APPLICATION OF FUZZY TOPSIS METHOD AND DOPGA ALGORITHM IN PURCHASING DECISION PROCESS: 3D TELEVISION EXAMPLE. Engin Ufuk ERGUL

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1 S.Ü. Müh. Blm ve Tekn. Derg., c.6, s.4, ss , 208 Selcuk Unv. J. Eng. Sc. Tech., v.6, n.4, pp , 208 ISSN: (Electronc) DOI: 0.537/Sctech APPLICATION OF FUZZY TOPSIS METHOD AND DOPGA ALGORITHM IN PURCHASING DECISION PROCESS: 3D TELEVISION EXAMPLE Engn Ufuk ERGUL Amasya Unversty, Faculty of Technology, Department of Electrcal and Electroncs Engneerng, 0500, Amasya, Turkey (Gelş/Receved: ; Kabul/Accepted n Revsed Form: ) ABSTRACT: Nowadays, televson s at the head of devces where people can have fun n ther homes, and people spend most of ther tme on televson. So, they want to purchase a pretty good televson. At the head of these televsons are three-dmenson televsons (3D TVs). Snce purchasng a televson s a long-term shoppng, t s necessary to evaluate t wth great care and to determne ts best before purchasng t. Evaluaton of the best 3D TV s a complex problem wth no defnte structure and t depends on the features of the 3D TVs. There are many performance ndcators as dynamc contrast rato, refresh rate, power consumpton on mode, depth, weght, cost, response tme etc. affectng the decson-makng process about 3D TVs. Therefore, evaluaton of 3D TVs s a complex decson- makng process. Hence, ths problem can be solved by mult-crtera decson-makng (MCDM) methods. In ths paper, eght popular 3D TVs consdered to be purchased n technology markets are compared by usng sx specfcatons obtaned from ts catalog and the experts opnon. In ths study, n order to determne the best 3D TV, alternatve 3D TVs were frst sorted by usng Fuzzy Technques for Order Preference by Smlarty to Ideal Soluton (TOPSIS) method, and then, the senstvty analyss was performed. Secondly, to compare the 3D TV rankng obtaned from Fuzzy TOPSIS method, 3D TVs were ordered by usng Domnaton Power of an Indvdual Genetc Algorthm (DOPGA) method and necessary comparsons and evaluatons were made and the best 3D TV was selected. Key Words: Decson makng, Evolutonary algorthms, Fuzzy TOPSIS, MCDM, Senstvty analyss, 3D TV Satın Alma Karar Sürecnde Bulanık Topss Yöntem ve Dopga Algortmasının Uygulanması: 3D Televzyon Örneğ ÖZ: Günümüzde nsanların evlernde eğlencel vakt geçreblecekler aygıtların başında televzyon gelmektedr ve vaktlernn çoğunu televzyon karşısında geçrmektedrler. Bu yüzden oldukça y br televzyon satın almak sterler. Bu televzyonların başında üç boyutlu televzyonlar (3D TV) gelmektedr. Televzyon satın alma, uzun vadel br alışverş olduğu çn satın almadan önce oldukça ttz br çalışmayla değerlendrme yapmak ve en ysn belrlemek gerekmektedr. En y 3D TV nn değerlendrlmes, kesn br yapıya sahp olmayan karmaşık br problemdr ve 3D TV nn özellklerne bağlıdır. 3D TV nn karar verme sürecn etkleyen, dnamk karşıtlık oranı, tazeleme oranı, güç tüketm, dernlk, ağırlık, malyet, cevap süres gb brçok performans gösterges vardır. Bu yüzden, 3D TV nn değerlendrlmes karmaşık br karar verme sürecdr. Bu nedenle, bu problem çok krterl karar verme (ÇKKV) yöntemler le çözüleblr. Bu makalede, teknolo marketlernden satın alınması düşünülen sekz popüler 3D TV, kataloglarından ve uzmanların görüşlernden elde edlen altı özellğ kullanılarak kararlaştırılmıştır. En y 3D televzyonu belrlemek çn yapılan çalışmada lk olarak Bulanık İdeal Çözüme Benzerlkle Terch Sıralama Teknğ

