INTEGRATED PRODUCTION AND MAINTENANCE PLANNING
Modeling Corrective Maintenance
Veronique Limère, Jasper Deschacht and El-Houssaine Aghezzaf
Department of Industrial Management, Faculty of Engineering and Architecture
Ghent University, Technologiepark 903, 9052 Zwijnaarde, Belgium
Keywords: Production Planning, Preventive Maintenance, Multi-Machine Planning, Integrated Planning.
Abstract: We are given a production system composed of several parallel machines subject to random failures. A set
of items are to be produced in lots on these machines. To prevent failure production system must be
maintained. We assume that these maintenance actions have an effect on the available production capacity
of each machine. The objective is to generate an integrated production and preventive maintenance plan that
optimizes the total costs for the system. In this paper we first discuss an existing mathematical formulation
of the problem and then propose an extension and illustrate it with an example.
1 INTRODUCTION
The recent past years have witnessed a very strong
increase in competition between manufacturing
companies worldwide. To cope with a tough
competition from immerging low-wage countries
and to insure their market position, most of the
western companies invested in highly automated
quality machinery. These machines typically require
fewer operators, produce high quality products, but
are also more expensive. Therefore, a company with
such modern machinery has to optimize the
utilization of the production capacity in order be
profitable (Aghezzaf et al., 2008).
In such a system the production capacity depends
on two processes: the production process itself and
the maintenance process. In most companies, these
two processes are planned independently. The result
is that conflicts may arise between both plans. It is
clear that both processes have a large influence on
each other. Therefore, it is useful to develop a
planning model that integrates both production and
maintenance.
2 STATE OF THE ART
As the importance of integrating production and
maintenance started growing over the recent past
few years, some studies have tried to study the
integrated problem. Ashayeri et al. (1996)
investigated this problem by performing a case study
in the process industry. They worked out an
integrated model for a multi-machine production
system but at the operational level. A large
disadvantage was the use of discrete chances to
simulate machine failure instead of the normally
used failure rate function. Graves & Lee (1999)
investigated an integrated planning for a single
machine also at an operational level. They used the
total weighted completion time as criteria for the
solution. The drawback here is the fact that only one
maintenance activity was allowed during the time
horizon. Later, Lee & Chen (2000) extended this
model to several parallel machines and then to job
shops.
While the above studies focus on the operational
level, Wienstein & Chung (1999) have proposed a
mixed integer program to evaluate the maintenance
policy of a company at the aggregate planning level.
They minimize the sum of the production costs, the
labor costs and the maintenance costs. Another
integrated model at the aggregate level, developed
by Cassady and Kutanoglu (2005), minimizes the
weighted completion time of the jobs. Both
aforementioned aggregate planning models take
into account preventive maintenance actions, but
they ignore reactive maintenance. Aghezzaf et al.
(2007) satisfy this lack by presenting a model that
explicitly takes into consideration the reliability
parameters of the system. The objective of this
204
Limère V., Deschacht J. and Aghezzaf E..
INTEGRATED PRODUCTION AND MAINTENANCE PLANNING - Modeling Corrective Maintenance.
DOI: 10.5220/0003729702040207
In Proceedings of the 1st International Conference on Operations Research and Enterprise Systems (ICORES-2012), pages 204-207
ISBN: 978-989-8425-97-3
Copyright
c
2012 SCITEPRESS (Science and Technology Publications, Lda.)
single machine model is the minimization of fixed
and variable production costs, inventory costs, and
costs related to preventive and corrective
maintenance activities. An extension to parallel
machines is given by Aghezzaf and Najid (2008).
Berrichi et al. (2009) also consider the parallel
machine problem. They propose two genetic
algorithms to solve a bi-objective model for joint
production and maintenance scheduling. The first
objective is related to production scheduling and is
the minimization of the makespan. The second
objective is related to maintenance scheduling and is
the minimization of the system unavailability.
Berrichi et al. (2010) present an ant colony based
heuristic to solve the latter problem leading to
superior results.
