Applied Mathematical Modelling 35 (2011) 152–164
Contents lists available at ScienceDirect
Applied Mathematical Modelling
journal homepage: www.elsevier.com/locate/apm
Selecting optimum maintenance strategy by fuzzy interactive linear
assignment method
Mahdi Bashiri *, Hossein Badri, Taha Hossein Hejazi
Department of Industrial Engineering, Shahed University, P.O. Box 18155/159, Tehran, Iran
a r t i c l e
i n f o
Article history:
Received 18 August 2009
Received in revised form 10 May 2010
Accepted 24 May 2010
Available online 27 May 2010
Keywords:
Maintenance strategy selection problem
(MSSP)
Linear assignment method
Interactive analysis
Fuzzy programming
a b s t r a c t
In the current competitive environment managers of manufacturing and service organizations try to make their organizations competitive by providing timely delivery of high quality products. Maintenance, as a system, plays a key role in reducing cost, minimizing
equipment downtime, improving quality, increasing productivity and providing reliable
equipment and as a result achieving organizational goals and objectives. This paper presents a new approach for selecting optimum maintenance strategy using qualitative and
quantitative data through interaction with the maintenance experts. This approach has
been based on linear assignment method (LAM) with some modifications to develop interactive fuzzy linear assignment method (IFLAM).
The proposed approach is an interactive method which uses qualitative and quantitative
data to rank the maintenance strategies. This method helps managers to find the best
maintenance strategy based on the determined criteria. Maintenance experts also can provide and modify their preference information gradually within the interaction process so as
to make the result more reasonable. The proposed method has been illustrated by a numerical example.
Ó 2010 Elsevier Inc. All rights reserved.
1. Introduction
Manufacturing environments recently have changed so fast that manufacturing system competitiveness has increased.
Manufacturing firms have been investing a lot to improve their manufacturing performance in terms of cost, quality, and
flexibility, in an effort to compete with other firms in the global marketplace [1]. In manufacturing firms there are varieties
of problems that can affect on the manufacturing cost, product quality and delivery time of products to customers; such as
manufacturing technology selection, maintenance strategy selection, machine location, evaluation of quality function. Maintenance, as a system, plays a key role in reducing cost, minimizing equipment downtime, improving quality, increasing productivity and providing reliable equipment and as a result achieving organizational goals and objectives.
One of the main expenditure items for the manufacturing firms is maintenance cost which can reach 15–70% of production costs, varying according to the type of industry [2]. On the other hand one third of all maintenance costs is wasted as the
result of unnecessary or improper maintenance activities [3]. Therefore, selection of optimum maintenance strategy can
highly affect on the manufacturing expenditures.
In the literature, maintenance can be classified into two main types: corrective and preventive [4,5]. Corrective maintenance is the maintenance that occurs after systems failure, and it means all actions resulting from failure; preventive maintenance is the maintenance that is performed before systems failure in order to retain equipment in specified condition by
* Corresponding author.
E-mail addresses: [email protected], [email protected] (M. Bashiri), [email protected] (H. Badri), [email protected] (T.H. Hejazi).
0307-904X/$ - see front matter Ó 2010 Elsevier Inc. All rights reserved.
doi:10.1016/j.apm.2010.05.014
M. Bashiri et al. / Applied Mathematical Modelling 35 (2011) 152–164
153
providing systematic inspections, detection, and prevention of incipient failure [6]. The most popular alternative maintenance strategies in the literature are as following:
(1) Corrective maintenance: This alternative maintenance strategy is also named as fire-fighting maintenance, failure
based maintenance or breakdown maintenance. When the corrective maintenance strategy is applied, maintenance
is not implemented until failure occurs [7].
(2) Time-based preventive maintenance: According to reliability characteristics of equipment, maintenance is planned and
performed periodically to reduce frequent and sudden failure. This maintenance strategy is called time-based preventive maintenance, where the term ‘‘time” may refer to calendar time, operating time or age [8].
(3) Condition-based maintenance: Maintenance decision is made depending on the measured data from a set of sensors
system when using the condition-based maintenance strategy. To date a number of monitoring techniques are already
available, such as vibration monitoring, lubricating analysis, and ultrasonic testing [8].
(4) Predictive maintenance: Predictive maintenance is the maintenance strategy that is able to forecast the temporary
trend of performance degradation and predict faults of machines by analyzing the observed parameters data.
The multiple-criteria decision-making (MCDM) provides instruments for finding the best option among the possible alternatives based on the evaluation of multiple conflict criteria. MCDM has been one of the fastest growing areas of operational
research, as it is often realized that many concrete problems can be represented by several (conflicting) criteria. Several qualitative and quantitative criteria may affect each other mutually when evaluating alternatives, which may make the selection
process complex and challenging [9]. In many cases, the decision maker (DM) has inexact information about the alternatives
with respect to an attribute.
The classical MCDM methods cannot effectively handle problems when information is imprecise. These classical methods,
both deterministic and random processes, tend to be less effective in conveying the imprecision and fuzziness characteristics. Fuzzy set theory which has been proposed by Zadeh [10] is a powerful tool to handle imprecise data.
The MCDM problems may be divided into two kinds of problem. One is the classical MCDM problems, among which the
ratings and the weights of criteria are measured in crisp numbers. Another is the fuzzy multi-criteria decision-making
(FMCDM) problems, among which the ratings and the weights of criteria evaluated on imprecision, subjective and vagueness
are usually expressed by linguistic terms and then set into fuzzy numbers [11].
