REAL-TIME VEHICLE ROUTING PROBLEM WITH TIME WINDOWS AND SIMULTANEOUS DELIVERY/PICKUP DEMANDS

Transcription

1 REAL-TIME VEHICLE ROUTING PROBLEM WITH TIME WINDOWS AND SIMULTANEOUS DELIVERY/PICKUP DEMANDS Me-Shang CHANG Assocate Professor Department of Bsness Admnstraton Chng Ha Unversty No.30, Tng Shang, Hsn Ch, Tawan Fax: E-mal: Che-F HSUEH Ph.D. Stdent Department of Cvl Engneerng Natonal Central Unversty No. 300, Jng-Da Road, W-Chan L, Chng-L, Tawan E-mal: Shyang-Rey CHEN Master Department of Bsness Admnstraton Chng Ha Unversty No.30, Tng Shang, Hsn Ch, Tawan Fax: Abstract: The real-tme vehcle rotng problem wth tme wndows and smltaneos delvery/pckp demands (RT-VRPTWDP) s formlated as a mxed nteger programmng model whch s repeatedly solved n the rollng tme horzon. The real-tme delvery/pckp demands are served by capactated vehcles wth lmted ntal loads. Moreover, pckp servces aren t necessarly done after delvery servces n each rote. A herstc comprsng of rote constrcton, rote mprovement and tab search s proposed. The rote mprovement procedre follows the general gdelnes of anytme algorthm. Nmercal examples made p by Gélnas were taken wth modfcaton for valdaton. Based on Tagch orthogonal arrays approach, the optmal parameter settng for tab search s set throgh expermentatons on the RT-VRPTWDP. The reslts show that the proposed algorthm can effcently decrease the total rote cost. Key Words: vehcle rotng problem wth tme wndow, real tme demands, anytme algorthm, Tagch orthogonal arrays approach 1. INTRODUCTION The vehcle rotng problem wth tme wndows (VRPTW) s a well-known NP-hard problem. Almost all VRPTW methods proposed are devoted to a statc problem where all data are known before the rote s constrcted and do not change thereafter. The advancement of commncaton and nformaton technology makes entrepreners more aware of the mportance of jst-n-tme manageral strateges. In the past decade, express transshpment actvtes and e-commerce bsness have experenced a rapd growth. These developments have led to a gradal growth of a new class of problems, known as real-tme rotng and schedlng problems, where problem sze and parameters change after the vehcle rotes are constrcted. Thogh strateges to deal wth real tme demands have been wdely dscssed, relevant models and algorthms for the VRPTW are scarce n the lteratre. In or research, we descrbe a real-tme vehcle rotng problem wth tme wndows and smltaneos delvery/pckp demands (RT-VRPTWDP), an extenson to tradtonal VRPTW. Some reqests are made after the rotes are constrcted. Each of the new reqests mst be assgned to an approprate vehcle n real tme. The ncertanty comes from the occrrence of the new servce reqests. There s no knowledge of frther ncomng reqests. The problem sze of RT-VRPTWDP changes therefore n real tme. Frthermore, mxed vehcle rotes wth both delvery and pckp servces are constrcted n or stdy. Ths paper s organzed as follows. After ntrodcton, a bref lteratre revew s presented n Secton 2. The RT-VRPTWDP s formlated as a mxed nteger programmng model n Secton 3. Secton 4 elaborates a proposed herstc, whch ncldes methods of rote

