Question 5 (a)
Hydroponic farm: formulating the profit-maximising LPP
6 marks
- Time to write
- About 12 minutes
Question
Question 5 (a) · 6 marksAn agriculturist has a hydroponic farm of 1200 acres. He grows spinach, cucumbers, and tomato, and sells everything he raises in the market. He gets ₹12 per kg for spinach, ₹60 per kg for cucumbers and ₹45 per kg for tomato. The average yield per acre is 4000 kg of spinach, 600 kg of cucumbers and 900 kg of tomato. For every 250 kg of spinach and cucumber each, and for 120 kg of tomato produced, ₹300 is spent on manure. The labour required to raise the crop on each acre is 120 man-days for spinach and 60 man-days each for cucumbers and tomatoes. A total of 2400 man-days of labour is available at a rate of ₹250 per man-day. Formulate this linear programming model so as to maximize the agriculturist's total profit. 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
What the paper leaves open
Q5 is parts (a) and (b), or the single 18-mark question printed after the OR.
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Decision variables 1 mark 1 min
- 2 Profit per acre objective coefficients 2 marks 4 min
- 3 Objective function 1 mark 2 min
- 4 Land and labour constraints, non-negativity 2 marks 3 min
- Check: arithmetic, units (₹, %), final answer underlined 1 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, x₂ and x₃ be the acres put under spinach, cucumbers and tomatoes respectively (decision variables of a linear programming problem).
Profit per acre (all produce is sold; costs are manure and labour)
| Per acre | Spinach | Cucumbers | Tomatoes |
|---|---|---|---|
| Yield (kg) | 4,000 | 600 | 900 |
| Price (₹ per kg) | 12 | 60 | 45 |
| Revenue (₹) | 48,000 | 36,000 | 40,500 |
| Manure (₹) | (4,000/250) × 300 = 4,800 | (600/250) × 300 = 720 | (900/120) × 300 = 2,250 |
| Labour (man-days) | 120 | 60 | 60 |
| Labour cost at ₹250 (₹) | 30,000 | 15,000 | 15,000 |
| Profit (₹) | 13,200 | 20,280 | 23,250 |
| Check: manure + labour cost + profit = revenue (₹) | 48,000 | 36,000 | 40,500 |
(Profit = revenue − manure − labour cost, e.g. spinach: 48,000 − 4,800 − 30,000 = 13,200.)
Maximise Z = 13,200x₁ + 20,280x₂ + 23,250x₃ (total profit, ₹)
subject to
x₁ + x₂ + x₃ ≤ 1,200 (land, acres) 120x₁ + 60x₂ + 60x₃ ≤ 2,400 (labour, man-days), i.e. 2x₁ + x₂ + x₃ ≤ 40 x₁, x₂, x₃ ≥ 0 (non-negativity).
The objective and the constraints are linear in the decision variables, which is what makes this an LPP. As the question directs, the model is not solved.
Where the 6 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Decision variables | 1 |
| 2 | Profit per acre (objective coefficients) | 2 |
| 3 | Objective function | 1 |
| 4 | Land and labour constraints, non-negativity | 2 |
| Total | 6 |
Report a problemopens an email to admin@getkunji.com with the question ID1402-Q5aalready filled in.