International Construction Specialty Conference of the Canadian Society for Civil Engineering (ICSC) (5th : 2015)

Modeling subway risk assessment using fuzzy logic Abouhamad, Mona; Zayed, Tarek Jun 30, 2015

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5th International/11th Construction Specialty Conference 5e International/11e Conférence spécialisée sur la construction    Vancouver, British Columbia June 8 to June 10, 2015 / 8 juin au 10 juin 2015   MODELING SUBWAY RISK ASSESSMENT USING FUZZY LOGIC Mona. Abouhamad1, 2 and Tarek. Zayed1  1 Concordia University, Montreal, Canada 2 monabuhamd@yahoo.com Abstract: According to the Canadian Urban Transit Association (CUTA 2012), 140 Billion CAD is required to maintain, rehabilitate, and replace subway infrastructure between the years 2010 and 2014. However, transit authorities are faced by a fund scarcity problem which is hindering them from addressing all the network rehabilitation requirements in an efficient manner. The solution according to the 2013 America’s infrastructure report card is to adopt a comprehensive asset management system to maximize investments. This research develops a risk assessment model for subway stations. Probability of failure of different subway elements are developed using Weibull reliability curves. Consequences of failure are measured against three predefined attributes these are financial, operational, and social impacts of failure. Finally, a criticality index measures the respective station criticality derived from its particular size, location in proximity to different attraction types, and, nature of use. A qualitative approach with the help of expert judgment is adopted to integrate the indices using the Fuzzy Analytic Network Process with application to Fuzzy Preference Programming. The three models are integrated into a fuzzy rule based risk index model to compute element and station expected risk index. The output of the model is a comprehensive risk index that can be used to prioritize elements across stations for rehabilitation. The model is verified through an actual case study comparing elements across six stations and computing probability of failure, consequence of failure, criticality and the risk index. This paper illustrates the general framework of the proposed methodology which will help decision makers prioritize stations and elements across stations for rehabilitation based upon their risk index.  1 INTRODUCTION Subway systems failure is associated with consequences like multiple fatalities or injuries, partial or complete loss of service, major traffic disruptions, and, different socio-economic effects. A subway network is composed of diverse components and systems operating simultaneously to deliver the required service. The component diversity causes a level of complexity which complicates the process of assessing and maintaining the network at the desired level of service. In addition, the problem of fund scarcity faced by most public authorities converts it into a tough task. According to (Semaan 2011), “Société de Transport de Montréal” estimated a required amount of 5.1 Billion CAD for maintenance of its subway system for the next ten years. Different systems operating in a subway network compete for rehabilitation priorities while having various consequences of failure and multiple failure modes which turns the prioritization process into a tough task. Moreover, elements operating in a subway network pose diverse rehabilitation and maintenance needs based on their role in the network hierarchy and operation. The current method used for prioritizing subway stations is visual inspection. Hastak and Baim (2001) stated that in the subway stations context, inspections are used to identify the needed assessment for rehabilitation work. However, since no federal or state regulatory is used for inspections; the development and the implementation of the inspection standards in mainly the transit management responsibility 225-1 (Russel et al. 1997). This research aims at developing a risk assessment model on a network level based upon the visual inspection reports subway structural elements. This is a four-phased model in which sub-models for measuring the components of a risk equation, namely probability and consequence of failure, are first developed. The paper proceeds to suggest an addition to the classical risk equation to be better suited for the case of subway networks. The risk equation components are then combined using the fuzzy inference system. The following section presents a background for the current practices adopted in addition to the available researches. The background is followed by the methodology section in which the developed models are explained in details. Finally, a case study is presented to demonstrate how the model works and its validation. 1.1 Background 1.1.1 Subway Assessment Efforts The literature demonstrates research and industrial efforts to assess the condition of subway stations and rank stations for maintenance and rehabilitation. California transit authority developed an evaluation system for stations and ranked them on a scale from excellent to poor based on predefined criteria combined using a weighted average technique (Abu-Mallouh 1999). Whereas, Metropolitan Transit Authority of New York Transit developed a ranking system for condition assessment by assigning points to different functional factors (Abu-Mallouh 1999). London Transit developed the Key Performance Indicator to evaluate the performance of stations from its customers’ point of view using a direct evaluation of customer satisfaction through surveys and interviews. The Paris Rapid Transit Authority worked on developing a selection procedure for stations in need of rehabilitation, the model used a seven functional criteria selection procedure.  Subways domain was shyly researched in academia with only a handful models assessing subway stations. Abu-Mallouh (1999) developed a model to optimize the number of stations accommodated within a given capital program for full and partial rehabilitation. Semaan (2009) developed a condition assessment model to diagnose specific subway stations and assess their conditions using an index (0-10). In a corresponding effort, (Farran 2006) developed a model to address life cycle costing for a single infrastructure element with probabilistic and condition rating approach for condition state. And finally, (Semaan 2011) developed a model to evaluate structural performance of different components in a subway network using performance curves for components and the entire network using reliability-based cumulative Weibull function.  It is noted that the reported transit management practices adopted a qualitative functional perspective to inspect and prioritize subway stations for rehabilitation. On the other hand, the academia focused mainly on structural quantitative models through condition assessment and deterioration models. While these two perspectives of assessment are vital; none of the reported literature integrates the functional and structural aspects of a subway station into a single model.  1.2 Methodology The developed methodology aims at combining structural and functional perspectives of a subway network into a single risk assessment model. The structural integrity is assessed through a probability of failure sub-model whereas the functional perspective is assessed through the consequence of failure and criticality index sub-models. The output of the three sub-models is then integrated into a risk index model using 30 rules extracted from experts’ knowledge. This section starts by presenting the network hierarchy used through the analysis and proceeds with the sub-models and model development.  1.2.1 Subway Hierarchy A generic subway network hierarchy is presented in Figure 1. A typical subway line is composed of a number of station buildings. They operate by means of their composing systems such as electrical, mechanical, security and communication, and, structural. This research focuses only on the operational 225-2 risk failure derived from the structural systems in a network. Therefore, the structural system is identified as a composition of stations, tunnels and auxiliary structures. These are composed of the elements located at the lowest level of the hierarchy. This hierarchy will be the basis of calculations through model development and its associated sub-models.   