Tube Steel Incorporated (TSI) is optimizing production at its 4 hot mills. TSI makes 8 types of tubular products which are either solid or hollow and come in 4 diameters. The following two tables show production costs (in dollars) per tube of each product at each mill and the extrusion times (in minutes) for each allowed combination.

Missing values indicate product–mill combinations that are not feasible.

Unit Cost Product Mill 1 Mill 2 Mill 3 Mill 4 0.5 in. solid 0.10 0.10 — 0.15 1 in. solid 0.15 0.18 — 0.20 2 in. solid 0.25 0.15 — 0.30 4 in. solid 0.55 0.50 — —

0.5 in. hollow — 0.20 0.13 0.25 1 in. hollow — 0.30 0.18 0.35 2 in. hollow — 0.50 0.28 0.55 4 in. hollow — 1.0 0.60 —

Unit Time Product Mill 1 Mill 2 Mill 3 Mill 4 0.5 in. solid 0.50 0.50 — 0.10 1 in. solid 0.60 0.60 — 0.30 2 in. solid 0.80 0.60 — 0.60 4 in. solid 0.10 1.0 — —

0.5 in. hollow — 1.0 0.50 0.50 1 in. hollow — 1.2 0.60 0.60 2 in. hollow — 1.6 0.80 0.80 4 in. hollow — 2.0 1.0 —

Yearly minimum requirements for the solid sizes

(in thousands) are 250, 150, 150, and 80, respectively.

For the hollow sizes they are 190, 190, 160, and 150. The mills can operate up to three 40-hour shifts per week, 50 weeks a year. Present policy is that each mill must operate at least one shift.

(a) Formulate a linear program to meet demand and shift requirements at minimum total cost using the decision variables xp, m!thousands of units of product p produced annually at mill m Main constraints should have a system of 4 minimum time constraints, followed by a system of 4 maximum time constraints, followed by a system of 8 demand constraints.

(b) State the dual of your primal LP model.

(c) Enter and solve your primal linear program with the class optimization software.

(d) Use your computer output to determine a corresponding optimal dual solution.

(e) Verify that your computer dual solution is feasible in the stated dual and that it has the same optimal solution value as the primal.

(f) Use your computer results to determine the marginal cost of producing each of the eight products.

(g) Use your computer results to explain why the policy of operating all mills at least one shift is costing the company money.

(h) Two options being considered would open mills 3 or 4 on weekends (i.e., add up to 16 extra hours to each of 3 shifts over 50 weeks). Taking each option separately, determine or bound as well as possible from your computer results the impact these changes would have on total production cost.

(i) Another option being considered is to hire young industrial engineers to find ways of reducing the unit costs of production at high-cost mill 4. For each of the 6 products there taken separately, use your computer results to determine to what level unit costs would have to be reduced before there could be any change in the optimal production plan.

(j) A final pair of options being considered is to install equipment to produce 4-inch solid and 4-inch hollow tubes at mill 4.

The new equipment would produce either product in 1 minute per unit. Taking each product separately, determine the unit production cost that would have to be achieved to make it economical to use the new facilities.