Module MA3484: Hilary Term 2015
Problems
1. Let ci,j be the coefficient in the ith row and jth column of the cost
matrix C, where


8 4 6
 3 7 2 

C=
 13 8 6  .
5 7 2
and let
s1 = 13, s2 = 11, s3 = 11, s4 = 15,
d1 = 21,
d2 = 12,
d3 = 17.
Determine non-negative real numbers xi,j for i = 1, 2, 3, 4 and j = 1, 2, 3
4 P
3
P
that minimize
ci,j xi,j subject to the following constraints:
i=1 j=1
3
X
xi,j = si
for i = 1, 2, 3, 4,
j=1
4
X
xi,j = dj
for j = 1, 2, 3,
i=1
and xi,j ≥ 0 for all i and j.
Also verify that the solution to this problem is indeed optimal.
2. Consider the following linear programming problem:—
find real numbers x1 , x2 , x3 , x4 , x5 , x6 so as to minimize the
objective function
8x1 + 4x2 + 2x3 + 4x4 + 3x5 + 3x6
subject to the following constraints:
3x1 + 4x2 + 7x3 + 3x4 + 5x5 + 4x6 = 10;
2x1 + 5x2 + 3x3 + 4x4 + 2x5 + 6x6 = 9;
xj ≥ 0 for j = 1, 2, 3, 4, 5, 6.
A feasible solution to this problem is the following:
x1 = 2,
x2 = 1,
x3 = 0,
x4 = 0,
x5 = 0,
x6 = 0.
(This feasible solution satisfies the constraints but does not necessarily
minimize the objective function.) Find an optimal solution to the linear
programming problem and verify that it is indeed optimal.
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