1. Shoe Manufacture

A small shoe manufacturer makes two styles of trainers, a cross-training shoe and a specialist running shoe. A pair of cross-training shoes requires 1 hour machining and 2 hours assembly and makes a profit of £10. A pair of running shoes requires 1.5 hours machining and 1.5 hours assembly, and makes a profit of £15. The company has 210 hours machining time available per week and 300 hours assembly time per week.

Let x denote the number of pairs of cross-trainers and y denote the number of pairs of running shoes the manufacturer can produce per week.

(a) What is the weekly profit of the manufacturer, assuming that the demand for trainers means that all pairs will be sold?

(b) Write down all the constraints for the problem.

(c) Solve the linear programming problem graphically to find the number of trainers of each type that maximise the revenue. On your graph indicate clearly the feasible region, the optimal point, an arbitrary isoprofit line, and the isoprofit line corresponding to optimal profit.

(d) Hence state the optimum number of trainers of each type the manufacturer should produce each week, and the maximum profit the manufacturer can make.