Directory UMM :Data Elmu:jurnal:I:International Journal of Production Economics:Vol68.Issue3.Dec2000:
Int. J. Production Economics 68 (2000) 307}317
The (R, Q) inventory policy subject to a compound Poisson
demand pattern
Peter Matheus, Ludo Gelders*
K.U. Leuven } Centre for Industrial Management, Celestijnenlaan 300 A, 3001 Heverlee, Belgium
Received 26 March 1998; accepted 13 September 1999
Abstract
Most inventory management models are based upon rather restrictive assumptions, e.g. unit sized demands and the
normal distribution for total demand during replenishment time. In a majority of inventory management systems,
circumstances seem to allow these simpli"cations, and inventory policies based upon these assumptions yield satisfying
results. However, in some particular cases, these simpli"cations di!er fundamentally from the actual conditions and
particle. Therefore, application of the models mentioned above can result in an overinvestment in inventory or in an
unacceptable low service level. One of the situations in which we cannot rely on these simpli"ed inventory models is
studied in this paper. We consider an inventory subject to a probabilistic non-unit sized demand pattern, and we propose
an exact and an approximate reorder point calculation method for the (R, Q) inventory policy. The exact algorithm
involves formulas for the discrete distributions of the total demand during replenishment time and the undershoot. The
approximate method is based on the use of continuous distributions, and will be more appropriate when historical data
are sparse. Results of both approaches are compared. The algorithms proposed in this study are simple, fast and easy to
implement in a variety of (existing) inventory management systems. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Inventory management; (R, Q) inventory policy; Reorder point calculation method
1. Introduction
The situation studied in this paper is a very
familiar one in inventory management: we consider
a central stock with orders arriving from a variety
of customers (Fig. 1). The customer order size is not
"xed, but follows a discrete probability distribution. The central stock could be anything, from
a central inventory of spare parts in a chemical
plant to a supermarket.
* Corresponding author. Tel.: #32-16-32-25-66; fax: #3216-32-29-86.
E-mail address: [email protected] (L. Gelders).
Normally, this kind of inventory is managed by
an order point and order up to level system, the
(s, S) policy (e.g. [1,2]). In general, this policy o!ers
the best way to deal with large demands crossing
the reorder point s, causing substantial undershoots. However, in some circumstances, the inventory manager has to use "xed reorder quantities Q,
e.g. if the supplier uses standard packings containing a "xed number of units, or if the `suppliera is a
internal production process with a "xed lotsize.
In this case the inventory may be managed by a
(R, Q) policy, which will be determined in this
paper.
We propose calculation methods for the reorder
point R of a (R, Q) inventory policy with a target
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 1 1 0 - 3
308
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
Fig. 1.
service level, subject to an non-unit sized demand
pattern. The reorder quantity is known (e.g. the
EOQ or a "xed quantity determined by other considerations). The demand pattern is compound
Poisson. This means that demand arrivals constitute a Poisson process (the interarrival times are
exponentially distributed), and that the individual
demand size follows some unspeci"ed discrete distribution. The Poisson assumption is appropriate if
the customer population consists of a large group
of individuals acting independently. Furthermore,
we assume the leadtime ¸ between supplier and
inventory to be "xed. In case of stockouts backlogging is applied, and the inventory level is continuously monitored.
The service level b, used in this study, is de"ned
as the steady-state percentage of the total number
of units requested met directly from stock on hand.
Assume that a cycle is the time elapsed between two
consecutive moments at which a replenishment order is received. Then the following basic formula
may be used for the calculation of b, though standard theory of regenerative processes cannot be
applied [2]:
b"1!
E(demand that goes short in one cycle)
E(total demand in one cycle)
(1)
with E denoting the expected (or average) value of
the expression between brackets.
As the backordering situation is considered in
this study, the average demand per cycle equals the
average amount received per cycle. Thus,
E(total demand in one cycle)"Q.
Let us de"ne following quantities:
B "E(shortage present at the beginning of a
1
cycle),
B "E(shortage at the end of a cycle).
2
This means that B is the expected shortage just
1
after a replenishment order has been received and
B is the expected shortage just prior to the arrival
2
of a replenishment order. Thus,
b"1!
(B !B )
2
1 .
Q
(2)
With u being the stochastic quantity representing
the undershoot (the amount by which the reorder
point is crossed when the reordering is triggered),
and d being the stochastic quantity representing
L
the leadtime demand, it is clear that
B "E([u#d !R!Q]`),
1
L
(3)
B "E([u#d !R]`).
2
L
(4)
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
An exact algorithm for the computation of B and
1
B (based on a discrete stochastic model and tested
2
by means of simulation), and an approximate
method (based on a continuous stochastic model)
will be proposed. Results obtained by both
methods will be compared for some testcases.
2. The discrete model
The basic problem in developing closed expressions for B and B is to "nd the discrete probabil1
2
ity distributions for the undershoot u and the
demand during leadtime d .
L
For the calculation of the distribution of d ,
L
Adelson's recursion scheme [3], can be applied. Let
j be the arrival rate of the demands, ¸ the ("xed)
leadtime, and U the probability of receiving a dei
mand of size i (i"1, 2,2, m, with m the maximum
size). The probability r of having zero demands
0
during the leadtime is given by
(5)
r "exp(!j¸).
0
The probability r of having a total leadtime dek
mand of k units can be obtained by the following
recursive formula:
j¸ k~1
r "
+ (k!j)U r ,
k
k~j j
k
j/0
309
Now that closed expressions for the discrete distributions of d and u are available, only one probL
lem remains to be solved: in formulas (3) and (4)
a summation of the two stochastic quantities
d and u appears. This means one needs the convoL
lution of these two distributions. Therefore we calculate f , the probability that the summation of the
k
undershoot and the total demand during leadtime
in a stockcycle equals k:
k
f " + m r , k"0, 1, 2,2, R.
(8)
k
j k~j
j/0
For reasons that become clear later, one only has to
calculate this distribution for values of k from 0 to
R#Q.
Now it is clear that
=
B " +
(k!(R#Q)) f ,
1
k
k/R`Q`1
=
B " + (k!R) f .
2
k
k/R`1
Since
b"1!
(9)
(10)
(B !B )
2
1 ,
Q
some algebra leads to
k"1, 2,2, R.
(6)
This recursion scheme o!ers an e$cient and numerically stable calculation method.
For the determination of the undershoot distribution, we use the formula proposed by Karlin [4]
and Silver et al. [5]. Let E(d) denote the average
demand size, then m , the probability of having an
k
undershoot of k units during stock cycle, can be
calculated by:
R`Q
R`Q
(11)
b" + f !1 + (k!R) f .
k
k Q
k/R`1
k/0
The procedure described above, resulting in this
simple formula, o!ers a calculation method for the
service level b for a given reorder point R.
The exactness of this algorithm was tested
by means of a discrete event simulation [6].
The following assumptions were made:
j"1,
1
m "
k E(d)
m
+ U , k"0, 1, 2,2, m!1.
j
j/k`1
(7)
Though this formula was proposed as an approximation of the undershoot distribution in an inventory controlled by a (s, S) policy, we proved
it to be exact in the case of the (R, Q) policy
(Appendix A).
¸"5 (except for the `outliersa case, then ¸"2),
Q"200.
