Replacement Model-3 calculator

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1. A computer contains 10,000 resistors. When any resistor fails, it is replaced. The cost of replacing a resistor individually is Rs 1 only. If all the resistors are replaced at the same time, the cost per resistor would be reduced to 35 paise. The percentage of surviving resistors say S(t) at the end of month t and the probability of failure P(t) during the month t are as follows:

t	0	1	2	3	4	5	6
P(t)	0	0.03	0.07	0.20	0.40	0.15	0.15

What is the optimal replacement plan?

2. The following mortality rates have been observed for a certain type of fuse:

t	0	1	2	3	4	5
P(t)	0	0.05	0.10	0.20	0.40	0.25

There are 1,000 fuses in use and it costs Rs 5 to replace an individual fuse. If all fuses were replaced simultaneously it would cost Rs 1.25 per fuse. It is proposed to replace all fuses at fixed intervals of time, whether or not they have burnt out, and to contiune replacing burnt out fuses as they fail. At what time intervals should the group replacement be made? Also prove that this optimal policy is superior to the straight forward policy of replacing each fuse only when it fails.

3. A computer has a large number of electronics tubes. They are subject to the following mortality rates:

t	0	1	2	3	4	5
P(t)	0	0.10	0.26	0.35	0.22	0.07

If the tubes are group replaced, the cost of replacment is Rs 15 per tube. Group replacement can be done at fixed intervals in the night shift when the computer is normally not used. Replacement of individual tube that fail in service, costs Rs 60 per tube. How frequently should the tubes be replaced?