Itertiol Jourl of Modelig d Optimiztio, Vol. No. April 202 Applyig Fuzzy Alytic Hierrchy Process to Evlute d Select Product of Noteook Computers Phrut Srichett d Wsiri Thurcho Astrct The ility, portility d moility of oteook computer re importt fctors tht cuses the oteook computers to e used widely. To uy oteook computer, oe should look for product tht pcks together est fetures t ffordle price. However, highly competitive usiess of oteook computers mkes the difficulty for uyers to determie. Therefore, usig the experts decisio mkig i evlutig d selectig the ltertive mog the curret products of oteook computers is the eeficil wy to help the uyers choose the est oe. The oective of this pper is to pply the fuzzy lytic hierrchy process (fuzzy AHP i determiig the reltive importce of the decisio criterio i order to evetully select the est product of oteook computers. The rel umericl fidig results hve lso ee demostrted. Both the theoreticl d prcticl ckgroud of this pper hve show tht fuzzy AHP is cple to efficietly hdle the fuzziess of the dt ivolved i the multi-criteri decisio mkig prolem of this study. Idex Terms Alytic hierrchicl process, decisio lysis, fuzzy logic, multi-criteri decisio mkig, oteook computer selectio I. INTRODUCTION Noteook computers c e cosidered s the importt roles i hum life i this er of techology ecuse of their ility, portility, d moility. Therefore, the selectio of effective oteook computers to suit the eeds of uyers is essetil. Nowdys, my iformtio sources hve preseted out choosig suitle oteook computer. They hve mostly preseted the fetures, prices, d pros d cos of ech product d model of oteook computers. Most of them hve t decided tht which oe is the est or the worst to e ought or ot ought, ut they hve ust give iformtio d let the uyers compre d decide y themselves. I prctice eviromet, the uyers hve to fce with vriety of oteook computers iformtio types tht re difficult to determie the decisio ltertives. This prolem c e multi-criteri decisio mkig (MCDM prolem. MCDM refers to fid the est ltertive from ll of the fesile ltertives i the presece of multiple decisio criteri []. Therefore, usig the experts decisio mkig i evlutig d selectig the ltertive mog the curret products of oteook computers uder severl qulittive d qutittive criteri is the eeficil wy to help the uyers choose the est oe. Muscript received Mrch 5, 202; revised April 3, 202. This work ws supported y the Reserch d Developmet Istitute, Udo Thi Rht Uiversity. The uthors re with the Deprtmet of Computer Sciece d Iformtio Techology, Fculty of Sciece, Udo Thi Rht Uiversity, Udo Thi, Thild 4000 (e-mil: phrut_sct@yhoo.com; kitock@hotmil.com. There hve ee differet methods o MCDM prolems, i.e., Alytic Hierrchy Process (AHP [2]-[3], Techique for Order Preferece y Similrity to Idel Solutio (TOPSIS [4], Preferece Rkig Orgiztio METHod for Erichmet of Evlutios (PROMETHEE [5], etc. The AHP is ccepted to e powerful d flexile method for rkig decisio ltertives d selectig the est oes whe decisio mker hs multiple criteri [6]. Its mi dvtges re hdlig multiple criteri, esy to uderstd, d effectively hdlig oth qulittive d qutittive dt. However durig