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A Linear Goal Programming Priority Method for Fuzzy Analytic Hierarchy Process and Its Applications in New Product Screening

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International Journal of Approximate Reasoning 49 (2008) 451–465 Contents lists available at ScienceDirect International Journal of Approximate Reasoning journal homepage: www.elsevier.com/locate/ijar A linear goal programming priority method for fuzzy analytic hierarchy process and its applications in new product screening q Ying-Ming Wang a,b,*, Kwai-Sang Chin b a b School of Public Administration, Fuzhou University, Fuzhou 350002, PR China Department of Manufacturing Engineering and Engin
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  A linear goal programming priority method for fuzzy analytichierarchy process and its applications in new product screening q Ying-Ming Wang a,b, * , Kwai-Sang Chin b a School of Public Administration, Fuzhou University, Fuzhou 350002, PR China b Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong  a r t i c l e i n f o  Article history: Received 23 November 2007Received in revised form 12 April 2008Accepted 20 April 2008Available online 29 April 2008 Keywords: Fuzzy analytic hierarchy processMultiple criteria decision makingNormalized fuzzy weightsLinear goal programmingNew product developmentProject screening a b s t r a c t Fuzzy analytic hierarchy process (AHP) has been widely used for a variety of applicationssuchassupplierselection,customerrequirementsassessmentandthelike.Thevastmajor-ity of the applications, however, were found avoiding the use of sophisticated approachesfor fuzzy AHP such as fuzzy least squares method while using a simple extent analysis forthe sake of simplicity. The extent analysis proves to be incorrect and may lead to a wrongdecision being made. This paper proposes a sound yet simple priority method for fuzzyAHP which utilizes a linear goal programming (LGP) model to derive normalized fuzzyweights for fuzzy pairwise comparison matrices. The proposed LGP priority method istested with three numerical examples including an application of fuzzy AHP to new prod-uct development (NDP) project screening decision making. Ó 2008 Elsevier Inc. All rights reserved. 1. Introduction Analytic hierarchy process (AHP) developed by Saaty[35]has been extensively applied in many areas such as selection,evaluation, planning and development, decision making, forecasting, and the like[39]. The traditional AHP requires precise judgments from decision makers (DMs). Precise judgments, however, are not always available and may sometimes be dif-ficult to be elicited from DMs. To overcome this difficulty, fuzzy judgments have been suggested and fuzzy AHP has beenapplied in a wide variety of applications[1–4,8–11,13–15,19–32,34,36–38].ThekeyissueoftheuseoffuzzyAHPfordecisionmakingishowtoderivepriorityvectorsfromfuzzypairwisecomparisonmatrices.AnumberofprioritymethodshavebeensuggestedforfuzzyAHPintheliterature.Forexample,VanLaarhovenandPedrycz[40]utilizedtriangularfuzzyjudgmentsinsteadofprecisejudgmentsandsuggestedalogarithmicleastsquaresmeth-od(LLSM) for fuzzyAHP. Boender et al.