Branch-and-Cut Algorithms for the Vehicle Routing Problem with Trailers and Transshipments

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1 Branch-and-Cut Agorithms for the Vece Routing Probem with Traiers and Transspments Technica Report LM Michae Dre Chair of Logistics Management, Gutenberg Schoo of Management and Economics, Johannes Gutenberg University, Mainz and Fraunhofer Centre for Appied Research on Suppy Chain Services SCS, Nuremberg E-mai: 25th September 2012 Abstract Ts paper studies the vece routing probem with traiers and transspments (VRPTT), a practicay reevant, but chaenging, generaization of the cassica vece routing probem. The paper maes three contributions: (i) Buiding on a non-trivia networ representation, two mied-integer programming formuations for the VRPTT are proposed. (ii) Based on these formuations, five different branch-and-cut agorithms are deveoped and impemented. (iii) The computationa behaviour of the agorithms is anayzed in an etensive computationa study, using a arge number of test instances designed to resembe rea-word VRPTTs. Keywords: Branch-and-cut; Vece Routing; Synchronization; Traier; Transspment 1 Introduction The vece routing probem with traiers and transspments (VRPTT) was introduced by Dre [11] in the contet of raw mi coection at farmyards. It is a generaization of the cassica vece routing probem (VRP, Dantzig and Ramser [9], Toth and Vigo [23], Goden et a. [14]). The version of the VRPTT studied in ts paper can be described as foows. There is a set of customers with a given suppy. To coect the suppy, a imited feet of heterogeneous veces (orries and traiers) stationed at a centra depot is avaiabe. Besides unequa fied and variabe costs and capacities, the veces differ in that orries are autonomous and abe to move in time and space on their own, whereas traiers are non-autonomous and can move in time on their own (that is, wait), but must be pued by a orry to move in space. In addition to the depot and the customer ocations, there are so-caed transspment ocations (TLs), where traiers can be pared and where oad transfers from orries to traiers can be performed. Some customers can ony be visited by a orry without a traier (a singe orry) and are hence caed orry customers. The other customers can be visited by a orry with or without a traier and are caed traier customers. There are time windows at the customers as we as at the TLs. A veces start and end their routes at the depot. There is no fied assignment of a traier to a orry. Any traier may be pued, on the whoe or on a part of its itinerary, by any compatibe orry. What is more, any orry may transfer part or a of its oad to any traier at any TL. The time a transspment taes depends on the amount of oad transferred. Lorries need not bring bac a traier, neither one they may have pued when eaving the depot, nor any other. The probem is to determine routes for orries and routes for traiers so that tota costs are minimized, the compete suppy of a customers is deivered to the depot, and oading capacities and time windows are maintained. Moreover, it must be ensured that the routes are synchronized 1

2 with respect to space, time and oad, so that traiers move in space ony when accompanied by compatibe orries and that the veces invoved in a oad transfer operation visit the pertinent ocation at the right time and transfer and receive the right amount of oad. An eampe route pan is depicted in Figure 1. Depot Lorry customer Traier customer Transspment ocation Lorry 1 Lorry 2 Lorry 3 Traier Figure 1: VRPTT eampe route pan In the eampe, orry 1 pus the traier from the depot to a TL and decoupes the traier there. Lorry 1 then visits two orry customers, returns to the traier, transfers oad, eaves the traier at the TL and returns to the depot via two orry and two traier customers. Lorry 2 starts at the depot and visits two orry customers before couping the traier, after orry 1 has performed its oad transfer. Lorry 2 then visits a traier customer, decoupes the traier at another TL, performs a oad transfer, visits some orry customers, returns to the traier, re-coupes it and pus it bac to the depot via a traier customer. Lorry 3 starts at the depot, visits some orry customers, and transfers some oad to the traier we orry 2 is visiting the three rightmost orry customers. After that, orry 3 returns to the depot via another orry customer. The two TLs in the centre of the figure are not used. The justification for studying the VRPTT is that the probem, apart from being interesting and chaenging in its own right, has numerous direct practica appications and, in addition, constitutes an archetypa, generic representative of the cass of vece routing probems with mutipe synchronization constraints (VRPMSs). Contrary to cassica VRPs, where a synchronization between veces is necessary ony with respect to wch vece visits wch customer, VRPMSs ebit additiona synchronization requirements with regard to spacia, tempora, and oad aspects. Ts is discussed in more detai in Section 2. The contribution of the present paper is three-fod: (i) Buiding on a non-trivia networ representation, two mied-integer programming (MIP) formuations for the VRPTT are proposed. (ii) Based on these formuations, five different branch-and-cut agorithms are deveoped and impemented. (iii) The computationa behaviour of the agorithms is anayzed in an etensive computationa study, using a arge number of test instances designed to resembe rea-word VRPTTs. The rest of the paper is structured as foows. The net section provides an overview of reated wor, incuding other types of VRPMSs. Section 3 presents a networ representation for the VRPTT as a basis for the two arc-variabe based formuations deveoped in Section 4. Section 5 describes the soution approaches that were impemented to sove these formuations, and Section 6 presents computationa eperiments performed with the impementations. The fina Section 7 concudes the paper with a research outoo 2 Literature review The avaiabe iterature on the VRPTT and reated VRPMSs is rather imited. As stated, the VRPTT was introduced in Dre [11], where formuations based on arc, path, and so-caed turn variabes were deveoped and a branch-and-cut agorithm for an arc variabe formuation 2