By generating seven di!erent demand size distributions, each with a di!erent target service level (for
three cases 99%, for two cases 95% and for to other
cases 90%), seven di!erent testcases were obtained.
310
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
The distributions were generated as follows: in
four cases, we took a well-known continuous probability distribution (normal, exponential, gamma
and Weibull). They were truncated at 0.5 and 20.5,
normalized and discretized. For two other cases,
the `bimodala and the `outliersa, a normalized
superposition of the probability density functions
of two normal distributions was used (representing
two types of customers). In the `bimodala case
these distributions were rather close to each other
(the two types of customers have only a slightly
di!erent average demand size). In the `outliersa
case there was a big gap between both distributions
(two very di!erent types of customers). The resulting distribution again was truncated (the `bimodala at 0.5 and 20.5, the `outliersa at 0.5 and
105.5), normalized and discretized. In the `randoma
case, a sequence of 20 normalized random numbers
was generated, representing the probability of having a demand size of 1}20 units.
After the determination of these discrete distributions, we calculated (in an enumerative way) the
reorder point R, required to attain the predetermined target service level b.
Since the reorder point must be integer, the
actual service level always was a bit higher than
the target value. Afterwards these results were
tested by means of a discrete simulation program.
For every case 30 simulation runs were performed
(except for the `normala case where we used
173 runs), and every run consisted of 4000 stock
cycles. An average service level and a con"dence
interval (90%) were calculated. As one can see
in Table 1, the results obtained by application
of the algorithm proposed above, fall very well
within the (narrow) limits of these con"dence
intervals.
The results from three simulations con"rm that
the calculation method described above is exact.
Furthermore, this algorithm is simple, straightforward and numerically stable.
However, even though the formulas are easy to
program, the implementation of this inventory control model in new or existing stock management
systems could cause some problems. The management of data, the necessary input for the algorithm,
could be fairly complex. The inventory management software needs support and input from
a database containing the arrival rate j, the
leadtime ¸, the reorder quantity Q, and the discrete
order size distributions for all the products in the
inventory under consideration. This database
should be updated with every transactions, and as
demand sizes may vary considerably, such a
database could be very memory consuming.
So, though the model proposed above could be
very useful for critical items, or items with rather
compact order size distributions, or simply (because of its exactness) for benchmark calculations
in tests for approximate algorithms, we should try
to "nd a good approximate model with less data
requirements. In the next section an approximation
based upon a continuous stochastic model is proposed.
3. The continuous model
As stated previously, the main drawback of
the discrete model is the storage of the discrete
Table 1
Case
Normal
Exponential
Gamma
Weibull
Bimodal
Random
Outliers
Demand size distribution
Average
value
Standard
deviation
10.00
7.38
10.71
9.88
10.50
11.77
19.00
2.02
5.25
4.42
0.33
4.59
5.05
27.15
Target
service
level (%)
Reorder
point
Discrete
model
(%)
Simulation
Average
value (%)
90%-con"dence
interval (%)
99
95
90
99
90
99
95
80
39
44
78
42
102
106
99.009
95.164
90.316
99.029
90.024
99.026
95.073
99.011
95.175
90.327
99.040
90.016
99.031
95.063
98.928
95.024
90.018
98.957
89.700
98.922
94.727
99.094
95.326
90.636
99.123
90.332
99.139
95.398
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
probability distributions. Therefore, we build an
approximate continuous model, using approximate
continuous probability distributions. For reasons
that will become clear, the normal and the gamma
distributions are applied. For these distributions,
one only has to store the average value and the
standard deviation. This study is based on results
obtained by Tijms [2] for an inventory under periodic review, managed by a (s, S) policy.
Again closed expressions for B and B have to
1
2
be developed. This means one has to "nd the distribution of the summation of the undershoot u and
the total leadtime demand d .
L
Assume that k is the average demand size, p is the
standard deviation of demand size, and D is the
L
stochastic quantity denoting the summation of total leadtime demand, d , and the size of the demand
L
that triggers the reordering-decision. Let f (x) be the
approximated continuous probability density function (pdf) of u#d , g(x) the pdf of d and h(x) the
L
L
pdf of D . Then one can easily prove by extension
L
of results obtained by Tijms [2] that
1
f (x)+ (P(d )x)!P(D )x)), x*0
L
L
k
(12)
(with P denoting the probability of the logical expression between brackets), and assuming that the
pdf of the demand size has a "nite third moment:
P
=
C
1
(x!C) f (x) dx"
2k
AP
P
!
=
(x!C)2h(x) dx
C
=
(x!C)2g(x) dx
C
B
(13)
with C some constant positive real number (C*0).
Thus,
B "E([u#d !R!Q]`)
1
L
P
"
=
(x!R!Q) f (x) dx
AP
=
(x!R!Q)2h(x) dx
R`Q
P
!
=
R`Q
and
1
B "
2 2k
AP
=
(x!R)2h(x) dx
R
P
!
=
B
(x!R)2g(x) dx .
R
(15)
The only remaining problem is the calculation of
the de"nite integrals. By a proper choice of the
distributions g(x) and h(x), this turns out to be an
easy task. In this study the normal distribution and
the gamma distribution (for which the integrals
become incomplete gamma integrals) are used. In
both cases fast codes are widely available. The
parameters of both approximate distributions remain to be determined. This can be done by matching of the "rst two moments, the average value and
the standard deviation. Thus, g(x) is de"ned by
E(d )"jk¸,
L
p2(d )"j¸(p2#k2)
L
(16)
and h(x) is de"ned by
E(D )"k#E(d ),
L
L
p2(D )"p2#p2(d ).
L
L
(17)
This approximate algorithm was tested for several
cases.
Again discrete demand size distributions were
generated in the same way as in the previous section, and for these distributions p and k were calculated. The following assumptions were made:
j"1,
¸"5 (except for the `outliersa case, then ¸"2),
Q"200.
R`Q
1
"
2k
311
B
(x!R!Q)2g(x) dx
(14)
With the exact discrete algorithm, the reorder point
R, required for the target service level b (90%, 95%
and 99%), and the actual service level were calculated. For this reorder point R, the service level
b again was determined by means of the approximate continuous algorithm described in this section.
The results of these tests can be found in Appendix B. For the approximation with the normal
312
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
distribution, the relative error lies between a minimum of 0.017% and a maximum of 2.204% and
reaches an average value of 0.393%. The worst
results were obtained in the `outliersa case. The
gamma approximation seems to yield better results:
the relative error lies between 0.001% and 0.769%
with an average value of 0.145%. Again the worst
cases can be found among the `outliersa. The inferior performance of the Normal approximation
may be due to the `taila of this distribution in the
negative domain.
In general, the results are satisfying and certainly su$cient for most practical purposes.
Therefore this algorithm, based on a continuous
stochastic model, forms an interesting alternative
for the exact, but memory consuming, discrete
algorithm.
m
+ U , u"0, 1, 2,2, m!1, (A.1)
j
j/u`1
with E(d) the average demand size, U the probability
j
that a demand size equals j units ( for j"1 to m), and
m the maximum demand size (with m(Q).
4. Conclusion
Proof. The inventory is modelled as a continuous
Markov chain (see Fig. 2).
