Scientia Iranica E (2017) 24(6), 3409{3422

Sharif University of Technology Scientia Iranica Transactions E: Industrial Engineering www.scientiairanica.com

An integrated scheduling and development model in railway networks

M. Shakibayifara, E. Hassannayebib;, H. Mirzahosseinc, Sh. Zohrabniad and A. Shahabie a. Department of Transportation Engineering and Planning, Iran University of Science & Technology, Tehran, Iran. b. Department of Industrial Engineering, Tarbiat Modares University, Tehran, Iran. c. Department of Civil Engineering, Faculty of Engineering, Imam Khomeini International University (IKIU), 34149, Qazvin, Iran. d. Faculty of Management and Accounting, Allameh Tabataba'i University (ATU), Tehran, Iran. e. School of Industrial Engineering, Islamic Azad University, Tehran South Branch, Tehran, Iran.

Received 7 April 2016; received in revised form 28 August 2016; accepted 1 October 2016

KEYWORDS Abstract. The evaluation of the railway infrastructure capacity is an important task of railway companies. The goal is to nd the best infrastructure development plan for Train scheduling; scheduling new train services. The question addressed by the present study is how the Railway infrastructure existing railway infrastructure can be upgraded to decrease the total delay of existing and development; new with minimum cost. To answer this question, a mixed-integer programming Capacity expansion; formulation is extended for the integrated train scheduling and infrastructure development Network problem. The train timetabling model deals with the optimum schedule of trains on a decomposition; railway network and determines the best stop locations for both the technical and religious Con ict resolution. services. We developed two heuristics based on variable xing strategies to reduce the complexity of the problem. To evaluate the e ect of railway infrastructure development on the schedule of new trains, a sequential decomposition is adopted for Iranian railway network. The outcomes of the empirical analysis performed in this study allow to gain bene cial insights by identifying the bottleneck corridors. The result of the proposed methodology shows that it can signi cantly decrease the total delay of new trains with the most emphasis on the bottleneck sections.

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1. Introduction handle this future demand. Constructing or upgrading railway infrastructure is an option for increasing the The passengers' demand on railway transportation is capacity. The railway capacity varies according to expected to increase signi cantly in the future. Hence, di erent factors, e.g. train heterogeneity, train speed, the railway network capacity has to be improved to stop patterns, infrastructure layout, and schedule ro- bustness [1]. Train stop time de nes the amount of *. Corresponding author. time trains spend stopped at a location. The stopping E-mail addresses: m [email protected] (M. pattern is a decisive factor to determine the operational Shakibayifar); [email protected] (E. Hassannayebi); capacity of the network. In Iranian railway network, [email protected] (H. Mirzahossein); [email protected] (Sh. Zohrabnia); trains stop at intermediate stations for load/unloading St a [email protected] (A. Shahabi) passenger, technical services as well as praying services. The last factor is a speci c religious related constraint doi: 10.24200/sci.2017.4397 that signi cantly a ects the train timetables. Based 3410 M. Shakibayifar et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 3409{3422 on religious regulation, during prede ned praying time capacity and resource constraints [3]. A majority of the windows, train must stop to perform praying services. approaches to train timetabling are analytical meth- Passengers get o the train and pray in the station ods. Moreover, the application of exible simulation mosque during the time window. However, the praying approaches for railway systems has been recognized by time windows change over time and location due to the previous studies [4]. Huisman et al. [5] provided an local geographic position. Thus, the optimal pattern excellent state-of-the-art review of railway optimization of stop for praying service has direct in uence on the problems. In the literature, numerous mathemati- capacity of the Iranian railway system. The optimiza- cal formulations are presented for train timetabling tion of the railway lines' capacity plays an important problem. Train timetabling problem is known to be role in railway transportation industry. The e ective NP-hard with respect to the number of con icts in utilization of a railway network results in the avoidance the schedule [6,7]. Thus, it is dicult to determine of resource con icts, and, at the same time, nding the optimum solutions to industry-sized problems in a an appropriate balance between capacity utilization reasonable time, and this raises the need for ecient and level of service [2]. This paper is concerned with heuristic and meta-heuristic algorithms. In the related Railway Network Infrastructure Development Problem literature, several sophisticated search procedures are (called RNIDP). The proposed approach combines the introduced, such as look-ahead search [8], backtrack- scheduling of new trains and re-scheduling of existing ing search [9], and meta-heuristics algorithms [10- trains by choosing the best railway infrastructure devel- 12]. D'ariano et al. [13] addressed the real-time train opment scenario for decreasing train delays. The focus scheduling problem in a railway network using an of this work is on the combinatorial analysis of infras- ecient Branch and Bound (B&B) algorithm. The em- tructure con guration optimization and timetable gen- pirical test cases make evident that the B&B algorithm eration. In other words, our work addresses the railway outperforms the commonly-used dispatching in terms infrastructure planning and train timetabling problem of both average and maximum secondary delays. A in one integrated model. To the best of our knowledge, decision support system, called ROMA, was designed this approach has not been extensively investigated in by D'Ariano [14] based on Alternative Graph (AG) literature, and if so, not to the equivalent scope to techniques to cope with real-time train rescheduling which it is taken in this study. In this paper, RNIDP problem with multiple delays. The aim was to improve is formulated as a Mixed-Integer Linear Programming punctuality index through better utilization of the (MILP) model. The key contribution of the study is the railway infrastructure. The system was extended by integrated investigation of the train timetabling and Corman et al. [15] to present an innovative distributed infrastructure development problems. The previous approach to manage train operations more e ectively in studies in the literature present an independent study multi-area dispatching areas. The performance of the of train timetabling and capacity planning problems. distributed approach was compared with the existing To deal with such a complex optimization problem, models in terms of computation time and reducing total a novel network decomposition approach is presented. delay. Hassannayebi and Kiaynfar [16] proposed three Furthermore, in order to solve the resulting sub- meta-heuristic algorithms based on Greedy Random- problem eciently, a fast heuristic based on variable ized Adaptive Search Procedure (GRASP) to nd the xing strategy is proposed. near-to-optimal train timetable in double- railway The rest of the paper is as follows. In Section 2, lines. The output results show the e ectiveness of a literature review of train timetabling problem and the proposed meta-heuristic algorithm in solving large- related issues is discussed. In Section 3, the problem sized instances of the train timetabling problem. is described in detail. In Section 4, the mathematical In a vast majority of the previous studies, the programming formulation for RNIDP is described. In train schedule is an input for railway capacity planning Section 5, heuristic algorithms are presented in detail. model. Some recent and related capacity planning liter- In Section 6, numerical analysis and the results are ature that is associated with aspects of train scheduling described. Finally, the discussion and conclusion are problems is as follows: proposing a scheduling model stated in Sections 7 and 8, respectively. for inserting additional trains into the existing time table without any railway infrastructure expansion 2. Literature review plan. Sajedinejad et al. [17] designed a simulation- based decision support framework called SIMARAIL Train timetabling is one of the most interesting for scheduling trains in railway networks. The proposed problems in transportation planning systems. Train framework was used as a tool to assess the di erent timetabling problem basically consists of determining train dispatching policies. Lai and Shih [18] developed the train's departure and arrival times in each sta- a capacity planning procedure for evaluation of pos- tion. The assessment of train timetable is a complex sible expansion scenarios and determine the optimal procedure due to the fact that it is subjected to the network investment plan to meet the future demand. M. Shakibayifar et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 3409{3422 3411

