Increasing the resilience level of a vulnerable rail network: The strategy of location and allocation of emergency relief trains

Transportation Research Part E 119 (2018) 110–128

Contents lists available at ScienceDirect

Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

Increasing the resilience level of a vulnerable rail network: The strategy of location and allocation of emergency relief trains

T

Mostafa Bababeika, Navid Khademia, , Anthony Chenb ⁎

a b

School of Civil Engineering, College of Engineering, University of Tehran, Tehran, Iran Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong

ARTICLE INFO

ABSTRACT

Keywords: Resilience Vulnerability Relief trains Cooperative coverage Railway systems Location and allocation model

This paper examines the optimal location and allocation of relief trains (RTs) to enhance the resilience level of the rail network. Unlike probabilistic approaches, the priority of demand is handled by link exposure measure which considers the operational attributes of links and accessibility to road system. We formulate the proposed model using a bi-objective programming and solve it using an augmented e-constraint method (AUGMECON) combined with a fuzzy-logic approach. The proposed framework is employed to a real-world case study, and analytical results reveal the superiority of the proposed model in providing an economical and effective layout compared to conventional maximal covering model.

1. Introduction Railway systems are part of critical infrastructures and the main backbone of economic development in every country. During a rail accident, some parts of infrastructure might be damaged, and some passengers injured. In such case, acceleration in the rate of recovery through the movement of relief equipment to disrupted lines, particularly those located in low access areas, is vital. Delayed arrival of relief equipment at disaster sites leads to delayed restoration of rail traffic causing diversion and cancellation of trains. For example, delayed relief operation in a low access tunnel of the south corridor in Iran, between Tang-haft and Andymeshk stations, caused a 41-h line blockage and many train cancellations (Kachoueyan et al., 2016). In Iran, rescue and relief equipment of railway is provided by a specific emergency rail-vehicle called relief train (RT). Such trains are equipped with medicine and first aid supplies, track components, a crane, re-railing and cutting equipment along with some technicians working with them. An RT in Iran consists of open wagons containing ballast, flat wagons holding traverses, a tank wagon for firefighting, a heavy rail crane, and a passenger wagon for technicians (see Fig. 1). In case of emergency, RT should be available at the disaster site with the least response time. However, locating RTs in stations is subjected to operational constraints such as preference of operating personnel, availability of crews, and nearness to repair depots. Moreover, RTs should be allocated to demand locations with high emergency priority where the relief coverage for the whole network is provided. In this paper, we address the problem of location and allocation of RTs in the rail network to reduce the response time to vulnerable points of the network. Through such strategy, our aim is to make the rail system resilient enough to mitigate the disastrous impact of extreme events and to restore it to normal status immediately. Resiliency is defined as the ability of a transportation network to recover from a disruption and return to normal function within a “reasonable” time frame (Snelder et al., 2012). Optimal location and allocation of emergency relief equipment throughout the rail network can greatly reduce the response time to vulnerable links after an extreme event and consequently the time needed to return to normal functioning, and thus enhance the resilience of the

⁎

Corresponding author.

https://doi.org/10.1016/j.tre.2018.09.009 Received 15 December 2017; Received in revised form 30 September 2018; Accepted 30 September 2018 1366-5545/ © 2018 Elsevier Ltd. All rights reserved.

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

Fig. 1. Relief train and its components (Source: own pictures).

system. The vulnerability analysis concerns the consequences of failing components of the system where the estimation of demand probability is not feasible. The consequences usually concern life, health and environment. If these issues are assigned monetary values, consequences can also be expressed as costs. Among vulnerability studies, Nagurney and Qiang (2012) introduced the network performance efficiency measure as the average throughput on a network with a given demand vector. With such measure, they investigate the importance of network components by studying their impact on the network efficiency through their removal. Khademi et al. (2015) developed a vulnerability analysis for transportation system at a major population center in a geographic area prone to earthquakes. Bababeik et al. (2017) also proposed the increased cost of routing and scheduling in case of a multi-link blockage in a railway network. Mattsson and Jenelius (2015) introduced the concept of conditional vulnerability to show how resilience and vulnerability can be framed. In our study, we suggest an alternative term, link exposure, for vulnerability. Link exposure is defined as how much an important link of the network is in urgent need to be provided by emergency facilities when disruption occurs. In our application, the accessibility of an important link to road as a backup system to transfer relief facilities determines its criticality. Link importance, in turn, reflects the crucial role of any link in the performance of the network. In our application, it is measured by a set of attributes influencing the performance of the rail network. A major difference of our study with previous vulnerability works is that they evaluated critical components by calculating the general costs after the removal of those elements while the current study utilizes link importance index to identify critical components of the network in terms of the need for emergency facilities when disruption occurs. In literature, quantitative research in emergency response has tackled two distinct areas: (1) models for emergency logistics planning, and (2) models for routine emergency relief. The first category deals with location and distribution of facilities which come into use after a rare event like earthquake or hurricane while the next category demands high frequent facilities such as ambulance, police, or fire services. In the first category, Caunhye et al. (2012) provided a comprehensive survey on emergency logistics operations and classified models further into two main sets: (a) facility location, and (b) relief distribution and casualty transportation. One of the most popular techniques to deal with the emergency facility location problem is the applications of the covering location models (Li et al., 2011). The first emergency facility location covering model was the location set covering problem (LSCP) proposed by Toregas et al. (1971). Belardo et al. (1984) used a maximal covering location problem (MCLP) to site resources for maritime oil spills. Balcik and Beamon (2008) also used an MCLP to preposition relief supplies, concerning capacity constraints. However, the LSCP and MCLP have two drawbacks: first, when a facility is called for service, demand points under its coverage are not covered by it anymore; second, they assume that a demand point is covered only when it can be reached within a specific predefined distance by at least one facility. Recently, some researchers have proposed gradual covering models to relax the assumption of coverage within a fixed distance. Berman et al. (2003) applied a level of partial coverage by considering the coverage decay function for the formulation of uncapacitated facility location problem. Karasakal and Karasakal (2004) developed a partial coverage version of maximal covering location problem and applied Lagrangian Relaxation to optimize it. Gradual coverage models still have a limiting assumption as each demand point is covered only by one facility which leads to solutions that require more facilities to cover the same amount of demands. In other words, it causes facilities to be concentrated in specific parts of the network. To cope with this challenge, the cooperative behavior of facility location in covering demand has recently been considered by researchers. Cooperative coverage framework is a new trend which aims to combine (cooperate) facilities to provide coverage (Li et al., 2011). In this paper, we replaced 111

