Linear Programming Solution Techniques: Enhancing Efficiency and Accuracy, Mastering the Essence of Problem Solving

# Introduction to Linear Programming: Mastering Techniques for Efficiency and Accuracy

**1.1 Linear Programming Overview**

Linear programming is a mathematical optimization technique used to solve problems related to resource allocation. It aims to find the optimal value of a target function under a set of constraints.

**1.1 Elements of Linear Programming**

A linear programming problem consists of the following elements:

- **Objective Function:** The linear function to be optimized (maximized or minimized).
- **Constraints:** A series of linear inequalities or equalities that limit the range of values for the decision variables.
- **Decision Variables:** The unknowns to be determined, representing values assigned to different resources.

**1.2 Applications of Linear Programming**

Linear programming is widely used in various fields, including:

- Resource Allocation: Optimizing the distribution of resources (such as time, money, materials) to achieve specific goals.
- Production Planning: Determining the best production plan to meet demands and maximize profits.
- Transportation Problems: Optimizing the transport of goods to minimize costs or time.

# 2. Linear Programming Solution Methods

Linear programming solution methods are primarily divided into three types: graphical method, simplex method, and dual simplex method. Each method has its unique features and applicable scenarios.

### 2.1 Graphical Method

The graphical method is suitable for small-scale linear programming problems, i.e., with fewer variables and constraints.

#### 2.1.1 Feasible Region Drawing

The feasible region is the set of solutions that satisfy all constraints. For a linear programming problem, the feasible region is a convex polyhedron in a multidimensional space. In two dimensions, the feasible region is a polygon.

The steps to draw the feasible region are as follows:

1. Represent each constraint as a linear equation.
2. Draw all linear equations on the same coordinate system.
3. Identify the region that satisfies all constraints, i.e., the feasible region.

#### 2.1.2 Determining the Optimal Solution

Within the feasible region, the objective function can reach a maximum or minimum value. The optimal solution is the solution at which the objective function attains its extreme value (maximum or minimum) in the feasible region.

The steps to determine the optimal solution are as follows:

1. Represent the objective function as a linear equation.
2. Intersect the objective function line with the boundary of the feasible region.
3. Find the coordinates of the intersection points of the objective function line and the feasible region boundary, i.e., the optimal solution.

### 2.2 Simplex Method

The simplex method is an iterative algorithm suitable for medium-scale linear programming problems.

#### 2.2.1 Creating a Simplex Table

A simplex table is a chart that records the objective function, constraints, and variable information of a linear programming problem. The steps to create a simplex table are as follows:

1. Convert the linear programming problem into standard form.
2. Write the objective function and constraints as an augmented matrix.
3. Convert the augmented matrix into a simplex table.

#### 2.2.2 Iterative Solution Process

The simplex method approximates the optimal solution through iterative calculations. Each iteration includes the following steps:

1. Choose a non-basic variable to enter the basis.
2. Choose a basic variable to leave the basis.
3. Update the simplex table.

The iteration continues until all non-basic variable coefficients are non-negative, at which point the values of the basic variables in the simplex table represent the optimal solution.

#### 2.2.3 Termination Conditions of the Algorithm

There are two termination conditions for the simplex method:

1. All non-basic variable coefficients are non-negative, indicating that the optimal solution has been found.
2. No feasible solution can be found, indicating that the linear programming problem is unsolvable.

### 2.3 Dual Simplex Method

The dual simplex method is an algorithm specifically designed for solvi