iBusiness, 2010, 2, 232-237
doi:10.4236/ib.2010.23029 Published Online September 2010 (http://www.SciRP.org/journal/ib)
A Proposal for Estimating the Order Level for
Slow Moving Spare Parts Subject to Obsolescence
Marcello Fera1, Alfredo Lambiase2, Maria Elena Nenni1
1Department of Engineering Management and Economics, Federico II University, Naples, Italy; 2Department of Mechanical
Engineering, University of Salerno, Salerno, Italy.
Received April 18th, 2010; revised June 10th, 2010; accepted August 1st, 2010.
ABSTRACT
During the last decade technologies have had a significant development in many areas, as military and civil protection,
telecommunication and electrical distribution and production. Particularly in the mentioned areas we can find very
complex products, with a cycle life generally longer than their components. Companies have thus the need to better
manage the replacement of spare parts in order to reduce the holding costs and to satisfy the service level. In this paper
authors analyse the state of the art about the spare parts logistic (SPL) problem for products characterized by a long
cycle life and by slow moving spare parts subject to obsolescence. A new model to estimate the spare part order level is
then proposed and tested on a simulated case.
Keywords: Spare Parts Replenishment, Order Level, Obsolescence
1. Introduction
Since the late Fifties, the industrial and academic re-
search has concerned in the topic of the spare parts logis-
tic (SPL). During these years many authors have pro-
posed different approaches for analysing and managing
the SPL problem, but it remains an open issue. Moreover
it has been impossible to individuate a general answer to
the problem that is very specialized and depending on the
field. In this paper the authors have chosen to focus on
the area of military and civil protection, telecommunica-
tion and electrical distribution and production. In these
fields, companies generally must ensure a very high ser-
vice level and manage high tech products, as radar or
other complex systems, whose components are particu-
larly subject to obsolescence. These features make the
SPL problem really critic and strongly linked to other
topics as distribution centre localization (DC) [1] and [2]
and maintenance management (MM) [3].
So far companies have addressed the problem retain-
ing many spare parts or incurring significant costs of
redesigning or finally ignoring it. A structured approach
including the obsolescence problem seems to be actually
missing. Therefore we propose a new model for spare
parts order level estimating as a first step for a new trend
on the SPL topic.
Geisler [4], who developed in 1956 a model to plan
very efficiently the military supplies, gave one of the first
contributions on SPL topic. The model was based on the
knowledge of demand, quantity on hand and priority for
each item. A second contribution by Wiggins in 1967 [3]
is very interesting because he has proposed an inventory
control policy aimed to keep in stock maximum one
spare part for each item. He was thus focussed to under-
stand exactly when a spare part is needed. The main as-
sumption is that a spare part must be in inventory at time
t* resulting from the trade-off between the cost related to
the waiting time and the cost related to the holding time.
In 1972 Hausmann [5] proposed a complex formula-
tion of the total cost for spare part management to indi-
viduate the optimal inventory policy.
In 1978 some authors [6] introduced the concept of
slow moving spare parts, including for them a differenti-
ate cost in a knapsack-type optimization model, which is
solvable in exact way only if the total number of parts is
under 10.000.
In 1997 the vision about the SPL topic changed [7].
The problem began to be seen not only related to the in-
ventory, but involving reliability and the whole produc-
tion management. A model for n parallel machines, lo-
cated in a production bottleneck and requiring spare parts
with a long lead time of supply was developed [8]. The

