Download Linear algebra with applications The Simplex Method

University of Leeds
School of Mathematics
MATH 1331
Linear algebra with applications
The Simplex Method for linear programming problems in
standard form.
A linear programming problem is said to be in standard form if
• the objective function is to be maximised ;
• each variable has to be > 0;
• all other inequalities are of the form
(linear combination of variables) 6 (positive constant).
A simplex tableau is an augmented m × (n + 1) matrix, with n > m, in which m of the
first n columns are the columns of the identity matrix Im (in some order). These m
columns (and the associated variables) are said to be of Type II; the remainder are of
Type I.
Step 1. Introduce slack variables and restate the problem in terms of a system of linear
equations.
Step 2. Construct the simplex tableau corresponding to the system, with the bottom
row of the matrix corresponding to the objective equation.
Step 3. If there are no negative entries in the bottom row of the unaugmented matrix, then the solution corresponding to the tableau (obtained by setting all the Type I
variables to 0) yields a maximum and the problem is solved.
Step 4. If there are negative entries, construct a new simplex tableau as follows.
(a) Choose the pivot column to be the one containing the most negative element on
the bottom row of the matrix.
(b) Choose the pivot element by computing ratios associated with the positive entries
in the pivot column. The ratio is the element in the augmented column divided by the
corresponding element in the pivot column. The pivot element is the one corresponding
to the smallest positive ratio.
(c) Construct the new simplex tableau by pivoting about the selected element.
Step 5. Repeat Steps 3 and 4 as many times as necessary to find a maximum.
ECL/11–10–2004
How to deal with problems in nonstandard form.
A. If the objective function is to be minimised, replace it by its negative and maximise
that (not forgetting to change the sign back again when the solution has been found).
B. If the inequalities are not of the form
(linear combination of variables) 6 (positive constant),
proceed as follows.
Step 1. Convert all inequalities (except those saying that the variables are positive)
into the form
(linear combination of variables) 6 (constant)
and construct the simplex tableau for the problem.
Step 2. If a negative number appears in the upper part of the last column of the
simplex tableau, remove it by pivoting:
(a) Select a negative entry in its row. The column containing that entry will be the
pivot column.
(b) Select the pivot element by determining the least of the positive ratios associated
with entries in the pivot column (except the bottom entry).
(c) Pivot.
Step 3. Repeat Step 2 until there are no negative entries in the upper part of the right
hand column of the simplex tableau.
Step 4. Proceed to apply the simplex method for a tableau in standard form.
ECL/11–10–2004