Determining Reorder Points When Demand Is Lumpy
Author(s): J. B. Ward
Source: Management Science , Feb., 1978, Vol. 24, No. 6 (Feb., 1978), pp. 623-632
Published by: INFORMS
Stable URL: https://www.jstor.org/stable/2630837
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MANAGEMENT SCIENCE
Vol. 24, No. 6, February 1978
Printed in U.S.A.
DETERMINING REORDER POINTS WHEN
DEMAND IS LUMPY*
J. B. WARDt
Lumpy or sporadic demand patterns with highly skewed distributions are common in parts
and supplies types of stockholdings, and much of available inventory control methodology is
not appropriate for such items. This paper presents a simple, easily used regression model to
calculate order points for lumpy items from knowledge of demand parameters and the desired
service level. It is derived from the particular compound Poisson distribution commonly
called 'stuttering Poisson.' Results are derived and presented in the following framework,
though application is not limited to these conditions. The control discipline is the order
quantity, order point (Q, R) approach with continuous review. Lead time is assumed to be
known and constant. An assigned service level is assumed based on fraction of demand
supplied without backorder. Joint optimization of Q and R is not addressed, but rather order
point is based on an independently calculated order quantity. Forecasting methods are not
addressed, but the mean and variance of lead time demand forecasts are assumed available.
1. Introduction
The methods described in this paper were developed in the course of implementing
a computerized materials management system to administer stocks of parts and
supplies required for the maintenance and expansion of a utility system. A total of
some 30,000 stockkeeping units carried at 50 storing locations are involved. Former
methods involved a relatively simple accounting system, with stock replenishment
handled by individual storekeepers from manually maintained stock cards. Procedures specified crude guidelines for reordering based on month's of supply rules as
applied to last year's average activity. The new system involves a centralized data
base, more timely transaction processing, on-line inquiry capability, and, where
possible, au.tomatic requisitioning for stock replenishment from individual stock item
demand forecasts.
Analysis of historical accounting records brought into focus the problem of dealing
with erratic or lumpy demand patterns. Approximately half of the stock units report
no activity for a full year and are considered unforecastable. Of the remainder, 75% to
80% have such sparse and sporadic demand patterns that widely published methods
based on the assumption of normally distributed demand in a lead time cannot be
applied safely. In the total stockholding, approximately 90% of the annual dollar
volume is accounted for by about 10% of the stockkeeping units, and most of these
have demand characteristics which are amenable to well established forecasting
models and reorder policies. The main attention in this paper is devoted to the
troublesome 'lumpy majority.'
An excellent review paper by Silver [9] points out: "Most useable inventory control
procedures are based upon assumptions about the demand distribution (e.g., unit
sized transactions or normally distributed demand in a replenishment lead time) that
are invalid in the case of an erratic item. If this is not the case, the procedures tend to
be computationally intractable." When the simplicity of the normal distribution
cannot be utilized, most treatments of order point determination assume knowledge of
the discrete probability distribution of demand for each stock item. Such methods
were eliminated in the present project by lack of enough demand data as well as
computational requirements.
* Accepted by Edward J. Ignall, former Departmental Editor; received April, 1976. This paper was with
the author 8 month4, for 1 revision.
t Pacific Power & Light Company, Portland, Oregon.
623
Copyright ?) 1978, The Institute of Management Sciences
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624
J.
B.
WARD
The main contribution of this paper is a simple empirical formula to be used in
calculating order point, given in equation (16) below wherein the coefficients are
estimated by multiple regression. Lacking actual distributions to determine order
point values for the regression, demand is modeled by the stuttering Poisson distribution. This distribution results when customer arrivals are Poisson and the batch size of
individual customer requests is geometric. For convenience, stuttering Poisson will be
abbreviated by 'sP' in what follows. Discussion of this distribution and its application
in inventory control can be found, for example, in Feeney and Sherbrooke [3];
Gallagher [4]; Galliher, Morse, and Simond [5]; Jewell [7]; Sherbrooke [8]; Silver [9];
Silver, Ho, and Deemer [10].
Although this new order point formula may not be so limited, its presentation here
is restricted to the following framework:
1. Captive demand is assumed and all demands not satisfied are backordered.
2. The stock replenishment discipline is the order quantity, order point method,
(Q, R), with continuous review.
