A Mathematical Model for Train Routing and Scheduling Problem with Fuzzy Approach

Transcription

1 Proceedings of the 202 International Conference on Industrial Engineering and Oerations Management Istanbul, Turey, July 3 6, 202 A Mathematical Model for Train Routing and Scheduling Problem with Fuzzy Aroach Amin Jamili Deartment of Railway Transortation MAPNA Grou, Tehran , Iran Abstract Routing and Maeu roblems, determine the routing and frequency of trains & the assignment of blocs to trains, but do not tae scheduling into consideration. Consequently later on, it may be difficult to find a timetable accommodating all lanned trains and satisfying line and station caacity. As an attemt to settle this issue, both the routing & the scheduling of freight transortation have been merged in the resent aer. The roosed model embodies three different objectives of: the cost of train formation, the cost of idle time of waiting wagons in stations, & the cost of wagon classifications in shunting stations. It is assumed that, the maximum tonnage and length of trains in each bloc-section is finite, and is deendent on the critical gradient of the route and also maximum length of internal tracs of stations. In real world alication, it is out of reality to assign cris values to the admissible maximum tonnage of trains in the lanning stage. A more alicable method is to consider this arameter as a fuzzy number. The aer intends to resent a ure 0- model with fuzzy aroach, & offers a decomosition-based heuristic algorithm to solve the large-scale roblems in a limited amount of time. Keywords Train Formation Problem-TFP, Train Routing and scheduling, Pure 0- Programming, Fuzzy aroach.. Introduction In the most researches, the routing and scheduling roblems are investigated individually, and therefore, the final solution may not be otimum. In this aer, both the routing and scheduling of trains are discussed together. Moreover, in real alications, the maximum allowable tonnage of trains in blocs could not be a certain amount and therefore, in this aer it is considered as a fuzzy number. The Train Formation Problem-TFP and the Train Scheduling Problem- TSP are the major arts of middle term lanning. The main goals of TFP are as follows. To minimize classifying oerations in shunting stations. To minimize the train formation costs. To minimize the idle time of wagons waiting for trains in shunting stations. To maximize the railroad trac caacity for train movements. To share almost equal wagon classification oerations in all shunting stations. To yield the otimum scheduling for each wagon Furthermore the following strategic lanning also would be obtained: To develo the critical shunting stations with the high wagon classification oerations. To build new shunting stations if required. To rocure more locomotives if required. On the other hand, the general objective of TSP is to obtain the least delays of trains in the mid-stations, and the arrival and dearture times of trains to/from stations are determined. Based on the survey conducted by Cordeau, et al [], TFP, sometimes called Routing and Maeu Problem0F, is a sub-class of networ routing roblems. The other sub-classes are Blocing ProblemF2 and Comound Routing and Scheduling Problems2F3. The followings are the definitions of these three classes resented in the literature: RMP 90

