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Closed loop repairable parts inventory system: a literature review.
Kapoor, Rohit ; Ambekar, Sudhir
Table 1 below gives the overview of the work done in METRIC.
Table 1 Various METRIC Models and Their Extensions
Source METRIC Modifications / Description
Improvements
Simon [1971] Inclusion of non The model follows a
-recoverable failures one--for--one
replenishment policy
between the bases and
the depot, wherein depot
uses a continuous review
policy for studying the
stationary properties of
a two--echelon
repairable item system
Muckstadt Multi--indenture Developed the multi--
[1973] indenture problem (MOD--
METRIC) where each item
is assumed to be of the
first indenture or Line
Repairable Units and
composed of
subcomponents or Shop
Replaceable Units
Sherbrooke Two--parameter Assumes that aggregate
[1986] approximation for the outstanding orders of
Graves [1985] distribution of all the bases at any
outstanding orders at time equates to the
the bases addition of the number
of back--orders in the
depot for a specified
target stock at the
beginning of the time
period and aggregate
failure at all bases
over the time interval
equivalent to the
delivery time from the
repair depot to the
base.
Moinzadeh Continuous review (r, Q) In this policy the base
and Lee [1986] policy both at the base sends items to the
as well as the depot depots for repair after
there are Q numbers of
failed items and base
also places an order of
same size to the depots
simultaneously to
replenish the stock.
Upon receiving the
failed items from each
base, the depot sends
the items to the base in
batches of Q quantity if
it has sufficient
inventory on hand,
otherwise, a back--
order is made at the
depot
Lee [1987]; Lateral transshipment Pooling groups are
between bases formed by grouping
identical bases. It
assumes that each pool
has identical bases. In
case of non fulfillment
of the demand from the
stock on hand at a base,
an emergency lateral
transshipment is done
from the stock--on--
hand at another base in
the same pooling group
to fill this demand,
otherwise the demand is
back--ordered.
Axsater [1990]
Jung [1993] Non--homogeneous failure The mean failure rate of
process at the base items is assumed to vary
with time and is based
on the concept of
reliability improvement.
Increased usage and
failures will lead to
design or repair
improvement and the mean
time between failures of
items is an increasing
function of the total
operational time. Spare
stock levels at the
bases and at the repair
facility are estimated
using the approach of
Graves [1985],
Kim et al. Algorithm for stock The stock level which
[1996] level at the base minimizes the total
expected holding and
shortage--cost function
is found by the
bisection method.
Another stock level
which satisfies the
fill--rate criterion is
also obtained. The
maximum of these is
chosen to be the optimal
stock level.
Diaz and Capacity constraint and Three cases of limited
Fu [1997] different priority repair capacity are
classes for repair considered, viz., M/M/c,
M/G/c single class
and M/G/c--multi -class
priority. It has been
shown that the
performance levels of
spare items at bases
significantly differ
under the assumption of
finite repair capacity.
Wang et al. Base--dependent The steady--state
[2000] distribution of transit probability distribution
times from the base to function of transit
the depot times for each base is
derived and from this
the distribution of
outstanding orders is
estimated. Through the
analytical results, it
has been shown that
there is a significant
difference in the
service levels at the
bases when they differ
in their transit times
to the depots.
Rustenberg Application of VARI- The study observes
et al. [2001] METRIC on a complex-- through an extensive
technology organization literature review that
the VARI--METRIC method
requires some
modifications with
respect to the
capacitated systems and
the hybrid product
structures with both,
repairable and non--
repairable parts.
Wang et al. Priority class of Exact steady--state
[2002] service differing in probability
replenishment lead times distributions of random
base delays are derived
for both the services.
There is a significant
reduction in inventory
when the service is
changed from emergency
to non -emergency type.
Table 2
General Queuing Network Models
Source Queuing Environment Description
Jackson [1963] Poisson arrival, Under the product--form
exponential service structure, the system is
times, multi--stages, solved by analyzing each
FCFS, open queuing node separately and then
network, exact analysis the results are
combined. In this case,
the joint probability
distribution of queue
lengths at each node in
the system is equal to
the product of the
probability distribution
of queue lengths at each
node
Gordon and Exponential service Suggested approximate
Newell [1967] time, multi--stages, expressions for the
closed queuing networks, marginal probability
exact and approximate distribution of items in
analysis. the system. An
asymptotic analysis for
closed systems with very
large number of stages
is also carried out.
Baskett Service time with Joint probability
et al. [1975] rational Laplace distribution (steady-
transformation; open, state) of queue lengths
closed and mixed is derived for multi--
network. node, multi--class
items. Four cases--
those of queue
discipline, first--
come--first--served
(FCFS), processor
sharing, no queuing, and
last--come--first--
served (LCFS)--are
analyzed.
