A manufacturer has one key machine at the core of one of its production processes. Because of heavy use, the machine deteriorates rapidly in both quality and output. Therefore, at the end of each week, a thorough inspection is done that results in classifying the condition of the machine into one of four possible states:

State Condition

0 Good as new

1 Operableminor deterioration

2 Operablemajor deterioration

3 Inoperableoutput of unacceptable quality

After historical data on these inspection results are gathered, statistical analysis is done on how the state of the machine evolves from month to month. The following matrix shows the relative frequency of each possible transition from the state in one month) to the state in the following month.

State 0 1 2 3

0 0 p₀₁ p₀₂ p₀₃

1 0 p₁₁ p₁₂ p₁₃

2 0 0 p₂₂ p₂₃

3 0 0 0 1

Once the machine becomes inoperable it remains inoperable. Leaving the machine in this state would be intolerable, since this would shut down the production process, so the machine must be replaced. The new machine then will start off in state 0. The replacement process takes 1 week to complete so that production is lost for this period. The cost of the lost production (lost profit) is $c1, and the cost of replacing the machine is $c2, so the total cost incurred whenever the current machine enters state 3 is $c1 plus $c2. Even before the machine reaches state 3, costs may be incurred from the production of defective items. The expected costs per week from this source are as follows:

State Expected Cost Due to Defective Items, $

0 0

1 c3

2 c4

However, there also are other maintenance policies that should be considered and compared with this one. For example, perhaps the machine should be replaced before it reaches state 3. Another alternative is to overhaul the machine at a cost of $c5. This option is not feasible in state 3 and does not improve the machine while in state 0 or 1, so it is of interest only in state 2. In this state, an overhaul would return the machine to state 1. A week is required, so another consequence is $c1 in lost profit from lost production.

It is desired to find the optimal maintenance policy? In other words, determine the optimal decisions to make when the machine is in state 0, 1, 2 and 3 that will keep the process running at a minimum cost?

P01=0.8710, P02=0.0349, P11=0.7726, P12=0.1479, P22=0.4539, c1=2156, c2=4089, c3=1137, c4=2842, c5=1800