Approximation of the probability distribution of the
customer waiting time under an (r,s,q) inventory policy
in discrete time
Horst Tempelmeier, Lars Fischer
To cite this version:
Horst Tempelmeier, Lars Fischer. Approximation of the probability distribution of the customer waiting time under an (r,s,q) inventory policy in discrete time. International Journal of
Production Research, Taylor & Francis, 2009, pp.1. .
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International Journal of Production Research
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Manuscript ID:
Manuscript Type:
Keywords (user):
04-Sep-2009
Tempelmeier, Horst; University of Cologne, Dep. of Production
Management
Fischer, Lars; University of Cologne, Dep. of Supply Chain
Management and Production
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Keywords:
Original Manuscript
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Complete List of Authors:
TPRS-2009-IJPR-0273.R2
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Date Submitted by the
Author:
International Journal of Production Research
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Journal:
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Approximation
of the probability distribution of the customer waiting time
under an (r,s,q)
inventory policy in discrete time
INVENTORY MANAGEMENT, PERFORMANCE MEASURES
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Page 1 of 22
International Journal of Production Research
RESEARCH ARTICLE
Approximation of the probability distribution of the
customer waiting time under an (r, s, q) inventory policy in
discrete time
Horst Tempelmeiera∗ and Lars Fischerb
a,b
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Seminar für Supply Chain Management und Produktion, Universität zu Köln,
Albertus Magnus-Platz, 50932 Köln, Germany
(Received 00 Month 200x; final version received 00 Month 200x)
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We study a single-item periodic review (r, s, q) inventory policy. Customer demands arrive on a discrete (e. g. daily) time axis. The replenishment lead times
are discrete random variables. This is the time model underlying the majority
of the Advanced Planning Software systems used for supply chain management
in industrial practice. We present an approximation of the probability distribution of the customer waiting time which is a customer-oriented performance
criterion that captures supplier-customer-relationships of adjacent nodes in
supply chains. The quality of the approximation is demonstrated with the help
of a simulation experiment.
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Keywords: inventory management; customer waiting time
2nd Revision, September 2009
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Vol. 00, No. 00, 00 Month 200x, 1–22
∗ Corresponding
author. Email: [email protected]
ISSN: 0020-7543 print/ISSN 1366-588X online
c 200x Taylor & Francis
DOI: 10.1080/00207540xxxxxxxxx
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1.
Introduction
In this paper, we consider the (r, s, q) inventory policy which is a version of the continuous
review reorder point-order quantity (s, q) inventory policy. Under the (r, s, q) policy, the
inventory position is reviewed every r periods. If it has reached or has fallen below
the reorder point s, a multiple n of the minimum order size q is ordered such that
the inventory position rises to a level between s and s + q. We consider the (r, s, q)
inventory policy in discrete time, which means that not only the review is periodic, but
also the demand model assumes that there is a discrete time axis with a period length of,
say, one day and all demands arriving during the day are aggregated to a single period
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demand quantity. This inventory policy has the virtue that replenishments are only made
at predefined points in time which allows the inventory replenishment processes to be
coordinated for different products that are purchased from the same supplier. In addition,
the inventory process for a product can be easily coordinated with the other inbound
logistical processes, such as material handling or transportation.
The (r, s, q) policy combines the advantages of the (s, q) policy with those of the (r, S)
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policy. Although the effective order quantity n · q is a random variable, from the (s, q)
policy the (r, s, q) policy inherits, at least for sufficiently large lot sizes, the stability of
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the order sizes. In addition, with the (s, q) it has the characteristic in common that a replenishment order is released only if the inventory position has reached the reorder point.
From the (r, S) policy it inherits the positive characteristic that the replenishments are
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restricted to discrete points in time and thus can be coordinated with the replenishments
of other products as well as with the execution of subsequent logistical processes.
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The performance of an inventory node within a supply chain is usually measured in
terms of costs and customer service. While there is a common understanding how to
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define the main components of the relevant costs, the term customer service is subject
to many interpretations. With respect to the output performance of an inventory node,
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in the literature several types of inventory-related measures are discussed (for example
Schneider (1981), Silver et al. (1998)). The majority of publications use some type of
service level such as the stockout probability or the fill rate to quantify the performance
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of an inventory node. By contrast, in management-oriented textbooks on logistics and
supply chain management, such as Bloomberg et al. (2002) or Christopher (2005), the
time dimension of the service is emphasized. In this interpretation, a logistic system (and
an inventory node being a part thereof) performs well if it has the ability to respond to
customers’ requirements in a short time frame, i. e. with short or zero waiting times.
From a managerial perspective, the customer waiting time of an inventory node may be a
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Page 3 of 22
valuable criterion in the optimization of the safety stock. If the decision maker is able to
predict the probability distribution of the customer waiting time associated with given
values of the decision variables of the inventory policy used, then several optimization
problems may be formulated and solved:
• Specification of a chance constraint with respect to the waiting time.
