Integrated Production and Imperfect Preventive Maintenance Planning
An Effective MILP-based Relax-and-Fix/Fix-and-Optimize Method
Phuoc Le Tam
1
, El-Houssaine Aghezzaf
1
, Abdelhakim Khatab
2
and Chi Hieu Le
3
1
Department of Industrial Systems Engineering and Product Design, Faculty of Engineering, Ghent University,
Technologiepark 903, B-9052 Zwijnaarde, Belgium
2
Industrial Engineering and Production Laboratory, National School of Engineering, Metz, France
3
Faculty of Engineering and Science, University of Greenwich, Kent, U.K.
Keywords:
Production Planning, Imperfect Preventive Maintenance, Optimization, Integrated Strategies.
Abstract:
This paper investigates the integrated production and imperfect preventive maintenance planning problem. The
main objective is to determine an optimal combined production and maintenance strategy that concurrently
minimizes production as well as maintenance costs during a given finite planning horizon. To enhance the
quality of the solution and improve the computational time, we reconsider the reformulation of the problem
proposed in (Aghezzaf et al., 2016) and then solved it with an effective MILP-based Relax-and-Fix/Fix-and-
Optimize method (RFFO). The results of this Relax-and-Fix/Fix-and-Optimize technique were also compared
to those obtained by a Dantzig-Wolfe Decomposition (DWD) technique applied to this same reformulation of
the problem. The results of this analysis show that the RFFO technique provides quite good solutions to the
test problems with a noticeable improvement in computational time. DWD on the other hand exhibits a good
improvement in terms of computational times, however, the quality of the solution still requires some more
improvements.
1 INTRODUCTION
Even though managed by two different departments in
some factories, the production planning and mainte-
nance planning are two closely interrelated functions.
In the major modern factories effort is done to also
carry these two planning functions in an integrated
manner. Researchers have also proposed strategies
and developed models to integrate the production and
maintenance planning decisions both at the tactical
as well as at the operational levels. Various mathe-
matical models focusing on coordinating production
and maintenance plans are proposed in (Lin et al.,
1992; Gurevich et al., 1996; Agogino et al., 1997;
Ben-Daya and Rahim, 2000; El-Amin et al., 2000;
Kiyoshi et al., 2002; Chattopadhyay, 2004; Martorell
et al., 2005; Aghezzaf et al., 2007; Dahal and Chakpi-
tak, 2007; El-Ferik, 2008; Fitouhi and Nourelfath,
2012; Wang, 2013). A wide variety of solution tech-
niques and algorithms including the whole spectrum
of heuristic techniques, dynamic programming, tabu-
search multi-objective optimization, expert systems
and many other hybrid techniques are also proposed,
see for example (Lin et al., 1992; Gurevich et al.,
1996; Agogino et al., 1997; Ben-Daya and Rahim,
2000; Kiyoshi et al., 2002; Chattopadhyay, 2004;
Martorell et al., 2005; Dahal and Chakpitak, 2007).
Integrated production and imperfect preventive main-
tenance planning models were also proposed, see for
example (Chung and Krajewski, 1984; Ben-Daya and
Rahim, 2000; Sana and Chaudhuri, 2010; Fitouhi and
Nourelfath, 2012; Aghezzaf et al., 2016). Imperfect
preventive maintenance, when performed, brings the
manufacturing system to an operating state that is be-
tween as bad as old and as good as new. The result-
ing mathematical models are naturally non-linear and
involve many binary variables. In (Aghezzaf et al.,
2016), the authors proposed a reformulation for the
natural integrated production and imperfect preven-
tive maintenance planning problem. The resulting op-
timization model is a mixed-integer linear program-
ming problem which is solved using a MILP-based
approximation method. The current paper proposes
to adopt a Relax-and-Fix/Fix-and-Optimize approach
and analyse its results.
This Relax-and-Fix/Fix-and-Optimize approach
results in quite good solutions. However, it still re-
quires a large amount of the computational time for
medium and large scale instances of the problem. To
deal with these large scale instances, some heuris-
Le Tam P., Aghezzaf E., Khatab A. and Le C.
