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Spare parts inventory pooling games
F.J.P. Karsten, M. Slikker∗, G.J. van Houtum
School of Industrial Engineering, Eindhoven University of Technology,
P.O. Box 513, 5600 MB, Eindhoven, The Netherlands
December 15, 2009
Abstract
We study a situation where nindependent companies separately stock spare parts
of the same item for a technically advanced machine. They may reduce expected joint
holding and downtime costs by pooling inventory. We analyze these situations by
defining a cooperative cost game. We examine the conditions under which such
a game has a nonempty core, i.e. a stable cost allocation exists. For situations
allowing companies to have non-identical demand rates and base stock levels and for
situations allowing companies to have non-identical downtime costs, we show that
the core of the associated game is non-empty. However, when companies have non-
identical downtime costs along with non-identical base stock levels or demand rates,
the associated game may have an empty core.
Keywords: Supply chain management, Game theory, Balancedness, Spare parts,
Inventory pooling.
1 Introduction
Equipment-intensive high-tech companies such as airlines, nuclear power plants and med-
ical equipment manufacturers are often confronted with the difficult task of maintaining
high availability of their technically advanced systems. A random failure of just one crit-
ical component can cause a complex machine to break down. To prevent long and costly
∗Corresponding author. Tel.: +31-40-247-3940. E-mail address: m.slikker@tue.nl.
1
downtimes, spare parts are kept on stock, such that the failed component can be quickly
replaced by a spare one from inventory.
Inventory pooling can be an effective strategy to improve system availability while
reducing total costs. Inventory pooling refers to an arrangement where demand at a stock-
point that is out of stock is satisfied from another stockpoint with a positive on-hand
inventory. The use of inventory pooling can considerably reduce spare parts provisioning
costs. For example, a case study by Kranenburg and van Houtum (2009) at ASML, an
OEM in the semiconductor industry, showed a cost reduction of 50% for a full pooling
scenario in comparison to a no pooling scenario. Relative cost reductions translate to mas-
sive amounts in terms of dollars, as a lot of capital is typically tied up in these expensive
spare parts. For example, the commercial aviation industry alone has as much as 44 billion
dollars worth of spare parts on stock (Harrington,2007).
Spare parts inventory models have been analyzed quite extensively in the literature.
Alfredsson and Verrijdt (1999), Axs¨ater (1990), Grahovac and Chakravarty (2001), Kra-
nenburg and van Houtum (2009), Kukreja et al. (2001), Kutanoglu (2008), Lee (1987),
Reijnen et al. (2009), Van Wijk et al. (2009) and Wong et al. (2005,2006) present mathe-
matical models where multiple stockpoints pool their inventory by using lateral transship-
ments. All consider centralized inventory systems with characteristics that are reasonable
for spare parts stockpoints, such as continuous Poisson demands processes and one-for-one
replenishments. Paterson et al. (2009) provide a literature overview and classification of
papers.
In this paper, we consider situations where several independent decision makers stock
spare parts of the same item. They may cooperate by pooling inventory. Consider, for
instance, multiple business units within a high-tech company, e.g. several independently
operating power-generating plants, each with their own stockpoints for spare parts, within
a large electric utility company (which was the motivating case for the study of Kukreja
et al.,2001). Another example setting would be multiple airline companies that are not
competitors of each other, use the same type of aircrafts and independently stock spare
engines at possibly separate locations, which motivated the paper of Wong et al. (2005). As
a final illustrative example, consider a single airport with several concourses, each of which
has its own unique bagage handling system. Every bagage handling system is maintained
by a different maintenance company with a separate stock of spare parts and some items
are stocked by all maintenance companies.
In the remainder of this paper, we will simply refer to the independent decision makers
as ”companies”. In all examples given in the preceding paragraph, it is clear that the
2
companies can cooperate by pooling their inventories, i.e. keeping their own independent
stockpoints but allowing the stockpoint of another company to satisfy demand in case of a
stock-out. However, each individually rational company will only agree to pool their spare
parts with other companies if doing so will bring more profits to itself. So, before any inter-
company inventory pooling arrangement will be implemented, the participating companies
will first have to be convinced that the arrangement is beneficial for everyone and that no
group of companies is merely subsidizing another. These cost allocation intricacies add
another layer of difficulty to the spare parts inventory pooling process.
In order to obtain insights into these issues, we will use cooperative game theory. The
context of cooperative games is appropriate for us, since it deals with joint profits or
costs that can be obtained by groups of decision makers if they coordinate their actions.
Our application of cooperative game theory to an inventory-based problem falls into a
growing stream of literature. An overview of the applications of cooperative game theory
to Operations Research problems is given in Borm et al. (2001). Of specific interest to us
are papers that have studied similar games arising from inventory control. Often, these
papers investigate whether a stable cost allocation exists or, in cooperative game theoretical
terminology, whether the core is non-empty.
A number of authors have focused on newsvendor games. In this setting, nindependent
retailers, each with single-period stochastic demands for the same item, face a newsvendor
problem. Groups of retailers might improve their expected joint profit by coordinating
their orders, followed by transshipments after demand realization is known. Hartman
et al. (2000), M¨uller et al. (2002), Slikker et al. (2005), ¨
Ozen et al. (2006), ¨
Ozen and Soˇsi´c
(2006) and ¨
Ozen et al. (2008) have studied this setting from a cooperative game theoretical
point of view. The authors show non-emptiness of the core under certain assumptions
on the joint demand distribution and/or when extensions such as asymmetric retail and
wholesale prices, non-negligible transshipment costs, delivery restrictions, updated demand
information and supplying via warehouses are allowed.