3. No optimal joint determination of order point and order quantity is attempted.
Precalculation of order quantity by the classical Wilson EOQ formula, or some
empirical adjustment of same, is assumed.
4. The control criterion used is an administratively set service level defined in terms
of 'fraction of demand satisfied without backorder'.
5. Replenishment lead times are known and constant.
6. Forecasting methodology is not treated. The mean and variance of lead time
demand are assumed to be available.
In what follows, ?2 reviews the properties of the sP distribution and introduces
some notation and terminology. ?3 indicates how reorder points are calculated from
tabulation of the lead time demand distribution and discusses problems introduced by
batch demands and overshoot of the reorder point. ?4 introduces the notion of a
limiting distribution and then develops the regression model and assesses its accuracy.
?5 compares the sP with the normal distribution and ?6 concludes the paper.
2. The Stuttering Poisson Distribution
This is a two parameter distribution. With a customer arrival rate of X customers
per unit of time, then the average number of customers appearing in an interval t is
Xt. Each customer, upon arriving for service, requests one or more units of the item,
the amount being given by the geometric distribution with parameter p. The average
batch size of the customer request is 1/(1 - p) units. Salient properties of the resulting
sP distribution of demand over an interval of length t are summarized briefly:
(1)
(2)
(3)
(4)
Mean value: m = Xt/(l - p),
Variance: v = Xt(1 + p)/(l _-p)2,
Coefficient of variation: C = ((1 + p)/Xt)1/2,
Ratio of variance to mean: C2m = (1 + p)/(1 -p),
where: X > 0, and 0 < p < l.
A useful recurrence relation given by Adelson [1] permits generation of an sP
distribution with given parameters. This is shown in equation (5), where Rn is the
probability that n units are demanded in a time interval of length t.
Ro= e-Xt
R = la jP 'Rn_j (S)
j=1
(When p = 0, the sum is replaced by Rn_ -1)
Given a time series of demand history accumulated over a sequence of equal time
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DETERMINING REORDER POINTS WHEN DEMAND IS LUMPY 625
intervals of length T, the parameters of an sP model can be estimated from the mean
and variance of the observations. Variance will be incorporated in the form of the
dimensionless coefficient of variation because this is a key parameter in the new order
point formula.
C2M ~~~~~~~~~~~(6)
C2rn- 1
C2M + 1
_
2m
T(C2m
+
(7)
1)
These parameters can then be used to produce an sP distribution giving the probabilities of demand occurrences during replenishment lead time L, which in general
will be different than the forecast update interval, T. In the lead time demand
distribution, relative to that associated with the update interval, the mean is directly
proportional to (L/ T) and the coefficient of variation is inversely proportional to the
square root of (L/ T). The distribution itself can be generated by substituting L for t
in (5).
In addition to the probability distribution, R, several related quantities are define
as follows:
Let Kn denote the cumulative probability that the demand during time interval t is
greater than n.
j=00 j=fn
Kn = E Rj= I1- , Rj; n = O, 1, 2, 3 . ... (8)
j=n+l
j=O
For example, given a reorder point of n units, then the probability of a stockout each
lead time is Kn as determined from the lead time demand distribution. This measure
ignores whether the possible shortage is only one unit or many.
An alternate service criterion considers the amount by which available stock may
be short of filling requests during the replenishment lead time. Let Sn denote th
expected value of demand in excess of n, or equivalently, the average amount short
each lead time for an order point of n units.
j=00
Sn = E (ji-n)Rj. (9)
j=n+ 1
This can be rewritten as:
j=n-1
Sn = m- n(1-R) + , (n-j)R,; n =1, 2, 3 ... (10)
j=1
SO= m.
It follows that a simple recurrence relation for Sn is:
Sn
=
S.-
I
n-
1(11)
For the purpose of comparing sP distributions having different mean values, it is
convenient to define two additional quantities which normalize Sn and n to the me
n=
Pn
With
order
Sn/M,
=
(12)
n/m.
point
equal
(13)
to
This follows since there is a one to one correspondence between number of lead times
and number of order cycles. It will be understood that here the word 'fraction' refers
to fraction of mean lead time demand. The companion value, Pn, will be called "order
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n,
626
J.
B.
