Based on McBride and Zufryden (1988). A company is trying to determine which of five possible products to include in its product line. The fixed cost of producing each product and the unit profit for each product are listed in the file P06_96.xlsx. There are five customer segments. The number of customers in each segment and the utility each customer segment associates with each product are also listed in this file. If a consumer believes that all available products have a negative utility, this customer will buy nothing. Otherwise, each customer will buy the available product that has the largest utility. For example, if products 1, 2, and 3 are available, customer segment 4 will purchase product 3. Determine which products the company should produce to maximize its profit, assuming that it will produce exactly enough to meet customer demand. (Use a binary decision variable for each product and a binary decision variable for each customer segment-product combination. To ensure that a customer buys only the product with the largest utility, include the following constraint for each combination of product and customer segment:

U_cj x_j ≥ U_ci x_i – M(1 – y_cj) for each i, j, c

Here, U_cj is the utility for customer segment c buying product j, x_j is a binary for product j being offered, y_cj is a binary for customer segment c buying product j, and M is a large number (M equal to the largest product utility will work). This constraint ensures that the y_cj binary can equal 1 only if the binary x_j equals 1, that is, customer segment c can buy product j only if it is included in the product line. Note that if y_cj is 0, then this inequality is automatically satisfied.)