Integrated Production and Imperfect Preventive Maintenance Planning - An Effective MILP-based Relax-and-Fix/Fix-and-Optimize Method.
DOI: 10.5220/0006285504830490
In Proceedings of the 6th International Conference on Operations Research and Enterprise Systems (ICORES 2017), pages 483-490
ISBN: 978-989-758-218-9
Copyright
c
2017 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
483
tics based on the Dantzig-Wolfe decomposition tech-
niques and a new version of the Relax-and-Fix/Fix-
and-Optimize method are investigated and developed.
The main goal is to obtain good quality solutions
within a reasonable amount of the computational time
frame.
The remainder of this paper is organized as fol-
lows. In section 2, a slightly modified version of
the mathematical model, for integrated production
planning and imperfect preventive maintenance plan-
ning, propose in (Aghezzaf et al., 2016) is presented.
In section 3, the developed Relax-and-Fix/Fix-and-
Optimize techniques (RFFO) is introduced and pre-
sented in details. Section 4 presents the Dantzig-
Wolfe (DWD) decomposition to solve the reformu-
lated production and maintenance planning model.
Computational results of a set of benchmark cases are
presented and discussed in Section 5. Finally, Section
6 summarizes the main findings of this research work
and discusses some possible research directions.
2 THE INTEGRATED
PRODUCTION AND
IMPERFECT PREVENTIVE
MAINTENANCE MODEL
In this section, the mathematical optimization model
for integrated production and imperfect preventive
maintenance problem described in (Aghezzaf et al.,
2016) is briefly summarized. Then, the reformulation
proposed by the authors for this production and im-
perfect preventive maintenance problem is shown and
used as the underlying optimization model for the de-
tailed subsequent discussion.
2.1 A Mathematical Optimization
Model for the Integrated Production
and Imperfect Preventive
Maintenance Problem (IPImPMP)
In the IPImPMP problems, it is assumed that the
system
´
s operating state is stochastically predictable,
in terms of its operating age, and that it can accord-
ingly be preventively maintained during preplanned
periods. The preventive maintenance is assumed
to be imperfect, so that after each maintenance
action the manufacturing system is at an operating
state that is between as bad as old and as good as new.
Along the same lines as in (Aghezzaf et al., 2016),
we consider a planning horizon H = {1,...T } of T pe-
riods, each having a duration τ, and a set of products
j P = {1, ...N} to be planned during this horizon.
Let d
jt
be the demand for item j in period t, f
jt
be
the fixed cost of producing item j in period t, p
jt
be
the variable cost of producing item j in period t, and
h
jt
be the variable holding cost of item j in period t.
The production system has a known maximum con-
stant production capacity κ
max
(given in time units)
and the processing time of each unit of item j is given
by ρ
j
. The system can be maintained preventively or
correctively when a failure occurs. The cost of carry-
ing out a k
th
preventive maintenance action is denoted
by C
k
PM
and the cost of performing a corrective main-
tenance action on the system when a failure occurred,
right after k
th
preventive maintenance, is denoted by
C
k
CM
. Finally, let δ
k
PM
be the expected time required
for the k
th
preventive maintenance action, and δ
k
CM
the
expected time required to perform a corrective main-
tenance action on the system when a failure occurred,
right after k
th
preventive maintenance.
