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1A. Write the general dual problem associated with the given LP. (Do not transform or rewrite the primal problem before writing the general dual) Maximize -4x1 + 2x2 Subject To 4x, +x, + X3 = 20 2x1 – X2 26 X1 – X, + 5x; 2-5 -3x, + 2x, + X3 S4 Xị <0, x2 2 0, x3 unrestricted IB. Given the following information for a product-mix problem with three products and three resources. Primal Decision Variables: x1 = number of unit 1 produced; x, = # of unit 2 produced; x; = # of unit 3 produced Primal Formulation: Dual Formulation: Max Z (Rev.) = 25x1 8x1 4x1 2x1 + 30x, + 20x3 + 6x2 + 2x2 + X2 X1, X2, X3 + 20n, +25n3 + 4n, +2n2 25 + 2n, +ng 2 30 + 3n, +2732 20 Min W = Subject To < 50 + 3x3 <20 + 2x; $25 20 (Res. 1 constraint) (Res. 2 constraint) (Res. 3 constraint) (Nonnegativity) + X3 Subject To T1, T2, T3 20 Optimal Solution: Optimal Z = Revenue = $268.75 XỊ = 0 (Number of unit 1) X2 = 8.125 (Number of unit 2) X; = 1.25 (Number of unit 3) Resource Constraints: Dual Var. Optimal Value = 22.5 (Surplus variable in 1t dual constraint) Dual Var. Optimal Value = 0 (Surplus variable in 25d dual constraint) Dual Var. Optimal Value = 0 (Surplus variable in 3rd dual constraint) Resource 1 = 0 leftover units Resource 2 = 0 leftover units Dual Var. Optimal Value = 3.125 = T1 Dual Var. Optimal Value = 5.625 = T2 Dual Var. Optimal Value = 0 = T; Resource 3 = 14.375 leftover units 1Bi. What is the fair-market price for one unit of Resource 3? 1Bi. What is the meaning of the surplus variable value of 22.5 in the 1 dual constraint with respect to the primal problem? AI

1A. Write the general dual problem associated with the given LP. (Do not transform or rewrite the primal problem before writing the general dual) Maximize -4x1 + 2x2 Subject To 4x, +x, + X3 = 20 2x1 – X2 26 X1 – X, + 5x; 2-5 -3x, + 2x, + X3 S4 Xị <0, x2 2 0, x3 unrestricted IB. Given the following information for a product-mix problem with three products and three resources. Primal Decision Variables: x1 = number of unit 1 produced; x, = # of unit 2 produced; x; = # of unit 3 produced Primal Formulation: Dual Formulation: Max Z (Rev.) = 25x1 8x1 4x1 2x1 + 30x, + 20x3 + 6x2 + 2x2 + X2 X1, X2, X3 + 20n, +25n3 + 4n, +2n2 25 + 2n, +ng 2 30 + 3n, +2732 20 Min W = Subject To < 50 + 3x3 <20 + 2x; $25 20 (Res. 1 constraint) (Res. 2 constraint) (Res. 3 constraint) (Nonnegativity) + X3 Subject To T1, T2, T3 20 Optimal Solution: Optimal Z = Revenue = $268.75 XỊ = 0 (Number of unit 1) X2 = 8.125 (Number of unit 2) X; = 1.25 (Number of unit 3) Resource Constraints: Dual Var. Optimal Value = 22.5 (Surplus variable in 1t dual constraint) Dual Var. Optimal Value = 0 (Surplus variable in 25d dual constraint) Dual Var. Optimal Value = 0 (Surplus variable in 3rd dual constraint) Resource 1 = 0 leftover units Resource 2 = 0 leftover units Dual Var. Optimal Value = 3.125 = T1 Dual Var. Optimal Value = 5.625 = T2 Dual Var. Optimal Value = 0 = T; Resource 3 = 14.375 leftover units 1Bi. What is the fair-market price for one unit of Resource 3? 1Bi. What is the meaning of the surplus variable value of 22.5 in the 1 dual constraint with respect to the primal problem? AI

Practical Management Science
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
Section: Chapter Questions
Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...
