Grade 12
  1. Algebra  1.1. Matrices  1.1.1. Definition and Types of Matrices  1.1.2. Operations on Matrices  1.1.3. Transpose of a matrix  1.1.4. Determinants and their properties  1.1.5. Inverse of a Matrix  1.1.6. Solving Linear Equations Using Matrices  1.1.7. Applications of Matrices  1.2. Complex Numbers  1.2.1. Definition and representation  1.2.2. Algebra of complex numbers  1.2.3. Modulus and argument  1.2.4. Polar and Exponential Forms  1.2.5. De Moivre's Theorem  1.2.6. Roots of complex numbers  1.2.7. Applications in Geometry  1.3. Vectors and 3D Geometry  1.3.1. Vector Algebra  1.3.2. Scalar and Vector Products  1.3.3. Equation of a Line in Space  1.3.4. Equation of a Plane  1.3.5. Distance between lines and planes  1.3.6. Angle between Lines and Planes
  2. Calculus  2.1. Limits and Continuity  2.1.1. Concept of Limits  2.1.2. Continuity of Functions  2.1.3. Types of Discontinuities  2.1.4. Left-hand and Right-hand Limits  2.2. Differentiation  2.2.1. Derivative as a Rate of Change  2.2.2. Techniques of Differentiation  2.2.3. Higher Order Derivatives  2.2.4. Applications of Derivatives  2.2.5. Tangents and Normals  2.2.6. Increasing and Decreasing Functions  2.3. Integration  2.3.1. Indefinite Integrals  2.3.2. Definite Integrals  2.3.3. Integration Techniques  2.3.4. Area under a curve  2.3.5. Applications of Integrals in Geometry  2.4. Differential Equations  2.4.1. Formation of Differential Equations  2.4.2. Order and Degree of Differential Equations  2.4.3. Solutions of Differential Equations  2.4.4. Applications of Differential Equations  2.4.5. Solving First-Order Linear Differential Equations
  3. Probability and Statistics  3.1. Probability  3.1.1. Conditional Probability  3.1.2. Bayes' Theorem  3.1.3. Random Variables  3.1.4. Probability Distributions  3.1.5. Binomial and Normal Distributions  3.2. Statistics  3.2.1. Mean and Standard Deviation  3.2.2. Variance and Covariance  3.2.3. Correlation and Regression  3.2.4. Hypothesis Testing
  4. Linear Programming  4.1. Graphical Method  4.1.1. Feasible and Infeasible Regions  4.1.2. Optimal solutions  4.1.3. Sensitivity Analysis in Graphical Method  4.2. Simplex Method  4.2.1. Formulation of Problems  4.2.2. Solving linear programming problems  4.2.3. Dual Simplex Method
  5. Relations and Functions  5.1. Types of Relations  5.1.1. Reflexive Relations  5.1.2. Symmetric Relations  5.1.3. Transitive Relations  5.1.4. Equivalence Relations  5.2. Types of Functions  5.2.1. Onto Functions  5.2.2. One-to-One Functions  5.2.3. Inverse Functions  5.2.4. Composite Functions  5.3. Inverse Trigonometric Functions  5.3.1. Principal Values in Inverse Trigonometric Functions  5.3.2. Properties and Graphs of Inverse Trigonometric Functions  5.3.3. Applications in Calculus
  6. Advanced Trigonometry  6.1. Trigonometric Equations  6.1.1. General solutions of equations  6.1.2. Transformation formulae  6.2. Hyperbolic Functions  6.2.1. Definitions and properties  6.2.2. Relationships between hyperbolic and trigonometric functions
  7. Mathematical Reasoning  7.1. Logical Reasoning  7.1.1. Statements and Logical Connectives  7.1.2. Tautologies and Contradictions  7.2. Proofs  7.2.1. Direct proof  7.2.2. Indirect Proof  7.2.3. Proof by contradiction  7.2.4. Proof by Mathematical Induction
  8. Applications of Mathematics  8.1. Applications of Calculus  8.1.1. Maxima and Minima Problems  8.1.2. Economics and Biology Applications  8.1.3. Rate of Change in Physics  8.2. Applications of Probability  8.2.1. Risk Analysis  8.2.2. Decision Making  8.2.3. Game Theory

Grade 12 → Linear Programming → Graphical Method ↓


Sensitivity Analysis in Graphical Method


Sensitivity analysis is an important part of linear programming, especially when we deal with real-world problems where certain parameters such as cost, resource availability or market demand may change. In the field of mathematics and operations research, sensitivity analysis helps us understand how the solution to a linear programming problem changes when there are changes to the coefficients of the objective function or the values on the right-hand side of the constraints.

