You are in the market for a new car and a new car costs $10,000. The annual operating costs and resale value of a used car are shown in the Table below. Age of Car (Years) 1 2 3 Resale Value ($) Operating Costs ($) 8000 300 (year 1) 7000 500 (year 2) 6000 800 (year 3) 5000 1200 (year 4) 3000 1600 (year 5) 1500 2200 (year 6) 4 5 6 Assuming that you now have a new car, determine a replacement policy that minimizes the net costs of owning and operating a car for the next six years. a. We will formulate this problem as a shortest-path problem by drawing a network where seven nodes represent whether you buy a new car in each of the years. Node 1 represents the beginning of year 1, then node 7 represents the end of year 6 (or the beginning of year 7). The length of the arcs are defined based on the costs. For example, the arc from year 1 to year 2 represents that you own your car for 1 year and buy a new car in year 2. Thus the costs associated with arc 1-2 is thus 2300 (which is 300 to operate for 1 year, 10,000 to buy a new car, minus the salvage value for a 1-year old car of -8000). Provide the values of the costs for the remaining open cells. Hint, there is a pattern. [7 points] Arc Costs 1 2 3 4 5 6 7 1 $2300 2 3 4 -------- 5 6 7 b. Apply an algorithm (such as Dijkstra's algorithm) to solve the shortest path problem. What is the optimal car replacement policy? How much does it costs to own and operate your car for the next six years? (8 points) You are in the market for a new car and a new car costs $10,000. The annual operating costs and resale value of a used car are shown in the Table below. Age of Car (Years) 1 2 3 Resale Value ($) Operating Costs ($) 8000 300 (year 1) 7000 500 (year 2) 6000 800 (year 3) 5000 1200 (year 4) 3000 1600 (year 5) 1500 2200 (year 6) 4 5 6 Assuming that you now have a new car, determine a replacement policy that minimizes the net costs of owning and operating a car for the next six years. a. We will formulate this problem as a shortest-path problem by drawing a network where seven nodes represent whether you buy a new car in each of the years. Node 1 represents the beginning of year 1, then node 7 represents the end of year 6 (or the beginning of year 7). The length of the arcs are defined based on the costs. For example, the arc from year 1 to year 2 represents that you own your car for 1 year and buy a new car in year 2. Thus the costs associated with arc 1-2 is thus 2300 (which is 300 to operate for 1 year, 10,000 to buy a new car, minus the salvage value for a 1-year old car of -8000). Provide the values of the costs for the remaining open cells. Hint, there is a pattern. [7 points] Arc Costs 1 2 3 4 5 6 7 1 $2300 2 3 4 -------- 5 6 7 b. Apply an algorithm (such as Dijkstra's algorithm) to solve the shortest path problem. What is the optimal car replacement policy? How much does it costs to own and operate your car for the next six years? (8 points)
