Applications of the Vehicle Routing Problem with Trailers and Transshipments

Transcription

1 Applications of the Vehicle Routing Poblem with Tailes and Tansshipments Technical Repot LM Michael Dexl Chai of Logistics Management, Gutenbeg School of Management and Economics, Johannes Gutenbeg Univesity Mainz and Faunhofe Cente fo Applied Reseach on Supply Chain Sevices SCS 25th Novembe 2011 Abstact The vehicle outing poblem with tailes and tansshipments (VRPTT) is a ecent and challenging extension of the well-known vehicle outing poblem. The VRPTT constitutes an achetypal epesentative of the class of vehicle outing poblems with multiple synchonization constaints (VRPMSs). In addition to the usual task coveing constaints, VRPMSs equie futhe synchonization between vehicles, concening spatial, tempoal, and load aspects. VRPMSs possess consideable pactical elevance, but limited coveage in the scientific liteatue. The pupose of the pesent pape is to descibe how seveal impotant types of VRPMSs, such as multi-echelon location-outing poblems and simultaneous vehicle and cew outing poblems, can be modelled as VRPTTs. Keywods: Vehicle outing poblem with tailes and tansshipments; Vehicle outing poblems with multiple synchonization constaints; Synchonization; Coodination; Models; Applications 1 Intoduction Vehicle outing poblems (VRPs) ae fundamental planning poblems in logistics and tanspot, and they have been the subject of intensive study fo moe than half a centuy now (Toth/ Vigo [31], Golden et al. [12], Lapote [17]). A ecent and challenging extension of the VRP is the vehicle outing poblem with tailes and tansshipments (VRPTT). The VRPTT constitutes an achetypal epesentative of the class of vehicle outing poblems with multiple synchonization constaints (VRPMSs). VRPMSs ae a boad class of VRPs which, despite thei consideable pactical elevance, have attacted compaatively little attention on the pat of science so fa. In classical VRPs, synchonization between vehicles is necessay only with espect to which vehicle visits which custome. VRPMSs ae VRPs which exhibit additional synchonization equiements with egad to spacial, tempoal, and load aspects. A ecent suvey of synchonization in vehicle outing (Dexl [10]) has shown that pactical applications of VRPMSs abound and that the solution of seveal types of VRPMSs is still a eseach issue. The fundamental diffeence between classical VRPs and VRPMSs is that the latte, contay to the fome, featue the so-called intedependence poblem: In standad VRPs, vehicles ae independent of one anothe in that a change in one oute does not affect any othe oute. In VRPMSs, by contast, a change in one oute may have effects on othe outes, due to the additional synchonization equiements. In the wost case, a change in one oute may ende all othe outes infeasible. This has consideable implications on potential solution appoaches. In fact, most exact and heuistic algoithms fo classical VRPs ely on the fact that outes 1

2 ae mutually independent. Consequently, these algoithms cannot diectly be applied to solve VRPMSs. A fist step towad solving poblems is popely modelling them. Theefoe, the contibution of the pesent pape, simila to the woks of Noon/Bean [21], Cainic et al. [6], and Baldacci et al. [1] fo othe outing poblems, is to popose the VRPTT as a unified modelling tool fo VRPMSs in geneal. To this end, it is descibed how seveal impotant types of VRPMSs can be epesented as VRPTTs. This demonstates the vesatility of the VRPTT as a epesentational famewok fo many types of ich VRPs and points out that the VRPTT needs and deseves futhe study. The development of exact o heuistic solution algoithms fo the VRPTT is beyond the scope of the pape. The next section descibes the VRPTT and develops a gaph-theoetic modelling famewok which can seve as a basis fo algoithmic solution appoaches. Subsequent sections descibe tansfomations of classical VRPs and of seveal types of VRPMSs that wee identified as paticulaly elevant and challenging in the above-mentioned suvey. The pape ends with a shot conclusion. 2 The vehicle outing poblem with tailes and tansshipments 2.1 Poblem desciption The VRPTT is a eal-wold poblem aising in aw milk collection at famyads (Dexl [9]). Basically, it can be descibed as follows. Thee is a set of customes with a given supply. To collect the supply, a set of heteogeneous vehicles stationed at one o seveal depots is available. In addition to potentially unequal costs, capacity, and tempoal availability, the vehicles diffe with espect to two othogonal citeia: Fist, thee ae autonomous vehicles able to move in time and space on thei own (loies) and non-autonomous vehicles, which can move in time on thei own, but must be pulled by a compatible autonomous vehicle to move in space (tailes). (Note that autonomy depends on the application context. The same eal-wold object may be consideed autonomous in one application and non-autonomous in anothe.) Second, thee ae task vehicles, which ae technically equipped to visit customes and collect supply, and thee ae suppot vehicles, which cannot visit customes, but can be used as mobile depots to which the task vehicles tansfe load. The load tansfes ae caied out at tansshipment locations (TLs) such as paking places o custome pemises. Using such locations may incu one-time fixed costs as well as fixed costs pe tansshipment opeation. The numbe of vehicles allowed to use a cetain TL duing the planning hoizon and the numbe of tansshipment opeations at a cetain TL duing the planning hoizon may be limited. Thee may also be limits on the numbe of vehicles allowed to be pesent and/o the numbe of tansshipment opeations allowed to be pefomed simultaneously at a TL. Moeove, single o multiple time windows ae associated with the customes as well as with the TLs. Some customes can only be visited by a loy without a taile and ae hence called loy customes. The othe customes can be visited by a loy with o without a taile and ae called taile customes. Geneally speaking, the vehicles ae subject to accessibility constaints, that is, not evey vehicle is necessaily able to visit evey location, due to physical estictions (e.g., limited manoeuving space) o technical constaints (e.g., lack of petinent equipment to pefom a sevice). Thee is no fixed assignment of a taile to a loy. Any non-autonomous vehicle may be pulled, on the whole o on a pat of its itineay, by any compatible autonomous vehicle. What is moe, any task o suppot vehicle is pemitted to tansfe its load patially o completely to any othe compatible task o suppot vehicle at any TL abitaily often. Vehicles need not cay any load when etuning to the depot. Loies need not bing back a taile, neithe one they might have pulled when leaving the depot, no any othe. Not all vehicles have to be used. 2

