Worked Examples for Chapter 9
Example for Section 9.3
Sarah has just graduated from high school. As a graduation present, her parents have
given her a car fund of $21,000 to help purchase and maintain a certain three-year-old
used car for college. Since operating and maintenance costs go up rapidly as the car ages,
Sarah's parents tell her that she will be welcome to trade in her car on another three-year-
old car one or more times during the next three summers if she determines that this would
minimize her total net cost. They also inform her that they will give her a
new
car in four
years as a college graduation present, so she should definitely plan to trade in her car
then. (These are pretty nice parents!)
The table
gives the relevant data for
each
time Sarah purchases a three-year-old
car. For example, if she trades in her car after two years, the next car will be in ownership
year 1 during her junior year, etc.
Sarah's Data Each Time She Purchases a Three-Year Old Car
Purchase
Operating and Maintenance Costs
for Ownership Year
Trade-in Value at End
of Ownership Year
Price
1
2
3
4
1
2
3
4
$12,000
$2,000
$3,000 $4,500
$6,500
$8,500 $6,500
$4,500 $3,000
When should Sarah trade in her car (if at all) during the next three summers to
minimize her total net cost of purchasing, operating, and maintaining the cars over her
four years of college?
(a) Formulate this problem as a shortest-path problem.
The following figure shows the network formulation of this problem as a shortest
path problem. Nodes 1, 2, 3, and 4 are the end of Sarah's year 1, 2, 3, and 4 of college,
respectively. Node 0 is now, before starting college. Each arc from one node to a second
node corresponds to the activity of purchasing a car at the time indicated by the first of
these two nodes and then trading it in at the time indicated by the second node. Sarah
begins by purchasing a car now, and she ends by trading in a car at the end of year 4, so
node 0 is the
origin
and node 4 is the
destination
.