Planning of Optimal Fuel Supply of International Transport Activity

Transcription

1 P Perodca Polytechnca Transportaton Engneerng P Plannng of Optmal Fuel Supply of Internatonal Transport Actvty 45(4), pp , 27 Creatve Commons Attrbuton b György Kovács *, Mklós Gubán 2 research artcle Receved 9 December 26; accepted 6 June 27 Abstract The selecton of the petrol staton for refuel and the amount of reflled fuel are not determned centrally at most transport companes, but depend on the ndvdual decson of the drver, so that the total cost of the burned fuel s not mnmal. The goal of ths study s to elaborate a precse and relable mathematcal model and method for the determnaton of the optmal refuel ponts and the amount of loaded fuel to fulfl specfc transport tasks. The model s a mxed valued non-lnear programmng model, whch can be handled by optmzaton procedures. Based on the elaborated model and method, a decson-supportng software was developed, whch provdes the requred nformaton that s necessary for the economcal fulflment of transport trps. Keywords transport trp, petrol staton, refuelng, optmzaton, software development Insttute of Logstcs, Faculty of Mechancal Engneerng and Informatcs, Unversty of Mskolc, -355 Mskolc, Mskolc-Egyetemváros, ungary 2 Department of Management and Informaton Technology of Economcs, Zalaegerszeg Faculty of Busness Admnstraton, Budapest Busness School -89 Zalaegerszeg, Gasparch u. 8/a, ungary * Correspondng author, e-mal: altkovac@un-mskolc.hu Introducton of nternatonal road transport The volume of the transport actvtes connected to producton and servce s ever ncreasng due to the growth of the effcency of the economy and the long nternatonal supply chans. In Europe the rato of road transport n the total transport volume s 78 % (data from (Fraunhofer, 24)), furthermore ths rato s constantly ncreasng. The remanng 22 % breaks up to ralway (7 %), waterway (3 %), sea (8 %), ppelne (4 %), whle ar freght transport s almost neglgble (8 mllon tons per year). Road transport s economcal manly for regonal range, but due to ts many advantages t s also used n case of long-dstance transport as well. Its hgh rato compared to the other transport modes s orgnated from ts hgh densty of road network, short transport tme, hgh level of adaptablty to the customer s demands and provdes door-to-door servce. Approxmately 3 % of the cost of the whole supply chan comes from transportaton, therefore every producton and servce companes put large emphass on the optmzaton of transportaton. The ncrease of the effcency of nternatonal transport routes and the decrease of the greenhouse gas emsson can be realzed by the followng ways: modernzaton of the vehcle fleet, the use of engnes wth lower fuel consumpton and greenhouse gas emsson (Zöldy and Török, 25), development of the drvers drvng sklls, applcaton of multmodal transport methods, where road, ral, and water transportaton modes are combned, optmzaton of the transport task: ntegraton of multple transportaton tasks wthn one transport trp, elmnaton of dle runs, maxmzaton of the load effcency of the vehcle, optmal formaton of transport routes, refuelng the optmal amount of fuel at the deal petrol staton. 86 Perod. Polytech. Transp. Eng. Gy. Kovács, M. Gubán

2 Goal of ths research s the optmzaton of the road freght transport actvty by the selecton of optmal petrol staton (dependng on the fuel prce and dstance to the orgnal transport way), and by the determnaton of the optmal amount of reflled fuel (only the requred amount of fuel, not hgher). The result of wrong decsons s that the cost of the burnt fuel s not mnmal. Ths s true n case of nland, but t has even more sgnfcance n case of nternatonal road freght transport routes, whch case hgh amount of fuel s burnt. Another goal s to develop a software based on the elaborated theoretcal method that would plan the optmal fuel supply, therefore the total cost of the transportaton actvty would be reduced. 