3 The poblem is to detemine outes fo loies and outes fo tailes so that total costs ae minimized, the complete supply of all customes is collected and deliveed to a depot, and loading capacities, accessibility constaints, and time windows ae maintained. Moeove, it must be ensued that the outes ae synchonized with espect to space, time and load, so that non-autonomous vehicles move in space only when accompanied by compatible othe vehicles and that the vehicles involved in a load tansfe opeation visit the petinent location at the ight time and tansfe and eceive the ight amount of load. The decisive point is that in a solution, a oute is detemined fo each vehicle which is actually used, be it a loy o a taile, an autonomous o a non-autonomous one. An example oute plan, which, fo simplicity, does not contain suppot vehicles, is depicted in Figue 1. Depot Loy custome Taile custome Tansshipment location Loy 1 Loy 2 Loy 3 Taile Figue 1: VRPTT example oute plan In the example, loy 1 couples the taile at the depot and goes to a TL, whee it decouples the taile. Loy 1 then visits two loy customes, etuns to the taile, tansfes some load, leaves the taile at the TL and etuns to the depot via two loy and two taile customes. Loy 2 stats at the depot and visits two loy customes befoe coupling the taile (afte loy 1 has pefomed its load tansfe). Loy 2 then visits a taile custome, decouples the taile at anothe TL, possibly pefoms a load tansfe, visits some loy customes, etuns to the taile, e-couples it and pulls it back to the depot via a taile custome. Loy 3 also stats at the depot, visits some loy customes, and tansfes some load to the taile while loy 2 is visiting the thee ightmost loy customes. Afte that, loy 3 etuns to the depot via anothe loy custome. The two TLs in the cente of the figue ae not used. The cental question in the VRPTT can be stated as: Which vehicle tansfes how much load when whee into which othe vehicle? A tansshipment o (de)coupling opeation is defined by: The location whee the opeation takes place The point in time when the opeation begins The passive vehicle, which povides capacity, that is, may eceive load, and/o may be coupled to o decoupled fom anothe vehicle The active vehicle, which equests capacity, that is, may tansfe load, and/o may couple o decouple anothe vehicle The amount of load tansfeed into the passive vehicle, which is zeo if only coupling o decoupling is pefomed, and which is negative if the passive vehicle supplies load fo the active one Hence, the decisive modelling issues esulting fom the cental question ae the following: How to ensue that a taile is accompanied by a compatible loy on an ac, that is, how to synchonize the movements of vehicles? How to synchonize the visiting times of vehicles at tansshipment locations? How to balance the load tansfe amounts of vehicles exchanging load? 3

4 2.2 A netwok epesentation of the VRPTT Fo a paticula vehicle outing poblem, thee is a boad spectum of options how to epesent the infomation and data on the elevant objects and thei elationships, such as the thee synchonization issues just mentioned. These options ange fom model the undelying poblem logic completely by means of decision vaiables and constaints to ceate a highly involved netwok that by itself ensues feasibility and synchonization. Thee is no silve bullet; it depends on the concete poblem type and the aveage instance data whee a model fo a cetain VRP should be positioned on this continuum. In what follows, a desciptive, gaph-based modelling famewok is pesented, which can seve to epesent and model VRPTTs (and, as will be demonstated, othe impotant VRPMSs), and which can fom the basis fo a concete solution appoach, be it a banch-cut-pice algoithm using a mixed intege pogamming fomulation (Desaulnies et al. [7]), a constaint pogamming method (Rousseau et al. [27]), o a (meta-)heuistic using constuction and impovement pocedues with local and lage neighbouhood seach (Gendeau/Potvin [11]) The basic modelling pinciples In pactice, that is, on eal oad netwoks, the distinction between autonomous and nonautonomous vehicles is shap: Each vehicle is eithe of the one o the othe type. The same holds fo the distinction between task and suppot vehicles. As will be shown, howeve, fo modelling puposes using gaphs, it is sensible to take a moe flexible appoach. Essentially, thee fundamental modelling pinciples will be employed: An adequate definition of subnetwoks fo vehicles, that is, the definition of which vetices and acs may be visited and tavesed by a cetain vehicle o vehicle class with o without othe vehicles (this means that (non-)autonomy is not necessaily a global, netwok-wide popety of a vehicle, but a local, ac-specific one) The specification of compatibilities between vehicles with espect to the ability of common movement (fomation of autonomous composite vehicles) and the ability to pefom load tansfes The setting of limits on the amount of load tansfeed o eceived by a paticula vehicle at a paticula location Using these ideas, the VRPTT can be defined on a diected netwok D = (V, A) with an associated set F of vehicles (the fleet) The netwok The vetex set V is patitioned into fou subsets: V VD, the set of vitual depot vetices, V D, the set of eal depot vetices, V C, the set of custome vetices, and V T, the set of tansshipment vetices. l(i) denotes the eal-wold location coesponding to i fo each i V \ V VD. V VD = {s, e}, whee s (e) is the vitual stat (end) depot vetex. Both have no coesponding eal-wold location. V D = V SD V ED, whee V SD (V ED ) is the set of stat (end) depot vetices. Each vetex i V has a time window [a i, b i ] indicating the ealiest and latest point in time fo the beginning of an opeation to be pefomed at i. This time window may have zeo length (a i = b i ), that is, epesent a concete point in time. The time window of each i V \ V VD is completely contained in one of l(i) s time windows. Each i V C has an associated supply s i. F i F is the set of vehicles allowed to visit i V The fleet The fleet F is patitioned into K classes o vehicle types. The vehicles in a class ae identical with espect to all elevant attibutes, such as those mentioned at the beginning of Section 2.1. In paticula, they have the same potential stat and end depots. All vehicles ae initially stationed at the vitual stat depot vetex s and must end thei outes at the vitual end depot vetex e. 4

