REVISTA DE MÉTODOS CUANTITATIVOS PARA LA ECONOMÍA Y LA EMPRESA (12) Págnas 5 38 Dcembre de 2011 ISSN: 1886-516X DL: SE-2927-06 URL: http://wwwupoes/revmetcuant/artphp?d=51 Branch-and-Prce and Heurstc Coumn Generaton for the Generazed Truck-and-Traer Routng Probem Drex, Mchae Gutenberg Schoo of Management and Economcs, Johannes Gutenberg Unversty Manz Fraunhofer Centre for Apped Research on Suppy Chan Servces SCS, Nuremberg Correo eectrónco: drex@un-manzde ABSTRACT The generazed truck-and-traer routng probem (GTTRP) consttutes a unfed mode for vehce routng probems wth traers and a fxed orrytraer assgnment The GTTRP s a generazaton of the truck-and-traer routng probem (TTRP), whch tsef s an extenson of the we-known vehce routng probem (VRP) In the GTTRP, the vehce feet conssts of snge orres and orry-traer combnatons Some customers may be vsted ony by a snge orry or by a orry wthout ts traer, some may aso be vsted by a orry-traer combnaton In addton to the customer ocatons, there s another reevant type of ocaton, caed transshpment ocaton, where traers can be parked and where a oad transfer from a orry to ts traer can be performed In ths paper, two mxed-nteger programmng (MIP) formuatons for the GTTRP are presented Moreover, an exact souton procedure for the probem, a branch-and-prce agorthm, and heurstc varants of ths agorthm are descrbed Computatona experments wth the agorthms are presented and dscussed The experments are performed on randomy generated nstances structured to resembe rea-word stuatons and on TTRP benchmark nstances from the terature The resuts of the experments show that nstances of reastc structure and sze can be soved n short tme and wth hgh souton quaty by a heurstc agorthm based on coumn generaton Keywords: vehce routng; transshpment; feet pannng; eementary shortest path probem wth resource constrants JEL cassfcaton: C610 MSC2010: 90B06; 90C11; 90C90 Artícuo recbdo e 22 de febrero de 2011 y aceptado e 8 de uo de 2011 5
Branch-and-Prce y generacón heurístca de coumnas para e probema generazado de rutas de trenes de carretera RESUMEN E probema generazado de rutas de trenes de carretera (generazed truckand-traer routng probem, GTTRP) consttuye un modeo unfcado para probemas de rutas de vehícuos con remoques y asgnacón fa camónremoque E GTTRP es una generazacón de truck-and-traer routng probem (TTRP), que es una extensón de conocdo probema de rutas de vehícuos (vehce routng probem, VRP) En e GTTRP, a fota de vehícuos consste en camones sn remoque (camones soos) y trenes de carretera Agunos centes pueden ser vstados excusvamente por un camón soo o un camón sn su remoque, otros pueden ser vstados tambén por un tren de carretera Además de as ubcacones de os centes hay otro tpo de ocazacón amada ubcacón de trasbordo Aí os remoques pueden ser aparcados, y es posbe efectuar un trasbordo de carga desde un camón a su remoque En este trabao se presentan dos modeos de programacón nea entero mxto (MIP) Además, se descrben un agortmo exacto branch-and-prce y varantes heurístcas de este agortmo Se presentan y anazan estudos computaconaes con os agortmos Se usan probemas generados aeatoramente, dseñados para semear stuacones reaes, y probemas TTRP de a teratura Los resutados muestran que, utzando un agortmo heurístco basado en generacón de coumnas, se pueden resover probemas de estructura y tamaño rea en poco tempo y con soucón de ata cadad Paabras cave: rutas de vehícuos; trasbordo; panfcacón de fotas; probema de camno más corto con mtacones de recursos Casfcacón JEL: C610 MSC2010: 90B06; 90C11; 90C90 6
1 Introducton The generazed truck-and-traer routng probem (GTTRP) s a generazaton of the truckand-traer routng probem (TTRP) (a term ntroduced by [8]), whch tsef s an extenson of the we-known vehce routng probem (VRP) ([10], [43], [19]) The GTTRP consttutes a unfed mode for vehce routng probems wth traers and a fxed orry-traer assgnment The GTTRP can be descrbed as foows There s a set of customers wth a known, determnstc suppy of a snge good, and a set of transshpment ocatons used for parkng and/or oad transfer Vstng a transshpment ocaton ony ncurs the dstance-dependent cost for the resutng detour We assume that both the parkng and oad transfer poses no addtona cost A customers and a transshpment ocatons may have a tme wndow assocated wth them Each customer must be vsted exacty once The customers suppes must be coected and transported to a centra depot To ths end, a feet of heterogeneous vehces wth mted oadng capacty s avaabe There are four man crtera n whch the vehces dffer: Frst of a, the feet s comprsed of snge orres and of orry-traer combnatons (LTCs) (see Fgure 1) Snge orry Lorry-traer combnaton (LTC) LTC orry Traer Fgure 1: GTTRP feet There s a fxed assgnment of an LTC orry to ts traer (and vce versa), e, each traer may be pued by ony one orry, and ony ths orry may perform a oad transfer nto ths traer Second, the vehces may have dfferent oadng capactes: Evdenty, snge orres have fewer oadng capacty than orry-traer combnatons In addton, aso vehces of the same type (that s, snge orres or LTCs) may have dfferent capactes For exampe, a snge orry may have two axes, mtng the oadng capacty (payoad) to approxmatey 10 tons, another may have three axes, eadng to a oadng capacty of about 15 tons (see aso Secton 52) For LTCs, two capactes are reevant, the orry capacty and the traer capacty Thrd, because of dfferent szes and tota weghts, the vehces may have dfferent fxed and dstance-dependent costs Fourth, the vehces may be subect to accessbty constrants, e, they may not be aowed to vst some customers Such constrants are mosty due to mted manoeuvrng space on customer premses or to bad road condtons, especay n wnter The drvng speeds are assumed to be dentca for a orres, whether or not they pu a traer, and whether or not they carry oad An LTC orry need not use ts traer at a and may smpy eave t at the depot Aso, t can decoupe and re-coupe ts traer at any transshpment ocaton A vehces are ntay ocated at the depot, and a vehces must return to the depot at the end of ther tour That s, a traer cannot be eft behnd at a transshpment ocaton by ts orry, t must 7
aso be pued back to the depot Each vehce may perform at most one tour If, n a souton, a vehce s not used, the vehce s fxed cost s not added to the obectve functon vaue of the souton Ths s for two reasons On the one hand, for a tactca pannng probem, where the feet sze and mx are to be determned, not usng a vehce n a souton means that the vehce need not be acqured or rented, so that the fxed cost s actuay not ncurred On the other hand, for an operatve pannng probem, not usng a vehce n a souton means that the vehce s avaabe for other purposes Moreover, n both a tactca and an operatve settng, usng fxed vehce costs s a way to reduce the overa number of vehces used, an obectve often pursued n vehce routng probems Customers whch can ony be vsted by a orry wthout a traer are caed orry customers The other customers, whch can be vsted by a snge orry, an LTC orry, or an LTC, are caed traer customers Spt coecton s not aowed, so there are no customers wth a suppy exceedng the capacty of the argest snge or LTC orry (or LTC, n the case of traer customers) The snge orres are used as n a usua VRP The LTCs are used as foows To vst orry customers, the traer s parked at a transshpment ocaton and the LTC orry does some coectng It then returns to ts traer and ether transfers ts oad to the traer and contnues coectng, or t re-coupes the traer, parks t esewhere, decoupes and does some more coectng before transferrng oad When a traer customer s vsted by an LTC, the suppy of the customer can drecty be oaded competey or party nto the traer In essence, the traers are used as mobe depots and extend the capacty of ther orres A oad transfer between two orres or between two traers s not aowed Ths means that snge orres never vst transshpment ocatons Fgure 2 depcts a possbe route pan wth one snge orry and one LTC The snge orry route s evdent The LTC orry starts at the depot wth ts traer, vsts a traer customer, goes on to a transshpment ocaton, decoupes the traer, vsts two customers, returns to the transshpment ocaton, re-coupes the traer, pus t to a second transshpment ocaton, decoupes t agan (and perhaps performs a oad transfer), vsts two more customers, returns to the second transshpment ocaton, re-coupes the traer and goes back to the depot Two transshpment ocatons n the centre of the fgure are not used Depot Lorry customer Traer customer Transshpment ocaton Snge orry LTC orry LTC Fgure 2: Exampe GTTRP route pan 8
