Razorback Tailgate (RT) makes tents for football game parking lot cookouts at its two different facilities, with two processes that may be employed in each facility. All facilities and processes produce the same ultimate product, and Razorback wants to build 22 in the next 80 business hours. Unit costs at facility 1 are $200 and

$350 for the two processes and consume 5 and 20 production hours per unit, respectively. Similarly, at facility 2 unit costs are $150 and $450 with 11 and 23 hours required, respectively, for the two processes. RT wants to meet demand within the 80 available hours at each facility at minimum total cost.

(a) Briefly justify how RT’s problem can be formulated as the following LP:

min 200x1 + 350x2 + 150x3 + 450x4 s.t. x1 + x2 + x3 + x4 Ú 22 5x1 + 20x2 … 80 11x3 + 23x4 … 80 x1, x2, x3, x4 Ú 0

(b) State the dual of the LP in part (a)

using variables v1, v2, v3 on the 3 main constraints.

(c) Demonstrate that primal solution x* = 114.73, 0.0, 7.27, 0.02 and v* =

1200.0, 0.0, -4.552 are optimal by showing they are both feasible and have the same solution value.

(d) Using only parts (a)–(c), what is the marginal cost of production at optimality?

(e) Using only parts (a)–(c), what would be the marginal savings from another hour of production time at facility 1?

(f) Using only parts (a)–(c), and assuming it had some impact, would a decrease in the constraint coefficient 11 of this LP increase or decrease the optimal solution value, and would the rate of change lessen or steepen with the magnitude of the decrease?

(g) Using only parts (a)–(c), and assuming it had some impact, would an increase in the objective function coefficient 150 of this LP increase or decrease the optimal solution value, and would the rate of change steepen or lessen with the magnitude of the increase?

(h) Facility 2, process 2 (i.e., x4) is not used in the current optimum. Using only parts (a)–

(c), at what lowered unit cost would it begin to be attractive to enter the solution?