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Introduction 2 In this study, fuzzy logic (FL), multiple criteria decision making (MCDM) and maintenance management (MM) are integrated into one subject.

Introduction 3  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

Maintenance Management 4 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.

Fuzzy Logic 5 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.

Fuzzy Logic 6 L M U X 1 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.

Multiple Criteria Decision Making (MCDM) 7 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.

Analytic Hierarchy Process (AHP) 8 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.

Analytic Hierarchy Process (AHP) 9 Alternative n Alternative 2 Alternative 1 Objective Criterion1 Criterion Criterion n Alternative 3 Numerical ValueExplanation 1Equally Important 3Slightly Important 5Reasonably Important 7Highly Important 9Definitely Important 2,4,6,8Intermediate Values

TOPSIS 10 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.

Fuzzy AHP 11 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.

Application Model 12 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.

Application Model 13 Objective definition Pairwise comparisons of alternatives and criteria accordance using fuzzy linguistic variables by every decision maker Conversion of linguistic comparisons to fuzzy triangular numbers Taking the average of the pairwise comparisons made by the decision makers Calculation of relative fuzzy performance points for the alternatives Fuzzy weighted performance measurement of the alternatives Defining criteria and subcriteria Pairwise comparisons of criteria and subcriteria importance using fuzzy linguistic variables by every decision maker Conversion of linguistic comparisons to fuzzy triangular numbers Taking the average of the pairwise comparisons made by the decision makers Calculation of relative fuzzy performance points for criteria and subcriteria Fuzzy weighted performance measurement of the criteria Fuzzy weighted performance measurement of the alternatives criteria accordance Fuzzy AHP Defining alternatives

Application Model 14 Calculation of the fuzzy negative and fuzzy positive ideal solutions Calculation of the fuzzy distances between ideal (negative/positive) solutions and alternatives Finding the relative distances of the alternatives to the ideal solution and ordering. Defuzzification Fuzzy TOPSIS Calculation of the classic distances between ideal (negative/positive) solutions and alternatives

Mathematical Model 15

Mathematical Model 16

Mathematical Model 17

Mathematical Model 18

Mathematical Model 19

Mathematical Model 20

Mathematical Model 21

Mathematical Model 22

Fuzzy Comparison Scale 23 Linguistic Variable Linguistic Variable’s Inverse LOMOROLTMTRT Slightly UnimportantSlightly Important1/ UnimportantImportant1/51/31135 Reasonably Unimportant Reasonably Important1/71/51/3357 Highly UnimportantHighly Important1/91/71/5579 Definitely UnimportantDefinitely Important1/9 1/7799 Equally Important Slightly ImportantSlightly Unimportant1131/311 ImportantUnimportant1351/51/31 Reasonably ImportantReasonably Unimportant3571/71/51/3 Highly ImportantHighly Unimportant5791/91/71/5 Definitely ImportantDefinitely Unimportant7991/9 1/7 (LO, MO, RO): Values of fuzzy linguistic variables, (LT, MT, RT): Values of inverse fuzzy linguistic variables.

Criteria List 24 A Windows based software has been developed for the application

Criteria Comparisons 25

Rank Ordering 26 Fuzzy Weighted Scores

Sensitivity Analysis 27 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.

Sensitivity Analysis for Cost 28 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.

Sensitivity Analysis for Cost 29 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.

Sensitivity Analysis for Safety 30 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.

Sensitivity Analysis for Safety 31 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.

Sensitivity Analysis 32 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.

Conclusion 33 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.