2 constrcton, rote mprovement and tab search. In Secton 5, 15 testng problems made p by Gélnas are taken wth mnor modfcatons for demonstraton and valdaton. Fnally, concldng remarks are gven n the end. 2. LITERATURE REVIEW The vehcle rotng problem has been and s stll an enrched research topc for researchers and practtoners. A large fracton of ths work s concerned wth statc problems, that s, all order for all cstomers are known a pror. For a descrpton of statc vehcle rotng problem, please see the recent srvey of the rotng problem (Ball et al., 1995; Fsher, 1995; Desrosers et al., 1995). Drng the past decade the nmber of pblshed papers dealng wth dynamc vehcle rotng problem has been growng. Psarafts (1995) examnes the man sses n ths area and provdes a srvey of the reslts fond for varos dynamc vehcle rotng problems. Relevant models and algorthms for the real-tme VRPTW are scarce n the lteratre. One of the earlest work on dynamc vehcle rotng problem was from Bertsmas and van Ryzn (1991, 1993) that s essentally a generc mathematcal model wth watng tme as an objectve fncton. A recent srvey of dynamc vehcle rotng s gven by Psarafts (1995), ncldng the delvery of petrolem prodcts or ndstral gases, corer servces, ntermodal servces, tramp shp operatons, pckp and delvery servces, management of contaner termnals. Powell et al. (1995) provde an excellent srvey on varos dynamc vehcle rotng problems sch as the dynamc traffc assgnment problem whch conssts n fndng the optma rotng of some goods from orgn to the destnaton throgh a network of lnks whch cold have tme-dependent capactes. Bertsmas and Smch-Lev (1996) provde a srvey of determnstc and statc as well as dynamc and stochastc vehcle rotng problems for whch they examne the worst and average-case behavors of the known algorthms for dynamc rotng problems. Gendrea and Potvn (1998) s the most recent srvey on the dynamc vehcle rotng problem. They pont ot that t s relevant to consder several sorces of ncertanty lke cancellaton of reqests and servce delays rather jst to focs on ncertanty n the tme-space occrrence of servce reqests. Gendrea et al. (1999) employed tab search wth parallel processng technqe to solve a problem wth real tme demands and soft tme wndow. The tab search herstc sed n ths work was orgnally desgned for the statc verson of ths problem and was therefore modfed n order to deal wth dynamc verson. Sheh and May (1998) treated a VRPTW problem wth real tme demands and demonstrated wth nmercal examples. L (2000) tackled a travelng salesman problem wth both tme dependent travel tmes and real tme demands. The tme-dependency acconts for varatons n travel speed cased by congeston. Larson (2001) provdes the lteratre revew dealng wth the dynamc vehcle rotng problem and related problems. The dynamc travelng reparman problem s extended to embrace advance reqest cstomers as well as mmedate reqest cstomer. The capactated vehcle rotng problem wth tme wndows s examned nder varyng levels of dynamsm. 3. MODEL FORMULATION Whenever the real tme demands are generated, the rotng schedle mst change n response to new or altered reqests. In the model we consder, there s an ntal rotng schedle that ncorporates all works crrently known. Ths rotng schedle s adjsted as new work arrves, and can be mproved provdng ths does not nterfere wth decsons that have already been commtted to. To clarfy the scope of the research, necessary assmptons are stated as follows: 1. There s a commncaton and transmsson systems between the dspatcher and drvers. Throgh the commncaton system, the dspatcher nforms the drvers whch demand to serve next only when commttng to that decson. Throgh the transmsson system, the dspatcher faxes delvery/pckp notes to the drvers. Once a drver s en rote to the next destnaton, however, he mst necessarly serve ths node. No dverson s allowed. 2. Real tme demands are generated n the rollng tme horzon. The real-tme rotng schedle needs to be solved wth estmated comptaton tme r nts for ntal solton. Real tme demand means that the plannng tme span to transfer demand nformaton from dspatcher to drver s short.

