Web Based Fuzzy Multicriteria Decision Making Tool

Transcription

1 Web Bsed Fuzzy Multicriteri Decision Mking Tool Lee Hu Jie, Mk Chee Meng, Chin Wen Cheong Fculty of Informtion Technology, Multimedi University, Cyberjy, Selngor, Mlysi Emil: Abstrct In this pper, we developed web-bsed decision mking tool which utilizes fuzzy nlytic hierrchy process (AHP) methodology to solve dily complicted decision mking problems Fuzzy concepts re used to enhnce the trditionl AHP, which is minly pplied in crisps decision environment Fuzzy linguistic term pproch is pplied to cpture the fuzziness nd subjectiveness of decision mkers judgements Due to simplicity nd effectiveness, we selected tringulr fuzzy numbers s reference to indicte the influence strength of ech element in the hierrchy structure α-cut-bsed method hs been utilized to prevent the controversil of fuzzy number rnking process The confidence level nd the optimistic level of decision mker re cptured by using α-cutbsed fuzzy number results A numericl emple is demonstrted to illustrte the fuzzy pproch The fuzzy pproch ehibits more pproprite nd fleible result compred to the trditionl pproch Keywords: Decision Mking,, Anlytic Hierrchy Process (AHP), Fuzzy Linguistic, α-cut-bsed method 1 Introduction Decision mking tool ims to relize conflicts tht occur due to vrious different opinions nd subjective ssessments by decision mkers Unlike simple decision mking problem (where it involves one criteri most of the time), in rel world, we re considering more thn one criterion nd lterntive s well With this considertion, our decision mking tool is bsed on one of the multicriteri decision mking (MCDM) in Turbn (1991); nmely Anlyticl Hierrchy Process (AHP) AHP is ble to hndle these typicl scenrios With this tool, it helps us mke better decisions nd improve the decision mking process However, AHP fce wekness in cpturing the vgue, uncertinty nd imprecise judgment by different users This my cused by level of eperiences, lcking of eperimentl dt nd other undecided fctors Therefore, vrition of AHP nmed Fuzzy AHP comes into implementtion It is being eplored to overcome the compenstory pproch nd the inbility of the AHP in hndling proper linguistic vribles Inherently, fuzzy set theory hs proven dvntges within vgue, imprecise nd uncertin contets In ZY Yng (2003), one bsic ppliction of fuzzy set theory is fuzzy synthetic evlution (FSE), which is decision-mking pproch within fuzzy environment At the sme time, AHP cn give comprehensive nd consistent nlysis on the weights of ll fctors; therefore, mny Interntionl Journl of The Computer, the Internet nd Mngement Vol 14No2 (My - August, 2006) pp 1-14

2 Lee Hu Jie, Mk Chee Meng, Chin Wen Cheong works on the integrtion of FSE nd AHP hve been performed to obtin the benefits from both In Bojdziev (1997), some reserchers use membership function to describe how the lterntives stisfy the criteri (fctors) nd then use n AHP to mke the decision Some works use the AHP to get the weights of the fctor only, fuzzify the weights, nd then use fuzzy synthesis evlution (FSE) or rnking s the decisionmking strtegy Vn Lrhoven nd Pedrycz (1983) employed tringulr fuzzy to represent pirwise comprison rtio in AHP insted of ect numbers nd new decision-mking methodology clled fuzzy AHP cme to eist Bsed on the brief fuzzy eplntion of previous prgrph, we integrte the fuzzy set theory pproch into our Decision Mking tool specificlly in one of the common re of humn decision mking problem By pplying the fuzzy set theory, we re ble to come up with more precise nd relible result thus is fundmentl of decision mking tool The proposed decision mking tool fetures the trditionl AHP nd Fuzzy AHP End user cn optimize the decision benefit from these two pproches to del with the problem of inconsistency s well s vgueness in humn judgment It lso let the end user to compre the result nd performnce from these two methods in one glnce In ddition, the decision mking tool will lso provide n integrted domin reference chnnel vi dtbse connection to ssist the end user obtin some updted informtion regrding the problem domin before construct the problem hierrchy in AHP Hence, our decision mking tool combines the chrcteristic of rel time informtion retrievl through internet nd MCDM problem nlyticl processing logic 2 Methodology of Fuzzy AHP 21 Introduction to Fuzzy AHP Inbility of trditionl AHP to del with the imprecision nd subjectiveness in the pirwise comprison process hve been improved in Fuzzy AHP Insted of single crisp vlue, Fuzzy AHP used rnge of vlue to incorporte decision mker s uncertinty From this rnge, decision mker cn select the vlue tht reflects his confidence nd lso he cn specify his ttitude like optimistic, pessimistic or moderte (Jegnthn, 2003) Optimistic ttitude is represented by the highest vlue of rnge, moderte ttitude is represented by the middle vlue of the rnge nd pessimistic ttitude is represented by the lowest vlue of the rnge In the fuzzy set terminology, the rtio supplied by the decision mker is fuzzy number described by membership function Here, membership function describes the degree with which elements in the judgment intervl belong to the preference set In norm, tringulr fuzzy number is used to represent the decision mker s ssessment on lterntives with respect to ech criterion The concept of fuzzy etent nlysis is pplied to solve the fuzzy reciprocl mtri for determining the criteri importnce nd lterntive performnce In Prksh (2003), the lph-cut nlysis is used to trnsform the fuzzy performnce mtri representing the overll performnce of ll lterntives with respect to ech criterion into n intervl performnce mtri, to void the comple nd unrelible process of compring fuzzy utilities 22 A Brief Introduction to Fuzzy Set Theory Fuzzy set theory is mthemticl theory designed to model the vgueness or 2