Complexity of manufacturing systems makes it difficult to decide about maintenance strategy. Due to this situation a well
designed decision process is needed to help managers in reduction of decision failures. In this paper interactive fuzzy linear
assignment method (IFLAM) has been proposed for maintenance strategy selection whose two main features are the utilization of both qualitative and quantitative data and decision making through an interactive process with the maintenance
experts.
The main features of the proposed approach in contrast with those of other existing methods are as follows:
– In the proposed approach several decision makers can state their opinions about both importance of criteria and evaluation of alternatives.
– The proposed approach has capabilities to handle both qualitative and quantitative data.
– Imprecise statements of decision makers can be analyzed by the proposed approach via fuzzy theory and related
operators.
– The decision makers can interact with the intermediate solutions in order to improve mathematical results with consideration of their experiences so an intermediate solution will be final optimal solution if the decision makers have been
satisfied.
The remainder of this paper is organized as following. Some of the proposed approaches in the field of MSSP will be presented in Section 2. Section 3 presents a brief review of fuzzy set theory. Section 4 introduces and describes the proposed
method. Section 5 illustrates the procedures in the proposed method using a numerical example. Conclusions are drawn
in Section 6.
2. Literature review
Since selection of the optimum maintenance strategy for each equipment; is a vital decision for manufacturing companies, many studies have been devoted to this area. Almedia and Bohoris [12] present a review of some basic decision
theory concepts and discussed their applicability in the selection of maintenance strategies. Triantaphyllou et al. [13]
proposed a method to find the criticality of each criteria dealing with maintenance strategies in which deals with
the simplifying of the complex maintenance criteria. Azadivar and Shu [14] presented a new approach to select the
optimum maintenance strategy for each class of systems in a just-in-time environment. In this paper they considered
16 characteristic factors that could play a role in maintenance strategy selection. Murthy and Asgharizade [15] proposed an approach for decision-making when the company out sources the maintenance. They used game theory to
conduct a decision when the customer (the receptionist of maintenance) wants to decide whether having a service contract or not.
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M. Bashiri et al. / Applied Mathematical Modelling 35 (2011) 152–164
Luce [16], Okumura and Okino [17] proposed a method to select the most effective maintenance strategy according to
different production loss and maintenance costs of each maintenance strategies. Löfsten [18] proposed a model based on cost
analysis to choose between corrective or preventive maintenance. Bevilacqua and Barglia [2] used AHP coupled with a sensitivity analysis for maintenance strategy selection in an Italian oil refinery. Ivy and Nembhard [19] integrate statistical quality control (SQC) and partially observable Markov decision processes (POMDPs) for the evaluation of maintenance policies
under conditions of limited information. Bertolini and Bevilacqua [20] used a combined AHP-GP model for maintenance
selection policy problem and in a case study used it for identifying the optimal maintenance policy for a set of centrifugal
pumps operating in the process and service plants of an Italian oil refinery.
[21] uses a new maintenance optimization model carry out the computations for calculating frequency of failures and
downtime as the maintenance data problems using decision-making grid (DMG) with fuzzy logic in maintenance decision
support system (DSS). Jafari et al. [22] proposed a new approach to the MSSP which can determine the best maintenance
strategy by considering the uncertainty level and also all the variety in maintenance criteria and their importance. Saumil
et al. [23] developed a continuous time Markov chain degradation model and a cost model to quantify the effects of maintenance on a multiple machine system. An optimal maintenance policy for a multiple machine system in the absence of resource constraints is obtained, in the presence of resource constraints, two prioritization methods are proposed to obtain
effective maintenance policies for a multiple machine system. Also a case study focusing on a section of an automotive
assembly line has been used to illustrate the effectiveness of the proposed method. Li et al. [24] calculated a reliability-based
dynamic maintenance threshold (DMT) based on the updated equipment status. In this paper the benefits of the DMT are
demonstrated in a numerical case study on a drilling process.
There are always a variety of criteria in selecting the most suitable maintenance strategy. Some of these criteria are quantitative such as hardware and software costs, training costs, time between failures, equipment reliability. There are also a lot
of qualitative criteria that must be considered in the selection of maintenance strategy, such as safety, flexibility, acceptance
by labor, high product quality. In real-world situation, because of incomplete or non-obtainable information, data are often
not so deterministic and the majority of these data can be assessed by human perception and human judgment. Therefore,
they usually are fuzzy imprecise and so fuzzy theory can be applied in this problem to analyze qualitative verbal assessments. Fuzzy linguistic models permit the translation of verbal expression into numerical terms, thereby dealing quantitatively with the expression of the importance of various objectives. These quantities can then be used to assess the optimum
degree of investment in various maintenance strategies [25]. Many maintenance goals or comparing criteria must be taken
into consideration, e.g. safety and cost in the selection of the suitable maintenance strategies. Therefore, multiple-criteria
decision-making methodology can be used for the maintenance strategy selection. Many researchers have implemented
MCDM methods for maintenance strategy selection. Sharma et al. [26] assessed the most popular maintenance strategies
using the fuzzy inference theory and MCDM evaluation methodology in fuzzy environment. Al-Najjar and Alsyouf [27] uses
past data and technical analysis of processes machines and components to identify the criteria for an MCDM problem. They
used fuzzy inference system (FIS) to assess the capability of each maintenance approach. Finally by utilizing simple additive
weighting (SAW) method, the efficient maintenance approach was selected.
As mentioned above there are a lot of qualitative criteria in the selection of most appropriate maintenance strategy and
fuzzy theory is a good solution in this regard. The fuzzy methodology based on qualitative verbal assessment inputs is more
practical, because many of the overall maintenance objectives of the organization are intangible [8]. Mechefske and Wang
[25] proposed to evaluate and select the optimum maintenance strategy using fuzzy linguistics. In this proposed approach,
firstly the organizational goal are determined then by interviewing the managers and employs the importance of each goal
and the capability of each maintenance strategy to satisfy each goal is captured, then by utilizing some equations in the fuzzy
environment the optimum maintenance strategy will be selected. Wang et al. [8] proposed a method for the evaluation of the
different maintenance strategies based on fuzzy analytic hierarchy process (AHP).
