TRUCK ROUTE PLANNING IN NON- STATIONARY STOCHASTIC NETWORKS WITH TIME-WINDOWS AT CUSTOMER LOCATIONS

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1 TRUCK ROUTE PLANNING IN NON- STATIONARY STOCHASTIC NETWORKS WITH TIME-WINDOWS AT CUSTOMER LOCATIONS Hossen Jula α, Maged Dessouky β, and Petos Ioannou γ α School of Scence, Engneeng and Technology, Pennsylvana State Unvesty - Hasbug, Mddletown, PA β Danel J. Epsten Depatment of Industal and Systems Engneeng, Unvesty of Southen Calfona, Los Angeles, CA γ Depatment of Electcal Engneeng, Unvesty of Southen Calfona, Los Angeles, CA Coespondng autho: Emal: oannou@usc.edu, Tel: (13) Abstact: Most exstng methods fo tuck oute plannng assume known statc data n an envonment that s tme vayng and uncetan by natue whch lmts the wdespead applcablty. The development of ntellgent tanspotaton systems such as the use of nfomaton technologes educes the level of uncetantes and makes the use of moe appopate dynamc fomulatons and solutons feasble. In ths pape, a tuck oute plannng poblem called Stochastc Tavelng Salesman Poblem wth Tme Wndows (STSPTW) n whch tavelng tmes along the oads and sevce tmes at the custome locatons ae stochastc pocesses s nvestgated. A methodology s developed to estmate the tuck aval tme at each custome locaton. Usng estmated aval tmes, an appoxmate soluton method based on dynamc pogammng s poposed. The algothm fnds the best oute wth mnmum expected cost whle t guaantees cetan levels of sevce ae met. Smulaton esults ae used to demonstate the effcency of the poposed algothm. Keywods: Tuck outng, Stochastc tavelng salesman poblem, Tme wndows, Contane movement, Uncetan envonment. I. INTRODUCTION In the tuckng ndusty, tme s money. The ablty of a tuckng company to succeed economcally ests on ts ablty to move goods elably and effcently, wth mnmal delay. In many taffc netwoks especally n majo ctes, taffc congeston has aleady educed moblty and system elablty, and has nceased tanspotaton costs. In addton to contbutng to tuck-dves neffcency, taffc congeston s a majo souce of a polluton (especally desel toxns), wasted enegy, nceased mantenance cost caused by the volume of tucks on oadways, etc. (Baton, 001). 1

2 Wth the expected substantal ncease n the volume of ntenatonal and natonal cagos enteng and movng though the U.S. hghway system, togethe wth the antcpated gowth n the numbe of pesonal vehcles n use, t s expected that the condton of taffc congeston wll only get wose, unless caeful plannng s ntated. Fo nstance, accodng to the Fedeal Hghway Admnstaton, cuently nealy half of the Calfona s uban hghways ae congested. It s expected that, between 000 and 05 n Calfona, pesonal vehcle tps wll be nceased by 38 pecent (CalTans 004), and the volume of contanes movng n and out of the majo Calfona pots wll be tpled (Mallon and Magaddno, 001). Thee ae numeous ways to mpove taffc congeston, and theefoe educe tanspot tmes assocated wth goods movements. Optons nclude developng new and expandng cuent facltes, deployng advanced technologes, and mpovng opeatonal chaactestcs and system management pactces. It should be noted that the scacty of land n majo ctes has made the opton of developng new facltes, f not nfeasble, sgnfcantly costly. Goods movements, n natue, contan uncetantes. Fo nstance, the custome demands, tavel costs, and tavel tmes ae uncetan, tme-dependent vaables. In the pesence of a hgh degee of uncetantes, t s wdely expected that optmal solutons fo goods movements wll be outpefomed, ove tme, by algothms that ae moe local n natue (Powell et al., 000). Howeve, the deployment of advanced technologes such as the use of nfomaton technologes can educe the level of uncetantes to a manageable level and makes the use of dynamc fomulatons and solutons feasble. The focus of ths pape s to nvestgate methods to mpove the opeatonal chaactestcs of goods movements by developng technques that can be easly mplemented usng new but cuently avalable compute and nfomaton technologes. The objectve of ths pape s to develop methods fo outng and schedulng of tucks n uncetan envonments whee they ae used to tansfe contanes between sevce statons (e.g., mane temnals, ntemodal facltes, and waehouses) and end customes. We assume that each of these customes/facltes may have mposed tme-wndow constants on pck-up/dop-off contanes. Jula et al. (005) has shown that the contane movement poblem by tucks wth tme-wndow constants at both ogns and destnatons could be modeled as an asymmetc Tavelng Salesmen Poblem wth Tme Wndows (TSPTW). In the TSPTW poblem, a vehcle, ntally located at the depot, must seve

3 a numbe of geogaphcally dspesed customes such that each custome s seved wthn a specfed tme wndow. The objectve s to fnd the optmum oute wth mnmum total cost of tavel. The TSPTW poblem s a specal case of the Vehcle Routng Poblem wth Tme Wndows (VRPTW) n whch the capacty constants ae elaxed. Dung the last two decades, the Vehcle Routng Poblem (VRP) and a vaety of ts pactcal applcatons have been the subject of a wde body of eseach (Golden and Assad 1988, Lapote 199, Fshe 1995, Toth and Vgo 00). The Stochastc Vehcle Routng Poblem (SVRP) ases wheneve some o all elements of the VRP poblem ae andom. The most studed aea n SVRP has been the VRP poblem wth Stochastc Demands (VRPSD), and wth Stochastc Customes (VRPSC) (Betsmas 199, Gendeau et al. 1995, Gendeau et al. 1996, Betsmas and Smch-Lev 1996, Secomand 001, Banke et al., 005). Despte ts mpotance and pactcal applcaton, especally n majo ctes wth taffc congeston, eseach effots on the stochastc VRP wth Stochastc Tavel Tmes (VRPST) have been lmted (Lapote et al. 199, Lambet et al. 1993, Km et al. 005). The objectve n the VRPST poblem s to fnd the optmal vehcle outes n the pesence of andom tavel and sevce tmes. Most VRPST soluton methods eque the knowledge of the dstbuton of the sum of the tavel and sevce tmes along the outes (Lapote et al. 199, Gendeau et al. 1996). In ths pape, the Stochastc TSPTW poblem wth non-statonay stochastc tavel and sevce tmes s nvestgated. Ths poblem heeafte wll be called Stochastc TSPTW (STSPTW) fo convenence. In the detemnstc tavelng salesman poblem wth had tme wndows, one can clealy goup outes nto feasble o nfeasble outes. A oute s feasble f t vsts all custome locatons befoe the demanded latest tmes. In the STSPTW, howeve, such a black and whte defnton cannot be easly put fowad. To avod any confuson, n ths pape, we wll defne and use the tems acceptable outes and sevce level n the stochastc scenao. A oute s consdeed acceptable f the pobablty of avng to each custome locaton n the oute wthn the tme wndow s geate than the sevce level. To detemne the acceptable outes, we develop a methodology to estmate the fst and second moments of aval tme at each custome locaton. One of the majo dffcultes n estmatng the aval tme at each node s the exstence of the non-lneaty fomed by tme wndows. In ths pape, ths nonlneaty s thooughly nvestgated and an appoxmate methodology s developed to addess the exstng tme wndows n the STSPTW poblem. Futhemoe, we popose an appoxmate soluton 3

