Two Phase Simplex Tableau
The Acme Bicycle Company problem has been altered for this example. The metal finishing machine limit has been changed to the equality constraint x₁ + x₂ = 4. The model is no longer a standard form linear program. This means that we must use the two-phase simplex method. Phase 1 uses the added W objective row to iterate to a solution that is feasible for the original problem, at which point we can solve the Phase 2 problem using the Z objective row.
The two-phase tableau for the altered ABC problem is given below. Note the "-1" in the W column: in Phase 1 we must minimize W.
Solve the problem as follows:
Select the pivot column (the entering basic variable) by clicking on a radio button below a column.
Press the Calculate button to calculate the ratios needed for the Minimum Ratio Test.
Select the pivot row (the leaving basic variable) by clicking a radio button to the right of a row
Repeat until the problem is solved.
Phase 1 is finished when all of the artificial variables are forced to zero, meaning that you have found a point that is feasible for the original problem. Now you ignore the W row and use the Z row to continue iterating. Phase 2 (and the entire solution) is finished when there are no more negative values in the Z objective function row.
Note: There is a tie for the first entering basic variable. If you select x₁ as the first entering basic variable then phase 2 finishes at the same time as phase 1. However, if you select x₂ as the first entering basic variable, then you must also iterate in phase 2 before the optimum is reached. Try both!