Between the theory and practice of inventory management – part 2
On the consequences resulting from an incorrect assumption regarding the type of demand distribution in the replenishment cycle.
This article is another in a series presenting typical errors made when determining the parameters of basic inventory replenishment systems and the consequences these errors have for meeting the assumed levels of two key evaluation indicators: service level and costs.
This time, we will look at the consequences of incorrect assumptions regarding the type of probability distribution of demand in the replenishment cycle. Demand is treated as a random variable, and knowledge of its distribution and the parameters of this distribution forms the basis for determining the relationship: safety stock – service level (probability of meeting demand). These relationships can be expressed in two ways:
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by determining the expected service level (probability of meeting demand) with a known level of safety stock
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by determining the necessary level of safety stock to ensure the required service level.
In practice, demand distribution is represented by a distribution function. In the case where demand quantities belong to the set of non-negative integers {0; 1; 2; 3, ….}, this function is also called a probability mass function (of demand occurrence). A distribution of this type is used, for example, in the case of demand for unit items, especially slow-moving ones (e.g., spare parts). In the case of continuous functions, i.e., when demand can take any value, we speak of a probability density function (of demand occurrence).
Distributions of demand observed in a given unit of time
As mentioned above, to determine the relationship between safety stock and service level, it is necessary to know the demand distribution in the replenishment cycle, i.e., in the period between the occurrence of the need for replenishment and the moment the stock is made available for use (processing, sale). In practice, this distribution is determined based on knowledge of the demand distribution in the adopted unit of time in which demand is observed and recorded (daily, weekly, monthly demand) and the replenishment cycle time – its mean value and standard deviation (in the most general terms – the distribution function of this time). Therefore, the starting point and condition for the correctness of calculations related to demand in the replenishment cycle is the correct determination of the demand distribution in the adopted observation period. Usually, three basic distributions are distinguished that are used in demand analysis:
1. Normal distribution, which provides a good description of the demand distribution for fast-moving items, classified (within the XYZ classification) in group X, and sometimes in Y – with higher values of the coefficient of variation,
2. Poisson distribution, used mainly to describe the demand distribution of rather slow-moving items (groups Y and Z), with quantities being non-negative integers. The condition for its applicability is the following relationship between the mean demand value Pavg and the standard deviation σP: Pavg ≈ σ2P,
3. Exponential distribution, suitable for describing the demand distribution of slow-moving items (group Z), in the case where Pavg ≈ σP.
Figure 1 shows an example of a demand distribution consistent with the normal distribution, and Figure 2 compares the Poisson and exponential distributions. Figure 2 shows that while the profile of the Poisson distribution changes its shape with an increase in the mean value, the basic shape of the exponential distribution profile remains unchanged.
Figures 1 and 2 clearly show that the normal distribution will not provide a good description of the demand distribution for slow-moving items. Only in the case of the Poisson distribution, with slightly larger demand quantities, does the distribution profile show some convergence with the model normal distribution.
The article comes from the journal “Logistyka” 1/2009.