Explanation Given Information: A payoff matrix, [ 8 2 5 1 8 4 ] . Formula used: Value of the game is: v = 1 z 1 + z 2 + ⋯ z n Calculation: Consider the payoff matrix, [ 8 2 5 1 8 4 ] . First, set the tableaux to find the optimal strategy of C then find the dual solution to get the optimal strategy of R . Let z 1 , z 2 and z 3 be chosen so as to maximize z 1 + z 2 + z 3 subject to the constraints, { 8 z 1 + 2 z 2 + 5 z 3 ≤ 1 z 1 + 8 z 2 + 4 z 3 ≤ 1 z 1 ≥ 0 , z 2 ≥ 0 , z 3 ≥ 0 Hence, the linear programming problem corresponding to the provided payoff matrix is, Maximize z 1 + z 2 + z 3 Subjects to the constraints { 8 z 1 + 2 z 2 + 5 z 3 ≤ 1 z 1 + 8 z 2 + 4 z 3 ≤ 1 z 1 ≥ 0 , z 2 ≥ 0 , z 3 ≥ 0 Now, solve the above problem as below: Introduce the slack variables t and u and write the problem in terms of a system of linear equations. { 8 z 1 + 2 z 2 + 5 z 3 + t = 1 z 1 + 8 z 2 + 4 z 3 + u = 1 z 1 ≥ 0 , z 2 ≥ 0 , z 3 ≥ 0 The objective function z 1 + z 2 + z 3 gives the equation, M = z 1 + z 2 + z 3 . So, the corresponding system of linear equation with slack variables is, { 8 z 1 + 2 z 2 + 5 z 3 + t = 1 z 1 + 8 z 2 + 4 z 3 + u = 1 − z 1 − z 2 − z 3 + M = 0 Where, M is as large as possible. Thus, the initial simplex table corresponding to above system is: z 1 z 2 z 3 t u M t 8 2 5 1 0 0 1 u 1 8 4 0 1 0 1 M − 1 − 1 − 1 0 0 1 0 As the left part of the bottom row has negative entries − 1 , the above table does not give the maximum value of the objective function. So, create a new simplex table by pivoting. Since, − 1 is the most negative entry in the last row, the pivot column can be chosen arbitrarily. Let first column becomes the pivot column. Now, to determine the pivot element, compute ratios as below: z 1 z 2 z 3 t u M Ratios t 8 2 5 1 0 0 1 1 8 u 1 8 4 0 1 0 1 1 M − 1 − 1 − 1 0 0 1 0 Since, the smallest non-negative ratio is 1 8 , so the pivot element is 8. Now, the new simplex table is constructed by pivoting around the element 8. Therefore, the new simplex table is, z 1 z 2 z 3 t u M z 1 1 1 4 5 8 1 8 0 0 1 8 u 0 31 4 27 8 − 1 8 1 0 7 8 M 0 − 3 4 − 3 8 1 8 0 1 1 8 The left part of the bottom row has negative entries ( − 3 4 , − 3 8 ) . Thus, the above table does not give the maximum value of the objective function. So, construct a new simplex table by pivoting. Since, − 3 4 is the most negative entry in the last row, the second column becomes the pivot column. Now, to determine the pivot element, compute ratios as below: z 1 z 2 z 3 t u M Ratios z 1 1 1 4 5 8 1 8 0 0 1 8 1 2 u 0 31 4 27 8 − 1 8 1 0 7 8 7 62 M 0 − 3 4 − 3 8 1 8 0 1 1 8 Since, the smallest non-negative ratio is 7 62 , so the pivot element is 31 4

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Author: Goldstein

Publisher: PEARSON

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Chapter 9.3, Problem 16E

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Chapter 9.3, Problem 15E

Chapter 9.3, Problem 17E

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Chapter 9 Solutions

FINITE MATHEMATICS & ITS APPLICATIONS

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The value of the game and optimal strategy for R for the payoff matrix, [ 8 2 5 1 8 4 ] .

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Chapter 9 Solutions