2 64 E. U. ERGUL (TOPSIS) metodu uygulanarak 3D TV ler sıralanmış ve daha sonra duyarlılık analz yapılmıştır. İknc olarak, Fuzzy TOPSIS yöntemnden elde edlen 3D TV sıralamasını karşılaştırmak çn Breyn Baskınlık Gücü Genetk Algortma (DOPGA) yöntem uygulanarak 3D TV ler sıralanarak gerekl karşılaştırmalar ve değerlendrmeler yapılmış ve en y 3D TV seçlmştr. Anahtar Kelmeler: Karar verme, Evrmsel algortmalar, Bulanık TOPSIS, MCDM, Duyarlılık analz, 3D TV INTRODUCTION TV technologes are developed very fast n every day. Therefore, more specfc TVs appear n technology markets. Every year mllons of televsons are sold n the technology markets. Accordng to Che (2009), Due to the changes of the dgtal sgnal transmsson mode, a dsplay technque revoluton has begun n TV market, among the LCD TV havng the advantages of slght, electrcty-savng, radaton-free and so on. LCD TV products are very popular nowadays. So, the LCD TVs are the best-sellng TVs n technology markets. When customers are buyng a TV, generally they look ts appearance, ts smartness and whether t has 3D or not except ts performance parameters. Therefore, TV sellers start not to share the real performance parameters and also they use varous terms nstead of ts (hgh dynamc contrast rato, ultra-dynamc contrast rato etc.). So, customers must consder a lot of parameters when purchasng a TV. The parameters taken nto consderaton n TV purchasng are: dynamc contrast rato, refresh rate, power consumpton on mode, depth, weght, cost, response tme, brghtness, statc contrast rato, number of connectons (HDMI, DVI etc.), smartness (nternet connecton), type (passve or actve) and number of 3D glasses. Because of TV sellers not sharng the real TV parameters wth the customers, people who consder buyng a TV are ndecsve durng the purchase process. Because of varous TVs and a lot of parameters, TV customers have to make a mult-crtera selecton. So, TV purchasng process must be thought a MCDM problem and MCDM methods must be used to solve t. Determnng the optmal soluton among alternatves s a very dffcult stuaton for the decson makers (Grobbelaar and Vsser, 205). Conventonal methods cannot acheve a certan soluton n ths selecton operaton. Nevertheless, Mult-Crtera Decson-Makng (MCDM) methods can solve the problem easly. Fuzzy logc and fuzzy set theory are very effectve methods to solve mult-crtera decson-makng ssues. Technques for Order Preference by Smlarty to Ideal Soluton (TOPSIS), Fuzzy TOPSIS, Analytc Network Process (ANP), Fuzzy ANP (FANP), Analytc Herarchy Process (AHP), ELECTRE, Data Envelopment Analyss (DEA) and genetc algorthms (GA) are often used for MCDM (Yayla and Yldz, 203). A comprehensve lterature revew about MCDM methods can be found n (Yldz and Yayla, 205). Shahanagh and Yazdan (2009) appled the fuzzy TOPSIS method for the best suppler selecton accordng to the determned crtera, n terms of purchasng of man components from alternatve supplers n the applcaton performed n the automotve company and the best suppler was selected at the end of the calculatons performed after the assessment of four alternatve supplers by three decson makers, accordng to the determned crtera. In the study of Wang et al. (2009), three supplers were assessed and arranged by three decson makers accordng to four selecton crtera n the study of suppler selecton, whch was performed accordng to herarchcal fuzzy TOPSIS. It was ndcated that ths method was more reasonable than other methods and could be appled for the calculaton of the weghts. Yayla et al. (202) used fuzzy TOPSIS method to select the most sutable suppler n a textle frm n Turkey. Accordng to results, rankng of three supplers determned by n terms of closeness ndex values. Roshandel et al. (203) used fuzzy TOPSIS to select the best suppler from four supplers that supply a bg healthcare products manufacturer n Iran. Vnodh et al. (204) developed a model and technque based on TOPSIS and AHP to evaluate the performance n fuzzy envronment. In ths study, t was amed to select the best plastc recyclng between

3 Applcaton Of Fuzzy Topss Method And Dopga Algorthm In Purchasng Decson Process: 3d Televson Example 65 varous processes of plastc recyclng and the combned approach whch was determned the best one. Cheng et al. (2006) used genetc algorthms and TOPSIS for calbraton of conceptual ranfall-runoff models n a new framework and they compared the results of two methods. Ic (204) studed the applcaton of a new approach, Desgn of Experment and TOPSIS method (DoE TOPSIS) together make company rankng as frequently encountered n the real-tme fnancal envronment. Bas (203) proposed an ntegrated framework for the analyss of an electrcty supply chan usng an ntegrated SWOT-fuzzy TOPSIS methodology combned wth AHP. Km et al. (203) proposed a new framework that prortzed the best stes for treated wastewater (TWW) n stream use by fuzzy TOPSIS. Rouyendegh and Saputro (204) provded an overvew of the fuzzy TOPSIS and Mult-Choce Goal Programmng (MCGP) methods for MCDM problem under uncertan envronments. Kannan et al. (204) used ths method to select the best green supplers for an electroncs company. Lee et al. (204) used fuzzy TOPSIS to mprove the general flood vulnerablty approach. Taylan et al. (204) combned fuzzy decson tree, fuzzy TOPSIS, and DEA for selectng Ar Traffc Controllers (ATCs). Yeh and Chuang (20) used two MOEAs to fnd a mathematcal plannng model for green partner selecton and they used the weghted sum approach to obtan much more Pareto-optmal solutons. Chan and Chung (2004) proposed a mult-crteron genetc algorthm for an orderly dstrbuton n a demand-drven supply chan. Yan (2009) proposed an ntegrated approach adoptng a genetc algorthm and AHP to accomplsh green suppler optmzaton. Altparmak et al. (2006) proposed a new soluton procedure based on genetc algorthms to fnd a Pareto-optmal set for mult-obectve suppler chan desgn problem. Chang (202) used a MOEA to search for the optmal desgn chan partner combnaton to mnmze product development cost and tme and maxmze product relablty and employed t for a dgtal TV box. Bandyopadhyay and Bhattacharya (203) used a NSGA-II based MOEA and appled t to a novel mult-obectve problem for a two-echelon seral supply chan. Ozcan et al. (207) used ANP and TOPSIS methods for evaluatng alternatve renewable energy nvestments n Turkey. In ths study, Fuzzy TOPSIS and DOPGA (Domnaton power of an ndvdual genetc algorthm) whch s one of the MOEA algorthms are used to evaluate the best 3D TV through the alternatve 3D TVs. Afterwards, the results are compared and the best 3D TV s determned. The rest of ths paper s organzed as follows. In Secton 2, general nformaton about fuzzy TOPSIS, MOEA and DOPGA are gven. In Secton 3, these methods are appled to the evaluaton of the best 3D TV problem separately, and the results are gven. Fnally, Secton 4 contans concludng remarks, assessments, and future drectons. MATERIALS AND METHODS Fuzzy Topss Technques for Order Preference by Smlarty to Ideal Soluton(TOPSIS) s a method that ranks the preferences by usng smlarty to the optmum soluton. Ths method s currently one of the most useful MCDM methods (Dymova et al.,203; Chu et al.,205). TOPSIS ams to reach the closest truth (Chen and Tsa, 2007). Km et al. (203) stated that TOPSIS method s used to solve multple-attrbute decson makng problems n whch preference nformaton s not artculated. In ths method, the deal alternatve has the best values for all attrbutes, whereas the negatve deal s the alternatve wth all of the worst attrbute values. In TOPSIS method, frst, the postve and negatve deal solutons are defned and after that, the dstance of the alternatves from the deal solutons are found. TOPSIS method dentfes a closeness coeffcent ndex to the postve deal soluton (PIS) and remoteness to the negatve deal soluton (NIS). TOPSIS selects the best alternatve accordng to maxmum closeness coeffcent to the PIS (Kannan et al., 204).

