Question

Question 1 [50 marks] The XYZ manuf one tonne of product B the following quantities of each material are used respectively Product company produces chemical products. Three materials are used to acture the A and B products. For the manufacturing of one tonne of product A and Material 1 2/5 Material2 0 1/5 Material 3 3/5 3/10 For the current production period, the company XYZ has at its disposal the following quantities (in tonnes) of each material: Material Material 1 Material 2 Material 3 Available quantity in tonnes 20 5 21 e contribution to the company's profit from the production of one tonne of product A is 40 Euro while from the production of one tonne of product B is 30 Euro. a) Formalize the above problem in a linear programming model. (10 marks) b) Consider how many tonnes of each product should be produced so that XYZ can maximize its profit. Answer the question using the graphical solution. (20 marks) c) Determine the value of the objective function at the optimum solution. (7 marks) d) Consider whether there is unused material for the production of products A and B. If so, determine the quantities of unused materials. (7 marks) f) Give the definition of the feasible area. (6 marks)

Answer 1

1). Let the firm produces X1 units of product A and X2 units of B.

Objective Function:

The Contribution per unit of Product A and B is 40 and 30 Euro Respectively.

so the objective function is

Maximise Z = 40(x1) + 30(x2)

Subject to, Condition 1: 2/5(x1) + 1/2(x2) <=20,

Condition 2: 1/5(x2) <=5 and

Condition 3: 3/5(x1) + 3/10(x2)<=21.

2). see attachments

point	Product A	Product B	Amount A 40 Euro	Amount B 30 Euro	Total Contribution
A	0	0.3	0	9	9
B	0.2	0.2	8	6	14
C	0.275	0.175	11	5.25	16.25
D	0.4	0	16	0	16

3. value of objective function at optimum slution is: 40(0.275) + 30(0.175) = 16.25Euro