The Application of Mathematical Methods to the Determination of Transport Flows

KLJUČNE RIJEČI metode operacijske analize rješavanje problema optimizacije javni autobusni prijevoz donošenje odluka model Detroit Ladislav Bartuška Department of Transport and Logistics Institute of Technology and Business České Budějovice, Czech Republic e-mail: bartuska@mail.vstecb.cz Jiří Čejka Department of Transport and Logistics Institute of Technology and Business České Budějovice, Czech Republic e-mail: cejka@mail.vstecb.cz Zdeněk Caha Department of Foreign Languages, Institute of Technology and Business České Budějovice, Czech Republic e-mail: caha@mail.vstecb.cz


INTRODUCTION
Transport planning and operational transport management is a field where the application of simulation models based on mathematical methods is the first step in the determination of an efficient solution.Mathematics is applied in transport system modelling as a basic tool, whether it deals with a network of airlines or a network of railway routes.This article outlines the application of basic mathematical methods particularly in the creation and optimization of a network of municipal public transport lines.The optimization or rationalization of already existing transport systems is actually more usual under the present conditions.
We can distinguish between strategic and operational planning in the public transport sphere.In the first phase of strategic planning, we can collect the basic data, e.g. on passenger intensities (O/D matrix) and set the strategic goals of transport service in relation to the city and its inhabitants.Numerous statistical and stochastic methods are applied in this phase, but also basic mathematical methods are applied as well, e. g. particularly for the determination of transport flows within a system.In the next phase of operational planning, the obtained data are processed to the best possible effectiveness of the transport system with the stress on cost minimization.The proposal of line routes within the given territory (that includes the calculation of vehicle turns on these lines and other processes linked to the optimization) can be included into this phase of the transport system optimization.Mathematical methods have to necessarily be integrated into these processes.The determination of the optimum number of vehicles and planning their turns is a typical case where the basic methods of operational research are applied, e. g. the "Travelling Salesman Problem" [1].
Nowadays, when there are still new demands arising in transport and transport systems are on the increase, mathematics is even more necessary and important for decision making problems in transport planning.Operational research and optimizing mathematical methods have a long history in air transport; for example, decision making problems are supported by operational analysis methods and have already been developing for 30 years.Airlines use optimization techniques for daily, weekly, and monthly planning such as fleet assignment, crew scheduling, and crew fostering [1].
Fleet assignment in forwarding companies may similarly be optimized or linear programming methods can be used for the determination of traffic flow utilization on roads and motorways, for thecreation of a future development model to predict congestion occurrence.The application opportunities are numerous.So let us focus on the application of certain mathematical methods to the determination of the size of the transport flows between individual areas within a municipal public transport system.

THE GENERATION OF TRANSPORT FLOWS BETWEEN THE AREAS OF A TRANSPORT SYSTEM
The complex technique of transport system modelling only developed in the times when problems requiring complex and coordinated solutions were growing on one side and sufficiently powerful computing technology that enabled such solutions was available on the other side.The whole process of transport system modelling can be split into four parts (the traditional four-step transport model) [2]: 1.The generation of transportation needs (volumes of origin and destination transport).2. The distribution of transportation relations (directions of transport flows).

The classification of transportation relations according to
the means of transport used (division of transportation labour).4. The utilization assignment to routes and transportation network sections.
Network transport research has to be done in order to obtain the intensities of passengers within a public transport system.The territory where the transport services within the transport network are provided , has to be divided into areas (zones).A better solution is to divide the territory to as many zones as there are nods in V.Each zone is thus allocated to a nod.Transport intensities q ij b between two different zones Q i and Q j may then be determined right from the research performed.Unless inter-zonal transport links are known, it is also possible to take the transport volume into account q i getting out of the zone (and thus also getting in) in the calculation and calculate these values, for example by multiple regression analysis in linear form.
The basic formula for calculation and determination of interzonal transport links may be considered as follows: , where x ij is a variable, which may be for example the distance between the i-th and the j-th zones, the price of transport between the zones or the travel time between the zones, n is the number of considered zones.The following conditions however have to be met at the same time:

TRANSPORTATION FORECASTING
The application of the four-step transportation model to transport forecasting requires transport research, evaluation of the exact values of the transport process, the derivation of the dependences valid for the present conditions and the prognosis of the expected transport flows may be calculated based upon these data.However, this includes the integration of various socio-demographic, developmental and other data and the prospective transport volumes may be based on them.

