Multi Responses Optimization Through Game Theory Approach

Transcription

1 Iteratoal Joural of Idustral Egeerg & Producto Research September 04, Volume 5, Number 3 pp. 5-4 pissn: Mult Resposes Optmzato Through Game Theory Approach H.R. Navd, A. Amr *, R. Kamrarad Hamdreza Navd, Assocate Professor, Departmet of Appled Mathematcs, Shahed Uversty, Tehra, Ira, avd@shahed.ac.r Amrhosse Amr, Assstat Professor, Idustral Egeerg Departmet, Shahed Uversty, Tehra, Ira, amr@shahed.ac.r Reza Kamrarad, Ph.D. studet, Idustral Egeerg Departmet, Shahed Uversty, Tehra, Ira, r.kamrarad@shahed.ac.r KEYWORDS Game theory; Mult resposes; Sgal to ose rato; Nash equlbrum; VIKOR ABSTRACT I ths paper, a ew game theoretc-based approach s proposed for mult-respose optmzato problem. Game theory s a useful tool for decso makg the coflct of terests betwee tellget players order to select the best ot strategy for them through selectg the best ot desrablty. Preset research uses the game theory approach va defto of each respose as each player ad factors as strateges of each player. Ths approach ca determe the best predctor factor sets order to obta the best ot desrablty of resposes. For ths am, the sgal to ose rato(sn) dex for each respose have bee calculated wth cosderg the ot values of strateges; the obtaed SN ratos for each strategy s modeled the game theory table. Fally, usg Nash Equlbrum, the best strategy whch s the best values of predctor factors s determed. A real case ad a umercal example are gve to show the effcecy of the proposed method. I addto, the performace of the proposed method s compared wth the VIKOR method. 04 IUST Publcato, IJIEPR, Vol. 5, No. 3, All Rghts Reserved.. Itroducto Game theory s a brach of operato research scope whch s useful for decso makg the coflct of terests betwee tellget players (decso makers). Ths method has wde applcatos the socal lfe, ecoomy, polcy, egeerg sceces, bologcal sceces ad so o (Navd et al. [] ad Osbore []). There are may deftos of game theory; for example Osbore [] beleves that game s a descrpto of the strategc teractos that occur betwee players. Also Myerso says that game theory s a study of mathematcal models of coflct ad cooperato * Correspodg author: Amrhosse Amr Emal: amr@shahed.ac.r Paper frst receved Jue 8, 03, ad accepted form Aprl,, 04. betwee wse players; for ths reaso, there are other ttles for the game theory such as coflct aalyss or teractve decso makg theory [3]. I the game, there are at least two players wth adverse deas or targets whch have smlar or dfferet strateges. Each player wats to acheve the maxmum desrablty. For example, f the results of the game are defed as wg or losg, the each player wsh to maxmze wg ad mmze losg. For ths problem, game theory ca propose the useful method for obtag the best ot desrablty such that each player ca receve hs target (Navd et al. [] ad Osbore []). O the other had, some of problems the metoed sceces have more tha oe respose varables whch are affected by oe or more predctor factors. These cases are called mult respose problems. Desg of expermets (DOE) s a method whch uses tools such as desrablty fucto, Taguch s loss fucto, Sgal

2 H.R. Navd, A. Amr, R. Kamrarad Mult Resposes Optmzato Through Game Theory Approach 6 to Nose rato, respose surface methodology ad so o to aalyze ad optmze these problems. The purpose of the DOE s to obta the best set of predctor varables order to determe the best value or rak of respose varable (s) (Poroch-Serta et al. [4]). Ths research s a ew ovato the optmzato of mult respose problems through game theory approach (GTA) DOE. Note that, oe of the most mportat dffereces betwee the DOE ad the GTA s that the DOE approach has a geeral or global vew to determe the optmum pot, meawhle the GTA searches equlbrum pot (EP) wth partal or local vew. Ths paper s orgazed as follows: the ext secto prevous related studes have bee revewed ad dscussed. I the thrd secto, a ovel approach has bee proposed for optmzato of mult respose problems through usg the GTA. I secto 4, a real case as well as a umercal example has bee llustrated to