Solution of ectngulr Fuzzy Gmes by Principle of Dominnce Using -type Trpezoidl Fuzzy Numbers Arindm Chudhuri ecturer Memtics nd Computer Science), Meghnd Sh Institute of Technology, Nzirbd, Uchchepot, Kolkt, Indi Emil: rindm_chu@yhoocoin ABSTACT Fuzzy Set Theory hs been pplied in mny fields such s Opertions eserch, Control Theory, nd Mngement Sciences etc In prticulr, n ppliction of is eory in Mngeril Decision Mking Problems hs remrkble significnce In is Pper, we consider solution of ectngulr Fuzzy gme wi py-off s imprecise numbers insted of crisp numbers viz, intervl nd -type Trpezoidl Fuzzy Numbers The solution of such Fuzzy gmes wi pure strtegies by minimx-mximin principle is discussed The Algebric Meod to solve Fuzzy gmes wiout sddle point by using mixed strtegies is lso illustrted Here, m n py-off mtrix is reduced to py-off mtrix by Dominnce Meod This fct is illustrted by mens of Numericl Exmple Keywords Intervl Number, -type Trpezoidl Fuzzy Number, Fuzzy gme, Dominnce INTODUCTION The problem of Gme Theory [, ] is defined s body of knowledge t dels wi mking decisions when two or more intelligent nd rtionl opponents re involved under conditions of conflict nd competition In prcticl life, it is required to tke decisions in competing sitution when ere re two or more opposite tems wi conflicting interests nd e outcome is controlled by e decisions of ll prties concerned Such problems occur frequently in Economics, Business Administrtion, Sociology, Politicl Science nd Militry Opertions [] In ll e bove problems of different Sciences where e competitive situtions re involved, one cts in rtionl mnner nd tries to resolve e conflict of interests in his fvour It is in is context t e Gme Theory cn prove useful decision mking tool Insted of mking inferences from e pst behvior of e opponent, e pproch of Gme Theory is to seek to determine rivl s most profitble counter strtegy to one s own best moves nd formulte e pproprite defensive mesures For Exmple, if two firms re locked up in wr to mintin eir mrket shre, en price-out by e first firm will invite rection from e second firm in e nture of price-cut This will, in turn, ffect e sles nd profits of e first firm which will gin hve to develop counter strtegy to meet e chllenge from e second firm The gme will us go on until one of e firms emerges s winner The memticl tretment of e Gme Theory ws mde vilble in 944, when John Von Newmnn nd Oscr Morgenstern published e fmous rticle Theory of Gmes nd Economic Behvior [] The Von Newmnn s pproch to solve e Gme Theory problems ws bsed on e principle of best out of e worst ie, he utilized e ide of minimiztion of e mximum losses Most of e competitive problems cn be hndled by is principle However, in rel life situtions, e informtion vilble is of imprecise nture
nd ere is n inherent degree of vgueness or uncertinty present in e system under considertion In order to tckle is uncertinty e concept of Fuzzy Sets cn be used s n importnt decision mking tool Imprecision here is ment in e sense of vgueness rer n e lck of knowledge bout e prmeters present in e system The Fuzzy Set Theory us provides strict memticl frmework in which vgue conceptul phenomen cn be precisely nd rigorously studied In is work, we hve concentrted on e solution of ectngulr Gmes by Principle of Dominnce Using -type Trpezoidl Fuzzy Numbers The -type Trpezoidl Fuzzy Numbers re defined by e Trpezoidl Membership Functions They re chrcterized by eir simple formultions nd computtionl efficiency nd us hve been used extensively to solve different problems in Engineering nd Mngement The solution of Fuzzy gmes wi Py-off s imprecise number is generlly given by e minimx-mximin Principle In e pst, work hs lso been done on Algebric Meod of solving m n ectngulr Fuzzy gmes wi Pyoff s Intervl Numbers hving no Sddle Point Determintion of Fuzzy gmes from ectngulr m n Fuzzy gme wiout Sddle Point is fundmentl problem