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Citation: Giat, Y. Stock Levels and
Repair Sourcing in a Periodic Review
Exchangeable Item Repair System.
Logistics 2024,8, 34. https://doi.org/
10.3390/logistics8020034
Academic Editor: Robert Handfield
Received: 1 January 2024
Revised: 27 February 2024
Accepted: 4 March 2024
Published: 22 March 2024
Copyright: © 2024 by the author.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
logistics
Article
Stock Levels and Repair Sourcing in a Periodic Review
Exchangeable Item Repair System
Yahel Giat
Department of Industrial Engineering, Jerusalem College of Tehnology, Jerusalem 9116001, Israel; yahel@jct.ac.il
Abstract: Background: Exchangeable item repair systems are inventory systems. A nonfunctional item
is exchanged for a functional item and returns to the system after being repaired. In our periodic
review setting, repair is performed either in-house or outsourced. When repair is in-house, a repaired
item is returned to stock regardless of the repair status of the other items in its order. In contrast,
with outsourced repair, the entire order must be repaired for it to return to stock. Methods: We
develop formulas for the window fill rate (probability for a customer to be served within a given time
window) to measure the system’s performance and compute it for each repair model. The cost of
outsourcing is the difference between the number of spares needed to maintain a target performance
level when repair is internal and when it is outsourced. Results and Conclusions: In our numerical
example, we show that the window fill rate in both models is S-shaped in the number of spares
and show how the graph shifts to the right when customer tolerance decreases and order cycle time
increases. Further, we show that the cost of outsourcing is increasing with customer tolerance and
with the target performance level.
Keywords: inventory; spares allocation; window fill rate; customer patience; order crossover;
optimization
1. Introduction
Periodic review models are employed in many logistical systems in which continuous
review is restrictively expensive or infeasible for lack of necessary resources [
1
]. When
such review is conducted and some items are found to be nonfunctional, depending on the
system, replacement parts may be ordered or the faulty units themselves may be repaired.
In the former—the order model—the supplier of these parts typically has these items
readily available and therefore once an order is issued for a batch of items, the entire batch
is shipped to the ordering facility. It is generally assumed that order shipping times are
i.i.d. and therefore each order may take a different time to arrive and possibly, orders may
even crossover. However, within an order, all the items will arrive at the same time. In
the latter setting—the repair model—items must be repaired at a repair facility with the
standard assumption that item repair times are i.i.d. The key difference between the two
models is what element forms the basic stochastic variable of the model—the order (as in
the order model) or the item (as in the repair model).
Repair models are an important component of sustainable logistics, hence their impor-
tance. Examples abound in the practice of operations and logistics. Closed-loop supply
chains encourage repair to recycle materials and resources used in the manufacturing
process (e.g., [
2
]). Other examples include hybrid manufacturing–remanufacturing and
green products remanufacturing [3,4].
This paper analyses a repair model under two inherently different settings, in-house
repair (IR model) and outsourced repair (OR model). In the first, items are repaired on-
site and therefore once each item is repaired, it can immediately be returned to stock. In
contrast, when repair is outsourced, the repair facility may be reluctant to ship each item
Logistics 2024,8, 34. https://doi.org/10.3390/logistics8020034 https://www.mdpi.com/journal/logistics
Logistics 2024,8, 34 2 of 19
individually, and therefore only once all the items that compose a single order are repaired,
then the order is shipped to the ordering facility.
Many inventory models use the fill rate to measure their performance. In many
settings of practical interest, however, the fill rate is not a reasonable approximation
of the customer–inventory-system relationship. The reason for this is that the fill rate
tacitly assumes that the reputation cost to the system due to customer waiting begins
immediately when the customer arrives. However, it is often that firms are obliged,
whether by government regulation or contractual commitment, to reduce the waiting time
to be below a predetermined time threshold. From the customers’ standpoint, too, there is
usually a certain tolerable or acceptable period of waiting, which may depend on their level
of patience or expectation. In these cases, the time from which the firm suffers reputation
costs begins only after this tolerable wait has passed [
5
]. Accordingly, the performance
measure we use for our facility is the window fill rate, which is the probability that each
item is replaced within a predetermined window of time.