In this paper we present a new method for selecting the optimum maintenance strategy through interaction with the
maintenance experts. This approach has been based on linear assignment method (LAM) with some modifications to develop
interactive fuzzy linear assignment method (IFLAM).
The proposed approach is an interactive method which uses qualitative and quantitative data to rank the maintenance
strategies. This method helps managers to find the best maintenance strategy based on the determined criteria. Maintenance
experts also can provide and modify their preference information gradually within the interaction process so as to make the
result more reasonable.
3. Preliminary definitions of fuzzy data
In the following, we briefly review some basic definitions of fuzzy sets which will be used throughout the paper.
Let X be a classical set of objects, called the universe, whose generic elements are denoted by x. The membership in a crisp
subset of X is often viewed as characteristic function lA from X to {0, 1} such that:
leA ðxÞ ¼
1 if and only if X 2 A
0
otherwise
;
ð1Þ
e is called a fuzzy set and
where {0, 1} is called a valuation set. If the valuation set is allowed to be the real interval [0, 1], A
e
e
denoted by A and leA ðxÞ is the degree of membership of x in A.
M. Bashiri et al. / Applied Mathematical Modelling 35 (2011) 152–164
155
Definition 1. A linguistic variable is a variable whose values are linguistic terms [28]. The concept of linguistic variable is
very useful in dealing with situations which are too complex or too ill-defined to be reasonably described in conventional
quantitative expressions [10]. These linguistic values can also be represented by fuzzy numbers.
e in a universe of discourse X is characterized by a membership function l
Definition 2. A fuzzy set A
eAðxÞ which associates
with each element x in X a real number in the interval [0, 1]. The function value le is termed the grade of membership
AðxÞ
e [29].
of x in A
e ¼ ða; b; c; dÞ, where b and c are the cene ¼ fðx; l Þjx 2 Xg fuzzy number can be denoted as A
Definition 3. The trapezoidal A
eAðxÞ
tral values ðle ðb 6 x 6 cÞ ¼ 1Þ, a is the left spread and d is the right spread (see Fig. 1).
A
~ ¼ ðn1 ; n2 ; n3 ; n4 Þ be two trapezoidal fuzzy numbers. If m
~ ¼n
~ , then
~ ¼ ðm1 ; m2 ; m3 ; m4 Þ and n
Definition 4. Let m
m1 = n1,m2 = n2, m3 = n3, and m4 = n4.
e is called a fuzzy matrix, if at least an entry in D
e is a fuzzy number [30].
Definition 5. D
e can be defined by A
e ¼ ða; b; c; dÞ shown in Fig. 1. The membership function l
Definition 6. A trapezoidal fuzzy number A
eAðxÞ
is defined as:
8 ðxaÞ
;
>
>
ðbaÞ
>
>
< 1;
leA ðxÞ ¼ ðxdÞ
>
> ðcdÞ ;
>
>
:
0;
axb
bxc
cxd
ð2Þ
:
otherwise
4. Proposed method
In this paper we propose a new approach for selecting the optimum maintenance strategy. This approach has been developed based on linear assignment method (LAM) with some modifications. Here we aimed to develop a new approach which
considers both quantitative and qualitative criteria in the selection process of maintenance strategy as well as keeps an interaction with the maintenance experts. This interaction allows to maintenance experts to provide and modify their preference
information gradually within the selection so as to make the results more reasonable. For this purpose an interactive fuzzy
linear assignment method (IFLAM) has been developed and applied to select the optimum maintenance strategy. The algorithm for the proposed approach will be developed in the following three major phases:
Firstly and after making a list of maintenance selection criteria, an expert committee is constituted to evaluate different
maintenance strategies. There are varieties of qualitative and quantitative criteria for maintenance strategy selection, the
expert team should screen out some criteria based on organizational goals and objectives. Rating of each maintenance strategy under quantitative criteria such as the mean time between failures (MTBF), equipment costs simply can be assessed and
computed. But under the qualitative criteria rating of each maintenance strategy should be assessed by the expert team
using linguistic variables. The expert team also uses linguistic variables to assess the importance weight of the selection criteria (phase 1).
By converting the linguistic evaluation into trapezoidal fuzzy numbers and aggregation of the rating and the weights, we
get the initial ranking of the maintenance strategies using fuzzy linear assignment method. This model is a modified and
combinational version of linear assignment method which uses both quantitative and qualitative data to rank the maintenance strategies under fuzzy environment (phase 2). The next phase is an interaction process which uses the initial ranking
gained in phase 2 as input and tries to improve it iteratively. For this purpose firstly the tight constraint set is identified by
µ n~ (x)
1
a
b
c
e
Fig. 1. Trapezoidal fuzzy number A.
d
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M. Bashiri et al. / Applied Mathematical Modelling 35 (2011) 152–164
solving a linear programming model and then the list of pairs which determine these boundaries is presented to the expert
team to modify the ranking (phase 3). The proposed method is illustrated in Fig. 2 and its procedure based on above conceptual model is as following:
Phase 1:
Step 1: Constitute a committee of maintenance experts. Assume that there is a committee of k experts, (Dt, t = 1, 2, . . . , k)
who are responsible for assessing m maintenance strategy (Ai, i = 1, 2, . . . , m) under each of the n selection criteria (Cj,
j = 1, 2, . . . , n) as well as the importance of the criteria.
