- 20 Marks
Question
Twins Nig. Ltd, a company based in Aba, produces industrial vanish. The selling prices and associated unit variable costs for vanish Grade A and Grade B are shown below:
| Grade A | Grade B | |
|---|---|---|
| N | N | |
| Selling price | 2,100 | 1,500 |
| Material X (N240 per kg) | 480 | 240 |
| Skilled labour (N144 per hour) | 720 | 288 |
| Unskilled labour (N60 per hour) | 120 | 180 |
| Variable overhead (N84 per machine hour) | 168 | 338 |
The fixed overhead cost is N600,000 per month and the company plans to maximise profits.
Available resources for the following month is as follows:
| Material X | 2,500 kg |
| Skilled labour | 4,800 hours |
| Unskilled labour | 3,900 hours |
| Machine hours | 5,000 hours |
Required:
a.
(i) Identify the objective function and the constraints to be used in a linear programming model to determine the optimum production plan for the following month. (5 Marks)
(ii) Produce, using simultaneous equations, the optimum production plan and the resulting profit for the following month. (5 Marks)
b.
Management is now considering negotiating overtime work with both the skilled and unskilled labour. Advise management what maximum amount should be offered per hour of overtime, separately, for skilled labour and for unskilled labour. (Show calculations). (5 Marks)
c. Assume that the company has the choice of producing either Grade A of the vanish or Grade B, but not both. Calculate which Grade of the vanish should be produced in order to maximise contribution. What quantity should be produced and the resulting total contribution? (5 Marks)
Answer
a)
i) * Contribution per unit
| Grade A | Grade B | |
|---|---|---|
| Selling price | 2,100 | 1,500 |
| Variable cost per unit | (1,488) | (1,044) |
| Contribution | 612 | 456 |
- LP Model
Objective function: Maximise contribution (C) = 612A + 456B Subject to constraints:
i) Material X: 2A + B ≤ 2,500
ii) Skilled labour: 5A + 2B ≤ 4,800
iii) Unskilled labour: 2A + 3B ≤ 3,900
iv) Machine hours: 2A + 4B ≤ 5,000
v) Non-negativity: A, B ≥ 0
ii) The equation of the binding constraints are:
5A + 2B = 4,800 …… (1)
2A + 3B = 3,900 …… (2)
Eq. (2) x 2.5: 5A + 7.5B = 9,750 …… (3)
Eq. (3) – Eq. (1): 0 + 5.5B = 4,950
B= 900 units
Substitute for B in Eq. (2): 2A + 3(900) = 3,900 A = 600 units
Thus, the optimum production plan is to produce 600 units of A and 900 units of B.
Total contribution = 612 (600) + 456 (900) = N777,600
Less fixed costs
N600,000 Profit
N177,600
b) We need to first compute the shadow prices. Using the dual method:
Let x = shadow price per skilled labour
y = shadow price per unskilled labour
5x + 2y = 612 ….. (1)
2x + 3y = 456 ….. (2)
Eq. (2) x 2.5: 5x + 7.5y = 1,140
Eq. (3) – Eq. (1): 0 + 5.5y = 528
y = 96
Substitute for y in Eq. (2): 2x + 3 (96) = 456
x = 84 Thus, the shadow price (i.e. the overtime premium) per hour of skilled labour is N84 and N96 for unskilled labour.
To get the maximum hourly rate we simply add the normal hourly rate:
Skilled labour = N144 + N84 = N228 Unskilled labour =N60 + N96 = N156
c) Maximum production of each grade is as follows:
| A | B | |
|---|---|---|
| Material X | (2,500/2) 1,250 | (2,500/1) 2,500 |
| Skilled labour | (4,800/5) 960 | (4,800/2) 2,400 |
| Unskilled labour | (3,900/2) 1,900 | (3,900/3) 1,300 |
| Machine hours | (5,000/2) 2,500 | (5,000/4) 1,250 |
To avoid violating the other constraints, the maximum of each grade of the vanish that can be produced is the lowest under each of the two grades.
The total maximum contribution is: Grade A 960 units x N612 = N587,520 Grade B 1,250 units x N456 = N570,000
- Topic: Decision-making techniques, Pricing Decisions
- Series: MAY 2025
- Uploader: Samuel Duah