2 Lterature revew Before the elaboraton of the optmzaton method for refuelng, the natonal and nternatonal lterature of the topc was revewed. A wde range of lterature s avalable that ntroduces the general logstcs costs (e.g. Ross, 25), and logstcs lterature often deals wth the ntroducton of cost components of road transportaton (e.g. Brge and Lnetsky, 27; Anbuudayasankar et al., 24). Vehcle route plannng (VRP) s an mportant soluton to reduce the fuel consumpton of delvery vehcles. VRP ams at plannng the routes of a fleet of vehcles on a gven freght transport network to serve a set of clents under constrants. Furthermore a reasonable route plan could effectvely reduce the transportaton costs and energy consumpton. (Zhfeng et al., 24). The lterature often dscusses route optmzaton and optmzaton of transport trps and networks (Brge and Lnetsky, 27; Carama and Dell Olmo, 28; Ehmke, 22; Anbuudayasankar et al., 24; Bohács et al., 26). There are several avalable mathematcal optmzaton algorthms to solve VRP problems, e.g. heurstc algorthms, such as genetc algorthm (Ombuk et al., 26) or ant colony algorthm (Yu et al., 29; Zhfeng et al., 24). There are very few lteratures that deal wth refuelng polces of transport trps, e.g. Ln et al. (27) show a lnear-tme greedy algorthm for fndng optmal refuelng strateges n a fxed route. Ln (22) later mproved the prevous study and proved the effcency of the best algorthm to mnmze fuel costs n case of a mult route problem. Tegelberg (25) optmzed the fuel expenses of long nternatonal trps by ntroducng a soluton of refuelng at cheap gas statons and refuelng less but more often, thus keepng the fuel consumpton low. Dynamc programmng was used to solve ths optmzaton problem. There s gap n the lterature n the feld of fuel optmzaton that takes nto consderaton the optmal route, the fuel prces at the dfferent statons, the fuel consumpton of dfferent vehcles on dfferent routes and the optmal amount of reflled fuel. The fuel optmzaton task to be solved has not been covered before, therefore ths topc s absolutely actual and unque. In order to organze road transportaton three dfferent categores have to be defned: smple trps, shuttle trps and round trps (Smch-Lev et al., 24; Anbuudayasankar et al., 24). An mportant characterstc of road freght transportaton s that t conssts of transport trps. The man goal of the organzaton of the transport routes s to mnmze the cost and dle runs, namely the unt cost of transportaton per product (Brge and Lnetsky, 27; Ehmke, 22). Every transport trp departure from the depot of the company, then goes to the frst product dspatch pont, then t proceeds to the frst dscharge staton, then contnues to the followng dspatch and dscharge statons etc. (Fg. ). After the last dscharge staton the vehcle fnshes ts route at the depot. The number of dspatch/dscharge statons can be arbtrary chosen n a transport route, furthermore any dspatch staton can be a dscharge staton as well. (Kovács and Cselény, 26, Kovács et al., 27). Fg. Structure of road transport trps 3 Problem formulaton In case of transport trps the selecton of the petrol staton for refuel and the amount of reflled fuel are not determned centrally at most transport companes, but depend on the ndvdual decson of the drver. Therefore many tmes the refuelng does not take place at the deal petrol staton, and not the optmal amount of fuel s reflled or n some cases the drver refuels unnecessarly. The result of these wrong decsons s that the cost of the burnt fuel s not optmal. Ths s true n case of nland transportaton tasks, but t has even more sgnfcance n case of nternatonal routes, where hgh amount of fuel s used. There are many avalable petrol statons, but the dfference between the unt prce of the fuel can be 5-7 UF/lter between the dfferent petrol statons, whch results n 35, UF dfference n case of the refll of only 5 lters. Another common mstake s that the drver refuels an unnecessarly hgh amount of fuel n the end of a transport route before arrvng to the depot of the company, sometmes at an expensve petrol staton. Ths s a bg loss, snce most transportaton companes have the opportunty to refll at the depot as well wth more reasonably prced fuel. Plannng of Optmal Fuel Supply of Internatonal Transport Actvty

3 The above examples justfy the am of the authors to determne an optmzaton method for refuelng that s essental for transportaton companes for reducton of the transport costs. The goal of ths study s to elaborate a mathematcal model that would elmnate these wastes, and would reduce the losses. The task s to determne the most effcent petrol staton amongst the preferred one by the company ( t ), n case of a transportaton task from staton F+ to destnaton L defned + by GPS coordnates, furthermore to determne the amount of reflled fuel wth a known ntal level of fuel (Fg. 2). s max s 2max Depot/L F + s mn t L + Q f Q fs s real Q(L,F+) Q(F+, t ) s s Q T Q B s 2real Depot/L F + t L + Q B t k Q( t,l+) Fg. 2 Determnaton of possble petrol statons, and the change of fuel level on the road Selecton of petrol statons and the optmal amount of reflled fuel durng long road transport trps s depends on the ndvdual decson of the camon