3 was presented. Ts author is not aware of other papers containing a soution approach for the VRPTT. A quite we-studied probem deaing with traiers in vece routing is the truc-and-traier routing probem (TTRP) (see, for eampe, Semet and Taiard [22], Chao [4], Scheuerer [21]). The TTRP is a specia case of the VRPTT with a fied orry-traier assignment, that is, each traier can be pued by ony one orry, and ony ts orry can transfer oad into the traier. Ts maes the probem much easier. In fact, the TTRP is not a VRPMS; there are no synchronization requirements other than customer covering. A recent survey of synchronization in vece routing is given in Dre [12]. In ts paper, five different types of synchronization are identified: (i) tas synchronization, referring to the decision wch vece(s) fufi(s) wch tas(s); (ii) operation synchronization, wch decides about time and ocation of some interaction between veces; (iii) movement synchronization, that is, the decision wch veces join to move in space; (iv) oad synchronization, wch determines the amounts of oad to be coected, deivered, or transspped; and (v) resource synchronization, wch decides about wch vece(s) use(s) a scarce resource at what time. The most important casses of VRPMSs identified in ts survey are muti-echeon vece and ocation-routing probems with tempora synchronization of veces of different echeons (Crainic et a. [8]), simutaneous vece and driver routing probems (as opposed to integrated vece and crew scheduing probems, where the tass to perform usuay have a given fied schedue) (Hois et a. [15]), and picup-and-deivery probems with transspments and simutaneous oad transfers (Cortés et a. [7]). Wors that study probems where different types of autonomous and non-autonomous objects that may join and separate en route (and not ony at a centra depot) are used to fufi tass are Bürcert et a. [3], Hois et a. [15], Cheung et a. [5], Kim et a. [16], and Dre et a. [13]. A of these papers deveop heuristic soution approaches: Bürcert et a. [3] consider a dynamic picup-and-deivery probem using orries, traiers, and swap-body patforms in the contet of ong-hau transport and sove their probem with a hoonic muti-agent system. Both Hois et a. [15] and Dre et a. [13] sove a simutaneous vece and driver routing probem (with appication in posta ogistics in Austraia and ong-hau road transport in Europe respectivey) with two-stage agorithms. Hois et a. [15] first compute abstract vece routes by soving a rich picup-and-deivery probem by heuristic coumn generation. Then, concrete veces and drivers are assigned by taing an integrated vece and crew scheduing approach, again done by soving an MIP by heuristic coumn generation. Dre et a. [13] determine routes for concrete veces in the first stage, and then compute routes for drivers, based on the vece routes from the first stage. They use the same arge neighbourhood agorithm in both stages. Cheung et a. [5] aso describe a picup-and-deivery probem, but for short-distance container transport, using drivers, tractors, and semi-traiers. Their probem is soved by an attribute-decision mode. Finay, Kim et a. [16] deveop a simpe but very effective heuristic for synchronizing service teams that are transported by veces between tas ocations: The teams, the veces, and the net tass are stored in three ists, and in each iteration, a tripet (team, vece, tas) is seected from the ists, using a best-fit criterion. Then the ists are updated to refect the situation when the seected vece transports the seected team to the ocation of the seected tas. Note that a of these wors assume mandatory synchronization of objects. Ts means that, to fufi tass, it is aways required to use a or severa object types (veces, drivers, equipment). In the VRPTT, by contrast, using traiers and performing transspments are optiona operations. Ts adds an additiona degree of freedom to the probem. Moreover, none of the above papers considers a probem where a five types of synchronization described above are reevant, as is the case for the VRPTT. 3

4 3 A networ representation In contrast to standard vece routing probems, for wch an adequate networ representation is straightforward, ts is not the case for the VRPTT, where the foowing critica modeing questions must be addressed: How to ensure that a traier is accompanied by a compatibe orry on an arc, that is, how to synchronize the movements of veces? How to synchronize the visiting times of veces at transspment ocations? How to baance the oad transfer amounts of veces echanging oad? A decouping, transspment or recouping operation is defined by (i) the ocation where the operation taes pace, (ii) the point in time when the operation begins, (iii) the passive vece (traier), wch provides capacity, that is, may receive oad, and/or may be decouped from or (re)couped to a orry, (iv) the active vece (orry), wch requests capacity, that is, may transfer oad, and/or may decoupe or (re)coupe a traier, and (v) the amount of oad transferred into the passive vece, wch is zero if ony decouping or recouping is performed. To answer the above modeing questions and hande the ogic of transspment operations, a space-timevece cass-operation networ D = (V, A) with an associated set F of veces (the feet), is proposed. The feet F is partitioned into two sets F L F T of casses of veces. F L is the set of orry casses, and F T is the set of traier casses. F denotes the number of veces of cass. For a F, q is the capacity of a vece of cass. Henceforth, is used to denote orry casses, and is used to denote traier casses. For a orry cass F L, C() is the set of compatibe casses of traiers, that is, the set of casses of traiers that a orry of cass can pu; for a traier cass F T, it is the set of compatibe casses of orries, that is, the set of casses of orries that can pu a traier of cass. τ is the oad transfer time per unit of oad, wch is assumed to be the same for a vece casses. It is assumed for simpicity that any orry can visit any customer, and that any traier can visit any traier customer. Each verte in V corresponds to a ocation in space, an absoute and/or reative period of time, a type of operation, and a cass of passive vece. L = {Depot} L C L I is the set of reevant rea-word ocations. L C = L CL L CLT is the set of customer ocations, wch is partitioned into L CL, the set of orry customers, and L CLT, the set of traier customers. L I is the set of pure transspment ocations. For each i V, L(i) L denotes the ocation corresponding to i. s i is the suppy of verte i, wch is zero for depot and transspment vertices. For S V, s S := i S s i. There is one start depot verte o and one end depot verte e. For each customer, there is one customer verte. Let V C = V CL V CLT be the set of customer vertices, where V CL is the set of orry customers, and where V CLT is the set of traier customers. Foowing Chao [4] and Scheuerer [21], it is assumed that the ocations corresponding to traier customers can aso be used for paring and transspment operations. Each verte i V has a time window [a i, b i ], where 0 a i b i T, and T is the ength of the panning horizon. o and e both have a time window of [0, T ]. Usuay, in the iterature, a i and b i respectivey denote the eariest and atest point in time when the service to be performed at i can start. In the VRPTT, a i has the same meaning, but, because of the oad-dependent oad transfer times, b i denotes the atest point in time when i must be eft, that is, the service at i must be finished no ater than b i. To provide a unified treatment, ts convention is adopted aso for the depot and customer vertices. For each combination of physica transspment ocation (pure transspment ocation or traier customer ocation) and traier cass, there are F verte tupes representing the operations decouping, transspment, and recouping. To be precise, for each transspment ocation and each traier cass, there are F tupes v d p, vt p, vr p, p = 1,..., F. V d is the set of decouping vertices, V t is the set of transspment vertices, and V r is the set of recouping vertices. V I = V d V t V r. The idea bend ts separation of transspment ocations and processes is the foowing: A verte v d p V d can ony be reached by a orry puing a traier of 4