the decisio mkig, the experts my e imprecise ecuse of the icomplete iformtio of the cosidered oteook computers, the vgueess of the hum thought process, d the iheret complexity d ucertity of the decisio eviromet. The oective of this pper is to pply the extesio of AHP, mely Fuzzy AHP [7]-[9], i order to hdle the fuzziess of dt ivolved i MCDM prolem of this study. The fuzzy APH hs ee pplied i vriety of computer sciece d iformtio techology res i literture for evlutig d selectig, e.g., the product of oteook computers [0], the moile phoe ltertives [], the opertig system [], the computer itegrted i mufcturig systems [2], the softwre qulity of vedors [3], the est techicl istitutios [4], d so o. I this reserch, the Fuzzy AHP will e employed to determie the reltive importce of the decisio criteri i order to evetully select the est product of oteook computers. The rest of this pper is orgized s follows. The cocept of AHP d fuzzy AHP re preseted i sectio II d III, respectively. Sectio VI descries the dt gtherig d preprocessig. Sectio V shows d explis the fidig results i the selectio prolem of oteook computers. Filly, sectio VI is the coclusio d discussio of this reserch. II. CONCEPT OF ANALYTIC HIERARCHY PROCESS Alytic Hierrchy Process (AHP, proposed y Sty [2]-[3], is trditiol powerful decisio-mkig methodology i order to determie the priorities mog differet criteri, comprig ltertives for ech criterio, d determiig overll rkig of the ltertives. The fil outcome of the AHP is the est choice mog decisio ltertives. The sic procedure to crry out the AHP cosists of the followig steps: Decomposig the decisio prolem ito hierrchy. The top level of the hierrchy represets the overll gol of the decisio prolem, the itermedite levels represet the criteri d su-criteri ffectig the decisio, d the ottom level represets the possile ltertives. 68
Itertiol Jourl of Modelig d Optimiztio, Vol. No. April 202 2 Clcultig the reltive importce weights of decisio criteri i ech level of the hierrchy usig pir-wise comprisos. I this step, the decisio mker uses the fudmetl scle or weight etwee (equl importce d 9 (extreme importce defied y Sty [2] to ssess the priority score for ech pir of criteri i the sme level. Tht is, the pir-wise compriso mtrix is costructed i which the elemets i iside the mtrix c e iterpreted s the degree of the precedece of the i th criterio over the th criterio. The, the verge weight for ech ormlized criterio is computed. 3 Evlutig the decisio ltertives tkig ito ccout the weights of decisio criteri. The ltertive scores re comied with the criterio weights to produce overll score for ech ltertive. The AHP provides cosistecy rte (CR to mesure the cosistecy of udgmet of the decisio mker tht will e preseted i the sectio of fuzzy AHP. III. FUZZY AHP The covetiol AHP is idequte for delig with the imprecise or vgue ture of liguistic ssessmet. I fuzzy AHP, commo sese liguistic sttemets hve ee used i the pir-wise compriso which c e represeted y the trigulr fuzzy umers [5]. Afterwrds, the step of ggregtig the pir-wise compriso d the sythesis of the priorities to determie the overll priorities of the decisio ltertives will e