Demand and distribution

The demand data used for the model is the historic weekly demand for the past 52 weeks (at 12-04-2017). The demand is defined as the transfer from items in the warehouse to the factory. Before using the demand data the data has to be cleansed. Therefore, products with a total demand less than one, products other than purchase items (P-items) (e.g. work-in-progress items, M-parts), KANBAN items, inactive items, products with a cost equal to zero and products for which no accurate lead time is available were removed from the dataset. After this data cleansing a demand data set of 4011 SKUs remains. The items in this data set are quite diverse, as indicated in Table 9, with demand ranging from 1 to 19,000 per year and prices ranging from €0.01 to €488,862.75. An overview of the cumulative demand is shown in Figure 30.

Table 9 Summary of demand data set

Demand per year per SKU Price (€) Lead time (days) MOQ

Min 1 0.01 13 1

Average 168.93 2,014.17 78.60 35.64

Max 19,000 488,862.75 270 10000

Figure 30 Cumulative Demand graph

4.2.1 Regression and detrending

To be able to determine the service levels for the ABC classification, the demand distribution needs to be known. Before being able to fit a distribution to the demand, it needs to be checked if the demand shows a (positive or negative) trend. And if so, the demand for those items should be detrended before fitting a possible distribution.

For this reason, linear regression will be performed on every item in the demand data set. Looking at the overall growth in the aggregate number of systems, a linear trend shows a better fit (higher R²-value) than an exponential fit. However, due to the phasing-in and-out of new and obsolete items, some new items have a long period with zero demand after which suddenly a weekly nonzero demand takes place,

0%

20%

40%

60%

80%

100%

1 251 501 751 1001 1251 1501 1751 2001 2251 2501 2751 3001 3251 3501 3751 4001

CDF

SKUs (sorted on ascending total demand)

Cumulative Demand

45

and vice versa for obsolete items. A long period of zero demand at the beginning or end of the 52-week period may cause an (in reality non-existing) significant trend. The data set has no properties in which can be seen when the item was phased-in/-out, it is therefore, especially for items with very low demand hard to identify if an item had a long period of zero demand or that this was caused by phasing-in/-out. Besides the chance of identifying non-existing trends, detrending the data subsequently may lead to high negative demand values, which is something that should be prevented.

Therefore, for the regression analysis and further classifications, only the demand values from the first to the last nonzero element are taken in case the first and/or last 16 weeks of the 52-week period have a total demand of zero. However, this is done for all items except for the first 2000 items in the data set (sorted on total demand) which have long periods of no demand because of the low demand pattern of the item in general. To clarify this, two exemplary scenarios are depicted in Figure 31 and Figure 32. The first graph shows the undesirable situation where a trend is identified due to a zero demand period for the first 12 weeks. The second graph shows the same graph, but now excluding the first 12 weeks of no demand.

Next, to prevent the identification of any non-existing trends even more, the significance level will be set at 1%. This leads to a total of 1094 items (27%) with a significant trend out of 4011, where the first significant trend is identified for item 509. The linear trend model is represented by the following formula, where 𝐷_𝑡 is the demand in period 𝑡:

𝐷_𝑡= 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 + 𝑡𝑟𝑒𝑛𝑑𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 ∗ 𝑡 = 𝑎 + 𝑏𝑡

For all items with a significant trend, the demand data will be detrended following the following formula:

𝐷_𝑡,𝑑= 𝐷_𝑡,𝑡− 𝑡 ∗ 𝑏

Where 𝐷_𝑡,𝑑 and 𝐷_𝑡,𝑡 represent the detrended demand and demand with trend in period 𝑡, respectively.

Figure 31 Example graph with 12 weeks of zero demand Figure 32 Example graph excluding the first 12 weeks without demand

46 4.2.2 Distribution fitting

Since the demand data shows high variation in demand levels, different demand distributions may be appropriate for slow and fast moving products. Looking at the slow moving items, the demand can be characterized as intermittent demand, meaning that demand occurs sporadically, with some time periods having no demand at all (Snyder, Ord, & Beaumont, 2012; Teunter, Syntetos, & Babai, 2011).

The demand quantities itself for these items are rather constant (usually 1), meaning it is not characterized as lumpy demand (i.e. if there is also a high variability in the nonzero demand (Willemain, Smart, & Schwarz, 2004). Particularly SKUs with intermittent demand are often subject to the greatest risk for obsolescence (Syntetos, Babai, Lengu, & Altay, 2011). For low demand and small demand sizes, using a discrete distribution is general more appropriate and gives often better results than using a continuous distribution (Broekmeulen & Van Donselaar, 2014b; Snyder et al., 2012). Moreover, Axsäter (2011) showed that using a continuous distribution like the Normal distribution leads to substantial errors in case of (very) low demand. Therefore, for the slow moving items the fit with discrete distributions will be tested. Based on the cumulative demand graph in Figure 30, the boundaries between the slower and faster moving items will be set at 80%, when demand is sorted on the sum of demands from week 1-52.

For slow moving items the Poisson distribution is regarded as a very well fitting distribution (Syntetos et al., 2011). Furthermore, the Poisson distribution is an often used distribution in the maintenance and spare parts industry, characterized by intermittent demand as well (Van Wingerden et al., 2016).