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

the individual coverage assumption with a mechanism where all facilities contribute to the coverage of each vulnerable point. Application of this approach helps the rail authority to dispatch RT from any other station if the nearby RT station is out of service. According to data type, facility location models may use deterministic or stochastic parameters. The approach for considering the uncertainty of demand is to include stochastic parameters using probabilistic distribution or disruption scenarios. For example, Rawls and Turnquist (2010) utilize a two-stage stochastic mixed integer program (SMIP) that considers uncertainty in demand for the stocked supplies as well as uncertainty regarding transportation network availability after a natural event. Mete and Zabinsky (2010) proposed a stochastic optimization approach for the storage and distribution problem of medical supplies to be used for disaster management. Their model captures the disaster-specific information and possible effects of disasters through the use of disaster scenarios. In a different approach, Bell et al. (2014) alternatively proposed robust optimization approach in the case when no reliable probability distribution of damages exists. In robust optimization, the objective is typically to minimize the cost or the regret in the worst situation, called best-worst models. Our proposed work is distinct from the above studies as existing emergency location models are limited to general relief operation following a “rare” event in which the amount of demand is mostly uncertain while its location is predetermined. In current study, however, the demand is scattered throughout the network while its amount is given (i.e., relief demand is determined). Moreover, the probability of incident occurrence in railway due to human error or malicious attacks can hardly be achieved, meaning that an objective distribution function for emergency demand does not exist. Then, applying probabilistic approaches is not appropriate for this purpose. Furthermore, robust optimization is not applicable in our problem as it is more reasonable in dealing with emergencies where a high performance is required from a system in the worst situation (Snyder, 2006), which makes it more suitable for dealing with natural disasters. Thus, it is unrealistic to consider best-worst situation for single-link failures due to human error or attacks as they might occur in railway. In addition, best-worst location problems are difficult to solve, thus exact solutions using this approach have been provided only for networks with special structures or for single supply facility cases (Bell et al., 2014). The second group of studies focused on the location and allocation of routine emergency facilities concerning high demand emergency facilities like emergency medical service (EMS) or fire vehicles. The main characteristics which have already been considered by researchers for location and allocation of these facilities include the availability of ambulance or fire vehicle, the frequency of calls to determine the amount of demand, providing maximum coverage to ensure efficiency, and considering the same response time to ensure fairness. The earlier works done by Hogan and ReVelle (1986) and Marianov and ReVelle (1996) were static and deterministic models and neglected stochastic considerations. The ambulance might be unavailable, causing some demand points not to be covered while there are idle ambulances in other locations. To cope with such issue, the researchers introduced stochastic parameters to consider fluctuations in demand and availability of ambulance service. Rajagopalan et al. (2008) offered a model to achieve the dynamic redeployment (relocation) of ambulances based on fluctuating demand over time. Ingolfsson et al. (2008) determined the number of ambulances needed to provide a specified service level by incorporating the randomness in the ambulance availability and in the delays and the travel times. Another approach for determining the location of routine emergency facilities is to maximize risk coverage. The accident risk levels of various areas can be calculated by the accident occurrence probability multiplied by the accident’s severity. In the work by Tzeng and Chen (1999), the risk has been addressed by minimizing the distance from any fire station to the high-risk area. Yang et al. (2007) introduced various fire risk categories to establish a fire station location model. In the railway application, Cheng and Liang (2014) applied a risk assessment approach by evaluating the accident occurrence frequency and the expected number of injuries to determine the location of ambulances for urban and railway emergency system. This work is the only model we have found that considered locating emergency vehicles for rail system while using probabilistic approaches. Our proposed work is also distinct from the above studies in some ways. First, the demand for relief train is not as frequent as high demand emergencies like medical or fire services as the incident happens occasionally in the rail network. However, it may occur in every part of the network. To address the occasional unavailability of RTs, we proposed the cooperative behavior of relief equipment where the coverage of critical links is met by multiple RT stations. Cooperative coverage of critical links results in covering these elements by a combined effort within a specified time. This manner is vital as the first RT station may be out of service for a critical link. In this situation, the RT is dispatched from the next RT station to cover its demand. Second, applying risk assessment is not recommended in our problem as it is commonly applied where the probability of failure can be evaluated. As mentioned previously, the probability of incident occurrence in railway due to human error or malicious attacks can hardly be achieved, meaning that an objective distribution function for emergency demand does not exist. Then, applying probabilistic approaches like risk assessment are not appropriate for this purpose. Briefly, despite the urgent need for developing a relief equipment layout in railway, no study is available to facilitate it. On the other hand, an efficient approach for location and allocation of relief trains can greatly reduce the response time and enhance the resilience of the system. Specifically, RTs are seen as the sole emergency vehicle to reach inaccessible sites by other transport modes. For example, part of the south rail corridor in Iran is located between two mountains which is not accessible even by helicopter. In such cases, an appropriate layout of relief trains can guarantee relief equipment arrives as quickly as possible and prevent secondary and derivative incidents. This study has the following contributions which distinguish it from the previous studies:

• We utilize a novel approach to recognize the high priority components of rail network based on the concepts of link importance

and exposure. We employ a multi-attribute decision making (MADM) technique to calculate link importance index. Then, based on the link importance, we suggest a measure called link exposure which represents the criticality of that link with respect to its accessibility to the road system. Unlike probabilistic approach, this measure focuses on the consequences of link closure to regard 112

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

• • •

the vulnerability of network components. Through this study, we propose a bi-objective location and allocation model for relief trains in the rail network. The objectives are (i) maximizing the cooperative coverage of link exposure by RT stations, and (ii) minimizing the total travel time from RT stations to the whole components of the network. Within this framework, we provide a cooperative coverage of RTs for vulnerable links and perform routing RTs through allocation of RT stations to any link of the network. Emergency facility location problems commonly represent demand at a few localized points over the area under study. In this paper, however, we consider the demand space to be distributed over all components of the network. In other words, as accidents may occur at every part of the network, each link is treated as a demand point. For this purpose, a heuristic algorithm is applied to make modifications on the initial network so that every link is converted to a demand point. The present study has combined the augmented e-constraint (AUGMECON) technique and a fuzzy-logic-based approach to provide optimal solution. Computation results are analyzed by applying the model to a real-world railway network to verify the capability of model in location and allocation of emergency vehicles in the rail network.

The paper is organized as follows. A description of the problem under study is provided in Section 2. The next Section presents network modification algorithm to consider demand space distributed over the whole links. Section 4 proposes the methodology of the location and allocation model of RTs. Section 5 provides the solution approach used to solve the model. Section 6 is devoted to a real-world case study of railway network to test the efficiency of the proposed model and the solution approach. Finally, Section 7 finishes the article through conclusions and directions for further works. 2. Problem description Let G = (N , L) represent a railway network with stations denoted by a node set N = {N1, N2 , , Nn} , and tracks denoted by a link set L = {L1, L2 , , Ll } . Relief trains (RTs) are required to be located only in hub stations due to operational issues. We specify them as a subset of stations called potential stations denoted by K = {K1, K2, , Kk} . Note that each potential station can accommodate only one relief train. In emergency situations, RT is dispatched from RT station to the interrupted link to provide relief facilities and release it. In our problem, we consider the emergency demand for every link of the network as it is prone to degradation due to accident, so we assume that every link has a fixed amount of emergency demand which requires being treated by RTs. The problem is to locate RTs throughout the railway network regarding budget concern and vulnerability of links, and then allocate opened RT stations to the whole network. We first determine vulnerable links of the network by addressing the concept of link exposure. In our application, this is defined as how critical an important link is because of weak accessibility to road as a backup system to transfer emergency relief. Link importance, in turn, is measured by a set of attributes reflecting the freight/passenger load and the serviceability of that link. We also consider the accessibility of rail lines to road system as an alternative way to deliver emergency facilities. Regarding both the accessibility and the link importance, we measure the link exposure. Accordingly, an important link with low accessibility to road system is regarded as high exposure. Section 4.2 describes determining these values. With the above description, we develop a bi-objective mathematical programming for location and allocation of RTs called LART such that the two aims are met: (i) the most coverage of link exposure is provided, and (ii) the total travel time from RT stations to the whole links is minimized. The model is based on the cooperative coverage of links as described in Section 4.3. Cooperative coverage of link exposure contributes a combined coverage of vulnerable links which further leads to immediate response time for such elements. Besides the decision on where to locate RTs, our model assigns each opened RT station to the links so that the rapidity of the whole system to return to initial state after disruption is increased. Bruneau et al. (2003) specify rapidity as a main component of four-dimensional framework (4R) for system resilience, which are “Robustness”, “Redundancy”, “Resourcefulness”, and “Rapidity”. Haimes (2009) also introduces immediate response as a key property of resilience. According to the literature, resilience represents the ability of a system to absorb perturbations/shocks, and the speed at which it returns to its equilibrium after a shock (Reggiani et al., 2015; Bruneau et al., 2003). With such definition, two key properties can be identified for measuring resilience as proposed by McDaniels et al. (2008): (i) robustness: the ability of the system to withstand a given level of stress or demand without suffering or loss of function, and (ii) rapidity: the ability of the system to quickly recover a level of performance at least as good as its original one or an acceptable performance after a perturbation. Fig. 2 provides a general illustration of these properties of resilience. This figure is based on the resilience triangle proposed by McDaniels et al. (2008) which is widely considered in resilience quantification. As the figure shows, a disruption such as an accident leads to a rapid decrease in performance. In our application, the performance of the system is set by the amount of passenger/freight load carried on the link per hour. The extent to which the performance is maintained (i.e., the extent to which system function is not driven to zero) reflects the system’s robustness. Over time, following the disruption, the system regains some level of stability or equilibrium. The speed with which this recovery of function is achieved reflects the system’s rapidity. The loss of resilience can be seen as the aggregate consequences of a disruption represented by the area between the dotted line in the figure corresponding to the full system performance and the relevant curve representing the reduced level of function. The figure illustrates that rapidity can be improved by the appropriate response activities following the event. The dashed curve represents the situation when the response time is reduced by taking such activities. Optimal location and allocation of emergency relief equipment throughout a transportation network is a strategy which can greatly reduce the response time for recovery after an event. In this paper, we address the problem of location and allocation of RTs in the rail network to reduce the response time to 113