The task s to devse tours mnmzng the sum of fxed and dstance-dependent, that s, varabe, cost over a vehces (some of whch may not be needed), such that the compete suppy of a customers s coected and devered to the depot whe respectng vehce capactes, accessbty constrants, and ocaton tme wndows To see when and why t s sensbe to use LTCs, magne that the suppy of the customers vsted by the LTC orry exceeds the capacty of the argest snge orry Then, wthout LTCs, two snge orry tours are necessary to serve these customers, so that overa, three orres (and, consequenty, three drvers) are requred nstead of two as n the above fgure Ths w ncrease overa fxed costs Moreover, n genera, the overa dstance traveed w ncrease as we, thus ncreasng varabe costs The GTTRP s of consderabe practca reevance Actua and potenta appcatons are fue o devery to prvate househods, food dstrbuton to supermarkets ([42]), raw mk coecton at farmyards ([6], [44], [18], [39]), ntermoda contaner transportaton ([22]), and the so-caed park-and-oop probem encountered n posta devery ([2], [4]) Moreover, many ocaton-routng probems (LRPs, see [28], pp 169 ff, [11], pp 339 ff, [33], [34]) can be modeed as GTTRPs The probem ust descrbed s denoted generazed TTRP, because t generazes the TTRPs consdered n the terature (see next secton) n the foowng ways: Varabe costs for the traers and fxed costs for the vehces are consdered Tme wndows are consdered Traer customer ocatons as we as pure transshpment ocatons can be used for parkng traers and performng oad transfers No other paper cted beow consders these three aspects smutaneousy In partcuar, the presence of pure transshpment ocatons adds an addtona degree of freedom, because vstng a pure transshpment ocaton s optona Hence, a seecton has to be made as to whch transshpment ocatons to use In most varants of the VRP or the VRP wth tme wndows (VRPTW) documented n the terature, wth the notabe excepton of ocaton-routng probems, such a seecton component s not present The remander of the paper s structured as foows In the next secton, the exstng terature s revewed In Secton 3, a mxed-nteger programmng (MIP) mode for the GTTRP, based on bnary arc fow and contnuous resource varabes, s deveoped In Secton 4, branch-andprce agorthms based on a path fow reformuaton of ths mode are presented Computatona resuts wth mpementatons of these agorthms are presented and anayzed n Secton 5 The paper ends wth a bref summary and a research outook n Secton 6 9
2 Lterature Revew The practca reevance of the probem s not adequatey refected by the exstng terature The foowng s a bref revew of the few contrbutons there are [6], [44], and [18] consder ony one transshpment ocaton per traer [6] deveops a heurstc sequenta souton approach for ths type of probem (custerng of customers, determnaton of one transshpment ocaton per traer, routng) [44] proceeds smary [18] presents severa constructon heurstcs and ntraand nter-tour exchange mprovement procedures Aso, the paper mentons an unpubshed workng paper contanng an MIP formuaton [42], [41], [8], and [39] (see aso [40]) aow that traers be parked severa tmes at dfferent ocatons However, they equate the potenta traer parkng ocatons wth the traer customers Ths means that a traer can be parked at any traer customer ocaton but nowhere ese, and that ony one traer can be parked at such a customer ocaton, because the correspondng orry s then assumed to servce the customer [42] consder tme wndows and heterogeneous orres, but dentca traers [42] and [41] aso consder accessbty constrants for the orres [8] and [39] mt the number of avaabe vehces and the ength or duraton of a tour The term Truckand-Traer Routng Probem (TTRP) was coned by [8] and s aso used by [39], [40], [20], [21], [31], [32] [39] extends the approaches of [41] and [8] and consders a mut-perod and a mutdepot verson of the probem [32] consder the case of an unmted number of vehces Contrary to a other papers, [20] and [21] consder the stuaton where the transshpment ocatons are separate ocatons and do not concde wth the traer customer ocatons A authors sove ther respectve probems by sophstcated heurstc procedures, mosty tabu search and smuated anneang [41] and [39] aso gve mathematca programmng formuatons of the probems they consder (0-1 IP modes), but they do not sove ther modes wth an exact method The present paper s based on the technca report [14] In the next subsecton, a network representaton for the GTTRP s presented Interestngy, a smar network was used n [29] for the souton of a ocaton-routng probem 3 A Mxed-Integer Programmng Mode for the GTTRP 31 Network Representaton The subsequent formuaton s based on a tme-space-operaton network D = (V, A) Each vertex n V corresponds to a ocaton n space, an absoute and/or reatve perod of tme, and a type of operaton L:={Depot} L C L T s the set of reevant rea-word ocatons L C := L CL L CLT s the set of customer ocatons, whch s parttoned nto L CL, the set of orry customers, and L CLT, the set of traer customers L T s the set of pure transshpment ocatons For each V, oc L denotes the ocaton correspondng to, [a, b ] [0, T ], s 10
the arrva tme wndow of, where T s the ength of the pannng horzon, and s s the suppy of vertex (whch s zero for depot and transshpment vertces) V s comprsed as foows There s one start depot vertex o and one end depot vertex d, both wth a tme wndow of [0, T ] For each customer, there s one customer vertex Let V C := V CL V CLT be the set of customer vertces, where V CL s the set of orry customers, and where V CLT s the set of traer customers Foowng [8] and [39], t s assumed that the ocatons correspondng to traer customers can aso be used for parkng and transshpment operatons Hence, for each traer customer ocaton and for each (pure) transshpment ocaton, there are n T S 3 vertces v dec v trans,1,, v trans,n T S 2, v coup Let V dec be the set of decoupng vertces, et V trans be the set of transfer vertces, and et V coup be the set of coupng vertces Let V I := V dec V trans V coup be the set of transshpment vertces The dea behnd ths separaton of transshpment ocatons and processes s the foowng: At a vertex n V dec, an LTC orry may decoupe ts traer, and t may aso perform a oad transfer Decoupng vertces can be reached ony wth a traer and eft wthout a traer At a vertex n V trans, an LTC orry may perform a oad transfer to ts traer Transfer vertces can be reached and eft ony wthout a traer At a vertex n V coup, t may re-coupe ts traer, and t may agan perform a oad transfer Coupng vertces can be reached ony wthout a traer and eft ony wth a traer It s assumed that each vertex s vsted by each vehce at most once If an LTC wants to use transshpment ocaton, the LTC must frst vst v dec The traer then moves n tme to the vertces v trans,1,, v trans,n T S 2, v coup, whe the LTC orry vsts customers and fnay re-coupes the traer at v coup The LTC orry need not vst any of the transfer vertces of The LTC orry must not vst any other vertex n V I before havng re-couped ts traer at v coup The formuaton beow takes ths nto account F := F L F LT denotes the set of vehces F L s the set of snge orres, whch do not have a traer F LT s the set of orry-traer combnatons For a k F, qk tota s the tota capacty of a vehce, e, t s the capacty of snge orry k, or, respectvey, the capacty of LTC orry k and ts traer For a k F LT, q orry k s the capacty of LTC orry k, and qk traer s the capacty of LTC orry k s traer A vehces are ntay at the start depot vertex o and end ther tour at the end depot vertex d Snge orres are aowed, n prncpe, to vst orry and traer customer vertces, and LTC orres may, n prncpe, vst a vertces of D Traers can ony reach the transshpment vertces and the traer customers However, as mentoned above, there may be addtona accessbty constrants for certan vehces at certan customers (eg, traer customers wth too much suppy for a snge orry) It s assumed that each LTC uses each transshpment ocaton at most once Thus, for each traer, there are at most n T S transshpment operatons at each transshpment ocaton As for the correct choce of n T S, note that n the worst case, a feasbe souton to the mode may exst ony when, for each traer, there are as many ntermedate and coupng vertces as there are orry customers, because t may be necessary to perform a oad transfer after each vst to a orry customer Consequenty, an optma souton to the mode may ony be an optma souton to an underyng probem nstance when there are as many opportuntes for a oad transfer at, 11