4 algothm based on a modfed dynamc pogammng method to fnd the least-cost oute fo the STSPTW poblem, whch meets the equed sevce level at the custome locatons. The pape s oganzed as follows. In Secton II, the poblem of stochastc TSPTW wth andom tavel and sevce tmes s descbed. In Secton III, we explan why t s dffcult to obtan the pobablty densty functon (PDF) of aval tmes at custome locatons even fo a smplfed STSPTW poblem. Theefoe, we develop a method to estmate the mean and vaance of the aval tmes at custome locatons. The concept of confdence coeffcent at a node, and the appoxmate soluton method fo STSPTW ae poposed n Secton IV. The expemental esults ae gven n Secton V, and Secton VI pesents the concluson. II. PROBLEM DESCRIPTION Let G=(ND,A) be a gaph wth node set ND = { o} U{ d} U N and ac set ( ) {,, } A= j j ND. The nodes o and d epesent the sngle depot (the ogn-depot and destnaton-depot), and { 1,,..., n} N = s the set of customes. Assocated wth each ac (,j) A, s a stochastc pocess X j (t) wth agument t epesentng the tavel tme on that ac. The agument t ndcates the tme when a vehcle entes ac (,j). We assume that the tavel tmes on the ndvdual lnks at any patcula tme t ae statstcally ndependent. A cost coeffcent denoted by c j s assocated to each ac (,j) A. The cost c j can be ethe a detemnstc numbe (e.g., the dstance between nodes and j), o a andom vaable (e.g., the tavel tme on the ac (,j)). Assocated wth each node ND s a sevce tme s epesentng the duaton of tme fo a vehcle to be seved at that node. We assume that the sevce tme s s a andom vaable, whch s ndependent of the tme sevce stats at node. Futhemoe, to each node ND a tme wndow a, ] s assocated whee a and b ae the ealest and the latest tme to vst node, [ b espectvely. We assume that f a vehcle aves at the custome locaton at any tme eale than a, t has to wat tll a to stat sevcng. If t aves at any tme late than b, t cannot be seved. We defne a oute n gaph G as an odeed set of nodes ={o,w 1,w,,w k,d}, whee w N, =1,...,k, {, j, j,, j, j } wth assocated ac set ( ) ( ) A = w w w w w w A w s vsted mmedately afte w. 4

5 Let Y o be the depatue tme fom node o. Gven the depatue tme,.e. Yo = y, the andom vaable o Y denotes the aval tme at node takng the path statng at node o, passng though the nodes n oute, n the specfed ode, and endng at node. The oute s acceptable f the pobablty of vstng evey node on oute befoe ts latest tme, b, s geate than an abtay constant ϒ, whch s called the sevce level at node. Pecsely speakng, the confdence coeffcent at node on oute, denoted by γ, s defned as the pobablty of avng at node no late than b,.e. { b} γ = PY. ( 1 ) We say that the aval tme at node takng oute meets the equed sevce level at ths node, say ϒ, f the confdence coeffcent at node, γ, s geate than o equal to ϒ. The cost of oute, denoted by C, s the cost of tavelng between nodes o and d n the specfed ode of nodes on oute. As dscussed, the cost C can be ethe a detemnstc numbe (e.g., the dstance) o a andom vaable (e.g., the tavel tme). The objectve of ths pape s to fnd the least-cost (.e., mnmum C, fo detemnstc numbes, and mnmum E[C ] fo andom vaables) acceptable oute statng fom ogn o, vstng all nodes n N and endng at destnaton d, such that each node s vsted exactly once. III. ESTIMATING THE FIRST TWO MOMENTS OF THE ARRIVAL TIME Let ={o,1,...,,j,...,m,d} be a oute n gaph G wth assocated ac set A. Fgue 1 gaphcally llustates a typcal oute. t, j j o t o,1 1 d t m, d m Fgue 1: A gaphcal epesentaton of a typcal oute. 5

6 Let f ( x, t) X j j be the non-negatve fst-ode pobablty densty functon (PDF) of the stochastc pocess X j (t) (the tavel tme on ac (, j) A ), and let f ( s ) be the PDF of the sevce tme at node. Gven the depatue tme fom the ogn, the tme wndow at node, f ( s ), and f X j ( xj, t), we ae nteested n fndng the PDF of the aval tme at node on oute,.e., f ( y ) S S Y. Wthout sevce tme and tme wndows In ths subsecton, we assume that thee ae no tme wndows and sevce tmes assocated wth the nodes of oute,.e., [ a, ] = [0, ) and s = 0. In othe wods, as soon as a vehcle aves b at node t mmedately contnues ts tavel towad node j. Suppose the aval tme at node on oute, Y, s gven. Snce thee s no watng tme at node, also ndcates the depatue tme fom ths node. Hence, the aval tme at successo node j on oute can be computed by Y j Y = Y + Z, ( ) j whee the andom vaable Z = X ( y ). j j Z j s the tavel tme on lnk (,j) gven the depatue tme fom node,.e., Equaton ( ) ndcates that gven the aval tme at node (the pesent), the aval tme at node j (the futue) s not nfluenced by the past but only by the pesent state. In othe wods, equaton ( ) mples a Makovan pocess (Papouls, 1991). Let Yo = y be the depatue tme fom the ogn o. Usng the Makov chan ule and the Chapman- o Kolmogooff equaton, the condtonal PDF of tavel tme between the ogn and node j on oute can be computed ecusvely by m 0 m m 1 m 1 o m 1 0 ( ) ( ) ( ) f y y = f y y f y y dy, ( 3 ) 6

