# How to formulate constraints given the following information

The following question was given in one of my class but none of us got the use of the market requirements in the problem:

A form produces and sells three products namely Product1, Product2 and Product3 . The respective unit profit earned are 4.00dollars for Product1 , 3.00dollars for Product2 and 5.00dollars for Product3. Each product is processed on two machines, Machine1 and Machine2 . A unit of Product1 requires 4minutes of processing time on Machine1 and 2minutes of processing time on Machine2. A unit of Product2 requires 3minutes of processing time on Machine1 and 2minutes of processing time on Machine2. A unit of Product3 requires 5minutes of processing time on Machine1 and 4minutes of processing time on Machine2. The available capacity on Machine1 is 2000 machine minutes and that on Machine2 is 2500 machine minutes.
For market requirement, the firm must produce at least 100 units of Product1 but not more than 150 units of the product, at least 200 units of Product2, and at least 50 units of Product3.

How do you formulate constraints from this information?

Let

$x_1$: Amount of product 1

$x_2$: Amount of product 2

$x_3$: Amount of product 3

In my opinion defining the variables is the most important step.

The objective function expresses the total profit:

$\texttt{max} \ \ 4x_1+3x_2+5x_3$

Constraint for capacity of machine 1:

Product 1 requires 4 minutes on machine 1, product 2 requires 3 minutes of processing time on machine 1 and a unit of product 3 requires 5 minutes.

The available capacity on machine 1 is 2000 machine minutes.

The total required machine minutes have to be smaller or equal to 2,000.

$4x_1+3x_2+5x_3 \leq 2,000$

Can you formulate the constraint for the second machine ?

• Yes I can,no where do the market requirements come in? Do they also form more constraints? @calculus – Manny265 May 16 '15 at 11:34
• @Manny264 Yes. For example this constraint "the firm must produce at least 100 units of product 1" is in mathematical terms $x_1\geq 100$.. In general all amounts have to be greater or equal to zero. But with the given constraints this condition is included. – callculus May 16 '15 at 11:39
• So there will be 4more constraints with regards to the market conditions on top of the other two machine constraints right? @calculus – Manny265 May 16 '15 at 22:13
• @Manny264 Yes. The two conditions for machine 1 can be summarized by $100 \leq x_1 \leq 150$. – callculus May 17 '15 at 2:38