of Fuzzy Gme Theory In Clssicl Gme Theory e Py-off is Crisp Number Prcticlly, it my hppen t Py-off is not necessrily fixed rel number Here, e Py-off is considered s -type Trpezoidl Fuzzy Number nd e m n mtrix is reduced to mtrix The Orgniztion of is Pper is s follows In section, we define e Trpezoidl Membership Function In e next section, we consider e concept of e Intervl Numbers In e section 4, we give e bsic definitions of e Two Person Zero Sum Gmes nd e Py-off Mtrix The section 5 discusses Solution of Gmes wi mixed strtegies In e next section, we discuss bout Gmes hving no Sddle Point The section 7 illustrtes e Concept of Dominnce This is followed by Simultion Exmple Finlly, in section 8 conclusions re given TAPEZOIDA MEMBESHIP FUNCTION A trpezoidl membership function is defined by four prmeters viz,, c, d s follows [3] trpezoid x, b, c, d) 3 INTEVA NUMBES 0, x x, xb b x c d x, cxd d c 0, d x x An intervl number [4] is defined s I [ X, X ] { x : X x X, x } Anoer wy of representing n intervl number in terms of midpoint is, I m i), w i) where, m i) midpoint of I X X ) / nd w i) hlf wid of I X ) / X Addition of two intervl numbers I [ X, X ] nd J [ Y, Y ] is I J [ X Y, X Y ] Using men wid nottions, if I m, w nd J m, w en I J m m, w w Similrly, e oer binry opertions on intervl numbers re defined [4] 3 ODEED EATION AMONG INTEVAS If I [, b], J [ c, d] en [, b] [ c, d], iff b c nd is denoted by I J I is contined in J iff c d nd is is denoted by I J Definition The dominted index DI) to proposition I is dominted over J s I I J ) m m ) / w ) w
Using DI e following rnking order is defined Definition If DI I J ), en I is sid to be totlly dominting over J in e sense of minimiztion nd J is sid to be totlly dominting over J in e sense of minimiztion This is denoted by I J Definition 3 If 0 DI I J ), en I is sid to be prtilly dominting over J in e sense of minimiztion nd J is sid to be prtilly dominting over J in e sense of minimiztion This is denoted by I J When DI I J) 0, en m m, it my be emphsized on e wid of intervl numbers I nd J If w w, en left end point of I is less n t of J nd ere is chnce t on finding minimum distnce, e distnce my be on I But t e sme time, since e right end point of I is greter n t of J, if one prefers I over J in minimiztion en in worst cse, he my be looser n one who prefer J over I Numericl Exmple: I [0,0] 5, 5, J [50,55 ] 5 5, 5, DI I J) 5 5 5) / So in minimiztion I is totlly dominting over J Definition 4 The dominted index DI) of proposition A,, ) is dominted over B, ) is given by DI A B) b ) / ) Using DI index e following rnking order is defined Definition 5 If DI A B), A is sid to be totlly dominting over B in e sense of minimiztion nd B is sid to be totlly dominting over A in e sense of mximiztion, it is lso denoted by A B Definition 6 If 0 DI A B), en A is sid to be prtilly dominting over B in e sense of minimiztion nd B is sid to be prtilly dominting over A in e sense of mximiztion, it is lso denoted by A B emm:if DI A B) 0 en A nd B re sid to be non comprble nd is denoted by A B In is cse A is preferred over B if ) nd or b) nd, oerwise pessimistic decision mker would prefer e number wi smller leng of support wheres n optimistic decision mker would do e converse Numericl Exmple: )If A 6,0, B,0 en DI A B ) 6 ) /0 0) Thus, A is totlly dominting over B in e sense of minimiztion nd B is sid to be totlly dominting over A in e sense of mximiztion )If A 04,0 5, B 03,0 5 en DI A B) 04 03) /05 05) 0 DI A B) 04 03) /05 05) 0 So, A is sid to be prtilly dominting over B 4 TWO PESON ZEO SUM GAMES AND PAY-OFF MATIX In is section we give some bsic definitions of e Two Person Zero Sum Gmes nd py-off mtrix These concepts form e bsic building blocks of Gme Theory 4 TWO PESON ZEO SUM GAMES A gme of two persons in which gins of one plyer re losses of oer plyer is clled two person zero sum gme, ie, in two person zero sum e lgebric sum of gins to bo plyers fter ply is bound to be zero Gmes relting to pure strtegies tken