3 MODEL AND SOLUTION
ALGORITHM
In this section we propose some alterations for the
mathematical formulation proposed by Aghezzaf
and Najid (2008). We will recapitulate this
formulation and extend the solution algorithm as
proposed by Aghezzaf et al. (2007) for a single
machine system. We will immediately refer to the
rewritten model which can be solved in CPLEX, for
the original model formulation we refer to Aghezzaf
and Najid (2008).
The goal of the model is to generate a production
planning so that each product ∈ fulfills the
demand
. Each machine has a limited capacity
that is consumed by the production process, but also
by the maintenance activities. In order to simulate
machine failure, a failure rate distribution is used.
Assume that preventive maintenance will restore the
machine to ‘as-good-as-new’. Reactive maintenance
will return the machine to an ‘as-good-as-old’ status.
This means only minimal repair is performed at
failure. The PM policy has to be determined by the
preventive maintenance cycle =
. The model
will, besides the production planning, also return the
optimal values of
.
Sets and Parameters
H Set of all the periods in the planning horizon
P Set of all the products
M
Set of all the machines, =1,…,
Cost of each preventive maintenance on
machine m
Cost to carry out a corrective maintenance
action on machine m
≤
Demand for item i in period t
Fixed cost of producing item i in period t on
machine m
ℎ
Variable holding cost of item i in period t
Nominal capacity (given in time units) of
machine
Variable cost of producing item i in period t on
machine m
Process time for each unit i on machine m
Failure rate distribution for machine m
The basic planning period duration
Number of periods of fixed length within the
planning horizon
Number of periods of fixed length within the
preventive maintenance cycle of machine m
Number of preventive maintenance activities
for machine m during the time horizon
=
⁄
Capacity usage because of preventive
maintenance on machine m
Capacity usage because of reactive
maintenance on machine m
Variables
Quantity of item i produced in period on
machine
Inventory of item i at the end of period
Binary variable
=1 if item i is produced in period on
machine
=0 otherwise
Available capacity (given in time units) of
machine in period
Mathematical Model
Minimize
,
,…,
=
+
,
∈
+
−
−1
−1
+
+
+ℎ
∈∈
(1)
Subject to,
∈
+
,
−
=
∀∈,∈
(2)
≤
∈,
∀∈,∈,∈
(3)
INTEGRATED PRODUCTION AND MAINTENANCE PLANNING - Modeling Corrective Maintenance
205
∈
≤
with,
=
−
−
+
−1
if =
−1
+1, i.e. if preventive
maintenance is done in period t on
machine m
=
−
+
−1
if
−1
+2≤≤
∀∈,1≤≤
∀
−1
≤≤
,≤
(4)
,
≥0;
∈;
∈
0,1
∀∈,∈,∈
(5)
We argue that the model as presented above does
not accurately calculate the reactive maintenance
costs. In periods in which no production is planned
on one or both machines, reactive maintenance is
still counted. To avoid this cost miscalculation, we
introduce a new binary variable
. The objective
function can now be changed as in (1’) whereby
reactive maintenance on a machine m will only be
incurred if production is planned in that certain
period t. By adding a new constraint (6) we ensure
that all variables
are assigned the correct values.
Binary variable
=1 if there is production in period on
machine
=0 otherwise
Minimize
,
,…,
=
+
,
∈
+
−
−1
−1
+
+
+ℎ
∈∈
(1’)
≥
∀∈,∈,∈
(6)
The model can now be solved by the solution
algorithm proposed in Aghezzaf et al. (2007),
extended for parallel machines.
4 COMPUTATIONAL RESULTS
The model for parallel machines proposed in this
paper is now compared with the original model
introduced by Aghezzaf and Najid (2008). It is
investigated if the new model generates significantly
different solutions and the impact on total costs is
evaluated.
Assume the time horizon consists of 8τ periods.