Hasannayebi et al. [19] developed an event-driven rail simulation tools to improve capacity utilization. simulation model for train timetabling problem with The developed system has the capability of automatic consideration of train stops for praying services. Path resource con ict resolution and train scheduling tools relinking algorithm was implemented to obtain good for multiple-track corridors. The methodology was quality solutions. Goverde et al. [20] introduced the validated through comparison with RailSys simulation new indicator for dynamic infrastructure occupation package. Hassannayebi et al. [26] proposed a robust assessment and analyzed infrastructure capacity under multi-objective stochastic programming approach for disturbances. The capacity assessment study was train scheduling at rapid rail transit lines. The performed on Dutch railway corridor with di erent mathematical models were developed to minimize the signalling systems under both scheduled and perturbed expected waiting times and the cost of overcrowding. trac circumstances. For scheduled trac, the UIC The e ectiveness of the proposed model was validated standard was used to compute infrastructure capac- through the application to Tehran underground railway ity, whereas Monte Carlo simulation technique was network. Pohleand Feil [27] conducted a research employed to evaluate rail capacity under disrupted based on integration of train scheduling and infrastruc- conditions. Schobel et al. [21] proposed an optimiza- ture planning. tion model for integrated timetable and infrastructure Corman et al. [28] addressed the integrated train design of a railway system. The problem was for- scheduling and delay management in real-time railway mulated as a combinatorial optimization problem and trac condition. The delay management model focuses solved using mathematical programming techniques on the e ect of rescheduling actions on the quality and metaheuristics. Hassannayebi et al. [22] proposed of service. The result of empirical test experiments a simulation-based optimization approach for train based on Dutch railway network con rms that good timetabling in urban rail. The problem was solved quality solutions can be obtained within a reasonable using genetic algorithm and the outcomes verify the computation time. Sama et al. [10] applied meta- e ectiveness of the solution method. heuristic algorithms to the real-time train timetabling The previous studies on capacity planning in problem in complex and congested railway networks. railway are mostly dedicated to the maximization of The proposed methodology is initiated with generating the throughput without making new infrastructure a good quality solution via a Branch and Bound investments. In the literature, two aspects of railway (B&B) algorithm. Subsequently, Variable Neighbor- capacity planning problem exist. The rst is the short- hood Search (VNS) and Tabu search algorithms were term planning of scheduling additional trains and the implemented to improve the solution by re-routing second is the long-term planning of railway infrastruc- of trains. The result indicated that the proposed ture upgrading. Railway infrastructure development algorithm outperforms the solutions attained by a is strongly related to a famous railway optimization commercial optimization package in terms of reduced problem called railway saturation problem. In the computation times and search performance. Qi et railway saturation problem, the main objective is to al. [29] designed an integrated multi-track station insert a maximum number of additional trains in a layout design and train scheduling model on railway prede ned train set. A notable point is that the railway corridors. The optimization model decides the number saturation problem does not deal directly with train of tracks or platforms within the budget con- scheduling problem. Delorme et al. [23] developed two straints. The total construction cost and total train heuristic approaches to evaluate the railway infrastruc- travel time are assumed as performance metrics. A bi- ture capacity. These models do not consider cost-based level programming model was developed to solve a train objective function and also service level constraints. scheduling model along with track assignment problem. Shih et al. [24] proposed an optimization model for the The optimization models were solved almost eciently optimal siding locations for single-track railway lines by commercial software GAMS with CPLEX solver subject to the infrastructure and trac characteristics. and local searching-based heuristic. As a concluding The experimental results proved that the optimal plan remark, a wide range of papers are devoted to the train can maximize the return on investment and achieve timetabling problem in the last decades, but there is the desired service level. Vansteenwegen et al. [25] still lack of integrated models which combine infras- proposed a method for adjusting an initial in case tructure upgrading issues with timetable generation. of planned temporary infrastructure unavailability. A trade-o was made between the level of service and 3. Problem description the capacity. The proposed model incorporates the timetable robustness to rescheduling procedure. The The train timetabling problem deals with the opti- developed procedure also aims to minimize the number mum dispatching of trains on a railway network and of cancelations. Pouryousef et al. [2] proposed a multi- determines the best station to stop for technical and objective linear programming model together with service-based purposes including religious requests. In 3412 M. Shakibayifar et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 3409{3422 the following sections, we describe the assumptions, no- Parameters: tation, introduction of special constraints, and railway (j; k) Index of the kth traveling segment in infrastructure terms in detail. the route of train j 3.1. Assumptions Kj Total number of segments on route of The problem is concerned with a railway network with train j a set of linked corridors. Railway network consists jk 1 if train j is an inbound train in of single, double, and multiple-track routes. A train segment k, 0 otherwise service is de ned as a trip of a train that travels from 'j1;j2;k 1 if segment k exist in route of train j1 its origin station to its destination station. Each train and j2 is assumed to have a pre-speci ed traveling route in the ij 1 if train j needs to have access network. Furthermore, the free running times of trains to platform for loading/unloading at segments are assumed to be constant. Train services passengers in station i, 0 otherwise are determined by a set of planned and unplanned pjk Free running time of train j at segment stops. There are three categories of trains' stops: k 1. Scheduled stops with xed stop time; dwij Minimum dwell time of train j in station i 2. Unscheduled stops with xed stop time; Hj1:j2:i The minimum headway between 3. Unscheduled stops with variable stop time. arrival/departure time of train j1 and j2 at station i Train technical services are planned during the sts Stopping time required for technical operation cycle, but religious services occur during stop type s dynamic and time-dependent intervals. Scheduled pt Passenger load and unloading time for stops are permitted at any intermediate stations for ij train j in station i any train. But, unscheduled stops are permitted in the limited set of stations which have the required facilities, spr The required time for praying type r e.g. mosque. We consider a track section between two msij Scheduled stop time for train j at adjacent stations. To prevent this scheduled delay, a station i minimum headway time needs to be considered. A edj Earliest departure time of train j from minimum headway time is the minimum time di erence its origin station (j 2 J2) between dispatching of two following trains that should ldj Latest departure time of train j from be kept in order to satisfy trac safety regulations. its origin station (j 2 J2) 3.2. Notation sdj Scheduled departure time of train j The notations of indices and sets used in the mathe- from its origin station (j 2 J1) matical model, the notations of parameters used in the i 1 if the location of mosque facility is mathematical model, and the decision variables used in in the inbound lane at station i, 0 the mathematical formulation are presented as follows: otherwise (i 2 I1) 1 if there is an facility at kth traveling Indices: v(j;k) segment of train j for technical stop, 0 j Train index otherwise k Segment index i 1 if there is an overpass facility in i Station index station i, 0 otherwise s index of technical stop types i 1 if there is a mosque facility in station (s = 1; 2; 3; :::; S) i, 0 otherwise r index of praying service r (r = NLi Number of tracks which is not 1; 2; 3; :::; R) connected with any platform in station i J1 Set of existing trains (J1 2 J) Npi Number of tracks which is connected J2 Set of new unscheduled trains (J2 2 J) with a platform in station i

