Set up the following problem in a LP framework. A business firm produces two lines of products, A and B, with a plant that consists of three production departments: cutting, mixing, and packaging. The equipment in each department can be used for 8 hours per day; hence we shall regard 8 hours as the daily capacity in each department. The process of production can be summarized as follows:

Product A is first cut, then packaged. Each ton of this product uses 2 hours of the cutting capacity and 1/3 hour of the packaging capacity.

Product B is first mixed, then packaged. Each ton of this product uses 1 hour of the mixing capacity and 2/3 hour of the packaging capacity.

Finally, products A and B can be sold at prices of $40 and $30 per ton, respectively.

a) What output combination should the firm choose in order to maximize total profits? Solve this problem by graphical methods.

b) Write down the dual specification of the above problem, solve it by using the information derived from the primal and dual complementary slackness conditions, and give the dual LP problem a meaningful economic interpretation.