Applied Mathematical Modelling 35 (2011) 152–164

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Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Selecting optimum maintenance strategy by fuzzy interactive linearassignment method

Mahdi Bashiri *, Hossein Badri, Taha Hossein HejaziDepartment 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 2009Received in revised form 10 May 2010Accepted 24 May 2010Available online 27 May 2010

Keywords:Maintenance strategy selection problem(MSSP)Linear assignment methodInteractive analysisFuzzy programming

0307-904X/$ - see front matter � 2010 Elsevier Incdoi:10.1016/j.apm.2010.05.014

* Corresponding author.E-mail addresses: Bashiri@shahed.ac.ir, Bashiri.m

a b s t r a c t

In the current competitive environment managers of manufacturing and service organiza-tions try to make their organizations competitive by providing timely delivery of high qual-ity products. Maintenance, as a system, plays a key role in reducing cost, minimizingequipment downtime, improving quality, increasing productivity and providing reliableequipment and as a result achieving organizational goals and objectives. This paper pre-sents a new approach for selecting optimum maintenance strategy using qualitative andquantitative data through interaction with the maintenance experts. This approach hasbeen based on linear assignment method (LAM) with some modifications to develop inter-active fuzzy linear assignment method (IFLAM).

The proposed approach is an interactive method which uses qualitative and quantitativedata to rank the maintenance strategies. This method helps managers to find the bestmaintenance strategy based on the determined criteria. Maintenance experts also can pro-vide and modify their preference information gradually within the interaction process so asto make the result more reasonable. The proposed method has been illustrated by a numer-ical example.

� 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, andflexibility, in an effort to compete with other firms in the global marketplace [1]. In manufacturing firms there are varietiesof problems that can affect on the manufacturing cost, product quality and delivery time of products to customers; such asmanufacturing technology selection, maintenance strategy selection, machine location, evaluation of quality function. Main-tenance, as a system, plays a key role in reducing cost, minimizing equipment downtime, improving quality, increasing pro-ductivity 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 produc-tion costs, varying according to the type of industry [2]. On the other hand one third of all maintenance costs is wasted as theresult of unnecessary or improper maintenance activities [3]. Therefore, selection of optimum maintenance strategy canhighly affect on the manufacturing expenditures.

In the literature, maintenance can be classified into two main types: corrective and preventive [4,5]. Corrective mainte-nance is the maintenance that occurs after systems failure, and it means all actions resulting from failure; preventive main-tenance is the maintenance that is performed before systems failure in order to retain equipment in specified condition by

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@gmail.com (M. Bashiri), ie.hbadri@gmail.com (H. Badri), Hejazi@shahed.ac.ir (T.H. Hejazi).

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 mainte-nance strategies in the literature are as following:

(1) Corrective maintenance: This alternative maintenance strategy is also named as fire-fighting maintenance, failurebased maintenance or breakdown maintenance. When the corrective maintenance strategy is applied, maintenanceis not implemented until failure occurs [7].

(2) Time-based preventive maintenance: According to reliability characteristics of equipment, maintenance is planned andperformed periodically to reduce frequent and sudden failure. This maintenance strategy is called time-based preven-tive 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 sensorssystem when using the condition-based maintenance strategy. To date a number of monitoring techniques are alreadyavailable, 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 temporarytrend 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 alter-natives based on the evaluation of multiple conflict criteria. MCDM has been one of the fastest growing areas of operationalresearch, as it is often realized that many concrete problems can be represented by several (conflicting) criteria. Several qual-itative and quantitative criteria may affect each other mutually when evaluating alternatives, which may make the selectionprocess complex and challenging [9]. In many cases, the decision maker (DM) has inexact information about the alternativeswith 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 characteris-tics. 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 theratings 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 vaguenessare 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 welldesigned decision process is needed to help managers in reduction of decision failures. In this paper interactive fuzzy linearassignment method (IFLAM) has been proposed for maintenance strategy selection whose two main features are the utili-zation of both qualitative and quantitative data and decision making through an interactive process with the maintenanceexperts.

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 evalu-ation 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 consid-

eration of their experiences so an intermediate solution will be final optimal solution if the decision makers have beensatisfied.

The remainder of this paper is organized as following. Some of the proposed approaches in the field of MSSP will be pre-sented in Section 2. Section 3 presents a brief review of fuzzy set theory. Section 4 introduces and describes the proposedmethod. Section 5 illustrates the procedures in the proposed method using a numerical example. Conclusions are drawnin Section 6.

2. Literature review

Since selection of the optimum maintenance strategy for each equipment; is a vital decision for manufacturing com-panies, many studies have been devoted to this area. Almedia and Bohoris [12] present a review of some basic decisiontheory 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 withthe simplifying of the complex maintenance criteria. Azadivar and Shu [14] presented a new approach to select theoptimum maintenance strategy for each class of systems in a just-in-time environment. In this paper they considered16 characteristic factors that could play a role in maintenance strategy selection. Murthy and Asgharizade [15] pro-posed an approach for decision-making when the company out sources the maintenance. They used game theory toconduct a decision when the customer (the receptionist of maintenance) wants to decide whether having a service con-tract or not.