4 66 E. U. ERGUL Fuzzy TOPSIS method, whch has demonstrated very successful applcaton examples of real-world problems n whch especally personal udgment are expressed wth lngustc data, can be utlzed n makng group decsons n fuzzy envronments. Classcal TOPSIS method s gven superorty because the fuzzy TOPSIS method ensures freedom for expressng experts thoughts n a certan range, as well as beng a mathematcal method handlng the udgments of TOPSIS method experts wth quanttatve data (Chen, 2000). Chen (2000) extended TOPSIS to the fuzzy case. The lngustc expressons of the fuzzy theory are consdered as natural presentatons of preferences, the fuzzy mportance weghts of the crtera and the fuzzy ratng of alternatves at crtera are nputs and located n a matrx form n the fuzzy TOPSIS approach. The TOPSIS algorthm s gven below (Chen, 2000): Step : Inputs are gven n a decson matrx format: D ~ W ~ ~ x ~ x 2 ~ x ~ x 2 22 ~ x ~ m x m2 w ~, w~,, w~ 2 n ~ x n ~ x 2n ~ x mn () (2) Here, x~ are fuzzy ratngs of the alternatves, number of alternatves, n s the number of crtera. ( Step 2: Compute the normalzed fuzzy decson matrx R ~ : w ~ are fuzzy mportance weghts of the crtera, m s the, 2,..., m,, 2,..., n ) R ~ = ~ r mxn (3) For the beneft crtera, the normalzed value of r~ s computed as: ~ l m r,, u u where; u u (4) u max u Normalzed value of r~ for the cost crtera s computed as: ~ l l l r,, u m l (5) (6) Here; l mn l (7)

5 Applcaton Of Fuzzy Topss Method And Dopga Algorthm In Purchasng Decson Process: 3d Televson Example 67 Step 3: Compute the weghted normalzed fuzzy decson matrx s V ~ : V ~ ~ v m n The weghted normalzed fuzzy value v ~ s computed as: (8) Here v ~ ~ r w~ (9) w ~ s the fuzzy weght of th crteron. Step 4: Defne the fuzzy PIS (FPIS) and the fuzzy NIS (FNIS): A v~, v~,..., v~ 2 n A v~, v~,..., v~ 2 n (0) () In Eqs(0)-(), v~,, and v ~ 0, 0, 0. Step 5: Compute the dstances of each alternatve to the FPIS and FNIS by usng: d d n n d ~ ~ v v (2) d v~ v~ In Eqs(2)-(3), d v~, v ~ ) shows the dstance between two fuzzy numbers. a b Step 6: Compute the relatve closeness to the deal soluton. The relatve closeness of the alternatve computed: (3) A s C d _ (d _ d ) (4) Here 0 C, an alternatve s closer to the FPIS and far from the FNIS as C approaches to. Step 7: Rank the alternatves accordng to C. Then, an alternatve whch has maxmum C s selected. Mult-Obectve Evolutonary Algorthm (MOEA) Evolutonary Algorthm (EA) s a global search method that mmcs the process of natural evoluton. EAs have been successful to solve many mult-obectve real-world optmzaton problems that have several conflctng obectves. Solvng a mult-obectve optmzaton problem (MOP), t can be found many Pareto-optmal solutons. The set of all the Pareto-optmal solutons s called the Pareto set (PS) and ts mage n the obectve space s the Pareto front (PF). MOEAs am to fnd reasonable numbers of solutons to approxmate the PF. Fndng PF s not enough for solvng a MOP all alone, after that, a decson maker must choose a sngle soluton wthn PF. There are a lot of MOEA methods proposed n the lterature. Interested readers may refer to Deb (200), Ghosh and Dehur (2004) and Coello et al. (2007) for detaled nformaton regardng the MOEA methods.