7 whee m=, K, j, and ( m, m 1) A. Geneally speakng, t s vey had and tedous to compute ( 3 ) fo evey node on oute and fo an abtay PDF f ( ). Instead, t s moe pactcal to fnd the fst two moments of the tavel tme between the ogn and node j. In the est of ths secton, we shall descbe how the fst and second moments of the aval tme at each node of oute can be estmated. Let η (t) and σ ( ) denote the mean and vaance of the stochastc pocess X j (t) coespondng to j j t ts fst-ode pobablty densty functon, espectvely. These paametes ae defned by () = () = (, ) η t E X t x f x t dx, ( 4 ) j j j Xj j j 0 j () t = E ( Xj () t j () t ) = ( xj () t j () t ) fx ( x, ) j j t dxj 0 σ η η. ( 5 ) Accodng to ( ), gven the mean aval tme at node, the mean aval tme at node j can be computed by (Papouls, 1991) E Yj E Y E E Zj Y y = + =. ( 6 ) = E Y + E E Xj( Y ) Usng ( 4 ), equaton ( 6 ) becomes whee ( ) E Y = E Y + E η Y j j, ( 7 ) j ( ) ηj ( ) Y ( ) 0 E η Y = y f y dy. ( 8 ) Usng Taylo's sees expanson and assumng that the functon η (t) s dffeentable at and aound E Y, the fst ode appoxmaton of ( 8 ) can be obtaned by (Fu and Rlett, 1998) j 7

8 ( ) η ( ) E ηj Y j E Y. ( 9 ) Substtutng ( 9 ) n ( 7 ), we get ( ) E Y E Y + η E Y j j. ( 10 ) Smlaly, by usng Taylo's sees expanson, the fst ode appoxmaton of the second moment of Y j n ( ) can be obtaned by (see also, Fu and Rlett 1998, and Papouls 1991) ( ) ( ) σ ( ) ( ) η ( ) va Y 1 va j + j E Y Y + j E Y. ( 11 ) whee η (). s the fst devatve of η (.). Theefoe, gven the depatue tme Yo = yo and the mean j j and vaance of the tavel tme on each ac (, j) A, the mean and vaance of the aval tme at each node can be calculated usng equatons ( 10 ) and ( 11 ), ecusvely. The fst ode appoxmaton model gven by ( 10 ) and ( 11 ) ae obtaned by assumng that the second and hghe devatves of η ( t) and ( t) j σ ae equal to zeo. Obvously, the fst ode j appoxmaton can be mpoved f hghe ode tems of the Taylo s sees ae also ncluded. Fndng the thd and fouth cental moments of these stochastc vaables ae nethe pactcally no computatonally feasble n many stuatons. In ths pape, we wll use only the fst ode appoxmaton. Wth sevce tme In ths subsecton, we assume that thee s no tme wndow assocated wth the nodes of oute. Howeve, as soon as a vehcle vsts a node, t wll be seved fo a peod of tme befoe depatng fo the next node. We assume that the sevce tmes at some o all the nodes on oute ae andom vaables, whch ae ndependent of the tme the sevce stats at those nodes. Let andom vaable S denote the sevce tme at node wth non-negatve PDF s ). Gven the aval tme at node takng oute, f S ( Y, the depatue tme fom node, W, s obtaned by 8

9 W = Y + S. ( 1 ) The aval tme at node j, Y j, s gven by j Y = W + Z. ( 13 ) j whee the andom vaable Z j s the tavel tme on lnk (,j) gven the depatue tme fom node,.e. Z = X ( w ). Accodng to ( 1 ), the fst moment of j j W can be calculated as [ ] E W = E Y + E S. ( 14 ) Snce andom vaable S s ndependent of the aval tme Y, we have cov( Y, S ) = 0. Hence, the second moment of W can be computed by ( W ) va( Y ) va( S ) va = +. ( 15 ) Gven the mean and vaance of the depatue tme W n ( 14 ) and ( 15 ), the fst ode appoxmaton model of the mean and vaance of and ( 11 ) by va Y j can be obtaned accodng to equatons ( 10 ) ( ) E Y j E W + ηj E W ( Y ) ( 1+ η ( E[ W ] ) va( W ) + σ ( E[ W ]) j j, and ( 16 ), ( 17 ) j whee now W depends on the sevce tme as descbed by equatons ( 14 ) and ( 15 ). Hence, gven the depatue tme Y o = y, the mean and vaance of the tavel tme on each ac o (, j) A, and the mean and vaance of the sevce tme at each node, the fst two moments of the aval tme at each node can be calculated usng equatons ( 14 ) to ( 17 ), ecusvely. 9

10 Wth tme wndows Hee, we consde the case n whch some o all the nodes on oute have tme wndows. At each node, we ae nteested n nvestgatng the effect of the exstng tme wndow on the chaactestcs of the aval tme. In othe wods, gven the fst and second moments of the aval tme at each node wth tme wndow, we would lke to appoxmate the fst and second moments of the depatue tme. We stat by assumng that thee s no sevce tme assocated wth nodes of oute. We also assume that the non-lnea functon g y ), llustated gaphcally n Fgue, denotes the depatue tme as a ( functon of the aval tme y at each node. g(y ) b a a b y Fgue : The gaphcal epesentaton of the tme wndow at node consdeed n ths pape. As shown n Fgue, f a vehcle aves at node eale than a, the vehcle wats tll tme a befoe depatng. Howeve, f the aval tme at node s geate than b the depatue tme wll be nfnte, whch ndcates that the vehcle wll not be seved f t aves late than b. Gven the aval tme Y at node takng oute, the depatue tme W s gven by W ( ) = g Y. ( 18 ) Note that fo W n ( 18 ) to be a andom vaable, the functon g (.) must have the followng popetes (Papouls, 1991): 1. Its doman must nclude the ange of the andom vaable Y. 10

