An integrated logarithmic fuzzy preference programming based methodology for optimum maintenance strategies selection

Accepted Manuscript Title: An integrated logarithmic fuzzy preference programming based methodology for optimum maintenance strategies selection Authors: Yawei Ge, Mingqing Xiao, Zhao Yang, Lei Zhang, Zewen Hu, Delong Feng PII: DOI: Reference:

S1568-4946(17)30432-5 http://dx.doi.org/doi:10.1016/j.asoc.2017.07.021 ASOC 4349

To appear in:

Applied Soft Computing

Received date: Revised date: Accepted date:

11-1-2017 21-6-2017 9-7-2017

Please cite this article as: Yawei Ge, Mingqing Xiao, Zhao Yang, Lei Zhang, Zewen Hu, Delong Feng, An integrated logarithmic fuzzy preference programming based methodology for optimum maintenance strategies selection, Applied Soft Computing Journalhttp://dx.doi.org/10.1016/j.asoc.2017.07.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Title: An integrated logarithmic fuzzy preference programming based methodology for optimum maintenance strategies selection Authors: Yawei Ge, Mingqing Xiao, Zhao Yang, Lei Zhang, Zewen Hu, Delong Feng E-mail addresses: [email protected] (Y.W. Ge), [email protected] (M.Q. Xiao) Affiliation: Aeronautics and Astronautics Engineering College, Air Force Engineering University. No. 1 Baling Road, Baqiao district, Shannxi Xi’an, 710038, P. R. China *Corresponding Author: Yawei Ge E-mail: [email protected] Telephone: +86 (29) 8478 7142 Graphical abstract

Highlights   

 

A logarithmic fuzzy preference programming based methodology is proposed to solve the optimum maintenance strategies selection problem. Both qualitative and quantitative data is utilized in the methodology. The methodology is free from the significant drawbacks of the existing methods with applying multiplicative constraints and deviation variables to process the upper and lower triangular fuzzy judgments. The methodology is proved to render a unique optimum solution and consensus results. The methodology processes the global comparison matrices simultaneously and directly to derive the optimum priorities in the form of super-matrix.

Abstract: Selecting optimum maintenance strategies plays a key role in saving cost, and improving the system reliability and availability. Analytic hierarchical process (AHP) is widely used for maintenance strategies selection in the Multiple Criteria Decision-Making (MCDM) field. But the traditional or hybrid AHP methods either produce multiple, even conflict priority results, or have complicated algorithm structures which are unstable to obtain the optimum solution. Therefore, this paper proposes an integrated 1

Logarithmic Fuzzy Preference Programming (LFPP) based methodology in AHP to solve the optimum maintenance strategies selection problem. The multiplicative constraints and deviation variables are applied instead of additive ones to utilize both qualitative and quantitative data, and process the upper and lower triangular fuzzy judgments to obtain the same priorities. The proposed methodology can produce the unique normalized optimal priority vector for fuzzy pairwise comparison matrices, and it is capable of processing all comparison matrices to obtain the global priorities simultaneously and directly in the form of supermatrix according to the different requirements and judgments of decision-makers. Finally, an example is provided to demonstrate the feasibility and validity of the proposed methodology. Keywords: Maintenance strategies selection, Analytic hierarchical process, Multiple-criteria decisionmaking, Integrated logarithmic fuzzy preference programming, Super-matrix

1. Introduction The maintenance activities have become increasingly important as the manufacturers face more advanced and more complex systems in the manufacturing and production process [1, 2]. Maintenance activities improve system availability and performance efficiency through technical actions and management measures [3, 4]. However, the maintenance cost occupies a large proportion in the product life cycle cost which ranges from 15% to 70% [5]. Selecting the optimum maintenance which can retain the system in the normal operating conditions with the lowest possible maintenance loss is one of the main concerns of manufacturing firms [6]. There are varieties of factors that affect maintenance time, cost and quality, such as maintenance technical actions and maintenance strategies selection et al., among which the maintenance strategies selection plays a key role in saving cost, minimizing system mean downtime, increasing system reliability and availability [7, 8]. In the literature, the maintenance is usually divided into three main types: Corrective Maintenance, Preventive Maintenance and Predictive Maintenance [9, 10]. And there are four main popular maintenance strategies: breakdown maintenance, time-based maintenance, condition-based maintenance and predictive maintenance [7], which have numerous applications in various fields. In recent years, many agencies and researchers have managed to develop and select maintenance strategies for port infrastructures [11], naval ships [12], pulse-code modulation circuit switching [13], leased equipment [14], multi-component systems [15], wind turbines [16], mining industries [17-19] and petroleum pipeline system [20]. Taking many decision goals and comparing criteria into consideration, the optimum maintenance strategies selection is considered as a complex Multiple-Criteria Decision-Making (MCDM) problem [21]. MCDM is a theory that applies several methods to obtain the optimum solution from the possible alternatives (e.g. maintenance strategies) among which decision-makers need to rank or select with the global analysis and evaluation of multiple conflict criteria (e.g. economic, environmental, etc.). From the latter half of the 2000s, the MCDM approach has gained momentum in the field of maintenance strategies selection [22]. However, the literature on the methods of selecting optimum maintenance strategies is limited. Bertolini and Bevilacqua [23] presented a combined method of “Lexicographic” Goal Programming (GP) and classical Analytic Hierarchy Process (AHP) to define the optimum maintenance strategies for critical centrifugal pumps in an oil refinery. Suresh et al. [24] and Al-Najjar and Alsyouf [25] introduced the fuzzy inference to select the optimum maintenance policy, which developed the classical AHP to fuzzy AHP. Wang et al. [26] used a fuzzy modification of AHP as an evaluation tool to solve the optimization problem with non-linear constraints and consistent or inconsistent fuzzy judgment matrix, and then, to select the optimum maintenance strategies. Pariazar et al. [27] presented an improved AHP method with 2