by plyers re considered here bsed on two ssumptions [, ] Plyer A is in better position nd is clled mximiztion plyer or row plyer) nd plyer B is clled minimizing plyer or column plyer) Totl gin of one plyer is exctly equl to totl loss of oer plyer In generl, if plyer A tkes m pure strtegies nd B tkes n pure strtegies, en e gme is clled two person zero sum gme or m n rectngulr gme 4 PAY-OFF MATIX Two person zero sum gmes re known s rectngulr gmes since ey re represented by rectngulr py-off mtrix A py-off mtrix is lwys written for mximizing plyer Considering e generl m n rectngulr gme, e py-off mtrix of A wi m pure strtegies A,, Am nd B wi n pure strtegies B,, Bn is given by [3], n m, bm mn n mn The elements re -type trpezoidl fuzzy numbers nd for crisp gme ey my be positive, negtive or zero When plyer A chooses strtegy A i nd plyer B selects B j, it results in py-off of -type trpezoidl fuzzy number to plyer A 5 SOUTION OF GAMES WITH MIXED STATEGIES Consider e fuzzy gme [3, 5] of plyers A strtegies represented horizontlly) nd B strtegies represented verticlly) whose py-off is given by following mtrix nd for which ere is no sddle point where, py-off re symmetric -type trpezoidl fuzzy numbers such t b If xi nd y j be e probbilities by which A chooses i strtegy nd B chooses j strtegy en: x y x y ) / ) ) / ) ) / ) ) / ) which re crisp numbers nd vlue of e gme cn be esily computed s V where, nd b re left nd right spreds of -type trpezoidl fuzzy numbers given by: ) / ) b b b b b ) / b b b ) b 6 GAMES WITH NO SADDE POINT The simplest cse is Fuzzy gme wi no sddle point Here, we consider m n Fuzzy gme Now we discuss prticulr meod In is meod, e pyoff cn be reduced to gmes so t it cn be solved by using e Fuzzy gme meod The meod of reduction of e pyoff mtrix by is process is clled e Dominnce property [, ] of e rows nd columns of e py-off mtrix
7 CONCEPT OF DOMINANCE If one pure strtegy of plyer is better for him or s good s noer, for ll possible pure strtegies of opponent en first is sid to dominte e second [, ] The dominted strtegy cn simply be discrded from py-off mtrix since it hs no vlue When is is done, optiml strtegies for e reduced mtrix re lso optiml for e originl mtrix wi zero probbility for discrded strtegies When ere is no sddle point in py-off mtrix, en size of e gme cn be reduced by dominnceefore e problem is solved Definition 7 If ll elements of e i row of py-off mtrix of m n rectngulr gme re dominting over r row in e sense of mximiztion, r row is discrded nd deletion of r row from mtrix does not chnge e set of optiml strtegies of mximizing plyer Numericl Exmple Consider e fuzzy gme of two plyers A strtegies represented horizontlly) nd B strtegies represented verticlly) wi e following py-off mtrix Plyer A is mximizing plyer nd plyer B is minimizing plyer,0 7,03,0 6,0,0 7,03 0,0,0 6,0 DI A3 A) 6 0) /0 0) DI A A ) ) /04 0) DI 3 A3 A) 7 6) /03 0) Thus A is dominting over A3 in e sense of mximiztion nd row A 3 is deleted The reduced mtrix is given by,,0 7,03,0 6,0,0 7,03 Definition 8 If ll elements of j column re dominting over s column in e sense of minimiztion e s column is deleted nd e deletion of s column from e mtrix does not chnge e set of optiml strtegies of minimizing plyer Numericl Exmple Considering e bove py-off mtrix, DI B B3) ) /0 03) DI B B3 ) 7 6) /0 03) Here B is totlly dominting over B3 in e sense of minimiztion nd e resultnt pyoff mtrix is given by,,0 7,03 6,0,0 Definition 9 If e liner combintion of p nd q rows domintes ll elements of e s row in e sense of minimiztion, s row is discrded nd e deletion of s row from mtrix does not chnge e set of optiml strtegies of mximizing plyer Numericl Exmple Considering prticulr py-off mtrix of two plyers A strtegies represented horizontlly) nd B strtegies represented verticlly) s follows:,04 3,05,0,0,03 3,04,0,0,04 The convex combintion of second nd ird row gives A A ) A 0 4 3 3