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

Fig. 2. Effects of decision making on resilience.

vulnerable points of the network and consequently the time needed to return to normal functioning, and thus enhance the resilience of the system. As Fig. 2 illustrates, immediate departure of RTs to link l at time tl instead of tl0 can reduce the response time to disruption, and alternatively improve the resilience of the system as shown by crossed hatch. We can quantify the improvement of resilience achieved by this action as:

(

POD tl0| l

Rimp = R 0 Rl = l L OD

rOD

tl | l

rOD

)

(1)

where R 0 and Rl are loss of resilience before and after taking action, respectively. POD is the passenger/freight load of origin-destination OD , and tl0| l rOD and tl | l rOD are response time to emergency demand of link l before and after taking action, respectively, when it is located on the route of origin destination OD . Proposed model helps to increase the resilience level of the network as it reduces the travel time from emergency facility locations (RT stations) to demand points (links). As noted earlier, a distinct feature of our model is that it considers the demand is distributed over the whole network instead of some localized points. Accordingly, a procedure is developed to modify the network topology as every link is treated as a demand component. The next section describes this procedure. 3. Network modification In our problem, each link of the network may demand RT. The topology of the network is then modified in such a way that every link is treated as a demand point. We represent the demand point related to any link with a dummy node in the middle of that link such that a potential RT station may be assigned to it. To clarify the procedure, we explain it through a simple network depicted in Fig. 3. This network contains 4 links {L1, L2 , L3 , L4 } , and 4 nodes {N1, N2, N3, N4} , from which two nodes, N2 and N4 , are considered as potential stations K1 and K2 , respectively (Fig. 3a). First, let us put a dummy node in the middle of the links to represent demand points (i.e. dots in the middle of links in Fig. 3b). If the links end at any of the potential stations, the dummy nodes are connected to the related potential RT station through a new link like L1' in Fig. 3c. The travel time of this link is half that of the original link. If two adjacent links end at a non-potential station, their dummy nodes are connected with a new link like the dashed link L14 " in Fig. 3c, and the related non-potential station (N1) is then removed. The travel time of the new link is the average of both initial links, e.g. the travel time of L14 " is half the sum of travel times of links L1and L4 . In the modified network, the set of nodes is updated to include both potential stations and dummy nodes, i.e. L , K N . With extending the above procedure, we propose Algorithm 1 to generate the modified network:

Fig. 3. Example of network modification: (a) original network, (b) dummy nodes on links, (c) modified network. 114

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

Algorithm 1 (Generating the modified network). 1. Take the inputs of the initial network as below: Set of stations (N ); Set of potential stations (K ); Set of links (L ). 2. Put a dummy node in the middle of each link and denote it with the same name of that link, e.g. dummy node L1 for link L1. 3. Repeat for each dummy node L i (i L ): Does dummy node Li end at a potential station k K ? 3.a: Yes; define a new link, L'i , between the dummy node L i and potential station k with travel timet L'i = t Li /2 . 3.b: No; repeat for any dummy node Lj (i j ) : Are dummy nodes Li and Lj adjacent? 3.b.a: Yes; join both dummy nodes with a new link, Lij", with travel time t Lij" = (t Li + t Lj )/2 ; e.g. link L14 " in Fig. 3c

3.b.b: No; go to Step 3.b. 4. Keep the potential stations and remove the other ones.

4. Methodology In this section, we first introduce the notations, second, explain the procedure of determining the link importance and link exposure measure, and last, formulate the problem of location and allocation of RTs. 4.1. Notation system The definition of sets, parameters, and variables used in the proposed model are summarized as follows: Sets N Set of nodes in the modified network indexed by n, including potential stations and dummy nodes L Set of dummy nodes in the modified network indexed by l (representing the links of initial network), L N K Set of potential stations indexed by k, K N Parameters tmn Travel time from node m N to n N exl Exposure of link l ck Supply cost of potential station k with RT a Cooperative coverage threshold (tkl ) Coverage function in terms of travel time from potential station k K to dummy node of link l L P Budget limit pl Importance index of link l exl Exposure of link l Variables kl Binary variable indicating whether link (m , n) of the modified network is located on the path from potential station k Xmn to link l L Yl Binary variable indicating whether link l L is covered Wk Binary variable indicating whether RT is located in potential station k K Ukl Binary variable indicating whether the potential station k K is assigned to link l L

K

4.2. Calculating the link importance and link exposure In our study, link importance is measured in terms of a set of attributes reflecting the crucial role of that link in the performance of the network. These attributes were determined by several interviews with railway experts in the field of relief train as specified in the following. A multi-attribute decision making (MADM) technique is then applied to determine the link importance. Also, we consider the accessibility of rail lines to road system as an alternative way to deliver emergency facilities. Accordingly, an important link with low accessibility to road system is highly critical to being treated by RTs. Regarding both the accessibility and link importance, we define a new measure called link exposure. The following steps describe determining these values: Step 1. Determining attributes and their weights: According to the definition of link importance, the contribution of any links of the network in the performance of the network is indicated by link importance. Accordingly, we wish to distinguish attributes of links that affect the performance of rail network. Such attributes might differ from application to application. To address these attributes comprehensively, we first reviewed vulnerability 115

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

Table 1 Attributes influencing the link importance. Attribute

Description

PT FT T U

Monthly passenger traffic of the track Monthly freight traffic of the track Monthly export/import transit of the track Direct serviceability of track to industry (factories, refinery plants, and customs)

literature for critical features of link influencing the performance of the network. Tampère et al. (2007) and Knoop et al. (2012) considered three attributes for calculating the link vulnerability indices in a national road network: traffic flow on the link, the capacity of the link, and the number of lanes of the link. El-Rashidy and Grant-Muller (2014) considered the capacity of link relative to the maximum capacity of all network links to reflect relative link importance. They also used the link length and the number of times a link is a component of the shortest path between different OD pairs as two properties representing the level of importance of the link. Ukkusuri and Yushimito (2009) developed a methodology for identifying critical links based on the system travel time as the performance measure before and after disruptive event. In Murray-Tuite (2006), the amount of time a link has an average speed lower than a threshold speed limit, and the link volume-to-capacity ratio were considered to reflect the importance of link. The attributes presented above are more applicable to road transportation networks. However, they cannot directly be applied to measure the importance of link in a rail network. To realize the main attributes in our application, we made a further effort through holding several brainstorming sessions with the railway experts from October to December 2016 (Bababeik, 2016). The details regarding the selection of such attributes are included in Appendix A. Table 1 shows four attributes we realized in the problem under study. They include the number of passenger-kilometer and freight tonne-kilometer passing through a rail link per month. In addition, some rail lines are important since they provide transit paths for import or export purposes. Finally, those tracks that provide accessibility to the industry like factories and refinery plants are considered significant in terms of network performance. Each of the above four attributes affects the performance of the network with different weight. To determine the relative normalized weights of the attributes, we quantify the subjective judgment of railway specialists through the geometric mean method of the Analytic Hierarchy Process (AHP). This method can efficiently deal with objective (tangible) as well as subjective (non-tangible) attributes, where the subjective judgment of different individuals constitutes an important part of the decision process. According to Saaty (1994), we first construct a pairwise comparison matrix for these four attributes. Elements of this matrix are entered based on the judgment of the decision makers and using the fundamental scale of AHP. Then, we find the relative normalized weight of each attribute by normalizing the geometric means of rows in the comparison matrix. The consistency of judgment is also evaluated by the consistency index. The details regarding the calculation of consistency ration is included in Appendix A. The attributes are further divided into two categories: beneficial and non-beneficial (Saaty, 1994). A beneficial attribute means its higher measures are more desirable in terms of link importance. A non-beneficial attribute is one in which the lower attribute value is desirable. In our problem, however, all attributes are considered as beneficial ones. Step 2. Determining the normalized decision matrix: We construct a decision matrix in which each row is assigned to one link and each column to one attribute. That is, any element mlj of this matrix gives the value of the attribute j for the link l in terms of original real unit (i.e. non-normalized value). Then, the normalized decision matrix, Rlj , is calculated using the following equation:

Rlj =

mlj L m2 l = 1 ij

(2)

where L is the number of links. Step 3. Constructing the weighted normalized decision matrix: This is done by the multiplication of each element of the column of the matrix Rij with its associated weight (wj ) obtained in Step 1. Therefore, the weighted normalized decision matrix Vlj is equal to: (3)

Vlj = Wj Rlj

Step 4. Determining the ideal and negative ideal solutions: We utilize the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) as a MADM technique to determine the priority of links (Tzeng and Huang, 2011). TOPSIS is a useful technique in dealing with multi-attribute decision making (MADM) problems in the real world. It helps the decision maker to carry out appropriate comparisons and rank the alternatives. In our application, alternatives are the network components (links) which should be prioritized according to their importance. Accordingly, a suitable ranking of links in terms of their importance will be made. The ranking of alternatives is based on the concept that the chosen alternative should have the shortest geometric distance from the positive ideal solution and the longest geometric distance from the negative ideal solution. The positive ideal (best) and negative ideal (worst) solutions are the links with the most and least link importance index, respectively, and can be determined as:

V + = {(maxVlj |j l

J ), (minVlj |j l

J ')|l = 1, 2,

, L} = {V1+, V 2+, 116

+ , VM }

(4)

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

V = {(minVlj |j l

J ), (maxVlj |j l

J ')|l = 1, 2,

, L} = {V1 , V 2 ,

, VM}

(5)

where M is the number of attributes, J = (j = 1, 2, , M ) is associated with beneficial attributes, and J '= (j = 1, 2, , M ) is associated with non-beneficial attributes. It should be mentioned that Eqs. (4) and (5) are general expressions; however, all attributes applied in this study are beneficial ones, so Eq. (5) remains inapplicable. Step 5. Determining the separation measure: The separation of each link from the link with the highest index of link importance is given by the Euclidean distance in the following equation: M

Sl+ = {

(Vlj V +j )2}0.5, l = 1, 2,

,L

(Vlj V j ) 2}0.5, l = 1, 2,

,L

(6)

j=1

M

Sl = {

(7)

j=1

where Sl+ and Sl are the distance between the importance of link l and the most and least important link, respectively. Step 6. Calculating the link importance index: The relative closeness of each link to ideal solution (most important link) explains the importance of that link. We consider such relative closeness as the link importance index and express it as follows:

pl =

Sl (Sl+ + Sl )

(8)

Step 7. Calculating the link exposure: Technical analysis of some accidents has confirmed that the inaccessibility of the railway line to the road in some cases made rescue operations difficult as far as the approach of facilities for handling the emergency is concerned. Therefore, we consider the accessibility of railway line to road system as an influencing factor for a link to be covered by RTs. Based on the link importance we suggest a measure called link exposure which represents the criticality of an important link with respect to accessibility to the road system. Particularly, an important link is critical when it has low accessibility to the road system. Mathematically, we can say:

exl =

pl acl

(9)

where exl is the exposure of link l , pl the importance of the link l , and acl the value of accessibility of rail line to road system. As mentioned previously, we consider the accessibility of rail lines to road system as an alternative way to deliver emergency facilities. Parameter acl may be evaluated by quantitative models which have already been developed to analyze the accessibility of a railway line in emergency conditions using the road system (Borghetti and Malavasi, 2016). Alternatively, we determine this value by measuring the relative length of a given rail line that can be reached by the road system. In other words, the ratio of the stretch of the rail line which can be approached by road system is considered as accessibility parameter. 4.3. Model formulation We utilize the idea of cooperative coverage of links to build the core model of location and allocation of RTs. The concept of cooperative coverage was first introduced in Cooperative Location Set Covering Problem (CLSCP) and Cooperative Maximum Covering Location Problem (CMCLP) by Berman et al. (2011). This concept is based on the mechanism that each facility at site k emits a “signal” that decays over distance according to a decay function. A demand point receives signals from all facility stations and is covered only if the “signal” it receives exceeds a certain threshold. In our application, the signal emitted by any potential RT station is determined by a mathematical function to model the gradual decline of coverage along with the increase in distance. If the cumulative effect of signals for any link of the network exceeds a given value, it is considered as covered. We define our gradual coverage function as a monotone decreasing function, as shown by Eq. (10). This function represents a measure of the quality of service as the distance with RT stations increases. For a review on decay coverage functions, one may refer to Eiselt and Marianov (2009).

1 (tkl ) =

tkl ta

if tkl tb tb

0

ta

if ta < tkl < tb (10)

otherwise

where ta is the threshold of travel time for the full coverage and tb is the maximum travel time beyond which the link is considered uncovered by that RT station. In reality, ta represents the threshold that the travel time to disrupted link is less than a satisfactory level. This level may be determined by local or regional standards regarding emergency operation. As the travel time exceeds the standard level, the quality of service drops off until no RT is dispatched to disrupted link in reality. This travel time is usually adopted by the railway authority and it determines the value of tb .

117

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

It is assumed that the coverage declines linearly along with the increase of the travel from RT station k to link l, denoted as tkl . We determine the travel time through a preprocessing stage in which the routes with least travel time from every potential RT station to any link of the network are calculated based on the shortest path problem. The preprocessing stage is formulated as follows: Preprocessing stage: (A) Solve:

Minimize k K l L (m , n ) N '

kl tmn Xmn

(11)

s.t. kl Xmn (m, n) N

kl Xnm =0

l

L,

k

K , m, n

kl Xnm

l

L,

k

K , m or n = k

l, k

(m, n ) N

kl Xmn (m, n) N

=1

(m, n ) N kl Xmn

kl Xnm =

(m, n) N

1

l

L,

k

K , m or n = l

(m, n ) N

(12) (13) (14)

(B) Calculate travel time: kl tmn Xmn

tkl =

l

L,

k

K

(15)

(m, n) N

The objective function (11) minimizes the total travel time from potential RT stations to the links of the network. Constraints (12)–(14) keep flow conservation at all nodes of the modified network. Eq. (15) determines total travel time from potential station kl k K to link l L . It should be noted that although Xmn is a binary variable, there is no need to include integrality constraint in the above formulation. Since the coefficient matrix of model (11)–(15) maintains the sufficient condition for total unimodularity, every basic feasible solution is ensured as automatically being integer (Chen et al., 2011). Therefore, it can be solved by a regular LP solver. We propose the model of location and allocation of relief trains (LART) as a bi-objective integer linear program with considering the two aims: (i) maximizing the cooperative coverage of link exposure by RT stations, and (ii) minimizing the total travel time from RT stations to the whole of the links. This model is proposed as the following: LART model

max f1 =

exl Yl

(16)

l L

min f2 =

tkl Ukl

(17)

k K l L

s.t.