3 Web Bsed Fuzzy Multicriteri Decision Mking Tool imprecision of humn cognitive processes tht pioneered by Zdeh (Lootsm, 1997) This theory is bsiclly theory of clsses with unshrp boundries Wht is importnt to recognize is tht ny crisp theory cn be fuzzified by generlizing the concept of set within tht theory to the concept of fuzzy set The stimulus for the trnsition from crisp theory to fuzzy one derives from the fct tht both the generlity of theory nd its pplicbility to rel world problems re enhnced by replcing the concept of crisp set with fuzzy set (Zdeh, 1994) Generlly, the fuzzy sets re defined by the membership functions The fuzzy sets represent the grde of ny element of X tht hve the prtil membership to A The degree to which n element belongs to set is defined by the vlue between 0 nd 1 If n element relly belongs to A if µ A () =1 nd clerly not if µ A() =0 Higher is the membership vlue, µ A(), greter is the belongingness of n element to set A In norm, fuzzy number is represented by cp on top The Fuzzy AHP presented in this pper pplied the tringulr fuzzy number through symmetric tringulr membership function A tringulr fuzzy number is the specil clss of fuzzy number whose membership defined by three rel numbers, epressed s (l, m, u) According to Te-heon Moon (1999), the tringulr fuzzy numbers is represented s follows µ A ( l)/( m l), l m, ( ) = ( u )/( u m), m u, 0 otherwise, Figure 1-1 Tringulr Membership Function Figure 1-2 Symmetric Tringulr Membership Function Interntionl Journl of The Computer, the Internet nd Mngement Vol 14No2 (My - August, 2006) pp

4 Lee Hu Jie, Mk Chee Meng, Chin Wen Cheong 23 Fuzzy AHP Workflow Figure 1-3 Fuzzy Anlyticl Hierrchy Process (AHP) Workflow (Step 1) Acquisition of Crisp PCM nd Fuzzyfying the Crisp PCM to Fuzzy PCM In the fuzzy AHP, the tringulr fuzzy number is used for the fuzzifiction of the crisp PCM Given crisp PCM A, hving the vlues rging from 1/9 to 9, the crisp PCM is fuzzified using the tringulr fuzzy number f = (l,m,u), which fuzzified the originl PCM using the conversion number s indicted in the tble below The l (lower bound) nd u (upper bound) represents the uncertin rnge tht might eist in the preferences epressed by the decision mker or eperts Tble 1- Conversion of Crisp PCM to Fuzzy PCM Crisp PCM vlue Fuzzy PCM vlue Crisp PCM vlue Fuzzy PCM vlue 1 (1,1,1) if digonl; (1,1,1) if digonl; 1/1 (1,1,3) otherwise (1,1,3) otherwise 2 (1,2,4) 1/2 (1/4,1/2,1/1) 3 (1,3,5) 1/3 (1/5,1/3,1/1) 4 (2,4,6) 1/4 (1/6,1/4,1/2) 5 (3,5,7) 1/5 (1/7,1/5,1/3) 6 (4,6,8) 1/6 (1/8,1/6,1/4) 7 (5,7,9) 1/7 (1/9,1/7,1/5) 8 (6,8,10) 1/8 (1/10,1/8,1/6) 9 (7,9,11) 1/9 (1/11,1/9,1/7) 4