rough set theory for the maintenance strategy selection. Bashiri et al. [7] presented the interactive fuzzy linear assignment method (IFFLAM) for ranking the maintenance strategies. Hinow and Mevissen [28] introduced the Genetic Algorithm (GA) to handle parameters diversity of Life Cycle Cost (LCC), in order to optimize maintenance strategies for the entire substation system. Uysal and Ömür [29] presented a multiattribute decision-making methodology—hierarchical fuzzy Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to define the best-suited selection of computerized maintenance management system (CMMS). Li et al. [30] and Ishizaka et al. [31] developed a new sorting method—Elimination and Choice Expressing Reality Sort (ELECTRE-SORT) to assign more precise and flexible strategies to different machines, which could handle an unlimited number of criteria. Maletič et al. [32] presented AHP to select the most appropriate maintenance policy for the case of Slovenian paper mill company, and used sensitivity analysis to evaluate the stability of the solution. Makinde et al. [33] presented decision making techniques of weighted decision matrix (WDM) and AHP to establish the optimal maintenance strategy for Reconfigurable Vibrating Screen (RVS). From the literature mentioned above, it is obvious that the AHP method is widely used in the problem of maintenance strategies selection. But in most cases of the problem, it is impossible to provide exact judgments between the criteria for the crisp AHP because of the complexity, vagueness and uncertainty of criteria. Laarhoven and Pedrycz [34] introduced fuzzy set theory to the AHP decision process and allowed decision-makers (DM) to handle the vagueness of meaning of linguistic terms to express their-essentially fuzzy-opinions in fuzzy numbers, and the method provided more realistic results than the original nonfuzzy method. Through comparison studies between fuzzy AHP methods and crisp AHP methods with numerical examples [35-39], the methods developed with fuzzy AHP have their advantages: (a) better modelling of the uncertainty, imprecision and inconsistence associated with pairwise comparison process; (b) cognitively less demanding on the DM; (c) adequate reflection of the DM’s attitude to risk and their degrees of confidence in the subjective assessments. Fuzzy AHP is more appropriate to be applied instead of crisp AHP in the MCDM problem of optimum maintenance strategies selection. However, the popular existing methods, such as fuzzy preference programming (FPP) based nonlinear priority method and extent analysis method used in fuzzy AHP, may produce multiple, even conflict priority results, leading to distinct conclusions and turn out to be subject to some significant drawbacks [40], such as: firstly, negative membership degree makes no sense; secondly, the model of FPP method produces multiple optimal solutions when there exists strong inconsistency among the fuzzy judgments; thirdly, the priority vectors derived by using the upper and lower triangular elements of a fuzzy pairwise comparison matrix in the FPP method are not the same, even significantly different. Furthermore, when the method is being used in combined forms, it will be complicated and unstable to obtain the optimum solution. In this paper, an integrated Logarithmic Fuzzy Preference Programming (LFPP) based methodology is proposed to solve the optimum maintenance strategies selection problem to obtain the final priorities with AHP, which has the following advantages in contrast with the existing methods:  The proposed approach can utilize both qualitative and quantitative data, and produce the unique normalized optimal priority vector for fuzzy pairwise comparison matrices;  The proposed approach applies multiplicative constraints and deviation variables to process the upper and lower triangular fuzzy judgments to obtain the same priorities, which is free from the significant drawbacks of original methods;  All comparison matrices obtained in AHP of optimum maintenance strategies selection are processed simultaneously and directly in the form of global super-matrix to obtain the final priorities. The reminder of the paper is organized as follows. Section 2 gives the related theoretical background of alternative maintenance strategies and the evaluation criteria. Section 3 introduces and describes the proposed LFPP method. Section 4 introduces the integrated LFPP based methodology with a two-stage 3

process. In Section 5, comparative results of an example are discussed. Conclusions are offered in Section 6. 2. Theoretical background 2.1. Alternative maintenance strategies As mentioned above, maintenance is usually divided into three types: Corrective Maintenance, Preventive Maintenance and Predictive Maintenance.  Corrective Maintenance: implemented after the failure occurs. It is a maintenance type resulting from failure.  Preventive Maintenance: performed before the system failure occurs. It is a maintenance type which implements preventive scheduled activities to detect the error state and repair or replace the fault components.  Predictive Maintenance: performed before the system failure occurs. It is implemented with fault prognostics and health management to retain system healthy by continuous and intermittent (periodic or noncyclical) inspections and detection of system state. According to the types of maintenance, the most popular alternative maintenance strategies are analysed as follows: (1) Breakdown Maintenance (BM): This alternative maintenance strategy is also called as failure based maintenance and run-to-breakdown maintenance, which belongs to corrective maintenance. When system failure occurs or system degradation state reaches to the threshold, BM will be implemented. Breakdown maintenance is the original maintenance strategy appeared in industry [41]. It has the characteristics of passivity and dynamic randomness, and it has no fees of health detection but with a high spare parts expense. Usually, when system failure occurs, the related facilities and equipment will also suffer from serious damage. Considering the increasing competition and great wasting of maintenance resource, the manufacturing firms apply great efforts to explore more effective maintenance strategies. (2) Time-based Maintenance (TBM): The maintenance strategy is based on “time”, which means the system operating time or system age [42]. It is based on reliability characteristics of equipment, and the maintenance activities are planned and performed on schedule to reduce the impact caused by frequent or abrupt failures. However, time-based maintenance need huge costs and specialist labours, and the breakdown maintenance is appreciated in the aspect of economy. Some researchers proposed an integrated maintenance strategy to preserve the system operational efficiency, which is called Reliability Centered Maintenance (RCM) [43]. (3) Condition-based Maintenance (CBM): CBM is a maintenance strategy when need arises. It is performed with one or more indicators showing that the system is degenerating or the system failure is going to occur. The maintenance decision is made based on real-time data obtained from the condition monitoring technologies by observing the system state to prioritize and optimize maintenance resource. An ideal condition-based maintenance allows the labour to do the only right thing, minimizes spare parts cost, reduces system downtime and effectively prolongs system life-span. While CBM has great advantages, it still needs to overcome several challenges. CBM requires a high initial cost and a large amount of supporting facilities, and also requires the maintenance staff with additional training. But with CBM coming into service gradually, the LCC will decrease significantly [25]. (4) Predictive Maintenance (PdM): A maintenance of “right information at right time”. Predictive maintenance predicts when to implement the maintenance according to the monitoring data of system operating states. It has some similarities with the CBM, but the differences are that it combines convenient scheduling of BM with state monitoring of CBM, and applies the prognostics and health management (PHM) 4

