The Simplex Method

Fundamental Concepts of Simplex Method

Simplex method is an algorithm applied to solve linear programming problems.
The method utilizes iterative procedures, starting from a feasible solution and attempting to improve it step by step.
It helps in finding the maximum or minimum of an objective function in a feasible region defined by linear inequalities.
The solution lies at the vertices or corners of the feasible region, hence the term ‘vertex method’.

Start with an initial feasible solution.
Identify the entering variable, which is the variable with the most negative coefficient in the objective function.
Identify the departing variable, usually decided by performing the ‘minimum ratio test’.
Perform row operations to create zeros in the column of the entering variable, except the pivot row.
The pivot row must be normalized so that the pivot element becomes 1.
Repeat this process until the optimal solution is reached.

The Simplex method only provides solutions if an optimal solution exists.
If the problem is unbounded (no maximum or minimum value), Simplex cannot provide a solution.
Circumstances can lead to multiple optimal solutions, which happens when more than one point provides the optimal value.
Degeneracy can occur when there are redundant constraints, causing a pivot operation that doesn’t improve the solution.

One major challenge faced while using this method is the cycling issue, where the algorithm keeps repeating the same set of points without finding the optimal solution. This can be tackled by using techniques like Bland’s Rule.
Another challenge is large-scale problems, as the Simplex method requires more computational resources. Advanced variations of the Simplex method, like the Revised Simplex Method or Barrier Method, can be used for these cases.