Tonya C. Brooks
This
whole semester, I have been wracking my brain for a situation in which people
use mathematics outside of school. The problem is not that people donŐt use
mathematics, but how they use mathematics. My husband uses mathematics all the
time in his job as a carpenter/deck builder. My mother uses mathematics often
in her job as a GM seat manufacturer for Lear Corporation. Everyone uses
mathematics in their everyday activities: paying the bills, balancing the
checkbook, calculating up how much items will cost at the grocery store, etc. I
looked through some previous essays that were written for this class. Some of
the previous essays are extremely interesting. Especially the essay about farm
irrigation; my grandparents are farmers in Missouri, and they have several
different irrigation systems. However, my grandparents didnŐt really worry
about the mathematics involved in watering their fields. Most of their fields
are not large enough to need more than one system, and many times, they water
parts of the gravel roads surrounding the fields. I wanted to write about a
type of mathematics that is used for everyday purposes but that requires
something more than the basic arithmetic functions. This was the hard part.
Then
I got to thinking about some of the problems that my father, brother, and uncle
continuously deal with. You see, they drive for a living. As a matter of fact
they all work for a company in Missouri called Jack Cooper, and Jack Cooper
hauls all different kinds of vehicles (mostly GM vehicles though).
Our
country runs on semi trucks. No matter where you go, you will see them. They
run 24 hours a day, 7 days a week. Open any newspaper and you will see ads
looking for truck drivers. I constantly see TV ads advertising truck driving
jobs for companies. Today, you can see the truck driving trade being talked
about on the news all the time due to high fuel prices, and many privately
owned trucking companies cannot continue to run due to high fuel prices, high
insurance, and low pay.
Now,
to the real problem that my father, brother, uncle, and thousands of other
truck drivers constantly struggle with. When you drive a truck for a living,
you have to constantly worry about road access for the truck, especially with
car haulers. Semi trucks are one of the tallest and longest vehicles on the
road, which means that low bridges, bridges that have maximum weight
capacities, narrow streets and many other issues cause major problems for semi
trucks. Car haulers have to worry about this even more than other truck drivers
because they are generally taller than most other trucks because of the
vehicles that they carry (especially when they are carrying vans).
Due
to all these obstacles, my father, brother, and uncle have to constantly
monitor what obstacles they are going to run into on their way to a drop site.
They have to be aware of whether there are low bridges on certain roads,
whether they are too heavy for bridges on a certain route, and if they get
lost, they have to know whether there is a route that will allow them to turn
around. I know how important this is because there were several times that
other Jack Cooper drivers were not aware of obstacles on their chosen route and
did not realize that they would not fit under a bridge until they were almost
there. By that time, it was too late to take a detour, and the driver had to
pull over and wait for help to come or had to unload the truck, drive under the
overpass and then reload the truck. In other words, mistakes like these cause
major disasters and cost quite a bit of money for the company and employees. It
is the driverŐs responsibility to make sure that mistakes like this do not
happen.
In
order to look at a possible way to solve these types of problems, I have
decided to use graphs to see possible routes that might be taken. I let towns
be the vertices in my graph and let roads between the towns be edges connecting
my different vertices together. In other words, an edge connects two vertices
if and only if there is a highway connecting those two cities.
LetŐs
take a look at one particular example. LetŐs say that we must run a load of
vans from Wentzville, MO and our last drop is in Dallas, TX. We must also make
other drops along the way in Little Rock, AR, and Austin, TX. Below you will
see several possible routes to the different cities as well as a few other
smaller towns along the way.
LetŐs
say that we are hauling a load of vans and that the height of the truck with
the vans is 13Ő 8Ó. In other words, we need to be aware of routes in which we
encounter bridges that we cannot pass beneath. We can see in the example above
that we have several different routes to choose from. We only see two roads
that we cannot take, the one from P to Little Rock and P2 to Dallas.
Now,
out of all the different routes that we have to choose from, we might prefer
some over others. If we know that there is construction happening on certain
highways, then depending upon the time that we will be passing through those
areas, we might decide to go around and take a longer route because we think it
might be faster and we will not have as much traffic. Also, I know that several
times, my father and brother have taken longer routes simply because they do
not want to have to pass through particular cities, such as Atlanta. They often
plan their trips as they go along and try to schedule when they leave specific
terminals or hotel rooms so that they reduce the amount of time that they spend
in traffic, not only because they do not get paid for time in traffic, but
because it has a higher risk for accidents, and many other reasons. On top of
this, they also have to be mindful of other obstacles such as hills and
mountainous areas due to the desire to keep their mileage rates high. Their
trucks are regulated to run at most 62 miles an hour, which means that if they
run into areas that require them to run up and down hills, or make lots of
turns, their truck cannot keep the speed up and they end up doing a
ridiculously slow speed. The company expects them to get so many miles per
gallon of fuel, so they try to make sure that the routes they take are not
going to jeopardize that. All of these factors go into the decisions that truck
drivers make (and this just covers the issues if the truck is under the legal
weight, log books are filled out correctly, and all other laws are obeyed).
Now,
I know that if I said anything to my family about the use of graphs in order to
find the best route to different places, they would all look at me as though I
had lost my mind. However, what is nice about using graphs is that they are
generally pretty simple and people use the ideas behind them all the time but
do not realize it. People can use graphs such as the one above to determine the
best way to deliver the mail and for work with circuits. Graphs can be used in
todayŐs society by companies that help people find their soul mates, such as
Match.com.
Graphs
can also be used to determine how a company should fill the jobs that are open
with the applicants. For example, letŐs say that a company is looking for a
carpenter, a plumber, an electrician, a landscaper, two painters, and two
drywallers. They get applicants from 10 people. See the applicants below and
the jobs that they are qualified for.
Applicant Qualified
For
A Painter,
drywaller
B Plumber
C Carpenter,
landscaper
D Painter,
plumber
E Electrician
F Electrician,
plumber
G Drywaller
H Carpenter,
Drywaller
I Landscaper,
painter, electrician
J Electrician,
drywaller
We
can use graphs in order to show this situation. LetŐs call our jobs J1 to J8.
Our graph would look like:
This
looks pretty complicated, but we can test to determine if there is a possible
way to cover all the jobs with the applicants. We will use the HallŐs Marriage
Theorem which states: Let G be a bipartite graph with bipartition V(G) = X U Y.
The following are equivalent:
1.
The graph G has a
maximum matching M that covers all of X.
2.
For all S X, we have |S| ˛
|N(S)|.
In
other words, we will be comparing the cardinality of S, with the cardinality of
the neighbors of S. If the cardinality of S is larger than the cardinality of
the neighbors of S for any subset S, then we cannot fill all the positions.
LetŐs look at an example. LetŐs say that S consists of jobs J1 and J2. The
neighbors of J1 and J2 are B, C, D, F, and H. Then |S| = 2 and |N(S)| = 5.
If
we continue this for all the subsets of our jobs, we see that this is the case for
all of our subsets of our set of jobs. We can go through in this case very
easily and assign jobs to applicants. One such assignment is (J1, C), (J2, B),
(J3, E), (J4, I), (J5, A), (J6, D), (J7, G), and (J8, H). Can you find any
others?
LetŐs
take a look at a case in which we canŐt find a matching that covers all of the
jobs.
In
this case, if we let our set S = {Applicant 1, Applicant 2}, then the neighbors
of S = {J1}. In this case, |S| > |N(S)| and we cannot fill all of our jobs
with the applicants.