technologies to achieve better scheduled and high cost-effective maintenance. PdM can make what have been “unplanned stops” transform to shorter and fewer “planned stops” and have advantages of prolonging system lifetime, improving system safety, reducing accidents with negative impacts on the whole system and optimizing the cost of spare parts [44]. In the life cycle of a system, different maintenance strategies would be performed under different conditions. Selecting optimum maintenance strategies for different system life stages depends on the evaluation criteria of the system. Next part will present the criteria of different maintenance goals. 2.2. Evaluation criteria The maintenance strategies are selected based on the maintenance goals. Different maintenance goals include kinds of evaluation criteria. In most cases, the evaluation criteria can be divided into four types:  Safety: The state of being “safe”. In the selection of optimum maintenance strategies, it represents the conditions of keeping away from non-desirable outcomes such as failures, errors, accidents or other events. And it also refers to the controllability of reducing the known threats to an acceptable level. Here, we choose the relevant factors of safety as the evaluation criteria for optimum maintenance strategies selection, which are Personnel Safety, Facilities Safety and Environment Safety.  Cost: More generalized in the field of economics, cost is a metric that is totalling up as a result of a process or as a differential for the result of a decision [45]. In the maintenance domain, the relevant factors of cost are Hardware Cost, Software Cost, and Personnel Training Cost.  Added-value: In the economic field, added-value is the difference between final selling price and the input of producing a particular product. In the maintenance domain, the added-value means the benefits and payback of maintenance activities. Usually, the more the added-value is, the more payback will be obtained with higher maintenance effectiveness and lower input. The relevant evaluation factors of addedvalue are Spare Parts Inventories, Production Loss and Fault Identification.  Feasibility: The criteria is applied to evaluate whether the maintenance strategy is suitable for the system. According to the different requirements of labours and techniques for maintenance strategies, the feasibility criteria can be divided into two relevant evaluation factors: Acceptance by Labours and Technique Reliability. According to the weights and priorities of evaluation criteria, we can select the optimum maintenance strategy at the right time. 3. LFPP method First developed by Satty [46], AHP is a powerful and understandable method that allows groups or individuals to combine qualitative and quantitative factors in decision-making process. It is a MCDM method to solve complicated and unstructured problems, which applies a hierarchical model with levels of goals, criteria, sub-criteria, and alternatives to derive priorities from judgments and make decisions much more accurate. The following are the steps to apply the AHP to the selection of maintenance strategies: Step1: Construct a network structure for the problem according to the decision-making goals; Step2: Identify and determine the relations among criteria in the constructed network structure; Step3: Build fuzzy pairwise comparison matrices using the triangular fuzzy judgments; Step4: Obtain local priorities at different levels with some specific methods; Step5: Form an unweighted super-matrix with each set of local priorities obtained above, and then implement normalization process;

5

Super-matrix is a matrix with each element represented at one row and one respective column. In step5, all the computed and normalized local priorities of the sub-elements with respect to parent elements are placed in one matrix, which is called as a weighted super-matrix. Step6: Obtain the final priorities by calculating the weights of criteria with the weighted super-matrix. The weighted super-matrix is raised to a significantly large power to converge at stable values, which are the desired priorities of the elements with respect to the decision-making goals of optimum maintenance strategies [47]. According to the structure built based on the relations among criteria, the fuzzy judgments are made by the decision makers and the LFPP method is applied to derive priorities in the AHP of optimum maintenance strategies selection. H is defined as the fuzzy pairwise comparison matrix, which is denoted as follows [40]:

 1 h12   h21 1 H   h31 h32   h  n1 hn 2







h13 h23 1 hn 3

h1n   h2 n  h3n    1 

(1)



Where, hij  lij , mij , uij , and h ji  1 uij ,1 mij ,1 lij , for 0  lij  mij  uij , with i, j  1,2,..., n; i  j . lij, mij, and uij each represents the lower, center and upper bounds of fuzzy uncertainty judgments, which have the meaning that the criterion i is between lij and uij times as important as the criterion j with mij being the most likely times. Considering the drawback that using additive constraints may cause one same fuzzy pairwise comparison matrix with different priority rankings from upper and lower triangular judgments, we take the multiplicative constraints to obtain the weight vector. Let w=(w1, w2,…, wn)T represents the weight vector. The elements of the weight vector satisfy the inequalities, and the transformation of the inequalities are as follows.

lij  wi w j  uij  ln lij  ln  wi w j   ln uij  ln lij  ln wi  ln w j  ln uij

(2)

Here, α-cut is introduced for each level to the Eq. (2) with 0 ≤ α ≤ 1 [40], where the transformation is:

ln lij    ln wi  ln w j  ln uij  









ln lij    ln lij    mij  lij 

ln uij    ln uij    mij  uij 

(3) (4) (5)

with i = 1, 2, …, n-1, and j = i+1, i+2, …, n. The membership function is defined to measure the satisfaction degree of fuzzy judgments [48].



 ln  wi w j 



 ln  wi w j   ln lij    ln mij  ln lij    ln uij    ln  wi w j   ln uij  ln mij 

ln  wi w j   ln mij

(6) ln  wi w j   ln mij

6



Where, the function  ln  wi w j 





is the membership degree of ln  wi w j  for the triangular fuzzy



judgments hij  lij , mij , uij . Here, let xi  ln wi and x j  ln w j to simplify the Eq. (6), which is:

 xi  x j  ln mij 1  ln u  ln m ij ij    xi  x j    1   xi  x j  ln mij  ln mij  ln lij 

xi  x j  ln mij  0 (7)

 xi  x j  ln mij  0

 xi  x j  ln uij  0 Considering ln lij  xi  x j  ln uij   , we can express it as w  R  0 , where   xi  x j  ln lij  0 n n1

R 



[40]. Then, the linear member function can be simplified with ln uij mij





and ln mij lij



obtained from the decision-makers. The function is adopted as follows:

w  Rk  1  ln u m  ij ij   k  w  Rk    1  w  Rk  ln  mij lij  

w  Rk  ln  uij mij  (8)

w  Rk  ln  mij lij 

With the Eq. (8), we can define the highest membership degree from the minimum priority vector as described below:



n





i 1



  max  min 1  w  R1  , 2  w  R2  , , n  w  Rn 1   wi  1 w

(9)

Using Max λ(s) (s is the number of comparison matrices) as the objective function, we take the LFPP method to derive the weights of elements. The model(1) of the prioritization problem for the upper or lower triangular judgments is as follows: Objective Function(1): Maximize λ Subject to: μp(λp) ≥ λ xpi – xpj + λ ln (upij/mpij) ≤ ln upij(α) –xpi + xpj + λ ln (mpij/lpij) ≤ – ln lpij(α) xp1 = 0

(10) (11)

with p = 1, 2, …, s, i = 1, 2, …, n-1, and j = i+1, i+2, …, n. In the original model of the existing methods [48], –wi + lijwj+ λ (mij–lij) wj ≤ 0 and wi –uijwj+ λ (uij– mij) wj ≤ 0 cannot hold at the same time with a negative value for λ. Similarly, not all the inequalities xpi – xpj + λ ln (upij/mpij) ≤ ln upij(α) or –xpi + xpj + λ ln (mpij/lpij) ≤ – ln lpij(α) can hold at the same time with a negative value for λ. And the λ is produced with a negative value, which means that there are no weights that meet all the fuzzy judgments in H . That’s to say, the negative value of λ makes no sense. For our proposed method, to avoid λ from taking a negative value, the nonnegative variables δij and ηij are applied to the model(1), which can ensure the results against obtaining a negative membership degree. The model(1) is rebuilt to produce the model(2), which is shown as follows [49]: 7

n 1

Objective Function(2): Minimize 1     M    ij2  ij2  2

n

i 1 j i 1

Subject to: μp(λp) ≥ λ xpi – xpj + λ ln (upij/mpij) ≤ ln upij(α) + δij –xpi + xpj + λ ln (mpij/lpij) ≤ – ln lpij(α) + ηij xp1 = 0, λ, xpi, xpj, δij, ηij ≥ 0.