154 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 todifferent production loss and maintenance costs of each maintenance strategies. Löfsten [18] proposed a model based on costanalysis to choose between corrective or preventive maintenance. Bevilacqua and Barglia [2] used AHP coupled with a sen-sitivity analysis for maintenance strategy selection in an Italian oil refinery. Ivy and Nembhard [19] integrate statistical qual-ity control (SQC) and partially observable Markov decision processes (POMDPs) for the evaluation of maintenance policiesunder conditions of limited information. Bertolini and Bevilacqua [20] used a combined AHP-GP model for maintenanceselection policy problem and in a case study used it for identifying the optimal maintenance policy for a set of centrifugalpumps 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 anddowntime as the maintenance data problems using decision-making grid (DMG) with fuzzy logic in maintenance decisionsupport system (DSS). Jafari et al. [22] proposed a new approach to the MSSP which can determine the best maintenancestrategy by considering the uncertainty level and also all the variety in maintenance criteria and their importance. Saumilet al. [23] developed a continuous time Markov chain degradation model and a cost model to quantify the effects of main-tenance on a multiple machine system. An optimal maintenance policy for a multiple machine system in the absence of re-source constraints is obtained, in the presence of resource constraints, two prioritization methods are proposed to obtaineffective maintenance policies for a multiple machine system. Also a case study focusing on a section of an automotiveassembly line has been used to illustrate the effectiveness of the proposed method. Li et al. [24] calculated a reliability-baseddynamic maintenance threshold (DMT) based on the updated equipment status. In this paper the benefits of the DMT aredemonstrated 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 quan-titative such as hardware and software costs, training costs, time between failures, equipment reliability. There are also a lotof qualitative criteria that must be considered in the selection of maintenance strategy, such as safety, flexibility, acceptanceby labor, high product quality. In real-world situation, because of incomplete or non-obtainable information, data are oftennot 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 assess-ments. Fuzzy linguistic models permit the translation of verbal expression into numerical terms, thereby dealing quantita-tively with the expression of the importance of various objectives. These quantities can then be used to assess the optimumdegree of investment in various maintenance strategies [25]. Many maintenance goals or comparing criteria must be takeninto consideration, e.g. safety and cost in the selection of the suitable maintenance strategies. Therefore, multiple-criteriadecision-making methodology can be used for the maintenance strategy selection. Many researchers have implementedMCDM methods for maintenance strategy selection. Sharma et al. [26] assessed the most popular maintenance strategiesusing the fuzzy inference theory and MCDM evaluation methodology in fuzzy environment. Al-Najjar and Alsyouf [27] usespast data and technical analysis of processes machines and components to identify the criteria for an MCDM problem. Theyused fuzzy inference system (FIS) to assess the capability of each maintenance approach. Finally by utilizing simple additiveweighting (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 andfuzzy theory is a good solution in this regard. The fuzzy methodology based on qualitative verbal assessment inputs is morepractical, 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 goaland the capability of each maintenance strategy to satisfy each goal is captured, then by utilizing some equations in the fuzzyenvironment the optimum maintenance strategy will be selected. Wang et al. [8] proposed a method for the evaluation of thedifferent 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 themaintenance experts. This approach has been based on linear assignment method (LAM) with some modifications to developinteractive fuzzy linear assignment method (IFLAM).

The proposed approach is an interactive method which uses qualitative and quantitative data to rank the maintenancestrategies. This method helps managers to find the best maintenance strategy based on the determined criteria. Maintenanceexperts also can provide and modify their preference information gradually within the interaction process so as to make theresult 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:

leAðxÞ ¼ 1 if and only if X 2 A

0 otherwise

�; ð1Þ

where {0,1} is called a valuation set. If the valuation set is allowed to be the real interval [0,1], eA is called a fuzzy set anddenoted by eA and leAðxÞ is the degree of membership of x in eA.

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 isvery useful in dealing with situations which are too complex or too ill-defined to be reasonably described in conventionalquantitative expressions [10]. These linguistic values can also be represented by fuzzy numbers.

Definition 2. A fuzzy set eA in a universe of discourse X is characterized by a membership function leAðxÞ which associateswith each element x in X a real number in the interval [0,1]. The function value leAðxÞ is termed the grade of membershipof x in eA [29].

Definition 3. The trapezoidal eA ¼ fðx;leAðxÞÞjx 2 Xg fuzzy number can be denoted as eA ¼ ða; b; c; dÞ, where b and c are the cen-tral values ðleAðb 6 x 6 cÞ ¼ 1Þ, a is the left spread and d is the right spread (see Fig. 1).

