Lecture Notes: Solving Maximization Problems Using Simplex Tableau

Introduction

• Objective: Solve a maximization problem using the simplex tableau method.
• Initial Setup:
  • Convert the problem into standard form.
  • Add slack variables to convert inequalities to equalities.
  • Ensure all variables are non-negative.

Simplex Method Overview

• Variables:
  • Two decision variables, additional slack variables.
• Basic Solution:
  • Set two variables to zero and solve for the others.
  • Example: X2 and S1 set to 0 results in X1 = 8 and S2 = -12 (not feasible due to non-negativity).
• Basic Feasible Solution:
  • Set X1 and X2 to 0 to make S1 = 16 and S2 = 12.
  • Feasible as all variables are non-negative.

Constructing the Simplex Tableau

• Components:
  • Cj Row: Objective coefficient row.
  • A Matrix: Coefficients of variables in constraints.
  • B Column: Right-side quantities (initially for basic variables).
• Basic Variables:
  • Slack variables S1 and S2, considered unit columns.
  • Basis column indicates which are basic variables.
• Zj Calculation:
  • Multiply Cb values with corresponding column values, sum the results.
  • Represents unit contribution to the objective function.

Net Evaluation Row (Cj - Zj)

• Indicates net change in objective function if a variable enters the solution.
• Selection:
  • Choose non-basic variable with highest positive value in Cj - Zj row as new basic variable.
  • Key column corresponds to this variable.

Iterations

• Initial Basis:
  • Basic feasible solution with X1, X2 = 0; S1 = 16; S2 = 12.
• Pivot Process:
  • Calculate ratios of B column values to pivot column values.
  • Select row with minimum ratio as pivot row.
  • Perform row operations to create unit column for new basic variable.
• Objective Function Value:
  • Calculated from Zj row.

Conclusion

• Optimal Solution:
  • Achieved when all Cj - Zj values are non-positive.
  • Final solution values: X1 = 2, X2 = 3, Objective Function = 32.
• Graphical Representation:
  • Basic solutions correspond to feasible region's extreme points.
• Steps Recap:
  1. Convert to standard form with slack variables.
  2. Develop initial tableau.
  3. Select non-basic variable with largest Cj - Zj.
  4. Calculate pivot row and perform row operations.
  5. Repeat until optimal solution is achieved.

Summary

• Successfully solved the maximization problem through systematic iterations and tableau adjustments.

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• End of Lecture