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1 Introduction In this study, fuzzy logic (FL), multiple criteria decision making (MCDM) and maintenance management (MM) are integrated into one subject. 2 Introduction Every system has a life and needs maintenance during its life cycle. Maintenance is the key factor to maintain a system under specified conditions. Since most of the systems are very complex and are affected by a lot of mutually exclusive criteria and parameters, selection of an appropriate maintenance strategy is an important problem of the maintenance management. Having a lot of parameters affecting the system on hand, it is necessary to use a multiple criteria evaluation technique. Also, if we take account the vague and fuzzy characteristics of the parameters, it is important to add fuzzy logic approach to the system 3 Maintenance Management Maintenance includes all the activities carried out for retaining a system in a desired operational state. Maintenance management refers to the application of the appropriate planning, organization and staffing, program implementation and control methods to a maintenance activity. Maintenance management contains all activities including defining works to be done, planning, resource allocation, performing maintenance and reporting. 4 Fuzzy Logic Fuzzy logic (FL) was developed by Zadeh in 1965 to the problems involving vagueness. Fuzziness is explained in terms of vagueness. Linguistic variables are used to explain vagueness in FL. If one can not define the boundaries of information precisely, then vagueness occurs.Characteristics of FL are defined as follows: Fuzzy cluster is defined as membership function which takes values in the interval of [0,1]. Information is given by linguistic variables. FL is suitable for the systems which are difficult to model mathematically. 5 Fuzzy Logic Fuzzy numbers are used in FL. In case of ease-of-use, appropriateness of decision making approaches, and widely usage, fuzzy triangular numbers are used in this study. A~ ( x) xL M L , L x M U x A~ ( x) ,M x U U M 0, Other 1 L M U X 6 Multiple Criteria Decision Making (MCDM) Decision making is the selection of the best activities which simultaneously satisfy goals and constraints. Decision making may be characterized as a process of choosing or selecting 'sufficiently good' alternative(s) or course(s) of action, from a set of alternatives, to attain a goal or goals. Decision making, which includes uncertainties, is a subjective process changing from one person to another. Fuzzy decision making models can be used under uncertainties and vagueness since classical decision making can not be used in those situations. MCDM consists of a finite set of alternatives among which a decision-maker has to select or rank; a finite set of criteria weighted according to their importance. In addition a decision matrix consisting of the rating of each alternative with respect to each criterion using a suitable measure is formed. The evaluation ratings are, then, aggregated taking into account the weights of the criteria, to get a global evaluation for each alternative and a total ranking of the alternatives. 7 Analytic Hierarchy Process (AHP) AHP which was developed by Saaty in 1980 is became one of the most widely used methods to solve MCDM problems practically. AHP solves a problem by structuring it in hierarchic orders. AHP uses those steps below to solve a problem: Decomposition: First problem is divided into small parts and structured as hierarchically. Saaty and Vargas (1991) stated that a decision maker can not simultaneously compare more than 7 ± 2 elements, and offer hierarchical decomposition for solving MCDM problems. We construct the structure of the problem according to its main components: goal/objective set of criteria for evaluation, and the decision alternatives. Pairwise comparison: The relative importance of criteria is established through pairwise comparisons using a square matrix. Hence, judgments are to be made on the importance of criteria which is done with the aid of Saaty's nine-point scale . Synthesis of priorities: Criteria weightings are calculated by using decision matrixes. Finally, relative weightings are synthesized by adding each other to select /sort alternatives. 