Question

In a factory concerned with the production of soft drinks you have been asked to analyse the production of two drinks, X and Y. Each litre of X that is produced can (if required) be further processed into Y. Producing one litre of X costs £0.12 and the further processing to produce Y costs an additional £0.03. Wholesalers have in the forthcoming month ordered 150,000 litres of X and 175,000 litres of Y.

The production capacity allocated to X and Y for the next month means that no more than 250,000 litres (X and Y combined) can be produced. There is none of X or Y currently in stock. A subcontractor can supply X to you for £0.40 per litre and Y for £0.63 per litre. Technological restrictions mean that the ratio of the number of litres of X produced to the number of litres of Y produced must be at least 0.33.

Formulate this decision problem as a linear program.

Assuming that the factory cannot store excess production how much of each drink should it produce next month?

Answer 1

Let amount of X produced be Xp, Y produced be Yp, X bought from supplier be Xb and Y be Yb

Production Cost for X = 0.12

Production Cost for Y = 0.12 + 0.03 = 0.15

Total Cost = 0.12*Xp + 0.15*Yp + 0.40*Xb + 0.63*Yb

We have to minimize this cost

Subject to Constraints:

Xp + Xb = 150,000............Constraint for order for X by wholesalers

Yp + Yb = 175,000.............Constraint for order for Y by wholesalers

Xp / Yp >= 0.33..................Constraint for ratio of Xp & Yp

Hence, Xp >= 0.33*Yp

Xp + Yp <= 250,000........... Constraint for production capacity available per month

Xp, Yp, Xb, Yb >= 0.............Non-negativity Constraint as amount of drink produced cannot be negative

We solve the above formulation in Excel as shown below:

The above solution in the form of formulas along with Excel Solver extract is shown below for better understanding and reference:

As seen from above solution,

Amount of X produced = 75000

Amount of Y produced = 175000

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