Linear Programming (LP) is one of the most important mathematical techniques used in Operations Research (OR) to determine the best possible allocation of limited resources. It is commonly used to maximize profit, minimize cost, optimize production, allocate manpower, schedule activities, and make other quantitative decisions.

Among the different techniques for solving Linear Programming Problems, the Simplex Method is one of the most widely used. Developed by George B. Dantzig in the 1940s, the Simplex Method provides a systematic procedure for moving from one feasible solution to another until the optimal solution is reached.

However, the Simplex Method is not limited to a single procedure. Depending on the structure of the problem, the type of constraints, and the form of the initial solution, several approaches can be used.

This article discusses the major ways of solving Linear Programming Problems using the Simplex Method and explains when each approach is appropriate.

What Is the Simplex Method?

The Simplex Method is an iterative algorithm for solving Linear Programming Problems.

The Simplex Method starts with an initial basic feasible solution (BFS) and systematically moves toward a better solution.

The process continues until there is no further improvement possible.

Simplex stopping rule for maximization

When this condition is satisfied, the current basic feasible solution is optimal.

Important: The stopping rule depends on the tableau convention. Some textbooks use Zj−Cj, in which case the signs are reversed. Always follow the convention used in your table.

Graphical Method vs. Simplex Method

Before discussing the different Simplex approaches, it is useful to distinguish the Graphical Method from the Simplex Method.

The Graphical Method can solve Linear Programming Problems with only two decision variables, because the constraints can be represented on a two-dimensional coordinate system.

The feasible region can be drawn graphically, and the optimal solution can be identified from the corner points.

The Simplex Method, however, can handle problems with many decision variables and constraints, making it much more practical for larger Operations Research problems.

1. The Standard Simplex Method

The Standard Simplex Method is the most commonly taught form of the algorithm.

It is particularly convenient when the constraints are of the form ≤ and the right-hand sides are non-negative.

Step 1: Formulate the LP

Identify:

• Decision variables
• Objective function
• Constraints
• Non-negativity restrictions

Step 2: Convert inequalities into equations

Because the constraints contain ≤, introduce slack variables.

The slack variables convert the inequalities into equations.

Step 3: Construct the initial Simplex tableau

The initial basic variables are normally the slack variables.

Step 4: Determine the entering variable

For a maximization problem using the Cj−Zj convention, identify the largest positive value in the Cj−Zj row.

That variable enters the basis.

Step 5: Determine the leaving variable

Use the minimum positive ratio:

RHS/pivot column.

The smallest positive ratio determines the leaving variable.

Step 6: Perform the pivot operation

The intersection of the entering-variable column and leaving-variable row is the pivot element.

Use row operations to make the pivot element equal to 1 and all other values in the pivot column equal to 0.

Step 7: Repeat

Continue the process for every variable.

The resulting solution is optimal.

2. Big M Method

The Big M Method is used when the initial Simplex tableau cannot be formed directly using only slack variables.

This commonly happens when the LP contains:

• ≥ constraints
• == constraints

These constraints may require artificial variables.

But this does not provide a basic variable that can immediately form the initial basis. Therefore, an artificial variable A1 is introduced:

The Big M Method assigns a very large penalty M to artificial variables.

For a maximization problem, an artificial variable is generally assigned:

−MA1 in the objective function.

The purpose of the large penalty is to force artificial variables out of the final solution.

Advantages of the Big M Method

• Can handle ≥ constraints.
• Can handle equality constraints.
• Can be applied directly using the Simplex tableau.

Disadvantages

• Calculations involving M can become complicated.
• Arithmetic errors are relatively common.
• Large symbolic values can make the tableau difficult to interpret.

3. Two-Phase Simplex Method

The Two-Phase Method is another approach for problems requiring artificial variables.

Instead of assigning a very large penalty M, the method solves the problem in two separate phases.

Phase I: Eliminate artificial variables

The first objective is to minimize the sum of the artificial variables.

If the minimum value of W is greater than zero, the original LP is infeasible.

If: W=0 then a feasible solution to the original problem has been found.

Phase II: Optimize the original objective

After artificial variables have been eliminated, restore the original objective function.

The Simplex process is then continued until the optimal solution is obtained.

Advantages of the Two-Phase Method

• Avoids the large MM calculations.
• Provides a clear feasibility test.
• Generally easier to implement computationally.
• Often preferred for numerical algorithms.

Disadvantage

It requires two stages of computation instead of one.

4. Revised Simplex Method

The Revised Simplex Method is mathematically equivalent to the ordinary Simplex Method, but it does not repeatedly manipulate the entire tableau.

Instead, it focuses on the important matrix components associated with the current basis.

The reduced costs can then be calculated using the current basis.

The Revised Simplex Method is particularly useful for large-scale Linear Programming Problems, because storing and manipulating the entire tableau can be inefficient.

Advantages

• More computationally efficient for large problems.
• Requires less memory than the full tableau approach.
• Forms the basis of many professional LP solvers.

Disadvantage

The mathematics involving matrices and basis inverses can be more difficult for beginners.

5. Dual Simplex Method

The Dual Simplex Method works differently from the ordinary Primal Simplex Method.

In the regular Simplex Method, we generally maintain primal feasibility while improving the objective value.

In the Dual Simplex Method, the algorithm maintains dual feasibility while working toward primal feasibility.

This method is especially useful when a problem begins with a tableau where:

• The solution is not primal feasible.
• The optimality conditions are already satisfied or can be maintained.

A common situation occurs when a constraint is modified after an LP has already been solved.