5 Each vehicle which is used moves fom s to some vetex in the set of its potential stat depot vetices, VSD k, at the beginning of its oute, and fom some vetex in the set of its potential end depot vetices, VED k, to e at the end of its oute. Unused vehicles move diectly fom s to e. If the stat and/o end depot of a vehicle k is/ae known in advance, VSD k and/o V ED k contain only one element. If the single-depot case is consideed, that is, if each vehicle is assigned to one stat and one end depot ex ante, the vitual depot vetices s and e ae not necessay. q k indicates the loading capacity of vehicle o vehicle class k. Ftans,comp k is the set of vehicles with which vehicle k can exchange load The subnetwoks V k V (A k A) is the subset of vetices (acs) that vehicles of class k ae allowed to visit (tavese). Consequently, fo each vehicle class k, thee is a subnetwok D k = (V k, A k ) descibing k s possible movements. Fo all k F, A k single is the set of acs k is able to tavese without being accompanied by anothe vehicle. Fmove,comp k,(i,j) is the set of vehicles with which vehicle k F can move along ac (i, j) A k \A k single, that is, fo a loy k, the set of tailes k can pull along (i, j), and fo a taile k, the set of loies which can pull k along (i, j). VC,0 k is the set of custome vetices a task vehicle is allowed to visit without being allowed to collect any load thee. Moeove, with each ac (i, j) A k ae associated vehicle-dependent costs c k ij and a vehicledependent tavel time t k ij. The usual cost types ae fixed, distance-, time-, and stop-dependent costs. Fixed vehicle costs, that is, one-time costs incued fo making a vehicle available, ae consideed on the acs leading fom e to eal stat depot vetices. Distance- and stop-dependent costs ae consideed on the espective acs. Since waiting times ae possible in the pesence of time windows, time-dependent costs must be consideed globally in an objective function by measuing oveall oute duation, which may be moe than the sum of tavel and sevice times. Each subnetwok fo a vehicle k F contains only vetices i with k F i, that is, V k = {i V : k F i }. Loies cannot ente the stat and end depot vetices of othe loies; tailes cannot ente any stat o end depot vetices othe than thei own. Unless othewise specified fo cetain modelling aspects, each subnetwok contains an ac between each pai of vetices with the following exceptions: No acs ente (leave) the vitual stat (end) depot vetex. k s eal stat (end) depot vetex/vetices can be eached only fom s (left only to e). Thee ae no acs fom taile stat depot vetices to loy custome vetices and fom loy custome vetices to taile end depot vetices. Of couse, thee ae also no acs (i, j) whee a i + t k ij > b j Modelling tansshipments In eal-wold VRPTTs, the following estictions on tansshipments ae possible at any potential tansshipment location l: A limit on the numbe of times a vehicle is allowed to visit l the numbe of diffeent vehicles allowed to visit l the numbe of diffeent vehicles allowed to tansfe o eceive load at l, that is, the numbe of potential active and passive vehicles the numbe of tansshipment opeations pefomed simultaneously at l the oveall numbe of tansshipment opeations pefomed at l the total amount of load tansfeed at l Intevals within which the amount of load that a vehicle tansfes o eceives at l duing one tansshipment opeation and oveall must lie the oveall amount of load tansfeed at l must lie 5

