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 1,080 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:

Min	10AM	+	6BM	+	14AP	+	9BP
s.t.
	1AM	+		+	1AP	+		=	100	Demand for model A
			1BM	+			1BP	=	150	Demand for model B
	4AM	+	3BM						600	Injection molding time
	6AM	+	8BM						1,080	Assembly time

AM, BM, AP, BP 0

Refer to the computer solution below.

Optimal Objective Value = 2170.00000

Variable	Value	Reduced Cost
AM	100.00000	0.00000
BM	60.00000	0.00000
AP	0.00000	1.75000
BP	90.00000	0.00000

Constraint	Slack/Surplus	Dual Value
1	0.00000	12.25000
2	0.00000	9.00000
3	20.00000	0.00000
4	0.00000	0.37500

Variable	Objective Coefficient	Allowable Increase	Allowable Decrease
AM	10.00000	1.75000	Infinite
BM	6.00000	3.00000	2.33333
AP	14.00000	Infinite	1.75000
BP	9.00000	2.33333	3.00000

Constraint	RHS Value	Allowable Increase	Allowable Decrease
1	100.00000	11.42857	100.00000
2	150.00000	Infinite	90.00000
3	600.00000	Infinite	20.00000
4	1,080.00000	53.33333	480.00000

(a)

Interpret the ranges of optimality for the objective function coefficients.

If a single change to the injection molding time or assembly time for either model case is within the allowable range, the optimal solution will not change. If a single change to the manufacturing cost or purchase cost for either model case is within the allowable range, the optimal solution will not change. If multiple changes are made to the manufacturing or purchase costs for either model case within their respective allowable ranges, the optimal solution will not change. If multiple changes are made to the injection molding time or assembly time for either model case within their respective allowable ranges, the optimal solution will not change.

(b)

Suppose that the manufacturing cost increases to $11.50 per case for model A. Would the optimal solution change?

Yes, it is necessary to solve the model again. No, the optimal solution remains the same.

What is the new optimal solution?

at (AM, BM, AP, BP) =

(c)

Suppose that the manufacturing cost increases to $11.50 per case for model A and the manufacturing cost for model B decreases to $4 per unit. Would the optimal solution change?

Yes, it is necessary to solve the model again. No, the optimal solution remains the same.

What is the new optimal solution?

at (AM, BM, AP, BP) =