The variables of the model are: Q
jt
the quantity
of item j produced during period t; I
jt
the inventory
of item j at the end of period t; x
jt
a binary variable
set to 1 if item j is produced during period t and 0
otherwise; y
t
a binary decision variable set to 1 if the
machine is setup to production during period t and 0
otherwise; and finally z
k
st
a binary variable set to 1 if
the last preventive maintenance of the system before
the time period t is the k
th
one and has taken place dur-
ing the time period s and 0 otherwise. By convention
we assume that the manufacturing system is preven-
tively maintained in the beginning of period 1, that is
z
1
11
= 1 for k s t. The optimization model for
the Integrated Production and Imperfect Preventive
Maintenance Planning Problem (IPImPMP) is given
by:
Minimize
Z
IP
ImPMP
=
T
t=1
N
j=1
( f
jt
x
jt
+ p
jt
Q
jt
+ h
jt
I
jt
)
+
T
t=1
t
k=1
C
k
PM
z
k
tt
+
T
t=1
t
s=1
s
k=1
C
k
st
(y)y
t
z
k
st
subject to:
Q
jt
+
I
j,t1
i f t > 1
0, i f t = 1
I
jt
= d
jt
,
j P,t H
(1)
Q
jt
κ
max
x
jt
0, j P,t H (2)
x
jt
y
t
0, j P,t H (3)
ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems
484
N
j=1
ρ
j
Q
jt
+
t
k=1
δ
k
PM
z
k
tt
+
t
s=1
s
k=1
κ
k
st
(y)y
t
z
k
st
κ
max
,t H
(4)
t
s=1
s
k=1
z
k
st
= 1, t H (5)
z
k
st
z
k
s,t+1
0, k, s,t H,k s t T 1 (6)
z
k
tt
t1
s=k1
z
k1
s,t1
0, t,k\{1} H, 1 < k t (7)
T
t=2
t
s=2
z
1
st
= 0 s,t 2 H,s t (8)
t
k=1
z
k
tt
y
t
, t H (9)
Q
jt
,I
jt
0, x
jt
,y
t
,z
k
st
{0, 1}
j P,s,t H,k s t
Constraints (1) are the flow conservation constraints.
They guarantee that the available inventory aug-
mented with the quantity produced in period t is suffi-
cient to satisfy the demand d
jt
of item j in that period.
The remainder is stocked for the subsequent periods.
Constraints (2) make sure that, when the production
of an item is scheduled in certain period, the system
is setup accordingly to produce that item in that pe-
riod. These constraints force also disbursement of the
fixed costs. Constraints (3) indicate whether the sys-
tem is operating or not in each period, in which it is
setup to produce some products. Constraints (4) are
the capacity restrictions which are defined for each
period t H. They guarantee that the quantity which
is produced in a period t does not exceed the avail-
able capacity of the system, given its status in terms
of the expected capacity loss during that period. Con-
straints (5) determine the periods during which the
preventive maintenance activities take place. In order
to assure the consistency, constraints (6) are estab-
lished to guarantee that if the last preventive mainte-
nance action, before a time period t + 1, takes place in
a period s < t and it is the k
th
one, then this preventive
maintenance action must also be the k
th
and the last
one before a period t. Again, in order to keep the con-
sistency, constraints (7) assure that the k
th
preventive
maintenance takes place in some period t k, only
if the (k 1)
th
preventive maintenance took place in
some period before t. Constraints (8) assures that the
system is always maintained for the first time in the
first period. Constraints (9) ensure that the k
th
pre-
ventive maintenance takes place only once and when
the system is setup to production.
As in (Aghezzaf et al., 2016), the function κ
k
ts
(y)
and C
k
ts
(y) are the expected production capacity loss
and expected maintenance cost during the time period
t when the k
th
and the last preventive maintenance ac-
tion before time period t has taken place in the time
period s, with k s t. These parameters depend on
the system’s setup vector y and are given by
κ
k
st
(y) =
δ
k
CM
τ
R
0
β
k
r
0
u + α
k
"
s1
t
0
=1
y
t
0
#
τ
!
du
if t = s,k s,
δ
k
CM
τ
R
0
β
k
r
0
u + α
k
"
s1
t
0
=1
y
t
0
#
τ
+
"
t1
t
0
=s
y
t
0
#
τ
!
du
if s t T.
(10)
C
k
st
(y) =
C
k
CM
τ
R
0
β
k
r
0
u + α
k
"
s1
t
0
=1
y
t
0
#
τ
!
du
if t = s,k s,
C
k
CM
τ
R
0
β
k
r
0
u + α
k
"
s1
t
0
=1
y
t
0
#
τ
+
"
t1
t
0
=s
y
t
0
#
τ
!
du
if s t T.
