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88 CHAPTER 3 LINEAR PROGRAMMING: SENSITIVITY ANALYSIS AND INTERPRETATION OF SOLUTION

analysis can help determine how much each additional hour of production time is
worth and how many hours can be added before diminishing returns set in.

3.2 Graphical Sensitivity Analysis

For linear programming problems with two decision variables, graphical solution
methods can be used to perform sensitivity analysis on the objective function
coefficients and the right-hand side values for the constraints.

Objective Function Coefficients

Let us consider how changes in the objective function coefficients might affect the
optimal solution to the GulfGolf problem. The current contribution to profit is $10
per unit for the standard bag and $9 per unit for the deluxe bag. It seems obvious
that an increase in the profit contribution for one of the bags might lead manage-
ment to increase production of that bag, and a decrease in the profit contribution for
one of the bags might lead management to decrease production of that bag. It is not
obvious, however, how much the profit contribution would have to change before
management would want to change the production quantities.
The current optimal solution to this problem calls for producing 540 standard golf
bags and 252 deluxe golf bags. The range of optimality for each objective function
coefficient provides the range of values over which the current solution will remain
optimal. In other words, it will show a lower value and an upper value and between
these values the current optimal solution will remain as the optimal solution.
Managerial attention needs to be focused on those objective function coefficients
that have a narrow range of optimality and coefficients near the end points of the
range. With these coefficients, a small change can mean modifying the optimal
solution. Let us now calculate the ranges of optimality for this problem.
Figure 3.1 shows the graphical solution. A careful inspection of this graph shows
that as long as the slope of the objective function line is between the slope of line A
(which coincides with the cutting and dyeing constraint line) and the slope of line B
(which coincides with the finishing constraint line), extreme point fi with S ¼ 540
and D ¼ 252 will be optimal. Changing an objective function coefficient for S or D
will cause the slope of the objective function line to change. In Figure 3.1 we see that
such changes cause the objective function line to rotate around extreme point fi.
However, as long as the objective function line stays within the shaded region,
extreme point fi will remain optimal.
Rotating the objective function line anticlockwise causes the slope to become
less negative, and the slope increases. When the objective function line rotates
anticlockwise (slope increased) enough to coincide with line A, we obtain alter-
native optimal solutions between extreme points fi and fl. Any further anticlock-
wise rotation of the objective function line will cause extreme point fi to be
nonoptimal. Hence, the slope of line A provides an upper limit for the slope of
the objective function line.
The slope of the Rotating the objective function line clockwise causes the slope to become
objective function line is more negative, and the slope decreases. When the objective function line rotates
negative; hence, rotating clockwise(slopedecreases)enoughtocoincidewithlineB,weobtainalternative
the objective function line
clockwise makes the line optimal solutions between extreme points fi and ›. Any further clockwise
steeper even though the rotation of the objective function line will cause extreme point fi to be nonoptimal.
slope is getting smaller Hence, the slope of line B provides a lower limit for the slope of the objective
(more negative).
function line.

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