Multiple Attribute Decision Making Based on Interval Number Aggregation Operators Hui LI* and Bing-jiang ZHANG

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1 206 Iteratoal Coferece o Power, Eergy Egeerg ad Maageet (PEEM 206) ISBN: Multple Attrbute Decso Makg Based o Iterval Nuber Aggregato Operators Hu LI* ad Bg-jag ZHANG School of Appled Scece, Beg Iforato Scece ad Techology Uversty, Beg 0092, Cha *Correspodg author Keywords: MADM, Aggregato operators, Fallg shadows ethod, Posto weght. Abstract. Based o the fallg shadows ethod, a desred value of the sae attrbute s gve. I the coputg process, oe stuato that the dfferece betwee eda ad the desred value ay equal wll appear. For the partcular case, the rato of devato s used to reassg respectve weght. The weghts obey oral dstrbuto. Cobed terval uber aggregato operators, we ca obta alteratve values ad take a order the ultple attrbute decso akg proble. Itroducto Sce fuzzy decso akg s proposed by Zadeh [], t develops rapdly. Multple attrbute decso akg (MADM) probles are also portat research parts of fuzzy decso theory. Sce the object thgs are fuzzy ad ucerta, the attrbutes volved the decso probles are ot always expressed as crsp ubers ad soetes t s ore covet to deote the by fuzzy ubers, such as terval uber, lgustc varable, tutostc fuzzy uber etc. I ths paper, we use terval uber to express fuzzess ad ucertaty. However, A kd of sythess of all dvdual ca be acheved by a approprate aggregato. We cosder aggregato operator to perfor the operato aog terval ubers. Aggregato operator s a terestg research topc ad also a rapdly developg atheatcal doa. Yager [2] troduced the ordered weghted averagg (OWA) operator whch s a useful tool for aggregatg the exact puts. The key of the OWA operator s to detere ts assocated weghts. O Haga [3] was the frst to detere OWA operator weghts ad suggested a u etropy ethod, whch forulated the OWA operator weght proble as a costraed olear optzato odel wth a predefed degree of oress as ts costrat ad the etropy as ts objectve fucto. Flev ad Yager [4,5] suggested a learg approach usg observed data ad a expoetal soothg ethod. Fullér ad Majleder [6] showed that the u etropy odel could be trasfored to a polyoal equato whch could be solved aalytcally. For a u varace ethod to obta the al varablty OWA operator weghts, Fullér ad Majleder [7] also suggested ther dea. Majleder [8] exteded the u etropy ethod to Rey etropy ad proposed a al Rey etropy ethod that produced al Rey etropy OWA weghts for a gve level of oress. Wag ad Parka [9] troduced a dsparty approach whch detered the OWA operator weghts by zg the u dfferece betwee two adjacet weghts uder a gve level of oress. Wag ad Luo [0] troduced two ew ethods for deterg the OWA operator weghts: the least squares ethod ad the ch-square ethod. Xu ad Da [] establshed a lear objectve-prograg procedure for obtag the OWA weghts fro observatoal data uder partal weght forato. Wth respect to the hybrd aggregato (HA) operator, they are also used to deal wth the proble of a kd of sythess of all dvdual. I ths paper, we utlze these tools to deal wth MADM proble preseted fuzzess for of terval uber. I Secto 2, we troduce soe basc deftos ad aggregato operators. The, Secto 3, we kow evaluate results of decso-aker to object thgs whch are geerally oral dstrbuto. Here we troduce oe ethod how to obta posto weght of alteratves the sae attrbute. We also utlze fallg shadows ethod to express the overall desred value ad respectve devato.

2 Algorth steps are preseted Secto 4. I Secto 5, a uercal experet that we evaluate the level of soe uverstes s llustrated how to use the above-etoed operators to slove practcal probles. Fally, Secto 6 suarzes the paper. Iterval Nuber ad Aggregato Operator Noral Dstrbuto Iterval Nubers Defto (See [2]) Supposed terval uber ab,, f attrbute value ra, b 2 r N,, the ab, s called oral dstrbuto terval uber, wrtte, ad, where accordg to a oral dstrbuto 3 prcple, the desred value ad the varace are as follows: 2 ab () 6 ba (2),, Defto 2 (See [2]) Supposed arbtrary two oral dstrbuto terval ubers,, the ) If 2, the 2. 2) If 2, 2, the , the , the 2. Aggregato Operator Cosder the oral dstrbuto terval uber oe attrbute, such as the th attrbute, 2,,,, j,2,,,, j,2,, to,, ad sort,, j,2,, accordace wth oral dstrbuto. Assue v v, v, j,2,,,.e., j, j j, j j, j, here whe t coes to the sae posto, adjust the orgal posto weght. That s to say, the orgal posto weght v v T, v2,, v has chaged as the adjusted posto weght T v v, v,, v a certa for. Ad, (), () (2), (2) ( ), ( ) 2 2,,,,,, aggregato operator. Defto 3 The terval ubers ordered weghted averagg (INOWA) s a appg:, such that s oe replaceet of, 2,, INOWA,,, w w w w j 2 j j j 2 2 j j. Hece, we defte a ew where w w w w T s the attrbute weght vector wth w 0,, 2,,,,, 2, we call t the terval uber ordered weghted averagg (INOWA) operator. (3) w ;