Contrary to the standard inventory service levels discussed in the literature such as the
fill rate, which are supplier-focused, the waiting time is a customer-oriented measure.
Indeed, from the perspective of a customer knowledge of the waiting time probability
distribution may have a significant value. As noted by Lee and Billington (1992) and
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demonstrated numerically with respect to an (r, S) policy by Tempelmeier (2000), the
waiting time distribution provides valuable information for supplier selection which
cannot be deduced from the fill rate. If customers use the waiting time as a criterion for
supplier differentiation, then a supplier can make his inventory decisions with respect
to the constraint that the probability of delivery within a maximum time window wmax
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must be greater than or equal to a given target value pmin Wang et al. (2005). This is
a type of constraint that is often used in industrial practice.
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• Guarantee of a fixed delivery time to the customer.
Here, a possibly poor performance of the inventory system which is associated with
an unfavorable probability distribution of the customer waiting time can be offset
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through the use of transportation modes with different speeds or through different
order processing and material handling procedures with different flow times (such as
faster order processing). In this case, the decision maker may seek the optimal com-
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bination of response times of the different logistical processes, including the inventory
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response time, thereby ensuring a prespecified delivery lead time to the customer.
• Multi-level safety stock optimization.
In a locally controlled multi-level (e. g. One-Warehouse-N-Retailer) inventory system,
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among others, the waiting time caused by stock-outs in the upstream node may be a
significant part of the replenishment lead time seen by the downstream nodes. Thus,
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International Journal of Production Research
poor inventory performance of the upstream node can be compensated by increased
safety stock to be held in the downstream nodes’ inventory without affecting the end
customer service level. In order to find the optimum allocation of the total safety stock
to the nodes in the supply network, the replenishment lead time of a downstream
node must be quantified as a function of the decision variables of the upstream node.
Most approaches available for the analysis of multilevel supply networks apply decomposition. Hereby, the nodes of the supply network are analyzed in isolation with the
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missing link between the nodes being the waiting time observed by a replenishment
order at the upstream node Graves and Willems (2003). Although there are a number
of models that use the expected value of the waiting time or its first two moments,
for the discrete time model that is considered in this paper, we propose the use of the
complete probability distribution of the waiting time.
In the following, we propose a procedure for the approximation of the probability distribution of the waiting times for an (r, s, q) policy in discrete time. By contrast, as will
be noted in the sequent literature overview, most single-level inventory models using the
waiting time as an inventory performance indicator consider different inventory policies
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or assume a different demand model where the orders arrive on a continuous time axis.
2.
Literature review
In the literature, only relatively few papers deal with the (r, s, q) inventory policy. A
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detailed treatment of the (r, s, q) inventory policy with backorder costs is presented by
Hadley and Whitin (1963) under Poisson and normally distributed demands in continuous
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time. Janssen et al. (1998) analyze the (r, s, q) policy in discrete time under compound
Bernoulli demand. As a performance criterion they use the fill rate and assume that at
each review at most the quantity q is reordered. They improve the work of Dunsmuir
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and Snyder (1989) by taking the undershoot of the inventory position under the reorder
point into account. Johansen and Hill (2000) consider the (r, s, q) policy with lost sales
under the assumption that at most one replenishment order may be outstanding. In
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addition, the lead time is an integral multiple of the review interval. Recently, Kiesmüller
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and de Kok (2006) considered an (r, s, q) policy with customers arriving in continuous
time according to a compound renewal demand process with both interarrival times and
customer order sizes assumed to be mixed Erlang distributed. Partial backordering is
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not allowed and the maximum waiting time allowed is the length of the replenishment
lead time. The authors propose an approximation procedure in which the demands are
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approximated by mixed Erlang distributions. In a simulation experiment with r = 1
they found that their procedure provides good approximations when the assumption
is met that demands arrive in continuous time. It is not clear if this approximation
is also applicable for the case that the review period r is significantly longer than the
average interarrival time of the demands, for example with daily demands and monthly
reviews. In a second simulation experiment, again with continuous-time customer arrivals,
they apply an approximation method by Tempelmeier (1985) specifically designed for
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Page 5 of 22
a discrete time demand process and, as could be expected, found that this leads to
non-satisfactory results. From this the conclusion can be drawn that the approximation
procedure for the waiting time distribution is sensitive to the assumptions with respect to
the characteristics of the demand process. Our paper differs from Kiesmüller and de Kok
(2006) in that we assume a discrete-time demand process, which is the correct model for
many industrial situations. In addition, we do not use the mixed Erlang distribution to
fit the demand probabilities but instead we use the demand probabilities associated with
the real-life case under consideration which may be gamma, normal, or even empirical
discrete. Other than Kiesmüller and de Kok (2006) we do not restrict the maximum
waiting time to the replenishment lead time. This allows us to consider also (r, s, q)
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inventory policies with extreme long waiting times which are significantly longer than
the replenishment lead time.
