Research Article Correlated Analytic Hierarchy Process

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1 Mathematical Problems i Egieerig, Article ID , 7 pages Research Article Correlated Aalytic Hierarchy Process Hsiag-Hsi Liu, 1 Yeog-Yuh Yeh, 2 adjih-jeghuag 3 1 Graduate Istitute of Iteratioal Busiess, Natioal Taipei Uiversity, Saxia, Taiwa 2 Departmet of Busiess Maagemet, Aletheia Uiversity, New Taipei City, Taiwa 3 Departmet of Computer Sciece & Iformatio Maagemet, Soochow Uiversity, Taipei, Taiwa Correspodece should be addressed to Jih-Jeg Huag; jjhuag@scuedutw Received 8 May 2014; Accepted 29 July 2014; Published 18 August 2014 Academic Editor: Tofigh Allahviraloo Copyright 2014 Hsiag-Hsi Liu et al This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited The aalytic hierarchy process (AHP) has bee the most popular tool for the field of decisio makig i the past 30 years, because of its simplicity ad ratioality Costruct a hierarchy system for evaluatio by decisio makers Hece, oly the effect of outerdepedece ca be cosidered i the AHP However, besides outer-depedece, correlatio is aother commo effect betwee criteria which caot be accouted for either by the AHP or by the aalytic etwork process (ANP) Hece, i this paper, we exted the AHP to cosider the correlatio effect I additio, a biobjective programmig model is proposed to derive the result Furthermore, the traditioal AHP ca be cosidered as the special case of the proposed model whe the correlatio effect betwee criteria is igored Fially, a umerical example is give to justify the proposed method ad compare the result with the AHP 1 Itroductio Sice Beroulli [1] proposed the cocept of utility fuctio to reflect huma persuadig, such as maximum satisfactory, ad vo Neuma ad Morgester [2]preseted the theory of game ad ecoomic behavior model, which expaded the studies o huma beig ecoomic behavior for multiple attribute decisio makig (MADM) problems, from that momet o, more ad more literatures egaged i this field Roughly speakig, the procedures of MADM ca be comprised of five mai steps as follows [3] Step 1 Defie the ature of problem Step 2 Costruct a hierarchy system for evaluatio (as show i Figure 1) Step 3 Select the appropriate evaluatig model Step 4 Obtaitherelativeweightsadperformacescoreof each attribute with respect to each alterative Step 5 Determie the best alterative accordig to the sythetic utility values, which are the aggregatio value of relative weights ad performace scores correspodig to alteratives O the basis of dealig with MADM problems, the aalytic hierarchy process (AHP) was proposed by Saaty [4, 5] to derive the relative weights accordig to the appropriate hierarchical system The AHP should be the most popular method used i dealig with multicriteria decisio makig (MCDM) problems i the field of operatios research/maagemet sciece (OR/MS), accordig to the published joural papers To exted the capability of the AHP for dealig with more sophisticated problems, may methods have bee proposed to revise or modify the limitatios ad defects of the AHP For example, Yu [6] proposed a revised model to release the assumptio of reciprocal matrix i the AHP ad Huag [7] exteded the AHP to cosider the fuzzy eviromets Although the past papers elaborated much work o the extesio of the AHP, they seem to igore the issue of correlatio betwee criteria As we kow, the AHP is used to derive the weights of criteria whe they are idepedet ad outerdepedet by the upper-level criteria However, there are other effects, such as ier-depedece ad feedback, which should be cosidered to preset the relatioship betwee