Inventory situations with an infinite time horizon, rather than a single period, have been
studied by several other authors. Gerchak and Gupta (1991), Robinson (1993) and Hart-
man and Dror (1996) analyzed a setting with stochastic demand and backordering, where
a stockpoint can profitably combine safety stocks of the same item meant for different cus-
tomers. The authors examined the problem of allocating the joint costs to the customers
and showed the non-emptiness of the core for this situation. Meca et al. (2004) and Anily
and Haviv (2007) show core non-emptiness for settings with deterministic continuous de-
mand, where firms can reduce ordering costs by cooperating through joint replenishments.
3
The specific characteristics of spare parts inventory systems, such as the focus on ex-
pensive low-demand items with high service requirements, make them distinctly different
from the aforementioned newsvendor or continuous review inventory systems. There are,
however, only few papers that have analyzed spare parts inventory systems from a game
theoretical point of view. Zhao et al. (2005,2006) examine inventory sharing in a decen-
tralized spare parts inventory system with non-cooperative game theoretical models and
Satir et al. (2009) characterize the optimal operating policy of a capacitated independent
service center in such a decentralized system. Wong et al. (2007) is, to the best of our
knowledge, the first study of spare parts inventory systems in the context of cooperative
game theory. They propose four cost allocation policies and show that in a three-company
numerical example all four policies yield cost allocations that are in the core of the game.
They also consider non-cooperative aspects. They left an investigation of (conditions on)
non-emptiness of the core in general as a future research direction. Kilpi et al. (2009)
specify a framework of cooperative strategies, each with a different degree of contractual
integration, for the availability of repairable aircraft components. They also focus on shar-
ing of pooling benefits but, again, a more extensive investigation of cooperation using game
theoretical models is left as a future research direction.
The dearth of insights into existence of stable cost allocations in the context of spare
parts inventory pooling is striking, as this lack of knowledge may impede a type of collabo-
ration that can potentially bring significant cost savings. Moreover, recent business trends
may make pooling of repairable spare parts more prevalent. Companies are recognizing
that the after-sales market is very profitable with high margins and that pooling of spare
parts is the best way to realize economies of scale (Cohen et al.,2006). Furthermore,
decreased costs of networked databases and transportation may make the implementation
of pooling arrangements easier (Grahovac and Chakravarty,2001) and the increase in en-
vironmental awareness may lead industries to adopt repairable spare parts rather than
consumable ones (Kilpi et al.,2009).
Our paper intends to fill the outlined void; we investigate non-emptiness of the core
for spare parts inventory pooling games in general. We formulate a cooperative game
associated with a simple spare parts inventory situation, which is quite similar to the
one introduced by Wong et al. (2007). The main difference is that they consider non-zero
lateral transshipment costs, a finite source of failures, and the possibility of partial pooling,
whereas we restrict ourselves to free transshipments, an infinite source of failures and full
pooling, in order to ensure analytical tractability of the model. This allows us to derive
structural results, which is our main interest.
4
The main contribution of this paper is that we show that for situations where compa-
nies may have non-identical demand rates and base stock levels and for situations where
companies may have non-identical downtime costs, the core of the associated game is non-
empty.
The structure of this paper is as follows. We start in Section 2 with some prelimi-
naries on cooperative game theory and the Erlang loss function. Then, in Section 3, we
describe the spare parts inventory model, discuss assumptions, and introduce cooperative
cost games associated with spare parts inventory situations. Section 4 concentrates on
cores of these games. We first look at a base setting where companies are (almost) iden-
tical and subsequently look at several generalizations. Finally, conclusions and directions
for future research are drawn in Section 5.
2 Preliminaries
For reasons of self-containedness, we first give a brief introduction to cooperative game
theory. Subsequently, we present the well-known Erlang loss function, which will be im-
portant in our inventory model. We will state several interesting properties of this function
that are mainly due to others.
2.1 Preliminaries cooperative game theory
Let us assume we have ndifferent players, with Nthe set of players. For convenience,
we number the players such that the player set is N={1,2, ..., n}. A subset of Nis
called a coalition and is denoted by M. The grand coalition refers to M=N. We are
interested in various coalitions M, and particularly to what extent a specific coalition can
reach their common objective without the players who are not part of the coalition. In
this paper we focus on cost games rather than benefit games. The function c: 2N→R
that assigns to every coalition M⊆Ncosts c(M) is called the characteristic cost function.
By convention, c(∅) = 0. We assume that the costs of coalition Mare freely transferable
between the players of M, i.e. players can make transfer payments to each other. A pair
(N, c) constitutes a cooperative cost game. In the remainder of this paper, we will simply
refer to this as game.
Two interesting properties that a game might satisfy are subadditivity and concavity.
A game is called subadditive if it is always beneficial to combine coalitions, i.e. for any two
disjoint coalitions Mand Lit holds that c(M)+ c(L)≥c(M∪L). A game is called concave
5
if any player would rather join a large coalition than a small one, i.e. if for each i∈Nand
for all M, L ⊆N\{i}with M⊆Lit holds that c(M∪ {i})−c(M)≥c(L∪ {i})−c(L).
An important issue is how to distribute the costs of the grand coalition over all players.
An allocation can be represented as a vector x= (xi)i∈N∈RN, which specifies for each
player i∈Nthe costs that this player has to pay if he cooperates with all the other
players. An allocation vector is called efficient if all the expected total costs of the grand
coalition are in fact split fully among all players, i.e. Pi∈Nxi=c(N). An allocation vector
is called stable if no non-empty subset of players is allocated more costs than what they
could expect by only cooperating together, i.e. Pi∈Mxi≤c(M) for all M∈2N
−, where
2N
−= 2N\{∅} denotes the set consisting of all non-empty subsets of N. The set of all stable
and efficient allocations, Core(N, c), is called the core. If the core of a game is non-empty,
then costs can be distributed to each player in such a way that no coalition is allocated
more costs than this coalition would have had to pay while acting independently. So, if
the costs are split according to a core element then no coalition has an incentive to leave
the grand coalition and form a smaller coalition on its own.