the rght transshpment ocaton as there are orry customers Dependng on the nstance data, better vaues for n T S can be computed However, as w become cear n the next secton, n the branch-and-prce agorthm, the numerca vaue of n T S need not be fxed n advance; rather, ths vaue s determned durng the souton of the prcng probems The arc set conssts of the foowng arcs: (o, d) (o, v c ) and (v c, d) for a customer vertces v c V C (o, v dec ) for a transshpment ocatons L CLT L T (v coup, d) for a transshpment ocatons L CLT L T (v c, v c ) for a customer vertces v c, v c V C wth c c (v dec, v c ) for a transshpment ocatons L CLT L T and a customer vertces v c V C (v trans,, v c ) for a transshpment ocatons L CLT L T, a {1,, n T S 2} and a customer vertces v c V C (v coup, ) for a transshpment ocatons L CLT L T and a vertces (V dec \ {v dec }) V CLT (, ) for a traer customer vertces V CLT and a transshpment vertces V I (, ) for a orry customer vertces V CL and a transfer and coupng vertces V trans V coup For each arc (, ) A, τ tr s the traversa tme τ tr ncudes the servce tme at vertex For a k F, et V k (A k ) be the set of vertces (arcs) that can be reached (traversed) by vehce k For a subsets of V (A) defned hereafter, et the superscrpt k denote the ntersecton of the respectve subset wth V k (A k ) In partcuar, A k LT :=Ak \({(, ) A : V CL A : V CL c orry,k V dec } {(, ) V coup }) s the set of arcs LTC orry k can traverse wth ts traer attached denotes the cost of traversng arc (, ) wth orry k F (for LTC orres, wthout ts traer attached), and c traer,k denotes the addtona cost of pung LTC orry k s traer over arc (, ) On arcs emanatng from the start depot o, except for the arc (o, d), the fxed vehce cost s ncuded n c orry,k o, respectvey, c traer,k o Smar to [8] and [39], a fxed tme for oad transfer s assumed, ndependent of the actua amount of oad transferred Ths s a sensbe assumpton when the setup cost s the determnng factor for the duraton of a transshpment operaton Fgure 3 vsuazes the subnetworks for the dfferent vehce types To keep the fgure cear and concse, there s ony one arc for each arc type present n the subnetwork of the respectve vehce type For exampe, n the LTC subnetwork, one arc from the eft traer customer to the rght one s depcted to ndcate that LTCs can freey move from any traer customer to any other traer customer (uness capacty, tme wndow, or accessbty constrants prohbt ths) The absence of an arc from the rght traer customer to the eft one n the fgure does not mean that there s no such arc n the network 12
Lorry customers Lorry customers o Decoupe Transfer Coupe Traer customers d o Decoupe Transfer Coupe d Traer customers Network vertces Subnetwork for LTCs o Lorry customers Lorry customers Decoupe Trans- Coupe d o Decoupe Transfer Coupe fer Traer customers Traer customers d Subnetwork for LTC orres when traer s at transshpment ocaton Subnetwork for snge orres and LTC orres when traer s not used Fgure 3: Network structure 32 Formuaton The subsequent formuaton uses the foowng varabes: x k {0, 1} k F, (, ) Ak x k = { 1, orry k traverses arc (, ) 0, otherwse y k {0, 1} k F LT, (, ) A k LT y k = { 1, LTC orry k traverses arc (, ) wth ts traer attached 0, LTC orry k does not traverse arc (, ) wth ts traer attached co,k R + k F, V k The tota amount of customer suppes that orry k has coected when reachng vertex, before k starts ts servce at trans,k R + k F LT, V k The tota amount of customer suppes that LTC orry k has transferred to ts traer when reachng vertex, or, equvaenty, the oad of LTC orry k s traer when k reaches vertex (wth or wthout ts traer attached), before k starts ts servce at t k R + k F, V k The pont n tme when orry k starts ts servce at vertex The resutng mode s: 13
(GTTRP): (h,) A k x k h mn k F (v coup,) A k LT (v coup,) A k LT c orry,k (,) A k k F x k + k F LT (,) A k LT c traer,k y k subect to (1) (h,) A k x k h = 1 V C (2) (h,) A k x k h 1 k F LT, V dec (3a) x k v coup (h,v dec ) A k LT x k hv dec = 0 k F LT, L CLT L T (3b) x k v coup 0 k F LT, L CLT L T, V trans, oc = (3c) x k d = 1 k F (3d) (,d) A k yd k = 1 k F LT (3e) (,d) A k LT x k h x k = 0 k F, V k \ {o, d} (3f) (h,) A k (,) A k yh k y k = 0 k F LT, VC k LT (3g) (h,) A k LT (,) A k LT y k x k k F LT, (, ) A k LT \ {(o, d)}, / V coup, / V dec (4a) x k h = yk h k F LT, V dec, (h, ) A k LT (4b) x k = y k k F LT, V coup, (, ) A k LT (4c) co,k trans,k trans,k x k = 1 co,k x k = 1 trans,k x k = 1 trans,k x k = 1 y k = 0 trans,k y k = 1 trans,k co,k k F LT, V k (5a) q orry k k F LT, V k (5b) + s co,k k F, (, ) A k (5c) trans,k k F LT, (, ) A k (5d) trans,k k F LT, V CL, (, ) A k (5e) trans,k k F LT, V CLT, (, ) A k (5f) trans,k + s k F LT, V CLT, (, ) A k LT (5g) x k = 1 t k + τ tr t k k F, (, ) A k (6a) t k v trans, t k v dec t k v trans,+1 t k v trans,n T S 2 t k v trans,1 k F LT, L CLT L T (6b) k F LT, {1,, n T S 3}, L CLT L T (6c) t k v coup k F LT, L CLT L T (6d) 14
t k v coup T (h,v dec ) A k LT x k hv dec k F LT, L CLT L T (6e) x k {0, 1} k F, (, ) A k (7a) y k {0, 1} k F LT, (, ) A k LT (7b) 0 co,k 0 trans,k q tota k k F, V k (7c) q traer k k F LT, V k (7d) a t k b k F, V k (7e) The obectve functon (1) mnmzes tota fxed and varabe costs over a orres and a traers Seen from a coumn generaton perspectve, constrants (2) are nkng constrants affectng more than one vehce wthn each snge constrant, (3) and (4) are non-nkng or ndependent fow constrants, (5) and (6) are non-nkng constrants specfyng the update of the oad and tme resource varabes respectvey, and (7) determne the ranges of the varabes Wth regard to contents, the constrants have the foowng meanng: (2) are the customer coverng constrants statng that each customer must be vsted exacty once (3a) (3c) make sure that each transshpment ocaton s used at most once by each k F LT, and that ony ntermedate and coupng vertces of those transshpment ocatons are used whose decoupng vertex s vsted (3d) and (3e) requre that each vehce reach the end depot (possby va the arc (o, d)) (3f) and (3g) are fow conservaton constrants: They ensure that f a vehce enters a vertex (except for the start and the end depot vertex), the vehce aso eaves ths vertex (4a) (4c) are the constrants nkng the routng of the traer to that of ts orry Note that t s possbe that an LTC orry does not use ts traer, n whch case the atter moves drecty from o to d (5a) requre that the oad of the traer be at most equa to the tota amount of suppy coected (5b) requre that the amount of suppy coected be not greater than the orry capacty and the amount of suppy aready transferred to the traer Wthout constrants (5a), trans,k coud aways be set equa to the traer capacty Because of constrants (5b), co,k woud then be bounded from above ony by the overa vehce capacty, and not by the orry capacty and the amount of suppy aready transferred Ths woud cty enarge the orry capacty A orry coud then park ts traer before dong any coectng and coect as much suppy as the overa vehce capacty permts, wthout havng to transfer any oad (5c) are the oad update constrants The tota amount of oad coected ncreases (at east) by the suppy of each vertex vsted Smary, constrants (5d) (5g) are for the update of the transshpment varabes, respectvey, the traer oad varabes (5d) state that the traer oad s non-decreasng (5e) state that the traer oad s non-ncreasng at orry customer vertces (5f) state that at traer customer vertces, the traer oad of an LTC k can ony ncrease f the vertex s vsted by the orry and ts traer (5g) state that at a traer customer vertex, the traer oad does not ncrease by more than the customer s suppy Constrants (6a) are the constrants for the update of the tmng varabes: The overa tme en route for a vehce after 15
traversng an arc ncreases (at east) by the trave tme aong the arc (6b) (6e) are needed to make sure that the vertces of a transshpment ocaton are vsted n the correct order, e, that decoupng vertces of a transshpment ocaton are vsted before ther correspondng coupng vertex Wthout these constrants, t woud be possbe to vst decoupng vertex v dec 1, go to coupng vertex v coup 2, then to decoupng vertex v dec 2 and then to v coup 1 Note that f an LTC orry does not vst a transshpment ntermedate vertex for some, the tme varabe for ths vertex can be fxed appropratey; t s not requred to have a vstng tme of zero for non-vsted vertces To mprove readabty, constrants (5c) (6a) are wrtten as ogca mpcatons However, these can easy be nearzed usng a Bg-M technque wthout ntroducng addtona varabes (see [46]) Hence, the convex hu of a ponts fufng (2) (7) s a bounded poyhedron The ony exstng formuatons for the TTRP, by [41] and [39], are conceptuay dfferent from the above formuaton [39] uses bnary three-ndex arc fow varabes smar to the above y k varabes, and fve-ndex varabes ndcatng whether a certan orry traverses a certan arc on the nth subtour startng at a certan traer customer or at the depot, where n s the number of customers (n the worst case, as many snge orry (sub-)tours startng at a traer customer or at the depot are necessary as there are customers) [41] uses smar varabes, but requres that at most one subtour orgnate at each traer customer, and hence the second varabe type has ony four ndces n hs mode Nether author uses resource varabes; rather, both formuatons are pure 0-1 IPs In the GTTRP, severa orry-traer combnatons may use the same transshpment ocaton Hence, the GTTRP formuaton (1) (7), usng the arc set defned n Secton 31, s a reaxaton of the TTRP modes from the terature To make sure that ony the orry whch vsts a certan traer customer vertex h uses the correspondng ocaton for parkng and oad transfer, the arc set can be restrcted by removng a arcs enterng the correspondng decoupng vertex, except for the arc (h, ) 4 Branch-and-Prce Agorthms for the GTTRP For capactated vehce routng probems (wth tme wndows), many authors have devsed formuatons and souton approaches based on the nteger programmng verson of Dantzg-Wofe decomposton, e, branch-and-prce Standard references focused on the methodoogy ncude [3], [45], and [30] [12] present a unfed mode for the souton of tme-constraned vehce routng and schedung probems by branch-and-prce In ths secton, a reformuaton of (1) (7) for use n a branch-and-prce agorthm s deveoped The subsequent exposton foows [12] 16
