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230 CHAPTER 5 LINEAR PROGRAMMING: THE SIMPLEX METHOD

NOTES AND COMMENTS

he entries in the net evaluation row provide the improve before it would be possible for the corre-
T reduced costs that appear in the computer sol- sponding variable to assume a positive value in the
ution to a linear programme. Recall that in Chapter 3 optimal solution. In general, the reduced costs are
we defined the reduced cost as the amount by which the absolute values of the entries in the net evalua-
an objective function coefficient would have to tion row.

5.6 Tableau Form: The General Case

This section explains When a linear programme contains all less-than-or-equal-to constraints with nonneg-
how to get started with
the Simplex method for ative right-hand side values, it is easy to set up the tableau form; we simply add a slack
problems with greater- variable to each constraint. However, obtaining the tableau form is more complex if the
than-or-equal-to and linear programme contains greater-than-or-equal-to constraints, equality constraints
equality constraints.
and/or negative right-hand side values. In this section we describe how to develop
tableau form for each of these situations and also how to solve linear programmes
involving equality and greater-than-or-equal-to constraints using the Simplex method.

Greater-Than-or-Equal-to Constraints (‡)

Suppose that in the HighTech Industries problem, management wanted to ensure that
the combined total production for both models would be at least 25 units. This require-
ment means that the following constraint must be added to the current linear programme:
1x 1 þ 1x 2 25
Adding this constraint results in the following modified problem:

Max 50x 1 þ 40x 2
s:t:
3x 1 þ 5x 2 150 Assembly time
1x 2 20 UltraPortable display
8x 1 þ 5x 2 300 Warehouse space
1x 1 þ 1x 2 25 Minimum total production
x 1 ; x 2 0
First, we use three slack variables and one surplus variable to write the problem in
standard form. This provides the following:

Max 50x 1 þ 40x 2 þ 0s 1 þ 0s 2 þ 0s 3 þ 0s 4
s:t:
3x 1 þ 5x 2 þ 1s 1 ¼ 150 (5:9)
1x 2 þ 1s 2 ¼ 20 (5:10)
8x 1 þ 5x 2 þ 1s 3 ¼ 300 (5:11)
1x 1 þ 1x 2 1s 4 ¼ 25 (5:12)
x 1 ; x 2 ; s 1 ; s 2 ; s 3 ; s 4 0

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