Bondareva (1963) and Shapley (1967) independently identified the class of games that
have non-empty cores as the class of balanced games. To describe this class, we define for
all M⊆Nthe vector eMby eM
i= 1 for all i∈Mand eM
i= 0 for all i∈N\M. A
map κ: 2N
−→[0,1] is called a balanced map if PM∈2N
−
κ(M)eM=eN. A game (N, c) is
called balanced if for every balanced map κit holds that PM∈2N
−
κ(M)c(M)≥c(N). The
following theorem is due to Bondareva (1963) and Shapley (1967).
Theorem 2.1. Let (N, c)be a game. Then the core of (N, c)is non-empty if and only if
(N, c)is balanced.
A sub-game is constructed by restricting the characteristic cost function to player set
M⊂N, denoted by c|M, i.e. then (M, c|M) is a sub-game of (N, c). A game (N , c) is called
totally balanced if it is balanced and each of its sub-games is balanced as well.
In the following lemma, we will state a property of balanced maps, which will be used
extensively in this paper.
Lemma 2.2. Let Nbe a player set, let κ: 2N
−→[0,1] be a balanced map and let f:N→R.
Then:
X
M∈2N
−
κ(M)·X
i∈M
f(i) = X
i∈N
f(i).
6
Proof. By the definition of a balanced map, for all i∈N
X
M∈2N
−
κ(M)·eM
i= 1.(1)
Multiplying both sides of Equation (1) by f(i), subsequently summing both sides over all
i∈Nand finally reversing the order of summation gives
X
M∈2N
−
κ(M)·X
i∈N
eM
if(i) = X
i∈N
f(i).
Rewriting the term Pi∈NeM
if(i) to Pi∈Mf(i) completes the proof.
2.2 Preliminaries Erlang loss model
The Erlang loss model describes an M/G/s/s queuing model with shomogeneous servers
(s∈N0:= N∪ {0}) and no additional waiting buffer. Jobs arrive according to a Poisson
process with rate ˆ
λand have a mean service time of 1/ˆµ. The offered load is ρ=ˆ
λ/ˆµ
(ρ > 0). If a job arrives when all sservers are busy, it is lost and not served. In such
a system, the steady-state probability of being in the state where all sservers are busy,
π0(s, ρ), is given by the well-known Erlang loss function (Jagerman,1974); for any s∈N0
and ρ > 0:
π0(s, ρ) = ρs/s!
s
X
y=0
ρy/y!
.(2)
The Erlang loss function can be extended to non-integral values of sby the following
continuous function; for any s∈[0,∞) and ρ > 0:
B(s, ρ) = ρZ∞
0
e−ρx(1 + x)sdx−1
.(3)
The following lemma shows that Equations (2) and (3) coincide for integer values of sand
is due to Jagerman (1974).
Lemma 2.3. B(s, ρ) = π0(s, ρ) for all s∈N0and ρ > 0.
These functions have several useful properties, some of which are captured in the following
two theorems. Theorem 2.4 is due to Jagers and Van Doorn (1986). Theorem 2.5 is based
on (an argument in the appendix of) Smith and Whitt (1981), simultaneously taking out
some of their inaccuracies.
7
Theorem 2.4. For each fixed ρ > 0,B(s, ρ)is a convex function of son [0,∞).
Theorem 2.5. For each fixed ρ > 0and s∈[0,∞),B(ts, tρ)is non-increasing in tfor
t > 0.
Proof. First, let t > 0, ρ > 0 and s∈[0,∞). By letting w=tρx in Equation (3), we
obtain:
B(tS, tρ) = tρ Z∞
0
e−tρx(1 + x)ts dx−1
(4)
=Z∞
0
e−w(1 + w
tρ)tsdw−1
.
For w≥0:
d
dt 1 + w
tρts
=d
dtets·ln(1+ w
tρ )
=1 + w
tρts
·s·ln 1 + w
tρ−w
tρ +w≥0,(5)
where the inequality is obtained by the relation ln(1+ h)≥h/(1 + h) for all h≥0. Hence,
the integral in (4) is non-decreasing in t, from which the theorem follows.
We remark that for s > 0, we can replace in Theorem 2.5 ”non-increasing” with ”strictly
decreasing”, since the inequality in (5) is strict for s > 0 and w > 0.
3 Model description
We first describe our single-echelon, multi-location, single-item inventory model in Section
3.1 and discuss our main assumptions in Section 3.2. Subsequently, in Section 3.3 we
introduce the associated game and define its characteristic cost function.
3.1 The spare parts inventory system
Consider a single company that stocks spare parts in order to combat costly downtimes
of its machines. We limit ourselves to one type of machine and to one critical component,
which is subject to failures. A failure leads directly to a demand for a spare part. This
demand occurs according to a Poisson process with rate λ(for all machines together). We
assume that the demand rate is constant over time. We have an infinite time horizon.
8
If the companies’ stockpoint has a spare part on hand when a demand occurs, then
this demand is fulfilled from stock. Subsequently, the failed part is sent into repair. Repair
lead times are i.i.d. with mean 1/µ. Repaired parts return to stock as ready-for-use spare
parts. There is no condemnation and ample repair capacity.
