 Original Paper
 Open Access
Sensitivity analysis of train schedule of a railway track network using an optimization modeling technique
 Malik Muneeb Abid^{1}Email author and
 Muhammad Babar Khan^{2}
https://doi.org/10.1007/s1254401401531
© The Author(s) 2015
 Received: 14 April 2014
 Accepted: 9 November 2014
 Published: 25 January 2015
Abstract
Introduction
Modeling of train operations is helpful to plan railway corridor activities and optimize the use of resources. This paper models the train scheduling problem of a single line railway network with main concern to minimize total operational cost and delays with relation to position of sidings.
Method
Problem is modeled as jobshop scheduling problem and solved using Branch and Bound (B&B) technique. Sensitivity analysis is done with changing the number of sidings and trains. Approach used here has quantified the relatedness between the number of sidings and train conflict delays. Objective was to show the impact of number and position of sidings on timetable.
Results
Results are quantified as conflict delays and computational effort involved for each option of sensitivity analysis. Some of sidings which don’t have any impact on scheduling of this track are mentioned using this analysis.
Conclusion
This technique can be used to determine the investments in terms of location and number of sidings because conflict related delays has direct relation with number and position of sidings.
Keywords
 Train
 Railway
 Scheduling
 Optimum
 Siding location
 Optimization techniques
1 Introduction
The Train scheduling is the art of finding arrival and departure time to and from each station. Efficient design of schedule can improve the level of service of railways. It is one of the initial steps in management of complex railway operations. Even attraction of traveler’s and freight carriers in traveling mode selection is also dependant on the schedule and its level of service. Railways system operations are scheduled based on the train schedule so it also effects the utilization of scarce resources such as cars, engines and crew. In order to optimize the use of infrastructure capacity, some helping tools such as Mathematical programming and simulation techniques are being used to solve long and complex scheduling problem. Albeit, train scheduling had been done manually for more than a century however to prepare train schedules for a rail network without any supporting tools and resolving conflicts “by hand” is rather a slow process. To manually schedule trains even over a small size rail network, a large number of experienced schedulers and ample amount of time is required. Nowadays, in the age of computer it cannot be conceived without the help of a computer [1].
Location of siding is a critical issue in the planning of rail line. It involves determining the number, length, position and vertical and horizontal alignment of line. Main concern in this problem is to place sidings in such a way which minimize total operating cost and total delay. Impact on the schedule, in terms of flexibility and feasibility of schedules, can be assessed by using these models if you want to extend or remove some sidings. The problem of sensitivity analysis in mathematical programming has been discussed by many authors, for example, Higgins et al. [3], Vanderplaats [4], Sobiesky et al. [5], Enevoldsen [6], Castillo et al. [7–10], etc. Burdett and Kozan [11] finds the effect of deviations in sectional running time and additional dwell time on three objectives, namely total train delay, makespan and total time window violation. Burdett and Kozan [12] use sensitivity analysis to find the affected operations by delay in the predictive train schedule and find that this sensitivity analysis is also a measure of robustness and provide information how to control the delays when they occur in real time.
Here considered problem is that improper position and abundant sidings on the network can also cause unreasonable conflict delays. Hence, in order to optimize the investment plan, only those sidings having impact on the timetable should be improved (by adding more tracks to cater more conflicts at that sidings etc.). Others do not have impact in decreasing conflict delays or their presence or absence has same worth, should not be further considered for investment.
This paper is organized as follows; in Section 2 literature review is presented. Section 3 gives details of modeling of train operations with objective functions and constraints. Solution strategy is given in Section 4 and experimental arrangements as well as results are shown in Section 5. Finally, in Section 6 conclusions and recommendations are presented.
2 Literature review
From the very beginning scheduling has been an active area of research in railways. Even before eighties mostly train scheduling was based on optimization and simulation techniques. Assad [13] is providing a review of the models developed for rail transportation before eighties. Amit and Goldfarb [14] seems to be first to apply the mathematical programming technique to train scheduling problem. Cordeau et al. [15] presented a detailed survey of train routing and scheduling optimization modeling techniques. Sahin [16] and Ghoseiri et al. [17] divided the techniques of railway traffic control techniques into three families: expert systems, simulations and mathematical programming. Over view of rail traffic management provided by D’Ariano [18] classifies the railroad management operations into three categories: Offline timetabling, real time traffic management and dynamic traffic management. Main focus of offline tabling is to design robust timetable such that propagation of delays should be minimum but in case of large delays and blocking of tracks no reasonable plan is robust or reliable [19]. Most of recent studies (D’Ariano [18], Montigel et al. [20], Corman [21–23] and Tornquist [24]) focused on the real time problem to find feasible deadlock free and conflict free solutions within a short time available to take real time dispatching decisions.
A wide range of studies in train scheduling problem have used branch and bound (B& B) technique to solve this problem. Szpigel [25] was first to formulate a model for train scheduling problem and applied Greenberg’s solution [26] framework based on branch and bound technique to solve it. Jovanovic and Harker [27] modeled the problem to minimize the deviation between actual and planned schedules. They used nonlinear integer programming to generate feasible meet pass plan using branch and bound technique. Higgins et al. [28] also used nonlinear integer programming to model the problem with objective to minimize total delay. They developed lower bound estimate based on the number of remaining conflicts in remaining partial schedule. Zhou and Zhong [29]) modeled to minimize total train travel time with three techniques to reduce the search space; i) Lagrangian lower bound estimate, ii) exact lower bound based on remaining conflicts in the partial schedule and, iii) Upper bound using beam search heuristic. D’Ariano et al. [30] modeled real time traffic control as huge job shop scheduling problem with no store constraints. Shafia et al. [31]) formulated to find the robust time table on single line track. They used branch and bound with beam search heuristic to solve the problem. Comparison of results of branch and bound technique with Lingo shows optimality of solution.
Trains scheduling problem is formulated here as resource constrained job shop scheduling problem. To be more specific, it is an optimization problem with set of trains running on single line track. Next sections provide the detail about the modeling and solution strategies used.
3 Mathematical modeling
Single line train scheduling problem is consideration of this model, in which trains are considered as tasks which are assigned to tracks (considered as machines). In this way jobshop scheduling problem is formulated with set of single line track segments and a set of trains having predefined traveling directions and fixed running times.
3.1 Assumptions and inputs

