# Linear Programming: Optimization

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A furniture manufacturer makes chairs and sofas. Each chair can be sold for a profit of Â£15 and each sofa for a profit of Â£5. It takes four hours to make a chair and five hours to make a sofa. The manufacturer has enough workers to provide 200 hours per week producing the furniture. Customer demand requires that at least seven times as many chairs as sofas arc made. Chairs each take up 1 m3 of storage space and sofas take up 3 m3. There is a total of 90 m3 of storage space available in the factory per week.

Let x denote the number of chairs and y denote the number of sofas the manufacturer can produce per week.

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

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

(c) Solve the linear programming problem graphically to find the number of chairs and sofas that maximize 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 chairs and sofas the manufacturer should produce each week, and the maximum profit that can be made.

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