5 the corresponding cass. The orry then eaves the verte singy, the traier moves on to v t p, the pertinent transspment verte. A verte v t p V t can ony be reached and eft by a singe orry and by a traier of the corresponding cass, where the atter comes from v d p. A verte v r p V r can ony be reached by a singe orry and by a traier of the corresponding cass, where the atter comes from the pertinent transspment verte v t p, and be eft by the orry puing that traier. At any verte of a transspment verte tupe, that is, aso at the decouping and the recouping verte, a orry visiting the verte can transfer oad to a traier. A veces are initiay at the start depot verte o and a veces that are used end their route at the end depot verte e. Lorries can visit a vertices of D, ecept for decouping and recouping vertices of incompatibe traiers. Traiers can ony visit their corresponding transspment vertices, traier customers, and, of course, the start and the end depot verte. F (i), for a i V I, is the cass of passive vece associated with i. For (i, j) A, F L and F T respectivey denote the set of orry and traier casses that can traverse (i, j). For a F, V (A ) is the set of vertices (arcs) that can be reached (traversed) by veces of cass. Moreover, for S V, et A (S) := {(i, j) A : i, j S}, δ (S) := {(i, j) A : i / S j}, and δ + (S) := {(i, j) A : i S j}. The arc set A contains the foowing eements: (o, i) and (i, e) for a customer vertices i V C (o, i) for a decouping and recouping vertices i V d V r (i, j) for a customer vertices i, j V C with i j (v d p, j) for a decouping vertices vd p V d and a vertices j V \ ({o} V d ) (v t p, j) for a transspment vertices vt p V t and a other vertices j V \ ({o, e} V d ) (v r p, j) for a recouping vertices vr p V r and a other vertices j V d \ {v d p : p p} V CLT {e} (i, j) for a i V CLT, j V I (i, j) for a i V CL, j V t V r Figure 2 visuaizes the verte types present in the networ and the arc types that can be traversed by orry-traier combinations, singe orries, and singe traiers. To eep the figure concise, there is ony one arc for each arc type present in the networ. For eampe, in the subfigure for LTCs, an arc from the eft traier customer to the right one is depicted to indicate that LTCs can freey move from any traier customer to any other traier customer (uness capacity or time window restrictions probit ts). The absence of an arc from the right traier customer to the eft one in the subfigure does not mean that there is no such arc. The arc from the recouping to the decouping verte in the subfigure for LTCs eists ony if the two transspment verte tupes (TVTs) represent the same traier cass, and if they ie at different ocations or the upper TVT has a gher p vaue. The indicated arcs to, from, and between transspment vertices in the subfigure for singe orries eist between a types of TVTs, that is, those representing the same or different traiers at the same or different ocations. For TVTs representing the same ocation, the arcs eist ony if the upper TVT has a gher p vaue. A cost coefficient c indicates the distance-dependent costs of a vece of cass F traversing arc (i, j) A. For the arcs emanating from the start depot verte, fied costs for using a vece of cass are aso contained in c oj. It is assumed that a arc costs and arc trave times are non-negative. For each arc (i, j) A, τ is the traversa time, wch is assumed to be the same for a vece casses. Moreover, it is assumed that τ oi a i for a i V \ {o}, and that b i + τ ie T for a i V \ {e}. 5

6 Lorry customers o Transspment verte tupes Decoupe Recoupe Transfer d Traier customers Networ vertices Arc types for orry-traier combinations Arc types for singe orries Arc types for singe traiers Figure 2: Networ vertices and arc types Lorries cannot traverse arcs (v d p, vt p ) and (vt p, vr p ). The corresponding traiers of cass can traverse these arcs, wch, consequenty, form the set A,s. An arc (v d p, vr p ) modes the possibiity that a orry can wait for a traier at a transspment ocation we another orry performs a oad transfer. If a orry currenty puing a traier wants to transfer oad to another traier at a certain ocation, the orry decoupes at a decouping verte at, transfers oad to, and re-coupes. 4 Two arc-variabe based formuations Both subsequent formuations are modifications of a formuation deveoped in Dre [11]. For both formuations, the foowing three fundamenta assumptions are made: (i) (ii) Each orry customer verte is visited by eacty one orry. Each traier customer verte is visited by eacty one orry and at most one traier. (iii) Each transspment verte is reached by at most one orry and at most one traier. These assumptions ensure that there is at most one transspment operation at each transspment verte, and that it is cear between wch orry and traier casses each of these operations is performed. In ts way, the three modeing issues mentioned at the beginning of Section 3 are addressed. With the networ defined as above, that is, with one transspment verte between a decouping and a recouping verte, these assumptions aow at most 3 F oad transfers per traier cass at each transspment ocation. These oad transfers can a be made into one and the same traier of cass, if ts traier is pued from v r p to vd,p+1 for p = 1,..., F 1, or into severa or a traiers of cass. 6

