An inventory management model for an inventory consisting of lumpy items, slow movers and fast...

international journal of

production economics

ELSEVIER Int. J. Production Economics 35 (1994) 379-389

An inventory management model for an inventory consisting of lumpy items, slow movers and fast movers

Ann Vereecke *qa Peter Verstraetenb

a De l&rick School for Management, University of Gent, Bellevue 6, 9050 Ledeherg. Gent, Belgium

bS & V Management consultants, Maaltecenter, Blok G. Derbystraat 51, 9051, Sint-Denijs- Westrem. Gent. Belgium

Abstract

This paper describes an algorithm developed for the implementation of a new computerized inventory management system for spare parts in a large chemical plant, located in Belgium. The spare parts inventory contains about 34000 different types of items; 90% of the items are classified as lumpy items, as their demand frequency is less than 4 times

a year. Empirical research by means of computer simulations has resulted in the proposal of the “Package Poisson” as a good

fitting general distribution function for the demand pattern of the spare parts. This “Package Poisson” forecasts the demand occurrences by means of a Poisson distribution, taking into account that on each occurrence a package of several pieces is asked for. The quantity f of this package can be calculated from the variance and the average of the demand history data of each item.

The performance of the algorithm has been measured by comparing the resulting reorder points with the reorder points calculated on the basis of other theoretical distribution functions. In all test cases the “Package Poisson” algorithm gives satisfying results.

1. Introduction

The algorithm described in this paper has been developed for the implementation of a new computerized inventory management system for spare parts in a large chemical plant. The spare parts inventory contains about 34000 different types of items, and has a value of over 550000000 BF (approximately $17 740000). 90% of the items, as well in numbers as in value, were classified as

* Corresponding author.

lumpy items, as their demand frequency was less than 4 times a year.

The existing inventory management system was based on the assumption that demand follows a Poisson distribution as far as the lumpy items are concerned. Slow and fast movers were controlled by an inventory model based on the assumption of normality of demand. The use of these two different models for one stock caused a number of problems, which has lead to the introduction of an inventory model based on a single algorithm for the inventory control of all items.

This paper will describe the concept of this inventory model, and will show its performance

0925-5273/94/$07.00 CQ 1994 Elsevier Science B.V. All rights reserved.

SSDI 0925-5273(93)00144-K

compared to models based on other distribution functions.

2. Probabilistic description of demand: A literature overview

2.1. Normal distribution qf demand

The issue of lumpy demand in inventory management is too often neglected. Handbooks describe reorder point models based on the normal distribution of demand. Many standard software packages for inventory management and control take normality of demand for granted. The reorder point calculation based on this normal probability distribution states that

reorder point = average demand during the lead time + safety stock

where the safety stock is the product of the standard deviation of demand during the lead time and a factor related to the desired service level.

A service level can be defined in many ways. In this article the service level is defined as the probability of no stockout, during an order cycle i.e. in the time interval between two consecutive supply orders. Equally, it is the percentage of order cycles without stockout, compared to the total number of order cycles. This service level definition does not take into account the number of deficient items in case of a stockout, nor does it consider the aspect of undershoot, i.e., the fact that with nonunit demand the inventory level may be lower than the reorder point at the moment an order is placed (see also Section 5).

The simplicity of the reorder point formula based on normality of demand and its relatively easy implementation probably explain why it is widely used in many inventory management systems, with fairly good results.

Difficulty arises when the underlying assumption of normality of demand is false. Fig. 1 shows a demand pattern of an item, with zero demand during several months over the year. The corresponding demand frequencies are also shown graphically.

2.2. Poisson distribution ~f’demand

In the case which is illustrated in Fig. 1, the probability of demand occurences can be approximated by a Poisson distribution. Given the average demand during the lead time p, the attained service level at a reorder point level ROP is the probability that the demand 6 during the lead time does not exceed the reorder point, or

RO? P(6 & ROP) = 1 pL6* exp ( - p)/6!

6=0

Knowing this, the reorder point for a given service level can be calculated iteratively: one calcu- lates the attained service level for a low reorder point, and compares this with the desired service level. If the attained service level is too low, one increases the reorder point with one unit, and recal- culates the resulting service level. This process is repeated until one finds a reorder point that results in the desired service level.