The usua decomposton approach for VRPs can aso be apped to the GTTRP The nkng constrants, e, the customer coverng constrants (2), go n the master program, the non-nkng constrants (3) (7) defne the sub- or prcng probems 41 The Master Program The mode presented n Secton 3 uses arc fow varabes for each reevant combnaton of orry or traer k and arc (, ) Branch-and-prce approaches for vehce routng probems use path fow varabes n the master program: There s one varabe for each feasbe o-d-path of each vehce As for VRPs wthout traers, aso for the GTTRP, each feasbe tour of a snge orry s an eementary path from the start depot vertex to the end depot vertex through the respectve subnetwork Ths s aso true for the LTC orres However, the movements of a traer n the network D of Secton 3 do not consttute a path: There are no traer fow varabes eavng decoupng vertces or enterng coupng vertces Ths, though, s ony to avod unnecessary varabes and constrants In the rea word, the tnerary of a snge orry or a traer resutng from a souton to the above formuaton s an eementary path, too, whereas the tnerary of an LTC orry usng ts traer s not eementary; t w contan cyces startng and endng at the transshpment ocatons where the traer s parked It foows from the fow decomposton theorem (see, for exampe, [1], p 80 f) that the extreme ponts of the convex hu of a ponts fufng the prcng probem constrants (3) (7) correspond to orry paths from o to d n D, some of whch may be non-eementary, because D s not acycc For LTCs, constrants (3d) (4c) addtonay mpy that the extreme ponts represent a path-ke structure consstng of an o-d-path for the orry and one or more paths for ts traer The unon of these traer paths s a subset of the orry path These extreme ponts are descrbed by fow and resource vectors (x k p, y k p, co,k p, trans,k p, t k p) = (x k p, yp, k co,k p, trans,k p, t k p) k F, p P k, (, ) A k, (8a) where P k s the set of extreme ponts for snge orry or LTC k, and where, for smpcty of notaton, n the foowng y k p = y k p := 0 k F L, and trans,k p = trans,k p := 0 k F L Any souton satsfyng (3) (7) can be expressed as a convex combnaton of these extreme ponts: x k = p P k x k pλ k p k F, (, ) A k (8b) x k {0, 1} k F, (, ) A k (8c) y k = ypλ k k p p P k k F LT, (, ) A k LT (8d) 17
y k {0, 1} k F LT, (, ) A k LT (8e) co,k = p λ k p k F, V k (8f) trans,k = p P k co,k p λ k p k F LT, V k (8g) p P k trans,k t k = p P k t k pλ k p k F, V k (8h) p P k λ k p 1 k F (8) λ k p 0 k F, p P k (8) The nteger master program (IMP) s then: λ k p + k F p P k c orry,k x k p (,) A k k F LT p P k (,) A k LT c traer,k y k p λ k p mn (9a) k F L p P k subect to λ k p = 1 L C (9b) x k hp VC (h,) A k λ k p 1 k F (9c) p P k λ k p 0 k F, p P k (9d) λ k p {0, 1} k F, p P k (9e) In genera, t s not true that bnary restrctons on the orgna x k arc fow varabes can be repaced by bnary restrctons on the λ k p path fow varabes However, ths statement hods for the above GTTRP reformuaton, snce the defnton of the orgna master probem souton space, e, the constrant set (2), nvoves ony the x k varabes Hence, bnary requrements on the path varabes are equvaent to bnary requrements on the x k varabes, see [12], p 75 Moreover, f the x k varabes are bnary, the yk varabes w be bnary, too Ths s because for any path of an LTC orry k represented by the vaues of the x k varabes, the vaues of the pertnent y k varabes are unequvocay determned: If the path vsts transshpment vertces, a parta paths endng at a decoupng vertex or startng at a coupng vertex have a y k varabes equa to one, and a parta paths between a decoupng and ts correspondng coupng vertex have a y k varabes equa to zero Paths vstng ony customers and at east one orry customer trvay have a y k varabes equa to zero, and paths vstng ony traer customers have a yk varabes equa to one, uness the tota demand of the customers on the path does not exceed the orry capacty, n whch case the correspondng traer w not be used 18
42 The Prcng Probems As t s the case for a vehce routng probems, aso the prcng probems for the GTTRP are (eementary) shortest path probems wth resource constrants ((E)SPPRCs) (see [24]) When the feet s heterogeneous, n each prcng step, a prcng probem must be soved for each vehce Because of the dua prces comng from the master probem, the prcng probem networks usuay have negatve cost cyces Ths makes the probem N P-hard n the strong sense (bd, p 38) There s a consderabe amount of terature on the souton of (E)SPPRCs It s beyond the scope of ths paper to gve a detaed revew, but some mportant contrbutons are [16], [37], [38], [36], [5], [7], [25] The standard souton technque for (E)SPPRCs s a abeng agorthm based on dynamc programmng In prncpe, such an agorthm works smar to a abeng agorthm for shortest path probems wthout resource constrants, eg, the Dkstra agorthm (see [1], pp 93 ff) The basc concepts used n such an agorthm are the foowng (see [24]) A resource s an arbtrary scaed one-dmensona quantty that can be determned or computed at the vertces of a drected wak n a network Exampes are cost, tme, oad, or the nformaton Is LTC orry k currenty pung ts traer? The vaue of a resource at a vertex s stored n a resource varabe An arbtrary scaed resource s constraned f there s at east one vertex n the network where the assocated resource varabe must not take a possbe vaues A cardnay scaed resource s constraned f there s at east one vertex n the network wth a fnte upper or ower bound on the vaue of the resource The resource wndow of a nomnay scaed resource r at a vertex s the set of aowed vaues of r at ths vertex The resource wndow of a cardnay scaed resource r at a vertex s the nterva [b r, ubr ] ], + [ A resource extenson functon (REF ) s defned on each arc n a network for each resource consdered An REF for a resource r maps the set of a possbe vectors of resource vaues at the ta of an arc to the set of possbe vaues of r at the head of the arc More precsey, et R := (σ 1,, σ R ) T be a vector of (vaues of) resource varabes Then, an REF for a resource r s a functon f r : A R R R For smpcty, et f r (R) := f r (((, ), R)) An REF for a cardnay scaed resource r ndcates (ower bounds on) the consumptons of r aong the arcs When seekng a path from an orgn vertex o to a destnaton vertex d, parta paths from o to a vertex d are extended aong a arcs (, ) emanatng from to create new, extended paths For each o--path, there s a correspondng abe resdent at that stores the vaues of a resource varabes at for ts path, aong wth the nformaton on how t was created: the arc (h, ) over whch was reached and (a reference to) the abe of the o-h-path whose extenson aong (h, ) yeded the o--path (ths makes t easy to reconstruct the path correspondng to a abe) Intay, there s exacty one abe correspondng to the path (o) Wog, the vaues of the resource varabes of the nta abe are a set to the ower bounds of ther respectve resource wndows at o A abe s feasbe ff the vaue of each resource varabe n s wthn the resource wndow of ts respectve resource If a abe s not feasbe, t s dscarded An extenson of a path/abe aong an arc (, ) s feasbe ff the resutng abe at s feasbe An h--path s feasbe ff, for each arc (, ) n the path, a 19