3 Fro forula (3), we kow the result of above-etoed aggregato operator s stll a terval uber ad the aggregato result s 2 2 INOWA w j, 2 j,, jw, w j, 2,, (4) Fally, assue that INOWA w j, 2 j,, j, ad accordg to Defto 2, copare the ad take a order MADM probles. Fallg Shadows Method ad Coputg Method of Posto Weght Fallg Shadows Method I the coprehesve evaluato of large scale, the terval uber wth people fuzzess ad ucertaty s cosstet, whch ca copesate the defceces of deterstc values to a greater extet. But the terval uber ca ot be drectly apples to the exstg coprehesve evaluato ethods. However, the set-value statstcs theory s a effectve way to solve ths proble. Fuzzy Sets ad fallg shadows of Rado Sets are troduced Wag Pezhuag [3]. I ths book, we ca uderstad the set-value statstcs theory, fallg shadows fucto ad the easure to fallg shadows. It s appled to hadle wth fuzzy forato ad the estato of fallg shadows fucto etc. I geeral, fallg shadows ethod s by evaluate group decso akg proble. However, ths paper, we ake evaluatve alteratve the sae attrbute as a sere ad use posto weght to sort alteratves. At the sae te, we also use fallg shadows fucto solve the total desred value ad devato every attrbute. For the judget atrx A ( a ),,2,,, j,2,,, f the eleet a s fuzzy uber, the we ca use terval ubers a [ a, a ] to dcate that t s the coparatve result betwee evaluato alteratve ad j wth respect to every attrbute. The the judget atrx A s a terval uber judget atrx, where a a. Thus, the dfferet alteratves the sae attrbute have fored a sequece,.e., {[ a, a ],[ a 2, a 2],,[ a, a ]},,2,, (5) where a { a }, a { a },, 2,,, j, 2,,, the the coparatve result aog evaluato alteratve j the sae attrbute s a rado dstrbuto o [ a, a ]. For arbtrary pot a, the fuzzy coverage scale s defed as where f ( a ) f ( a ) a j (6), a [ a, a ] fa ( a ) 0, other (7) The desred value of dfferet alteratves oe attrbute s aˆ a a a a a f ( a ) da f ( a ) da (8)

4 Accordg to the algorths of terval uber, by the Eq. 3 ad Eq. 4, we ca be derved ad a a f ( a ) da [ a a ] j (9) a a a f a da a a 2 2 ( ) [( ) ( ) ] 2 j Fro, the Eq. 0 Eq. Eq. 2 ca be wrtte as (0) aˆ 2 2 [( a ) ( a ) ] j 2 [ a a ] j ad ts rado devato s () s ( a a ) da [( a a ) ( a a ) ] a ˆ ˆ ˆ a 3 j Coputg Method of Posto Weght The evaluato result of decso-aker to objectve attrbute s geerally oral dstrbuto. We should cosder respectve posto weght betwee attrbute values. Defto 4 (See [4]) Let x be the cotuous rado varable, ad defe ts probablty desty fucto as x 2 2 f x e 2, x where ad 0 2 are kow quatty. The x s orally dstrbuted wth a ea of ad a stadard devato of. The oral dstrbuto provdes a realstc approxato to the dstrbuto of devatos ay experetal stuatos, especally for the cetral porto of the devatos. Assue vj 0, j, 2,,, v j ad the collecto of, 2,,, we have j (2) e v e,,2,, j 2 2 e where j, If the codto that the eda of attrbute value are equal appears, the weght of the sae posto also s equal. Hece, we eed a ethod to do the secod allocato ter of rato of correspodg varace. The weghts obey oral dstrbuto.. (3) Algorths Based o dscrpto of aggregato operators, fallg shadows ethod ad posto weght, we put forward to algorth steps about slovg MADM probles.