The Poisson distribution is often proposed as an alternative of the normal distribution for slow moving items. The assumption of Poisson distributed demand can easily be checked as the mean and variance of a Poisson process are equal. Silver and Peterson [l] accept the use of the Poisson distribution as long as the variance falls within the range of the mean plus or minus 10%.

Many examples of items can be found where this test fails. Fig. 2 shows the demand history and the corresponding demand frequencies of a spare part with a lumpy demand pattern, with a mean of 0.75 and a variance of 2.85. In this case the Poisson distribution is definitely not valid.

2.3. Other probability distributions qf’demand

Many other probability distributions have been proposed in literature. Listed by Silver and Peter- son [l] are the Exponential, Lognormal, Logistic, Negative Binomial, Pearson, and Tukey’s Lambda. Ehrhardt [2] proposes the Power Approximation as an approximately optimal policy for reorder point calculation.

A. Vereeckr. P. Versrraeten/Int, J. Production Economics 35 11994) 379-389 381

jan feb mar apr may june july a4 SBP act llOV dec

tima

0 1 2 3 4 5 6 7

units/demand

Fig. 1

I

2.4. Probability distributions ?f demand size and demand frequency

An alternative approach is the use of separate estimates of demand occurence and demand size. Sev- eral studies based on this approach can be found in literature. Dunsmuir and Snyder [3] developed a model that emphasizes periods with positive (non- zero) demand. The gap between successive demands is also estimated, and positive monthly demand is assumed to be gamma distributed.

Croston [4], e.g., describes the use of exponential smoothing techniques for estimating demand size and demand frequency seperately. He reports that this approach results in a lower stock level and in fewer out of stock situations, when compared to the use of exponential smoothing techniques for the demand as such.

Another example of this approach is the use of a Poisson distribution for demand occurence, combined with another probability distribution for demand size

382 A. Vrrreckr, P. Versiraeten/fnt. J. Production Economics 35 (1994) 379-389

jan feb mar aPr may june jW au9 SeP

time

OCt "0" dec

(b)

I

6 7

0

0 1 2 3 4 5

units/demand

Fig. 2.

(1) Friend [S] briefly describes the use of a Poisson distribution ,for demand occurence, combined with demands of constant size.

(2) Hadley and Within [6] suggest the use of a Stut- tering Poisson distribution, which is a combination of a Poisson distribution ,for demand occurence, and a geometric distribution for demand size.

(3)Another possibility is the combination of a Pois- son distribution jh demand occurence, and a normal distribution .for demand size.

The combinations with the geometric or the normal distribution are quite complicated to work with although they are amongst the simplest combinations of Poisson occurence and any probability distribution of demand size.

Adelson [7] shows that the Stuttering Poisson is a special case of a whole class of compound Poisson distributions. He presents a recurrence relation for calculating the individual terms of these distributions.

Care must be taken in choosing the distribution function, as the use of a reorder point calculation

A. Vererckr, P. Verstraeimjlnt. J. Produclion Economics 35 (19941 379-389 383

based on false assumptions can have serious nega- according to their own judgment. As a result the

tive consequences. system had lost its credibility. Companies usually have different types of inven-

tories, such as raw material for production purposes, end items or spare parts, An analysis of a sample of all types of inventory items is necessary in order to take the right decisions.

Brown [S] states that, within one type of inventory, the form of the distribution function tends to be similar for most of the items. The inventory management system in use in the chemical plant on the contrary was based on the assumption that the spare parts had to be divided in two classes of items, with fundamentally different demand patterns.

We tried to improve the system by replacing the queuing line algorithm by a Poisson distribution and by experimenting with item classifications, based on different demand frequencies. The prob- lem with this experiment is that, in cases where the demand frequency of an item decreases but with its total yearly demand remaining unchanged, the Poisson distribution does not result in a new reorder point. The users did not agree with this outcome, as they realized that the average quantity per demand had grown. Furthermore, we experienced similar illogical results when the spare parts moved from one class of items to another.

3. The existing inventory management system

The existing inventory management system for the spare parts in the chemical plant classified all spare parts on the basis of their demand frequency. Spare parts with a low demand frequency (less than 4 times a year), the so-called lumpy items, were entered into a model based on queuing line theories for the calculation of the reorder point. The reorder point calculation for the slow and fast movers (with a demand frequency of 4 or more a year) was based on the normal probability distribution.