doe. A. Trigulr Fuzzy Numers (TFNs The TFNs used i the pir-wise compriso re defied y three rel umers expressed s triple (l, m, u where l m u for descriig fuzzy evet. From umer of TFNs tht hve ee proposed i literture, the oe tht seems to correspod etter to the prefereces scle of the crisp AHP is summrized i Tle I. TABLE I: TRIANGULAR FUZZY CONVERSION SCALE Liguistic Scle TFNs Reciprocl TFNs Eqully importt ( ( Wekly more importt (2/3, 3/2 (2/3, 3/2 Strog more importt (3/ 5/2 (2/5, / 2/3 Very strog more importt (5/ 3, 7/2 (2/7, /3, 2/5 Asolutely more importt (7/ 4, 9/2 (2/9, /4, 2/7 B. Costruct the Fuzzy Pir-Wise Compriso Mtrix To costruct the fuzzy udgmet mtrix Ã={ã i } of criteri or ltertives vi pir-wise compriso, the TFNs re used s follows. ã ã ã Ã ã ã ã ( x l/( m l, l x m μ ( x = ( u x/( u m, m x u 0, otherwise The opertios o TFNs c e dditio, multiplictio, d iverse. Suppose M d M 2 re TFNs where M =(l, m, u d M 2 =(l 2, m 2, u 2, the Additio: M M 2 = (l + l 2, m + m 2, u + u 2 (2 Multiplictio: M M 2 = (l l 2, m m 2, u u 2 (3 Iverse: M - = (l, m, u - (/u, /m, /l (4 C. Aggregte the Group Decisios After collectig the fuzzy udgmet mtrices from ll decisio mkers, these mtrices c e ggregted y usig the fuzzy geometric me method of Buckley [6]-[7]. The ggregted TFN of decisio mkers udgmet i certi cse ũ i = (l i, m i, u i is: i= ( u~ / i = ( ~ ik (5 where ã ik is the reltive importce i form of TFN of the k th decisio mker s view, d is the totl umer of decisio mkers. D. Compute the Vlue of Fuzzy Sythetic Extet Bsed o the ggregted pir-wise compriso mtrix, Ũ={ũ i }, the vlue of fuzzy sythetic extet S i with respect to the i th criterio c e computed s (6 y mkig use of the lgeric opertios o TFNs s descried i (2 (4. m m ~ m ~ S i = ui u (6 i = i= = m m m u ~ = i l, m, u d m u ~ i = li, mi, ui. i= = i= i= i= where = = = = E. Approximte the Fuzzy Priorities Bsed o the fuzzy sythetic extet vlues, the o-fuzzy vlues tht represet the reltive preferece or weight of oe criterio over others re eeded. Therefore, this pper firstly uses Chg s method [2] to fid the degree of possiility tht S S s follows: V ( S S = ( m 0 l u u ( m l, if m m,, if l u otherwise where d is the ordite of the highest itersectio etwee μ S d μ S s show i Fig. Tht is, it c e expressed tht V ( S S = hight( S S = μ ( d. S (7 where ã i is fuzzy trigulr umer, ã i =(l i, m i, u i, d ã i = /ã i. For ech TFN, ã i or M = (l, m, u, its memership fuctio μ ã (x or μ M (x is cotiuous mppig from rel umer - x to the closed itervl [0, ] d c e defied y equtio (. Fig.. The itersectio etwee S d S d their degree of possiility 69
Itertiol Jourl of Modelig d Optimiztio, Vol. No. April 202 It is oted tht oth vlues of V(S S d V(S S re required. The degree of possiility for TFN S i to e greter th the umer of TFNs S k c e give y the use of opertio mi proposed y Duois d Prde [8]: V S S, S,..., S = mi V ( S S, = w ( S (8 ( i 2 k i k i where k=, d k # i, d is the umer of criteri descried previously. Ech w (S vlue represets the reltive preferece or weight, o-fuzzy umer, of oe criterio over others. However, these weights hve to e ormlized i order to llow it to e logous to weights defied