2 . Blocing Problem (BP) Wagons at station i, which are destined for station j must be added to a bloc that will next be shied to station, ossibly transiting by other intermediate stations. Wagons in a bloc will not be reclassified until the bloc reaches its final destination. The solution should indicate the routing of freight through the networ and the distribution of classification wor among stations, but does not secify the trains to be run or the assignment of blocs to trains. It is worth to note that this roblem is different from the so-called bloc-to-train assignment roblem. For more detail refer to Krishna, et al [2]. Newton, et al [3] rovided a mathematical model for BP which are minimizing the oeration costs. Ahuja, et al [4] roosed an algorithm using an emerging technique nown as very large-scale neighborhood search that is able to solve the blocing roblem to near otimality. Yue, et al [5] introduced a new mathematic model as well as an ant colony algorithm which describe the blocing strategy and various combinations of multi-route O D airs in large scale railway networ..2 Routing and Maeu Problem (RMP) In this roblem the routing, the frequency of trains and the assignment of blocs to trains should be determined. Furthermore the blocing olicy may be either determined endogenously, or be given as an inut Verma, et al [6] resented an otimization methodology for the railroad tactical lanning roblem with ris and cost objectives to determine the routes to be used for each shiment, the yard activities, and the number of trains of different tyes. Ahavan, and Yaghini [7] roosed a metaheuristic solution method based on elitist ant system to solve Combined Blocing and Train Maeu Problem. Keaton [8] resented a linear 0- IP model and a heuristic method based on Lagrangian Relaxation. In addition, Keaton [9], considered a ure strategy constraints for blocing and maximum transit times for each origin-destination airs and used a dual adjustment method for imlementing lagrangian relaxation. Lin [0] introduced two aroaches: first a lagrangian one and the second, an imlicit enumeration algorithm with ε-otimality. This model minimizes the sum of handling and transortation costs. Marin and Salmeron []-[2] roosed a local search heuristic for the Tactical Design of rail freight networs, in a series of two aers. Their objectives are minimizing the oerating and time costs. Lin [0] resented an imlicit enumeration algorithm with ε-otimality to solve the TFP model. Shafia et al. [3] introduced a non-linear mathematical model which is abstracted to an integer model. They roosed a robust mathematical model, and a heuristic algorithm to solve the roblem..3 Comound Routing and Scheduling Problem (CRSP) The routing roblems, do not tae scheduling into consideration, it may be difficult to later find a timetable accommodating all lanned trains and satisfying line and station caacity. As a result, in CRSP, both the routing and the scheduling asects of freight transortation are merged. Zhang, Li [4] develoed an integer rogramming model to determine otimal oerations in minimizing the most significant cost figures involved in such oerations. Huntley et al. [5] resented a non-linear MIP model and used simulated annealing algorithm. Gorman [6] offered an alication of genetic and tabu search algorithms for this roblem. Godwin, et al [7], roosed a heuristic algorithm for freight train routing and scheduling in a assenger rail networ. Caimi, et al. [8] addressed the roblem of routing trains through railway stations for a given timetable and outline two algorithms for this secific roblem. Flamini and Pacciarelli [9] addressed a scheduling roblem arising in the real time management of a metro rail terminus. It mainly consists in routing incoming trains through the station and scheduling their deartures with two objectives, in lexicograhical order, the minimization of tardiness/earliness and the headway otimization..4 Outline The resent aer could be classified in CRSP class. The current aer is organized as follows: In section 2 the roblem definition is resented along with the roosed mathematical model. In section 3, a fuzzy aroach is used, and the resulted mathematical model is roosed. In section 4, a novel heuristic method is roosed to solve the introduced roblem. Finally, the concluding remars are given at the end to summarize the contribution of the aer. 2 BP 3 CRSP 9

3 2. Problem Definition and Mathematical Model A railroad networ can be considered as a grah consisting of shunting stations as nodes and the otential aths as arcs. Trains move between stations transorting a limited number of wagons. Each train should select a single arc between all arcs originating a articular station. A simle railroad networ is shown in Fig.. This samle consists of 4 shunting stations. The arc (,3) shows a ath which starts from station no. to station no.3 with no sto in station 2. In this examle, a train which is lanned to move from station to station 4 has four otions as follows: It can ass arcs numbered (,2) (2,3) (3,4) It can ass arcs numbered (,2) (2,4) It can ass arcs numbered (,3) (3,4) It can ass arc numbered (,4) Figure : A railroad samle Passing through each arc, results a articular cost for trains. It is obvious that the cost related to assing arcs (,2) and (2,3) is equal to assing arc (,3). The difference between these two cases is the idle time cost of wagons and locomotives in station 2 for the first ath which should wait a articular amount of time nown as idle time. Trains are two tyes, direct and indirect. Direct ones travel from their origins to their destinations without stoing in the middle stations. Indirect trains sto in the middle stations waiting for connecting and disconnecting the lanned wagons. In order to model the roblem, in each arc, some otential trains are defined to travel. For each train the redetermined starting time from origin, is nown. As the urose of scheduling is to determine the arrival and dearture time of trains and wagons in their assing stations, some of these otential trains are selected according to the arrival time of wagons of each consignment to the stations. The roosed model is exhibited on the basis of the following assumtions: The unit of each consignment is wagon. Each train can transort a limited number of wagons according to the toograhy of each ath, and the internal length of stations. The arrival time of wagons of each consignment to their origin is redetermined. The travel time of trains in bloc-sections, i.e. distances amongst stations are re-estimated. In the roosed model the notations shown in table are utilized: Table : The used notation Indices Index for consignments a Index for each wagon Index for train i, j Indices for stations Parameters θ Required time for assing arc ( i, j) (, a) ϕ Dearture time of train in arc ( i, j) Predetermined resence time of wagon a of consignment in its origin station α Ct Train formation cost for arc ( i, j) The cost of one hour idle or being in service related to each wagon of consignment Cw Cc Shunting oeration cost for each wagon u Maximum allowable wagons hauled by a locomotive in arc ( i, j) 92