[1]pointedout amistakeof VanLaarhovenandPedrycz’s LLSMinnormalizingfuzzyweightsandpresentedamodifiednormalizationmethod.ThemodifiednormalizationmethodforLLSMwasalsofoundincor-rectbyWangetal.[43]whothereforeproposedamodifiedfuzzyLLSMforfuzzyAHP.XuandZhai[47]alsopresentedanLLSM forfuzzyjudgmentmatrices,buttheirLLSMwasbasedonadifferentEuclideandistancemetricthatwasdefinedastheintegralof the distance of every t  -level set. As a result, the weights derivedby their LLSMwereintervals characterized withdifferent t  -levels. Xu[46]usedthesamedistancemetrictodevelopafuzzyleast-squareprioritymethodforfuzzyjudgmentmatrices. 0888-613X/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.ijar.2008.04.004 q The work described in this paper was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China[Grant No. CityU 111906], and also partially supported by the Natural Science Foundation of Fujian Province of China under the Project No. A0710005. * Corresponding author. Address: School of Public Administration, Fuzhou University, Fuzhou 350002, PR China. Tel.: +86 591 87893307; fax: +86 59187892545. E-mail address: msymwang@hotmail.com(Y.-M. Wang).International Journal of Approximate Reasoning 49 (2008) 451–465 Contents lists available atScienceDirect International Journal of Approximate Reasoning journal homepage:www.elsevier.com/locate/ijar  Buckley[6]employedthegeometricmeanmethodtocalculatefuzzyweightsforeachfuzzymatrixandcombinedthemintheusualmannertodeterminethefinalfuzzyweightsfordecisionalternatives.Chang[16]proposedanextentanalysismethodtoderivecrispweightsfromfuzzycomparisonmatrices.Buckleyetal.[7]directlyfuzzifiedSaaty’ssrcinalprocedureofcom-puting weights in hierarchical analysis to get fuzzy weights in fuzzy hierarchical analysis. Such a method proved to be quitecomplicatedandrequiredanevolutionaryalgorithmtobedesignedforobtainingfuzzyweights.CsutoraandBuckley[18]pro-posedaLambda–Maxmethodtofindfuzzyweights,whichwasalsothedirectfuzzificationofthe k max method.TheLambda–Maxmethodwasfoundonlyabletogeneratenonnormalizedfuzzyweights,whichmakenosensebecausesomeofthemaretoowide to be true. To derive a set of normalized fuzzy weights, Wang and Chin[44]came up with an eigenvector method, butfound that not every fuzzy pairwise comparison matrix could derive a normalized fuzzy weight vector. Mikhailov[32–34]developedafuzzypreference programming(FPP) method, whichderivescrispweights fromfuzzycomparisonmatrices.Among the above approaches, the extent analysis was found to be the most widely used approach due to its computa-tional simplicity. However, it proves to be incorrect and may result in a wrong decision to be made[44]. Detailed analysison the extent analysis can be found in Wang et al.