from the AHP method. The, the ormlized weight w(s i will e formed i terms of weight vector s follows: W T = ( w( S, w( S..., w( S (9 Oce the weights of criteri re evluted, it is required to clculte the scores of ltertives with respect to ech criterio d the determie the composite weights of the decisio ltertives y ggregtig the weights through hierrchy. F. Cosistecy Test of the Compriso Mtrix To ssure certi qulity level of decisio, we hve to lyze the cosistecy of evlutio. I order to test the vlue of cosistecy of the compriso mtrix depeded o, the cosistecy rte (CR hve to e computed. The CR is defied i (0 s rtio etwee the cosistecy of cosistecy idex (CI d the cosistecy of rdom cosistecy idex (RI. Its vlue should ot exceed 0. for mtrix lrger th 4x4. For pir-wise compriso mtrix eig comptile, upper-oud of CR should e like wht is show i Tle II [2]-[3]. CR = CI / RI (0 determied y ggregtig the weights through the hierrchy. IV. DATA GATHERING AND PREPROCESSING This sectio presets the wy to collect the dt of oteook computers, the costructio of hierrchicl structure to e lyzed, d the steps of geertig the fuzzy pir-wise compriso mtrix. A. Collectig Dt of Noteook Computers At this step, we prepre the questioire out the preferece of the product of oteook computers to sk te experts who hve computer skill more th five yers. I this study, the idetifictio of the criteri set for selectig the product of oteook computers hs ee performed y comitio of commo fetures of oteook computers offered i differet rochures, mgzies, d wesites. The questioire cosists of two sectios. The first sectio lists the fetures of oteook computers d llows the experts to give the preferece with rtig scles: highest, high, medium, low, d lowest. The secod sectio provides the list of te well-kow products of oteook computers, i.e., Acer, Asus, Dell, BeQ, Fugitzu, HP, Levoo, Sumsug, Soy, d Toshi. Te up-to-dte models of ech product offered durig the yer 200-20 re lso provided. Therefore, there re hudred oteook computers the experts hve to give the suective udgmet sed o the focused fetures of oteook computers. The experts were requested to evlute whether or ot to uy these oteook computers y cosiderig their fetures. Eight fetures d te products of oteook computers re cosidered s criteri d ltertives of the hierrchy respectively. TABLE II: UPPER BOUND FOR PAIR-WISE COMPARISON MATRIX TO BE COMPATIBLE 3 x 3 4 x 4 > 4 CR 0.58 0.90.2 The CI is used to mesure the icosistecy pir-wise compriso s show i ( where the eigevlue λ mx c e computed y vergig ll eigevlues of the pir-wise compriso mtrix (2. Tle III shows vlues of RI i differet vlues of. CI = ( λ mx /( ( W λ mx = i =, i, =..., (2 W = i TABLE III: VALUES OF RANDOM CONSISTENCY INDEX (RI PER DIFFERENT NUMBER OF CRITERIA RI RI 3 0.58 8.4 4 0.90 9.45 5.2 0.49 6.24.5 7.32 2.48 Next, the process moves o to the phse i which reltive weights re derived for vrious decisio criteri. The composite weights of the decisio ltertives re the Fig. 2. Hierrchicl structure of selectig products of oteook computers B. Geertig the Pir-Wise Compriso Mtrices The ext phse fter the questioires re swered y experts is to estlish the