5 Step Based o the fallg shadows ethod, copute the overall desred value ad devato value. Step 2 Accordg to a practcal ssue, we should aalyse dfferet attrbute of the ssue. For every attrbute, calculate the eda ad respectve devato value ad copare the dfferece betwee the eda ad desred value. If occur oe stuato that two or ore dfferece value are equal, we eed reallocate posto weght for of the rato of respectve devato. Step 3 Calculate the posto weght ad adjust the proporto of the sae weght part. Step 4 Uder every attrbute, copute the desred value,.e. the desred value s that ultply by eda of terval ubers ad respectve adjustve posto weght. Step 5 Based o the INOWA operator, ake use of the result of Step 4 to calculate every alteratve value. If alteratve values are very close to each other, the we wll eed copare devato both the. Step 6 Accordg to the result of Step 5 ad the rule of Defto 2 take a order wth every alteratve. Nuercal Experets Soe uverstes are evaluated as the top, the ordary ad the low. We ca use teachg, scetfc research ad servce as assesset crtera ad attrbute weght w 0.3, 0.4, 0.3 T. Accordg to assesset crtera, we eed to ark wth fve sttutes A, 2, 3, 4. Evaluato forato of every attrbute presets ter of terval uber as follows. Table. Decso-akg Matrx. Alteratves A A 2 A 3 A 4 A 5 Attrbute I I 2 I 3 [73.4,77.6] [79.,80.9] [77.9,79.7] [77.,78.9] [78.0,8.0] [83.8,89.2] [80.7,84.3] [8.9,83.] [8.4,85.6] [74.0,77.0] [8.7,85.3] [79.,83.9] [87.,9.9] [82.9,87.7] [79.0,82.0] Frstly, the data of Table have bee adjusted ad sorted for of oral dstrbute. Accordg to the fallg shadows ethod,.e. Eq. ad Eq. 2, see table 2. Table 2. The desred value ad devato value. Desred value Attrbute /Devato value I I 2 I 3 a ˆ Next, For attrbute I, every eda ad respectve devato fro Eq. ad Eq. 2 are as follows. Iterval uber j j, Table 3. Meda ad devato of attrbute I. Alteratve A A 2 A 3 A 4 A , , , , ,0.5 So, the dfferece d ˆ a are respectvely 5.569,3.069,.43,5.569,8.43 Fgure. Obvously, we ca fd two dfferece values equal. It s ecessary for us to allocate correspodg posto weght value ter of devato.

6 Dfferece Posto Fgure. The dffereces I. Cosder the posto, 2,3, 4,5, fro Eq. 3, the adjustve weght value of all attrbutes Table 4. Table 4. The adjustve weght. Attrbute A A 2 Alteratve A 3 A 4 A 5 I I 2 I Fro the INOWA operator calculato,.e. Eq. 4, we have table 5 as follows. Table 5. The alteratve values. Alteratve A, A A A 2 A 3 A 4 A 5 {9.2758,0.0807} {6.4226,0.34} {25.38,0.580} {6.9680,0.427} {4.42, 0.36} But, because the desred value of A2 ad A4 s qute close, we should aalyse respectve devato. Fally, based o Defto 2, The orderd result s A3 A2 A4 A5 A. Cocluso I ths paper, wth respect to ultple attrbute decso akg probles whch both the attrbute weght ad posto weghts take the for of terval ubers, the approach of deterg posto weght s obtaed by oe rule that evaluate object thgs always follow oral dstrbuto. O the oe had, we utlze fallg shadows ethod to acqure the desred value of the sae attrbute ad copare the dfferece. The approach s to ake evaluato alteratves as oe seres a attrbute. O the other had, we aalyse several class of aggregato operator to ga alteratve values dfferet attrbute. I addto, the paper also has soe proveets, for exaple, the dfferece betwee eda ad the desred value ay obey Posso dstrbuto; the dstrbuto of posto weghts ay obey Posso dstrbuto. Ackowledgeets The authors would lke to express ther grattude to the Edtor ad the aoyous referees for ther careful atteto ad precous suggestos to prove the auscrpt. The research was supported

7 by the Natoal Natural Scece Foudato Project (609725) ad Beg ucpal educato cosso research project (SM ). Refereces [] Zadeh, L.A., Fuzzy set, Iforato ad Cotrol, 965, 8: [2] Yager, R.R., O OWA aggregato operators ultcrtera decso akg, IEEE Trasactos o Systes, Ma ad Cyberetcs, 8 (988) [3] O Haga, M., Aggregatg teplate or rule atecedets real-te expert systes wth fuzzy set logc, : Proceedgs 22d Aual IEEE Asloar Coferece o Sgals, Systes ad Coputers, Pacfc Grove, 988, pp [4] Flev, D. ad Yager, R.R, Aalytc propertes of al etropy OWA operators, Iforato Sceces, 85 (995) -27. [5] Flev, D., Yager, R.R, O the ssue of obtag OWA operator weghts, Fuzzy Sets ad Systes, 94 (998) [6] Fuller, R., Majleder, P., A aalytc approach for obtag al etropy OWA operator weghts, Fuzzy Sets ad Systes, 24 (200) [7] Fuller, R., Majleder, P., O obtag al varablty OWA operator weghts, Fuzzy Sets ad Systes, 36 (2003) [8] Majleder, P., OWA operators wth al Rey etropy, Fuzzy Sets ad Systes, 55 (2005) [9] Wag, Y.M., Parka, C., A dsparty approach for obtag OWA operator weghts, Iforato Sceces, 75 (2005) [0] Wag, Y.M., Luo, Y., Two ew odels for deterg OWA operator weghts, Coputers ad Idustral Egeerg, 52 (2007) [] Xu Zeshu, Da Q.L., The ucerta OWA operator, It J Fuzzy Itell Syst, 7 (2002) [2] Wag Xfa, Xao Masheg, Approach of group decso based o oral dstrbuto terval uber wth coplete forato, Cotrol ad Decso, 25 (200) [3] Wag Pe Zhuag: Fuzzy Sets ad fallg shadows of Rado Sets, Beg Noral Uversty Press, Beg, 985. [4] Xu Zeshu, A overvew of ethods for deterg OWA weghts, Wley IterScece, Iteratoal Joural of tellget systes, 20 (2005)