6 This infomation must be captued by paametes and consideed in a concete model and solution appoach, fo example by petinent constaints in a mixed intege pogam. Details on the pactical elevance of these paametes in eal-wold VRPTTs ae povided by Dexl [9]. The definitions given in Sections delibeately leave some feedom egading the pecise design of the netwok fo a paticula poblem. To illustate this scope fo modelling tansshipments, conside the following two exteme situations: It can be decided to ceate one vetex fo each time window of each physical location and to allow that all vehicles visit all tansshipment vetices moe than once and tansfe o eceive abitay amounts of load. Then, in addition to the outing decisions fo each vehicle, the following decisions must be modelled outside the netwok, that is, fo example, by decision vaiables in a mixed intege pogam: The maximal numbe of visits of each vehicle at each vetex, the point in time when each visit of each vehicle takes place, and the amount of load tansfeed o eceived duing each visit. This coesponds to the appoach model the poblem logic by means of decision vaiables and constaints mentioned at the beginning of Section 2.2. Altenatively, it can be decided to define a fixed-schedule space-time-opeation-vehicle netwok in which each tansshipment vetex coesponds to a concete location, a concete point in time when the tansshipment stats, a concete passive vehicle, and a concete load tansfe amount. Then, thee ae no decisions to take in addition to the outing decisions fo each vehicle. This coesponds to the appoach ceate a netwok that by itself ensues feasibility and synchonization. In this way, a specific configuation of the netwok lies the foundation fo an optimization model. The decision how to configue the netwok has to take the concete aspects of a specific application into account, and diffeent set-ups ae possible. The necessity to find a poblem-adequate modelling configuation is a constitutive popety of VRPTTs and VRPMSs in geneal. Howeve, the basic ideas of subnetwoks and compatibility specifications allow the consideation of a vey boad ange of VRPTT vaiants. In the following sections, it will be shown how othe impotant basic classes of VRPMSs can be epesented with the descibed VRPTT modelling famewok. 3 Extensions of the VRP It is evident that standad vehicle outing poblems without multiple synchonization constaints, such as the capacitated VRP and the VRP with time windows, ae special cases of the VRPTT. By its definition, the VRPTT aleady encompasses the heteogeneous fleet vesions of these poblems. Real-wold constaints such as multiple capacity constaints, vehicle-specific time windows etc. ae also easily accommodated. Moeove, the concepts of subnetwoks and compatibilities between vehicles with espect to movement and load tansfe allow modelling multiple-depot VRPs, VRPs with multiple use of vehicles, VRPs with multiple planning peiods, the genealized VRP (GVRP), the open VRP, capacitated ac outing poblems (CARPs), and the tuck-and-taile outing poblem (TTRP). This is explained in the following. The multiple-depot VRP (Codeau et al. [5]), whee, fo each vehicle, one out of seveal potential eal depots must be selected as stat and end depot, can be modelled with the VRPTT famewok as follows: Fo each eal vehicle (loy) k, a vitual vehicle (taile) k v is intoduced. k v is non-autonomous and able to leave the vitual stat depot vetex s only togethe with k, even along the ac (s, e) fo unused vehicles. k v can only visit s, e, and k s eal stat and end depot vetices, that is, the vetices in VSD k and V ED k. Fom a eal stat depot vetex, a vitual vehicle must move diectly to the coesponding eal end depot vetex, and it is able to do so without being accompanied by its associated eal vehicle. To each e, k v must again be pulled by k. In this way, it is ensued that the eal vehicle, at the end of its oute, will visit the coect eal end depot vetex. 6

7 VRPs with multiple use of vehicles (Taillad et al. [30]), whee one vehicle may pefom seveal outes stating and ending at a cetain depot, can also be modelled. To this end, tansshipment vetices fo each eal depot ae intoduced. Vitual vehicles as defined in the pevious paagaph ae allowed to move on thei own fom the stat depot vetices to these special tansshipment vetices and between the latte, but must be accompanied by thei associated eal vehicles when moving fom these tansshipment vetices to the coesponding eal end depot vetex. The vitual vehicles ae uncapacitated. Real vehicles can only tansfe load into thei coesponding vitual vehicle, and vitual vehicles can only eceive load fom thei coesponding eal vehicle. Thus, the eal vehicles will unload completely at one of the tansshipment vetices and/o the selected eal end depot vetex. Of couse, these special tansshipment vetices may have associated time windows epesenting multiple peiods (fo example, fo each depot, thee may be one such vetex pe weekday). In this way, multiple-peiod VRPs (Zäpfel/Bögl [32]) can be modelled. In the genealized VRP (Baldacci et al. [1]), the set of customes is patitioned into a set of clustes, and instead of equiing that each custome be visited exactly once, it is stipulated that exactly one custome fom each cluste be visited exactly once. Since the VRPTT as modelled above aleady coves the case whee thee ae multiple vetices fo one custome and exactly one of these vetices must be visited, the GVRP can be modelled as a VRPTT, too. Also the (single- as well as multiple-depot) open VRP (Bandão [2]), whee the outes of the vehicles need not end whee they stated, can easily be handled with the above model: It is sufficient to emove all eal end depot vetices and intoduce an ac with zeo costs and tavel time fom each custome o tansshipment vetex diectly to the vitual end depot vetex e. In addition, it is well known that capacitated ac outing poblems (Golden/Wong [13]) can be tansfomed into vehicle outing poblems (Pean et al. [24], Longo et al. [19]); consequently, it is possible to model CARPs as VRPTTs. Finally, the tuck-and-taile outing poblem (Semet/Taillad [29], Chao [4], Scheuee [28]) is a athe well-studied vehicle outing poblem which, as its name implies, also consides tailes, and which can be modelled as a VRPTT. In fact, the TTRP is a special case of the VRPTT whee thee is a fixed loy-taile assignment. This means that each taile can be pulled by a unique associated loy, and only this loy is pemitted to tansfe load into the taile. 4 N-echelon vehicle and location-outing poblems Gonzalez Feliu et al. [15] and Peboli et al. [25] fomally intoduce the class of multi-echelon (o N-echelon) vehicle outing poblems and ae the fist to use these tems. The basic idea behind this poblem class is that customes ae not deliveed diectly fom a cental depot, but via N legs in an N-stage distibution netwok. An N-stage distibution netwok contains N +1 levels of location. Echelon n {1,..., N} consides tanspots fom location level n 1 to n, see Figue 2. Fo each echelon n, thee ae dedicated vehicles which can only visit the locations o facilities defining echelon n. This means that only the vehicles of echelon N ae task vehicles, that is, ae allowed to visit customes; all othe vehicles ae suppot vehicles. Load tansfes ae only possible between vehicles of diffeent echelons. The diffeence to distibution netwok design poblems lies in the fact that fo each vehicle in the poblem, a complete oute is computed. Gonzalez-Feliu [14] studies the geneal N-echelon location-outing poblem (LRP). The diffeence between the N-echelon vehicle outing poblem and the N-echelon location-outing poblem is that the latte consides fixed costs fo opening a facility, contay to the fome. Thee ae many diffeent vaiants of N-echelon outing poblems. Subsequently, it is fist descibed how a basic poblem can be modelled as a VRPTT; extensions ae consideed aftewads. Fo simplicity, the desciption is based on the 2-echelon LRP, but the elaboations genealize diectly to abitay N N. 7

