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A MAXIMIZATION PROBLEM 37

Describe the Objective The objective here is to maximize the total contribution to
profit.
Describe Each Constraint Four constraints relate to the number of hours of
manufacturing time available; they restrict the number of standard bags and the
number of deluxe bags that can be produced.

Constraint 1 Number of hours of cutting and dyeing time used must be less
than or equal to the number of hours of cutting and dyeing time available.
Constraint 2 Number of hours of sewing time used must be less than or equal to
the number of hours of sewing time available.
Constraint 3 Number of hours of finishing time used must be less than or equal
to the number of hours of finishing time available.
Constraint 4 Number of hours of inspection and packaging time used must be less
than or equal to the number of hours of inspection and packaging time available.

Define the Decision Variables A decision variable is one where management can
control its value and therefore decide what value the variable should take. The
controllable values for GulfGolf are (1) the number of standard bags produced, and
(2) the number of deluxe bags produced. Let:

S ¼ number of standard bags
D ¼ number of deluxe bags

In linear programming terminology, S and D are referred to as the decision variables.

Write the Objective in Terms of the Decision Variables The company’s profit
contribution comes from two sources: (1) the profit contribution made by producing
S standard bags, and (2) the profit contribution made by producing D deluxe bags.
If the company makes $10 for every standard bag, the company will make $10S if
S standard bags are produced. Also, if the company makes $9 for every deluxe bag,
the company will make $9D if D deluxe bags are produced. So, we have:
Total Profit Contribution ¼ 10S þ 9D

Because the objective – maximize total profit contribution – is a function of the
decision variables S and D, we refer to 10S +9D as the objective function. Using
‘Max’ as an abbreviation for maximize, we write GulfGolf’s objective as follows:
Max 10S þ 9D

Write the Constraints in Terms of the Decision Variables
Constraint 1:

Hours of cutting and Hours of cutting and

dyeing time used dyeing time available
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Every standard bag GulfGolf produces will use / 10 hour cutting and dyeing time;
therefore, the total number of hours of cutting and dyeing time used in the manu-
7
facture of S standard bags is / 10 S. In addition, because every deluxe bag produced
uses one hour of cutting and dyeing time, the production of D deluxe bags will use
1D hours of cutting and dyeing time. So, the total cutting and dyeing time required
for the production of S standard bags and D deluxe bags is given by:
The units of Total hours of cutting and dyeing time used ¼ / S þ 1D
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measurement on the 10
left-hand side of the The director of manufacturing stated that GulfGolf has at most 630 hours of cutting
constraint must match
the units of measurement and dyeing time available. Therefore, the production combination we select must
on the right-hand side. satisfy the requirement:

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