LinesStation Building 1Station Building 2Station Building 3Station Building ...Mechanical Structural Electrical Station Building NSecurity and CommunicationTunnel Auxiliary Structure StationTop SlabsDomeWalls Top SlabsWallsBottom SlabInterior StairsInterior WallsBottom SlabBottom Slab Exterior StairsExterior Walls Figure 1: A Generic subway network hierarchy 1.2.2 Probability of Failure Sub-Model The Probability of Failure (PoF) sub-model builds upon the performance model developed by (Semaan 2011). Semaan (2011) used reliability-based cumulative Weibull function to evaluate the structural performance of different components in a subway network and develop performance curves for subway components and the entire network. Reliability-based cumulative Weibull function takes a probabilistic approach that yields a reliability index, which is the inverse of the PoF. Therefore, PoF can be estimated as the inverse of the reliability and is shown in Equation [1] [1]                   Where, R (T) = Reliability, t = Time,  = deterioration parameter, α = location parameter, τ= scale parameter, δ= and e = exponential.   Different system configuration requires different calculations for PoF values. The series-parallel reliability technique (Hillier and Lieberman 1972) entitles that any system is composed of components outlined in parallel, in series, or, in a combination of both. A system in parallel is a redundant system where components work simultaneously; hence, it can operate even if one of its components fails. This is the logic used to calculate the different PoF values. A subway network is composed of lines, stations, and auxiliary structures, the PoF is calculated for each system based on the configuration shown in Figure 2.    225-3 Station System (STA): In a subway station system, the slab and stairs are redundant systems and can be considered as a parallel system. The wall system is a series system in which if any wall “fails” to perform, the whole station becomes unsafe, and thus does not perform.  PoF of a station system can be computed using equation [2]  [2] PSTA = 1- [(1- )*(1- )*(1- )]    Where,  PSTAj = Probability of station j failure, = Probability of exterior stairs failure, = Probability of interior stairs failure, = Probability of external slab failure, = Probability of internal slab failure, = Probability of internal wall failure, = Probability of external wall failure, and, i=1, 2 … n = station floor.    Tunnel System (TUN): A tunnel system operates in series in which it fails if any of its components fail, therefore, PoF values are calculated using equation [3] [3] PTUN =1 -          Where;  PTUN = Probability of tunnel failure,  = Probability of Dome failure,  = Probability of wall failure,  = Probability of slab failure.   Auxiliary structures System (AS): These systems operate in series in which it fails if any of its components fail, therefore, PoF are calculated using equation [4] [4] P Aux St = 1 –  (1-   )          Where;  P Aux St = Probability of auxiliary structure failure,  = Probability of walls failure,  = Probability of top slab failure, and,   = Probability of bottom slab failure.  A Line System: is composed of all stations, tunnel, and auxiliary structure systems operating on the line. These systems together operate in series whereas; the composition of each system operates in parallel. The stations systems are redundant system, they operate in parallel and will fail to operate when all stations in a line fail. Likewise, a line failure occurs when all tunnels on the line fail to operate. Same applies for the auxiliary structure, operating is parallel in a line systems. On the other hand, the three systems operate in series. If any of the systems fails entirely that means the subway line is in a failure status and can no more function effectively. The line hierarchy is shown in Figure 2 (a) and is computed using equation [5] 225-4 [5] Pline i = 1 – [ * * ]      Where;  Pline = Probability of line failure, PSTA = Probability of station failure, PTUN = Probability of tunnel failure, PAux St = Probability of auxiliary structure failure, and i=1, 2 … n = number of systems in a line.   Subway Network; a subway network is composed of all the lines operating in the network. It can be concluded that the lines in a network operate in parallel. Hence, the network only fails when all the lines operating in the network fail. This can be computed using equation [6] and concluded from Figure 2 (b).  [6] PNet =             Where;  PNet = Probability of network failure, = Probability of line failure.  Station 1 Station 2 Station 3 Tunnel 1 Station n Tunnel 2 Tunnel3 Tunnel nAux Structure 1 Aux Structure  2 Aux Structure 3 Aux Structure  n Line 1 Line 2 Line 3 Line n  (a) Network Hierarchy    (b) Line Hierarchy Figure 2: Schematic diagrams for network and line hierarchy.  1.2.3 Consequences of Failure Sub-Model  A generic risk management system should identify PoF and Consequences of Failure (CoF) to be combined later to produce a representative risk index. A formal review of failure consequences diverts attention away from maintenance tasks having little or no effects and focuses on maintenance tasks that are more effective. This ensures the maintenance spending is optimized and guarantees the inherent reliability of equipment is enhanced (Gonzalez et al. 2006). Indirect impacts of failure of a subway station include, but are not limited to, service disruption, passenger delay, loss of reputation, loss of revenue in addition to other socio-economic impacts reflected as the extent to which the failure affects adjacent services and customers benefiting from the service and the ease of providing an alternative service. However, only a fraction of the expected CoF can be monetized whereas most of the expected indirect CoF are difficult to monetize and measure (Muhlbauer and W Kent. 2004). One way to overcome the difficulty inherent in these calculations is measuring CoF using indices, which facilitates comparing between expected CoF and highlights areas of higher failure impacts. This research determined factors affecting CoF calculations in terms of tangible and intangible impacts using the Triple Bottom line approach. This revealed a wide spectrum of consequences occurring at element and station levels. A station is composed of a number of elements operating simultaneously; based on the location of the element and its nature, the element failure might cause total, partial, or no station closure. This suggests CoF are element-dependent, Figure 3 outlines the CoF model. Based on 225-5 literature and expert opinion, CoF are broadly grouped into financial, social, and, operational impacts of failure. It is noted that some factors could fall under two different perspectives simultaneously.  Local and global weightsConsequence of failure (CFi)CFi = Cwi * SsiSeverity Score (Ssi)Financial Effects Social Effects Operational EffectsLiterature review Expert OpinionIdentify Subway station Consequences of failure Questionnaire SurveyConsequence scoresHistorical DataConsequence of Failure Weight (Cwi)Fuzzy ANP Expert OpinionInspection Reports Figure 3: Consequences of failure model outline. The defined impacts of failure along different categories are interdependent; hence, the effect of a single impact cannot be measured independently without considering how other impacts affect and are affected by its occurrence. Therefore, Fuzzy Analytical Network Process (FANP) is selected to obtain relative weights of these factors. FANP addresses the interdependency inherent in the relation between these factors and accounts for the uncertainty caused by using expert opinions. The reader is referred to (Abouhamad and Zayed 2013a) for further model details. For each subway element, the consequence of failure index (  is computed using equation [7] [7]              Where; = Consequence of Failure Index,  = Criteria weight obtained using questionnaire survey and FANP,  = Severity score calculated from network data and inspection reports, i= elements operating per station.   