solution

A company produces two products made from aluminum and copper. The table below gives the unit requirements, the unit production man-hours required, the unit profit and availability of the resources (in tons).

	Aluminum	Copper	Man-hours	Unit profit
Product 1 (X1)	1	0	2	50
Product 2 (X2)	1	1	3	60
Available	10	6	24

The problem was solved by the Solver giving the following output:

Optimal Objective Value = 540.00000

Variable	Value	Reduced Cost
X1	6.00000	0.00000
X2	4.00000	0.00000

Constraint	Slack/Surplus	Dual Value
1	0.00000	30.0000
2	2.00000	0.0000
3	0.00000	10.0000
Variable	Objective Coefficient	Allowable Increase	Allowable Decrease
X1	50.00000	10.0000	10.0000
X2	60.00000	15.0000	10.0000
Constraint	RHS Value	Allowable Increase	Allowable Decrease
1	10.00000	2.00000	1.00000
2	6.00000	Infinite	2.00000
3	24.00000	2.00000	4.00000

1. What is the optimal production schedule?
2. Within what range for the profit on product 2 will the solution in (a) remain optimal? What is the optimal profit when C2 = 70?
3. Suppose that the unit profits on X1 and X2 changed simultaneously from 50 to 55 and 60 to 65, respectively. Would the optimal solution change? Explain.
4. Explain the meaning of the “Dual Values” column. Given the optimal solution, why should the dual value for copper be 0?
5. What is the increase in the value of the objective function for an extra unit of aluminum?
6. Man-hours were not figured into the unit profit as it must pay three workers for eight hours of work regardless of the number of man-hour used. What is the dual price for man-hours? Interpret.
7. On the other hand, aluminum and copper are resources that are ordered as needed. The unit profit coefficients were determined by: Selling price per unit – Cost of the resources per unit. The 10 units of aluminum cost the company $100. What is the most the company should be willing to pay for extra aluminum?