# 11.3.3 Safety Stock Calculation with Continuous Demand

### Intended learning outcomes: Describe different techniques for determining safety stock. Identify different patterns of the deviation of demand from forecast. Explain safety stock in relation to service level. Disclose the normal integral distribution function (service function) and the Poisson distribution integral function. Present the formula for safety stock.

Figure 11.3.1.1 indicates that without safety stock, there will be a stockout in half of the cycles defined by the saw-toothed curve. This results in backorders.

Safety stock or buffer stock serves to cushion the impact of forecast errors or deviations in the lead time as well as in the demand during the lead time.

Anticipation inventories is a similar term, used in the management of distribution inventory. It means additional inventory above basic pipeline stock to cover projected trends of increasing sales, planned sales promotion programs, seasonal fluctuations, plant shutdowns, and vacations ([APIC16]).

Figure 11.3.3.1 shows different techniques for determining safety stock depending on the nature of the item.

Fig. 11.3.3.1       Different techniques for determining safety stock.

The first two techniques determine safety stock in a largely intuitive manner. For the statisti­cal derivation, however, there are formal techniques available, as described in the following:

1. Statistical Fluctuations in the Lead Time

Fluctuations in the lead time due to unplanned delays in production or procurement, for example, are absorbed by a safety lead time.

The safety lead time is an element of time added to normal lead time to protect against fluctuations. Order release and order completion are planned for earlier dates (before real need dates), according to the time added.

Safety stock due to fluctuations in lead time is calculated simply as the demand forecast during this safety lead time. This technique is often used, because it is easily understood.

2. Statistical Fluctuations in Demand

For purposes of absorbing demand fluctuations, safety lead time is not a sufficient basis for calculation.

Fluctuation inventory, or fluctuation stock, is inventory that is carried as a cushion to protect against forecast error ([APIC16]).

Figure 11.3.3.2 shows the pattern of demand for two items with the same demand forecast, but different demand fluctuations.

Fig. 11.3.3.2       Different patterns of the deviation of demand from forecast.

The fluctuation inventory for the item in Situation B must be larger than that for the item in Situation A. A pattern of demand that has only a small dispersion around the demand forecast will result in a smaller quantity of safety stock; one with large variation will require a larger quantity of safety stock.

The service level, or cycle service level, or level of service, is the percentage of order cycles that the firm will go through without stockout, meaning that inventory is sufficient to cover demand.

The probability of stockout is the probability that a stockout will occur during each order cycle before a replenishment order arrives.

According to these definitions, the following relationship holds (see Figure 11.3.3.3).

Fig. 11.3.3.3       Service level expressed as the complement of probability of stockout.

With the order point technique, fluctuating demand can be satisfied from stock even without safety stock in about half of all cases. For this reason, the service level using this technique can be assumed to be at least 50%.

Safety stock — and with it carrying cost — grows quantitatively in dependency upon service level, as Figure 11.3.3.4 shows. Once the desired service level is set, safety stock can be estimated accurately through statistical derivation.

Fig. 11.3.3.4       Safety stock — and thus carrying cost — in relation to service level.

The safety factor is the numerical value, a particular multiplier, for the standard deviation of demand.

The service function is the integral distribution function, for which the integral under the distribution curve for demand up to a particular safety factor s corresponds to the service level.

If demand follows a normal distribution, or a bell-shaped curve, the service level corresponding to the safety factor s is the area shown in gray in Figure 11.3.3.5.

Fig. 11.3.3.5       Normal integral distribution function (service function).

Therefore, the safety factor is also the inverse function of the integral distribution function. It is the numerical value used in the service function (based on the standard deviation of the forecast) to provide a given level of service.

Figure 11.3.3.6 reproduces examples for corresponding values of the service level and the safety factor. They can be read from tables, such as the following table from [Eilo62], p. 26.

Fig. 11.3.3.6       Service level and safety factor when demand follows a normal distribution. (From [Eilo64], p. 26.)

Figure 11.3.3.7 shows the resulting formula for safety stock. With a normal distribu­tion, it is possible to use 1.25 * MAD (mean absolute deviation) instead of the standard deviation.

Fig. 11.3.3.7       Formula for safety stock.

Exercise: Get used to the influence of the safety stock on order point calculation by chosing different values for the service level.

In particular for small demand quantities, we cannot always assume that demand is normally distributed. Sometimes, we could assume a Poisson distribution instead. However, with a mean value (average demand quan­tity) of merely 9 units, the upper part of the Poisson distribution curve is very close to the curve of the normal distribution. This is particularly true for larger safety factors and high service levels. See also Figure 10.5.5.1.

Figure 11.3.3.8 shows an example of the Poisson distribution and its integral function. De­pen­ding upon the mean value λ, a different curve and a different inverse function will result.

Fig. 11.3.3.8       Poisson distribution integral function.

Figures 11.3.3.9 and 11.3.3.10 show pairs of values of the service level and safety factor for means of λ = 4 and λ = 9, respectively.

For small consumption quantities, the cost of a stockout often does not depend so much on the quantity not delivered as upon the fact that there is a failure to meet the full quantity. Thus, with small usage quantities the ten­dency is to choose a high service level, which in turn results in a high safety factor. The calculated safety factor that is based on a Poisson distribution is then generally fairly equivalent to the one based on a normal distribution.

Fig. 11.3.3.9       Table of values for the Poisson cumulative distribution with a mean demand value of λ = 4 and standard deviation SQRT(λ)  = 2 units per period. (From [Eilo62], p. 84 ff.)

Fig. 11.3.3.10    Table of values for the Poisson cumulative distribution with a mean demand value of λ = 9 and standard deviation SQRT(λ) = 3 units per period. (From [Eilo62], p. 84 ff.)

However, based on probability of stockout alone, we cannot say anything about the stockout quantity, the stockout percentage, or backorder percent-age. Thus, service level is not the same as fill rate, which only measures what actually happens when demand occurs. See also [Bern99], [Chap06].

Like fill rate (see the definition in Section 5.3.1), service level is the quantitative application of the answer to the following question: What are the costs of not meeting customer demands from stock? Both measures, fill rate and service level, are thus estimates of opportunity cost. To achieve a specific fill rate, however, it is generally sufficient to set a smaller number as the service level, or desired probability that demand can be met from stock. The relationship between the two measures, fill rate and service level, as well as ways of determining the appropriate service level, are examined in Section 11.3.4.

## Course section 11.3: Subsections and their intended learning outcomes

• ##### 11.3.4 Determining the Service Level and the Relation of Service Level to Fill Rate

Intended learning outcomes: Describe the order point technique where the length of order cycle provided by the batch size is a multiple of the lead time. Explain the probability of stockout in dependency on stockout costs per unit. Present the service function (of the stockout quantity coefficient) P(s) in dependency upon the safety factor s. Produce an overview on and examples of the relation between fill rate and service level.