8 Analytic Hierarchy Process (AHP) Objective Criterion1 Alternative 1 Criterion 2 Alternative 2 .......... Alternative 3 Criterion n ......... Numerical Value Explanation 1 Equally Important 3 Slightly Important 5 Reasonably Important 7 Highly Important 9 Definitely Important 2,4,6,8 Intermediate Values Alternative n 9 TOPSIS TOPSIS (Technique for Order Preferences by Similarity to an Ideal Solution) which was developed by Hwang and Yoon (1981) is used to order alternatives. TOPSIS sorts alternatives by calculating the distances between ideal solution and alternatives. For this, first positive and negative ideal solutions are defined separately. Positive ideal solution is called as maximum benefit solution and includes the best values of the criteria. Negative ideal solution is known as minimum benefit solution and includes the worst values of the criteria. Solutions are defined as points which are the nearest to the positive ideal solution and the farthest to the negative ideal solution at the same time in TOPSIS. Optimum alternative is the one which is the nearest to the positive ideal solution and the farthest to the negative ideal solution. TOPSIS calculation process is given below: Obtaining normalized decision matrix for alternatives, Obtaining weighted decision matrix for alternatives, Calculating positive and negative ideal solutions, Calculating distances to the positive and negative ideal solution for each alternative, Calculating the relative closeness to the ideal solution for each alternative, Sorting the alternatives. 10 ~ 91x Fuzzy AHP To overcome the difficulties faced in classic MCDM, FAHP is offered FAHP expands AHP by using the fuzzy cluster theory. Deciding the relative importance of the criteria and making the fuzzy decision matrix, a fuzzy ratio scale is used in FAHP. Fuzzy Number Explanation ~ 1 (1,1,3) ~x (x-2,x,x+2), x=3,5,7 için ~ 9 (7,9,9) 11 Application Model Criteria comparisons, normalized decision matrix for alternatives, and weighted decision matrix are prepared using the FAHP. Fuzzy TOPSIS is used to order alternatives. Fuzzy comparison scale (FCS) is used to compare alternatives and criteria. Fuzzy AHP-Fuzzy TOPSIS method which is developed in this study is used for maintenance strategy selection problem. Developed method is used for Istanbul Metro maintenance applications including fixed installations for electronics and electro mechanic systems comprised of signaling, SCADA, telecommunications, public announces, CCTV, escalators, elevators, fire detection and extinguishing. Currently, corrective and periodic maintenance techniques are used for those equipments. First objective is defined and then related criteria and alternatives are defined hierarchically to achieve this goal. FAHP method is used for comparisons and fuzzy TOPSIS is used for alternative ordering. 12 Application Model Objective definition Defining criteria and subcriteria Pairwise comparisons of criteria and subcriteria importance using fuzzy linguistic variables by every decision maker Pairwise comparisons of alternatives and criteria accordance using fuzzy linguistic variables by every decision maker Conversion of linguistic comparisons to fuzzy triangular numbers Conversion of linguistic comparisons to fuzzy triangular numbers Taking the average of the pairwise comparisons made by the decision makers Taking the average of the pairwise comparisons made by the decision makers Calculation of relative fuzzy performance points for criteria and subcriteria Calculation of relative fuzzy performance points for the alternatives Fuzzy weighted performance measurement of the criteria Fuzzy weighted performance measurement of the alternatives Fuzzy AHP Defining alternatives Fuzzy weighted performance measurement of the alternatives criteria accordance 13 Application Model Calculation of the fuzzy negative and fuzzy positive ideal solutions Fuzzy TOPSIS Calculation of the fuzzy distances between ideal (negative/positive) solutions and alternatives Calculation of the classic distances between ideal (negative/positive) solutions and alternatives Defuzzification Finding the relative distances of the alternatives to the ideal solution and ordering. 