Question

ABC Electricity owns a coal gasifier plant. The coal gasifiers on the plant can use three types of coal (A, B and C) to produce two qualities of producer gas (X and Y). There are two processes; a new process and an old process, which are available to use blended coal. For each production run, the old process uses 20 units of coal A, 28 units of coal B and 8 units of coal C to produce 24 units of X and 20 units of Y. For the new process, 12 units of coal A, 36 units of coal B and 16 units of coal C are used to produce 20 units of X and 28 units of Y. The gasifier plant is required to produce at least 1800 units of X and at least 1600 units of Y. However, it has only 2000 units of coal A, 2500 units of coal B and 1500 units of coal C. For each unit of X, the company earns $4000 and for Y it receives $6000 per unit. Formulate the linear programming model which determines the required number of production runs to maximise ABC Electricity’s revenue for the required production.

Answer 1

Step-by-step explanation:

Let's define the decision variables:

Let x be the number of production runs using the old process.

Let y be the number of production runs using the new process.

The objective is to maximize the revenue, which is given by:

Revenue = 4000 * (24x + 20y) + 6000 * (20x + 28y)

Subject to the following constraints:

1. Coal A constraint: 20x + 12y <= 2000 (available units of coal A)

2. Coal B constraint: 28x + 36y <= 2500 (available units of coal B)

3. Coal C constraint: 8x + 16y <= 1500 (available units of coal C)

4. X production constraint: 24x + 20y >= 1800 (required units of X)

5. Y production constraint: 20x + 28y >= 1600 (required units of Y)

6. Non-negativity constraint: x >= 0, y >= 0

Thus, the linear programming model can be formulated as follows:

Maximize:

Z = 4000(24x + 20y) + 6000(20x + 28y)

Subject to:

20x + 12y <= 2000

28x + 36y <= 2500

8x + 16y <= 1500

24x + 20y >= 1800

20x + 28y >= 1600

x >= 0, y >= 0

This model can be solved using various linear programming techniques to find the optimal values of x and y that maximize the revenue for the required production.