6 68 E. U. ERGUL In ths paper, a newly proposed ftness assgnment scheme s used for rankng some 3D TV brands usng ther sx specfcatons (refresh rate, dynamc contrast rato, power consumpton on mode, weght, depth, and cost). DOmnaton Power of an ndvdual Genetc Algorthm (DOPGA) ranks among all the solutons (Ergul and Emnoglu, 204). Furthermore, DOPGA does not use ftness sharng, nchng and crowdng dstances etc. to rank the solutons. In lterature, there are a lot of ftness assgnment scheme for rankng the Pareto-optmal solutons (Fonseca and Flemng, 993; Srnvas and Deb, 994; Ztzler and Thele, 999; Ztzler et al., 200; Deb et al., 2002). They ranked solutons usng ther schemes to fnd the best solutons to the mult-obectve problems. These methods dd not use only ftness assgnment to solve MOPs but also used GA operators (crossover and mutaton), selecton mechansms, eltsm strateges, and archve management. Detaled surveys of MOEAs can be accessed n Coello(2006) and Zhou et al.(20). The ftness values of the solutons are calculated by usng a two-phased algorthm n DOPGA. In the frst phase, the populaton s separated nto several subpopulatons and the rank values of the solutons n the subpopulatons are determned by MOGA (Mult-obectve Genetc Algorthm) (Fonseca and Flemng, 993). In MOGA, the rank of a soluton s found by the number of solutons by whch t s domnated. All Pareto solutons are assgned rank, whle domnated ones are penalzed accordng to the populaton densty of the correspondng regon of the trade-off surface. In the second phase, the solutons n each subpopulaton are re-ranked accordng to the domnaton power. The ftness assgnment approach of DOPGA only uses the neghborhood of the solutons. Therefore, a decson maker can easly select the best soluton. Because, DOPGA can order the solutons from the best one to the worst one n a unque order. In ths paper, DOPGA s only used to rank the solutons, so we use only ftness assgnment (or rankng) scheme. After the DOPGA, solutons have a unque rank and DOPGA can sort them the best one to the worst one. In the MOEA lterature, the use of pror methods n decson makng s very popular. In these methods, MOPs are transformed to SOPs (Sngle Obectve Optmzaton Problems) usng aggregaton methods lke weghted sum approaches. Domnaton power of an ndvdual genetc algorthm (DOPGA) DOPGA s a recently proposed ftness assgnment scheme for MOEAs (Ergul and Emnoglu, 204). DOPGA assgns a ftness value to a soluton n a two-phased algorthm. In the frst phase, the man populaton s separated nto several subpopulatons by usng MOGA (Fonseca and Flemng, 993). In the next phase, solutons are re-ranked accordng to the domnaton power algorthm. In ths paper, we use only the ftness assgnment (or rankng) scheme of DOPGA. The whole algorthm of DOPGA can be found n Ergul and Emnoglu(204). The rankng or ftness assgnment algorthm of DOPGA s gven as: Step : Assgn dummy ftness to each soluton nversely proportonal to ther sub-populaton number (obtaned from MOGA algorthm): df ( sol ) sub populaton( ) (5) where sol sub populaton( ),,, p, and. means the number of elements n the set. Step 2: Calculate relatve domnaton power of a soluton. To do so, fnd out how many solutons are domnated by ths partcular soluton and sum all the dummy ftnesses of domnated solutons. Ths wll determne: rdp( sol ) df ( sol ) (6) r k k

7 Applcaton Of Fuzzy Topss Method And Dopga Algorthm In Purchasng Decson Process: 3d Televson Example 69 where sol k s domnated by sol and k,, r, k. Step 3: Repeat Step 2 for the rest of the solutons wthn the same subpopulaton. Sum the rdp of all solutons wthn the same sub-populaton and obtan total domnaton power for that sub-populaton as formulated below: p tdp( sub populaton( )) rdpsol (7) n represents all solutons nsde the sub populaton( ). where,, p Step 4: Calculate the real ftness or rank for the -th soluton wth the followng formula (where e-6 s added to the denomnator to avod dvde by zero): rf ( ) sub populaton( ) rdp( ) tdp( sub populaton( )) e 6 (8) Step 5: Repeat Step 2 to Step 4 untl all solutons are ranked. For the sake of clarty, there s a populaton of solutons ranked by DOPGA n Fg.. Frst, solutons are ranked by usng MOGA. So, the solutons have dentcal ranks. In Fg. a; soluton A, B, C, and D are n the frst Pareto-optmal front and ther ftness values are one (). Soluton E and F are on the second front and ther ftness values are two (2). Soluton G has ftness value of three (3) and Soluton H has ftness value of eght (8). Second, each soluton n all subpopulatons s re-ranked by usng the domnaton power concept. The fnal (or real) rank values of the solutons are found by usng the above algorthm. An example populaton s depcted n Fg. b. For a better understandng, the rank of soluton B s computed step by step below. Step : Soluton B s on the frst Pareto-optmal front (the frst subpopulaton) df ( B) sub populaton( B) Step 2: Soluton B domnates three solutons (F, G, and H), therefore the relatve domnaton power of B s calculated as: rdp( B) df ( F) df ( G) df ( H) rdp( B) Step 3: Total domnaton power of the frst subpopulaton s found by summng the relatve domnaton powers of all solutons on ths front. tdp( sub pop(st)) rdp( A, B, C, D) Step 4: The real ftness or rank of soluton B s calculated as:

8 620 E. U. ERGUL rdp( B) rf( B) sub pop( B) tdp( sub pop( st)) e rf( B) e 6 Solutons n the frst Pareto-optmal front have dfferent ftness values n Fg. b. (a) (b) THE CASE STUDY Fgure. (a) Raw ftness values and, (b) ranks or real ftness (rf) values In ths paper, Fuzzy TOPSIS and DOPGA are mplemented for evaluatng the best 3D TV through the alternatve 3D TVs. Eght 55 nch Full HD 3D LED TVs are used n the evaluaton/decson process. TV catalogs have been utlzed to determne the crtera for evaluatng the best 3D TV. It s benefted from three experts opnon to determne the TV parameters used n ths paper. The TV parameters used n ths study are dynamc contrast rato, refresh rate, weght, power consumpton, depth, and cost. Herarchy of the 3D TV evaluaton problem s depcted n Fg. 2. Fgure 2. Herarchy of the 3D TV selecton problem

9 Applcaton Of Fuzzy Topss Method And Dopga Algorthm In Purchasng Decson Process: 3d Televson Example 62 Flowchart of the methods appled to the 3D TV evaluaton problem s shown n Fg. 3. Fgure 3. Flowchart of the methods appled to the TV selecton problem. Applcaton of Fuzzy TOPSIS Method to 3D TV Evaluaton Problem The catalog values of selected parameters of eght alternatve 3D TVs are gven n Table. Snce the crtera of weght, power consumpton on mode, depth, and cost ($) are generally known; only the dynamc contrast rato and refresh rate crtera are brefly explaned below.

10 622 E. U. ERGUL CRITERIA Table. The catalogue values of alternatve 3D TVs ALTERNATIVE 3D TVs TV TV2 TV3 TV4 TV5 TV6 TV7 TV8 Dynamc Contrast Rato : : : : : : : : Refresh Rate (Hz) Weght (kg) , ,4 22, Power Consumpton on Mode (W) Depth (Fneness-mm) , Cost ($).259, ,99.598,00.695, , ,00.60,00.639,00 Dynamc contrast rato s a ratng used to determne the mage qualty of a TV. The statc contrast rato denotes the lumnance of the brghtest possble whte colors on the screen versus the deepness and darkness of black shades. The refresh rate s the number of tmes n a second that dsplay hardware updates ts buffer. After determnng the crtera and herarchy of the problem, people who are consderng the purchase of 3D TV have used the lngustc varables n Table 2 to determne mportance weghts of the 3D TV selecton crtera (Table 3). Table 2. Lngustc varables for the mportance weght of crtera Trangular Fuzzy Numbers Lngustc varable (TFNs) Very low (VL) (0,0,0.) Low (L) (0,0.,0.3) Medum low (ML) (0.,0.3,0.5) Medum (M) (0.3,0.5,0.7) Medum hgh (MH) (0.5,0.7,0.9) Hgh (H) (0.7,0.9,) Very hgh (VH) (0.9,,) Table 3. Importance weghts of crtera Crtera Decson Fuzzy Dynamc Contrast Rato C VH (0.9,,) Refresh Rate (Hz) C2 H (0.7,0.9,) Weght (Kg) C3 MH (0.5,0.7,0.9) Power Consumpton on Mode (W) H (0.7,0.9,) Depth (Fneness-mm) C5 H (0.7,0.9,) Cost ($) C6 VH (0.9,,) Then, the lngustc varables n Table 4 are determned by usng a transformaton calculaton of the catalog values n Table. Afterward, the evaluatons n Table 5 are done by usng the lngustc varables n Table 4.