In this paper an exact and e$cient algorithm is
proposed for the reorder point calculation of an
inventory managed by an (R, Q) policy, subject to
a compound Poisson demand. This algorithm is
based upon Adelson's recursion scheme [3] and an
undershoot distribution formula that was proved
to be correct. Though the algorithm could easily be
implemented in a new or existing inventory management system, the data requirements could pose
some practical problems. So, when dealing with
very critical items, or in cases where vast demand
size data are available, or simply when an exact
benchmark calculation is needed, this discrete
model yields a simple and fast solution.
Whenever historical data are nor present and the
inventory manager can only give a rough estimation of the form of the demand size distribution (the
average value and the standard deviation), or
whenever one cannot dispose of a su$ciently large
database, the approximate continuous model, proposed in the last section of this paper, can be
applied. This model is an extension of results obtained by Tijms [2] for the periodic review (s, S)
policy. It is based upon an approximation of the
discrete distributions with normal or gamma probability density functions. Results from a variety of
tests clearly indicate that the performance of this
continuous model is su$cient for most practical
situations.
Appendix A
Theorem. In a continuously monitored inventory,
controlled by a (R, Q) policy and subject to a compound Poisson demand pattern, the probability m of
u
having in a stock cycle an undershoot of u units
equals:
1
m "
u E(d)
In this chain three types of states are de"ned:
(z) for z"R#1 to R#Q!m,
(z)@ and (z)A for z"R#Q!m#1 to R#Q
with z denoting the stock position. So every stock
position between R#1 and R#Q!m de"nes one
state, the stock positions between R#Q!m#1
and R#Q de"ne two separate states. The di!erence between the states (z)@ and (z)A will become
clear later. The steady-state probabilities are p(z),
p((z)@) and p((z)A).
The transitions in this chain are de"ned by stock
movements. Two types of stock movements are
considered. The "rst type is caused by the arrival of
a demand that does not cross the reorder point.
This means that the original stock position is
lowered by the demand size. The second type is
caused by a demand that crosses the reorder point
R, and thus increases the stock position with the
reorder quantity Q minus the demand size.
The demands causing these stock movements
(and transitions), arrive as a Poisson stream with
parameter j. This stream is split into m separate
streams, according to the actual demand size. The
Markov property still holds after splitting, since
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
313
Fig. 2.
splitting of a Poisson stream (under probabilities
U , U ,2, U ) generates m new Poisson streams
1 2
m
with parameters U j (i"1, 2,2, m). These parai
meters U j de"ne the transition rates in the coni
tinuous Markov chain (see Fig. 2).
Now the di!erence between the states (z)@ and (z)A
can be explained.
A state (z)@ can only be reached by a stock movement of the second kind (thus with a reordering
decision). If the original stock position was x, the
size d of the demand causing the transition must be
equal to or larger than x!R. The resulting state is
(z)@, with
z"x#Q!d.
(A.2)
Whenever state (z)@ is entered, an undershoot of size
u occurs, with
u"d!(x!R)"Q#R!z.
(A.3)
Because of the PASTA property of Poisson
processes (Poisson arrivals see time average),
p((z)@) not only equals the fraction of time that
the system remains in state (z)@, but also equals
the probability that a stock movement enters
this state, and thus results in an undershoot of
size u. Therefore, the probability that a stock
movement causes an undershoot of size u equals
p((Q#R!u)@).
On the other hand, a state (z)A can only be the
result of stock movement of the "rst kind. This
means that (z)A can only be reached from a state (x)@
or (x)A if x is strictly larger than z.
The probability that a demand causes a reordering-decision equals
R`Q
+
p(( j)@).
j/R`Q~m`1
314
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
Now, one can compute the conditional probability
m that a demand that triggered a reordering causes
u
an undershoot of size u:
p((R#Q!u)@)
, u"0, 1, 2,2, m!1.
m "
u +R`Q
p(( j)@)
j/R`Q~m`1
(A.4)
The only problem that remains to be solved is the
computation of the steady-state probabilities p((z)@).
Let us de"ne a reduced continuous Markov
chain. In this chain the states (z)@ and (z)A are combined into one state (z) (for z"R#Q!m#1 to
R#Q). The rest of the chain remains unchanged.
Let p (z) be the steady-state probability in this
r
Fig. 3.
Table 2
Overview of testables and reorder points
Case
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Order size distribution
Reorder point for target service level
Based upon
Average
value
Standard
deviation
90%
95%
99%
Normal
Normal
Normal
Normal
Exponential
Exponential
Exponential
Exponential
Gamma
Gamma
Gamma
Gamma
Weibull
Weibull
Weibull
Weibull
Bimodal
Bimodal
Bimodal
Bimodal
Random
Random
Random
Random
Outliers
Outliers
Outliers
Outliers
10.00
10.01
10.05
10.35
8.86
7.38
5.14
2.54
4.10
5.92
7.87
10.71
9.18
9.51
9.73
9.88
10.32
10.50
10.32
10.50
10.63
11.77
10.32
10.60
55.00
37.00
19.00
10.91
2.02
3.00
3.82
5.39
5.63
5.25
4.16
1.97
2.78
3.89
3.76
4.42
2.12
1.18
0.65
0.33
5.10
4.59
4.77
5.14
5.46
5.05
5.61
5.66
45.05
41.32
27.15
9.44
38
38
39
42
33
24
10
0
4
14
26
44
33
34
36
37
42
42
41
43
44
51
42
44
216
152
73
12
54
54
55
60
50
39
23
5
15
27
40
61
48
50
51
52
59
60
58
60
62
69
60
62
271
198
106
29
80
82
84
90
78
65
43
17
31
48
64
91
73
75
77
78
89
90
88
91
93
102
90
93
383
294
180
77
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
315
Table 3
Results of the normal approximation
Case
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Results for the discrete algorithm
Results for the continuous approximation Relative errors
90%
95%
99%
90%
95%
99%
90%
95%
99%
90.284
90.107
90.196
90.093
90.093
90.095
90.218
90.127
92.880
90.487
90.089
90.393
90.136
90.331
90.018
90.303
90.272
90.024
90.026
90.207
90.149
90.179
90.100
90.162
90.091
90.093
90.153
90.052
95.240
95.059
95.037
95.198
95.226
95.164
95.233
95.309
95.356
95.075
95.132
95.153
95.207
95.225
95.121
95.108
95.131
95.216
95.006
95.059
95.172
95.033
95.185
95.170
95.021
95.046
95.073
95.208
99.009
99.047
99.051
99.015
99.029
99.069
99.021
99.050
99.069
99.002
99.007
99.027
99.039
99.030
99.051
99.029
99.021
99.059
99.005
99.048
99.033
99.026
99.001
99.027
99.013
99.015
99.016
99.016
89.879
89.685
89.766
89.649
89.624
89.737
89.692
92.600
90.157
89.653
89.950
89.894
89.922
89.164
89.909
89.938
89.838
89.588
89.582
89.776
89.713
89.771
89.655
89.725
90.794
91.253
91.975
89.289
95.097
94.909
94.896
95.112
95.100
94.973
94.938
94.974
95.019
94.787
94.919
95.058
95.032
95.060
94.954
94.945
95.032
95.122
94.892
94.954
95.096
94.971
95.100
95.091
96.312
96.604
97.168
95.141
99.171
99.224
99.244
99.244
99.255
99.283
99.191
99.034
99.146
99.155
99.176
99.244
99.187
99.174
99.196
99.176
99.244
99.273
99.277
99.263
99.267
99.258
99.233
99.258
99.688
99.764
99.908
99.933
0.449
0.468
0.476
0.492
0.523
0.533
0.483
0.301
0.365
0.484
0.491
0.467
0.452
0.449
0.436
0.434
0.481
0.485
0.493
0.478
0.484
0.453
0.494
0.484
0.780
1.287
2.021
0.847
0.150
0.158
0.148
0.090
0.132
0.201
0.310
0.351
0.354
0.303
0.224
0.100
0.284
0.173
0.175
0.171
0.105
0.099
0.120
0.110
0.080
0.065
0.089
0.082
1.359
1.639
2.204
0.070
0.164
0.180
0.196
0.231
0.229
0.216
0.171
0.017
0.077
0.154
0.170
0.219
0.150
0.145
0.146
0.148
0.225
0.216
0.225
0.217
0.236
0.235
0.234
0.233
0.681
0.757
0.900
0.926
reduced chain. Then it is clear that
p (z)"p(z), z"R#1,2, R#Q!m,
r
p (z)"p((z)@)#p((z)A)
r
z"R#Q!m#1,2, R#Q.