7 4.1 First formuation The foowing variabes are used: {0, 1} F, (i, j) A. = { 1, a vece of cass traverses arc (i, j) 0, otherwise i R0 + F, i V \ {o}. The amount of oad a vece of cass is carrying when reacng verte i. t i R 0 + i V \ {o}. The unique point in time when verte i is reached. The resuting formuation is: (VRPTT1): c min (1) F (i,j) A subject to = 1 i V C (2a) (h,i) A F L F (i) (h,i) A F (i) 1 F L F L F L i V I = F (i) i V d, (i, j) A F (i) (2c) (h,i) A F (i) i V t, (i, j) A F (i) (2d) (h,i) A = F (i) h i i V r, (h, i) A F (i) (2e) (h,i) A (2b) C( ) F L F T, i {o} V CLT, (i, j) A (2f) F (i) = C( ) F L i V r, (i, j) A F (i) (2g) ie (i,e) A F F (2h) = F, i V \ {o, e} (2i) (h,i) A (i,j) A = 1 = 1 i + i + s i j + j ) (i j (i) = 1 F i + i V CLT, F L, C(), (i, j) A A (3a) F (i) j i V d, C(F (i)), (i, j) A F (i), (i, j ) A (3b) 7

8 ) (i) = 1 F i + (i j F (i) j i V t, F L, (i, j) A F (i), (i, j ) A, j e (3c) ( ) = 1 F (i) i + i j F (i) j i V r, C(F (i)), (i, j) A F (i) A, j e (3d) = 1 i + s i j F L, i V C, (i, j) A, F T = (3e) = 1 i j = 1 F, i V CLT, (i, j) A, F T = 0 i + s i j F L, i V CLT, (i, j) A, F T (3f) (3g) C() F T = 1 j i F L, i V I, (i, j) A (3h) ( F (i) = 1 t i + F (i) j ) F (i) i τ b i i V r, (i, j) A F (i) (3i) = 1 t i + s i τ + τ t j F L, i V C, (i, j) A (3j) ( ) = 1 t i + i j τ + τ t j F L, i V I, (i, j) A (3) F (i) i t i + F (i) j i V t, (i, j) A F (i) (4a) ( ) F (i) j F (i) i τ + τ t j i V d V t, (i, j) A F (i) (4b) {0, 1} F, (i, j) A (5a) 0 i q F, i V \ {o} (5b) a i t i b i s i τ i V \ {o} (5c) (1) is the objective function, wch minimizes tota costs. Constraints (2) are those invoving ony fow variabes. (2a) (2g) are the routing synchronization constraints, and (2h) (2i) are vece-cass specific routing constraints. Constraints (3) are ogica impications ining routing and resource variabes. (3a) (3d) are oad synchronization constraints, (3e) (3h) are vece-cass specific oad update constraints, and (3i) (3) are time update constraints. Constraints (4) are resource update constraints not invoving fow variabes, and (5) determine the ranges of the variabes. (2a) are the usua customer covering constraints. (2b) ensure that each transspment verte is visited at most once by a corresponding traier. (2c) (2e) mae sure that a transspment verte i is ony visited by a orry if the corresponding traier visits ts verte, and, hence, that at most one orry visits a transspment verte. Observe that, for i V d, the ony arc in A F (i) eaving i is the arc to the subsequent transspment verte. Simiary, for i V t, the ony arc in A F (i) eaving i is the arc to the subsequent recouping verte. (2a) (2e) impy that each verte in V \ {o, e} is visited by at most one orry and one traier. (2f) and (2g) guarantee that a traier is pued by a compatibe orry if traverses a spacia arc. For the remaining constraints, note that a orry that brings a traier to a decouping verte and then moves on to the depot wi not transfer any oad at ts verte. Liewise, a orry that pus a traier from a recouping verte to the depot wi aso not transfer any oad. Moreover, a orry wi never visit a transspment verte i V t directy before moving to the depot; thus, as stated above, no arcs (i, e) with i V t eist. For each vece cass, (2h) imit the number of veces reacng the end depot to the number of veces of each cass. (2i) are the fow conservation constraints. 8