4 s e (, a), Origin station of consignment Destination station of consignment Variables δ Equals, if wagon a of consignment is transorted by train in arc ( i, j) γ Equals, if train is formed in arc ( i, j), and 0, otherwise. λ (, a) i Equals, if wagon a of consignment is classified in station i, and 0, otherwise., and 0, otherwise. 2. The cost of train formation The objective of this model includes three arts: The model minimizes the cost of each train formation. This cost consists of train ersonnel wages (including locomotive driver, locomotive driver assistant, reairman, and train chief), consumed oil and gasoline, amortization of locomotive. This cost is equal to γ Ct. i j 2.2 The cost of idle time of wagons in stations waiting for trains As the number of trains leaving a articular station for a same destination rises, the waiting time of wagons dros. ), (, a) This cost is equal to: ( δ ( ϕ + θ α ) Cw ) a i ie ie ie 2.3 The Cost of wagon classifications in shunting stations This cost includes wagon searation wors from arrived trains and wagon connection wors to leaving trains. This cost contains the oil and gasoline of shunting locomotives, lus the wages of locomotive driver, locomotive driver assistant, reairman and shunting oerators. P) This cost is equal to λ Cc. i a i As all the objectives are defined in cost, they are simly added to mae a single objective function. This function is ), ( P Min Z = Ct + + Cw + a, ) γ ( δ ( ϕ θ ) ) λ Cc () i e i i j a i i a 2.4 Constraints. In order to ensure that all consignments leave their origins, Eq. 2 is alied to the model. ), =, a, s < j < e j δ s j (2) 2. To ensure that all consignments ass the middle stations successively to arrive their destinations, Eq. 3 is used. ), ), δ = 0, a, s < j < e, s i, h e (3) i δ jh h This equation ensures that if wagon a of consignment, arrives to a middle station, e.g. i, where s < i < e, then it must be dearted by a train from the station. 3. To ensure that all consignments reach their destinations. Eq. 4 is alied. ), δ =, a & s i < e (4) i ie 4. Inequality 5 revents assigning wagons more than the maximum caacity of trains. ), δ u. γ i, j, & s i < e, s < j e (5) a 93

5 5. Inequality 6 shows that no wagon leaves its origin earlier than the related "redetermined resence time in, ) origin", i.e. ( a α. ), ) ϕs j. δ s j α a & s < j e j, (6) 6. Inequality 7 ensures that all wagons travel from their origins to their destinations successively. ), ), ϕ. δ δ ( ϕ + θ hi hi hi ) a,, i, j & s i < e & s < j e (7) h 7. Inequality 8 secifies that if a wagon stos in a station, the necessary classification wors should be done, and therefore, the relevant costs are added to the objective function. ) ), ), λ j δ δ a,, j, s i jl i < e, s < j < e (8) l To clarify the roosed model, the following examle is resented: Examle: Consider the rail networ shown in Fig.. This networ consists of four stations. Assume that all consignments should transort from left to right. Three consignments are to transort from their origins to their destinations at this networ. The relevant data are secified in table 2. Table 2: The consignments data Consignment Origin Destination 4 The quantity of wagons 8 2 resence time of wagons at origin (hour) 5 Consignment Consignment The objective is to determine: - The number of required trains to transort consignments from their origin to their destinations 2- The timetable for each train containing the dearture and arrival time in each station 3- The timetable for each wagon containing the dearture and arrival time in addition to the idle time of wagons in each station. The related costs for this examle are considered as follows: The wagon classification cost for each wagon: 3 units The cost for one hour idle or being in service time for each wagon: 2 units Train formation costs in all arcs are shown in Fig. 2. Figure 2: Train Formation costs for examle As an examle, in Fig. 2, the train formation cost from station 2 to station 4 is equal to 5000 units. Moreover, 6 otential trains for each station are defined. The redetermined starting time for each train are indicated in table 3. Table 3: The dearture time of trains Train number 2 3 Starting time of trains from station (hour) 2 3 Starting time of trains from station 2 (hour) Starting time of trains from station 3 (hour)