Tking 0 5 e elements of A4 re,0 35,,0 35 nd,0 30 Now A 4 is dominting over A in e sense of mximiztion nd row A is discrded such t e resulting py-off mtrix is given by, 3,05,03,0,0 3,04,04 Definition 0 If j column is dominted by e convex combintion of m nd n column, j column is discrded in sense of minimiztion nd deletion of j column from mtrix does not chnge e set of optiml strtegies of e minimizing plyer Numericl Exmple Considering bove py-off mtrix, e convex combintion of B nd B ie, B B ) B 0 4 Tking 0 5 elements of B4 re,,04,03 NowB 4 is totlly dominting over B3 nd us e ird column is removed such t e resulting mtrix is given by, 3,05,0 3,04,03 Definition When ere is no sddle point nd no course of ction domintes ny oer e vlues for different sub gmes re computed As A is mximizing plyer he will definitely select t pir strtegies which will give e best vlue of sub gmes nd e corresponding sub mtrix provides optiml solution Similrly, B is minimizing plyer he will definitely select t pir of courses, which will give e lest vlue of sub gmes, e corresponding sub mtrix will provide optiml solution to e fuzzy problem Numericl Exmple Consider e fuzzy gme whose py-off mtrix is given by, 9,0 5,04 6,0 0,0 0,04 5,04 There is no sddle point nd no course of ction domintes ny oer The vlues V, V, V3 re computed from e following ree sub gmes s obtined from given mtrix 9,0 5,04 Sub gme : 0,0 0,04 95 7 The corresponding vlue V, 6 5 9,0 6,0 Sub gme : 0,0 5,04 The corresponding vlue V 6,0 5,04 6,0 Sub gme 3: 0,04 5,04 45 The corresponding vluev 3, 6 5 Here, min{ V, V, V3} V3 such e resulting py-off mtrix is: 5,04 0,04 6,0 5,04 8 NUMEICA SIMUATION To illustrte dominnce meod, we consider e following py-off mtrix:
8,03 9,0 0,03 5,04 5,05 0,0 4,0 7,04 5,05,04 6,0 5,04 All DI A A ), so A is totlly dominting over A in e sense of minimiztion nd row A is deleted, such t e resulting py-off mtrix is: 9,0 003 5,05 0,0 7,04 5,05 6,0 5,04 Agin ll DI B4 B3), so B4 is totlly dominting over B3 in e sense of minimiztion nd e column B 3 is deleted such t e resulting py-off mtrix is: 9,0 003 5,05 0,0 6,0 5,04 This is no course of ction which domintes ny oer nd ere is no sddle point The vlues for different pir of strtegies re computed Since, B is minimizing plyer e minimum vlue is considered nd corresponding py-off mtrix provides optiml solution to fuzzy problem The previous lest vlue of { V, V, V3} isv 3 so optiml strtegies of A re A, ) nd A3 of B re B, B3) The finl py-off mtrix is given by, 5,05 0,0 6,0 5,04 The probbilities re 5 x 0, x, x3, y 0, y, 6 6 6 5 y3 0, y4 6 45 nd vlue of e gme is V, 6 5 Hence, optiml solution of e complete gme is 0, x, ) for A nd x3 0, y,0, y4 ) for B e vlue of gme beingv 9 CONCUSION We considered e solution of ectngulr Fuzzy gmes using -type Trpezoidl Fuzzy Numbers Here py-off is considered s imprecise numbers insted of crisp numbers which tkes cre of e uncertinty nd vgueness inherent in such problems -type Trpezoidl Fuzzy Numbers re used becuse of eir simplicity nd computtionl efficiency We discuss solution of Fuzzy gmes wi pure strtegies by minimx-mximin principle nd lso Algebric Meod to solve Fuzzy gmes wiout sddle point by using mixed strtegies The Concept of Dominnce Meod is lso illustrted -type Trpezoidl Fuzzy Numbers genertes optiml solutions which re fesible in nture nd lso tkes cre of e impreciseness spect EFEENCES Gupt P K nd Mn Mohn, Problems in Opertions eserch, Sultn Chnd nd Sons, 006 Kpoor V K nd Kpoor S, Opertions eserch-techniques for Mngement, Sultn Chnd nd Sons, 006 3 Nryn A, Meenkshi A nd msmy A M S, Fuzzy Gmes, The Journl of Fuzzy Memtics, 0, 00 4 Sengupt A nd Pl T K, On Compring Intervl Numbers, Europen Journl of Opertionl eserch, 7, 000 5 Zimmermnn H J, Using Fuzzy Sets in Opertionl eserch, Europen Journl of Opertionl eserch, 3, 983