5 Web Bsed Fuzzy Multicriteri Decision Mking Tool Crisp PCM, A, A = m m2 1n 2n mn The fuzzy PCM, A ~ will be s follows, A ~ = ( 11l 11m ( 21l 21m ( m 1l m1m 11u 21u ) ) m1u ) ( ( ( 12l 22l m2l 12m 22m m2m 12u 22u ) ) m2u ) ( ( ( 1nl 2nl mnl 1nm 2nm mnm lnu 2nu mnu ) ) ) (Step 2) Fuzzy Etent Anlysis for Clcultion of Performnce Rtings, Weight Multipliction nd Summtion Then, the fuzzy etent nlysis is pplied on the bove fuzzy PCM to obtin the fuzzy performnce mtri The purpose of fuzzy etent nlysis is to obtin the criteri importnce nd lterntive performnce by solving these fuzzified reciprocl PCMs X ~ i or W ~ j k j= 1 = k ~ k j i= 1 j= 1 where i= 1,2,3 p, j= 1,2,3 q nd k=p, or k=q, depending upon the element under opertion, whether it is n lterntive or criteri (the number of rows nd columns in the PCM) X ~ i = ( ( 11l 21l ( ijl 11m 21m ijm 11 u ) u ) ) 21 iju ~ ij Fuzzy etent nlysis is pplied to get the fuzzy decision or performnce mtri ( X ~ i ) nd fuzzy weights ( W ~ ) After tht, fuzzy weighted performnce mtri ( P ~ ) cn thus be obtined by multiplying the weight vector with the decision mtri P ~ = X ~ i *W ~ = = ( wl ( w m ( w m 11l 21l ijl w w p1l p1m p p2l p2m p pil pim p m 11m w iu m 1u m 2u 21m ijm w w m 11u w m m 21 iju ) u ) ) The net step will be weight summtion where the weighted performnce mtri ( P ~ ) for ech lterntive under ech criteri contet is sum up to obtin totl weighted performnce mtri for ech lterntive (Step 3) Check Fuzzy Rnking with Alph- Cuts-Bsed Method 1 According to Wng (1997), in order to mke crisp choice mong the lterntives, lph-cuts-bsed method 1 is needed for checking nd compring fuzzy number The lph-cuts-bsed method 1 stted tht if let A nd B be fuzzy numbers with α-cuts, A α = - [ α, + - α ] nd B α = [b α, b + α ] It sy A is - - smller thn B, denoted by A B, if α < b α + + nd α < b α for ll α (01] The dvntge of this method is the conclusion is less controversil Here, pply the lph cut nlysis to the totl weighted performnce mtrices for ech lterntive, nd checking for the rnking Interntionl Journl of The Computer, the Internet nd Mngement Vol 14No2 (My - August, 2006) pp