The general outcome of our experiments can be stated as follows: the definition of the different classes of items according to the demand pattern is arbitrary. Which item is to be regarded as a slow mover and which item is not? Moreover, the classification of items inherently causes problems for items which are ‘crossing the borders’ between classes. The Brown statement that the form of the distribution function tends to be similar for all items in the inventory has convinced us to look for an overall fitting distribution function.

This inventory management system showed illogical results for items which were crossing the classification border. When for example the yearly demand frequency of an item decreased from 4 to 3 (which implies that the item moved from the ‘slow class’ to the ‘lumpy class’ of items) with at the same time a decrease of total yearly demand the user expected the system to suggest a lower reorder point, but the system sometimes imposed an increasing reorder point.

4. The proposed demand distribution: A ‘Package Poisson’

4.1. The choice qf a distribution ,function

In case of a stockout the system recalculated the reorder point. It often happened that one unex- pectedly high demand caused a stockout for a fast mover, although the total yearly demand of the item was decreasing. In this case the recalculation resulted in a lower reorder point.

The best solution for this case probably was to gather enough demand history to create a personal frequency table for each spare part. However, the required data were not available. Moreover, such an inventory management model for 34000 items would certainly consume incredibly large quantities of CPU time. This was not a feasible solution. Therefore, we started searching for a single distribution function that could approximate the demand pattern of each spare part.

The users of the inventory system experienced these outcomes as illogical. As their disbelief in this inventory management system was growing they were often overwriting the system’s decisions

Analysis of the demand history data shows that the demand variance often largely differs from the average demand more specifically that the demand variance is a multiple of the average demand. This shows that the Poisson distribution is definitely not

3x4 A. Verrrckr. P. Ve~raeten~lnt. J. Production Economics 35 i 1994) 379-389

a good approximation of the demand but at the same time it suggests that the demand occurrence can be Poisson distributed. taking into account that on each occurrence a package of several units is asked for. AX’ Goodness-of-Fit test has proven that the combination of a Poisson distribution for demand occurrence and a constant demand size _ as proposed by Friend [S] - is indeed a useful distribution function for the demand pattern of the spare parts in this chemical company. We named this combination the ‘Package Poisson’.

4.2. Calculation of the reorder point using the ‘Package Poisson ’

Assume that the number of units in the package isf: The average demand expressed in numbers of packages (prackage) isftimes lower than the original value (pUnit), which was based on demand per unit (see Eq. (1)). The variance (cJ&,_), calculated on the basis of a number of packages isf’ times lower than the variance (Ou2nil) calculated on the basis of unit demands (see Eq. (b)). The average and the variance of the unit demand are calculated over a period of twelve months or more depending on the available data.

/-knit =.f*Ppackage (1)

02unit =.P*~pZackage~ (2)

The eqs. (1) and (2) lead to the calculation of

.f(Eq. (3))

f= (T,Znif/Punit. (3)

Demand quantities are expressed as a number of ,f-packages. The average demand value (ppackage), which is to be used as a parameter in the Poisson distribution is then found by dividing the average of the observational demand data by the factorf

&ackage = Punidf (4)

i.e., we are now calculating the Poisson probability of a number of demand packages instead of a number of units.

A correction has to be made to this &ackage as, in reality, demands do not occur in an integer number of packages. Rather unit demands can take any value in between two consecutive numbers of packages; Assuming that each unit demand in between two packages has the same probability of occurrence, we can use the iteration formula (5) to adapt the p-value in this respect. The iteration is stopped when the new value pi does not differ by more than 0.001 from the previous value pi_ 1

Pi = /&nit/f + (f - 1 )/2f*(l - exp( - Pi- 11). (5)

In order to calculate the probability of stockout, the average demand is to be considered over the lead time. Therefore, this final p-value is multiplied by the lead time and used as the mean value in the formula of the Poisson probability.

It is worth noting that the lead time is assumed to be constant in this model: this simplification was justified by the efforts made in this company to work together with the suppliers in trying to make lead times more reliable, and thus more constant. Thus lead time was considered a parameter to be influenced through managerial actions, rather than a statistically fluctuating variable.

The reorder point calculation is now performed in exactly the same way as is described in Section 2.2., i.e., by iteratively increasing its value until the required service level is provided.

In case this service level is not exactly met, inter- polation is used to find the exact number of packages to be used as reorder point. Suppose for example that a service level of 95% is required, and that calculations show that a reorder point of 10 packages provides a service level of 94%, and a reorder point of 11 packages provides a service level of 96%. In this case the reorder point will result in 10.5 packages.

