Question 5 OR (a)
Two types of pens: formulating the profit-maximising LPP
6 marks
- Time to write
- About 12 minutes
Question
Question 5 OR (a) · 6 marksA company makes two types of pens, A and B. Pen A is of superior quality, while Pen B is of lower quality. The net profits on pen A and pen B are ₹60 and ₹40 respectively. The raw material needed for pen A is twice that needed for pen B. The supply of raw material is enough for only 1200 pens of B per day. Pen A needs a special nib, and only 450 such nibs are available. Only 800 nibs are available in a day for pen B. Formulate this as a linear programming problem. DO NOT SOLVE.
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Read from the question
- two typesThe question fixes the number: two types. Give that many, each under its own heading.
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Decision variables and objective 2 marks 3 min
- 2 Raw-material constraint 2 marks 3 min
- 3 Nib constraints and non-negativity 2 marks 3 min
- Check: arithmetic, units (₹, %), final answer underlined 2 min
Try it first. About 12 minutes for 6 marks. Write it out, then open the solution.
Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
Decision variables. Let x₁ = number of pens of type A and x₂ = number of pens of type B made per day.
Objective. Net profit is ₹60 on each A and ₹40 on each B:
Maximise Z = 60x₁ + 40x₂.
- Raw material. A pen of type A needs twice the raw material of a pen of type B. Measuring raw material in "B-pen units", the day's supply is enough for 1,200 pens of B, so 2x₁ + x₂ ≤ 1,200.
- Special nibs for A. Only 450 are available: x₁ ≤ 450.
- Nibs for B. Only 800 are available a day: x₂ ≤ 800.
- Non-negativity: x₁, x₂ ≥ 0.
Maximise Z = 60x₁ + 40x₂ subject to 2x₁ + x₂ ≤ 1,200; x₁ ≤ 450; x₂ ≤ 800; x₁, x₂ ≥ 0.
| Resource | Used by one A | Used by one B | Available |
|---|---|---|---|
| Raw material (B-pen units) | 2 | 1 | 1,200 |
| Nibs for A | 1 | 0 | 450 |
| Nibs for B | 0 | 1 | 800 |
| Profit per pen (₹) | 60 | 40 |
The objective and all constraints are linear in x₁ and x₂. As directed, the problem is not solved.
Where the 6 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Decision variables and objective | 2 |
| 2 | Raw-material constraint | 2 |
| 3 | Nib constraints and non-negativity | 2 |
| Total | 6 |
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