11 . It must be a Bae functon,.e. fo evey w the value of consst of the unon of a countable numbe of ntevals. y such that g( y ) w must 3. The events { ( ) = ± } g must have zeo pobablty. Y The depatue tme wndow g (.), shown n Fgue, volates popety 3. That s, wheneve y > b the depatue tme becomes nfnty, whch mples that the pobablty of the events { ( ) = + } g s Y not zeo. To ovecome ths poblem, we assume that f a vehcle aves at node at any tme y > b, the depatue tme wll be functon g (.). M y, whee M s a vey lage numbe. Fgue 3 llustates the modfed g(y ) b Slope= M a Note that the value of M s chosen to be vey lage such that t would be almost mpossble fo a vehcle avng at node at tme a b y Fgue 3: The modfed tme wndow g(.). y > b to contnue ts path towad the next node. In othe wods, the value of M does not penalze (as n the case of soft tme wndows) the late vehcles, but pohbts them n eachng the othe nodes on the oute wthn a easonable amount of tme. Hence, one may thnk of selectng the value of M asymptotcally close to nfnty. Let f ( y ) Y be the PDF of the andom vaable Y. The fst moment of the depatue tme W = g( Y ) can be obtaned by ( ). Y ( ) 0 E W = g y f y dy. ( 19 ) 11

12 As dscussed eale, t s vey dffcult to obtan the PDF of the andom vaable Y at each node. Hence, t s moe pactcal to fnd means to estmate the fst and second moments of the andom vaable W n tems of the functon g (.) and moments of Y. These estmates can be obtaned by Y appoxmatng the functon g ( ) at o aound E Y usng Taylo's sees expanson as follows 1 g( y ) ( ) ( ) ( ) ( ) ( ) = g E Y +g E Y y -E Y + g E Y y -E Y + L. ( 0 ) Note that fo the Taylo's sees expanson n ( 0 ) to be vald, the mean aval tme at node, E Y, should be fa enough fom ponts y j=a j and y j =b j, whee the functon (.) g s not dffeentable. In the followng, we fst stat by assumng that the above condton holds. Late, we popose an appoxmaton method to deal wth cases n whch E Y s ae close to the knk ponts. Fom Fgue 3, one can easly obseve that the second and hghe devatves of functon g (.) ae zeo,.e. (n) g =0 fo n. Thus, accodng to ( 0 ) the fst cental moment of the andom vaable W can be obtaned by ( ) E W = g E Y, ( 1 ) and the vaance of W can be calculated by ( ) = va( g( Y ) =E[ g ( Y )] ( E[ g( Y )]) va W. ( ) Lkewse, the functon ( ) sees expanson as follows g Y n ( ) can be appoxmated at o aound E Y usng Taylo's g 1 ( y ) =g ( E[ Y ]) + ( g ) ( E[ Y ]) ( y -E[ Y ]) + ( g ) ( E[ Y ]) ( y -E[ Y ]) + L. ( 3 ) Knowng that (n) g =0 fo n, we have 1

13 ( g ) = gg ( g ) = ( gg ) = ( g ) ( n) ( g ) = 0, n 3. ( 4 ) Theefoe, accodng to ( 3 ) and ( 4 ), E g ( Y ) n ( ) can be obtaned by ( ) = ( ) + ( ) ( ) va ( ) E g Y g E Y g E Y Y Substtutng ( 5 ) and ( 1 ) n ( ), we obtan va ( W ) = ( g ) ( E Y ) va ( Y ). ( 5 ). ( 6 ) Equatons ( 1 ) and ( 6 ) lead to estmatng the mean and vaance of the depatue tme n tems of the functon g (.) and the mean and vaance of the aval tme W at node Y. Howeve, t should be noted that by usng equatons ( 1 ) and ( 6 ) we may face the followng poblem: a slght dffeence n the mean aval tme may lead to a lage dffeence n the vaance of the depatue tme. By means of an example, we addess ths poblem. Consde the gaph G shown n Fgue 4 n whch two statonay yet stochastc outes, labeled 1 and, connect node to node j. We assume that the mean and standad devaton of the tavel tme between nodes and j on oute 1 ae η 1 =1:55 hou and σ 1 =45 mnutes and those of oute ae η =:05 hou and σ =10 mnutes, espectvely. We also assume that the tme wndow at node j s [11:00, 1:30], and the depatue tme at node s 9:00 a.m. η 1 =1:55 h σ 1 =30 mn j l η =1:50 h σ =10 mn Fgue 4: Two outes wth dffeent stochastc chaactestcs connectng node to node j. 13

14 Accodng to ( 1 ) and ( 6 ), the mean and vaance of the depatue tme havng taken oute 1 wll be 1 =11:00 and 1 va ( W j ) E W j =0, and fo the oute ae =11:05 and va ( W j ) E W j =100. That s, a slght dffeence n the mean aval tmes may lead to a lage dffeence n the vaance of the depatue tme. It should be noted that at ponts y j =a j and y j =b j n Fgue 3, the ght and left devatves of g (.) ae not equal. Moe pecsely, Equaton ( 6 ) s vald as long as ( ) ( ) E Y a >> σ Y j j j E Y b >> σ Y j j j, and ( 7 ) hold. Fo the cases n whch the nequaltes n ( 7 ) do not hold, the dffeences n the slope of g (.) aound ponts y j =a j and y j =b j should also be taken nto account. Hee, we appoxmate the value of σ by the standad devaton ( W j ) whee (.) g s gven n Fgue 3 and va( W j ) σ ( W ) EY j + σ ( Y j ) 1 σ ( Wj ) g ( x) dx, ( 8 ) EY j σ ( Yj ) =. j Fo the above dscussed example, we notce that tavelng on any oute 1 o esults n ( ) E Y a <σ Y j j j fo =1,, whch volates the nequaltes n ( 7 ). Appoxmatng the standad 1 devaton of the depatue tme usng ( 8 ), we get σ ( W j ) =0 mnutes, and σ ( W j ) whch shows the advantage of takng oute ove oute 1 as fa as the mnmum ( ) =7.5 mnutes va s concened. Recall that we stated ths subsecton by assumng that thee s no sevce tme assocated wth the nodes of oute. Now f the andom vaable S, whch denotes the sevce tme at node ND, s gven, the tme when sevce stats at node can be appoxmated by equatons ( 1 ) and ( 8 ), espectvely. Hence, gven the mean and vaance of the sevce tme at each node, the mean and W j 14