drver, not defned by a central decson makng process at forwardng companes. Therefore the total cost of the burned fuel s not mnmal. The goal of the study s to elaborate a mathematcal model that allows the elmnaton of lost arsng from the above mentoned problems. Therefore the task s to defne the locaton of the optmal petrol staton from the preferred (t ) petrol statons f t s requred n case of a transport task from pont F+ (dspatch staton) to pont L+ (dscharge staton, Fg. 2), and to defne the volume of fuel to be reflled (Q T ). Therefore the goal s to determne whether the vehcle s able to fulfl the whole transport task wth the known ntal level of fuel, or t has to be reflled between the dspatch and dscharge statons. In case of refuelng s necessary, the deal refll pont and the amount of fuel has to be determned. Most of the transport companes due to the huge amount of fuel consumpton can buy fuel on a dscounted prce at a contracted fuel sellng company. Transport companes prefer those fuel supplers whch have a world-wde patrol staton network. The contracted fuel sellng company provdes the GPS coordnates of the avalable petrol statons and the actual fuel prce at s the ndvdual statons. Accordngly the GPS coordnates of the petrol statons can always be defned. Dgtal maps convolute the road network to short, straght sectons (Gubán and Gubán, 2). Ths allows the determnaton of the length of the shortest way between any two chosen ponts (between dspatch/dscharge staton and petrol staton, between the dspatch and the dscharge staton, etc.). Ths can be acheved by the use of an A* algorthm (Russel, 2; Bazaraa, 27; Zhang and Yap, 2), or by the applcaton of roadmap sheets (data sheets) (Gubán and Gubán, 2). Furthermore topographcal nformaton s also provded, whch s also necessary snce fuel consumpton s dfferent on flat roads than uphll or downhll condtons. In the followng sectons the possble petrol statons wll be chosen from every possble petrol statons, whch can contrbute to soluton of the task. Ths sgnfcantly decreases the number of varables n the fnal model. Durng the calculatons t s advantageous to dvde the transport trps nto sectons, whch means the road between a dspatch staton and a dscharge staton. Ths s necessary because the value of each cost category vares due to the dfferent carred loads, road condtons, etc. The total cost of a transport trp s the sum of costs of sectons: Cα = sαβ cαβ [ euro ], () β where: sαβ - the dstance of β -th secton of α -th transport trp [km]; cαβ - specfc cost of the β -th secton of α-th transport trp [euro/km]; β - secton dentfer, α - transport trp dentfer. The fuel consumpton of a vehcle depends on the consumpton of vehcle wthout useful freght load (general characterstcs of the engne), the weght of the transported freght load and the topography. These above mentoned nfluencng factors can be calculated or estmated, and as a consequence they can be taken nto consderaton durng the elaboraton of our model. There are further factors, whch also have nfluence on the fuel consumpton, e.g. the traffc stuaton on the road, the weather condtons, the road condtons, etc. These factors are absolutely stochastc, therefore t s not easy takng nto consderaton. These stochastc effects are handled n our model by the applcaton of an emergency amount of fuel ( QB ), whch covers the extra consumpton resultng from these non-predctable condtons (Fg. 2). Dfferent alternatves for dfferent refllng petrol statons because of the hgh densty of petrol staton network can not result n hgh dfferences n transport dstances and transport tmes especally n case of long nternatonal transport trps. We assume that the selecton of optmal petrol staton does not have any nfluence on the tme constrant of the transportaton, so the allowed tme wndow of the transport tasks can be satsfed. 88 Perod. Polytech. Transp. Eng. Gy. Kovács, M. Gubán

4 The specfc cost can be calculated by Eq. (2). ( ) D T euro cαβ = pαβ f f + f f εαβ + ε qαβ km (2) where: pαβ - prce of fuel [ euro lter] ; ff - specfc fuel consumpton n case of empty vehcle lter [ km ]; D εαβ - correcton factor for fuel consumpton dependng on topography, vares between (flatland),.3 (downy),.6 (mountan); T ε - correcton factor for dfferent loadng condtons (every addtonal tons of useful load results.5 lter extra fuel lter consumpton) [ ton km ] ; qαβ - transported useful load [ton]. The determnaton of correcton factors s based on a prevous research of authors of these paper (Kovács and Cselény, 26, Kovács et al., 27). The fuel level of the vehcle s known at startng pont ( Qfs ), the fuel consumpton can be calculated contnuously between dspatch- and dscharge statons (Fg. 2). There s a constrant relatng to the emergency amount of fuel capacty ( QB ) whch has to be avalable. Q emergency B amount covers the extra consumpton resultng from traffc jam, dverson, mssed way, etc. between ponts L and t. Q B emergency amount at the end of the secton also covers the before mentoned stochastc effects and the fuel consumpton of the vehcle to fnd the next petrol staton after the dscharge staton L+. Ths amount of fuel ensures that the vehcle can refuel on the end pont or the nearest deal petrol staton to the end pont. The maxmal dstance (s max ) whch can be completed by a vehcle s calculated as follows (Fg. 2): αβ max s Q QB = T f + f ε + ε q f f fs D αβ Ths radus arch ( smax Δ s ) defnes the coordnates of the t possble petrol statons, from whch the deal one should be selected. The value of Δ s can be freely defned, and t refers to the dstance from whch a sutable petrol staton has to be selected before runnng out of fuel. 4 Determnaton of the optmal petrol staton and the amount of reflled fuel The optmal petrol statons wll be selected from the deal petrol statons. Although the model does not requre prevous selecton, ths selecton has some practcal values, snce t reduces the model sze. The prevously defned varables wll be specfed for the model. A specfc road plan ncludes some statons that must be crossed to dspatch and/or dscharge the load. Ther order s predetermned and used to dentfy each dspatch and dscharge staton. N notes the number of dspatch and dscharge statons αβ (3) ncludng the departure and fnal destnaton statons as well (whch can be the same n case of transport trps, but n ths case they are noted wth dfferent ndexes). Each dspatch and dscharge staton s noted by a natural number between and N. From ths pont on the dspatch and dscharge statons wll not be dfferentated (both of them are called nterchanges). Dspatch and dscharge statons wll dffer by weght of the load durng the road-secton departng from the gven nterchange ( n case of dscharge, and hgher than n case of dspatch). 4. Defntons of used terms Road-secton: the set of roads between the gven dspatch or dscharge staton and the next nterchange accordng to the planned transport route s called a road-secton. Usually there are more alternatve drectons to the next nterchange, whch wll be defned at the petrol statons n ths model. Road-sectons wll be noted by ndexes. Therefore the road-secton β of trp α can be easly dentfed by the nterchanges of the trp and the denttes of the assgned road-sectons (road-sectons wll be noted by k ndex). Ths dentfcaton s more approprate n case of computerzed solutons. Set of fuel statons, probable road-secton: T stands for the set of fuel statons between departure staton and destnaton staton +. In ths case τ = T notes the number of petrol statons on road-secton. A specfc road of the road-secton belongng to the fuel statons s called a probable road-secton. Probable road-sectons are noted by, k ndexes. Probable road-sectons must nclude only one petrol staton. If there are more than one, the model can be changed to a new model n whch there s only one petrol staton on each road-secton (see also Remark 4). Notaton of petrol statons: tk T notes petrol staton k (petrol staton that belongs to probable road-secton k) on road-secton. Maxmum number of petrol statons on a gven road-secton: ( = max τ ) notes the maxmum number of the petrol statons on a road-secton. If road-secton ncludes fewer than M petrol statons, then fctve probable road-sectons are to be defned between τ + and. The length of the probable road-secton should be hgh enough to ensure that t wll not be chosen durng the optmzaton process. Road (wth ndex) s a road-secton wthout petrol refll, whch means the vehcle does not stop on ths probable road-secton. Topography: There wll be two topography-related varables assgned to the probable road-sectons n ths model: the frst one s the topography factor assgned to the secton from the departure staton to the petrol staton; the second one s assgned to the secton from the petrol staton to the destnaton. DL εk notes the topography factor n relaton, + of the secton to the petrol staton k. DM εk notes the topography factor n relaton, + of the secton from petrol staton k to the destnaton. Plannng of Optmal Fuel Supply of Internatonal Transport Actvty