(A.5)
(A.6)
The function f (x) is de"ned by
G
x
if R#1)x)R#Q,
f (x)" x!Q
if x'R#Q,
x#Q
if x(R#1.
(A.7)
Fig. 3 shows only a part of the reduced continuous
Markov chain, namely those states that are connected with state (z) (for z"R#1 to R#Q). This
reduced chain clearly constitutes a closed loop,
with the following set of steady-state equations:
m
m
+ jU p ( f (z#j))"p (z) + jU
j 3
3
j
j/1
j/1
for z"R#1,2, R#Q,
R`Q
+ p (z)"1.
(A.8)
3
z/R`1
The structure of this set of equations clearly implies
that the steady-state probabilities of all these
Q states must be equal, thus,
1
p (z)" , z"R#1,2, R#Q.
3
Q
(A.9)
316
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
Table 4
Results of gamma approximation
Case
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Results for the discrete algorithm
Results for the continuous approximation Relative errors
90%
95%
99%
90%
95%
99%
90%
95%
99%
90.284
90.107
90.196
90.093
90.095
90.218
90.127
92.880
90.487
90.089
90.393
90.136
90.331
90.018
90.303
90.330
90.272
90.024
90.026
90.207
90.149
90.179
90.100
90.162
90.091
90.093
90.153
90.052
95.240
95.059
95.037
95.198
95.226
95.164
95.223
95.309
95.356
95.075
95.132
95.153
95.207
95.225
95.121
95.108
95.131
95.216
95.006
95.059
95.172
95.033
95.185
95.170
95.021
95.046
95.073
95.208
99.009
99.047
99.051
99.015
99.029
99.069
99.021
99.050
99.069
99.002
99.007
99.027
99.039
99.030
99.051
99.029
99.021
99.059
99.005
99.048
99.033
99.026
99.001
99.027
99.013
99.015
99.016
99.016
90.423
90.246
90.345
90.268
90.199
90.235
89.960
92.630
90.251
89.967
90.413
90.504
90.421
90.116
90.424
90.461
90.447
90.200
90.191
90.392
90.338
90.416
90.275
90.351
90.474
90.245
90.871
89.968
95.414
95.230
95.202
95.340
95.334
95.245
95.241
95.103
95.291
95.112
95.241
95.321
95.362
95.392
95.295
95.286
95.281
95.376
95.168
95.208
95.139
95.208
95.321
95.307
94.882
94.587
95.124
95.413
99.017
99.045
99.040
98.978
98.979
99.022
98.993
99.019
99.054
98.986
99.002
99.013
99.044
99.040
99.061
99.040
98.990
99.037
98.988
99.006
98.995
98.995
98.957
98.979
98.689
98.512
98.862
99.691
0.154
0.154
0.165
0.914
0.116
0.018
0.185
0.269
0.263
0.136
0.022
0.208
0.100
0.108
0.134
0.145
0.193
0.195
0.183
0.205
0.210
0.263
0.194
0.210
0.425
0.168
0.796
0.093
0.182
0.180
0.173
0.149
0.114
0.085
0.008
0.215
0.068
0.039
0.115
0.177
0.163
0.175
0.183
0.188
0.157
0.168
0.171
0.157
0.155
0.184
0.143
0.145
0.146
0.484
0.053
0.215
0.008
0.001
0.011
0.038
0.050
0.047
0.028
0.032
0.016
0.016
0.006
0.014
0.005
0.010
0.010
0.011
0.031
0.023
0.017
0.042
0.038
0.031
0.044
0.049
0.328
0.508
0.155
0.682
This means that the inventory position z at steady
state is uniformly distributed on the integers
[R#1; R#Q]. The same result was obtained in
a recent study of AxsaK ter [7].
Now, the steady-state probabilities of the states
(R#Q!u)@ in the original continuous Markov
chain can be calculated by solving next equation
(for u"0, 1,2, m!1):
m
m
+ jU p (R#j!u)"p((R#Q!u)@) + jU .
j 3
j
j/u`1
j/1
(A.10)
Some algebra yields the following result:
1
p((R#Q!u)@)"
Q
m
+ U.
j
j/u`1
(A.11)
Thus,
R`Q
+
p(( j)@)
j/R`Q~m`1
m
1 m
+ U # + U #2#U
"
j
j
m
Q
j/1
j/2
1 m
1
"
+ U " E(d)
j
Q
Q
j/1
and
A
A
B
B
(A.12)
U
(1/Q)+m
m
1
j/u`1 j "
m "
+ U.
(A.13)
u
j
(1/Q) E(d)
E(d)
j/u`1
For reasons of clarity this proof is limited to
values m of the maximum demand size smaller than
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
the reorder quantity Q, though it can be extended
to cases for which m is larger than Q.
Appendix B
For the overview of testcases and reorder points
see Table 2.
For the results of the normal approximation see
Table 3.
For the results of the gamma approximation see
Table 4.
References
[1] R.H. Hollier, K.L. Mak, C.L. Lam, An inventory model for
items with demands satis"ed from stock or by special deliveries, International Journal of Production Economics
42 (1995) 229}236.
317
[2] H.C. Tijms, Stochastic Models, an Algorithmic Approach,
Wiley, Chichester, 1995.
[3] R.M. Adelson, Compound poisson distribution, Operations
Research Quarterly 17 (1996) 73}75.
[4] S. Karlin, The application of renewal theory to the study of
inventory policies, in: K. Arrow, S. Karlin, H. Scarf (Eds.),
Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, Stanford, CA, 1958.
[5] E.A. Silver, D.F. Pyke, R. Peterson, Inventory Management
and Production Planning and Scheduling, 3rd Edition,
Wiley, New York, 1998.
[6] A.M. Law, W.D. Kelton, Simulation Modeling and Analysis, 2nd Edition, McGraw-Hill, New York, 1991.
[7] S. AxsaK ter, Simple evaluation of echelon stock (R, Q) policies for two-inventory level systems, IIE-Transactions 29
(1997) 661}669.