15 vaance of the depatue tme fom node, 15 ) by whee E g( Y ) and ( ) va ( ) W, can be calculated accodng to equatons ( 14 ) and ( ( ) [ ] E W E = g Y + E S ( ) ( ) ( ) ( ), and ( 9 ) va W = va g Y + va S, ( 30 ) g Y ae the mean and vaance of the tme when sevce stats at node and ae gven by ( 1 ) and ( 8 ), espectvely. Consequently, the fst two moments of the aval tme at node j on oute can be computed usng equatons ( 16 ) and ( 17 ). In summay, gven the depatue tme Y o = y, the mean and vaance of the tavel tme on each ac o (, j) A, the mean and vaance of the sevce tme, and the tme wndow at each node, the mean and vaance of the aval tme at each node takng oute can be ecusvely calculated usng equatons ( 16 ) - ( 17 ), ( 1 ), ( 8 ) - ( 30 ). IV. PROPOSED SOLUTION METHOD FOR THE STSPTW Consde the gaph G as defned n Secton III. Recall fom Secton II that the aval tme at node takng oute s sad to meet the equed sevce level at ths node, say ϒ, f the confdence coeffcent at node, γ, s geate than o equal to ϒ. Fo the sake of smplcty, heeafte, we assume that the sevce level at all the nodes ae equal,.e., ϒ =ϒ,. Thus, oute s sad to be acceptable f the aval tmes at all nodes ND meet the sevce level ϒ. Recall also that, n the STSPTW poblem, the objectve s to fnd the least-cost acceptable oute statng fom ogn o, vstng all nodes n N and endng at destnaton d, such that each node s vsted exactly once. In ths secton, we popose an appoxmate soluton method based on a modfed dynamc pogammng method to fnd the least-cost acceptable oute n the STSPTW. We assume that the mean and vaance of the tavelng tme X j (t) on ac (,j) A, denoted by η (t) and σ ( t ) and j j 15

16 defned n ( 4 ) and ( 5 ), ae gven. We also assume that the mean and vaance of the sevce tme s at each node ND denoted by E[s ] and va(s ), espectvely, ae known a po. We assume that the cost of tavelng on a oute equals to the mean tavel tme on that oute. In ode to apply dynamc pogammng, we defne a state by a -tuple (S,), whee S N s an unodeed set of vsted nodes, and S s the last vsted node. Assocated to each state ae: - The mean, E Y S, and vaance, S va ( Y ), of the aval tme at node takng the path statng fom the ogn passng though evey node of S exactly once and endng at node, - A cost denoted by S C, defned as the cost of tavelng on the path descbed above, and S - A confdence coeffcent γ. Defnton (acceptable ac): Let the aval tme at node meet the equed sevce level,.e., ϒ. An ac (,j) A s sad to be acceptable f the aval tme at node j tavelng on ac (,j) also meets that sevce level. Defnton (state expandablty): Let node S be the last vsted node of the state (S,). The state S s sad to be expandable to node j, j ND and j {S,o}, f the ac (,j) A s acceptable. The modfed dynamc pogammng method s befly descbed n the followng. We note that the wost-case computatonal complexty of the algothm s exponental n the numbe of nodes n gaph G. At each state (S,), the appoxmate soluton algothm fo the STSPTW looks fo uncoveed nodes (.e., j ND and j {S,o}), whch could be added to the set of nodes n S. In ode to educe the computatonal tme, two types of elmnaton tests ae pefomed: ac elmnaton and state elmnaton. 1) Ac elmnaton test: The ac elmnaton test looks one step ahead to educe the numbe of states. Assume that the state (S,) s expandable to node j. The state (S,) wll be expanded to node j f all acs (j,k), k ND/{S, j, o}, ae acceptable. Othewse the state wll be elmnated. ) State elmnaton test. Ths test mplements the dynamc pogammng algothm to educe the numbe of states. Gven states (S 1,) and (S,), whee S 1 and S cove the same set of nodes n 16

17 S S1 S ND, the second state s elmnated f C C (o fo andom vaables E C E C ) S 1 S 1 S1 S and the andom vaable Y domnants Y (e.g., E Y E Y and va S ( ) ( ) 1 S Y va Y ). The followng s the summay of the appoxmate soluton algothm fo the STSPTW. Algothm: S Step 1 (Intalzaton): Level l=1, EW [ ] = Y, va ( ) = 0 o o W ; Fo all expandable nodes N/{o} fom ogn o: o 1) Geneate (S,): S={o,}, ) Compute E Y va( S Y, S, ) S C accodng to ( 16 ) and ( 17 ). Step : (Dung): Level l=l+1, Fo all states (S,) at level l-1, and Fo all expandable nodes j N / S fom state (S,): Pefom Ac and State elmnaton tests, Fo the emanng nodes: 1) Geneate (S {j},j): ) Compute ( 30 ). E Y j Y, S, va( S ) j S C j accodng to ( 16 )-( 17 ), ( 1 ), ( 8 ) - Step 3 (Temnaton): If l< ND go to step. If l= ND, go to step and set j={d}. 17