6 The required time for assing each arc is deicted in Fig. 3. E Figure 3: required time for assing each arc for examle As an examle the required time for assing arc (,2) is equal to 5 hours. It should be noted that, in this examle it is assumed that, a is the index for each set of three wagons. It means that each set of three wagons connect to each other at their origin and wouldn't disconnect till arriving to their destinations. For all arcs, the arameter u is considered to be equal to 36 wagons. The required time for connecting and disconnecting wagons to/from trains in stations is assumed to be hour. The otimal solution is stated in table 4. Table 4: The otimum solution of examle The final otimum value for objective function: 865 The formed trains: γ 2 = γ 3 = γ = γ = γ = The wagons should sto in stations: (2,) (2,2) (2,3) (2,4) (2,5) (2,6) λ 2 = λ2 = λ2 = λ2 = λ2 = λ2 = (3,) (3,2) (3,3) (3,4) (3,5) (3,6) (3,7) λ 3 = The assigned trains and arcs to wagons: (,),5 (,2),5 (,3),5 (,4),5 (,5),5 (,6),5 (,7),5 (,8),5 (,9),5 (,0),5 δ (2,3),4 (2,4),4 (2,5),4 (2,6),4 (2,),5 (2,2),5 (2,3),6 (2,4),6 (2,5),6 δ (3,),6 (3,2),6 (3,3),6 (3,4),6 (3,5),6 (3,6),6 (3,7),6 δ = (,2),5 (,3),5 (,4),5 (,5),5 (,6),5 (,7),5 (,8),5 (,9),5 (,0),5 δ = (3,),7 (3,2),7 (3,3),7 (3,4),7 (3,5),7 (3,6),7 (3,7),5 δ 3 = (2,6),6 2 = = Table 4 can be interreted as follows: The first train dearts station at time 4:00 and arrives to station 2 at time 9:00. This train contains 2 wagons of consignment 2. At station 2, 2 wagons of consignment 3 connect to the train, then this train leaves station 2 at time 2:00 and arrives to station 3 after 8 hours, i.e. at 20:00. At this station, consignment 2 is searated, and finally the train dearts station 3 at 2:00 hauling just remained 2 wagons of consignment 3. The second train dearts station at 5:00 and taes 2 wagons of consignment 2 and also 30 wagons of consignment. It arrives to station 3 at 8:00 and the wagons of consignment 2 are disconnected in this station. Finally this train leaves station 3 at 9: Alying the Fuzzy Aroach to the Model In classical linear rogramming, the violation of any constraint renders the solution infeasible, but in real alicable cases, the role of constraints can be different, where the decision maer might accet small violation of constraints but might also attach different degrees of imortance to violations of different constraints. In this aer, the arameter u of the roosed model, is suosed to be imrecise. This arameter affects the right-hand side of inequality 5. As the objective function assumed to be cris, the Werner's aroach is alicable. Alying directly this aroach to inequality 5 results a non-linear model, as a result, this inequality is decomosed in to inequalities 9 and 0. a δ (, ), u i j,, s i< e, s < j e a, (9) 95

7 ), δ M. γ i j,, s i< e, s < j e a where, M is a big number. We next resent how this aroach can be alied to inequality 9. Consider inequality 9, the fuzzy triangular number of arameter u is deicted in Fig. 4., (0) Figure 4: The fuzzy number of arameter u The membershi function is equal to Eq.. a δ (, ), u a = () u 96 Furthermore, the membershi function of the objective function, is equal to Eq. 2. c T x f G = (2) f f 0 Where, c T x equals the objective function value, and f is the otimum objective value of the roosed model in cris mode and f is the otimum objective value of the roosed cris model when the arameter 0 u is relaced by u + u. Finally, by introducing one new variable, η, the mathematical model with fuzzy constraint transforms to cris model (3). Max η Subject to : + ), a ( a, u η δ u + u (3) T ( f f 0 ) η + c x f Constraints 2,3,4,6,7,8 Examle 2: Consider examle under the introduced fuzzy condition, where, u = 3. At the first ste, examle is solved with the assumtion of u = u + u which yields f In the next ste, using model 3, the following solution is achieved. The otimum objective function is: η = and the solution is interreted as follows: A direct train hauls 30 wagons of consignment from station to 4, starting at 5:00. The second one dearts station at 6:00 and taes 8 wagons of consignment 2. This train arrives to station 2 at :00. In this station, 2 wagons of consignment 3 are connected to the train and finally this train leaves station 2 at 2:00, and arrives to station 3 at 20:00. In this station, wagons of consignment 2 are searated from the train. At the end, the train leaves this station at 2:00. It can be concluded that the solution resulted by the alied fuzzy aroach is more alicable so that the related cost is substantially reduces in comarison with the cris mode. 4. Heuristic Method to Solve the Problem As stated, the secified roblem which is a ure binary one, merges the scheduling roblem with the routing roblem, each of which nown to be NP-Hard. Therefore, it could be resulted that the roosed model is comutationally very hard to solve by exact software acages, lie GAMS, LINGO, CPLEX, etc. Therefore, in order to find a good solution for large-scale roblems, a heuristic method is roosed. The roosed heuristic method is as follows: 0 =