Another option is the use of the Negative Binomial Distribution. There is many empirical evidence that support the use of the Negative Binomial distribution and it is often recommended to be applied in practice (Syntetos et al., 2011).

For the medium- and fast(er)-moving items, continuous distributions will be tested as well. The Normal distribution is a commonly used method, both in theory and practice. A disadvantage of this distribution is the high probability of negative demand if the coefficient of variation is high and that it may be less suitable for slower moving items with an intermittent demand pattern, in which case the coefficient of variation is typically high. Moreover, Axsäter (2011) argues that the Normal distribution is usually not reasonable to use if the demand during lead time is less than ten, making it generally unsuitable for slow moving items. Furthermore, because the Normal distribution is symmetrical, while demand in reality is usually right-skewed (Axsäter, 2011), this distribution may be less appropriate. An often suggested alternative to this situation with intermittent demand is to use the non-negative Gamma distribution (Teunter et al., 2010; Teunter & Duncan, 2009). It is argued that the use of a Gamma distribution performs better for a positive or right-skewed density than the more often used Normal distribution (Dunsmuir & Snyder, 1989). It has also been argued that for intermittent demand with many nonzero demand and a positive skewness for positive demand, the Gamma distribution is a good distribution to consider (Nenes, Panagiotidou, & Tagaras, 2010). Moreover, the Gamma distribution can represent a wide range of distribution shapes (Boylan, 1997), is only defined for non-negative values and has the advantage its generally mathematically tractable within inventory control (Syntetos et al., 2011). The

47

Gamma distribution can be regarded as the continuous analogue of the Negative Binomial Distribution (Teunter, Syntetos, & Babai, 2009).

4.2.2.1 Goodness-of-fit test

To test if there is a significant statistical fit for a certain distribution the Chi-Square test will be used, which is a goodness-of-fit test that measures the fit between the observed and expected demand. The Chi-Square test groups data together in bins/categories and checks if the observed and expected demands fall in the same category. Another option is the use of the Kolmogorov-Smirnov test or the Anderson-Darling test. The advantage of the latter methods are the fact they do not require the grouping of data, but they are only accurate if demand is strictly continuous (Syntetos et al., 2011), which make these methods unsuitable. The null-hypothesis of the Chi-Square test is that the observed data comes from the specified distribution with a certain significance level (in this thesis 5%), i.e. there is a significant fit is the hypothesis is not rejected. The corresponding Chi -Square test statistic is calculated as follows:

𝜒²= ∑(𝑂_𝑖− 𝐸_𝑖)² 𝐸_𝑖

𝑁

𝑖=1

where 𝑂_𝑖 are the observed counts and 𝐸_𝑖 the expected counts, based on the hypothesized distribution.

Next to the Chi-Square test, Adan, Van Eenige, & Resing (1995) developed a fitting procedure. They presented a method to fit a discrete distribution based on the mean and variance (the first two moments) of a random variable. To determine the appropriate distribution, they defined the variable 𝑎 = 𝑐_𝑣²−¹_𝜇=^𝜎_𝜇²₂−¹_𝜇=

𝜎2 𝜇−1

𝜇 . Looking at the data, for the first about 8% of the items this value is (very close to) zero, after which it starts to vary. A value of 𝑎 = 0 implies that the mean is equal to the variance, 𝜇 = 𝜎², which is exactly the characteristic of the Poisson distribution. For this reason and since the Poisson distribution is commonly used for slow movers as indicated above, for these items the Poisson distribution will be used. Since after these items the variable 𝑎 starts to vary, the Chi-Square test will be used for the remaining items.

The Chi-Square goodness-of-fit test will be performed in MATLAB, which contains a function (chi2gof) to apply the test and automatically indicates p-value and whether the null-hypothesis is rejected or not at the specified significance level. To use this function only the parameters of the distributions have to be specified:

Poisson distribution

 𝜆 = 𝜇

Negative Binomial distribution

 𝑝 = 𝑠𝑢𝑐𝑐𝑒𝑠𝑠 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 =_𝜇+𝜎^𝜇₂

 𝑟 = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑒𝑠 = 𝑝 ∗ 𝜇

48 Normal distribution

 𝑚𝑒𝑎𝑛 = 𝜇

 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 = 𝜎 Gamma

 𝛼 = 𝑠ℎ𝑎𝑝𝑒 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 = (^𝜇_𝜎)²

 𝛽 = 𝑠𝑐𝑎𝑙𝑒 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 =^𝜎_𝜇² Exponential

 𝑚𝑒𝑎𝑛 = 𝜇

The results of the test are depicted in Table 10. As can be seen, for both the slower and faster moving items, the Gamma distribution provide the best fit.

Table 10 Goodness-of-fit results

Slower movers 8-80% Faster movers 80-100%

Poisson 68.7 41.0

Neg. Binomial 56.8 35.9

Normal 20.5 68.5

Gamma 83.1 83.5

Exponential 55.0 34.5

Based on these results, two distributions will be used for the remaining analysis, the Poisson and the Gamma distribution.