In linear programming, we often deal with maximizing or minimizing a linear objective function subject to a set of linear inequalities or equation constraints. Graphical method is one of the ways to solve two-variable linear programming problem, where we can visually identify the feasible region and find the optimal solution by analyzing the vertices or corners of this region.

Understanding the basics of the graphical method

In the graphical method, you will typically work on problems with two decision variables that can be represented in a two-dimensional space. Here is a quick refresher on the basic steps involved:

  1. Define the objective function that is to be maximized or minimized.
  2. Set up the system of linear inequalities that represent the constraints.
  3. Graph each linear inequality on a two-dimensional plane, using the appropriate scale.
  4. Determine the feasible region by identifying the overlapping region where all constraints are satisfied.
  5. Identify the vertices (corners) of the feasible region.
  6. Evaluate the objective function at each vertex to find the optimal solution.

The concept of sensitivity analysis

Sensitivity analysis in the context of the graphical method is about understanding how changes in the parameters of a linear programming problem affect the optimal solution. It has two main areas:

  • Objective function coefficients: Change in the coefficients of the decision variables in the objective function.
  • Right-hand side of constraints (RHS): Variation in the limitations of resources or constraints.

By performing a sensitivity analysis, you can answer the following key questions:

  • How sensitive is the optimal solution to changes in the objective function coefficients?
  • How much can we increase or decrease the availability of resources before the current solution is no longer optimal?
  • What effect does adding or removing constraints have on the solution?

Sensitivity analysis of objective function coefficients

Suppose you have a linear programming problem, where the objective function is given as:

Maximize Z = c1*x1 + c2*x2

Here, c1 and c2 are the coefficients of the decision variables x1 and x2. Sensitivity analysis involves varying these coefficients to see how the optimal solution changes.

Example:

Suppose the objective function is:

Maximize Z = 3x + 4y

With constraints:

2x + y ≤ 10
x + 2y ≤ 12
x ≥ 0
y ≥ 0

The feasible region for the constraints is graphed, and the vertices are calculated for evaluation.

feasible region

Let us find and label the vertices for evaluating the objective function on the graph.

  • Vertex A is (0,0): Z = 3*0 + 4*0 = 0
  • Vertex B is the intersection point of 2x + y = 10 and x = 0, which is (0,10): Z = 3*0 + 4*10 = 40
  • Vertex C is the point of intersection of x + 2y = 12 and y = 0, which is (12,0): Z = 3*12 + 4*0 = 36
  • Vertex D is the intersection of both constraints: Solving the simultaneous equations, x=4, y=3, i.e., point (4,3): Z = 3*4 + 4*3 = 24

The highest Z value is 40 at vertex B. Let's see how B remains optimal by changing c2 in our objective function.

If c2 increases from 4 to 5, then:

Maximize Z = 3x + 5y

Result obtained from vertex B: Z = 3*0 + 5*10 = 50

Sensitivity analysis of the right-hand side (RHS) of the constraints

Consider the same example. Now, we will investigate how changes in the RHS values affect the feasible region and the optimal solution.

Basic constraints:

2x + y ≤ 10
x + 2y ≤ 12

Let us increase the RHS of the first condition from 10 to 14:

2x + y ≤ 14

See graphically how the feasible region expands with this new restriction:

Extended feasible region

New top counts:

  • Updated changes to vertex D calculation. Solving new equations.

Finding new intersections, calculating objective function results...

Conclusion

Sensitivity analysis broadens our ability to make informed decisions by not only focusing on an optimal solution but also allowing us to explore how robust that solution is to changes in the problem parameters. Especially in graphical methods, it gives us intuitive insight and understanding of the underlying dynamics. Practice with different vertex configurations and constraint coefficients as you go deeper. Linear programming combined with sensitivity analysis provides a powerful toolkit for tackling optimization problems in a wide variety of areas.


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