1. Give a matrix formulation of the Cutting Edge Knife Company problem of Example 1.
The exclusive Cutting Edge Knife Company manufactures chef's knives and pocket knives. Each chef's knife requires 3 labor-hours, 7 units of steel, and 4 units of wood. Each pocket knife requires 6 labor-hours, 5 units of steel, and 3 units of wood. The profit on each chef's knife is $30, and the profit on each pocket knife is $50. Each day, the company has available 90 labor-hours, 138 units of steel, and 120 units of wood.
(a) How many of each type of knife should the Cutting Edge Knife Company manufacture daily to maximize its profits?
(b) Suppose that an additional 18 units of steel were available each day. What effect would this have on the optimal solution?
(c) Generalize the result in part (b) to the case in which the increase in the number of units of steel available each day is h. (The value of h can be positive or negative.) For what range of values will the result be valid?
2. Does every linear programming problem have a matrix formulation? If not, under what conditions will a linear programming problem have a matrix formulation?