Question

(a) A firm produces four products: P, Q, R, and S. Each unit of product P requires two hours of milling, one hour of assembly, and $10 worth of in-process inventory. Each unit of product Q requires one hour of milling, three hours of assembly, and $5 worth of in-process inventory. Each unit of R requires 2.5 hours of milling, 2.5 hours of assembly, and $2 worth of in-process inventory. Finally, each unit of product S requires five hours of milling, no assembly, and $12 of in-process inventory.

The firm has 120 thousand hours of milling time and 160 thousand hours of assembly time available. In addition, not more than $1 million may be tied up in in-process inventory.

Each unit of product P returns a profit of $40; each unit of Q returns a profit of $24; each unit of product R returns a profit of $36; and each unit of product S returns a profit of $23. Not more than 20 000 units of product P can be sold; not more than 16 000 units of product R can be sold; and any number of units of products P and S may be sold. However, at least 10 000 units of product S must be produced and sold to satisfy a contract requirement.

The objective of the firm is to maximize the profit resulting from the sale of the four products. Formulate the above as a linear programming problem.

(b) Use the simplex method to maximize

                                               z = x₁ + x₂

                            subject to

                                              x₁ + 5x₂ ≤ 5

                                              2x₁ + x₂ ≤ 4

                             with

                                                x₁, x₂ ≥ 0