A machine produces a great, good or defective set of parts. If a defective set o

Question

A machine produces a great, good or defective set of parts. If a defective set of parts is produced, then the process is stopped for the time it takes to create one set of parts and the machine is readjusted during this time. When the machine is back up and running the next set will be good and great with equal probability. If the machine produced a great set of parts, then the machine is equally likely to produce either a good set or a great set next. If the machine produces a good set of parts, then it will produce a good set 70% and a bad set 30% of the time.

What proportion of time is the machine getting reset?

If a great set of parts is worth $300, and a good set is worth $250. A defective set is worth $100 with a probability .3 and $50 with probability .7. Resetting the machine costs $300. What is the expected payoff per time unit?

Explanation / Answer

After first step, P(Great) = 0.5 , P(Good) = 0.5

At second step, P(Great/Great) = 0.5*0.5= 0.25,P(Good/Great) = 0.5*0.5= 0.25 P(Good/Good) =0.7*0.5 = 0.35, P(Bad/Good) = 0.15

Machine will get reset with a 15% probability and will take 15% of the time it takes to produce the parts.

Hence the proportion of the time machine is getting reset is 15/[100+15] = 15/115=3/23

For 115 time units, the payoffs are

Great : 25*300 = 7500

Good: (25+35)*250 = 15000

Bad : 15 * [100*0.3+50*0.7) = 15*(65) = 975

Resetting time: 15* - 300 = -4500

Expected payoff per unit time = [7500+15000+975-4500]/115= 165

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