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A MINIMIZATION PROBLEM 61

The standard form requires a slack variable for the constraint and a surplus
variable for the constraint. However, neither a slack nor a surplus variable is
required for the third constraint since it is already in equality form.
When solving linear programmes graphically, it is not necessary to write the
problem in its standard form. Nevertheless, you should be able to calculate the
values of the slack and surplus variables and understand what they mean,
because the values of slack and surplus variables are included in the computer
solution of linear programmes and have an important management use. In
Chapter 5 we will introduce an algebraic solution procedure, the simplex
method, which can be used to find optimal extreme-point solutions for linear
programming problems with as many as several thousand decision variables. The
mathematical steps of the simplex method involve solving simultaneous equa-
tions that represent the constraints of the linear programme. Thus, in setting up
alinear programmefor solution by thesimplexmethod,wemusthaveonelinear
equation for each constraint in the problem; therefore, the problem must be in
its standard form.
A final point: the standard form of the linear programming problem is equivalent
to the original formulation of the problem. That is, the optimal solution to any linear
programming problem is the same as the optimal solution to the standard form of
the problem. The standard form has not changed the basic problem; it has only
changed how we write the constraints for the problem.

Computer Solution of the M&D Chemicals Problem
The solution obtained using Excel Solver is presented in Figure 2.17. The computer
output shows that the minimum-cost solution yields an objective function value of
E800. The values of the decision variables show that 250 litres of product A and 100
litres of product B provide the minimum-cost solution.
From the Constraints section of the solution, we can see that constraint 2 and 3
are binding whilst constraint 1 has a surplus of 125 units.

Figure 2.17 Excel solution for M&D Chemicals

Target Cell (Min)
Original
Name Value Final Value
Minimize Cost Product A 0 800
EXCEL file
M&D
Adjustable Cells
Original
Name Value Final Value
Litres produced Product A 0 250
Litres produced Product B 0 100

Constraints
Name Cell Value Status Slack
Not
Demand for product A (LHS)
250 Binding 125
Total production (LHS) 350 Binding 0
Processing time (LHS) 600 Binding 0

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