Financial impacts are twofold; repair/replacement cost defined as the direct cost of repair or replacement and loss of revenue defined as the profit loss due to service interruption. Operational impacts is measured in terms of ease of providing alternative and time to repair. The ease of providing alternative is 225-6 measured by means of available bus stops and reroutes in case of no service whereas, the time to repair is the time required to return the component to a full functioning state. The social impacts are measured by user traffic frequency, maximum allowable interruptions per year per station and the degree of service interruption whether partial, total or no interruption at all.  1.2.4 Criticality Index Model  This research introduces criticality for the scope of subway networks as the Criticality index. The subway network breakdown structure is assessed differently, the element is selected such that its criticality level is dominant and diverse enough to prevail over other network components. Consequently, subway stations are selected to be the focus of criticality analysis. Systems and subsystems share the same major role of delivering the service; however, their criticality is derived from their respective locations in stations that vary in criticality according to several factors. From this discussion, the concept of criticality propagation is introduced; criticality level propagates upwards and downwards in a hierarchy of a subway network such that they acquire the same criticality level as stations where they operate. Similarly, a line criticality is computed as the sum of criticality indices of stations existing on this line. For interconnecting systems such as tunnels and auxiliary structures, CR is computed as the higher index of the two corresponding stations through which this system connects.   Factors contributing to station CR are identified through historical data, expert opinion and by consulting current structure and map of several subway networks. Station criticality is a complex decision based on different attributes defined as; number of lines, number of levels, station use whether end or intermodal, and station proximity to different attraction locations. CR factors defining a station differ in significance, thus, a weight component is introduced to the CR equation to accommodate the subjective variability in attributes weight. Attribute scores are computed based upon the network under examination and individual station information. Station criticality is defined in terms of three main factors and seven sub factors or attributes. Amongst attributes identified, the station location is the most diverse. For further details about this model, the reader is referred to (Abouhamad and Zayed 2013b). Station criticality attributes are strongly connected, hence, cause and effects loops flow between them. Therefore, FANP with application to Fuzzy Preference Programming is used to compute the attributes weight. The Criticality Index model is outlined in Figure 4.    Criticality Index CiiCRi = CRwi * CRSiCriticality Score (CRsi)Station LocationStation Size Station Nature of useLiterature review Expert OpinionIdentify Station Criticality attributesQuestionnaire Subway MapStation DataCriticality Weight (CRwi)FANP Figure 4: Criticality Index model outline. Criticality Index per station (  is computed using equation [8] [8]            225-7 Where; CR= Criticality Index per station, = Criticality attributes weights calculated using questionnaire surveys and FANP,  = Criticality scores calculated using current network data, and i=1,2, …,n, n= criticality attributes 1.2.5 Risk Index Model  Risk by definition is a combination of PoF and the severity of adverse effects (Lowrance 1967). When studying the risk level, it should be noted that elements with similar PoF might show wide variation in terms of consequences of failure and vice versa. In addition, critical elements with high consequences of failure usually compose a smaller portion of the overall network. Accordingly, focusing only on these elements would result in an unbalanced management practices since unexpected failures may occur in less-critical elements, which constitute the majority of the network. Furthermore, a comprehensive risk assessment should consider the relative importance of different components and systems of a subway network. A criticality index is introduced to measure the relative importance and consider it in the risk index development. Consequently, a new term is added to the risk equation, named as the criticality index ( ). Several methods exist to compute the risk index value, ranging from simple straightforward multiplication to more sophisticated computation of risk matrix.   The Fuzzy Rule Based (FRB) technique was selected to compute the risk index in this research. This method permits users to integrate their experience into the decision support system through using “if-then” rules. Fuzzy sets allow for a more precise presentation of element’s membership particularly when it is difficult to determine the boundary of the set as crisp values. An FRB consists of a set of if-then rules defined over fuzzy sets (Masulli et al. 2007). The rules are usually created using “expert knowledge” (Castillo et al. 2008). The relationship between different fuzzy variables is represented by if-then rules of the form “If antecedent…… Then Consequent”. In cases where the antecedent has more than one part, the fuzzy operator is applied to obtain one number representing the consequence for the antecedents of that rule. This is the number used afterwards to obtain the output function. The Mamdani fuzzy inference system (Mamdani and Assilian 1975) uses the min-max composition as defined in equation [9] [9]        Where; , ,  are the membership functions for output “z” for rule “k”,X and y are inputs. Whereas in our case, the antecedent and the consequent are fuzzy propositions. The proposed model is performed using MATLAB® fuzzy logic toolbox. Mamdani algorithm based on experts’ knowledge is used to construct the rule base. The model combines PoF, CoF, and CR expressed as triangular membership functions. The min-max composition is used whereas the defuzzification was done using the Centre of Area method. The fuzzy risk equation solves equation [10] and is shown in equation [11]   [10] Risk Index = Probability of Failure * Consequence of Failure * Criticality Index   [11] Ri: IF PoF is Xi and CoF is Yi and CR is Zi then Risk is Li        Where, i= 1, 2, 3 ….k , Xi. Yi, Zi, and Li are linguistic constants as defined in model, k = number of rules  225-8 The threshold for risk values are set based on the maximum allowable PoF and CoF values. This eliminates the major drawback of a risk matrix in differentiating between the two extreme cases of high PoF with low CoF and vice versa. It also ensures the highest priority is given to elements with most emerging rehabilitation need whether derived from high PoF or high CoF. Based upon feedback from experts, CoF is categorized into three levels based upon the combined effect of failure on financial, social, and operational levels. Criticality serves to define stations into normal stations with moderate importance and critical stations with higher criticality. All data incorporated in the risk index calculations is reserved for a detailed analysis of each station. The membership functions were selected based on literature review and unstructured interview with subway experts. A set of 30 rules (5 rules for PoF, 3 for CoF and 2 for CR) is generated to develop the Risk Index.  1.3 Case Study and Model Implementation An actual case study was conducted on a sub network in Montreal metro to validate the model and proof its robustness. Montreal subway is one of the oldest networks in North America, with 68 stations spreading on four lines and covering the north, east, and centre of the Island of Montreal. Six stations (SEG 1 to SEG 6) on three different lines are analysed in the model with one station being the interconnecting station. SEG 1 to SEG 3 fall on the same line given the name Line A, SEG 4 and SEG5 both fall on the second line B. SEG 6 falls on line C whereas, SEG 2 is the interconnecting station for the three lines. Stations were selected from literature review (Semaan 2011) and based upon availability of inspection reports for different indices calculations. 1.3.1 Sub-Models output PoF is calculated using year 2014 as the base for calculations. The subway system hierarchy together with the equations presented earlier were used to compute PoF values for elements at the lowest level of the hierarchy then aggregated upwards to compute the integrated PoF values for stations, tunnels and auxiliary structures, identified as a segment (SEG).Sample output PoF values are illustrated in Figure 5.  