Q1 (a) Four products are processed successively on two machines. The manufacturing times in hours per unit of each product are tabulated below for the two machines.
The total cost of producing 1 unit of each product is based directly on the machine time. Assume that the costs per hour for machines 1 and 2 are $10 and $15, respectively. The total hours budgeted for all the products on machines 1 and 2 are 500 and 380. If the sales prices per unit for products 1, 2, 3 and 4 are $65, $70, $55 and $45 respectively, formulate the problem as a linear programming model to maximize total net profit.
(b) (i) Find the optimal solution for the following Linear Programming problem, using Simplex method.
Maximize 2 x1 + x2 + x3
subject to x1 + 5 x2 + x3 ≤ 5
3 x1 + 2 x2 + x3 ≤ 9
5 x1 + 4 x2 + 2 x3 ≤ 10
x1, x2, x3 ≥ 0
(ii) Which of the constraints (1) - (3) are binding, and which of them are non-binding?
(iii) Write down the shadow prices of constraints (1) - (3).
(iv) Write down the reduced costs for x1, x2 and x3.