Simplex Tableaux

Optimization · Axiom Academy

Understanding the matrix representation of linear programs A simplex tableau is organized with: Columns for decision variables and slack variables RHS (right-hand side) column for constant values Objective row at the bottom showing reduced costs Basic variables correspond to identity columns in the tableau (columns with exactly one 1 and all other entries 0). The row with the 1 determines which constraint defines that variable's value. 3 Reading the Current Solution To extract the current solution from a tableau: For each basic variable, read the RHS value in its identity column row All non-basic variables equal zero The objective value is in the bottom-right corner (with sign convention) The pivot operation transforms the tableau to move to an adjacent vertex by swapping a basic and non-basic variable. Steps: Step 1: Identify the pivot element (entering column, leaving row) Step 2: Divide the pivot row by the pivot element Step 3: Use row operations to zero out other entries in the pivot column For a maximization problem , the current solution is optimal when: All reduced costs (entries in the objective row, excluding RHS) are non-positive. If any entry is positive, the solution can be improved.

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