ΠΜΣ: Μαθηματικά των Υπολογιστών και των Αποφάσεων

ΠΜΣ: Μαζεκαηηθά ηωλ Υπνινγηζηώλ θαη ηωλ Απνθάζεωλ.
Μάζεκα: Επηρεηξεζηαθή Έξεπλα
Αθαδεκαϊθό Έηνο: 2010-11
Δηδάζθωλ: Ν. Τζάληαο
Homework 2
1. Αζθήζεηο: δείηε ηηο ζειίδεο 2-11 ηνπ παξόληνο.
2. Πξόθεηηαη γηα αηνκηθέο αζθήζεηο νη νπνίεο ζπλεηζθέξνπλ ην 25% ηνπ ηειηθνύ ζαο βαζκνύ.
3. Τν παξαδνηέν είλαη νιόζωκν θείκελν Word όιωλ ηωλ αζθήζεωλ καδί ζε ειιεληθή γιώζζα καδί
κε ηα αξρεία δεδνκέλωλ ηωλ κνληέιωλ πνπ θαηαζθεπάζαηε ζην QSB (.net, ή .cpm).
4. Όια ηα παξαπάλω αξρεία ζα πξέπεη λα ζπκπηπρζνύλ ζε έλα zip αξρείν θαη λα απνζηαινύλ
ειεθηξνληθά (e-mail: tsantas@upatras.gr) ζηνλ δηδάζθνληα κέρξη ηελ 24.00 ηεο 12εο Ινπλίνπ
2011. Σην κήλπκα πξέπεη λα αλαθέξεηε απαξαίηεηα ην νλνκαηεπώλπκό ζαο. Εξγαζίεο πνπ
παξαιακβάλνληαη εθπξόζεζκα επηζύξνπλ βαζκνινγηθέο θπξώζεηο (0,5 βαζκό γηα θάζε εκεξνινγηαθή
εκέξα θαζπζηέξεζεο). Εξγαζίεο πνπ ππνβάιινληαη κε θαζπζηέξεζε κεγαιύηεξε
από 7 εκέξεο δελ γίλνληαη δεθηέο.
5. Τν αξρείν Word ζα ην νλνκάζεηε κε ιαηηληθνύο ραξαθηήξεο "eponymo_onoma_HW2.doc".
Τα αξρεία δεδνκέλωλ ζα ηα νλνκάζεηε κε ηνλ αύμνληα αξηζκό ηεο άζθεζεο πνπ αληηζηνηρνύλ
(π.ρ. 01.net, 06.cpm, θιπ).
6. Σην θείκελν ηεο εξγαζίαο, γηα θάζε άζθεζε, πξέπεη λα ππάξρεη κηα ελόηεηα κε ηελ ελλνηνινγηθή
πξνζέγγηζε, δειαδή ηελ πεξηγξαθή ηνπ πξνβιήκαηνο θαη ηωλ ππνζέζεώλ ηνπ. Σηε
ζπλέρεηα κηα ελόηεηα κε ηελ αλάπηπμε ηνπ θαηάιιεινπ κνληέινπ θαη ηέινο, κηα ελόηεηα κε ηηο
απαληήζεηο ζηα ππόινηπα εξωηήκαηα ηεο άζθεζεο.
7. Πίλαθεο απνηειεζκάηωλ ηνπ QSB πξέπεη λα είλαη κέξνο ηνπ θεηκέλνπ ωο εηθόλεο.
8. Σηελ πξώηε ζειίδα ηεο εξγαζίαο πξέπεη λα αλαθέξεηε ην όλνκά ζαο. Οη ζειίδεο είλαη Α4,
όια ηα margins 2.5, ην spacing 1.5 θαη ε γξακκαηνζεηξά Arial 12.