In the literature, there are a number of papers that propose exact or approximative
procedures for the computation of the waiting time distribution. These publications
either assume a continuous time axis for the demand arrivals (such as Kiesmüller and
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de Kok (2006)) or they consider inventory policies other than the (r, s, q) policy. Kruse
(1980, 1981) derived the waiting time distribution for an (S − 1, S) policy and an (s, S)
policy with poisson demand. Tempelmeier (1985) proposed an approximation procedure
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for an (s, q) policy in discrete time with a review period r = 1 and the complete demand
in a period treated as a single order. Our model differs from Tempelmeier (1985) in that
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the waiting time probabilities are computed based on the demand units and not on the
customer orders, which usually are larger than one. In addition, our approach allows for
arbitrary values of the reorder point, which may also be negative, and for review periods
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which may be significantly greater than 1. Finally our model is much more robust with
respect to small order quantities.
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Van der Heijden and de Kok (1992) study the customer waiting time distribution for
an (r, S) policy with a compound poisson demand process. Tempelmeier (2000) approx-
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imates the waiting time distribution for an (r, S) policy with discrete-time demand arrivals. Chen and Zheng (1992) consider an (r, S) policy with a compound renewal demand
process. Hausman et al. (1998) derives the waiting time distribution for a base-stock pol-
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International Journal of Production Research
icy with discrete-time demand arrivals. Wang et al. (2005) consider an (s, S) policy with
discrete-time demand arrivals and a constant replenishment lead time.
In the current paper, we analyze the (r, s, q) policy where the review period is r ≥ 1. We
consider customer arrivals on a discrete time axis. This is the way, the demand process is
monitored in standard materials management software systems. Modeling such a demand
process as a continuous-time arrival process can be expected to result in approximation
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International Journal of Production Research
errors with the same order of magnitude that has been observed by Kiesmüller and de
Kok (2006) when using the model by Tempelmeier (1985) with continuous-time demand
arrivals.
Based on our observations of the development of logistical processes in industrial practice
we also allow for discrete stochastic lead times. However, we assume that order crossing
does not occur. We present an approximation of the probability distribution of the customer waiting time that is precise also for negative reorder points and for small order
sizes, i. e. when several replenishment orders are simultaneously outstanding.
3.
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Model description and analysis
Consider an inventory node using an (r, s, q) policy where customers arrive periodically
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on a discrete time axis. The period demands D are i. i. d. random variables (usually nonunit sized) with mean E{D} and variance Var{D}. Unfilled demands are backordered
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with partial demand fulfillment allowed. Inventory data are gathered at the end of each
period. Every r periods the inventory position is compared to the reorder point s. If at
the review time t the inventory position has reached or dropped below the reorder point,
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a replenishment order of size n · q is triggered which arrives at the beginning of period
t+L+1. Replenishment lead times are discrete random variables. However, it is assumed
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that replenishment orders do not cross in time. If the replenishments are made by a single
supplier who processes orders in the sequence of their arrivals, then this assumption will
usually be met.
Arrival of a replenishment order from the supplier.
Delivery of backordered demands according to the FCFS discipline.
If possible, delivery of the actual demand of period t, else the demand is backordered.
Inventory review and placement of a replenishment order, if necessary.
Calculation of average inventory and service level.
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The sequence of events in period t is as follows:
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We focus on the computation of the probability distribution of the customer waiting
time.
The waiting time is measured for each individual demand unit. It starts in the period of
the demand arrival and ends in the period of delivery.
In the remainder of this paper we use the following notation:
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V
W
Y (t)
∆
µ3
β
demand per period
International
Production
Research
net inventory position
at theJournal
end of of
period
t
replenishment lead time
risk period, i. e. number of periods to by covered by safety stock
order size
reorder interval
reorder point
undershoot, i. e. the difference between the reorder point and the
inventory position at the point in time when a replenishment order
is released
difference between net inventory and reorder point in the
last period before the reorder point is hit
customer waiting time
demand during t periods; Y (0) = 0
reordering delay
= E{(D − E{D})3 }, third central moment of the demand distribution
fill rate
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The development of the inventory in a system with a large order size q is illustrated in
Figure 1. At some point in time the inventory position hits the reorder point s. In contrast
to a standard (s, q) policy, where a replenishment order would be released immediately,
under a (r, s, q) policy the order is launched only after the complete actual review cycle
r has elapsed. Thus, a reordering delay ∆ occurs which randomly increases the length
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of the risk period that must be covered by the reorder point. The risk period is now a
discrete random variable equal to L = ∆ + L, as opposed to the replenishment lead time
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L which is observed under the (s, q) policy.