A Proposal for Estimating the Order Level for Slow Moving Spare Parts Subject to Obsolescence
233
authors emphasized the strategic impact of not having
spare parts when one of the n machines fails and they
included in the cost model parameters from inventory,
operations and maintenance management (i.e., holding
costs, lead time and downtime cost).
Some authors [9] addressed the SPL problem when the
time for waiting spare parts is instead short, including in
their model parameters as 1) time, a continuous variable
in [0,L], where L is the lifetime of the machine, 2) the
lifetime of the part in the machine, exponentially distrib-
uted with parameter λ, 3) the purchasing cost, 4) the
holding cost and 5) the backorder cost. Particularly they
design a model to choose the best policy in function of a
factor R, which denotes the smallest remaining lifetime
for which the stock is increased to 1, with the assumption
it is equal to zero before. The model is aimed to estimat-
ing the optimal spare parts order level.
The last two contributions had the meaning to empha-
size that delivery time can be a key factor for choosing
the optimization model.
In 1999 the SPL problem in the specific case of two-
echelon spare parts distribution centre was deeply inves-
tigated and linked to the DC topic [1]. Mehrotra et al. [10]
used a nonlinear objective function to reduce overall in-
ventory, in case of more than one distribution centre.
In 2000 Strijbosch et al. [11] proposed to use a  dis-
tribution for modelling the spare parts’ demand.
As a matter of fact, the SPL problem has been investi-
gated for long time and from different points of view.
Otherwise many issues are still open just in the field un-
der our focus.
Many problems in the field of military and civil pro-
tection, telecommunication and electrical distribution and
production are related to a strategic vision of the spare
parts management. In fact companies operating on com-
plex systems are increasingly requiring for a maintenance
service included in the purchase contract often during the
whole cycle-life. For sellers it means planning and opti-
mizing the spare parts management for a horizon of even
35 years. In addition we find the just mentioned critical-
ities as 1) high service level, 2) slow moving spare parts,
3) variable and often very long time for supplying spare
parts (delivery time), 4) unknown spare parts’ demand, 5)
spare parts’ obsolescence.
From literature review we have selected the Aka’s
model as more fitting with our requirements. However it
doesn’t take in the right considerations points 2) and 4)
and it completely lacks of a strategic view. Even the ob-
solescence problem is addressed just as an item in the
holding cost. Regarding the spare parts’ demand, the
most frequent approach is based on the historical data but
in our opinion the link between demand and spare parts’
reliability is not thorough enough.
Generally all models solve the SLP problem from a
specific point of view, but no one fully embraces our
objective.
The proposal is then to set a new specific approach in
which the cornerstone and aim of this first paper is the
model for estimating the order level.
2. The Spare Parts Order Level
The main problem in managing spare parts of complex
systems, as radars, satellites, ships, etc., is knowing ex-
actly how many and how long parts have to be main-
tained in stock. The right answer is linked to the 1) de-
mand rate and to the 2) delivery time for each item.
Regarding 1), it strictly depends on item failure rate.
For electric and electronic items, which are the most and
more critic in the systems into consideration, the bathtub
function is typically as in Figure 1.
As well known, in the first part of the curve the Reli-
ability is described by (1) and the probability density
function by (2):












t
IetR )( (1)


























t
Ie
t
tf
1
)( (2)
where
β is a shape parameter; it is expressed by a pure number,
α is the expected life span; it is expressed in days/item,
t is the time; it is expressed in days.
In the second part the failure rate is constant. The reli-
ability is thus by (3) and the probability density function
by (4):
kt
II etR 
)( (3)
λ (t)
t
Ⅰ Ⅱ
Figure 1. The failure rate (bathtub) function for electric
and electronic components.

A Proposal for Estimating the Order Level for Slow Moving Spare Parts Subject to Obsolescence
234
kt
II ektf 
)( (4)
where
k is the constant failure rate expressed in items/day,
t is the time, expressed in days.
In both cases the failure rate function is well-known
and described respectively by (5) and (6):
1
)(
)(
)(


















t
tR
tf
t
I
I
I (5)
k
tR
tf
t
II
II
II  )(
)(
)( 
(6)
Concerning the delivery time, it is generally calculated
through any forecasting model based on historical data.
However in case of obsolescence, the delivery time func-
tion could have a discontinuity because of changing of
item specification or supply condition. The new delivery
time should be initially estimated while the forecasting
model should be applicable only if new or appropriate
historical data are available. Moreover the model will be
fitting until obsolescence occurs again. We are now de-
veloping a new model based on knowing the approach
for the obsolescence managing and on assessing eco-
nomic and technical parameters, in order to estimate
when obsolescence is expected. A first contribution of
our study is already reported in Fera, 2008 [12]. In this
paper we are limited to use only main result from our
studies and we divide the cycle life into parts of length
L(
 ), which is the expected time for occurring obsoles-
cence and we consider a delivery time (DT) as a piece-
wise continuous function in the range L(
 ):





else
known is item the for data historical the If
II
I
DT
LDT
DT
))(( 
(7)
where
Θ (L(τ)) calculates the expected delivery time in days and
depends on L(
 );
Ω is an estimated DT if historical data are unavailable.
According to an order level, order quantity policy (r,
Q), the minimum stock level r should be equal to the
product between the failure’s rate functions and DT.
Combining the different functions for the  and L, we
have four cases: 1) the historical data for delivery time
are known and the failure rate is described by a
Weibull’s function, 2) the historical data for delivery
time are known and the failure rate is constant, 3) the
historical data for delivery time are unknown and the
failure rate is constant, 4) the historical data for delivery
time are unknown and the failure rate is described by a
Weibull’s function.
Minimum stock level for all cases is expressed by (8,9,
10,11):
 ))(()(
1
tL
t
ntDTtr III 







 


 (8)
 ))(()( tLkntDTtr IIIII 



(9)
 

kntDTtr IIIIIII )(

(10)
 