The (R, Q) inventory policy subject to a compound Poisson
demand pattern
Peter Matheus, Ludo Gelders*
K.U. Leuven } Centre for Industrial Management, Celestijnenlaan 300 A, 3001 Heverlee, Belgium
Received 26 March 1998; accepted 13 September 1999
Abstract
Most inventory management models are based upon rather restrictive assumptions, e.g. unit sized demands and the
normal distribution for total demand during replenishment time. In a majority of inventory management systems,
circumstances seem to allow these simpli"cations, and inventory policies based upon these assumptions yield satisfying
results. However, in some particular cases, these simpli"cations di!er fundamentally from the actual conditions and
particle. Therefore, application of the models mentioned above can result in an overinvestment in inventory or in an
unacceptable low service level. One of the situations in which we cannot rely on these simpli"ed inventory models is
studied in this paper. We consider an inventory subject to a probabilistic non-unit sized demand pattern, and we propose
an exact and an approximate reorder point calculation method for the (R, Q) inventory policy. The exact algorithm
involves formulas for the discrete distributions of the total demand during replenishment time and the undershoot. The
approximate method is based on the use of continuous distributions, and will be more appropriate when historical data
are sparse. Results of both approaches are compared. The algorithms proposed in this study are simple, fast and easy to
implement in a variety of (existing) inventory management systems. ( 2000 Elsevier Science B.V. All rights reserved.
Keywords: Inventory management; (R, Q) inventory policy; Reorder point calculation method
1. Introduction
The situation studied in this paper is a very
familiar one in inventory management: we consider
a central stock with orders arriving from a variety
of customers (Fig. 1). The customer order size is not
"xed, but follows a discrete probability distribution. The central stock could be anything, from
a central inventory of spare parts in a chemical
plant to a supermarket.
* Corresponding author. Tel.: #32-16-32-25-66; fax: #3216-32-29-86.
E-mail address: [email protected] (L. Gelders).
Normally, this kind of inventory is managed by
an order point and order up to level system, the
(s, S) policy (e.g. [1,2]). In general, this policy o!ers
the best way to deal with large demands crossing
the reorder point s, causing substantial undershoots. However, in some circumstances, the inventory manager has to use "xed reorder quantities Q,
e.g. if the supplier uses standard packings containing a "xed number of units, or if the `suppliera is a
internal production process with a "xed lotsize.
In this case the inventory may be managed by a
(R, Q) policy, which will be determined in this
paper.
We propose calculation methods for the reorder
point R of a (R, Q) inventory policy with a target
0925-5273/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 9 2 5 - 5 2 7 3 ( 9 9 ) 0 0 1 1 0 - 3
308
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
Fig. 1.
service level, subject to an non-unit sized demand
pattern. The reorder quantity is known (e.g. the
EOQ or a "xed quantity determined by other considerations). The demand pattern is compound
Poisson. This means that demand arrivals constitute a Poisson process (the interarrival times are
exponentially distributed), and that the individual
demand size follows some unspeci"ed discrete distribution. The Poisson assumption is appropriate if
the customer population consists of a large group
of individuals acting independently. Furthermore,
we assume the leadtime ¸ between supplier and
inventory to be "xed. In case of stockouts backlogging is applied, and the inventory level is continuously monitored.
The service level b, used in this study, is de"ned
as the steady-state percentage of the total number
of units requested met directly from stock on hand.
Assume that a cycle is the time elapsed between two
consecutive moments at which a replenishment order is received. Then the following basic formula
may be used for the calculation of b, though standard theory of regenerative processes cannot be
applied [2]:
b"1!
E(demand that goes short in one cycle)
E(total demand in one cycle)
(1)
with E denoting the expected (or average) value of
the expression between brackets.
As the backordering situation is considered in
this study, the average demand per cycle equals the
average amount received per cycle. Thus,
E(total demand in one cycle)"Q.
Let us de"ne following quantities:
B "E(shortage present at the beginning of a
1
cycle),
B "E(shortage at the end of a cycle).
2
This means that B is the expected shortage just
1
after a replenishment order has been received and
B is the expected shortage just prior to the arrival
2
of a replenishment order. Thus,
b"1!
(B !B )
2
1 .
Q
(2)
With u being the stochastic quantity representing
the undershoot (the amount by which the reorder
point is crossed when the reordering is triggered),
and d being the stochastic quantity representing
L
the leadtime demand, it is clear that
B "E([u#d !R!Q]`),
1
L
(3)
B "E([u#d !R]`).
2
L
(4)
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
An exact algorithm for the computation of B and
1
B (based on a discrete stochastic model and tested
2
by means of simulation), and an approximate
method (based on a continuous stochastic model)
will be proposed. Results obtained by both
methods will be compared for some testcases.
2. The discrete model
The basic problem in developing closed expressions for B and B is to "nd the discrete probabil1
2
ity distributions for the undershoot u and the
demand during leadtime d .
L
For the calculation of the distribution of d ,
L
Adelson's recursion scheme [3], can be applied. Let
j be the arrival rate of the demands, ¸ the ("xed)
leadtime, and U the probability of receiving a dei
mand of size i (i"1, 2,2, m, with m the maximum
size). The probability r of having zero demands
0
during the leadtime is given by
(5)
r "exp(!j¸).
0
The probability r of having a total leadtime dek
mand of k units can be obtained by the following
recursive formula:
j¸ k~1
r "
+ (k!j)U r ,
k
k~j j
k
j/0
309
Now that closed expressions for the discrete distributions of d and u are available, only one probL
lem remains to be solved: in formulas (3) and (4)
a summation of the two stochastic quantities
d and u appears. This means one needs the convoL
lution of these two distributions. Therefore we calculate f , the probability that the summation of the
k
undershoot and the total demand during leadtime
in a stockcycle equals k:
k
f " + m r , k"0, 1, 2,2, R.
(8)
k
j k~j
j/0
For reasons that become clear later, one only has to
calculate this distribution for values of k from 0 to
R#Q.
Now it is clear that
=
B " +
(k!(R#Q)) f ,
1
k
k/R`Q`1
=
B " + (k!R) f .
2
k
k/R`1
Since
b"1!
(9)
(10)
(B !B )
2
1 ,
Q
some algebra leads to
k"1, 2,2, R.
(6)
This recursion scheme o!ers an e$cient and numerically stable calculation method.
For the determination of the undershoot distribution, we use the formula proposed by Karlin [4]
and Silver et al. [5]. Let E(d) denote the average
demand size, then m , the probability of having an
k
undershoot of k units during stock cycle, can be
calculated by:
R`Q
R`Q
(11)
b" + f !1 + (k!R) f .
k
k Q
k/R`1
k/0
The procedure described above, resulting in this
simple formula, o!ers a calculation method for the
service level b for a given reorder point R.
The exactness of this algorithm was tested
by means of a discrete event simulation [6].
The following assumptions were made:
j"1,
1
m "
k E(d)
m
+ U , k"0, 1, 2,2, m!1.
j
j/k`1
(7)
Though this formula was proposed as an approximation of the undershoot distribution in an inventory controlled by a (s, S) policy, we proved
it to be exact in the case of the (R, Q) policy
(Appendix A).
¸"5 (except for the `outliersa case, then ¸"2),
Q"200.
By generating seven di!erent demand size distributions, each with a di!erent target service level (for
three cases 99%, for two cases 95% and for to other
cases 90%), seven di!erent testcases were obtained.