(12) (13)

with p = 1, 2, …, s, i = 1, 2, …, n-1, and j = i+1, i+2, …, n. When λ=1, it shows that the priorities match the fuzzy judgments perfectly well. But if λ=0, it represents that there exists strong inconsistency in the fuzzy judgments unless

  n 1

n

i 1

j i 1



2 ij

 ij2   0 .

It is the most desirable that the values of the deviation variables are the smaller the better, which can be treated as an inconsistency measure for fuzzy pairwise comparison matrices H . In this way, it can ensure that the value of λ is free from taking a nonnegative value. Meanwhile, in order to be free from the drawbacks, we also introduce a specified sufficiently large constant M to the model(2), such as M=1000, which has the purpose to minimize the violations of the fuzzy judgments and maximize the decision-makers’ satisfaction (the value of λ) when finding weights within the support intervals of fuzzy judgments, and also to ensure the unique optimality of the solution when there exists strong inconsistency among the fuzzy judgments. Making an assumption that x*pi  i  1,2,..., n, p  1, 2, , s  is the optimal solution of the model(2), we can obtain the normalized priorities for the i-th weight in p-th comparison matrix with the function: x*pi  ln w*pi , w*pi 

exp  x*pi 



exp  x*pj  j 1

n

(14)

with p = 1, 2, …, s, i = 1, 2, …, n. With regard to the advantages of LFPP method over the existing methods, the following theorems are given. Theorem 1: The proposed LFPP method can always obtain the unique optimal priority vector for any fuzzy pairwise comparison matrix. Proof. The constraints of model(2) are all linear inequalities, which form a non-empty open convex set S in Rn. The Objective Function(2) is a twice differentiable function defined in S. And the Hessian matrix of the Objective Function(2) is positively definite (The proof is given in Appendix). Then, the Objective Function(2) is a strict convex function for the non-empty open convex set S. Therefore, the computing process of the model is a convex programming. According to the optimization theory, when the objective function of the convex programming is a strict convex function, the minimal point of the objective function exists is the unique one. And meanwhile, the local minimal point of objective function in the convex set S is the global minimal point. As a result, the optimal priority vector for any fuzzy pairwise comparison matrix obtained by the proposed LFPP method is unique and global optimal. Proof completes. Theorem 2: The priorities derived by the LFPP method with multiplicative constraints from the upper triangular elements and the lower triangular elements of a fuzzy pairwise comparison matrix are exactly the same.

8

Proof. For any pair of fuzzy judgments hij   lij , mij , uij  and h ji  1 uij ,1 mij ,1 lij  , the constraints of the model(2) derived from h ji can be written as:

  x pj – x pi   ln  u pji / m pji   lnu pji      – x pj  x pi   ln  m pji / l pji   lnl pji   Where for h ji , l pji  1 / u pij , m pji  1 / m pij , and u pji  1 / l pij , the inequalities can be expressed as:

   x pi + x pj   ln  m pij / l pij    ln l pij      x pi  x pj   ln  u pij / m pij   ln u pij   It is clearly to see that the constraints mentioned above are exactly the same with those of the model (2) derived from hij . In other words, the constraints derived by the model from h ji and hij are always same, which proves that when applying the LFPP method for weight derivation, the priorities derived from the upper triangular elements are always the same with those from the lower triangular elements. Proof completes. 4. Integrated LFPP based methodology On the basis of logarithmic fuzzy preference programming method proposed above, this section introduces an integrated LFPP based methodology with two-stage process, in order to achieve that all fuzzy comparison matrices of AHP can be processed simultaneously and directly in the form of super-matrix. Since the LFPP method has the same procedure and produces the same priorities for the upper and lower triangular judgments, the two-stage process for both of the upper and lower triangular judgments is the same. Using the upper triangular judgments to obtain the priority vector, the two-stage process of the integrated LFPP based methodology is as follow.: Stage 1:



Objective Function: Minimize 1     1



2

n 1

 M    ij2  ij2  n

i 1 j i 1

Subject to: μp(λp) ≥ λ(1) xpi – xpj + λ(1) ln (upij/mpij) ≤ ln upij(α) + δij –xpi + xpj + λ(1) ln (mpij/lpij) ≤ – ln lpij(α) + ηij xp1 = 0, λ(1), xpi, xpj, δij, ηij ≥ 0.

(15) (16)

with p = 1, 2, …, s, i = 1, 2, …, n-1, and j = i+1, i+2, …, n. In stage 1, min operator is applied to obtain λ(1), which represents the worst case scenario. Stage 2: Objective Function: Minimize

n 1 n 1 s * 2 1    M ij2  ij2   p    s p 1 i 1 j i 1

Subject to: λp* ≥ λ(1) 9

xpi – xpj + λp* ln (upij/mpij) ≤ ln upij(α) + δij –xpi + xpj +λp* ln (mpij/lpij) ≤ – ln lpij(α) + ηij xp1 = 0, λp*, xpi, xpj, δij, ηij ≥ 0.

(17) (18)

with p = 1, 2, …, s, i = 1, 2, …, n-1, and j = i+1, i+2, …, n. In stage 2, the arithmetic mean method is applied to deal with all s comparison matrices in the AHP of optimum maintenance strategies selection, which makes the deviations of all achievements balance better. According to the Step5 mentioned in section 3, an unweighted super-matrix with each set of local priorities is built. With the Eq. (14), we can obtain the weighted super-matrix. With the Step6, the weighted supermatrix is raised to a significantly large power to converge at stable values. Then, we can obtain each set of local priorities at each level through handling all comparison matrices directly and simultaneously in the form of super-matrix. For the lower triangular judgments, the objective function of the lower triangular judgments is similar with the upper triangular judgments, which is

