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O Consider the following linear program: minimise x1 + 2x2 - 6x3 subject to 4₁ - 8x2 + x3 = 10, 3x19x2 ≤ 20, 2x1 + 10x2 + 2x3 ≥ 30, I1 ≤ 40, X1, X₂ ≥ 0, T3 unrestricted (i) Convert program (1) to standard inequality form, and give the value of the constraint matrix A, the vector b (which gives the right hand side of the constraints), and the vector c (which gives the coefficients for the objective function). (ii) Give the dual of program (1).

O Consider the following linear program: minimise x1 + 2x2 - 6x3 subject to 4₁ - 8x2 + x3 = 10, 3x19x2 ≤ 20, 2x1 + 10x2 + 2x3 ≥ 30, I1 ≤ 40, X1, X₂ ≥ 0, T3 unrestricted (i) Convert program (1) to standard inequality form, and give the value of the constraint matrix A, the vector b (which gives the right hand side of the constraints), and the vector c (which gives the coefficients for the objective function). (ii) Give the dual of program (1).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section9.3: Systems Of Inequalities
Problem 15E
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(a) Consider the following linear program: minimise x1 + 2x2 673 HILING subject to 4x1 8x2 + x3 = 10, 73 3x1 9.12 ≤20, 2.r1 + 10.x2 + 2.73 30, I1 ≤ 40, I1, I₂ > 0. T3 unrestricted (i) Convert program (1) to standard inequality form, and give the value of the constraint matrix A, the vector b (which gives the right hand side of the constraints), and the vector c (which gives the coefficients for the objective function). (ii) Give the dual of program (1). (b) Consider a linear program in standard inequality form: maximise cx subject to Ax ≤ b, X>0 Let t > 0) be some constant and suppose we modify the program by multiplying the right-hand side of each constraint by t maximise cTx subject to Ax ≤ tb, x>0 (i) Show that x is a feasible solution of program (2) if and only if tx is a feasible solution of program (3). (ii) Show that if program (2) is unbounded then the program (3) is unbounded. (iii) Show that if tx is an optimal solution of program (3) then x is an optimal solution of program (2). Hint: it may be easier to prove the contrapositive: that if x is not an optimal solution for (2) then tx is not an optimal solution to (3).
Transcribed Image Text:(a) Consider the following linear program: minimise x1 + 2x2 673 HILING subject to 4x1 8x2 + x3 = 10, 73 3x1 9.12 ≤20, 2.r1 + 10.x2 + 2.73 30, I1 ≤ 40, I1, I₂ > 0. T3 unrestricted (i) Convert program (1) to standard inequality form, and give the value of the constraint matrix A, the vector b (which gives the right hand side of the constraints), and the vector c (which gives the coefficients for the objective function). (ii) Give the dual of program (1). (b) Consider a linear program in standard inequality form: maximise cx subject to Ax ≤ b, X>0 Let t > 0) be some constant and suppose we modify the program by multiplying the right-hand side of each constraint by t maximise cTx subject to Ax ≤ tb, x>0 (i) Show that x is a feasible solution of program (2) if and only if tx is a feasible solution of program (3). (ii) Show that if program (2) is unbounded then the program (3) is unbounded. (iii) Show that if tx is an optimal solution of program (3) then x is an optimal solution of program (2). Hint: it may be easier to prove the contrapositive: that if x is not an optimal solution for (2) then tx is not an optimal solution to (3).
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