1. OLD DOMINION ENERGY
The United States is the biggest consumer of natural gas, and the second largest natural gas producer in
the world. According to the U.S. Energy Information Administration (EIA), in 2001 the U.S. consumed
22.7 trillion cubic feet of natural gas. Stemming from phased deregulation, the transportation and delivery
of natural gas from wellheads has grown since the '80s and there are now more than 278,000 miles of gas
pipeline nationwide. With more electric power companies turning to natural gas as a cleaner-burning fuel,
natural gas is expected to grow even more quickly over the next 20 years.
To ensure an adequate supply of natural gas, gas storage facilities have been built in numerous
places along the pipeline. Energy companies can buy gas when prices are low and store it in these
facilities for use or sale at a later date. Because energy consumption is influenced greatly by the weather
(which is not entirely predictable), imbalances often arise in the supply and demand for gas in different
parts of the country. Gas traders constantly monitor these market conditions and look for opportunities to
sell gas from storage facilities when the price offered at a certain location is high enough. This decision is
complicated by the fact that is costs different amounts of money to transport gas through different
segments of the nationwide pipeline and the capacity available in different parts of the pipeline is
constantly changing. Thus, when a trader sees an opportunity to sell at a favorable price, he or she must
quickly see how much capacity is available in the network and create deals with individual pipeline
operators for the necessary capacity to move gas from storage to the buyer.
Bruce McDaniel is a gas trader for Old Dominion Energy (ODE), Inc. The network in Figure 5.41
represents a portion of the gas pipeline where ODE does business. The values next to each arc in this
network are of the form (x,y) where x is the cost per thousand cubic feet (cf) of transporting gas along the
arc, and y is the available transmission capacity of the arc in thousands of cubic feet. Note that the arcs in
this network are bidirectional (i.e., gas can flow in either direction at the prices and capacities listed).
FIGURE 5.41
Gas pipeline network for Old
Dominion Energy
Bruce currently has 100,000 cf of gas in storage at Katy. Industrial customers in Joliet are offering
$4.35 per thousand cf for up to 35,000 cf of gas. Buyers in Leidy are offering $4.63 per thousand cf for up
to 60,000 cf of gas. Create a spreadsheet model to help Bruce answer the following questions.
Given the available capacity in the network, how much gas can be shipped from Katy to Leidy?
From Katy to Joliet?
How much gas should Bruce offer to sell to Joliet and Leidy if he wants to maximize profits?
Is Bruce able to meet all the demand from both customers? If not, why not?
If Bruce wanted to try to pay more to obtain additional capacity on some of the pipelines, which
ones should he investigate and why?
2

2. US EXPRESS
US Express is an overnight package delivery company based in Atlanta, Georgia. Jet fuel is one of the
largest operating costs incurred by the company and they want your assistance in managing this cost. The
price of jet fuel varies considerably at different airports around the country. As a result, it seems that it
might be wise to 'fill up' on jet fuel at airports where it is least expensive. However, the amount of fuel an
airliner burns depends, in part, on the weight of the plane—and excess fuel makes an airplane heavier and,
therefore, less fuel-efficient. Similarly, more fuel is burned on flights from the east coast to the west coast
(going against the jet stream) than from the west coast to the east coast (going with the jet stream). The
following table summarizes the flight schedule (or rotation) flown nightly by one of the company's planes.
For each flight segment, the table summarizes the minimum required and maximum allowable amount of
fuel on board at takeoff and the cost of fuel at each point of departure. The final column provides a linear
function relating fuel consumption to the amount of fuel on board at takeoff.
For instance, if the plane leaves Atlanta for San Francisco with 25,000 gallons on board, it should
arrive in San Francisco with approximately 25 - (3.2 + 0.45 X 25) = 10.55 thousand gallons of fuel.
The company has many other planes that fly different schedules each night, so the potential cost
savings from efficient fuel purchasing is quite significant. But before turning you loose on all of their
flight schedules, the company wants you to create a spreadsheet model to determine the most economical
fuel purchasing plan for the previous schedule. (Hint: Keep in mind that the most fuel you would purchase
at any departure point is the maximum allowable fuel level for takeoff at that point. Also, assume that
whatever fuel is on board when the plane returns to Atlanta at the end of the rotation will still be on board
when the plane leaves Atlanta the next evening.)
Draw the network diagram for this problem.
Implement the model for this problem in software and solve it.
How much fuel should US Express purchase at each departure point and what is the cost of this
purchasing plan?
3