Definition 4. Let ~m ¼ ðm1;m2;m3;m4Þ and ~n ¼ ðn1;n2;n3;n4Þ be two trapezoidal fuzzy numbers. If ~m ¼ ~n, thenm1 = n1,m2 = n2, m3 = n3, and m4 = n4.

Definition 5. eD is called a fuzzy matrix, if at least an entry in eD is a fuzzy number [30].

Definition 6. A trapezoidal fuzzy number eA can be defined by eA ¼ ða; b; c; dÞ shown in Fig. 1. The membership function leAðxÞis defined as:

leAðxÞ ¼ðx�aÞðb�aÞ ; a � x � b

1; b � x � cðx�dÞðc�dÞ ; c � x � d

0; otherwise

8>>>><>>>>:

: ð2Þ

4. Proposed method

In this paper we propose a new approach for selecting the optimum maintenance strategy. This approach has been devel-oped based on linear assignment method (LAM) with some modifications. Here we aimed to develop a new approach whichconsiders both quantitative and qualitative criteria in the selection process of maintenance strategy as well as keeps an inter-action with the maintenance experts. This interaction allows to maintenance experts to provide and modify their preferenceinformation gradually within the selection so as to make the results more reasonable. For this purpose an interactive fuzzylinear assignment method (IFLAM) has been developed and applied to select the optimum maintenance strategy. The algo-rithm 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 differentmaintenance strategies. There are varieties of qualitative and quantitative criteria for maintenance strategy selection, theexpert team should screen out some criteria based on organizational goals and objectives. Rating of each maintenance strat-egy under quantitative criteria such as the mean time between failures (MTBF), equipment costs simply can be assessed andcomputed. But under the qualitative criteria rating of each maintenance strategy should be assessed by the expert teamusing linguistic variables. The expert team also uses linguistic variables to assess the importance weight of the selection cri-teria (phase 1).

By converting the linguistic evaluation into trapezoidal fuzzy numbers and aggregation of the rating and the weights, weget the initial ranking of the maintenance strategies using fuzzy linear assignment method. This model is a modified andcombinational version of linear assignment method which uses both quantitative and qualitative data to rank the mainte-nance strategies under fuzzy environment (phase 2). The next phase is an interaction process which uses the initial rankinggained in phase 2 as input and tries to improve it iteratively. For this purpose firstly the tight constraint set is identified by

~(x)nµ

a db

1

c

Fig. 1. Trapezoidal fuzzy number eA.

156 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 expertteam to modify the ranking (phase 3). The proposed method is illustrated in Fig. 2 and its procedure based on above con-ceptual 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 linguisticratings.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 usinglinguistic variables.

Fig. 2. The proposed method.

M. Bashiri et al. / Applied Mathematical Modelling 35 (2011) 152–164 157

Phase 2:

Step 1: Convert the linguistic terms into trapezoidal fuzzy numbers. Let ~wjt ¼ ðajt; bjt; cjt; djtÞ; j ¼ 1;2; . . . ;n; t ¼ 1;2; . . . ; kbe the linguistic weight of jth criteria given to maintenance strategies by expert Dt. And Let ~xijt ¼ ðoijt ;

pijt ; qijt ; rijtÞ 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.Step 2: Pool the experts’ opinions to get the aggregated fuzzy weight ~wj of criterion Cj and to get the aggregated fuzzyrating of strategy Ai under criterion Cj. If maintenance expert committee has k persons, the aggregated fuzzy weight ofcriterion Cj can be calculated as [31]:

~wj ¼1k

~wj1ðþÞ~wj2ðþÞ � � � ðþÞ~wjk

� �: ð3Þ

Also the aggregated rating score of maintenance strategies under criterion Cj can be calculated as

~xij ¼1k

~xij1ðþÞ~xij2ðþÞ � � � ðþÞ~xijk

� �: ð4Þ

Step 3: Solve a fuzzy linear assignment programming as described more in following sentences to get the initial ranking ofthe strategies. Here we study the solutions of fuzzy linear assignment programming. Our proposed method to formulateand solve fuzzy linear assignment method (FLAM) is based on the method presented by Zhang et al. [32] and Hwang andYoon [33] which is illustrated in Eqs. (5)–(9). They developed a number of theorems so as to convert the fuzzy linearmethod (FLM) to a multi-objective optimization problem with four-objective functions.Let us define a matrix ~p as a square m �m nonnegative matrix whose element represent the summation of the fuzzyweights 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, andyip = 0 otherwise.Since we defined our weights as trapezoidal fuzzy numbers, thus we show ~pip matrix as ~pij ¼ ð~pijð1Þ; ~pijð2Þ; ~pijð3Þ; ~pijð4ÞÞ.The fuzzy linear assignment problem can be written by the following linear programming (LP) format:

MaxX4

r¼1

kr

Xm¼m1þm2

i¼1

Xm1�1

p¼1

~pipðrÞxip þXm

i¼1

Xm2�1

p¼5

pipxip ð5Þ

Subject toXm

p¼1

xip ¼ 1; i ¼ 1;2; . . . ;m; ð6Þ

Xm

i¼1

xip ¼ 1; p ¼ 1;2; . . . ;m; ð7Þ

X4

r¼1

kr ¼ 1 ð8Þ

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 isdetermined by experts and m1, m2 are number of the qualitative and quantitative criteria, respectively. Constraints 6 and7 ensure that each alternative belong to exactly one position.