Reiser and Closed networks, mean Through--put time (when
Lavenberg value analysis an item is processed at
[1980] a node), throughput
rates and queue lengths
are updated sequentially
according to Little's
Law. The convergence of
queue lengths at several
nodes is used as a
stopping criterion.
Whitt [1983]; General arrival, general Uses the decomposition
Bitran and service time, multi-- approach for analyzing
Tirupati [1988] stages, Open networks, the complex open queuing
approximate analysis networks. In the
using decomposition decomposition approach,
approach. the interaction between
the various nodes is
analyzed first. Then the
network is decomposed
into sub--systems of
individual stations and
analyzed. The results
are then recomposed to
obtain the network
performance
Table 3
Queuing Models Applied to Repairable Item Inventory Problem
Source Problem Characteristics/ Description
Solution Methodology
Gross et al. Multi-echelon, Poisson The model consists of a
[1983] failure, Exponential system which has a single
service time, FCFS, base with a base repair
Implicit enumeration facility and a depot
repair facility. The
system here is viewed as
a network with three
nodes, one for operating
and spare machines at the
base, one for machines in
base repair, and one for
machines in depot repair.
An optimization problem
is also solved where the
decision variables are
the base and depot repair
capacities along with the
spare items. The cost is
minimized subject to the
constraint of operational
availability
Gross et al. Multi-echelon, Poisson The number of items owed
[1987]; failure, Exponential by the depot to the bases
Gupta and service time, FCFS, (back--orders) is fixed
Albright Decomposition approach so that the problem can
[1992] be decomposed. After
decomposing, problem at
each base is solved using
one--dimensional birth--
and--death process. The
problem at depot is
solved using an n--
dimensional birth--and--
death model
Gross et al. Multi--echelon, Poisson The model simplifies the
[1993] failure, Exponential inversion and a solution
service time, FCFS, is obtained by
Iterative procedure iterations. Methods for
carrying out iterations
include the Jacobi
iteration, Gauss--Seidel
and the bi-conjugate
gradient.
Daryanani and Multi--echelon, Poisson The model gives
Miller [2002] failure, Exponential computational formula for
service time, Dynamic the steady--state
backorder filling probabilities.
policy, iterative
procedure involving the
taboo structure of state
space.
Table 4
LORA Models
Authors Tools used to model LORA Description
Alfredsson Mathematical framework A mathematical framework
[1997] using both LORA and for solving the problems
METRIC related to decision about
the optimizing the amount
of spare part to be
stocked, level of test
equipment, tools and
repair man power to be
installed and the
locations of these
installations and spares
using C programming.
Barros [1998] Integer programming This model is used for
maintenance planning as a
tool for deciding among
the options available for
level of repair. The
model divides the time
dependent life-cycle
maintenance cost in two
fixed and variable costs
for getting the optimum
solution.
Saranga and Genetic Algorithm Optimization model for
Kumar [2006] maintenance of aircraft
engine. This algorithm
can be effectively used
for deciding about
allocation of
repair/reject option
among different echelons.
This allocation will help
in minimizing total
maintenance cost where in
total life cycle cost
(LCC) will also be
minimum at the design
stage.
Basten et al. Integer programming by Specific problem solved
[2009] removing the integrality in this study is a NP
constraints hard problem. The
integrality constraints
on most of the variables
are removed. The problem
is a linear programming
problem after removal of
the integrality
constraints and can be
solved in polynomial
time. The computational
time is dependent on
number of components in
the system, indenture and
echelon levels and number
of fixed cost sets of the
components.
Brick and Mixed integer programming The model considered
Uchoa [2009] discrete location of
facilities and
installation of
capacitated resources and
applied to 15 real world
problem which has
distinct maintenance
polices. This technique
is considered as more
comprehensive as compared
to other methods and can
be solved in reasonable
time using any commercial
solvers.
Basten et al. Minimum cost flow problem The modeling consist of
[2011] with side constraints four nodes-source node,
decision node,
transformation node and
sink node and the use of
resources act as side
constraints. An
occurrence of failures of
a certain subsystem at a
certain system location
as a source node is
modeled. From these
source nodes it is moved
to Decision nodes which
consists of three
decisions namely, move
the component to next
level, repair the
component or reject the
component. If the repair
option is chosen at
decision node then parts
are moved to the
transformation node which
represents the repair of
a parent component. If
the decision process end
with any of the decision
then parts are moved to a
sink node.