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L
∆
100
60
s
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q
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r
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D
IN (t)
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r
s
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Inventory
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10 11 12 13 14 15
Period
Figure 1. Development of the inventory
In order to find the probability distribution of the reordering delay, assume that the last
review has taken place in period 0 and that the reorder point will be hit in the current
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International Journal of Production Research
review interval. Define the virtual inventory position as (inventory on hand – backorders
+ actual outstanding orders + delayed orders). This virtual inventory position observed
under the (r, s, q)-policy is identical to the inventory position that would be seen if no
reordering delays would occur and it exhibits the same stochastic behavior. It is wellknown that in the standard (s, q)-policy the inventory position at any time (and also at
time 0) is uniformly distributed between s and s + q.
Caused by the demand arrivals, the virtual inventory position hits the reorder point in
a random period τ > 0, which can be any period up to the next review period r. As
the period demands are stationary, period τ is uniformly distributed between 1 and r.
The time (r − τ ) that a demand unit that caused the virtual inventory position to fall
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below the reorder point has to wait until its associated order quantity is launched, is
thus uniformly distributed over the interval [0, r − 1]. This is true for large orders as
well as for small order quantities, in case that a multiple n of the basic order size q is
ordered. As an illustration, when q = 1, in each period τ the observed demand Dτ is
added to the virtual inventory position and has to wait (r − τ ) periods which means that
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the reordering delays are uniformly distributed over the interval [r − 1, r − 2, . . . , 0].
If the replenishment lead time has the discrete probability distribution P {L = ℓ} (ℓ =
Lmin , Lmin + 1, . . . , Lmax ), then the risk period L can take on discrete values between
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Lmin = Lmin and Lmax = Lmax + r − 1. The probability distribution of L can then be
found through the convolution of the probability distributions of the reordering delay ∆
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and the replenishment lead time L, which according to the above assumptions are both
discrete random variables. As a consequence of the above transformation, the (r, s, q)policy with lead time L and possible reordering delays is equivalent to an (r = 1, s, q)-
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policy with lead time L without reordering delays. Thus, we can derive the desired results
by considering the (r = 1, s, q)-policy with leadtime L.
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In the sequel, we focus on the waiting times of the individual demands units. Each
demand unit is uniquely associated to an order from which it is served. Waiting starts
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with the arrival of the demand and ends with the arrival of the associated order. In order
to determine the probability distribution of the waiting times, for each demand unit the
expected waiting time is computed.
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Summing up the expected demand experiencing the given waiting time w and dividing
this by the order size q we get the waiting time probability P {W = w}.
Due to the periodic review and the non-unit demand sizes, an undershoot U occurs in
the time period when the inventory position drops below the reorder point.
The considered demand process can be modelled as a renewal process Karlin (1958),
where the demand quantities are equivalent to the inter-event times, and the undershoot
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Page 9 of 22
is equivalent to the forward-recurrence time. Using asymptotic results from renewal theory for q → ∞ (see, for example, Tijms (1994), Baganha et al. (1996), Silver et al. (1998)),
the expectation and the variance of the undershoot can be approximated as follows:
E{U } ≃
E{D}2 + Var{D}
2 · E{D}
(1)
E{D}2
µ3
Var{D}
Var{D}
+
· 1−
+
Var{U } ≃
3 · E{D}
2
2 · E{D}2
12
(2)
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In addition to the undershoot, the demand during the risk period plays an important
role in the evaluation of an inventory policy, e. g. in the determination of the expected
backorders. The probability distribution of the demand during the risk period plus the
undershoot is found by convolution. Hereby we assume that the undershoot has the same
type of distribution as the demand, which is only true in case of exponential demands.
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For example, with gamma distributed demands we compute the first two moments of the
demand during the risk period plus the undershoot and then fit a gamma distribution
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to these moments. Although this approximation is based on asymptotic considerations
for q → ∞, numerical tests have shown that for the most common demand distributions
this approximation is good even for small order sizes (see Baganha et al. (1996)).
Probability distribution of the customer waiting time
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3.1.
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The customer waiting time W is the time between the arrival of a customer order and its
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delivery. In order to determine the waiting time probabilities, several cases are examined.
Case 1. Consider first the case with s ≥ 0 and assume that the order quantity q is
large enough to fill all outstanding backorders. Let period t = 0 be the period when the
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inventory position drops below the reorder point s. At the end of this period (immediately
before the inventory review) the net inventory is IN (0) = s − U . If IN (0) < 0, then a
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backorder occurs already in the review period which has to wait for delivery until period
ℓ + 1. The expected amount backordered is G(s, U ), where G(s, Z) = E{max(Z − s, 0)}
is the first-order loss function with respect to the random variable Z. In the following
periods t (t = 1, 2, . . . , ℓ) additional backorders possibly accumulate. The additional
amount backordered in period t is G(s, U + Y (t) ) − G(s, U + Y (t−1) ) .