14 Mathematical Model Ai: Alternative i, i=1, 2,…, n Cj: Criterion j, j=1, 2,...,m a~i j : Fuzzy accordance point for alternative i to criterion j ~ B : Fuzzy comparison matrix for criteria ~ U : Alternative-criteria accordance matrix obtained by evaluating alternative i for criterion j. ~ N : Final fuzzy weighted evaluation matrix calculated by evaluating for alternative i to criterion j. djep: Linguistic variable that decision maker p makes pairwise comparison for criteria j and e , e=1, 2, ..., m. ~ d jep : Fuzzy triangular numbers for djep linguistic variable. (Ljep, Mjep, Rjep): Fuzzy triangular numbers’ left, middle, and right side values in order for djep linguistic variable. ~ d je : Fuzzy pairwise evaluation point for criteria j and e. ~ : Relative fuzzy weighting for criterion Cj. w j 15 Mathematical Model ~ W : Fuzzy weighting vector. ~ : Fuzzy eigenvalue vector. (i , j , k ) : Left, middle, and right side values for fuzzy eigenvalue vector in order. ~ m ij : Fuzzy weighted performance point for alternative i to criterion j. (Lmij , Mmij , Rmij ) : Left, middle, and right side values for the ~ fuzzy weighted performance point. m ij ~ k j : Maximum fuzzy performance point which alternative i takes for criterion j (fuzzy positive ideal solution). ~ k j : Minimum fuzzy performance point which alternative i takes for criterion j (fuzzy negative ideal solution). ~ d i : Fuzzy distance to positive ideal solution for i alternative’s performance point. ~ d i : Fuzzy distance to negative ideal solution for i alternative’s performance point. d i : Distance to positive ideal solution for i alternative’s performance point. d i : Distance to negative ideal solution for i alternative’s performance point. (Ldi , Mdi , Rdi ) : Left, middle, and right side values for fuzzy distance (Ldi , Mdi , Rdi ) : Left, middle, and right side values for fuzzy distance : Optimism index. Ri: Relative closeness to ideal solution for alternative i. ~ d i in order. ~ d i in order. 16 Mathematical Model ~ 1. djep linguistic variable is converted to fuzzy triangular numbers ( d jep ) using fuzzy comparison scale. ~ d jep =(Ljep,Mjep,Rjep), j=1,2,...,m, e=1,2,...,m, p=1,2,...,t (2) 2. Fuzzy decision making matrix is calculated as below using the decision makers’ ~ d comparisons which is converted to fuzzy triangular numbers ( jep ) p Lje= Mje= p L je1 L je2 ... L jep , j=1,2,...,m; e=1,2,...,m; p=1,2,...,t M je1 M je2 ... M jep , j=1,2,...,m; e=1,2,...,m; p=1,2,...,t (3) (4) p Uje= U je1 U je2 ... U jep , j=1,2,...,m; e=1,2,...,m; p=1,2,..,t (5) ~ d je =( Lje, Mje, Rje), j=1,2,...,m; e=1,2,...,m (6) ~ d ej =( 1/Rje, 1/Mje, 1/Lje), j=1,2,...,m; e=1,2,...,m (7) 17 Mathematical Model Then, fuzzy comparison matrix ( B~ ) is become as shown below: C1 C1 ~ C B 2 ... Cm C2 ... ~ d 11 ~ d 21 ... ~ d m1 ~ d 12 ~ d 22 ... ~ d m2 Cm ... ... ... ... ~ d 1m ~ d 2m ... ~ d mm (8) ~ ~ 1. After fuzzy comparison is obtained, relative fuzzy weightings ( w j ) and fuzzy weighted vector ( W ) related to criteria are calculated. m ~ w j ~ d je e 1 m m ~ d je , j=1,2,...,m; e=1,2,...,m; (9) j 1 e 1 ~ ~ ,w ~ ,...,w ~ ) W (w 1 2 m T (10) 18 Mathematical Model ~ ~ W 4. Having fuzzy weighting vector ( ) is found, fuzzy eigenvalue vector ( ), classic eigenvalue vector ( ), and maximum eigenvalue ( max ) are calculated. ~ (i , j , k ) ~ ~ (11) ~ ~ B W / W (12) 3 i j k (13) max max( ) (14) 5. Consistency index (CI) and consistency ratio (CR) are calculated. If consistency ratio is smaller than 0.1 then matrix is accepted to be consistent, otherwise evaluations should be revised. Randomness index (RI) values are shown in Table 4. max n n 1 CI CR RI CI (15) (16) Randomness index (Saaty and Vargas, 1991) n Randomness Index n: Matrix size 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 0 0,58 0,9 1,12 1,24 1,32 1,41 1,45 1,49 1,51 1,48 1,56 1,57 1,59 19 Mathematical Model ~ ~ 6. Using equations (2)–(10) alternative-criteria accordance matrix ( U ) is formed similarly. U is subjected to consistency test defined in equations (11)-(16). C1 C2 ... A1 a~11 ~ ~ A2 a 21 U ... ... Am a~m1 a~12 a~ 22 ... a~ m2 Cm ... a~1m ... a~2 m ... ... ... a~mm (17) ~ 7. Final fuzzy evaluation matrix ( N ) is formed by multiplying equations (10) and (17) for every criterion separately. ~ ~ ~ A1 a~11 w~1, a~12 w~2 ,...,a~1m w~m A1 m11 m12 ... m1m ~ ~ ~ A2 a~21 w~1 , a~22 w~2 ,...,a~2 m w~m A2 m 21 m 22 ... m 2 m ~ N ... ............................................. = ... ... (18) ... ... ... ~ ~ ~ m ... m Am a~m1 w~1, a~m 2 w~2 ,...,a~mm w~m Am m m1 m2 mm 8. After having final fuzzy evaluation matrix is calculated, fuzzy TOPSIS method is used to order alternatives. ~ ~ The best ( k j ) and the worst ( k j ) fuzzy performance points are calculated using max min ~ k j ~ , i 1,2,.., n; j 1,2,...,m m ij ~ k j ~ , i 1,2,..,n; j 1,2,...,m m ij ~ N. (19) 20 (20) Mathematical Model 9. Since positive and negative ideal solutions are found, fuzzy distances to positive and negative ideal solutions are calculated using equations (21) and (22). 