6 Lee Hu Jie, Mk Chee Meng, Chin Wen Cheong consistency for ech lterntive under different lph level circumstnces (Step 4) Alph Cut Anlysis for Confidence Level Representtion The lph cut nlysis is pplied to trnsform the totl weighted performnce mtrices into intervl performnce mtrices The lph cut is to ccount for the uncertinty in the fuzzy rnge chosen In this cse, the decision mker epressed personl confidence bout this rnge The confidence vlue rnges between 0 nd 1, from the lest confidence to the most confidence Alph Cuts Anlysis α Left = [α * ( Middle_fuzzy Left_fuzzy)] + Left_fuzzy α Right = Right_fuzzy [[α * ( Right_fuzzy Middle_fuzzy)] P ~ α = [ p [ p 2lα [ p 1lα ilα, p, p, p 1rα 2rα irα ] ] ] where l nd r represent the left nd right vlue of the intervl set (Step 5) Lmbd Function nd Crisp Vlues Normliztion Through the lph cut nlysis, it will get two vlues nmely Alph_Right (mimum rnge) nd Alph_Left (minimum rnge) which need to been converted into crisp vlue It is done by pplying the Lmbd function which represents the ttitude of the decision mker The ttitude of the decision mker is mybe optimistic, moderte or pessimistic Decision mker with optimistic ttitude will tke the mimum vlues of the rnge; the moderte person will tke the medium vlue nd the pessimistic person will tke the minimum vlue of the rnge Here, the concept of optimism inde, λ, is introduced to obtin the crisp output Crisp_vlue = λ * α Right + [( 1 λ) * α Left ] C λ = λ * p rα where λ = [0,1] C λ C1 C2 = C i λ λ λ + (1 λ) * P Finlly, the crisp vlues need to be normlized, becuse the elements of the PCM do not hve the sme scle It is importnt to note tht elements cn be compred if they hve the uniform scle C iλ = C iλ C iλ 3 Decision Mking Tool We developed web bse decision mking tool using web technology; ASPNET In our decision mking tool, we used C#NET for supporting the lgorithm When users re first directed to the webpge (Figure 1-4), they re required to input their problem domin, gol, criteri nd lterntives lα, 6

7 Web Bsed Fuzzy Multicriteri Decision Mking Tool Figure 1-4 Input Pge for Decision Mking Tool After key-in the input, users re directed to net pge which is computtion for pirwise comprison (Figure 1-5) Figure 1-5 Pirwise Comprison mong Criteri nd Alterntives Interntionl Journl of The Computer, the Internet nd Mngement Vol 14No2 (My - August, 2006) pp

8 Lee Hu Jie, Mk Chee Meng, Chin Wen Cheong Users re required to do evlution on the criteri nd lterntives They re ble to evlute either numericlly or verblly For numericl pproch, users re llowed to select from the drop down bo to rnk them (rnges from 1 to 9) ccordingly Besides, users re lso provided to ssess verblly (eg more importnt, eqully importnt nd less importnt) on them After the evlution, users re required to choose mode of opertion; trditionl AHP or Fuzzy AHP (Figure 1-6) Then, users will be directed to result pge In the result pge, confidence nd optimistic level of Fuzzy AHP opertion re provided for user customiztion From there, users re ble to see the different output generted which shown in grphicl view Most importntly, this tool generte best lterntive mong ll the others which users hve provided t the first plce Figure 1-6 MCDM Approch Option 4 Numericl Evlution A numericl emple is demonstrted in this section Consider fresh grdute student would like to choose job tht cn provide overll stisfction in term of benefits, collegues, loction nd reputtion The vilble jobs re job A, B nd C The hierrchy for the job selection problem is depicted s below Then, the trditionl nd Fuzzy AHP re pplied respectively 8

9 Web Bsed Fuzzy Multicriteri Decision Mking Tool 41 Assessment using Trditionl AHP From the fuzzified PCM (Tble 5 9), we cn convert to the crisp PCM for the criteri, nd lterntive under ech criteri contet Then, perform the trditionl AHP opertions, nd we cn obtin the finl score nd rnking for ech lterntives Tble 2- Score of ech Alterntive Alterntive Score Rnk A B C In order to check for the consistency of the pirwise comprison mtri, we need to perform consistency inde (CI) nd consistency rtio (CR) computtion Given the pirwise comprison mtri of criteri Criteri Benefits Collegues Loction Reputtion Benefits Collegues 1/ Loction 1/6 1/3 1 2 Reputtion 1/8 1/4 1/2 1 Locl weight A w Aw λ m = Consistency Inde, CI = Given rndom Inde, RI, n = 4; RI = 090, Consistency Rtio, CR = 00139/090 = Since the Consistency Rtio (CI/CR) < 010, so the degree of consistency is stisfctory The decision mker s comprison is probbly consistent enough to be useful 42 Assessment using Fuzzy AHP To perform ssessment using Fuzzy AHP, the originl crisp PCM should fuzzified by referring to the fuzzy number conversion tble Tble 3- Fuzzified Pirwise Comprison of Criteri Criteri Benefits Collegues Loction Reputtion Benefits (1,1,1) (1,3,5) (4,6,8) (6,8,10) Collegues (1/5,1/3,1/1) (1,1,1) (1,3,5) (2,4,6) Loction (1/8,1/6,1/4) (1/5,1/3,1/1) (1,1,1) (1,2,4) Reputtion (1/10,1/8,1/6) (1/6,1/4,1/2) (1/4,1/2,1/1) (1,1,1) Interntionl Journl of The Computer, the Internet nd Mngement Vol 14No2 (My - August, 2006) pp