feasbe abe at exsts whch can be extended aong (, ) to a feasbe abe at To keep the number of abes as sma as possbe, t s very mportant to perform a domnance procedure to emnate feasbe but unnecessary abes A abe 1 domnates a abe 2 ff both resde at the same vertex, the vaue of the resource varabe of each nomnay scaed resource n 1 s equa to the correspondng vaue n 2, the vaue of the resource varabe of each cardnay scaed resource n 1 s better (ess or greater, dependng on the resource) than or equa to the correspondng vaue n 2, and the vaue of the resource varabe of at east one cardnay scaed resource n 1 s strcty better than the correspondng vaue n 2 Domnated abes are dscarded as we The resources used n the abeng agorthm for vehce k are: an unconstraned, cardnay scaed resource for cost one cardnay scaed resource for each of the three resource varabes used n (1) (7), e, for coected oad, transferred oad, and tme one cardnay scaed vstaton counter resource for each customer one nomnay scaed auxary resource for each traer In the foowng, these resources are descrbed n deta The auxary resource s presented frst, because t s needed to descrbe the REFs for cost and oad The auxary resource s prmary needed for the correct modeng of the routng ogc at transshpment ocatons It must be consdered that, on ts tnerary, an LTC orry must vst the decoupng, transfer, and coupng vertces of each transshpment ocaton n the correct order (or not at a), and that, after an LTC orry has vsted the decoupng vertex of transshpment ocaton, t must not vst any decoupng, transfer or coupng vertex of any other transshpment ocaton before havng vsted the coupng vertex of To ths end, one nomnay scaed resource r tp s used to mode the traer ogc r tp ( traer poston ) records the current poston of the traer by means of a resource varabe σ tp, whch ndcates the poston of the traer when the LTC orry reaches vertex σ tp s ether zero, meanng that the LTC orry s currenty pung ts traer, or equa to the vertex number of the decoupng vertex where the traer was parked The correspondng REF s f rtp wth σ tp, V C V trans {d} f rtp (σ tp ) =, V dec (, ) A, (10) 0, V coup and the resource wndows at a vertex are as shown n Tabe 1 20
Resource wndow for {0} {o, d} V dec {1,, V } V CL {0,, V } V CLT {} V trans V coup, V dec, oc = oc Tabe 1: Resource wndows for r tp at vertex Addtonay, r tp serves the foowng purposes: It s used to mode the oad transfer ogc at traer customer and transshpment ocatons (see beow) It s used for the update of the vstaton counter resources (see beow) It does not ony determne the feasbty of an extenson of a path aong an arc, t s aso reevant for (exact) domnance consderatons, as a abe can ony domnate another abe f ther respectve current traer postons are equa (see beow) It s used to consder the traer costs n the prcng probems for LTCs, e, to express the y k varabes: If LTC orry k reaches vertex va arc (, ) wth ts traer attached, the c traer,k costs are ncurred Ths s regstered n the prcng probems by checkng whether σ tp = 0 For each of the resource varabes used n formuaton (1) (7), there s one constraned resource For the tota amount of customer suppes that orry k has coected when reachng vertex, the resource r co wth resource varabe σ co and REF f rco wth s used f rco f rco modes constrants (5c) and sets σ co (σ co ) = σ co + s (, ) A (11) these constrants The resource wndow at a vertex s [0, q tota k ] to the owest nonnegatve vaue fufng For the tota amount of customer suppes that LTC orry k has transferred when reachng vertex, the resource r trans wth resource varabe σ trans and REF f rtrans wth f rtrans (σ trans, σ co, σ tp ) = σ trans, V CL ( V CLT σ tp ) σ trans σ trans + mn{s, q traer k + mn{σ co σ trans }, V CLT σ tp = σ trans, qk traer σ trans }, V I (, ) A (12) s used f rtrans modes constrants (5a), (5b), and (5d) (5g) and sets σ trans to the hghest possbe vaue that fufs these constrants The resource wndow at a vertex s [0, q traer k ] 21
Ths means that an LTC orry aways transfers as much oad as possbe to ts traer (ts compete oad or an amount of oad equa to the resdua capacty of ts traer, whchever s ess) Ths s a sensbe stpuaton, as t was assumed above that the tme needed for a oad transfer s fxed, ndependent of the actua amount of oad transferred If ths assumpton s abandoned, there s a trade-off between transferrng few oad and savng tme on the one hand, and transferrng much oad and ganng orry capacty on the other hand In the former case, t s possbe to vst more customers before the end of ther tme wndows, but the orry capacty may be nsuffcent to do so In the atter case, t s possbe to vst more customers before havng to perform the next oad transfer, but perhaps t s no onger possbe to arrve at some customers before the end of ther tme wndows Correcty treatng such a trade-off s non-trva It s beyond the scope of ths paper to eaborate further on ths ssue The reader s referred to [23] For the pont n tme when orry k begns ts servce at vertex, the resource r tme wth resource varabe σ tme and REF f rtme wth f rtme (σ tme ) = σ tme + τ tr (, ) A (13) s used f rtme modes constrants (6a) and sets σ tme to the owest nonnegatve vaue fufng these constrants The resource wndow at a vertex s [a, b ] The branch-and-prce agorthm n the present paper soves the eementary SPPRC n each prcng step For the usua VRP or VRPTW, to sove the ESPPRC by dynamc programmng, t s necessary to have a bnary vstaton counter resource for each customer (vertex) Ths s aso the case for the GTTRP In the course of the agorthm, for a abe resdng at a vertex v, ths resource s set to zero for a customer c f t s not or no onger possbe to extend such that c s vsted after v on any extenson of Ths w mosty be the case because c has aready been vsted, but possby aso because of tme wndow, capacty and accessbty constrants To be precse, for each v c V C, there s one bnary resource r vst,v c wth resource varabe σ vc at each vertex V and REF f rvst,vc wth f rvst,vc (σ v c, σ co, σ trans, σ tme, σ tp ) = 0, v c = not enough capacty not enough tme σ v c, otherwse (, ) A (14) where σ vc = 0 means that customer v c cannot be vsted any more, and where not enough capacty means σ co + s > q tota k (v c σ co (σ tp σ co (v c σ tp + s + s vc > q tota k ) σ trans σ co + s > q orry k ) σ trans + s + s vc > q orry k ) 22
and not enough tme means σ tme + τv tr c > b vc, where τhh tr := 0 h V The resource wndow for vertex at vertex s [1, 1], and the resource wndow for a vertex at vertex s [0, 1] When the prcng probems are soved by dynamc programmng, t s not necessary to ntroduce more than one ntermedate vertex n the LTC subnetworks It s suffcent to have ony one, and to aow mutpe vsts to ths vertex Vstaton counters are mantaned ony for the customers, not for the transshpment vertces Cyce emnaton can be performed wth vstaton counters ony for the customers, as every possbe cyce n the subnetworks (as we as n the network correspondng to the compact formuaton) contans a customer It s then aso possbe to use a transshpment ocaton more than once, whch may be reevant dependng on the tme wndows Ths s an addtona degree of freedom compared to the compact formuaton Indeed, when the prcng probems are soved by dynamc programmng, t woud be possbe to use ony one vertex per transshpment ocaton Athough ths woud reduce the subnetwork sze for the LTC subprobems, the number of possbe abes, whch determnes the dffcuty of an (E)SPPRC, woud reman the same Fnay, there s one unconstraned resource r cost measurng the cost The correspondng resource varabe s σ cost, and the REF s f rcost wth f rcost (σ cost ) = σ cost + c k (, ) A, (15) where c k s the reduced cost of traversa of arc (, ) for vehce k Introducng dua varabes (α T, γ T ) := (α 1,, α L C, γ 1,, γ F ) (16) for constrants (9b) and (9c), the obectve functon of the prcng probem for vehce k can be wrtten as (,) A k c orry,k x k + (,) A k LT c traer,k y k L C V C (h,) A k α x k h γk, (17) and the reduced costs of the arcs n the prcng probem for a orry-traer combnaton k can be stated as n Tabe 2 The reduced cost of the arcs n the prcng probem for a snge orry k are as shown n Tabe 3 c k for c orry,k α (, ) / A k LT, V k C, oc = c orry,k + c traer,k max(1 σ tp, 0) α (, ) A k LT, V k C LT, oc = (contnued on next page) 23
(contnued from prevous page) c k for c orry,k + c traer,k V dec c orry,k V trans V coup c orry,k d + c traer,k d γ k = d Tabe 2: Reduced cost of arcs n the GTTRP prcng probem for a orry-traer combnaton k c k for c orry,k α (, ) A k, V k C, oc = c orry,k d γ k = d Tabe 3: Reduced cost of arcs n the GTTRP prcng probem for a snge orry k Wth the above resource specfcatons, a feasbe abe 1 domnates a feasbe abe 2 f and ony f both resde at the same vertex, σ tp ( 1) = σ tp ( 2), σ cost ( 1 ) σ cost ( 2 ), σ co ( 1 ) σ co ( 2 ), σ trans ( 1 ) σ trans ( 2 ), σ tme ( 1 ) σ tme ( 2 ), σ v c ( 1 ) σ v c ( 2 ) v c V C, and at east one of the above nequates s strct, where σ r () denotes the vaue of the resource varabe σr for a abe resdent at vertex The vaue of the cost resource of a abe resdent at the end depot vertex d ndcates the reduced cost of the path represented by the abe The reduced cost of a path p s c k p(α, γ) = c orry,k x k p + c traer,k yp k (,) A k L C (h,) A (,) A k LT V C x k hp α γ k (18) If a path p s feasbe and f (18) s negatve, a new coumn correspondng to p can be added to the restrcted master probem The path tsef s recursvey reconstructed from the, because each abe stores ts drect predecessor arc and predecessor abe As expaned n Secton 41, the traer (sub)path(s) of a path of an LTC orry s/are aso unequvoca and can easy be 24