18 If l= ND +1, fnd the mnmum cost S C d among all states (S,d), that s the best oute. V. EXPERIMENTAL RESULTS To evaluate the effcency of the poposed STSPTW algothm, expements ae pefomed n ths secton on gaphs wth statonay as well as non-statonay tavel and sevce tmes. The algothm s coded n Matlab 6.5 developed by MathWoks, Inc. Expement 1 (Statonay Stochastc Netwok): In ths expement, we consde gaph G n Fgue 5 consstng of 5 nodes n whch node 1 epesents the sngle depot (ogn-depot and destnatondepot) Fgue 5: The gaph G conssts of 5 nodes n Expement 1. We assume that the tavel tme between evey pa of nodes and j n gaph G s statonay,.e., the tavel tme X j s ndependent of tme t, the tme when the vehcle entes the ac (,j) A. The mean and standad devaton of the tavel tme n gaph G s gven by η = , ( 31 ) σ = ( 3 )

19 n matx fom, whee η j and σ j ae the mean and standad devaton of the tavel tme between nodes and j n mnutes, espectvely. Fo the sake of smplcty, we assume that gaph G s symmetc. It should be noted that the poposed algothm s geneal and ts applcaton s not lmted to symmetc netwoks. The mean and standad devaton of the sevce tme of the nodes of gaph G ae gven by [ ] µ =, ( 33 ) [ ] Σ = ( 34 ) n vecto foms, whee µ and Σ ae the mean and standad devaton of the sevce tme at node n mnutes, espectvely. Lkewse, the ealest and latest tme to vst nodes of gaph G ae gven by [ ] α =, ( 35 ) [ ] β = ( 36 ) whee α and β ae the ealest and latest hous of the day to vst node, espectvely. The developed algothm s appled on gaph G wth dffeent values of sevce level ϒ. The lowe bound fo the confdence coeffcent n ( 1 ) s found by applyng the Tchebycheff and Chenoff bounds (see also Papouls 1991) assumng a nomal dstbuton fo all tavel and sevce tmes n gaph G. Table 1 summazes the computatonal esults fo each sevce level. The cost s equal to the mean tavel tme. Table 1: Computatonal esults fo statonay, STSPTW n Expement 1. Tchebycheff Bound Chenoff Bound Sevce level Cost of the Route Cost of the Route Best Route Best Route [mn.] [mn.] ϒ=80% ={ } a 465 ={ } 495 ϒ=90% ={ } 495 ={ } 495 ϒ=95% ={ } 510 ={ } 495 ϒ=97.5% ={ } 510 ={ } 510 ϒ=99% NA b NA ={ } 510 a) The optmum oute fo detemnstc gaph,.e. δ=0 and Σ=0, see also (Dumas et al., 1995). b) NA: Not Avalable,.e. no oute could be found. 19

20 Table 1 ndcates that as the value of the sevce level nceases, the cost of the best oute nceases too. That s, wth a hgh value of ϒ, moe outes become unacceptable, thus, educng the numbe of acceptable low-cost outes n the algothm. The estmated mean and standad devaton of the aval tmes at the nodes of the best outes n Table 1 ae shown n Table ows and 3. These estmated values ae found usng the developed method dscussed n Secton III. To evaluate the estmated aval tmes at the nodes and to valdate the esults of Table 1, the best outes obtaned n Table 1 ae smulated 10,000 tmes usng the same paametes gven n ( 31 ) to ( 36 ). In each tal, the tavel tme on the lnks and the sevce tme at the nodes of the best outes ae geneated usng a nomal andom numbe geneato wth the mean and standad devaton gven n ( 31 ) to ( 34 ). Table ows 4 and 5 pesent the mean and standad devaton of the aval tmes at nodes of the smulated best outes. Table ow 6 shows the pecentage of tmes that the smulated outes met the tme wndow constants at the nodes gven n ( 35 ) and ( 36 ). Ths pecentage s called the oute success ate n Table. Note that the mean and standad devaton of the aval tmes at the nodes gven n ows 4 and 5 ae calculated fo smulated outes whch managed to meet all tme wndow constants. Table : Estmated vs. smulated fst two moments of aval tmes at the nodes of the best outes n Expement 1. Route ={ } ={ } ={ } Mean Estmated {8 9:30 11:30 1:45 14:30 15:57} {8 9:30 10:45 1:45 14:0 16:15} {8 9:0 11:15 1:30 13:45 16:30} [h] aval tmes Std at nodes a { } { } { } [mn] Mean Smulated {8 9:9.5 11:8.8 1: :8.0 15:43.1} {8 9:9.9 10:44.8 1: : :14.4} {8 9:0 11:15.1 1:30 13:45 16:30.1} [h] aval tmes Std at nodes { } { } { } [mn] Route success ate 95.61% 99.39% 100.0% a) Std: Standad Devaton. Compason between Table ows and 3 wth ows 4 and 5 shows how good the developed estmaton method s. In all cases, ou estmaton method s close to the smulated values, especally fo ={ }. 0

21 Table ow 6 also ndcates that the poposed appoxmate soluton method was able to fnd a good soluton to the STSPTW poblem. Moeove, t shows that the Tchebycheff bound geneates tghte lowe bounds fo the lowe values of the sevce level,.e. ϒ=80%; the Chenoff bound s much tghte fo the hghe values of the sevce level,.e. ϒ {95%, 97.5%, 99%}. Expement (Statonay STSPTW): In ths expement, we ae nteested n evaluatng the computatonal aspects of the poposed algothm. Hee, the expemental test conssts of a Eucldean plane n whch the nodes coodnates ae unfomly dstbuted between 0 and 50, and the coodnates of the depot s at 0 and 0. The mean tavel tme equals dstance, and the standad devaton of the tavel tme s set to be one tenth of the mean tavel tme. We assume that thee s no sevce tme assocated wth the nodes. The tme wndow at each node s geneated aound the tme to begn sevce at that node accodng to the fst neaest neghbo TSP tou (Renelt, 1994). That s, assumng tavel tme equals dstance, we found the best detemnstc TSP tou based on the fst neaest neghbo heustc algothm. Accodngly, the tme to each node takng the geneated tou, say T, s calculated and the tme wndow a, ] s geneated aound ths tme by [ b whee w=0, 30, 40, 60, and 80 mnutes. w a = max 0, T, ( 37 ) b w = T, ( 38 ) + The poposed soluton method s appled to the geneated gaph fo a sevce level ϒ=90% usng the Tchebycheff bound. Table 3 pesents the expemental esults wth a dffeent numbe of nodes (customes), N, and dffeent tme wndow wdths, w. The cost s equal to the mean tavel tme. The second column n Table 3 shows the detemnstc cost of takng the fst neaest neghbo TSP soluton. Ths soluton, howeve, may not be acceptable fo the STSPTW. The expements ae tested on an Intel Pentum M, 1.6 GHZ. In Table 3, each set of nodes (ow) s bult upon the pevous ow. Fo nstance, fo the numbe of nodes equal to 30, we kept the same 1