A questionnaire survey was launched to gather the required data for CoF and Criticality models’ development. The questionnaire conducted pairwise comparisons between attributes, sub-attributes and goals for each of the two sub-models. It also contained open ended questions for experts to provide their opinion on model development and suggest any required modifications. The output of the questionnaires are local and global weights for attributes in CoF and CR models. Further details about the resultant weights can be found in (Abouhamad and Zayed 2014). FANP calculations are done using MATLAB® software and FPP as a prioritization tool. Scores for CoF attributes were obtained from literature review (Farran, 2006) and current information of Montreal subway. Sample calculations for CoF index are seen in Figure 5. As stated earlier, CoF are calculated for elements at the lower level of the hierarchy then aggregated upwards. It is evident that CoF are highly affected by PoF value for each system since all the factors accounted for in the model are directly proportional with PoF value. Calculations for CR were done for the entire Montréal subway network (68 stations). Two stations with maximum and minimum criticality levels were set as thresholds for normalizing the index for the six stations under study. Criticality index is defined as the functional role a station plays and thus is calculated on stations level. A tunnel criticality index is taken as the higher value for the two connecting stations, while auxiliary structures acquire the criticality index of corresponding subway station. This explains the constant CR value per segment as seen in Figure 5. Unlike PoF and Cof where values are upwards aggregated, for an element level analysis, CR values for a given element are the same as the station where it operates.   225-9 Abouhamad, Mona and Zayed, Tarek. 2013a. Multiple perspective consequence of failure estimation of subway stations, Canadian Society of Civil Engineers (CSCE), 4th Construction Specialty Conference. Montreal, Canada. Abouhamad, Mona and Zayed, Tarek . 2013b, Criticality-based model for rehabilitationg subway stations. 30th International Symposium of Automation and Robotics in Construction and Mining (ISARC 2013). Montreal, Canada. Abu-Mallouh, M., 1999. Model for station rehabilitation and planning (MSRP), PhD Dissertation,  Polytechnic University, Civil Engineering, USA.  Castillo, Oscar and Melin, Patricia. 2008, Type-2 Fuzzy Logic: Theory and Applications. Springer Publishing Company, Incorporated. Canadian Urban Transit Association, CUTA. 2012. Transit Infrastructure Needs for the Period 2012-2016, < http://www.cutaactu.ca/en/index.asp> (March 27, 2012). Farran, Mazen. 2006. Life cycle cost for rehabilitation of public infrastructures:application to Montreal metro system. Master Dissertation. Concordia University, Montreal, Canada.  González, Javier, Rosario Romera, Jesús Carretero Pérez, and José M. Pérez. 2006. Optimal railway infrastructure maintenance and repair policies to manage risk under uncertainty with adaptive control. Working Papers. Statistics and Econometrics Series, Vol. 05. Hastak, M. and Baim, E. 2001. Risk Factors Affecting Management and Maintenance Cost of Urban Infrastructure. Journal of Infrastructure Systems, 7(2), 67–76. Hillier, F.S. and Lieberman, G.J. 1972. Introduction to Operation Research. Holden-Day. Lowrance, W. 1967. Of acceptable risk: Science and the determination of safety. Los Altos: William Kaufman Inc. Mamdani, E. H., & Assilian, S.1975. An experiment in linguistic synthesis with a fuzzy logic controller. International Journal of Man Machine Studies, 7(1), 1-13. Masulli, Francesco, Sushmita Mitra and Pasi, Gabriella. 2007,  Applications of Fuzzy Sets Theory. 7th International Workshop on Fuzzy Logic and Applications. Camogli, Italy ,Vol. 4578. Springer, Muhlbauer, W Kent. 2004. Pipeline risk management manual: ideas, techniques, and resources. Gulf Professional Publishing.  Russel, H., Gilmore, J., and TCRP. 1997. Inspection policy and procedures for rail transit tunnels and underground structures—Synthesis of transit practice 23, National Research Council, Washington, D.C. Semaan, Nabil. 2006. Subway station diagnosis index (SSDI) : a condition assessment model. Master Dissertation. Concordia University, Montreal, Canada.  Semaan, Nabil. 2011. Structural Performance Model for Subway Network. PhD Dissertation, Concordia University, Montreal, Canada.    225-12  5th International/11th Construction Specialty Conference 5e International/11e Conférence spécialisée sur la construction    Vancouver, British Columbia June 8 to June 10, 2015 / 8 juin au 10 juin 2015   MODELING SUBWAY RISK ASSESSMENT USING FUZZY LOGIC Mona. Abouhamad1, 2 and Tarek. Zayed1  1 Concordia University, Montreal, Canada 2 monabuhamd@yahoo.com Abstract: According to the Canadian Urban Transit Association (CUTA 2012), 140 Billion CAD is required to maintain, rehabilitate, and replace subway infrastructure between the years 2010 and 2014. However, transit authorities are faced by a fund scarcity problem which is hindering them from addressing all the network rehabilitation requirements in an efficient manner. The solution according to the 2013 America’s infrastructure report card is to adopt a comprehensive asset management system to maximize investments. This research develops a risk assessment model for subway stations. Probability of failure of different subway elements are developed using Weibull reliability curves. Consequences of failure are measured against three predefined attributes these are financial, operational, and social impacts of failure. Finally, a criticality index measures the respective station criticality derived from its particular size, location in proximity to different attraction types, and, nature of use. A qualitative approach with the help of expert judgment is adopted to integrate the indices using the Fuzzy Analytic Network Process with application to Fuzzy Preference Programming. The three models are integrated into a fuzzy rule based risk index model to compute element and station expected risk index. The output of the model is a comprehensive risk index that can be used to prioritize elements across stations for rehabilitation. The model is verified through an actual case study comparing elements across six stations and computing probability of failure, consequence of failure, criticality and the risk index. This paper illustrates the general framework of the proposed methodology which will help decision makers prioritize stations and elements across stations for rehabilitation based upon their risk index.  1 INTRODUCTION Subway systems failure is associated with consequences like multiple fatalities or injuries, partial or complete loss of service, major traffic disruptions, and, different socio-economic effects. A subway network is composed of diverse components and systems operating simultaneously to deliver the required service. The component diversity causes a level of complexity which complicates the process of assessing and maintaining the network at the desired level of service. In addition, the problem of fund scarcity faced by most public authorities converts it into a tough task. According to (Semaan 2011), “Société de Transport de Montréal” estimated a required amount of 5.1 Billion CAD for maintenance of its subway system for the next ten years. Different systems operating in a subway network compete for rehabilitation priorities while having various consequences of failure and multiple failure modes which turns the prioritization process into a tough task. Moreover, elements operating in a subway network pose diverse rehabilitation and maintenance needs based on their role in the network hierarchy and operation. The current method used for prioritizing subway stations is visual inspection. Hastak and Baim (2001) stated that in the subway stations context, inspections are used to identify the needed assessment for rehabilitation work. However, since no federal or state regulatory is used for inspections; the development and the implementation of the inspection standards in mainly the transit management responsibility 225-1 (Russel et al. 1997). This research aims at developing a risk assessment model on a network level based upon the visual inspection reports subway structural elements. This is a four-phased model in which sub-models for measuring the components of a risk equation, namely probability and consequence of failure, are first developed. The paper proceeds to suggest an addition to the classical risk equation to be better suited for the case of subway networks. The risk equation components are then combined using the fuzzy inference system. The following section presents a background for the current practices adopted in addition to the available researches. The background is followed by the methodology section in which the developed models are explained in details. Finally, a case study is presented to demonstrate how the model works and its validation. 