One obvious way to improve performance is keeping a large stock of spare parts. Since
this may be a very expensive solution, it is of special interest to managers to determine
how spares affect the window fill rate, thereby keeping the minimum inventory to meet
their target window fill rate. We use our analysis of the window fill rate of both models
to examine the cost of outsourcing in terms of additional inventory. This is performed by
comparing between the number of spares needed in each model (i.e., the IR and OR models)
to meet a required window fill rate level. Prima facie, the OR model requires more spares
than the IR model. This is because, all else being equal, the OR model requires waiting for
all the items in an order to be repaired before they are returned to the inventory, whereas in
the IR models, each item returns to stock as soon as it is repaired.
We begin by deriving the stationary window fill rate for the IR model by tracking
the demand and supply for spares and computing the probability that within
w
units of
time from a random customer’s arrival there will be a functional item available for the
customer. Since this entails finding the difference between supply and demand, each of
which are Poisson random variables, our formula makes repeated use of Skellam random
variables [6], whose distribution is nowadays ubiquitously available.
Next, we derive the stationary window fill rate for the OR model. Here, however,
there are two major complications. First, in contrast to the IR model, the delivery times of
repaired items in the OR model do not follow a Poisson process, since the delivery times of
items in a particular order are dependent on each other. To overcome this, we track each
order’s status, that is, whether it was delivered or not, and for each state in the delivery
state space, we derive the probability of the state and the expected window fill rate. The
weighted average of these expected window fill rates is the system’s window fill rate. The
second complication in the OR model is with respect to the evaluation of the window fill
rate’s formula, which requires conditioning on the values of numerous random variables
and therefore cannot be executed within a reasonable time and accuracy. This is overcome
by identifying the components of the formula whose evaluation can be expedited through
simulation. This hybrid approach of using simulations and analytical evaluation allows
us to evaluate the window fill rate accurately (within less than 1% margin of error) within
reasonable time.
We complete our mathematical analysis with a numerical illustration that illustrates
functional form of the window fill rate. In both models, the window fill rate is S-shaped in
the number of spares and the implication of this to managers is discussed. The difference
between the graphs of the IR and OR model reveals that all else being equal, a shift from
internal repair to external repair results in a lower window fill rate. Thus, to maintain
the same level of performance, more spares must be procured when outsourcing repair
compared to in-house repair. We estimate this cost in our numerical example for different
levels of tolerable waiting and target performance.
Our study, therefore, ignores managerial and operational costs of in-house and out-
sourced repair. See, in contrast, [
7
] that optimizes the stock management and ignores the
Logistics 2024,8, 34 3 of 19
inventory cost and [
8
] that considers maintenance and inventory jointly. Therefore, the
practical decision of whether to repair in-house or outsource repair should weigh possible
managerial advantages of outsourcing against the cost of additional spares needed to
maintain the required system performance.
The contributions of our model are therefore both theoretical and applicable in their
nature. We contribute to theory by developing a formula for the window fill rate in the
IR model and an efficient algorithm for the estimation of the window fill rate in the OR
model. Our contribution to the practice of inventory modeling is threefold. We provide
a tool for managers to quantify how many spares are needed to meet their performance
objectives. Second, it allows managers to assess the cost—in terms of spares—of changing
contract terms such as service times and performance guarantees. Third, it allows managers
to be informed about inventory requirements when deciding whether to outsource their
repair operations.
The remainder of this paper is as follows: In the next section, we review the litera-
ture focusing on inventory systems and performance measures related to our study. In
Sec
tions 3and 4
, we develop the formula and algorithm for the window fill rate in the IR
and OR model, respectively.
Section 5
is a numerical illustration that demonstrates the cost
of outsourcing in terms of spare capacity and Sect
ion 6
concludes and details future paths
for research.
