# Thread: Formulating an Integer Programming problem

1. ## 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).

2. 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