fuzzy pir-wise comprisio mtrices. The dt preprocessig steps re s follows: Step : From the experts swers of the first sectio i the questioire, the preferece of fetures will e compred. The compred results c e five differet scles. Therefore, we trsform these scles to the trigulr fuzzy umers. Step 2: For ech feture vlue of ech product, we compute the rtio of the rought oteook computers with respect to te models of ech product. Step 3: Compre the rtio results of ech product otied 70
Itertiol Jourl of Modelig d Optimiztio, Vol. No. April 202 from step 2 with other products. The differet or distce vlues will e trsformed ito the liguistic scle, tht is, we will get the trigulr fuzzy umers (TFNs. Step 4: Costruct the fuzzy pir-wise compriso mtrices sed o the trsformed TFNs. Accordig to these steps, te fuzzy pir-wise compriso mtrices will e costructed ccordig to te experts. These te costructed mtrices will e susequetly used to determie the product of oteook computers. Tle VI illustrtes oly the origil fuzzy pir-wise compriso mtrix of the first expert evlutio. TABLE VI: THE PAIR-WISE COMPARISON MATRIX OF THE FIRST EXPERT EVALUATION IN CRITERIA LEVEL C C2 C3 C4 C5 C6 C7 C8 C ( C2 C3 C4 C5 ( C6 C7 C8 ( ( ( ( ( 0.67 ( ( ( ( ( 0.67 ( ( ( ( ( 0.67 ( ( ( ( ( ( ( 0.67 ( ( ( ( ( 0.67 (.50, 2.50 (.50, 2.50 (.50, 2.50 (.50, 2.50 (.50, 2.50 ( 0.80, 0.90 C6 (0.7 0.80,.08 C7.5,.4 C8.28 (0.60, 0.83 (0.29, (0.54, 0.6 0.7 (0.29, 0.6 0.80, 0.98 (0.77,.08 (0.59, (0.7.38 (..43,.67 (0.83,.20 (.5,.4 (0.65,.8 (0.7.08 ( (0.53, 0.66, 0.85 TABLE VI: THE SUMS OF HORIZONTAL AND VERTICAL DIRECTIONS Criteri Row Sums Colum Sums C (6.55, 7.89, 9.6 (6.77, 8.29, 0.20 C2 (7.46, 8.88, 0.58 (6.23, 7.3 8.72 C3 (7.89, 9.43,.9 (5.98, 6.99, 8.33 C4 (7.36, 8.7 0.32 (6.36, 7.43, 8.79 C5 (5.77, 6.93, 8.47 (7.74, 9.93,.40 C6 (6.23, 7.50, 9.7 (7.4, 8.6 0.40 C7 (7.78, 9.24, 0.9 (6.0, 7.0, 8.39 C8 (5.53, 6.60, 8.04 (8.26, 0. 2. Sum of row or colum sums (54.57, 65.9, 78.3 2.50.5, 2.53 (.8,.5.88 ( B. The Computed Fuzzy Sythetic Extet Vlues The fuzzy sythetic extet vlue S i with respect to the i th criterio c e computed with (6. The exmple of clcultig this vlue for the criterio C is show elow. For other criteri, their fuzzy sythetic extet vlues re show i Tle VII. S C = (6.55, 7.89, 9.6 = (0.0836, 0.2 0.762 (54.57, 65.9, 78.3 V. FINDING RESULTS IN THE SELECTION PROBLEM OF PRODUCT OF NOTEBOOK COMPUTERS This sectio presets the fidig results i the selectio prolem of product of oteook computers icludig the ggregted fuzzy pir-wise mtrix, the computed fuzzy sythetic extet vlues, the pproximted fuzzy priorities for criteri, d the pproximted fuzzy priorities for ltertives. A. The Aggregted Fuzzy Pir-wise Mtrix After the experts fuzzy pir-wise compriso mtrices re costructed, the ggregted fuzzy pir-wise compriso mtrix is computed ccordig to (5 s show i Tle V. The sums of horizotl d verticl directios from Tle V re illustrted i Tle VI. The sum of row or colum sums will e used to compute the fuzzy sythetic extet vlues. TABLE V: THE AGGREGATED FUZZY PAIR-WISE COMPARISON IN CRITERIA LEVEL C C2 C3 C4 C5 C6 C7 C8 C ( C2.5,.53 C3 (.3.70 (0.65,.8 (.08 (0.59,.08 (.28.8.28 (..25,.39.5,.53 (.3.7.5,.4.5,.4 (.5,.3 (0.7.08.8.08.28 (.4.64,.85 (.3.7 C4.28 (0.65, 0.87.8.28 ( (.0.25,.5 (0.7.38 (0.60, 0.83 (0.83,.2 C5 (0.7 (0.77, (0.59, (0.66, ( (0.29, TABLE VII: THE FUZZY SYNTHETIC EXTENT OF EACH CRITERION Criteri Fuzzy Sythetic Extet Vlue ( S i C (0.0836, 0.2 0.762 C2 (0.095 0.36 0.938 C3 (0.008, 0.448, 0.2052 C4 (0.0940, 0.338, 0.89 C5 (0.0737, 0.063, 0.553 C6 (0.0796, 0.5 0.680 C7 (0.0993, 0.48, 0.999 C8 (0.0706, 0.0 0.474 C. The Approximted Fuzzy Priorities for Criteri From the fuzzy sythetic extet vlues, the o-fuzzy vlues tht represet the reltive prefereces or weights of oe criterio over other criteri will e pproximted. Ech of them is the degree of possiility computed s follows (show oly the weight of C criterio over others: V( S C S C 2 = (0.0952 0.762/(0.2 0.762 (0.362 0.0952 = 0.8429 V( S C S C 3 = (0.008 0.762/(0.2 0.762 (0.447 0.008 = 0.768 V( S C S C 4 = (0.0940 0.762/(0.2 0.762 (0.338 0.0940 = 0.8672 V ( S C S C 5 V ( S C S 6 = C = 7
Itertiol Jourl of Modelig d Optimiztio, Vol. No. April 202 V( S C S 7 = (0.0993 0.762/(0.2 0.762 (0.48 0.0993 C V ( S C S 8 C = = 0.7878 Hece, the reltive weight of the criterio C is: V( S S,... S = mi( 0.8428, 0.768, 0.867 0.7878, C C2 C8 = 0.768= w ( S The reltive weights of other criteri (w (S C2 util w (S C8 re computed d illustrted i Tle VIII. These reltive weights hve to e ormlized i order to llow them to e logous to weights defied from the AHP method. The ormlized weight w(s i is show i Tle VIII. TABLE VIII: THE NORMALIZED WEIGHT VALUES OF EACH CRITERION Criteri Reltive Weight Normlized Weight (w (S i (w(s i C 0.768 0.202 C2 0.962 0.446 C3 0.578 C4 0.8892 0.403 C5 0.5872 0.0927 C6 0.6943 0.095 C7 0.978 0.533 C8 0.573 0.086 Sum of W (S i 6.3380 From the weights of criteri, the criterio C3 (CPU speed hs the highest weight followed y the criterio C7 (durility. It c e iterpreted tht the experts give precedece to the CPU speed over every criteri i the prolem. The durility is the secod preferece criterio d so o. The cosistecy rte we got is 0.0028 which is less th 0. therefore, the mtrix c e cosidered to e cosistet. Now we kow the decisio criterio tht is most importt to select the product of oteook computers. Other criteri my ffect the product selectio of oteook computers lso. The ext step is to select the most importce product of oteook computers. D. The Approximted Fuzzy Priorities for Altertives Similrity, the, the trsformtio procedures for compriso etwee criteri sed o ech ltertive will e clculted. The reltive weights of criteri sed o ech ltertive re show i Tle XI. Filly, the fil results of ormlized weights from this tle with respect to the overll criteri weights re computed d show i Tle X. TABLE XI: WEIGHTS OF CRITERIA BASED ON EACH ALTERNATIVE C C2 C3 C4 C5 C6 C7 C8 A 0.085 0.05 0.096 0.04 0.30 0.09 0.6 0.30 A2 0.27 0.6 0.03 0.04 0.23 0.34 0.8 0.2 A3 0.072 0.055 0.039 0.065 0.005 0.097 0.074 0.037 A4 0.62 0.244 0.3 0.9 0.290 0.39 0.29 0.263 A5 0.3 0.79 0.225 0.33 0.30 0.0 0.8 0.83 A6 0.082 0.055 0.066 0.059 0.057 0.083 0.078 0.058 A7 0.07 0.062 0.073 0.8 0.046 0.089 0.084 0.057 A8 0.063 0.054 0.042 0.046 0.0 0.07 0.099 0.039 A9 0.08 0.052 