MATH 340: Linear Programming
Suggested Time: 135 minutes
Optional Assignment
You must show your work. Quote names of main LP theorems used, as appropriate.
1. Solve the following linear programming problem, using our standard two phase method and
using Anstee’s rule. Find all optimal solutions.
Maximize
2x1
−2x1
x1
−x2 +x3
−x2 −x3 ≤ −4
−2x2 +x3 ≤ −3
+x3 ≤
2
x1 , x2 , x3 ≥ 0
2. Consider the following linear program:
Maximize −2x1 +3x2
+x2
−x1 +x2
+x1 +x2
+7x3
+2x3 ≤ 2
+4x3 ≤ 3
+4x3 ≤ 4
x1 , x2 , x3 ≥ 0
a) Give the Dual Linear Program of the above Primal Linear Program.
b) You are given that an optimal primal solution has x∗1 = 0, x∗2 = 1, x∗3 = 21 . Determine an
optimal dual solution (without pivoting), stating which theorems you have used.
c) Find all optimal primal and dual solutions. Show that they are indeed optimal solutions.
3. Given A, b, c, current basis (and B −1 for your computational ease), use our Revised Simplex
method and Anstee’s rule to determine the next entering variable (if there is one), the next
leaving variable (if there is one), and the new basic feasible solution after the pivot as well as
the new basis matrix B (if there is both an entering and leaving variable). The current basis
is {x7 , x2 , x3 }.
x1
x5 2
x6 
 2
x7 1

T
c =
x1
1
x2 x3 x4 x5
0 −1 −1 1
1 −1 1
0
−1 −2 −1 0
x2 x3
−1 1
x4
2
x5
0
x6
0
1
0
x6
0
x7
b



0 x5 −4


0
 x6  −3 
1 x7 −7
x7 0
4. The manufacturing plan is in the table below:
Hat Jersey Scarf Availability
Expert Labour
3
3
2
9
Fibre
4
5
3
13
Unskilled Labour
5
6
4
16
Net Profit
11
13
8
x5 x6
x7 −3 1
= x2 
 −1 1
x3 −1 0

B −1
x7

1
0

0
Let x1 denote the units of Hat, x2 denote the units of Jersey and x3 denote the units of Scarf
and let x3+i denote the ith slack for i = 1, 2, 3. We allow fractional values for our variables.
The final dictionary is:
x1
x2
x3
z
= 2 −2x4
+x6
= 1 +x4 −2x5 +x6
= 0 +x4 +3x5 −3x6
= 35 −x4 −2x5
x4 x5 x6

x1 2
0 −1


= x2  −1 2 −1 
x3 −1 −3 3

B −1
NOTE: All questions are independent of one another. They are not cumulative. You need not
complete all of the very final dictionary, merely the variables in the basis and the constants
and all the entries in the z row.
a) Give the marginal values for each of the resources: Expert Labour, Fibre and Unskilled
Labour.
b) Is the current solution degenerate?
c) Give the range on c1 so that the current solution remains optimal.
d) Give the reduced cost of Jersey.
e) Consider the possibility of changing the Expert Labour to 9 + p hours, the Fibre to 13 + p
and the Unskilled Labour to 16 + p. Determined the range on p so that the current basis
{x1 , x2 , x3 } remains optimal and report the profit as a function of p in that interval.
f) Change the availability of Expert Labour, Fibre and Unskilled Labour to 10, 14 and 17
respectively. Determine the new optimal solution using the Dual Simplex method. Report
the new solution as well as the new marginal values.
g) When considering the current optimal solution, it is decided to add a new counter space
constraint 2x1 + 3x2 ≤ 6 to our original problem. Solve using the Dual Simplex method.
Report the new solution as well as the new marginal values.
5. Show that either
i) There exist an x with Ax ≥ 0, c · x < 0,
or
ii) There exists y with AT y = c, y ≥ 0,
but not both.
6. Consider the game given by payoff matrix A below (the payoff to the row player).


2 1

A= 1 3 

2 0
State the LP for the row player and column player. Find the optimal strategies of the row
player and the column player.