I1 Set of one-lane stations (I1 2 I) Lri Lower bound of time window of praying service type r in station i I2 Set of two-lane stations (I2 2 I) U Upper bound of time window of K Set of single track segments ri 1 praying service type r in station i K2 Set of double track segments tp Minimum time to prepare for praying CO Set of corridors in the railway network and performing praying service M. Shakibayifar et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 3409{3422 3413

C Desirable value for summation of trains depends on the location of site and daily time horizon traveling time changes. Praying times change daily according to M A large positive constant the seasons and train exact location in the railway network. Performing religious services for trains also LCi Construction cost of a new track in station i depends on the dispatching time of the trains from their origin stations and the arrival time to destination PCi Construction cost of a new platform in station i station. Finding the best station for stopping to perform religious service is a challenging optimization Decision variables: problem added to train timetabling problem. Eqs. (1) xdjk Departure time of train j from and (2) state the condition which forces the trains to beginning of the segment k stop for praying service in the intermediate stations. xajk Arrival time of train j at the end of We denote the dispatching time of train j from the the segment k origin station by xd and arrival time of this train j;(j;1) to destination station by xaj;(j;K ). Every praying zj1;j2;k 1 if train j1 is entered before train j2 j in the same direction on segment k, 0 service type needs to be checked before determining otherwise the best station to stop. If the following conditions are satis ed, then train j should stop in one station (which uj1;j2;k 1 if train j1 is entered before train j2 in the opposite direction on segment k, have mosque facility) on its route for praying service 0 otherwise type r. The two conditions of praying service activation can be expressed as the following two equations: ysij 1 if train j stopped for sth technical service at station i, 0 otherwise xdj;(j;1)  Lr;(j;1) + tp wijt 1 if train j is assigned to tth track at station i, 0 otherwise 8j 2 J; 8r = 1; 2; 3; (1) oijt 1 if train j occupies tth platform at station i, 0 otherwise xa  U t j;(j;Kj ) r;(j;Kj +1) p grij 1 if train j stopped for rth praying service at station i, 0 otherwise 8j 2 J; 8r = 1; 2; 3; (2) Ali Number of additional tracks (do not connected with platform) need to be where (Lri;Uri) is the praying time interval in station i, constructed in station i when trains are allowed to stop for performing praying

Api Number of additional tracks (connected service with type r. Note that, in Eq. (1), Lr;(j;1) is with platform) need to be constructed de ned as a lower bound of praying interval for praying in station i service r in origin station of train j on its route. Trains may have to stop for all praying services, or none of Construction of new platform and track in the them. station is considered as infrastructure development options in the mathematical model. The optimization model determines which section of the railway network 4. Mathematical formulations needs to be upgraded with what type of capacity In this section, we formulate the train timetabling improvement scenario. Construction of new lines problem as a mixed-integer linear programming. In the and platforms in each station has a prede ned cost following, we state the details of the integrated math- which depends on the station infrastructure character- ematical model for generating an extended timetable istics. according to the best railway infrastructure upgrading 3.3. Religious-related constraints plan. The objective function is to minimize Eq. (3) sub- We considered several religious constraints in our ject to constraints given by Eqs. (4)-(20). The objective model that appear in Iranian railway network. Train function is the minimization of railway infrastructure movements on the railway network in Iran are regulated development cost: by the three daily prays. All trains should stop during the praying time window for a period of 20- XjjIjj 25 minutes. Each praying service has a dynamic time min Z = (Ali:LCi + Api:P Ci) ; (3) window to be performed. This time window depends i=1 on geographic location of station and canonical time xd = sd 8j 2 J ; (4) horizon. These time horizons consist of morning, noon, j:(j:1) j 1 and sunset. We assume that each station has a local ed  xd  ld ; 8j 2 J ; (5) canonical time horizon. So, the start time of praying j j;(j;1) j 2 3414 M. Shakibayifar et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 3409{3422