Step 2: Make a list of maintenance strategies and the selection criteria.
Step 3: Screen out some criteria according to organizational goals and objectives.
Step 4: Choose appropriate linguistic variables for the importance weights of the selection criteria and the linguistic
ratings.
Step 5: Assess the importance of each criterion Cj by experts, using linguistic variables.
Step 6: Evaluate the rating score of each maintenance strategy under each criterion by maintenance experts and using
linguistic variables.
Fig. 2. The proposed method.
M. Bashiri et al. / Applied Mathematical Modelling 35 (2011) 152–164
157
Phase 2:
~ jt ¼ ðajt ; bjt ; cjt ; djt Þ; j ¼ 1; 2; . . . ; n; t ¼ 1; 2; . . . ; k
Step 1: Convert the linguistic terms into trapezoidal fuzzy numbers. Let w
be the linguistic weight of jth criteria given to maintenance strategies by expert Dt. And Let ~
xijt ¼ ðoijt ;
pijt ; qijt ; rijt Þ i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n; t ¼ 1; 2; . . . ; k be the linguistic rating assigned to the maintenance strategy Ai
for criterion Cj by expert Dt.
~ j of criterion Cj and to get the aggregated fuzzy
Step 2: Pool the experts’ opinions to get the aggregated fuzzy weight w
rating of strategy Ai under criterion Cj. If maintenance expert committee has k persons, the aggregated fuzzy weight of
criterion Cj can be calculated as [31]:
~j ¼
w
1
~ j2 ðþÞ ðþÞw
~ jk :
~ j1 ðþÞw
w
k
ð3Þ
Also the aggregated rating score of maintenance strategies under criterion Cj can be calculated as
~xij ¼
1
~xij1 ðþÞ~xij2 ðþÞ ðþÞ~xijk :
k
ð4Þ
Step 3: Solve a fuzzy linear assignment programming as described more in following sentences to get the initial ranking of
the strategies. Here we study the solutions of fuzzy linear assignment programming. Our proposed method to formulate
and solve fuzzy linear assignment method (FLAM) is based on the method presented by Zhang et al. [32] and Hwang and
Yoon [33] which is illustrated in Eqs. (5)–(9). They developed a number of theorems so as to convert the fuzzy linear
method (FLM) to a multi-objective optimization problem with four-objective functions.
~ as a square m m nonnegative matrix whose element represent the summation of the fuzzy
Let us define a matrix p
weights of the criteria in which Ai is ranked the pth attribute-wise ranking.
Let us define permutation matrix X as m m square matrix whose element yip = 1 if Ai is assigned to overall rank p, and
yip = 0 otherwise.
~ ip matrix as p
~ ij ¼ ðp
~ ijð1Þ ; p
~ ijð2Þ ; p
~ ijð3Þ ; p
~ ijð4Þ Þ.
Since we defined our weights as trapezoidal fuzzy numbers, thus we show p
The fuzzy linear assignment problem can be written by the following linear programming (LP) format:
4
X
Max
r¼1
Subject to
m
X
kr
m¼m
1 þm2 m
1 1
X
X
i¼1
p~ ip ðrÞxip þ
p¼1
m m
2 1
X
X
i¼1
pip xip
ð5Þ
p¼5
xip ¼ 1;
i ¼ 1; 2; . . . ; m;
ð6Þ
xip ¼ 1;
p ¼ 1; 2; . . . ; m;
ð7Þ
p¼1
m
X
i¼1
4
X
kr ¼ 1
ð8Þ
r¼1
xip 2 f0; 1g for all i and j;
ð9Þ
where kr is the importance weight of rth objective function (r = 1, . . . , 4 because the fuzzy number is trapezoidal) and is
determined by experts and m1, m2 are number of the qualitative and quantitative criteria, respectively. Constraints 6 and
7 ensure that each alternative belong to exactly one position.
Phase 3:
~ X matrix based on the ranking identified from FLAM.
Step 1: Construct m
Let X be a permutation of m strategies gained from FLAM, which represents a preference order of the m strategies, and X
(p) be the strategy in the pth position of the order X.
~ X matrix and the importance weights w.
~
Step 2: Defuzzify the m
~ X be the (m 1) m matrix whose rows m
~ XðpÞ are given by ~
Getting the initial feasible order X, let m
xXðpÞ ~
xXðpþ1Þ . Here
~ X and get mX matrix by the following equation:
we defuzzify the fuzzy numbers of m
MXðpÞ ¼
1
ða þ b þ c þ dÞ:
4
ð10Þ
~ X matrix directly and after linear normalization.
Quantitative scores are fed into m
Step 3: Solve the L1 model to identify the binding constraints of WX. This step is based on interactive simple additive
weighting method [33].
Now the problem is to identify the tight constraint set. The boundary of WX is determined by those rows of Eq. (12) which
are tight (active) constraints for some value of w. For such rows the optimum value of the following LP problem (L1) gives
zero:
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M. Bashiri et al. / Applied Mathematical Modelling 35 (2011) 152–164
Lj : Min M XðjÞ w 8j ¼ 1; . . . ; m 1;
MX w P 0;
n
X
wi ¼ 1;
ð11Þ
ð12Þ
ð13Þ
i¼1
wi P 0;
i ¼ 1; 2; . . . ; n:
ð14Þ
Step 4: Present the expert committee with the list of pairs in X which determine the boundary of WX. Stop if there are not
any binding constraints or if this order is accepted by the committee, otherwise ask the experts to revise and change the
rating of one pair from the list and return to step1 in phase 3.
