MODEL TO CALCULATE THE OPTIMAL MODE OF TRAIN LOCOMOTIVES TURNOVER Summary

A model to calculate the optimal work modes of train locomotives while serving train flows is proposed. The model is a further development of the dynamic transportation problem. Parameters of train movement and locomotive utilization are published. Both locomotive turnover and train schedules are produced. Useful utilization of locomotives increases from 70% to 90–95%. This allows the reduction of several thousands of train locomotives.


INTRODUCTION
The problem of rational utilization of locomotives is quite relevant at the present time.It is necessary to find the best trade off.When there is an excess of locomotives, we have less train delays, but high expenditures, connected with locomotive servicing.When there is a lack of locomotives, the converse is true.
Considering only operational planning covering several hours, it is impossible to produce from separate time intervals the effective work mode as a whole.Therefore, in practice, locomotive turnover schedules are developed for 7-10 days considering the service area structure and the volume of train flow.Usually, such calculations are accompanied by hand-made locomotive turnover schedules.Considering a service area with several hundred locomotives and a period of several days, however, there will be a great number of variants of train locomotive turnover schedules.This is the reason the optimization model is required.

LITERATURE REVIEW
Since the 1980s an extensive number of articles have been dedicated to the problem of managing train locomotives.In [1] Booler tries to solve a scheduling model using linear programming but he tests the method only on small instances (10-50 trains).Wright in [2] underlines that the approach in [1] is not suitable for more realistic situations (100-500 trains).In 1999 Canadian scientists assigned multiple types of locomotives to a fixed timetable [3].They developed the branch-first, cut-second method to solve the integer programming model.Cordeau et al. in [4] describe a Benders' decomposition method for the assignment of locomotives and cars.An alternative approach to solve the problem of locomotive assigning was proposed by a group of scientists from Princeton University.Their idea is based on the formulation of the original problem as a dynamic programming problem [5][6][7][8][9][10].In 2012 Powell et al.,in [11], describe an application called Princeton Locomotive And Shop Management system (PLASMA), which claimed to solve the problem of assigning locomotives to trains over a planning horizon capturing a high level of detail.Important improvements in building realistic locomotive assigning models are described by authors in [12][13][14].Ahuja et al. in [12] describe a heuristic based on very large-scale neighborhoods to find near-optimal schedules for locomotives, which considers breakups and the desire for weekly patterns in the flows of locomotives.In [13,14] authors provide models to solve a locomotive routing problem capturing a number of constraints with an adaptation of their large neighborhood search strategy.
Currently, in Russia, it is rather difficult to use foreign train locomotive optimization models and systems because of the following problems: 1. the interface of foreign models and systems is not adopted to Russian users.2. the high cost of optimization systems (e.g.PLASMA).
3. the necessity to adopt a model for the Russian railway features of infrastructure, technological process, and the information environment, which, in real life, can create a large number of problems to be solved for the full reconstruction of a model.To overcome these difficulties in Russia a domestic optimization system is developed , which is adapted to the features and volume of work in the Russian railways.

DESCRIPTION OF THE MODEL USED IN THE OPTIMIZATION SYSTEM
This article describes the system of train locomotive turnover optimization «Labyrinth».The optimization model in this system is based on the dynamic transportation problem.The solution of the dynamic transportation problem is reduced to a static one with reproduction in time [15].The problem has been already discussed by the authors in the scientific press [16].This article presents a further development of the proposed approach.The model described in [16] turned into a system of the locomotive turnover optimization that can become an optimizing unit in the corresponding automatic control systems.
The railway station layout represented in the model consists of three parts -the input sector (index i), the output sector (index j), and the makeup origin point (index q).
Trains (makeups with locomotives) or reserve locomotives arrive at the input sector and depart from the output sector.At the origin point makeups appear according to the scheduled train departures.The connection of locomotives and makeups also takes place at this point.Fig. 1 shows the process of locomotive and makeup connection.In this case the reserve locomotive arriving from the previous station is denoted as  The transit passing of a station is displayed as a joint movement of the locomotive and the makeup (Fig. 2).(t y de ) or waits in the input or output sector (variables ) (t y ii or ) (t y jj , see Fig. 1).
In the dynamic transportation problem the source and the drain should be given.For the flows that have not reached the drains up to time T, artificial drains are introduced (variables ) ( S T ).

Basic equations
The dynamics of the makeups in the input sector are as follows: The dynamics of the makeups in the output sector are as follows: The display of the train delays is as follows: is the number of trains being late at the moment t .
The representation of waiting for the locomotive is as follows: Model to calculate the optimal mode of train locomotives turnover 129 The balance of the locomotives in the input sector is as follows: The new locomotive either arrives from the previous station or is taken from the pool.Dynamics of the locomotives in the output sector are as follows: Locomotive balance in the makeup origin point is as follows:

Additional restrictions
The impossibility for the makeup to move into the output sector without a locomotive is as follows: The makeup in the output sector can only be with a locomotive, as follows: The impossibility to dispatch a makeup without a locomotive is as follows: where η is service area number, k is railway station number, c c is hour cost of the makeup idle time, о c is hour cost of the train idle time, с c β is hour cost of locomotive utilization.The costs c c , о c , and с c β are set for one simulation time-step; η c is cost of one locomotive-kilometer, multiplied by the length of the service area η .

The meaning of the terms of the functional
is cost of the makeup idle time (all costs are measured during one simulation time-step) is penalty for the train delay.
is cost of locomotive utilization.Technical operations at the station are shown by is cost of the locomotive when it is empty running.In this equation, costs, connected with the makeup movement, are subtracted from the whole train costs.

RESULTS OBTAINED USING THE OPTIMIZATION SYSTEM
Testing of the optimization system was carried out on the Gorkovskaya Railway Druzhinino-Vekovka service area (Fig. 4).First of all, locomotive utilization parameters are provided (Fig. 5) Fig. 5. Locomotive utilization Fig. 5 shows that empty running is minimized.It is impossible to achieve such results using manual control because there are billions of variants of locomotive movements during several days.Idle time of the train includes time for technical operations and the idle time connected with overtakings.
It is possible to see the work parameters of each locomotive (Tab.1).
The contents of the table can be sorted by any column.It is also possible to open the locomotive turnover schedule from each line (Fig. 6).
Moreover, a considerable number of locomotives do not have empty runs at all (Fig. 7).Useful employment does not include idle time of the train because of overtakings.Similar information about train movement is also provided (Fig. 8).
As can be seen, waiting for a locomotive is also minimized.The optimization system provides information about train movement in each direction (Fig. 9).
The system also produces the train schedule and provides the parameters for each train (Fig. 10).

CONCLUSION
The optimization system allows carrying out different experiments including the calculation of the optimal number of locomotives in conditions of given cost parameters, determination of the best locomotive location at the beginning of the calculation period with the given train flow structure, the evaluation of the influence on train movement depending on the number of locomotives, and so on.
If there is a need, it is possible to connect the optimization system to the corresponding automatic control systems to use in operative planning.

Fig. 3 .
Fig. 3. Scheme of a train arrival at the terminal station: ) (t х i ∆ -delay

Fig. 6 .Fig. 7 .
Fig. 6.Locomotive turnover schedule (the black line shows the idle time and empty running)

Table 1
Locomotives work parameters