1.1 Background 1.1.1 Subway Assessment Efforts The literature demonstrates research and industrial efforts to assess the condition of subway stations and rank stations for maintenance and rehabilitation. California transit authority developed an evaluation system for stations and ranked them on a scale from excellent to poor based on predefined criteria combined using a weighted average technique (Abu-Mallouh 1999). Whereas, Metropolitan Transit Authority of New York Transit developed a ranking system for condition assessment by assigning points to different functional factors (Abu-Mallouh 1999). London Transit developed the Key Performance Indicator to evaluate the performance of stations from its customers’ point of view using a direct evaluation of customer satisfaction through surveys and interviews. The Paris Rapid Transit Authority worked on developing a selection procedure for stations in need of rehabilitation, the model used a seven functional criteria selection procedure.  Subways domain was shyly researched in academia with only a handful models assessing subway stations. Abu-Mallouh (1999) developed a model to optimize the number of stations accommodated within a given capital program for full and partial rehabilitation. Semaan (2009) developed a condition assessment model to diagnose specific subway stations and assess their conditions using an index (0-10). In a corresponding effort, (Farran 2006) developed a model to address life cycle costing for a single infrastructure element with probabilistic and condition rating approach for condition state. And finally, (Semaan 2011) developed a model to evaluate structural performance of different components in a subway network using performance curves for components and the entire network using reliability-based cumulative Weibull function.  It is noted that the reported transit management practices adopted a qualitative functional perspective to inspect and prioritize subway stations for rehabilitation. On the other hand, the academia focused mainly on structural quantitative models through condition assessment and deterioration models. While these two perspectives of assessment are vital; none of the reported literature integrates the functional and structural aspects of a subway station into a single model.  1.2 Methodology The developed methodology aims at combining structural and functional perspectives of a subway network into a single risk assessment model. The structural integrity is assessed through a probability of failure sub-model whereas the functional perspective is assessed through the consequence of failure and criticality index sub-models. The output of the three sub-models is then integrated into a risk index model using 30 rules extracted from experts’ knowledge. This section starts by presenting the network hierarchy used through the analysis and proceeds with the sub-models and model development.  1.2.1 Subway Hierarchy A generic subway network hierarchy is presented in Figure 1. A typical subway line is composed of a number of station buildings. They operate by means of their composing systems such as electrical, mechanical, security and communication, and, structural. This research focuses only on the operational 225-2 risk failure derived from the structural systems in a network. Therefore, the structural system is identified as a composition of stations, tunnels and auxiliary structures. These are composed of the elements located at the lowest level of the hierarchy. This hierarchy will be the basis of calculations through model development and its associated sub-models.   LinesStation Building 1Station Building 2Station Building 3Station Building ...Mechanical Structural Electrical Station Building NSecurity and CommunicationTunnel Auxiliary Structure StationTop SlabsDomeWalls Top SlabsWallsBottom SlabInterior StairsInterior WallsBottom SlabBottom Slab Exterior StairsExterior Walls Figure 1: A Generic subway network hierarchy 1.2.2 Probability of Failure Sub-Model The Probability of Failure (PoF) sub-model builds upon the performance model developed by (Semaan 2011). Semaan (2011) used reliability-based cumulative Weibull function to evaluate the structural performance of different components in a subway network and develop performance curves for subway components and the entire network. Reliability-based cumulative Weibull function takes a probabilistic approach that yields a reliability index, which is the inverse of the PoF. Therefore, PoF can be estimated as the inverse of the reliability and is shown in Equation [1] [1]                   Where, R (T) = Reliability, t = Time,  = deterioration parameter, α = location parameter, τ= scale parameter, δ= and e = exponential.   Different system configuration requires different calculations for PoF values. The series-parallel reliability technique (Hillier and Lieberman 1972) entitles that any system is composed of components outlined in parallel, in series, or, in a combination of both. A system in parallel is a redundant system where components work simultaneously; hence, it can operate even if one of its components fails. This is the logic used to calculate the different PoF values. A subway network is composed of lines, stations, and auxiliary structures, the PoF is calculated for each system based on the configuration shown in Figure 2.    225-3 Station System (STA): In a subway station system, the slab and stairs are redundant systems and can be considered as a parallel system. The wall system is a series system in which if any wall “fails” to perform, the whole station becomes unsafe, and thus does not perform.  PoF of a station system can be computed using equation [2]  [2] PSTA = 1- [(1- )*(1- )*(1- )]    Where,  PSTAj = Probability of station j failure, = Probability of exterior stairs failure, = Probability of interior stairs failure, = Probability of external slab failure, = Probability of internal slab failure, = Probability of internal wall failure, = Probability of external wall failure, and, i=1, 2 … n = station floor.    Tunnel System (TUN): A tunnel system operates in series in which it fails if any of its components fail, therefore, PoF values are calculated using equation [3] [3] PTUN =1 -          Where;  PTUN = Probability of tunnel failure,  = Probability of Dome failure,  = Probability of wall failure,  = Probability of slab failure.   Auxiliary structures System (AS): These systems operate in series in which it fails if any of its components fail, therefore, PoF are calculated using equation [4] [4] P Aux St = 1 –  (1-   )          Where;  P Aux St = Probability of auxiliary structure failure,  = Probability of walls failure,  = Probability of top slab failure, and,   = Probability of bottom slab failure.  A Line System: is composed of all stations, tunnel, and auxiliary structure systems operating on the line. These systems together operate in series whereas; the composition of each system operates in parallel. The stations systems are redundant system, they operate in parallel and will fail to operate when all stations in a line fail. Likewise, a line failure occurs when all tunnels on the line fail to operate. Same applies for the auxiliary structure, operating is parallel in a line systems. On the other hand, the three systems operate in series. If any of the systems fails entirely that means the subway line is in a failure status and can no more function effectively. The line hierarchy is shown in Figure 2 (a) and is computed using equation [5] 225-4 [5] Pline i = 1 – [ * * ]      Where;  Pline = Probability of line failure, PSTA = Probability of station failure, PTUN = Probability of tunnel failure, PAux St = Probability of auxiliary structure failure, and i=1, 2 … n = number of systems in a line.   Subway Network; a subway network is composed of all the lines operating in the network. It can be concluded that the lines in a network operate in parallel. Hence, the network only fails when all the lines operating in the network fail. This can be computed using equation [6] and concluded from Figure 2 (b).  [6] PNet =             Where;  PNet = Probability of network failure, = Probability of line failure.  Station 1 Station 2 Station 3 Tunnel 1 Station n Tunnel 2 Tunnel3 Tunnel nAux Structure 1 Aux Structure  2 Aux Structure 3 Aux Structure  n Line 1 Line 2 Line 3 Line n  (a) Network Hierarchy    (b) Line Hierarchy Figure 2: Schematic diagrams for network and line hierarchy.  1.2.3 Consequences of Failure Sub-Model  A generic risk management system should identify PoF and Consequences of Failure (CoF) to be combined later to produce a representative risk index. A formal review of failure consequences diverts attention away from maintenance tasks having little or no effects and focuses on maintenance tasks that are more effective. This ensures the maintenance spending is optimized and guarantees the inherent reliability of equipment is enhanced (Gonzalez et al. 2006). Indirect impacts of failure of a subway station include, but are not limited to, service disruption, passenger delay, loss of reputation, loss of revenue in addition to other socio-economic impacts reflected as the extent to which the failure affects adjacent services and customers benefiting from the service and the ease of providing an alternative service. However, only a fraction of the expected CoF can be monetized whereas most of the expected indirect CoF are difficult to monetize and measure (Muhlbauer and W Kent. 2004). One way to overcome the difficulty inherent in these calculations is measuring CoF using indices, which facilitates comparing between expected CoF and highlights areas of higher failure impacts. This research determined factors affecting CoF calculations in terms of tangible and intangible impacts using the Triple Bottom line approach. This revealed a wide spectrum of consequences occurring at element and station levels. A station is composed of a number of elements operating simultaneously; based on the location of the element and its nature, the element failure might cause total, partial, or no station closure. This suggests CoF are element-dependent, Figure 3 outlines the CoF model. Based on 225-5 literature and expert opinion, CoF are broadly grouped into financial, social, and, operational impacts of failure. It is noted that some factors could fall under two different perspectives simultaneously.  Local and global weightsConsequence of failure (CFi)CFi = Cwi * SsiSeverity Score (Ssi)Financial Effects Social Effects Operational EffectsLiterature review Expert OpinionIdentify Subway station Consequences of failure Questionnaire SurveyConsequence scoresHistorical DataConsequence of Failure Weight (Cwi)Fuzzy ANP Expert OpinionInspection Reports Figure 3: Consequences of failure model outline. The defined impacts of failure along different categories are interdependent; hence, the effect of a single impact cannot be measured independently without considering how other impacts affect and are affected by its occurrence. Therefore, Fuzzy Analytical Network Process (FANP) is selected to obtain relative weights of these factors. FANP addresses the interdependency inherent in the relation between these factors and accounts for the uncertainty caused by using expert opinions. The reader is referred to (Abouhamad and Zayed 2013a) for further model details. For each subway element, the consequence of failure index (  is computed using equation [7] [7]              Where; = Consequence of Failure Index,  = Criteria weight obtained using questionnaire survey and FANP,  = Severity score calculated from network data and inspection reports, i= elements operating per station.   Financial impacts are twofold; repair/replacement cost defined as the direct cost of repair or replacement and loss of revenue defined as the profit loss due to service interruption. Operational impacts is measured in terms of ease of providing alternative and time to repair. The ease of providing alternative is 225-6 measured by means of available bus stops and reroutes in case of no service whereas, the time to repair is the time required to return the component to a full functioning state. The social impacts are measured by user traffic frequency, maximum allowable interruptions per year per station and the degree of service interruption whether partial, total or no interruption at all.  1.2.4 Criticality Index Model  This research introduces criticality for the scope of subway networks as the Criticality index. The subway network breakdown structure is assessed differently, the element is selected such that its criticality level is dominant and diverse enough to prevail over other network components. Consequently, subway stations are selected to be the focus of criticality analysis. Systems and subsystems share the same major role of delivering the service; however, their criticality is derived from their respective locations in stations that vary in criticality according to several factors. From this discussion, the concept of criticality propagation is introduced; criticality level propagates upwards and downwards in a hierarchy of a subway network such that they acquire the same criticality level as stations where they operate. Similarly, a line criticality is computed as the sum of criticality indices of stations existing on this line. For interconnecting systems such as tunnels and auxiliary structures, CR is computed as the higher index of the two corresponding stations through which this system connects.   Factors contributing to station CR are identified through historical data, expert opinion and by consulting current structure and map of several subway networks. Station criticality is a complex decision based on different attributes defined as; number of lines, number of levels, station use whether end or intermodal, and station proximity to different attraction locations. CR factors defining a station differ in significance, thus, a weight component is introduced to the CR equation to accommodate the subjective variability in attributes weight. Attribute scores are computed based upon the network under examination and individual station information. Station criticality is defined in terms of three main factors and seven sub factors or attributes. Amongst attributes identified, the station location is the most diverse. For further details about this model, the reader is referred to (Abouhamad and Zayed 2013b). Station criticality attributes are strongly connected, hence, cause and effects loops flow between them. Therefore, FANP with application to Fuzzy Preference Programming is used to compute the attributes weight. The Criticality Index model is outlined in Figure 4.    Criticality Index CiiCRi = CRwi * CRSiCriticality Score (CRsi)Station LocationStation Size Station Nature of useLiterature review Expert OpinionIdentify Station Criticality attributesQuestionnaire Subway MapStation DataCriticality Weight (CRwi)FANP Figure 4: Criticality Index model outline. Criticality Index per station (  is computed using equation [8] [8]            225-7 Where; CR= Criticality Index per station, = Criticality attributes weights calculated using questionnaire surveys and FANP,  = Criticality scores calculated using current network data, and i=1,2, …,n, n= criticality attributes 1.2.5 Risk Index Model  Risk by definition is a combination of PoF and the severity of adverse effects (Lowrance 1967). When studying the risk level, it should be noted that elements with similar PoF might show wide variation in terms of consequences of failure and vice versa. In addition, critical elements with high consequences of failure usually compose a smaller portion of the overall network. Accordingly, focusing only on these elements would result in an unbalanced management practices since unexpected failures may occur in less-critical elements, which constitute the majority of the network. Furthermore, a comprehensive risk assessment should consider the relative importance of different components and systems of a subway network. A criticality index is introduced to measure the relative importance and consider it in the risk index development. Consequently, a new term is added to the risk equation, named as the criticality index ( ). Several methods exist to compute the risk index value, ranging from simple straightforward multiplication to more sophisticated computation of risk matrix.   