3e2a1b56-dafb-454d-87ad-86adea3e7b86

Chapter 12 Inventory planning and control 359
the distributions which describe lead-time variation and the demand rate during the lead time.
If safety stock is set below the lower limit of this distribution then there will be shortages
every single replenishment cycle. If safety stock is set above the upper limit of the distribution,
there is no chance of stock-outs occurring. Usually, safety stock is set to give a predetermined
likelihood that stock-outs will not occur. Figure 12.12 shows that, in this case, the first replenishment
order arrived after t 1 , resulting in a lead-time usage of d 1 . The second replenishment
order took longer, t 2 , and demand rate was also higher, resulting in a lead-time usage of d 2 .
The third order cycle shows several possible inventory profiles for different conditions of
lead-time usage and demand rate.
Worked example
A company which imports running shoes for sale in its sports shops can never be certain
of how long, after placing an order, the delivery will take. Examination of previous orders
reveals that out of ten orders: one took one week, two took two weeks, four took three
weeks, two took four weeks and one took five weeks. The rate of demand for the shoes
also varies between 110 pairs per week and 140 pairs per week. There is a 0.2 probability
of the demand rate being either 110 or 140 pairs per week, and a 0.3 chance of demand
being either 120 or 130 pairs per week. The company needs to decide when it should place
replenishment orders if the probability of a stock-out is to be less than 10 per cent.
Both lead time and the demand rate during the lead time will contribute to the leadtime
usage. So the distributions which describe each will need to be combined. Figure 12.13
and Table 12.2 show how this can be done. Taking lead time to be one, two, three, four
or five weeks, and demand rate to be 110, 120, 130 or 140 pairs per week, and also assuming
the two variables to be independent, the distributions can be combined as shown in
Table 12.2. Each element in the matrix shows a possible lead-time usage with the probability
of its occurrence. So if the lead time is one week and the demand rate is 110 pairs
per week, the actual lead-time usage will be 1 × 110 = 110 pairs. Since there is a 0.1 chance
of the lead time being one week, and a 0.2 chance of demand rate being 110 pairs per
week, the probability of both these events occurring is 0.1 × 0.2 = 0.02.
Figure 12.13 The probability distributions for order lead time and demand rate combine
to give the lead-time usage distribution
➔