Phase 4:
In this step, following model will be solved for all binding constraints to get most reliable weights of criteria whereas final
satisfying ranking remained unchanged:
Maximize L;
M X w P L;
ð15Þ
ð16Þ
ai þ aðbi ai Þ 6 wi 6 di aðdi ci Þ;
ð17Þ
wi P 0;
ð18Þ
i ¼ 1; 2; . . . ; n:
Eqs. (15) and (16) ensure that the current ranking will remain unchanged and Eq. (17) is used to find most reliable values
of weights with respect to predefined membership value (a). Where (ai, bi, ci, di) are definition parameters for ith fuzzy
weight (see formula (2) and Fig. 1).
5. Numerical example
To illustrate the proposed method, a hypothetical numerical example has been presented. Suppose that a company desires to select the most appropriate maintenance strategy that requires different alternatives to be assessed for a range of
criteria. There is a variety of strategies in maintenance management systems depending on the type of applied industry
in a company. These strategies can be categorized into two main groups of corrective and preventive ones [34]. In corrective
maintenance strategy, no maintenance activity is carried out until a failure occurs. Small profit margins along with ascending
competition trend make the maintenance managers apply more reliable maintenance strategies. Conversely, preventive
maintenance strategy is the strategy which is utilized before system failure to retain the system in the expected condition.
Preventive strategies can be divided into different subfolders like time-based, condition-based, and predictive maintenance
strategies [34].
These strategies would be five custom policies of corrective maintenance (CM), preventive maintenance (PM), time-based
maintenance (TBM), condition-based maintenance (CBM) and predictive maintenance (PDM) strategies which the decisionmaking group has to choose one among them. According to the explained steps in Section 3, the proposed method is applied
to handle the subjective judgments of decision makers.
Phase 1:
Step 1: A committee of three maintenance experts, D1, D2 and D3 has been formed to conduct the evaluation process and
to select the most suitable maintenance strategy.
Step 2: The committee makes an initial list of the selection criteria and five custom policies CM,PM, TBM, CBM and PDM
are selected to be evaluated.
Step 3: After screening four benefit criteria are considered as qualitative measures: C1, C2, C3 and C4. Also two cost criteria
are considered as quantitative measures: C5 (total cost) and C6 (mean time between failure – MTBF).
Step 4: Seven linguistic variables have been used for weighting which are shown in Table 1. Also we choose seven linguistic variables for rating strategies which are shown in Table 2.
Step 5: The decision makers use the weighting linguistic variables (Table 1) to assess the importance of the criteria which
are presented in Table 3.
Table 1
Linguistic variables for the importance weight of each criterion.
Very low (VL)
Low (L)
Medium low (ML)
Medium (M)
Medium high (MH)
High (H)
Very high (VH)
(0, 0, 0.1, 0.2)
(0, 0.1, 0.2, 0.3)
(0.1, 0.2, 0.4, 0.5)
(0.3, 0.4, 0.6, 0.7)
(0.5, 0.6, 0.8, 0.9)
(0.7, 0.8, 0.9, 1)
(0.8, 0.9, 1, 1)
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M. Bashiri et al. / Applied Mathematical Modelling 35 (2011) 152–164
Table 2
Linguistic variables for the ratings.
Very poor (VP)
Poor (P)
Medium poor (MP)
Fair (F)
Medium good (MG)
Good (G)
Very good (VG)
(0, 0, 1, 2)
(0, 1, 2, 3)
(1, 2, 4, 5)
(3, 4, 6, 7)
(5, 6, 8, 9)
(7, 8, 9, 10)
(8, 9, 10, 10)
Table 3
Importance values of each criterion by DMs.
D1
D2
D3
C1
C2
C3
C4
VH
H
H
H
MH
MH
M
MH
H
VH
VH
MH
Table 4
Rating of alternatives under each criterion in terms of linguistic variables determined by DMs.
Alternatives
CM
Criteria
C1
C2
C3
C4
C1
PM
C2
C3
C4
C1
TBM
C2
C3
C4
C1
CBM
C2
C3
C4
C1
PDM
C2
C3
C4
D1
D2
D3
MG
VG
G
G
VG
F
G
F
MG
MG
G
VG
G
G
MG
VG
MG
G
MG
G
MG
VG
F
G
MG
G
MG
G
MG
G
G
G
MG
G
MG
G
VG
VG
MG
VG
MG
VG
F
F
G
G
VG
MG
G
F
MG
VG
F
G
G
MG
VG
MG
G
MG
Step 6: Also they use the linguistic variables (Table 2) to rate the alternatives. Rating of alternatives is shown in Table 4.
Phase 2:
Step 1: Convert the linguistic terms into trapezoidal fuzzy numbers (Tables 5 and 6).
For example (0.8, 0.9, 1,1) in Table 5 is the value of VH for the weight of C1 according to D1. And (5, 6, 8, 9) is the value of
MG for the rating of alternative A under criterion C1 according to D1.
f j of criterion Cj and the aggregated fuzzy rating of alternative Ai under
Step 2: Calculate the aggregated fuzzy weight W
criterion Cj by using Eqs. (3) and (4)
f 11 ¼ 1 ½0:8 þ 0:7 þ 0:7 ¼ 0:73;
W
3
1
f
W 12 ¼ ½0:9 þ 0:8 þ 0:8 ¼ 0:83;
3
f 13 ¼ 1 ½1 þ 0:9 þ 0:9 ¼ 0:93;
W
3
1
f
W 14 ¼ ½1 þ 1 þ 1 ¼ 1:
3
f 1 ¼ ½0:73; 0:83; 0:93; 1And for alternative CM under criterion C1:
Thus: W
1
½5 þ 8 þ 7 ¼ 6:67;
3
1
¼ ½6 þ 9 þ 8 ¼ 7:67;
3
1
¼ ½8 þ 10 þ 9 ¼ 9;
3
1
¼ ½9 þ 10 þ 10 ¼ 9:67:
3
~x11 ¼
~x12
~x13
~x14
Table 5
Importance weights of criteria in terms of fuzzy numbers for each DMs.