reconstructed, so, the obectve functon coeffcent of a new coumn n the restrcted master probem can be determned effcenty, too The network D for the compact formuaton (1) (7) contans an arc (o, d) that can be traversed by a traer wthout beng pued by ts orry to mode the fact that an LTC orry may aso be used wthout ts traer Ths must be taken nto account by sovng a prcng probem ESPPRC for each LTC, for each snge orry, and for each LTC orry for whch there s no snge orry wth dentca propertes However, f severa vehces are dentca, the resutng symmetry n the probem can be expoted Ths can be done as descrbed n [12], p 84 ff 5 Computatona Experments 51 Agorthms Extensve studes of possbe set-ups for branch-and-prce agorthms were performed In partcuar, many computatona experments were performed to determne usefu strateges for the souton of the prcng probems The am was to fnd successfu procedures for the exact as we as the heurstc souton of the prcng probems, and, hence, the overa probem For standard VRP(TW)s, t has proved usefu for computatona purposes to perform heurstc prcng, e, to sove the ESPPRC heurstcay as ong as negatve reduced cost paths are found, and to sove the ESPPRC exacty ony when heurstc procedures fa to fnd any more negatve reduced cost paths (see, for exampe, [37]) In vew of ths, the foowng exact approach turned out to be acceptabe The prcng probems are soved n three stages: () Intay, the ESPPRC s soved heurstcay consderng the vstaton counters ony n the REFs (to ensure that ony eementary paths are returned), but competey dsregardng the vstaton counters n the domnance procedure In ths way, fewer abes are created, because the domnance procedure becomes much stronger (ndeed, t becomes too strong, and not a undomnated negatve reduced cost paths w be found: It s possbe that a path s now domnated by another path athough both cover dfferent customers) Ths frst stage s performed for each vehce cass on a reduced subnetwork where ony the m s shortest arcs n the forward star of each vertex are consdered (m s = 3 for nstances wth ess than 15 customers, and m s = 5 otherwse) () If no negatve reduced cost path s found, the ESPPRC s soved heurstcay (e, gnorng the vstaton counters n the domnance step as before) for each vehce cass on the entre network () Ony f ths aso does not yed a negatve reduced cost path, the ESPPRC s soved exacty (e, wth the vstaton counter resources of a customers consdered n the domnance step) for each vehce cass on the entre network 25
By performng each stage for each vehce cass, emphass s put on qucky returnng many negatve reduced cost coumns To exacty sove the prcng probems, aso bounded bdrectona dynamc programmng as descrbed n [38] was used Ths yeded moderatey better resuts than the undrectona approach for the arger nstances of the test bed descrbed beow Incrementa state space augmentaton as descrbed n [38] and [5] was aso tred, but dd not ead to good resuts For more detas on experments wth exact agorthms see [13] The foowng branchng strategy was used: () () branch on the number of tours branch on an aggregated arc fow varabe (e, aggregated over a vehces or vehce casses) for an arc whose head and/or ta s a customer vertex () branch on an arc fow varabe for a vehce or vehce cass for an arc whose head and/or ta s a customer vertex It s suffcent to consder the x k varabes for the branchng decsons As descrbed n Secton 41, once these are a bnary, the path varabes w be bnary as we It s aready suffcent that a x k varabes where or are customer vertces are ntegra Ths s because each customer s vsted exacty once Ths aso hods f there are dentca prcng probems, agan because of the constrant that each customer s vsted exacty once For the souton of the GTTRP wth a heurstc approach based on branch-and-prce or coumn generaton, aso many dfferent settngs were nvestgated These settngs were combnatons of heurstc soutons to the prcng probems Many dfferent heurstc souton strateges for ESPPRCs are descrbed, for exampe, n [37], p 207 f Most of the heurstc strateges tred yeded sma computaton tmes, but a qute bad souton quaty The foowng three strateges (henceforth referred to as H-1, H-2 and H-3) turned out to offer the best trade-off between these two confctng obectves: () () Sove the subprobems by competey dsregardng the vstaton counter resources for a customers n the domnance procedure, as n the frst two stages of the three-stage exact approach Sove the subprobems by dsregardng the vstaton counter resources for a customers and aso gnorng the traer ocatons of the abes, e, gnorng the requrement that a abe 1 can ony domnate a abe 2 f σ tp ( 1) = σ tp ( 2) () Use the same domnance as n (), but sove the probem ony at the root vertex of the branch-and-bound tree, e, perform heurstc coumn generaton After no more coumns are found at the root vertex, sove a set coverng probem wth the generated coumns If, n the souton of the set coverng probem, a customer s vsted more than once, keep the customer ony n the tour where the savng n tour ength obtaned by removng the customer from the tour s mnma over a tours where the customer s vsted, and remove the customer from a other tours 26
A three heurstc approaches aso frst soved the subprobems on thn graphs, as n the exact approach 52 Test Instances The appcaton context that motvated ths research s, as n severa other papers descrbed n the terature revew, raw mk coecton at farmyards (n ths case n southern Bavara, Germany) The am was to be abe to sove reastc nstances Therefore, the test nstances were devsed so as to resembe rea-word stuatons wth respect to number of customers and transshpment ocatons, customer suppes, and vehce costs and capactes In practce, the foowng types of vehce are used: Snge orres: snge orry wth two axes and a capacty of 10 tons snge orry wth three axes and a capacty of 15 tons Lorry-traer combnatons: two-axe orry wth three-axe traer, wth a capacty of 10 and 15 tons, respectvey ( 2/3- combnaton ) three-axe orry wth two-axe traer, wth a capacty of 15 and 10 tons, respectvey ( 3/2- combnaton ) The cost data used are shown n Tabe 4 These data approxmatey refect absoute vaues of the dfferent cost types for each vehce type as we as ratos of cost of one vehce type to another Two- and three-axe snge orres have the same cost data as two- and three-axe orres of a orry-traer combnaton Cost type Vehce type Fxed Dstance-dependent Two-axe orry 18,000 65 Three-axe traer 2,500 6 Three-axe orry 20,000 70 Two-axe traer 2,000 4 Tabe 4: Cost data for computatona experments The transfer of mk from one tank s fast, comfortabe and cean and requres no specfc ocaton for ther operaton, t s performed by the drver hmsef wthout mposng any addtona cost Therefore, oad transfer performance ony nvoves tme costs The customer and transshpment ocatons were randomy seected on a 100 100 km grd wth the depot ocated n the centre The resutng Eucdean dstances between each par of 27
vertces were mutped by a dstance factor of 13 The customer suppes were chosen randomy from [1,000; 10,000] As oad transfer tme, 2 mnutes per 1,000 unts of suppy were assumed throughout The ength of the pannng horzon was assumed to be 12 hours or 1,320 mnutes, respectvey A customers and pure transshpment ocatons have one tme wndow: [0; 1,320] Ths s supposed to mode the fact that n the targeted appcaton context, most customers and transshpment ocatons do not have tme wndows at a, or at east very ong ones (hence, competey gnorng tme wndows woud not be ustfed) For a code capabe of sovng probems wth tme wndows, such nstances represent the worst case and thus consttute a stress test It s cear that wth tghter tme wndows, arger nstances can be soved For each n {6,, 10, 20, 25}, 30 so-caed x y z -nstances were created wth the vehce, cost and dstance data specfed above x stands for the number of orry customers, y stands for the number of traer customers, and z stands for the number of pure transshpment ocatons, and x = y = z = n Ths means that an x y z-nstance has 1 + x + y + 3 (y + z) + 1 vertces: One for the start depot, x for the orry customers, y for the traer customers, three for each transshpment ocaton (of whch there are y +z) to represent decoupng, transfer, and coupng, and one for the end depot The foowng tabe shows the szes of the nstances and the resutng subnetworks The coumn No arcs n VRP ndcates the number of arcs n a standard VRP wth as many vertces as the LTC subnetwork n the same coumn Consderng the number of arcs, a 10 10 10-nstance of the GTTRP hence corresponds to a VRP nstance wth 53 customers Instance Type 6 6 6 10 10 10 20 20 20 25 25 25 No vertces n snge orry subnetworks 14 22 42 52 No arcs n snge orry subnetworks 157 421 1,641 2,551 No vertces n LTC subnetworks 50 82 162 202 No arcs n LTC subnetworks 1,033 2,841 11,281 17,601 No arcs n VRP 2,353 6,481 25,761 40,201 Arc rato VRP/GTTRP 228 228 228 228 Tabe 5: Test nstance szes 53 Computatona Resuts The agorthms of Secton 51 were mpemented n C++ wth the Vsua C++ comper usng the Boost Graph brary (BGL, boostorg), the ABACUS branch-and-cut-and-prce-framework (wwwnformatkun-koende/abacus) wth CPLEX (wwwogcom/products/cpex) as LP sover and the r c shortest paths code from the BGL for the souton of the prcng probems The tests were performed on a 216 GHz processor wth 2 GB of man memory under 28
