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 1. Example-1 (Previous example)

### 2. Example-2

2. The following mortality rates have been observed for a certain type of fuse:
 t P(t) 0 1 2 3 4 5 0 0.05 0.1 0.2 0.4 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.

Solution:
The percentage of survivers resistors at the end of month t and the probability of failure are as follows,

 t P(t) 0 1 2 3 4 5 0 0.05 0.1 0.2 0.4 0.25

Let N_i be that number of resistors replaced at the end of the i^(th) month.

The different value of N_i can then be calculated as follows

N_0 = number of resistors in the beginning = 1000

N_1 = number of resistors being replaced by the end of 1^(st) month

= N_(0) P_(1)

= 1000 * 0.05

= 50

N_2 = number of resistors being replaced by the end of 2^(nd) month

= N_(0) P_(2) + N_(1) P_(1)

= 1000 * 0.1 + 50 * 0.05

= 103

N_3 = number of resistors being replaced by the end of 3^(rd) month

= N_(0) P_(3) + N_(1) P_(2) + N_(2) P_(1)

= 1000 * 0.2 + 50 * 0.1 + 103 * 0.05

= 210

N_4 = number of resistors being replaced by the end of 4^(th) month

= N_(0) P_(4) + N_(1) P_(3) + N_(2) P_(2) + N_(3) P_(1)

= 1000 * 0.4 + 50 * 0.2 + 103 * 0.1 + 210 * 0.05

= 431

N_5 = number of resistors being replaced by the end of 5^(th) month

= N_(0) P_(5) + N_(1) P_(4) + N_(2) P_(3) + N_(3) P_(2) + N_(4) P_(1)

= 1000 * 0.25 + 50 * 0.4 + 103 * 0.2 + 210 * 0.1 + 431 * 0.05

= 333

The expected life of each resistor is given by
= sum_(i=1)^6 x_i p(x_i)

= 1 * 0.05 + 2 * 0.1 + 3 * 0.2 + 4 * 0.4 + 5 * 0.25

= 3.7 months.

Average number of failures per month is given by
N_0 / "(Mean age)" = 1000/3.7 = 270.27

 = 270 resistors (approx.)

Hence, the total cost of individual replacement at the cost of Rs 5 per resistor will be

Rs (270 * 5) = Rs 1350

The cost replacemnet of all the resistors at the same time can be calculated as follows:
 End of month Total Cost of Group Replacement (Rs) Average Cost per Month (Rs) 1 (50) * 5 + 1000 * 1.25=1500 1500 1500 = 1500 / 1 2 (50 + 103) * 5 + 1000 * 1.25=2015 1500 = 1500 / 1 1007.5 1007.5 = 2015 / 2 3 (50 + 103 + 210) * 5 + 1000 * 1.25=3065 1007.5 = 2015 / 2 1021.67 1021.67 = 3065 / 3 4 (50 + 103 + 210 + 431) * 5 + 1000 * 1.25=5220 1021.67 = 3065 / 3 1305 1305 = 5220 / 4 5 (50 + 103 + 210 + 431 + 333) * 5 + 1000 * 1.25=6885 1305 = 5220 / 4 1377 1377 = 6885 / 5

Since the average cost per month of Rs 1,007.5 is obtained in the 2^(nd) month,

it is optimal to have a group replacement after every 2^(nd) month

This material is intended as a summary. Use your textbook for detail explanation.
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 1. Example-1 (Previous example)

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