Integrated Production and Imperfect Preventive Maintenance Planning

An Effective MILP-based Relax-and-Fix/Fix-and-Optimize Method

Phuoc Le Tam

, El-Houssaine Aghezzaf

, Abdelhakim Khatab

and Chi Hieu Le

Department of Industrial Systems Engineering and Product Design, Faculty of Engineering, Ghent University,

Technologiepark 903, B-9052 Zwijnaarde, Belgium

Industrial Engineering and Production Laboratory, National School of Engineering, Metz, France

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

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

be the demand for item j in period t, f

the ﬁxed cost of producing item j in period t, p

the variable cost of producing item j in period t, and

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

. The system can be maintained preventively or

correctively when a failure occurs. The cost of carry-

ing out a k

preventive maintenance action is denoted

by C

and the cost of performing a corrective main-

tenance action on the system when a failure occurred,

right after k

preventive maintenance, is denoted by

. Finally, let δ

be the expected time required

for the k

preventive maintenance action, and δ

the

expected time required to perform a corrective main-

tenance action on the system when a failure occurred,

right after k

preventive maintenance.

The variables of the model are: Q

the quantity

of item j produced during period t; I

the inventory

of item j at the end of period t; x

a binary variable

set to 1 if item j is produced during period t and 0

otherwise; y

a binary decision variable set to 1 if the

machine is setup to production during period t and 0

otherwise; and ﬁnally z

a binary variable set to 1 if

the last preventive maintenance of the system before

the time period t is the k

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

= 1 for k ≤ s ≤ t. The optimization model for

the Integrated Production and Imperfect Preventive

Maintenance Planning Problem (IPImPMP) is given

by:

Minimize

ImPMP

∑

t=1

∑

j=1

( f

+ p

+ h

)

∑

t=1

∑

k=1

∑

t=1

∑

s=1

∑

k=1

(y)y

subject to:



j,t−1

i f t > 1

0, i f t = 1

− I

= d

∀ j ∈ P,∀t ∈ H

(1)

− κ

max

≤ 0, ∀ j ∈ P,∀t ∈ H (2)

− y

≤ 0, ∀ j ∈ P,∀t ∈ H (3)

ICORES 2017 - 6th International Conference on Operations Research and Enterprise Systems

484

∑

j=1

∑

k=1

∑

s=1

∑

k=1

(y)y

≤ κ

max

,∀t ∈ H

(4)

∑

s=1

∑

k=1

= 1, ∀t ∈ H (5)

− z

s,t+1

≥ 0, ∀k, s,t ∈ H,k ≤ s ≤ t ≤ T − 1 (6)

−

t−1

∑

s=k−1

k−1

s,t−1

≤ 0, ∀t,k\{1} ∈ H, 1 < k ≤ t (7)

∑

t=2

∑

s=2

= 0 ∀s,t ≥ 2 ∈ H,s ≤ t (8)

∑

k=1

≤ y

, ∀t ∈ H (9)

≥ 0, x

∈ {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

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

one, then this preventive

maintenance action must also be the k

and the last

one before a period t. Again, in order to keep the con-

sistency, constraints (7) assure that the k

preventive

maintenance takes place in some period t ≥ k, only

if the (k − 1)

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

pre-

ventive maintenance takes place only once and when

the system is setup to production.

As in (Aghezzaf et al., 2016), the function κ

(y)

and C

(y) are the expected production capacity loss

and expected maintenance cost during the time period

t when the k

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

(y) =











u + α

s−1

∑

if t = s,k ≤ s,

u + α

s−1

∑

t−1

∑

if s ≤ t ≤ T.

(10)

(y) =











u + α

s−1

∑

if t = s,k ≤ s,

u + α

s−1

∑

t−1

∑

if s ≤ t ≤ T.

(11)

We adopt a hybrid failure rate model which was

deﬁned in (Aghezzaf et al., 2016). If after k

pre-

ventive maintenance the failure rate function remains

below some threshold function, r

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

(t). If the k

preventive maintenance takes place

τ units of time after an overhaul, that is in the be-

ginning of period T

having ﬁxed length τ, the hazard

rate function r

(t) of the system is then deﬁned as:

(t) = β

(t +α

AOT

t ∈ [0,(T

k+1

− T

)τ],∀k,1 ≤ k ≤ k

max

(12)

where T

AOT

is the actual operating time of the

system since the beginning of the planning hori-

zon until the beginning of period T

, the period

during which the k

preventive maintenance is

taking place. The parameters α

and β

stand,

respectively, for the age reduction coefﬁcient and

the hazard rate increasing coefﬁcient (adjustment

factor) such that 0 ≤ α

≤ α

≤ . .. ≤ α

kmax

≤ 1 and

1 ≤ β

≤ β

≤ .. . ≤ 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

(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

main-

tenance takes place in period s and, w

(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

maintenance

takes place in period s and the system must be pro-

duced at period t.

