Use a simplex tableau

Use a simplex tableau

Simplex Tableau

  • Understand that a simplex tableau is a matrix representation of a linear programming problem. This tool helps you visually manage and solve these problems.

  • Remember that each row in the tableau represents a constraint from the original problem, and each column represents a variable. The last row is an artificial row used to compute profit/loss and to check optimality of the solution found so far.

  • Know how to initialise the simplex tableau. Your objective function becomes the last row of the tableau, and your constraints become the rest of the rows.

Process of Using Simplex Tableau

  • Recognise that the first step involves identifying the pivot column. This is usually the column with the most negative number in the last row.

  • Move on to identifying the pivot row. Do this by dividing each number in the last column of the constraint rows by the corresponding number in the pivot column. The pivot row is usually the row whose resulting number is smallest and positive.

  • Manipulate the pivot row so that the pivot element (the intersection of pivot column and row) becomes 1. This process is called scaling.

  • Use the row operations to make all the other elements in the pivot column to be 0. This can be achieved by subtracting multiples of the pivot row from the other rows.

  • Continue this process in cycling iterations until all the numbers in the last row are nonnegative. When the last row contains only nonnegative figures, it signifies that an optimal solution has been found.

Interpretation of Simplex Tableau

  • Recognise that when the tableau is in optimal form, the values in the bottom row corresponding to the original variables, tell you the optimal values of the original problem.

  • Realise that the last column of the tableau gives the maximum or minimum value of the objective function.

  • Know that multiple or infinite optimal solutions exists, if there are other nonnegative coefficients in the bottom row, besides the one already chosen as pivot column.

  • Understand that no optimal solution exists, if ratio test fails and a pivot column cannot be established.

  • Realise unbounded solutions exist if, in row operation stages, all elements in pivot column are nonpositive.

Pitfalls and Troubleshooting

  • Be cautious and avoid cyclic repetitions of the same tableaus. Look out for signs such as returning to a previously seen tableau.

  • To avoid repetitive cycling, employ strategies like Bland’s Rule, which recommends always choosing the leftmost column as pivot column and amongst the ties in the ratio test, taking the uppermost row as pivot row.