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Class 12 Science (PCM)

Mathematics Part 2

Linear Programming

Linear Programming

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Study Action Plan (Sorted by Difficulty)
Bounded and Unbounded Regions
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Corner Points and Boundary Lines
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Evaluating Objective Function at Vertices
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Maximise vs Minimise LPP
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Multiple Optimal Solutions
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No Feasible Region and No Solution
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Objective Function and Constraints
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Optimal Feasible Solution
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Resource Allocation Problems
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Decision Variables in LPP
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Feasible Solution and Infeasible Solution
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Graphing Linear Inequalities for LPP
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Mathematical Formulation Steps in LPP
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Optimisation Problem and Linear Programming
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Theorems on Optimal Solutions
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Corner Point Method
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Unbounded Feasible Region Optimality Check
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Feasible and Infeasible Regions
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Graphical Method for Linear Programming
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Linear Constraints from Word Problems
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Linear Objective Function
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Linear Programming Problem Formulation
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Non-Negative Restrictions
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Learning Objectives

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Learning Objectives

  • Understand the concept of linear programming and its applications.
  • Formulate a linear programming problem mathematically.
  • Identify and define the objective function and constraints in a linear programming problem.
  • Solve linear programming problems using the graphical method.
  • Determine the feasible region for a linear programming problem.
  • Identify corner points of the feasible region and evaluate the objective function at these points.
  • Apply the Corner Point Method to find optimal solutions in linear programming problems.
  • Recognize the significance of bounded and unbounded feasible regions in linear programming.

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Revision Notes & Summary

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Chapter 12: Linear Programming

12.1 Introduction

  • Linear programming involves optimizing a linear objective function subject to constraints.
  • Example: A furniture dealer wants to maximize profit from tables and chairs.
    • Investment: Rs 50,000
    • Storage: Maximum of 60 pieces
    • Costs: Table = Rs 2500, Chair = Rs 500
    • Profits: Table = Rs 250, Chair = Rs 75

12.2 Linear Programming Problem and its Mathematical Formulation

12.2.1 Mathematical Formulation

  • Let:
    • x = number of tables
    • y = number of chairs
  • Constraints:
    1. Investment: 2500x + 500y ≤ 50000 (or 5x + y ≤ 100)
    2. Storage: x + y ≤ 60
    3. Non-negativity: x ≥ 0, y ≥ 0
  • Objective function: Z = 250x + 75y (maximize)

12.2.2 Graphical Method of Solving Linear Programming Problems

  • Graph the constraints:
    1. 5x + y ≤ 100
    2. x + y ≤ 60
    3. x ≥ 0
    4. y ≥ 0
  • Feasible Region: The area that satisfies all constraints.
  • Feasible Solutions: Points within or on the boundary of the feasible region.
  • Infeasible Solutions: Points outside the feasible region.

Important Theorems

  1. Theorem 1: Optimal values occur at corner points of the feasible region.
  2. Theorem 2: If the feasible region is bounded, both maximum and minimum values exist at corner points.

Example of Corner Points and Values

Corner PointCorresponding value of Z
(0, 0)0
(30, 0)120 (Maximum)
(20, 30)110
(0, 50)50

Conclusion

  • Linear programming is crucial for optimizing resources in various fields such as industry and management.
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Practice Test – MCQs, True/False

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Multiple Choice Questions

A.

Check if the open half-plane ax+by>Max + by > M has no point in common with the feasible region.

B.

Check if the open half-plane ax+by<max + by < m has no point in common with the feasible region.

C.

Verify if the feasible region is bounded.

D.

Ensure that the objective function is linear.
Correct Answer: A

Solution:

For an unbounded feasible region, the maximum value of the objective function exists if the open half-plane ax+by>Max + by > M has no point in common with the feasible region.

Chapter Concept:

Unbounded Feasible Region Optimality Check

A.

Rs 4500

B.

Rs 5000

C.

Rs 6250

D.

Rs 7000
Correct Answer: C

Solution:

The maximum profit occurs at the vertex (10, 50) of the feasible region, giving Z=250(10)+75(50)=6250Z = 250(10) + 75(50) = 6250.

Chapter Concept:

Evaluating Objective Function at Vertices

A.

x+y≤60x + y \leq 60

B.

5x+y≤1005x + y \leq 100

C.

2500x+500y≤500002500x + 500y \leq 50000

D.

x≥0,y≥0x \geq 0, y \geq 0
Correct Answer: C

Solution:

The investment constraint is 2500x+500y≤500002500x + 500y \leq 50000, representing the maximum amount he can invest.

Chapter Concept:

Objective Function and Constraints

A.

They all give different optimal values.

B.

They all give the same optimal value.

C.

They give optimal values only if the line segment is horizontal.

D.

They do not contribute to the optimal solution.
Correct Answer: B

Solution:

If two adjacent corner points give the same optimal value, every point on the line segment joining them also gives the same optimal value, as stated in the concept of multiple optimal solutions.

Chapter Concept:

Multiple Optimal Solutions

A.

