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Engineering Computer Science Operations Research : Applications and Algorithms

To use the Dantzig-Wolfe decomposition algorithm to solve the given problem. Given problem: max z= 7x 1 +5x 2 +3x 3 s .t . x 1 +2x 2 +x 3 ≤ 10 x 2 +x 3 ≤ 5 x 1 ≤ 3 2x 2 +x 3 ≤ 8 x 1 ,x 2 ,x 3 ≥ 0 Explanation: Dantzig-Wolfe decomposition algorithm is a method used for solving large-scale linear programming problems. This approach is particularly useful when the problem has a specific structure. To apply the Dantzig-Wolfe decomposition algorithm, rewrite the given linear programming problem in the block-angular form, which allows us to break the problem into smaller sub-problems. Now, rewrite the problem in block-angular form by introducing new variables: Let x 1 = x 1 ' , x 2 = x 2 ' + x 2 " , x 3 = x 3 ' + x 3 " The new problem becomes: max z = 7 x 1 ' + 5 ( x 2 ' + x 2 " ) + 3 ( x 3 ' + x 3 " ) s . t . x 1 ' + 2 ( x 2 ' + x 2 " ) + ( x 3 ' + x 3 " ) ≤ 10 ( x 2 ' + x 2 " ) + ( x 3 ' + x 3 " ) ≤ 5 x 1 ' ≤ 3 2 ( x 2 ' + x 2 " ) + ( x 3 ' + x 3 " ) ≤ 8 x 1 ' , x 2 ' , x 2 " , x 3 ' , x 3 " ≥ 0 Now, decompose the problem into subproblems: Subproblem 1: max 7 x 1 ' + 5 x 2 ' + 3 x 3 ' s . t . x 1 ' + 2 x 2 ' + x 3 ' ≤ 10 x 2 ' + x 3 ' ≤ 5 x 1 ' ≤ 3 x 1 ' , x 2 ' , x 3 ' ≥ 0 Subproblem 2: max 5 x 2 " + 3 x 3 " s . t .2 x 2 " + x 3 " ≤ 8 x 2 " , x 3 " ≥ 0 Now, solve these subproblems independently: Subproblem 1: Using the simplex method or any other method, obtain the optimal solution as x 1 ' =3, x 2 ' =1 .5, x 3 ' =2 .5, and z 1 =28 .5 Subproblem 2: Using the simplex method or any other method, obtain the optimal solution as x 2 " =4, x 3 " =0, and z 2 =20 Now, combine the solutions: x 1 = x 1 ' = 3 x 2 = x 2 ' + x 2 " = 1.5 + 4 = 5.5 x 3 = x 3 ' + x 3 " = 2.5 + 0 = 2.5 Conclusion: The optimal solution for the original problem is: x 1 = 3, x 2 = 5 .5, x 3 = 2 .5 , and the optimal objective value is z = 7(3) + 5(5 .5) + 3(2 .5) = 21 + 27 .5 + 7 .5 = 56 .

To use the Dantzig-Wolfe decomposition algorithm to solve the given problem. Given problem: max z= 7x 1 +5x 2 +3x 3 s .t . x 1 +2x 2 +x 3 ≤ 10 x 2 +x 3 ≤ 5 x 1 ≤ 3 2x 2 +x 3 ≤ 8 x 1 ,x 2 ,x 3 ≥ 0 Explanation: Dantzig-Wolfe decomposition algorithm is a method used for solving large-scale linear programming problems. This approach is particularly useful when the problem has a specific structure. To apply the Dantzig-Wolfe decomposition algorithm, rewrite the given linear programming problem in the block-angular form, which allows us to break the problem into smaller sub-problems. Now, rewrite the problem in block-angular form by introducing new variables: Let x 1 = x 1 ' , x 2 = x 2 ' + x 2 " , x 3 = x 3 ' + x 3 " The new problem becomes: max z = 7 x 1 ' + 5 ( x 2 ' + x 2 " ) + 3 ( x 3 ' + x 3 " ) s . t . x 1 ' + 2 ( x 2 ' + x 2 " ) + ( x 3 ' + x 3 " ) ≤ 10 ( x 2 ' + x 2 " ) + ( x 3 ' + x 3 " ) ≤ 5 x 1 ' ≤ 3 2 ( x 2 ' + x 2 " ) + ( x 3 ' + x 3 " ) ≤ 8 x 1 ' , x 2 ' , x 2 " , x 3 ' , x 3 " ≥ 0 Now, decompose the problem into subproblems: Subproblem 1: max 7 x 1 ' + 5 x 2 ' + 3 x 3 ' s . t . x 1 ' + 2 x 2 ' + x 3 ' ≤ 10 x 2 ' + x 3 ' ≤ 5 x 1 ' ≤ 3 x 1 ' , x 2 ' , x 3 ' ≥ 0 Subproblem 2: max 5 x 2 " + 3 x 3 " s . t .2 x 2 " + x 3 " ≤ 8 x 2 " , x 3 " ≥ 0 Now, solve these subproblems independently: Subproblem 1: Using the simplex method or any other method, obtain the optimal solution as x 1 ' =3, x 2 ' =1 .5, x 3 ' =2 .5, and z 1 =28 .5 Subproblem 2: Using the simplex method or any other method, obtain the optimal solution as x 2 " =4, x 3 " =0, and z 2 =20 Now, combine the solutions: x 1 = x 1 ' = 3 x 2 = x 2 ' + x 2 " = 1.5 + 4 = 5.5 x 3 = x 3 ' + x 3 " = 2.5 + 0 = 2.5 Conclusion: The optimal solution for the original problem is: x 1 = 3, x 2 = 5 .5, x 3 = 2 .5 , and the optimal objective value is z = 7(3) + 5(5 .5) + 3(2 .5) = 21 + 27 .5 + 7 .5 = 56 .

Operations Research : Applications and Algorithms

Operations Research : Applications and Algorithms

4th Edition

ISBN: 9780534380588

Author: Wayne L. Winston

Publisher: Brooks Cole

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