Equipment Replacement: Dynamic programming formulation.

Equipment Replacement: Dynamic programming formulation. Suppose a shop needs to have a certain machine over the next five-year period. Each new machine costs $1000. The cost of maintaining the machine during its ith year of operation is as follows: m1= $60 m2=$80 and m3=$120. A machine may be kept up to three years before being traded in. The trade in value after i years is s1=$800 s2=$600 s3=$500. Given that a new machine must be purchased now (time 0) the shop wants to determine a replacement and trade-in Policy that minimizes the net costs = (maintenance cost)+( replacement cost)-(salvage value received) during the next 5 years. How can the shop minimize costs over the five-year period? Let the stages correspond to each year. The state is the age of the machine for that year. The decisions are whether to keep the machine or trade it in for a new one. Let ft(x) be the minimum cost incurred from time t to time 5 given the machine is x years old in time t. Since we have to trade in at time 5
f (x) ? ?sx 5 Now consider other time periods. If you have a three year old machine in time t you must trade in so If you have a two-year-old machine you can either trade or keep. Trade costs you: . Keep costs you . So the best thing to do with a two-year-old machine is the minimum of the two. Similarly Finally at time zero we have to buy so This is solved with backwards recursion as follows: Stage 5.
Stage 4. Stage 3. Stage 2. Stage 1. Stage 0.
So the cost is 1280 and one solution is to trade in years 1 and 2. There are other optimal solutions.
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