Explain the concept and computational steps of the simplex method for solving linear programming problems

Explain the concept and computational steps of the simplex method for solving linear programming problems. How would you identify whether an optimal solution to a problem obtained using simplex algorithm is unique or not?

a) What is the difference between a feasible solution, a basic feasible solution, and an optimal solution of a linear programming problem?

b) What is the difference between simplex solution procedure for a maximization’ and aminimization’ problem?

c) Using the concept of net contribution, provide an intuitive explanation of why the criterion for optimality for maximization problem is different from that of minimization problems. Outline the steps involved in the simplex algorithm for solving a linear programming maximization problem. Also define the technical terms used therein.

The correct answer and explanation is :

Simplex Method for Solving Linear Programming Problems

The simplex method is an iterative procedure used to solve linear programming problems (LPPs). It optimizes a linear objective function subject to linear equality and inequality constraints. The computational steps involve:

1. Formulating the Problem: Convert constraints into standard form with slack, surplus, and artificial variables.
2. Setting Up the Initial Tableau: Construct the simplex tableau, representing the system of equations in tabular form.
3. Checking Optimality: Identify the entering variable (with the most negative coefficient in the objective row).
4. Determining the Leaving Variable: Compute the ratio test to determine the exiting variable.
5. Pivoting: Perform row operations to update the tableau and maintain feasibility.
6. Iterate Until Optimality: Repeat steps 3-5 until all coefficients in the objective row are non-negative (for maximization problems).
7. Identifying Uniqueness: If a non-basic variable has a zero reduced cost, multiple optimal solutions exist.

(a) Types of Solutions

• Feasible Solution: Satisfies all constraints but may not be optimal.
• Basic Feasible Solution (BFS): A solution corresponding to a basic variable set (equal to the number of constraints).
• Optimal Solution: The BFS that maximizes or minimizes the objective function.

(b) Maximization vs. Minimization

• For maximization, the entering variable is the one with the most negative coefficient in the objective row.
• For minimization, the most positive coefficient is chosen.

(c) Net Contribution and Optimality Criterion

For maximization, a variable enters the basis if it improves the objective function (negative reduced cost). For minimization, a variable enters if it decreases cost (positive reduced cost).

Steps of the Simplex Algorithm (Maximization)

1. Convert constraints into standard form.
2. Construct the initial tableau.
3. Identify the entering variable (most negative cost coefficient).
4. Determine the leaving variable (minimum ratio test).
5. Perform pivoting to update the tableau.
6. Repeat until all cost coefficients are non-negative.
7. Read off the optimal solution.

Key Terms

• Pivot Element: The element used to normalize the tableau.
• Reduced Cost: Measures potential improvement in the objective function.
• Basis Variables: Variables currently included in the solution.

This method ensures a stepwise improvement towards the optimal solution.