D1
D2
D3
C1
C2
C3
C4
C5
C6
(0.8, 0.9, 1, 1)
(0.7, 0.8, 0.9, 1)
(0.7, 0.8, 0.9, 1)
(0.7, 0.8, 0.9, 1)
(0.5, 0.6, 0.8, 0.9)
(0.5, 0.6, 0.8, 0.9)
(0.3, 0.4, 0.6, 0.7)
(0.5, 0.6, 0.8, 0.9)
(0.7, 0.8, 0.9, 1)
(0.8, 0.9, 1, 1)
(0.8, 0.9, 1, 1)
(0.5, 0.6, 0.8, 0.9)
(0.5, 0.6, 0.8, 0.9)
(0.7, 0.8, 0.9, 1)
(0.8, 0.9, 1, 1)
(0.5, 0.6, 0.8, 0.9)
(0.8, 0.9, 1, 1)
(0.7, 0.8, 0.9, 1)
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Table 6
Rating of alternatives under each criterion in terms of fuzzy numbers.
Alternative
CM
Criteria
C1
C2
C3
C4
D1
D2
D3
(5, 6, 8, 9)
(8, 9, 10, 10)
(7, 8, 9, 10)
(7, 8, 9, 10)
(8, 9, 10, 10)
(3, 4, 6, 7)
(7, 8, 9, 10)
(3, 4, 6, 7)
(5, 6, 8, 9)
(5, 6, 8, 9)
(7, 8, 9, 10)
(8, 9, 10, 10)
Alternative
PM
Criteria
C1
C2
C3
C4
D1
D2
D3
(7, 8, 9, 10)
(7, 8, 9, 10)
(5, 6, 8, 9)
(8, 9, 10, 10)
(5, 6, 8, 9)
(7, 8, 9, 10)
(5, 6, 8, 9)
(7, 8, 9, 10)
(5, 6, 8, 9)
(8, 9, 10, 10)
(3, 4, 6, 7)
(7, 8, 9, 10)
Alternative
TBM
Criteria
C1
C2
C3
C4
D1
D2
D3
(5, 6, 8, 9)
(7, 8, 9, 10)
(5, 6, 8, 9)
(7, 8, 9, 10)
(5, 6, 8, 9)
(7, 8, 9, 10)
(7, 8, 9, 10)
(7, 8, 9, 10)
(5, 6, 8, 9)
(7, 8, 9, 10)
(5, 6, 8, 9)
(7, 8, 9, 10)
Alternative
CBM
Criteria
C1
C2
C3
C4
D1
D2
D3
(8, 9, 10, 10)
(5, 6, 8, 9)
(8, 9, 10, 10)
(8, 9, 10, 10)
(8, 9, 10, 10)
(5, 6, 8, 9)
(3, 4, 6, 7)
(3, 4, 6, 7)
(7, 8, 9, 10)
(7, 8, 9, 10)
(8, 9, 10, 10)
(5, 6, 8, 9)
Alternative
PDM
Criteria
C1
C2
C3
C4
D1
D2
D3
(7, 8, 9, 10)
(3, 4, 6, 7)
(5, 6, 8, 9)
(8, 9, 10, 10)
(3, 4, 6, 7)
(7, 8, 9, 10)
(7, 8, 9, 10)
(5, 6, 8, 9)
(8, 9, 10, 10)
(5, 6, 8, 9)
(7, 8, 9, 10)
(5, 6, 8, 9)
Thus: ~
x1 ¼ ½6:67; 7:67; 9; 9:67
Step 3: Construct the fuzzy decision matrix with fuzzy weight of each criterion (Table 7).
Step 4: Get the initial ranking of alternatives by solving a fuzzy linear assignment programming. Note that before any
other attempt, since in criteria C2 alternatives CM and PDM and in criteria C4 alternatives CM and CBM have the same
ranking, we took one more criteria with half of the original weight value. Thus the weights will be:
f 1 ¼ ð0:73; 0:83; 0:93; 1Þ;
W
f 21 ¼ ð0:285; 0:335; 0:415; 0:465Þ;
W
f 22 ¼ ð0:285; 0:335; 0:415; 0:465Þ;
W
f 3 ¼ ð0:5; 0:6; 0:77; 0:87Þ;
W
f 41 ¼ ð0:35; 0:4; 0:465; 0:485Þ;
W
f 42 ¼ ð0:35; 0:4; 0:465; 0:485Þ;
W
f 5 ¼ ð0:66; 0:76; 0:9; 0:96Þ;
W
f 6 ¼ ð0:66; 0:76; 0:83; 0:96Þ:
W
Table 8 shows all alternatives and relative ranking under each criterion.
Table 7
Fuzzy decision matrix.