3. THE MAJOR ELECTRIC CORPORATION
Henry Lee is the Vice President of Purchasing for the consumer electronics division of the Major Electric
Corporation (MEC). The company recently introduced a new type of video camcorder that has taken the
market by storm. Although Henry is pleased with the strong demand for this product in the market place,
it has been a challenge to keep up with MEC's distributors' orders of this camcorder. His current challenge
is how to meet requests from MEC's major distributors in Pittsburgh, Denver, Baltimore, and Houston
who have placed orders of 10,000, 20,000, 30,000, and 25,000 units, respectively, for delivery in two
months (there is a one-month manufacturing and one-month shipping lead time for this product). MEC has
contracts with companies in Hong Kong, Korea, and Singapore who manufacture camcorders for the
company under the MEC label. These contracts require MEC to order a specified minimum number of
units each month at a guaranteed per unit cost. The contracts also specify the maximum number of units
that may be ordered at this price. The following table summarizes these contracts:
MEC also has a standing contract with a shipping company to transport product from each of these
suppliers to ports in San Francisco and San Diego. The cost of shipping from each supplier to each port is
given in the following table along with the minimum required and maximum allowable number of
shipping cartons each month:
Under the terms of this contract, MEC guarantees it will send at least 20 but no more than 65
shipping containers to San Francisco each month and at least 30 but no more than 70 shipping containers
to San Diego each month.
Each shipping container can hold 1,000 video cameras and will ultimately be trucked from the
seaports on to the distributors. Again, MEC has a standing contract with a trucking company to provide
trucking services each month. The cost of trucking a shipping container from each port to each distributor
is summarized in the following table.
As with the other contracts, to obtain the prices given above, MEC is required to use a certain
minimum amount of trucking capacity on each route each month and may not exceed certain maximum
shipping amounts without incurring cost penalties. These minimum and maximum shipping restrictions
are summarized in the following table.
4

Henry is left with the task of sorting through all this information to determine the least purchasing
and distribution plan to fill the distributor's requests. But because he and his wife have tickets to the
symphony for this evening, he has asked you to take a look at this problem and give him your
recommendations at 9:00 tomorrow morning.
Create a network flow model for this problem. (Hint: Consider inserting intermediate nodes in
your network to assist in meeting the minimum monthly purchase restrictions for each supplier
and the minimum monthly shipping requirements for each port.)
Implement the model for this problem in software and solve it.
What is the optimal solution?
5

4. TURNER AIRLINES
Fred Turner is a successful entrepreneur in Atlanta, Georgia who has just hit upon his next moneymaking
idea. Recently, Fred was planning to fly to Washington, DC early one morning for a business meeting.
Fred's own personal Leer jet happened to have a mechanical problem that morning, which forced Fred to
try to book a flight on a commercial carrier. Much to his dismay, Fred learned that all the flights to DC
were overbooked but he was told might be able to get a seat if he waited at the gate as a standby
passenger. When Fred arrived at the gate, he was surprised to find several people waiting, hoping to get on
a flight as a standby passenger.
When Fred saw this situation, he immediately saw a business opportunity. He knew that the route
from Atlanta to Washington, DC was a popular one as these airports are two of the busiest in the world.
And the frustration he saw on the faces of those waiting in line with him convinced him that there was
money to be made here. Fred knew that it would be difficult to compete head to head against the
established, full-service, international airlines. But he figured he could begin a flight service to fill the
niche to business travelers wanting to travel between Atlanta and DC.
Fred called some of his business associates and put some people to work on this idea. A month
later, their marketing research revealed that there were several times during every business day when
travelers want to hop a plane going between Atlanta and DC. They also discovered that many business
travelers would prefer to fly to Dulles Airport in Washington, whereas most airlines fly to Washington's
Reagan Airport. The following table summarizes the flight schedules where there is considerable unmet
demand and the estimate revenues and variable costs that would be incurred in operating these flights.
In addition to the costs given, the airport in Atlanta charges a $600 servicing charge for planes left there
overnight, whereas Dulles Airport charges $700. Of course, there are also the costs of buying or leasing
aircraft, hiring pilots, etc. Before pursuing this any further, Fred wants to know if the proposed flight
schedule is profitable and how many planes it would take to implement it. And he has turned to you for
help.
Draw the graph of the network flow problem Fred should solve to determine the minimum
number of planes required to implement the proposed flight schedule in the most efficient manner.
Implement the model for this problem in software and solve it.
Describe the optimal solution to the problem.
Suppose there was adequate demand to run two flights from Atlanta to Washington leaving Atlanta
at 4 p.m. and arriving at Dulles at 6 p.m. Would this be more profitable than the current schedule?
Should Fred be willing to sacrifice any of the current flights to add this second flight out of
Atlanta at 4 P.M.?
6

