Question

A company can extract silver and gold from a particular site. It receives $8000 per unit of silver extracted and $12000 per unit of gold extracted. It has a fixed number of workers and machines that can

extract up to 12 units of silver a day or up to 9 units of gold, or a combination of silver and gold, e.g. 6 units of silver and 4.5 units of gold. It can transport at most 140 tonnes of silver and gold combined per day, where each unit of silver weighs 10 tonnes and each unit of gold weighs 20 tonnes.

You will construct this as a linear optimisation problem, and find the maximum profit the company can make.

(a) Write down the variables and the profit function, and express the constraints on the variables.

(b) Sketch the feasible region and write down its vertices.

(c) Solve the optimisation problem to find the maximum profit.

Answer 1

To formulate an LPP first we ahve to determine the decision variables.

In this case the decision variables are number of units of silver and gols.

Let the number of units of silver =

The number of units of gold =

Profit on 1 unit silver =

Profit on units of silver =

Profit one 1 unit of gold =

Profit on units of gold =

Total profit

This our objective function or profit function.

We have to maximize this according to the constraints.

Maximum number of silver units that can be extracted per day =

Therefore

Maximum number of units of gold extracted per day =

Therefore

Total weight of silver transported per day = tonnes.

Total weight of gold transported per day = tonnes.

Maximum weight or combined weight that can be transported per day = tonnes

Therefore .

Reducung this to the simple form we get

Now our constraints are . Since the units of gold and silver cannot be negative . Therefore

Setting the constraints into equations and drawing the graph of we get the above graph in which the shaded region indicates the feasible solutions. The maximum and minimum solution exist at the corners of the vertices

fROM THE FIGURE

and solving the equation by putting we get

Therefore

Now by calculating the value of the profit function at all the vertices to check the profit

Profit at

Profit at

Profit at

Profit at  

We observe that the maximum profit is at that is . For this point .

Therefore the company has to produce units of silver and unit of gold to get the maximum profit.