(11)
We adopt a hybrid failure rate model which was
defined in (Aghezzaf et al., 2016). If after k
th
pre-
ventive maintenance the failure rate function remains
below some threshold function, r
k
max
(t), the system
can again be preventively maintained. However, if
it reaches or exceeds this threshold level, it is over-
hauled and will be returned to an ”as-good-as-new”
state. We considers a lifetime of a whole system is
randomly distributed and for which the correspond-
ing initial hazard rate function is given by the function
r
0
(t). If the k
th
preventive maintenance takes place
T
k
τ units of time after an overhaul, that is in the be-
ginning of period T
k
having fixed length τ, the hazard
rate function r
k
(t) of the system is then defined as:
r
k
(t) = β
k
r
0
(t +α
k
T
k
AOT
),
t [0,(T
k+1
T
k
)τ],k,1 k k
max
(12)
where T
k
AOT
is the actual operating time of the
system since the beginning of the planning hori-
zon until the beginning of period T
k
, the period
during which the k
th
preventive maintenance is
taking place. The parameters α
k
and β
k
stand,
respectively, for the age reduction coefficient and
the hazard rate increasing coefficient (adjustment
factor) such that 0 α
1
α
2
. .. α
kmax
1 and
1 β
1
β
2
.. . β
kmax
.
Integrated Production and Imperfect Preventive Maintenance Planning - An Effective MILP-based Relax-and-Fix/Fix-and-Optimize Method
485
2.2 Reformulation of the Problem
(Re IPImPMP)
The natural formulation of problem (IPImPMP) is
nonlinear. It can be reformulated and modeled as a
mixed-integer linear program as is shown in (Aghez-
zaf et al., 2016). We propose a slight variation of
the mathematical reformulation proposed in (Aghez-
zaf et al., 2016) by adding the variables v
k
st
(p,q), with
p s t,k s and q t s + 1, to be a binary
variable assuming value 1 if the system is setup to
production with p time during the horizon 1, ...,s 1
and q time during the period s,..,t, and the k
th
main-
tenance takes place in period s and, w
k
st
(p,q), with
p s t,k s and q t s + 1, to be a binary vari-
able assuming value 1 if the system is setup for pro-
duction p time during the horizon 1, ...,s 1 and q
time during the period s, ..,t and the k
th
maintenance
takes place in period s and the system must be pro-
duced at period t.
(Re IPImPMP) : Minimize Z
Re IP
ImPMP
=
T
t=1
N
j=1
f
jt
x
jt
+ p
jt
Q
jt
+ h
jt
I
jt
+
T
t=1
t
k=1
C
k
PM
z
k
tt
+
T
t=1
t
s=1
s
k=1
s1
p=0
ts
q=0
C
k
st
(p,q)v
k
st
(p,q)
subject to :
Eq. (1) - (3) and (5) - (9)
N
j=1
ρ
j
Q
jt
+
t
k=1
δ
k
PM
z
k
tt
+
t
s=1
s
k=1
s1
p=0
ts
q=0
c
k
st
(p,q)v
k
st
(p,q) κ
max
,t H
(13)
s1
p=0
ts
q=0
p.u
st
(p,q)
s1
s
0
=1
y
s
0
, i f s > 1
0, i f s = 1
0,
t,s H, 1 s t
(14)
s1
p=0
ts
q=0
q.u
st
(p,q)
t1
s
0
=s
y
s
0
, i f t > 1
0, i f t = 1
0,
t,s H, 1 s t
(15)
s1
p=0
ts
q=0
u
st
(p,q) = 1, t,s H,1 s t (16)
z
k
st
+ u
st
(p,q) v
k
st
(p,q) 1,
k, s,t, p, q H, p s 1,q t s
(17)
y
t
+ v
k
st
(p,q) w
k
st
(p,q) 1,
k, s,t, p, q H, p s 1,q t s
(18)
Q
jt
,I
jt
0, x
jt
,y
t
, z
k
st
, u
st
{p,q},
v
k
st
{p,q}, w
k
st
{p,q} {0, 1}
j P, s,t H, k s t, p s 1, q t s
where u
st
(p,q), with p s t and 0 q t s + 1,
to be a binary variable assuming value 1 if the system
is setup to production p times during the horizon 1,...,
s-1 and q times during the periods s,..., t-1.