Basten et al. Integrated Algorithm for The model solves the
[2012 A] jointly solving of LORA problem of maintenance of
and spare parts stocking capital goods like MRI-
problems scanner at hospital or
baggage handling system
at airport. It has shown
significant amount of
cost reduction. It
developed an integrated
algorithm which jointly
solved LORA and spare
parts stocking problems
based on Alfredsson,
[1997] mathematical
model. The spare part
stocking problem was
solved by using a METRIC
method. This integrated
algorithm was effective
for solving two-echelon
(or single-echelon),
single-indenture problems
Table 5
below presents the classification of the recent papers.
Source METRIC Modifications / Problem
Improvements Characteristics
Jung [2003] algorithm to find the Multi-echelon repairable
spare inventory level at inventory system with
each base so as to emergency lateral
minimize total expected transshipments
cost
Caglar et al. minimize the system- two-echelon, multi-item
[2004] wide inventory cost spare parts inventory
subject to a response system; Poisson process
time constraint ateach for part failure; highly
field depot reliable and very
expensive parts
Kilpi and balanced inventory Standard statistical
Veps.al.ainen pooling arrangements model of component
[2004] among various airlines availability showing
relations between the
four factors of
availability
(reliability, turnaround
time, service level and
the number of units
supported)
Kim et al. An algorithm to find general repair time
[2007] spare inventory level to distribution; M/G/c
minimize the total queuing system for both
expected cost and base and depots repair
simultaneously to
satisfy a specified
minimum service rate
Kim et al. Performance based Poisson process for part
[2007 A] contract using multitask failure; repair facility
principal-agent model with infinite capacity
modeled as an M/G/"
queue; each supplier is
compensated based on his
total realized cost and
realized backorder level
Wong et al. cost allocation problem Game theoretic models;
[2007] in the context of two games-1) games with
repairable spare parts full cooperation; 2)
pooling games with competition
Kutanoglu and An algorithm to find Two-echelon distribution
Mahajan [2009] inventory level at local system with one central
warehouses that meet all warehouse (depot) and
the time-based service large number of local
level constraints at warehouses (bases);
minimal costs with infinite capacity at
emergency lateral depots; Poisson demands;
transshipments warehouses share their
inventory;
Kilpi et al. Impact of various type Four types of
[2009] of co-operative co-operative strategies
strategies on inventory considered namely solo,
levels and overall cost ad-hoc cooperation,
using a game theoretical co-operative pooling and
setting commercial pooling
Mirzahosseinian inventory model for a Poisson process for part
and Piplani repairable parts system failure; exponential
[2011] operating under distribution for repair
Performance based time; M/M/m queue
contract inventory system;
Jin and Trade-offs between Renewal equation and
Tian [2012] reliability design and Poisson process for
inventory level estimating the aggregate
fleet failures
Basten et al. iterative algorithm to Poisson process for part
[2012 A] solve the joint problem failure; repair lead
of LORA and spare parts time are IID
stocking
Tracht et al. cost-optimal inventory Poisson process for part
[2013] levels subject to budget failure; repair time is
and inventory level assumed to be constant;
limitations ample repair capacity
Ruan et al. Configuration and Four stage process-1)
[2014] optimization method of forecast spares demand
partial repairable rate; 2) optimize spare
spares stock based on spares
model and algorithm; 3)
determine the support
constraint targets; 4)
calculate spares reorder
point and order quantity
Tracht et al. impact of varying repair Single item system; no
[2014] capacity on a system for item condemnations; the
repairable items repair shop uses first-
come-first-serve (FCFS)
prioritization; Poisson
process for part
failure; exponential
distribution for repair
time
Source Solution Models Inventory Policy
Method
Jung [2003] Approximate Stochastic Continuous review
Caglar et al. Heuristic Continuous review, base
[2004] stock policy
Kilpi and Simulation
Veps.al.ainen model
[2004]
Kim et al. Approximate Stochastic Continuous review
[2007]
Kim et al. Exact One-for-one base stock
[2007 A] policy
Wong et al. Exact Stochastic One-for-one base stock
[2007]
Kutanoglu and Implicit Integer One-for-one base stock
Mahajan [2009] enumeration nonlinear policy
program
Kilpi et al. Simulation
[2009] model
Mirzahosseinian Exact Stochastic One-for-one base stock
and Piplani
[2011]
Jin and Heuristic Multi-phase adaptive
Tian [2012] inventory control policy
Basten et al. Heuristic One-for-one base stock
[2012 A]
Tracht et al. simulation One-for-one base stock
[2013] model
Ruan et al. Exact Combination of (s-1, s)
[2014] and (R, Q) inventory
policy
Tracht et al. Simulation Stochastic Continuous review
[2014] model