THE CALCUATION OF THE PROSPECTIVE TRANSPORT VOLUMES
Multifactor analysis procedures, namely multiple regression analysis, were commonly used for the determination of the prospective transport volumes of the individual areas.The transport volume was calculated as a variable in dependence on various socio-demographic data of the area as relevant independently variable structural quantities: -is a dependent variable (transport volume of the i-th area) a -regression equation constant b 1, b 2,…, b n -partial regression coefficients X 1, X 2,…, X n -independently variable structural quantities with decisive influence on transport volume Double regression analysis was used for the calculation of the prospective transport volumes as the simplest method, in which numbers of inhabitants and labour opportunities were used as structural quantities.Linear regression analysis was examined in Slovakian Bratislava within the Czechoslovak state research task of P-13 order with the introduction of eight independently variable quantities that were available in the detailed territorial division: the active and passive inhabitants, labour opportunities in sectors I and II, labour opportunities in sector III divided into active and passive, the numbers of places at primary schools, secondary and tertiary schools and other equipment of the area in m 2 of usable area [3].
The calculations were performed for models with fixed input of structural quantities in a certain order, with the classification of the individual structural quantities according to the link strength always for equations with an absolute term (with regression equation constant) and for equations with a limiting condition without an absolute term.
Although the results were very good, there were still some doubts whether linear regression analysis is the right procedure for the determination of the transport volumes.There is a serious reservation that it does not provide a view of establishment and expiry of the transport links from the transport engineering point of view.The regression analysis was gradually replaced in calculation of transport volumes by different methods, particularly the Specific Momentum Method, which however requires more detailed calculation inputs.

TRANSPORT FLOW ROUTING
After the phase of the determination of the transport volumes between the individual zones within a transport network, the determination of directivity of these transport flows may follow.This means the determination of the size of each transport link D ij between two zones i for starting trips and j for ending trips and thus create a complete matrix of links for all n districts the territory is divided into.
The calculation of the prospective directing of the transport flows has also developed from the simplest one-factor procedures, via the average growth factor to more complex analogical and synthetic procedures.
The requirement for the sums of the volumes of the source and destination transport (that are to be equal and to equal the total transport volume of the monitored territory) is the basic condition for all these procedures.There are further marginal conditions, mainly the requirements for the sum of all trips from area i to the other areas j to equal the volume of the source transport of area i and for the sum of all trips to area j from all areas to equal the volume of destination transport of area j.These conditions cannot usually be met for the first time; so the calculation is repeated with gradual approximation (iteration) and after the achievement of certain accuracy, the procedure is finished.
Analogical procedures determine the prospective transport links by the analogy with the present situation, while the simplest of them only takes account of the basic requirements (the destination and source transport volumes, the sourcedestination distance etc.).The oldest one (the uniform growth factor procedure) assumes that all inter-zonal relations grow uniformly, i.e. that the growth factor is identical for the whole monitored territory and is defined by the quotient of the prospective and the present transport volumes.In the average growth factor procedure, the prospective transport link is determined by the product of the present link and the arithmetic average of the growth factor of source transport i and destination transport j of the area.
More complex analogical procedures, like the Detroit and Fratar Models already take account of not only the pair of monitored areas, but also the mutual influence of the other areas of the territory.The Detroit Model is based on the assumption that the prospective transport link is directly proportional to the present link and the growth factor of both the monitored areas and is indirectly proportional to the growth factor of the whole city [2]: , where -prospective transport link, -present transport link, -growth factor of source transport in areai, -growth factor of destination transport in area j, -whole city growth factor Synthetic procedures seek various ways of expressing the individual factors for the future, as they significantly influence the size of the prospective transport link D ij .The gravity model in various modifications is particularly used in transport engineering.

