In an L.P.P., the function to be optimised is the:

Objective function. Z is maximised or minimised.

The constraints x ≥ 0 and y ≥ 0 are called:

Non-negativity constraints. Quantities cannot be negative.

The region common to all the constraints is the:

Feasible region. Every point in it satisfies all constraints.

For a bounded feasible region, the optimum value of Z occurs at:

A corner point. Corner point theorem.

Z = 3x + 2y has feasible corner points (0, 0), (4, 0), (3, 2) and (0, 4). The maximum value of Z is:

13. Values are 0, 12, 13 and 8.

Maharashtra State Board · HSC Commerce · Class 12 (Std XII)

Class 12 Maths and Statistics Part 2 Chapter 6: Linear Programming

Short notes, key concepts, quick revision and practice MCQs with answers

Linear programming finds the best (maximum profit or minimum cost) use of limited resources. The chapter explains the terms of a linear programming problem (L.P.P.), formulates problems mathematically and solves two-variable problems graphically by the corner point method.

Key concepts: Linear Programming

Linear programming problem: A problem of optimising (maximising or minimising) a linear function subject to linear constraints and non-negativity restrictions.
Decision variables: The unknown quantities to be decided, such as the numbers x and y of two products to make.
Objective function: The linear function Z = c₁x + c₂y to be maximised (profit, revenue) or minimised (cost, time).
Constraints: Linear inequations expressing limited resources or requirements: 'at most' gives ≤ and 'at least' gives ≥. x ≥ 0 and y ≥ 0 are the non-negativity constraints.
Feasible region and solution: The region common to all constraints is the feasible region; any point in it is a feasible solution. The feasible solution giving the optimum value of Z is the optimal solution.
Corner point method: Draw the constraint lines, shade the feasible region, find its corner points and evaluate Z at each. For a bounded region, the maximum and minimum occur at corner points.
Special cases: If two corner points give the same optimum, every point on the segment joining them is optimal (infinitely many solutions). The feasible region may be unbounded, in which case a maximum may not exist.
Formulation steps: Identify the decision variables, write the objective function, express each restriction as a linear inequation, and add non-negativity constraints.

Quick revision: important points

The graphical method is used when there are two decision variables.
Test a point (usually the origin) to decide which side of a line to shade.
The optimum of a bounded L.P.P. lies at a vertex of the feasible region.
Profit ₹a per unit of A and ₹b per unit of B gives Z = ax + by.

Practice MCQs with answers

In an L.P.P., the function to be optimised is the:
1. Constraint
2. Objective function
3. Feasible region
4. Decision variable
Show answer
Answer: B. Objective function
Z is maximised or minimised.
The constraints x ≥ 0 and y ≥ 0 are called:
1. Objective constraints
2. Non-negativity constraints
3. Equality constraints
4. Optimal constraints
Show answer
Answer: B. Non-negativity constraints
Quantities cannot be negative.
The region common to all the constraints is the:
1. Optimal region
2. Feasible region
3. Infeasible region
4. Objective region
Show answer
Answer: B. Feasible region
Every point in it satisfies all constraints.
For a bounded feasible region, the optimum value of Z occurs at:
1. The origin only
2. A corner point
3. The centre of the region
4. Any interior point
Show answer
Answer: B. A corner point
Corner point theorem.
Z = 3x + 2y has feasible corner points (0, 0), (4, 0), (3, 2) and (0, 4). The maximum value of Z is:
1. 12
2. 13
3. 8
4. 10
Show answer
Answer: B. 13
Values are 0, 12, 13 and 8.

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