2 2 Mathematical Problems i Egieerig Goal Overall objective 1st level Goal Aspect Dimesio 1 Dimesio j Dimesio k Criteria C 11 C 1r C j1 C js C k1 C kt 2d level Criterio 1 Criterio 2 Criterio Alteratives A 1 A i A Figure 1: Hierarchical system for MADM 3rd level Sub-C 11 Sub-C 1m Sub-C 21 Sub-C 2m Sub-C 1 Sub-C m criteria Hece, Saaty [8] proposed the aalytic etwork process to accout for the effects of outer-depedece, ierdepedece, ad feedback Hereafter, the AHP/ANP is the most commo tool i the field of decisio makig to deal with various effects betwee criteria The correlatio betwee criteria is commoly observed i the realistic problems Let us cosider the followig ivestmet situatio to describe the isufficiecy of the AHP/ANP as follows Assume a ivestor is cosiderig costructig a portfolio i a stock market Two major criteria, that is, risk ad retur, are cosidered ad ca be divided ito may idepedet subcriteria, respectively The problem above ca be solved by the AHP, because of the idepedet subcriteria However, it is irratioal to assume that risk ad retur are idepedet of each other i the real world I additio, there is o evidece that risk ca affect retur ad vice visa All we ca say is that risk ad retur are correlated with each other Hece, the AHP/ANP caot be used i this situatio The purpose of this paper is to propose the correlated aalytic hierarchy process (CAHP) which ca accout for the correlatio betwee criteria i the AHP A biobjective programmig model is proposed to derive the result I additio, we propose a umerical example to demostrate the proposed model ad compare the result with the traditioal AHP It should be highlighted that the traditio AHP ca be cosidered as the special case whe we reduce our model to sigle-objective programmig model The rest of this paper is orgaized as follows I Sectio 2, we preset the procedure of the AHP ad ucover the limitatio o the issue of correlatio betwee criteria The way to develop the CAHP is give i Sectio 3Aapplicatioused here to demostrate the proposed method is i Sectio 4 Discussios are preseted i Sectio 5 ad coclusios are i the last sectio 2 Aalytic Hierarchy Process Aalytic hierarchy process was proposed by Saaty [4, 5] to model subjective decisio makig processes based o multiple attributes i a hierarchical system From that momet o, it has bee widely used i corporate plaig, portfolio selectio, ad beefit/cost aalysis by govermet agecies for resource allocatio purposes It should be highlighted Figure 2: The hierarchical structure of the AHP that all decisio problems are cosidered as a hierarchical structureitheahpthefirstlevelidicatesthegoalfor the specific decisio problem I the secod level, the goal is decomposed of several criteria ad the lower levels ca follow this pricipal to divide ito other subcriteria Therefore, the geeral form of AHP ca be depicted as sho Figure 2 The mai four steps of the AHP ca be summarized as follows Step 1 Set up the hierarchical system by decomposig the problem ito a hierarchy of iterrelated elemets/criteria Step 2 Compare the comparative weight betwee the attributesofthedecisioelemetstoformthereciprocal matrix Step 3 Sythesize the idividual subjective judgmet ad estimate the relative weight Step 4 Aggregate the relative weights of the elemets to determie the best alteratives/strategies I this sectio, the method of the AHP is firstly reviewed as follows If we wish to compare a set of attributes pairwise accordig to their relative weights (importace), where the weights are deoted by,w 2,,w, the the matrix of weight ratios ca be represeted as W =[j ], (1) where j =w 1 ji, j =k w kj,adj = / Multiplyig W by the weight vector, w, yields Ww = [ [ w w w = =w [ ] [ ] ] [ w ] [ w ] ] (2) w w w

