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


Linear programming is a fascinating area of mathematics that deals with optimizing a particular outcome based on some set of linear relationships. It is an important topic, especially in fields such as economics, business, engineering, and military applications. At its core, linear programming attempts to maximize or minimize a linear objective function subject to a set of linear inequality or equality constraints.

Understanding linear programming

To put it simply, imagine you run a factory that makes two kinds of products: widgets and gadgets. You make a profit from both, but due to resource limitations, you can't produce unlimited quantities of each. How can you decide the best way to maximize your profit? This is where linear programming can help.

Basic concepts

Objective function

The objective function is the formula we want to maximize or minimize. In our factory example, if you make a profit of $20 per widget and $30 per gadget, your objective function would be:

Z = 20W + 30G

Here, Z represents the total profit, W represents the number of widgets, and G represents the number of gadgets.

Restrictions

Constraints are conditions that must be met for a solution to be feasible. In a mathematical sense, these are inequalities or equalities. For example, suppose you have a limit on machine hours or raw materials. Suppose your machine can only run 40 hours a week, and it takes 1 hour to make a widget and 2 hours to make a gadget. The constraints would be:

1W + 2G ≤ 40

You may also have more constraints such as budgetary limits or material limits. These constraints limit the number of widgets and gadgets you can produce.

Feasible region

The feasible region is the area in the graph where all constraints overlap. It shows all possible combinations of W and G that satisfy all the conditions.

Graphical method

Let's illustrate the concept of linear programming using a simple graphical method. For two-variable problems like our widget and gadget example, graphical methods are particularly useful.

1W+2G≤40

In this example, the x-axis represents W (number of widgets) and the y-axis represents G (number of gadgets). The line 1W + 2G = 40 represents a constraint. The shaded region is the feasible region.

Solving linear programming problems

Now that we have a basic understanding of lines and constraints, let's solve a simple linear programming problem using the graphical method.

Example problem

Consider our previous example with some additional restrictions:

Objective Function: Maximize Z = 20W + 30G Subject to: 1W + 2G ≤ 40 W ≤ 15 G ≤ 10
Objective Function: Maximize Z = 20W + 30G Subject to: 1W + 2G ≤ 40 W ≤ 15 G ≤ 10

Step-by-step solution

  1. Plot the constraints on the graph.
  2. Shade the feasible region, which is bounded by the obstacles.
  3. Identify the corner points of the feasible region.
  4. Calculate the value of the objective function at each corner point.
  5. The maximum or minimum value will solve the problem.

Graph for the example problem

(15,0) (10,10) (0,10)

The feasible region is shaded blue, and the corner points are: (0,0), (15,0), (10,10), and (0,10).

Calculations at corner points

  • At (0,0): Z = 20*0 + 30*0 = 0
  • At (15,0): Z = 20*15 + 30*0 = 300
  • At (10,10): Z = 20*10 + 30*10 = 500
  • At (0,10): Z = 20*0 + 30*10 = 300

The maximum value of Z is 500, which lies at the point (10,10). Therefore, producing 10 widgets and 10 gadgets maximizes profit under the given constraints.

Applications of linear programming

Linear programming is widely used in various industries and fields. Here are some examples where linear programming can be applied:

  • Manufacturing: Deciding the mix of different products to be produced in a factory in order to maximize profits or minimize costs, while taking into account resources, labor, and machine time.
  • Transportation: Optimizing transportation routes to minimize the cost or time in delivering goods from warehouses to various locations.
  • Dietary problems: determining the most cost-effective combination of foods that will meet all nutritional requirements.
  • Investing: Allocating a portfolio to balance risk and return by maximizing returns within given constraints such as financial horizons.

Advanced methods beyond the graphical method

While the graphical method is excellent for solving two-variable problems, real-world linear programming problems often involve more variables and restrictions. For these, more advanced computational methods are used:

Simplex method

The simplex method is an algorithmic approach to solve linear programming problems with more than two variables. It works on the principle of evaluating the vertex points in the feasible region and visiting adjacent vertices to find the optimal solution.

Dual simplex method

Similar to the simplex method, the dual simplex method is used when the problem is not initially feasible, but becomes feasible through adjustment of the constraints.

Interior-point method

An alternative to simplex methods, the interior-point method is efficient in handling large-scale linear programming problems. Instead of traversing the feasible region edge-by-edge, it traverses through the interior of the feasible regions.

Important things to remember

Here are some important points when dealing with linear programming:

  • The objective function and constraints must be linear expressions.
  • The slope of the objective function and constraints play an important role in determining the feasible region.
  • Infeasibility means that no solution can satisfy all the constraints.
  • Unbounded solutions occur when the feasible region extends indefinitely, leading to infinite maximization or minimization.

Conclusion

Linear programming is a powerful technique in mathematical optimization that is used in a variety of real-life problems. It provides a systematic and efficient way to allocate resources in the best possible way. Full understanding requires mastering both the geometry of feasible regions and the algebra of constraints and objective functions. Practicing with various examples strengthens the understanding and builds problem-solving skills.


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