8 Level 0 Level 1 Level 2 Level 3 1st Echelon 2nd Echelon 3d Echelon Depot, Factoy Facility (Waehouse, Hub, Coss-dock) Custome Figue 2: Example of a 3-echelon outing poblem The basic ideas ae that (i) vitual depots, eal depots, and customes etain thei meaning, wheeas tansshipment locations ae used to model facilities, and that (ii) echelons ae epesented by an adequate definition of subnetwoks fo suppot and task vehicles. Moe pecisely, if thee ae n 0 and n 1 potential facilities of level 0 and 1 espectively, each facility is epesented by one tansshipment vetex t 0 1, t0 2,..., t0 n 0 and t 1 1, t1 2,..., t1 n 1. If no tempoal aspects ae consideed, it is sufficient to have only one vetex pe facility/tansshipment location. To define the subnetwoks, fo each suppot vehicle class, thee is one suppot vehicle fo the fist echelon, and fo each task vehicle class, thee is one task vehicle fo the second echelon. Suppot vehicles may only move between depots and facilities of the fist echelon. Task vehicles may only move between depots and vetices of the second echelon. Fo the facilities of level 0, it is sufficient to define one vitual suppot vehicle, say, k 0. The oute of k 0 only defines which level-0 facilities to open. Theefoe, to avoid symmeties, it is sufficient to define the subnetwok fo k 0 with acs fom s to each level-0 facility, fom each level-0 facility to e, and acs (t 0 1, t0 2 ), (t0 1, t0 3 )... (t0 1, t0 n 0 ), (t 0 2, t0 3 ), (t0 2, t0 4 )... (t0 2, t0 n 0 )... between level-0 facilities. k 0 is uncapacitated and is hence able to delive the necessay amount of load fom the vitual depot to each of the selected level-0 facilities. A suppot vehicle seving the fist echelon, that is, moving between the facilities of levels 0 and 1, stats its oute at the vitual stat depot, visits its selected eal depot, then necessaily one of the open facilities of level 0 (since the vehicle is initially empty and must load befoe visiting a level-1 facility), then one o moe level-1 facilities, then again a level-0 facility etc. Finally, it visits its assigned end depot and etuns to the vitual end depot. A custome vehicle opeates analogously between level-1 facilities and customes. The following extensions occu in pactice: Facilities with limited capacity The estiction that, on echelon n, only one out of seveal potential vehicles is allowed to actually visit a facility of level n A fixed assignment of vehicles to facilities, that is, each vehicle on echelon n is pemitted to visit only one paticula facility of level n 1 o, fo echelons n = 1,..., N 1, each vehicle on echelon n is pemitted to visit only one paticula facility of level n A fixed assignment of facilities of level n = 1,..., N to facilities of level n 1, that is, the equiement that a cetain facility be seved by a vehicle assigned to one paticula othe facility Time windows at exactly one echelon (without the equiement of tempoal synchonization of vehicles of diffeent echelons) Necessity of tempoal synchonization of vehicles of diffeent echelons (this includes time windows at facilities of all echelons) Ways to handle these extensions ae descibed in what follows. Capacitated facilities l can be modelled by setting the allowable inteval fo the oveall amount of load tansfeed to [0, q l ] fo all facility vetices i V T, whee q l is the capacity of facility l and l(i) = l. 8