3 Mathematical Problems i Egieerig 3 or (W I) w = 0 (3) Next, i order to estimate the weight ratio j by a ij,where A = [a ij ], we ca calculate the approximate weights by fidig the eigevector w with respect to λ max which satisfies Aw =λ max w, (4) where λ max is the largest eigevalue of the matrix A I additio, sice A is a approximate for W, we should calculate the cosistecy idexes (CI) to check if the cosistecy coditio is almost satisfied for A usig the followig equatio: CI = λ max 1, (5) where λ max is the largest eigevalue ad deotes the umbers of the attributes Saaty [5] suggested that the value of the CI should ot exceed 01 for a cofidet result O the other had, for the AHP, a ear cosistet matrix A with a small reciprocal multiplicative perturbatio of a cosistet matrix is give by [9] A = W E, (6) where deotes the Hadamard product, W =[j ] is the matrix of weight ratios, ad E [ε ij ] is the perturbatio matrix, where ε ij =ε 1 ji From (4)ad(6),it ca be see that λ max = a ij λ max =0, (7) a ij = ε ij (8) O the other had, the multiplicative perturbatio ca be trasformed to a additive perturbatio of a cosistet matrix such that ε j = + ] j j, (9) j where ] ij is the additive perturbatio Sice a ij / = ε ij,wecarewrite(8)as ( a ij )= ] ij = ( ε ij )= (a ij ) ( + ] ij ), (10) O the basis of (8) (10),itcabeseethatλ max =if ad oly if all ε ij =1or ] ij =0, which is equivalet to havig all a ij = /, idicates the cosistet situatio Therefore, the problem of derivig the relative weights amog criteria i the AHP is equivalet to solvig the followig mathematical programmig problem to obtai : mi st a ij p i=1 =1, 1 i<j, (11) where p deotes the p-orm ad p {1,2,}Notethat, i this paper, we set p=2i our model As metioed previously, although the AHP is widely used i the field of decisio makig, it caot deal with the situatio of correlatio betwee criteria Hece, we propose the extesio of the AHP by cosiderig the correlatio betwee criteria i the followig sectio 3 Correlated Aalytic Hierarchy Process I this sectio, we will propose a way to exted the AHP to cosider the correlatio betwee criteria I order to avoid the cofusio betwee the proposed model ad AHP/ANP, we first itroduce the followig four effects which could happe betwee criteria as follows (1) Outer-depedece ( ): a criterio ifluecig aother criterio is called outer-depedece (2) Ier-depedece loop ( ): a criterio has a ierdepedece loop if its elemets are to deped o each other (3) Feedback ( ): two criteria depedig o each other are called feedback (4) Correlatio ( ): a criterio correlated to aother criterio is called correlatio It should be highlighted that correlatio does ot imply causatio betwee criteria We should highlight that the AHP ca accout for the outer-depedece effect from upper criteria but claims each subcriterio is idepedet of the other Although the ANP ca accout for the effects of outer-depedece, ierdepedece loop, ad feedback, it caot deal with the correlatio betwee criteria Let us first cosider Figure 3 to describe the preseted problem which is cosidered i this paper Accordig to the presetatio of Figure 3, it ca be see that Criteria 1 ad 2 are cosidered to affect the decisio of the problem Criterio 1 ca be divided ito 3 idepedet subcriteria ad so ca Criterio 2 We should highlight that sice Criteria 1 ad 2 are correlated with each other, this problemcaotbesolvedeitherbytheahporbytheanp I order to cosider the correlatio effect i the AHP, we should first quatify the correlatio matrix betwee criteria