Phase 3:

Step 1: Construct ~mX matrix based on the ranking identified from FLAM.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.Step 2: Defuzzify the ~mX matrix and the importance weights ~w.Getting the initial feasible order X, let ~mX be the (m � 1) �m matrix whose rows ~mXðpÞ are given by ~xXðpÞ � ~xXðpþ1Þ . Herewe defuzzify the fuzzy numbers of ~mX and get mX matrix by the following equation:

MXðpÞ ¼14

aþ bþ c þ dð Þ: ð10Þ

Quantitative scores are fed into ~mX matrix directly and after linear normalization.Step 3: Solve the L1 model to identify the binding constraints of WX. This step is based on interactive simple additiveweighting method [33].Now the problem is to identify the tight constraint set. The boundary of WX is determined by those rows of Eq. (12) whichare tight (active) constraints for some value of w. For such rows the optimum value of the following LP problem (L1) giveszero:

158 M. Bashiri et al. / Applied Mathematical Modelling 35 (2011) 152–164

Lj : Min MXðjÞw 8j ¼ 1; . . . ;m� 1; ð11ÞMXw P 0; ð12ÞXn

i¼1

wi ¼ 1; ð13Þ

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 notany binding constraints or if this order is accepted by the committee, otherwise ask the experts to revise and change therating 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 finalsatisfying ranking remained unchanged:

Table 1Linguis

VeryLowMedMedMedHighVery

Maximize L; ð15ÞMXw P L; ð16Þai þ aðbi � aiÞ 6 wi 6 di � aðdi � ciÞ; ð17Þwi P 0; i ¼ 1;2; . . . ;n: ð18Þ

Eqs. (15) and (16) ensure that the current ranking will remain unchanged and Eq. (17) is used to find most reliable valuesof weights with respect to predefined membership value (a). Where (ai,bi,ci,di) are definition parameters for ith fuzzyweight (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 de-sires to select the most appropriate maintenance strategy that requires different alternatives to be assessed for a range ofcriteria. There is a variety of strategies in maintenance management systems depending on the type of applied industryin a company. These strategies can be categorized into two main groups of corrective and preventive ones [34]. In correctivemaintenance strategy, no maintenance activity is carried out until a failure occurs. Small profit margins along with ascendingcompetition trend make the maintenance managers apply more reliable maintenance strategies. Conversely, preventivemaintenance 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 maintenancestrategies [34].

These strategies would be five custom policies of corrective maintenance (CM), preventive maintenance (PM), time-basedmaintenance (TBM), condition-based maintenance (CBM) and predictive maintenance (PDM) strategies which the decision-making group has to choose one among them. According to the explained steps in Section 3, the proposed method is appliedto 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 andto 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 PDMare selected to be evaluated.Step 3: After screening four benefit criteria are considered as qualitative measures: C1, C2, C3 and C4. Also two cost criteriaare 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 linguis-tic 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 whichare presented in Table 3.

tic variables for the importance weight of each criterion.

low (VL) (0,0,0.1,0.2)(L) (0,0.1,0.2,0.3)ium low (ML) (0.1,0.2,0.4,0.5)ium (M) (0.3,0.4,0.6,0.7)ium high (MH) (0.5,0.6,0.8,0.9)(H) (0.7,0.8,0.9,1)high (VH) (0.8,0.9,1,1)

Table 2Linguistic variables for the ratings.

Very poor (VP) (0,0,1,2)Poor (P) (0,1,2,3)Medium poor (MP) (1,2,4,5)Fair (F) (3,4,6,7)Medium good (MG) (5,6,8,9)Good (G) (7,8,9,10)Very good (VG) (8,9,10,10)

Table 3Importance values of each criterion by DMs.

C1 C2 C3 C4

D1 VH H M VHD2 H MH MH VHD3 H MH H MH

Table 4Rating of alternatives under each criterion in terms of linguistic variables determined by DMs.

