The Applicatio of Mathematical Methods to the Determiatio of Trasport Flows Primjea matematičkih metoda kod određivaja prometih tokova Ladislav Bartuška Departmet of Trasport ad Logistics Istitute of Techology ad Busiess České Budějovice, Czech Republic e-mail: bartuska@mail.vstecb.cz Jiří Čejka Departmet of Trasport ad Logistics Istitute of Techology ad Busiess České Budějovice, Czech Republic e-mail: cejka@mail.vstecb.cz Zdeěk Caha Departmet of Foreig Laguages, Istitute of Techology ad Busiess České Budějovice, Czech Republic e-mail: caha@mail.vstecb.cz DOI 10.17818/NM/2015/SI1 UDK 519:656 Origial scietific paper / Izvori zastvei rad Paper accepted / Rukopis primlje: 6. 3. 2015. Summary This article deals with the basic mathematical methods applied to the determiatio of the size of traffic flows i a urba area trasport etwork. The methods were applied to the desig of a ew system of public bus trasport i the city of České Budějovice. This article represets the partial outcome of the research carried out by the Istitute of Techology ad Busiesses i České Budějovice. Sažetak U ovom radu govori se o osovim matematičkim metodama koje se primjejuju kod određivaja veličie prometih tokova u urbaoj prometoj mreži. Metode su primijejee a dizaj ovog sustava javog autobusog prijevoza u gradu Česke Budejovice. U radu je izlože dio rezultata istraživaja koje je proveo Istitut tehologije i poslovaja u Českim Budejovicama. KEY WORDS operatioal aalysis methods solvig optimizatio problems public bustrasport decisio makig Detroit model KLJUČNE RIJEČI metode operacijske aalize rješavaje problema optimizacije javi autobusi prijevoz doošeje odluka model Detroit INTRODUCTION Trasport plaig ad operatioal trasport maagemet is a field where the applicatio of simulatio models based o mathematical methods is the first step i the determiatio of a efficiet solutio. Mathematics is applied i trasport system modellig as a basic tool, whether it deals with a etwork of airlies or a etwork of railway routes. This article outlies the applicatio of basic mathematical methods particularly i the creatio ad optimizatio of a etwork of muicipal public trasport lies. The optimizatio or ratioalizatio of already existig trasport systems is actually more usual uder the preset coditios. We ca distiguish betwee strategic ad operatioal plaig i the public trasport sphere. I the first phase of strategic plaig, we ca collect the basic data, e.g. o passeger itesities (O/D matrix) ad set the strategic goals of trasport service i relatio to the city ad its ihabitats. Numerous statistical ad stochastic methods are applied i this phase, but also basic mathematical methods are applied as well, e. g. particularly for the determiatio of trasport flows withi a system. I the ext phase of operatioal plaig, the obtaied data are processed to the best possible effectiveess of the trasport system with the stress o cost miimizatio. The proposal of lie routes withi the give territory (that icludes the calculatio of vehicle turs o these lies ad other processes liked to the optimizatio) ca be icluded ito this phase of the trasport system optimizatio. Mathematical methods have to ecessarily be itegrated ito these processes. The determiatio of the optimum umber of vehicles ad plaig their turs is a typical case where the basic methods of operatioal research are applied, e. g. the Travellig Salesma Problem [1]. Nowadays, whe there are still ew demads arisig i trasport ad trasport systems are o the icrease, mathematics is eve more ecessary ad importat for decisio makig problems i trasport plaig. Operatioal research ad optimizig mathematical methods have a log history i air trasport; for example, decisio makig problems are supported by operatioal aalysis methods ad have already bee developig for 30 years. Airlies use optimizatio techiques for daily, weekly, ad mothly plaig such as fleet assigmet, crew schedulig, ad crew fosterig [1]. Fleet assigmet i forwardig compaies may similarly be optimized or liear programmig methods ca be used for the determiatio of traffic flow utilizatio o roads ad Naše more, Special Issue, 62(3)/2015., pp. 91-96 91