evaluate the performace of the proposed approach. Fally our cocludg remarks have bee preseted secto 5.. Revew of Lterature I recet years, some procedures have bee developed to determe the best factor settg whch optmzes multple resposes smultaeously. I may researches, expermet optmzato approaches have bee categorzed to two geeral groups; approaches wth cosderg ad gorg the correlato betwee resposes. There are several methods the frst group whch ca be set three categores. Some of these methods attempt to optmze the mult respose problems usg complcated mathematcal models. Refer to Khur ad Colo [5], Lagothets ad Hagh [6] ad Tag ad Su [7] for more formato about these methods. Due to the complexty, these methods are ot applcable for all mult resposes problems. Other category of frst group uses heurstc ad Meta heurstc algorthms to optmze the metoed problems. For example, JeyaPaul et al. has bee proposed a tegrated approach to mult resposes optmzato usg SN rato ad geetc algorthm [8]. Also Tag ad Hseh used eural etwork techque for ther problem. These approaches also have drawbacks because they do ot guaratee optmal solutos [9]. I the thrd category, at frst all of resposes are coverted to a process performace dex (PPI) ad the the process has bee optmzed accordg to the calculated dex. Pa et al. [0] ad Haq et al. used the dark depedecy aalyss method to obta the PPI []. Ramakrsha ad Karuamoorthy have bee proposed the mult respose SN rato []. Tag et al. proposed the VIKOR (Vlse Krterumsk Optmzaca Kompromso Resee, a Serba ame), approach that s a cotgecy ratg method to solve mult resposes optmzato [3]. Lao used the prcpal compoet aalyss (PCA) techque to determe the PPI dex [4]. Also, Datta et al. proposed the applcato of athrophy measurg techque o the bass of Taguch approach for solvg the correlated optmzato problems [5]. I ther research, they used the PCA techque for elmatg the avalable correlato amog resposes ad covertg them to depedet resposes. Oe of the other statstcal tools for aalyzg the correlated mult resposes problem s the multvarate aalyss of varace (MANOVA). The secod group has bee addressed by Bashr ad Heaz whch mult respose optmzato problem has bee studed through the Techque for Order Preferece by Smlarty to Ideal Soluto (TOPSIS) approach [6]. Pal & Gaur examed the effcecy of dfferet cotuous methods for optmzg the depedet mult-resposes problem usg multomal regresso [7]. He et al. proposed a robust desrablty fucto method to smultaeously optmze multple resposes wth cosderg the ucertaty assocated wth the ftted respose surface model. Ther method takes to accout all values the cofdece terval rather tha a sgle predcted value for each respose ad the defes the robustess measure for the tradtoal desrablty fucto usg the worst case strategy. Also, they developed a hybrd geetc algorthm to fd the robust optma [8]. Heaz et al. assessed the qualty egeerg problems whch several qualty characterstcs ad factors are to be aalyzed through a smultaeous equatos system. Besdes, they cosdered several probablstc covarates the proposed model. Fally, they could detfy terrelatos amog exogeous ad edogeous varables, whch gve mportat sght for systematc mprovemets of qualty [9]. Zadbood et al. proposed a Harmoy search meta-heurstc algorthm to optmze the mult respose surface problem. Ths algorthm ca fd the best set of cotrol varables whch optmze the multple resposes. Fally, they dcated that the effcecy ad performace of ther proposed algorthm s better tha some of the well kow meta-heurstc algorthm [0]. Bashr ad Bagher proposed a Imperalst Compettve Metaheurstc algorthm to optmze the o-lear mult resposes whch the best cotrollable factor s levels are obtaed such that mmum devatos of the resposes meas from ther targets are acheved. Moreover, they used the Pareto optmal soluto to release the aggregatve fucto. The, they compared the multple respose mperalst compettve algorthm wth multple obectve geetc algorthm. The results showed the effcecy of the proposed approach both aggregatve ad o aggregatve methods for optmzato of the olear mult respose programmg []. Iteratoal Joural of Idustral Egeerg & Producto Research, September 04, Vol. 5, No. 3