If no part is available when a demand occurs, an emergency supply is instigated from
an outside infinite source and the machine with the failed component goes down until the
emergency part arrives. The failed part is sent to the emergency supplier and does not
return to the inventory system. When such an emergency order is needed, the company
incurs expected costs cem, which encompasses costs of downtime, lost production and fast
transportation.
Based on these assumptions, the inventory system at this company can be seen as being
controlled by a base stock policy with base stock level S∈N0, where Sis the total number
of spare parts, both on stock and in repair.
Now, consider nindependent companies that each face an i.i.d. failure, repair and
emergency process as outlined above. Furthermore, all companies have chosen the same
base stock levels. We call this the base setting. These symmetry assumptions will be
gradually relaxed later in this paper. We do, however, allow non-identical holding cost
rates in the base setting. Holding costs, which encompass capital and storage costs at a
company i, are incurred at a rate of hiper spare part per unit of time. These costs are
incurred when spare parts are in the on-hand inventory as well as when they are in repair.
The stockpoints of these companies are assumed to be at negligible distance from each
other, i.e. transshipment between stockpoints is free and happens instantaneously.
Aspare parts inventory situation is defined as a tuple (N, S, λ, µ, (hi)i∈N, cem ), where N
denotes the set of companies (independent stockpoints) and Sthe base stock level at each
company, respectively. Furthermore, λdenotes the demand rate and 1/µ the expected
repair lead time at each company. Lastly, hidenotes the holding cost rate at company
i∈Nand cem denotes expected emergency procedure costs at each company. Throughout
this work we assume S∈N0,λ > 0, µ > 0, cem >0 and hi≥0 for all i∈N. With Γ we
shall denote the set of all spare parts inventory situations.
3.2 Discussion of critical assumptions
We have made several assumptions in the formulation of our model. In this section, we will
justify why our assumptions are reasonable for settings with expensive spare parts meant
for technologically advanced machines. Our main assumptions are as follows:
9
(i) Demands at each stockpoint occur according to independent Poisson processes with
constant rates. Many real-life complex machines are employed at close-to constant rates
and have components with long and (close to) exponentially distributed lifetimes (Wong
et al.,2006). Furthermore, a company typically employs a large amount of these machines.
Consider, for example, a commercial airline company, which employs many airplanes, each
of which is flying a comparable number of hours per week. We remark that when a machine
is down, no failures occur. However, the fraction of time during which a machine is down
is negligible, since failure rates are low and, due to the use of emergency procedures,
downtimes are never long. Hence it is reasonable to assume a Poisson failure process with
constant rates for all machines at a company together, as is often done in the spare parts
literature. Independence holds since each company will serve a disjoint set of machines.
(ii) A failed component is immediately sent into repair and is perfectly repairable. One-
for-one repairs are reasonable if the setup and transportation costs associated with initiat-
ing a batch of repairs are small relative to the price of the spare part and if time between
successive demands is long. This is the case for expensive low-demand spare parts. Note
that for our model it is in fact only essential that the inventory positions be kept at a
constant level. Therefore, the model is also applicable for spare parts under condemnation
and even for consumable spare parts, if a new spare part is immediately procured in case
a part cannot be repaired and the expected lead time of obtaining a ready-for-use spare
part remains 1/µ. Such an (S-1,S) inventory policy is reasonable for low-demand spare
parts for which the fixed ordering costs are small relative to the price of the spare part.
(iii) There is ample repair capacity and the repair processes of the companies are i.i.d..
In most cases, all companies will be using the same repair facility. If each company is
a separate business unit of a parent company, or if the OEM is doing the repairs for all
companies, then this is definitely the case. It is common business practice for a repair
facility to agree on a certain fixed repair lead time with its customers, in which case our
assumption is justified.
(iv) We have an infinite time horizon. Real-life complex machines typically have life-
times of several decades, which is long enough to justify the use of an infinite time horizon.
(v) In case of a stock-out, an emergency supplier sources the spare part, the machine
goes down until it arrives, and the failed component is sent to the emergency supplier.
Availability of an outside infinite source is often assumed in spare parts literature; see
e.g. Alfredsson and Verrijdt (1999), Wong et al. (2005), Kutanoglu (2008), Kranenburg
and van Houtum (2009) and Reijnen et al. (2009). It is reasonable when hourly downtime
costs are very high, in which case one does not want to wait for a normal replenishment
10
of a spare part or on a direct repair of the failed component. In case of a stock-out,
the machine goes down because the failed component is critical. If not, then it is still
reasonable to assume that the failed component has a penalizing effect on the performance
of the machine, which is then reflected in cem. We stated that the failed component is sent
to the emergency supplier, since our model requires inventory positions to stay at constant
levels. Another sensible way to retain constant inventory positions would be for companies
to lease a spare part from the emergency supplier until the failed component is repaired,
which is common practice in the airline industry (Wong et al.,2006).
(vi) Stockpoints are at negligible distance from each other. For e.g. airline companies
with adjacent maintenance facilities and spare parts stockpoints at an airport, this is
a reasonable assumption. And when we consider heavy components such as engines or
turbines, then the assumption of negligible transshipment time may even be justified if
stockpoints are slightly more geographically dispersed, since for such heavy components
substantial time is needed to take off the failed component from the machine and prepare
it to receive the spare part, during which the transshipment can take place (as mentioned
by Kukreja et al.,2001).
(vii) The chosen base stock level is fixed. This implies that after coalition formation the
combined base stock level will not be altered, even if a change may lead to lower expected
costs. This assumption is reasonable when companies consider a cooperation during only
a certain part of the operational life of a machine and buying/selling spare parts to jointly
optimize base stock levels for this time period is unwieldy or too expensive. Such a type of
cooperation is interesting, since cooperative arrangements over the entire life cycle are not
widely employed in practice yet and/or it may be hard to sustain the trust needed for a
deep integration in a rapidly changing business environment (Kilpi and Veps¨al¨ainen,2004;
Kilpi et al.,2009). Furthermore, in case the spare part in question is not in production
anymore, changing stocking levels is also practically impossible.