(tkl ) Wk

a × Yl

l

L

(18)

k K

ck Wk

P

(19)

k K

Ukl

Wk

Wk

l

L,

Ukl

k k

(20)

K

K

(21)

l L

Ukl = 1

l

L

(22)

k K

Yl , Wk , Ukl

(23)

{0, 1}

The first objective function, Eq. (16), maximizes the total coverage of the link exposure measure throughout the network. The second objective, Eq. (17), is to assign opened RT stations to the links of the network such that the total emergency travel time (travel time from RT stations to links) is minimized. Constraint set (18) ensures that a link is covered if its cooperative coverage of all potential RT stations is greater than the given threshold a. This parameter specifies when such aggregation reaches a certain value, the link is considered as covered. In other words, it indicates how large the cumulative effect of RT stations should be for any link to consider that link as covered. Each potential RT station requires a fixed cost to be equipped with RT facilities. Constraint set (19) limits the total supply cost of RT stations to an available budget. Constraint set (20) ensures that only opened RT stations are assigned to the links. Constraint set (21) makes certain that at least one link is assigned to any opened RT station. Constraint set (22) makes

118

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

sure that every link is provided by one opened RT station. Finally, constraint set (23) keeps the integrality condition for variables. The proposed model is a bi-objective mathematical programming in which there is no single optimal solution that simultaneously optimizes both objective functions. The first objective aims at maximizing a combined coverage of high exposure link while the second objective aims to assign relief trains to the whole links of the network with minimum emergency travel time (travel time from RT stations to links). In fact, each objective has its own meaning and contribution to the problem under study. The first objective ( f1) seeks to provide cooperative coverage of the links of the network such that the total measure of link exposure is maximized. In other words, it concerns the coverage of critical links with one or more relief trains through a combined effort and within the specified threshold of time. This manner is vital as the first RT station may be out of service for the critical links. In this situation, the RT is dispatched from the second RT station to cover its demand. The second objective ( f2 ), however, is related to allocation of relief trains to all components of the network so that the minimum emergency travel time is provided over the whole network. Paying careful attention, we realize that the bi-objective formulation provides backup coverage for critical components of the network within a specified threshold. Moreover, it distributes relief facilities fairly over the network while the travel time to some less critical links might exceed the specified threshold. 5. Solution approach In this section, we propose the solution approach of LART model. This model is categorized in multi-objective optimization problems. In a multi-objective mathematical programming, there is no single optimal solution that simultaneously optimizes all the objective functions. In these cases, the concept of optimality is replaced with that of noninferiority or Pareto optimality. A feasible solution to a multi-objective programming problem is noninferior if there exists no other feasible solution that will yield an improvement in one objective without causing a degradation in at least one other objective. A comprehensive survey of solution techniques to multi-objective mathematical programming can be found in the book of Ehrgott (2006). We utilized augmented e-constraint (AUGMECON) technique in our problem to find efficient solutions for LART model. This method is proposed by Mavrotas and Florios (2013) and eliminates some of the weak points in the conventional e-constraint method. In fact, the conventional e-constraint method has two points that require attention: the range of the objective functions over the efficient set (mainly the calculation of nadir values) and the guarantee of the efficiency of the obtained solution. To overcome the first ambiguity, they proposed the use of lexicographic optimization for every objective function to construct the payoff table (the table with the results from the individual optimization of the p objective functions) with only efficient solutions. To overcome the second ambiguity, the objective function constraints in a regular e-constraint method are transformed to equalities by explicitly incorporating the appropriate slack or surplus variables. At the same time, the sum of these slack or surplus variables is used as a second term (with lower priority) in the objective function forcing the program to produce only efficient solutions. The new problem which is called augmented e-constraint method becomes:

max(f1 (x ) +

× (s2 + s3+

+sp))

s.t.

f2 (x ) s2 = e2

f3 (x ) s3 = e3 …

fp (x ) sp = ep

x

S and si

R+

where fp (x ) is the objective function p, sp is the slack or surplus variable associated to this objective function, ep is the RHS of objective function p in conventional e-constraint method, S is the feasible region, and is a small number (usually between 10 3 and 10 6 ). By parametrical variation in the RHS of objective function (ep ), the efficient solutions of the problem are obtained. The range of every objective function is also calculated through lexicographic optimization to construct payoff table. Proof demonstrating the above formulation produces only optimal solutions is provided by Mavrotas and Florios (2013). After generation of the Pareto optimal solutions, the next step is to find the most appropriate solution, i.e. preferred solution. In the literature, there are several approaches related to the selection of preferred solutions, such as the k-mean clustering procedure, the weighted-sum approach and the fuzzy-logic-based approach. Cluster analysis can classify data into groups in which individuals are similar to each other. This method chooses a set of solutions rather than a single solution; moreover, it is usually used together with evolution algorithms such as genetic algorithms. The decision maker can choose a preferred solution by the weighted-sum approach when a preferred weight vector of objectives is provided. While the method gives the absolute weighted sum of objective values of a solution, it fails to indicate the degree of optimality of a solution. The degree of optimality is a measure which indicates the degree to which each Pareto optimal solution satisfies the decision maker viewpoint. The fuzzy-logic-based approach not only provides the most preferred solution but also indicates its degree of optimality. Hence, we adopt this method to choose a preferred

119

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

solution among Pareto optimal solutions of LART model. In this respect, we develop the following algorithm based on a fuzzy logic approach to find the preferred solution: Algorithm 2 (Finding the preferred solution in LART model). 1. Convert the maximum objective function f1 to a minimum one by a negative coefficient. 2. Find the optimum values of each objective function by solving it individually through problems P1 and P2, and find the ideal objective vector for them: Problem P1: min f1 = Problem P2: min f2 = k K l L tkl Ukl s.t. Constraints exl Yl s.t. Constraints l L

(18)–(23) (18)–(23) Optimal (Yl 1, Wl 1, Ukl1) (Yl 2, Wl 2, Ukl2 ) values Ideal f1I f2I objective 3. Find the Nadir objective vector. It represents the upper limit of each objective in the entire Pareto set but not in the entire objective space. To calculate the upper limit of each objective, solve each objective with new constraints according to problems P3 and P4: exl Yl s.t. Ukl = Ukl2 Problem P3: min f1 = Problem P4: min f2 = k K l L tkl Ukl s.t. YL = Yl 1 l L Constraints (18)–(23) Constraints (18)–(23) Nadir f1N f2N objective 4. Consider there are J noninferior points after solving the LART model by AUGMECON method. We then define the membership function to indicate the degree of optimality for every objective function of each noninferior point through Eq. (24). Let MF (fi j )

be the membership function of objective function p in the noninferior solution j, so we

fi j

1 have: MF (fi j ) =

fiN fiN

fi

j

fi I

0

fi I

f i I < fi j < fiN fi

j

fi

(24)

N

5. Calculate the membership degree for each noninferior solution j based on its individual membership functions as follows:

j

=

j 2 i = 1 MF (fi ) j J 2 j = 1 i = 1 MF (f i )

(25)

6. If decision maker can offer a preferred weight for each objective, another way of calculating membership degree is provided, i.e.,

j

=

j 2 i = 1 i MF (f i ) 2 i=1 i

(26)where

i is

the weight of the objective function p.

7. Select the solution with the maximum value of

j

as the most preferred solution.

6. Case study In this section, we present a numerical example based on the railway network of Iran and its operation data to examine the practicability of the developed methodology and solution approach. 6.1. Characteristics of the case network The railway network of Iran is a national state-owned railway system which manages 35 million tons of goods and 25 million passengers annually, accounting for 12.5 and 10 percent of the whole transport system in Iran, respectively (RAI, 2016). The network consists of 446 links and 440 stations covering a length of 10,000 km across the country. In terms of strategic importance, this network has a western extension to Turkey, a North-South corridor from Azerbaijan and Russia to the Persian Gulf, and an eastern connection to the bogie-changing station at Sarakhs (see Fig. 4). Therefore, such widespread network demands a great deal of attention to an efficient emergency management plan. Currently, RT has a major contribution to providing relief facilities to disrupted tracks in case of emergency. Fig. 4 illustrates the position of existing RT stations with yellow triangles while the green ones indicate the location of potential RT stations. 120

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

Fig. 4. Case study network. Table 2 Pairwise comparison matrix and results obtained with AHP.

Monthly passenger traffic of the track (PT) Monthly freight traffic of the track (FT) Monthly export/import transit of the track (T) Direct serviceability of track to industry (U)

PT

FT

T

U

Weight

1 0.2 0.5 0.143

5 1 1 0.2

5 1 1 0.2

7 5 5 1

0.61 0.17 0.17 0.05

max

4.21

CI

RI

CR

0.07

0.9

0.077

6.2. Determining the measure of link exposure According to the structure of multi-criteria decision making described in Section 4.2, the exposure of each link should be determined. After interviews with the railway experts (Bababeik, 2016), the comparison matrix is obtained based on the pairwise comparison of the attributes. The relative normalized weights of the attributes are then obtained using the geometric mean method of AHP method. Since the comparison was carried out through personal judgments, some degree of inconsistency could occur, and therefore consistency verification was conducted to ensure consistency. The computational results of AHP calculation are presented in Table 2. The consistency ratio for the pairwise comparison matrix was 0.077 < 0.1, and thus there exists a good consistency in the judgment of analyst regarding the problem under study (Saaty, 1994). It is shown that the passenger traffic (PT) is considered to be the most important attribute for determining the link importance index whereas the attribute of direct serviceability to industry (U) is considered to be of least concern. 121

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

Table 3 Pareto optimal solutions of case study. Pareto optimal solution No.