3 3. There s a tme lag n the on-lne dspatch system. Sppose t s the tme at whch the deployment of vehcle rotng begns and eqal to the crrent wall-clock tme pls r nts. 4. Delvery problems (delverng goods from a depot to the cstomer) and pckp problems (pckng p goods at the cstomer and brng back to a depot) are consdered smltaneosly. Moreover, pckp servces aren t necessarly done after delvery servces n each rote. 5. Delvery demand and pckp demand are rrelevant. The pckp goods cannot be drectly delvered to the cstomers. 6. Uncertanty comes from a sngle sorce, namely the occrrence of new reqests. There s no ncertanty assocated wth the cstomer locatons and travel tmes. 7. In a least commtment strategy, the drvers are asked to wat at ther crrent locaton f some watng tme s expected at the next cstomer. The latest possble tme allows last changes to the planned rotng schedle. 8. Demand forecast s not tackled by ths research. Expected qanttes and occrrence tmes of orders can mprove the solton qalty of the real-tme rotng problem, bt ncrease the complexty of the real-tme rotng problem. 3.1 Notaton 1. Parameters and constants a b r t e G k G k l M q ' q Q k Q k (t ) s c j : weght assocated wth lnk travel tme n the objectve fncton : weght assocated wth watng tme n the objectve fncton : estmated comptaton tme for ntal solton : tme to mplement the reslts compted from the RT-VRPTWDP, whch s sally set as the tme of occrrence of new demands : lower end of the tme wndow at node : ntal loads of vehcle k at depot : remanng loads to delver of vehcle k on arrval at crtcal node at tmet : pper end of the tme wndow at node : a very bg nmber : delvery demands at node : pckp demands at node : capacty of vehcle k : free capacty of vehcle k on arrval node at tmet : servce tme at node : travel tme between nodes and j 2. Set of Nodes {0} : depot N c : set of crtcal nodes at tme t ; crtcal node s defned as the last node beng served or schedled to be served by each vehcle : set of nassgned nodes at tme t N N N N P k c 0 c0 ( h) 3. Set of vehcles : set of crtcal and nassgned nodes at tme t : set of depot and nassgned nodes at tme t : set of depot, crtcal and nassgned nodes at tme t : set of nodes whch are served after node h by vehcle k

4 K K 0 K 0 : set of all vehcles : set of vehcles n the depot at tme t : set of dspatched vehcles from the depot at tme t 4. Sperscrpts and sbscrpts,j,h k k : node desgnaton : vehcle desgnaton : vehcle headng to or crrently at crtcal node 5. Varable a a 0 k ( h : tme arrvng at node : tme for vehcle k to retrn the depot c,, : ncreased travel tme after the nserton of node h between nodes and j C : generalzed cost accred from nodes and j j ( 0, h,0) ( h C( h) C : total cost for servng node h from the depot C,, : total ncreased cost after the nserton of node h between nodes and j D : total ncreased cost for servng node h : tme to depart from node d d 0 k G Q x k k w ( h : tme for vehcle k to depart from the depot : remanng loads to delver of vehcle k on arrval at node : free capacty of vehcle k on arrval at node : 1 f vehcle k departs node toward node j; 0, otherwse : watng tme before departre at node w,, : ncreased watng tme after the nserton of node h between nodes and j 3.2 Mathematcal Model At tme t, the real-tme vehcle rotng problem wth tme wndows and delvery/pckp demands can be formlated as a mxed nteger problem as follows: mn = a   Âcj x + b  Z w (1) ŒN jœn t ) k Œ c 0 0 ( N c The feasble regon s represented by the followng constrants. Flow Conservaton Constrants:  x = 1 " Œ Nc (2) j k  x = 1 " j Œ N (3) k ( ) k Œ K  xhk xh = 0 " h Œ N t, (4) - j

5  x = 1 " Œ Nc (5) j  x0 1 " k Œ K 0 (6) x j Œ {,1} " Œ N ( ), j Œ N 0 c 0 Tme Wndow and Departre Tme Constrants: Vehcle Capacty Constrants: e 0 t, k Œ K (7) a l " Œ N (8) a 0 k l 0 " k Œ K (9) d - a + s 0 " Œ N t (10) d d ( ) ( ) c - t 0 " Œ N (11) c 0k t G Q Defntonal Constrants: d k k [ - x ] M 0 " j Œ N ( ) k Œ K - t, (12) ( ), j Œ N ( ) k Œ K ( ), j Œ N k Œ K q f x = 1 " Œ N t t, (13) q ' f x = 1 " Œ N t, (14) ( ), j Œ N ( ) k Œ K ( ) k Œ K c ( ) k Œ K ( ), j Œ N ( ) k Œ K + c = a f x = 1 " Œ N t t, (15) j j d 0k + cj = a j f x0 = 1 " j Œ N t, (16) d + c0 = a0k f x0k = 1 " Œ N t, (17) G = G - q f x = 1 " Œ N t t, (18) G G Q k ( )- q f x = 1 " Œ N c, j Œ N k Œ K f x = 1 " j Œ N ( ) k Œ K - q ' + q f x = 1 " Œ N ( ), j Œ N k Œ K ( )- q ' + q f x = 1 " Œ N c, j Œ N k Œ K - Gk f x = 1 " j Œ N ( ) k Œ K ( a + s ) " Œ N = G k t, (19) = Gk 