Computations for the recorder point (ROP) and safety or buffer stocks are simplified considerably if we can justify the assumption that the probability distribution for the demand during lead time follows some particular, well defined distribution, such as a normal, Poisson, or negative exponential distribution.
First let us recall that buffer stock is defined as the difference between the reorder point and the expected demand during lead time.
B = ROP – D
Where B is the buffer stock, ROP is the reorder point, and D is the expressed demand during lead time.
We now define ROP = D + nσD; that is, the reorder point is the average demand D plus some number of standard deviation units n, that is associated with the probability of occurrence of that demand. (In practice, n is often called the safety factor). Substituting this statement of ROP in our definition of B we have
B = ROP – D = (D + nσD) – D = nσD
This simple statement allows us to determine easily those buffer stocks that meet risk requirements when we know the mathematical form of the demand distribution. The procedure is as follows:
1) Determine whether a normal, Poisson, or negative exponential distribution approximately describes demand during lead time for the case under consideration. This determination is critically important and requires well known statistical methodology.
2) Set a service level based on (a) managerial policy, (b) an assessment of the balance of incremental inventory and stock out costs, or (c) n assessment of the manager’s trade off between service level and inventory cost when stock out costs are not known.
3) Using the service level, define ROP in terms of the appropriate distribution
4) Compute the required buffer stock from Equation where n is the safety factor and σD is the standard deviation for the demand distribution.
We will illustrate this approach in the context of the normal distribution.
Buffer Stocks for the Normal Distribution:
The normal distribution has been found to describe many demand functions adequately, particularly at the factory level of the supply – production – distribution system. Given the assumption of normality and a service level of, perhaps, 95 percent, we can determine B by referring to the normal distribution tables, a small part of which has been reproduced as table. The normal distribution is a two parameter distribution that is described completely by its mean D and the standard deviation σD. Implementing a service level of 95 percent means that we are willing to accept a 5 percent risk of running out of stock. Table shows that demand exceeds D + (nσD) with a probability of .05, or 5 percent of the time when n = 1.645; therefore, this policy is implemented when B = 1.645 σD. As an example, if the estimate of σD is 300 units and D=1500 units, assuming a normal distribution, a buffer stock to implement a 95 percent service level would be B=1.645 x 300 = 494 units. Such a policy would protect against the occurrence of demands up to ROP = 1500 + 494 = 1994 units during lead time. Obviously, any other service level policy could be implemented in a similar way.
Determination of demand during lead time:
The probability distribution for the demand during lead time can be estimated by observing actual demand over lead time for several instances. These observations can be used to compare the expected value, D, and the standard deviation, σD, of demand during lead time.
If the lead time is approximately constant but the daily demand rate is normally distributed, then D and σD for the demand during lead time can be derived easily. We assume that daily demands are independent. First, using past observations we determine the average daily demand and its variance. Suppose the average daily demand is 10 units and its variance is 9 units. Further suppose the lead time is 4 days. Now, to calculate D, we simply multiply the lead time by the average daily demand or DS = 4 x 10 = 40 units. The variance of the demand during lead time, σ2D, is computed by multiplying the lead time and the variance of the daily demand. In our example, σ2D = 4 x 9 = 36 units. The σD = 6 units. The demand during lead time is normally distributed with the mean D = 40 units and the standard deviation σD = 6 units.
When both the lead time, L, and the daily demand rates are variable, then we need to estimate four quantities: the average daily demand, the standard deviation of daily demand, the average lead time, and the standard deviation of lead time. If daily demand as well as lead time is normally distributed, then D and σD can be easily computed. Suppose the average daily demand is 10 units and the average lead time is 4 days. Then,
D = Average daily demand x Average lead time
= 10 x 4
= 40 units.