The production system of a company consists of two
machines, a new one (M1) and an old one (M2). Due
to the age difference, some of the parameters for the
two machines are different. These parameters are
summarized in Table 1. The inventory costs
ℎ
are independent of the product and the time period
and are equal to 2. Finally the failure function is the
same for both machines. It is a Weibull distribution
with a shape and scale parameter both equal to 1.5.
The values for the failure function are given in Table
2. Demand for both products is given in Table 3.
Table 1: Machine dependent parameters.
Machine 1 Machine 2
=20
=15
=40
=45
=35
=40
=
=1
=5
=6
=
=25;∀∈,∈
=
=5;∀∈,∈
Table 2: Failure function. Table 3: Product
demand.
Age machine Expected #
of failures
t d
1t
d
2t
t=1: [0, 1τ[ 0.544 1 13 6
t=2: [1τ, 2τ[ 0.995 2 8 3
t=3: [2τ, 3τ [ 1.289 3 10 11
t=4: [3τ, 4τ [ 1.526 4 4 6
t=5: [4τ, 5τ [ 1.731 5 7 9
t=6: [5τ, 6τ [ 1.914 6 12 7
t=7: [6τ, 7τ [ 2.081 7 5 9
t=8: [7τ, 8τ [ 2.236 8 8 6
Results for this problem instance are shown in
Table 4. When comparing results for the old and
new model, optimal costs of the different
maintenance policies differ relatively between 8 and
30%. These cost differences are significant and we
therefore conclude that it is worthwhile using our
extended model to obtain realistic cost results. In
this example the optimal solution is for both models
the same, i.e.
=
=5, but the costs differ with
17%. In other cases the new model might even lead
to a different optimal maintenance policy.
ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems
206
Table 4: Cost matrix: Comparison old model with new model.
Limère et al.
(2012)
Maintenance policy Machine 1
k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
Maintenance policy
Machine 2
k=1
1799,1 1691,8 1388,8 1693,6 1222,3 1415,1 1584,1 1758,2
k=2
1619,1 1519,2 1225,7 1544,3 1060,4 1285,4 1450,5 1664,0
k=3
1357,0 1258,4 1135,7 1245,8 989,7 1216,3 1286,3 1358,3
k=4
1529,1 1438,2 1135,7 1497,3 1003,7 1203,3 1437,8 1703,1
k=5
1253,0 1126,6 975,6 1163,0 956,7
1073,5 1154,5 1291,4
k=6
1312,0 1213,4 1090,7 1252,8 993,5 1255,5 1330,1 1403,4
k=7
1379,0 1272,4 1090,7 1344,9 993,5 1255,5 1506,9 1620,0
k=8
1484,1 1393,2 1090,7 1495,3 953,5 1255,5 1506,9 1820,2
Aghezzaf and
Najid (2008)
Maintenance policy Machine 1
k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8
Maintenance policy
Machine 2
k=1
1951,4 1857,5 1553,0 1885,9 1408,1 1607,1 1799,9 1998,1
k=2
1843,6 1749,7 1445,2 1778,0 1300,2 1499,2 1692,0 1890,3
k=3
1588,6 1493,7 1336,2 1512,0 1189,2 1397,2 1500,0 1623,3
k=4
1855,6 1764,8 1462,3 1798,1 1315,3 1519,3 1712,1 1915,4
k=5
1512,7 1417,9 1228,4 1438,2 1149,4
1281,4 1397,2 1535,5
k=6
1637,3 1542,4 1388,9 1562,7 1239,9 1472,9 1585,8 1699,0
k=7
1779,5 1688,7 1472,2 1718,0 1323,2 1556,2 1775,0 1893,3
k=8
1955,0 1864,1 1561,6 1910,4 1414,6 1645,6 1864,4 2120,7
5 CONCLUSIONS
We made a change to the model of Aghezzaf and
Najid (2008) and have shown that our model more
accurately represents the real situation. In the future,
the model can be further extended. For instance, a
production system with machines in series can be
investigated. Moreover, integration of this model at
the aggregate planning level with operational
scheduling models offers a new research direction.
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