The maximum waiting time ℓ + 1 is observed by (the portion of) the demand of period
0 that is backordered, if the undershoot is greater than s. This happens with probability
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P {W = ℓ + 1|L = ℓ} =
G(s, U )
q
(3)
The backorders occuring in period t (t = 1, 2, . . . , ℓ−1), G(s, U +Y (t) )−G(s, U +Y (t−1) )
will have to wait for w (w = ℓ, ℓ − 1, . . . , 2) periods, and finally the backorders that occur
in period ℓ, G(s, U + Y (ℓ) ) − G(s, U + Y (ℓ−1) ) have to wait for one period. Thus, the
waiting probabilities are
P {W = w|L = ℓ} =
G(s, U + Y (ℓ+1−w) )
q
(4)
G(s, U + Y (ℓ+1−w−1) )
−
q
rP
Fo
w = 1, 2, . . . , ℓ
All remaing demand units are delivered without waiting (w = 0). The probability is
P {W = 0|L = ℓ} = 1 −
G(s, U + Y (ℓ) )
q
(5)
Case 2. Consider now the case that s ≥ 0 and that the order quantity is very small and
not sufficient to fill all outstanding backorders. Let again period t = 0 be the period when
ee
the inventory position drops below s. But now assume that in this period a backorder
occurs that is greater than q, i. e. IN (0) = s − U < −q. In this case, in t = 0 one
rR
or more additional replenishment orders must be released. The number of backorders
that occur in period t = 0 and which are associated to each replenishment order is
G(s, U ) − G(s + q, U ). Thus, we have
P {W = ℓ + 1|L = ℓ} =
ev
G(s, U ) G(s + q, U )
−
q
q
(6)
Similar to Case 1, the number of backorders that occur in period t (t = 1, 2, . . . , ℓ −
1), which are delivered by the currently considered order, is G(s, U + Y (t) ) − G(s +
q, U + Y (t) ) − G(s, U + Y (t−1) ) − G(s + q, U + Y (t−1) ) . These backorders must wait
w
ie
w (w = ℓ, ℓ − 1, . . . , 2) periods, and finally the backorders that occured in period ℓ,
G(s, U + Y (ℓ) ) − G(s + q, U + Y (ℓ) ) − G(s, U + Y (ℓ−1) ) − G(s + q, U + Y (ℓ−1) ) ,
On
observe a waiting time of one period. Thus, the waiting probabilities are
P {W = w|L = ℓ} =
[G(s, U + Y (ℓ+1−w) ) − G(s + q, U + Y (ℓ+1−w) )]
q
−
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[G(s, U + Y (ℓ+1−w−1) ) − G(s + q, U + Y (ℓ+1−w−1) )]
q
(7)
w = 1, 2, . . . , ℓ
and
"
G(s, U + Y (ℓ) ) G(s + q, U + Y (ℓ) )
P {W = 0|L = ℓ} = 1 −
−
q
q
#
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(8)
Page 11 of 22
Note that for large order sizes the terms involving the function G(s + q, . . .) in equations
(6) – (8) disappear and (6) – (8) cover also Case 1.
Case 3. Finally, we consider the situation that the reorder point s < 0, which means
that at least one demand unit observes a waiting time.
0
s+V+Y
(3)
w=7
w=6
s+V+Y (2)
rP
Fo
w=5
s+V+D
w=4
s+V
w=3
s
s-U
ee
w=2
s-U-D
w=1
s-U-Y(2)
-4
-3
-2
rR
-1
0
1
l=2
delivery
2
3
Time
Figure 2. Inventory development
ev
Figure 2 shows the development of the net inventory in the last periods before the arrival
of the replenishment order under focus. Let period 0 be the period when the inventory
ie
position drops below s. All demands that arrive when the net inventory is less than
w
zero are backordered. Note that this happens also to demands that occur earlier than
period 0. As noted above, the development of the net inventory which is the result of the
demand arrivals is a renewal process, where the undershoot U = s − IN (0) is equivalent
On
to the forward recurrence time (residual life) and the difference V = IN (−1) − s is
equivalent to the backward recurrence time (age). Note that a well-known result from
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renewal theory states that these variables have the same probability distribution (see Cox
(1962)). Therefore, we assume that U and V have asymptotically the same probability
distribution.
In order to find the waiting time probabilities, we again start with considering the demand
that arrives in period t = 0, when the net inventory drops below the reorder point s.
All or a portion of the demand units that arrive in this period are backordered during
ℓ + 1 periods. The amount backordered in this period is equal to the difference of the net
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International Journal of Production Research
inventory at the end of period 0, IN (0) = s − U , and the net inventory at the beginning
of period 0, IN (−1) = s + V .
The probability of the waiting time W = ℓ + 1 is then
P {W = ℓ + 1|L = ℓ} =
G(s, U ) − G(s + q, U )
q
(9)
G(s, −V ) − G(s + q, −V )
−
q
The remaining probabilities of W ≤ ℓ are
"
#
G(s, U + Y (ℓ+1−w) ) G(s + q, U + Y (ℓ+1−w) )
P {W = w|L = ℓ} =
−
q
q
#
"
G(s, U + Y (ℓ+1−w−1) ) G(s + q, U + Y (ℓ+1−w−1) )
−
−
q
q
rP
Fo
(10)
w = 1, 2, . . . , ℓ
"
G(s, U + Y (ℓ) ) G(s + q, U + Y (ℓ) )
P {W = 0|L = ℓ} = 1 −
−
q
q
ee
#
(11)
Note that (10) and (11) are the same as (7) and (8), respectively.
