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 booklet is to advance a collection of potent and effective linear programming types and techniques for fixing matrix video games with payoffs in fuzzy numbers. Divided into six chapters, it discusses the strategies of suggestions of matrix video games with payoffs of durations, besides their linear programming types and strategies. in addition, it really is without delay proper to the examine box of matrix video games below doubtful fiscal administration. The e-book deals a necessary source for readers considering theoretical study and functional functions from a variety of varied fields together with video game concept, operational study, administration technological know-how, fuzzy mathematical programming, fuzzy arithmetic, commercial engineering, company and social economics.

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0 ðj ¼ 1; 2; . 4 Acceptability-Degree-Based Linear Programming Models … 33 which may be rewritten as the following bi-objective linear programming model: minfxR g xL þ xR g minf 8 n 2 P > > aRij zj xR ði ¼ 1; 2; . ; 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 > > z ¼1 > > j¼1 j > > > > > > zj ! 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.

Value of the interval-valued matrix game A (2) Computational results obtained by different methods and analysis In this subsection, the aforementioned numerical example is solved by the linear programming method proposed in the above Sect. 2 and other similar methods [18, 19, 28, 29]. The computational results are analyzed and compared to show the validity, applicability, and superiority of the developed linear programming method. (2a) Computational results obtained by the developed linear programming method According to the linear programming method proposed in the above Sect.

0 ðj ¼ 1; 2; . ; nÞ > > Rj : xR unrestricted in sign, where zRj (j ¼ 1; 2; . ; n) and xR are decision variables. To solve Eq. 48), let zRj ðj ¼ 1; 2; . 2, without loss of generality, assume that xR [ 0. Then, tRj ! 0 (j ¼ 1; 2; . ; n) and n X j¼1 tRj ¼ n X zRj j¼1 xR ¼ 1 : xR ð1:50Þ 46 1 Interval-Valued Matrix Games Hence, Eq. 50) can be transformed into the linear programming model as follows: maxf n X tRj g j¼1 8P n < aRij tRj 1 ði ¼ 1; 2; . ; mÞ s:t: j¼1 : tRj ! 0 ðj ¼ 1; 2; . ; nÞ; ð1:51Þ where tRj (j ¼ 1; 2; .