5 T Correcton factor for dfferent loadng condtons: ε notes the correcton factor for dfferent loadng condtons (See also (2)). Unt prce matrx: pk notes the unt prce of the fuel at the petrol staton tk ( P = [ pk ] ). p = ( = ; ; N). N + Dstance matrces: Dstance matrces smlarly to topography factors are dvded nto two parts: the frst one s assgned to the way of the secton from the departure staton to the petrol staton, and the second one s assgned to the way of the secton from the petrol staton to the destnaton. lk notes the length of the road-secton from the departure staton (wth destnaton staton + ) to the petrol staton k ( L = [ lk ] ). N + mk notes the length of the road-secton between departure staton to destnaton staton + from the petrol staton k to the destnaton staton ( M = [ mk ] ). N + The defnton of the total length of road-secton s also necessary: sk = l + m ( S = k k [ sk ] ). N + Vector of the amount of load q notes the amount of load n the relaton, +. It s n case of empty vehcle. Ths way the loaded and dle transfer trps can be handled unformly ( q = [ q ] ). N Specfc fuel consumpton: ff notes specfc fuel consumpton (lter/km) see also at (2). Capacty of fuel tank: Qmax notes the maxmum capacty of the fuel tank. Remarks. In case of L, M, S, P matrces due to computatonal reasons ndexes starts from and ends at. 2. Because of practcal reasons n case of road the total length assgned to l s m = ( l = s, m = ). 3. When t s possble to refll fuel on the hghway as well, whch s road, then ths road must be defned agan, but n ths case wth l and m ndexes assgned to the petrol staton. Ths probable road-secton appears twce, but the secton wth ndex dfferent than s a probable road-secton where fuel s reflled. In case of there are more than one petrol statons on the hghway, the procedure descrbed by Statement should be executed. 4. There are two possble ways to proceed from one ntersecton to the next one: a) Wth free choce, that means t s allowed to choose freely from the probable road-sectons. b) Mandatory proceedng, whch means that f we arrve from probable road-secton k then the next ntersecton also must be approached on road-secton k. The sgnfcance of t les n the followng Statement. In the followng the gven mathematcal model, for the sake of smplcty, can take nto account only one petrol staton per road secton. Although, practcally t s possble to have more petrol staton wthn one secton. Therefore a statement s gven whch prove that any gven practcal task (whch contans more than one petrol statons on a probable road secton) can be modelled n a way that allows only one patrol staton on every probable road sectons. Statement. The orgnal problem can always be converted nto a task, n whch there s only one petrol staton between two ntersectons. The convoluton of a probable road-secton s necessary when there are more petrol statons on the same probable road-secton. In the orgnal problem each ntersecton allows free choce (Fg. 3). Case. Let us convolute a probable road-secton nto as many probable road-sectons, as the number of the petrol statons. Ths convoluton s complete when all road-secton fulfl the requrement that the dstance between the prevous probable petrol staton (or the departure staton n case of the frst road-secton) and the petrol staton of the partcular road-secton can be reached wth one full tank wthout ntermttent refll, furthermore the petrol staton of the next road secton (or the destnaton staton n case of the last road-secton) can also be reached wth the same condtons (Fg. 4). Two petrol statons are on the same possble road secton accordng to condtons of Case (Fg. 3) staton k staton k+ + Fg. 3 Case. Two statons are n a probable road-secton between nodes, +. Base case. staton k staton k+ + Fg. 4 The probable road secton s substtuted two probably road sessons as descrbed n Case. Case 2. When Case. cannot be fulflled, then a fctve ntersecton that only exsts n the model has to be nserted nto road-secton at the ntal phase, so that t stands between the two petrol statons of the probable road-secton and t ensures the fulflment of the crteron for the probable road-sectons (Fg. 5). Then n the same manner fctve ntersectons must be defned 9 Perod. Polytech. Transp. Eng. Gy. Kovács, M. Gubán