0.027 0.06 0.27 0.069 0.095 0.080 A0 0.082 0.077 0.08 0.073 0.080 0.098 0.089 0.033 C TABLE X: THE WEIGHTS OF CRITERIA BASED ON EACH ALTERNATIVE WITH RESPECT TO THE OVERALL WEIGHTS OF CRITERIA C C2 C3 C4 C5 C6 C7 C8 A 0.00 0.03 0.02 0.03 0.06 0.03 0.04 0.06 A2 0.05 0.04 0.02 0.03 0.05 0.06 0.04 0.05 A3 0.009 0.007 0.005 0.008 0.00 0.02 0.009 0.004 A4 0.09 0.029 0.037 0.023 0.035 0.07 0.06 0.032 A5 0.04 0.022 0.027 0.06 0.06 0.03 0.04 0.022 A6 0.00 0.007 0.008 0.007 0.007 0.00 0.009 0.007 A7 0.03 0.007 0.009 0.04 0.006 0.00 0.00 0.007 A8 0.008 0.007 0.005 0.006 0.00 0.009 0.02 0.005 A9 0.03 0.006 0.003 0.03 0.05 0.008 0.0 0.00 A0 0.00 0.009 0.002 0.009 0.00 0.02 0.0 0.004 From Tle X, the priority of ech ltertive ws doe y cosiderig criteri. Bsed o the weights of criterio C3 (CPU speed, the product A4 is the highest d the product A5 is the secod. Weights of ll criteri of the product A4 lso e the highest compred with other products. The grph i Fig. 3 illustrtes the percetge of weight sums of ech ltertive. Fig. 3. The grph of weight percetge of ech ltertive From Fig. 3, the product A4 lso hs the highest ltertive weight. It revels tht the product A4 is the most preferle ltertive over the others with A5 the secod. Therefore for selectig product of oteook computers from my experts, vriety of products d multiple criteri with fuzzy AHP method, the product A4 is the est oe. VI. CONCLUSION This reserch used the fuzzy AHP to solve the prolem of evlutig d selectig product of oteook computers mog the others. It is utilized due to its ility for tkig ito ccout oth the qulittive d qutittive mesures. Eight decisio criteri hve ee used for ssessig te differet products of oteook computers. I this reserch, the trigulr fuzzy umers re utilized i estlishig the pir-wise comprisos of criteri d ltertives through liguistic scles. Further, group-sed fuzzy lyticl hierrchy process ws used i geertig criteri weights for the evlutio of products of oteook computers. By usig fuzzy AHP, the qulittive udgmet c e qulified to mke compriso more perceptio d reduce ssessmet is i pir wise compriso process. This fidig result will help the uyers to select the est product of oteook computers. However, uyig oteook computer is ctully sed o the 72
Itertiol Jourl of Modelig d Optimiztio, Vol. No. April 202 uyers ecuse of their udgets i hd d the differet stisfctio i desiged style of product. ACKNOWLEDGMENT Fudig for this reserch ws supported y the Reserch d Developmet Istitute, Udo Thi Rht Uiversity. We would like to thk oymous reviewers for their vlule commets d recommedtio. REFERENCES [] G. Işklr d G. Bűyűkőzk, Usig Multi-criteri Decisio Mkig Approch to Evlute Moile Phoe Altertives, Computer Stdrds d Iterfce, vol. 29, o. pp. 265-274, Fe. 2007. [2] T. L. Sty, the Alytic Hierrchy Process, Plig, Priority Settig, Resource Alloctio, New York: McGrw-Hill, 980. [3] T. L. Sty, How to Mke Decisio: The Alytic Hierrchy Process, Iterfces, vol. 24, o. 6, pp. 9-43, Nov.