Kj 1 xaj; = xdj; + pj; ; X (j;k) (j;k) (j;k) :y = 1; 8j 2 J; 8s 2 S; (17) (j:k) s:(j:k):j 8j; k = 1; 2; :::; kj; (6) k=1

xd  xa + dw + ms ij  msij ; (18) j;(j;k+1) j;(j;k) (j;k);j (j;k);j

jjJjj 8j; k = 1; 2; :::; kj; i 2 I; (7) X oijt:ij  Npi + Api; 8i 2 I; 8t 2 T; (19)  Xjjsjj j=1 ms = max pt : st ; y (j:k);j (j;k);j s s:(j;k);j s=1 oijt + wijt  1; 8i 2 I; 8j 2 J; 8t 2 T: (20) Constraints (4) and (5) are related to the departure Xjjrjj  : sp ; q ; i 2 I ; (8) time constraints. Constraints (4) de ne the departure r r;(j;k);j 1 r=1 time of the train at the rst segment. Constraints (5) ensure that the departure time of the train from the  Xjjsjj origin is within the allowed range. Eq. (6) states the ms = max pt : st :y (j;k);j (j;k);j s s;(j;k);j running time constraints on segments. Eq. (7) de nes s=1 the minimum station dwell time constraint. The train technical, passengers load/unload and religious jjrjj  X services are parallel tasks which are performed in : (sp +t : : ):g ; r 0 (j;k) j;(j;k+1) r;(j;k);j the stations. So, the maximum stopping time of a r=1 train in a station is the maximum time of all tasks above. The calculated stopping time in each station i 2 I2; (9) is distinguished in Constraints (8) and (9) according ( xd + p + H  xd + (1 z ):M to the station characteristics. According to Eq. (8), j1k j1k j1:j2:K j2;k j1;j2;k the scheduled stop time for train j (running in the xd + p + H  xd + z :M j2:k j2k j1;j2;k j1:k j1;j2;k inbound direction) in station is the maximum time of all possible technical operations or religious services. 8j :j :k; ' = 1;  +  = f0:2g (10) 1 2 j1:j2;k j1:k j2:k Likewise, the scheduled stop time of trains at outbound ( route is formulated in Eq. (9). The minimum headway xd +p +H  xd + (1 z ):M j1k j1k j1:j2:K j2:k j1:j2:k Constraints (10)-(12) also describe the minimum safety

xdj1:k + pj1k + Hj1:j2:K  xdj2:k + zj1:j2:k:M headway requirements between the departure times and arrival times of successive trains at the same

8j1:j2:k; 'j1:j2:k = 1; j1:k + j2:k = f0:2g (11) segment. More speci cally, Constraints (10) and (11) ( state the overtaking con ict for trains. Constraint (12)

xdj1k +pj1k +Hj1:j2:K  xdj2:k +(1 uj1:j2:k):M states the crossing con ict for inbound and outbound trains. Constraint (13) states that the total traveling xdj2:k + pj2k + Hj1:j2:K  xdj1:k + uj1:j2:k:M time of trains should be less than a desired value. Constraint (14) ensures that the number of trains 8j1:j2:k; 'j1:j2:k = 1; j1:k + j2:k = 1 (12) occupying a station in the same time is less than jjXJjj   maximum capacity of the station. Eqs. (15) and xaj: xdj:  C; (13) (16) are related to the situation that trains stop for (j:kj ) (j:1) j=1 praying services. Constraints (15) ensure that only one station is assigned to each train to perform praying jjXJjj services. Constraints (16) state that the departure w  1 + 8i 2 I; 8t 2 T; (14) ijt i time of trains is between the allowed praying time j=1 windows. Constraint (17) de nes the stopping pattern of trains for technical services. Trains are permitted to jjXKj jj stop at a set of eligible stations for technical services.  :gr: = 1; 8j 2 J; 8r 2 R; (15) (j:k) (j:k):j Constraint (18) states that a free platform should k=1 be allocated to a train which stops in the station. L M(1 g )  xd  U r:(j:k) rij j:(j:k+1) r:(j:k) Constraint (19) ensures that the number of trains which needs free platform, at any given time, is less than the + M(1 grij); total number of existing and additional platforms in the stations. Eq. (20) states the logical constraint about 8j 2 J; 8k 2 K; 8r 2 R; (16) occupation of a platform in the stations. M. Shakibayifar et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 3409{3422 3415

NDrj Set of trains which their conditions for praying service type r are not de nitely active

eaj Earliest possible arrival time of train j to its destination station

APrj 1 if the praying service type r is active for train j, 0 otherwise

rj 1 if the rst condition of praying service is true, 0 otherwise

rj 1 if the second condition of praying service is true, 0 otherwise

eaj can be calculated as follows:

KXj +1 XKj ea =ed + (pt +dw )+ p ; j j (j;k);j (j;k);j j;(j;k) Figure 1. The overview of the proposed heuristic k=1 k=1 algorithm. 8j 2 J: (21) 5. Solution methodology The praying service activation conditions can be stated In this section, we present two heuristics and a de- by parameters eaj and ldj in the following equations: composition approach to reduce the complexity of the problem. The overview of the proposed decomposition- ld  L + t ; (22) j r:(j;1) p based heuristic algorithm to solve train timetabling problem in network is illustrated in Figure 1. Heuristic eaj  Ur: tp: (23) algorithms contributes to a decrease in the number of (j;Kj +1) decision variables and constraints via variable xing Next, we describe the main components of our prepro- strategies. The decomposition approach is also used for cessing in Algorithm 1. dividing the complete optimization problem into sub- We proposed a mathematical optimization model problems. Each sub-problem is associated with a train which determines the departure time of trains from the timetabling model on a separate route. origin station to minimize the number of active praying services. This algorithm is a class of heuristics which 5.1. Variable xing strategies is based on constraint satisfaction techniques. In the Given the mathematical model presented in the previ- following, we propose the mathematical model for the ous sections, the problem turns into a large-scale mix MAPS algorithm: integer program. As is well known, this type of model is far more dicult to solve optimally. Here, to handle the above mixed-integer programming model with an jjXJjj XR exponential number of variables and constraints, we min Z = APrj; (24) present a preprocessing algorithm for decreasing the j=1 r=1 number of active praying services. By applying this procedure, we can determine the condition of praying 2APrj  rj + rj 8j 2 J; 8r 2 R; (25) service activation before optimizing the mathematical model. By changing the dispatching time of a train rj + rj  1 + APrj 8j 2 J; 8r 2 R; (26) between its earliest and latest departure times, the praying services may become active or inactive. The xd L t  M(1 ); goal is to assign the trains into a set of de nite j:(j:1) r:(j:1) p rj and inde nite trains according to their praying service activation. The notations used to describe the heuristic 8j 2 J; 8r 2 R; (27) procedures are presented as follows: xd L t > M: ; j;(j:1) r:(j:1) p rj Drj Set of trains which their conditions forpraying service type r are de nitely 8j 2 J; 8r 2 R; (28) active 3416 M. Shakibayifar et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 3409{3422