Alternatives CM PM TBM CBM PDM

Criteria C1 C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C4 C1 C2 C3 C4

D1 MG G G MG G VG MG VG MG G G G VG VG F G G VG G MGD2 VG VG F G G MG G F G MG G MG VG MG F VG F F MG GD3 G F MG VG MG G MG G MG G MG G MG VG G MG MG G VG MG

M. Bashiri et al. / Applied Mathematical Modelling 35 (2011) 152–164 159

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 ofMG for the rating of alternative A under criterion C1 according to D1.Step 2: Calculate the aggregated fuzzy weight fW j of criterion Cj and the aggregated fuzzy rating of alternative Ai undercriterion Cj by using Eqs. (3) and (4)

Table 5Importa

D1

D2

D3

fW 11 ¼13½0:8þ 0:7þ 0:7� ¼ 0:73;

fW 12 ¼13½0:9þ 0:8þ 0:8� ¼ 0:83;

fW 13 ¼13½1þ 0:9þ 0:9� ¼ 0:93;

fW 14 ¼13½1þ 1þ 1� ¼ 1:

Thus: fW 1 ¼ ½0:73;0:83;0:93;1�And for alternative CM under criterion C1:

~x11 ¼13½5þ 8þ 7� ¼ 6:67;

~x12 ¼13½6þ 9þ 8� ¼ 7:67;

~x13 ¼13½8þ 10þ 9� ¼ 9;

~x14 ¼13½9þ 10þ 10� ¼ 9:67:

nce weights of criteria in terms of fuzzy numbers for each DMs.

C1 C2 C3 C4 C5 C6

(0.8,0.9,1,1) (0.7,0.8,0.9,1) (0.3,0.4,0.6,0.7) (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) (0.5,0.6,0.8,0.9) (0.5,0.6,0.8,0.9) (0.8,0.9,1,1) (0.7,0.8,0.9,1) (0.8,0.9,1,1)(0.7,0.8,0.9,1) (0.5,0.6,0.8,0.9) (0.7,0.8,0.9,1) (0.5,0.6,0.8, 0.9) (0.8,0.9,1,1) (0.7,0.8,0.9,1)

Table 6Rating of alternatives under each criterion in terms of fuzzy numbers.

Alternative CM

Criteria C1 C2 C3 C4

D1 (5,6,8,9) (7,8,9,10) (7,8,9,10) (5,6,8,9)D2 (8,9,10,10) (8,9,10,10) (3,4,6,7) (7,8,9,10)D3 (7,8,9,10) (3,4,6,7) (5,6,8,9) (8,9,10,10)

Alternative PM

Criteria C1 C2 C3 C4

D1 (7,8,9,10) (8,9,10,10) (5,6,8,9) (8,9,10,10)D2 (7,8,9,10) (5,6,8,9) (7,8,9,10) (3,4,6,7)D3 (5,6,8,9) (7,8,9,10) (5,6,8,9) (7,8,9,10)

Alternative TBM

Criteria C1 C2 C3 C4

D1 (5,6,8,9) (7,8,9,10) (7,8,9,10) (7,8,9,10)D2 (7,8,9,10) (5,6,8,9) (7,8,9,10) (5,6,8,9)D3 (5,6,8,9) (7,8,9,10) (5,6,8,9) (7,8,9,10)

Alternative CBM

Criteria C1 C2 C3 C4

D1 (8,9,10,10) (8,9,10,10) (3,4,6,7) (7,8,9,10)D2 (5,6,8,9) (8,9,10,10) (3,4,6,7) (8,9,10,10)D3 (8,9,10,10) (5,6,8,9) (7,8,9,10) (5,6,8,9)

Alternative PDM

Criteria C1 C2 C3 C4

D1 (7,8,9,10) (8,9,10,10) (7,8,9,10) (5,6,8,9)D2 (3,4,6,7) (3,4,6,7) (5,6,8,9) (7,8,9,10)D3 (5,6,8,9) (7,8,9,10) (8,9,10,10) (5,6,8,9)

160 M. Bashiri et al. / Applied Mathematical Modelling 35 (2011) 152–164

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 anyother attempt, since in criteria C2 alternatives CM and PDM and in criteria C4 alternatives CM and CBM have the sameranking, we took one more criteria with half of the original weight value. Thus the weights will be:

Table 7Fuzzy d

Crite

CMPMTBMCBMPDM

Weig

fW 1 ¼ ð0:73;0:83;0:93;1Þ;fW 21 ¼ ð0:285;0:335;0:415; 0:465Þ;fW 22 ¼ ð0:285;0:335;0:415; 0:465Þ;fW 3 ¼ ð0:5;0:6;0:77;0:87Þ;fW 41 ¼ ð0:35;0:4;0:465;0:485Þ;fW 42 ¼ ð0:35;0:4;0:465;0:485Þ;fW 5 ¼ ð0:66;0:76;0:9;0:96Þ;fW 6 ¼ ð0:66;0:76;0:83; 0:96Þ:

Table 8 shows all alternatives and relative ranking under each criterion.

ecision matrix.

