Problem 11.1-2: A young couple, Eve and Steven, want to divide their main household chores (marketing, cooking, dishwashing, and laundering) between them so that each has two tasks but the total time they spend on household duties is kept to a minimum. Their efficiencies on these tasks differ, where the time each would need to perform the task is given by the following table:

1. Formulate a BIP model for this problem.
2. Display this model on an Excel spreadsheet.
3. Use the computer to solve this model.

11.3-1: 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 \(x_{1}, x_{2}, x_{3}\), and \(x_{4}\) 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 \(5 x_{1}+3 x_{2}+6 x_{3}+4 x_{4} \leq 6,000\)

or \(\quad 4 x_{1}+6 x_{2}+3 x_{3}+5 x_{4} \leq 6,000\).

1. Introduce auxiliary binary variables to formulate a mixed BIP model for this problem.
2. Use the computer to solve this model.

11. 3-4: The Toys-R-4-U Company has developed two new toys for possible inclusion in its product line for the upcoming Christmas season. Setting up the production facilities to begin production would cost $50,000 for toy 1 and $80,000 for toy 2 . Once these costs are covered, the toys would generate a unit profit of $10 for toy 1 and $15 for toy 2 .

The company has two factories that are capable of producing these toys. However, to avoid doubling the start-up costs, just one factory would be used, where the choice would be based on maximizing profit. For administrative reasons, the same factory would be used for both new toys if both are produced.

Toy 1 can be produced at the rate of 50 per hour in factory 1 and 40 per hour in factory 2. Toy 2 can be produced at the rate of 40 per hour in factory 1 and 25 per hour in factory 2. Factories 1 and 2 , respectively, have 500 hours and 700 hours of production time available before Christmas that could be used to produce these toys. It is not known whether these two toys would be continued after Christmas. Therefore, the problem is to determine how many units (if any) of each new toy should be produced before Christmas to maximize the total profit.

1. Formulate an MIP model for this problem.
2. Use the computer to solve this model.

Price: $16.2

Solution: The downloadable solution consists of 9 pages, 720 words and 1 charts.
Deliverable: Word Document