310
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
The distributions were generated as follows: in
four cases, we took a well-known continuous probability distribution (normal, exponential, gamma
and Weibull). They were truncated at 0.5 and 20.5,
normalized and discretized. For two other cases,
the `bimodala and the `outliersa, a normalized
superposition of the probability density functions
of two normal distributions was used (representing
two types of customers). In the `bimodala case
these distributions were rather close to each other
(the two types of customers have only a slightly
di!erent average demand size). In the `outliersa
case there was a big gap between both distributions
(two very di!erent types of customers). The resulting distribution again was truncated (the `bimodala at 0.5 and 20.5, the `outliersa at 0.5 and
105.5), normalized and discretized. In the `randoma
case, a sequence of 20 normalized random numbers
was generated, representing the probability of having a demand size of 1}20 units.
After the determination of these discrete distributions, we calculated (in an enumerative way) the
reorder point R, required to attain the predetermined target service level b.
Since the reorder point must be integer, the
actual service level always was a bit higher than
the target value. Afterwards these results were
tested by means of a discrete simulation program.
For every case 30 simulation runs were performed
(except for the `normala case where we used
173 runs), and every run consisted of 4000 stock
cycles. An average service level and a con"dence
interval (90%) were calculated. As one can see
in Table 1, the results obtained by application
of the algorithm proposed above, fall very well
within the (narrow) limits of these con"dence
intervals.
The results from three simulations con"rm that
the calculation method described above is exact.
Furthermore, this algorithm is simple, straightforward and numerically stable.
However, even though the formulas are easy to
program, the implementation of this inventory control model in new or existing stock management
systems could cause some problems. The management of data, the necessary input for the algorithm,
could be fairly complex. The inventory management software needs support and input from
a database containing the arrival rate j, the
leadtime ¸, the reorder quantity Q, and the discrete
order size distributions for all the products in the
inventory under consideration. This database
should be updated with every transactions, and as
demand sizes may vary considerably, such a
database could be very memory consuming.
So, though the model proposed above could be
very useful for critical items, or items with rather
compact order size distributions, or simply (because of its exactness) for benchmark calculations
in tests for approximate algorithms, we should try
to "nd a good approximate model with less data
requirements. In the next section an approximation
based upon a continuous stochastic model is proposed.
3. The continuous model
As stated previously, the main drawback of
the discrete model is the storage of the discrete
Table 1
Case
Normal
Exponential
Gamma
Weibull
Bimodal
Random
Outliers
Demand size distribution
Average
value
Standard
deviation
10.00
7.38
10.71
9.88
10.50
11.77
19.00
2.02
5.25
4.42
0.33
4.59
5.05
27.15
Target
service
level (%)
Reorder
point
Discrete
model
(%)
Simulation
Average
value (%)
90%-con"dence
interval (%)
99
95
90
99
90
99
95
80
39
44
78
42
102
106
99.009
95.164
90.316
99.029
90.024
99.026
95.073
99.011
95.175
90.327
99.040
90.016
99.031
95.063
98.928
95.024
90.018
98.957
89.700
98.922
94.727
99.094
95.326
90.636
99.123
90.332
99.139
95.398
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
probability distributions. Therefore, we build an
approximate continuous model, using approximate
continuous probability distributions. For reasons
that will become clear, the normal and the gamma
distributions are applied. For these distributions,
one only has to store the average value and the
standard deviation. This study is based on results
obtained by Tijms [2] for an inventory under periodic review, managed by a (s, S) policy.
Again closed expressions for B and B have to
1
2
be developed. This means one has to "nd the distribution of the summation of the undershoot u and
the total leadtime demand d .
L
Assume that k is the average demand size, p is the
standard deviation of demand size, and D is the
L
stochastic quantity denoting the summation of total leadtime demand, d , and the size of the demand
L
that triggers the reordering-decision. Let f (x) be the
approximated continuous probability density function (pdf) of u#d , g(x) the pdf of d and h(x) the
L
L
pdf of D . Then one can easily prove by extension
L
of results obtained by Tijms [2] that
1
f (x)+ (P(d )x)!P(D )x)), x*0
L
L
k
(12)
(with P denoting the probability of the logical expression between brackets), and assuming that the
pdf of the demand size has a "nite third moment:
P
=
C
1
(x!C) f (x) dx"
2k
AP
P
!
=
(x!C)2h(x) dx
C
=
(x!C)2g(x) dx
C
B
(13)
with C some constant positive real number (C*0).
Thus,
B "E([u#d !R!Q]`)
1
L
P
"
=
(x!R!Q) f (x) dx
AP
=
(x!R!Q)2h(x) dx
R`Q
P
!
=
R`Q
and
1
B "
2 2k
AP
=
(x!R)2h(x) dx
R
P
!
=
B
(x!R)2g(x) dx .
R
(15)
The only remaining problem is the calculation of
the de"nite integrals. By a proper choice of the
distributions g(x) and h(x), this turns out to be an
easy task. In this study the normal distribution and
the gamma distribution (for which the integrals
become incomplete gamma integrals) are used. In
both cases fast codes are widely available. The
parameters of both approximate distributions remain to be determined. This can be done by matching of the "rst two moments, the average value and
the standard deviation. Thus, g(x) is de"ned by
E(d )"jk¸,
L
p2(d )"j¸(p2#k2)
L
(16)
and h(x) is de"ned by
E(D )"k#E(d ),
L
L
p2(D )"p2#p2(d ).
L
L
(17)
This approximate algorithm was tested for several
cases.
Again discrete demand size distributions were
generated in the same way as in the previous section, and for these distributions p and k were calculated. The following assumptions were made:
j"1,
¸"5 (except for the `outliersa case, then ¸"2),
Q"200.
R`Q
1
"
2k
311
B
(x!R!Q)2g(x) dx
(14)
With the exact discrete algorithm, the reorder point
R, required for the target service level b (90%, 95%
and 99%), and the actual service level were calculated. For this reorder point R, the service level
b again was determined by means of the approximate continuous algorithm described in this section.
The results of these tests can be found in Appendix B. For the approximation with the normal
312
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
distribution, the relative error lies between a minimum of 0.017% and a maximum of 2.204% and
reaches an average value of 0.393%. The worst
results were obtained in the `outliersa case. The
gamma approximation seems to yield better results:
the relative error lies between 0.001% and 0.769%
with an average value of 0.145%. Again the worst
cases can be found among the `outliersa. The inferior performance of the Normal approximation
may be due to the `taila of this distribution in the
negative domain.
In general, the results are satisfying and certainly su$cient for most practical purposes.
Therefore this algorithm, based on a continuous
stochastic model, forms an interesting alternative
for the exact, but memory consuming, discrete
algorithm.
m
+ U , u"0, 1, 2,2, m!1, (A.1)
j
j/u`1
with E(d) the average demand size, U the probability
j
that a demand size equals j units ( for j"1 to m), and
m the maximum demand size (with m(Q).
4. Conclusion
Proof. The inventory is modelled as a continuous
Markov chain (see Fig. 2).