Constraints (1) - (3) and (5) - (9) are the same
as before. Constraints (13) are revisited from (4).
However constraints (14), (15), (16) determine the
values of the variables u and v. The constraints (17)
relate the variables z with u and v, meaning that if
the k
th
and last preventive maintenance before t takes
places in period s t and if the system is setup to pro-
duction p times during the horizon {1, ...,s 1} and
q times during the periods {s,...t 1} then v
k
st
(p,q)
=1. The constraints (18) relate the variables v with
y and w, meaning that if both the k
th
and last preven-
tive maintenance before t takes places in period s t
and if the system is setup to production p times dur-
ing the horizon {1,...,s 1} and q times during the
periods {s,...t 1} and the system must be produced
at period t then w
k
st
(p,q) =1.
Here again, as reported in (Aghezzaf et al., 2016),
we let C
k
st
(p,q) and κ
k
st
(p,q) be respectively the ex-
pected maintenance cost and expected loss in produc-
tion capacity of the system during period t, when the
last preventive maintenance action before time period
t has taken place in the beginning of period s, s t.
These parameters are given by:
C
k
st
(p,q) =
C
k
CM
τ
R
0
β
k
r
0
(u + α
k
pτ)du
i f t = s,
C
k
CM
τ
R
0
β
k
r
0
(u + α
k
pτ + qτ)du
i f s t T.
(19)
and
κ
k
st
(p,q) =
δ
k
CM
τ
R
0
β
k
r
0
(u + α
k
pτ)du
i f t = s,
δ
k
CM
τ
R
0
β
k
r
0
(u + α
k
pτ + qτ)du
i f s t T.
(20)
3 RELAX-AND-FIX WITH
FIX-AND-OPTIMIZE
HEURISTIC (RFFO) TO SOLVE
THE Re IPImPMP MODEL
The Relax-and-Fix (RF) heuristic solves MIP prob-
lem by sequentially resolving sub-problems in which
some variable are fixed and others are relaxed. To
ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems
486
solve production planning problems, for example,
the planning horizon is partitioned and the setup
variables are fixed backward or forward. Fix-and-
Optimize is an improvement heuristic also based on
decompositions of the original problem into multi-
ple sub-problems with smaller number of binary vari-
ables. The RFFO method is a framework designed to
combine the Relax-and-fix (RF) and fix-and-optimize
(FO) heuristics. In (Toledo et al., 2015), the authors
propose an approach in which most binary variables
are fixed or relaxed and only few of them are forced to
be integer and are optimized. They named this small
set of integer variables a ”window” and suggested
three window type strategies: row-wise, in which the
window moves along rows; column-wise, in which
the window moves along columns; and value-wise, in
which the window selects the variables with relaxed
values closest to 0.5. As in (Toledo et al., 2015), For
both heuristics applied to the Re IPImPMP model, we
consider a matrix Y where each of its entries is the
binary varaible y
t
. The inputs of the RF are the set
of binary variables (sol.y), the number of binary vari-
ables (windowSize) to be chosen, the selection criteria
to choose variables (windowType), the overlap rate
of binary variables to be re-optimized (overlap) and
the execution time limit (timeLimit). Initially, all bi-
nary variables in the RF solution (sol.y) are relaxed,
and a window is defined as a set that includes a fixed
amount of (windowSize) variables. Then, those vari-
ables which are inside the window are enforced to
be integer in the set y
MIP
, while the others are kept
in y
LP
. We solve the problem to get the results of
MIP. Next, a new set of variables (window) is de-
fined by a subset of fixed integers (y
f ixed
), sets of
optimized integer (y
LP
) and relaxed variables (y
MIP
).