2. Literature Review
2.1. Exchangeable-Item Inventory Systems
The subject of our study lies within the research field of exchangeable-item (also called
recoverable-item, repairable-item) inventory management. This is a vast field of research
that is described in many professional books and is a main topic in top operations research,
management, and logistics journals [
9
–
11
]. In the classical model, customers arrive at
the inventory system (i.e., a logistics warehouse) with a failed item and exchange it for a
serviceable item from the system’s stock. The failed item itself is repaired on- or off-site,
after which it is returned to the system’s stock. Since repair is time-consuming, spare
items are placed in stock in order to increase the number of items available for arriving
customers. We follow two assumptions followed by most studies in the field. The first is
that customers are served according to a first-come–first-serve policy (FCFS). Models that
pivot away from the FCFS service rules assume customer priority (e.g., [
12
,
13
]). The second
standard assumption is that there is ample repair capacity. This then allows us to assume
that repair times are i.i.d. (see, in contrast, ref. [14], for a limited capacity model).
The review policy in a single location can be grouped into two main approaches.
The continuous review—typically an
(S
,
S−
1
)
model—is one of the assumptions of
Sherbrooke’s seminal paper [
15
] and more recent studies such as [
5
,
16
]. The second
grou
p—wh
ich this paper belongs to—comprises models with periodic review policies. We
now focus our attention to studies belonging to this group.
Periodic review policies may be further grouped to fixed order size policies and the
“up-to” policies. In the former, notated by
(t
,
s
,
q)
, inventory is reviewed every
t
periods,
and if it falls below
s
, then an order of
q
(or its multiples) is issued. Our model belongs to
the latter group in which order sizes are designed to raise the current inventory level up to
a target level. This group can be further split into
(r
,
s
,
S)
and
(r
,
S)
policies. With
(r
,
s
,
S)
,
inventory is reviewed every r periods, and if the order falls below the threshold
s
, then an
order is issued so that the inventory level will reach
S
(e.g., [
17
,
18
]). Our model assumes an
(r
,
S)
policy such that the inventory is reviewed every
r
periods and upon review, an order
is issued so that the inventory level will reach
S
. Recent studies based on this approach
include [19–21].
To complete this review, we note that certain operations may be carried out continu-
ously while others are executed periodically. Recent examples include [
22
], a continuous
(s
,
S)
model whose parameters are updated periodically, and [
23
] that analyzes a network
Logistics 2024,8, 34 4 of 19
of remote locations in which the inventory review is continuous, but order delivery is
constrained by periodical shipments.
2.2. Performance Measures
Among a logistic system’s performance measures, the expected waiting time and
the fill rate are probably the most prominent. Our research is related to the second main
performance measure, the fill rate, which is defined as the portion of demand that is satisfied
upon arrival [
24
]. In contrast to the expected waiting time, the fill rate does not penalize
the system for the length of the customer’s weight but on whether the customer had to
wait. See [25] [Section 2.1] for a brief review of fill-rate-associated performance measures.
The notion of a tolerable wait, also termed as “reasonable wait” or “acceptable
w
ait” [26],
was extensively researched in service-related fields such as medical ser
vice [27]
,
tr
ansit [28],
supplier selection [
29
], and call centers [
30
]. In contrast, within inventory
sciences, there is surprisingly little research that considers such willingness to wait. This
lacuna is clearly unjustified since many inventory contracts actually do state time windows
for demand to be supplied. As there are no losses to the inventory systems as long as they
meet these time windows, these contracts effectively imply that customers have a tolerable
wait. In a similar vein, ref. [
31
] (p. 744) complained about the failure of inventory models
“to capture the time-based aspects of service agreements as they are actually written”.
The term “window fill rate” denotes the probability to wait less than a predetermined
window of time. Mathematically, the window fill rate can be derived by assigning the
tolerable wait in the waiting time distribution. Ref. [
5
] characterized the functional form of
the window fill rate and used this to develop an efficient spares allocation algorithm that
optimizes the window fill rate of a multiple-location inventory system. In a two-echelon
system, optimizing the window fill rate requires varying degrees of resource pooling [
32
]
and this was later applied to the battery swapping problem [33,34].
In a periodic review setting—the subject of this paper—there have been very few
studies that consider the window fill rate [
35
]. In the
(r
,
S)
model, the assumption that
repair times are independent stochastic poses a considerable mathematical challenge since
to determine accurately when a customer is expected to leave the system, the delivery
status of all past orders must be considered. The literature offers solutions to this problem.