(Re IPImPMP) : Minimize Z

Re IP

ImPMP

∑

t=1

∑

j=1



+ p

+ h



∑

t=1

∑

k=1

∑

t=1

∑

s=1

∑

k=1

s−1

∑

p=0

t−s

∑

q=0

(p,q)v

(p,q)

subject to :

Eq. (1) - (3) and (5) - (9)

∑

j=1

∑

k=1

∑

s=1

∑

k=1

s−1

∑

p=0

t−s

∑

q=0

(p,q)v

(p,q) ≤ κ

max

,∀t ∈ H

(13)

s−1

∑

p=0

t−s

∑

q=0

p.u

(p,q) −







s−1

∑

, i f s > 1

0, i f s = 1

≤ 0,

∀t,s ∈ H, 1 ≤ s ≤ t

(14)

s−1

∑

p=0

t−s

∑

q=0

q.u

(p,q) −







t−1

∑

, i f t > 1

0, i f t = 1

≤ 0,

∀t,s ∈ H, 1 ≤ s ≤ t

(15)

s−1

∑

p=0

t−s

∑

q=0

(p,q) = 1, ∀t,s ∈ H,1 ≤ s ≤ t (16)

+ u

(p,q) − v

(p,q) ≤ 1,

∀k, s,t, p, q ∈ H, p ≤ s − 1,q ≤ t − s

(17)

+ v

(p,q) − w

(p,q) ≤ 1,

∀k, s,t, p, q ∈ H, p ≤ s − 1,q ≤ t − s

(18)

≥ 0, x

, z

, u

{p,q},

{p,q}, w

{p,q} ∈ {0, 1}

∀ j ∈ P, s,t ∈ H, k ≤ s ≤ t, p ≤ s − 1, q ≤ t − s

where u

(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

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

(p,q)

=1. The constraints (18) relate the variables v with

y and w, meaning that if both the k

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

(p,q) =1.

Here again, as reported in (Aghezzaf et al., 2016),

we let C

(p,q) and κ

(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:

(p,q) =











(u + α

pτ)du

i f t = s,

(u + α

pτ + qτ)du

i f s ≤ t ≤ T.

(19)

and

(p,q) =











(u + α

pτ)du

i f t = s,

(u + α

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

. 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

. 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

) 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

, corresponding to

the items production plans m generated by the sub-

problems for products j, and z

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

∑

j=1

M j

∑

m=1

∑

t=1

( f

¯x

+ p

+ h

)

∑

t=1

∑

k=1

∑

t=1

∑

s=1

∑

k=1

( ¯y). ¯y

subject to:

∑

j=1

M j

∑

m=1

∑

t=1

∑

k=1

∑

s=1

∑

k=1

( ¯y)z

≤ κ

max

,∀t ∈ H(µ

)

(21)

M j

∑

m=1

= 1, ∀ j ∈ N(π

) (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

<= ¯y

, ∀ j ∈ N;t ∈ H (η

) (23)

≥ 0, ∀ j ∈ N (24)

≥ 0, ∀ j ∈ N;t, s ∈ H (25)

and Eq. (5) - (9)

where ¯y

= max

j,m

¯x

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

and z

satisfying the following conditions:

∈ {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

) from a solution of master problem

(MPJ PPM) as follows:

M j

∑

m=1

, ∀ j ∈ N, ∀t ∈ T (26)

M j

∑

m=1

¯x

, ∀ j ∈ N, ∀t ∈ T (27)

4.1.1 Subproblem

Assuming that µ

is the vector of dual variables as-

sociated with the constraints (21), indexed by t, the

the vector of dual variables associated with the set

of convexity constraints (22) and the η

is a dual

associated variables with constraints (23). Each sub-

problem is one of following types:

SPJ

IPImPMP

∑

t=1

( f

+ p

+ h

)

−

∑

t=1

− µ

−

∑

t=1

,∀ j ∈ N

SPJ

IPImPMP

∑

t=1

[( f

− η

+ p

+ h

]

−

∑

t=1

− µ

,∀ j ∈ N

subject to:

Eq. (1) - (2)

≥ 0, x

∈ {0,1}

∀ j ∈ N, s,t ∈ H,k ≤ s ≤ t, p ≤ s − 1, q ≤ t − s

variables :Q

≥ 0, x

parameters: d

,µ

,π

,η

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/(3k +7) and

= (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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