(viii) Complete pooling is applied. This is an assumption on the cooperation process,
which will be described in the following section. It is commonly made in the spare parts
literature (see Paterson et al.,2009). This assumption will be discussed further in Section
4.4.
3.3 Cooperation between companies
Let ϕ= (N, S, λ, µ, (hi)i∈N, cem ) be a spare parts inventory situation. Consider M∈2N
−,
i.e. a coalition of m=|M|companies (1 ≤m≤ |N|). We assume that these companies
11
&%
'$
0&%
'$
1 ... &%
'$
mS
-
-
-
mS ·µ(mS −1) ·µ µ
mλ mλ mλ
Figure 1: A Markov chain representation of the inventory process with mcompanies and expo-
nentially distributed repair lead times. A state is defined by the number of spare parts on hand.
cooperate by fully pooling their inventory of spare parts. In this arrangement, companies
do not hold back any stock for themselves. They will always honor a spare part request
of another company when that other company faces a demand while being out-of-stock.
As emergency procedure costs can only be incurred when none of the members of the
pooling group have a spare part available and all companies have the same repair lead
time distribution, it is effectively irrelevant which company in Msources a demand for a
spare part. Hence, the stockpoints of all companies in the coalition together can be seen
as one joint warehouse with an aggregate base stock level of mS facing a Poisson demand
process with rate mλ.
It follows from our assumptions that the stock-on-hand process of the spare parts
inventory is identical to the process for the number of free servers in an Erlang loss system
with an arrival rate mλ, mean service time µand mS servers. As a result, the steady-
state probability that no part is available on stock when a demand comes in, i.e. the
probability that mS parts are in repair, is equal to π0(mS, mλ/µ) as described by the
Erlang loss function (2). For the specific situation where repair lead times are exponentially
distributed, we would obtain an M/M/mS/mS queue, for which the Markovian inventory
process is depicted in Figure 1.
We will now formulate a game associated with the spare parts inventory situation ϕ.
Coalition Mfaces holding costs of Pi∈MhiSper unit of time and, during the fraction of
time in which total on-hand inventory in the coalition is zero, expected costs of |M|λcem
per unit of time for emergency procedures. Each company is interested in the expected
costs per unit of time. Hence, the game associated with ϕis defined by
cϕ(M) = X
i∈M
hiS+π0|M|S, |M|λ
µ· |M|λcem for all M∈2N
−.(6)
Example 3.1. Consider the 3-company spare parts inventory situation ϕ= (N,S,λ,µ,
(hi)i∈N,cem) with player set N={1,2,3}, base stock level S= 1, yearly demand rate
12
λ= 5, expected years of repair lead time 1/µ = 1/25, holding dollar costs per spare part
per year h1=h2=h3= 4000, and dollar costs per emergency procedure cem = 13000. By
Equation (2), the steady-state probability that a single company, working alone, has no
part on hand is π0(1,5
25 ) = 1/6. For two companies pooling inventory, π0(2,10
25 ) = 2/37 and
for the grand coalition, π0(3,15
25 )=9/454. Finally, by Equation (6) the associated game is
described by:
cϕ(M) =
0 if M=∅;
148331
3if |M|= 1;
15027 1
37 if |M|= 2;
15865145
227 if M=N.
Note that this game is balanced, i.e. it has a non-empty core, since for example x=
(cϕ(N)/3, cϕ(N)/3, cϕ(N)/3) is a core element. However, it is not concave, since for exam-
ple c({1}∪{3})−c({1})< c({1,2}∪{3})−c({1,2}). ♦
We remark that in our example, the overall stock-out probability did not increase when
more identical companies joined a pooling group. This holds in general, which was shown
in Theorem 2.5. Non-concavity in this example is in line with the observation of Kilpi and
Veps¨al¨ainen (2004) that ”the more similar the sizes of the cooperating airlines are, the
higher is the savings potential of the total cooperative effort”.
4 Analysis of the core
In this section we investigate balancedness of spare parts inventory pooling games. First
we do this for the base setting in Section 4.1. Afterwards, we consider two complementary
generalizations that relax some of the symmetry assumptions: in Section 4.2 we allow
companies to have non-identical emergency costs and in Section 4.3 we allow companies to
have non-identical base stock levels as well as non-identical demand rates. In both cases,
we show that the associated games are totally balanced. Finally, in Section 4.4 we consider
the combination of both generalizations, i.e. situations where companies have non-identical
emergency costs and moreover non-identical base stock levels and/or demand rates. For
such situations, we show that the associated game may have an empty core.
13
4.1 The base setting
Let ϕ= (N, S, λ, µ, (hi)i∈N, cem )∈Γ be a spare parts inventory situation. We now define
the allocation vector ”No Transfer Payments”, NTPϕ∈RN, as follows for all i∈N:
N T P ϕ
i=hiS+π0|N|S, |N|λ
µ·λcem.(7)
This particular allocation NTPϕis easy to compute and easy to administer, as each
company simply pays its own holding costs and its own local emergency procedure costs.
The following theorem shows that NTPϕis a core element of the game associated with ϕ.
Theorem 4.1. For all spare parts inventory situations ϕ∈Γ,NTPϕ∈Core(N , cϕ).
Proof. Let ϕ= (N, S, λ, µ, (hi)i∈N, cem )∈Γ. It is easily seen that NTPϕis efficient, i.e.