1st obj. function

2nd obj. function

MF of 1st obj.

MF of 2nd obj.

Membership degree

1 2 3 4 5 6 7 8

104.99 103.67 102.78 102.71 101.39 99.91 97.45 84.06

88564.86 86225.25 80416.84 72746.63 70407.02 69869.88 67689.87 65830.07

1 0.94 0.89 0.89 0.83 0.76 0.64 0

0 0.1 0.36 0.7 0.8 0.82 0.92 1

0.5 0.52 0.63 0.79 0.81 0.79 0.78 0.5

The TOPSIS approach is applied to calculate the link importance in terms of four attributes, as described in Section 4.2. The exposure of each link is then determined by dividing the importance of each link to its road accessibility measure. Fig. 4 illustrates the result of calculating link exposure for the whole set of links. 6.3. Solution results We employ the LART model for the case network using the data obtained in previous stages and solve it with AUGMECON approach to find the Pareto optimal solutions. The processing time was less than two minutes. The results are recorded in Table 3. There are 8 noninferior solutions obtained by this approach. The second and third columns show the value of objective functions f1 and f2 in LART model, respectively. The results denote that it is impossible to make any reduction on the total travel time (second objective function) without decreasing the total coverage of link exposure (first objective function) or vice versa. To offer a suitable solution to a real application, we perform the fuzzy-logic approach using Algorithm 2. The results are provided in the last three columns of Table 3 for each noninferior solution. The obtained values of membership degree of different solutions show that the optimal solution No. 5 is among the most preferred ones. 6.4. Analysis of the results In this analysis, the same weight is considered for both objectives, but the decision-maker can consider the priority of objectives by assigning different weights to them and calculating membership degree through Eq. (26). Accordingly, a further experiment is carried out to investigate the effect of choosing different weights of objectives on the preferred solution in the fuzzy logic selection procedure. For this purpose, we compared the normalized value of objective functions in preferred solution when the different weights are selected for them. The result has been depicted in Fig. 5. As it can be seen, when the weight of objective function f1 (maximizing cooperative coverage of link exposure) increases, its value increases continuously while the negative value of objective function f2 (minimizing total emergency travel time) decreases continuously. As the lower value of total travel time is preferable, its negative value has been used. This figure reveals the weights in which the most cooperative coverage is reached whereas the least total travel time provided occurs in the weights of 1 = 0.57 and 2 = 0.43. The result of solving the case network with such weights has been depicted in Fig. 6. According to budget limit of 16000, there will be 16 opened stations which are then allocated to links. The green circles indicate opened RT stations, and the links assigned to each one are depicted with the same color. The resulted

Fig. 5. The effect of different weight factors of the objective functions in the preferred solution. 122

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

Fig. 6. Opened RT stations and allocation layout of the preferred solution.

Total link exposure ratio

80% 70% 60% 50% 40%

Considerable increase in coverage

30% 20%

Negligible increase in coverage

10% 0% 12000

14000

16000

18000

20000

22000

24000

Budget level

Fig. 7. The effect of weight factor on preferred solutions.

layout indicates the travel time from RT station to assigned links is reduced 7.6% throughout the network compared to existing layout (Fig. 4). Therefore, according to Eq. (1), the LART model contributes to 27.7% and 26.7% increase in resilience in terms of passenger and freight load, respectively. A sensitive analysis is also used to determine how different values of budget level impact the preferred solution. Fig. 7 provides a visual guideline to the decision makers as they illustrate how different values of budget level may impact the coverage of link exposure throughout the network. The figure depicts the ratio of total link exposure which is covered over the network increases monotonically by enhancing the level of budget. However, when the budget level exceeds 18,000 units, the increase in this ratio is negligible. Hence, the decision maker might choose this level of the budget as an economical option while he considers providing satisfactory coverage of high exposure elements. It can also be seen that even when the highest budget level is available, the total link exposure ratio is 69%, which means that some links remain uncovered in the preferred solution within the maximum allowable emergency time (tb = 120 min). It implies the fact that the available potential RT stations are not properly spread throughout the network to cover all links. Therefore, the potential RT stations should be scattered throughout the network or additional ones should be considered by the railway authority. The result of the analysis for different budget level provides the network authority with important insight into the priority of opening new RT stations. Table 4 shows the opened RT stations for each budget level increasing from 16,000 unit. Since some of the available RT stations in the current network are not among preferred solutions, the results of LART model suggests opening new ones 123

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

Table 4 The priority of new RT stations for different budget levels. Budget level

Required RT stations

New opened or relocated by existing stations

16,000 17,000 18,000 19,000 20,000 21,000 22,000

S1, S3, S1, S2, S1, S2, S1, S2, S1, S2, S1, S2, S1, S2, S24

S3, S2, S2, S2, S2, S2, S2,

S4, S3, S3, S3, S3, S3, S3,

S5, S7, S8, S9, S11, S12, S13, S14, S15, S16, S18, S21, S23 S4, S5, S7, S8, S9, S11, S12, S13, S14, S15, S16, S18, S21, S23 S4, S5, S7, S8, S9, S11, S12, S13, S14, S15, S16, S18, S20, S21, S23 S4, S5, S7, S8, S9, S11, S12, S13, S14, S15, S16, S17, S18, S20, S21, S23 S4, S5, S6, S7, S8, S9, S11, S12, S13, S14, S15, S16, S17, S18, S20, S21, S23 S4, S5, S6, S7, S8, S9, S11, S12, S13, S14, S15, S16, S17, S18, S20, S21, S23, S24 S4, S5, S6, S7, S8, S9, S10, S11, S12, S13, S14, S15, S16, S17, S18, S20, S21, S23,

S9, S3, S3, S3, S3, S3, S3,

S16, S21 S9, S16, S21 S9, S16, S20, S21 S9, S16, S17, S20, S21 S6, S9, S16, S17, S20, S21 S6, S9, S16, S17, S20, S21, S24 S6, S9, S10, S16, S17, S20, S21, S24

Fig. 8. Comparison of optimal solutions in LART and MCLP models.

or relocating those existing stations. The last column suggests new RT stations which should be newly opened or considered by relocating existing RT stations. To highlight the role of cooperative coverage, we compared the preferred optimal solution in the proposed model (LART) with the optimal solutions of a maximal covering location problem (MCLP) for various budgetary level, as illustrated in Fig. 8. We observe that the LART model provides a higher coverage of link exposure over the network in comparison with MCLP with the same level of budget. Hence, the developed cooperative model can achieve a better performance than the conventional MCLP in providing coverage of critical elements in the network. Also, as the budget level for supplying RT increases, the solution by LART model provides a higher coverage of link exposure over the network than the one by MCLP. We also compared the solutions obtained by LART and MCLP models in terms of emergency travel time (travel time from the RT