20 gadgets and 40 widgets

B.

30 gadgets and 20 widgets

C.

25 gadgets and 30 widgets

D.

35 gadgets and 15 widgets
Correct Answer: B

Solution:

The constraints are: 3x+2y≤1203x + 2y \leq 120 (machine time) and 2x+y≤802x + y \leq 80 (labor). Solving with the objective function Z=30x+20yZ = 30x + 20y, the optimal solution is x=30x = 30, y=20y = 20 for maximum profit.

Chapter Concept:

Linear Constraints from Word Problems

A.

To find the feasible region

B.

To evaluate the objective function at all vertices of the feasible region

C.

To determine the linear constraints

D.

To graph the system of linear inequalities
Correct Answer: B

Solution:

The Corner Point Method evaluates the objective function at all vertices of the feasible region to find maximum and minimum values.

Chapter Concept:

Corner Point Method

A.

There is a feasible region but no optimal solution.

B.

There is no feasible region and hence no feasible solution.

C.

There is a feasible region with multiple optimal solutions.

D.

There is a feasible region with a unique optimal solution.
Correct Answer: B

Solution:

If no point satisfies all constraints simultaneously, the linear programming problem has no feasible region and hence no feasible solution or optimal solution.

Chapter Concept:

No Feasible Region and No Solution

A.

5x+y≤1005x + y \leq 100

B.

x+y≤60x + y \leq 60

C.

2500x+500y≤500002500x + 500y \leq 50000

D.

x≥0,y≥0x \geq 0, y \geq 0
Correct Answer: B

Solution:

The constraint x+y≤60x + y \leq 60 represents the storage limitation, as the dealer can store at most 60 pieces.

Chapter Concept:

Mathematical Formulation Steps in LPP

A.

40,000 people

B.

50,000 people

C.

60,000 people

D.

70,000 people
Correct Answer: C

Solution:

Let xx be the number of online ads and yy be the number of television ads. The constraints are 200x+500y≤10,000200x + 500y \leq 10,000. The objective function to maximize is R=1000x+2000yR = 1000x + 2000y. Solving this using the corner point method, the maximum reach is achieved at x=25x = 25, y=10y = 10, giving R=1000(25)+2000(10)=60,000R = 1000(25) + 2000(10) = 60,000 people.

Chapter Concept:

Graphical Method for Linear Programming

A.

Theorem of Boundedness

B.

Theorem of Unboundedness

C.

Corner Point Theorem

D.

Theorem of Optimal Solutions
Correct Answer: D

Solution:

Theorem of Optimal Solutions states that the optimal value of the objective function occurs at a corner point of the feasible region.

Chapter Concept:

Theorems on Optimal Solutions

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True or False

Correct Answer: True

Solution:

In linear programming, the decision variables are subject to non-negative constraints, meaning they must be greater than or equal to zero. This is a fundamental aspect of defining the feasible region for the problem.

Chapter Concept :

Objective Function and Constraints

Correct Answer: True

Solution:

A linear programming problem involves maximizing or minimizing a linear objective function subject to constraints in the form of linear inequalities, and the decision variables must be non-negative.

Chapter Concept :

Linear Programming Problem Formulation

Correct Answer: True

Solution:

According to Theorem 1, when the objective function has an optimal value, it must occur at a corner point of the feasible region.

Chapter Concept :

Theorems on Optimal Solutions

Correct Answer: True

Solution:

In a linear programming problem, the objective function is a linear function of the form Z=ax+byZ = ax + by, where aa and bb are constants, and it is either maximized or minimized subject to constraints.

Chapter Concept :

Objective Function and Constraints

Correct Answer: True

Solution:

In linear programming, decision variables represent quantities such as the number of items to produce or purchase, which cannot be negative. Therefore, constraints like x≥0x \geq 0 and y≥0y \geq 0 are included to ensure non-negative values.

Chapter Concept :

Non-Negative Restrictions

Correct Answer: True

Solution:

According to theorems in linear programming, when the objective function has an optimal value, it must occur at a corner point (vertex) of the feasible region.

Chapter Concept :

Graphical Method for Linear Programming

Correct Answer: False

Solution:

The Corner Point Method states that the optimal value of the objective function occurs at a corner point of the feasible region only if the region is bounded. If the region is unbounded, the optimal value may not exist.

Chapter Concept :

Corner Point Method

Correct Answer: True

Solution:

According to Theorem 1, when the objective function has an optimal value, it must occur at a corner point (vertex) of the feasible region.

Chapter Concept :

Graphical Method for Linear Programming

Correct Answer: True

Solution:

In linear programming, the objective function is indeed subject to a set of linear inequalities, which are known as constraints, and the decision variables are restricted to be non-negative.

Chapter Concept :

Mathematical Formulation Steps in LPP

Correct Answer: True

Solution:

A feasible region is defined as bounded if it can be enclosed within a circle, meaning it does not extend indefinitely in any direction.

Chapter Concept :

Bounded and Unbounded Regions

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