Pi∈NN T P ϕ
i=cϕ(N). We will now show that NTPϕis stable. Let M∈2N
−. Then:
X
i∈M
N T P ϕ
i=X
i∈M
hiS+π0|N| · S, |N| · λ
µ·X
i∈M
λcem
≤X
i∈M
hiS+π0|M| · S, |M| · λ
µ·X
i∈M
λcem =cϕ(M),
where the inequality follows by Theorem 2.5 and Lemma 2.3. We conclude that NTPϕis
stable. This implies that NTPϕis a core element, which completes the proof.
As ϕwas an arbitrarily chosen element of Γ, it follows that for any spare parts inventory
situation, its associated game will have a non-empty core. This result can be strengthened
by noting that every sub-game of (N, cϕ) is a game associated with a spare parts inventory
situation itself. Hence, the following corollary follows immediately from Theorem 4.1.
Corollary 4.2. For all spare parts inventory situations ϕ∈Γ, its associated game (N, cϕ)
is totally balanced.
4.2 Asymmetric emergency procedure costs
In this section we relax the assumption that companies have identical emergency procedure
costs. After all, companies may face different downtime costs or one company may have a
different emergency supplier than another. A spare parts inventory situation allowing for
nonidentical emergency procedure costs is a tuple ϕ= (N, S, λ, µ, (hi)i∈N,(cem
i)i∈N), where
cem
idenotes the emergency costs at company i∈Nand the other parameters are as before.
14
With ΓEwe shall denote the set of all such situations. Hence, the game associated with
ϕ∈ΓEis defined by
cϕ(M) = X
i∈M
hiS+π0|M|S, |M|λ
µ·X
i∈M
λcem
ifor all M∈2N
−.(8)
Analogously to Section 4.1, we define the allocation vector NTPEϕ∈RNas follows for
all i∈N:
N T P Eϕ
i=hiS+π0|N|S, |N|λ
µ·λcem
i.(9)
The following theorem generalizes Theorem 4.1 to the set ΓE. The proof is omitted, as it is
identical to the proof of Theorem 4.1 after replacing every instance of N T P with N T P E
and cem with cem
ithere.
Theorem 4.3. For all spare parts inventory situations ϕ∈ΓE,NTPEϕ∈Core(N , cϕ).
Analogously to the explanation given for Corollary 4.2, we immediately obtain the following
corollary.
Corollary 4.4. For all spare parts inventory situations ϕ∈ΓE, its associated game (N , cϕ)
is totally balanced.
4.3 Asymmetric base stock levels and demand rates
In this section we will consider situations in which companies are allowed to have non-
identical base stock levels as well as nonidentical demand rates. Companies may have
individually optimized their own base stock levels based on different service requirements,
or the OEM may have recommended an erroneous stocking level to one company, leading to
different base stock levels. Companies could face asymmetric demand rates if they employ
a different number of machines or use them in a different setting with different tempera-
ture or humidity. Another possibility is that one company may be using the machines less
intensively than another, implying a lower overall failure rate.
In the process of proving balancedness for such situations, we first need two interme-
diate results. Firstly, we show balancedness for spare parts inventory situations allowing
for nonidentical base stock levels, but with identical demand rates. Secondly, we make
use of this first result in order to prove balancedness for spare parts inventory situations
allowing for nonidentical base stock levels and demand rates, but restricted to situations
with rational-valued demand rates. We conclude this section by generalizing that second
auxiliary result, allowing real-valued demand rates.
15
We begin by defining a spare parts inventory situation allowing for nonidentical base
stock levels, which is a tuple ϕ= (N, (Si)i∈N, λ, µ, (hi)i∈N, cem ), where Sidenotes the base
stock level at company i∈Nand the other parameters are as before. With ΓBwe shall
denote the set of all such situations. Hence, the game associated with ϕ∈ΓBis defined
by
cϕ(M) = X
i∈M
hiSi+π0 X
i∈M
Si,|M|λ
µ!· |M|λcem for all M∈2N
−(10)
We now show that even though companies may have different base stock levels, the asso-
ciated game still has a non-empty core. In the process of proving this, we will use Lemma
4.5, which considers balanced combinations of stock-out probabilities.
Lemma 4.5. Let ϕ= (N, (Si)i∈N, λ, µ, (hi)i∈N, cem )∈ΓBbe a spare parts inventory situ-
ation and let κ: 2N
−→[0,1] be a balanced map. Then:
|N| · π0 X
i∈N
Si,|N|λ
µ!≤X
M∈2N
−
κ(M)· |M| · π0 X
i∈M
Si,|M|λ
µ!.(11)
Proof.
|N| · π0 X
i∈N
Si,|N|λ
µ!=|N| · π0
X
M∈2N
−
κ(M)·X
i∈M
Si,|N|λ
µ
=|N| · B
X
M∈2N
−
κ(M)·|M|
|N|·|N|
|M|X
i∈M
Si,|N|λ
µ
≤ |N| · X
M∈2N
−
κ(M)·|M|
|N|·B |N|
|M|·X
i∈M
Si,|N|λ
µ!
≤ |N| · X
M∈2N
−
κ(M)·|M|
|N|·B X
i∈M
Si,|M|λ
µ!
=X
M∈2N
−
κ(M)· |M| · π0 X
i∈M
Si,|M|λ
µ!,
where the first equality holds by Lemma 2.2 (taking f(i) = Sifor all i∈N) and the second
equality by Lemma 2.3. The first inequality holds by Theorem 2.4, since PM∈2N
−
κ(M)|M|
|N|=
1 (taking f(i) = 1 for all i∈N) by Lemma 2.2. The second inequality holds by Theorem
2.5. The final equality holds by Lemma 2.3 again.
16
From the following lemma, it is inferred that games associated with a spare parts inventory
situation allowing for nonidentical base stock levels have a non-empty core.