Fig. 9. Comparison of emergency travel time in MCLP and proposed model. 124

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

station to the link) under two assumptions. In the first one, it is assumed that all opened RT stations are ready to provide service to any disrupted link in the network. In the second one, it is assumed that the nearest RT station to each link is busy or unavailable to respond to the emergency request, so the RT should be dispatched from the next nearest station. Fig. 9 shows that the total emergency travel time in both models reduces as the budget level increases under two assumptions. However, we see a drastic decrease in travel time obtained by LART model compared to the one by MCLP for every budget level when all opened RT stations are available (LART1st and MCLP-1st). This fact implies that optimizing two objectives separately (maximizing coverage by MCLP and then minimizing travel time by assignment problem) does not necessarily lead to the least emergency response time while the bi-objective LART model makes sure that the least response time is obtained by combining two objectives. In a similar fashion, Fig. 9 illustrates the LART model provides less travel time than the MCLP model under the second assumption although the difference is not significance (LART2nd and MCLP-2nd). In fact, the cooperative coverage provides more effective coverage than maximal covering when the nearest RT station is out of service. This is attributed to the distinctive feature of cooperative coverage models when a link is covered by considering the cumulative effect of nearby RT stations. 7. Conclusion Location and allocation of emergency vehicles can be an effective mechanism for improving the capability of a system to recover from an incident immediately after it occurs. In this paper, we addressed the optimal location and allocation of Relief Trains (RTs) in the rail network to enhance the resilience of the system. In this respect, we propose a mathematical programming model (LART) which consists of two conflicting objectives: (i) maximizing the cooperative coverage of the link exposure by RT stations, and (ii) minimizing the total travel time of relief trains from RT stations to the whole links of the network. The first objective incorporates the measure of link exposure which numerically indicates the criticality of the link to be covered by RTs. This measure is determined by a multi-attribute decision making (MADM) technique and the accessibility of links to road system as a backup service in case of emergency. The second objective assigns opened RT stations to links throughout the network such that the total travel time of RTs is minimized. The proposed problem is solved with a modified version of an e-constraint method called AUGMECON to produce Pareto optimal solutions. A fuzzy logic approach is then applied to select the preferred solution. A case study addressing the railway network of Iran illustrates the application of the proposed model and solution approach. Here, we used the same weight for both objectives, but deciding about proper weights is a tradeoff between maximizing coverage and minimizing total travel time of RTs to the links. Specifically, we performed an experiment to investigate the effect of adopting various weight factors on the preferred solution. The result of experiment reveals the weights in which the highest total coverage is reached whereas the least total travel time is provided. Based on the obtained result of this experiment, we made a location and allocation layout for RTs in the case network. In our study, we also developed a formulation to quantify the improvement in resilience in terms of the rapidity in return to normal performance when a strategy of rapid recovery is adopted. Appropriate action of location and allocation of RTs can play a crucial role in lessening post-disaster delays and improves the resilience of the rail network. The layout obtained by LART model contributes to a significant increase in resilience in terms of passenger and freight load. The sensitive analysis of different values of budget level provides some insight into optimal investment on location and allocation of relief train. The result of this analysis shows that when the budget level exceeds a certain level, the increase in coverage of link exposure throughout the network is negligible. Hence, this analysis might help the decision maker to choose an economical decision while he considers providing maximum coverage of critical links. We also observe that about one-third of the total measure of link exposure is still not satisfied in case of opening all potential ERT stations, so the railway authority should scatter the potential stations over the network to provide efficient coverage. A comparison between the results of LART model and conventional maximal covering location problem (MCLP) also reveals that the proposed model provides a better performance in terms of covering the high exposure links with the same level of budget. Regarding emergency travel time, we observe a drastic decrease in travel time is obtained by LART model compared to the one by MCLP for any budget level when all opened RT stations are available. This fact implies that optimizing two objectives separately (maximizing coverage by MCLP and then minimizing travel time by assignment problem) does not necessarily lead to the least emergency response time while the bi-objective LART model makes sure that least response time is obtained by combining two objectives. When the first RT station is not available, LART model provides less response time than the MCLP although the difference is not significant. Several avenues present themselves as directions for further work. Different categories of emergency vehicles are required to provide cooperative emergency services in a rail incident. Especially, rail-road rescue vehicles provide more flexibility in case of emergency. A generalization that treats location and allocation of such categories is certainly worth developing. As another avenue, adding new RT stations to existing railway rescue centers is expected to be developed. Appendix A In our study, we conducted several interviews with railway experts to distinguish main attributes affecting the importance of link and link exposure. Initial interviews provided useful clues about factors influencing the location and allocation of relief trains in the railway network. In the next step, we are referred to experts who were exactly engaged in the field of rail emergency. It should be mentioned that relief trains in Iran are currently regulated under the supervision of Fleet and Safety Departments. Those experts were the holders of knowledge in the area of emergency logistics in railway which was intended to be investigated in our problem. The details of experts have been included in Table A.1. The last two columns of the table represent the main topics and findings discussed 125

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

Table A.1 Details of interviews with experts. Expert No.

Department (Discipline)

Experience (year)

1 2 3 4

Fleet dep. (wagon) Fleet dep. (wagon) Fleet dep. (RT) Safety dep.

12 10 12 9

Main topics discussed

Main findings

emergency situations in railway • handling influencing the location and • factors allocation of RTs affecting the vulnerability of rail • factors network elements • organizational chart of fleet and safety

layout of relief train • the role of track accessibility in rail • the emergencies role of various attributes affecting the • the importance of link instructions regarding location and • practical allocation of RTs of rail disasters and emergency • types response

Table A.2 The list of attributes considered in the study. Attribute

Description

Rank

Monthly passenger traffic Monthly freight traffic Monthly export/import transit Serviceability to industry Connected OD pairs Link capacity Distance with main stations. Structures link length

The The The The The The The The The

1 2 3 4 5 6 7 8 9

average volume of passenger traffic passing through the link within a month average volume of freight traffic passing through the link within a month amount of freight which is passed through the link on a transit corridor for export/import purposes amount of product passing through the link from industrial places to terminal stations/cities number of OD pairs which is connected by the link capacity of the link distance of the link to the nearest hub station number of structures on the link length of the link

through the interviews. We distinguished the main attributes for determining the importance of link based on both the survey of previous studies and comprehensive discussion through interviews. As discussed in Section 4.2, a review on related literature distinguished some important features of link in transportation networks. Moreover, the result of interviews with the railway experts revealed specific features of rail lines which should be considered during the location and allocation of emergency facilities. Concluding from both literature and the interviews, a set of attributes was prepared so as to be comprehensive and mutually exclusive as displayed in Table A.2. The experts were then asked to score each attribute with the range of 0 to 10 with the highest number as highly important while the lowest number as least important. The third column in Table A.2 shows the rank of each attribute based on the mean score value. After scoring process, those attributes with the higher mean score were determined for calculating the link importance in our study. Such attributes are listed in Table 1. It should be emphasized that the attributes should be determined based on the purpose of the problem under study and may differ from application to application. To obtain the weight of selected attributes, a separate questionnaire was prepared to provide a pairwise comparison between any two attributes. We set the questions in such a way that the purpose of our problem (optimal location and allocation of RTs) is regarded to extract the weights appropriately when the comparison is made by decision maker. For example, the following question is set to make a comparison between the attributes of passenger and freight traffic: “if one hour delay in delivery of a passenger due to link closure costs one unit, how much more (less) does one hour delay in delivery of one ton of freight cost?” The qualitative answers were then converted to quantitative weights according to the fundamental scale of AHP as proposed by (Saaty, 1994). Based on the resulted weights, the pairwise comparison matrix is obtained. To measure the logical rationality of pairwise comparison, the consistency ratio CR = CI /RI is evaluated, where CI is consistency index and can be obtained as follows:

CI =

M M 1

max

(A.1)

where max is the maximum Eigen value of pairwise comparison matrix and M is the number of attributes. Also, RI is the random index for the number of attributes used in decision making and it is determined according to Table A.3. Usually, a CR of 0.1 or less is considered as acceptable, and it reflects an informed judgment attributable to the knowledge of the analyst regarding the problem Table A.3 Random Index (RI) values (Saaty, 1994). Number of attributes

3

4

5

6

7

8

RI

0.52

0.89

1.11

1.25

1.35

1.4

126

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

under study. The relative normalized weight (wi ) of each attribute is obtained by calculating the geometric mean of i-th row of the pairwise comparison matrix (Eq. (A.2)), and then normalizing the geometric means of rows in the comparison matrix by Eq. (A.3): 1/ M

M

GMi =

bij

(A.2)

j=1

wi = GMi /

M i=1

GMi

(A.3)

where bij is an element of pairwise comparison matrix. After determination of the weights of each attribute, the decision matrix is built in which any element, mlj , gives the value of the attribute j for the link l in terms of original real unit. The first row of this matrix, as a case in point, is depicted in the following. The values indicate the monthly passenger traffic (PT), freight traffic (FT), export/ import transit (T), and serviceability to industry (U) of the first link in the case network.

1st link

PT (passenger)

FT (ton)

T (ton)

U (ton)

10,427,000

1,076,342

50,640

483,756

Appendix B. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.tre.2018.09.009.