THE USE OF THE MATHEMATICAL BASIC METHOD TO PROPOSE LINK LINES
Let us now have a look at an example of the application of the basic mathematical methods to the preparation of a transport model.It is namely a transport problem solving algorithm applied to an assignment problem solution.This method has been chosen particularly to show that we are able to save a substantial part of costs, even by means of a principally simple mathematical method, which would have been many times higher if the same situation had been solved by for example the VISEVA, VISUM (Modal Split) model [4].
The transport model designed this way was used within the research also for optimization of public transport routing in České Budějovice.The solution itself was preceded by the transport research on the public transport lines, from which the transport volume data were used.

THE MATHEMATICAL MODEL OF THE ASSIGNMENT PROBLEM
The optimization of the transport links in public transport can be formulated and solved as an assignment in which the individual districts are regarded both as sources and destinations.The rates are sums of the identified source and destination traffic intensities between any two districts.In order to avoid assigning the same two districts, so called prohibitive rates were chosen on the main diagonal (see table 1).The criterion of optimality is the maximization of the total number of passengers transported without transfer.
The mathematical model of the problem has the following form [5], [6]: To maximize: (6) x ij = 1, if the i-th transport district is assigned to the j-th transport district x ij = 0, in the opposite case In case of restrictions: The classical task assignment is characterized by minimization.However, in our case it is necessary to maximize the traffic flows.In this case,we only need to replace the objective function f with the function [12], [17], [18]:

THE COMPREHENSIVE PROPOSAL FOR THE ROUTING OF THE LINKS OF URBAN AND SUBURBAN TRANSPORT LINES
Based on the results of the mathematical model the distribution of direct transport links was determined.It was necessary to move from the stage when the individual flows are proposed using a mathematical model to the stage where from a number of proposed options such options must be selected which follow on from the transport significance of a district (stated by the decision maker), from the stretch intensity between districts and from basic transport principles (especially the classical theory of transport).
This procedure also requires the use of common sense, experience and the knowledge of the history of the transport system.It is therefore necessary to also apply the personal approach of a decision maker.It should be noted that as part of the decision making process.It is also necessary to take into account the system of trolleybus transport, especially the existing trolleybus network.
The methodical solution applied in the research of the transport model proposed with the support of mathematical methods represents a modern approach to designing the routes of the lines in České Budějovice and it can be applied to any city public transport system.The implemented changes represent a significant positive contribution to the overall public transport system.Compared to 2000, the number of passengers carried has increased by at least 30% (on the Máj -Nádraží route).The system is gradually becoming more "user friendly" due to the introduction of the routes that passengers use [8], [9].
Another benefit is the expansion of the trolleybus network in České Budějovice.A new trolleybus line to České Vrbné was completed in 2007.
From the point of view of scientific knowledge, this study is an example of creative research applied to a purely practical problem with the use of exact mathematical methods.Furthermore, this research has confirmed that it is necessary to know its substance as well as modern methods for its solution for a successful solution to any problem [10]- [13].

CONCLUSION
The theoretical part of this article outlines the mathematical methods that can be used for the determination of the transport links between individual areas within a territory, or actually the transport flows and their directionality.The practical part of this article contains the determination of the transport flows in the territory of České Budějovice from the input data by means of basic mathematic methods.The new municipal public transport lines were also proposed and the existing ones were optimized upon the determined transport flows.
The benefits cited in the previous chapter are quite significant and it is undeniable that research brings not only new insight into some mathematical methods, but also into the possible marketing or economic aspects when designing or modifying transport models, and not only in public transport sector.