A farmer just purchased (at time 0) a new tractor for $12,000. The cost of maintaining...

 

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A farmer just purchased (at time 0) a new tractor for $12,000. The cost of maintaining a tractor during a year depends on its age at the beginning of the year, as given in Table 1. To avoid the high maintenance costs associated with an older tractor, the farmer may trade in the tractor and purchase a new tractor. The price that the farmer receives on a trade-in depends on the age of the tractor at the time of trade-in (see Table 2). To simplify the computations, we assume that at any time, it costs $12,000 to purchase a new tractor. The goal is to minimize the net cost (purchasing costs + maintenance costs - money received in trade-ins) incurred during the next five years. a) Sketch the directed network associated with finding the minimum net cost during the next five years as a shortest path problem. Clearly describe in words what your nodes and arcs represent, and what the "costs" of your arcs correspond to. b) Use the shortest path algorithm to solve the problem in part a). Table 1 Tractor Maintenance Cost ($) Table 2 Tractor Trade-In Prices Age of Tractor Annual Maintenance Age of Tractor Trade-In Price (Years) Cost ($) (Years) ($) 0 2,000 1 7,000 1 4,000 2 6,000 2 5,000 3 2,000 3 9,000 4 1,000 4 12,000 S 0 A farmer just purchased (at time 0) a new tractor for $12,000. The cost of maintaining a tractor during a year depends on its age at the beginning of the year, as given in Table 1. To avoid the high maintenance costs associated with an older tractor, the farmer may trade in the tractor and purchase a new tractor. The price that the farmer receives on a trade-in depends on the age of the tractor at the time of trade-in (see Table 2). To simplify the computations, we assume that at any time, it costs $12,000 to purchase a new tractor. The goal is to minimize the net cost (purchasing costs + maintenance costs - money received in trade-ins) incurred during the next five years. a) Sketch the directed network associated with finding the minimum net cost during the next five years as a shortest path problem. Clearly describe in words what your nodes and arcs represent, and what the "costs" of your arcs correspond to. b) Use the shortest path algorithm to solve the problem in part a). Table 1 Tractor Maintenance Cost ($) Table 2 Tractor Trade-In Prices Age of Tractor Annual Maintenance Age of Tractor Trade-In Price (Years) Cost ($) (Years) ($) 0 2,000 1 7,000 1 4,000 2 6,000 2 5,000 3 2,000 3 9,000 4 1,000 4 12,000 S 0