4 4 Mathematical Problems i Egieerig φ f 2 f (ideal poit) Criterio 1 Criterio 2 f 2 Subcriterio 11 Subcriterio 12 Subcriterio 13 Subcriterio 21 Subcriterio 22 Subcriterio23 Figure 3: The cocept of correlatio betwee criteria Flexible solutios Pareto solutios Figure 4: Cocept of the compromise solutios f 1 f 1 which is give by a expertise Take Figure 3 as the example, we ca obtai the followig correlatio matrix: The 2-level correlated aalytic hierarchy process (CAHP) ca be formulated by the followig biobjective programmig model: max w Rw, r 11 r 12 r 13 r 14 r 15 r 16 r 21 r 22 r 23 r 24 r 25 r 26 R = r 31 r 32 r 33 r 34 r 35 r 36 r 41 r 42 r 43 r 44 r 45 r 46 (12) [ r 51 r 52 r 53 r 54 r 55 r 56 ] [ r 61 r 62 r 63 r 64 r 65 r 66 ] mi st a IJ w I + a j, J p p L I=1 w I =1, (15) Or we ca rewrite the above correlatio matrix as where R =[ R 11 R 12 R 21 R 22 ], (13) r 11 r 12 r 13 r 14 r 15 r 16 R 11 = [ r 21 r 22 r 23 ], R 12 = [ r 24 r 25 r 26 ], [ r 31 r 32 r 33 ] [ r 34 r 35 r 36 ] r 41 r 42 r 43 r 44 r 45 r 46 R 21 = [ r 51 r 52 r 53 ], R 22 = [ r 54 r 55 r 56 ] [ r 61 r 62 r 63 ] [ r 64 r 65 r 66 ] (14) I this example, R 11 = R 22 = 0 sice they do ot have the self-correlatio effect Note that the self-correlatio effect could happe i other situatios I additio, it should be highlighted that R ji = R ij, i, j, because the correlatio effect is symmetric The, we assume that if Criterio i is highly correlated to Criterio j, they have similar weights or ifluece to the problem Hece, if we obtai the correlatio matrix betwee criteria, we ca derive the first objective to maximize the correlatio, that is, w Rw The,weicorporate(11) tothe proposed model to form the biobjective programmig model w I = k i i=1, w I, [0, 1], I=1,,L; i=1,,l, I=1,,L; i=1,,l, where a IJ deotes the give estimated weight ratio of the upper-level Criteria I ad J, w I deotes the true weight of the upper-level Ith criterio, a ij deotes the give estimated weight ratio of the lower-level Criteria i ad j,ad deotes the true weight of the lower-level ith criterio I additio, the upper-level Ith criterio ca be divided ito k i lower-level criteria I order to solve the above biobjective programmig, we itroduce the compromise solutio as follows I a multiple objective programmig (MOP) problem, a ideal (or utopia) poit is usually ot attaiable while trade-offs betwee objectives exist Hece, Yu [10] proposed the compromise solutios to determie the optimal solutio, which is closest to the ideal poit, amog Pareto solutios based o the L p - orm distace The cocept of the compromise solutios ca be depicted as sho Figure 4 The L p -orm distace betwee a poit ad the ideal poit ca be defied as d p = f f p, p=1,, (16) I a geeralized optimal problem, the distace measured by the L p -orm betwee a poit ad the ideal poit ca be preseted as sho Figure 5 The left-lower square belogs to the maximized problems (maximize all the objective fuctios)adisthecasedealtwithithissectiofrom

5 Mathematical Problems i Egieerig 5 p= φ p=2 Fuctio Subsidiary f p=1 Speed Moitor resolutio Camera resolutio Price Service Appearace Figure 6: The correlated hierarchical structure of the problem Figure 5: Cocept of the L p -orm distace Figure 5, it ca be see that the shape of p=1is a square diamod, p = 2 is a circle, ad p = is a square The differet shapes of the L p -orm may result i the differet result of the optimal solutio I additio, other kids of the L p -orm are less discussed because they have o cocrete meaig i practice The procedures of the compromise solutios ca be demostrated by cosiderig a multiple objective programmig (MOP) problem as follows: max z (x) =[z 1 (x),z 2 (x),,z (x)] st g(x) b, x 0 (17) The first step of the compromise solutios is to determie the ideal poit of each objective This ca be doe by optimizig each objective as follows: max z j (x) st (x) b, x 0 (18) The, we ca obtai the ideal poit as z =(z 1,z 2,,z ) Next, we wat to determie which poit located o the Pareto solutios is closest to the ideal poit as the optimal solutio Hece, we ca use the cocept of the L p -orm to measure thedistacebetweeobjectivevaluesadtheidealpoitad formulate the compromise solutio method as [10] mi d p = { { w p j [z j (x) z j (x)] p } } { } st g(x) b, x 0, 1/p, p=1,, (19) where deotes the importace of the jth objective Besides usig the traditioal L p -orm, Gersho ad Duckstei [11] proposed the ormalized L p -orm ad formulated the compromise solutios as mi d p ={ w p j [ z j (x) z p 1/p j (x) zj (x) z j (x)] }, p=1,, st g(x) b, x 0, (20) where z j (x) deotes the miimum value of the jth goal I this paper, we adopt Duckstei s method ad trasform the first objective to miimizatio by multiply 1 Next, we propose a umerical example to demostrate the proposed model as follows 4 A Numerical Example Let us cosider a purchasig decisio of a smart phoe to illustrate the proposed method as follows Assume that two criteria, for example, Fuctio ad Subsidiary, are cosidered to determie a smart phoe Fuctio ca be divided ito three subcriteria, Speed, Moitor Resolutio, ad Camera Resolutio O the other had, Subsidiary ca be divided ito Price, Service, ad Appearace I additio, Fuctio ad Subsidiary are cosidered to be correlated with each other Hece, the hierarchical structure of the problem ca beshowasifigure6 Next, we assume that Criterio 1 is two times more importat tha Criterio 2 ad cosider the followig compariso matrices of subcriteria: W 11 = [ 1 1, W 22 = 3 1 1, [ ] [ 2] (21) ] [ 4 2 1] W 12 = W 12 = 0 The, we ca give the correlatio matrices as follows: R 12 = R 21 = [ ], R 11 = R 22 = 0 (22) [ ]

6 6 Mathematical Problems i Egieerig Table 1: The compariso of the weight betwee the AHP ad CAHP Method w w 21 w 22 w 23 AHP CAHP Table 2: The stability aalysis of the proposed method Δ w w 21 w 22 w 23 Δ= Δ= Δ= Δ= Next, we ca employ (15)toderivetheresultoftheCAHP by solvig the biobjective programmig as follows: mi (021 w w w w 21 w 2 =w 21 +w 22 +w 23,,w 2,1,2,3,w 21,w 22,w 23 0 (23) mi w w w w w 23 ) (2 w 2 )+( 1 2 w 2 ) + (2 1 )++(2 w 23 ) 2 w 22 st +w 2 =1, = , The, we ca use the compromise solutio to solve the above problem ad derive the result of the CANP I additio, if we take off the first objective, we ca derive the result of the traditioal AHP The results of the AHP ad CAHP ca be preseted as sho Table 1 The result of CAHP is slightly differet from that of the AHP due to the correlatio effect betwee criteria I order to uderstad the stability of the proposed method, we add some deviatios, that is, radom errors, ito the previous correlatio matrix such that 02 + δ 1 02 Δ% 03 + δ 2 03 Δ% 02 + δ 3 02 Δ% R 12 = R 21 = [ 04 + δ 4 04 Δ% 05 + δ 5 05 Δ% 03 + δ 6 03 Δ% ], (24) [ 02 + δ 7 02 Δ% 04 + δ 8 04 Δ% 03 + δ 9 03 Δ% ] where δ i, i = 1,,9, is a radom dummy variable deoted by { 1, 1} ad Δ is a user-defied value ragig from [0, 100] to idicate the possible fluctuatio For example, we specify Δ=10ad ru δ i oce to obtai the correlatio matrix as % % % R 12 = R 21 = [ % % % ] (25) [ % % % ] The, we ca simulate 3 times the stability aalysis of the proposed method with respect to differet Δ values ad compare these results with the origial solutio as sho Table 2 From the results of Table 2, itcabeshowthatthe proposed method presets the excellet stability ad prevets the rak reversal of the criteria here; eve the fluctuatio or expert s error is equal to 30% Next, we modify the previous origial correlatio matrix as follows: R 12 = R 21 = [ ] (26) [ ] With the same procedure above, we ca obtai the modified result of the CAHP, as sho Table 3