As this reorder point is now expressed as a number of packages, one has to multiply it withf and round it up to the nearest integer to find the reorder point expressed in units.

The automatic inventory management system is conceptually based on this algorithm. However, multiple adaptations were necessary to obtain the final programming logic that performed well its inventory management goal in the real life case.

A. Vrrercke. P. V~~rstrrrc,tc~tl!Int. J. Production Gonornics 35 i 1994) 379 389 385

5. Measuring the performance of the algorithm: Methodology

The methodology for the performance measurement has been chosen from a practical real life point of view. In the end one could build a ‘personal’ frequency table for the demand pattern of each individual spare part, by recording all its historical demands. Derived from that frequency table one can then calculate an appropriate reorder point for each individual item. This concept of a ‘personal’ distribution function is optimally suited from a theoretical point of view but unpractical to realize due to excessive storage requirements to record large histories for thousands of items. Moreover, this method would consume high quantities of CPU. This reasoning has been maintained though to prove that the ‘Package Poisson’ described above can be used as an approximation of several other theoretical distribution functions, for inventory control purposes. In this way the performance measurement methodology chosen to evaluate the theoretical proposition, put forward in this paper, reflects the approach of the real life case.

Frequency tables of normal distributions with different means and variances have been generated. A series of 100 demands have been drawn random- ly from this demand distribution. These 100 demands have been used as the historical demand series (over 100 periods) for the calculation of the mean and variance, and hence of the package sizef: The corresponding reorder point for a chosen service level has been calculated. This reorder point has then been compared with the theoretical reorder point that can be derived directly from the frequency table as it is the demand level with a cumulative probability equal to the chosen service level.

For this demand series another 100 demands have been drawn from the frequency table, and have been considered as the demand during the next 100 order cycles in the future. The performance of the calculated reorder point is measured against this new demand series by counting the number of periods with demand greater than the calculated reorder point. We called this the ‘performance of the algorithm’, which represents the service level we could obtain by using the algorithm.

The same performance has been measured for the theoretical reorder point. We called this the ‘theoretical performance’, which will slightly differ from the desired service level due to the random effects in the sampling process.

This performance measurement process has been repeated a 100 times and the average and the standard deviation have been calculated for the algorithm’s reorder point and both the theoretical performance and the algorithm’s performance.

Service levels have been chosen within a wide range of realistic levels: 60%, 80%, 90% and 95%. As has been noted in section 2.1 the model does not take into consideration the effect of undershoot. As a consequence the service levels will be slightly overestimated. Two remarks have to be made here - In the case of the chemical company, the spare

parts have been classified in 4 groups, ranging from critical parts to parts of little importance. Each class has been attributed one service level (from 95% for the critical items, to 60% for those items that caused only minor trouble in case of stockout). This means that the exact degree of the service level was not a critical issue for this company.

~ The performance of the algorithm is being measured by comparing the reorder point calculated through the ‘Package Poisson’ with a reorder point calculated through another distribution function. The effect of undershoot impacts the service level in both cases in the same way, so the comparison remains valid. The same comparison as described above has

been made with probability distributions other than the normal distributions (a) A second choice has been the Poisson distrihu-

tion of demand. (b) A Stuttering Poisson distribution, the combina-

tion of a Poisson distribution for demand occurrence and the geometric distribution for demand size, has also been considered.

(c) Finally the algorithm has been tested against the combination of a Poisson distribution ,for

demand occurrence and a normal distribution ,fhr

demand size.

The results of these comparisons of the ‘Package Poisson’ with the other theoretical distributions can be found in Tables 1- 13.

6. Performance

The algorithm gives very satisfying results when the demand is generated by a Poisson distribution. Comparisons were made with average demand varying from IL = 1 to p = 25. The service levels generated by the algorithm do not deviate by more than 3% from the desired service levels. (Except for the lower service level of 60% and the small average (,u = 1). The cause of this deviation is that the difference between a reorder point of 0 and 1 auto- matically generates a service level of 74%.) See Tables l-4.

The reorder points do not deviate by more than 1 unit from the theoretical reorder points.

In situations with a demand generated by a normal distribution, the algorithm tends to give better results for the higher variance. Service levels do not deviate by more than 1% from the desired service levels. For the low variance and high service levels the calculated reorder points are too high when compared to the theoretical reorder points. For service levels of 80% or lower the algorithm provides good results (see Tables 5, 6).