[44]and its misapplications are provided in Refs.[2–4,8–11,13–15,19– 31,36–38,48].ThepurposeofthispaperistodevelopasoundyetsimpleprioritymethodforfuzzyAHPsothatthefuzzyweightsoffuzzypairwisecomparisonmatricescanbederivedmoreeasilythanthosesophisticatedapproachessuchasfuzzyLLSM,fuzzypref-erenceprogrammingmethodandthelike.Theproposedprioritymethodusesalineargoalprogramming(LGP)modeltoderivenormalized fuzzy weights for triangular fuzzy pairwise comparison matrices and will be tested with a number of numericalexamples including an application of fuzzy AHP to newproduct development (NDP) project screening decision making.The rest of the paper is organized as follows. Section2develops the LGP priority method for fuzzy AHP. Section3dis- cusses the aggregationof local fuzzy weights to global fuzzy weights. Numerical examples are tested in Section4to providean application of fuzzy AHP in NPD project screening. The paper concludes in Section5. 2. The LGP priority method for fuzzy AHP Consider a fuzzy pairwise comparison matrix: e  A ¼ ð ~ a ij Þ n  n ¼ 1 ð l 12 ; m 12 ; u 12 Þ ÁÁÁ ð l 1 n ; m 1 n ; u 1 n Þð l 21 ; m 21 ; u 21 Þ 1 ÁÁÁ ð l 2 n ; m 2 n ; u 2 n Þ ............ ð l n 1 ; m n 1 ; u n 1 Þ ð l n 2 ; m n 2 ; u n 2 Þ ÁÁÁ 1 266664377775 ; ð 1 Þ where l ij =1/ u  ji , m ij =1/ m  ji and u ij =1/ l  ji for all i , j =1, ... , n ; j 6¼ i . The above fuzzy comparison matrix can be split into threecrisp nonnegative matrices:  A L ¼ 1 l 12 ÁÁÁ l 1 n l 21 1 ÁÁÁ l 2 n ...... ÁÁÁ ... l n 1 l n 2 ÁÁÁ 1 266664377775 ; A M  ¼ 1 m 12 ÁÁÁ m 1 n m 21 1 ÁÁÁ m 2 n ...... ÁÁÁ ... m n 1 m n 2 ÁÁÁ 1 2666437775 and A U  ¼ 1 u 12 ÁÁÁ u 1 n u 21 1 ÁÁÁ u 2 n ...... ÁÁÁ ... u n 1 u n 2 ÁÁÁ 1 2666437775 ; ð 2 Þ where e  A ¼ ð  A L ;  A M  ;  A U  Þ . Note that A L and A U  are no longer reciprocal matrices.For the fuzzy comparison matrix e  A , there should exist a normalized fuzzy weight vector, f W  ¼ ðð w L 1 ; w M  1 ; w U  1 Þ ; ... ; ð w Ln ; w M n ; w U n ÞÞ T , which is close to e  A in the sense that ~ a ij ¼ ð l ij ; m ij ; u ij Þ % ð w Li ; w M i ; w U i Þ = ð w L j ; w M  j ; w U  j Þ for all i , j =1, ... , n ; j 6¼ i .According to Wang and Elhag[42], the fuzzy weight vector f W  is normalized if and only if  X ni ¼ 1 w U i À max  j ð w U  j À w L j Þ P 1 ; ð 3 Þ X ni ¼ 1 w Li þ max  j ð w U  j À w L j Þ 6 1 ; ð 4 Þ X ni ¼ 1 w M i ¼ 1 ; ð 5 Þ which can be equivalently rewritten as w Li þ X n j ¼ 1 ;  j 6¼ i w U  j P 1 ; i ¼ 1 ; ... ; n ; ð 6 Þ w U i þ X n j ¼ 1 ;  j 6¼ i w L j 6 1 ; i ¼ 1 ; ... ; n ; ð 7 Þ X ni ¼ 1 w M i ¼ 1 : ð 8 Þ 452 Y.-M. Wang, K.-S. Chin/International Journal of Approximate Reasoning 49 (2008) 451–465  Ifthefuzzycomparisonmatrix e  A definedbyEq.