The window moves forward by the step parameter at
each iteration on which each step = round(|overlap
windowSize|), overlap [0,1]. All variables that
leave the window are fixed in the next iteration, and
the same number of relaxed variables are enforced to
be integer. The algorithm proceed in this way until all
variables are fixed. After the RF phase is complete,
FO tries to improve this initial solution until the time
limit has been reached. If the improvement achieved
by a FO solution is not satisfactory, the window size
is increased. The MIP subproblems become larger as
an attempt to find better solutions.
The pseudo-code of the method can be provided to
interested researchers upon request.
4 SINGLE DANTZIG-WOLFE BY
PRODUCT DECOMPOSITION
WITH FIX AND OPTIMIZE TO
SOLVE THE MINLP IPImPMP
MODEL
Dantzig-Wolfe decomposition (DWD) method is used
in (Pimentel et al., 2010) to solve the multi-item ca-
pacitated lot sizing problem with setup times. The
authors applied the standard DantzigWolfe decompo-
sition (DWD) in three different ways. In the first the
subproblems are defined by items (PIDWD), in the
second they are defined by periods (PJDWD) and a
third decomposition in which the subproblems of both
types are integrated in the same model (MDWD). The
three approaches were tested on the IPImPMP model
defined in Section 2.1. Based on the results, the suit-
able approach seems to be the production decompo-
sition which is then selected for the comparison of
the results. We consider the capacity constraints 4,
linking constraints the the variables associated with
different products as the master problem for this de-
composition.
4.1 Master Problem
The master problem includes the collections of pro-
duction plans of each items and the maintenance plan-
ning. The decision variables are ϑ
m
j
, corresponding to
the items production plans m generated by the sub-
problems for products j, and z
k
st
are the variables cor-
responding to the maintenance plan developed at the
master problem level. The linear programming relax-
ation of the master problem of the production decom-
position is given below:
(MPJ-PPM) : Minimize Z
MPJ
IPImPPM
=
N
j=1
M j
m=1
"
T
t=1
( f
jt
¯x
m
jt
+ p
jt
¯
Q
m
jt
+ h
jt
¯
I
m
jt
)
#
ϑ
m
j
+
T
t=1
t
k=1
C
k
PM
z
k
tt
+
T
t=1
t
s=1
s
k=1
c
k
st
( ¯y). ¯y
t
z
k
st
subject to:
N
j=1
M j
m=1
ρ
j
¯
Q
m
jt
ϑ
m
j
+
T
t=1
t
k=1
δ
k
PM
z
k
tt
+
t
s=1
s
k=1
κ
k
st
( ¯y)z
k
st
κ
max
,t H(µ
t
)
(21)
M j
m=1
ϑ
m
j
= 1, j N(π
j
) (22)
Integrated Production and Imperfect Preventive Maintenance Planning - An Effective MILP-based Relax-and-Fix/Fix-and-Optimize Method
487
M j
m=1
¯x
m
jt
ϑ
m
j
<= ¯y
t
, j N;t H (η
jt
) (23)
ϑ
m
j
0, j N (24)
z
k
st
0, j N;t, s H (25)
and Eq. (5) - (9)
where ¯y
t
= max
j,m
n
¯x
m
jt
o
and the objective function
minimizes the overall production and maintenance
costs. The set of constraints (21) are capacity con-
straint. These constraints impose that the combina-
tion of the chosen production plans satisfies the avail-
able capacity in each period based on the maintenance
plans. Constraints (22) are the convexity constraints.
The mixing of the chosen production plans is forced
by constraints (23) to satisfy the production setup re-
quirements. Constraints (24) and (25) force the de-
cision variables to take nonnegative values.