-Dec. 994. [4] C. L. Hwg d K. Yoo, Multiple Attriutes Decisio Mkig Methods d Applictios, Berli Heidelerg: Spriger-Verlg, 98. [5] J. P. Brs d P. Vicke, A Preferece Rkig Orgistio Method: The PROMETHEE Method for MCDM, Mgemet Sciece, vol. 3 o. 6, pp. 647-656, 985. [6] B. W. Tylor, Itroductio to Mgemet Sciece, New Jersey, Perso Eductio, 2004. [7] D. Chg, Applictios of the Extet Alysis Method o Fuzzy AHP, Europe Jourl of Opertiol Reserch, vol. 95, o. 3, pp. 649-655, Dec. 996. [8] P. J. M. V Lrhove d W. Pedrycz, A Fuzzy Extesio of Sty s Priority Theory, Fuzzy Sets d Systems, vol. o.3, pp. 99-227, 983. [9] T. Chg d T. Wg, Usig the Fuzzy Multi-Criteri Decisio Mkig Approch for Mesurig the Possiility of Successful Kowledge Mgemet, Iformtio Scieces, vol. 79, o. 4, pp. 355-370, Fe. 2009. [0] P. Srichett d W. Thurcho, Group Decisio Alysis for the Product Selectio Prolem of Noteook Computers usig Fuzzy AHP, i Proc. It. Cof. Computer d Computtiol Itelligece, Bgkok, 20 pp. 85-90. [] E. Tolg, M. L. Demirc, d C. Khrm, Opertig System Selectio Usig Fuzzy Replcemet Alysis d Alytic Hierrchy Process, Itertiol Jourl Productio Ecoomics, vol. 97, o. pp. 89-7, July 2005. [2] C. E. Bozdg, C. Khrm, d D. Ru, Fuzzy Group Decisio Mkig for Selectio mog Computer Itegrted Mufcturig Systems, Computers i Idustry, vol. 5 o. pp. 3-29, My 2003. [3] K. F. Kevi d C. W. Lu Hery, Evlutig Softwre Qulity of Vedors usig Fuzzy Alytic Hierrchy Process, i Proc. It. Multi Cof. Egieers d Computer Scietist, 2008, pp. 26-30. [4] D. Chtteree d B. Mukheree, Study of Fuzzy-AHP Model to Serch the Criterio i the Evlutio of the Best Techicl Istitutios: A Cse Study, Itertiol Jourl Egieerig Sciece d Techology, vol. o.7, pp. 2499-250, July 200. [5] Y. C. Eresl, T. Öc, d M. L. Derirc, Determiig Key Cpilities i Techology Mgemet usig Fuzzy Alytic Hierrchy Process: A Cse Study of Turkey, Iformtio Sciece, vol. 76, o. 8, pp. 2755-2770, Sep. 2006. [6] J. J. Buckley, Rkig Altertives usig Fuzzy Numers, Fuzzy Sets d Systems, vol. 5, o. pp. 2-3 Fe. 985. [7] J. J. Buckley, Fuzzy Hierrchicl Alysis, Fuzzy Sets d Systems, vol. 7, o. 3, pp. 233-247, Dec. 985. [8] D. Duois d H. Prde, Fuzzy Sets d Systems: Theory d Applictios, New York: Acdemic Press, 980.. Phrut Srichett is memer of IACSIT d IAENG. This uthor ws or t Skol Nkor provice, Thild. The dte of irth is Octoer 5, 970. The uthor s eductiol ckgroud is listed s follows: B.Sc. i mthemtics from Chigmi Uiversity, Chigmi, Thild i 992 d M.Sc. i techology of iformtio system mgemet from Mhidol Uiversity, Bgkok, Thild i 2000. Curretly, the uthor is the Ph.D. cdidte of Mhidol Uiversity, Thild. She worked s get sttistic officer, mthemtic sectio, Bgkok Life Assurce Co. Ltd., durig 992-996. Curretly, she is the lecturer t Udo Thi Rht Uiversity. The reserch res she iterests i d works o re softwre egieerig, dt miig, decisio lysis, d recommeder system. Wsiri Thurcho ws or t Udo Thi provice, Thild. The dte of irth is Mrch 3 972. The uthor s eductiol ckgroud is listed s follows: B.Sc. i computer sciece from Lmpg Rht Uiversity, Lmpg, Thild i 996 d M.Sc. i iformtio techology from Nresu Uiversity, Pitsulok, Thild i 2004. She worked s the lecturer i Lmpg Rht Uiversity durig 996-2007. The, she moved to Udo Thi Rht Uiversity d worked s lecturer t this uiversity util preset. The curret reserch iterests re e-commerce d decisio lysis 73