n 1 n 2 1 s 1  * p   M    ji2   2ji  .   s p 1 i 1 j i 1

Where, with the Theorem 2 in the Section 3, we can acquire the equation λ*p = λp* by the reason of reciprocal relation hij = 1/hji. Therefore, the objective function of the lower triangular judgment is the same as the upper one, and meanwhile, the integrated model of the upper triangular judgments can be applied for the lower triangular judgments. With the aforementioned integrated LFPP based methodology, we can process all comparison matrices directly and simultaneously to obtain the priorities of criteria for the AHP of optimum maintenance strategies selection in the form of super-matrix. 5. Case Study In this section, a maintenance strategies selection example investigated by Wang et al. [26] is studied. With the criteria discussed in the section 2, we can build the hierarchical structure of the problem, shown in Fig.1. In the hierarchical structure, C1, C2, C3 and C4 represent the selected criteria: Safety, Cost, Added-value and Feasibility, and meanwhile, the criteria have their sub-criteria. BM, TBM, CBM and PdM represent the selection alternatives of different potential optimum maintenance strategies for the investigators. Here, we take the fuzzy comparison matrices from the Wang et al. [26] to derive the priorities for optimum maintenance strategies selection problem with the comparison between the integrated LFPP based methodology and the existing fuzzy AHP prioritization method. The data is used directly without any changes for the fuzzy judgments. Thus, this section will present the process step by step to verify its feasibility and give the global super-matrix to show that the methodology can process all the comparison matrices directly and simultaneously. Table 1 shows the rules of translating the imprecise and uncertain assessment of the criteria to the triangular fuzzy judgments [50]. With x, y, z  R, x>1, y>z. With Table 1, the decision-makers make the importance assessments between the relation elements to decide the priority in the hierarchical structure, and produce the fuzzy judgments to build the fuzzy comparison judgment matrices. With the fuzzy comparison judgment matrices at different levels in the hierarchical structure of optimum maintenance strategies selection, the results obtained by using the proposed integrated LFPP

10

based methodology and the existing fuzzy AHP prioritization method (fuzzy preference programming, FPP) are displayed in Table 2~7, which are shown as follows. * From Table 8, the priority of integrated LFPP are WLFPP   0.3473,0.2436,0.1890,0.2201 . However, *(1) with the FPP method, the priorities have three optimal results wFPP   0.4545,0.1818,0.1818,0.1818 , *(2) *(3) wFPP   0.25,0.25,0.25,0.25 and wFPP   0.2855,0.2382,0.2382,0.2382  .

According to the results, the priority of integrated LFPP proves again that there is one and only one optimal result, which is the absolute optimal solution and safety is the most important criteria. However, for the FPP method, there are three optimal results, which represent three different priorities among the four criteria in the optimum maintenance strategies selection, which are the suboptimal results. In the view of results, the proposed integrated LFPP based methodology ensures the uniqueness of the priorities, which as a result ensure the uniqueness of the final results. However, the FPP method may lead to multiple, even conflict priority results. Furthermore, the proposed methodology has the advantages of processing both the integer and fractional judgments of transition rules. Furthermore, the integrated LFPP based methodology is capable of processing all comparison matrices directly and simultaneously in the form of super-matrix with the fuzzy pairwise comparison matrices of Table 2~7, which is shown as follows. Table 9 and Fig.2 show the weighted super-matrix and the results obtained from the whole fuzzy pairwise comparison priorities using the integrated LFPP based methodology. The blank areas in the table show the fuzzy judgments between the related criteria. Due to the limited space in the table, the specific values of fuzzy judgments are given in Table 2~7. The grey areas of the table are filled with the priorities obtained from the fuzzy judgments at each level. With the integrated LFPP based methodology in Section 4, the global weights of criteria at each level in the super-matrix can be obtained directly and simultaneously which are shown at the last row in the table. According to the results, safety is more important than the other three criteria at the first level. And for the four maintenance strategies, the predictive maintenance is the best alternative under the conditions of given fuzzy judgments. According to different requirements and different judgments of decision-makers, the integrated LFPP based methodology can be processed to get the global priorities directly and simultaneously, and ensure the uniqueness of the global optimum results. 6. Conclusions and Discussions Considering optimum maintenance strategies selection as a problem of MCDM, the paper uses AHP as the hierarchical analysis method and proposes an integrated logarithmic fuzzy preference programming (LFPP) based methodology. The proposed methodology utilizes both qualitative and quantitative data, and applies multiplicative constraints and deviation variables to ensure the same priority rankings for upper and lower triangular judgments with the same fuzzy pairwise comparison matrix, which is free from the significant drawbacks of the existing methods. The computational process demonstrates that the proposed methodology can always obtain the unique optimal priority vector for any fuzzy pairwise comparison matrix. Moreover, the proposed methodology is capable of obtaining the global priorities in the form of global super-matrix through handling all comparison matrices directly and simultaneously. An example is provided to show the methodology in more details. The comparison between the FPP method and the integrated LFPP based methodology in Table 2~7 demonstrates the validity and feasibility. Furthermore, Table 8 and 9 show that the methodology works better than the existing methods in more exact and complicated applications. 11

The integrated LFPP based methodology proposed in this paper can be applied to many other MCDM problems in many fields, which contain both qualitative and quantitative data. Also, the mathematical formulation in Section 3 can be extended to solve more practical problems with priority rankings. Future research will be concentrated on developing a faster and self-adaption method, which selects the global optimum results according the different requirements and decision-making environments. Acknowledgement This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. The authors are grateful to the reviewers for the precious suggestions and feedbacks for improving this paper and to the Editors for their meticulous processing. Appendix Theorem: The Hessian matrix of the Objective Function(2) is positively definite. Proof. n 1

Objective Function(2): Minimize 1     M    ij2  ij2  2

n

i 1 j i 1

The constraint conditions for Objective Function(2) can be simplified as follows: μ(λ) ≥ λ xi – xj + λ ln (uij/mij) ≤ ln uij + δij (19) –xi + xj + λ ln (mij/lij) ≤ – ln lij + ηij (20) λ, xi, xj, δij, ηij ≥ 0, M is a specified sufficiently large constant. with i = 1, 2, …, n-1, and j = i+1, i+2, …, n, n ≥ 2. The inequation (19) and (20) can be equivalently expressed as follows, – xj + λ ln (uij/mij) – ln uij ≤ xi – xj + λ ln (uij/mij) – ln uij ≤ δij (21) –xi + λ ln (mij/lij) + ln lij ≤ –xi + xj + λ ln (mij/lij) + ln lij ≤ ηij (22) As mentioned above, uij, mij, and lij are given by the decision makers, which are known. So  rij, sij ≥ 0, which makes the equations δij = – xj + λ ln (uij/mij) – ln uij + rij and ηij = –xi + λ ln (mij/lij) + ln lij + sij hold. Here, aij, bij, cij, and dij are used to denote (ln (uij/mij)), (– ln uij + rij), (ln (mij/lij)), and (ln lij + sij), which the equations can be expressed as follows. δij = –xj + aijλ + bij (23) ηij = –xi + cijλ+ dij (24) With the equations (23) and (24), the Objective Function(2) can be rewritten as

1   

2

n 1

n

 M 

i 1 j i 1

  x  a   b     x  c   d   . 2

j

ij

ij

2

i

ij

ij

The Hessian matrix and its principal minor determinants of the Objective Function, which is f  x1 , x2 ,

n 1

n

, xn ,    1     M   2

i 1 j i 1

  x  a   b     x  c   d   . 2

j

ij

ij

2

i

ij

ij

12

The Hessian matrix Z of the Objective Function f  x1 , x2 ,   n 1    0     n 1  Z   2M   0    0    k 1 k   aij  i 1 j  k