motorways, for thecreatio of a future developmet model to predict cogestio occurrece. The applicatio opportuities are umerous. So let us focus o the applicatio of certai mathematical methods to the determiatio of the size of the trasport flows betwee idividual areas withi a muicipal public trasport system. THE GENERATION OF TRANSPORT FLOWS BETWEEN THE AREAS OF A TRANSPORT SYSTEM The complex techique of trasport system modellig oly developed i the times whe problems requirig complex ad coordiated solutios were growig o oe side ad sufficietly powerful computig techology that eabled such solutios was available o the other side. The whole process of trasport system modellig ca be split ito four parts (the traditioal four-step trasport model)[2]: 1. The geeratio of trasportatio eeds (volumes of origi ad destiatio trasport). 2. The distributio of trasportatio relatios (directios of trasport flows). 3. The classificatio of trasportatio relatios accordig to the meas of trasport used (divisio of trasportatio labour). 4. The utilizatio assigmet to routes ad trasportatio etwork sectios. Network trasport research has to be doe i order to obtai the itesities of passegers withi a public trasport system. The territory where the trasport services withi the trasport etwork are provided S = ( V, H ), has to be divided ito areas (zoes). A better solutio is to divide the territory to as may zoes as there are ods v V i V. Each zoe is thus allocated to a od. Trasport itesities q ij b betwee two differet zoes Q i ad Q j may the be determied right from the research performed. Uless iter-zoal trasport liks are kow, it is also possible to take the trasport volume ito accout q i gettig out of the zoe (ad thus also gettig i) i the calculatio ad calculate these values, for example by multiple regressio aalysis i liear form. The basic formula for calculatio ad determiatio of iterzoal trasport liks may be cosidered as follows: f = i= 1 j= 1 q ij x ij, (1) where x ij is a variable, which may be for example the distace betwee the i-th ad the j-th zoes, the price of trasport betwee the zoes or the travel time betwee the zoes, is the umber of cosidered zoes. The followig coditios however have to be met at the same time: i= 1 i= 1 q ij = q i i = 1,.. (2) q ij = q j j = 1,.. (3) TRANSPORTATION FORECASTING The applicatio of the four-step trasportatio model to trasport forecastig requires trasport research, evaluatio of the exact values of the trasport process, the derivatio of the depedeces valid for the preset coditios ad the progosis of the expected trasport flows may be calculated based upo these data. However, this icludes the itegratio of various socio-demographic, developmetal ad other data ad the prospective trasport volumes may be based o them. THE CALCUATION OF THE PROSPECTIVE TRANSPORT VOLUMES Multifactor aalysis procedures, amely multiple regressio aalysis, were commoly used for the determiatio of the prospective trasport volumes of the idividual areas. The trasport volume was calculated as a variable i depedece o various socio-demographic data of the area as relevat idepedetly variable structural quatities: D i = a + b1 X1 + b2 X 2 +... + b X, (4) where D i - is a depedet variable (trasport volume of the i-th area) a - regressio equatio costat b 1, b 2,, b - partial regressio coefficiets X 1, X 2,, X - idepedetly variable structural quatities with decisive ifluece o trasport volume Double regressio aalysis was used for the calculatio of the prospective trasport volumes as the simplest method, i which umbers of ihabitats ad labour opportuities were used as structural quatities. Liear regressio aalysis was examied i Slovakia Bratislava withi the Czechoslovak state research task of P-13 order with the itroductio of eight idepedetly variable quatities that were available i the detailed territorial divisio: the active ad passive ihabitats, labour opportuities i sectors I ad II, labour opportuities i sector III divided ito active ad passive, the umbers of places at primary schools, secodary ad tertiary schools ad other equipmet of the area i m 2 of usable area [3]. The calculatios were performed for models with fixed iput of structural quatities i a certai order, with the classificatio of the idividual structural quatities accordig to the lik stregth always for equatios with a absolute term (with regressio equatio costat) ad for equatios with a limitig coditio without a absolute term. Although the results were very good, there were still some doubts whether liear regressio aalysis is the right procedure for the determiatio of the trasport volumes. There is a serious reservatio that it does ot provide a view of establishmet ad expiry of the trasport liks from the trasport egieerig poit of view. The regressio aalysis was gradually replaced i calculatio of trasport volumes by differet methods, particularly the Specific Mometum Method, which however requires more detailed calculatio iputs. TRANSPORT FLOW ROUTING After the phase of the determiatio of the trasport volumes betwee the idividual zoes withi a trasport etwork, the determiatio of directivity of these trasport flows may follow. This meas the determiatio of the size of each trasport lik D ij betwee two zoes i for startig trips ad j for edig trips ad thus create a complete matrix of liks for all districts the territory is divided ito. 92 L. Bartuška et al: The Applicatio of Mathematical Methods...