As depicted in Figure 2, it may happen that the waiting time of a demand is longer than
rR
the replenishment lead time. This situation is likely to occur when the reorder point is
less than zero. To account for this case, we consider the development of the net inventory
in the periods -1, -2, . . . . Based on the same arguments as above, the probabilities for
the waiting times W ≥ ℓ + 2 are
G(s, −V − Y (w−ℓ−2) )
q
ie
P {W = w|L = ℓ} =
"
ev
G(s + q, −V − Y (w−ℓ−2) )
−
q
"
(w+1−ℓ−2)
G(s, −V − Y
)
−
q
#
w
(12)
On
G(s + q, −V − Y (w+1−ℓ−2) )
−
q
w = ℓ + 2, ℓ + 3, . . .
#
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Formulas (9), (11), (10) and (12) include the above cases 1 and 2, as V is not associated
to backorders for s ≥ 0.
There is no upper bound on the waiting time, as s may be arbitrarily small. However,
with increasing w the numerators in (12) converge to zero. Once this has happened,
the evaluation of (12) can be stopped and the maximum waiting time with positive
probability has been found.
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Page 13 of 22
Note that the fill rate is equal to P {W = 0}. However, as there are many possible shapes
of waiting time distributions with the same value of P {W = 0}, the fill rate does not
provide the required information to discriminate among these distributions.
The evaluation of the above formulas has been implemented as a prototype in Visual
Basic. For the computation of the probabilities for the gamma distribution and the
normal distribution we used standard routines. The computational requirements are
very small, usually a few milliseconds on a standard PC. It is also easy to implement the
equations with the help of spreadsheet software, such as MS-Excel, using the available
standard functions for computing the first-order loss function.
Next, we state that (9), (10), (11) and (12) define a probability distribution.
rP
Fo
Proposition.
∞
P
P {W = w|L = ℓ} = 1
(13)
w=0
Proof.
rR
Rewriting (13) as
ee
P {W = 0|L = ℓ} +
∞
P
P {W = w|L = ℓ} = 1
(14)
w=1
and using (11) we obtain
ie
ev
#
∞
P
G(s, U + Y (ℓ) ) G(s + q, U + Y (ℓ) )
−
=1−
P {W = w|L = ℓ}
1−
q
q
w=1
"
w
(15)
On
Referring to (10), (9), and (12) equation (15) can be rewritten as
G(s, U + Y (ℓ) ) G(s + q, U + Y (ℓ) )
−
=
q
q
ℓ
P
w=1
P {W = w|L = ℓ} + P {W = ℓ + 1|L = ℓ} +
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∞
P
(16)
P {W = w|L = ℓ}
w=ℓ+2
Now consider the elements of the right side of (16) separately. Note that
Pℓ
w=1 P {W = w|L = ℓ} is a telescoping series that can be expressed as follows:
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International Journal of Production Research
ℓ
P
P {W = w|L = ℓ}
w=1
G(s, U + Y (ℓ+1−1) ) G(s + q, U + Y (ℓ+1−1) )
−
q
q
=
(17)
G(s, U + Y (ℓ+1−ℓ−1) ) G(s + q, U + Y (ℓ+1−ℓ−1) )
−
+
q
q
G(s, U + Y (ℓ) ) G(s + q, U + Y (ℓ) ) G(s, U ) G(s + q, U )
−
−
+
q
q
q
q
P
Analogously we can proceed with ∞
w=ℓ+2 P {W = w|L = ℓ}.
=
rP
Fo
∞
P
P {W = w|L = ℓ}
w=ℓ+2
=
G(s, −V − Y (ℓ+2−ℓ−2) ) G(s + q, −V − Y (ℓ+2−ℓ−2) )
−
q
q
G(s, −V − Y (t+1−ℓ−2) ) − G(s + q, −V − Y (t+1−ℓ−2) )
− lim
t→∞
q
=
(18)
ee
G(s, −V ) G(s + q, −V )
−
q
q
rR
Inserting equations (17), (9), and (18) into (16), we obtain
G(s, U + Y (ℓ) ) G(s + q, U + Y (ℓ) )
−
q
q
ev
=
G(s, U + Y (ℓ) ) G(s + q, U + Y (ℓ) ) G(s, U ) G(s + q, U )
−
−
+
q
q
q
q
ie
(19)
w
+
G(s, U ) G(s + q, U ) G(s, −V ) G(s + q, −V )
−
−
+
q
q
q
q
+
G(s, −V ) G(s + q, −V )
−
q
q
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We complete the proof noting that P {W = w|L = ℓ} ≥ 0 must hold for all w which
is true regarding the according equations. With random discrete lead times, the unconditional probability distribution of the customer waiting time is then
P{W = w} =
L
max
X
P{W = w|L = ℓ} · P{L = ℓ}w = 0, 1, . . .