Lemma 4.6. For all spare parts inventory situations ϕ∈ΓB, its associated game (N, cϕ)
is totally balanced.
Proof. Let ϕ= (N, (Si)i∈N, λ, µ, (hi)i∈N, cem )∈ΓBand let κ: 2N
−→[0,1] be a balanced
map. We start by exploiting the result of Lemma 4.5. In Equation (11), multiply both
sides by λcem and subsequently add Pi∈NhiSito both sides to obtain:
cϕ(N)≤X
i∈N
hiSi+X
M∈2N
−
κ(M)·π0 X
i∈M
Si,|M|λ
µ!· |M|λcem (12)
By Lemma 2.2 (taking f(i) = Sifor all i∈N), we can rewrite Pi∈NhiSito PM∈2N
−
κ(M)·
Pi∈MhiSi. Then it is easily seen that Equation (12) is equivalent to cϕ(N)≤PM∈2N
−
κ(M)·
cϕ(M). Hence, the game is balanced. Noting that every sub-game of (N, cϕ) is a game
associated with an element of ΓBitself completes the proof.
We shall now generalize the result of Lemma 4.6; in addition to allowing nonidentical base
stock levels, we will also allow companies to have nonidentical demand rates. Now, an
inventory situation allowing for nonidentical demand rates and base stock levels is a tuple
ϕ= (N, (Si)i∈N,(λi)i∈N, µ, (hi)i∈N, cem ), where λidenotes the demand rate at company
i∈Nand the other parameters are as before. With ΓDwe shall denote the set of all such
situations. Hence, the game associated with ϕ∈ΓDis defined by
cϕ(M) = X
i∈M
hiSi+π0 X
i∈M
Si,X
i∈M
λi
µ!·X
i∈M
λicem for all M∈2N
−.(13)
In order to prove nonemptiness of the core of such games in general, we find it convenient
to first restrict ourselves to the subset of ΓDwith rational-valued demand rates, which we
denote by ΓD
Q.
Lemma 4.7. For all spare parts inventory situations ϕ∈ΓD
Q, its associated game (N, cϕ)
is totally balanced.
Proof. Let ϕ= (N, (Si)i∈N,(λi)i∈N, µ, (hi)i∈N, cem )∈ΓD
Q. For all i∈N, we know that
λi∈Qand hence we can pick ai, bi∈Nsuch that λi=ai
bi. Let `= (Qi∈Nbi)−1and for all
i∈Nlet Ki=ai
bi·1
`.
17
Now, we will construct a spare parts inventory situation allowing for nonidentical base
stock levels ϕB∈ΓB, by splitting each company i∈Ninto Ki(sub)companies such that
each (sub)company has a demand rate of `. We define ϕB= ( ¯
N, ¯
S,¯
λ, ¯µ, ¯
h,¯cem) by
•¯
N=N1∪N2∪. . . ∪Nnwith Ni={j1
i, j2
i, . . . , jKi
i}for all i∈N;
•¯
S= ( ¯
Sk
i)jk
i∈¯
N, where for all i∈N:¯
S1
i=Siand ¯
Sk
i= 0 for all k∈ {2, ..., Ki};
•¯
h= (¯
hk
i)jk
i∈¯
N, where ¯
hk
i=hifor all i∈Nand all k∈ {1, ...Ki};
•¯
λ=`;
•¯cem =cem;
•¯µ=µ.
For all M∈2N
−, define L(M) to be the set of (sub)companies in ¯
Ncreated from companies
in M, i.e. L(M) = Si∈MNi. Let M∈2N
−. Then
cϕB(L(M)) = X
jk
i∈L(M)
¯
hk
i¯
Sk
i+π0
X
jk
i∈L(M)
¯
Sk
i,|L(M)|`
µ
· |L(M)|`cem
=X
i∈M
hiSi+π0 X
i∈M
Si,X
i∈M
λi
µ!·X
i∈M
λicem =cϕ(M).
Now, let y= (yk
i)jk
i∈¯
Nbe an element of the core of game ( ¯
N, cϕB), which exists by Lemma
4.6. For all i∈Ndefine cost allocation xi=PKi
k=1 yk
i. By making use of cϕB(L(M)) =
cϕ(M), as derived above, and y∈Core(¯
N, cϕB), we obtain
X
i∈N
xi=X
i∈N
Ki
X
k=1
yk
i=cϕB(¯
N) = cϕ(N) (efficiency); and
X
i∈M
xi=X
i∈M
Ki
X
k=1
yk
i=X
jk
i∈L(M)
yk
i≤cϕB(L(M)) = cϕ(M) (stability).
We conclude that (xi)i∈N∈Core(N, cϕ). Therefore, Core(N, cϕ)6=∅. Finally, noting that
every sub-game of (N, cϕ) is a game associated with an element of ΓD
Qitself completes the
proof.
18
In the following theorem we generalize the result of Lemma 4.7 to the set ΓD, allowing
real-valued demand rates.
Theorem 4.8. For all spare parts inventory situations ϕ∈ΓD, its associated game (N , cϕ)
is totally balanced.
Proof. We employ a straightforward continuity argument. Let ϕ= (N, (Si)i∈N,(λi)i∈N, µ,
(hi)i∈N, cem)∈ΓD. Pick, for all i∈N, a sequence {Λn
i}∞
n=1 such that Λn
i∈Q+for all n∈N
and limn→∞ Λn
i=λi. For all m∈Nwe define, by replacing demand rates (λi)i∈Nin ϕby
(Λm
i)i∈N, a new spare parts inventory situation ϕm= (N, (Si)i∈N,(Λm
i)i∈N, µ, (hi)i∈N, cem).
