(a) What are the ranges of optimality for the profit of Product 1, Product 2, Product 3, and Product 4?

(b) Find the dual prices of the three constraints and interpret their meanings. What are the ranges in which each of these dual prices is valid?

(c) If the profit contribution of Product 3 changes from $24 per unit to $50 per unit, what will be the optimal solution? What will be the new total profit? (Note: Answer this question by using the ranging results. Do not solve the problem again).

(d) Which resource should be obtained in larger quantity to increase the profit most? (Note: Answer this question using the ranging results given above. Do not solve the problem again).

Let P1 = number of Product 1 to be produced

P2 = number of Product 2 to be produced

P3 = number of Product 3 to be produced

P4 = number of Product 4 to be produced

Maximize 15P1 + 20P2 + 24P3 + 15P4 Total profit

Subject to

8P1 + 12P2 + 10P3 + 8P4 3000 Material requirement constraint

4P1 + 3P2 + 2P3 + 3P4 1000 Labor hours constraint

P2 > 120 Minimum quantity needed for Product 2 constraint

And P1, P2, P3, P4 0 Non-negativity constraints

The QM for Windows output for this problem is given below.

Linear Programming Results:

Variable	Status	Value
P1	Non Basic	0
P2	Basic	120
P3	Basic	156
P4	Non Basic	0
slack 1	Non Basic	0
slack 2	Basic	328
surplus 3	Non Basic	0
optimal Value (Z)		6144

Ranging Results:

Variable	Value	Reduced Cost	Original Val	Lower Bound	Upper Bound
P1	0	4.2	15	- infinity	19.2
P2	120	0	20	- infinity	28.8
P3	156	0	24	18.75	Infinity
P4	0	4.2	15	- infinity	19.2

Constraint	Dual Value	Slack/Surplus	Original Val	Lower Bound	Upper Bound
Material Constraint	2.4	0	3000	1440	4640
Labor Constraint	0	328	1000	672	infinity
Product 2 Constraint	-8.8	0	120	0	250