Problem Definition

The company Tetravaal located in Johannesburg manufactures two types of robots, Model \(P_{1}\) and Model \(P_{2}\). The production plant is consisted of four different sections: metal machining, plastic moulding, electrical work and assembly. The metal machining section has a capacity of 7500 units of \(P_{1}\) or 6000 units of \(P_{2}\) per month.

Plastic moulding can process 5000 units of \(P_{1}\) or 9000 units of \(P_{2}\) per month.

Electrical work can process 6000 units of \(P_{1}\) or 7000 units of \(P_{2}\) per month.

In Assembly, there are two assembly lines that work in parallel, one per each robot model.

The first assembly line can process 4000 units of \(P_{1}\) per month

The second assembly line can process 5000 units of \(P_{2}\) per month

Knowing that the unitary profit of \(P_{1}\) is 500€ and that the unitary profit of \(P_{2}\) is 600€, and that both robots have a great demand and therefore all manufactured robots are sold, Michelle Bradley, CEO of Tetravaal, asks his engineering team:

Calculate the number of units of each robot that needs to be manufactured to maximise profit for the company.

Model

We want to maximise the company profits:

\(\max z = 500x_{1} + 600x_{2}\)

where z represents the profits (€). The decision variables are:

\(x_{1}:\) units of \(P_{1}\) per month \(x_{2}:\) units of \(P_{2}\) per month

The objective function is subject to the following constraints:

\(x_{1}/7500+x_{2}/6000 \leq 1\) Metal machining constraint

\(x_{1}/5000+x_{2}/9000 \leq 1\) Plastic moulding constraint

\(x_{1}/6000 + x_{2}/7000 \leq 1\) Electrical work constraint

\(x_{1} \leq 4000\) First assembly line constraint

\(x_{2} \leq 5000\) Second assembly line constraint