37242 Optimisation

37242 Optimisation in Quantitative
Management
Assignment
Students can do this Assignment either individually or in group. The
number of students in a group cannot exceed four.
QUESTION 1
This question is based on the material in the pdf file Pineapple Canners
that has been emailed to you.
You must
• formulate a linear programming problem that determines the optimal
production plan for the Pineapple Canners;
• solve this linear program using LINGO;
• present the results of your work as a written report.
The written report must
• clearly describe each variable and each constraint;
• present the entire linear programming formulation;
• present the LINGO code which was used to solve the linear program;
• present the LINGO printouts with the results;
• present the optimal production plan.
The LINGO code (the linear program in LINGO) must
• use sections SETS and DATA;
• use the commands @FOR and @SUM.
1
QUESTION 2
This question is based on the material in the section 5.5.2 The Big-M
Method in S.G. Nash and A. Sofer, Linear and Nonlinear Programming.
McGraw-Hill, 1996. A copy of this section has been emailed to you as the
pdf file Nash and Sofer Linear and Nonlinear Programming.
• Study the section 5.5.2 The Big-M Method in S.G. Nash and A. Sofer,
Linear and Nonlinear Programming. McGraw-Hill, 1996.
• Using the Big-M Method, solve the linear programming problem
min −4x1 − 5x2 + 3x3
subject to
x1 + 2x2 + x3 = 10
x1 − x2 ≥ 6
x1 + 3x2 + x3 ≤ 14
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.
Show your working.
QUESTION 3
Let A be an m×n matrix of rank m, m < n, and b ∈ Em.
and z and w be the optimal values of the objective functions of the linear
programs
min xn
subject to
Ax = b
x ≥ 0
and
max xn
subject to
Ax = b
x ≥ 0
respectively. Prove that for any a ∈ [z, w] there exists
x ∈ {y : Ay = b, y ≥ 0, y ∈ En}
such that xn = a.
2
QUESTION 4
Consider the linear programming problem
After introducing slack variables x3 and x4, the simplex method produced
the following final tableau
(a) Find d, e, f, g, h, k, and w. Show your working.
(b) Find c1, c2, b1, b2, a1,1, a1,2, a2,1, a2,2. Show your working.
QUESTION 5
Consider the linear program
min c
T x
subject to
Ax ≤ b
x ≥ 0
where c ∈ En is a nonzero vector, b ∈ Em, m < n, and A is m × n matrix of
rank m. Prove that if
Ax
0 < b and x
0 > 0,
then x
0
cannot be an optimal solution.
3
QUESTION 6
Consider the linear programming problem
min c
T x
subject to Ax = b
x ≥ 0
where A is an m × n matrix of rank m, m < n, b ∈ Em, c ∈ En. Suppose
that in the optimal basic feasible solution, obtained according to the Phase
I of the two-phase method, all basic variables are non-artificial variables.
(a) What is the value of the objective function for this solution? Justify
your answer.
(b) What is the reduced cost of each basic variable in this solution? Justify
your answer.
(c) What is the reduced cost of each artificial variable for this solution?
Justify your answer.
(d) What is the reduced cost of each non-artificial nonbasic variable in this
solution? Justify your answer.
QUESTION 7
Consider the linear programming problem
(a) Prove that the feasible region of this linear program has no extreme
points.
(b) Convert this linear program into an equivalent linear programming
problem in standard form.
(c) Show that the feasible region of the linear program obtained in (b) has
extreme points.