0 t, (20) = Q t, (21) k Q = Q k t, (22) Q = Qk 0 t, (23) w = d - (24) c The objectve of the RT-VRPTWDP, as shown n Eqn (1), s constrcted as a weghted fncton of travel tmes for all lnks and watng tmes before departre at all nodes. The respectve weghts are a and b wth the relatonshps of a > b whch s determned de to the fact: for each vehcle movng cost s generally hgher than stoppng cost becase the former needs to pay for gasolne, deprecaton, and addtonal socal costs sch as traffc congeston, ar pollton and rsk of traffc ncdents whereas the latter only acconts for the deprecaton. Eqn (2) reqres one vehcle leavng from crtcal or nassgned node once. Eqn (3) denotes only one vehcle can arrve at nassgned node j once. Eqn (4) states for each nassgned node h, the enterng vehcle mst leave the node eventally. Eqn (5) reqres that vehcle k that arrved or s approachng crtcal node mst also leave that node once. Note that vehcle k s known at tme t. Eqn (6) desgnates each vehcle can leave the depot at most once. Eqn (7) desgnates x as 0-1 ntegers; set x eqal 1 f vehcle k departs node toward node j, 0, otherwse. Eqn (8) reqres that for each node, the arrval tme mst be wthn the tme wndow. Eqn (9) c

6 ndcates that all vehcles mst retrn back before the depot s closed. Eqn (10) reqres that for crtcal or nassgned node, departre tme d mst be greater than or eqal to the completon tme of servce, a + s. The followng two Eqns are abot the constrants of dspatch tmng. Eqn (11) ndcates that for crtcal node, departre tme d mst be greater than or eqal to t. Eqn (12) ndcates that vehcles cannot leave the depot before tme t. Eqn (13) states that for each vehcle, the remanng loads to delver on arrval at node mst be greater than or eqal to the delvery demands at ths node. Eqn (14) ndcates that for each vehcle, the free capacty on arrval at node mst be greater than or eqal to the pckp demands at ths node. Eqns (15)~(17) defnes that for each node (ncldng the depot) the arrval tme s eqal to the departre tme from the prevos node pls the travel tme. Eqns (18)~(20) defnes that the conservaton of the delvery loads for each node. Eqns (21)~(23) defnes that the conservaton of the free capacty for each node. Eqn (24) comptes the watng tme before departre at node.. Note that real tme demands may sometmes reslt n a staton where not all cstomers can be served wthn ther tme wndows. Here we smply delete the cstomers that volate the tme wndow constrants from set N and solve the real-tme vehcle rotng problem wth delvery/pckp demands and tme wndows, problem (1), for the rest of cstomers. 4. SOLUTION ALGORITHM The VRPTW problem s known to be NP-hard and when temporal dmenson s ncorporated, t becomes more dffclt to be solved n a reasonable perod by an exact solton especally for large problems. To take care of both comptatonal effcency and real tme response reqrement, a herstc comprsng of rote constrcton and rote mprovement s proposed for the RT-VRPTWDP. In the followng sectons, prelmnares abot nserton cost and tab search are descrbed n Secton 4.1. In Secton 4.2 a systematc solton procedre embeddng rote constrcton, rote mprovement and tab search s llstrated. Effcent strategy proposed for rote constrcton s elaborated n Secton 4.3. The method of rote mprovement wth tab search s llstrated n Secton Pelmnares Calclaton of Inserton Cost Accordng to the objectve fncton shown n Eqn (1), the generalzed cost C j accred from nodes and j can be defned below as a weghted fncton of travel tme c j and watng tme for departre from node, w. C j j = a c + bw (25) Consder a vehcle rote,..,j,j+1,j+2,.,j+n,0, where 0 represents the depot. If we nsert D C h can be expressed n terms of nodes, h, and j as node h, the total ncreased cost ( ) C ( h) = mn( C(, h,, C( 0, h,0) ) C (, h, a c(, h, + bw(, h, c( h, = ch + chj - cj where C ( h, and j, ( 0, h,0) D (26) = (27), (28) ' ' ( h, = w - w + Â( w j - w j ) w, (29) ( 0, h,0) c0h ch0 jœpk ( h) C = + (30), represents the total ncreased cost after the nserton of node h between nodes ' C denotes the total cost for servng node h from the depot, and w, w denote the watng tme for departre from node before and after the nserton of node h. The total ncreased cost C (, h, s a weghted sm of ncreased travel tme c (, h, and ncreased watng tme w (, h,. The defntons of c (, h, and w (, h, can be expressed n Eqn (28) and (29). The last term of Eqn (29) s the total ncreased watng cost along the rote after the nserted node h.

