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109
T
EACHING
S
UGGESTIONS
Teaching Suggestion 9.1:
Meaning of Slack Variables.
Slack variables have an important physical interpretation and rep
resent a valuable commodity, such as unused labor, machine time,
money, space, and so forth.
Teaching Suggestion 9.2:
Initial Solutions to LP Problems.
Explain that all initial solutions begin with
X
1
5
0,
X
2
5
0 (that is,
the real variables set to zero), and the slacks are the variables with
nonzero values. Variables with values of zero are called
nonbasic
and those with nonzero values are said to be
basic.
Teaching Suggestion 9.3:
Substitution Rates in a Simplex Tableau.
Perhaps the most confusing pieces of information to interpret in a
simplex tableau are “substitution rates.” These numbers should be
explained very clearly for the Frst tableau because they will have a
clear physical meaning. Warn the students that in subsequent
tableaus the interpretation is the same but will not be as clear be
cause we are dealing with
marginal
rates of substitution.
Teaching Suggestion 9.4:
Hand Calculations in a
Simplex Tableau.
It is almost impossible to walk through even a small simplex prob
lem (two variables, two constraints) without making at least one
arithmetic error. This can be maddening for students who know
what the correct solution should be but can’t reach it. We suggest
two tips:
1.
Encourage students to also solve the assigned problem
by computer and to request the detailed simplex output.
They can now check their work at each iteration.
2.
Stress the importance of interpreting the numbers in the
tableau at each iteration. The 0s and 1s in the columns of
the variables in the solutions are arithmetic checks and
balances at each step.
Teaching Suggestion 9.5:
Infeasibility Is a Major Problem in
Large LP Problems.
As we noted in Teaching Suggestion 7.6, students should be aware
that infeasibility commonly arises in large, realworldsized prob
lems. This chapter deals with how to spot the problem (and is very
straightforward), but the real issue is how to correct the improper
formulation. This is often a management issue.
A
LTERNATIVE
E
XAMPLES
Alternative Example 9.1:
Simplex Solution to Alternative Ex
ample 7.1 (see Chapter 7 of Solutions Manual for formulation and
graphical solution).
This is not an optimum solution since the
X
1
column contains a
positive value. More proFt remains ($
C\v
per #1).
This is an optimum solution since there are no positive values in
the
C
j
2
Z
j
row. This says to make 4 of item #2 and 8 of item #1 to
get a proFt of $60.
Alternative Example 9.2:
Set up an initial simplex tableau,
given the following two constraints and objective function:
Minimize
Z
5
8
X
1
1
6
X
2
Subject to:
2
X
1
1
4
X
2
ù
8
3
X
1
1
2
X
2
ù
6
The constraints and objective function may be rewritten as:
Minimize
5
8
X
1
1
6
X
2
1
0
S
1
1
0
S
2
1
MA
1
1
MA
2
2
X
1
1
4
X
2
2
1
S
1
1
0
S
2
1
1
A
1
1
0
A
2
5
8
3
X
1
1
2
X
2
1
0
S
1
2
1
S
2
1
0
A
1
1
1
A
2
5
6
9
CHAPTER
Linear Programming: The Simplex Method
1st Iteration
C
j
l
Solution
3
9
0
0
b
Mix
X
1
X
2
S
1
S
2
Quantity
0
S
1
14
1 0
2
4
0
S
2
12
0 1
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 Spring '11
 MichaelHanna

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