The Dynamic Traveling Repair Problem: Examination of An Asymptotically Optimal Algorithm

Transcription

1 The Dyamc Travelg Repar Problem: Examato of symptotcally Optmal lgorthm Xagwe Lu Isttute of Trasportato Studes Uversty of Calfora, Irve mela Rega Departmet of Cvl ad Evrometal Egeerg d, Graduate School of Maagemet Uversty of Calfora, Irve Sadra Ira Departmet of Iformato ad Computer Scece Uversty of Calfora, Irve bstract The dyamc travelg repar problem volves provdg servce to customers whose locatos are uformly dstrbuted over a bouded area the Eucldea plae We assume that customer requests arrve accordg to a Posso pot process Earler research provded a coecture that the asymptotcally optmal algorthm for ths problem uder very heavy traffc testy volves the followg: partto the bouded area to sub-regos, wat for suffcet demad to accumulate the sub-regos, serve the demads the sub-regos accordg to the optmal TSP tour, ad vst the sub-regos frst-come frst-served order as a GI/G/m queue Further, the researchers coectured that the optmal algorthm for the sgle server case ca be exteded to the m-server problem by smply parttog the servce rego to m sub-regos of the same sze ad assgg oe vehcle to a sub-rego I ths paper we defe a class of algorthms whch cludes the above algorthm We the demostrate the asymptotc optmalty of a algorthm ths class ad that the above algorthm s optmal amog the class Therefore, we prove the frst coecture made by the researchers Fally, we argue that thecoecturemadeaboutthemultpleservercasesalsotrue Keywords: Dyamc ad stochastc routg ad schedulg, Dyamc Travelg Repar Problems, Probablstc alyss of lgorthms, symptotc Optmalty July, 00

2 Itroducto The dyamc travelg repar problem (DTRP) s the followg: m moble servers are postoed wth a bouded rego the Eucldea plae The servers travel at a fxed, costat speed v per tme ut Servce requests arrve over tme accordg to a Posso process wth arrval rate λ Whe requests arrve, they are dstrbuted to the bouded rego depedetly accordg to a uform dstrbuto The server must sped some tme travelg to the customer locatos ad t must sped some tme provdg o-ste servce The o-ste servce tme for each customer s depedetly ad detcally dstrbuted accordg to a dstrbuto wth mea s ad varace σ s The goal s to mmze the average watg tme of all customers I ths paper we exame ths problem ad address a coecture about the asymptotc optmalty of a parttog algorthm for sequecg servce to customers made by earler researchers The examato of the asymptotc performace of heurstc algorthms for vehcle routg ad schedulg s a commo aalyss techque symptotc performace s thought to provde better sght to a heurstcs typcal performace tha worst case aalyss (See for example, Bramel ad Smch-Lev, 997) I a paper ttled Stochastc ad Dyamc Vehcle Routg wth Geeral Demad ad Iterarrval Tme Dstrbutos, Bertsmas ad va Ryz (993, p 96) examed the followg / / GI G m servce polcy, whch we refer as the BvR (, k ) algorthm:

3 For a fxed teger k, the servce area s parttoed arbtrarly to k areas of equal sze Servce requests are dstrbuted uformly over the whole area Whe batches of k customer requests accumulate a partto, these are deposted to a queue a frst-come frst-served maer as a GI / G/ m queue I each partto, customer requests are served accordg to the optmal TSP tour across ther locatos We defe the expected fracto of tme the vehcle speds provdg o-ste servce as follows: ρ = λs / m The researchers coectured that there exsts a fucto g( k, ρ ) whch determes, ( g( k, ρ )) = such that as ρ ad k, the ( (, ρ ), k ) algorthm s asymptotcally optmal We refer to BvR ( g ( k, ρ ), k ) BvR g k as the optmal BvR (, k ) algorthm ad use BvR (, k ) * to deote t We cosder a class of Geeral Partto lgorthms whch clude the BvR (, k ) algorthms as sub-class ad develop a lower boud o the average wat tme uder ths class of algorthms We develop the lower boud for two dfferet systems cofguratos I the frst, the sze of the parttos depeds upo the umber of customers the system I the other, the parttos are fxed a pror We refer to these as small partto ad fxed partto cases We obta exactly the same boud for geeral partto algorthms appled to these systems Ths lower boud matches the