6 n case of each road-sectons of the, + relaton. Ths fctve staton could be any pont of the relaton except due to computatonal reasons the pont of the actual petrol staton of the probable road-secton. The convoluted road-sectons must be ndexed so that the k ndex of the road secton that arrves to the vrtual ntersecton s the same as of the outgong ndex of the convoluted road-secton. Then the nserted ntersecton must be defned as an ntersecton wth mandatory proceedng (Fg. 6). If the crteron stll cannot be fulflled, then Case 2. must be repeated untl t s fulflled, or a new road-secton s created on whch there are not any petrol statons and the road-secton cannot be solved (but at the same tme the task can be completed). Two petrol statons are on the same possble road secton accordng to condtons of Case 2. (Fg. 5). staton k staton k+ + Fg. 5 Two statons are n a probable road-secton between nodes, + accordng to Case 2. Base case. Remark. Fg. 3 and 5 are same forms, but the dstance of staton k and k + n the Fg. 3 s less or equal than the dstance of a full tank can be done, as n the Fg. 5 s greater. staton k staton k+ + Fg. 6 Node s nserted between nodes, +. Node becomes an ntersecton wth mandatory proceedng 5 Constrants and objectve functon of the optmzaton process for the determnaton of the deal refuelng ponts and the amount of fuel to be reflled The followng matrx contans the frst group of the varables. The elements of t defne the amount of reflled fuel at each road-secton. It s noted by: x k X = [ x ], k, N (4) 5. Constrants 5.. Constrant for the determnaton of the probable route Note f the probable road-secton k s chosen, otherwse. If the drver refuels n probable road-secton k then xk >. Then sgn ( xk ) = so the probable road-secton k s chosen. One probable road-secton has to be chosen. Ths leads to the followng: τ sgn( x k )=, (5) as a consequence one of the road-sectons has to be chosen Constrant for the total length of the transport way Based on formulas (5), the total length of the road can be determned. The model does not contan the total length of road, but t s very mportant other calculatons (for example rest perod, calculatng of requred number of drvers, etc.). sgn( xk ) sk = sgn( xk ) ( lk + mk ) (6) means the length of the road-secton between and, f we refll at petrol staton k. Based on these and takng nto account the bypasses necessary for refuelng, the total length of the drve s: n k s = sgn( xk ) s = = k (7) 5..3 Constrant for the mnmal amount of fuel for the proceedng of the vehcle from a dspatch or dscharge staton Q denotes the amount of fuel n the vehcle at ntersecton. Q = [ ] Q N where Q notes the amount of fuel n the tank of the vehcle at ntersecton. Then the amount of fuel after the refuelng ntersecton can be defned as: + ( Q = Q + sgn( x ) ( l + m ) + x, DM ff + ff εk + ε q ) k k k k T (8) (9) the amount of reflled fuel at petrol staton k that s between departure staton and destnaton staton +. In case of k = no fuel loadng takes place on the gven road-secton, so that the amount s a fctve quantty x =. Ths does not cause any problems, snce ths varable wll not appear n the cost of the fuel consumpton. whch means to subtract the consumed amount of fuel (dependng on the load) from the avalable amount of fuel at the prevous ntersecton and addng the amount of reflled fuel. Plannng of Optmal Fuel Supply of Internatonal Transport Actvty

7 5..4 Constrant for the amount of remanng fuel upon arrval to the last dscharge staton or depot after the fulflment of the transport task An mportant crteron s that the amount of fuel cannot drop under a mnmal lmtng value. The tank of the vehcle must be flled wth ths mnmal amount of fuel when arrvng to the petrol staton: DL T Q sgn( xk ) l f f q Q k k ( f + f εk + ε ) = B () 5..5 Constrant for the maxmum amount of loaded fuel The amount of reflled fuel ( xk ) cannot exceed the capacty of the tank: DL T k max k ( f f k ) x Q Q + l f + f ε + ε q. () 5..6 Constrant for the oblgatory proceedng drecton n case of a round trp O notes the set of the ndexes of mandatory ntersectons. Then the followng crteron belongs to the mandatory ntersectons: ( )= ( ) = sgn xk sgn x+, k, O, k,,. (2) 5..7 Constrant for the amount of remanng fuel upon arrval to the depot after the fulflment of the transport task Upon arrval to the destnaton ( = N ) a certan amount of fuel ( QZ ) has to reman n the vehcle n order to prepare for the departure of the next route: Q N Q. (3) C t notes the cost of loaded fuel at the relaton ( n case there were no refuelng), furthermore C T = Z N = The cost of refuelng at the gven fuel staton s: C t Ct = pk x k.. (4) (5) 5.2 Cost objectve functon Objectve functon means the cost of the ntal loadng of the fuel tank (fx cost), and the cost of the refuelng durng the drve decreased by the cost of the remanng fuel (varable cost): C = C( XQ, )= C + C C = C + p x k = sgn( x ) p N fs T r fs = k k Q N, k N,k N (6) The prevous crteron means that the cost of the remanng fuel has to be calculated wth the unt prce of the last fuel