7 4.1.2 Tab Search The tab search herstc was frst ntrodced n Glover (1986). Startng from some ntal solton, a neghborhood of the solton s generated throgh dfferent classes of transformatons. Then, the best solton n ths neghborhood s selected as the new crrent solton, even f t s worse than the crrent solton. Snce the crrent solton may deterorate drng the search, ant-cyclng rles mst be mplemented. Ths, a memory (tab lst) s sed to remember the recent search trajectory. In addton, dversfcaton mechansms can be sed to help the method to escape from local optma and explore a broad porton of the search space. The detals of tab search are dscssed n Secton Unfed Framework of Solton Procedre Ths Secton descrbes a nfed framework of solton procedre for the RT-VRPTWDP. The man concept s to take care of real tme nformaton and n the meantme to mprove the qalty of the solton n respondng to the ever-changng envronment along the rollng tme horzon. By constantly checkng whether (1) departre tme for crtcal node s p; (2) new demands have been generated; strateges to dspatchng en rote and/or on-call vehcles wth the rght tme to the assgned cstomers, to reconstrctng rotes and to mprovng the qalty of the exstng rotes are repeatedly appled. Note that the herstcs for RT-VRPTWDP mst be nterrpted at checkponts de to the reqrement of real tme response (Sheh and May, 1998). The real-tme rote mprovement procedre follows the general gdelnes of anytme algorthm (Zlbersten and Rssell, 1996). If the earlest departre tme of the crtcal nodes has arrved, the herstc s nterrpted at the checkponts and otpts the crrent solton to the dspatcher. The man components of ths solton procedre are brefly ntrodced: Frst, we make the ntal rotng schedle accordng to known cstomers beforehand. All the statc vehcle rotng algorthms wth tme wndows and smltaneos delvery/pckp demands can be adopted to fnd the ntal solton. If new demands appear, we nsert the new demands to the ntal rotng schedle. Fnally, a better real-tme rotng schedle s repeatedly searched before earlest departre tme of the crtcal nodes. In the real-tme rote mprovement procedre, Or-opt, 3-opt*/2-opt*, and Swap-opt algorthms are adopted for obtanng an nferor mprovement. The basc dea of these rote mprovement methods are stated as follows: Or-opt: Ths exchange procedre s descrbed n Or (1976). For a seqence of three consectve cstomers, two consectve cstomers or a sngle cstomer n a rote s removed and nserted at another locaton wthn the same or wthn another rote. However, some restrctons apply de to the constrants of vehcle capacty, tme wndow, delvery loads and dspatch tmng. 2-opt*: Ths s an extenson of the 2-opt neghborhood for problem wth mltple rotes. Consderng a lnk (, j 1 1 ) on a rote, a lnk (, j 2 2 ) on another rote, we replace them by two new lnks (, j 1 2 ) and (, j ) 2 1 f ths feasble exchange can reslt n a lower cost. Note that a feasble exchange stated heren means that feasblty wth respect to all the above constrants. Swap-opt: Ths exchange procedre s descrbed n Dhamel et al. (1997). Two cstomers are selected n the dame rote or n two dfferent rotes, and exchange ther poston. For the same reason, feasble move are constraned by the sde constrants. For the sake of frther mprovement solton, the tab search s ntegrated wth Or-opt, 3- opt*/2-opt*, and Swap-opt algorthms. 4.3 Method for Real-Tme Rote Constrcton For VRPTW problem, the nserton-type method has been proven effectve n constrctng statc rotes (Solomon, 1987). We assme t performs eqally effectve for the RT- VRPTWDP problems and therefore s adopted wth modfcatons as follows. Steps for Real-Tme Rote Constrcton

8 Step 1: Inpt data Inpt real tme delvery/pckp demands { q ' } q and estmated tme for explotaton, t., Step 2: Classfy cstomers and calclate relevant data Step 2.1: Classfy cstomers nto crtcal nodes N (t ) and nassgned nodes N (t ). Step 2.2: For dspatched vehcle k, calclate the remanng loads and free capacty at crtcal node, G and Q, sng Eqn (18) and Eqn (21). k k Step 3: Fnd the optmal place and departre tme for each nassgned node and calclate ts accred cost For each nassgned node ΠN (t ), fnd the optmal place and departre tme for nserton and calclate ts accred cost. Step 3.1: Set k=1. Step 3.2: If vehcle k s en rote, Step 3.2.1: Calclate nserton cost, C ( h,,, after checkng the