9 (3a) (3d) are the oad update constraints for both orries and traiers at traier customer and transspment vertices. (3a) state that the suppy of a traier customer is divided up arbitrariy between the orry and the traier visiting the customer. The net constraints, (3e), are the oad update constraints for the orries at customer vertices. Constraints (3f) state that, at a traier customer verte, no oad transfer is possibe. Ts is a sensibe requirement, because, as stated above, for each traier customer ocation, there are aso tupes of transspment vertices for each traier cass, and movements between vertices corresponding to the same physica ocation incur virtuay no costs and tae virtuay no time. Constraints (3g) are for the correct update of the oad variabes at traier customer vertices visited by a singe orry. Constraints (3h) mae sure that no oad transfer from a traier to a orry (passive to active vece) is possibe, respectivey, that the amount of oad transferred from a orry to a traier is non-negative. Without ts constraint, negative oad transfer times can resut. (3i) mae sure that the oad transfer at transspment vertices is finished by the end of the time window. Note that, because of (3a) (3d), it is sufficient to require (3i) for F (i). Moreover, it is sufficient to require (3i) for i V r, since then the time windows wi be maintained aso for the preceding transspment vertices. Constraints (3j) and (3) are for the timing update for orries on arcs emanating from customer vertices and at transspment vertices respectivey. (4a) mae sure that the oad of a traier does not decrease at transspment intermediate vertices not visited by a orry, and (4b) are for the timing update of traiers at their transspment vertices. The atter two constraint types are no impications, because these constraints can be fufied aso when the arc (i, j) A F (i) for i V d V t is not used. Synchronization constraints. In standard VRPs such as the capacitated VRP (CVRP) or the VRP with time windows (VRPTW), the customer covering constraints (2a) are the ony ogicay couping (ining, joint) ones. In the VRPTT, the cose interdependency between the veces must be deat with by severa additiona types of ogicay couping constraints, namey, (2b) (2g) and (3a) (3d). Infating resource variabe vaues. At the end depot verte, there is ony one oad variabe, e, for a veces of each cass F. Ts means that if one vece of a cass reaches the end depot fuy oaded (according to the variabe vaues), any other vece beonging to the same cass does so aso, even if ts vece actuay (in reaity) carries ess oad. The constraints aow to infate oad variabes, meaning that the oad variabe of a vece cass at the vertices on a tour may be set gher than the rea oad, we sti not eceeding the rea vece capacity. In ts way, the differences between the vaues of the oad variabes aong a tour refect the rea situation, so that the correct synchronization of oad transfer amounts and visit times, wch depends on differences in oad variabe vaues, remains possibe. Ts technique of infating the vaues of resource variabes aows to avoid introducing routing and resource variabes for each vece, and, when using variabes for vece casses instead of individua veces, maes the creation of start and end depot vertices for each vece unnecessary. 4.2 Second formuation The second formuation is based on the same networ as the first formuation, and it uses the same type of binary variabes. However, foowing an idea introduced independenty by van E [24] and Maffioi and Sciomachen [18], it uses different types of continuous resource variabes: R0 + F, (i, j) A. If the arc (i, j) is used by a vece of cass, the amount of oad the vece is carrying when eaving verte i; zero otherwise. t R 0 + (i, j) A. If the arc (i, j) is used by any vece, the unique point in time when verte i is eft; zero otherwise. The resuting formuation is: 9

10 (VRPTT2): (1) subject to (2) and ( ) + s i + + (h,i) A (i,j) A i V CLT (6a) (h,i) A F L C(F (i)) + F L (h,i) A + C(F (i)) (h,i) A F L (h,i) A F T + F (i) C(F (i)) F (i) h i (h,i) A F (i) (i,j ) A + + F L (i,j ) A F L F (i) h i (h,i) A F (i) (i,j) A C(F (i)) F L F L F T F (i) i V d (6b) (i,j) A F (i) F (i) i V t (6c) (i,j) A F (i) + F (i) i V r (6d) ( ) + s i i V CL (6e) (i,j) A F T (h,i) A F L (h,i) A F T ( ) + s i F L (i,j) A F L i V CLT (6f) (i,j) A s i F T h i (h,i) A F L i V CLT (6g) (i,j) A i V I (6h) (h,i) A F (i) F (i) i V t (6i) (h,i) A F (i) (i,j) A F (i) (h,i) A ( t + τ + s i τ t t + τ F (i) F L ) (h,i) A F (i) + F (i) (i,j) A F (i) (i,j) A F (i) (h,i) A F (i) i V C (6j) τ t i V I (6) (i,j) A F (i) t = t i V d (6) (i,j) A F (i) (i,j ) A\A F (i) t (i,j) A F (i) b i 1 t (i,j ) A\A F (i) t + τ F L (i,j ) A\A F (i) i V t (6m) (h,i) A\A F (i) F L 10

11 + (h,i) A \A F (i) F L (h,i) A\A F (i) (i,j ) A \A F (i) F L τ t i V t (6n) (i,j) A F (i) t + τ + τ (h,i) A (i,j) A t (i,j) A F L F L F L i V r {0, 1} F, (i, j) A (7a) 0 q F, (i, j) A (7b) ( a i + s i τ ) t b i (i, j) A, i {o} V C (6o) (7c) F L F L a i t b i i V I, (i, j) A \ A F (i),s (7d) F L F L a i F (i) t b i F (i) i V d V t, (i, j) A F (i) (7e) t ie b e τ ie (i, e) A (7f) Constraints (6a) (6i) and (6j) (6n) are for the update of the oad and time variabes respectivey; constraints (7) specify the ranges of the variabes. Constraints (6a) (6h) correspond to (3a) (3h) in formuation (VRPTT1). Simiary, (6i) corresponds to (4a), and (6j) (6) correspond to (3j) (3). (6) (6o) ensure the correct update of the timing variabes at decouping, transspment intermediate, and recouping vertices respectivey. The corresponding constraint to (3i), wch maes sure that the oad transfer at transspment vertices is finished by the end of the time window, is (7c). (7b) (7e) coupe the fow and resource variabes. These constraints aow to avoid impications in formuation (VRPTT2). 4.3 Vaid inequaities The foowing vaid inequaities were used in the computationa eperiments described in the net section. Lorry fow cut: ie n L,min (i,e) A F L (8) Ts inequaity requires that the fow of orries into the end depot be at east equa to the minimum number of necessary orries, n L,min. As the tota customer suppy must be brought to the depot, the inequaity is vaid. A ower bound on n L,min can be determined as foows. A possibe orry-traier combinations and singe orries are sorted by non-increasing tota capacity and the capacities are summed up unti the sum eceeds the tota customer suppy. For each summand, n L,min is increased by one. As traiers are usuay compatibe with more than one orry, a orry-traier combinations that are no onger possibe because the respective orry or traier or both has/have aready been used with another orry or traier must be deeted during the summation. A simiar cut is possibe for traiers, but was not usefu in the computationa eperiments described beow. 11

