Ch09_109-124 - 72106 CH09 GGS 3:15 PM Page 109 C H A P T E...

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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 0, X 2 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 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, real-world-sized 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 profit remains ($ C\v per #1). This is an optimum solution since there are no positive values in the C j Z j 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 8 X 1 6 X 2 Subject to: 2 X 1 4 X 2 8 3 X 1 2 X 2 6 The constraints and objective function may be rewritten as: Minimize 8 X 1 6 X 2 0 S 1 0 S 2 MA 1 MA 2 2 X 1 4 X 2 1 S 1 0 S 2 1 A 1 0 A 2 8 3 X 1 2 X 2 0 S 1 1 S 2 0 A 1 1 A 2 6 9 C H A P T E R 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 1 4 1 0 24 0 S 2 1 2 0 1 16 Z j 0 0 0 0 0 C j Z j 3 9 0 0 2nd Iteration C j l Solution 3
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