Simulations of a stuttering Poisson process of demand generation were made for a Poisson mean of 3, and a parameter c for the geometric distribution of 0.3, 0.5. 0.7 and 0.8. All calculated reorder points do not deviate by more than 1 unit from the theoretical reorder points. Calculated and theoretical service levels lay within a range of the desired service levels plus or minus 3%. (see Tables 7710)

Table I Poisson: 11 = I (average values in bold, standard dewations in

italics)

Service ROP Performance

level Theoretical Algorithm Theoretxxl Algorithm

6O:‘o 1 1 74.05 74.05 0 5.17 5.17

80% 2 1.2 92.60 77.30 0.4 2.71 MY

90 %, 2 2 92.2 92.2 0 xx _7..?K

9 5 41, 3 2.35 97.85 93.25

0.4H 1.49 3.96

Table 2

Poisson: p = 5 (average values in bold, standard deviations In italics)


level Theoretical Algorithm Theoretical Algorithm

50% 5 4.95

(I.-72

80% 7 6.2 0.40

90% 8 7.7 0.46

95% 9 8.45 0.50

61.85 3.43

85.35 3.90

93.35 ,7.3.F

97.4

I.-74

61.30 4.42

77.70 6.40

90.75 S.36

95.05 3.43

Table 3 Poisson: p = IO (average values in bold, standard deviations in

italics)


level TheoretIcal Algorithm Theoretical Algorithm

60% II 10.2 69.6 59.9

0.4 4.X1 7.Y4

80% 13 12.00 86.1 78.9 0.45 -7.74 4%

90% 14 13.75 91.7 90.80 0.43 -7.5 3.-7x

95% 15 15.05 95.4 95.55

6.38 -7.-79 I .Hd

Table 4 Poisson: p = 5 (average values in bold. standard deviations In

italics)



50% 26 25.55 62.9 59.10 0.50 4.Y4 6YY

&O”/r 29 28.50 81.90 78.55

0.59 M9 4 Y4

90% 32 31.00 91.85 89.35 0.55 -7.37 -7.8.5

9 5 V0 33 32.85 96.00 95.65

0.48 1.x0 .? -76

A. Vereecke, P. Verstraeten/Int. J. Production Economics 35 (1994) 379-389 387

Table 5 Table 8

Normal: p = 25, c = 2 (average values in bold, standard devi- Stuttering Poisson: i. = 3, 11 = 0.5 (average values in bold, stan-

ations in italics) dard deviations in italics)



60% 26 25.85 79.15 76.4

0.36 4.15 7.16

80% 27 28.8 90.3 96.6

0.4 2.47 1.39

90% 28 31.05 96.95 100

0.22 1.47 0

95% 28 33 96.25 100

0 1.81 0



60% 6 5.90 60.85 60.05

0.54 3.34 5.42

80% 9 9.15 82.35 82.70

0.73 3.53 4.04

90% 12 11.75 92.40 91.20

0.89 2.15 3.89

95% 14 14.15 96.00 95.75

1.24 1.87 2.38

Table 6

Normal: p = 25, CT = 10 (average values in bold, standard devi-

ations in italics)

Table 9

Stuttering Poisson: 1 = 3, 18 = 0.7 (average values in bold, stan-

dard deviations in italics)



60% 28 27.25 63.9 61.00

1.04 3.65 5.24

80% 34 33.55 81.00 79.65

1.47 3.61 4.92

90% 38 38.05 89.5 89.1

1.94 2.38 4.19

95% 42 43.15 94.8 95.9

2. IO 1.73 2.47



60% 10 10.65 58.85 61.60

0.79 5.00 6.34

80% 16 15.90 82.75 82.70

1.04 3.10 4.10

90% 20 20.65 91.30 91.90

1.68 2.53 3.00

95% 24 24.70 95.60 95.80

1.98 1.88 2.29

Table 7

Stuttering Poisson: i = 3, v = 0.3 (average values in bold, standard deviations in italics)

Table 10

Stuttering Poisson: i. = 3, v = 0.8 (average values in bold, standard deviations in italics)



