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

available in each of four production departments. The information shown in the
Slack/Surplus column provides the value of the slack variable for each of the depart-
ments. This information (after rounding) is summarized here:

Constraint Number Constraint Name Slack
1 Cutting and dyeing 0
2 Sewing 120
3 Finishing 0
4 Inspection and packaging 18

From this information, we see that the binding constraints (the cutting and dyeing
and the finishing constraints) have zero slack at the optimal solution. The sewing
department has 120 hours of slack, or unused capacity, and the inspection and
packaging department has 18 hours of slack or unused capacity.
The Dual Prices column contains information about the marginal value of each of
the four resources at the optimal solution. In Section 3.2 we defined the dual price as
follows:

The dual price associated with a constraint is the change in the value of the
solution per unit change in the right-hand side of the constraint.
Try Problem 10 to test So, the nonzero dual prices of 4.37496 for constraint 1 (cutting and dyeing con-
your ability to use straint) and 6.93753 for constraint 3 (finishing constraint) tell us that an additional
computer output to
determine the optimal hour of cutting and dyeing time increases the value of the optimal solution by $4.37
solution and to interpret and an additional hour of finishing time increases the value of the optimal solution
the values of the dual by $6.94. So, if the cutting and dyeing time were increased from 630 to 631 hours,
prices.
with all other coefficients in the problem remaining the same, the company’s profit
would be increased by $4.37 from $7668 to $7668 + $4.37 ¼ $7672.37. A similar
interpretation for the finishing constraint implies that an increase from 708 to 709
hours of available finishing time, with all other coefficients in the problem remaining
the same, would increase the company’s profit to $7668 + $6.94 ¼ $7674.94.
Because the sewing and the inspection and packaging constraints both have slack
or unused capacity available, the dual prices of zero show that additional hours of
these two resources will not improve the value of the objective function. The output
confirms the calculations we did earlier.
Referring again to the computer output in Figure 3.4, we see that after providing
the constraint information on slack/surplus variables and dual prices, the solution
provides ranges for the objective function coefficients and the right-hand sides of the
constraints.
Considering the information provided under the computer output heading
labelled OBJECTIVE COEFFICIENT RANGES, we see that variable S,which
has a current profit coefficient of 10, has the following range of optimality
for C S :

6:30 C S 13:49993

So, as long as the profit contribution associated with the standard bag is
between $6.30 and $13.50, the production of S ¼ 540 standard bags and D ¼ 252
deluxe bags will remain the optimal solution. Note that the range of optimality

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