, xn ,   is    2 n  1 2      aij    cij   i  2 j  i 1  i 1 j  2     n 1 n  n  2 n 1      aij    cij   i  n 1 j i 1  i 1 j  n 1   n 1 n   aij  i 1 j  n   1 n 1 n 2 2     aij  cij   M i 1 j i 1  1

0

0

n

  cij

0

i 1 j  i 1

n 1

0

0

0

n 1

0

0

0

n 1

2 n  1 2      aij    cij  i  2 j  i 1  i 1 j  2 

n 1 n n 1 n  n  2 n 1      aij    cij   aij i  n 1 j i 1 i 1 j  n  i 1 j  n 1 

According to the theory “The sufficient and necessary condition when a symmetric matrix is positively definite is that each order principal minor determinant of the symmetric matrix is positive.”, we need calculate each order principal minor determinant. The principal minor determinants of one order to n order are obviously positive. And the n+1 order principal minor determinant is





 c c 2   c c 2  c c 2   c c 2   c c 2    12 1n   13 14   13 1n  1n 1 n1   12 13     2 2 2 2 2   a12  c23     a12  c2 n    c23  c24     c23  c2 n    c2 n1  c2 n     2  n1  2 2 2 2 2 2 2  a  a  a  c   a  c  a  c   a  c  c  c   c  c  c  c                 n1  n  1 13 23 13 34 13 3 n 23 34 13 3 n 34 35 34 3 n 3 n 3 n  1    .  2M   M    2 2 2 2 2 2 2    a1 n1  a2 n1   a1 n1  a n2 n1  a2 n1  a3 n1   a2 n1  a n2 n1   an3 n1  a n2 n1  a1 n1  c n1n   an2n1  cn1n     2 2 2   a1n  a2 n 2   a1n  a n1n   a2 n  a3n 2   a2 n  a n1n   a n2n  a n1n +  n  1   















 



















 









For the n+1 order principal minor determinant, each row in the bracket of the result is the sum of squared differences between the elements, which are contained in the corresponding position Zi(n+1) (i=1, 2, …, n) of Hessian matrix Z. And obviously, the n+1 order principal minor determinant is positive. The Hessian matrix of the Objective Function(2) is positively definite. This completes the proof.

13

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16

Personnel Safety (C11) Safety Safety(C (C11))

Facilities Safety (C12) Environment Safety (C13) Hardware Cost (C21)

Cost Cost(C (C22)) Selection of Optimum Maintenance Strategies Added-value Added-value(C (C33))

Breakdown Maintenance (BM)

Software Cost (C22) Personnel Training Cost (C23)

Time-based Maintenance (TBM)

Spare Parts Inventories (C31)

Condition-based Maintenance (CBM)

Production Loss (C32) Fault Identification (C33)

Predictive Maintenance (PdM)

Acceptance by Labours (C41) Feasibility Feasibility(C (C44)) Technique Reliability (C42)

Fig.1. The hierarchical structure of optimum maintenance strategies selection

17

The weights of optimum maintenance strategies with integrated LFPP methodology 0

0.1

0.2

0.3

0.4

0.5

0.6

0.45 Alternatives, PdM, 0.3866

Alternatives, CBM, 0.2383 Criteria, C3, 0.2772

Weights

0.35 0.3

C11, 0.2882

Alternatives, TBM, 0.2021 Criteria, C2, 0.1046

0.25 Alternatives, BM, 0.173 Criteria, C1, 0.4488

0.2 0.15

C32, 0.1935

C42, 0.1482

Criteria & Alternatives

Criteria, C4, 0.1694

0.4

C12, 0.1029 0.1 C33, 0.0584

0.05

C13, 0.0578

C22, 0.0503 C31, 0.0254

C41, 0.0212

C21, 0.0329

C23, 0.0215

Sub-criteria

0

C42

C41

C33

C32

C31

C23

C22

C21

C13

C12

C11

Sub-criteria

Alternatives Criteria

Fig.2. The weights obtained from the super-matrix of optimum maintenance strategies selection

18

Table 1 The transition relation from the uncertain assessment to fuzzy judgment Uncertain Assessment

Fuzzy Judgment

Equal or subequal

(1/2, 1, 2), (1, 1, 1)

Most likely x times more important

(x-1, x, x+1)

Most likely x times less important

(1/(x+1), 1/x, 1/(x-1))

Between y and z times more important

(y, (y+z)/2, z)

Between y and z times less important

(1/z, 2/(y+z), 1/y)

19

Table 2 The priorities of fuzzy pairwise comparison matrices at the criteria level Selection of optimum

Added-value

Cost (C2)

Safety (C1)

(1, 1, 1)

(3, 4, 5)

(1, 2, 3)

(2, 3, 4)

0.4488

0.4487

Cost (C2)

(1/5, 1/4, 1/3)

(1, 1, 1)

(1/4, 1/3, 1/2)

(1/3, 1/2, 1)

0.1046

0.1044

Added-value (C3)

(1/3, 1/2, 1)

(2, 3, 4)

(1, 1, 1)

(1, 2, 3)

0.2772

0.2783

Feasibility (C4)

(1/4, 1/3, 1/2)

(1, 2, 3)

(1/3, 1/2, 1)

(1, 1, 1)

0.1694

0.1686

maintenance strategies

(C3)

Feasibility (C4)

Integrated

Safety (C1)

LFPP

FPP

20

Table 3 The comparison results of fuzzy lodgement matrices at the sub-criteria level Personnel Safety

Facilities Safety

Environment Safety

Integrated

(C11)

(C12)

(C13)

LFPP

Personnel Safety (C11)

(1, 1, 1)

(2, 3, 4)

(4, 5, 6)

0.6421

0.6458

Facilities Safety (C12)

(1/4, 1/3, 1/2)

(1, 1, 1)

(1, 2, 3)

0.2292

0.2285

(1/6, 1/5, 1/4)

(1/3, 1/2, 1)

(1, 1, 1)

0.1287

0.1258

Cost (C2) (λ = 0.6167)

Hardware Cost (C21)

Software Cost (C22)

Personnel training

Integrated

Cost (C23)

LFPP

Hardware Cost (C21)

(1, 1, 1)

(1/3, 1/2, 1)

(1, 2, 3)

0.3141

0.3141

Software Cost (C22)

(1, 2, 3)

(1, 1, 1)

(1, 2, 3)

0.4806

0.4808

Personnel training (C23)

(1/3, 1/2, 1)

(1/3, 1/2, 1)

(1, 1, 1)

0.2053

0.2052

Added-value (C3)

Spare parts

Production loss

Fault identification

Integrated

(λ = 0.6666)

Inventories (C31)

(C32)

(C33)

LFPP

(1, 1, 1)

(1/9, 1/8, 1/7)