ria C1 C2 C3 C4 C5 C6

(6.67,7.67,9,9.67) (6,7,8.33,9) (5,6,8.33,8.67) (6.67,7.67,9,9.67) 362 3576(6.33,7.33,8.67,9.67) (6.67,7.67,9,9.67) (5.67,6.67,8.33,9.33) (6,7,8.33,9) 135 3283(5.67,6.67,8.33,9.33) (6.33,7.33, 8.67,9.7) (6.33,7.33,8.67,9.67) (6.33,7.33,8.67,9.67) 245 2683(7,8,9.33,9.67) (7,8,9.33,9.67) (4.33,5.33,7,8) (6.67,7.67,9,9.67) 167 2958(5,6,8.33,8.67) (6,7,8.33,9) (6.67,7.67,9,9.67) (5.67,6.67,8.33,9.33) 177 2845

ht (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)

Table 8Attribut

Rank

1st2nd3rd4th5th

M. Bashiri et al. / Applied Mathematical Modelling 35 (2011) 152–164 161

for example ~p11 ¼ fW 41 ¼ ð0:35;0:4;0:47;0:49Þ which shows that alternative CM has 1st rank in criterion C41 and~p12 ¼ fW 1 þ fW 42 ¼ ð1:08;1:23;1:4;1:49Þ which shows that alternative CM has 2nd rank in criterion C1 and C42.

Thus the FLAP model based on Eqs. (5)–(9) will be:

MaxX4

r¼1

kr

X5

i¼1

X4

p¼1

~pipðrÞxip þX5

i¼1

X6

p¼5

pipðrÞxip

S:t:X5

p¼1

xip ¼ 1; i ¼ 1; . . . ;5;

X5

i¼1

xip ¼ 1; p ¼ 1; . . . ;5;

xip P 0 for all i and p;

where

X4

r¼1

kr ¼ 1:

Assuming equal importance for each element of fuzzy numbers, here kr = 0.25 is considered for all r. Solving this linear modelusing Software LINGO 8.0, the optimal permutation matrix X* is

X� ¼

0 1 0 0 00 0 1 0 00 0 0 1 01 0 0 0 00 0 0 0 1

26666664

37777775:

Applying the optimal permutation matrix X* to the alternatives, we find the optimal order:

X1 ¼ ½CBM;CM; PM; TBM;PDM�:

Phase 3:

Step 1: Construct eMX matrix based on the ranking identified from FLAP.The associated eM1 matrix is:

eM1¼

eM11eM12eM13eM14

266664

377775¼

CBM�CM

CM�PMPM�TBMTBM�PDM

26664

37775¼

ð�2:67;�1;1:66;3Þ ð�2;�0:33;2:33;3Þ ð�4:34;�3;1;3Þ ð�3;�1:33;1:33;3Þð�3;�1;1:67;3:28Þ ð�3:67;�2;0:66;2:33Þ ð�4:33;�2:33;1:66;3Þ ð�2:33;�0:66;2;3:67Þð�3;�1;2;4Þ ð�3;�1;1:67;3:34Þ ð�4;�2;1;3Þ ð�3:67;�1:67;1;2:67Þð�3;�1:67;2:33;4:33Þ ð�2:67;�1;1:67;3:67Þ ð�3:34;�1:67;1;3Þ ð�3;�1;2;4Þ

26664

37775:

Step 2: Defuzzify the eMX matrix and the importance weights based on Eq. (10)

e-wise preference.

C1 C21 C22 C3 C41 C42 C5 C6

CBM CBM CBM PDM CM CBM PM TBMCM PM PM TBM CBM CM CBM PDMPM TBM TBM PM TBM TBM PDM CBMTBM PDM CM CM PM PM TBM PMPDM CM PDM CBM PDM PDM CM CM

Table 9Final re

Rank

Crite

W*

162 M. Bashiri et al. / Applied Mathematical Modelling 35 (2011) 152–164

M1 ¼

M11

M12

M13

M14

26664

37775 ¼

0:24 0:75 �0:83 0 0:86 0:690:24 �0:67 �0:5 0:67 �1 �0:3280:5 0:25 �0:25 �0:42 0:48 �0:670:49 0:42 �0:5 0:5 �0:299 0:181

26664

37775:

For example under criterion C1

MXð11Þ ¼14�2:67� 1þ 1:66þ 3ð Þ ¼ 0:24:

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 0:24W1 þ 0:75W2 � 0:83W3 þ 0W4 þ 0:86W5 þ 0:69W6

Subject to : 0:24W1 þ 0:75W2 � 0:83W3 þ 0W4 þ 0:86W5 þ 0:69W6 P 0;

0:24W1 � 0:67W2 � 0:5W3 þ 0:67W4 �W5 � 0:328W6 P 0;