11 Applcaton Of Fuzzy Topss Method And Dopga Algorthm In Purchasng Decson Process: 3d Televson Example 623 Table 4. Lngustc varables used n the evaluaton of alternatve 3D TVs Lngustc Trangular Fuzzy Numbers Very poor (VP) (0,0,) Poor (P) (0,,3) Medum poor (,3,5) Far (F) (3,5,7) Medum good (5,7,9) Good (G) (7,9,0) Very good (VG) (9,0,0 Table 5. Evaluaton Results of Alternatve 3D TVs Accordng to Sx Crtera Crtera Alternatves TV TV2 TV3 TV4 TV5 TV6 TV7 TV8 C VG P P MP MG VP P G C2 MP MP MP MP MG P VG MG C3 MG VG G G VP G MG MG C4 P G VG MG F MG F VP C5 MG MG MP VG VP MG MG F C6 G MG G G VP F G G Fuzzy decson matrx s formed by convertng the lngustc evaluatons to trangular fuzzy numbers (Table 6). Table 6. Fuzzy decson matrx and fuzzy weghts of eght alternatves Crtera Alternatves TV TV2 TV3 TV4 TV5 TV6 TV7 TV8 C (9,0,0 (0,,3) (0,,3) (,3,5) (5,7,9) (0,0,) (0,,3) (7,9,0) C2 (,3,5) (,3,5) (,3,5) (,3,5) (5,7,9) (0,,3) (9,0,0 (5,7,9) C3 (5,7,9) (9,0,0 (7,9,0) (7,9,0) (0,0,) (7,9,0) (5,7,9) (5,7,9) C4 (0,,3) (7,9,0) (9,0,0 (5,7,9) (3,5,7) (5,7,9) (3,5,7) (0,0,) C5 (5,7,9) (5,7,9) (,3,5) (9,0,0 (0,0,) (5,7,9) (5,7,9) (3,5,7) C6 (7,9,0) (5,7,9) (7,9,0) (7,9,0) (0,0,) (3,5,7) (7,9,0) (7,9,0) Fuzzy decson matrx s normalzed as n Table 7, then weghted normalzed fuzzy decson matrx s obtaned as gven n Table 8. Table 7. Normalzed fuzzy decson matrx Alternatves Crtera TV TV2 TV3 TV4 TV5 TV6 TV7 TV8 C (0.9,,) (0,0.,0.3) (0,0.,0.3) (0.07,0.2,0.4) (0.5,0.7,0.9) (0,0,0.) (0,0.,0.3) (0.7,0.9,) C2 (0.,0.3,0.5) (0.,0.3,0.5) (0.,0.3,0.5) (0.,0.3,0.5) (0.5,0.7,0.9) (0,0.,0.3) (0.9,,) (0.5,0.7,0.9) C3 (0.5,0.7,0.9) (0.9,,) (0.7,0.9,) (0.7,0.9,) (0,0,0.) (0.7,0.9,) (0.5,0.7,0.9) (0.5,0.7,0.9) C4 (0,0.,0.3) (0.7,0.9,) (0.9,,) (0.5,0.7,0.9) (0.3,0.5,0.7) (0.5,0.7,0.9) (0.3,0.5,0.7) (0,0,0.) C5 (0.5,0.7,0.9) (0.5,0.7,0.9) (0.,0.3,0.5) (0.9,,) (0,0,0.) (0.5,0.7,0.9) (0.5,0.7,0.9) (0.3,0.5,0.7) C6 (0.7,0.9,) (0.5,0.7,0.9) (0.7,0.9,) (0.7,0.9,) (0,0,0.) (0.3,0.5,0.7) (0.7,0.9,) (0.7,0.9,)

12 624 E. U. ERGUL Table 8. Weghted normalzed fuzzy decson matrx Alternatves Crtera TV TV2 TV3 TV4 TV5 TV6 TV7 TV8 C (0.8,,) (0,0.,0.3) (0,0.,0.3) (0.06,0.23,0.43) (0.45,0.7,0.9) (0,0,0.) (0,0.,0.3) (0.63,0.9,) C2 (0.07,0.27,0.5) (0.07,0.27,0.5) (0.07,0.27,0.5) (0.07,0.27,0.5) (0.35,0.63,0.9) (0,0.09,0.3) (0.63,0.9,) (0.35,0.63,0.9) C3 (0.25,0.49,0.8) (0.45,0.7,0.9) (0.35,0.63,0.9) (0.35,0.63,0.9) (0,0,0.09) (0.35,0.63,0.9) (0.25,0.49,0.8) (0.25,0.49,0.8) C4 (0,0.09,0.3) (0.49,0.8,) (0.63,0.9,) (0.35,0.63,0.9) (0.2,0.45,0.7) (0.35,0.63,0.9) (0.2,0.45,0.7) (0,0,0.) C5 (0.35,0.63,0.9) (0.35,0.63,0.9) (0.07,0.27,0.5) (0.63,0.9,) (0,0,0.) (0.35,0.63,0.9) (0.35,0.63,0.9) (0.2,0.45,0.7) C6 (0.7,0.9,) (0.5,0.7,0.9) (0.7,0.9,) (0.7,0.9,) (0,0,0.) (0.3,0.5,0.7) (0.7,0.9,) (0.7,0.9,) Dstance from FPIS ( d from FNIS ( d ) values of each alternatve 3D TVs are calculated by Equaton 0 and ther dstance ) values are computed by Equaton. Then, closeness ndex s computed for each alternatve by usng Equaton 4 consderng d and Table 9. d values. The results are gven n Table 9. d, d and C values TV TV2 TV3 TV4 TV5 TV6 TV7 TV8 d d C Due to closeness coeffcents, the order of the 3DTV alternatves s as TV4> TV7> TV> TV8> TV2> TV3> TV6> TV5 based on Table 9. Accordng to ths result, t wll be the best to choose TV4 whch has the hghest closeness coeffcent value among alternatve 3D TVs. A senstvty analyss s appled to show how well the alternatves perform wth each obectve and how senstve the alternatves are to changes n the mportance of the obectves. After the order of the alternatves determned, t must be evaluated that how senstve the order of alternatves and the fnal decson aganst the changes n the udgments n order to check the results of the establshed model. By changng the weghts of the selecton crtera n the senstvty analyss, the varaton of the closeness coeffcents and the order of the alternatves under the total of 9 dfferent cases shown n Table 0 are examned and the obtaned results are shown n Fg. 4 and Fg. 5.