The track is considered to be composed of segments, separated by sidings.

It is assumed that each train has pre specified direction and route.

Free running times are assumed to be constant for a track segment.

Travelling of trains are assumed as tasks to be assigned to machines (here tracks and stations are taken as machines).

Sidings are places where trains can cross each other.

A minimum headway will be maintained for trains to follow each other on a track segment.

Trains can have maximum 30 min conflict delay otherwise this option will be neglected.

A station may have capacity to be occupied by more than one trains but only one train capacity per station is considered in here.

Fixed running times are calculated for each segment.

Planned departure times of all trains to be scheduled.

Maximum time a train can wait for other train at a siding.
3.2 Definition of variables
Definition of variables
Definition  Symbol 

Train index  t 
Segment index  s 
Segment sequence number in train route  j 
Station index  i 
Set of trains  T 
Set of segments S = m  S 
Set of stations J = m + 1  J 
Direction indicator for train t, p(t) = 0 for an inbound train and p(t) = 1 for an outbound train  p(t) 
Segment index of the jth traveling segment in a route for train i, σ (t,j) = j for outbound trains, σ (t,j) = m + 1j for inbound trains  σ(t, j) 
Downstream station number of the jth traveling segment in a given route for train t, b(t,j) = j foroutbound trains, b(t,j) = m j for inbound trains  b(t, j) 
Planned departure time for train t at its first station  k_{t} 
Free running time for train t at segment s  f _{(t,s)} 
Minimum required station dwell time before train t entering segment s  d _{(t,s)} 
Maximum allowed station dwell time before train i entering segment j  \( {\overline{d}}_{\left(t,s\right)} \) 
Minimum headway between arrival and departure times of two consecutive trains at segment j  h _{ s } 
Minimum headway between arrival times of two consecutive trains at station u  g _{ i } 
Entering time for train t at segment s, i.e., start time for job t on machine s  o _{ t,s } 
Leaving time for train t at segment s, i.e., end time for job t on machine s  c _{ t,s } 
Binary Variable,1 if train t is scheduled before train t’ on segment s, 0 otherwise  \( {A}_{t,{t}^{\prime },s} \) 
Sufficiently large constant  M 
3.3 The proposed model
3.3.1 Objective function
Inequality (2) is departure time constraint, which imposes a limit on the actual departure time of trains from starting station. This ensures that actual departure time at starting station is always equal or greater than the planned time of that train at starting station. Equation (3) states that leaving time (finish time of job) of a section must be equal to the entering time plus free running time. Constraint (4) is ensuring that scheduled stop is more than minimum dwell time which is practically required to load and unload passengers and freight trains. Constraints (5) and (6) are headway constraints to ensure safe operations. Constraint (5) is imposing minimum headway requirement for safe operation of trains running, in opposite or same direction, on the same track. While, inequality (6) imposes that minimum headway requirement on two consecutive trains approaching at same station. Constraint (7) gives the upper bound of time which a train can wait for other train at a station.
4 Solution procedure
Branch and bound is applied here, which is a general algorithm applied to many areas to find optimum solution of various optimization problems. It consists of enumeration of all candidate solution systematically while a large amount of fruitless candidates are discarded based on upper and lower bound estimation of quantity to optimized.

