Question

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 case for model B. Depending upon demand and the time available at the Newpark 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 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 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

The sensitivity report is shown in the figure 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		1080.00000		53.33333		480.00000

1. Interpret the ranges of optimality for the objective function coefficients. If there is limit, then enter the text "NA" as your answer. If required, round your answers to two decimal place.
   

   	Ranges of Optimality
   Decision Variable	lower limit	upper limit
   AM
   BM		9
   AP
   BP

   
   
   If one  of the objective function coefficients is/are changed within the ranges above the optimal solution will not  change.
   
2. Suppose that the manufacturing cost increases to $11.20 per case for model A. If your answer is zero then enter "0".
   

   Model Varaible	Optimal Solution
   AM	100
   BM	60
   AP	0
   BP	90

   
   Total Cost: $ 2290.00
   
3. Suppose that the manufacturing cost increases to $11.20 per case for model A and the manufacturing cost for model B decreases to $5 per unit. Would the optimal solution change? If your answer is zero then enter "0".
   
   Yes

   What is the new optimal solution?

   Model Varaible	Optimal Solution
   AM
   BM
   AP
   BP

   
   Total Cost: $  

Answer 1

Solution:

A.  

Decision Variable	Ranges of Optimality
AM	No lower limit to 11.75
BM	3.667 to 9
AP	12.25 to No Upper Limit
BP	6 to 11.333

             Provided a single change of an objective function coefficient is within its above range, the optimal solution

            

AM	100
BM	60
AP	0
BP	90

             AM = 100, BM = 60, AP = 0, and BP = 90 will not change.

       B.   This change is within the range of optimality.

             The optimal solution remains AM = 100, BM = 60, AP = 0, and BP = 90.

             The $11.20 - $10.00 = $1.20 per unit cost increase will increase the total cost to $2170 = $1.20(100) = $2290.

       C.  

Variable	Cost	Change	Allowable Increase/Decrease	Percentage Change
AM	10	Increase 1.20	11.75 - 10 = 1.75	(1.20/1.75)100 = 68.57
BM	6	Decrease 1	6.0 - 3.667 = 2.333	(1/2.333)100 = 42.86
				111.43

             111.43% exceeds 100%; therefore, we must resolve the problem.

             Resolving the problem provides the new optimal solution:            

AM	0
BM	135
AP	100
BP	15

       AM = 0, BM = 135, AP = 100, and BP = 15; the total cost is $22,100.