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Question 1. 1A. Write the general dual problem associated with the given LP. (Do not transform or rewrite the primal problem before writing the general dual) Maximize -4x1 + 2x2 Subject To 4x, +x2 + x; = 20 2x1 – X2 26 X1 – x, + 5x; 2-5 -3x, + 2x, + x; <4 Xị <0, x2 20, x3 unrestricted IB. Given the following information for a product-mix problem with three products and three resources. Primal Decision Variables: x1 = number of unit 1 produced; x, = # of unit 2 produced; x; = # of unit 3 produced Primal Formulation: Dual Formulation: Min W = Max Z (Rev.) = 25x1 50т + 20n, +25n3 + 47, +2n32 25 + 2n, +n3 2 30 + 3n, +2n;2 20 20 + 30x, +20x3 Subject To 8x: + 6х < 50 (Res. 1 constraint) (Res. 2 constraint) (Res. 3 constraint) (Nonnegativity) + X3 Subject To + 2x, + 3x3 S20 + 2x3 <25 20 4x1 2x1 + x2 X1, X2, X3 T1, 72, 73 Optimal Solution: Optimal Z = Revenue = $268.75 XỊ = 0 (Number of unit 1) X2 = 8.125 (Number of unit 2) X; = 1.25 (Number of unit 3) Resource Constraints: Dual Var. Optimal Value = 22.5 (Surplus variable in 1*t dual constraint) Dual Var. Optimal Value = 0 (Surplus variable in 2nd dual constraint) Dual Var. Optimal Value = 0 (Surplus variable in 3rd dual constraint) Resource 1 = 0 leftover units Dual Var. Optimal Value = 3.125 = T Dual Var. Optimal Value = 5.625 = 1, Dual Var. Optimal Value = 0 = n3 Resource 2 = 0 leftover units Resource 3 = 14.375 leftover units IBi. What is the fair-market price for one unit of Resource 3? 1Bü. What is the meaning of the surplus variable value of 22.5 in the 1* dual constraint with respect to the primal problem?
Transcribed Image Text:Question 1. 1A. Write the general dual problem associated with the given LP. (Do not transform or rewrite the primal problem before writing the general dual) Maximize -4x1 + 2x2 Subject To 4x, +x2 + x; = 20 2x1 – X2 26 X1 – x, + 5x; 2-5 -3x, + 2x, + x; <4 Xị <0, x2 20, x3 unrestricted IB. Given the following information for a product-mix problem with three products and three resources. Primal Decision Variables: x1 = number of unit 1 produced; x, = # of unit 2 produced; x; = # of unit 3 produced Primal Formulation: Dual Formulation: Min W = Max Z (Rev.) = 25x1 50т + 20n, +25n3 + 47, +2n32 25 + 2n, +n3 2 30 + 3n, +2n;2 20 20 + 30x, +20x3 Subject To 8x: + 6х < 50 (Res. 1 constraint) (Res. 2 constraint) (Res. 3 constraint) (Nonnegativity) + X3 Subject To + 2x, + 3x3 S20 + 2x3 <25 20 4x1 2x1 + x2 X1, X2, X3 T1, 72, 73 Optimal Solution: Optimal Z = Revenue = $268.75 XỊ = 0 (Number of unit 1) X2 = 8.125 (Number of unit 2) X; = 1.25 (Number of unit 3) Resource Constraints: Dual Var. Optimal Value = 22.5 (Surplus variable in 1*t dual constraint) Dual Var. Optimal Value = 0 (Surplus variable in 2nd dual constraint) Dual Var. Optimal Value = 0 (Surplus variable in 3rd dual constraint) Resource 1 = 0 leftover units Dual Var. Optimal Value = 3.125 = T Dual Var. Optimal Value = 5.625 = 1, Dual Var. Optimal Value = 0 = n3 Resource 2 = 0 leftover units Resource 3 = 14.375 leftover units IBi. What is the fair-market price for one unit of Resource 3? 1Bü. What is the meaning of the surplus variable value of 22.5 in the 1* dual constraint with respect to the primal problem?
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