Muthothi Ltd$.$ Operates a conventional stock control system based on re$-$order levels and Economic Order Quantities $($EOQ$).$ The various control levels were set originally based on estimates which did not allow for any uncertainty and this has caused difficulties because, in practice, lead times, demands and other factors to vary.

As part of a review of the system, typical stock item, part no$.$ X $206,$ has been studied in detail as follows:

Data for Part No$.$ X $206$

Lead times

Probability

Demand

Probability

$($Days$)$

$($units$)$

$15$

$0\mathrm{.}2$

$5000$

$0\mathrm{.}4$

$20$

$0\mathrm{.}5$

$7000$

$0\mathrm{.}6$

$25$

$0\mathrm{.}3$

The company works for $360$ days per year and it costs Sh$\mathrm{.}1,000$ to place an order. The holding cost is estimated at Sh$\mathrm{.}0\mathrm{.}025$ for storage plus $10\%$ opportunity cost of capital. Each unit is purchased at Sh$\mathrm{.}2.$ The re$-$order level for this part is currently $150,000$ units and it can be assumed that the demands would apply for the whole of the appropriate lead$-$time.

Required:

a$)$ Calculate the level of buffer stock implicit in a re$-$order level of $150,000$ units. $(3$ marks$)$

b$)$ Calculate the probability of stock$-$outs. $(2$ marks$)$

c$)$ Calculate the expected annual stock$-$outs in units. $(3$ marks$)$

d$)$ Compute the stock$-$out costs per unit at which it would be worthwhile raising the re$-$order level to $175,000$ units. $(2$ marks$)$

e$)$ Discuss the possible alternatives to a re$-$order level EOQ inventory system and their advantages and disadvantages. $(5$ marks$)$