The National Steel Corporation (NSC) produces a special-purpose steel that is used in the aircraft and aerospace industries.
• The marketing department of NSC has received orders for 2400, 2200, 2700, and 2500 tons of steel during each of the next four months.
• NSC can meet these demands by producing the steel, by drawing from its inventory, or by a combination of both.
• The production costs per ton of steel during each of the next four months are projected to be $7400, $7500, $7600, and $7800.
• Production capacity can never exceed 4000 tons in any month.
• All production takes place at the beginning of the month and immediately thereafter the demand is met. The remaining steel is then stored in inventory at a holding cost of $120/ton for each month that it remains there.
• NSC currently has an inventory of 1000 tons of steel.
• The inventory level at the end of the fourth month must be at least 1500 tons.
The NSC wants to find the production plan that will minimize the total cost (production + holding) over the next four months while satisfying the demand. Below is the linear programming formulation for this problem:
Min z = 7400P1 + 7500P2 + 7600P3 + 7800P4 + 120I1 + 120I2 + 120I3 + 120I4
s.t. P1 + I0 ≥ 2400. (D1)
P2 + I1 ≥ 2200 (D2)
P3 + I2 ≥ 2700 (D3)
P4 + I3 ≥ 2500 (D4)
I0 = 1000 (E0)
I0 + P1 – I1 = 2400 (E1)
I1 + P2 – I2 = 2200 (E2)
I2 + P3 – I3 = 2700 (E3)
I3 + P4 – I4 = 2500 (E4)
I4 ≥ 1500 (E5)
P1 ≤ 4000 (C1)
P2 ≤ 4000 (C2)
P3 ≤ 4000 (C3)
P4 ≤ 4000 (C4)
P1 ≥ 0 (N1)
P2 ≥ 0 (N2)
P3 ≥ 0 (N3)
P4 ≥ 0 (N4)
I1 ≥ 0 (N5)
I2 ≥ 0 (N6)
I3 ≥ 0 (N7)
I4 ≥ 0 (N8)
Here the decision variables P1, P2, P3, and P4 stand for the tonnage of steel produced at the beginning of the months 1, 2, 3, and 4 respectively, I0 is the initial inventory, and I1, I2, I3, and I4 stand for the tonnage of inventory at the end of the months 1, 2, 3, and 4 respectively.
a. Use the Excel solver to find the optimal solution to the above LP and interpret it. That is, answer the questions: what production plan will minimize the NSC total cost (production + holding) over the next four months while satisfying the demand? What is the total cost associated with that production plan?
b. Which constraints are binding? What is the slack of each constraint? For this question, you can ignore the non-negativity constraints.
