Write the objective function which represents the total profit from the sale of total bags of both types.

Linear Programming

LPP Formulation — Objective Function

Official Answer

The objective is to maximise the total profit from x bags of rice and y bags of wheat.

Objective function:

Profit per bag of rice = Rs 100, so profit from rice $= 100x$ .
Profit per bag of wheat = Rs 90, so profit from wheat $= 90y$ .
Maximise $Z = 100x + 90y$ (in rupees).

objective functionmaximise ZZ = 100x + 90yprofit per bagdecision variables x and ylinear programmingtotal profit

Marking Scheme

11 mark: writing the objective function $Z = 100x + 90y$ (with 'Maximise' stated or clearly implied).
2Accept full marks even if the word 'Maximise' is omitted, provided the expression $100x + 90y$ is correct.

Hint

Multiply each decision variable by its per-bag profit and add; state it as 'Maximise'.

Quick Oral Answer

The total profit is Rs 100 per bag of rice plus Rs 90 per bag of wheat, so the objective function is Maximise $Z = 100x + 90y$ .

Analysis & Explanation

This is the first step of formulating any Linear Programming Problem — identifying the quantity to be optimised.

Concept:

The decision variables are already fixed by the stem: x = bags of rice, y = bags of wheat.
The objective function is a linear combination of the decision variables with the per-unit profit as coefficients.

Exam trap:

Do not confuse the profit coefficients (Rs 100, Rs 90) with the cost coefficients (Rs 1,800, Rs 1,200) — the latter belong to the constraint, not the objective.
Always state whether Z is to be maximised or minimised; here it is a profit, so it is maximised.

Common Mistakes

1Using the cost figures ( $1800x + 1200y$ ) in the objective function instead of the profit figures.
2Forgetting to state that Z is to be maximised, or writing it as a minimisation.
3Swapping the coefficients — writing $90x + 100y$ instead of $100x + 90y$ .

Interesting Facts

The objective function and constraints of an LPP were formalised by Soviet mathematician Leonid Kantorovich in 1939 and, independently, by George Dantzig, who devised the simplex method in 1947.

In every LPP exactly one linear objective function is optimised, but there can be many constraints — separating 'what to optimise' from 'what limits you' is the core modelling skill.

Kantorovich shared the 1975 Nobel Prize in Economics for these optimal-allocation ideas, which power problems exactly like this rice-and-wheat purchase decision.

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Frequently Asked Questions

What is an objective function in an LPP?

It is the linear expression in the decision variables that we want to optimise (maximise or minimise). Here it represents total profit, $Z = 100x + 90y$ , where x and y are the numbers of rice and wheat bags.

Why do we use profit and not cost in the objective function?

The man's goal is to maximise his earnings, so the objective must measure profit (Rs 100 and Rs 90 per bag). The costs (Rs 1,800 and Rs 1,200) limit how much he can buy and therefore appear in a constraint, not in the objective.