WARD
point factor," i.e., the factor by which m is multiplied to equal the reorder point in
units.
3. Reorder Point Calculation
In preparation for introducing an empirical formula for P, the order point factor,
this quantity will first be derived exactly from the tabular form of an sP demand
distribution. For purposes of illustration, the unit of time is taken as a day, and
transactions are accumulated over 4-week (28 day) update intervals. A sample
distribution produced by equation (5) is shown in Table 1 in which the mean and
coefficient of variation are both equal to 2.0. The list is arbitrarily truncated at n = 30.
In this distribution if the time interval is 28 days or one update period, it models the
period demand of a customer stream with an average interarrival time of 63 days and
an average request per customer of 4.5 units. In what follows, the time interval, t is
assumed to be the lead time, and hence the distribution is appropriate to use in
establishing reorder level.
In Table 1, the third column lists the product of mean times the probability Rn for
all values of demand beyond n = 0. The reason for calculating this quantity is
deferred to the next section. The remaining columns display quantities previously
defined: K shows the cumulative probability of demand greater than n as in equation
(5); S is the expected value of demand in excess of n from equation (10), and F and P
are defined in (12) and (13).
TABLE 1
Sample Stuttering Poisson Distribution
Mean = 2, Coefficient of variation = 2 Parameters: p = 7/9, Xt = 4/9
n
R
m-R
K
S
F
P
0 0.64118 ******* 0.35882 2.00000 1.00000 0.0
1 0.06333 0.12665 0.29549 1.64118 0.82059 0.5
2 0.05238 0.10476 0.24311 1.34569 0.67284 1.0
3 0.04328 0.08655 0.19984 1.10258 0.55129 1.5
4 0.03571 0.07143 0.16412 0.90274 0.45137 2.0
5 0.02944 0.05888 0.13468 0.73862 0.36931 2.5
6 0.02425 0.04849 0.11043 0.60394 0.30197 3.0
7 0.01995 0.03990 0.09048 0.49350 0.24675 3.5
8 0.01640 0.03280 0.07409 0.40302 0.20151 4.0
9 0.01347 0.02694 0.06062 0.32893 0.16447 4.5
10 0.01105 0.02211 0.04957 0.26831 0.13416 5.0
11 0.00906 0.01813 0.04050 0.21875 0.10937 5.5
12 0.00743 0.01485 0.03308 0.17824 0.08912 6.0
13 0.00608 0.01216 0.02700 0.14517 0.07258 6.5
14 0.00497 0.00995 0.02202 0.11817 0.05909 7.0
15 0.00407 0.00814 0.01795 0.09615 0.04807 7.5
16 0.00332 0.00665 0.01463 0.07820 0.03910 8.0
17 0.00271 0.00543 0.01191 0.06357 0.03178 8.5
18 0.00222 0.00443 0.00970 0.05165 0.02583 9.0
19 0.00181 0.00361 0.00789 0.04195 0.02098 9.5
20 0.00147 0.00295 0.00642 0.03406 0.01703 10.0
21 0.00120 0.00240 0.00522 0.02764 0.01382 10.5
22 0.00098 0.00196 0.00424 0.02243 0.01121 11.0
23 0.00080 0.00159 0.00344 0.01819 0.00909 11.5
24 0.00065 0.00130 0.00280 0.01474 0.00737 12.0
25 0.00053 0.00105 0.00227 0.01195 0.00597 12.5
26 0.00043 0.00086 0.00184 0.00968 0.00484 13.0
27 0.00035 0.00070 0.00149 0.00784 0.00392 13.5
28 0.00028 0.00057 0.00121 0.00635 0.00317 14.0
29 0.00023 0.00046 0.00098 0.005 14 0.00257 14.5
30 0.00019 0.00037 0.00079 0.00416 0.00208 15.0
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DETERMINING REORDER POINTS WHEN DEMAND IS LUMPY 627
Given an sP model of the lead time demand distribution, such as the sample of
Table 1, the order point can be determined as follows. Assume that order quantity has
been established as Q units. Let H denote the assigned service level, defined as the
desired long run fraction of requests to be filled without back order. Its complement
(1 - H) is the corresponding target shortage fraction. This shortage fraction can be
equated to the ratio of average units short per order cycle divided by average demand
per order cycle, which is S/Q. This leads to:
S= (1-H)Q or F= (1-H)Q/m. (14)
In effect, this translates the assigned service level, H, into an equivalent target
shortage per order cycle, S, or shortage fraction per order cycle, F.