Algorithm 1. The main components of the proposed preprocessing.

xaj; Ur: + tp  M:(rj 1); for searching the infrastructure upgrading options will (j:Kj ) (j:Kj +1) decrease. 8j 2 J; 8r 2 R; (29) In order to account for the evaluation of di er- ent capacity expansion scenarios, we propose a novel xaj; Ur: + tp  M:rj ; heuristic algorithm, which decomposes the physical (j:Kj ) (j:Kj +1) network into bottleneck and underutilized corridors. 8j 2 J; 8r 2 R; (30) The de nition of bottlenecks can be based on the analysis of infrastructure, technical de nitions, opera- KXj +1  tional systems, economical, spatial and social context, xaj: =xdj: + pt + dw (j:Kj ) (j:1) (j:k):j (j:k):j and travel demand forecasts. Normally, the trac k=1 density through the bottleneck sections is expected to be very high. We de ne the bottleneck corridor as the XKj + p 8j 2 J; 8r 2 R; (31) demand for transport exceeding the available capacity j:(j:k) of infrastructure. The underutilized corridors are the k=1 other railway sections with lower demand. Bottlenecks rj = (0; 1); rj = (0; 1): (32) in railway transport system usually are concerned with: (1) the level of transport speed (maximum travel speed, In the formulation, Objective (24) de nes the number train travel time, passenger waiting time); and (2) the of total active praying services. Constraints (25) capacity (maximum number of operating trains per and (26) ensure that if the two conditions of praying railway section). The analysis to identify the existing services are true, then the praying service will be active. bottlenecks corridors come up with the necessary mea- Constraints (27) and (28) de ne the relation between sures. On the main stream of bottleneck analysis for the rst condition of praying service activation with the existing railway infrastructure is the evaluation of auxiliary variables. Similarly, Constraints (29) and the train timetables to calculate the average delay time (30) de ne the relation between the second condition of trains. The main idea of this heuristic algorithm of praying service activation with auxiliary variables. is decomposition of a railway network based on our Constraints (31) de ne the trains traveling time. de nition of the level of railway capacity utilization. 5.2. Physical railway network decomposition Di erent policies for generating train timetables are It should be noted that large-scale train scheduling then applied to the two types of corridors according problems with hundreds of trains, which move in to their properties. di erent routes along hundreds of stations, are far The scheduling strategy initiates by solving the more dicult to solve exactly. Based on a physical timetabling problem of the bottleneck corridors and decomposition of the railway network in sub-networks, then using the result of the solved problem to gen- the problem is divided into sub-problems corresponding erate timetable for other underutilized corridors. In to these sub-networks. These sub-problems are then bottleneck corridors, the track topology usually con- solved and the whole process is coordinated at a higher sists of single-track segments, hence there is a huge level in order to generate a global feasible solution. potential for railway infrastructure development. In This decomposition procedure is performed in a hierar- the infrastructure development problem, we test dif- chy framework. The proposed decomposition approach ferent scenarios to decrease the trac density of the consists of corridor-based decomposition, train route- bottleneck corridors. In order to clarify the de nition of based decomposition, and double-track segments de- the bottleneck corridors, we used the terms bottleneck composition. In this framework, computational e ort section and bottleneck station to quantify the trac M. Shakibayifar et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 3409{3422 3417 density on the railway resources. The bottleneck sec- tion and bottleneck station are de ned by a percentage of potential trac density. A saturation criterion is used to quantify the trac density of a railway resource. Delorme et al. [23] de ned the capacity of a segment of a rail system as the maximum number of trains that can be scheduled on it within a certain unit of time (e.g., an hour). This concept is extended here to determine the trac density ratio of the sections. The following notations are used for describing the saturation criteria:

Kc Set of segments exist in corridor c

Ic Set of stations exist in corridor c k The minimum trac density ratio of Figure 2. Total number of passengers carried in the track segment k Iranian rail network.

i The minimum trac density ratio of station i methods. Transport system of the Iranian railway is c The minimum utilization ratio of a growing fast now (Figure 2). The length of railway as corridor c well as expressway network have doubled during the LDjk Latest departure time of train j from last 15 years. The economic aspects of this expansion the beginning of the segment k plan have not been yet investigated extensively by the transport experts; this is the main motivation of the EAjk Earliest arrival time of train j at the end of the segment k preset study. By 2020, the Iranian railway expects to reach every regional capital by rail. Nevertheless, eaij Latest departure time of train j from station i the demand is much higher than the available trains, as new rolling stock is always one step behind the ld Earliest arrival time of train j at ij opening of new lines and the tracks are shared with station i heavy freight trac. The theoretical expressions of the capacity of a section and station noted k and i, respectively, can 6.1. Case data be de ned as: The decomposed mathematical formulation has been PjjJjj applied to the whole railway network which consists (EAjk LDjk) = j=1 8k 2 K ; (33) of long railway corridors. The example includes 132 k T c trains and 56 stations and the network is composed

PjjJjj of 30 double-tracked and 25 single-tracked segments ij : (eaij ldij)  = j=1 8i 2 I ; (34) (Figure 3). The Tabriz-Mashhad corridor is a double- i T c tracked line with segments separated by stations, where where T is the planning horizon. Now, we can calculate a mean of 90 daily long distance trains run from the weighted average of the station and section satura- Tabriz to Mashhad, or vice versa. Because of special tion ratios as utilization ratio of a railway corridor as infrastructure characteristics of the railway network in follows: Iran, there is a signi potential for increasing the P P railway capacity by upgrading railway infrastructure. jjIcjj i + jjKcjj k  = i2Ic k2kc 8c 2 CO: The information of the existing railway network is c jjI jj + jjk jj c c (35) presented in Table 1. Therefore, we can divide the set of corridors into the 6.2. Numerical analysis bottleneck corridors with the highest ratio of utilization To demonstrate the capabilities of the approaches and underutilized corridors. This separation is accord- developed in this paper, a real case study is solved. ing to a prede ned threshold ratio of utilization (i.e., The results of preprocessing algorithm are summarized 30%). for di erent train services. The output of the algorithm is the categorization of trains according to their praying 6. Computational results service activation status. In this case, there are three daily praying services and each praying service could In this section, some examples taken from the Iranian be de nitely active or not. The trains with de nite railway network are used to illustrate the proposed praying service status can help reduce the number 3418 M. Shakibayifar et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 3409{3422

Figure 3. The Iranian railway network (north corridor).