22 andomly geneated nodes fo N=0 and added 10 newly geneated ones. A new fst-neaest-neghbo TSP tou s then found and a tme wndow s assgned to each node accodng to ( 37 ) and ( 38 ). Table 3: Computatonal esults fo statonay, STSPTW n Expement usng Tchebycheff bound wth ϒ=90%. No of nodes a w=0 Mn w=30 Mn w=40 Mn w=60 Mn w=80 Mn Int Best b CPU c Best CPU Best CPU Best CPU Best CPU NA d NA NA NA NA NA NA NA NA NA NA NA a) The cost of the ntal oute n mnutes geneated by the detemnstc fst neaest neghbo TSP heustc. b) The cost of the best soluton n mnutes usng the poposed appoxmate STSPTW method. c) The CPU tme n seconds on a Pentum M (1.6 GHz) pesonal compute. d) NA: Not Avalable,.e., no oute could be found. The esults n Table 3 ndcate that the appoxmate algothm s successful n solvng poblems up to 80 nodes wth faly wde tme wndows. The esults also show that the CPU tme nceases wth the wdth of the tme wndows. It should be noted that as the wdth of the tme wndows and the numbe of customes ncease, the numbe of acceptable outes nceases shaply. Thus, the algothm needs moe tme to fnd the best oute. Expement 3 (Non-statonay Stochastc Netwok): Consde gaph G n Fgue 5 agan. In ths expement, we assume that the tavelng tme on some acs of gaph G ae non-statonay stochastc pocesses. Let η j (t) and σ j (t) n ( 40 ) and ( 41 ) be the mean and standad devaton of the stochastc pocess X j (t) (the tavel tme between nodes and j n mnutes),

23 0 xt ( ) yt ( ) xt ( ) 0 xt ( ) η() t = yt () xt () , ( 39 ) x( t) xt ( ) xt ( ) 0.1 yt ( ) xt ( ) xt ( ) σ () t = 0.1 yt ( ) 0.1 xt ( ) , ( 40 ) x( t) xt ( ) 0 whee the functons x(t) and y(t) ae gven gaphcally by Fgue 6 and Fgue 7, espectvely. In othe wods, we assume that the tavel tmes on acs (1,), (1,3), (,3), and (4,5) ae non-statonay andom pocesses. Snce gaph G s assumed to be symmetc, the tavel tmes on acs (,1), (3,1), (3,), and (5,4) ae also non-statonay wth the same chaactestcs as the countepats (1,), (1,3), (,3), and (4,5), espectvely. Thee dffeent scenaos ae consdeed n Fgue 6 and Fgue 7 fo x(t) and y(t): a) statonay tavel tme, b) non-statonay tavel tme wth elatvely low dynamcs, and c) non-statonay tavel tme wth elatvely hgh dynamcs. In Expement 1 scenao a was nvestgated. 3

24 90 85 Mean tavel tme (mn.) (a) (b) (c) Hou of the day Fgue 6: The mean tavel tme x(t) n mnutes vesus the tme of the day. (a) statonay, (b) nonstatonay wth low dynamcs, and (c) non-statonay wth hgh dynamcs Mean tavel tme (mn.) (a) (b) (c) Hou of the day Fgue 7: The mean tavel tme y(t) n mnutes vesus the tme of the day. (a) statonay, (b) nonstatonay wth low dynamcs, and (c) non-statonay wth hgh dynamcs. In ths expement, the poposed soluton method s appled to gaph G wth non-statonay tavel tmes on some acs as explaned above. Dffeent values of sevce level ϒ and dffeent lowe confdence bounds ae consdeed. Table 4 summazes the computatonal esults n whch the cost s equal to the mean tavel tme. 4

25 Table 4: Computatonal esults fo non-statonay, STSPTW n Expement 3. Sevce level Non-statonay wth hgh Non-statonay wth low dynamc Lowe dynamc bound Cost of the Cost of the Best Route Best Route Route Route ϒ=80% Tchebycheff ={ } ={ } 514. Chenoff ={ } 50 ={ } 514. ϒ=90% Tchebycheff ={ } 50.0 ={ } 56.3 Chenoff ={ } 50 ={ } 514. ϒ=95% Tchebycheff ={ } ={ } 56.3 Chenoff ={ } 50 ={ } 56.3 ϒ=97.5% Tchebycheff ={ } NA a NA Chenoff ={ } ={ } 56.3 ϒ=99% Tchebycheff NA NA NA NA Chenoff ={ } ={ } 56.3 a) NA: Not Avalable,.e. no oute could be found. Usng the developed estmaton method n Secton III, the estmated mean and standad devaton of the aval tmes at the nodes of the best outes of Table 4 ae calculated and shown n ows and 3 of Table 5 and Table 6. Table 5 shows the esults fo the non-statonay tavel tmes wth elatvely low dynamcs, whle Table 6 s fo the non-statonay tavel tmes wth hgh dynamcs. The esults n Table 4 and the estmated aval tmes n Table 5 and Table 6 ae valdated and smulated though unnng a smulaton pogam fo 10,000 tmes. In each tal, the tavel tmes on the lnks and the sevce tme at the nodes of gaph G ae geneated usng a nomal andom numbe geneato wth the mean and standad devaton gven above. The mean tavel tmes on acs (1,), (1,3), (,3) and (4,5) ae geneated accodng to Fgue 6 and Fgue 7 at each nstant of tme. Table 5 and Table 6 also pesent the pecentage of tmes (oute success ate) that the smulated outes met the tme wndow constants n the non-statonay, stochastc gaph G descbed above. Table 5: Estmated vs. smulated fst two moments of aval tmes at the nodes of the best outes n Expement 3, non-statonay netwok wth low dynamcs. Route ={ } ={ } ={ } Mean Estmated {8 9: :30.5 1: : :43.5} {8 9: :56.4 1: : :.0} {8 9: :4.4 1: : :35.6} [h] aval tmes Std at nodes a { } { } { } [mn] Mean Smulated {8 9: :9.5 1: :30. 15:4.0} {8 9: :56. 1: : :1.} {8 9: :4.4 1: : :35.6} [h] aval tmes Std at nodes { } { } { } [mn] Route success ate 95.49% 98.98% 100.0% 5