10. ~ d i ~ k~ m j ij ~ d i ~ k~ m j ij m j 1 m j 1 2 2 i 1,2,...,n (21) i 1,2,...,n (22) ~ ~ di and d i fuzzy numbers are defined as: ~ di (Ldi , Mdi , Rdi ) ~ d i ( Ldi , Mdi , Rdi ) (23) (24) 11. Calculating distances from fuzzy negative and fuzzy positive ideal solutions to alternatives, fuzzy negative and fuzzy positive ideal solutions are defuzzified by using the methods explained below. To compare alternative ordering obtained from different defuzzification methods more than one defuzzification method is used. 21 Mathematical Model i. Centroid method: Centroid method is one of the most widely used defuzzification methods. A lot of author is used centroid method for defuzzification (Opricovic and Tzeng, 2003; Kuo et al., 2002; Chen et al., 2005; Chan et al., 2003; Chiou et al., 2005), di (Ldi Mdi Rdi ) / 3 (25) di (Ldi Mdi Rdi ) / 3 (26) ii. Optimism based defuzzification methods: Optimism/pessimism level of decision makers is considered while defuzzification is made. a. Kaufmann and Gupta Method: In this method calculation is made as fuzzy numbers’ middle value is multiplied by a big coefficient. (Chan et al., 2003). d i ( Ld i 2Mdi Rdi ) / 4 (27) d i ( Ld i 2Mdi Rdi ) / 4 (28) a. Liou and Wang Method: Calculation is made using optimism index (Kaptanoglu and Ozok, 2006). d i ((1 )Ldi Mdi Rdi ) / 2 (29) d i ((1 )Ldi Mdi Rdi ) / 2 (30) 12. After defuzzification process, relative distances of alternatives to the ideal solution are calculated. The best alternative is the one which is the farthest from negative ideal solution and the closest to the positive ideal solution. d i Ri i 1,2,...,n d i d i (31) 22 Ri shows the final performance point. Alternatives are listed related to their Ri values from smaller to bigger. Fuzzy Comparison Scale Linguistic Variable Linguistic Variable’s Inverse LO MO RO LT MT RT Slightly Unimportant Slightly Important 1/3 1 1 1 1 3 Unimportant Important 1/5 1/3 1 1 3 5 Reasonably Important 1/7 1/5 1/3 3 5 7 Highly Unimportant Highly Important 1/9 1/7 1/5 5 7 9 Definitely Unimportant Definitely Important 1/9 1/9 1/7 7 9 9 Equally Important Equally Important 1 1 1 1 1 1 Slightly Important Slightly Unimportant 1 1 3 1/3 1 1 Important Unimportant 1 3 5 1/5 1/3 1 Reasonably Important Reasonably Unimportant 3 5 7 1/7 1/5 1/3 Highly Important Highly Unimportant 5 7 9 1/9 1/7 1/5 Definitely Important Definitely Unimportant 7 9 9 1/9 1/9 1/7 Reasonably Unimportant (LO, MO, RO): Values of fuzzy linguistic variables, (LT, MT, RT): Values of inverse fuzzy linguistic variables. 23 Criteria List A Windows based software has been developed for the application 24 Criteria Comparisons 25 Rank Ordering Fuzzy Weighted Scores 26 Sensitivity Analysis Sensitivity analysis has been made to investigate the changes of ranks of the alternatives according to the changes of criteria’s importance. The main criteria’s importance is changed assigning different fuzzy linguistic variables at the sensitivity analysis in fuzzy model and the effect of these changes on the result has been investigated. Also the changes of the result have been investigated assigning different values to optimism coefficient at the Liou and Wang method which uses optimism coefficient. The effect of changes in the importance of criteria on the changes of the ranks of alternatives has been investigated at sensitivity analysis. 