The calculatio of the prospective directig of the trasport flows has also developed from the simplest oe-factor procedures, via the average growth factor to more complex aalogical ad sythetic procedures. The requiremet for the sums of the volumes of the source ad destiatio trasport (that are to be equal ad to equal the total trasport volume of the moitored territory) is the basic coditio for all these procedures. There are further margial coditios, maily the requiremets for the sum of all trips from area i to the other areas j to equal the volume of the source trasport of area i ad for the sum of all trips to area j from all areas to equal the volume of destiatio trasport of area j. These coditios caot usually be met for the first time; so the calculatio is repeated with gradual approximatio (iteratio) ad after the achievemet of certai accuracy, the procedure is fiished. Aalogical procedures determie the prospective trasport liks by the aalogy with the preset situatio, while the simplest of them oly takes accout of the basic requiremets (the destiatio ad source trasport volumes, the sourcedestiatio distace etc.). The oldest oe (the uiform growth factor procedure) assumes that all iter-zoal relatios grow uiformly, i.e. that the growth factor is idetical for the whole moitored territory ad is defied by the quotiet of the prospective ad the preset trasport volumes. I the average growth factor procedure, the prospective trasport lik is determied by the product of the preset lik ad the arithmetic average of the growth factor of source trasport i ad destiatio trasport j of the area. More complex aalogical procedures, like the Detroit ad Fratar Models already take accout of ot oly the pair of moitored areas, but also the mutual ifluece of the other areas of the territory. The Detroit Model is based o the assumptio that the prospective trasport lik is directly proportioal to the preset lik ad the growth factor of both the moitored areas ad is idirectly proportioal to the growth factor of the whole city [2]:, (5) where - prospective trasport lik, - preset trasport lik, Z K i - growth factor of source trasport i areai, C K j - growth factor of destiatio trasport i area j, v D loc. district K = - whole city growth factor s D loc. district Sythetic procedures seek various ways of expressig the idividual factors for the future, as they sigificatly ifluece the size of the prospective trasport lik D ij. The gravity model i various modificatios is particularly used i trasport egieerig. THE USE OF THE MATHEMATICAL BASIC METHOD TO PROPOSE LINK LINES Let us ow have a look at a example of the applicatio of the basic mathematical methods to the preparatio of a trasport model. It is amely a trasport problem solvig algorithm applied to a assigmet problem solutio. This method has bee chose particularly to show that we are able to save a substatial part of costs, eve by meas of a pricipally simple mathematical method, which would have bee may times higher if the same situatio had bee solved by for example the VISEVA, VISUM (Modal Split) model[4]. The trasport model desiged this way was used withi the research also for optimizatio of public trasport routig i České Budějovice. The solutio itself was preceded by the trasport research o the public trasport lies, from which the trasport volume data were used. THE MATHEMATICAL MODEL OF THE ASSIGNMENT PROBLEM The optimizatio of the trasport liks i public trasport ca be formulated ad solved as a assigmet i which the idividual districts are regarded both as sources ad destiatios. The rates are sums of the idetified source ad destiatio traffic itesities betwee ay two districts. I order to avoid assigig the same two districts, so called prohibitive rates were chose o the mai diagoal (see table 1). The criterio of optimality is the maximizatio of the total umber of passegers trasported without trasfer. The mathematical model of the problem has the followig form [5], [6]: To maximize: x ij = 1, if the i-th trasport district is assiged to the j-th trasport district x ij = 0, i the opposite case I case of restrictios: The classical task assigmet is characterized by miimizatio. However, i our case it is ecessary to maximize the traffic flows. I this case,we oly eed to replace the objective fuctio f with the fuctio [12], [17], [18]: (6) (7) (8) Naše more, Special Issue, 62(3)/2015., pp. 91-96 93