60% 5 4.30 72.55 62.75

0.56 3.80 9.50

80% 6 6.00 78.80 78.75

0.45 4.02 5.79

90% 8 7.85 90.10 89.30

0.48 2.61 2.51

95% 10 8.90 96.95 94.45

0.89 1.56 3.76



60% 16 16.10 62.95 63.05

1.14 3.38 5.46

80% 24 24.75 81.80 82.15

1.97 3.64 4.70

90% 31 31.30 90.40 90.90

1.95 2.24 3.22

95% 37 37.10 95.10 94.75

3.74 2.77 3.10

Table I I Poisson: i. = 2; Normal: /l = 10. CT = 2 (average values in bold, standard deviations in italics)


level

Theoretical Algorithm Theoretical Algorithm

60”/,, 2’ 21.80 61.70 59.95

1.50 5.70 6.48

X0% 32 32.55 81.40 81.40 2.27 5.57 6.0:

90% 40 39.65 90.75 90.70 X6 3.05 3.x7

95?<, 46 47.45 94.70 95.15

2.4: 2.69 2.76

Table I2

Poisson: j. = 3: Normal: = 5. /i 0 = 2 (average values in bold,

standard deviations in itahcs)



60% 16 16.25 60.65 61.75

1.18 5.53 6.74

80% 23 23.00 82.15 81.35 I.H-7 3.15 4.w

90”/1, 28 28.90 91.30 91.95 -7.47 2.59 3.97

9 s % 32 32.95 95.60 95.3s

.?.HY 1-w -7.Y.5

Finally, the combination of a Poisson process for demand occurrence (with mean A) and a normal process for demand size (with mean 11 and standard deviation (T) give the same satisfying results. Calculations have been made for i, = 2, p = 10, 0 = 2; for i. = 3, kl = 5 rr = 2; and for j. = 5, ~1 = 2, a= 1.

No significant difference has been found between the results of these three different simulation setups (see Tables 1 1 - 13).

7. Conclusions

The inventory of spare parts in the chemical plant described in this paper consists of a large

Table I3

Poisson: i = 5; Normal: p = 2, r~ = I (average values in bold, standard deviations in italics)


level

Theoretical Algorithm Theoretical Algorithm

60% II 10.95 64.40 63.45

0.74 4.04 7.93

80% I4 14.55 81.60 82.85

1.1-7 3.170 6.74

90% I7 17.85 91.10 91.65

1.71 2.45 4.32

950/u 20 20.65 94.65 95.00

1.w 1.74 -'.-'a

number of items, which can be classified as lumpy items slow movers and fast movers. The existing inventory management system was based on different statistical models for each class of items. This approach caused some problems, mainly due to the nontransparency of the system.

The model which we propose, a ‘Package Pois- son’, is based on a single distribution function for all items: we assume the demand occurences to be Poisson distributed whereas the demand size is approximated by a constant number, a ‘package’.

The reorder points and service levels resulting from this approach have been compared to several distribution functions with satisfying results.

Acknowledgement

With thanks to Professor Doctor H. Steyaert and Mr. P. Verbeeck from the University of Ghent for the statistical analyses of the distribution patterns of the items and the proposition of the ‘Pack- age Poisson’.

References

[I] Silver, E.A. and Peterson. R., 1985. Decision Systems for

Inventory Management and Production Planning. Wiley,

New York.

[I] Ehrhardt, R., 1979. The power approximation for comput- ing (s, S) inventory policies. Mgmt. Sci., 25(8): 777-786.

A. Vereecke, P. Verstraetcw/lttt. J. Production Economics 35 (1994) 379-389 389

[S] Dunsmuir, W.T.M. and Snyder, R.D., 1989. Control of

inventories with intermittent demand. Eur. J. Oper. Res., 40(l): 16-21.

[4] Croston, J.D., 1972. Forecasting and stock control for inter-

mittent demands Oper. Res. Quart., 23(3): 2899303. [S] Friend, J.K., 1960. Stock control with random oppor-

tunities for replenishment. Oper. Res. Quart., I l(3): 130. 136.

[6] Hadley, G. and Within, T.M., 1963. Analysis of Inventory

Systems Prentice-Hall, Englewood Cliffs, N.J.

[7] Adelson, R.M., 1966. Compound Poisson distributions. Oper. Res. Quart., 17( 1): 73- 75.

[S] Brown, R.G., 1977. Materials Management Systems, Wiley, New York.

[9] Croston, J.D., 1974. Stock levels for slow-moving items.

Oper. Res. Quart., 25(l): 123- 130.

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