4 upper boud o the average watg tme provded by BvR (, k ) * Therefore, we show that BvR (, k ) * s optmal amog algorthms ths class Fally, we detfy a algorthm fallg to ths class whose asymptotc (heavy-traffc) performace s close to optmal Therefore, we also prove the asymptotc optmalty of BvR (, k ) * The dyamc travelg repar problem falls to a more geeral class of stochastc vehcle routg problems Ths class cludes the dyamc travelg salesma problem (DTSP), the probablstc travelg salesma problem (PTSP), the probablstc travelg salesma locato problem ad other problems such as the probablstc shortest path problem ad dyamc vehcle allocato problem excellet revew of these problems Powell, Jallet ad Odo (995) provde a I the DTSP, customer locatos are kow advace ad servce requests arrve accordg to a Posso pot process at each ode The obectve s to determe a dspatchg strategy whch mmzes customers expected watg tme That problem s dscussed Psarafts (988) I the PTSP, a a pror tour must be costructed for a etwork whch each ode has a gve probablty of requrg a vst (Jallet, 988) The probablstc travelg salesma faclty locato problem volves detfyg the optmal locato for a depot ode a etwork whch the probablty that customers wll requre a vst s kow (Berma ad Smch-Lev, 988, Bertsmas, 989) 3

5 lgorthms ad Propertes for the Sgle Server Case Defto ad Notato Bertsmas ad va Ryz showed that whe ρ s less tha oe, there exsts a fucto of (,, k ) g( k, ρ ) whch determes, = g( k, ρ ) such that such that the BvR g ( k ρ ) algorthm ca satsfy all servce requests Ths mples that N, the expected umber of requests queue s fte ssume for ow that there s ust a sgle server ad that the servce rego s a ut square We umber the demads accordg to the order whch they are served Let d be the dstace traveled from ( ) th demad to th demad Let s be the o-ste servce tme for demad The total servce tme cludes the travel tme d v ad the oste servce tme s If, for all tmes t, the umber of watg requests the system s bouded almost surely uder a specfc polcy, we call ths a stable polcy Usg the deftos ad otato preseted by Bertsmas ad va Ryz (99, 993), for a stable polcy, we let W deote the watg tme for demad The watg tme s the tme betwee the arrval of demad ad the arrval of the server at the locato of demad The lmtg expected values of these radom varables are defed as W lm EW [ ] ad, d lm Ed [ ] =, = N, the expected umber of requests the queue, s equal to λw s do Bertsmas ad va Ryz, we assume these lmts exst 4

6 Polces of Iterest Let W( x) be the expected watg tme for a radomly selected customer located at pot x d, let W be the average watg tme uder the algorthm of terest We oly cosder algorthms that satsfy the followg codto: there exsts ϖ ad ϖ,suchthat W x 0 < ϖ ϖ () W Ths s a techcal requremet for our proof If ϖ grows large ad ϖ grows small, costrat () becomes progressvely less tght ad the class of algorthms satsfyg the costrat creases 3 Geeral Partto lgorthms We ow defe a class of algorthms for the sgle vehcle DTRP whch we refer to as Geeral Partto lgorthms These work as follows: usg a grd, dvde the area to k parttos of equal sze geeral partto algorthm wll, for each rego, partto tme to perods Let t, deote the ed of rego 's th perod t,0 s defed to be 0 for each rego The server the serves the requests a sequece of vsts I each vst, the server selects a rego to serve based o the accumulated demad each rego Ths s the oly formato cosdered the sequecg decso I the th vst to rego, the server wll serve exactly those requests whch arrve rego the terval from t, through t, The order whch the regos are vsted ad the way 5

