Supplement
14
Operational Decision-Making Tools:
Linear Programming
LECTURE OUTLINE
A. Model Formulation
1. Decision variables
2. Objective function
3. Constraints
B. Graphical Solution Method
1. Identify the feasible solution space
2. Identify extreme points
3. Identify optimal solution
4. Sensitivity analysis
C. Linear Programming Model Solution
1. The Simplex Method
2. Slack and Surplus Variables
D. Solving Linear Programming Problems with Excel
E. Sensitivity Analysis
F. Summary
G. Summary of Key Terms
SUGGESTED VIDEOS
Graphical Solution
LP Graphical Solution: http://www.youtube.com/watch?v=M4K6HYLHREQ
LP Graphical Solution: http://www.youtube.com/watch?v=__wAxkYmhvY
LP Graphical Solution: http://www.youtube.com/watch?v=pzgnUCFNN7Q
Tomato LP Example
Example Part 1: http://www.youtube.com/watch?v=f0PS8OwXqcw&feature=related
Example Part 2: http://www.youtube.com/watch?v=LqWq2qNpGyI&feature=related
Example Part 3: http://www.youtube.com/watch?v=M_0jQJ-Ey6c&feature=related
EXCEL FILES
Topic
Reference
Exhibit S14.1.xls
Exhibit S14.4.xls
Example S14.1
Example S14.1 – Sensitivity Analysis
These Excel file, along with Web links, Internet Exercises, and Virtual Tours can be found on the text’s Web
site at http://www.wiley.com/college/russell.
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PAUSE AND REFLECT
1.
This exercise is related to getting students to “feel” what a linear programming solution should look like, to
help them sort out feasible combinations from non-feasible solutions. Give students a relatively
straightforward problem (but preferably with three products, not two). Problem S14-4, Pinewood Cabinet,
is a good starting point. Have them identify the decision variables, write out the objective function and the
constraints. Without having them solve the problem, have them compete to find “intuitive” answers—
solutions that are feasible, but not necessarily optimal. Such answers must be specific—detailing exact
amounts of each product involved. Award points for earlier responses, and for better responses. This can
be done with individual students, or it can be done in teams. Often, one or more students will stumble upon
the optimal solution. This is best done in class, not over a Web discussion, to prevent students from solving
the problem without the instructor’s knowledge. Solving the problem with Excel afterward is good
reinforcement.
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MAPPING LEARNING OBJECTIVES TO ASSIGNMENTS
#
Topic
Learning Objective
Problems
Other
Questions
S14.1
MODEL
FORMULATION
formulate a basic linear
programming model including
the definition of decision
variables, objective function,
and constraints
1, 2
3, 5, 6, 7
1, 7, 15, 16, 17,
18, 19, 20, 21
solved
problem 1,
case
problem 1
8
15
case
problem 1
S14.2
GRAPHICAL
SOLUTION METHOD
determine the optimal solution
for a linear programming
model with two decision
variables using the graphical
solution method
S14.3
LINEAR
PROGRAMMING
MODEL SOLUTION
calculate unused or surplus
resources given a model
solution and provide an
overview of the simplex
method
S14.4
SOLVING LINEAR
PROGRAMMING
PROBLEMS WITH
EXCEL
solve a linear programming
model using excel
S14.5
SENSITIVITY
ANALYSIS
(1 through 14)
(18 through 46)
(1 through 14),
(18 through 46)
solved
problem 1,
case
problems 1,
2, 3, 4, 5, 6
15, 20, 21, 25,
26, 27, 31, 32,
33, 34
case
problems 1,
5
16,
interpret the sensitivity report
from an excel linear
programming solution and
perform sensitivity analysis
4
solved
problem 1,
case
problems 1,
2, 3, 4, 5, 6
TEACHING NOTES
Teaching Note 1—Students may have seen linear programming in other courses
Students may have been exposed to some linear systems in their mathematics courses. They may have seen some
linear programming in a prior quantitative methods course. The instructor should inquire about prior learning in
this area.
Teaching Note 2—Linearity needs explaining
Students do not readily get the concept of “linearity,” especially in the case of constraints. Point out that linearity
in the objective means that “profit” per unit cannot vary, and that the products represented in the objective
function are fully independent of one another, except that they compete for the same resources. Point out that
linearity means no roots, powers, radicals, reciprocals, or ratios. Point out that linearity means that the “recipe”
for making one unit of a product does not change—that there’s no learning curve, no economies of scale, no
quantity discount, etc.
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Teaching Note 3—LP needs to be more applied, less abstract
Students often treat linear programming as an abstract, logical exercise. They need to be provided with real-world
uses of LP.
Teaching Note 4—Some LP outcomes are not intuitive; practice helps
Students are not prepared for some of the outcomes of linear programming problems. They expect all of every
resource to be used up. They expect all decision variables to be included in the mix. They expect the most
profitable product to be produced to its maximum quantity. They expect the number of constraints to
coincidentally match the number of variables. The instructor will need to explain why these things do not
necessarily occur.
Students are especially skeptical when the issue of what happens when two resources are scarce, but only one
resource is added to. Students want the answer to be “you can’t use the added resource because you’re still out of
the other.” The instructor has to emphasize the notion of altering the mix to accommodate the single added
ingredient. Letting students experiment with Excel can also help here.
And, finally, they definitely do not appreciate the quantum nature of LP solutions—that a physical solution holds
unchanged until a ranging crossover point is reached, at which time the solution “bounces” to another extreme
point. This is perhaps best illustrated with a two-dimensional problem done in Excel. Illustrate by making
changes in the objective function within, even close to, the ranging limits to show that while the objective
changes, the physical quantities do not. Then move barely beyond the crossover point, to show the sudden
movement to another corner.