(1)isaprecisecomparisonmatrixaboutthefuzzyweightvector f W  ,namely, ~ a ij ¼ ð l ij ; m ij ; u ij Þ ¼ ð w Li ; w M i ; w U i Þ = ð w L j ; w M  j ; w U  j Þ for all i , j =1, ... , n but j 6¼ i , then e  A must be able to be written as e  A ¼ 1 ð w L 1 ; w M  1 ; w U  1 Þð w L 2 ; w M  2 ; w U  2 Þ ÁÁÁ ð w L 1 ; w M  1 ; w U  1 Þð w Ln ; w M n ; w U n Þð w L 2 ; w M  2 ; w U  2 Þð w L 1 ; w M  1 ; w U  1 Þ 1 ÁÁÁ ð w L 2 ; w M  2 ; w U  2 Þð w Ln ; w M n ; w U n Þ ...... ÁÁÁ ð w Ln ; w M n ; w U n Þð w L 1 ; w M  1 ; w U  1 Þð w Ln ; w M n ; w U n Þð w L 2 ; w M  2 ; w U  2 Þ ÁÁÁ 1 2666666666437777777775 : ð 9 Þ According to the division operation rule of fuzzy arithmetic, i.e. ( b L , b M  , b U  )/( d L , d M  , d U  )=( b L / d U  , b M  / d M  , b U  / d L ), where ( b L , b M  , b U  )and ( d L , d M  , d U  ) are two positive triangular fuzzy numbers, the fuzzy comparison matrix e  A defined by Eq.(9)can be furtherexpressed as e  A ¼ 1 w L 1 w U  2 ; w M  1 w M  2 ; w U  1 w L 2   ÁÁÁ w L 1 w U n ; w M  1 w M n ; w U  1 w Ln   w L 2 w U  1 ; w M  2 w M  1 ; w U  2 w L 1   1 ÁÁÁ w L 2 w U n ; w M  2 w M n ; w U  2 w Ln   ...... ÁÁÁ ... w Ln w U  1 ; w M n w M  1 ; w U n w L 1   w Ln w U  2 ; w M n w M  2 ; w U n w L 2   ÁÁÁ 1 2666666666437777777775 ; ð 10 Þ which can be split into three crisp nonnegative matrices, as shown below:  A L ¼ 1 w L 1 w U  2 ÁÁÁ w L 1 w U n w L 2 w U  1 1 ÁÁÁ w L 2 w U n ...... ÁÁÁ ... w Ln w U  1 w Ln w U  2 ÁÁÁ 1 2666666666437777777775 ; A M  ¼ 1 w M  1 w M  2 ÁÁÁ w M  1 w M n w M  2 w M  1 1 ÁÁÁ w M  2 w M n ...... ÁÁÁ ... w M n w M  1 w M n w M  2 ÁÁÁ 1 2666666666437777777775 and A U  ¼ 1 w U  1 w L 2 ÁÁÁ w U  1 w Ln w U  2 w L 1 1 ÁÁÁ w U  2 w Ln ...... ÁÁÁ ... w U n w L 1 w U n w L 2 ÁÁÁ 1 2666666666437777777775 : It is easy to verify that  A L W  U  ¼ W  U  þ ð n À 1 Þ W  L ; ð 11 Þ  A U  W  L ¼ W  L þ ð n À 1 Þ W  U  ; ð 12 Þ  A M  W  M  ¼ nW  M  ; ð 13 Þ where W  L ¼ ð w L 1 ; ... ; w Ln Þ T , W  M  ¼ ð w M  1 ; ... ; w M n Þ T and W  U  ¼ ð w U  1 ; ... ; w U n Þ T are three crisp weight vectors, based on which thefuzzy weight vector f W  can be expressed as f W  ¼ ð W  L ; W  M  ; W  U  Þ . Eqs.(11) and (12)are important links between the lowerand upper bounds of the fuzzy weight vector f W  .It can also be verified from the above three crisp nonnegative matrices that a M ij ¼ a M ik a M kj and a Lij a U ij ¼ ð a Lik a U ik Þð a Lkj a U kj Þ for any i ;  j ; k ¼ 1 ; ... ; n ; i 6¼ j 6¼ k ; ð 14 Þ which is the perfectly consistent condition for a triangular fuzzy comparison matrix to be perfectly consistent. Consider thefuzzy pairwise comparison matrix e  A ¼ 1 ð 1 ; 5 = 3 ; 3 Þ ð 4 = 3 ; 5 = 2 ; 6 Þð 1 = 3 ; 3 = 5 ; 1 Þ 1 ð 2 = 3 ; 3 = 2 ; 4 Þð 1 = 6 ; 2 = 5 ; 3 = 4 Þ ð 1 = 4 ; 2 = 3 ; 3 = 2 Þ 1 2435 , which meets the condition of (14).It is therefore a perfectly consistent fuzzy pairwise comparison matrix. However, due to subjectivity and uncertainty in real judgments, DMs’ subjectivejudgmentscannotalwaysbe100 percentaccurate. Inother words, Eqs.