3 7 H.R. Navd, A. Amr, R. Kamrarad Mult Resposes Optmzato Through Game Theory Approach Zolghare et al. appled the Taguch desg ad prcple compoet aalyss mult resposes optmzato to fd the effectve parameters for achevg a hgher adsorpto capacty ad removal percetage of the bary mxture cotag alzar red ad yellow colors []. Sh et al. proposed a soluto framework based o dscrete-evet smulato, sequetal bfurcato ad respose surface methodology to aalyze a mult resposes optmzato problem heret a auto parts supply cha. They detfed the most effcet operatg settg that would maxmze the logstcs performace after the expaso of the assembly plat s capacty due to market growth. The, they appled the sequetal bfurcato to detfy the most mportat factors that fluece system performace. Also, order to determe the optmal levels of these key factors, they employed the respose surface methodology to develop Derrger Such s Meta models that best descrbe the relatoshp betwee key decso varables ad the multple system resposes. Fally, the results of ths paper showed that the proposed method eables the greatest mprovemet o system performace [3]. Also, there are some researches mult-obectve scope as the game optmzato. For example, L et al. proposed the game optmzato theory (GOT) to stablty optmzato aalyss, Dstace Etropy Mult-Obectve Partcle Swarm Optmzato ad Fuzzy Mult-weghts Decso-makg Method. They beleved that the GOT s a comprehesve system ot oly hadle mult-obectve optmzato problems effectvely but also could offset the dsadvatages of tradtoal optmzato theores, such as lack of framework ad the suffcet cosderato of relevat elemets. So, they used the GOT for solvg the dstrbuto systems plag ssue by mplemetg dstrbuted geerato. Ther proposed model tegrates costs, losses, ad voltage dex to acheve optmal sze ad ste of dstrbuted geerato. The model allows mmzg total system costs, system power losses ad maxmzg voltage mprovemet [4]. Rao preseted a cocept of Pareto-optmal soluto the cotext of a mult-obectve structural optmzato problem. They used the graphcal terpretatos of the o-cooperatve ad cooperatve game theory approaches for a two-crtera optmzato problem. Fally, they descrbed the relatoshp betwee Pareto-optmal solutos ad game theory optmzato [5]. Zamarrpa et al. appled the game theory optmzatobased tool for supply cha plag cooperatve/compettve evromets. Ther proposed model could mprove the tactcal decso-makg of a supply cha uder a ucerta competto scearo through the use of dfferet optmzato crtera. Also, They solved the mult-obectve optmzato problem usg the -costrat method order to approxmate the Pareto space of o-domated solutos whle a framework based o game theory s used as a reactve decso makg support tool to deal wth the ucertaty of the compettve scearo [6]. I ths paper, a ew approach based o coflctg betwee resposes as the Game theory has bee proposed to optmze the mult resposes problem. 3. Problem Defto, Methods ad Equatos I ths paper, a ew geeralzed game theory cocept has bee proposed to mult resposes problem optmzato. Optmzato cocept the DOE meas that the best set of predctor varables s selected order to determe the best value or rak of respose varable(s). For ths purpose, the metoed methods the prevous secto are used. Note that ths paper, two cocepts of the DOE ad the GTA have bee composed such that the ot desrablty (or each ot optmzato dex) of each respose (each player) s calculated usg the DOE tools ad the these desrablty values are modeled the game theory table. For ths am, at frst the desrablty of each strategy from each player wth cosderg the strategy of other player(s) should be calculated; other words, desrablty of each strategy s affected from the opposte player. After calculatg the ot desrablty of each player, the game theory model s proposed as desrablty table ad the a EP of each row ad colum of the game model s selected usg the NE. The EP of each row ad colum the game table s the best desrablty accordg to the target of related player. If the best desrablty from the