The tabular lead time demand distribution can be entered with either S or F to
determine the appropriate order point, n, directly. In this approach, P serves no
particular purpose, but the reason for defining it will emerge later. As an example, if
H = 0.95, Q = 5, and m = 2, then F= 0.125. Entering Table 1, order point is taken as
1, the smallest value of n having an F value equal to or less than the target value of
0.125.
This process of setting order point implicitly assumes smooth demand with items
being issued one at a time. With batch issues which occur in lumpy demand patterns,
two difficulties arise. These will be described briefly, together with suggestion of
approximate remedies.
First, equation (14) assumes that the order quantity, Q, produces separate, distinct
order cycles equal in average number per year to the annual demand divided by Q.
With batch demands it is possible that the available stock overshoots or is carried so
far below order point that some multiple of Q must be reordered to raise the stock
position above order point. This reduces the average number of order cycles and thus
the number of exposures to stockout, with a resulting alteration of the probabilities of
backorder on which the calculation of order point is based. To alleviate this distortion
it is suggested that a lower limit be imposed on the order quantity. An upper bound
on the expected value of order cycles per year is the expected number of customers
per year. Hence, a very conservative floor on order quantity would be the average
demand per customer, or 1/(1 - p). A multiple of 1.5 to 2.0 times this might be
appropriate. In this connection, there may be some merit in employing the "order up
to level" doctrine rather than that of fixed order quantity; however, this approach has
not been explored in the present study.
Even if (14) properly relates a target service level to the chance of shortage per
order cycle, another problem arises when lumpy demand overshoots order point. If
order point is based on unit issues, then when a batch issue overshoots order point the
stock is deficient at the very beginning of the replenishment lead time and the
probability of shortage is greater than that implied by the assigned service level. An
approximate remedy here is to raise order point and thus augment safety stock by an
amount equal to the average overshoot. Silver [9] points out that with geometric
distribution of batch size, the overshoot distribution is geometric with average value
one less than the average request per customer, i.e: p7(1 - p) or (C2m - 1)/2. (An
exact procedure would involve the convolution of the overshoot distribution with the
original sP distribution of demand in a lead time.) In the remaining discussion of
order point in this paper, such adjustment for overshoot is ignored for simplicity. It
can be easily reintroduced in any implementation of results described here.
In relatively common circumstances, equation (14) may lead to an order point less
than the mean lead time demand, resulting in negative safety stock, or even to a value
of F greater than unity. This might occur in the presence of a short lead time and a
relatively long order cycle. In any real sP distribution, the value of F cannot be
greater than unity, and such result arising from equation (14) means that the target
service level is satisfied by a reorder criterion of 'order when you run out'. It is
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628
J.
B.
WARD
assumed that materials management policy would override the probability algorithm
in this region, and perhaps impose a lower limit on order point at average lead time
demand, which would amount to specifying a minimum of zero safety stock.
4. Approximate Reorder Point Calculation
The use of an sP distribution model as produced by equation (5) on an item by item
basis would be impractical in a large stockholding. This difficulty is surmounted by an
empirical formula presented here, wherein a limited number of sP distributions
covering a range of parameter values serve to provide observations for a multiple
regression fitting process. The key to a tractable model is the discovery of a way to
essentially eliminate the mean value of the demand distribution except as it is
embedded in the calculation of the coefficient of variation. This is described next.
4.1 Evidence and Conjecture on a Limiting Distribution
Examination of a number of sP distributions led to the observation that for a fixed
coefficient of variation but with increasing mean value, a sequence of sP distributions
appears to converge to a fixed relative shape. That is, at a fixed value of P, given C,
the corresponding value of F, for example, tends toward a constant as m increases.
Examination of some data will assist in making more specific what is meant by the
suggestion of a limiting relative shape.
Table 2 summarizes data taken from six sP distributions having mean values
ranging from 1.0 to 160, but all with the same coefficient of variation of 1.0. General
observations and conclusions which will be discussed in relation to this set of data
have been found to prevail over a wide range of coefficient of variation values, both
greater than and less than the value of unity illustrated here.
