What is Linear Programming Class 12: Definition & Concepts Explained

Understanding Linear Programming in Class 12 Mathematics

Linear Programming is a branch of mathematics used to optimize (maximize or minimize) a linear objective function subject to a set of linear inequalities called constraints. In Class 12 NCERT Maths, this chapter introduces you to the basics of formulating and solving such problems.

Key points:

• Objective function: The function you want to optimize, e.g., profit or cost.
• Constraints: Linear inequalities representing limitations like resources or time.
• Feasible region: The set of all points satisfying constraints.

Linear Programming helps in decision-making where resources are limited, such as manufacturing, finance, and logistics.

Formulating Linear Programming Problems: Objective Function & Constraints

To solve a Linear Programming problem, first translate the real-world situation into mathematical terms:

1. Define variables: Assign variables to quantities to be determined. 2. Write the objective function: A linear function like $Z = ax + by$ to maximize or minimize. 3. Set constraints: Linear inequalities such as:

$$ \begin{cases} ax + by \leq c \\ dx + ey \geq f \end{cases} $$

4. Non-negativity conditions: Usually $x \geq 0$, $y \geq 0$.

Example:

A factory produces two products, $x$ and $y$. Profit per unit is Rs. 5 and Rs. 4 respectively. If production constraints are:

• $2x + y \leq 100$ (raw material limit)
• $x + 2y \leq 80$ (labour limit)

Objective function: Maximize profit $Z = 5x + 4y$.

This formulation is the first step in solving Linear Programming problems.

Graphical Method to Solve Linear Programming Problems

The graphical method is used when there are two variables. Steps include:

• Plot each constraint inequality on the coordinate plane.
• Identify the feasible region where all constraints overlap.
• Plot the objective function line for a chosen value.
• Move the objective line parallelly towards increasing (or decreasing) values to find the optimal point.
• The optimal solution lies at one of the corner points (vertices) of the feasible region.

Example:

Given constraints:

$$ \begin{cases} x + y \leq 6 \\ x \geq 0, y \geq 0 \end{cases} $$

Objective: Maximize $Z = 3x + 2y$

Plotting and checking corner points (0,0), (6,0), (0,6), (3,3) will give the maximum $Z$ at (3,3) with $Z = 15$.

This method is intuitive and helps visualize the solution.

Key Terminology in Linear Programming for Class 12 Students

Understanding terms is crucial:

• Feasible Solution: Any point satisfying all constraints.
• Feasible Region: The set of all feasible solutions, often a polygon.
• Infeasible Problem: No solution satisfies all constraints.
• Bounded/Unbounded Region: Whether the feasible region is enclosed or extends infinitely.
• Optimal Solution: The point in the feasible region where the objective function attains its max or min.

Knowing these terms helps in exam questions and problem-solving.

Worked Example: Solving a Linear Programming Problem

Problem:

A company produces two items, A and B. Each unit of A requires 2 hours, B requires 3 hours. Total available time is 12 hours. Profit per unit is Rs. 30 for A and Rs. 50 for B. How many units of each should be produced to maximize profit?

Step 1: Define variables

• Let $x$ = units of A
• Let $y$ = units of B

Step 2: Objective function

Maximize profit $Z = 30x + 50y$

Step 3: Constraints

• Time constraint: $2x + 3y \leq 12$
• Non-negativity: $x \geq 0$, $y \geq 0$

Step 4: Graphical solution

• Plot $2x + 3y = 12$.
• Find intercepts: $x=6$ (if $y=0$), $y=4$ (if $x=0$).
• Feasible region is below this line in the first quadrant.

Step 5: Corner points

• (0,0), (6,0), (0,4)

Calculate $Z$:

Maximum profit Rs. 200 at (0,4) means produce 0 units of A and 4 units of B.

This example illustrates the entire process clearly.

Common Mistakes to Avoid in Linear Programming Problems

Students often make these errors:

• Incorrectly formulating the objective function or constraints.
• Forgetting non-negativity conditions ($x, y \geq 0$).
• Plotting inequalities as equalities without shading the feasible region.
• Missing corner points when checking for optimal solutions.
• Confusing maximization with minimization problems.

Tips:

• Carefully read the problem and define variables.
• Double-check constraints and objective function.
• Use graph paper for accuracy.
• Always evaluate the objective function at all corner points.

Avoiding these mistakes improves accuracy and exam scores.