iven the following information for a product-mix problem with three products and three resources. Primal Decision Variables: x1 = number of unit 1 produced; x2 = # of unit 2 produced; x3 = # of unit 3 produced Primal Formulation:Dual Formulation: Max Z (Rev.) = 25x1 + 30x2 + 20x3Min W = 50π1+ 20π2+25π3 Subject To8x1+ 6x2+ x3≤ 50(Res. 1 constraint)Subject To8π1+ 4π2+2π3≥ 254x1+ 2x2+ 3x3≤ 20(Res. 2 constraint)6π1+ 2π2+π3 ≥ 302x1+ x2+ 2x3≤ 25(Res. 3 constraint)π1+ 3π2+2π3≥ 20x1, x2, x3≥ 0 (Nonnegativity)π1, π2, π3 ≥ 0 Optimal Solution: Optimal Z = Revenue = $268.75 x1 = 0 (Number of unit 1)Dual Var. Optimal Value = 22.5 (Surplus variable in 1st dual constraint) x2 = 8.125 (Number of unit 2)Dual Var. Optimal Value = 0 (Surplus variable in 2nd dual constraint) x3 = 1.25 (Number of unit 3)Dual Var. Optimal Value = 0 (Surplus variable in 3rd dual constraint) Resource Constraints: Resource 1 = 0 leftover unitsDual Var. Optimal Value = 3.125 = π1 Resource 2 = 0 leftover unitsDual Var. Optimal Value = 5.625 = π2 Resource 3 = 14.375 leftover unitsDual Var. Optimal Value = 0 = π3 1Bi. What is the fair-market price for one unit of Resource 3? 1Bii. What is the meaning of the surplus variable value of 22.5 in the 1st dual constraint with respect to the primal problem?
iven the following information for a product-mix problem with three products and three resources. Primal Decision Variables: x1 = number of unit 1 produced; x2 = # of unit 2 produced; x3 = # of unit 3 produced Primal Formulation:Dual Formulation: Max Z (Rev.) = 25x1 + 30x2 + 20x3Min W = 50π1+ 20π2+25π3 Subject To8x1+ 6x2+ x3≤ 50(Res. 1 constraint)Subject To8π1+ 4π2+2π3≥ 254x1+ 2x2+ 3x3≤ 20(Res. 2 constraint)6π1+ 2π2+π3 ≥ 302x1+ x2+ 2x3≤ 25(Res. 3 constraint)π1+ 3π2+2π3≥ 20x1, x2, x3≥ 0 (Nonnegativity)π1, π2, π3 ≥ 0 Optimal Solution: Optimal Z = Revenue = $268.75 x1 = 0 (Number of unit 1)Dual Var. Optimal Value = 22.5 (Surplus variable in 1st dual constraint) x2 = 8.125 (Number of unit 2)Dual Var. Optimal Value = 0 (Surplus variable in 2nd dual constraint) x3 = 1.25 (Number of unit 3)Dual Var. Optimal Value = 0 (Surplus variable in 3rd dual constraint) Resource Constraints: Resource 1 = 0 leftover unitsDual Var. Optimal Value = 3.125 = π1 Resource 2 = 0 leftover unitsDual Var. Optimal Value = 5.625 = π2 Resource 3 = 14.375 leftover unitsDual Var. Optimal Value = 0 = π3 1Bi. What is the fair-market price for one unit of Resource 3? 1Bii. What is the meaning of the surplus variable value of 22.5 in the 1st dual constraint with respect to the primal problem?
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter2: Introduction To Spreadsheet Modeling
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Question
Given the following information for a product-mix problem with three products and three resources.
Primal Decision Variables: x1 = number of unit 1 produced; x2 = # of unit 2 produced; x3 = # of unit 3 produced
Primal Formulation:Dual Formulation:
Max Z (Rev.) = 25x1 + 30x2 + 20x3Min W = 50π1+ 20π2+25π3
Subject To8x1+ 6x2+ x3≤ 50(Res. 1 constraint)Subject To8π1+ 4π2+2π3≥ 25
4x1+ 2x2+ 3x3≤ 20(Res. 2 constraint)6π1+ 2π2+π3 ≥ 30
2x1+ x2+ 2x3≤ 25(Res. 3 constraint)π1+ 3π2+2π3≥ 20
x1, x2, x3≥ 0 (Nonnegativity)π1, π2, π3 ≥ 0
Optimal Solution:
Optimal Z = Revenue = $268.75
x1 = 0 (Number of unit 1)Dual Var. Optimal Value = 22.5 (Surplus variable in 1st dual constraint)
x2 = 8.125 (Number of unit 2)Dual Var. Optimal Value = 0 (Surplus variable in 2nd dual constraint)
x3 = 1.25 (Number of unit 3)Dual Var. Optimal Value = 0 (Surplus variable in 3rd dual constraint)
Resource Constraints:
Resource 1 = 0 leftover unitsDual Var. Optimal Value = 3.125 = π1
Resource 2 = 0 leftover unitsDual Var. Optimal Value = 5.625 = π2
Resource 3 = 14.375 leftover unitsDual Var. Optimal Value = 0 = π3
1Bi. What is the fair-market price for one unit of Resource 3?
1Bii. What is the meaning of the surplus variable value of 22.5 in the 1st dual constraint with respect to the primal problem?
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