TABLE 2
Comparison of Six sP Distributions with C = 1.0 and Different Mean Values
Order Point Factor P
Quantity
Mean
0
0.5
1.0
2.0
3.0
5.0
R 1 0.36788 - 0.36788 0.18394 0.06131 0.00307
4 0.20190 0.11888 0.08986 0.03945 0.01417 0.00131
8 0.16901 0.05958 0.04475 0.01959 0.00707 0.00067
20 0.14886 0.02384 0.01788 0.00782 0.00283 0.00027
80 0.13872 0.00596 0.00447 0.00196 0.00071 0.00007
160 0.13703 0.00298 0.00223 0.00098 0.00035 0.00003
mR 1 - - 0.36788 0.18394 0.06131 0.00307
4 0.47551 0.35942 0.15779 0.05669 0.00526
8 0.47661 0.35799 0.15674 0.05655 0.00535
20 - 0.47688 0.35758 0.15646 0.05652 0.00537
80 0.47692 0.35751 0.15640 0.05651 0.00537
160 0.47693 0.35750 0.15640 0.05651 0.00537
K 1 0.63212 - 0.26424 0.08030 0.01899 0.00059
4 0.79810 0.55001 0.35520 0.12951 0.04170 0.00340
8 0.83099 0.57687 0.37503 0.13856 0.04524 0.00380
20 0.85114 0.59394 0.38770 0.14420 0.04736 0.00403
80 0.86128 0.60273 0.39427 0.14709 0.04842 0.00413
160 0.86297 0.60422 0.39538 0.14758 0.04859 0.00415
F 1 1.0 - 0.36788 0.10364 0.02334 0.00069
4 1.0 0.63325 0.38448 0.12877 0.03919 0.00297
8 1.0 0.63366 0.38543 0.12993 0.03991 0.00311
20 1.0 0.63377 0.38570 0.13025 0.04011 0.00314
80 1.0 0.63379 0.38575 0.13031 0.04015 0.00315
160 1.0 0.63380 0.38575 0.13031 0.04015 0.00315
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DETERMINING REORDER POINTS WHEN DEMAND IS LUMPY 629
As shown by column headings in Table 2, entries in the table are listed at fixed
values of P ranging from zero well out into the tail of the distributions. Labels along
the left hand border of the array show that four quantities are listed in the body of the
table, each from the same set of six different distributions with mean value arranged
in ascending order.
Examine first the values of R. The main interest here is in the first column which
lists the six values of Ro = exp( - Xt). The limiting values of Ro are easy to estab
At the low extreme of p= 0, Xt is equal to m, the mean. At the other extreme, as
p -* 1, then m -* oo and inspection of equation (7) shows that concurrently Xt - 2
For the case of Table 2, the limiting value of Ro for large mean is exp(-2)=
0.135335.
The next group lists the product of mean times R, except at values of Ro which were
discussed above. Although no use is made of the result in the application, as a matter
of interest the table indicates that at fixed P greater than zero, and with increasing
mean, the values of R tend to be inversely proportional to the mean.
In the two remaining groups, K and F, at fixed values of P the values tend toward
constants with increasing mean, the effect seeming to be more robust in the shortage
fraction per order cycle, F. (Note that in the distribution with mean value of unity, no
values exist for P = 0.5.) As the mean increases, the discrete probability values
become more dense between any two fixed values of P. It appears that the envelopes
of these distributions conform closely to the same curves for values of the variance to
mean ratio, C 2m, greater than about 8. It can be shown that the coefficient of
skewness of the sP distribution is:
7=C
(3_
-
(15)
As C2m increases this converges rapidly toward the constant 1.5C in a sequence of
distributions having the same coefficient of variation. As one measure of shape, this
performance of the relative skewness encourages the conjecture that the convergence
visible in Table 2 is dependent on C2m.
4.2 A Regression Model for Order Point Factor (P)
The results shown in Table 2 led to the concept of estimating a multiple regression
model for P as a function of F and C. A set of 62 observations were selected from sP
distributions having relatively large values of C2m, ranging from 40 to 80. Values of F
covering a range from 0.5 to 0.002 were selected, together with their corresponding
values of P, from seven distributions having values of C2 equal to 0.1, 0.25, 1.0, 4.0,
9.0, 16.0, and 25.0. The model finally selected is:
P = B1 + B2C + B3C2 + (B4 + B5C + B6C2)In F + (B7 + B8C)(ln F)2 (16)
The resulting Bi coefficients are given in Table 3.