7 that tme s parttoed for each rego wll deped o the partcular partto algorthm appled If the server travels across regos whch there are o watg customers, e route to a rego whch there are watg customers, we say that the empty rego has also bee vsted BvR (, k ) s a geeral partto algorthm whch a perod eds for a rego whe k ew requests have accumulated that rego The regos are vsted frst-come-frstserve order accordg to whe each perod eds Note that t s ot requred that the curret perod for a rego be completed whe the server arrves that rego For example, the class of Exhaustve Partto lgorthms wll vst a rego, wll serve all watg customers, ad wll also serve those customers that arrve whle servce s beg provded to watg customers Thus, the stoppg crtero for a perod s whe there are o outstadg requests the rego 4 Propertes of Geeral Partto lgorthms We wll ow show how a geeral partto algorthm gves rse to a dstrbuto fucto f whch descrbes the spatal dstrbuto of cosecutvely served requests Fx a geeral partto algorthm ad postve tegers M ad We are terested the th request through the ( M ) + request However, for coveece, we would lke to focus o a th Note that Bertsmas ad va Ryz used a sweep algorthm rather tha a grd y method to develop equal parttos wll do We use a grd to facltate the developmet of our proof 6

8 sequece of requests whch start ad ed at the boudares of vsts Suppose that the th request s served the mddle of the l th vst Let rb be the frst request served vst l (We use b for `beg') Now suppose that the ( M ) + request s served the mddle th of the th p vst Let re be the last request served vst p (Weusee for `ed') We wll focus o requests r, L, r b e Suppose that requests r,, r b L e comprse q cosecutve vsts Suppose that r dstct regos are vsted durg these q vsts We call these R,, R Fx these q vsts, let r v, v, L, vr be the frst vst to rego R,, L R ad v ' ' ' r, v, L, vr be the last vst to rego L R,, R r amog to these q vsts Rememberg that each vst cossts of a arrval perod ad that each vst clears oly the customers arrvg durg that perod For each L {,, r} L,let t beg, refer to the begg of v ad t, ed refer to the ed of ' v Therefore, t ed, t, beg refers to the tme terval assocated wth the Posso arrval process for rego R Let Z deote the umber of requests served rego R durg the terval t beg, to t ed, Z s exactly the umber of requests that arrve R durg the terval t beg, to t ed, Weuse R to deote the area of partto R Z s therefore a Posso radom varable wth mea R ( ted, tbeg, ) λ Scethe parttos are of equal sze, R s equal to k,where k s the umber of parttos Thus, Z s a Posso radom varable wth mea λ ( ted, tbeg, ) Further, { Z, =, L, r} k 7

9 are depedet Posso radom varables The depedece results from the fact that the regos are dsot Next we geerate a radom permutato x,, x of the set of cosecutvely served requests r,, r b L e L where s the umber of requests We observe the followg: We have r depedet Posso radom varables, each represetg the umber of customers rego R, wth mea λ ( ted, tbeg, ) k Further, the locatos of these requests are uformly dstrbuted rego x R Ι x = defe a R Let { R 0 otherwse dex fucto over R The locatos of the served requests, x,, x,arearealzato L of d radom varables wth probablty desty fucto of the form f ( x) = ( t ed, tbeg, ) ΙR ( x) = r (,, ) = t t R ed beg () 5 The Small Partto Case Let N represet the expected umber of customers queue s before, let R be the sze of the parttos For ay fxed θ ε ad θ ε, we cosder oly those geeral partto algorthms uder whch θ s NR θ ε Forayε >0, let M = Nε ad assume that M s a teger For a fxed geeral partto algorthm Γ, whe the system s steady-state, for ay radomly selected request, we are terested requests from request through the ext M cosecutvely served requests We expad the sequece of requests as descrbed the prevous sectos so that ths sequece begs ad eds at the boudares 8