5. MANEUVERS IN THE SENATE
Sherill Wiley, the only reporter from Channel Four covering United States congressional proceedings,
knows that the public's perception of partisan politics in the Senate is not entirely correct. Although
Republicans and Democrats are often at loggerheads over many national issues—such as the crime bill,
health care reform, and others—members of the two parties frequently vote together on bills that are
considered beneficial to both. Sherill was surprised at the level of cooperation between the parties when
passing legislation that benefits a particular region. For example, if a Democratic senator from the west
was sponsoring a bill that would provide funding to divert water from a river so that part of his sundrenched
state would eventually receive it, quite a few Republican senators would vote for such a bill. The
Republican senators know that when they need votes for a project that would be advantageous for their
states, the Democratic senators would return the favor and vote for that bill.
Richard Kennely, the leading Republican in the Senate and a strong conservative from a Midwest
state, has little if any direct interaction with Robin Doll. Doll, a senator from a New England state, is the
Senate's most liberal Democrat. The two senators have never served on a committee together or
participated in a joint task force. Yet the two often rely on each other and their respective parties when
voting on many fiscal issues.
Senator Doll recently added an amendment to a finance bill that would provide funding for research
on the effects of using a mouse when working on a computer. Several universities from Doll's state
submitted grant proposals to do this research, citing the widespread use of the mouse and the growing
number of people who complain of wrist and arm fatigue as a result. If the bill and the attached
amendment pass, the universities in Doll's state would be the main beneficiaries of the funding. Sherill
knows that Senator Kennely, who was recently on national television railing against what he referred to as
"excessive spending for trivial research," would be the main opponent to the bill. Senator Doll cannot
directly influence Senator Kennely's position or vote. But she can contact other senators with whom she
has influence, pleading her case and hoping that they, in turn, will influence others who might eventually
influence Senator Kennely to vote in favor of the bill.
After many years as a reporter in Washington, Sherill has developed a sense of the amount of
influence one senator has on another, based on party affiliation, school ties, past favors, mutual interests,
and so on. She even developed a numerical system, on a scale from 0 to 10 that quantifies the amount of
influence one senator has on another. A10 means that a particular senator is almost guaranteed to secure
the vote of another senator and a 0 means that there is no interaction at all between two senators. Sherill
created a table that shows the senators' names and the amount of influence each has on the other.
Sherill saw Senators Abbott, Bartle, Crumb, Dodge, and Fernandez leave Senator Doll's office one
morning last week. Looking at her influence table, she notes that Doll's influence on Abbott rates a 7, on
Bartle a 6, a 9 for Crumb, only a 3 for Dodge, and a 4 for Fernandez. Later that week, Senator Abbott
called in Senators Bartle and Dodge for a strategy meeting. Abbott's influence on Bartle is a 7 and on
7