Wndows XP Professona 32 SP 3 The resuts obtaned are shown n the subsequent Tabes 6 10 (The 20 20 20 and 25 25 25 nstances coud be soved ony wth the H-3 approach) The most mportant observatons to be made n Tabes 6 10 are: Souton quaty: In genera, the souton quaty of a heurstc agorthms s very good The H-1 approach was abe to sove to optmaty more than 95 % of the nstances for whch an optma souton s known The nstance defnng the worst reatve souton quaty (of 1137 %) for the H-1 approach s a unque outer There were ony two nstances where the H-1 approach computed a souton that was more than 025 % worse than the best souton found wth the exact agorthm The H-2 and H-3 approaches dd not fnd many optma soutons, but average devatons of 03 % and 13 % respectvey from the soutons found wth the exact agorthm are st hghy satsfactory For a consderabe percentage of nstances, the heurstc approaches found better soutons than the exact one Ths apparenty paradox resut s due to the fact that for such nstances, the exact agorthm was not abe to expore the branch-and-bound tree deep and wde enough wthn the avaabe computaton tme It s aso noteworthy that the average and maxma engths (n number of arcs) of the tours n the soutons decreased steady from the exact to the most heurstc agorthm Ths mpes that the heurstc agorthms were not abe to fnd the rather ong tours needed to sove some nstances to optmaty Computaton tmes: The computaton tme mts for each agorthm are ndcated n the tabes The maxmum tmes reported sometmes exceed these vaues, because the eapsed tme was checked at the end of each prcng step, not wthn the (E)SPPRC souton agorthm and not wth a second thread The computaton tmes vary wdey for a three branch-and-prce approaches For exampe, the shortest CPU tme for the exact agorthm for a 6 6 6-nstance was two seconds, the ongest was 2,630 seconds (whch s more than 1,000 tmes onger) For H-3, the coumn generaton heurstc, the varaton s much smaer The prcng step s by far the most tme consumng part n the agorthms The number of subprobems to be soved vares wdey for the branch-and-prce agorthms The number of (E)SPPRCs to be soved at each subprobem s much ess varabe Consequenty, the number of subprobems to be soved s the man determnant for the computaton tme of the branch-and-prce agorthms for an nstance 29
Instance Type 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 No nstances soved to optmaty 30 28 25 12 14 No nstances soved at root 12 12 8 5 7 COW tme [s] (Max: 11,100 s) 2 / 268 / 2,630 5 / 1,689 / 11,500 13 / 5,083 / 11,200 36 / 9,216 / 18,700 868 / 10,341 / 19,900 Prcng tme [% COW tme] 89 / 98 / 100 94 / 98 / 100 95 / 99 / 100 97 / 100 / 100 98 / 100 / 100 No subprobems 1 / 97 / 1,535 1 / 136 / 1,565 1 / 825 / 9,021 1 / 202 / 3,943 1 / 293 / 3,503 No (E)SPPRCs per subprobem 21 / 70 / 232 17 / 64 / 148 4 / 55 / 196 1 / 46 / 180 1 / 60 / 208 No abes per (E)SPPRC 3,055 / 9,272 / 23,641 5,730 / 17,215 / 97,205 7,003 / 28,558 / 85,050 14,124 / 72,370 / 147,280 42,289 / 97,557 / 246,438 No generated varabes 237 / 1,893 / 17,505 265 / 2,438 / 21,847 306 / 5,285 / 53,453 426 / 1,500 / 13,893 624 / 1,077 / 2,326 Hghest eve n tree 1 / 8 / 35 1 / 9 / 33 1 / 22 / 161 1 / 17 / 143 1 / 9 / 43 No tours 2 / 31 / 4 3 / 38 / 4 3 / 41 / 5 3 / 42 / 5 4 / 51 / 8 No LTC tours 2 / 25 / 3 2 / 31 / 4 3 / 37 / 5 3 / 39 / 5 3 / 43 / 6 Longest tour [No arcs] 6 / 90 / 12 7 / 90 / 11 7 / 98 / 13 8 / 107 / 14 8 / 103 / 18 Tabe 6: Computatona resuts for exact agorthm (mn / avg / max) Instance Type 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 No nstances soved to optmaty 29 26 24 12 13 No nstances soved at root 13 12 8 5 6 COW tme [s] (Max: 3,900 s) 1 / 30 / 389 3 / 251 / 2,930 5 / 1,236 / 3,910 13 / 1,750 / 3,900 33 / 2,408 / 13,700 Prcng tme [% COW tme] 80 / 91 / 98 89 / 93 / 100 81 / 94 / 100 86 / 97 / 100 88 / 97 / 100 No subprobems 1 / 95 / 2,015 1 / 123 / 1,463 1 / 1,980 / 11,855 1 / 929 / 6,637 1 / 931 / 4,233 No (E)SPPRCs per subprobem 20 / 61 / 164 21 / 59 / 144 8 / 49 / 148 6 / 46 / 156 2 / 47 / 188 No abes per (E)SPPRC 1,718 / 4,717 / 10,198 3,792 / 8,317 / 48,782 4,111 / 10,161 / 45,394 7,518 / 15,972 / 36,605 11,237 / 32,925 / 190,515 No generated varabes 235 / 1,693 / 24,983 265 / 2,754 / 36,791 306 / 16,921 / 75,016 426 / 13,689 / 54,831 625 / 8,582 / 34,054 Hghest eve n tree 1 / 7 / 34 1 / 8 / 33 1 / 20 / 87 1 / 38 / 233 1 / 23 / 203 No tours 2 / 31 / 4 3 / 38 / 4 3 / 40 / 5 3 / 42 / 5 4 / 48 / 6 No LTC tours 2 / 25 / 3 2 / 31 / 4 3 / 37 / 5 3 / 39 / 5 3 / 45 / 6 Longest tour [No arcs] 6 / 89 / 12 7 / 89 / 11 7 / 99 / 13 8 / 104 / 14 8 / 101 / 14 Tabe 7: Computatona resuts for H-1 agorthm (mn / avg / max) 30
Instance Type 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 No nstances soved to optmaty 0 1 0 2 0 No nstances soved at root 13 11 7 4 4 COW tme [s] (Max: 3,900 s) 0 / 1 / 8 0 / 3 / 31 1 / 7 / 36 1 / 158 / 3,900 2 / 404 / 3,900 Prcng tme [% COW tme] 30 / 47 / 81 39 / 54 / 94 44 / 60 / 87 4 / 61 / 88 6 / 62 / 98 No subprobems 1 / 10 / 81 1 / 13 / 79 1 / 35 / 165 1 / 145 / 1,773 1 / 304 / 2,861 No (E)SPPRCs per subprobem 15 / 57 / 116 15 / 58 / 120 14 / 48 / 152 13 / 44 / 160 12 / 47 / 148 No abes per (E)SPPRC 640 / 1,107 / 2,854 969 / 1,793 / 8,953 1,361 / 2,218 / 5,968 1,867 / 3,375 / 7,304 2,135 / 7,102 / 51,936 No generated varabes 154 / 395 / 1,825 233 / 476 / 2,577 268 / 813 / 3,525 371 / 2,405 / 17,520 413 / 5,076 / 47,280 Hghest eve n tree 1 / 4 / 17 1 / 5 / 29 1 / 7 / 18 1 / 17 / 119 1 / 16 / 105 No tours 1 / 30 / 4 2 / 37 / 4 3 / 39 / 5 3 / 40 / 5 4 / 47 / 6 No LTC tours 1 / 26 / 4 1 / 30 / 4 2 / 36 / 5 2 / 37 / 5 3 / 45 / 6 Longest tour [No arcs] 4 / 86 / 11 7 / 84 / 10 7 / 89 / 12 8 / 93 / 11 8 / 93 / 11 Tabe 8: Computatona resuts for H-2 agorthm (mn / avg / max) Instance Type 6 6 6 7 7 7 8 8 8 9 9 9 10 10 10 20 20 20 25 25 25 No nstances soved to optmaty 0 0 0 0 0?? No nstances soved at root 30 30 30 30 30 30 30 COW tme [s] (Max: 3,900 s) 0 / 0 / 1 0 / 1 / 1 0 / 1 / 1 1 / 1 / 3 1 / 2 / 3 16 / 27 / 43 35 / 63 / 107 Prcng tme [% COW tme] 33 / 53 / 75 39 / 59 / 77 51 / 65 / 76 57 / 70 / 76 60 / 69 / 75 70 / 76 / 83 64 / 79 / 85 No subprobems 1 / 1 / 1 1 / 1 / 1 1 / 1 / 1 1 / 1 / 1 1 / 1 / 1 1 / 1 / 1 1 / 1 / 1 No (E)SPPRCs per subprobem 68 / 90 / 116 76 / 99 / 124 68 / 108 / 156 84 / 121 / 160 88 / 122 / 184 148 / 182 / 232 164 / 193 / 236 No abes per (E)SPPRC 714 / 1,055 / 1,908 969 / 1,418 / 2,153 1,173 / 2,113 / 3,615 2,106 / 3,208 / 5,251 2,658 / 3,944 / 6,538 19,549 / 28,090 / 39,953 32,867 / 47,204 / 77,468 No generated varabes 154 / 259 / 438 212 / 291 / 486 226 / 365 / 536 320 / 491 / 805 363 / 560 / 921 1,297 / 1,638 / 2,065 1,558 / 2,128 / 3,428 Hghest eve n tree 1 / 1 / 1 1 / 1 / 1 1 / 1 / 1 1 / 1 / 1 1 / 1 / 1 1 / 1 / 1 1 / 1 / 1 No tours 2 / 31 / 4 3 / 39 / 4 3 / 41 / 5 3 / 43 / 5 4 / 49 / 6 8 / 93 / 11 11 / 118 / 13 No LTC tours 2 / 27 / 4 2 / 32 / 4 3 / 38 / 5 3 / 39 / 5 3 / 45 / 6 7 / 89 / 10 10 / 114 / 13 Longest tour [No arcs] 6 / 88 / 11 6 / 83 / 10 7 / 89 / 11 8 / 94 / 11 8 / 93 / 11 9 / 103 / 12 9 / 100 / 12 Tabe 9: Computatona resuts for H-3 agorthm (mn / avg / max) 31
Exact H-1 Rato H-1 / Exact H-2 Rato H-2 / Exact H-3 Rato H-3 / Exact No nstances soved to optmaty 109 104 954 % 3 28 % 0 00 % No nstances soved at root 44 44 1000 % 39 886 % na na Best reatve souton quaty 763 % 765 % 766 % Avg reatve souton quaty 990 % 1003 % 1013 % Worst reatve souton quaty 1137 % 1075 % 1083 % No tmes best souton was found 124 140 1129 % 8 65 % 4 32 % COW tme [s] 2 / 5,253 / 19,900 1 / 1,135 / 13,700 206 % 0 / 115 / 3,900 20 % 0 / 1 / 3 00 % Prcng tme [% CPU tme] 89 / 99 / 100 80 / 94 / 100 949 % 4 / 57 / 98 576 % 33 / 63 / 77 636 % No subprobems 1 / 307 / 9,021 1 / 811 / 11,855 2641 % 1 / 101 / 2,861 329 % 1 / 1 / 1 na No (E)SPPRCs per subprobem 1 / 59 / 232 2 / 52 / 188 881 % 12 / 51 / 160 864 % 68 / 108 / 184 1831 % No abes per SPPRC 3,055 / 44,391 / 246,438 1,718 / 14,419 / 190,515 325 % 640 / 3,119 / 51,936 70 % 714 / 2,348 / 6,538 53 % No generated varabes 237 / 2,438 / 53,453 235 / 8,728 / 75,016 3584 % 154 / 1,833 / 47,280 752 % 154 / 393 / 921 161 % Hghest eve n tree 1 / 13 / 161 1 / 19 / 233 1490 % 1 / 10 / 119 744 % 1 / 1 / 1 na No tours 2 / 40 / 8 2 / 40 / 6 1000 % 1 / 39 / 6 975 % 4 / 41 / 6 1025 % No LTC tours 2 / 35 / 6 2 / 35 / 6 1000 % 1 / 35 / 6 1000 % 2 / 36 / 6 1029 % Longest tour [No arcs] 6 / 97 / 18 6 / 96 / 14 990 % 4 / 89 / 12 918 % 6 / 89 / 11 918 % Tabe 10: Comparson of computatona resuts over dfferent approaches, nstance casses 6 6 6 10 10 10, Exact avg = 100 % Instance No orry No traer Best souton H-3 souton Rato number customers customers Cost Tme [s] No snge orres No LTCs Cost Tme [s] No snge orres No LTCs Cost Tme 1 12 38 55711 203 1 4 56997 134 4 3 10231 % 69 % 2 25 25 60822 200 1 4 63203 149 5 3 10391 % 75 % 3 37 13 61804 160 2 3 61974 118 4 3 10028 % 74 % 6 56 19 93064 378 4 5 98501 184 9 3 10584 % 58 % Tabe 11: Computatona resuts for Chao TTRP nstances 32
