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?
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 | 6 |
|---|
| P(t) | 0 | 0.03 | 0.07 | 0.2 | 0.4 | 0.15 | 0.15 |
|---|
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 = 10000
`N_1` = number of resistors being replaced by the end of `1^(st)` month
= `N_(0) P_(1)`
= `10000 * 0.03`
= `300`
`N_2` = number of resistors being replaced by the end of `2^(nd)` month
= `N_(0) P_(2) + N_(1) P_(1)`
= `10000 * 0.07 + 300 * 0.03`
= `709`
`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)`
= `10000 * 0.2 + 300 * 0.07 + 709 * 0.03`
= `2042`
`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)`
= `10000 * 0.4 + 300 * 0.2 + 709 * 0.07 + 2042 * 0.03`
= `4171`
`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)`
= `10000 * 0.15 + 300 * 0.4 + 709 * 0.2 + 2042 * 0.07 + 4171 * 0.03`
= `2030`
`N_6` = number of resistors being replaced by the end of `6^(th)` month
= `N_(0) P_(6) + N_(1) P_(5) + N_(2) P_(4) + N_(3) P_(3) + N_(4) P_(2) + N_(5) P_(1)`
= `10000 * 0.15 + 300 * 0.15 + 709 * 0.4 + 2042 * 0.2 + 4171 * 0.07 + 2030 * 0.03`
= `2590`
The expected life of each resistor is given by
`= sum_(i=1)^6 x_i p(x_i)`
= `1 * 0.03 + 2 * 0.07 + 3 * 0.2 + 4 * 0.4 + 5 * 0.15 + 6 * 0.15`
= `4.02` months.
Average number of failures per month is given by
`N_0 / "(Mean age)" = 10000/4.02 = 2487.56`
` = 2488` resistors (approx.)
Hence, the total cost of individual replacement at the cost of Rs `1` per resistor will be
Rs `(2488 * 1)` = Rs `2488`
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 | `(300) * 1 + 10000 * 0.35=``3800` | 3800 `3800 = 3800 / 1` |
| 2 | `(300 + 709) * 1 + 10000 * 0.35=``4509` `3800 = 3800 / 1` | 2254.5 `2254.5 = 4509 / 2` |
| 3 | `(300 + 709 + 2042) * 1 + 10000 * 0.35=``6551` `2254.5 = 4509 / 2` | 2183.67 `2183.67 = 6551 / 3` |
| 4 | `(300 + 709 + 2042 + 4171) * 1 + 10000 * 0.35=``10722` `2183.67 = 6551 / 3` | 2680.5 `2680.5 = 10722 / 4` |
| 5 | `(300 + 709 + 2042 + 4171 + 2030) * 1 + 10000 * 0.35=``12752` `2680.5 = 10722 / 4` | 2550.4 `2550.4 = 12752 / 5` |
| 6 | `(300 + 709 + 2042 + 4171 + 2030 + 2590) * 1 + 10000 * 0.35=``15342` `2550.4 = 12752 / 5` | 2557 `2557 = 15342 / 6` |
Since the average cost per month of Rs 2,183.67 is obtained in the `3^(rd)` month,
it is optimal to have a group replacement after every `3^(rd)` month
This material is intended as a summary. Use your textbook for detail explanation.
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