41. It is anticipated that steel production at a new plant in
Bethlehem, Pennsylvania, will generate approximately
50,000 gallons of raw sewage per hour that must be
treated at a local treatment facility. The plant plans to
use excess capacity on existing pipes. Will the existing
system of pipes between pumping stations be sufficient
42. The Wichita State University (WSU) baseball team is
preparing for the upcoming college world series. It has
two games left in the regular season, followed by at least
two games in the double elimination World Series
tournament. WSU has already played three of the four
teams—Texas (UT), Arizona State (ASU), and Florida
State (FSU)—and is very familiar with its other
opponent, California State University, Fullerton
(CSUF).
WSU has four starting pitchers and will start a
different one against each team. Based on past
performance, the WSU coach has compiled an
effectiveness statistic for each pitcher based on the
pitcher’s and the opponent team’s strengths and
weaknesses. He has used these statistics throughout the
year, which may account for his successful 45–12
won–lost record. The effectiveness statistics for these
opponents are as follows.
Effectiveness Factors
UT ASU FSU CSUF
Clyde Rollins 62 65 80 50
Carlos Pascual 76 70 82 55
Sid Thompson 75 40 77 57
Ted Quillici 45 48 50 36
a. Based on these factors, which pitcher should WSU
start against each team to maximize the total overall
effectiveness rating?
b. Suppose the WSU coach will let a pitcher start up to
two games. Modify the problem and solve first as an
assignment problem and then as a transportation
problem.
c. Use the transportation model format to determine
the starting pitchers if the WSU coach will allow a
pitcher to start as many as three games.
43. During the early 1970s, the political scandal Watergate
shook the United States and toppled a presidency. While
there were many aspects to the episode (robbery,
enemies lists, abuse of power, cover-ups, etc.), a key
component was the “laundering” of funds from big
money contributors to campaign coffers. This practice
consists of channeling a large “gift” of money through
various banks and individuals so that its source cannot be
traced. Unfortunately, such activities continue today as
evidenced by congressional investigations beginning in
1997.
Suppose millionaire I. S. Halverson has $5000 (in
reality, he would probably have 10 or 100 times this
amount) that he would like to donate “anonymously” to
the Independent National Party (INP). He might first
split the money up in smaller units and deposit the
money in several bank accounts spread throughout the
world. Money from these accounts could be mixed or
further divided and sent to other accounts or individuals,
who, in turn, would do the same, until several checks for
$1000 or less eventually arrive at party headquarters.
To avert suspicion, a limit has been placed on the
amount of each transaction between intermediaries.
These limits are given in the network at the top of the
next page depicting I. S. Halverson, the intermediaries,
and the INP. Given these limitations, how much of the
$5000 can I. S. Halverson launder to the INP?
(Note: The federal government employs management
scientists who also use such models to help determine
transaction limits that should be monitored.)
to support this operation, or will additional piping
capacity be required? (The numbers give the maximum
number of thousands of gallons per hour possible through
each pipe.) Sewage can flow in either direction between
Stations 1 and 2 and Stations 4 and 5.
Additional Problems/Cases CD-465
Chapter 4 Extra Problems/Cases
Steel Plant Treatment
Facility Station 3
Station 1 Station 4
Station 2 Station 5
35 5 55
25 15
10
10
45
35
20 10
80
Problem 41
44. Luxor Motorhomes has two plants, one in Riverside,
California, and the other in Des Moines, Iowa. Each
plant can produce three different models: the Grand
Cruiser, the Traveler, and the Weekender. Labor time
at the Riverside plant limits production to 600 models
per month, while the Des Moines plant can produce up
to 1000 models per month. The manufacturing costs and
monthly production capacities for each model vary,
depending on the plant. These costs are summarized in
the following table.
Manufacturing Costs and Maximum Monthly
Production Levels
Riverside Des Moines
Manufacturing Cost
Grand Cruiser $53,000 $50,000
Traveler $29,000 $27,000
Weekender $18,000 $17,000
Maximum Monthly Production
Grand Cruiser 200 400
Traveler 500 500
Weekender 600 900
Once the units are manufactured, they are shipped to
central distribution locations in Florida, Texas, and
California, where they are ultimately purchased by
retailers. The demand for motorhomes at the
distribution locations for this month’s production is as
follows.
Demand for Motorhomes
Florida Texas California
Grand Cruiser 100 50 150
Traveler 200 100 300
Weekender 225 175 250
The transportation costs for shipping a motorhome
from a plant to a distribution center are independent of
the model. These are given in the following table.
Motorhome Shipping Costs
Florida Texas California
Des Moines $1000 $800 $1200
Riverside $2000 $700 $ 300
Formulate this problem as a capacitated
transshipment problem and solve for the optimal
production and distribution of motorhomes during this
month.
(Hint: Define a set of nodes for the plants, a set for the
models, and a set for the models at the distribution
locations.)