0:5W1 þ 0:25W2 � 0:5W3 � 0:42W4 þ 0:48W5 � 0:67W6 P 0;

0:49W1 þ 0:42W2 � 0:25W3 þ 0:5W4 � 0:229W5 þ 0:181W6 P 0;

X4

i¼1

wi ¼ 1;

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 rankedabove CM. Now X2 = [CBM,PM,CM,TBM,PDM], return to step 2.

eM2¼

eM21eM22eM23eM24

2666664

3777775¼

CBM�PM

PM�CM

CM�TBM

TBM�PDM

266664

377775¼

ð�2:67;�0:67;2;3:34Þ ð�2:67;�0:67;2;3Þ ð�5;�3;0:33;2:33Þ ð�2:33;�0:67;2;3:67Þð�3:34;�1:67;1;3Þ ð�2:33;�0:67;2;3:67Þ ð�3;�1:67;2:33;4:33Þ ð�3:67;�2;0:67;2:33Þð�2:67;�0:67;2:33;4Þ ð�3:67;�1:67;1;2:67Þ ð�4:67;�2:67;1;2:34Þ ð�3;�1;1:67;3:34Þð�3;�1:67;2:33;4:33Þ ð�2:67;�1;1:67;3:67Þ ð�3:34;�1:67;1;3Þ ð�3;�1;2;4Þ

266664

377775;

M2 ¼

M21

M22

M23

M24

266664

377775 ¼

0:5 0:41 �1:33 0:67 �0:14 0:33

�0:25 0:67 �0:5 0:497 1 0:328

0:74 �0:42 �1 0:25 0:51 �1

0:497 0:42 �0:25 0:5 �0:299 0:181

266664

377775:

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.

sults for numerical example.

X2 = [CBM,PM,CM,TBM,PDM]

ria C1 C2 C3 C4 C5 C6

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 ofthe problem, since that assuming a = 0.2, the final model could be written as follows:

Maximize L

Subject to : 0:24W1 þ 0:75W2 � 0:83W3 þ 0W4 þ 0:86W5 þ 0:69W6 P L;

0:24W1 � 0:67W2 � 0:5W3 þ 0:67W4 �W5 � 0:328W6 P L;

0:5W1 þ 0:25W2 � 0:5W3 � 0:42W4 þ 0:48W5 � 0:67W6 P L;

0:49W1 þ 0:42W2 � 0:25W3 þ 0:5W4 � 0:229W5 þ 0:181W6 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:

0:73þ 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 qual-itative and quantitative data and interaction with the decision makers; the proposed approach could suggest the mostappropriate 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, cycletime, product quality, failure costs is one of the most important decision problems. Maintenance, as a system, plays a keyrole in reducing cost, minimizing equipment downtime, improving quality, increasing productivity and providing reliableequipment and as a result achieving organizational goals and objectives. This paper proposed an interactive method forthe selection of optimum maintenance strategy which uses both quantitative and qualitative measures. The proposed meth-od firstly gets an initial ranking by using the fuzzy linear assignment method. Then the algorithm tries to improve the rank-ing through interaction with the maintenance experts. In the proposed approach several decision makers can state theiropinions about both importance of criteria and evaluation of alternatives. The decision makers in the proposed approachcan interact with the intermediate solutions in order to improve mathematical results with consideration of their experi-ences so an intermediate solution will be final optimal solution if the decision makers have been satisfied. Finally applyingthe proposed method for determination of priorities for the failures can be as a future research. Also as the number of criteriacan decrease the method precision we suggest using a pre stage for criteria reduction before using the proposed approach asanother future research.

References

[1] E.E. Karsak, E. Tolga, Fuzzy multi-criteria decision making procedure for evaluating advanced manufacturing system investments, International Journalof Production Economics 69 (2001) 49–64.

[2] M. Bevilacqua, M. Braglia, The analytic hierarchy process applied to maintenance strategy selection, Reliability Engineering and System Safety 70(2000) 71–83.

[3] R.K. Mobley, An Introduction to Predictive Maintenance, Second ed., Elsevier Science, New York, 2002.[4] J.R. Li, L.P. Khoo, S.B. Tor, Generation of possible multiple components disassembly sequence for maintenance using a disassembly constraint graph,

International Journal of Production Economics 102 (2006) 51–65.[5] G. Waeyenbergh, L. Pintelon, A framework for maintenance concept development, International Journal of Production Economics 77 (2002) 299–313.[6] H. Wang, A survey of maintenance policies of deteriorating systems, European Journal of Operational Research 139 (2002) 469–489.[7] L. Swanson, Linking maintenance strategies to performance, International Journal of Production Economics 70 (2001) 237–244.[8] L. Wang, J. Chu, J. Wu, Selection of optimum maintenance strategies based on a fuzzy analytic hierarchy process, International Journal of Production

Economics 107 (2007) 151–163.[9] B. Vahdani, H. Hadipour, Extension of VIKOR method based on interval-valued fuzzy sets, International Journal of Advanced Manufacturing Technology

47 (2010) 1231–1239.[10] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Information Sciences 8 (1975). 199–249(I), 301–357(II).