13 Closeness Coeffcents Applcaton Of Fuzzy Topss Method And Dopga Algorthm In Purchasng Decson Process: 3d Televson Example 625 Table 0. The Case combnatons wth dfferent crtera weghts Cases Combnatons Case Current Case 2 All Low Case 3 All Medum Case 4 Dynamc Contrast Rato Hgh, The Rest Low Case 5 Refresh Rate Hgh, The Rest Low Case 6 Weght Hgh, The Rest Low Case 7 Power Consumpton on Mode Hgh, The Rest Low Case 8 Depth Hgh, The Rest Low Case 9 Cost Hgh, The Rest Low TV TV2 TV3 TV4 TV5 TV6 TV7 TV8 Cases Fgure 4. Closeness coeffcents changes based on senstvty analyss

14 Rankng 626 E. U. ERGUL TV4 TV7 TV TV8 TV2 TV3 TV6 TV5 Cases Fgure 5. The rankng changes of the alternatve 3DTVs based on the senstvty analyss When Fg. 4 and Fg.5 are examned, t s seen that TV4 has the hghest closeness coeffcent n all cases except Case 4 and Case 5, and TV5 has the lowest closeness coeffcent n all cases. Fgure 5 shows the order of the alternatves, t s seen that n the Case 4 and Case 5, the dynamc contrast rato and refresh rate (Hz) crtera have a negatve effect on the TV4 and the dynamc contrast rato crtera on TV7 n a smlar way. Also, t s seen that dynamc contrast rato and depth crtera have a postve effect on the TV. Applcaton of DOPGA Method to 3D TV Evaluaton Problem DOPGA s used as MOEA method to rank the alternatve 3D TVs. For the best graphcal vsualzatons, crtera are used n DOPGA two by two. Afterward, the real ftness/rank values of alternatve 3D TVs are found by rankng or ftness assgnment algorthm of DOPGA. The vsualzatons are shown n Fg. 6, Fg. 7 and Fg. 8. Then, DOPGA s worked by usng the real ftness/rank values founded by two by two applcatons. The results of DOPGA s depcted n Fg. 9 by usng the obtaned real ftness values of all solutons. The real ftness values or ranks calculated by usng Eq.8 are gven n Table.

15 Applcaton Of Fuzzy Topss Method And Dopga Algorthm In Purchasng Decson Process: 3d Televson Example 627 Fgure 6. Crtera values of depth and power consumpton on the mode of eght alternatve 3D TVs Fgure 7. Crtera values of cost and weght of eght alternatve 3D TVs

16 628 E. U. ERGUL Fgure 8. Crtera values of refresh rate and dynamc contrast rato of eght alternatve 3D TVs The result obtaned from DOPGA are gven n Table. Soluton,2,3,4 and 7 are Pareto-optmal solutons and DOPGA separated them perfectly. And the decson maker no longer confuses to separate them. DOPGA sorts the solutons from the most desred one to the least desred one. Fgure 9. Real ftness values/ranks found by DOPGA Table. Results obtaned from DOPGA Alternatve 3DTVs Real ftness (rf) values Rank TV TV TV TV4.270 TV TV TV TV

17 Applcaton Of Fuzzy Topss Method And Dopga Algorthm In Purchasng Decson Process: 3d Televson Example 629 Accordng to the real ftness values, alternatves are ranked as TV4> TV7> TV> TV2 > TV3 > TV8 > TV6 > TV5 n descendng order shown n Table. Accordng to ths result, t wll be the best to choose TV4 whch has the lowest real ftness value among alternatve 3D TVs. Table 2 s obtaned va the results derved by mplementng every two methods ndvdually at the 3D TV selecton problem. Table 2. The results of fuzzy TOPSIS and DOPGA Alternatve 3D TVs Rankng Fuzzy TOPSIS DOPGA TV 3 3 TV2 5 4 TV3 6 5 TV4 TV5 8 8 TV6 7 7 TV7 2 2 TV8 4 6 Table 2 shows the ranks of TVs obtaned by the closeness coeffcents of fuzzy TOPSIS and the real ftness values of DOPGA. Accordng to these results, TV4 s the frst order, TV7 s the second order and TV s the thrd order n both methods. Besdes, the last two 3D TVs are the same wth both methods. Therefore, t s detected that both methods have a smlar rankng. Accordng to these results, someone who purchases a 3D TV should select TV4. CONCLUSION Innovatve products are developed rapdly. One of them s TV. Wth the rapd advancement of technology and the contnuous ncrease of customers' needs, a huge ncrease n the varety of TVs. Thus, TV purchasng becomes a MCDM problem and TV customers confuse n selectng the correct TV among varous TVs that have a lot of features. Usng MCDM methods as n solvng other selecton problems wll contrbute to solvng the TV selecton problem. Fuzzy TOPSIS s a MDCM method used n the fuzzy envronment and t assgns dfferent mportance weght to the decson crtera. Therefore, t makes easy to select a more accurate and more relable choce among the alternatves for decson makers. In addton, because of fuzzy TOPSIS method gets decson makers thoughts for the preference artculaton; t can be avoded possble conflcts among decson makers. EAs are very popular and useful search tools for optmzaton. Unlke the classcal optmzaton methods, EAs use a populaton of solutons to fnd the optmum of a problem. EAs do not need dervatve nformaton, and they are very easy to mplement. In ths paper, newly proposed DOPGA method s appled to 3D TV selecton problem. DOPGA method can assgn a unque rank to all solutons. DOPGA can sort the solutons (TV alternatves) from the best one to the worst one. Due to these reasons, fuzzy TOPSIS and DOPGA are used to solve 3D TV selecton problem n ths study.

