medium difficulty10 min

Mathematical Formulation of an LPP

A linear programming problem is formulated by choosing decision variables, writing a linear objective function such as Z = ax + by, and expressing all given conditions as linear constraints.

In one line: Variables first, target next, limits last: VTL gives a clean formulation order.

Practice This Concept

Understand it

Formulation converts a word problem into mathematics. The unknown quantities are represented by variables, the quantity to be maximized or minimized becomes the objective function, and each limitation or requirement becomes a linear inequality. Non-negativity conditions such as x ≥ 0 and y ≥ 0 are included when variables represent quantities like units, hours, kilograms, or packets.

Example

If x chairs and y tables are produced, and profit is Rs 30 per chair and Rs 50 per table, then the objective function is Z = 30x + 50y. If each chair needs 2 hours and each table needs 4 hours, with at most 40 hours available, then 2x + 4y ≤ 40.

Formula and rule

General form for Class 12 graphical LPP: maximize or minimize Z = ax + by, subject to constraints a1x + b1y ≤, ≥, or = c1; a2x + b2y ≤, ≥, or = c2; and usually x ≥ 0, y ≥ 0.

Use ≤ for upper limits and ≥ for minimum requirements.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

A shopkeeper sells x packs of type A and y packs of type B.

1Profit is Rs 20 on A and Rs 30 on B.
2Storage allows at most 100 packs, and type B packs must be at least 20.
3Formulate the LPP.
4Objective: maximize Z = 20x + 30y.
5Storage constraint: x + y ≤ 100.
6Minimum B condition: y ≥ 20.
7Non-negativity: x ≥ 0, y ≥ 0.
8Therefore, maximize Z = 20x + 30y subject to x + y ≤ 100, y ≥ 20, x ≥ 0, y ≥ 0.

Method you can reuse

1First choose variables with units.
2Next identify whether the objective is maximum or minimum.
3Then convert each condition into an inequality or equation.
4Finally add x ≥ 0 and y ≥ 0 wherever negative values are impossible.

Visual tip

A diagram is not compulsory for formulation alone, but the formulated inequalities must be suitable for drawing half-planes in the graphical method.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

A factory makes x units of product A and y units of product B. Profit per unit is Rs 80 for A and Rs 120 for B. Write the objective function for maximum profit.

Think first, then show answer

The objective function is maximize Z = 80x + 120y.

Common wrong answer

Writing x + y ≥ 100 for a statement like storage can hold at most 100 packs is a realistic wrong step.

Don't mix this up

Students often write the objective function correctly but reverse a constraint inequality, such as using ≥ for an upper limit like available time or raw material.

Exam tip

Underline words such as at most, at least, not more than, minimum, maximum, profit, cost, and requirement before writing inequalities.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Usually asked as a short-answer or long-answer first part: define variables, write the objective function, list constraints, and mention non-negativity restrictions.

1
Enough for 1 mark
A linear programming problem is formulated by choosing decision variables, writing a linear objective function such as Z = ax + by, and expressing all given conditions as linear constraints.
2
Add for 2 marks
Example: If x chairs and y tables are produced, and profit is Rs 30 per chair and Rs 50 per table, then the objective function is Z = 30x + 50y. If each chair needs 2 hours and each table needs 4 hours, with at most 40 hours available, then 2x + 4y ≤ 40.
3
Add for 3 marks
Formulation converts a word problem into mathematics.

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