Introduction

A major regional retailer replenishes more than 60 shops from a central warehouse using a periodic review system, usually delivering two to four times per week to every shop in order to achieve the high service standard they have imposed. Table 1 shows the inventory profile of a typical shop where the items are classified depending on the item's average daily demand rate. Class III items are just 2% of the items but they are responsible for 29% of the units sold and 11% of the stock, while class I items are 69% of the items and account just for 14% of the units sold but 40% of the stock. Obviously, the retailer defined service target for every items depending on its demand rate. This retailer, as many others in the industry, has to cope with a critical task: to establish and maintain the periodic review parameters for literally thousands of items in every shop at least every 3 months looking for the right balance between stock and cycle service. Therefore, a reliable and efficient method is needed to process these massive data and maintain inventory levels at a minimum.

Table 1 - Inventory profile of a retailing shop based on a whole year and expressed as average per day, items classified per average daily demand rate.

Full table

The most widely analysed periodic review models in the literature share the simplifying hypothesis detailed by Silver et al (1998), including the following ones especially relevant to the purpose of this paper: (a) there is a negligible chance of no demand between two consecutive reviews, so every review places a replenishment order; and (b) the usual level of backorders is negligibly small when compared with the average stock level. This simplified approach is called classical approximation in the rest of this paper.

Unfortunately, the classical method is not suitable for many of the items shown in Table 1 as they violate the hypotheses (a) and (b). Moreover, van der Heijden and de Kok (1998) point out that cost optimal control policies in multi-echelon systems often imply low service levels at intermediate nodes, resulting in a violation of the (b) hypothesis. Additionally, even when a retail chain is chasing a high global service level, Cardós and García (2005) show that the service level required for slow-moving items is usually low, resulting in a violation of the (a) hypothesis. Different models have been developed to manage slow-moving items and sporadic-demand items (Williams, 1984). More recently, some interesting papers highlight the implications of size orders on slow-moving items (Johnston et al, 2003) and evaluate periodic review policies and forecasting methods applied to low and intermittent demand items (Sani and Kingsman, 1997). However, no reference could be found on the calculation of low cycle service levels.

Therefore, this paper is primarily concerned with the exact calculation of the cycle service level when a periodic review system is applied and the demand is intermittent, even for low cycle service levels. At the extent of the authors' knowledge, this is the first time the (R, S) system has been generalized to include intermittent demand and slow-moving items instead of developing specialized models, and the first time low cycle service levels can be suitably computed. The proposed method only needs discrete demand with a known probability function.

The periodic review system

Periodic review systems place replenishment orders every R fixed periods to restore a predefined level called the order up to level. The replenishment order is received r periods after being launched, therefore, at the end of that period the corresponding stock is available for cycle service during the next periods. The notation used in Figure 1 and in the rest of this paper is

Figure 1.

Example of a periodic review system.

Full figure and legend (9K)

S

order up to level (units),

R

review period corresponding to the time between two consecutive reviews (time units) and replenishment cycle corresponding to the time between two consecutive deliveries (time units),

r

lead time for the replenishment order (periods),

z_t

physical stock in time t from the first reception,

D_t

accumulated demand during t consecutive periods,

X⁺

maximum {X, 0} for any expression X.

The replenishment order is added to the inventory at the end of the period in which it is received; therefore, these products are ready to face the demand of the following period. Physical stock is considered in this paper since unfulfilled demand is lost. Otherwise, it should be replaced by the available inventory, modifying the results presented in this paper. Additionally, it is assumed that rR, since this is the most typical situation in retail chains of consumer goods; otherwise, the transit stock should be also included in the models.

The cycle service level concept with sporadic demand

One of the most usual indicators of the cycle service level is the cycle service level (CSL), and has been defined by Chopra and Meindl (2001) as 'the probability of not having a stockout in a replenishment cycle'. In fact, Silver et al (1998) evaluate this metric as the probability of the stock in the review was not exceeded by the demand during the review period plus the lead time.

In general terms, service metrics are defined for articles with demand in every period, which is not the case for the slow-moving items analysed in this document. For example, if there is a product without stock that faces an average demand of 1 unit per week, but this demand only appears 10% of the weeks, then the application of the previous definition leads to CSL=0,90. This CSL estimation is useless since there is no service; therefore, the usual definition of this service metric must be corrected when the demand is sporadic.

Moreover, the evaluation proposed by Silver et al (1998) implicitly assumes that the probability of stockout is very low. A more rigorous expression, would consider the probability of the demand during the review cycle being positive and lower than the stock at the beginning of the cycle

Stock levels at the replenishment

Based on (1), the probability of every stock level will be needed to compute CSL. In order to obtain them, we will apply an inductive reasoning to the inventory levels between: (a) the replenishment and the inventory review; and (b) the inventory review and the next replenishment. Without loss of generalization, it can be accepted that the inventory will never be upper than S. Additionally, the demand is supposed to be discrete and independent and identically distributed in successive periods.