1
)(




t
ntDTtr IIIIV (11)
where:
n is the size of the item.
3. The Experimental Campaign
In order to test the effectiveness of our proposal, we have
developed the order level and a comparison models both
through MATLAB software. By varying the model pa-
rameters, as the Table 1 shows, we have derived the ex-
perimental plan:
The total life cycle span (t) was fixed in 12500 days -
about 35 years - and divided into 100 steps. The first part
of the bathtub is instead over on the 1000th day. During
the whole life cycle span, r should be calculated through
models expressed by (8) and (9) both in case of historical
data are know (Model I-II) and by (10) and (11) both else
(Model III-IV). Models by (8) and (10) are valid until the
1000th day and models by (9) and (11) beyond.
Finally we have summarized results from models in 2
matrixes as in Figure 2 in which r is reported varying
parameters during the life cycle span.
In the last row the arithmetic means of the order levels
are reported (rI-II (t) and rIII-IV (t)).
Regarding the comparison model we have chosen to
use a “conservative approach”. We have thus compared
results from our proposal with the result from the same
model in the worst case, in which parameters assume
values to make order level higher. A “conservative ap-
proach” is very frequent for several companies in the
focused field because it reflects the need to meet the ser-
vice level agreement.
Table 1. Variation of the order level parameters.
Parameter Range Step
n (0, 1000] 10
 (0, 2] 0.02
 (0, 2] 0.02
L (0, 730] 7.30
k (0, 2] 0.02

A Proposal for Estimating the Order Level for Slow Moving Spare Parts Subject to Obsolescence
235
Model I-II Model III-IV
t (days) 0 … 1000 1125 …125000 …1000 1125 … 12500
rI = f (n,  , L) rII = f (n, k, L) rIII = f (n, k) rIV = f (n,  )
rI-II (t) rIII-IV (t)
Figure 2. Matrixes in which results from models are summarized.
Time [days]
Figure 3. Comparison between the conservative approach and the order level model.
Figure 4. Percentage improvement of the order level model respect to the conservative approach.
In Figures 3, 4, a comparison between the conserva-
tive approach and the results from the order level models
are reported respectively as size of spare parts and as
percentage improvement. Bearing in mind the low effect-
tiveness of the conservative approach, performance from
proposed model looks equally more than acceptable. In
fact using Model I-II the percentage improvement is al-
ways higher than 90%. In case of Model III-IV the im-
provement is initially about 90% but it decreases rapidly
during the life cycle span. The phenomenon is explain-
able considering that unknowing of delivery time is more
and more penalizing with the time.
Following the general test, we have investigated three
specific cases for which parameters are reported in the
Table 2.
Specific cases were chosen because particularly repre-
senting of many operative situations in the focused con-
text.
Results from Model I-II and Model III-IV and from
the conservative approach are reported partially in the
Table 3. From Figures 5, 6 and 7 performance of the
proposed model is confirmed.

A Proposal for Estimating the Order Level for Slow Moving Spare Parts Subject to Obsolescence 236
Table 2. Parameters for three specific experiments.
Experiment 1 Experiment 3 Experiment 3
 1.5 0.6 1.05
 365 100 200
n 500 1500 1000
L 120 10 65
k 2.7*E-03 8.0*E-04 1.75*E-03
Table 3. Results from Model I-II and Model III-IV and from the conservative approach.
Experiment 1 Experiment 2 Experiment 3
Time Mo.
I-II
Mo.
III-IV Cons. Mo. I-II Mo. III-IV Cons. Mo. I-II
Mo.
III-IV Cons.
1 13 16 500 567 1419 1500249 307 1000
125 143 180 500 82 205 1500317 390 1000
250 203 254 500 62 156 1500328 404 1000
375 249 311 500 53 132 1500335 412 1000
500 288 359 500 47 118 1500340 418 1000
625 162 202 500 12 30 1500114 140 1000
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
12500 162 202 500 12 30 1500114 140 1000
Figure 5. Results from experiment 1.
Figure 6. Results from experiment 2.

A Proposal for Estimating the Order Level for Slow Moving Spare Parts Subject to Obsolescence
237
Figure 7. Results from experiment 3.
4. Conclusions and Open Issues
In this paper the SPL problem was addressed for slow
moving spare parts subject to obsolescence.
Many contributions to the SPL problem have been
presented but no one has been recognised as fitting with
specific requirements. Authors have thus presented a
model for calculating an optimal order level, in which the
main idea is of using a delivery time function with a dis-
continuous trend because of the obsolescence problem.
A first test of effectiveness was conducted comparing
the proposal with a conservative approach. The test has
demonstrated that the proposed model provides an im-
provement of about 80-85% regarding the order level
size with the same service level.
Currently the authors are investigating the delivery
time function to get a better definition based on the per-
fect matching between spare parts specifications and ap-
proach for the obsolescence problem. But the solution
looks already quite good. Next step could be instead fo-
cused in introducing the implications of the distribution
centre localization on the delivery time.
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