For example, suppose a company changes the availability of a resource. Instead of solving the entire problem from scratch, the Dual Simplex Method may allow the existing optimal tableau to be efficiently re-optimized.

Advantages

• Useful for sensitivity and re-optimization.
• Efficient when constraints are modified.
• Particularly useful in computer-based LP algorithms.

Primal Simplex vs. Dual Simplex

The two methods can be compared as follows:

Feature	Primal Simplex	Dual Simplex
Main condition maintained	Primal feasibility	Dual feasibility
Starting solution	Usually primal feasible	May be primal infeasible
Main purpose	Improve objective	Restore primal feasibility
Common application	Original LP solution	Re-optimization
Typical learning level	Beginner	Intermediate/Advanced

A useful way to remember the distinction is:

Primal Simplex: Start feasible → improve until optimal.

Dual Simplex: Start optimality-compatible → restore feasibility while maintaining dual feasibility.

Which Simplex Method Should You Use?

The appropriate method depends largely on the structure of the problem.

Problem Situation	Recommended Method
All constraints are ≤\leq and RHS is nonnegative	Standard Simplex
≥\geq or == constraints require artificial variables	Big M or Two-Phase
Avoiding Big M calculations	Two-Phase
Very large LP problems	Revised Simplex
Starting solution is primal infeasible but dual feasible	Dual Simplex
Re-optimization after changing constraints	Dual Simplex
Only two decision variables	Graphical Method may be simpler

A Simple Decision Guide

You can use the following decision process when solving an LP:

Step 1: Look at the constraints

If all constraints are ≤ with nonnegative RHS values, the Standard Simplex Method is usually the easiest choice.

Step 2: Check for ≥ or ==

If the problem contains that, you may need artificial variables.

Then choose between:

• Big M Method
• Two-Phase Method

Step 3: Consider the size of the problem

For a small classroom problem, the tableau approach is usually convenient.

For a large computational problem, the Revised Simplex Method is more appropriate.

Step 4: Check whether the problem is already close to an optimal solution

If a previously solved LP has been modified and the existing solution becomes infeasible, the Dual Simplex Method may be particularly efficient.

Common Mistakes When Using the Simplex Method

Students often make mistakes not because the Simplex concept is difficult, but because several decisions must be made correctly at every iteration.

Mistake 1: Choosing the wrong entering variable

Always check whether your textbook uses:

Cj−Zj or Zj−Cj

The entering-variable rule changes depending on the convention.

Mistake 2: Using negative ratios

The minimum-ratio test should consider only positive coefficients in the pivot column.

Mistake 3: Forgetting the non-negativity condition

The standard LP assumption is xi≥0 unless the problem explicitly states otherwise.

Mistake 4: Stopping too early

For the Cj−Zj maximization convention, the solution is not optimal if there is still a positive value in the Cj−Zj row.

Mistake 5: Ignoring artificial variables

When using Big M or Two-Phase, artificial variables must not remain in the final solution with positive values.

How Do You Know When to Stop?

This is one of the most important concepts in the Simplex Method.

For a maximization problem using the Cj−Zj convention:

Stop when there are no positive values in Cj−Zj.

In other words:

Cj−Zj≤0 for every column.

At that point, the current basic feasible solution is optimal.

For a minimization problem, the stopping condition depends on the formulation and tableau convention being used. Therefore, it is important to establish the convention before applying the stopping rule.

What Happens If There Is No Optimal Solution?

The Simplex Method can also reveal special conditions.

Unbounded Solution

If an entering variable can improve the objective function but there are no positive coefficients in its pivot column, the problem may be unbounded.

Infeasible Solution

In the Big M or Two-Phase Method, if artificial variables cannot be eliminated appropriately, the original LP may be infeasible.

Multiple Optimal Solutions

If the optimal tableau contains a non-basic variable with zero reduced cost, there may be multiple optimal solutions.

Degeneracy

If the minimum-ratio test produces a zero or tied ratio, the problem may exhibit degeneracy.

These conditions are important because not every LP has one unique, finite optimal solution.

Why Learn Multiple Simplex Methods?

Learning multiple versions of the Simplex Method is valuable because different LP structures require different strategies.

The Standard Simplex Method is ideal for understanding the fundamental mechanics.

The Big M Method provides a way to handle artificial variables using penalty terms.

The Two-Phase Method handles artificial variables through a separate feasibility phase.

The Revised Simplex Method provides a more efficient computational formulation.

The Dual Simplex Method is particularly valuable for re-optimization and problems where primal feasibility is initially violated.

Together, these methods provide a much more complete understanding of how Linear Programming is solved in Operations Research.

Conclusion

The Simplex Method is more than a single table-based calculation. It is a family of related techniques designed to efficiently solve Linear Programming Problems under different conditions.

The most important approaches to remember are:

1. Standard Simplex Method – for straightforward LPs with ≤ constraints.
2. Big M Method – for problems requiring artificial variables using a large penalty M.
3. Two-Phase Method – separates feasibility from optimization.
4. Revised Simplex Method – uses matrix calculations and is suitable for large problems.
5. Dual Simplex Method – particularly useful for re-optimization and situations involving primal infeasibility.

The key is not simply to memorize the tableau operations. A good Operations Research practitioner should first understand why a particular Simplex approach is appropriate for a given Linear Programming Problem.

Once the structure of the LP is identified, choosing the correct method becomes much easier.

In short:

Understand the LP → Identify the appropriate Simplex approach → Construct the initial solution → Iterate → Check the optimality condition → Interpret the solution.

That systematic thinking is what makes the Simplex Method one of the most powerful tools in Operations Research.