# Formulating an Integer Programming problem

• 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 \$\displaystyle x\$ be the number of units of product 1 produced, and \$\displaystyle y\$ be the number of units of product 2.

The objective is the profit:

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

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

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

\$\displaystyle 3x+6y \le 120\$

CB