8 The roblem is divided in two individual roblems, one is routing, also nown as train formation roblem, and the other is train scheduling roblem. In other words, the roblem is solved in two stes: In the first ste, the routing and the frequency of trains are determined and the wagons are assigned to trains. Shafia, et al [3] roosed a linear mathematical model to solve the train formation roblem. Moreover, they roosed a heuristic algorithm which is based on decomosition of the main roblem, and is solved into two hases. The first hase is to find the routes of comartments and to allocate the necessary trains, such that the train formation cost is minimized. Then, the outut of the first hase is considered as the inut of the second hase. In hase 2, the exact assignment of the wagons to each train is determined. Note that as the scheduling asects are not involved with this roblem, it is assumed that, the entrance of trains has a Poisson distribution. For more details, refer to [3]. In the remaining arts of the aer, the heuristic algorithm roosed by Shafia, et al [3], is called RT algorithm. In the second ste, the scheduling art of the roblem is solved considering all the oututs of the first ste as the inut data of the second ste, i.e. scheduling art of the roblem. Shafia, et al [20] roosed a Branch & Bound as well as a Beam Search algorithm to solve a timetabling roblem. For the details, refer to [20]. In the remaining arts of the aer, this algorithm is called, SL algorithm. Moreover, Shafia, et al, [2] roosed a Simulated Annealing to solve a fuzzy train scheduling roblem. By these exlanations, the roosed algorithm is as follows: Algorithm. The heuristic method to solve the fuzzy train routing and scheduling roblem Ste. Let =, i, j. Find f, and o f. For this urose, at the first ste use the RT algorithm to determine the routing and the frequency of trains as well as the assignment of wagons to trains. In the second ste run the SL algorithm to find the aroriate time schedule. Ste 2. Considering the amount of, i, j, Find u, i, j, based on Eq.. Then, solve the first ste, i.e. the routing roblem, using RT algorithm. Considering the oututs of RT, run the SL algorithm. Ste 3. Find the, based on Eq. 2. If G G < α, ublish the best found solution and terminate the algorithm, otherwise, go to ste 4. Ste 4. If G <, then go to ste 5, otherwise, go to ste 6. Ste 5. Consider G =, and go to Ste 2. 2 Ste 6. Consider G = +, and go to Ste 2. 2 Note that the best solution secified in Ste 3, is the one that Min{,{, i j } Moreover,α, is the termination criterion. 97, is more than the other candidates. G, Examle 3: The Case Study In this section, a case study which is art of Iranian railway networ is considered. The details are secified in [3]. Briefly, the studied networ consists of 6 stations in which shunting yards among them can service shunting oerations. The toograhy of the line and maximum length of stations changes in different arts of the networ. There are 2 comartments which should be transorted by the trains from their origins to their destinations. The distances among all shunting yards, the characteristics of the comartments, the characteristics of studied railway networ, and all the related costs are secified in [3], and therefore, not reeated here. By solving the roblem in cris mode, the final solution includes forming trains to transort all the comartments from their origins to their destinations. In this case, the total cost equals 59 units. The solving iterations are shown in Table 5. Table 5. The iterations of solving the roblem by Algorithm. OFV G G