Criteria
C1
C2
C3
C4
C5
C6
CM
PM
TBM
CBM
PDM
(6.67, 7.67, 9, 9.67)
(6.33, 7.33, 8.67, 9.67)
(5.67, 6.67, 8.33, 9.33)
(7, 8, 9.33, 9.67)
(5, 6, 8.33, 8.67)
(6, 7, 8.33, 9)
(6.67, 7.67, 9, 9.67)
(6.33, 7.33, 8.67, 9.7)
(7, 8, 9.33, 9.67)
(6, 7, 8.33, 9)
(5, 6, 8.33, 8.67)
(5.67, 6.67, 8.33, 9.33)
(6.33, 7.33, 8.67, 9.67)
(4.33, 5.33, 7, 8)
(6.67, 7.67, 9, 9.67)
(6.67, 7.67, 9, 9.67)
(6, 7, 8.33, 9)
(6.33, 7.33, 8.67, 9.67)
(6.67, 7.67, 9, 9.67)
(5.67, 6.67, 8.33, 9.33)
362
135
245
167
177
3576
3283
2683
2958
2845
Weight
(0.73, 0.83, 0.93, 1)
(0.57, 0.67, 0.83, 0.93)
(0.5, 0.6, 0.77, 0.87)
(0.7, 0.8, 0.93, 0.97)
(0.66, 0.76, 0.9, 0.96)
(0.66, 0.76, 0.83, 0.96)
161
M. Bashiri et al. / Applied Mathematical Modelling 35 (2011) 152–164
f 41 ¼ ð0:35; 0:4; 0:47; 0:49Þ which shows that alternative CM has 1st rank in criterion C41 and
~ 11 ¼ W
for example p
f1 þ W
f 42 ¼ ð1:08; 1:23; 1:4; 1:49Þ which shows that alternative CM has 2nd rank in criterion C1 and C42.
p~ 12 ¼ W
Thus the FLAP model based on Eqs. (5)–(9) will be:
4
X
Max
kr
r¼1
5
X
S:t:
5 X
4
X
p~ ip ðrÞxip þ
5 X
6
X
xip ¼ 1;
pip ðrÞxip
i¼1 p¼5
i¼1 p¼1
i ¼ 1; . . . ; 5;
p¼1
5
X
xip ¼ 1;
p ¼ 1; . . . ; 5;
i¼1
xip P 0 for all i and p;
where
4
X
kr ¼ 1:
r¼1
Assuming equal importance for each element of fuzzy numbers, here kr = 0.25 is considered for all r. Solving this linear model
using Software LINGO 8.0, the optimal permutation matrix X* is
2
0
6
60
6
X ¼6
60
6
41
0
0
3
1 0
0
0
1 0
0
0
0
0
0
0
7
07
7
1 07
7:
7
0 05
0 1
Applying the optimal permutation matrix X* to the alternatives, we find the optimal order:
X1 ¼ ½CBM; CM; PM; TBM; PDM:
Phase 3:
e X matrix based on the ranking identified from FLAP.
Step 1: Construct M
e 1 matrix is:
The associated M
2
3 2
3 2
e 11
CBM CM
ð2:67;1;1:66;3Þ
M
6
7 6
6 ð3;1;1:67;3:28Þ
e
6 M 12 7 6 CM PM 7
6
7
e 1 ¼6
7
M
7¼6
6 e 7¼6
4 M 13 5 4 PM TBM 5 4 ð3;1;2;4Þ
e 14
ð3;1:67;2:33;4:33Þ
TBM PDM
M
3
ð2;0:33;2:33;3Þ
ð4:34;3;1;3Þ
ð3;1:33;1:33;3Þ
ð3:67;2;0:66;2:33Þ ð4:33;2:33;1:66;3Þ ð2:33;0:66;2;3:67Þ 7
7
7:
ð3;1;1:67;3:34Þ
ð4;2;1;3Þ
ð3:67;1:67;1;2:67Þ 5
ð2:67;1;1:67;3:67Þ ð3:34;1:67;1;3Þ
ð3;1;2;4Þ
e X matrix and the importance weights based on Eq. (10)
Step 2: Defuzzify the M
Table 8
Attribute-wise preference.
Rank
C1
C21
C22
C3
C41
C42
C5
C6
1st
2nd
3rd
4th
5th
CBM
CM
PM
TBM
PDM
CBM
PM
TBM
PDM
CM
CBM
PM
TBM
CM
PDM
PDM
TBM
PM
CM
CBM
CM
CBM
TBM
PM
PDM
CBM
CM
TBM
PM
PDM
PM
CBM
PDM
TBM
CM
TBM
PDM
CBM
PM
CM
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M. Bashiri et al. / Applied Mathematical Modelling 35 (2011) 152–164
2
M 11
3
2
0:24 0:75
0:83 0
0:86
6 M 7 6 0:24 0:67 0:5
0:67
1
6 12 7 6
M1 ¼ 6
7¼6
4 M 13 5 4 0:5
0:25
0:25 0:42 0:48
0:49 0:42
M 14
0:5
0:5
0:69
3
0:328 7
7
7:
0:67 5
0:299 0:181
For example under criterion C1
MXð11Þ ¼
1
ð2:67 1 þ 1:66 þ 3Þ ¼ 0:24:
4
Step 3: Identify the binding constraints of WX by solving the L1 mode.
For example model L1 in iteration No.1 based on Eqs. (11)–(14) will be as following:
L1 : Min
0:24W 1 þ 0:75W 2 0:83W 3 þ 0W 4 þ 0:86W 5 þ 0:69W 6
Subject to : 0:24W 1 þ 0:75W 2 0:83W 3 þ 0W 4 þ 0:86W 5 þ 0:69W 6 P 0;
0:24W 1 0:67W 2 0:5W 3 þ 0:67W 4 W 5 0:328W 6 P 0;
0:5W 1 þ 0:25W 2 0:5W 3 0:42W 4 þ 0:48W 5 0:67W 6 P 0;
0:49W 1 þ 0:42W 2 0:25W 3 þ 0:5W 4 0:229W 5 þ 0:181W 6 P 0;
4
X
wi ¼ 1;
i¼1
wi P 0;
i ¼ 1; . . . ; 4:
By solving the L1 model in 4 iterations, we find the following binding pairs:
Iteration No. 1: (CBM, CM), (CM, PM)
Iteration No. 2: (CBM, CM), (CM, PM)
Iteration No. 3: (CBM, CM), (CM, PM), (PM, TBM)
Iteration No. 4: (TBM, PDM)
Thus the list of our binding pairs will be (CBM, CM), (CM, PM), (PM, TBM) and (TBM, PDM).