Table 1. Infrastructure information of the existing railway network. Cost of constructing Cost of constructing Station # Track # Platform a track with platform a track without platform ($ per km) ($ per km) 1-5 4 2 8700 7600 6-15 3 1 6900 8700 16-20 5 3 8800 4200 21-56 4 2 6400 6600 of decision variables and constraints in mathematical result of each sub-problem and the average delay of model. The total number of decision variables de- the existing and new trains are summarized in Table 3. creases 13% after implementing these algorithms in a Each of the sub-problems generated from the decom- mathematical model. position approach was solved by using GAMS/CPLEX The purpose of decomposition approach is to 12.0 on a PC equipped with a 3.3 GHz Pentium evaluate how much the average delay can be decreased IV processor. Regarding the computation time, the by the suggested decomposition approach and compare decomposed sub-problems are terminated in less than the results to the outcome if the infrastructure stays 12 min for all instances. However, it should be noted unchanged. In this example, the number of the existing that this was due mainly to the use of the proposed trains in circulation is 100 before adding the additional decomposition method. According to the obtained train services. We plan to schedule 32 new trains to the result given in Table 3, the existing number of tracks railway network in the planning horizon to satisfy the and platform has been expanded, on average, by about constraints given in Section 4 by choosing the minimum 32% and 46%, respectively. The average delay of train cost infrastructure upgrading options. The result of the in the whole network after upgrading the infrastruc- decomposition algorithm is shown in Table 2. ture is about 6.9 minutes. Furthermore, the total Di erent capacity expansion scenarios have been cost of upgrading the existing infrastructure is about generated in the decomposed optimization models to 511000 ($). The objective value also shows a strong increase the utilization of the railway capacity. The relation between the construction of new platforms M. Shakibayifar et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 3409{3422 3419

Table 2. Physical decomposition of the railway network. Minimum utilization Saturation Number of generated Number of train Corridor name a ratio (c) status sub-problems sub-route Tehran-Mashhad 14% Ub 4 12 Tehran-Tabriz 40% Bc 5 7 Tehran-Ahvaz 35% B 3 5 Tehran-Sarri 11% U 4 8 aThreshold ratio of utilization = 30%; bU: under-utilize; cB: bottleneck.

Table 3. Size, speci cation, and optimal solution of the generated sub-problems. Infrastructure Sub- Infrastructure Infrastructure % Increase % Increase Average delay Computation development problem before upgrading after upgrading of NCT of NCP (min) time (sec) cost ($) NTa NSb NCTc NCPd 1 14 9 6 4 42.86% 44.44% 54000 13.4 540.44 2 12 10 4 3 33.33% 30.00% 24000 9.6 650.45 3 9 7 2 3 22.22% 42.86% 12000 10.3 300.35 4 10 8 4 4 40.00% 50.00% 42000 5.4 400.34 5 24 4 5 5 20.83% 125.00% 10000 4.2 760.76 6 15 9 4 6 26.67% 66.67% 20000 6.8 230.23 7 7 7 4 2 57.14% 28.57% 48000 4.7 320.35 8 21 11 3 1 14.29% 9.09% 28000 6.9 140.34 9 8 5 3 2 37.50% 40.00% 58000 8.3 300.45 10 9 4 1 1 11.11% 25.00% 70000 2.2 130.15 11 10 9 4 3 40.00% 33.33% 12000 4.5 100.65 12 8 10 2 3 25.00% 30.00% 20000 6.6 70.16 13 5 8 3 4 60.00% 50.00% 45000 7.2 60.54 14 8 9 2 4 25.00% 44.44% 30000 5.4 50.76 15 9 5 3 5 33.33% 100.00% 13000 3.1 110.55 16 10 8 2 2 20.00% 25.00% 25000 11.9 260.89 aNT: The number of scheduled train; bNS: The number of existing stations; cNCT: The number of new tracks; dNCP: The number of new platforms. and decreasing the average delay resulted from the derutilized corridors (Table 4). However, the aver- combination of the infrastructure development options. age computation time of the decomposition method Also, the obtained results indicate that adding a new for bottleneck instances is mostly obtained from the platform in the station is more e ective than adding decomposition method for underutilized instances. a new track in terms of decreasing the average train There is a need for upgrading railway infras- delay. tructure when the number of train increases. The The decomposition method obtains a better ob- decomposition approach helps nd the minimum cost jective value for the instances which belong to the scenario for developing infrastructure and decreasing bottleneck corridors rather than to those in the un- delay systematically. As stated before, the proposed