Table 1 Adjusted Table - The Sum of the Source ad Destiatio Traffic Itesities - prohibitive rates o the mai diagoal Source: authors THE SOLUTION OF THE ASSIGNMENT PROBLEM AND INTERPRETATION OF THE RESULTS The assigmet problem is usually solved by the so called Hugaria Method [4], [7]. Due to the software optios,the cosidered assigmet problem was solved as a trasportatio problem usig the Dumkosa program (a add-i XLA, created by the Departmet of Operatioal ad Systems Aalysis at PEF ČZU i Prague). Followig the theory of liear programmig, the solutio of a assigmet problem obtaied by the methods for solvig the trasport problem is greatly degeerated. Completig the umber of occupied boxes to the umber required for a o-degeerated solutio is usually doe by usig a egligibly small amout of EPS. The symbol ALT i some fields meas that by fillig this field we would obtai the equivalet optimal solutio with the same value of the objective fuctio (samples for first ad fourth grade directioal trasport flows ad are idicated i tables 2 ad 3). Usig the Dumkosa Program, first grade trasport flows were obtaied; the strogest liks are betwee the hubs (see table 2). However, the algorithm suggests such coectios so as the value of the objective fuctio (the sum of all trasported passegers without trasfer) is at its maximum at ay give momet [5], [6]. The procedure was the repeated four times. The procedure was always the same, but further prohibitive rates were put ito the places of the assigig rates. As a result, four trasport flows were obtaied (see tables 2-5) which are marked for the sake of simplicity. It is ecessary to poit out that the first grade is 94 the most sigificat oe. It is also very importat to realize that the obtaied grades of the trasport flows are ot comparable give the importace of the idividual districts. For this reaso, it is ecessary to cosider every trasport grade as sigificat i the case of a large district (estate), whereas it is possible to cosider oly the first ad possibly the secod grade trasport flows i the case of a weak trasport district (e. e. Havlickova Koloie). The proposed allocatio of the districts represets the best directliks accordig to the selected mathematical model [8]. A situatio where idividual trasport flows come together or coicide is very crucial i this cotext. A typical case of this occurs i districts A ad S (table 1). The proposed direct trasport liks represet the best direct lik lies of public trasport i České Budějovice while respectig the criterio of optimality where by the maximum umber of passegers travel direct ad without trasferrig. For those trasferrig, it is proposed to miimize waitig times based o the trasport theory: limitig parallel lies followig the same stretches to differet destiatios. Furthermore,suitable trasfer termials ca be read out of the model (followig tables 2 ad 3). Accordig to the above-metioed mathematical algorithm 4 trasport flow grades were idetified (samples for first ad fourth grade directioal trasport flows ad are idicated i tables 2 ad 3) [8]. L. Bartuška et al: The Applicatio of Mathematical Methods...
Table 2 Optimal Solutio of the Trasport Model - First Grade Directioal Trasport Flows Source: authors Table 3 The Optimal Solutio of the Trasport Model Fourth Grade Directioal Trasport Steams Source: authors Naše more, Special Issue, 62(3)/2015., pp. 91-96 95