Root node is initiated with an empty schedule. Departure time, segment free running time and train directions are input and upper bound is set to 999999.

Constraints are applied at each activity and conflicts (where two or more trains want to use same resource) are identified with position and trains involved in a conflict.

For each available option child nodes are generated.

Active node for further branching is selected based on Depth First Search (DFS).

Lower bound value is estimated for each node.

Node elimination rules are applied to move further with remaining child nodes.

Loop until no more nodes are remaining in list of active nodes or until stop condition of node selection rule is satisfied.
5 Calculations and results
In the sensitivity analysis of this track, we first consider two terminal stations with only 15 intermediate sidings, which are being used to resolve conflicts. Then, those sidings which have only one conflict are ignored one by one, starting from Lalamusa terminal. After that, we ignored the sidings which have two conflicts to get resolved.
Results of option testing
Sidings  8 Trains  9 Trains  10 Trains  11 trains  12 trains  

Nodes  Margin  Delay  Margin  Nodes  Margin  Delay  Margin  Nodes  Margin  Delay  Margin  Nodes  Margin  Delay  Margin  Nodes  Margin  Delay  Margin  
17  486  40  1716  57  14563  84  54389  99  179742  111  
16  301  −185  52  12  1021  −695  69  12  5633  −8930  95  11  17185  −37204  110  11  64014  −115728  127  16 
15  361  60  58  6  967  54  74  5  5335  −298  102  7  14048  −3137  116  6  66433  2419  138  11 
14  473  112  67  9  1271  −304  83  9  6664  1329  111  9  18385  4337  125  9  91798  25365  148  10 
13  253  −220  82  15  1383  −112  133  50  8710  2046  186  75  14479  −3906  202  77  64472  −27326  231  83 
12  197  −56  82  0  1105  278  133  0  6400  −2310  186  0  9188  −5291  202  0  35420  −29052  231  0 
11  197  0  82  0  1105  0  133  0  6400  0  186  0  8956  −232  202  0  46878  11458  242  11 
10  311  114  99  17  1029  76  141  8  5527  −873  194  8  8229  −727  210  8  35005  −11873  250  8 
9  409  98  138  39  837  192  174  33  3766  −1761  227  33  6017  −2212  242  32  24693  −10312  279  29 
6 Conclusions and recommendations
The main focus of this paper is to find the significance of position of sidings on railway network. The problem is formulated as resource constrained job shop scheduling problem, with operational safety constraints. The developed model is solved by branch and bound technique. The proposed methodology is demonstrated by sensitivity analysis of real world problem of single track section of Pakistan Railways. Results are quantified as conflict delays and computational effort involved for each option of sensitivity analysis. Results show that with decrease in the number of sidings delay increases but the computation effort decreases and this phenomenon become prominent for more trains and less sidings. Some of sidings which don’t have any impact on scheduling of this track are mentioned using this analysis. This shows the potential of this technique to get applied at midterm planning level for preparation of strategic investment plans.
The contribution of this paper is to help decision makers, responsible to inspect and maintain the network, to prioritize the elements of network (i.e., sidings, tracks etc.) for improvement. By using such type of techniques they can optimally allocate the available limited resources. Further is the extension of this work by identifying and incorporating the practical considerations of investment on network. Also the future consideration is the development of decision support system for network manager to find optimum strategy for investment.
Declarations
Acknowledgments
We are thankful to honorable, Dr. Muhammad Irfan, Assistant Professor at NIT, NUST, Islamabad and Dr. Xuesong Zhou, Assistant Professor of the Department of Civil and Environmental Engineering at the University of Utah, for their timely, indispensible continuous help, constructive critique and able guidance. We gratefully acknowledge Pakistan Railways (PR) department for providing us valuable data to complete this work. Especially, I owe my deepest gratitude to Mr. Umar Riaz and Mr. Ehtsham Khan, Divisional Planning Officer and Divisional Signal Engineer of Rawalpindi Division for their cooperation and guidance. We are also grateful to Zohaib Ahmed for his valuable assistance in computer programming.
Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Authors’ Affiliations
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