9 The equiement that only one vehicle be allowed to visit a level-n facility i can be modelled by setting the coesponding paamete fo the numbe of diffeent vehicles which may delive load at i to one (see Section 2.2.5). If sufficiently many vehicles of each class ae available, a fixed assignment of vehicles of echelon n to facilities of level n 1 can be modelled by intoducing, fo each potential facility l of level n 1, one vehicle of each class. This vehicle is capable of visiting, on echelon n, only the vetex coesponding to l. If the numbe of vehicles is limited, such a fixed assignment can be modelled as follows: All vehicles (loies) of echelon n ae non-autonomous and ae consideed to have a load tansfe amount of zeo at all vetices (by setting the intevals fo the amount of load each vehicle tansfes o eceives at any facility of level n 1 to [0, 0]). Fo each combination of potential facility l of level n 1 and loy k of echelon n, thee is a vitual non-autonomous taile k l,k which is pemitted to visit, on level n 1, only the vetex coesponding to l and which can only be pulled by k. k l,k might thus be called facility assignment taile. k l,k has the same capacity as k and is allowed to leave the vitual stat depot vetex s only togethe with k (except fo the ac (s, e) fo unused vehicles). Now, if k is to use only facility l on level n 1, k couples the coesponding facility assignment taile at the vitual stat depot vetex s, and keeps it coupled fo the complete oute. In this manne, the two vehicles can only visit the vetex coesponding to l on level n 1. Since k cannot eceive o tansfe any load, it will neve decouple k l,k en oute. The case whee each vehicle on echelon n may visit only one facility of level n is analogous. An ex-ante defined fixed assignment of a facility of level n = 1,..., N (egading customes as level-n facilities ) to a facility of level n 1 can also be modelled by facility assignment tailes. Fo example, if facility l n must be seved by a vehicle stationed at facility l n 1, only facility assignment tailes assigned to l n 1 ae allowed to tansfe a positive amount of load at l n. A combination of such assignment equiements may make it necessay that a loy pulls moe than one taile at a time. This is possible and is descibed in moe detail in Section 7. Time windows on one echelon can be consideed if one vehicle is intoduced fo each eal vehicle of the espective echelon. By intoducing one vehicle fo each eal vehicle of each echelon, tempoal synchonization of vehicles of the fist and second echelon (including the consideation of time windows at facilities of both echelons) can be modelled. Load tansfes ae then handled as descibed in the section on the VRPTT. The issues at tansshipment locations with espect to how to deal with the fact that a vehicle may want to tansfe load to anothe vehicle at the same location moe than once etc. ae exactly the same as fo VRPTTs. An advantage of modelling N-echelon outing poblems as VRPTTs lies in the fact that, contay to existing models, non-autonomous objects such as tailes and swap-body platfoms can be consideed. To this autho s expeience, these ae used in many eal-wold applications of N- echelon LRPs, but no petinent scientific publications ae known. Classical location-outing poblems as descibed in Nagy/Salhi [20] ae special cases of N-echelon LRPs with N = 1. In addition, Nagy/Salhi [20] descibe seveal applications of multi-echelon LRPs whee, on one o moe echelons, no oute planning is equied, but only a selection of the facilities to be used and an assignment of these facilities to those of the next highest echelon is sought. All such poblems ae also special cases of the N-echelon LRP, since they can be modelled using facility assignment tailes. Besides, the classical location-outing poblem with uncapacitated facilities, a fixed assignment of vehicles to facilities, and no time windows can be modelled as a VRPTT with one loy and one taile, as descibed in Dexl [9]. 9

10 5 Simultaneous vehicle and cew outing and scheduling poblems Fo the most pat, the VRP liteatue does not distinguish between a vehicle and its dive. In thei monogaph on VRPs, fo example, Toth/Vigo [31] state that thoughout, the constaints imposed on dives ae imbedded in those associated with the coesponding vehicles. Howeve, it is a fact that dives egulaly need beaks and ests and must obey the existing petinent social legislation and tade union ules egading diving, beak, and est times. On the othe hand, vehicles can essentially be used twenty-fou hous a day. Consequently, consideing a vehicle and a dive a fixed unit inevitably leads to a suboptimal tempoal utilization of vehicles. Simultaneous vehicle and cew outing and scheduling poblems (SVCRSPs) (Hollis et al. [16]) ae concened with the situation whee the equied tasks have no given timetable/no fixed schedule, and whee a dive-vehicle combination is not consideed an insepaable unit anymoe, so that outes have to be planned fo both vehicles and dives. Fo such poblems to make sense, thee must be a set of locations whee dives can change vehicles and vice vesa (elay stations). Essentially, SVCRSPs ae VRPTTs with the sole specific chaacteistic that all vehicles ae nonautonomous. Moe pecisely, such poblems can be modelled, o athe, intepeted, as VRPTTs in the following way: Fo each dive and each vehicle, thee is one non-autonomous vehicle. The vehicles coesponding to dives have zeo capacity. The elay stations coespond to the tansshipment locations. At the tansshipment locations/elay stations, a dive (vehicle) can uncouple a vehicle (dive) and couple a diffeent one. Tansshipments of load between vehicles may be allowed o not. In the latte case, the inteval fo the oveall amount of load tansfeed is simply set to [0, 0] at all tansshipment vetices i V T. If tansshipments ae allowed, thee may also be suppot vehicles, which ae non-autonomous as well. 6 Pesonnel dispatching poblems with spatio-tempoal synchonization constaints Thee is quite a numbe of applications whee pesons must be synchonized with espect to space and time to pefom some kind of sevice, but whee no load tansfes ae pefomed and no vehicle capacities ae elevant. Examples include the dispatching of sevice technicians with diffeent qualifications who have to meet to epai machines, o homecae staff outing, whee two nuses must visit a disabled peson at the same time fo lifting puposes o with a specified delay to apply medicine afte a meal etc. (see, fo example, Li et al. [18], Bedstöm/Rönnqvist [3], Dohn et al. [8]). Such poblems can be epesented as VRPTTs in the following way. Each vehicle coesponds to a peson, and all vehicles ae uncapacitated task vehicles. Thee ae no tansshipment locations, o, put diffeently, the sets of custome and tansshipment vetices happen to coincide. Qualifications of diffeent staff membes fo cetain types of sevice ae epesented by appopiate accessibility constaints at vetices. If two pesons with two diffeent qualifications u 1, u 2 must be pesent at a location at the same time to pefom a task, this is epesented by two custome vetices i and j, whee i (j) may be called task enty (task exit) vetex. All vehicles coesponding to pesons with qualification u 1 o u 2 can each i. The only ac leaving i is the ac (i, j). All vehicles ae non-autonomous along (i, j). To move along (i, j), a vehicle coesponding to a peson with qualification u 1 must be accompanied by a vehicle coesponding to a peson with qualification u 2 and vice vesa. The ac (i, j) is also the only ac enteing j. Fom j, all vehicles can move on thei own to othe task enty o end depot vetices. The paamete fo the numbe of allowed visits at i by all vehicles altogethe is set to two. In this way, it is ensued that exactly two compatible vehicles (qualified pesons) meet at the task location at exactly the same time. 10