18 630 E. U. ERGUL Accordng to the results, TV4 s selected the best 3D TV n fuzzy TOPSIS and DOPGA methods. In both methods, the order of alternatve 3D TVs s nearly the same. It can be seen that fuzzy TOPSIS and DOPGA can be used together to solve 3D TV selecton problem. Ths complex problem s solved by usng fuzzy TOPSIS and DOPGA for a customer who purchases a 3D TV. On ths count, the people who wll purchase 3D TV wll no longer be confused. Purchasng problems lke ths can easly be solved n ths way. Ths study s the frst study for solvng 3D TV selecton problems n the lterature. In a future work, t s planned to use fuzzy ANP and generalzed Choquet to solve ths problem and to compare the results of ths study. Therefore, t can be performed a more precse selecton. Also, the TV selecton model, the methods and the results used n ths study wll gude TV customers and researchers. REFERENCES Altparmak, F., Gen, M., Ln L., Paksoy, T.A., 2006, Genetc Algorthm Approach for Mult-Obectve Optmzaton of Supply Chan Networks, Computers & Industral Engneerng, Vol. 5(), pp Bandyopadhyay, S., Bhattacharya, R., 203, Applyng Modfed NSGA-II for B-Obectve Supply Chan Problem, Journal of Intellgent Manufacturng, Vol. 24(), pp Bas, E., 203, The Integrated Framework for Analyss of Electrcty Supply Chan Usng an Integrated SWOT-Fuzzy TOPSIS Methodology Combned wth AHP: The Case of Turkey, Electrcal Power and Energy Systems, Vol. 44, pp Chan, F.T.S., Chung, S.H., 2004, A Mult-Crteron Genetc Algorthm for Order Dstrbuton n a Demand Drven Supply Chan, Internatonal Journal of Computer Integrated Manufacturng, Vol. 7(4), pp Che, Z.H., 2009, Prcng Strategy and Reserved Capacty Plan Based on Product Lfe Cycle and Producton Functon on LCD TV Manufacturer, Expert Systems wth Applcatons, Vol. 36, pp Chen, C.F.,Tsa, D., 2007, How Destnaton Image and Evaluatve Factors Affect Behavoral Intentons?, Toursm Management, Vol. 28, pp Chen, C.T., 2000, Extensons of the TOPSIS for Group Decson-Makng Under Fuzzy Envronment, Fuzzy Sets and Systems, Vol. 4, pp. -9. Cheng, C.T., Zhao, M.Y., Chau, K.W., Wu, X.Y., 2006, Usng Genetc Algorthm and TOPSIS for Xnanang Model Calbraton wth a Sngle Procedure, Journal of Hydrology, Vol. 36, pp Chang, Tzu-An., 202, Mult-obectve Decson-Makng Methodology to Create an Optmal Desgn Chan Partner Combnaton, Computers & Industral Engneerng, Vol. 63, pp Chu, M. C., Kuo, M. Y., Kuo, T. C., 205, A Systematc Methodology to Develop Busness Model of A Product Servce System, Internatonal Journal of Industral Engneerng: Theory, Applcatons and Practce, Vol. 22(3). Coello, C.A.C., 2006, Evolutonary Mult-Obectve Optmzaton: A Hstorcal Vew of the Feld, IEEE Computatonal Intellgence Magazne, Vol. (), pp Coello, C. A. C., Lamont, G. B., Van Veldhuzen, D. A., 2007, Evolutonary Algorthms for Solvng Mult-Obectve Problems, Vol. 5, New York: Sprnger. Dymova, L., Sevastanov, P., Tkhonenko, A., 203, An Approach to Generalzaton of Fuzzy TOPSIS Method, Informaton Scences, Vol. 238, pp Deb, K., 200, Mult-obectve Optmzaton Usng Evolutonary Algorthms, Chchester, U.K, Wley. Deb, K., Pratap, A., Agarwal, S.,Meyarvan, T., 2002, A Fast and Eltst Mult-Obectve Genetc Algorthm: NSGA-II, IEEE Transactons on Evolutonary Computaton, Vol. 6(2), pp

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