Production Mix example

Problem Definition

Production Mix A company produces two kinds of products, P1 and P2.

The selling price of P1 is 300€ The selling price of P2 is 250€

  • P1 requires 2 operator man-hours and 1 hour of machine operating time

  • P2 requires 1 operator man-hours and 3 hours of machine time

  • Operator man-hours cannot be higher than 40 hours per day

  • Machine operating time cannot be higher than 45 hours per day

  • Marketing states that P1 units per day cannot be higher than 12

How many units of P1 and P2 should the company produce every day to maximise income?

Model

We want to find the maximum solution to:

\(\max Z = 300x_{1} + 250x_{2}\)

This is our objective function. The decision variables are:

\(x_{1}:\) units of Product 1

\(x_{2}:\) units of Product 2

The objective function is subject to the following constraints:

\(2x_{1} + x_{2} \leq 40\)
\(x_{1} + 3x_{2} \leq 45\)
\(x_{1} \leq 12\)

We define the problem in its standard form as:

\(max Z = 300x_{1} + 250x_{2} + 0s_{1} + 0s_{2} + 0s_{3}\)

And the constraints as:

\(2x_{1} + x_{2} + s_{1} = 40\)
\(x_{1} + 3x_{2} + s_{2} = 45\)
\(x_{1} + s_{3} = 12\)