11 If one visit must pecede the othe, it is sufficient to ceate two custome vetices i and j fo the task, to set the two paametes fo the numbe of allowed visits at i and j by all vehicles altogethe to one, and to allow that i (j) can only be eached by vehicles coesponding to pesons with qualification u 1 (u 2 ). No estictions with espect to enteing and leaving acs apply in this case. Tempoal elationships between the visiting times at i and j can then be established as descibed fo tansshipments in Section 2.2, that is, by appopiate constaints on visiting times o by ceating fixed-schedule space-time-opeation-vehicle netwoks. 7 Poblems with moe than two types of vehicle Afte the peceding explanations, it is easy to imagine the situation whee moe than two types of vehicle may be allowed o equied to join to move in space. To be pecise, two aspects must be distinguished: It is possible that (i) an autonomous elementay o composite object can pull moe than one elementay non-autonomous object, and that (ii) two o moe non-autonomous objects must join to fom an autonomous object. These extensions ae of pactical as well as theoetical elevance. A pactical example fo a VRPTT featuing the fist case is oad tanspot of agicultual poducts with special vehicles, whee it is allowed that one tacto pulls two tailes at the same time. A pactical example fo the second case is when dives and loies ae sepaate planning entities as descibed in Section 5. Then, a taile must join with a dive and a loy to move in space. Futhemoe, in the simultaneous vehicle and cew outing and scheduling poblems descibed in Section 5, it is possible to conside dive shuttle tanspots, whee small shuttle vans ae used to tanspot loy dives between elay stations. Doing so inceases the flexibility fo matching loies and dives, since a loy dive is enabled to leave a loy at a elay station and subsequently dive a loy stating fom a diffeent elay station. Each shuttle van is essentially an autonomous suppot vehicle with capacity zeo which can couple zeo o moe dives and is not compatible with task vehicles. Such situations ae epesented by the model developed in Section 2.2 as follows. Remembe that Fmove,comp k,(i,j) was defined as the set of vehicles with which vehicle k can move along ac (i, j): Fo a loy k, it is the set of tailes k can pull, and fo a taile k, it is the set of loies which can pull k. The definition of this set does not estict the numbe of tailes (non-autonomous vehicles) a loy (autonomous vehicle) may pull at the same time, so that the above case (i) is coveed. Moeove, the set of vehicles capable of moving k along an ac may of couse contain elementay as well as composite vehicles, so that case (ii) is also coveed. A vey impotant point is that the numbe of synchonization equiements gows only linealy when moe than two types of vehicle ae consideed: If, fo example, an autonomous loy k is supposed to pull the two tailes k and k along an ac (i, j), it is sufficient to synchonize k with k and with k. This will ensue that k and k ae also synchonized with each othe. An impotant application which can be modelled by using moe than two types of vehicle is descibed in the next section. 8 Pickup-and-delivey poblems with (tailes and) tansshipments A popula extension of the classical VRP is the class of vehicle outing poblems with pickups and deliveies, o pickup-and-delivey poblems (PDPs). In a classical VRP, eithe each custome has a cetain supply to be collected and bought to a depot, o each custome has a cetain demand to be fulfilled fom a depot. In PDPs, thee is a set of tanspot equests that must be fulfilled. A equest consists in the tanspot of a cetain amount of load fom a equest-specific pickup location to a equest-specific delivey location. (Thee ae othe types of PDPs. The 11

