Chap 9 WE - Worked Examples for Chapter 9 Example for...

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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 .
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The number of arcs on the path chosen from the origin to the destination indicates how many times Sarah will purchase and trade in a car. For example, consider the path This corresponds to purchasing a car now, then trading it in at the end of year 1 to purchase a second car, then trading in the second car at the end of year 3 to purchase a third car, and then trading in this third car at the end of year 4. Since Sarah wants to minimize her total net cost from now (node 0) to the end of year 4 (node 4), each arc length needs to measure the net cost of that arc's cycle of purchasing, maintaining, and trading in a car. Therefore, Arc length = purchase price + operating and maintenance costs - trade-in value. For example, consider the arc from node 1 to node 3. This arc corresponds to purchasing
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This note was uploaded on 04/25/2008 for the course ADG 111 taught by Professor Tomas during the Spring '08 term at Uni. Iceland.

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Chap 9 WE - Worked Examples for Chapter 9 Example for...

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