# Ch 9 solutions

**Topics:**Linear programming, Optimization, Operations research

**Pages:**157 (6381 words)

**Published:**September 21, 2013

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9

C H A P T E R

Linear Programming: The Simplex Method

TEACHING SUGGESTIONS

Teaching Suggestion 9.1: Meaning of Slack Variables.

Slack variables have an important physical interpretation and represent 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 X1 ϭ 0, X2 ϭ 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 ﬁrst 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 because 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 problem (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, real-world-sized problems. 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.

ALTERNATIVE EXAMPLES

Alternative Example 9.1: Simplex Solution to Alternative Example 7.1 (see Chapter 7 of Solutions Manual for formulation and graphical solution).

1st Iteration

Cj l

b

3

X1

9

X2

0

S1

0

S2

Quantity

S1

S2

1

1

4

2

1

0

0

1

24

16

Zj

Cj Ϫ Zj

0

0

Solution

Mix

0

3

0

9

0

0

0

0

0

3

X1

9

X2

0

S1

0

S2

⁄4

1

⁄2

1

0

⁄4

Ϫ1⁄2

0

1

6

4

⁄4

⁄4

9

0

9

⁄4

Ϫ9⁄4

0

0

54

2nd Iteration

Cj l

b

9

0

Solution

Mix

X2

S2

1

Zj

Cj Ϫ Zj

9

3

1

Quantity

This is not an optimum solution since the X1 column contains a positive value. More proﬁt remains ($C\v per #1).

3rd/Final Iteration

Cj l Solution

b

Mix

3

X1

9

X2

0

S1

0

S2

Quantity

9

3

X2

X1

0

1

1

0

⁄2

Ϫ13⁄2

Ϫ ⁄2

23⁄2

4

8

Zj

Cj Ϫ Zj

3

0

9

0

⁄2

Ϫ ⁄2

⁄2

Ϫ3⁄2

60

1

3

3

1

3

This is an optimum solution since there are no positive values in the Cj Ϫ Zj row. This says to make 4 of item #2 and 8 of item #1 to get a proﬁt of $60.

Alternative Example 9.2: Set up an initial simplex tableau,

given the following two constraints and objective function:

Minimize Z ϭ 8X1 ϩ 6X2

Subject to:

2X1 ϩ 4X2 ജ 8

3X1 ϩ 2X2 ജ 6

The constraints and objective function may be rewritten as:

Minimize ϭ 8X1 ϩ 6X2 ϩ 0S1 ϩ 0S2 ϩ MA1 ϩ MA2

2X1 ϩ 4X2 Ϫ 1S1 ϩ 0S2 ϩ 1A1 ϩ 0A2 ϭ 8

3X1 ϩ 2X2 ϩ 0S1 Ϫ 1S2 ϩ 0A1 ϩ 1A2 ϭ 6

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REVISED

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CHAPTER 9

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LINEAR PROGRAMMING: THE SIMPLEX METHOD

The ﬁrst tableau would be:

Cj l

b

Solution

Mix

8

X1

6

X2

0

S1

0

S2

M

A1...

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