The Fuzzy Rule Based (FRB) technique was selected to compute the risk index in this research. This method permits users to integrate their experience into the decision support system through using “if-then” rules. Fuzzy sets allow for a more precise presentation of element’s membership particularly when it is difficult to determine the boundary of the set as crisp values. An FRB consists of a set of if-then rules defined over fuzzy sets (Masulli et al. 2007). The rules are usually created using “expert knowledge” (Castillo et al. 2008). The relationship between different fuzzy variables is represented by if-then rules of the form “If antecedent…… Then Consequent”. In cases where the antecedent has more than one part, the fuzzy operator is applied to obtain one number representing the consequence for the antecedents of that rule. This is the number used afterwards to obtain the output function. The Mamdani fuzzy inference system (Mamdani and Assilian 1975) uses the min-max composition as defined in equation [9] [9]        Where; , ,  are the membership functions for output “z” for rule “k”,X and y are inputs. Whereas in our case, the antecedent and the consequent are fuzzy propositions. The proposed model is performed using MATLAB® fuzzy logic toolbox. Mamdani algorithm based on experts’ knowledge is used to construct the rule base. The model combines PoF, CoF, and CR expressed as triangular membership functions. The min-max composition is used whereas the defuzzification was done using the Centre of Area method. The fuzzy risk equation solves equation [10] and is shown in equation [11]   [10] Risk Index = Probability of Failure * Consequence of Failure * Criticality Index   [11] Ri: IF PoF is Xi and CoF is Yi and CR is Zi then Risk is Li        Where, i= 1, 2, 3 ….k , Xi. Yi, Zi, and Li are linguistic constants as defined in model, k = number of rules  225-8 The threshold for risk values are set based on the maximum allowable PoF and CoF values. This eliminates the major drawback of a risk matrix in differentiating between the two extreme cases of high PoF with low CoF and vice versa. It also ensures the highest priority is given to elements with most emerging rehabilitation need whether derived from high PoF or high CoF. Based upon feedback from experts, CoF is categorized into three levels based upon the combined effect of failure on financial, social, and operational levels. Criticality serves to define stations into normal stations with moderate importance and critical stations with higher criticality. All data incorporated in the risk index calculations is reserved for a detailed analysis of each station. The membership functions were selected based on literature review and unstructured interview with subway experts. A set of 30 rules (5 rules for PoF, 3 for CoF and 2 for CR) is generated to develop the Risk Index.  1.3 Case Study and Model Implementation An actual case study was conducted on a sub network in Montreal metro to validate the model and proof its robustness. Montreal subway is one of the oldest networks in North America, with 68 stations spreading on four lines and covering the north, east, and centre of the Island of Montreal. Six stations (SEG 1 to SEG 6) on three different lines are analysed in the model with one station being the interconnecting station. SEG 1 to SEG 3 fall on the same line given the name Line A, SEG 4 and SEG5 both fall on the second line B. SEG 6 falls on line C whereas, SEG 2 is the interconnecting station for the three lines. Stations were selected from literature review (Semaan 2011) and based upon availability of inspection reports for different indices calculations. 1.3.1 Sub-Models output PoF is calculated using year 2014 as the base for calculations. The subway system hierarchy together with the equations presented earlier were used to compute PoF values for elements at the lowest level of the hierarchy then aggregated upwards to compute the integrated PoF values for stations, tunnels and auxiliary structures, identified as a segment (SEG).Sample output PoF values are illustrated in Figure 5.  A questionnaire survey was launched to gather the required data for CoF and Criticality models’ development. The questionnaire conducted pairwise comparisons between attributes, sub-attributes and goals for each of the two sub-models. It also contained open ended questions for experts to provide their opinion on model development and suggest any required modifications. The output of the questionnaires are local and global weights for attributes in CoF and CR models. Further details about the resultant weights can be found in (Abouhamad and Zayed 2014). FANP calculations are done using MATLAB® software and FPP as a prioritization tool. Scores for CoF attributes were obtained from literature review (Farran, 2006) and current information of Montreal subway. Sample calculations for CoF index are seen in Figure 5. As stated earlier, CoF are calculated for elements at the lower level of the hierarchy then aggregated upwards. It is evident that CoF are highly affected by PoF value for each system since all the factors accounted for in the model are directly proportional with PoF value. Calculations for CR were done for the entire Montréal subway network (68 stations). Two stations with maximum and minimum criticality levels were set as thresholds for normalizing the index for the six stations under study. Criticality index is defined as the functional role a station plays and thus is calculated on stations level. A tunnel criticality index is taken as the higher value for the two connecting stations, while auxiliary structures acquire the criticality index of corresponding subway station. This explains the constant CR value per segment as seen in Figure 5. Unlike PoF and Cof where values are upwards aggregated, for an element level analysis, CR values for a given element are the same as the station where it operates.   225-9 Abouhamad, Mona and Zayed, Tarek. 2013a. Multiple perspective consequence of failure estimation of subway stations, Canadian Society of Civil Engineers (CSCE), 4th Construction Specialty Conference. Montreal, Canada. Abouhamad, Mona and Zayed, Tarek . 2013b, Criticality-based model for rehabilitationg subway stations. 30th International Symposium of Automation and Robotics in Construction and Mining (ISARC 2013). Montreal, Canada. Abu-Mallouh, M., 1999. Model for station rehabilitation and planning (MSRP), PhD Dissertation,  Polytechnic University, Civil Engineering, USA.  Castillo, Oscar and Melin, Patricia. 2008, Type-2 Fuzzy Logic: Theory and Applications. Springer Publishing Company, Incorporated. Canadian Urban Transit Association, CUTA. 2012. Transit Infrastructure Needs for the Period 2012-2016, < http://www.cutaactu.ca/en/index.asp> (March 27, 2012). Farran, Mazen. 2006. Life cycle cost for rehabilitation of public infrastructures:application to Montreal metro system. Master Dissertation. Concordia University, Montreal, Canada.  González, Javier, Rosario Romera, Jesús Carretero Pérez, and José M. Pérez. 2006. Optimal railway infrastructure maintenance and repair policies to manage risk under uncertainty with adaptive control. Working Papers. Statistics and Econometrics Series, Vol. 05. Hastak, M. and Baim, E. 2001. Risk Factors Affecting Management and Maintenance Cost of Urban Infrastructure. Journal of Infrastructure Systems, 7(2), 67–76. Hillier, F.S. and Lieberman, G.J. 1972. Introduction to Operation Research. Holden-Day. Lowrance, W. 1967. Of acceptable risk: Science and the determination of safety. Los Altos: William Kaufman Inc. Mamdani, E. H., & Assilian, S.1975. An experiment in linguistic synthesis with a fuzzy logic controller. International Journal of Man Machine Studies, 7(1), 1-13. Masulli, Francesco, Sushmita Mitra and Pasi, Gabriella. 2007,  Applications of Fuzzy Sets Theory. 7th International Workshop on Fuzzy Logic and Applications. Camogli, Italy ,Vol. 4578. Springer, Muhlbauer, W Kent. 2004. Pipeline risk management manual: ideas, techniques, and resources. Gulf Professional Publishing.  Russel, H., Gilmore, J., and TCRP. 1997. Inspection policy and procedures for rail transit tunnels and underground structures—Synthesis of transit practice 23, National Research Council, Washington, D.C. Semaan, Nabil. 2006. Subway station diagnosis index (SSDI) : a condition assessment model. Master Dissertation. Concordia University, Montreal, Canada.  Semaan, Nabil. 2011. Structural Performance Model for Subway Network. PhD Dissertation, Concordia University, Montreal, Canada.    