tme wndow constrants, free capacty constrants and remanng load constrants for nassgned node n all possble places along the movng rote of vehcle k. Step 3.2.2: Record the mnmal nserton cost and the assocated place for nassgned node along the movng rote of vehcle k. Step 3.3: If k=k, contne. Otherwse, set k=k+1 and go to Step 3.2. Step 3.4: If some vehcles are avalable for dspatchng n the depot, calclate the accred cost, C ( 0, h,0), and check tme wndow constrants for nassgned node for the newly dspatched vehcle. Step 3.5: Select the mnmal nserton cost and ts place for node n all possble rotes among vehcles ether en rote or beng dspatched. If ether capacty constrants or tme constrants cannot be flflled, exclde that cstomer from the servce lst. Step 3.6:If every nassgned node ΠN (t ) has been examned, contne. Otherwse, go back to Step 3.1 for next nassgned node. Step 4: Insert the newly accred cstomer and pdate the relevant data Update the system by nsertng node nto the place wth mnmal accred cost n an approprate rote (of vehcle k*) and redefne relevant data sch as departre tme, arrval tme, free capacty and remanng loads for each affected node. Once nserted, node s then removed from N (t ). Step 5: Stoppng Check for Assgnment If set N s empty, enter the nfed framework of solton procedre. Otherwse, do the followng: Step 5.1: For each nassgned node, compte the nserton cost and ts correspondng place along the rote of vehcle k*. Note that for nassgned nodes, ther correspondng cost and optmal place for nserton n all rotes other than that taken by vehcle k* are calclated already and not changed and hence need not be recompted. Step 5.2: Compare and select the optmal place for nsertng nassgned node wth the mnmal ncreased costs among all possble rotes. If nassgned node cannot be nclded for servce sbject to the crrent vehcle capacty and tme wndow constrants, exclde node N t s empty, enter the nfed framework of solton from the servce lst. When set ( ) c

9 procedre. Otherwse, go to Step Method for Real-Tme Rote Improvement wth Tab Search Or tab search herstc s nspred by a smlar work for VRPTW (Badea et al., 1997; Cordea et al., 2001; Potvn et al., 1996). We also se a tab lst of the fxed length. However, solton feasblty s always mantaned for the real-tme VPRTW (Dhamel et al., 1997). De to the dspatchng reqrements, the herstc can be nterrpted at checkponts. In addton, we always keep the best solton fond so far. The ntal set of rotes that are obtaned by real-tme rote constrcton and mprovement procedres mst be remembered ntl a better solton s fond by the tab search. The tab search herstc can be smmarzed as follows: Steps for Real-Tme Rote Improvement wth Tab Search Step 1: Intalzaton Set the crrent and best solton to the ntal rote schedlng that are obtaned by real-tme rote constrcton and mprovement procedres. Step 2: Tab search step Repeat Step 2 ntl maxmm nmber of teratons s performed, or ntl maxmm nmber of consectve teratons s performed wthot any mprovement to the best known solton. Step 2.1: Tab search wth Or-opt Repeat Step 2.1 ntl maxmm nmber of consectve teratons s performed wthot any mprovement to the best known solton. Step 2.1.1: Generate the neghborhood of the crrent solton by applyng Or-opt exchanges. Note that constrants mst be satsfed. Step 2.1.2: Select the best non-tab solton n ths neghborhood and defne ths solton to be the new crrent solton. Step 2.1.3: Update ts tab lst. The nverse move s declared tab for T teratons. Step 2.2: If the earlest departre tme of the crtcal nodes has arrved, go to Step 3. Otherwse, contne. Step 2.3: Tab search wth 3-opt and 2-opt Repeat Step 2.3 ntl maxmm nmber of consectve teratons s performed wthot any mprovement to the best known solton. Step 2.3.1: Generate the neghborhood of the crrent solton by applyng 3-opt and 2-opt exchanges. Check these exchanges mst satsfy all the constrants. Step 2.3.2: Select the best non-tab solton n ths neghborhood and defne ths solton to be the new crrent solton. Step 2.3.3: Update ts tab lst. The nverse move s declared tab for T teratons. Step 2.4: If the earlest departre tme of the crtcal nodes has arrved, go to Step 3. Otherwse, contne. Step 2.5: Tab search wth Swap-opt. Repeat Step 2.5 ntl maxmm nmber of consectve teratons s performed wthot any mprovement to the best known solton.