strategy of frst player ad other player(s) are occurred the ot cell of the game theory table, the ths pot s a NE pot. 3-. The Used Mult Resposes Optmzato Method As metoed the prevous secto, there are may effcet methods for optmzato of mult resposes problem. Oe of these methods s the Taguch s sgal to ose rato whch ca show mportace or effect of the resposes. The Taguch method of expermetal desg s a wdely used techque to accomplsh the task for optmzato of process parameters. Taguch used orthogoal arrays to perform expermets ad appled the sgal to ose (SN) rato as the qualty measuremet dex, wth smultaeous cosderato of the mea ad varablty of the qualty characterstc, to determe the optmal settg of process parameters (Wu & Yeh [7]). Ths method has dfferet formulas for ay type of resposes such as smaller the better (STB), omal the best (NTB) ad larger the better (LTB). The followg equatos are used to compute three metoed types of Iteratoal Joural of Idustral Egeerg & Producto Research, September 04, Vol. 5, No. 3

4 Game Model Lk Fucto H.R. Navd, A. Amr, R. Kamrarad Mult Resposes Optmzato Through Game Theory Approach 8 SN rato ( db ). Note that ths paper, two respose varables from types of STB ad NTB are studed. db 0 log( y ) ( STB type), () y db 0log( ) ( NTBtype), () s db 0log( ( )) ( LTBtype), (3) y where y s the value of qualty charasterstc, y s the samples mea of replcates ad s ( y ) y s the ubased estmator of process varace Wu & Yeh [7]. 3-. Proposed Method: Game Theory Approach for Mult Resposes Optmzato (GTA-MRO) I ths secto, GTA-MRO has bee proposed to select the EP (optmum strategy). I the proposed method, the umber of resposes determes the umber of players. I other words, each player s equvalet wth each respose. Also, the depedet factors as well as ther levels determe the strateges the game theory model. Based o ths defto, we provde the game theory table. The, the ot SN rato for each player wth defte strategy s calculated accordg to other player(s) strategy(s) ad s put the game theory table. Fally, the EP s obtaed usg the NE. The NE s the most commo soluto for solvg the strategc games. The output of the NE s a stable state of strategc teracto for all of players such that f players behave based o the NE, the they do ot have ay motvato for devato from ther decsos. I other words, the NE strategy of each player s the best aswer to the NE strateges of other players. Fgure shows the relatoshp betwee the DOE ad the GTA schematcally. x x x y y y NE OP Utlty Strateges Players SN Rato OP: Optmal Pot Fg.. Schematc relatoshp of the DOE ad the GTA The NE equato to determe the EP s gve equato (4). * * * u ( s, s ) u ( s, s ) s S,,..., (4) where: G N,( S ) N {,,..., } S S S N... S, ( u ) s ( s, s,..., s ) S N Iteratoal Joural of Idustral Egeerg & Producto Research, September 04, Vol. 5, No. 3

5 9 H.R. Navd, A. Amr, R. Kamrarad Mult Resposes Optmzato Through Game Theory Approach S s u : S S... S ( s,..., s S, s... S S,..., s... S ) R such that: G: Statc game N: Number of players S: Set of all strateges of each player s: Selected strategy by player S - : Set of all strateges whch s ot cluded the th set of strategy s - : Selected strateges by players except th strategy u : Utlty of selected ot strateges., * * * If ( s, s,..., s ) S s a NE, devato of each player from the NE leads to worse utlty for that player (Navd et al. [] ad Osbore []). Fgure demostrates the mplemetato of the proposed approach (GTA-MRO). Note that, the proposed approach always has at least oe NE for fte expermetal desgs. Because, Vo Neuma ad Morgester [8, pp ] showed that each fte matrx game has at least oe NE. Sce, all of the expermetal desgs wth fte umber of the treatmets ad the respose varables are defed as the fte matrx game. Hece, the proposed approach ca be used to aalyze all mult respose optmzato problems. Defe players based o respose varables Obta the strateges of each player based o the values of predctor varables Calculate the ot utlty for each player accordg to ot strategy Model these calculated utltes as game theory table Use the NE method to obta the equlbrum pot Optmal Pot s determed No Is there more tha oe NE? Yes Use the focal pot method Optmal Pot s determed Yes Is there a domat pot? NO Select the optmal pot Use the mxed strategy method Fg.. Mult resposes optmzato by the GTA Iteratoal Joural of Idustral Egeerg & Producto Research, September 04, Vol. 5, No. 3