Table 4. Improvement of the total delay by development of the bottleneck corridors. Corridor Saturation Improvement in Average computation name status total delay (%) time (sec) Tehran-Mashhad U 13 234 Tehran-Tabriz B 47 650 Tehran-Ahvaz B 54 925 Tehran-Sarri U 11 353 3420 M. Shakibayifar et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 3409{3422

task. Therefore, numerous capacity expansion scenar- ios should be evaluated in order to work out how many extra trains can be scheduled by the existing infrastruc- ture and how much investment will be required for new infrastructure. In other words, the railway capacity is extremely dependent on infrastructure. It is clear that the proposed decomposition approach does not guaran- tee for the optimal solution to complete mathematical model. As a consequence, the results lend further credence to our earlier suggestion that decomposition approach (especially route-based decomposition part) has short computation time as an advantage for large- scale problems. Specially, we conclusively recommend Figure 4. Average delay increases by scheduling additional train services. establishing a reductionism procedure for eliminating the extra decision variables before optimization. The preprocessing helps the decomposition approach to decomposition method identi es the potential infras- explore the regions in a more promising way in less tructure alternatives and chooses the near-optimum time than the exact approaches does. The proposed capacity expansion plan. Figure 4 shows how the method allows us to analyze the performance of the average delays increase exponentially when the number railway networks (e ect of infrastructure development) of trains exceeds the saturation level. The average on delays as it has been demonstrated in the real case. delay is computed as the ratio of the total train delay at destination to the total number of trains. This value is computed for each test problem and the associated 8. Conclusion best found solution. In order to obtain each point in The design and optimization of new railway infras- this graph, we solve the optimization model to nd the tructure is a complex long-term planning procedure best capacity upgrading plan. Also, the timetabling that involves several operational constraints. The new model is solved for the case when no capacity expansion upgraded rail infrastructure a ects the timetabling de- decisions are allowed. Then, the best found solutions cisions as well. Thus, it is worth analyzing both tactical are considered and their performances are compared in and strategic aspects of this problem in an integrated terms of average delay of trains at destinations. approach. Up to now, there is no optimization-based decision support tool to determine a minimum cost 7. Discussion infrastructure upgrading plan by taking into account all the constraints de ned by the operation of train The question addressed by the present study is how trac in the railway system. In this paper, we the railway infrastructure can be upgraded to decrease presented a mathematical programming model for the the total delay of the existing and new trains. The train scheduling and railway infrastructure develop- main nding of the study is that developing the infras- ment problems. The Iranian railway network was se- tructure of the bottleneck corridors is more e ective lected as the test benchmark. The religious constraints than upgrading underutilized corridors for reduction associated with the passenger praying services were of total delay. In our experiments, the average delay addressed in the timetabling model. The mathematical in generated timetable relates strongly to the railway model of the present paper was then extended to handle infrastructure. As expected, by increasing the number the infrastructure planning issues. In our framework, of platforms in the stations, the average delay becomes the complex mathematical model was simpli ed by two shorter. However, railway development problem makes heuristic methods based on variable xing techniques congestion issues that a ect the resulting timetable. in order to be solved eciently. We also presented a The experiments showed that there is a potential network-wide decomposition procedure with the aim of improvement for decreasing the average delay by imple- reducing the complexity of the complete mathematical menting the decomposition approach and construction model of the centralized instance. The result of imple- of new platforms in the bottleneck corridors. menting the proposed method shows the advantages Evidence that our model helps the railway oper- of the methodology to design the railway networks ators in infrastructure development decisions is that with minimum cost so that it makes us capable of the evaluation of several capacity expansion scenarios scheduling new trains eciently. We considered the takes a lot of time even in the simulation software construction of new tracks and platforms as infrastruc- products. It should be noted that optimizing the use of ture development options to upgrade the infrastructure railway infrastructure is a complex and time-consuming for decreasing the average delay. For the future studies, M. Shakibayifar et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 3409{3422 3421 the existing mathematical model can be combined with timetabling problem", Computers & Op- train routing model to cope with real assumptions. Our erations Research, 78, pp. 439-453 (2017). mathematical modeling approach is also capable to be 12. Zhang, L., Tang, Y., Hua, C. and Guan, X. \A new incorporated with detailed evaluation of the robustness particle swarm optimization algorithm with adaptive and reliability indexes. inertia weight based on Bayesian techniques", Applied Soft Computing, 28, pp. 138-149 (2015). References 13. D'ariano, A., Pacciarelli, D. and Pranzo, M. \A branch and bound algorithm for scheduling trains in a railway 1. Abril, M., Barber, F., Ingolotti, L., Salido, M.A., network", European Journal of Operational Research, Tormos, P. and Lova, A. \An assessment of railway 183(2), pp. 643-657 (2007). capacity", Transportation Research Part E: Logistics and Transportation Review, 44(5), pp. 774-806 (2008). 14. D'Ariano, A. \Innovative decision support system for railway trac control", IEEE Intelligent Transporta- 2. Pouryousef, H., Lautala, P. and Watkins, D. \De- tion Systems Magazine, 1(4), pp. 8-16 (2009). velopment of hybrid optimization of train schedules model for N-track rail corridors", Transportation Re- 15. Corman, F., D'Ariano, A., Pacciarelli, D. and Pranzo, search Part C: Emerging Technologies, 67, pp. 169-192 M. \Centralized versus distributed systems to resched- (2016). ule trains in two dispatching areas", Public Transport, 2(3), pp. 219-247 (2010). 3. Hassannayebi, E., Zegordi, S.H. and Yaghini, M., \Train timetabling for an urban rail transit line using a 16. Hassannayebi, E. and Kiaynfar, F. \A greedy random- Lagrangian relaxation approach", Applied Mathemati- ized adaptive search procedure to solve the train se- cal Modelling, 40(23), pp.9892-9913 (2016). quencing and stop scheduling problem in double track Hassannayebi, E., Zegordi, S.H. and Yaghini, M. railway lines", Journal of Transportation Research, \Train timetabling in urban rail transit line us- 9(3), pp. 235-257 (2012). ing Lagrangian relaxation approach", Applied Mathe- 17. Sajedinejad, A., Mardani, S., Hasannayebi, E. and matical Modelling, (2016). http://dx.doi.org/10.1016/ Kabirian, A. \SIMARAIL: simulation based optimiza- j.apm.2016.06.040. tion software for scheduling railway network", in Sim- 4. Hassannayebi, E., Sajedinejad, A. and Mardani, S. ulation Conference (WSC), Proceedings of the 2011 \Disruption management in urban rail transit sys- Winter. IEEE (2011). tem: A simulation based optimization approach", in 18. Lai, Y.C. and Shih, M.C. \A Robust Decision Sup- Handbook of Research on Emerging Innovations in Rail port Framework for Long-Term Railway Capacity Transportation Engineering, IgI-Global. pp. 420-450 Planning", In Proceedings of 9th World Congress on (2016). Railway Research, Lille, France (2011). 5. Huisman, D., Kroon, L.G., Lentink, R.M. and Vro- 19. Hasannayebi, E., Sajedinejad, A., Mardani, S. and mans, M.J. \Passenger railway optimization", Hand- Mohammadi, K.A.R.M. \An integrated simulation books in Operations Research and Management Sci- model and evolutionary algorithm for train timetabling ence, 14, pp. 129-187 (2006). problem with considering train stops for praying", In 6. Cai, X., Goh, C. and Mees, A.I. \Greedy heuristics Proceedings of the 2012 Winter Simulation Conference for rapid scheduling of trains on a single track", IIE (WSC), pp. 1-13, IEEE (2012). Transactions, 30(5), pp. 481-493 (1998). 20. Goverde, R.M., Corman, F. and D'Ariano, A. \Rail- 7. Caprara, A., Fischetti, M. and Toth, P. \Modeling way line capacity consumption of di erent railway and solving the train timetabling problem", Operations signalling systems under scheduled and disturbed Research, 50(5), pp. 851-861 (2002). conditions", Journal of Planning & 8. Sahin, I._ \Railway trac control and train scheduling Management, 3(3), pp. 78-94 (2013). based oninter-train con ict management", Transporta- 21. Schobel, A., Raidl, G.R., Grujicic, I., Besau, G. tion Research Part B: Methodological, 33(7), pp. 511- and Schuster, G. \An optimization model for inte- 534 (1999). grated timetable based design of railway infrastruc- 9. Adenso-Daz, B., Gonzalez,M.O. and Gonzalez-Torre, ture", in Proceedings of the 5th International Semi- P. \On-line timetable re-scheduling in regional train nar on Railway Operations Modelling and Analysis- services", Transportation Research Part B: Method- RailCopenhagen (2013). ological, 33(6), pp. 387-398 (1999). 22. Hassannayebi, E., Sajedinejad, A. and Mardani, 10. Sama,M., Corman, F. and Pacciarelli, D. \A variable S. \Urban rail transit planning using a two-stage neighbourhood search for fast train scheduling and simulation-based optimization approach", Simulation routing during disturbed railway trac situations", Modelling Practice and Theory, 49, pp. 151-166 (2014). Computers & Operations Research, 78, pp. 480-499 23. Delorme, X., Rodriguez, J. and Gandibleux, X. (2017). \Heuristics for railway infrastructure saturation", 11. Hassannayebi, E. and Zegordi, S.H. \Variable and Electronic Notes in Theoretical Computer Science, adaptive neighbourhood search algorithms for rail 50(1), pp. 39-53 (2001). 3422 M. Shakibayifar et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 3409{3422