26 Table 6: Estmated vs. smulated fst two moments of aval tmes at the nodes of the best outes n Expement 3, non-statonay netwok wth hgh dynamcs. Route ={ } ={ } ={ } Mean Estmated {8 9:31. 11:31. 1: : :44.9} {8 9:31. 11: : :19. 16:34.} {8 9: :39.4 1: : :46.3} [h] aval tmes Std at nodes a { } { } { } [mn] Mean Smulated {8 9: :9.8 1: : :4.7} {8 9:31. 11: :1. 14: :33.8} {8 9: :39.5 1: : :46.1} [h] aval tmes Std at nodes { } { } { } [mn] Route success ate 9.81% 97.89% 99.99% a) Std: Standad Devaton. Compason between Table 5 ows and 3 wth ows 4 and 5 shows that ou developed method was able to estmate the aval tmes at the nodes of the non-statoney stochastc netwok wth vey small eos. These eos ae slghtly hghe n Table 6 due to the hghe dynamcs of the netwok. Table 5 and Table 6 ows 6 ndcate that the poposed algothm s successful n fndng a good appoxmate soluton fo the non-statonay STSPTW poblem. As expected, the method tends towad fndng moe consevatve solutons fo hghly dynamc poblems. The esults also demonstate the advantage of usng the Chenoff bound ove the Tchebycheff bound fo hghly dynamc poblems. Expement 4 (Non-statonay STSPTW): Hee, we evaluate the computatonal aspects of the poposed algothm fo the non-statonay, STSPTW. In ths expement, we adopt the same geneated netwoks as descbed n Expement. We kept half of the lnks n that netwok statonay and alteed the othe half to non-statonay lnks. Moe pecsely, the tavel tme of the lnk (,j) s kept unchanged (statonay) f +j s an even numbe. If the sum s an odd numbe, the mean tavelng tme on the lnk (,j) s assumed to follow the scheme shown n Fgue 8. Shown n Fgue 8 s the mean tavel tme η j (t) nomalzed by ts statc countepat, η j, fom Expement. 6

27 Nomalzed mean tavel tme (mn.) Fgue 8: The nomalzed mean tavel tme on lnk (,j) n mnutes vesus the tme of the day. The mean tavel tme n nomalzed by ts statc countepat η j fom smulaton Expement. In ths expement, we assume that the standad devaton of the tavel tme at each tme s one tenth of the mean tavel tme at that tme, and that no sevce tme s assocated wth the nodes of the netwok. We also assume that the cost of the tavelng s equal to the mean tavel tme at that tme, and that the sevce level s ϒ=90%. Table 7 pesents the expemental esults fo dffeent numbe of nodes (customes), N, whee the Tchebycheff bound s used. The tme wndow wdths ae assumed to be w=30, 40, 60, and 80 mnutes Hou of the day Table 7: Computatonal esults fo non-statonay, STSPTW n Expement 4 usng Tchebycheff bound wth ϒ=90%. No of nodes w=30 Mn w=40 Mn w=60 Mn w=80 Mn Best a CPU b Best CPU Best CPU Best CPU NA c NA NA NA NA NA NA NA NA NA NA NA a) The cost of the best soluton n mnutes usng the poposed appoxmate STSPTW method. b) The CPU tme n seconds on a Pentum M (1.6 GHz) pesonal compute. c) NA: Not Avalable. 7

28 The esults n Table 7 ndcate that the appoxmate STSPTW algothm was able to solve nonstatonay stochastc poblems wth up to 80 nodes wth faly wde tme wndows. As expected, the CPU tme nceases wth the tme wndow wdth and poblem sze. VI. SUMMARY AND CONCLUSION In ths pape, a tuck oute plannng poblem called Stochastc Tavelng Salesman Poblem wth Tme Wndows (STSPTW) n whch tavelng tmes along the oads and sevce tmes at the custome locatons ae stochastc pocesses s nvestgated. The objectve s to fnd the least-cost oute among all outes whch meet the equed sevce level at the custome locatons. To do so, we developed a methodology to estmate the fst and second moments of aval tme at each custome locaton. In addton, we developed a methodology to addess the exstng tme wndows n the STSPTW poblem. We futhe poposed an appoxmate soluton methodology based on a modfed dynamc pogammng method to fnd the least-cost oute n the STSPTW. The esults fom vaous expements ndcate that ou developed method was able to estmate the mean and standad devaton of the aval tmes at the nodes of both statonay and non-statonay netwoks wth vey small eos. The esults also show that the appoxmate soluton method s effcent fo both statonay and non-statonay STSPTW poblems and that the appoxmate algothm s successful n solvng STSPTW poblems wth up to 80 nodes wth faly wde tme wndows. In ths pape, we compaed the use of the Tchebycheff and Chenoff bounds n fndng the confdence levels fo the appoxmate solutons. We obseved, though expemental esults, that the Tchebycheff bound geneates tghte lowe bounds fo low values of the sevce level, wheeas the Chenoff bound s much tghte fo the hghe ones. The expemental esults also demonstated the advantages of usng the Chenoff bound ove the Tchebycheff bound fo the hghly dynamc nonstatonay STSPTW poblem. Howeve, t should be noted that, n geneal, the Chenoff bound s moe dffcult to compute and knowledge of the chaactestc functon s needed a po. The fndngs of ths eseach have sgnfcant pactcal elevance. One of the pmay ctcsms of applyng stochastc optmzaton methods to solvng outng poblems s the lack of technques that ae 8

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