Step 4: The DM is unsatisfied with the relative positions of alternatives CM and PM and suggests that PM should be ranked
above CM. Now X2 = [CBM, PM, CM, TBM, PDM], return to step 2.
2
3 2
3 2
e 21
CBMPM
ð2:67;0:67;2;3:34Þ
M
6
7 6
6
6M
e 22 7 6 PMCM 7
7 6 ð3:34;1:67;1;3Þ
7 6
e 2 ¼6
7¼6
M
6
7¼
7 6
6M
e 23 7 6
CMTBM
4
5 4 ð2:67;0:67;2:33;4Þ
4
5
e
TBMPDM
ð3;1:67;2:33;4:33Þ
M 24
2
M 21
3
2
0:5
0:41
1:33 0:67
7 6
6
0:5
6 M 22 7 6 0:25 0:67
7¼6
M2 ¼ 6
7 6
6
0:74
0:42
1
M
4 23 5 4
M 24
0:497
0:42
0:25
ð2:67;0:67;2;3Þ
ð5;3;0:33;2:33Þ
ð2:33;0:67;2;3:67Þ
3
7
ð2:33;0:67;2;3:67Þ ð3;1:67;2:33;4:33Þ ð3:67;2;0:67;2:33Þ 7
7;
7
ð3:67;1:67;1;2:67Þ ð4:67;2:67;1;2:34Þ ð3;1;1:67;3:34Þ 5
ð2:67;1;1:67;3:67Þ ð3:34;1:67;1;3Þ
0:14
0:497 1
0:33
ð3;1;2;4Þ
3
7
0:328 7
7:
7
1 5
0:25
0:51
0:5
0:299 0:181
By solving the L1 model in four iterations, we find the following binding pairs:
Iteration No. 1: (CBM, PM) (PM, CM) (TBM, PDM)
Iteration No. 2: (CBM, PM) (PM, CM) (TBM, PDM)
Iteration No. 3: (CM, TBM) (TBM, PDM)
Iteration No. 4: (CBM, PM) (CM, TBM) (TBM, PDM)
Thus the list of our binding pairs will be (CBM, PM) (PM, CM) (TBM, PDM) and (CM, TBM).
The DM is satisfied with the current ranking.
Table 9
Final results for numerical example.
Rank
X2 = [CBM, PM, CM, TBM, PDM]
Criteria
C1
C2
C3
C4
C5
C6
W*
0.986
0.59
0.52
0.962
0.948
0.68
M. Bashiri et al. / Applied Mathematical Modelling 35 (2011) 152–164
163
Phase 4:
In this phase, final weights will be determined according to the above ranking and related membership functions. In this
numerical example it was concluded by some sensitivity analysis that the amount of a should be at least 0.2 to feasibility of
the problem, since that assuming a = 0.2, the final model could be written as follows:
Maximize
L
Subject to :
0:24W 1 þ 0:75W 2 0:83W 3 þ 0W 4 þ 0:86W 5 þ 0:69W 6 P L;
0:24W 1 0:67W 2 0:5W 3 þ 0:67W 4 W 5 0:328W 6 P L;
0:5W 1 þ 0:25W 2 0:5W 3 0:42W 4 þ 0:48W 5 0:67W 6 P L;
0:49W 1 þ 0:42W 2 0:25W 3 þ 0:5W 4 0:229W 5 þ 0:181W 6 P L;
0:75 6 w1 6 0:986;
0:59 6 w2 6 0:91;
0:52 6 w3 6 0:85;
0:72 6 w4 6 0:962;
0:68 6 w5 6 0:948;
0:68 6 w6 6 0:934:
For example, in fifth constraint we have:
0:73 þ 0:2ð0:83 0:73Þ ¼ 0:75 6 w1 6 0:986 ¼ 1 0:2ð1 0:93Þ:
Table 9 shows final results of IFLAM for this problem.
Thus we got ranking of the maintenance strategies through the proposed method which shows preventive maintenance is
the best maintenance strategy for this company.
The illustrated numerical example was to show applicability of the proposed approach. In this example four decision
makers stated their opinions about both importance of criteria and evaluation of alternatives. Then by the use of both qualitative and quantitative data and interaction with the decision makers; the proposed approach could suggest the most
appropriate maintenance strategy according to organizational goals and limitations.
6. Conclusion
Maintenance planning because of its high effects on manufacturing performance indices such as production rate, cycle
time, product quality, failure costs is one of the most important decision problems. Maintenance, as a system, plays a key
role in reducing cost, minimizing equipment downtime, improving quality, increasing productivity and providing reliable
equipment and as a result achieving organizational goals and objectives. This paper proposed an interactive method for
the selection of optimum maintenance strategy which uses both quantitative and qualitative measures. The proposed method firstly gets an initial ranking by using the fuzzy linear assignment method. Then the algorithm tries to improve the ranking through interaction with the maintenance experts. In the proposed approach several decision makers can state their
opinions about both importance of criteria and evaluation of alternatives. The decision makers in the proposed approach
can interact with the intermediate solutions in order to improve mathematical results with consideration of their experiences so an intermediate solution will be final optimal solution if the decision makers have been satisfied. Finally applying
the proposed method for determination of priorities for the failures can be as a future research. Also as the number of criteria
can decrease the method precision we suggest using a pre stage for criteria reduction before using the proposed approach as
another future research.
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