Of course at the end of the process, the problem
is solved again but then with the variables ϑ
m
j
and z
k
st
satisfying the following conditions:
ϑ
m
j
,z
k
st
{0,1}∀ j N,s,t H, k s t,m M j
To recover solution of problem IPImPMP, in
terms of the original variables, we can obtain the
value of (Q
jt
,x
jt
) from a solution of master problem
(MPJ PPM) as follows:
Q
jt
=
M j
m=1
¯
Q
m
jt
ϑ
m
j
, j N, t T (26)
x
jt
=
M j
m=1
¯x
m
jt
ϑ
m
j
, j N, t T (27)
4.1.1 Subproblem
Assuming that µ
t
is the vector of dual variables as-
sociated with the constraints (21), indexed by t, the
π
j
the vector of dual variables associated with the set
of convexity constraints (22) and the η
jt
is a dual
associated variables with constraints (23). Each sub-
problem is one of following types:
Z
SPJ
IPImPMP
=
T
t=1
( f
jt
x
jt
+ p
jt
Q
jt
+ h
jt
I
jt
)
T
t=1
ρ
j
Q
jt
π
j
µ
t
T
t=1
η
jt
x
jt
, j N
or
Z
SPJ
IPImPMP
=
T
t=1
[( f
jt
η
jt
)x
jt
+ p
jt
Q
jt
+ h
jt
I
jt
]
T
t=1
ρ
j
Q
jt
π
j
µ
t
, j N
subject to:
Eq. (1) - (2)
Q
jt
,I
jt
0, x
jt
,y
t
{0,1}
j N, s,t H,k s t, p s 1, q t s
variables :Q
jt
,I
jt
0, x
jt
parameters: d
jt
,µ
t
,π
j
,η
jt
The pseudo-code of the proposed DWD approach
can be provided to interested researchers upon re-
quest.
5 RESULTS AND DISCUSSIONS
In order to evaluate the effectiveness of the
Re IPImPMP model and the developed algorithms,
the following paragraphs present the results of the
computational experiments on some test instances,
available in the literature. In particular, a collection of
test instances from the LOTSIZELIB (Trigeiro, 1989)
is used to evaluate the performance of the model
(Re IPImPMP) and developed algorithm. Of course,
the instances from the LOTSIZELIB were extended
and adapted to the integrate maintenance optimization
aspect as done in (Aghezzaf et al., 2016).
5.1 The Test Instances
The algorithms presented above are coded in AMPL
using the callable CPLEX 12.6 library to solve the
MILP problems. The computation tests were car-
ried out on an Intel(R) Core(TM) i7-3770 CPU @
3.40 GHz, 3401 MHz, 4 Core(s), 8 Logical with 32
GB RAM. under windows 7. CPU times are given
in seconds. For the maintenance part, we assumed
that the machine is subject to the random failures ac-
cording to a Gamma distribution Γ(m=2, ν=2) with
a shape parameter m = 2 and a rate parameter ν = 2
as in (Aghezzaf et al., 2016). We also assume imper-
fect preventive maintenance with α
k
= k/(3k +7) and
β
k
= (12k + 1)/(11k + 1) for all k.
Table 1: Initial value of WindowS ize and Overlap chosen
in RFFO Algorithm.
Instances WindowSize Overlap %
A2007 5 60
tr6 15 8 60
tr6 30 12 60
tr12 15 8 60
tr12 30 12 60
tr24 15 8 60
tr24 30 12 60
set1ch 8 60
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Table 2: A Summary of the experimental results for comparision between the Re IPImPMP and RFFO Heuristic algorithm.
Instances Re PPImPMP CPU Time (sec) RFFO CPU Time(sec) GAP %
A2007 815.435 1.05 815.435 51.25 0.00
tr6 15 337,355.000 4,339.16 337,355.000 210.59 0.00
tr6 30 675,607.000 2,103,730.00 676,080.000 642,130.00 0.07
tr12 15 1,245,920.000 2,710.00 1,252,120.000 520.00 0.50
tr12 30 4,387,470.000 2,088,910.00 4,387,470.000 1,126,980.00 0.00
tr24 15 2,502,640.000 2,920.00 2,502,640.000 245.89 0.00
tr24 30 8,272,760.000 2,031,020.00 8,272,760.000 752,230.00 0.00
set1ch 107,532.000 130.00 107,532.000 244.69 0.00
Table 3: A summary of the experimental results of the DWD applied algorithm.