12 two-pape suvey by Paagh et al. [22], [23] on PDPs lists seveal poblem vaiants. All of these ae essentially special cases of the geneal PDP as just descibed.) An impotant obsevation is that, when no tansshipments ae allowed in VRPs and PDPs, a given solution in fom of a set of vehicle outes completely detemines the path each equest takes, be the latte a simple supply o demand equest o a pickup-and-delivey equest. This is because a equest is tanspoted by exactly one vehicle, and only this vehicle visits the coesponding equest location(s). When tansshipments ae possible, this is no longe the case, because the vehicle picking up a equest need not necessaily tanspot it to the depot/delivey location. PDPs with and without tansshipments can be epesented as follows within the modelling famewok intoduced above. Fo each equest, two custome vetices v +, v and one vitual vehicle, a dedicated equest vehicle k, ae intoduced. Request vehicles ae non-autonomous and have capacity zeo. Fo each custome vetex i, s i was defined as the supply of i. s i may take positive as well as negative values and thus epesent a supply as well as a demand. Hence, s v + > 0 (s v < 0) specifies the amount of load to be picked up at the pickup location (deliveed to the delivey location) of equest, and s v + = s v. Each equest vehicle k moves fom s, the vitual stat depot vetex, diectly to v +, and fom v diectly to e, the vitual end depot vetex. This is modelled by adding (s, v + ) and (v, e) to A k as the only acs emanating fom s and enteing e. Request vehicles cannot use the ac (s, e), since this would imply that the coesponding equest is not fulfilled. All compatible eal vehicles, that is, vehicles which ae technically equipped and allowed to tanspot, can each v +, but ae non-autonomous on all acs leaving v + and can leave v + only togethe with k. Similaly, all compatible eal vehicles can leave v, but ae non-autonomous on all acs enteing v and can each v only togethe with k. To model tansshipments of pickup-and-delivey equests, fo each equest, two vetices v t and v t+ ae intoduced fo each elevant time window of each desied potential tansshipment location. The two vetices ae linked by one ac (v t, v t+ ). The fist vetex, v t, is only eachable by a composite vehicle of which k is pat of. At this vetex, k is decoupled fom the vehicle(s) which has/have accompanied k to v t. This is modelled by equiing that k can leave v t only via the ac (v t, v t+ ), wheeas no othe vehicle is allowed to use this ac. Vehicles othe than k move fom v t to any othe vetex. At vetex v t+, k is coupled by a suitable elementay o composite vehicle. This is modelled by equiing that k can ente v t+ only via the ac (v t, v t+ ), wheeas all othe vehicles may ente v t+ by any othe ac, and that v t+ can only be left by a composite vehicle of which k is pat of. To ensue this, the allowable inteval fo the oveall amount of load tansfeed at both v t and v t+ is set to [s v +, s v + ], and the paametes fo the numbe of diffeent vehicles allowed to tansfe load at v t and to eceive load at v t+ as well as those fo the oveall numbe of tansshipment opeations pefomed at v t and v t+ ae set to one. Now, in ode to epesent the load tansfe, the equest vehicle k is assumed to have a loading capacity of s v +. The intevals fo the amount of load eceived by k at all vetices v t and tansfeed fom k at all vetices v t+ ae set to [s v +, s v + ]. The intevals fo the amount of load eceived by o tansfeed fom k at all othe tansshipment vetices ae set to [0, 0], and V k C,0 = V C, that is, k is not allowed to collect any load at custome vetices. Thus, the eal vehicle caying a equest to v t tansfes it to k thee, and k tansfes to the eal vehicle caying away fom v t+. In this way, the equest vehicle ensues that the ight load, the one picked up at v +, is deliveed to v. Note that it is possible that a poblem instance compises nomal custome vetices as in the VRPTT vesion descibed in Section 2 and pickup-and-delivey equests at the same time. Note futhe that the above epesentation of pickup-and-delivey equests implies that moe than two types of vehicle may join to move in space, as descibed in the pevious section. 12

13 9 Conclusion Vehicle outing poblems with multiple synchonization constaints ae challenging optimization poblems. In contast to many othe types of VRPs and despite thei pactical elevance, VRPMSs have only ecently enteed the focus of the scientific community. This may patly be owed to the fact that they ae difficult to solve, and that solution appoaches fo classical VRPs cannot diectly be applied to VRPMSs. This autho is unawae of any solution pocedue fo N-echelon vehicle o location-outing poblems with tempoal synchonization of vehicles of diffeent echelons. Liteatue on pickup-and-delivey poblems with tansshipments and simultaneous load tansfes, on simultaneous vehicle and cew outing and scheduling poblems, and on poblems with moe than two types of vehicle is still vey scace. The vehicle outing poblem with tailes and tansshipments is an achetypal example of a VRPMS. The pesent pape has demonstated the usefulness of the VRPTT as a geneal modelling tool by epesenting seveal classes of VRPMSs as VRPTTs. No claim is made that, even if poweful algoithms fo the VRPTT wee available (which is not yet the case), solving the descibed applications by tansfoming them into VRPTTs would necessaily be the method of choice. Howeve, as Pisinge/Ropke [26] have shown, geneal heuistics capable of solving a boad ange of vehicle outing poblems can indeed poduce high-quality solutions. Moeove, it is to be expected that futue algoithmic advances fo the VRPTT could yield valuable insights fo the solution of othe types of VRPMSs. Theefoe, the development of exact and heuistic solution pocedues fo the VRPTT with continuous load vaiables and volume-dependent load tansfe times constitutes a challenging as well as pomising aea of futue eseach. Acknowledgement This eseach was funded by the Deutsche Foschungsgemeinschaft (DFG) unde gant no. IR 122/5-1. Refeences [1] Baldacci R, Batolini E, Lapote G (2010): Some Applications of the Genealized Vehicle Routing Poblem Jounal of the Opeational Reseach Society 61: [2] Bandão J (2004): A Tabu Seach Algoithm fo the Open Vehicle Routing Poblem Euopean Jounal of Opeational Reseach 157: [3] Bedstöm D, Rönnqvist M (2008): Combined Vehicle Routing and Scheduling with Tempoal Pecedence and Synchonization Constaints Euopean Jounal of Opeational Reseach 191: [4] Chao I (2002): A Tabu Seach Method fo the Tuck and Taile Routing Poblem Computes & Opeations Reseach 29: [5] Codeau J, Gendeau M, Lapote G (1997): A Tabu Seach Heuistic fo Peiodic and Multi-Depot Vehicle Routing Poblems Netwoks 30: [6] Cainic T, Ricciadi N, Stochi G (2009): Models fo Evaluating and Planning City Logistics Systems Tanspotation Science 43: [7] Desaulnies G, Desosies J, Solomon M (eds) (2005): Column Geneation Spinge, New Yok 13

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