9 < In the best found solution, in fuzzy case, the required amount of trains reduces one, and the cost reduces by 378 units in comarison with the cris mode. 5. Conclusion The aer is delivered a new linear binary train routing and scheduling model. As it is out of reality to assign a constant amount to the maximum allowable wagons hauled by a locomotive, the fuzzy logic has been alied to the roosed cris model. Then, to illustrate the model, a simle examle was solved in both cris and fuzzy environments. The results show that, it is imossible to achieve an otimum solution for the large-size roblems, during a rational amount of time. Therefore, a heuristic algorithm is roosed based on decomosition of the roosed roblem in two individual roblems. The results demonstrate that the fuzzy model reduces the cost in comarison with the cris model. Acnowledgements This aer is financed by MAPNA GROUP. References [] Cordeau J., Toth P., and Vigo D., 998 A Survey of Otimization Models for Train Routing and Scheduling, Transortation Science, 32 (4). [2] Jha, K. C., Ahuja, R. K. and Şahin, G., New aroaches for solving the bloc-to-train assignment roblem, Networs, vol. 5, no., , [3] Newton, H. N., Barnhart, C., and Vance, P. H., Constructing Railroad Blocing Plans to Minimize Handling Costs, Transortation Science, vol. 32, , 998. [4] Ahuja, R. K., Jhu, K. C., and Liu J., Solving Real-Life Railroad Blocing Problems, Interfaces, vol. 37, no. 5, , [5] Yue, Y., Zhou, L., Tue, Q., Fan, Z., Multi-route railroad blocing roblem by imroved model and ant colony algorithm in real world, Comuters & Industrial Engineering, vol. 60 no.,. -42, 20. [6] Verma, M., Verter, V., Gendreau, M., A Tactical Planning Model for Railroad Transortation of Dangerous Goods, Transortation Science, vol. 45, no. 2, , 20. [7] Ahavan, M. R., Yaghini, Solving Combined Blocing and Train Maeu Problem with Ant Colony Otimization, 4th International Conference of Iranian Oerations Research Society, in ress, 20. [8] Keaton M. H., Designing Otimal Railroad Oerating Plans: Lagrangian Relaxation and Heuristic Aroaches, Transortation Research, vol. no. B, , 989. [9] Keaton M. H., Designing Railroad Oerating Plans: A Dual adjustment method for imlementing lagrangian relaxation, Transortation science, vol.26, , 992. [0] Lin C. C., The freight routing roblem of time-definite freight delivery common carriers, transortation research art B, vol. 35, , 200. [] Marin, A. and Salmeron, J., Tactical Design of Rail Freight Networs. Part : Exact and Heuristic Methods, Euroean Journal of Oerational Research, vol. 90, , 996. [2] Marin, A. and Salmeron, j., Tactical Design of Rail Freight Networs. Part 2: Local Search Method with Statistical Analysis, Euroean Journal of Oerational Research, vol. 94, , 996. [3] Shafia, M. A., Sadjadi, S. J., and Jamili, A., Robust Train Formation Planning, Proceedings of the IMechE Part F: Journal of Rail and raid transit, vol. 224, no. F2, , 200. [4] Zhang, Li, Solving railway routing and scheduling roblems in an intermodal freight transortation system, Master's Dissertation, Concordia University, [5] Huntley, C. L., Brown, D. E., Saington, D. E., and Marowicz, B. P., Freight Routing and Scheduling at CSX Transortation, Interfaces, vol. 25, no.3,. 58-7, 995. [6] Gorman, M. F., An Alication of Genetic and Tabu Searches to the Freight Railroad Oerating Plan Problem, Annals of Oerations Researches, vol. 78,. 5-69, 998. [7] Godwin, T., Goalan R., and Narendrant T. T., Freight Train Routing and Scheduling in a Passenger Rail Networ: Comutational Comlexity and the Stewise Disatching Heuristic, Asia-Pacific Journal of Oerational Research, vol. 24, no. 4, ,

10 [8] Caimi, G., Burolter, D., and Herrmann, T., Finding Delay-Tolerant Train Routings through Stations, Oerations Research Proceedings, Sringer Berlin Heidelberg, , [9] Flamini, M., Pacciarelli, D., Real time management of a metro rail terminus, Euroean Journal of Oerational Research, vol.89, no. 3, , [20] Shafia, M. A., Sadjadi, S. J., Jamili, A., Tavaoli-Moghaddam, R., and Pourseyed Aghaee, M., The eriodicity and robustness in a single-trac train scheduling roblem, Alied Soft Comuting Journal, DOI. 0.06/j.asoc , 202. [2] Shafia, M. A., Pourseyed Aghaee, M., Sadjadi, S. J., and Jamili, A., Robust Train Timetabling roblem: Mathematical model and Branch and bound algorithm, IEEE Transactions on Intelligent Transortation Syatems, DOI. 0.09/TITS ,