Dodge a 5.
Bartle next met with Senators Dodge, Evans, and Fernandez. Bartle's influence rating on these
senators is a 4 on Dodge and on Evans, and a 5 on Fernandez. The senators continued to hold meetings
with each other and with Senators Gene and Harris. Sherill summarized the influence rating of all the
senators in the following table:
Sherill believes that if all the senators (other than Doll) bear at least 20 total units of influence on
Senator Kennely, he would not be able to resist and he would have to vote for Senator Doll's bill. With
Kennely's support, the bill would definitely pass because the rest of the Republicans and Democrats will
follow their leaders.
Considering the amount of networking that takes place between the senators, Sherill thinks she can
build and solve a network model to predict if the bill will pass or not.
Formulate a network model for Sherill and solve it. Will the bill pass? (Hint: Make a node for
each senator.)
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6. THE IMAGINATION TOY CORPORATION
Amy White is the director of marketing for the Imagination Toy Corporation (ITC). She just received a
phone call from her boss indicating that the company's board of directors gave final approval for the
production and marketing of the Mighty Morphin' Monkeys—a new product line of action play toys for
ITC. Amy worked hard in developing the concept for this product line and is thrilled that her ideas will
become reality. But this news also means that she must get busy developing the marketing and sales force
training materials needed to launch this new product line successfully. Amy's boss wants to know how
soon she can have the sales staff trained and equipped to handle the new line.
The development of marketing materials and training of the sales staff for the new product line
constitute a project. Amy identifies ten specific project tasks that need to be accomplished before she can
roll out the marketing program for this product line. First, Amy needs to collect information about the
details of the decisions made by the board of directors. She can start this task (task A) immediately, and
she estimates that it will take five days to determine exactly which items and accessories will be included
in the first offering of the product line. After she completes this task, she will request prototypes of all
items from the engineering department (task B), which she expects will take ten working days. While
waiting for the prototypes, she can begin laying out the marketing program (task C). She expects this
activity to take eight days. After the prototypes (task B) are available, Amy estimates that it will take
seven days to prepare instructions (task D) on the operation and use of the items in the product line and
nine days to design its packaging (task E). When the marketing program (task C) is finished, it must be
approved (task F) by the president of the company. Amy expects this approval to take three days.
Amy plans to hold a two-day training course (task G) for the sales force once the operating
instructions (task D) and the packaging design (task E) are completed. When the operating instructions
(task D) are finished and the marketing plan is approved (task F), Amy will develop an information guide
(task H) that the sales force can distribute to retailers. Amy expects to take eight days to complete the
information guide. Also, as part of the marketing plan, Amy wants to hire a number of actors to portray
Mighty Morphin' Monkeys (task I) at various promotional events around the country. Hiring and training
these actors is expected to take eight days and can be done only after the marketing plan is approved (task
F). Finally, after the marketing plan (task F) has been approved and the packaging for the product has
been designed (task E), special point-of-sale display racks must be manufactured (task J). Amy expects
this activity to take twelve working days.
Develop an AON network for this problem.
Calculate the earliest and latest start and finish times, the slack for each activity, and the critical
activities.
Identify the critical path.
If Amy starts working immediately, how long will it take her to complete this project.
Suppose that the engineering department can create the prototypes in only eight days if the
engineers work overtime on this activity. Would this help reduce the length of time required to
complete this project?
9

7. A COSMETICS COMPANY
A cosmetics company has identified a set of activities that must performed in order to bring a new
product to market. The activities, their precedence relations, and time estimates to complete each activity
(in days) are shown in the following table.
Estimated Duration (days)
Activity Predecessors Optimist Most Pessimist
Likely
A -- 0 0 0
B A 10 15 20
C B 15 20 22
D C 21 26 30
E C 15 18 23
F E 13 15 17
G D, F 30 38 45
H G 20 25 30
I G 10 15 20
J G, O 11 18 22
K H, I, J 23 30 45
L K 22 28 39
M A 120 140 180
N M 13 18 22
O N 15 20 25
P O 10 15 20
Q L, P 30 33 44
R Q 5 8 11
S R 10 15 25
T R 13 17 19
U S, T 20 25 40
1. On average, how long should it take to finish the project?
2. Suppose a change can be made in activity J so that it no longer requires activity O to be completed
before it begins. What effect would this have on the project completion time and the criticality of
each activity?
Suppose that the maximum time listed for each activity is actually its normal completion time. Further
assume that the maximum number of crash days per activity and cost per crash day are as follows.
Activity
Maximum
Crash Days
Crash Cost
($)
per Day
A 0 0.0
B 10 8.4
C 7 3.3
D 9 24.4
E 8 17.0
F 4 31.3
G 15 13.5
H 10 3.0
10

I 10 10.4
J 11 6.3
K 22 4.5
L 17 13.1
M 60 18.5
N 9 8.9
O 10 10.9
P 10 2.0
Q 14 11.6
R 6 3.8
S 15 6.3
T 6 18.0
U 20 7.2
3. If crashing is allowed, how quickly could the project be completed? What is the optimal cost of
completing the project in the least amount of time?
4. What is the optimal cost of completing the project in 320 days?
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