medium difficulty11 min

Application-Based Linear Programming Problems

Application word problems use linear programming to model real situations such as diet planning, manufacturing, transport, and allocation of limited resources.

In one line: Table before inequalities: context becomes algebra with fewer mistakes.

Practice This Concept

Understand it

These problems require translating a context into variables, objective function, and constraints. The mathematical solution must be interpreted in the original context. Since Class 12 LPP is graphical, problems usually involve two decision variables and linear restrictions.

Example

In a manufacturing problem, x and y may represent numbers of two products. Machine-hour limits, material limits, and demand conditions form constraints, while profit or cost forms the objective function.

Formula and rule

Application modelling pattern: decision variables x, y; objective function Z = cost/profit expression; constraints from resource limits or minimum requirements; non-negativity x ≥ 0, y ≥ 0.

Use ≤ for limited availability and ≥ for required minimum intake, demand, or production.

See it worked out

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

Worked example

A small unit makes x standard bags and y deluxe bags.

1Profit is Rs 40 and Rs 60 respectively.
2Cutting time available is 80 hours, with each standard bag needing 2 hours and each deluxe bag needing 4 hours.
3Stitching time available is 60 hours, with each needing 3 hours and 2 hours respectively.
4Formulate and solve.
5Maximize Z = 40x + 60y subject to 2x + 4y ≤ 80, 3x + 2y ≤ 60, x ≥ 0, y ≥ 0.
6Simplify first constraint: x + 2y ≤ 40.
7Corner points are (0,0), (20,0), intersection of x + 2y = 40 and 3x + 2y = 60, and (0,20).
8For intersection, subtract: (3x + 2y) - (x + 2y) = 60 - 40, so 2x = 20, x = 10.
9Then 10 + 2y = 40, so y = 15.
10Evaluate Z: Z(0,0)=0, Z(20,0)=800, Z(10,15)=400+900=1300, Z(0,20)=1200.
11Maximum profit is Rs 1300 at x = 10, y = 15.

Method you can reuse

1Read the context and identify the two quantities to be decided.
2Create a table for profit or cost and resource use.
3Write the objective function.
4Convert each resource or requirement statement into a constraint.
5Add non-negativity conditions.
6Solve graphically and interpret the optimum in words with units.

Visual tip

The application itself may not need a special diagram, but the solved LPP normally requires a coordinate graph with feasible region and corner points when optimum is requested.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

In a diet problem, if food A contains 3 units of nutrient P and food B contains 5 units of nutrient P, and at least 30 units of P are needed, what constraint is formed?

Think first, then show answer

The constraint is 3x + 5y ≥ 30, where x and y are the quantities of foods A and B.

Common wrong answer

Leaving the final answer as only Z = 1300 without stating the required values of x and y loses the decision part of the problem.

Don't mix this up

Students often ignore units and mix quantities, such as adding kilograms of material directly to hours of labour in one constraint.

Exam tip

Make a small table of resource use per unit before writing constraints; it reduces sign errors and missing conditions.

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 appears as a long-answer or case-study problem where students must form the LPP, draw the feasible region, evaluate corner points, and write the final real-world decision.

1
Enough for 1 mark
Application word problems use linear programming to model real situations such as diet planning, manufacturing, transport, and allocation of limited resources.
2
Add for 2 marks
Example: In a manufacturing problem, x and y may represent numbers of two products. Machine-hour limits, material limits, and demand conditions form constraints, while profit or cost forms the objective function.
3
Add for 3 marks
These problems require translating a context into variables, objective function, and constraints.

MCQ Quiz

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