10 Step 2.5.1: Generate the neghborhood of the crrent solton by applyng Swap-opt exchanges. Check these exchanges mst satsfy all the constrants. Step 2.5.2: Select the best non-tab solton n ths neghborhood and defne ths solton to be the new crrent solton. Step 2.5.3: Update ts tab lst. The nverse move s declared tab for T teratons. Step 3: Retrn the best overall solton of the tab search If the new crrent solton of rotes s better than the ntal solton of rotes, set the best overall solton to the new crrent solton of rotes. Otherwse, the best overall solton s stll the ntal solton of rotes that are obtaned throgh real-tme rote constrcton and mprovement procedres. 5. NUMERICAL EXAMPLES 5.1 Problem Set The herstc was tested on the Edcldean problems of Gélnas et al. (1995). Gélnas problems were obtaned from Solomon s 100-cstomer Edcldean VRPTWs known as problem R101 to R105. To smplfy the demonstraton, we sppose each cstomer has ether delvery demands or pckp demands and randomly select 10%, 30% and 50% of the 100 cstomers as pckp cstomers. However, or formlaton has taken the delvery/pckp demands of the same cstomer nto accont. In test problems, the travel tmes are eqal to the correspondng Ecldean dstances. A fxed amont of 10 tme nts s needed to nload or load the goods at each cstomer locaton. The wdth of the tme wndow at the depot s set at 230 tme nts n all problems. In addton, a dscrete-tme smlator was developed to test or anytme algorthm. The smlator ses relevant data to prodce new servce reqests by the followng formla: max( 0, e q c0 -x ) (35) - where c 0 denotes the dstance between depot 0 and node, parameter x s a random nmber smaller than e - q c0, and adjstment parameter q s desgned to avod generatng new demands n the neghborhood of tme wndow at node. Here we set q = 1. The weghts for travel tme and watng tme are assmed as a = 0. 7, b = 0. 3, and the estmated comptaton tme for the ntal solton of the RT-VRPTWDP s set as D = 5 tme nts. 5.2 Parameter Settng of the Tab Search Based on Tagch orthogonal arrays approach, parameter settng for tab search was analyzed throgh expermentaton on the real-tme VRPTW problems. The best and least senstve parameter settng of tab search s selected by means of sgnal-to-nose rato (s/n). The sze of the Tagch orthogonal arrays s determned by the nmber of mportant parameters to be consdered (Bono, et al., 1995). Based on or jdgment, we defne for control factors: length of tab lst, consectve nmber of faled teratons, nmber of teratons and ntal loads. Each factor s dvded nto three levels. Ths, a Tagch orthogonal array corresponds to for factors and three levels. The combnatons of the parameter settng for tab search are reported n Table 1. De to the stochastc featres of RT-VRPTWDP, 10 ndependent rns were performed on each problem. In addton, the average length of tab lst s 4.2. Ths vale s slghtly smaller than the one for solvng statc VRP. The average of consectve nmber of faled teratons and total nmber of teratons s 10.3 and 51.3, respectvely. Roghly speakng, they obey the emprcal rles of parameter settng. The total nmber of teratons s more than ten tmes the length of tab lst. The consectve nmber of faled teratons s abot one ffth of total nmber of teratons. Fnally, the average of ntal loads s The rato of ntal loads to vehcle capacty s abot 0.72.

