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).
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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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