The Research and Development Division of the Progressive Company has been developing four possible new product lines. Management must now make a decision as to which of these four products actually will be produced and at what levels. Therefore, an operations research study has been requested to find the most profitable product mix.
A substantial cost is associated with beginning the production of any product, as given in the first row of the following table. Management€™s objective is to find the product mix that maximizes the total profit (total net revenue minus start-up costs).

Let the continuous decision variables x1, x2, x3, and x4 be the production levels of products 1, 2, 3, and 4, respectively. Management has imposed the following policy constraints on these variables:
1. No more than two of the products can be produced.
2. Either product 3 or 4 can be produced only if either product 1 or 2 is produced.
3. Either 5x1 + 3x2 + 6x3 + 4x4 ‰¤ 6,000 or 4x1 + 6x2 + 3x3 + 5x4 ‰¤ 6,000.
(a) Introduce auxiliary binary variables to formulate a mixed BIP model for this problem.
(b) Use the computer to solve this model.

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Product 2 4 Start-up cost Marginal revenue $50,000 $40,000 $70,000 $60,000 $70 $60 $90 $80