24. Shih, M.C., Lai, Y.C., Dick, C. and Wu, M.H. Erfan Hassannayebi is an Assistant Professor of \Optimization of siding location for single-track lines", Industrial Engineering at Department of Industrial Transportation Research Record: Journal of the Trans- Engineering, Faculty of Engineering, Central Tehran portation Research Board, pp. 71-79 (2014). Branch, Islamic Azad University, Tehran, Iran. He 25. Vansteenwegen, P., Dewilde, T., Burggraeve, S. and received his PhD degree in Industrial Engineering from Cattrysse, D. \An iterative approach for reducing Tarbiat Modares University in 2017. His research the impact of infrastructure maintenance on the per- interests are in the area of applied operational research, formance of railway systems", European Journal of simulation-based optimization, and metaheuristic algo- Operational Research, 252(1), pp. 39-53 (2016). rithms. He has several publications in international 26. Hassannayebi, E., Zegordi, S.H., Amin-Naseri, M.R. journals. and Yaghini, M. \Train timetabling at rapid rail transit lines: a robust multi-objective stochastic programming Hamid Mirzahossein received his BSc degree in approach", Operational Research, pp. 1-43 (2016). 2009, MSc degree in 2011, both from Imam Khomeini 27. Pohle,D. and Feil, M. \What are new insights using International University (IKIU), and PhD degree in optimized train path assignment for the development 2017 from Iran University of Science and Technology of railway infrastructure? ", in Operations Research (IUST). At present he is Assistant Professor at Imam Proceedings, Springer. pp. 457-463 (2016). Khomeini International University (IKIU). His research 28. Corman, F., D'Ariano, A., Marra, A.D., Pacciarelli, interests are mainly in the areas of land-use and trans- D. and Sama,M., \Integrating train scheduling and portation interaction with special focus on accessibility delay management in real-time railway trac control", and mathematical models and optimization techniques Transportation Research. Part E: Logistics and Trans- for congestion pricing and trac assignment, and portation Review, 105, pp. 213-239 (2017). more recently, on mathematical optimization of railway 29. Qi, J., Yang, L., Gao, Y., Li, S. and Gao, Z. systems. \Integrated multi-track station layout design and train scheduling models on railway corridors", Transporta- Shaghayegh Zohrabnia received her MSc degree tion Research Part C: Emerging Technologies, 69, pp. in Industrial Management from Allameh Tabatabee 91-119 (2016). University of Tehran, Iran. Her professional interests focus on multi-objective optimization, meta-heuristic Biographies methods in solving NP-hard problems{such as Tabu Search, Simulated Annealing and Genetic Algorithm{ Masoud Shakibayifar received his PhD degree in target costing and value engineering, balanced score- Industrial Engineering from Iran University of Science card, management information system, queuing system & Technology in 2017. He received BSc (1997-2001) optimization, nonparametric methods in operations and MSc (2004-2006) degrees in Civil Engineering at research such as Data Envelopment Analysis (DEA). Sharif University of Technology, Tehran, Iran. He Her current project includes timetabling optimization has managed more than 15 transportation planning of mixed double and single tracked railway network. projects in the Iranian Railway Company and the Ministry of Roads and Urban Development so far. Ali Shahabi is a PhD candidate at the Islamic Azad Recently, he has managed a research project related University, Tehran Branch, in the eld of Industrial to the design of railway trac simulation software Engineering. His research interests include simulation, for the Iranian Railway Company. His research in- reliability engineering, supply chain management and terests are railway electri cation, railway scheduling logistics, optimization methods, and advance quality and rescheduling, and optimization of transportation control. He has also several publications in interna- systems under uncertainty. tional journals.