TABLE 3
Regression Coefficients for Fitted Model,
Equation (16)
B1 = 0.322358 B5 = -0.149687
B2 = -0.212598 B6 = -0.475839
B3 = 0.0318138 B7 =-0.024474
B4 = -0.306230 88 = 0.0054646
Sum of squared residuals = 0.1663
Coefficient of determination = 0.999989
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630
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B.
WARD
The largest residuals in terms of per cent error were about 5.7%. These appeared in
the middle of the range of values selected from the distribution with C2 = 0.1. This
represents relatively smooth demand and is approaching normality. Over the rest of
the region, error is in the order of 2% or less, with the best relative fit in the more
lumpy area.
4.3 A ccuracy
More sP distributions were created to test the applicability of the regression model
over a wider range than the C2m of 40 to 80 used in the fitting process. Table 4
presents the results. The body of this table compares order points estimated from the
model with exact values read from the distributions. The column headings show the
range of C2m and companion values of p. As shown along the left margin, the
coefficient of variation ranges from C2 = 0.1 to C2 = 25. In each of the 24 distributions, the order point comparison is made at the three target values of F equal to 0.5,
0.05, and 0.005. The mean value of each test distribution is also listed.
TABLE 4
Accuracy Test
Comparison of order points calculated from regression model with exact values
obtained from sP distributions
Distribution Parameters
C2m=1
p
=
0
4
0.6
9
199
0.8
0.99
C2 F EXACT EST. EXACT EST. EXACT EST. EXACT EST.
0.1
0.5
0.05
6
6
13
0.005
21
13
17
21
49
18
46
51
67
48
110
70
1002
113
149
1046
2432
2500
3290
3474
157
m = 10 m = 40 m = 90 m = 1990
0.25
0.5
0.05
3
6
0.005
9
0.5
0.05
9
0.005
4
m=
4.0
0.5
0.05
0.005
2
3
16
3
4
1
19
1
2
8
4
2
8
13
=
7
7
2
13
1798
1801
36
=
796
155
561
9
4
4
550
925
m=
17
30
m
148
41
m=
446
1216
25
42
17
422
1221
82
m
26
19
m=
1
4
11
21
55
82
=
12
5
20
56
37
m
1
3
9
25
37
4
1
3
9
25
m=
1.0
3
7
905
199
87
371
30
646
88
370
644
m = 0.25 m = 1 m = 2.25 m = 49.75
9.0
0.5
0.05
0.005
2
2
1
1
2
7
2
3
12
7
2
15
12
4
4
15
27
77
331
27
583
77
332
585
m = 0.111 m = 0.444 m = 1 m = 22.111
25.0
0.5
0.05
0.005
1
2
1
1
2
7
3
11
2
7
2
14
11
4
14
25
4
72
310
25
548
78
310
548
m = 0.04 m = 0.16 m = 0.36 m = 7.96
In arriving at the exact order point from the tabular distributions, the smallest
integer n having a table value of F less than or equal to the target value of F is used.
Estimated order point is obtained by multiplying P by the mean, rounding to the first
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DETERMINING REORDER POINTS WHEN DEMAND IS LUMPY 631
decimal place, and if a fractional part remains, taking the next larger integer. The
model gives reasonably satisfactory results even in the low C2m range.
5. Comparison of sP and Normal Distribution
Brown [2] utilizes the normal partial expectation function E(k) together with service
level, H, as defined here, to establish order point and safety stock in terms of a safety
factor, k. It is of interest to reconcile this approach with the concept of "order point
factor" as defined earlier, and to indicate the general magnitude of differences which
can arise from applying the assumption of normality to skewed lead time demand
distributions.
By Brown's approach:
(1-H)Q
E(k)= - -(I - H ,Q (16)
OP
=
m
+
ka,
(17)
where: OP is order point in units, Q is order quantity in units, m is mean lead time
demand, a is standard deviation of lead time demand, k is the 'safety factor' as
derived from E(k) by tables or empirical formula, and E(k) is the normal partial
expectation function.