6 H.R. Navd, A. Amr, R. Kamrarad Mult Resposes Optmzato Through Game Theory Approach 0 4. Performace Evaluato ad Comparso 4-. Example : A Real Case Phadke (989) troduced a case o the optmzato of a polyslco deposto process (Wu & Yeh [7]). The expermet data of two qualty characterstcs cludg surface defect ad thckess are used to llustrate the GTA for optmzato of mult resposes problem. Ths example cotas three predctor varables called deposto temperature (A), deposto pressure (B) ad slece flow (C) whch have, ad 3 levels, respectvely; hece, total umbers of treatmets for ths example are. The level of predctor varables expermetal desg ad two respose varables data are summarzed Table. Tab.. Levels of predctor varables, ad surface defect ad thckess respose values Expermet No. Predctor Varables A B C Surface Defect Data Thckess Data Here, t should be oted that levels of A ad B factors are as the surface defect respose (secod player) strateges ad C factor levels are the other respose (frst player) strateges. I other words, the strateges of the secod player are as pars. Now, usg the data set Table ad equatos () ad (), the ot SN ratos for each treatmets of the above expermetal desg are calculated ad show Table. Tab.. Jot SN ratos for frst ad secod player Predctor Varables SN Ratos Expermet No. Frst player Secod Player A B C (STB) (NTB) Accordg to Table, the game theory table s modeled Table 3 as follows: Player wth strategy of (C) Tab.3. Game theory model usg ot SN ratos Player wth strateges of (A,B) Strategy (,) (,) (,) (,) The set of all NE s for game G s show as N (G), fact the NE s based o the best soluto. The steps of determg the NE are as follows:. If player selects the strategy umber ; player selects the (, ) strategy: maxmum ot SN rato for NTB type; the player wll select the Iteratoal Joural of Idustral Egeerg & Producto Research, September 04, Vol. 5, No. 3

7 H.R. Navd, A. Amr, R. Kamrarad Mult Resposes Optmzato Through Game Theory Approach strategy of the colum (, ): mmum ot SN rato for STB type. Sce the selected strateges of each player for frst treatmet are occurred commo pot, so NE for ths treatmet s [, (, )].. If player selects the strategy umber ; player selects the (, ) strategy; the, player wll select the strategy of the colum of (, ). Sce the selected strateges of each player for secod treatmet are ot occurred a commo pot, so there s ot NE for ths treatmet. 3. If player selects the strategy umber 3; player selects the (, ) strategy; the Player wll select the strategy of the colum of (, ). Sce the selected strateges of each player for thrd treatmet are ot occurred commo pot, there s ot NE for ths treatmet. Accordg to these steps, the oly EP of ths game s (, (, )). I other words, the NE of ths game s N (surface defect data ad thckess data) = (, (, )). It s possble that some games have more tha oe NE. A questo arses such games s whch NE s better to behave? Schellg suggests usg the cocept of focal pot effect to solve ths problem [9]. Usg ths cocept, t s possble that selectg expectato of oe NE s more tha dfferet NE s of other players. Ths case of NE s called as Focal Equlbrum (FE). There s aother method that s exteso of the NE referred to as Pareto optmal Nash equlbrum (PONE). PONE of a * * * * strategc game (G) s the NE of s ( s, s,..., ) such that there s o ay equlbrum lke '* '* '* '* s ( s, s,..., ) the N (G) whch satsfes the s followg codto (Navd et al. [] ad Osbore []): * '* u ( s ) u ( s ) N s (5) The attractveess of Pareto optmal Nash equlbrum strategc games s that players ot as oe-sded ad ot as coordate are reluctat to devate from ths equlbrum pot. The set of all PONE s for game G s show by N po (G). Aother case that may happe s that oe of the NE s ot domat over the other pots such that the