THE COMPREHENSIVE PROPOSAL FOR THE ROUTING OF THE LINKS OF URBAN AND SUBURBAN TRANSPORT LINES Based o the results of the mathematical model the distributio of direct trasport liks was determied. It was ecessary to move from the stage whe the idividual flows are proposed usig a mathematical model to the stage where from a umber of proposed optios such optios must be selected which follow o from the trasport sigificace of a district (stated by the decisio maker), from the stretch itesity betwee districts ad from basic trasport priciples (especially the classical theory of trasport). This procedure also requires the use of commo sese, experiece ad the kowledge of the history of the trasport system. It is therefore ecessary to also apply the persoal approach of a decisio maker. It should be oted that as part of the decisio makig process.it is also ecessary to take ito accout the system of trolleybus trasport, especially the existig trolleybus etwork. The methodical solutio applied i the research of the trasport model proposed with the support of mathematical methods represets a moder approach to desigig the routes of the lies i České Budějovice ad it ca be applied to ay city public trasport system. The implemeted chages represet a sigificat positive cotributio to the overall public trasport system. Compared to 2000, the umber of passegers carried has icreased by at least 30% (o the Máj Nádraží route). The system is gradually becomig more user friedly due to the itroductio of the routes that passegers use [8],[9]. Aother beefit is the expasio of the trolleybus etwork i České Budějovice. A ew trolleybus lie to České Vrbé was completed i 2007. From the poit of view of scietific kowledge, this study is a example of creative research applied to a purely practical problem with the use of exact mathematical methods. Furthermore, this research has cofirmed that it is ecessary to kow its substace as well as moder methods for its solutio for a successful solutio to ay problem [10]-[13]. CONCLUSION The theoretical part of this article outlies the mathematical methods that ca be used for the determiatio of the trasport liks betwee idividual areas withi a territory, or actually the trasport flows ad their directioality. The practical part of this article cotais the determiatio of the trasport flows i the territory of České Budějovice from the iput data by meas of basic mathematic methods. The ew muicipal public trasport lies were also proposed ad the existig oes were optimized upo the determied trasport flows. The beefits cited i the previous chapter are quite sigificat ad it is udeiable that research brigs ot oly ew isight ito some mathematical methods, but also ito the possible marketig or ecoomic aspects whe desigig or modifyig trasport models, ad ot oly i public trasport sector. REFERENCES [1] Bordörfer, R., Grötschel, M., Löbel, A. Optimizatio of Trasportatio Systems. ACTA FORUM ENGELBERG. 1998. [2] Kusierova, J., Hollarek, T. Metody modelováia a progózovaia prepravého a dopravého procesu. Žiliská uiverzita v Žiliě. Žilia. ISBN 80-7100-673-4. 2000. [3] Kusierova, J., Hollarek, T., Lilov, M. Teoretické podmieky a predpoklady pre overovaie dopravej sústavy. Závěrečá správa úlohy RVT P-13-580-271-02. ÚHA Bratislava. 1980. [4] Fiala, J., Jabloský, J., Maňas, M. Vícekriteriálí rozhodováí. Prague. VŠE. 1994. [5] Bierma, H., Boii, Ch. P., Hausma, W. H. Quatitative Aalysis for Busiess Decisios. Homewood, Irwi. 1986. [6] Čerá, A., Čerý, J. Teorie řízeí a rozhodováí v dopravích systémech. Pardubice, Istitute of Ja Perer. ISBN 80-86530-15-9. 2004. [7] Volek, J.: Operačí výzkum I., Pardubice, Uiversity of Pardubice, 2002, ISBN 80-7194-410-6. [8] Čejka J., Kovařík, J. Dopraví průzkum ve městě České Budějovice. Sborík DPMCB.. Dopraví podik města Českých Budějovic a.s. 2006. [9] Dittrich, T. Verkehrsplaerische Berechuge miot dem Modellkomplex VISEVA/VISUM zur Wirsamkeit eier Regiotram i der Regio Liberec. Istitut für Verkehrsplaug ud Straβeverkehr. 2005. [10] Kampf, R., Ližbeti, J., Ližbetiová, L. Requiremets of a Trasport System User. Commuicatios, Žilia: Uiversity of Zilia, Vol. 14 (4), pp. 106-108. ISSN 1335-4205. 2012. [11] Kubasáková, I., Kampf, R., Stopka, O. Logistics iformatio ad commuicatio techology. Commuicatios. Vol. 16 (2). pp. 9 13. ISSN 1335-4205. 2014. [12] Krile, S. Efficiet Heuristic for No-liear Trasportatio Problem o the Route with Multiple Ports, Polish Maritime Research, Gdask, Polad, Vol. 20 (4), pp. 80-86, ISSN 1233-2585. 2013. [13] Krile, S., Žagar D., Martiović G. Better Badwidth Utilizatio of Multiple Lik Capacities with Mutual Traffic Correlatio, Tehički vjesik - Techical Gazette,Vol. 16 (4), pp.11-18, ISSN 1330-3651. 2009. 96 L. Bartuška et al: The Applicatio of Mathematical Methods...