The first is to assume that orders do not crossover [
36
,
37
]. This assumption effectively
contradicts the independence of the lead times and [
38
] shows, using real data, that this
results in a poor approximation. The second approach is to overcome the challenge of the
evaluation of the exact formula by using simulation for key random variables of the window
fill rate [
38
]. In both studies, however, the lead times of the orders was given exogenously
and therefore the probability for order delivery was independent of the order size. In
contrast, our model mimics outsourced repair in a more realistic fashion by assuming
that item repair times are independent and therefore order repair times do depend on the
order size.
2.3. Outsourcing Repair
The decision whether to outsource repair or not depends on multiple considerations,
including financial, environmental, informational, and logistical [
29
,
39
]. Relevant papers
considering logistical aspects of this decision include [
40
] whose model outsources the
repair of non-conforming items. Similarly, ref. [
41
] set repair to be performed either exter-
nally or internally depending on whether the “internal repair shop has the authorization
(from the original equipment manufacturer) and the capability to repair the part (p. 412).”
Finally, ref. [
42
] characterized outsourced repair as one that does not require the system to
acquire repair infrastructure and therefore outsourced repair costs are only variable costs.
The closest study to our proposal with respect to repair outsourcing is [
43
] that
compared between in-house and outsourced recycling (comparable to the concept of repair
in our proposal) in a continuous
(r
,
Q)
model (i.e., when stock level reaches
r
order quantity
Q
). In addition to their model being continuous whereas the system in our proposal is
Logistics 2024,8, 34 5 of 19
under periodic review, their objective was minimizing costs while our paper focuses on the
window fill rate as the system’s objective.
3. IR Model: In-House Repair
3.1. Model Setup
Consider a warehouse into which each customer arrives with a single failed item
and expects to exchange it for a functional one. Customers’ arrivals are assumed to be
distributed Poisson with rate
λ
, and customers are served according to a first-come–first-
serve order. The warehouse is adjacent to a repair workshop so that any nonfunctional
item is repaired in-house. The repair workshop has ample identical and independent
workstations so that repair times are i.i.d. When repair is outsourced, it is usually the case
that the repair contractor will receive a batch of failed items, repair them, and return them
as a batch. This is not the case with on-site repair. Here, it is reasonable to assume that
individual items from the same repair order will be returned to stock once they are repaired.
Repair orders are managed periodically, and are issued every
r
units of time. We
denote the
i
’th order,
i= (
1, 2, ...
)
, as the order issued at time
ir
. Let Jane be a random
customer that arrives at time
ir +t
with a failed item. That is, Jane arrived
t
units of time
into the
i+
1’s cycle. The objective of the following derivations is to compute the probability
that within
w
units of time from Jane’s arrival (i.e.,
ir +t+w
), Jane has received a functional
item. To this end, we track the demand and supply of functional items and observe that
Jane will be served by
ir +t+w
if and only if at that date there is enough supply to meet
the demand up to and including her arrival. The supply is
S
plus all the parts that were
repaired on time, whereby on time we mean
ir +t+w
. Let
q
denote the number of complete
cycles in the tolerable wait, and therefore
(i+q)r<ir +t+w<(i+q+
2
)r
. That is,
ir +t+w
falls either in the
i+q+
1’th cycle or the
i+q+
2’th cycle, depending on when
Jane arrived during the i+1’th cycle.
Accordingly, to correctly account for all the orders that may contribute to the supply,
we distinguish between early and late arrival in the cycle. If
t
is sufficiently late in the cycle
such that
ir +t+w>(i+q+
1
)r
, then the order issued at
(i+q+
1
)r
possibly contributes
to the supply, since it may have returned before
ir +t+w
. In contrast, if
t
is sufficiently
early in the cycle such that
ir +t+w≤(i+q+
1
)r
, then that order (i.e., the order issued at
(i+q+
1
)r
) need not be considered for the supply. In Figure 1, we plot Jane’s arrival time
(the arrow’s tail) that determines the demand, and the time by which items must return to
count for the supply (the arrowhead).
Since late arrival in the cycle requires considering an additional order, it is convenient
to define a binary variable
a
to indicate whether
t
is late in the cycle. Formally,
a
is defined
as follows:
a=0 if t+w<(q+1)r,
1 if t+w≥(q+1)r.(1)
ir ir+r ir+qr (i+q+1)r(i+q+2)r
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