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60 CHAPTER 2 AN INTRODUCTION TO LINEAR PROGRAMMING

convert the constraint to equality form. Just as with slack variables, surplus variables
are given a coefficient of zero in the objective function because they have no effect
on its value. After including two surplus variables, S 1 and S 2 , for the constraints
and one slack variable, S 3 , for the constraint, the linear programming model of
the M&D Chemicals problem becomes:
Min 2A þ 3B þ 0S 1 þ 0S 2 þ 0S 3
s:t:
1A 1S 1 ¼ 125
1A þ 1B 1S 2 ¼ 350
2A þ 1B þ 1S 3 ¼ 600
A; B; S 1 ; S 2 ; S 3 0
Try Problem 20 to test All the constraints are now equalities. Hence, the preceding formulation is the stand-
your ability to use slack ard-form representation of the M&D Chemicals problem. At the optimal solution of
and surplus variables to
write a linear programme A ¼ 250 and B ¼ 100, the values of the surplus and slack variables are as follows:
in standard form.

Constraint Value of Surplus or Slack Variables
Demand for product A S 1 = 125
Total production S 2 =0
Processing time S 3 =0

Refer to Figures 2.15 and 2.16. Note that the zero surplus and slack variables are
associated with the constraints that are binding at the optimal solution – that is, the
total production and processing time constraints. The surplus of 125 units is asso-
ciated with the nonbinding constraint on the demand for product A.
In the GulfGolf problem all the constraints were of the type, and in the M&D
Chemicals problem the constraints were a mixture of and types. The number
and types of constraints encountered in a particular linear programming problem
depend on the specific conditions existing in the problem. Linear programming
problems may have some constraints, some constraints, and some ¼ constraints.
Try Problem 21 to
practice solving a linear For an equality constraint, feasible solutions must lie directly on the constraint line.
programme with all three An example of a linear programme with two decision variables, G and H, and all
constraint forms. three constraint forms is given here:
Min 2G þ 2H
s:t:
1G þ 3H 12
3G þ 1H 13
1G 1H ¼ 3
G; H 0
The standard-form representation of this problem is:

Min 2G þ 2H þ 0S 1 þ 0S 2
s:t:
1G þ 3H þ 1S 1 ¼ 12
3G þ 1H 1S 2 ¼ 13
1G 1H ¼ 3
G; H; S 1 ; S 2 0

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