# Graphical and algebraic interpretations of iterations

## Graphical and algebraic interpretations of iterations

### Graphical Interpretation of Iterations

Iteration:

• Recall that iteration in the simplex method is the process of moving from one feasible solution to another until reaching the optimal solution.
• Understand that each iteration corresponds to moving from one vertex (or corner point) of the feasible region to another on the graphical representation of the problem.

Basis Switch:

• Remember that this movement from vertex to vertex in each iteration is equivalent to a basis switch, where one variable enters the basis, and another leaves the basis.

Objective Function Line:

• Consider that during each iteration of the simplex method, the objective function line (line representing the value of the objective function in the graphical method) is moved away from the origin in the direction of improvement.

### Algebraic Interpretation of Iterations

Pivot Operations:

• Understand the algebraic process of each iteration as a series of pivot operations on the simplex tableau.

Basic and Non-basic Variables:

• Recognise that each iteration involves selecting a non-basic (non-pivot) variable to become basic (pivot), and a basic variable to become non-basic.

Feasible Solutions:

• Recognise that each basic feasible solution corresponds to a vertex of the feasible region in the graphical representation of the linear programming problem.

Ratio Test:

• Recall that the ratio test (or minimum ratio test) is used algebraically to determine which basic variable will become non-basic in each iteration.

Optimal Solution:

• Be aware that iteration will continue until the optimal solution is reached, where no further improvement to the objective function is possible. This corresponds to the objective function line being unable to move further away from origin without leaving the feasible region in the graphical interpretation.