Task $3$:

Case Study: Reliant Industries

LO Area: Group Replacement

Reliant Industries is using a special purpose computer as part of industrial automation. The computer has $140$ transistors in its motherboard which must be replaced as and when failure happens. The failure rates of transistors are summarized in Table $3.$

Table $3-$ Failure Rates of Transistors in Computers

$\backslash$table$[[$End of week,$1,2,3,4,5,6,7],[$Probability of failure to date,$0\mathrm{.}06,0\mathrm{.}19,0\mathrm{.}36,0\mathrm{.}47,0\mathrm{.}65,0\mathrm{.}85,1\mathrm{.}00]]$

The transistors could be replaced individually for a cost of OMR $16$ per transistor. If all the transistors are replaced simultaneously, it would cost OMR $4$ per transistor. Any one of the following two options can be followed to replace the transistors:

$($a$)$ Replace the transistors individually as and when they fail $($individual replacement policy$).$

$($b$)$ Replace all the transistors simultaneously at fixed intervals and replace the individual transistors as they fail in service during the fixed interval $($group replacement policy$).$

Find out the optimal replacement policy, i$.$e$.$ individual replacement policy or group replacement policy. If the group replacement policy is optimal, then find at what equal intervals all the transistors should be replaced.

$[$Total Marks: $25]$