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48 CHAPTER 2 AN INTRODUCTION TO LINEAR PROGRAMMING

Substituting this expression for S into Equation (2.8) and solving for D provides the
following:
1ð900 1:4286DÞþ 0:6667D ¼ 708
900 1:4286D þ 0:6667D ¼ 708
:7619D ¼ 192
192
D ¼ ¼ 252
:7619
Using D ¼ 252 in Equation (2.10) and solving for S, we obtain:

S ¼ 900 1:4286ð252Þ
¼ 900 360 ¼ 540

Although the optimal The exact location of the optimal solution point is S ¼ 540 and D ¼ 252. Hence, the
solution to the GulfGolf optimal production quantities are 540 standard bags and 252 deluxe bags, with a
problem consists of
integer values for the resulting profit contribution of 10(540) + 9(252) ¼ $7668. Consider what we have
decision variables, this done. We can now advise management that the maximum possible profit contribution
result will not always be that can be achieved is $7668 and that this is achieved by producing 540 standard golf
the case.
bags and 252 deluxe bags. This is a major finding of considerable importance to the
company and one that would have been very difficult to obtain any other way.
For a linear programming problem with two decision variables, the exact values of the
decision variables can be determined by first using the graphical solution procedure to
identify the optimal solution point and then solving the simultaneous constraint equa-
tions associated with it. In fact, the two constraints we have used to confirm the optimal
solution are referred to as binding constraints. Effectively at the optimal solution it is
these two constraints that bind the solution – they prevent a higher value for the
objective function. It is easy to see why. For the cutting and dyeing constraint we had:
0:7S þ 1D ¼ 630

That is, at the optimal solution point of S ¼ 540 and D ¼ 252 and in order to
produce this quantity of the two products, all of the 630 hours of cutting and dyeing
time available is required. Further production of S and D is not possible as we have
used all the available cutting and dyeing time. Similarly for the Finishing constraint
this is also binding, hence we need all the available finishing time to be able to
produce the optimal quantities of S and D. However, our other two constraints, for
Sewing and for Inspection and Packaging are non-binding. Non-binding constraints
do not directly affect the optimal solution. We shall see later that important manage-
ment information can be obtained through understanding binding and non-binding
constraints for an LP solution.

A Note on Graphing Lines

Try Problem 7 to test An important aspect of the graphical method is the ability to graph lines showing the
your ability to use the constraints and the objective function of the linear programme. The procedure we used
graphical solution
procedure to identify the for graphing the equation of a line is to find any two points satisfying the equation and
optimal solution and find then draw the line through the two points. For the GulfGolf constraints, the two points
the exact values of the were easily found by first setting S ¼ 0 and solving the constraint equation for D.Then
decision variables at the we set D ¼ 0 and solved for S.For thecuttingand dyeing constraint line:
optimal solution.
0:7S þ 1D ¼ 630
this procedure identified the two points (S ¼ 0, D ¼ 630) and (S ¼ 900, D ¼ 0).
The cutting and dyeing constraint line was then graphed by drawing a line through
these two points.
All constraints and objective function lines in two-variable linear programmes can
be graphed if two points on the line can be identified. However, finding the two points

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