225-12  5-level evaluation, 9 functional Criteria , 1“Physical and Structural” condition.  SUBWAY CURRENT PRACTICES  Literature review 2 Reno-Station I & II. Structural and architectural renovations.  Age and Visual inspection only.  KPI (0-10) evaluation scale based upon 23 items. From customer point of view . Ranking system based on 11 weighted factors (main 3). Structural evaluation based on field inspection. Constant weights per station.  Used 7 criteria selection procedure. •  Platform users •  Visual aspect of station •  Environment Daily rider satisfaction. Overview 3 4 State of Good repair = 2.5/5  Adopt comprehensive asset management systems to maximize investments Overview SUBWAY NETWORK EFFORTS  Literature review 5 • Improved MTA NYCT model & developed MSRP model • Used IP and AHP to optimize fund allocation . Abu-Mallouh (1999)  • Subway Station Diagnosis Index. • Diagnose subway station and assess its condition state using an index. Semaan (2006) • Developed M&RPPI model.  • Select the optimum rehabilitation action based on (LCC) for a single infrastructure element. Farran (2006)  • Developed SUPER model • Evaluates structural performance of different components using performance curves.  Semaan (2011)  Assess subway components Functional + Structural perspectives Network Level Easily adopted RESEARCH LIMITATIONS Problem Statement  6 Functional Perspective	Asset Level Condition Assessment Deterioration Models Asset/network level Risk Index NoProbability of Failure model Consequence of failure modelCriticality index  modelFuzzy Risk Index Model Research techniquesSubway networks assessmentStructural performance model Criticality factorsKnowledge AcquisitionModel development and AnalysisLiterature ReviewSubway Network HierarchyInfrastructure assets performance modelsBudget allocationRisk-based asset managementMaintenance optionsCost OptionsRisk IndexRisk Model DevelopmentOptimization Model DevelopmentRisk-Based Budget Allocation Model Failure consequencesModel Validated?YesProbability of Failure Sub-Model NETWORK HIERARCHY Probability of Failure Model  9 Probability of Failure Model  Probability of Failure Model 10 STATIONS Probability of failure Model 11 Str Ex Str Int Slb Ex Slb Int Wall Ex Wall Int. PSTAj = 1- [(1- ∏𝑖=1↑𝑛▒​𝑃↓𝑆𝑇𝐸𝑖  ​𝑃↓𝑆𝑇𝐼𝑖 )*(1- ∏𝑖=1↑𝑛▒​𝑃↓𝑆𝐸𝑖  ​𝑃↓𝑆𝐼𝑖 )*(1- ∏𝑖=1↑𝑛▒(1− ​𝑃↓𝑊𝐼𝑖 )(1− ​𝑃↓𝑊𝐸𝑖 ) )] TUNNELS Probability of failure Model 12 PTUN = 1 - (1− ​𝑃↓𝐷 )∗(1− ​𝑃↓𝑤 )∗(1− ​𝑃↓𝑠 )  AUXILIARY STRUCTURE Probability of failure Model 13 Top Sl Btm Slb Wall. PAux St = 1 – (1− ​𝑃↓𝑤 ) (1- ​𝑃↓𝑇𝑆  ∗ ​𝑃↓𝐵𝑆 )	LINES SYSTEMS Probability of failure Model 14 Tunnel 1 Tunnel n Station 1 Station n Aux Str 1 Aux Str n Tunnel .. Station .. Aux Str .. Pline z = 1 – [(1−∏𝑖=1↑𝑛▒​​P↓STA ↓𝑖  )*(1−∏𝑖=1↑𝑛▒​​P↓TUN ↓𝑖  )*(1−∏𝑖=1↑𝑛▒​​P↓AUX ↓𝑖  )] NETWORK Probability of failure Model 15 PNet = ∏𝑖=1↑𝑛▒​𝑃↓𝐿𝑖𝑛𝑒𝑖  	Consequence of Failure Sub-Model NETWORK HIERARCHY Consequences of failure Model  17 Consequences of failure Financial Impacts Social Impacts Operational Impacts  Revenue Loss Replacement/repair cost Service continuation Interruption Rate User traffic frequency Time to repair  Ease of providing alternative 18 §  Introduction 19          IMPACTS WEIGHTS   Consequence of failure Model 20        0.00 0.10 0.20 0.30 0.40 0.50 0.60 1. Financial Impacts 2. Social Impacts 3. Operational Impacts F1:Revenue Loss F2: Replacement repair cost S1: Service continuation S2: Interruption rate S3: User traffic frequency O1: Ease of Providing Alternative O2: Time to repair Global weight Local weight Criticality Index 22 Ridership 700,000 12 M Exits 1 9 # Lines Depth  4,3 m 29, 6 m 68 	End Station Intermodal Proximity NETWORK HIERARCHY Criticality Index Model 23 Criticality Index Model 24 Criticality Index Characteristics (C) Station Location (L) Nature of use (N) C1: Number of Lines C2: Number of exits N2: End Station N1: Intermodal Station Criticality Index Model 25 Attraction type Points of Interest Grouping Main Touristic Attractions Museums, Theatres, Centre Infotouriste, Old Montreal and Old Port, Parks, Historical Sites, Squares, Malls Recreational  Sports Arenas, Stadium, Clubs Culture China Town, Cinemas, Libraries, Cemetery Transportation Central Bus Station, inter-city rail station Businesses Commerce Chambers, Quartier International de Montréal Vitalities  Worship Places Churches, Mosques, Temples, Cathedral, Oratory Educational Schools, Universities, Colleges Governmental City Hall, Court Health Care Hospitals, CLSC’s, Health Institutes Residence  Areas of high, medium, and low residence  Residence Criticality Index Model 26 Criticality Index Characteristics (C) Station Location (L) Nature of use (N) C1: Number of Lines C2: Number of exits L3: Residence Locations L2: Vital Locations L1: Recreational N2: End Station N1: Intermodal Station 27 IMPACTS WEIGHTS Criticality Index Model 28 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 1. Station Characteristics (C)	2. Station Location (L)	3. Station nature of use (N)	C1: Number of exits	C2: Number of Levels	L1: Recreational	L2: Residence	 L3: Vitalities	 N1: End Station	N2: Intermodal Station	Global weight Local weight Risk Index Risk Index Model 30 Probability of Failure Consequences of Failure Criticality Index Fuzzy Sets Fuzzy Rules Knowledge Base Fuzzy Inference Engine Fuzzification Defuzzification Risk Index Probability of Failure Consequences of Failure Criticality Index Risk Index Model 31 Probability of Failure Consequences of Failure Criticality Index Risk Index Model 32 Risk Index RI: IF PoF is Xi and CoF is Yi and CR is Zi then Risk Index is Li Model Implementation 33 Organizational effects CONSEQUENCES OF FAILURE Critical (0.6,1, 1.4) Tolerable (0.2,0.6,0.8) Negligible (-0.4,0,0.4) Financial Financial cost will be high for repair and for giving alternative >5M$ Financial impact is a factor but usually the amount of money needed for this type of impact is easily absorbed during the current year or the following one 2M$-5M$ financial is not an impact it’s covered by operational cost <2M$ Social Reduction of customer satisfaction rate that causes their permanent loss Reduction of customer satisfaction rate that causes temporary shifting of service Customers are barely affected by service disruption Operational Failure causes a service outage affecting more than one metro line for more than 30 minutes. Failure causes a service outage affecting a subway line in full or partial interchange outage affecting more than one line for a maximum of 15 min Failure causing operation mode degradation for a time between 2 and 5 minutes. Risk Index Model 34 Risk level Membership function Significance Negligible -0.25,0,0.25 No intervention required Minor 0,0.25,0.5 Intervention required is optional, can be postponed. Significant 0.25,0.5,0.75 Intervention is required and should be planned. Critical 0.5,0.75,1 Obligatory intervention required, yet not urgent Catastrophic 0.75,1,1.75 Urgent and Obligatory intervention is required Model Implementation 35 CASE STUDY  Model Implementation 36 STA 1 STA 4 STA 6 STA 5 STA 3 STA 2 CASE STUDY  Model Implementation 37 STA TUN AUX STB 1 0.2501 0.1173 0.0859 STB 2 0.0000 0.1800 0.0000 STB 3 0.2685 0.1797 0.0000 PoF line A 0.0038 STB 4 0.6732 0.1644 0.2257 STB 5 0.2243 0.1489 0.0000 PoF line B 0.1718 STB 6 0.5130 0.0000 0.0000 PoF line C 0.5130 P Segment=∏3↑i=1▒​ P↓Linei  =0.0003  Model Implementation 38 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Station Tunnel Aux Structure Station Tunnel Aux Structure Station Tunnel Aux Structure Station Tunnel Aux Structure Station Tunnel Aux Structure Station Tunnel Aux Structure STB 1 STB2 STB3 STB4 STB5 STB6 PoF	   CoF	   Cri(cality	  Index	   Risk	  Index	  Risk Index Model 39 RISK REPORT Model Implementation 40 Station ü  Probability of Failure ü  Consequence of Failure ü  Criticality Index ü  Risk Index ü  Revenue Loss ($CAD) ü  Repair Cost ($CAD) ü  Service continuation ü  Interruption Rate ü  Time to repair (days) ü  User Traffic (annual) STA 4 0.673 0.343 0.743 0.821 $583,779 $225,000 Weekend Total (1) 65 1092714 STA 6 0.513 0.279 0.252 0.5 $526,706 $225,000 Weekend Total (1) 50 1281651 Research Contributions 41 RESEARCH CONTRIBUTIONS  Develop a network level risk-based asset management model Integrate the three models into a comprehensive risk index model Study subway network @ functional and structural perspectives Develop Probability of failure, Consequence of Failure and Criticality models 

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