Dividing both sides of (17) by m, we obtain an expression fol order point factor:
P= 1 + k/rm= 1 + kC. (18)
Herron [6] has published a convenient empirical formula for evaluating k, given
E(k):
k = 4.85 - 0.3924E(k) 3 - 5.359E(k)? 135. (19)
Comparing (16) with (14), we see that E(k) = F/C, and upon substituting this and
(19) into (18), equation (20) gives order point factor in terms of coefficient of variation
and shortage fraction per order cycle for normally distributed lead time demand.
P = 1 + (4.85 - 0.3924(F/C)'13 - 5.359(F/C )035)C. (20)
This is to be compared with equation (16) and the coefficients of Table 3 for the sP
distribution of lead time demand. Table 5 reveals this comparison over a region of C
and F. The normal distribution always produces a smaller order point factor, and can
be drastically deficient under certain conditions.
TABLE 5
Comparison of Order Point Factors from the Normal Partial Expectation
and the sP Distribution
(Upper Value from Normal, Lower from sP Distribution)
F
C
0.5
0.002
N
2.15
sP
1.0
N
0.01
1.84
2.49
3.53
0.05
1.45
2.07
2.97
0.1
1.25
1.53
2.67
0.5
1.26
-
1.90
sP 5.20 4.03 2.76 2.19
2.0
N
6.48
5.46
4.18
3.53
1.68
sP 15.09 11.29 7.42 5.73 1.76
3.0
N
sP
9.56
30.97
8.11
23.00
6.29
14.99
5.38
2.81
11.53
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3.46
632
J.
B.
WARD
6. Conclusion
The main contribution here is the regression model of Equation (16), with the
coefficients of Table 3, which is offered to facilitate the easy calculation of order
points. Clearly, the applicability of this result in practice rests on the matter of
whether the real-world demand distribution is sufficiently similar to the stuttering
Poisson model, from which the formula is derived. If the lead time demand distribution is known to be otherwise, and the item is financially important, such knowledge
should be exploited, if possible. Numerous circumstances might lead to a lumpy
demand pattern unlike that implied by the sP model with the same mean and
variance, but often data is sparse and little is known of the underlying structure of
demand distributions. It appears that the sP model and the results derived here offer
an intuitively plausible and easily usable approximate solution.
References
1. ADELSON, R. M., "Compound Poisson Distributions, " Operational Research Quarterly, Vol. 17, No. 1
(March 1966), pp.73-75.
2. BROWN, R. G., "Decision Rules for Inventory Management, " Dryden Press, Hinsdale, Illinois, 1967,
pp. 172-173.
3. FEENEY, G. J. AND SHERBROOKE, C. C., "The (s - 1, s) Inventory Policy Under Compound Poisson
Demand," Management Science, Vol. 12, No. 5 (January 1966), pp. 391-411.
4. GALLAGHER, D. J., "Two Periodic Review Inventory Models with Backorders and Stuttering Poisson
Demands," AIIE Transactions, Vol. 1, No. 2 (June 1969), pp. 164-171.
5. GALLIHER, H. P., MORSE, P. M., AND SIMOND, M., "Dynamics of Two Classes of Continuous Review
Inventory Systems, " Operations Research, Vol. 7, No. 3 (May-June 1959), pp. 362-384.
6. HERRON, DAVID, "Industrial Engineering Application of ABC Curves," American Institute of Industrial
Engineers, Vol. 8, No. 2 (June 1976).
7. JEWELL, W. S., "The Properties of Recurrent Event Processes, " Operations Research, Vol. 8, No. 4
(July-August 1960), pp. 446-472.
8. SHERBROOKE, C. C., "Discrete Compound Poisson Processes and Tables of the Geometric Poisson
Distribution, " Memorandum RM-4831-PR, RAND Corporation, Santa Monica, July, 1966.
9. SILVER, E. A., "Some Ideas Related to the Inventory Control of Items Having Erratic Demand
Patterns, " CORS Journal, Vol. 8, No. 2 (July 1970), pp. 87-100.
10. , Ho, C. M., AND DEEMER, R. L., "Cost-Minimizing Inventory Control of Items Having a
Special Type of Erratic Demand Pattern," INFOR, Vol. 9, No. 3 (November 1971), pp. 198-219.
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