Digital Controls, Inc. (DCI) manufactures two models of a radar gun used by police to monitor the speed of automobiles. Model A has an accuracy of plus or minus 1 mile per hour, whereas the smaller model B has an accuracy of plus or minus 3 miles per hour. For the next week, the company has orders for 100 units of model A and 150 units of model B. Although DCI purchases all the electronic components used in both models, the plastic cases for both models are manufactured at a DCI plant in Newark, New Jersey. Each model A case requires 4 minutes of injection-molding time and 6 minutes of assembly time. Each model B case requires 3 minutes of injection-molding time and 8 minutes of assembly time. For next week the Newark plant has 600 minutes of injection-molding time available and 1080 minutes of assembly time available. The manufacturing cost is $10 per case for model A and $6 per case for model B. Depending upon demand and the time available at the Newark plant, DCI occasionally purchases cases for one or both models from an outside supplier in order to fill customer orders that could not be filled otherwise. The purchase cost is $14 for each model A case and $9 for each model B case. Management wants to develop a minimum cost plan that will determine how many cases of each model should be produced at the Newark plant and how many cases of each model should be purchased. The following decision variables were used to formulate a linear programming model for this problem:

AM = number of cases of model A manufactured

BM = number of cases of model B manufactured

AP = number of cases of model A purchased

BP = number of cases of model B purchased

The linear programming model that can be used to solve this problem is as follows:

Min10AM + 6AM + 14AP + 9BP

s.t.

1AM + + 1AP + = 100Demand for model A

1BM + 1BP = 150Demand for model B

4AM + 3BM ≤ 600Injection molding time

6BM + 8BM ≤ 1080Assembly time

AM, BM, AP, BP ≥ 0

The computer solution developed using The Management Scientist is shown in Figure.

THE MANAGEMENT SCIENTIST SOLUTION FOR THE DIGITAL CONTROLS, INC., PROBLEM

a. What is the optimal solution and what is the optimal value of the objective function?
b. Which constraints are binding?
c. What are the dual prices? Interpret each.
d. If you could change the right-hand side of one constraint by one unit, which one would you choose?Why?

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Objective Function Value - 2170.000 Variable Value Reduced Costs AM HM AP BP 100.000 60.000 0.000 90.000 0.000 0.000 1.750 0.000 Constraint Slack/Surplus Dual Prices 0.000 0.000 20.000 0.000 -12.250 -9.000 0.000 0.375 OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit AM BM AP BP No Lower Limit 3.667 12.250 6.000 10.000 6.000 14.000 11.750 9.000 No Upper Limit 11.333 9.000 RIGHT HAND SIDE RANGES Constraint Lower Linit Current Value Upper Limit 0.000 60.000 580 000 600.000 100.000 150.000 600.000 1080.000 111.429 No Upper Limit No Upper Limit 1133.333