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 → Simplex Method ↓


Formulation of Problems


Linear programming is a mathematical method used to efficiently allocate scarce resources under certain constraints. The simplex method is one of the most popular techniques for solving linear programming problems. But before you can apply the simplex method, you need to learn how to formulate the problem correctly. Formulating a problem involves expressing it mathematically with an objective function and a set of constraints.

Understanding linear programming

Linear programming deals with the problem of maximizing or minimizing a linear objective function, subject to a set of linear constraints (equality or inequality). The general form of the linear programming problem can be written as:

Maximum (or minimum): c 1 x 1 + c 2 x 2 + ... + c n x n
Subject to constraints:
a 11 x 1 + a 12 x 2 + ... + a 1n x n <= b 1
a 21 x 1 + a 22 x 2 + ... + a 2n x n <= b 2
,
a m1 x 1 + a m2 x 2 + ... + a mn x n <= b m
and the non-negativity constraints:

x 1 , x 2 , ..., x n >= 0

Where:

  • c i represents the coefficients of the objective function.
  • a ij represent the coefficients of the constraints.
  • b i denotes the right-hand side values of the constraints.
  • x i are the decision variables.

Step-by-step problem formulation

To formulate a linear programming problem using the simplex method, follow these steps:

Identify the decision variables

The first step is to identify what quantities you are trying to determine, known as the decision variables. They must be logical and make economic sense. For example, if a company manufactures two products, A and B, the decision variables might be the number of units of A and B produced, represented as x A and x B, respectively.

Write the objective function

The objective function is the formula that must be maximized or minimized. It reflects the goal of the linear programming problem. For example, if each unit of product A brings a profit of $40 and product B brings a profit of $30, then the objective function for maximizing profit would be:

Max: 40x A + 30x B

Create obstacles

Constraints are conditions that must be met and are usually inequalities. They represent restrictions or limits on the decision variables. Continuing from the previous example, suppose there are resource constraints, such as limits on available labor hours and materials. These constraints can be expressed as:

2x A + 3x B <= 100 (labour hours)
x A + x B <= 75 (contents)

Ensure non-negativity

Decision variables must not have negative values. In economic scenarios, this condition is obvious because you cannot produce a negative number of products. The non-negativity constraints would be:

x A >= 0

x B >= 0

A practical example

Let's look at a practical example to clarify the steps in problem formulation:

Example: A manufacturer makes two types of gadgets: gizmos and gadgets. These require some resources to manufacture, and the objective is to maximize profits. Here is the available information:

  • Profit from Gizmo: $5 per unit
  • Profit from gadget: $7 per unit
  • Maximum available manufacturing hours: 40 hours per week
  • Maximum content available: 20 units per week
  • The gizmo takes 2 hours and 1 unit of materials to build
  • Gadget requires 1 hour and 3 units of materials

Formulation of the problem

Identify the decision variables:

x 1 = number of gizmos produced
x 2 = Number of gadgets produced

Write the objective function (profit function):

Maximum: 5x 1 + 7x 2

Create constraints based on resources:

2x 1 + 1x 2 <= 40 (manufacturing hours)
1x 1 + 3x 2 <= 20 (contents)

Non-negativity constraints:

x 1 >= 0

x 2 >= 0
x 1 x 2 2x 1 + x 2 = 40 x 1 + 3x 2 = 20

The feasible region, where all constraints are satisfied, is the intersection bounded by the shaded area in the graph.

Solving a formulated problem using the simplex method

Once the problem is formulated, it can be solved using the simplex method. However, the goal of this explanation is to focus on the formulation of linear programming problems, so we will not go into the specifics of the algorithm just yet. The simplex method moves from one vertex of the feasible region to another vertex with a more favorable value of the objective function until either the maximum or minimum value is reached.

The importance of problem formulation

Proper formulation is important because an incorrectly defined problem will lead to the wrong solution. The above steps ensure that every aspect of the scenario is considered and translated into mathematical expressions. Challenges in formulating problems often arise from misunderstanding real-world scenarios or ignoring constraints. Correctly identifying the decision variables, objective function, and constraints ensures that the simplex method can effectively find the best solution.

In addition, properly formulated problems can be easily communicated to others, ensuring stakeholders understand the problem and the proposed solution. This transparency is crucial for business decision making, where linear programming has wide applications, from manufacturing and transportation to finance and beyond.

It is also important to remember that not all problems can be fully modeled by linear functions due to the inherent complexities. In such instances, linear programming provides an approximation that can still yield practical decisions.

Completion of the learning journey

Formulation of problems in the simplex method of linear programming involves understanding the problem, transforming it into algebraic expressions, and understanding how these mathematical representations serve as the basis for advanced techniques that solve complex real-world problems. Mastering problem formulation is an essential step before applying algorithms or computational methods.

Learning and practicing these skills at a basic level will develop a deeper understanding and the ability to tackle more complex linear programming problems as you progress. Furthermore, these basic steps reflect careful research and analytical thinking that are invaluable in both academic and professional settings.


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Formulation of Problems

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