45. The Texas Education Association wishes to hold its
annual meeting in one of three cities: Dallas, Austin, or
Abilene. Representatives from 21 different school
districts will attend, including representatives from the
three possible host cities. The driving distances in miles
are given in the network for problem 45 on the next
page. Assume that each attendee will drive the shortest
route from his or her city to the meeting site.
a. Which site should be selected if the goal is to
minimize the maximum driving distance of any
attendee?
b. Which site should be selected if the goal is to
minimize the average driving distance of all the
attendees?
46. The small rural town of Campton has only one
elementary school. Beginning early every morning, a
school bus leaves the school, picks up children at four
stops, and returns to the school. The table for problem
46 gives the distances between the stops.
Problem 46 Miles Between Pickup Points
Willow General Old
Crossroad Creek Store Highway
School 6 29 24 10
Crossroad 19 21 20
Willow Creek 5 27
General Store 16
CD-466 Chapter 4 Extra Problems/Cases
6 1
2
2000
2000
250
1500
1000
1000
1000
1000
1000
2000
100
300
200
400
3
4
5
INP
I.S. Halverson
Numbers on the arcs represent the maximum amount that can be laundered in either direction.
Problem 43
49. Topless City is a small chain of car dealerships that
sells vintage convertibles throughout the Southern
United States. It is owned and managed by Brandon
and Kyle Winslow. Each month Brandon and Kyle
attend two car auctions, at which they purchase
convertibles: one in Atlanta, the other in Miami. The
cars are then shipped to one of three locations: Jackson,
Mississippi, Birmingham, Alabama, or Orlando,
Florida. There, the cars are refurbished, repainted,
safety inspected, and sold at the Topless City
dealership in that city.
In August, Brandon found 20 cars at the Atlanta
auction, and Kyle found 50 cars at the Miami auction
which met the needs of the company. Only 15 cars can
be worked on at each city during the month, however.
Another auction is coming up in September; thus, only
45 cars are to be purchased in August.
Topless City wishes to minimize its costs of
transporting the cars to the refurbishing locations. The
cost to transport cars between cities is shown in the table
for problem 49.
Problem 49
Jackson Birmingham Orlando
Atlanta $200 $100 $175
Miami $250 $200 $125
a. Give a linear programming formulation for this
problem.
b. Formulate the problem as a transportation problem
and solve.
c. Do the assumptions of the transportation model
appear to be valid for this problem? Comment.
50. Consider the situation faced by Topless City in problem
49. For some time now, Brandon and Kyle have been
considering converting their facilities in these three
cities to sales lots only and performing all refurbishing
operations in other cities. If they do so, they can actually
use all 70 cars: 15 in Jackson, 25 in Birmingham, and 30
in Orlando.
One plan under consideration is to contract out the
painting to shops in Tuscaloosa, Alabama, and
Columbus, Georgia, and then transport the cars for
mechanical work to shops in Montgomery, Alabama, and
Gainesville, Florida, before delivery to a Topless City
location. Alternatively, a full-service operation in
Jacksonville could handle both the painting and
mechanical work.
a. Given the tables for problem 50, which reflect the
average unit transportation costs per vehicle between
locations, formulate the problem as a transshipment
problem and solve for the optimal shipping patterns.
How many cars are painted and fixed mechanically in
each location? Explain.
Tables for Problem 50
To
Tuscaloosa Columbus Jacksonville
From
Atlanta $150 $ 75 $150
Miami $200 $175 $125
To
Montgomery Gainesville
From
Tuscaloosa $50 $100
Columbus $50 $ 75
CD-468 Chapter 4 Extra Problems/Cases
23,000
Buy 2-Year-Old Car
6000
Keep
Buy New Car
31,760
3000
Keep
16,000
Buy New Car
42,000
Buy New Car 1 1
2
4
3
Purchase +
Operating Cost
Year 1 Year 2 Year 3 Year 4 Year 5
Purchase +
Operating CostTrade-in
Age at
Beginning of
Year 2
Operating
Cost
START
Problem 48
To
Jackson Birmingham Orlando
From
Montgomery $130 $ 70 $110
Gainesville $150 $135 $ 45
Jacksonville $180 $130 $ 60
b. After painting and refurbishing the vehicles and
deducting other expenses (sales personnel, utilities,
etc.), the average gross profit is $x per car. Based on
the August auction figures, what breakeven value of x
would justify implementing the new plan of buying
and selling all 70 cars, rather than maintaining the
current policy of purchasing 45 cars and doing all the
work at Topless City locations?
c. Solve for the shortest path (in terms of cost) from
Atlanta to the Topless City locations; solve for the
shortest path from Miami to the Topless City
locations.
d. Use the results of part (c) to convert the
transshipment problem to a transportation problem.
Solve and show that the solutions are equivalent to
those found in part a.
Additional Problems/Cases CD-469
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