Reference Bababeik, Mostafa, 2016. Interview report for research project: Location and allocation of reilef trains in the railway network. Iran. Bababeik, Mostafa, Khademi, Navid, Chen, Anthony, Nasiri, M. Mahdi, 2017. Vulnerability analysis of railway networks in case of multi-link blockage. Transp. Res. Procedia 22, 275–284. Balcik, Burcu, Beamon, Benita M, 2008. Facility location in humanitarian relief. Int. J. Logist. 11 (2), 101–121. Belardo, Salvatore, Harrald, J., Wallace, William A, Ward, J., 1984. A partial covering approach to siting response resources for major maritime oil spills. Manage. Sci. 30 (10), 1184–1196. Bell, Michael G.H., Fonzone, Achille, Polyzoni, Chrisanthi, 2014. Depot location in degradable transport networks. Transport. Res. B: Methodol. 66, 148–161. Berman, Oded, Drezner, Zvi, Krass, Dmitry, 2011. Discrete cooperative covering problems. J. Operat. Res. Soc. 62 (11), 2002–2012. Berman, Oded, Krass, Dmitry, Drezner, Zvi, 2003. The gradual covering decay location problem on a network. Eur. J. Oper. Res. 151 (3), 474–480. Borghetti, Fabio, Malavasi, Gabriele, 2016. Road accessibility model to the rail network in emergency conditions. J. Rail Transp. Plann. Manage. 6 (3), 237–254. Bruneau, Michel, Chang, Stephanie E., Eguchi, Ronald T., Lee, George C., O’Rourke, Thomas D., Reinhorn, Andrei M., Shinozuka, Masanobu, Tierney, Kathleen, Wallace, William A., Von Winterfeldt, Detlof, 2003. A framework to quantitatively assess and enhance the seismic resilience of communities. Earthq Spectra 19 (4), 733–752. Caunhye, Aakil M, Nie, Xiaofeng, Pokharel, Shaligram, 2012. Optimization models in emergency logistics: a literature review. Socio-Econ. Plan. Sci. 46 (1), 4–13. Chen, Der-San, Batson, Robert G, Dang, Yu., 2011. Applied Integer Programming: Modeling and Solution. John Wiley & Sons. Cheng, Yung-Hsiang, Liang, Zheng-Xian, 2014. A strategic planning model for the railway system accident rescue problem. Transport. Res. E: Logist. Transport. Rev. 69, 75–96. Ehrgott, Matthias, 2006. Multicriteria Optimization. Springer Science & Business Media. Eiselt, H.A., Marianov, Vladimir, 2009. Gradual location set covering with service quality. Socio-Econ. Plan. Sci. 43 (2), 121–130. El-Rashidy, Rawia Ahmed, Grant-Muller, Susan M, 2014. An assessment method for highway network vulnerability. J. Transp. Geogr. 34, 34–43. Haimes, Yacov Y., 2009. On the definition of resilience in systems. Risk Anal. 29 (4), 498–501. Hogan, Kathleen, ReVelle, Charles, 1986. Concepts and applications of backup coverage. Manage. Sci. 32 (11), 1434–1444. Ingolfsson, Armann, Budge, Susan, Erkut, Erhan, 2008. Optimal ambulance location with random delays and travel times. Health Care Manage. Sci. 11 (3), 262–274. Kachoueyan, Hamid Reza, Nasiri, Hossein, Rahimi, Hamid, 2016. Rail Incident Analysis Vol. 1 Golbahar, Tehran, Iran. Karasakal, Orhan, Karasakal, Esra K, 2004. A maximal covering location model in the presence of partial coverage. Comput. Oper. Res. 31 (9), 1515–1526. Khademi, Navid, Balaei, Behrooz, Shahri, Matin, Mirzaei, Mojgan, Sarrafi, Behrang, Zahabiun, Moeid, Mohaymany, Afshin S, 2015. Transportation network vulnerability analysis for the case of a catastrophic earthquake. Int. J. Disaster Risk Reduct. 12, 234–254. Knoop, Victor L, Snelder, Maaike, van Zuylen, Henk J, Hoogendoorn, Serge P, 2012. Link-level vulnerability indicators for real-world networks. Transport. Res. A: Pol. Pract. 46 (5), 843–854. Li, Xueping, Zhao, Zhaoxia, Zhu, Xiaoyan, Wyatt, Tami, 2011. Covering models and optimization techniques for emergency response facility location and planning: a review. Math. Methods Oper. Res. 74 (3), 281–310. Marianov, Vladimir, ReVelle, Charles, 1996. The queueing maximal availability location problem: a model for the siting of emergency vehicles. Eur. J. Oper. Res. 93 (1), 110–120. Mattsson, Lars-Göran, Jenelius, Erik, 2015. Vulnerability and resilience of transport systems–a discussion of recent research. Transport. Res. A: Pol. Pract. 81, 16–34. Mavrotas, George, Florios, Kostas, 2013. An improved version of the augmented ε-constraint method (AUGMECON2) for finding the exact Pareto set in multi-objective integer programming problems. Appl. Math. Comput. 219 (18), 9652–9669. McDaniels, Timothy, Chang, Stephanie, Cole, Darren, Mikawoz, Joseph, Longstaff, Holly, 2008. Fostering resilience to extreme events within infrastructure systems: characterizing decision contexts for mitigation and adaptation. Global Environ. Change 18 (2), 310–318. Mete, Huseyin Onur, Zabinsky, Zelda B, 2010. Stochastic optimization of medical supply location and distribution in disaster management. Int. J. Prod. Econ. 126 (1), 76–84. Murray-Tuite, Pamela M., 2006. A comparison of transportation network resilience under simulated system optimum and user equilibrium conditions. Simulation Conference, 2006. WSC 06. Proceedings of the Winter. Nagurney, Anna, Qiang, Qiang, 2012. Fragile networks: identifying vulnerabilities and synergies in an uncertain age. Int. Trans. Oper. Res. 19 (1–2), 123–160. RAI, 2016. Year Book of Railway, Overview Report. Department of Transportation, Iran: Iran. Rajagopalan, Hari K, Saydam, Cem, Xiao, Jing, 2008. A multiperiod set covering location model for dynamic redeployment of ambulances. Comput. Oper. Res. 35 (3), 814–826. Rawls, Carmen G, Turnquist, Mark A, 2010. Pre-positioning of emergency supplies for disaster response. Transport. Res. B: Methodol. 44 (4), 521–534. Reggiani, Aura, Nijkamp, Peter, Lanzi, Diego, 2015. Transport resilience and vulnerability: the role of connectivity. Transport. Res. Part A: Pol. Pract. 81, 4–15. Saaty, Thomas L., 1994. Fundamentals of Decision Making and Priority Theory with the Analytic Hierarchy Process. RWS Publications. Snelder, M., Van Zuylen, H.J., Immers, L.H., 2012. A framework for robustness analysis of road networks for short term variations in supply. Transport. Res. A: Pol. Pract. 46 (5), 828–842. Snyder, Lawrence V., 2006. Facility location under uncertainty: a review. IIE Trans. 38 (7), 547–564.

127

Transportation Research Part E 119 (2018) 110–128

M. Bababeik et al.

Tampère, Chris, Stada, Jim, Immers, Ben, Peetermans, Els, Organe, Katia, 2007. Methodology for identifying vulnerable sections in a national road network. Transport. Res. Record: J. Transport. Res. Board 2012, 1–10. Toregas, Constantine, Swain, Ralph, ReVelle, Charles, Bergman, Lawrence, 1971. The location of emergency service facilities. Oper. Res. 19 (6), 1363–1373. Tzeng, Gwo-Hshiung, Huang, Jih-Jeng, 2011. Multiple Attribute Decision Making: Methods and Applications. CRC Press. Tzeng, Gwo-Hshiung, Chen, Yuh-Wen, 1999. The optimal location of airport fire stations: a fuzzy multi-objective programming and revised genetic algorithm approach. Transport. Plan. Technol. 23 (1), 37–55. Ukkusuri, Satish V, Yushimito, Wilfredo F, 2009. A methodology to assess the criticality of highway transportation networks. J. Transport. Security 2 (1–2), 29–46. Yang, Lili, Jones, Bryan F, Yang, Shuang-Hua, 2007. A fuzzy multi-objective programming for optimization of fire station locations through genetic algorithms. Eur. J. Oper. Res. 181 (2), 903–915.

128