Instance Value DWD CPU Time (sec) GAP %
A2007 875.520 1.95 7.37
tr6 15 337,787.000 1.65 0.13
tr6 30 859,051.000 1.29 27.15
tr12 15 2,159,410.000 1.05 73.32
tr12 30 8,917,970.000 3.59 103.26
tr24 15 4,829,470.000 1.01 92.98
tr24 30 16,945,000.000 3.76 104.83
set1ch 172,504.000 1.72 60.42
In this study, to evaluate the effect of windowSize
and overlap for the RFFO heuristic method, we tested
all the windowSize parameters from 1 to the last pe-
riod of the planning horizon, and increasing overlap
by step from 1 to WindowSide for instances A20007
and tr6
15 to get initial value of the problem as shown
in table 1.
5.2 Analysis and Discussions about the
Experiments
Table 2 summarizes the results of the experiments
which were carried out to compare the Re IPImPMP
(Aghezzaf et al., 2016) and the RFFO heuristic
method. The first column of the table identifies the
solved instances. The second column reports the op-
timal value of each instance which was obtained from
the Re IPImPMP model and the third column reports
the resulted CPU running time. The fourth column
describes the value of each instance which was ob-
tained by the proposed RFFO heuristic algorithm and
the fifth column presents the obtained CPU running
time. The last column shows the GAP between the
RFFO heuristic value and the Re IPImPMP value.
When comparing the results of the proposed RFFO
with the results of the Re IPImPMP model, it is clear
that the proposed RFFO algorithm has the same op-
timal value with a considerable saving of the CPU
(solve) time; it is faster from 4 to 10 times for the
cases of the medium and/or large scale problems.
However, in the small scale problems (A2007 and
set1ch instances) was increasing large CPU time by
the inner loop algorithm.
Table 3 summarizes the results of the computa-
tional experiments carried out for the DWD decom-
position method. The first column of the table identi-
fies the instances solved. The second column presents
the value of each instance which was obtained via
the proposed DWD decomposition algorithm; and
the third column reports the CPU (solve) time. The
last column describes the GAP between the DWD
decomposition value and the Re IPImPMP value by
GAP% =
(valueDW D valueRe IPImPMP)
valueRe IPImPMP
100.
As shown in Table 3, the proposed DWD pro-
vides solutions which are not so far from the op-
timal Re IPImPMP value, with the less CPU time
and memory which is used to reach the optimality
for the same instance. However, the GAP, which
is defined as the ratio of the difference between the
value of the DWD algorithm and the value of the
Re IPImPMP model, showing it is just suitable for
small and medium scale instances.
6 CONCLUSIONS
In this paper, we investigated the optimization model
of an integrated production planning and imperfect
preventive maintenance. The natural optimization
Integrated Production and Imperfect Preventive Maintenance Planning - An Effective MILP-based Relax-and-Fix/Fix-and-Optimize Method
489
model for this problem is as a nonlinear mixed in-
teger problem. We slightly modified the reformula-
tion (Re IPImPMP) the problem proposed in (Aghez-
zaf et al., 2016) that solves the problem as a lin-
ear mixed integer program. There are a few major
limitations for this reformulated model, including the
time consuming as well as the increased number of
variables and constraints for the large and medium-
sized problems. We applied the Relax-and-Fix/Fix-
and-Optimize heuristics and the Dantzig-Wolfe De-
composition (DWD) methods to select the suitable
strategies to solve the proposed optimization models.
The developed algorithm are tested and compared for
CPU time and gap. The results from the numerical
examples and computational experiments showed that
the developed algorithm for solving the Re IPImPMP
problem has a very good solution quality with reduced
computational time. Further studies are currently in-
vestigated to improve the DWD method in order to
obtain better quality solutions and increase computa-
tional time savings, especially for large scale, block
structured, and linear programming problems of inte-
grated production planning and imperfect preventive
maintenances.
ACKNOWLEDGMENTS
We would like to express our sincere appreciation
to the reviewers for their helpful comments. These
helped improve the quality of this research work.
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