12 Connectivity cuts: i j F, S V \ {e}, (i, j) A, i S (9) (i,j ) δ (S),i j These cuts require that, for each vece cass and for each subset of the verte set not containing the end depot verte, if a vece of ts cass uses an arc (i, j) whose tai is in the subset, there must be another arc (i, j ) entering ts subset whose tai must not be j. Ts is impied by the fow conservation constraints (2h) and (2i). An eact procedure for the separation of vioated connectivity cuts wors as foows. For each > 0, a maimum fow probem from j to e is soved in the support graph where the arc capacities are equa to the vaues of the fow variabes of vece cass, but where verte i, and, hence, a arcs incident to i, are deeted. Ts yieds a cut with j on one side and e on the other. If the vaue of the maimum fow is ess than, a vioated connectivity cut has been found. Ts procedure is correct, because in any feasibe soution, > 0 means = 1, and i is on the same side as j in any j-e-cut, because each verte is visited by each vece cass at most once. In other words, there is no feasibe soution where i is on the unique path of a vece of cass from j to e, if arc (i, j) is used by a vece of cass. Ts means that there must be a fow of at east from the j side of the cut to the e side of the cut, but ts fow must not pass through i. Cuts (9) can sti be ifted. If one of the incident vertices i, j of an arc (i, j) is a recouping verte, the corresponding decouping and intermediate vertices cannot be visited any more, and, hence, any potentia cut arc emanating from or eading to such a verte can be discarded. If i or j is an intermediate verte, the same hods for arcs incident to the corresponding decouping verte. κ-path cuts: κ-path cuts are we-nown vaid inequaities for the VRP(TW), cf. Koh et a. [17]. In the VRPTW with homogeneous feet (in particuar, without traiers), the idea of κ-path cuts for κ 1 is that, for any customer subset S for wch at east κ(s) veces are necessary to serve a customers in the subset, at east κ veces must visit ts subset, and, consequenty, the fow into ts subset must be at east κ. 2-path cuts for heterogeneous feet: Koh et a. [17] have very successfuy used 2-path cuts in a branch-and-price-and-cut agorithm for the VRPTW. For vece routing probems with heterogeneous feet as we as for the VRPTT, 2-path cuts can be generaized as foows. For each subset S V C, et κ (S) be a ower bound on the minima number of routes of veces of cass F necessary to coect the compete suppy of the customers in S when no transspments are aowed and no vece of a different cass is used (that is, when it is assumed that each vece cass can visit each customer and that traiers can visit customers without being pued by a orry). For eampe, if there are no time windows, the bound s S /q can be used. Then, the foowing inequaity is vaid: S V C. (10) { F :κ (S) 2} (i,j) δ (S) { F :κ (S)=1} (i,j) δ (S) The vaidity of (10) foows from the customer covering constraints (2a), the oad update constraints (3a) (3h), and the vece capacity constraints (5b). κ-path cuts for heterogeneous feet for genera κ 2: Setting κ := min F {κ (S)} for a S V C, a κ-path cut for genera κ 2 can be written as foows: κ S V C (11) F (i,j) δ (S) 12

13 Note that (10) and (11) require summation over a F, that is, over a orry and traier casses, and that (10) as we as (11) are vaid for sets S containing an arbitrary number of orry and/or traier customers. Lifted generaized arge mutistar cuts: Yaman [25] has shown that the foowing cuts are vaid for heterogeneous feet VRPs: F ξ i (i,j) δ (S) s S + F sj S V C, (12) (i,j) δ + (S) where ξ i := min{q s i, s S + ma (i,j) δ + (S){s j}}. Essentiay, ts type of cut requires that the remaining capacity of the veces visiting customers in S be at east equa to the suppy of the customers in S pus the suppy of those customers (if any) visited immediatey after eaving S. Symmetry-breaing cuts: A traier routes visiting the same ocations in the same order and receiving the same amount of oad at each ocation are equivaent, irrespective of wch tupe of transspment vertices is used at a certain ocation. That is, two otherwise identica traier routes r 1 and r 2, where r 1 uses the transspment vertices v d p 1, v t p 1, v r p 1 and r 2 uses v d p 2, v t p 2, v r p 2, are equivaent and incur the same costs, irrespective of whether or not p 1 = p 2, since they describe the same movements in reaity. Such symmetries can be removed by the foowing cuts: F L i, i V d, L(i) = L(i ), F (i) = F (i ), p(i) = p(i ) 1 (13) F L These cuts require that a transspment verte tupe with order number p is ony visited if a preceding transspment verte tupes, that is, those with ower order number, are aso visited. If two traiers 1 and 2 of the same cass use the same transspment ocation, simiar symmetries arise: It does not mae a difference whether 1 uses the transspment verte tupe with order number p 1 and 2 uses the tupe with order number p 2 or the other way round. To remove such symmetries, the foowing cuts can be used in formuation (VRPTT1): t i t i i, i V d, L(i) = L(i ), F (i) = F (i ), p(i) = p(i ) 1 (14) These cuts require that the service at the decouping verte of a transspment verte tupe with order number p starts not earier than the service at the decouping verte of the transspment verte tupe with order number p 1. For formuation (VRPTT2), the foowing cuts can be used instead of (14): t (b i + 1) F (i) (i,j) A F (i) ti j (b i + 1)F (i ) i j (i,j ) A F (i ) i, i V d, L(i) = L(i ), F (i) = F (i ), p(i) = p(i ) 1 (15) Some remars on vaid inequaities foow. The orry fow, connectivity, and 2-path cuts were aready used in Dre [11]. Badacci et a. [2] propose a ifting for the κ-path cuts of Koh et a. [17], but ts cannot be used for arc variabe formuations. Desauniers et a. [10] have introduced generaized κ-path cuts for the VRPTW with homogeneous feet. Athough these can be generaized to heterogeneous feet, they were not used in the computationa eperiments described beow, because the wide time windows in the test instances used essentiay made these cuts equivaent to cuts (11). Ascheuer et a. [1] have introduced, for the asymmetric TSP with time windows, the so-caed infeasibe path constraints. These cuts are vaid aso for the VRPTT, but are not usefu when time windows are non-restrictive. Finay, Pessoa et a. [19] describe cuts for a heterogeneous feet VRP, but these are based on a different formuation. 13

