2. Example-2
2. The following mortality rates have been observed for a certain type of fuse:
| t | 0 | 1 | 2 | 3 | 4 | 5 |
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| P(t) | 0 | 0.05 | 0.10 | 0.20 | 0.40 | 0.25 |
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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 | 0 | 1 | 2 | 3 | 4 | 5 |
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| P(t) | 0 | 0.05 | 0.1 | 0.2 | 0.4 | 0.25 |
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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.
Any bug, improvement, feedback then
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