7 Mathematical Problems i Egieerig 7 Table 3: The modified result of the CAHP Method w w 21 w 22 w 23 CAHP From Table 3, it ca be see that the fial result may chage depedig o the specific correlatio matrix However, the traditioal AHP ca oly obtai the same result, sice it igores the correlatio betwee criteria Next, we propose the discussios i the followig sectio accordig to the fidig of the umerical example 5 Discussios The AHP/ANP has bee the most popular tool for decisio makig i the recet 20 years, because of its simplicity ad reasoability The AHP is used whe we cosider the situatio of outer-depedece betwee idepedet criteria O the other had, the ANP ca accout for the effects of outerdepedece, ier-depedece, ad feedback i a etwork structure However, the traditioal AHP/ANP igores the situatio of correlatio betwee criteria which idicates that oe criterio is correlated to aother I realistic situatio, the correlatio betwee criteria is commo ad importat It may provide iformatio for a decisio maker to derive more accurate result I this paper, the CAHP is proposed to cosider the correlatio effect betwee criteria i the AHP First, we ask respoders to quatify the correlatio matrix The, we cosider a biobjective programmig model to derive the weights of criteria The first objective icorporates the iformatio of correlatio betwee criteria ad maximizes w Rw, where w ad R deote the weight vector ad correlatio matrix, respectively The philosophy above is to claim that the weights of two criteria are similar whe they have higher degree of correlatio ThesecodobjectivefollowsthepurposeoftheAHPto miimize the error betwee the give ad true weight ratios betwee criteria Fially, we use the compromise solutio to dealwiththemodeladobtaithefialresultitshouldbe highlighted that the AHP ca be cosidered as the special case of the proposed model whe the values of correlatio betwee criteria are zero The, the biobjective programmig is reduced to sigle-objective programmig Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper Refereces [1] D Beroulli, Specime Theoriae Novae de Mesura Sortis, Commetarri Academiae Scietiarum Imperialis Petropolitaae, vol 5, pp , 1738 [2] J vo Neuma ad O Morgester, Theory of Games ad Ecoomic Behavior, Priceto, NJ, USA, Priceto Uiversity Press, 2d editio, 1947 [3] D Dubois ad H Prade, Fuzzy Sets ad Systems, Academic Press, New York, NY, USA, 1980 [4] T L Saaty, A scalig method for priorities i hierarchical structures, JouralofMathematicalPsychology, vol 15, o 3, pp ,1977 [5] T L Saaty, The Aalytic Hierarchy Process, McGraw-Hill, New York, NY, USA, 1980 [6]PLYu,Multiple Objective Decisio Makig: Cocepts, Techiques ad Extesio,PleumPress,NewYork,NY,USA,1985 [7] J J Huag, A matrix method for the fuzzy aalytic hierarchy process, Iteratioal Joural of Ucertaity, Fuzziess ad Kowledge-Based Systems,vol19,o2,pp ,2011 [8] T L Saaty, Decisio Makig with Depedece ad Feedback, The Aalytic Network Process, RWS Publicatios, Pittsburgh, Pesylvaia, 1996 [9] T L Saaty, Decisio-makig with the AHP: why is the pricipal eigevector ecessary, Europea Joural of Operatioal Research,vol145,o1,pp85 91,2003 [10] P L Yu, A class of solutios for group decisio problems, Maagemet Sciece,vol19,pp ,1973 [11] M Gersho ad L Duckstei, A procedure for selectio of a multiobjective techique with applicatio to water ad mieral resources, Applied Mathematics ad Computatio, vol 14, o 3,pp , Coclusios I this paper, we exted the AHP to cosider the correlatio effect betwee criteria The, a biobjective programmig model is developed to derive the result of the CAHP I additio, we use the compromise solutio to solve the biobjectiveprogrammigmodelfromtheresultoftheumerical example, it ca be see that the correlatio effect betwee criteria ca be icorporated ito the traditioal AHP The proposed model ca be reduced ito the AHP whe we take off the first objective, that is, igore correlatio betwee criteria

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