ℓ=Lmin
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(20)
Page 15 of 22
3.2.
Numerical Experiment
In order to test the quality of the approximations, we developed an ARENA-simulation
model of the considered inventory system and performed two groups of simulation experiments. In the first group we considered deterministic replenishment lead times and
tested all combinations of the parameters shown in Table 1.
Table 1. Parameters (deterministic lead times)
Average Demand E{D}
Demand Variability CVD
Review Period r
Replenishment Lead time L
Order Quantity q
Reorder Point s
{100}
{0.3, 0.9, 1.5}
{1, 10, 20}
{10, 20}
{100, 1000, 2000}
{−1000, −900, . . . , 1900, 2000}
ee
rP
Fo
With a coefficient of variation CVD =0.3 the period demands were assumed to be normally
distributed and with CVD = {0.9, 1.5} gamma-distributed demands were considered.
These demand distributions represent a wide spectrum of demand patterns to be found
rR
in industrial practice, including demand for C-items with high variability. The other
parameters were chosen such that many situations observed in industry are covered.
Some combinations that are not expected to occur in reality (e. g. q = E{D}) are simply
ev
used to test the robustness of the approximations for extreme cases.
From these 1674 parameter combinations we eliminated 20 cases that would result in
ie
a fill rate β = 100%. In this case waiting does not occur at all. The remaining 1654
parameter combinations were simulated with 10 replications and 500000 periods each.
w
For each parameter combination we computed the probability distribution of the customer waiting time and compared it to its simulated counterpart mass point by mass
point. Thereby two subgroups of cases were considered.
• Cases with β > 0.
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The first subgroup of cases comprises 1158 parameter combinations that resulted in a
fill rate greater than zero, which means that at least one single demand unit observed
no waiting time. Table 2 shows the frequency distribution of the absolute differences
between simulated and computed waiting time probabilities. Among all these probability
distributions the maximum absolute deviation of any computed probability mass point
from its simulated counterpart was only 0.00369.
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Table 2. Frequency distribution of deviations between simulated and computed probabilities (cases with
β > 0)
International Journal of Production Research
%
99.68%
0.27%
0.04%
0.01%
It is interesting to see what happens with the shape of the probability distribution of
the waiting time when the reorder point s in increased. This is demonstrated with the
help of Figure 3 where for r = 10, q = 100, and s = {700, 900, 1100, 1300, 1500, 1700}
the simulated as well as the computed probabilities are depicted. Note that due to the
high quality of the approximations the computed values are not distinguishable from the
simulated values.
rP
Fo
r=10, s=700,q=100
0.12
0.10
0.10
0.08
0.06
0.04
0.02
0.00
0
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0.06
0.04
0.02
0.00
10 11 12 13 14 15 16
0
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9
0.16
0.14
P{W=w}
0.12
0.4
r=10, s=1300,q=100
0.3
0.2
ev
0.10
10 11 12 13 14 15 16
w
r=10, s=1100,q=100
0.18
P{W=w}
0.08
rR
0.20
9
r=10, s=900,q=100
0.14
0.12
P{W=w}
P{W=w}
0.14
ee
0.08
0.06
0.1
0.04
0.02
0.00
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0
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0
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0.6
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0.8
r=10, s=1500,q=100
0.7
0.5
3
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9
10 11 12 13 14 15 16
w
w
r=10, s=1700,q=100
0.6
0.5
P{W=w}
P{W=w}
0.4
0.3
0.4
0.3
0.2
0.2
0.1
0.1
0.0
0.0
0
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9
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Deviation
0.0000 – 0.0010
0.0010 – 0.0020
0.0020 – 0.0030
0.0030 – 0.0040
10 11 12 13 14 15 16
Figure 3. Deviations for r = 10, s = {700, 900, 1100, 1300, 1500, 1700}, q = 100
With increasing reorder point s the waiting time distribution shifts to the origin and
becomes more and more asymmetric. Obviously, this behavior must be taken into account in solution approaches for multi-level inventory management, where the customer
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Page 16 of 22
Page 17 of 22
waiting time observed in a warehouse is part of the replenishment lead time seen by the
downstream nodes in the supply chain. Particularly the use of the normal assumption
for the demand during the replenishment lead time may be questionable.
• Cases with β = 0.
The second subgroup of cases includes 496 parameter combinations which resulted in a
fill rate β = 0. This means that the complete demand is backordered and delivered only
after a waiting time. These cases show that there may be significant differences with
respect to the waiting time observed by a customer which are not reflected by the fill
rP
Fo
rate. In other words, from the point of view of the inventory decision maker all parameter
combinations perform equally well, whereas from the point of view of the customer there
are significant differences in performance.