In this paper an exact and e$cient algorithm is
proposed for the reorder point calculation of an
inventory managed by an (R, Q) policy, subject to
a compound Poisson demand. This algorithm is
based upon Adelson's recursion scheme [3] and an
undershoot distribution formula that was proved
to be correct. Though the algorithm could easily be
implemented in a new or existing inventory management system, the data requirements could pose
some practical problems. So, when dealing with
very critical items, or in cases where vast demand
size data are available, or simply when an exact
benchmark calculation is needed, this discrete
model yields a simple and fast solution.
Whenever historical data are nor present and the
inventory manager can only give a rough estimation of the form of the demand size distribution (the
average value and the standard deviation), or
whenever one cannot dispose of a su$ciently large
database, the approximate continuous model, proposed in the last section of this paper, can be
applied. This model is an extension of results obtained by Tijms [2] for the periodic review (s, S)
policy. It is based upon an approximation of the
discrete distributions with normal or gamma probability density functions. Results from a variety of
tests clearly indicate that the performance of this
continuous model is su$cient for most practical
situations.
Appendix A
Theorem. In a continuously monitored inventory,
controlled by a (R, Q) policy and subject to a compound Poisson demand pattern, the probability m of
u
having in a stock cycle an undershoot of u units
equals:
1
m "
u E(d)
In this chain three types of states are de"ned:
(z) for z"R#1 to R#Q!m,
(z)@ and (z)A for z"R#Q!m#1 to R#Q
with z denoting the stock position. So every stock
position between R#1 and R#Q!m de"nes one
state, the stock positions between R#Q!m#1
and R#Q de"ne two separate states. The di!erence between the states (z)@ and (z)A will become
clear later. The steady-state probabilities are p(z),
p((z)@) and p((z)A).
The transitions in this chain are de"ned by stock
movements. Two types of stock movements are
considered. The "rst type is caused by the arrival of
a demand that does not cross the reorder point.
This means that the original stock position is
lowered by the demand size. The second type is
caused by a demand that crosses the reorder point
R, and thus increases the stock position with the
reorder quantity Q minus the demand size.
The demands causing these stock movements
(and transitions), arrive as a Poisson stream with
parameter j. This stream is split into m separate
streams, according to the actual demand size. The
Markov property still holds after splitting, since
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
313
Fig. 2.
splitting of a Poisson stream (under probabilities
U , U ,2, U ) generates m new Poisson streams
1 2
m
with parameters U j (i"1, 2,2, m). These parai
meters U j de"ne the transition rates in the coni
tinuous Markov chain (see Fig. 2).
Now the di!erence between the states (z)@ and (z)A
can be explained.
A state (z)@ can only be reached by a stock movement of the second kind (thus with a reordering
decision). If the original stock position was x, the
size d of the demand causing the transition must be
equal to or larger than x!R. The resulting state is
(z)@, with
z"x#Q!d.
(A.2)
Whenever state (z)@ is entered, an undershoot of size
u occurs, with
u"d!(x!R)"Q#R!z.
(A.3)
Because of the PASTA property of Poisson
processes (Poisson arrivals see time average),
p((z)@) not only equals the fraction of time that
the system remains in state (z)@, but also equals
the probability that a stock movement enters
this state, and thus results in an undershoot of
size u. Therefore, the probability that a stock
movement causes an undershoot of size u equals
p((Q#R!u)@).
On the other hand, a state (z)A can only be the
result of stock movement of the "rst kind. This
means that (z)A can only be reached from a state (x)@
or (x)A if x is strictly larger than z.
The probability that a demand causes a reordering-decision equals
R`Q
+
p(( j)@).
j/R`Q~m`1
314
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
Now, one can compute the conditional probability
m that a demand that triggered a reordering causes
u
an undershoot of size u:
p((R#Q!u)@)
, u"0, 1, 2,2, m!1.
m "
u +R`Q
p(( j)@)
j/R`Q~m`1
(A.4)
The only problem that remains to be solved is the
computation of the steady-state probabilities p((z)@).
Let us de"ne a reduced continuous Markov
chain. In this chain the states (z)@ and (z)A are combined into one state (z) (for z"R#Q!m#1 to
R#Q). The rest of the chain remains unchanged.
Let p (z) be the steady-state probability in this
r
Fig. 3.
Table 2
Overview of testables and reorder points
Case
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Order size distribution
Reorder point for target service level
Based upon
Average
value
Standard
deviation
90%
95%
99%
Normal
Normal
Normal
Normal
Exponential
Exponential
Exponential
Exponential
Gamma
Gamma
Gamma
Gamma
Weibull
Weibull
Weibull
Weibull
Bimodal
Bimodal
Bimodal
Bimodal
Random
Random
Random
Random
Outliers
Outliers
Outliers
Outliers
10.00
10.01
10.05
10.35
8.86
7.38
5.14
2.54
4.10
5.92
7.87
10.71
9.18
9.51
9.73
9.88
10.32
10.50
10.32
10.50
10.63
11.77
10.32
10.60
55.00
37.00
19.00
10.91
2.02
3.00
3.82
5.39
5.63
5.25
4.16
1.97
2.78
3.89
3.76
4.42
2.12
1.18
0.65
0.33
5.10
4.59
4.77
5.14
5.46
5.05
5.61
5.66
45.05
41.32
27.15
9.44
38
38
39
42
33
24
10
0
4
14
26
44
33
34
36
37
42
42
41
43
44
51
42
44
216
152
73
12
54
54
55
60
50
39
23
5
15
27
40
61
48
50
51
52
59
60
58
60
62
69
60
62
271
198
106
29
80
82
84
90
78
65
43
17
31
48
64
91
73
75
77
78
89
90
88
91
93
102
90
93
383
294
180
77
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
315
Table 3
Results of the normal approximation
Case
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Results for the discrete algorithm
Results for the continuous approximation Relative errors
90%
95%
99%
90%
95%
99%
90%
95%
99%
90.284
90.107
90.196
90.093
90.093
90.095
90.218
90.127
92.880
90.487
90.089
90.393
90.136
90.331
90.018
90.303
90.272
90.024
90.026
90.207
90.149
90.179
90.100
90.162
90.091
90.093
90.153
90.052
95.240
95.059
95.037
95.198
95.226
95.164
95.233
95.309
95.356
95.075
95.132
95.153
95.207
95.225
95.121
95.108
95.131
95.216
95.006
95.059
95.172
95.033
95.185
95.170
95.021
95.046
95.073
95.208
99.009
99.047
99.051
99.015
99.029
99.069
99.021
99.050
99.069
99.002
99.007
99.027
99.039
99.030
99.051
99.029
99.021
99.059
99.005
99.048
99.033
99.026
99.001
99.027
99.013
99.015
99.016
99.016
89.879
89.685
89.766
89.649
89.624
89.737
89.692
92.600
90.157
89.653
89.950
89.894
89.922
89.164
89.909
89.938
89.838
89.588
89.582
89.776
89.713
89.771
89.655
89.725
90.794
91.253
91.975
89.289
95.097
94.909
94.896
95.112
95.100
94.973
94.938
94.974
95.019
94.787
94.919
95.058
95.032
95.060
94.954
94.945
95.032
95.122
94.892
94.954
95.096
94.971
95.100
95.091
96.312
96.604
97.168
95.141
99.171
99.224
99.244
99.244
99.255
99.283
99.191
99.034
99.146
99.155
99.176
99.244
99.187
99.174
99.196
99.176
99.244
99.273
99.277
99.263
99.267
99.258
99.233
99.258
99.688
99.764
99.908
99.933
0.449
0.468
0.476
0.492
0.523
0.533
0.483
0.301
0.365
0.484
0.491
0.467
0.452
0.449
0.436
0.434
0.481
0.485
0.493
0.478
0.484
0.453
0.494
0.484
0.780
1.287
2.021
0.847
0.150
0.158
0.148
0.090
0.132
0.201
0.310
0.351
0.354
0.303
0.224
0.100
0.284
0.173
0.175
0.171
0.105
0.099
0.120
0.110
0.080
0.065
0.089
0.082
1.359
1.639
2.204
0.070
0.164
0.180
0.196
0.231
0.229
0.216
0.171
0.017
0.077
0.154
0.170
0.219
0.150
0.145
0.146
0.148
0.225
0.216
0.225
0.217
0.236
0.235
0.234
0.233
0.681
0.757
0.900
0.926
reduced chain. Then it is clear that
p (z)"p(z), z"R#1,2, R#Q!m,
r
p (z)"p((z)@)#p((z)A)
r
z"R#Q!m#1,2, R#Q.