10 Lee Hu Jie, Mk Chee Meng, Chin Wen Cheong Tble 4- Fuzzified Pirwise Comprison of Alterntives in the Contet of Benefits Alterntives A B C A (1,1,1) (1,3,5) (4,6,8) B (1/5,1/3,1/1) (1,1,1) (2,4,6) C (1/8,1/6,1/4) (1/6,1/4,1/2) (1,1,1) Tble 5- Fuzzified Pirwise Comprison of Alterntives in the Contet of Collegues Alterntives A B C A (1,1,1) (1/5,1/3,1/1) (1,3,5) B (1,3,5) (1,1,1) (3,5,7) C (1/5,1/3,1/1) (1/7,1/5,1/3) (1,1,1) Tble 6- Fuzzified Pirwise Comprison of Alterntives in the Contet of Loction Alterntives A B C A (1,1,1) (4,6,8) (1,3,5) B (1/8,1/6,1/4) (1,1,1) (1/6,1/4,1/2) C (1/5,1/3,1/1) (2,4,6) (1,1,1) Tble 7- Fuzzified Pirwise Comprison of Alterntives in the Contet of Reputtion Alterntives A B C A (1,1,1) (1,2,4) (4,6,8) B (1/4,1/2,1/1) (1,1,1) (1,3,5) C (1/8,1/6,1/4) (1/5,1/3,1/1) (1,1,1) After obtining the fuzzified pirwise comprison mtrices, the Fuzzy etent nlysis is pplied s following this procedure: For the fuzzified pirwise comprison of criteri, Totl sum of the whole fuzzy PCM:- Left = ( / /8+1/ /10+1/6+1/4+1) = (b 1 ) Middle = ( / /6+1/ /8+1/4+1/2+1) = (b 2 ) Right = ( / /4+1/ /6+1/2+1/1+1) = (b 3 ) 10

11 Web Bsed Fuzzy Multicriteri Decision Mking Tool The first row sum (for Benefits) Left = ( ) = 12 ( 1 ) Middle = ( ) = 18 ( 2 ) Right = ( ) = 24 ( 3 ) First row sum / Totl sum Left = 1 /b 3 = 12/ = Middle = 2/b 2 = 18/ = Right = 3/b 1 = 24/ = The sme clcultion bove pplies to other criteri- collegues, loction nd reputtion Performnce Tble 8- Overll Weight of ech Criterion (fter Fuzzy Etent Anlysis) Criteri Overll Weight Left Middle Right Benefits Collegues Loction Reputtion Tble 9- Performnce of ech Alterntive (fter Fuzzy Etent Anlysis) Benefits Collegues Loction Reputtion A (02526,05970,13344) (00985,02915,08194) (02526,05970,13344) (02697,06000,13577) B (01347,03184,07625) (02239,06054,15217) (00544,00846,01668) (01011,03000,07311) C (00544,00846,01668) (00601,01031,02731) (01347,03184,07625) (00596,01000,02350) Tble 10- Weighted Performnce of ech Alterntive Weighted Performnce (P=X i *W) Benefits Collegues Loction Reputtion A (00660,03389,15979) (00090,00766,05315) (00128,00659,04161) (00089,00355,01807) B (00352,01808,09131) (00205,01591,09870) (00028,00093,00520) (00033,00177,00973) C (00142,00480,01997) (00055,00271,01771) (00068,00352,02377) (00020,00059,00313) Tble 11- Totl Weighted Performnce Totl Weighted Performnce Left Middle Right A B C Interntionl Journl of The Computer, the Internet nd Mngement Vol 14No2 (My - August, 2006) pp

