# Formulating an Integer Programming problem

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• Apr 24th 2009, 05:09 PM
jlt1209
Formulating an Integer Programming problem
A manufacturer can sell product 1 at a profit of \$2/unit and product 2 at a profit of \$5/unit. Three units of raw material are needed to manufacture 1 unit of product 1, and 6 units of raw material are needed to manufacture 1 unit of product 2. A total of 120 units of raw material are available. If any of product 1 is produced, a setup cost of \$10 is incurred, and if any of product 2 is produced, a setup cost of \$20 is incurred. Formulate an IP to maximize profits. For computational simplicity you can assume that it is allowed to produce fractions of product 1 or product 2 (so the numbers of products do not have to be integers).
• Apr 25th 2009, 12:11 AM
CaptainBlack
Quote:

Originally Posted by jlt1209
A manufacturer can sell product 1 at a profit of \$2/unit and product 2 at a profit of \$5/unit. Three units of raw material are needed to manufacture 1 unit of product 1, and 6 units of raw material are needed to manufacture 1 unit of product 2. A total of 120 units of raw material are available. If any of product 1 is produced, a setup cost of \$10 is incurred, and if any of product 2 is produced, a setup cost of \$20 is incurred. Formulate an IP to maximize profits. For computational simplicity you can assume that it is allowed to produce fractions of product 1 or product 2 (so the numbers of products do not have to be integers).

Let $x$ be the number of units of product 1 produced, and $y$ be the number of units of product 2.

The objective is the profit:

$O=2x+5y-10I(x)-20I(y)$

where $I(.)$ is the indicator function which is $1$ if its argument is greater than $0$ and $0$ otherwise (alternativly introduce two 0-1 variables to handle the set up costs).

The nonnegativity constraints are that $x\ge 0$ , $y\ge 0$ and the materials constraint is:

$3x+6y \le 120$

CB