(1/3, 1/2, 1)

0.0916

0.0912

Production loss (C32)

(7, 8, 9)

(1, 1, 1)

(2, 3, 4)

0.6979

0.6983

Fault identification (C33)

(1, 2, 3)

(1/4, 1/3, 1/2)

(1, 1, 1)

0.2105

0.2105

Feasibility (C4) (λ = 1)

Acceptance by labours (C41)

Technique reliability (C42)

(1, 1, 1)

(1/8, 1/7, 1/6)

0.1250

0.1250

(6, 7, 8)

(1, 1, 1)

0.8750

0.8750

Safety (C1) (λ = 0.8341)

Environment Safety (C13)

Spare parts Inventories (C31)

Acceptance by Labour (C41) Technique Reliability (C42)

Integrated LFPP

FPP

FPP

FPP

FPP

21

Table 4 The priorities of fuzzy pairwise comparison matrices for the maintenance strategy alternatives with Safety C1 Sub-criteria

Integrated

BM

TBM

CBM

PdM

BM

(1, 1, 1)

(1/4, 1/3, 1/2)

(1/6, 1/5, 1/4)

(1/9, 1/8, 1/7)

0.0612

0.0609

TBM

(2, 3, 4)

(1, 1, 1)

(1/3, 1/2, 1)

(1/4, 1/3, 1/2)

0.1443

0.1531

CBM

(4, 5, 6)

(1, 2, 3)

(1, 1, 1)

(1/4, 1/3, 1/2)

0.2516

0.2535

PdM

(7, 8, 9)

(2, 3, 4)

(2, 3, 4)

(1, 1, 1)

0.5429

0.5326

BM

TBM

CBM

PdM

BM

(1, 1, 1)

(1/4, 1/3, 1/2)

(1/6, 1/5, 1/4)

(1/9, 1/8, 1/7)

0.0612

0.0613

TBM

(2, 3, 4)

(1, 1, 1)

(1/3, 1/2, 1)

(1/5, 1/4, 1/3)

0.1443

0.1475

CBM

(4, 5, 6)

(1, 2, 3)

(1, 1, 1)

(1/4, 1/3, 1/2)

0.2516

0.2551

PdM

(7, 8, 9)

(3, 4, 5)

(2, 3, 4)

(1, 1, 1)

0.5429

0.5316

BM

TBM

CBM

PdM

BM

(1, 1, 1)

(1/4, 1/3, 1/2)

(1/5, 1/4, 1/3)

(1/9, 1/8, 1/7)

0.0648

0.0650

TBM

(2, 3, 4)

(1, 1, 1)

(1/3, 1/2, 1)

(1/5, 1/4, 1/3)

0.1588

0.1577

CBM

(4, 5, 6)

(1, 2, 3)

(1, 1, 1)

(1/4, 1/3, 1/2)

0.2251

0.2267

PdM

(7, 8, 9)

(3, 4, 5)

(2, 3, 4)

(1, 1, 1)

0.5513

0.5506

(Personnel C11)

Sub-criteria (Facilities C12)

Sub-criteria (Environment C13)

LFPP

Integrated LFPP

Integrated LFPP

FPP

FPP

FPP

22

Table 5 The priorities of fuzzy pairwise comparison matrices for the maintenance strategy alternatives with Cost C2 Sub-criteria

Integrated

BM

TBM

CBM

PdM

BM

(1, 1, 1)

(1/2, 1, 2)

(5, 6, 7)

(7, 8, 9)

0.4545

0.4507

TBM

(1/2, 1, 2)

(1, 1, 1)

(4, 5, 6)

(7, 8, 9)

0.4098

0.4146

CBM

(1/7, 1/6, 1/5)

(1/6, 1/5, 1/4)

(1, 1, 1)

(1, 2, 3)

0.0817

0.0808

PdM

(1/9, 1/8, 1/7)

(1/9, 1/8, 1/7)

(1/3, 1/2, 1)

(1, 1, 1)

0.0540

0.0539

BM

TBM

CBM

PdM

BM

(1, 1, 1)

(1, 2, 3)

(5, 6, 7)

(7, 8, 9)

0.4923

0.4849

TBM

(1/3, 1/2, 1)

(1, 1, 1)

(3, 4, 5)

(6, 7, 8)

0.3654

0.3707

CBM

(1/7, 1/6, 1/5)

(1/5, 1/4, 1/3)

(1, 1, 1)

(1, 2, 3)

0.0850

0.0872

PdM

(1/9, 1/8, 1/7)

(1/8, 1/7, 1/6)

(1/3, 1/2, 1)

(1, 1, 1)

0.0573

0.0572

BM

TBM

CBM

PdM

BM

(1, 1, 1)

(1, 2, 3)

(4, 5, 6)

(7, 8, 9)

0.5298

0.5195

TBM

(1/3, 1/2, 1)

(1, 1, 1)

(2, 3, 4)

(3, 4, 5)

0.2826

0.2857

CBM

(1/6, 1/5, 1/4)

(1/4, 1/3, 1/2)

(1, 1, 1)

(2, 3, 4)

0.1280

0.1292

PdM

(1/9, 1/8, 1/7)

(1/5, 1/4, 1/2)

(1/4, 1/3, 1/2)

(1, 1, 1)

0.0596

0.0656

(Hardware C21)

Sub-criteria (Software C22)

Sub-criteria (Personnel training C23)

LFPP

Integrated LFPP

Integrated LFPP

FPP

FPP

FPP

23

Table 6 The priorities of fuzzy pairwise comparison matrices for the maintenance strategy alternatives with Added-value C3 Sub-criteria(Spare parts

Integrated

BM

TBM

CBM

PdM

BM

(1, 1, 1)

(1/5, 1/4, 1/3)

(1/6, 1/5, 1/4)

(1/9, 1/8, 1/7)

0.0582

0.0580

TBM

(4, 5, 6)

(1, 1, 1)

(1/3, 1/2, 1)

(1/4, 1/3, 1/2)

0.1850

0.1926

CBM

(4, 5, 6)

(1, 2, 3)

(1, 1, 1)

(1/4, 1/3, 1/2)

0.2389

0.2418

PdM

(7, 8, 9)

(2, 3, 4)

(2, 3, 4)

(1, 1, 1)

0.5179

0.5076

BM

TBM

CBM

PdM

BM

(1, 1, 1)

(1/6, 1/5, 1/4)

(1/8, 1/7, 1/6)

(1/9, 1/8, 1/7)

0.0444

0.0479

TBM

(4, 5, 6)

(1, 1, 1)

(1/4, 1/3, 1/2)

(1/5, 1/4, 1/3)

0.1612

0.1726

CBM

(6, 7, 8)

(2, 3, 4)

(1, 1, 1)

(1/3, 1/2, 1)

