e pevious section to e data which is taken fom Shang and Sueyoshi(995) and alteed into 2 R&D pojects. Table shows e input and e output factos fo e R&D pojects.

7 Table. The input and e output factos fo e R&D pojects (Units : 0 million Won) Input Factos utput Factos Altenatives Investment costpoject temxpected pofitpevasive effect A B C D F G H I J K L Application of DA model Figue shows e esult of e oiginal DA model which assumes e data fom e Table to be not qualitative but quantitative. Figue. The esult fom e oiginal DA model

8 Howeve, e analysis above has e eo of consideing e qualitative facto 'pevasive effect' as a quantitative facto. In oe wods, fo a qualitative facto, '3' does not mean e.5 times as big as '2'. The values of a qualitative facto mean just pefeence. Consequently, it is impossible to apply e oiginal DA model to e data fom Table. Application of CKS model In e pevious subsection, it was pointed out at e analysis which ignoes e chaacteistics of odinal factos can lead decision makes to make big mistakes in selecting altenatives. Table 2 shows e esult of e CKS model which can incopoate odinal data into DA model. Howeve, is esult inheits e same poblem fom e oiginal DA model, which is at ee ae many DMUs wi e highest efficiency. In spite of e steng of incopoating odinal data, e CKS model is not e desiable model fo decision makes to select altenatives. Table 2. The esult fom e CKS model Altenative CKS fficiency A B C D F G H I J K L Application of DA/AHP model consideing odinal factos In e pevious subsections, we showed at e oiginal DA model and CKS model can not help decision makes to select altenatives wi odinal factos. In is subsection, we use e model pesented in is pape.

9 To get e lowe bounday ε fo each facto, we solve e optimization poblem AA and BA. As a esult, we can get e value of ε = Using is esult, we detemine e values of AA, BA, BB, AB by paiwise compaison fo each pai of DMUs. Using e q. (4), we can get e compaison matix A. Table 3. Compaison Matix To pefom single level AHP to ank altenatives, we nomalize e compaison matix fo each column and compute e aveages, which is e ank of each altenative, fo each ow of e esult matix. Finally e esult of DA/AHP model and e anking of altenatives can be obtained. Table 4. The esult fom e DA/AHP model AltenativesNomalized valueranking A B C D F G H I J K L

10 Test of e esults AHP can pefom a consistency test fo e esult afte analyzing altenatives, so at it can epesent e objectivity of e esult as a numeical value. Decision makes can select altenatives on e basis of is value. Saaty who developed AHP model fo e fist time, suggested at if e esult value of e consistency test wee less an 0%, it is a eliable esult fo e anking, if less an 20%, it is a acceptable esult, but, if geate an 20%, ee is no consistency and e initial data set should be fixed. He also ecognized at it is had to get e eigenvalues duing e consistency test and pesented e meod at can be used to pefom e consistency test easily by using geometic means. In is pape, to pefom e consistency test, we used e meod of using geometic means, pesented by Saaty. The esult value of e consistency test is 0.063% and is much less an e uppe bounday of 0% suggested by Saaty, so at we can ely on e esult of anking and select e altenative "I" as e best R&D pojects. CNCLUSIN This pape pesented a new model at combines bo CKS model and AHP model, as a meodology fo selecting R&D pojects, and incopoates odinal factos which was ignoed by e pevious CCR model. We got a compaison matix by paiwise compaisons of R&D pojects and used e compaison matix as e input data fo single level evaluation of AHP model, which gave us full anking of e R&D pojects. By pefoming consistency test, we estimated e confidence level of e esult of is pape. The special featues of e model pesented in is pape ae as follows; Fist, e model incopoates odinal factos. The pevious R&D poject selection models which use e oiginal DA model have e limitation at ey conside only quantitative factos. But, is model is based on e CKS model at incopoates odinal factos, so at odinal factos can be evaluated wi quantitative factos unde e same condition. Second, e model fully anks all e R&D pojects by using AHP model. It is impotant to select R&D pojects popely to suvive in e global competition. The full anking of e R&D pojects can be helpful to decision makes. The esult of e consistency test of e model satisfied e eliable condition of being less an 0%, suggested by Saaty. If e esult value of e consistency test wee moe an 0%, we should eset e initial data. Howeve, ee ae no geneal meods how to eset e initial data. Moe eseaches on is issue should be pefomed.

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