(A.5)
(A.6)
The function f (x) is de"ned by
G
x
if R#1)x)R#Q,
f (x)" x!Q
if x'R#Q,
x#Q
if x(R#1.
(A.7)
Fig. 3 shows only a part of the reduced continuous
Markov chain, namely those states that are connected with state (z) (for z"R#1 to R#Q). This
reduced chain clearly constitutes a closed loop,
with the following set of steady-state equations:
m
m
+ jU p ( f (z#j))"p (z) + jU
j 3
3
j
j/1
j/1
for z"R#1,2, R#Q,
R`Q
+ p (z)"1.
(A.8)
3
z/R`1
The structure of this set of equations clearly implies
that the steady-state probabilities of all these
Q states must be equal, thus,
1
p (z)" , z"R#1,2, R#Q.
3
Q
(A.9)
316
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
Table 4
Results of gamma approximation
Case
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
Results for the discrete algorithm
Results for the continuous approximation Relative errors
90%
95%
99%
90%
95%
99%
90%
95%
99%
90.284
90.107
90.196
90.093
90.095
90.218
90.127
92.880
90.487
90.089
90.393
90.136
90.331
90.018
90.303
90.330
90.272
90.024
90.026
90.207
90.149
90.179
90.100
90.162
90.091
90.093
90.153
90.052
95.240
95.059
95.037
95.198
95.226
95.164
95.223
95.309
95.356
95.075
95.132
95.153
95.207
95.225
95.121
95.108
95.131
95.216
95.006
95.059
95.172
95.033
95.185
95.170
95.021
95.046
95.073
95.208
99.009
99.047
99.051
99.015
99.029
99.069
99.021
99.050
99.069
99.002
99.007
99.027
99.039
99.030
99.051
99.029
99.021
99.059
99.005
99.048
99.033
99.026
99.001
99.027
99.013
99.015
99.016
99.016
90.423
90.246
90.345
90.268
90.199
90.235
89.960
92.630
90.251
89.967
90.413
90.504
90.421
90.116
90.424
90.461
90.447
90.200
90.191
90.392
90.338
90.416
90.275
90.351
90.474
90.245
90.871
89.968
95.414
95.230
95.202
95.340
95.334
95.245
95.241
95.103
95.291
95.112
95.241
95.321
95.362
95.392
95.295
95.286
95.281
95.376
95.168
95.208
95.139
95.208
95.321
95.307
94.882
94.587
95.124
95.413
99.017
99.045
99.040
98.978
98.979
99.022
98.993
99.019
99.054
98.986
99.002
99.013
99.044
99.040
99.061
99.040
98.990
99.037
98.988
99.006
98.995
98.995
98.957
98.979
98.689
98.512
98.862
99.691
0.154
0.154
0.165
0.914
0.116
0.018
0.185
0.269
0.263
0.136
0.022
0.208
0.100
0.108
0.134
0.145
0.193
0.195
0.183
0.205
0.210
0.263
0.194
0.210
0.425
0.168
0.796
0.093
0.182
0.180
0.173
0.149
0.114
0.085
0.008
0.215
0.068
0.039
0.115
0.177
0.163
0.175
0.183
0.188
0.157
0.168
0.171
0.157
0.155
0.184
0.143
0.145
0.146
0.484
0.053
0.215
0.008
0.001
0.011
0.038
0.050
0.047
0.028
0.032
0.016
0.016
0.006
0.014
0.005
0.010
0.010
0.011
0.031
0.023
0.017
0.042
0.038
0.031
0.044
0.049
0.328
0.508
0.155
0.682
This means that the inventory position z at steady
state is uniformly distributed on the integers
[R#1; R#Q]. The same result was obtained in
a recent study of AxsaK ter [7].
Now, the steady-state probabilities of the states
(R#Q!u)@ in the original continuous Markov
chain can be calculated by solving next equation
(for u"0, 1,2, m!1):
m
m
+ jU p (R#j!u)"p((R#Q!u)@) + jU .
j 3
j
j/u`1
j/1
(A.10)
Some algebra yields the following result:
1
p((R#Q!u)@)"
Q
m
+ U.
j
j/u`1
(A.11)
Thus,
R`Q
+
p(( j)@)
j/R`Q~m`1
m
1 m
+ U # + U #2#U
"
j
j
m
Q
j/1
j/2
1 m
1
"
+ U " E(d)
j
Q
Q
j/1
and
A
A
B
B
(A.12)
U
(1/Q)+m
m
1
j/u`1 j "
m "
+ U.
(A.13)
u
j
(1/Q) E(d)
E(d)
j/u`1
For reasons of clarity this proof is limited to
values m of the maximum demand size smaller than
P. Matheus, L. Gelders / Int. J. Production Economics 68 (2000) 307}317
the reorder quantity Q, though it can be extended
to cases for which m is larger than Q.
Appendix B
For the overview of testcases and reorder points
see Table 2.
For the results of the normal approximation see
Table 3.
For the results of the gamma approximation see
Table 4.
References
[1] R.H. Hollier, K.L. Mak, C.L. Lam, An inventory model for
items with demands satis"ed from stock or by special deliveries, International Journal of Production Economics
42 (1995) 229}236.
317
[2] H.C. Tijms, Stochastic Models, an Algorithmic Approach,
Wiley, Chichester, 1995.
[3] R.M. Adelson, Compound poisson distribution, Operations
Research Quarterly 17 (1996) 73}75.
[4] S. Karlin, The application of renewal theory to the study of
inventory policies, in: K. Arrow, S. Karlin, H. Scarf (Eds.),
Studies in the Mathematical Theory of Inventory and Production, Stanford University Press, Stanford, CA, 1958.
[5] E.A. Silver, D.F. Pyke, R. Peterson, Inventory Management
and Production Planning and Scheduling, 3rd Edition,
Wiley, New York, 1998.
[6] A.M. Law, W.D. Kelton, Simulation Modeling and Analysis, 2nd Edition, McGraw-Hill, New York, 1991.
[7] S. AxsaK ter, Simple evaluation of echelon stock (R, Q) policies for two-inventory level systems, IIE-Transactions 29
(1997) 661}669.