The review inventory level can be obtained from the probabilities of the inventory levels at the replenishment

and it can be also expressed as a matrix

where

Finally, considering the stock balance for this interval

it is found to be

where f_R-r() is the probability density function and F_R-r() is the cumulative distribution function of the demand during R-r periods.

Analogously, the inventory level at the next replenishment cycle can be expressed from the inventory review as

and the stock balance for this new interval is now

Hence, in this case the values of the matrix come by

From the expressions (2) and (3) we have that

where we define as the transition matrix between the inventory levels from the beginning of the replenishment cycle to its end. Therefore, it can be deduced that

The transition matrix and its powers have two properties: (a) all their elements are nonnegative; and (b) the sum of the elements of each row is always the unit. Particularly, if the powers of the transition matrix converge to there exists a probability distribution of the inventory at the replenishment whose value comes by

where is an arbitrary vector. In fact, the matrix is composed by equal rows to the vector . Therefore, taking one of its rows suffices to know the probability density function of the inventory.

Exact calculation of cycle service level

Being z₀ the inventory at the beginning of the review cycle, expression (1) leads to

Therefore,

The application of this expression requires a high computational effort since it is necessary to estimate the probabilities of each inventory level by means of (4).

Classical versus exact calculation

The already mentioned CSL definition proposed by Silver et al (1998) can be expressed as

As an example, Figure 2 shows the exact and classical approximation to the CSL for a Poisson demand with demand rate =1 and the other parameters being r=1, R=3 and S=1...7. Not surprisingly, these estimations are almost identical for high CSL but the gap increases for low values. The most important point is the consequences on the setting of the order up to level given a low target cycle service level. For example, a target CSL=0.50 can be accomplished with an order up to level S=3 but the classical approximation leads to S=4.

Figure 2.

Comparison between the exact and classical approximation to the cycle service level with Poisson demand with =1 and the other parameters being r=1 and R=3.

Full figure and legend (14K)

Another important problem with the classical estimation appears with slow-moving items and intermittent demand. Let the demand be Poisson with =0.1 and the same parameters as above. Let the target CSL be 0.50 as earlier, but now the lowest possible S is the unit and it results in a higher CSL=0.80 for R=3 as can be seen in Figure 3. Fortunately, we can stretch out the review period in order to adjust the resulting CSL to the target but it would result in R=15 using the classical approximation instead of the exact R=12. As a consequence, with the classical approximation CSL=0.42 and the target cycle service level is not guaranteed. In fact, this problem appears even with high CSL targets.

Figure 3.

Comparison between the exact and classical approximation to the cycle service level with Poisson demand with =0.1 and the other parameters being r=1 and S=1.

Full figure and legend (15K)

Conclusions

In practice, probably the most usual method to estimate the parameters of a periodic review system begins with a previously established review period and a cycle service target, so the order up to level is determined. The application of this approach to slow-moving items, for example in retailing, tends to result in a unit order up to level but still with a cycle service level higher than the target. Even then, the stock policy can be improved stretching out the review period, usually to a multiple of the basic review period applied to fast-moving items. Unfortunately, the classical approach gives poor results so a more reliable method should be used, such as the proposed exact method. The same applies to the calculation of low cycle service levels, another very usual situation in distribution.

However, the exact calculation requires a high computational effort. Therefore, further research will approach the definition and evaluation of alternative methods that could be applied to thousands of different items without significant deviations and in reasonable computational time.

References

1. Cardós M and García JP (2005). Designing a consumer products retail chain inventory replenishment policy with the consideration of transportation costs. Int J Prod Econ (in press): doi: 10.1016/j.ijpe.2004.12.022. | Article |
2. Chopra S and Meindl P (2001). Supply Chain Management. South-Western Publishing Co: Upper Saddle River.
3. Johnston FA, Boylan JE and Shale EA (2003). An examination of the size of orders from customers, their characterisation and the implications for inventory control of slow moving items. J Opl Res Soc 54: 833–837. | Article |
4. Sani B and Kingsman BG (1997). Selecting the best periodic inventory control and demand forecasting method for low demand items. J Opl Res Soc 48: 700–713. | Article |
5. Silver EA, Pyke DF and Peterson R (1998). Inventory Management and Production Planning and Scheduling, 3rd edn. John Wiley & Sons: New York.
6. van der Heijden MC and de Kok T (1998). Estimating stock levels in periodic review inventory systems. Opns Res Lett 22: 179–182. | Article |
7. Williams TM (1984). Stock control with sporadic and slow-moving demand. J Opl Res Soc 35: 939–948.