staton. The goal: C = C( XQ, ) mn. (7) The task s a mathematcal programmng task. Note: Cfs can be neglected durng the optmzaton, snce t s constant so t does not affect the optmum. Then the followng equaton can be used: N C = C ( XQ, )= C C. = T r (8) 6 Summary of constrants and objectve functon Cfs can be neglected durng the optmzaton, snce t s constant so t does not affect the optmum. The model can be smplfed accordng to (Gubán and Kovács, 26). The fnal model s the followng: xk, =,, N ; k =,, (9) y ;,,, N ; k,, (2) k { } = = yk =, =,, N (2) Q = Q fs (22) DL T Q yk lk ( fü + fü ε k + ε q ) Q, =,, N B (23) x y, =,, N ; k =,,τ (24) k k k ( ) DL T xk Qmax Q + lk f f + ff εk + ε q y, =,, N ; k =,, τ DM T yk ( lk + mk ) ff + ff εk + ε q + x + Q Q + =, =,, N k ( ) (25) (26) yk y+, k =, O, k =,, (27) Q N N Q (28) C = C ( XYQ,, )= C T C r = p k x k y N, k p N, k QN mn. Z = (29) 92 Perod. Polytech. Transp. Eng. Gy. Kovács, M. Gubán

8 [UF]: Amount of fuel at the end of secton [lter]: 7 Software applcaton for determnaton of the optmal petrol staton and the amount of fuel to be reflled Based on the elaborated theoretcal concept, a software was developed for optmzaton of refuelng process by the contrbuton of Norbert Czczer, an engneerng student. The software was wrtten n C# programmng language and the Mcrosoft Vsual Studo was used to develop the software (Czczer, 26). The developed software has two menu ponts whch are the Defnton of new data and Optmzaton. 7. Menu Defnton of new data In the menu Defnton of new data we can defne: new petrol statons (Fg. 7), new dspatch statons and dscharge statons (Fg. 8), and new transport vehcles (Fg. 9). 7.2 Menu Optmzaton Part.a of Fg. shows the possblty of selecton of a transport vehcle for a gven transport task. In ths menu the actual fuel level of the vehcle [lter] at the begnnng of the transport task and the fuel prce [UF/lter] of ths exstng fuel also can be defned. Part.b of Fg. provdes the possblty of determnaton of the transport trp, the dspatch statons and dscharge statons and the loadng condtons [ton] (transported weght and the changng of loadng condtons at the dspatch- or dscharge statons). Part.c of Fg. shows the results of the optmzaton. The total cost of the road transport trp [UF], the total fuel consumpton of the transport way [lter] and the remanng fuel volume [lter] at the end of the way are lsted. Part.d of Fg. descrbes the graphcal map of the transport way and the locaton of the optmal petrol staton where the drver has to refll. Part.e Fg. shows the name and locaton of the optmal petrol staton and the volume of fuel [lter] to be reflled. The actual fuel prce [UF/lter] of the deal staton and the total cost [UF] of the fuel to be reflled also lsted. Optmzaton.staton Selecton of wehcle:.a Load [ton]: 2. staton Changng of load [ton]: 3. staton Changng of load [ton]: Fuel volume at the begnnng [lter]: Fuel prce at the begnnng [UF/lter]:.b.c Fg. 7 Defnton of new petrol statons 4. staton Changng of load [ton]: 5. staton Changng of load [ton]: 6. staton Add new Delete staton staton Man menu Ext Results: Total cost of the whole transport way [UF]: Total volume of fuel usage on whole transport way [lter]: Volume of remanng fuel at the last staton [lter]: Optmzaton.e Name of the Fuel prce Statons: petrol staton: [UF]: Amount of fuel to be reflled [lter]: 4. Cost of fuel to be reflled Amount of actual loadng condton [ton]: 5..d Optmal petrol staton Fg. Screen of Optmzaton menu, result of the case study Fg. 8 Defnton of new dspatch statons and dscharge statons Fg. 9 Defnton of new transport vehcles 7.3 Case study applcaton of the software In our case study the transport trp s the followng (see Fg. ): Staton.: Mskolc, +3 ton; Staton 2.: Budapest, -3 ton +3 ton; Staton 3.: Győr, -3 ton +3 ton; Staton 4.: Pécs, -3 ton +3 ton; Staton 5.: Szeged, -3 ton +3 ton; Staton 6.: Mskolc, -3 ton. Dstances between the dspatch statons and dscharge statons can be seen n Fg.. The selected vehcle s a lght truck, fuel consumpton s 4 lter/km, the maxmal loadng capacty s 3,5 ton, the maxmal fuel tank capacty s 5 lter. The correcton factor for dfferent loadng condtons s.3 (every addtonal tons of addtonal freght load results.3 lter extra fuel consumpton). Plannng of Optmal Fuel Supply of Internatonal Transport Actvty