14 5 Soution approaches Formuation (VRPTT1), the origina formuation, contains a ot of ogica constraints (impications). To dea with these, the foowing five soution approaches were impemented in C++, using Visua C and IBM Iog Cpe Concert Technoogy, version 12.2: (i) Formuation (VRPTT1), inearizing the impications using taiored Big-M vaues, taing into account the vece capacities and the ength of the panning horizon. (ii) (iii) (iv) Formuation (VRPTT2), thereby avoiding ogica constraints atogether. Formuation (VRPTT1), empoying a Cpe feature that aows entering impications directy as such and etting the sover automaticay hande them. How ts is done eacty, though, is not documented. Formuation (VRPTT1), appying the combinatoria Benders cuts introduced by Codato and Fischetti [6] and successfuy used by Cortés et a. [7]. The basic idea of ts approach is to decompose the probem into a master probem containing ony the variabes and the constraints invoving ony these variabes, and a subprobem containing ony the continuous variabes and a constraints not invoving variabes. Then, after soving the LP reaation of the master probem, a impications activated by variabes that are one or zero in the soution of the LP reaation are added to the subprobem. If the subprobem, wch has no objective function, is feasibe, then evidenty the variabes soution to the master probem is feasibe and optima for the compete probem. Otherwise, at east one of the integer variabes must change its vaue. Ts is enforced by identifying an incusion-minima infeasibe subset (by means of a buit-in Cpe function) of the subprobem constraints and adding the constraint (1 ) S 1 + S 0 1 (16) to the master, where the set S 1 (S 0 ) contains those variabes that activate a constraint when they tae vaue one (zero) and do tae ts vaue in the current LP soution. (v) Formuation (VRPTT1), using deferred consideration of ogica constraints. Ts wors as foows. At the root node of the branch-and-bound tree, the LP reaation of a reaed probem is soved, where a impications are ignored. Then, at each subsequent node, it is checed for wch variabes the ower bound is greater than zero. A such variabes wi have vaue one in each integer feasibe soution (if any) at the subtree rooted at ts node. Simiary, it is checed wch variabes have an upper bound of ess than one. A such variabes wi have vaue zero in each integer feasibe soution at the subtree rooted at ts node. Then, instead of having inearized constraints containing Big-M or adding combinatoria Benders cuts, ony the right hand sides of the activated impications, that is, the respective inear constraints to the right of in (3), are added as ocay vaid constraints to the LP at the respective node. A approaches use the cuts described in Section 4.3. For formuation (VRPTT2), cuts (15) are used instead of (14). The orry fow cut as we as both types of symmetry-breaing cuts are directy added to the formuation as static cuts. The connectivity cuts are dynamicay separated in the course of the agorithm. As for the κ-path and mutistar cuts, it turned out that a instances that coud be soved with or without them are sma enough to aow compete enumeration of a cuts e ante. Therefore, a cuts (11) and (12) are aso added as static cuts. As brancng strategy, the foowing four-stage strategy outperformed the defaut best-bound singe variabe brancng strategy provided by Cpe for approaches (i), (ii), and (iv). (i) Branch on the sum of a arc variabes eaving the start depot verte. (ii) Branch on the sum of a orry cass arc variabes eaving the start depot verte. (iii) Branch on the sum of a traier cass arc variabes eaving the start depot verte. (iv) Use the defaut Cpe brancng strategy. For approaches (iii) and (v), the Cpe defaut brancng performed better. Strong brancng was used in a cases. 14

15 6 Computationa eperiments In ts section, the resuts of an etensive computationa study with the approaches described in the previous section are presented. 6.1 Test instances Dre [11] has created a set of VRPTT test instances structured to resembe the situation in raw mi coection. These use two orry and two traier casses as specified in Tabe 1. Capacity Fied costs Costs per m Compatibe traier casses Lorry cass 1 10, , 2 Lorry cass 2 15, Traier cass 1 10, Traier cass 2 15, Tabe 1: Feet data The customer and transspment ocations are randomy seected on a m grid with the depot ocated in the centre. The resuting Eucidean distances between each pair of vertices are mutipied by a distance factor of 1.3. The customer suppies are chosen randomy from [1,000; 10,000]. As oad transfer time, two minutes per 1,000 units of suppy are assumed. The ength of the panning horizon is 12 hours (1,320 minutes). A customers and transspment ocations have a time window of [0; 1,320]. Such non-restrictive time windows represent the situation in raw mi coection where ony very few customers or transspment ocations actuay have time windows. The test instances were created with enough veces of each cass such that the compete suppy coud be coected with either ony orries of cass 1 and traiers of cass 2 or ony orries of cass 2 and traiers of cass 1. With these parameters, a set of so-caed y z instances was created, where 2 20 stands for the number of customers, 2 y 20 for the number of potentia transspment ocations, and 4 z 24 for the number of veces. It is aways = y, and there are aways /2 orry customers and /2 traier customers. y/2 potentia transspment ocations correspond to the traier customer ocations, and y/2 are pure transspment ocations. For each of the four vece casses, there are z/4 veces. Ts means that an y z instance has 1++3 F T F y+1 vertices: one for the start depot, for the customers, three for each traier at each of the y potentia transspment ocations to represent decouping, transfer, and recouping, and one for the end depot. Tabe 2 shows detais on instance sizes. Instance No. of Vertices Arcs Variabes Constraints Cuts cass instances Binary Continuous Symmetry- κ-path Mutistar (VRPTT1) (VRPTT2) (VRPTT1) (VRPTT2) breaing 1 / 2 (avg.) / ,340 2,384 2,176 0 / ,516 2, ,116 7,408 6,344 8 / ,074 1, ,976 5,299 4,630 0 / ,390 5, ,216 16,597 13, / ,008 10, ,344 29,444 24, / Tabe 2: Instance sizes 6.2 Agorithm setup and system parameters Some remars on the setup of the agorithms are appropriate. In approach (iv), combinatoria Benders cuts can either be separated ony when the LP soution is integer or aso when some variabes are fractiona. Moreover, it is possibe to add the cuts as goba or oca ones, and either ony one or severa vioated cuts can be added in each iteration. Tests showed that the 15

