A TV company produces two types of TV sets, the Astro and the Cosmo. There are two production lines, one for each set, and there are two departments, both of which are used in the production of each set. The capacity of the Astro production line is 70 sets per day. The capacity of the Cosmo line is 50 sets per day. In department A, the major audio and video components for flat-panel television incorporating liquid-crystal display (LCD) are produced. In this department the Astro set requires 1 labor hour and the Cosmo set requires 2 labor hours. Presently in department A a maximum of 120 labor hours per day can be assigned to production of the two types of sets. In department B the chassis is constructed. In this department B the Astro set requires 1 labor hour and the Cosmo also requires 1 labor hour. Presently, in department B a maximum of 90 labor hours per day can be assigned to production of the two types of sets. The unit cost of an Astro is $210 and the unit cost of a Cosmo is $230. A marketing survey showed that the demand for the TV sets is price-sensitive. That is, more TV sets can be sold only if the selling price can be reduced. In other words, the company faces downward sloping demand curves for its products. These demand curves are quantified by the following equations:

PA = 0.01A^2 -1.9A + 314

PC = = -0.14C + 243

where = daily production of Astros

= selling price (per unit) of Astros

= daily production of Cosmos

= selling price (per unit) of Cosmos

For example, in the preceding expressions, is the price that the company must set for Astros in order to sell all of the Astros it produces. A similar interpretation applies to . a) Given the information outlined above, formulate a profit-maximizing nonlinear program (state the decision variables, the objective function and constraints of the model) and solve using the Excel Solver. What is the optimal solution (quantities produced and selling prices) and the maximum profit? b) Suppose in the Astro/Cosmo problem stated above that Astros and Cosmos are economic substitutes. This means that an increase in price of one causes an increase in demand for the other. More specifically, suppose that the demand equations for the two TV sets as a function of the prices for each are:

A = 1000 - 4.7PA + PC

C = 1000 + 2PA - 6.2PC

Given this new information, reformulate the problem (as a nonlinear programing model) and solve using the Excel Solver. What is the optimal solution and the maximum profit?