By Deng-Feng Li
This publication addresses two-person zero-sum finite video games during which the payoffs in any scenario are expressed with fuzzy numbers. the aim of this publication is to boost a set of potent and effective linear programming versions and techniques for fixing matrix video games with payoffs in fuzzy numbers. Divided into six chapters, it discusses the suggestions of options of matrix video games with payoffs of durations, in addition to their linear programming types and strategies. moreover, it's at once suitable to the learn box of matrix video games lower than doubtful financial administration. The booklet deals a useful source for readers fascinated about theoretical examine and functional functions from various various fields together with online game concept, operational learn, administration technological know-how, fuzzy mathematical programming, fuzzy arithmetic, business engineering, enterprise and social economics.
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Extra resources for Linear Programming Models and Methods of Matrix Games with Payoffs of Triangular Fuzzy Numbers
0 ðj ¼ 1; 2; . ; nÞ > > : xL and xR unrestricted in sign; ði ¼ 1; 2; . ; mÞ ð1:24Þ where zj (j ¼ 1; 2; Á Á Á ; n), xL , and xR are variables, and e 2 ½0; 1 is given a priori. Analogously, using the weighted average method, Eq. 24) may be aggregated into the linear programming model as follows: 3xR þ xL g minf 8 n 4 P > > aRij zj xR ði ¼ 1; 2; Á Á Á ; mÞ > > > j¼1 > > > > n n P P > > > ð1 À eÞ aLij zj þ e aRij zj ð1 À eÞxL þ exR > > > j¼1 j¼1 < s:t: xL xR > > n P > > > zj ¼ 1 > > > j¼1 > > > > > zj !
22), where tÃ ¼ ½tÃL ; tÃR is an interval. Thus, the maximin (or optimal) mixed strategy yÃ and the gain-floor tÃ for the player I can be obtained. 7, Eq. 21) can be transformed into the bi-objective programming model as follows: minfxR g xL þ xR g minf 8 n 2 P > > aRij zj xR ði ¼ 1; 2; . ; mÞ > > > j¼1 > > > n > P > > aLij zj ÀxL > > > j¼1 > e ði ¼ 1; 2; . ; mÞ > n n P P > > < ðxR ÀxL ÞÀð aRij zj À aLij zj Þ j¼1 j¼1 s:t: > xL xR > > > n > >P > > zj ¼ 1 > > > j¼1 > > > > > zj ! 0 ðj ¼ 1; 2; .
5), ðmR ; yÃR Þ is an optimal solution to the linear programming model as follows: maxftR g 8P m > > aRij yRi ! tR ðj ¼ 1; 2; . ; nÞ > > > i¼1 > >
i¼1 > > > > yRi ! 0 ði ¼ 1; 2; . ; mÞ > > : tR unrestricted in sign, ð1:36Þ where yRi (i ¼ 1; 2; . ; m) and tR are variables. To solve Eq. 36), let xRi ¼ yRi tR ði ¼ 1; 2; . 2, assume that tR [ 0. Then, xRi ! 0 (i ¼ 1; 2; . ; m) and m X xRi ¼ i¼1 m X yRi i¼1 tR ¼ 1 : tR ð1:38Þ Hence, Eq. 36) can be transformed into the linear programming model as follows: minf m X i¼1 8 m P < s:t: : xRi g aRij xRi !