164 M. Bashiri et al. / Applied Mathematical Modelling 35 (2011) 152–164

[11] Y.J. Wang, H.S. Lee, Generalizing TOPSIS for fuzzy multiple-criteria group decision-making, Computers and Mathematics with Applications 53 (2007)1762–1772.

[12] A.T. de Almedia, G.A. Bohoris, Decision theory in maintenance decision making, Quality in Maintenance Engineering 1 (1) (1995) 39–45.[13] E. Triantaphyllou, B. Kovalerchuk, L. Mann, G. Knapp, Determining the most important criteria in maintenance decision making, Journal of Quality in

Maintenance Engineering 3 (1) (1997) 16–28.[14] F. Azadivar, V. Shu, Maintenance policy selection for JIT production systems, International Journal of Production Research 37 (16) (1999) 3725–3738.[15] D.N.P. Murthy, E. Asgharizade, Optimal decision making in a maintenance service operation, European Journal of Operational Research 116 (1999)

259–273.[16] S. Luce, Choice criteria in conditional preventive maintenance, Mechanical Systems and Signal Processing 13 (1) (1999) 163–168.[17] S. Okumura, N. Okino, A maintenance policy selection method for a critical single-unit item in each workstation composing a FMS with CBM

optimization, International Journal of COMADEM 6 (2) (2003) 3–9.[18] H. Löfsten, Management of industrial maintenance – economic evaluation of maintenance policies, International Journal of Operations and Production

Management 19 (7) (1999) 71–73.[19] J.S. Ivy, H.B. Nembhard, A modeling approach to maintenance decisions using statistical quality control and organization, Quality and Reliability

Engineering International 21 (2005) 355–366.[20] M. Bertolini, M. Bevilacqua, A combined goal programming—AHP approach to maintenance selection problem, Reliability Engineering and System

Safety Roach to Maintenance Selection Problem 91 (7) (2006) 839–848.[21] Z. Tahir, A.S. Prabuwono, B.M. Aboobaider, Maintenance decision support system in small and medium industries: an approach to new optimization

model, International Journal of Computer Science and Network Security 8 (11) (2008).[22] A. Jafari, M. Jafarian, A. Zareei, F. Zaerpour, Using fuzzy Delphi method in maintenance strategy selection problem, Journal of Uncertain Systems 2 (4)

(2008) 289–298.[23] A. Saumil, L. Li, N. Jun, Condition-based maintenance decision-making for multiple machine systems, Journal of Manufacturing Science and

Engineering 131 (3) (2009). p031009:1–031009:9, 9p.[24] L. Li, M. You, N. Jun, Reliability-based dynamic maintenance threshold for failure prevention of continuously monitored degrading systems, Journal of

Manufacturing Science and Engineering 131 (3) (2009). p031010:1–031010:9, 9p.[25] C.K. Mechefske, Z. Wang, Using fuzzy linguistics to select optimum maintenance and condition monitoring strategies, Mechanical Systems and Signal

Processing 17 (2) (2003) 305–316.[26] R.K. Sharma, D. Kumar, P. Kumar, FLM to select suitable maintenance strategy in process industries using MISO model, Journal of Quality in

Maintenance Engineering 11 (4) (2005) 359–374.[27] B. Al-Najjar, I. Alsyouf, Selecting the most efficient maintenance approach using fuzzy multiple criteria decision making, International Journal of

Production Economics 84 (2003) 85–100.[28] E. Palaneeswaran, M. Kumaraswamy, Recent advances and proposed improvements in contractor prequalification methodologies, Building and

Environment 36 (2001) 73–87.[29] Y.I. Topcu, A decision model proposal for construction contractor selection in Turkey, Building and Environment 39 (2004) 469–481.[30] J.J. Buckley, Fuzzy hierarchical analysis, Fuzzy Sets and Systems 17 (1985) 233–247.[31] T.C. Chen, Extensions of the TOPSIS for group decision-making under fuzzy environment, Fuzzy Sets and Systems 114 (2000) 1–9.[32] G. Zhang, Y.H. Wu, M. Remias, J. Lu, Formulation of fuzzy linear programming problems as four-objective constrained optimization problems, Applied

Mathematics and Computation 139 (2003) 383–399.[33] C.L. Hwang, K. Yoon, Multiple Attribute Decision Making – Method and Applications, A State-of-the-Art Survey, Springer Verlag, New York, 1981.[34] Y.M. Wang, T.M.S. Elhag, Fuzzy TOPSIS method based on alpha level sets with an application to bridge risk assessment, Expert Systems with

Applications 31 (2) (2006) 309–319.