0.3550

0.3283

PdM

(7, 8, 9)

(3, 4, 5)

(1, 2, 3)

(1, 1, 1)

0.4394

0.4511

BM

TBM

CBM

PdM

BM

(1, 1, 1)

(1/2, 1, 2)

(1/8, 1/7, 1/6)

(1/9, 1/8, 1/7)

0.0590

0.0590

TBM

(1/2, 1, 2)

(1, 1, 1)

(1/8, 1/7, 1/6)

(1/9, 1/8, 1/7)

0.0589

0.0590

CBM

(6, 7, 8)

(6, 7, 8)

(1, 1, 1)

(1/3, 1/2, 1)

0.3772

0.3823

PdM

(7, 8, 9)

(7, 8, 9)

(1, 2, 3)

(1, 1, 1)

0.5049

0.4997

inventories C31)

Sub-criteria (Production loss C32)

Sub-criteria (Fault identification C33)

LFPP

Integrated LFPP

Integrated LFPP

FPP

FPP

FPP

24

Table 7 The priorities of fuzzy pairwise comparison matrices for the maintenance strategy alternatives with Feasibility C4 Sub-criteria(Acceptance

Integrated

BM

TBM

CBM

PdM

BM

(1, 1, 1)

(1, 2, 3)

(2, 3, 4)

(3, 4, 5)

0.4488

0.4480

TBM

(1/3, 1/2, 1)

(1, 1, 1)

(1, 2, 3)

(2, 3, 4)

0.2772

0.2297

CBM

(1/4, 1/3, 1/2)

(1/3, 1/2, 1)

(1, 1, 1)

(1, 2, 3)

0.1694

0.1681

PdM

(1/5, 1/4, 1/3)

(1/4, 1/3, 1/2)

(1/3, 1/2, 1)

(1, 1, 1)

0.1046

0.1042

BM

TBM

CBM

PdM

BM

(1, 1, 1)

(1, 2, 3)

(5, 6, 7)

(7, 8, 9)

0.4806

0.4977

TBM

(1/3, 1/2, 1)

(1, 1, 1)

(2, 3, 4)

(6, 7, 8)

0.3603

0.3354

CBM

(1/7, 1/6, 1/5)

(1/4, 1/3, 1/2)

(1, 1, 1)

(3, 4, 5)

0.1140

0.1169

PdM

(1/9, 1/8, 1/7)

(1/8, 1/7, 1/6)

(1/5, 1/4, 1/3)

(1, 1, 1)

0.0450

0.0501

by labours C41)

Sub-criteria(Technique reliability C42)

LFPP

Integrated LFPP

FPP

FPP

In Table 2~7, the fuzzy comparison judgment matrices are shown in the form of vectors, which are listed at the middle of the tables. And from the Table 2~7, the priorities obtained by using the integrated LFPP based methodology are listed in the column under the heading “Integrated LFPP”, and the column under the heading “FPP” is the results of FPP method. From the results obtained by the comparison of both methods, the priorities are very close, where the feasibility and validity of the proposed integrated LFPP based methodology is verified. However, the Wang et al. [26] only considered using the integer rules for transforming the uncertain assessment to fuzzy triangular judgments. So, Table 8 is given as an example to illustrate the validity and accurate of the proposed method with both the integer and fractional judgments in the transition rules, which are closer to the human’s judgments.

25

Table 8 The priorities of fuzzy pairwise comparison matrices with integer and fractional number at the criteria level Selection of optimum

Safety (C1)

Cost (C2)

Added-value (C3)

Feasibility (C4)

Safety (C1)

(1, 1, 1)

(3/2, 2, 5/2)

(3/2, 2, 5/2)

(2/3, 1, 3/2)

Cost (C2)

(2/5, 1/2, 2/3)

(1, 1, 1)

(1, 1, 1)

(3/2, 2, 5/2)

Added-value (C3)

(2/5, 1/2, 2/3)

(1, 1, 1)

(1, 1, 1)

(2/5, 1/2, 2/3)

Feasibility (C4)

(2/3, 1, 3/2)

(2/5, 1/2, 2/3)

(3/2, 2, 5/2)

(1, 1, 1)

maintenance strategies

26

Table 9 The weighted super-matrix for optimum maintenance strategies selection with integrated LFPP based methodology Criteria level C1

C2

Sub-criteria level C3

C4

C1 C2

The concrete fuzzy judgment

C3

from Table 2

C11

C12

C13

0.6421

0.2292

0.1287

Alternatives C21

0.3141

C22

0.4806

C23

C31

C32

C33

C41

0.6421

C12

0.2292

C13

0.1287 0.3141

C22

0.4806

C23

0.2053

CBM

PdM

0.0616 0.1462

0.2482

0.5440

0.4881 0.3624

0.0928

0.0567

0.0488 0.1418

0.3490

0.4604

0.1250 0.8750 0.4766 0.3499

0.1210

0.0525

0.0612 0.1443

0.2516

0.5429

0.0612 0.1443

0.2516

0.5429

0.0648 0.1588

0.2251

0.5513

0.4545 0.4098

0.0817

0.0540

0.4923 0.3654

0.0850

0.0573

0.5298 0.2826

0.1280

0.0596

0.0582 0.1850

0.2389

0.5179

0.0444 0.1612

0.3550

0.4394

0.0590 0.0589

0.3772

0.5049

0.4488 0.2772

0.1694

0.1046

in 0.4806 0.3603

0.1140

0.0450

0.2383

0.3866

0.0916 0.6979 0.2105

Fuzzy judgment in Table 3

0.0916 Fuzzy

C32

TBM

Fuzzy judgment in Table 3

C21

C31

BM

0.2053

C4 C11

C42

judgment

in

0.6979 Table3

C33

0.2105

C41

0.1250

Fuzzy

C42

0.8750

judgment Table3

BM

0.0616 0.4881 0.0488 0.4766 0.0612

0.0612

0.0648

0.4545

0.4923

0.5298

0.0582 0.0444 0.0590 0.4488 0.4806

TBM 0.1462 0.3624 0.1418 0.3499 0.1443 0.1443 0.1588 0.4098 0.3654 0.2826 0.1850 0.1612 0.0589 0.2772 0.3603 CBM 0.2482 0.0928 0.3490 0.1210 0.2516 0.2516 0.2251 0.0817 0.0850 0.1280 0.2389 0.3550 0.3772 0.1694 0.1140 PdM 0.5440 0.0567 0.4604 0.0525 0.5429 0.5429 0.5513 0.0540 0.0573 0.0596 0.5179 0.4394 0.5049 0.1046 0.0450

w*

0.4488 0.1046 0.2772 0.1694 0.2882

0.1029

0.0578

0.0329

0.0503

0.0215

0.0254 0.1935 0.0584 0.0212 0.1482 0.1730 0.2021

27