Overa, the computatona resuts show that the H-3 heurstc approach offers the best trade-off between souton quaty and computaton tme Evdenty, the vstaton counters and, even more nterestngy, the current traer ocaton act as fetters n the domnance procedures and sow down the souton progress wthout sgnfcanty mprovng souton quaty The computatona resuts obtaned are satsfactory nsofar as wth the H-3 approach, nstances wth reastc network and cost structure and wth reastc sze (50 customers and 50 pure transshpment ocatons) coud be soved n short tme and wth very hgh souton quaty The test nstances created for ths paper use a vehce cost functon contanng varabe and fxed components It s noteworthy that tests showed that when the fxed cost component was dsregarded, the nstances were consderaby easer to sove wth a four souton approaches: Not ony dd the overa computaton tmes decrease, but aso the dfference n computaton tme between the nstance that took ongest and the one that took shortest to sove became smaer 54 Computatona Experments wth Exstng TTRP Benchmark Instances [8] ntroduced a set of 21 benchmark nstances for the TTRP verson he consdered (no pure transshpment ocatons, homogeneous fxed feet, no tme wndows, Eucdean dstances) The nstances range from 38 orry and 12 traer customers to 50 orry and 149 traer customers Encouraged by the resuts obtaned wth the H-3 approach as we as by the paper by [9], who present a heurstc coumn generaton approach to the heterogeneous feet VRP and obtan very convncng resuts on benchmark nstances for ths probem, t was tred to sove the Chao benchmark nstances wth the H-3 approach (due to the sze of the Chao nstances, a souton wth the other approaches was not possbe) The H-3 approach aows an unmted number of vehces, and t cannot easy be modfed so as to use a fxed number In addton, as expaned n Secton 32, the GTTRP aows that a traer customer ocaton be used as a transshpment ocaton by more than one orry-traercombnaton Hence, the H-3 approach soves a reaxaton of the probem studed by Chao Unfortunatey, the resuts obtaned were rather dsappontng Athough a 21 nstances coud be soved wthn a few mnutes, ony the four smaest nstances coud be soved wth the unmodfed H-3 approach To sove the arger nstances, t was necessary to mt the number of abes to be created wthn one run of the SPPRC agorthm and to operate on thn graphs contanng ony the m s shortest arcs n each forward star The resuts for the four smaest nstances were as gven n Tabe 11 The best known soutons are a from [32] These authors obtaned ther resuts on a 15 GHz processor In the above tabe, the orgna tmes were dvded by two, yedng a generous ower bound for the computaton tmes had these experments been performed on the same hardware as the ones descrbed n ths paper In a four soutons, no transshpment ocaton was used by more than one vehce 33
The souton quaty for the other Chao nstances was very ow and decreased wth ncreasng nstance sze; however, the scaed computaton tmes were aways shorter than the ones reported n [32] The reason for these resuts s that the average rato of customer suppy to vehce capacty s much smaer n the Chao nstances Ths makes the souton of the subprobems much harder, as many more paths are possbe, and therefore many more abes have to be created The approach of sovng (E)SPPRCs wth a abeng agorthm reaches ts mts here As mentoned n the terature revew, very recenty [32] have studed the TTRP verson ntroduced by Chao aowng an unmted number of vehces They used a smuated anneang heurstc and were abe to mprove the best known soutons for 18 of the 21 Chao benchmark nstances They aso created 36 new test nstances, a of whch had at east 75 customers and at east 18 traer customers Due to the unsatsfactory performance of the H-3 approach for the Chao nstances, these nstances were not tacked Consderng that for the vehce routng probem wth tme wndows, 50 cents means 50 vertces and 2,550 arcs (and a homogeneous feet), the foowng quote from [17], p 492, puts the above computatona resuts n perspectve: It shoud be noted that coumn generaton has been the domnant approach for the Vehce Routng Probem wth Tme Wndows (VRPTW) Current branch-and-prce agorthms can consstenty sove tghty constraned nstances (those wth narrow tme wndows) wth up to 100 cents However, they often fa on ess constraned nstances wth ony 50 cents See aso the resuts obtaned by [38] for the capactated vehce routng probem (CVRP) 6 Summary and Research Outook Ths paper has studed the generazed truck-and-traer routng probem (GTTRP) and has presented an arc-fow-based and a path-fow-based formuaton for ths probem Moreover, the paper has descrbed an exact souton procedure for the probem, a branch-and-prce agorthm, and heurstc versons of ths agorthm A maor advantage of the proposed souton approach over a possbe branch-and-cut agorthm based on the arc fow formuaton s that feet pannng (decdng on the number of vehces to use of each possbe type) and vehce routng can be performed smutaneousy wthout addtona modeng or computatona effort Extensve computatona experments have been performed The experments used randomy generated nstances structured to resembe rea-word stuatons and TTRP benchmark nstances from the terature The resuts showed that wth a heurstc coumn generaton approach, reaword GTTRP nstances can be soved n short tme wth hgh souton quaty However, the resuts for the TTRP benchmark nstances were not so successfu due to the ow rato of customer 34
suppes to vehce capacty n these nstances, whch makes the use of abeng agorthms for the souton of the prcng probems hghy dffcut The mpementatons can st be mproved sgnfcanty As the computatona experments showed, the number of subprobems to be soved s the man factor nfuencng the computaton tmes Therefore, the most mportant refnements (whch may make the souton of the arger Chao nstances and the new nstances by [32] possbe) are: The ncuson of stabzaton n the coumn generaton process ([30], pp 1017 ff) The deveopment of better ower boundng procedures, e, the addton of cuttng panes It woud be partcuary nterestng to nvestgate (the possbty of) generazng the two-path cuts used for the VRPTW ([27]) to a heterogeneous feet wth traers The souton of the prcng probems by branch-and-cut ([26]) The souton of the prcng probems by (meta)heurstcs, and the embeddng of the branchand-prce framework nto a metaheurstc framework Recenty, [35] have obtaned very good resuts for the VRPTW wth a arge neghbourhood search agorthm that uses heurstc branch-and-prce (heurstc souton of the prcng probem by tabu search, depth-frst branchng wthout backtrackng) as the reconstructon procedure Ths appears to be a very promsng approach aso for the GTTRP The frst two ponts woud reduce the number of subprobems to be soved, the ast two ponts woud (hopefuy) reduce the average tme needed for sovng one subprobem A very nterestng and chaengng further area of research s the vehce routng probem wth traers and transshpments (VRPTT) ([13]) The GTTRP as descrbed above s a speca case of the VRPTT In the VRPTT, the assumpton of a fxed orry-traer assgnment s abandoned A traer may be pued by any compatbe orry on a part or on the whoe of ts tnerary, and any orry may perform a oad transfer to any traer Moreover, there may aso be socaed support vehces (orres as we as traers), whch are not techncay equpped to vst customers The vehces equpped to vst customers may use these support vehces as mobe depots for oad transfers (note that a ocaton-routng probems descrbed n [34] are speca cases of the VRPTT as shown n [13]) The dffcuty wth the VRPTT s that, by abandonng the fxed orry-traer assgnment, the so-caed nterdependence probem arses, that s, the tours of dfferent vehces become nterdependent: In the smpest case, f a orry 1 wants to transfer oad nto a traer at a certan transshpment ocaton, the traer must be at ths ocaton before the transfer can start If a dfferent orry 2 pus ths traer to the ocaton, 1 s tour depends on 2 s tour, and 1 may have to wat Moreover, the amount of oad that 1 can transfer s affected by the amount of oad other orres have transferred nto the traer before Such nterdependences do not usuay occur n VRPs studed n the terature and requre speca souton approaches It s beyond the scope of ths paper to eaborate further on ths ssue Suffce t to say that vehce routng probems wth nterdependent tours such as the VRPTT are an emergng topc n VRP research and to refer the reader to the recent survey [15] 35
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