16 best setup is to separate cuts aso when there are fractiona variabes, and to add one vioated cut per iteration as a goba one. Simiary, in approach (v), it is possibe to chec for newy fied variabes either ony when the LP soution is integer in the variabes or after each soution of the LP reaation. Here, tests showed that the former variant is sighty but consistenty better. In a approaches, connectivity cuts are added ony if the vioation is at east 0.3. In each iteration, a connectivity cuts that reach ts vioation threshod are added. Tests with adding ony one vioated connectivity cut per vece cass and iteration yieded a worse performance. The maimum fow probems for the separation of the connectivity cuts are soved with the Edmonds-Karp agorithm provided by the Boost Graph ibrary (boost.org). The resuts presented subsequenty were obtained on a computer with a 2.80 GHz CPU and 16 GB of main memory for instance casses , and on a macne with a 2.83 GHz CPU and 8 GB of main memory for instance cass 8 8 8, both running Win XP Sp 2, 64-bit. A CPU time imit of 10,800 seconds was set. 6.3 Evauation of the vaid inequaities Tabe 3 reports the average reative increase in the root ower bound for formuations (VRPTT1) and (VRPTT2) when the different casses of vaid inequaities are used. As can be seen, the root ower bounds of formuation (VRPTT2) are significanty gher than those of (VRPTT1). Moreover, the tabe shows that the orry fow cut is very important: It cuts off fractiona soutions where orries are used ony partiay, and the resuting better consideration of the fied costs raises the ower bounds consideraby. Formuation: (VRPTT1) (VRPTT2) (VRPTT2) Average reative increase compared to: (VRPTT1), no cuts (VRPTT1), no cuts (VRPTT2), no cuts No cuts n.a % n.a. Ony orry fow cut % % 36.8 % Ony connectivity cuts 67.8 % % 3.3 % Ony κ-path cuts 7.6 % % 0.4 % Ony mutistar cuts 17.7 % % 0.3 % Tabe 3: Comparison of root ower bounds (instances ) 6.4 Comparison of the different soution approaches Tabe 4 presents the computationa resuts obtained with the different approaches and the described settings. Coumns Optima and Feasibe respectivey indicate the percentage of instances that coud be soved optimay and for wch a feasibe soution coud be computed witn 10,800 seconds. Coumn Gap specifies the average reative gap between the upper and the ower bound, UB and LB respectivey, upon termination of the optimization, for those instances for wch a feasibe soution coud be computed, as (UB LB)/LB. Coumn CPU time indicates the average running time over a instances of a cass, whether or not a feasibe soution coud be found. Coumn None better reports the percentage of instances that no approach soved better. No approach soves an instance i better than a certain approach A if either A soves i to optimaity and no other approach that aso soves i optimay is faster, or if no approach finds the optima soution witn the time imit and no approach finds a better feasibe soution than A. Coumn Best indicates the percentage of instances that were soved best with the respective approach. An approach A is best in soving an instance i if A finds the optima soution faster than any other approach, or if it finds a better soution than any other approach. To account for measuring inaccuracies, computation times in seconds were rounded to integer mutipes of 10 seconds when computing the vaues in coumns None better and Best. Finay, coumn No. B & B nodes gives the average number of nodes in the branch-and-bound tree upon termination of the optimization, irrespective of the soution status. 16

17 Approach Instance Optima Feasibe Gap CPU time None better Best No. B & B cass [%] [%] [%] [secs.] [%] [%] nodes (i): (VRPTT1), inearizing the impications using taiored Big-M vaues , , , , , , ,221 Overa , ,382 (ii): (VRPTT2), avoiding ogica constraints atogether , , , Overa , (iii): (VRPTT1), empoying a Cpe feature that aows entering impications directy , , , , Overa , ,478 (iv): (VRPTT1), appying the combinatoria Benders cuts , , , , , , ,060 Overa , ,997 (v): (VRPTT1), using deferred consideration of ogica constraints , , , , , , , ,642 Overa , ,310 Tabe 4: Computationa resuts The foowing observations can be made in Tabe 4: Overa, there is no approach that outperforms a others with respect to every criterion. However, approach (v) performs best in four out of seven criteria, and, for one additiona criterion, there is no better one. Approach (iii) ceary performs worst. Instance casses were not even tried with ts approach. It seems that the automatic handing of impications buit into Cpe is inadequate for the probem structure at hand. Athough formuation (VRPTT2) has a significanty stronger LP reaation than formuation (VRPTT1), approach (ii) soves fewer instances to optimaity than approach (i), and it requires sighty more computation time on average. Apparenty, the soution of the LP reaations taes onger. Approach (v) soves the ghest number of instances to optimaity, 75 out of 109, wch corresponds to 68.8 %. It is the ony approach to sove an instance of cass to optimaity. However, it fais to find a feasibe soution for five instances, wch is worse than approaches (i), (ii), and (iv). There is one particuary hard instance in cass 4 4 8, wch gets soved optimay ony by approach (ii). The overa optimaity gap is comparativey gh for approach (i). In particuar, though feasibe soutions are found for a but two instances of cass 8 8 8, the gap for ts cass is amost twice as gh as for the other approaches. 17