9 +3 ton -3 ton +3 ton -3 ton 5. Szeged -3 ton 96 km 92 km 4. Pécs. Mskolc 72 km Fg. Case study +3 ton 84 km 3. Győr 2. Budapest 4 km -3 ton +3 ton -3 ton +3 ton The fuel volume at the begnnng of the transport trp s 8 lter (see part.a of Fg. ), the statons of the transport way and the loadng condtons can be defned n part.b of Fg.. As t can be seen on part.d of Fg.. the volume of the fuel wll be under the gven lmted value before the staton 5, so the drver has to refuel the vehcle between staton 4. and staton 5. The software defnes the optmal fuel staton n the searchng area (red rectangle n part.d of Fg. ) after the calculaton. The dentfer of the optmal refuelng staton s MOL_327 (where the fuel prce s mnmal) and the amount of fuel (83.4 lter) to be reflled are lsted n the part.e. The fnal result of the fuelng optmzaton can be seen n part.c of Fg.. The total cost of the whole transport way of the case study s 435 UF, the total volume of fuel usage on the whole transport way s 3.8 lter, the volume of remanng fuel at the last staton s only 3.6 lter. The developed software s capable for selecton of the optmal petrol staton and the determnaton of the optmal amount of reflled fuel durng a long transport trp. Based on ths nformaton the camon drvers can be supported by the requred refuelng nformaton and the total cost of the transport trp can be mnmzed. There are lot of possble petrol statons durng the transport way and n the practce the selecton of the petrol staton where the drver refll the vehcle s absolutely dependng on the ndvdual decson of the drver. Therefore the total cost of the burned fuel s not optmal, because the fuel prces are dfferent at dfferent petrol statons. In ungary for example the dfference of fuel prces at the cheapest and at the most expensve petrol statons can be 5-7 UF/lter. It can be concluded that f the decson makng of the drver could be supported by our software the total cost of the transport way can be reduced sgnfcantly. Ths reducton s 3.5-9% n case of one lter of fuel. It can be calculated that n case of a long transport way the cost reducton can be sgnfcant. Recently the software s under testng, but we hope that n the near future more and more companes wll apply t and the transport cost can be reduced n ths way. 8 Concluson The man goal of the study was to elaborate a precse and relable mathematcal model and method for the determnaton of optmal fuel refll locatons and the amount of loaded fuel for the executon of specfc transportaton tasks. There have not been any avalable solutons n the lterature for the complex handlng of our refuelng problem. Ths motvated the authors to create a model and method that s able to generally handle the refuelng problem and to also gve a general soluton for t. In the frst step the concepts were clarfed and dsambguated. Ths created a dctonary that made the creaton of model possble. In the followng step the probable (deal) petrol statons were chosen, where t s worth refuelng. It s necessary, snce countless petrol statons are avalable durng a longer (nternatonal) transport trp. A method was also gven whch determnes the length of the way between each ntersectons or petrol statons based on ther GPS coordnates and dgtal map parameters. After choosng the probable refuel ponts the mathematcal model of the problem was created takng nto account the condtons and constrants arsng from practcal tasks. A software was also developed for the optmzaton of refuelng problem, whch helped to demonstrate the applcablty of the model on a case study. References Anbuudayasankar, S. P., Ganesh, K., Mohapatra, S. (24). Models for Practcal Routng Problems n Logstcs: Desgn and Practces. Sprnger Internatonal Publshng, Swtzerland. Bazaraa, M. S., Sheral,. D., Shetty, C. M. (27). Nonlnear programmng: Theory and algorthm. John Wley & Son Inc., oboken, New Jersey. Brge, J. R., Lnetsky, V. (ed.) (27). andbooks n Operatons Research and Management Scence. Vol. 5. Elsever. Bohács, G., Gymes, A., Rózsa, Z. (26). Development of an ntellgent path plannng method for materals handlng machnery at constructon stes. Perodca Polytechnca Transportaton Engneerng. 44(), pp Carama, M., Dell Olmo, P. (28). Mult-objectve management n freght logstcs, ncreasng capacty, servce level and safety wth optmzaton algorthms. Sprnger-Verlag, London. Czczer, N. (26). (Plannng of optmal fuel supply of natonal road transport actvty.) Master Thess, Insttute of Logstcs, Unversty of Mskolc. (n ungaran) Ehmke, J. F. (22). Integraton of nformaton and optmzaton models for routng n cty logstcs. Sprnger-Verlag, New York. Fraunhofer Isttute (25). Executve summary. [Onlne]. Avalable from: haltgketsbercht-25/nachhaltgketsbercht-25-executve-sum- mary.pdf [Accessed: 5th September 26] 94 Perod. Polytech. Transp. Eng. Gy. Kovács, M. Gubán

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