The concept of linear programming and its solution method “simplex” was first introduced by George Dantzig.

The solution space is presented by m linear equations and n variables.

The majority of linear programs have a greater number of variables compared
equations n>m. This yields an infinite number of solutions, which creates the solution space. 

A slack or surplus is utilized to reconstruct a constraint to obtain equation format.

A slack variable represents the remainder of an expendable resource and applies to less than or equal constraints.

A surplus represents the excess amount of a restricted resource and applies to greater than or equal constraints.

If two corner-point feasible solutions are connected by a line segment, these cornerpoint feasible solutions are adjacent. Two corner-point feasible solutions are adjacent if all but one of their respective variables have the same value.

A corner-point feasible solution is the optimum corner-point feasible solution only if it has no adjacent corner-point feasible solutions with a more desirable Z value.

Iteration is a repetition of a mathematical procedure applied to the result of a previous application.

An iterative algorithm aims to obtain successively closer approximations to the
solution.

• The method solely focuses on CPF solutions, i.e., BF solutions.
• It is an iterative algorithm, as described in Figure 5.2.
• The preferred initial BF solution is the CPF solution at the origin, where all decision variables are set to zero.
• Optimality test subject changes from a BF solution to an adjacent one.
• The selection between the adjacent BF solutions is based on the improvement in Z.
• If there are many adjacent BF solutions that improve the Z, the one providing the largest
improvement is selected.
• If none of the adjacent BF solutions improve Z, the current BF solution is optimal.

Similar to the definition of adjacent CPF solutions, two BF solutions are adjacent if their basic variables are equal except one. As the basic and non-basic variables are interchangeable only with each other, two adjacent BF solutions have the same non-basic variables with a single exception. Inherently, all but one
of their basic variables are the same. 

The criteria is the rate of improvement of Z. The coefficients of the variables in the objective function refer to the effect of one unit increase of respective variable on the function output Z.

The one that first decreases to zero as the value of entering variable increases

• Initialization
• Test for optimality
• Determining the entering variable
• Determining the leaving variable

Initialization is an augmentation of the model by slack, surplus and artificial variables to obtain a system of equations.

As a compact form of the model, the simplex tableau provides the essential information required by the simplex method.

Identity matrix of n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. An identity matrix is also called a unity matrix.

The minimum value means limiting the entering variable the most.

1. Take each positive coefficient in the pivotal column.
2. Divide each of these into the right-hand side constant for the corresponding row.
3. Identify the row that has the minimum ratio.
4. Pivotal row is the row that is identified in the previous step.

The value at the intersection of the pivotal column and row is called the pivotal number. It is alternatively called as pivot number or pivot element.