Table 3 shows the frequency distribution of the absolute differences between simulated
and computed waiting time probabilities. Among all these probability distributions the
maximum absolute deviation of any computed probability mass point from its simulated
counterpart was 0.02067.
rR
ee
Table 3. Frequency distribution of deviations between simulated and computed probabilities (cases with
β = 0)
Deviation
0.0000 – 0.0010
0.0010 – 0.0020
0.0020 – 0.0030
0.0030 – 0.0040
0.0040 – 0.0060
0.0060 – 0.0080
0.0080 – 0.0100
0.0100 – 0.0200
0.0200 – 0.0300
%
98.92%
0.42%
0.16%
0.14%
0.13%
0.06%
0.08%
0.08%
0.02%
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It is again interesting to consider the development of the shape of the waiting time
distribution as a function of the reorder point. Figure 4 shows this for several parameter combinations under normally distributed demand with r = 1, q = 100, and
s = {−1000, −800, −600, −400, −200, −100, 0, 200}. Here only the computed probabilities are depicted, as the simulated values are almost identical (The maximum absolute
deviation from the simulated values for this set of parameter combinations was only
0.0015).
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s=0
International Journal of Production Research
0.6
P{W=w}
0.5
0.4
s=-100
0.3
s=-200
s=-400
0.2
s=-600
s=-800
s=-1000
s=200
0.1
0.0
0
2
4
6
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8 10 12 14 16 18 20 22 24 26 28 30 32 34
w
Figure 4. Waiting time distributions for normally distributed demand, r = 1, q = 100, s =
{−1000, −800, −600, −400, −200, −100, 0, 200}
In the second group of experiments we considered random replenishment lead times. In
particular the parameter combinations shown in Table 4 were used. Both replenishment
ee
lead time distributions have the same expected value (E{L} = 19). For the review period
r = 10 order crossing is impossible, while for r = 5 there is a small probability for order
crossing.
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Table 4. Parameters (random lead times)
Average Demand E{D}
Demand Variability CVD
Review Period r
P {L = ℓ}
(a)
(b)
0.25
0.35
0.25 0.24
0.13
0.25 0.05
0.05
0.25 0.05
0.05
0.05
0.02
0.01
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{1000, 2000}
{−1000, −900, . . . , 5900, 6000}
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Order Quantity q
Reorder Point s
ℓ
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Replenishment Lead time L
{100}
{1.5}
{5, 10}
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Page 18 of 22
Again, we constructed the individual simulation scenarios by combining these parameter
values which resulted in 568 combinations. From these we eliminated 32 instances that
resulted in a fill rate β = 100%. The simulations were run with 10 replications over
5000000 periods. Again we computed the probability distribution of the customer waiting
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Page 19 of 22
time and compared it to its simulated counterpart mass point by mass point. In all these
probability distributions the maximum absolute deviation of any computed probability
mass point from its simulated counterpart was only 0.0021.
Table 5 shows the frequency distribution of the absolute differences between simulated
and computed waiting time probabilities.
Table 5. Frequency distribution of deviations between simulated and computed probabilities
Deviation
0.0000 – 0.0010
0.0010 – 0.0020
0.0020 – 0.0030
4.
%
99.18%
0.81%
0.02%
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Concluding remarks
We have presented a very precise procedure to compute the probability distribution of
the waiting time observed by customers in an (r, s, q) inventory system in discrete time.
Our approach is an approximation for several reasons. First, if a single period demand is
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greater than q (which will happen if q is very small), then it is not possible to associate all
demand units of that period to a unique order. Second, the undershoot is approximated.
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Third, under random replenishment lead times, the assumption that orders will not cross
is made which may be violated in practice.
The probability distribution of the waiting time can be used for setting the safety stock
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with respect to a performance criterion that is truly customer-oriented as well as for
modelling supplier-customer-relationships in a supply chain.
While for a continuous time axis that is often considered in multi-level inventory models
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only in special cases the complete distribution of the lead time demand can be deter-
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mined, in the current case of a discrete time axis the replenishment lead time is a discrete
random variable that can be taken into account without significant computational effort.
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The availability of the probability distribution of the waiting time enables us to directly
characterize the performance measure that is relevant for the customer nodes of a supplying node in a logistic network. Therefore internal service levels that are often used
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as an intermediate criterion to control the performance of a supplying node in a supply
chain are not required.
Acknowledgement
We are grateful to an anonymous reviewer for his valuable comments that helped to
improve the paper.
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International Journal of Production Research
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Captions of the Figures
Figure 1: Development of the inventory
Figure 2: Inventory development with s < 0
Figure 3: Deviations for r = 10, s = {700, 900, 1100, 1300, 1500, 1700}, q = 100
Figure 4: Waiting time distributions for normally distributed demand, r = 1, q = 100,
s = {−1000, −800, −600, −400, −200, −100, 0, 200}
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