best EP caot be determed by Pareto optmal; hece, the mxed strategy method should be used to solve ths problem. For more formato refer to Navd et al. [] ad Osbore []. Followg example has bee llustrated for ths case. 4-. Example : A Numercal Example wth Two NE Pots Cosder prevous real case; ths secto the data related to frst ad the secod resposes have bee chaged order to show the two NE pots ad select the domate pot game theory model. The data related to two resposes are show Table 4. Tab. 4. Levels of predctor varables, other surface defect ad thckess data Expermet No. Predctor Varables A B C Y Y For ths example, the ot SN ratos are calculated ad put the game theory table that s show Table 5. Tab. 5. Game theory model usg ot SN ratos wth two NE pots Player wth strategy of (C) Player wth strateges of (A,B) Strategy (,) (,) (,) (,) As show Table 5, there are two NE of (, (, )) ad (3, (, )) or N (G) = {(, (, )), (3, (, ))} for the secod example. Domated pot should be selected usg the followg structo: sce the ot SN ratos of each player (3, (, )) strategy are better tha the (, (, )) strategy, so ths example the Iteratoal Joural of Idustral Egeerg & Producto Research, September 04, Vol. 5, No. 3

8 H.R. Navd, A. Amr, R. Kamrarad Mult Resposes Optmzato Through Game Theory Approach PONE s [3, (, )]. I other words, N ( G) {(,(,)); (3,(,))}. Hece, the Pareto optmal NE s N po ( G) (3,(,)) Comparso the Proposed Approach wth the VIKOR Method I ths secto, a decso makg method called VIKOR has bee used to accumulate the SN ratos of resposes. Ths comparso ca determe the effcecy of the proposed GTA for MRO. Note that the best levels of cotrollable factors are determed such that the best treatmet has mmum value of Q. The VIKOR dex s gve equato (8) (Tag et al. [3]). Z Z R R Q ( ), (8) Z Z R R where the varables of the equato (8) are defed Table 6 as follows: Tab. 6. Varables of the VIKOR dex Varable Ttle Varable Ttle Q Accumulato Idex Z + Mmum Z υ Weght Ga Group Z - Maxmum Z Z Utlty measure R + Mmum R R Regret measure R - Maxmum R We assumed the value of 0.5 for parameter υ. Also, the Z ad the R dces the equato (8) are computed by usg equatos (9) ad (0), respectvely: Z w ( NSN NSN NSN NSN w ( NSN NSN max[ NSN NSN ), (9) ) ] R, (0) where NSN s the Normalzed S/N rato of the th respose the th treatmet, the the NSN NSN ad are the maxmum ad mmum NSN, respectvely for the th respose[3]. I ths secto, the Q dces for the real case ad the umercal example are calculated ad show Table 7. Tab. 7. Accumulato dex varables for the examples ad Expermet No. Example Example Z R Q Z R Q As metoed the treatmet whch has a mmum of Q value s the best treatmet that cludes the best levels of predctor varables. As show Table 7, the mmum value of Q dex for the real case s zero; ad ths value s related to the frst treatmet whch has levels of,, for each predctor varable. The calculated accumulato dces of secod example show that last treatmet of ths example has mmum value of Q whch has cluded the levels of,, 3 for predctor varables, respectvely. Note that the frst Q value of secod example has a very low gap from last Q. Ths case ca be better terpreted wth the GTA. Through lookg at Table 5, t ca be uderstood that addto to the (3, (, )) NE, the (, (, )) strategy also s NE; but usg the Pareto optmal, the (3, (, )) s the best strategy of ths game. The accuracy of the GTA s determed by comparg the results of Tables 3 wth 7 ad 5 wth Valdato of the Proposed Game Theory- Based Approach I ths subsecto, we valdate the proposed game theory-based approach (heurstc approach) through smulato studes. For ths purpose, we compare the results of the proposed approach ad the VIKOR method uder 0,000 smulated problems. The smulato steps are gve as follows: Step : Geerate the three depedet factor levels wth specfed lower ad upper boud usg radom umbers betwee ad 4 Matlab software ad the geerate the correlated resduals of the regresso Iteratoal Joural of Idustral Egeerg & Producto Research, September 04, Vol. 5, No. 3

