4.1 Simplex algorithm and tableau

The Simplex algorithm is a powerful tool for solving linear programming problems. It uses a systematic approach to find optimal solutions by iteratively improving the objective function value. The algorithm relies on the Simplex tableau, a matrix representation that organizes all necessary information for efficient problem-solving.

The Simplex tableau provides a visual representation of the linear program, facilitating easy manipulation and analysis. It allows for tracking solution progress, identifying basic and non-basic variables, and detecting optimality conditions. Understanding the tableau structure and manipulation techniques is crucial for effectively applying the Simplex method.

Simplex Tableau Structure

Components and Layout

• Matrix representation of a linear programming problem containing all necessary information for solving with simplex algorithm
• Columns represent variables (decision, slack, and surplus variables)
• Rows represent constraints and objective function
• Rightmost column "right-hand side" (RHS) contains constant terms of constraints and current objective function value
• Bottom row "index row" or "z-row" represents coefficients of objective function used to determine optimality
• Body contains coefficients of variables in each constraint equation
• Identity matrix for slack/surplus variables initially occupies right side of tableau
• Column indicating basic variables currently in solution labeled as "basis" or "basic variable" column

Interpretation and Significance

• Provides visual representation of linear program facilitating easy manipulation and analysis
• Allows tracking of solution progress through iterations of simplex algorithm
• Enables quick identification of basic and non-basic variables
• Supports efficient pivoting operations to move between basic feasible solutions
• Simplifies detection of optimality, unboundedness, or infeasibility conditions
• Facilitates sensitivity analysis by providing shadow prices and reduced costs

Solving Linear Programs

Simplex Algorithm Steps

• Iterative process moving from one basic feasible solution to another improving objective function value
• Convert linear programming problem into standard form adding slack or surplus variables for equality constraints
• Identify initial basic feasible solution setting decision variables to zero and solving for slack/surplus variables
• Select entering variable identifying most negative coefficient in index row (maximization) or most positive (minimization)
• Determine leaving variable using minimum ratio test ensuring solution remains feasible after pivot operation
• Perform pivot operation updating tableau replacing leaving variable with entering variable in basis
• Repeat process of selecting entering and leaving variables and pivoting until reaching optimality or detecting unboundedness

Tableau Manipulation Techniques

• Gaussian elimination used for pivoting operations
• Maintain tableau in row echelon form throughout iterations
• Update RHS column to reflect changes in basic feasible solution
• Recalculate index row after each pivot to evaluate optimality conditions
• Track basis column changes to identify current basic variables
• Handle degeneracy by implementing anti-cycling rules (Bland's rule)

Basic vs Non-basic Variables

Characteristics and Identification

• Basic variables represented by columns with single '1' entry and all other entries '0' in tableau
• Non-basic variables have multiple non-zero coefficients in respective columns or not listed in basis column
• Number of basic variables always equal to number of constraints in original problem (excluding non-negativity constraints)
• Basic variables correspond to variables with non-zero values in current basic feasible solution
• Non-basic variables typically set to zero in current solution
• Pivoting exchanges basic variable with non-basic variable to improve objective function value

Significance in Simplex Method

• Basic variables form basis for current solution providing geometric interpretation of solution space
• Non-basic variables represent potential directions for improvement
• Entering variable selection focuses on non-basic variables with favorable reduced costs
• Leaving variable selection ensures feasibility maintained when basic variable exits basis
• Tracking basic and non-basic variables helps identify alternate optimal solutions
• Understanding relationship between basic and non-basic variables crucial for interpreting sensitivity analysis results

Optimality Conditions

Identifying Optimal Solutions

• Maximization problems reach optimality when all coefficients in index row (z-row) are non-negative
• Minimization problems achieve optimality when all coefficients in index row are non-positive
• Optimal solution values read directly from RHS column for basic variables listed in basis column
• Multiple optimal solutions exist if non-basic variable has zero coefficient in index row (reduced cost of zero)
• Unboundedness detected if column selected to enter basis has all non-positive entries in constraint rows indicating no upper bound on improvement
• Infeasibility identified if artificial variables remain in basis with non-zero values after first phase of two-phase simplex method

Interpreting Optimal Tableau

• Shadow prices (dual variables) obtained from coefficients of slack/surplus variables in final optimal tableau's index row
• Reduced costs of non-basic variables indicate potential improvement in objective function per unit increase
• Basis inverse (B^-1) can be extracted from columns corresponding to slack/surplus variables
• Sensitivity analysis performed using optimal tableau to determine allowable changes in objective function coefficients and RHS values
• Degeneracy at optimality indicated by presence of zero values in RHS column for basic variables
• Complementary slackness conditions verified by examining product of primal and dual variable values