11 5.3 Testng Reslts The herstc proposed n Secton 3 was mplemented wth the C programmng langage. The tests were rn on Pentm IV personal commter. De to the stochastc featres of RT- VRPTWDP, 30 ndependent rns were performed on each problem. Wth the above npt data, two modfcatons of or algorthm were frther hypotheszed for testng: (1) Partal algorthm 1: only real-tme rote constrcton procedre s performed; (2) Partal algorthm 2: real-tme rote constrcton procedre and real-tme rote mprovement procedre wthot tab search are performed. The dfferences of soltons obtaned by dfferent algorthms n terms of total rote costs and nmber of vehcles are smmarzed n left sdes of Table 4 and Table 5, respectvely. In rght sdes of Table 2 and Table 3, the percentages of mprovement of solton obtaned by the algorthms wth rote mprovement procedre over the solton obtaned by the algorthm wthot rote mprovement procedre are shown for each problem. In addton, the last colmns of Table 2 and Table 3 are the dfferences of soltons obtaned by complete algorthm and partal algorthm 2. As a reslt, complete algorthm and partal algorthm 2 are sperorly than partal algorthm 1 n terms of total rote costs. The total rote costs has been lowered p to 12.94% and 16.63% n average for partal algorthm 2 and complete algorthm, respectvely. We also observe that complete algorthm s sperorly than partal algorthm 2 n terms of total rote costs. The total rote costs has been lowered p to 3.69% n average. The percentage of mprovement wth respect to the nmber of rotes s rather erratc. The nmber of rotes has been lowered p to 8.80% and 7.73% n average for partal algorthm 2 and complete algorthm, respectvely. Bt the algorthm wth tab search sn t certanly sperorly than the algorthm wthot tab search n terms of the nmber of vehcles. Perhaps t s becase the nmber of vehcles s not taken nto accont the objectve fncton of rote mprovement. When mnmzng the total rote costs, the exchange that leads to the largest decrease n rote costs s preferred over all other exchanges, ndependently of the nmber of rotes. 6. CONCLUDING REMARKS In ths paper, the RT-VRPTWDP s stded and formlated as a mxed nteger programmng model. The decson varables nclde not only lnk flows x bt also departre tme at node, d, and at the depot 0 by vehcle k, d 0 k. The problem s more dffclt to handle, de to the smltaneos capacty constrants of the pckp and the delvery. Frthermore, the dspatcher wshes to know the solton to the crrent problem as soon as possble (preferably wthn mntes or seconds). The tme lmt of dspatchng mples that rerotng s often done by adoptng local mprovement herstcs. It s therefore essental that pdatng nformaton mechansm are ntegrated nto the solton method. The rote mprovement procedre follows the general gdelnes of anytme algorthm n or stdy. A nfed framework comprsng of ntal rote constrcton, real-tme rote constrcton, real-tme rote mprovement and tab search was proposed and valdated for the RT-VRPTWDP. 15 problems made p by Gélnas were taken wth mnor modfcatons. Based on Tagch orthogonal arrays approach, parameter settng for tab search was analyzed throgh expermentaton on the real-tme VRPTW problems. The average of length of tab lst was slghtly smaller than the one for solvng statc VRP. The algorthms wth rote mprovement procedre are sperorly than the algorthm wthot rote mprovement procedre. We also observe that the elaboraton of tab search n the anytme algorthm can frther redce the total rote costs. The RT-VRPTWDP can be mproved or extended n a nmber of ways. Frst, the comptaton of the new watng tme w' s the key. Comptaton of all the watng tmes n the rote s not practcal. Parallel algorthms shold be sefl for the fast comptaton. Second, the occrrence of new reqests s the only sorce of ncertanty n ths stdy. The stochastc propertes of demands, cstomers, servce tme and travel tme wold be consdered n an extended model. These research topcs are crrently ndergong by the athors.

12 Percentage of Pckp Cstomers 10% 30% 50% Table 1. Combnatons of Parameter Settngs for the Tab Search Level Length of Tab Lsts Consectve Nmber of Faled Iteratons Control Factor Total Nmber of Iteratons Intal Loads Table 2. Dfferences of Total Rote Costs Problem R101 R102 R103 R104 R105 Percentage of Pckp Cstomers Total Rote Costs Percentage of Improvement A B C D E F Partal Algorthm 1 Partal Algorthm 2 Complete Algorthm (A B)/A (A C)/A E D 10% % 26.13% 8.94% 30% % 22.15% 3.82% 50% % 21.16% 4.14% 10% % 18.58% 3.76% 30% % 18.26% 2.67% 50% % 17.10% 2.65% 10% % 11.15% 5.95% 30% % 12.25% 1.48% 50% % 11.35% 3.70% 10% % 9.40% 3.50% 30% % 11.16% 3.70% 50% % 11.11% 6.09% 10% % 22.02% 4.50% 30% % 18.43% 0.03% 50% % 19.26% 0.40%

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