9 3 H.R. Navd, A. Amr, R. Kamrarad Mult Resposes Optmzato Through Game Theory Approach models usg multvarate ormal dstrbuto wth μ [0,0] ad Σ, where ad ad are the varaces of two resposes s the covarace betwee the resposes. I our smulato, we use the, ad 0. respectvely for varaces of the resposes ad the covarace betwee resposes. Step : Calculate the values of two resposes usg the followg hypothetcal regresso models: y y 0.3 x 0.5 x 0. x 0.8 x 0.5 x 3 0. x 3 Step 3: Determe the S/N rato of these resposes ad desg the game theory model to select the NE pot. Step 4: Determe the VIKOR dex ad select the optmal settg. The smulato showed the detcal results of the proposed approach ad VIKOR method 9% of cases. Hece, the proposed approach performs effcet dfferet problems. 5. Cocluso ad Future Researches I ths paper, a ew method was proposed to mult resposes optmzato based o the game theory approach. GTA s a useful method for decso makg the coflct of terests betwee tellget players order to select the best ot strategy for them through selectg the best ot desrablty. May expermets have more tha oe respose varable whch may have coflctg obectves such as LTB, NTB ad STB. Ths paper cosdered two resposes wth the LTB ad NTB type the real case ad the umercal example. Frst, SN ratos of each respose (or each player) were calculated ad modeled as game theory table. The, usg the NE ad PONE, the PE of those examples were determed whch were the best ot utltes for selected strateges of each player. Also, the VIKOR method was used to evaluate the effcecy of the proposed GTA. The obtaed results represeted the accuracy of the proposed approach. Hece, the GTA ca be used effcetly for mult resposes optmzato problems. Geerally, the proposed approach s a ovel heurstc approach whch ca be easly appled stead of some complcated mult respose optmzato methods ad get the same results. As a future research, mult resposes optmzato problem wth more tha two resposes whch represets the games by three ad more players ca be vestgated. Also, some stuatos, the respose varables are lgustc ad characterzed by fuzzy radom umbers. I these cases, t seems the fuzzy game theory model should be developed to solve the mult respose optmzato problem whch s a frutful area for research. 6. Ackowledgemet The authors would lke to thak the aoymous revewers for hs valuable ad costructve commets whch led to mprovemet the paper. Refereces [] Navd, H., Ketabch, S., Mess, M., A Itroducto to Game Theory, ( Fars), Shahed Uversty, 0. [] Osbore, M.J., A Course Game Theory, Oxford Uversty Press, 003. [3] Myerso, R.B., Game Theory: Aalyss of Coflct, The MIT Press, 997. [4] Poroch-Serta, M., Gutt, S., Gutt, G., Crestescu, I., Coocaru, C., Sever, T., Desg of Expermets for Statstcal Modelg ad Mult-Respose Optmzato of Nckel Electroplatg Process, Chemcal Egeerg Research ad Desg, Vol. 89, 0, pp [5] Khur, A.I., Colo, M., Smultaeous Optmzato of Multple Resposes Represeted by Polyomal Regresso Fuctos, Techometrcs, Vol. 3, 98, pp [6] Logothets, N., Hagh, A., Characterzg ad Optmzg Mult-Respose Processes by the Taguch Method, Qualty ad Relablty Egeerg Iteratoal, Vol. 4, 988, pp [7] Tog, L.I., Su, C.T., Optmzg Mult-Respose Problems the Taguch Method by Fuzzy Multple Attrbute Decso Makg, Qualty ad Relablty Egeerg Iteratoal, Vol. 3, 997, pp [8] Jeyapaul, R., Shahabudee, P., Krshaah, K., Smultaeous Optmzato of Mult-Respose Problems the Taguch Method Usg Geetc Algorthm, The Iteratoal Joural of Advaced Maufacturg Techology, Vol. 30, 005, pp [9] Tog, L.I., Hseh, K.L., A ovel meas of applyg artfcal eural etworks to optmze mult-respose problem, Qualty Egeerg, Vol. 3, 000, pp [0] Pa, L.K., Haq., Optmzg Multple Qualty Characterstcs Va Taguch Merthod-Based Grey Aalyss, Joural of Materals Processg Techology, Vol. 8, 007, pp [] Haq, A.N., Marmuthu, P., ad Jeyapaul, R., Mult Respose Optmzato of Machg Parameters of Drllg Al/SC Metal Matrx Composte usg Grey Relatoal Aalyss the Taguch Method, The Iteratoal Joural of Advaced Maufacturg Techology, Vol. 37, 007, pp Iteratoal Joural of Idustral Egeerg & Producto Research, September 04, Vol. 5, No. 3

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