Learn & Review: Lec-6 Simplex Method | Maximization Problem | Mathematical Example Solution | Operations Research

Simplex Method for Linear Programming

This lecture series tutorial explains how to solve a linear programming problem (LPP) using the simplex method. The process involves converting the general form of an LPP into its standard form and then iteratively constructing simplex tables to find the optimal solution.

1. Converting General Form to Standard Form

The initial step is to transform the given LPP into a standard form. This involves:

• Objective Function: For a maximization problem, the objective function remains unchanged.
  • Example: Maximize Z = 3x₁ + 2x₂ + 5x₃
• Constraints:
  • Decision variables (x₁, x₂, x₃) are assumed to be non-negative (≥ 0).
  • Inequality constraints of the type '≤' are converted into equations by introducing slack variables. Slack variables represent the unused resources and are added to the left-hand side of the constraint.
    • Example: x₁ + 2x₂ + x₃ ≤ 430 becomes x₁ + 2x₂ + x₃ + s₁ = 430
  • Slack variables (s₁, s₂, s₃) are also added to the objective function with a coefficient of zero.
    • Example: + 0s₁ + 0s₂ + 0s₃
  • All variables (original decision variables and slack variables) must be non-negative.
    • Example: x₁, x₂, x₃, s₁, s₂, s₃ ≥ 0

2. Initial Basic Feasible Solution

After converting to standard form, the initial basic feasible solution is determined:

• The original decision variables (x₁, x₂, x₃) are set to zero.
• The values of the slack variables are determined from the right-hand side of the constraint equations.
  • Example: If x₁=0, x₂=0, x₃=0:
    • s₁ = 430
    • s₂ = 460
    • s₃ = 420

3. Matrix Form

The standard form LPP can be represented in matrix form as Ax = B, where:

• A: Matrix of coefficients of the variables in the constraints.
• x: Vector of variables (including original and slack variables).
• B: Vector of the right-hand side values of the constraints.

4. Constructing the Initial Simplex Table

The simplex table is a tabular representation used to solve the LPP. The initial table includes:

• Cⱼ Row: Coefficients of the variables in the objective function (3, 2, 5, 0, 0, 0 for x₁, x₂, x₃, s₁, s₂, s₃ respectively).
• Basic Variables Column (B): Initially populated with the slack variables (s₁, s₂, s₃).
• Cost of Basic Variables Column (C<0xE2><0x82><0x99>): Costs corresponding to the basic variables. For slack variables, this is 0.
• X<0xE2><0x82><0x99> Column: Right-hand side values of the constraints (430, 460, 420).
• Variable Columns (x₁, x₂, x₃, s₁, s₂, s₃): Coefficients of these variables from the constraint equations.

5. Calculating Zⱼ - Cⱼ and Checking for Optimality

This step involves calculating the Zⱼ - Cⱼ row to determine if the current solution is optimal.

• Formula: Zⱼ - Cⱼ = C<0xE2><0x82><0x99> * Xⱼ - Cⱼ
  • Where C<0xE2><0x82><0x99> is the cost of basic variables, Xⱼ is the column of coefficients for variable j, and Cⱼ is the cost of variable j from the objective function.
• Optimality Condition: The solution is optimal if all values in the Zⱼ - Cⱼ row are greater than or equal to zero (≥ 0).
• Identifying Key Column:
  • If there are negative values in the Zⱼ - Cⱼ row, the solution is not optimal.
  • The column with the most negative Zⱼ - Cⱼ value is identified as the key column. The variable corresponding to this column is the incoming vector.
  • Example: If Zⱼ - Cⱼ values are -3, -2, -5, 0, 0, 0, then -5 is the most negative, making the x₃ column the key column and x₃ the incoming vector.

6. Calculating Minimum Ratio

If the solution is not optimal, the minimum ratio is calculated to determine the key row and outgoing vector.

• Formula: Minimum Ratio = X<0xE2><0x82><0x99> / Key Column
• The calculation is performed for each row where the element in the key column is strictly greater than zero (> 0).
• The row with the minimum positive ratio is identified as the key row. The basic variable in this row is the outgoing vector.
• The element at the intersection of the key row and key column is the key element.
  • Example: If X<0xE2><0x82><0x99> = [430, 460, 420] and the key column (x₃) values are [1, 2, 0]:
    • Row 1 ratio: 430 / 1 = 430
    • Row 2 ratio: 460 / 2 = 230
    • Row 3 ratio: 420 / 0 (not calculated as key element is not > 0)
    • Minimum ratio is 230, making the second row the key row, s₂ the outgoing vector, and 2 the key element.

7. Constructing the New Simplex Table (Iteration)

A new simplex table is constructed for the next iteration:

• Basic Variables: The outgoing vector is replaced by the incoming vector in the Basic Variables column.
• C<0xE2><0x82><0x99> Column: Costs corresponding to the new basic variables are updated.
• Calculating New Values:
  • Key Row: All elements in the old key row are divided by the key element to form the new key row. This makes the key element '1'.
  • Other Rows: For other rows, a formula is used to calculate new values based on the old values, the key row, and the key element. This ensures that the elements in the key column (except for the key element) become zero.
    • Formula (Conceptual): New Value = Old Value - (Corresponding Value in Key Column * New Key Row Value)
    • Alternative Calculation Method (Cross-Multiplication): New Value = (Old Value * Key Element - Corresponding Value in Key Column * Corresponding Value in Old Key Row) / Key Element

8. Repeating the Process

Steps 5 (Calculating Zⱼ - Cⱼ and checking for optimality) and 6 (Calculating Minimum Ratio) are repeated with the new simplex table. This process continues until all values in the Zⱼ - Cⱼ row are non-negative, indicating that the optimal solution has been reached.

9. Optimal Solution

Once the optimal solution is found:

• The values of the original decision variables (x₁, x₂, x₃) are read from the X<0xE2><0x82><0x99> column corresponding to the basic variables.
• The maximum value of the objective function (Z) is calculated by multiplying the costs of the basic variables (C<0xE2><0x82><0x99>) with their corresponding values in the X<0xE2><0x82><0x99> column and summing them up (Z = C<0xE2><0x82><0x99> * X<0xE2><0x82><0x99>).

The lecture concludes by stating that the next session will cover solving minimization problems using the simplex method.