27 Sensitivity Analysis for Cost The opinion of the decision makers is that cost main criterion is more important than other criteria except safety criterion. Cost is a little less important than safety criterion. Accordingly when cost criterion’s importance is decreased against other criteria ranking doesn’t change at Centroid and Kaufmann-Gupta methods; and it doesn’t change when α≥0,29 at Liou and Wang method. RCM moves up to first rank when α<0,29 and TPM takes the second place. Kaufman Liou and Liou and Liou and Centroid Alternatives and Gupta Wang Wang Wang (Alfa=0,29) (Alfa=0,5) (Alfa=0,9) Ri CM 0,0044 0,0078 0,00120 0,0078 0,0047 PM 0,1817 0,1828 0,1834 0,1828 0,1824 CBM 0,3219 0,3172 0,3106 0,3172 0,3221 PDM 0,5802 0,5677 0,5506 0,5677 0,5804 RCM 0,7882 0,7905 0,7932 0,7905 0,7884 28 TPM 0,8125 0,8047 0,7932 0,8047 0,8134 Sensitivity Analysis for Cost When the importance of cost is decreased according to other criteria, TPM continues to go further from positive ideal solution but ranking doesn’t change. If the importance of cost is made equal to the others, ranking changes at all methods RCM moves up to the first rank, and TPM moves down to the second rank. Kaufman Liou and Centroid and Gupta Alternatives Liou and Liou and Wang Wang Wang (Alfa=0,1) (Alfa=0,5) (Alfa=0,9) Ri CM 0,0056 0,0097 0,0260 0,0097 0,0058 PM 0,1848 0,1858 0,1862 0,1858 0,1857 CBM 0,3184 0,3130 0,2881 0,3130 0,3188 PDM 0,5692 0,5550 0,4894 0,5550 0,5699 RCM 0,8083 0,8091 0,8113 0,8091 0,8086 TPM 0,8007 0,7924 0,7502 0,7924 0,8022 29 Sensitivity Analysis for Safety Decision makers believe that safety is much more important than other criteria. When the importance of safety is increased (making it much more important than all criteria) an improvement is seen at RCM but there is no change at the rankings of alternatives in Centroid and Kaufman Gupta methods, in Liou and Wang method there is no change at ranking when α>0.12, RCM moves up to the first rank and TPM moves down to the second rank when α≤0,12 (Ri=0,7722). The rankings of the alternatives don’t change when safety’s importance is turned to “highly important” at Centroid and Kaufmann-Gupta methods but Ri values of TPM and RCM alternatives approach each other very much. (Ri=0,7879 for TPM, Ri=0,7799 for RCM in Kaufmann-Gupta method). In Liou and Wang method ranking doesn’t change for α>0.4 , RCM moves up to first rank (Ri=0,7850), and TPM moves down to second (Ri=0,7825) for α≤0.4 . 30 Sensitivity Analysis for Safety The rankings of the alternatives don’t change when safety’s importance is turned to “definitely important” at Centroid method, ranking changes at Kaufmann-Gupta method and RCM moves up to first rank, and TPM moves down to second. At Liou-Wang method ranking doesn’t change for α>0.64 , TPM moves down to second rank, and RCM moves up to first rank for α≤0.64 in Liou-Wang method. When the importance of safety criterion is reduced, RCM goes further from positive ideal solution, and TPM approaches to positive ideal solution continuously. But ranks of the alternatives don’t change at any method. 31 Sensitivity Analysis Decision makers believe that applicability is less important than cost and safety, more important than competitive advantage and working morale. When the importance of applicability increases PDM approaches to RCM, RCM approaches to TPM but ranking doesn’t change. When the importance of applicability is decreased, RCM goes further from positive ideal solution, TPM approaches to the ideal solution. But the ranking of alternatives don’t change. Decision makers believe that competitive advantage is more important than working morale, and less important than the other criteria. There is no change at the ranking of alternatives when the importance of competitive advantage is increased or decreased. Decision makers believe that working morale is less important than all the other criteria. There is no change at the ranking of alternatives when the importance of working morale is increased or decreased. 32 Conclusion A fuzzy multiple criteria decision making model has been developed for choosing maintenance strategies and a study has been made for choosing maintenance strategies using fuzzy multiple criteria decision making approach. The method used in this study offers a systematic approach to the selection of maintenance strategies. Most used maintenance strategies at Istanbul Metro are corrective maintenance and periodic maintenance. But it has been understood that corrective and periodic maintenance used currently in Istanbul Metro is not suitable because the system of Istanbul Metro is complex, has high importance about safety and is directly related with the passengers. 33