(11)–(13)cannotalwayshold. In the case that they do not hold, we introduce the following deviation vectors: E ¼ ð  A L À I  Þ W  U  À ð n À 1 Þ W  L ; ð 15 Þ C ¼ ð  A U  À I  Þ W  L À ð n À 1 Þ W  U  ; ð 16 Þ D ¼ ð  A M  À nI  Þ W  M  ; ð 17 Þ where E  =( e 1 , ... , e n ) T , C =( c 1 , ... , c n ) T , D =( d 1 , ... , d n ) T , I  is an n  n unit matrix, e i , c i and d i for i =1, ... , n are all deviation vari-ables. Itismost desirablethattheabsolutevaluesofthedeviationvariablesbekeptassmallaspossible, whichenablesustoconstruct the following nonlinear goal programming (NGP) model for determining the fuzzy weight vector f W  : Y.-M. Wang, K.-S. Chin/International Journal of Approximate Reasoning 49 (2008) 451–465 453  Minimize J  ¼ X ni ¼ 1 ðj e i j þ j c i j þ j d i jÞ Subject to ð  A L À I  Þ W  U  À ð n À 1 Þ W  L À E ¼ 0 ; ð  A U  À I  Þ W  L À ð n À 1 Þ W  U  À C ¼ 0 ; ð  A M  À nI  Þ W  M  À D ¼ 0 ; w Li þ X n j ¼ 1 ;  j 6¼ i w U  j P 1 ; i ¼ 1 ; ... ; n ; w U i þ X n j ¼ 1 ;  j 6¼ i w L j 6 1 ; i ¼ 1 ; ... ; n ; X ni ¼ 1 w M i ¼ 1 ; W  U  À W  M  P 0 ; W  M  À W  L P 0 ; W  L P 0 ; ð 18 Þ where the first three constraints are Eqs.(15)–(17), the middle three constraints are the normalization constraints on thefuzzy weight vector f W  , and the last three constraints are those on the lower and upper bounds of  f W  and its nonnegativity.FromSaaty’seigenvectormethod(EM)[35], itisknownthat foranycrisppairwisecomparisonmatrix  A M  , therewillexista maximum principal right eigenvector c W  M  such that A M   c W  M  P n  c W  M  . So, deviation vector D can always be nonnegative.That is D P 0. There is no guarantee, however, that the deviation vectors E  and C can also be nonnegative.Let e þ i ¼ e i þ j e i j 2and e À i ¼À e i þ j e i j 2 ; i ¼ 1 ; ... ; n ; ð 19 Þ c þ i ¼ c i þ j c i j 2and c À i ¼À c i þ j c i j 2 ; i ¼ 1 ; ... ; n : ð 20 Þ Then E  þ ¼ ð e þ 1 ; ... ; e þ n Þ T P 0, E  À ¼ ð e À 1 ; ... ; e À n Þ T P 0, C þ ¼ ð c þ 1 ; ... ; c þ n Þ T P 0 and C À ¼ ð c À 1 ; ... ; c À n Þ T P 0. Based upon the defini-tions of  e þ i and e À i , e i and j e i j can be expressed as e i ¼ e þ i À e À i ; i ¼ 1 ; ... ; n ; ð 21 Þj e i j ¼ e þ i þ e À i ; i ¼ 1 ; ... ; n ; ð 22 Þ where e þ i Á e À i ¼ 0 for i =1 to n . As such, c i and j c i j can be expressed as c i ¼ c þ i À c À i ; i ¼ 1 ; ... ; n ; ð 23 Þj c i j ¼ c þ i þ c À i ; i ¼ 1 ; ... ; n ; ð 24 Þ where c þ i Á c À i ¼ 0 for i =1 to n . As a result, the NGP model(18)can be rewritten as Minimize J  ¼ X ni ¼ 1 ð e þ i þ e À i þ c þ i þ c À i þ d i Þ ¼ e T ð E þ þ E À þ C þ þ C À þ D Þ Subject to ð  A L À I  Þ W  U  À ð n À 1 Þ W  L À E þ þ E À ¼ 0 ; ð  A U  À I  Þ W  L À ð n À 1 Þ W  U  À C þ þ C À ¼ 0 ; ð  A M  À nI  Þ W  M  À D ¼ 0 ; w Li þ X n j ¼ 1 ;  j 6¼ i w U  j P 1 ; i ¼ 1 ; ... ; n ; w U i þ X n j ¼ 1 ;  j 6¼ i w L j 6 1 ; i ¼ 1 ; ... ; n ; X ni ¼ 1 w M i ¼ 1 ; W  U  À W  M  P 0 ; W  M  À W  L P 0 ; W  L ; E þ ; E À ; C þ ; C À ; D P 0 ; ð 25 Þ 454 Y.-M. Wang, K.-S. Chin/International Journal of Approximate Reasoning 49 (2008) 451–465
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