12 Lee Hu Jie, Mk Chee Meng, Chin Wen Cheong Tble 12- Fuzzy Rnking Check through α-cut-bsed Method 1 α Level A B C α Left α Right α Left α Right α Left α Right From the α-cut-bsed Method 1, the result is consistent enough to show job A hs the highest fuzzy rking t ll lph level following by job B nd C respectively Tble 13- Result of Fuzzy AHP through Alph Cut nd Lmbd Function Alph Cut (α = 05) Crisp Vlue Crisp Vlue (fter normliztion) Rnk α Left α Right λ = 05 λ = 07 λ = 05 λ = 07 A B C Tble 14- Comprison of Trditionl AHP nd Fuzzy AHP Results Trditionl AHP Fuzzy AHP (α = 05) λ = 05 λ = 07 A B C The score of ech lterntive under trditionl AHP nd Fuzzy AHP is consistent where job A, B nd C s finl score re pproimte 05, 03 nd 01 respectively Hence, both pproches indicted tht job A is the lterntive tht cn provide the highest job stisfction 12

13 Web Bsed Fuzzy Multicriteri Decision Mking Tool Pessimistic Decision Mker ( λ = 01) Crisp Vlue Job A Job B Job C Alph Level Figure 1-7 Pessimistic Decision Mker for the Job Selection Problem Moderte Decision Mker ( λ = 05) Crisp Vlue Job A Job B Job C Alph Level Figure 1-8 Moderte Decision Mker for the Job Selection Problem Optimistic Decision Mker ( λ = 09) Crisp Vlue Job A Job B Job C Alph Level Figure 1-9 Optimistic Decision Mker for the Job Selection Problem Interntionl Journl of The Computer, the Internet nd Mngement Vol 14No2 (My - August, 2006) pp

14 Lee Hu Jie, Mk Chee Meng, Chin Wen Cheong The three grphs bove indicted the fuzzy number rnking of Fuzzy AHP is consistent t different lph level for pessimistic, moderte nd optimistic decision mker The job A lwys obtined the highest score compred to job B nd job C 5 Conclusion In this pper, fuzzy set theory nd AHP hve been deployed to cpture the decision mking process by users to provide relible nd efficient decision Most importntly, user is spoilt with user friendly interfce provided in the tool This chieved progrm simplicity nd bstrction of the lgorithm tht works behind the scene User would hrdly need to hve ny mtri knowledge to run this tool to solve their decision mking Besides, this tool provides web bse feture for user to ccess this tool regrdless where they re which enhnces portbility We re lso integrting the domin informtion repository into the tool to provide user with the rw dt nd meningful ttributes (criteri nd lterntives) for severl common problem domin Hence, user who is first timer in using this tool cn refer to their specified problem domin which mtches with the eisting repository on possible vilble ttributes nd dt User cn obtined relible nd best desired decision conveniently nd efficiently by using this tool Indeed, our proposed pper will bring new brethe to decision mking field to the society overll References Bojdziev, George nd Bojdziev, Mri (1997) Fuzzy logic for business, finnce, nd mngement vol 12, British: World Scientific Publishing Niemir, Michel (2003), Industry nd Competitive Anlysis [Online] Avilble: mniemir/hp1pdf [2004, October 20] Prksh, TN, Lnd Suitbility Anlysis For Agriculturl Crops: A Fuzzy Multicriteri Decision Mking Approch, Interntionl Institute for Geo-informtion Science nd Erth Observtion, 2003 Sty, T L (1980) The Anlytic Hierrchy Process New York: McGrw-Hill Interntionl Sty, TL, Rnk, Normliztion nd Ideliztion in the Anlytic Hierrchy Process, The 7th Interntionl Symposium on Anlytic Hierrchy Process, 2003 Te-heon Moon, Woo-be Lee, Construction Of Supporting System For Decision Mking Process Of Zoning Designtion nd Chnge Tht hs Fuzziness, The 6 th Interntionl Conference Computers in Urbn Plnning nd Urbn Mngement, 1999 Turbn nd Meredith (1991) Fundmentls of mngement science (5th Ed) New York: Richrd D Irwin Vn Lrhoven, P J M nd W Pedrycz (1983) A Fuzzy etension of Sty's Priority Theory: Fuzzy Sets nd Systems, Volume: 11, pp Wng, Li-Xin (1997) A Course in Fuzzy Systems nd Control United Sttes of Americ: Prentice-Hll ZY Yng, YH, Chen nd WS, Sze, Using AHP nd Fuzzy Sets to Determine the Build Orienttion in Lyer-bsed Mchining, Interntionl Journl Computer Integrted Mnufcturing, 2003, Volume: 16, No 6, pp