Let κ: 2N
−→[0,1] be a balanced map. By Lemma 4.7, we have for all m∈N
X
M∈2N
−
κ(M)cϕm(M)≥cϕm(N).(14)
The Erlang loss function π0(s, ρ) is continuous in ρand therefore the characteristic cost
function is continuous in the demand rates of all companies. By combining this continuity
and Equation (14), we can obtain
X
M∈2N
−
κ(M)cϕ(M) = lim
m→∞ X
M∈2N
−
κ(M)cϕm(N)≥lim
m→∞ cϕm(N) = cϕ(N).
We conclude that (N, cϕ) is balanced. Noting that every sub-game of (N, cϕ) is a game
associated with an element of ΓDitself completes the proof.
4.4 Non-optimal full pooling
We now consider spare parts inventory situations where companies have different emergency
costs and moreover asymmetric base stock levels and/or demand rates. Then we can find
situations where full pooling is not always beneficial for all companies. We will show
two examples. The first example considers a situation with non-identical demand rates
and emergency costs and the second example considers a situation with non-identical base
stock levels and emergency costs.
Example 4.1. Consider ϕ1= (N, S, (λi)i∈N, µ, (hi)i∈N,(cem
i)i∈N), a 2-company inventory
situation that is asymmetric in demand rates and emergency procedure costs, with N=
{1,2},S= 1, λ1= 1, λ2= 24, µ= 25, h1=h2= 400, cem
1= 60000 and cem
2= 30. The
19
associated game is defined by cϕ1(M) = Pi∈MhiS+π0|M|S, Pi∈M
λi
µ·Pi∈Mλicem
ifor
all M∈2N
−. The actual values for our example game are:
cϕ1(M) =
0 if M=∅;
2707 9
13 if M={1};
75232
49 if M={2};
12944 if M=N.
Clearly, the core of this game is empty. In fact, the game is not subadditive. ♦
Example 4.2. Consider ϕ2= (N, (Si)i∈N, λ, µ, (hi)i∈N,(cem
i)i∈N), a 2-company inventory
situation asymmetric that is in base stock levels and emergency procedure costs, with
N={1,2},S1= 3, S2= 0, λ= 5, µ= 25, h1=h2= 400, cem
1= 60000 and cem
2= 30.
The associated game is defined by cϕ2(M) = Pi∈MhiSi+π0Pi∈MS, |M|λ
µ·Pi∈Mλcem
i
for all M∈2N
−. The actual values for our example game are:
cϕ2(M) =
0 if M=∅;
1527117
229 if M={1};
150 if M={2};
3347427
559 if M=N.
Again, we have a game that is not subadditive and that has an empty core. ♦
An intuitive explanation behind these (very similar) counter-examples would be as follows.
Company 2, with very low emergency costs, has relatively low costs by itself. Relative to
its base stock level, company 2 has a much higher demand rate than company 1, i.e. λ2/S2
is much larger than λ1/S1. If we combine both companies in a coalition, then company 2
adds a relatively high demand rate while not contributing sufficient stock to make up for
that. The result is that company 2 is requesting much more spare parts when out-of-stock
from company 1 than vice versa. This assymetry would not impede profitable collaboration
if companies 1 and 2 had identical emergency procedure costs (as shown in Theorem 4.8).
But since company 1 has much higher emergency costs than company 2 in the above two
examples, company 2 is now taking parts that would have better been saved for company
1. Hence, the main problem in these counter-examples lies in the full pooling approach
that is assumed, which is clearly not always optimal when companies are very different.
20
Van Wijk et al. (2009) derive (sufficient) conditions under which full pooling is the
optimal lateral transshipment policy for an inventory situation with two companies (stock-
points). If we apply their conditions to our model for 2 players, retaining our assumption
of zero lateral transshipment costs, we obtain that if λ1
µ+λ1cem
2< cem
1and λ2
µ+λ2cem
1< cem
2
then full pooling is optimal. In examples 4.1 and 4.2, these conditions are not satisfied. We
stress, however, that optimality of full pooling merely implies subadditivity of the game and
not necessarily nonemptiness of the core. On the other hand, we remark that optimality
of full pooling is not a necessary condition for subadditivity or core nonemptiness.
5 Conclusion
We have presented a model of a spare parts inventory system with nindependent companies
that stock the same type of repairable spare part for a technically advanced machine. They
can pool their inventory by keeping own stockpoints but allowing the stockpoint of another
company to satisfy demand in case of a stock-out. Our aim was to determine whether ex-
pected joint costs can be allocated in a stable way. We have proven that the core of the
cooperative cost game is not empty for the base setting in which companies are identical ex-
cept possibly for their holding cost rates. The managerial implication is that collaboration
between independent spare parts stockpoints will be a win-win situation. Furthermore,
this result of core nonemptiness can be generalized to situations allowing companies to
have non-identical base stock levels and demand rates or non-identical emergency costs.
However, when a combination of these asymmetries is considered, then example situations
have been found with empty cores, essentially due to non-optimality of the full pooling
approach.
Directions for future research are manifold and we think that further analysis of spare
parts inventory systems from a game theoretical point of view will prove to be relevant and
fruitful. Our current work can be extended in several ways; situations can be considered
with non-zero lateral transshipment costs, base stock levels that are (jointly) optimized
within a coalition, or a smarter partial pooling approach. A two-echelon structure, in-
corporating the manufacturer or a third-party pooling provider into the model, is also an
interesting extension.
21
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