Teaching Note 5—Active learning exercise to make LP more comfortable
This exercise is related to getting students to “feel” what a linear programming solution should look like, to help
them sort out feasible combinations from non-feasible solutions. Give students a relatively straightforward
problem (but preferably with three products, not two). Have them identify the decision variables, write out the
objective function and the constraints. Without having them solve the problem, have them compete to find
“intuitive” answers—solutions that are feasible, but not necessarily optimal. Such answers must be specific—
detailing exact amounts of each product involved. Award points for earlier responses, and for better responses.
This can be done with individual students, or it can be done in teams. Often, one or more students will stumble
upon the optimal solution. Solving the problem with Excel afterward is good reinforcement.
Teaching Note 6—Active learning to accompany Excel solutions
Students are skeptical of Excel outputs, whether it is the results or the ranging tables. Point out how they can
verify by hand calculation, how much of a resource has been used up, and how much is left over. Point out how
they can verify the amount of the objective function.
In ranging analysis, students will complain that they don’t understand what the upper and lower bounds mean.
Explain that the ranging table is really two tables in one. The first is a sub-table with one row for each decision
variable, for examining changes in physical output quantities when the objective coefficients are varied. The
second sub-table has one row per constraint, for examining changes in dual values when quantities of resources
are varied.
Also, invite students to perform sensitivity analysis, not by printing the ranging table, but by making single, small,
changes in the problem. If the ranging table states that a dual value rises when a resource is reduced to an amount
X, have the students see for themselves by editing the problem and re-solving.
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Teaching Note 7—Students need extra help with certain constraints
Students have a really hard time with proportionality constraints, or constraints that pit one decision variable
against another. Constraints like “At least 20 percent of all units produced must be nimnots,” or like, “The
number of widgets must be at least as large as the number of gadgets.” Students will stumble on the algebraic
representations of these. The instructor may need to remind them that in constraints, only variables occupy the
left side of an expression, and only constants occupy the right. The instructor may also need to remind students
that negative coefficients are OK
ALTERNATE EXAMPLE
Alternate Example 1
Noah Karks makes three varieties of moderate-priced table wines: Red, White, and Rosé. The raw materials,
labor, and profit contribution per case of each of these wines is summarized below.
Wine
Red
White
Rosé
Grapes - A
(bushels)
4
0
2
Grapes - B
(bushels)
0
4
2
Sugar
(lbs)
1
0
2
Labor
(man hours)
3
1
2
Contrib.
(per case)
$14
$27
$20
The winery has 1800 bushels of Variety A grapes, 840 bushels of Variety B grapes, 800 pounds of sugar, and 760
man-hours of labor available during the next season. The firm operates to achieve maximum contribution.
Problem Solution
This problem has three decision variables—the quantities of Red, White, and Rose to be produced this season,
and an objective function of maximizing profit.
The problem has four constraints, representing the four “ingredients” of the wines: Variety A grapes, Variety B
grapes, Sugar, and Labor. The problem is linear, because there is no interaction among variables, there are no
roots, powers, or ratios in the objective or the constraints. Also, the “recipe” does not change with quantity
produced.
So far, so good. The problem appears to be a linear programming problem—there are decision variables, an
objective function, constraints, and linearity. These expressions are detailed below.
Objective function
Max Z = $14xRed + $27xWhite +$20xRose
Subject to:
Variety A Grapes constraint
4xRed + 0xWhite + 2xRose <= 1200
Variety B Grapes constraint
0xRed + 4xWhite + 2xRose <= 840
Sugar constraint
1xRed + 0xWhite + 2xRose <= 800
Labor constraint
3xRed + 1xWhite + 2xRose <= 600
Since there are three variables in this problem, it was solved using Excel Solver. According to the solution, Noah,
if he is to maximize profit, should produce 130 cases of Red wine, 210 cases of White wine, but no cases (zero,
nada, zilch) of the Rose. Noah, if he implements this solution, will earn $7,490.
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The solution reveals what resources have not been used up. 680 bushels of variety A grapes are left over, but no
variety B grapes are left over. Also 670 pounds of sugar remain, while no labor is unused. (Remember that
there’s room for only one variable in the solution per constraint to the problem.)
Sensitivity analysis reveals what happens if small changes are made to the problem. It also reveals why the Rose
wine was not included in the solution. No Rose wine would be produced unless its contribution were larger than
$20.5. Twenty dollars is close, but not enough! (If you’re skeptical, change the contribution from $20 to $21 and
see for yourself!)
For Variety B grapes, if Noah could have one more bushel of Variety B grapes to use in determining the
optimal solution, that solution would have its contribution increase by $5.5833. (Don’t believe it? Make
the change and see for yourself!)
Note the following characteristics of the problem: Only two of the three decision variables were included
in the optimal solution. Not all of every resource was used up. The product with the lowest contribution
was NOT the variable omitted from the solution. And finally, more of Variety B grapes increases
contribution even though Noah also uses up all of the labor.
CHAPTER QUIZ
1.
A slack variable is added to a constraint to
a. take the place of a decision variable
b. show the profit contribution of a resource constraint
c. offset the use of a surplus variable
d. turn an inequality into an equality
2.
The dual value is the
a. marginal value of an additional unit of a resource
b. the cost of resources
c. the cost of a slack variable
d. the total cost of all resources
3.
The graphical solution method
a. is not a practical linear programming solution method
b. is limited to a problem with two decision variables
c. provides insight into how linear programming problems are solved
d. all of the above
4.
Which of the following is not true.
a. The objective function and constraints are linear relationships.
b. The model parameters are assumed to be known with certainty.
c. Problems with two decision variables can be solved graphically.
d. The solution values will be integers.
Answers:
d, a, d, d
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