10 of vsts Let Q deote the radom varable that dcates the umber of requests ths expaded sequece Note that f the area of each partto s proportoal to ( ρ ) ths esures that the probablty that the umber of request a radomly selected rego s more tha Nε s eglgble Wth very hgh probablty, Nε Q 3Nε,as ρ approaches oe Let x, x,, xq represet the locatos of Q cosecutvely served customers Let the dex represet the order whch servce s performed Now let y, y,, yq be a radom permutato of x, x,, x M Therefore, y, y,, ym are, dstrbuted depedetly accordg to a dstrbuto fγ ( x) where fγ ( x) s a pece-wse (dscotuous) fucto Let W( x) be the expected watg tme for a radom selected customer that s located at pot x We preset three propostos about f ( x) Γ ad W( x ) Let W be the average watg tme for fxed algorthm Γ, we show that that the expected watg tme for ay M demads ca be bouded by W Ths leads to a costrat (5) o f Γ ( x), the dstrbuto of customer locatos Proposto If θ s NR θ, the expected watg tme for a customer located x s W( x )Whe ρ s bg eough W ( x) ε ( x) QfΓ 4λ (3) λr 9

11 Proposto If we focus o polces uder whch W x 0 < ϖ ϖ,whe ρ s large W eough, the f ( x) 4 ϖ + (4) Γ ε εθ s Proposto 3 For ay costrat f ε >0, whe tme s suffcetly large, f ( x) N ( + ε ) Γ x dx Qω Γ satsfes the followg (5) where ω = λr s + / v To obta our lower boud for W, we aalyze the average dstace betwee cosecutve demads served Frst, we obta a lower boud for the dstace traveled to serve all of these selected M demads expressed f ( x) Γ whch leads a lower boud o the average dstace traveled per customer served Ths s show lemma We obta lemma by geeralzg the classcal TSP result of Beardwood et al(96) ad applyg a smoothg techque to the dstrbuto fucto f ( x) Γ Next, mmzg the lower boud obtaed uder costrat (5) leads to our lower boud for the average dstace traveled to serve each customer (6a) We use lemma 3 to obta the lower boud (6a) We the provde a lower boud o the average watg tme for servce (6b) based o (6a) For the proof of theorem ad the related lemmas, please see secto 5 ad the attached appedx Lemma Let d be the expected average dstace traveled per demad served ad let N be the average umber of customers awatg servce, R be the sze of the partto ad 0

12 Q represet the umber of cosecutvely served customers We cosder algorthms for whch for ay ε >0, we fx θ ε ad θ ε For ay geeral partto algorthms uder whch θ s NR θ ε, for ay ε > 0 ρ 0, such that whe ρ > ρ0, Qd f x dx,whereβs the TSP costat defed by Beardwood, Halto β ε Γ ad Hammersley (96) Lemma Lettg Z =m = x subect to = x + ε ad x =, = Z + ε Theorem Let d be the expected dstace traveled per demad served, N be the expected umber of customers awatg servce ad R be the sze of each partto For ay fxed θ ε ad θ ε, for ay geeral partto algorthm uder whch followg result holds: θ s NR θ ε,the lm N d β ρ (6a) { ρ W} lm (- ) λβ m v ρ (6b) Note that whe θ s s larger ad θ ε s smaller, the class of geeral partto algorthms cosdered becomes broader t some pot, all geeral partto algorthms obey the requred codtos

13 Fally, as ρ we fd that f we wat to mmze the average watg tme, uder the optmal algorthm amog the class of algorthms cosdered, the dstrbuto fucto f ( x) that descrbes the spatal dstrbuto of M cosecutvely served customers wll be almost uform a small area ad zero over all other areas I fact, the lower bouds (equatos 6a ad 6b) hold for a class of algorthms that satsfyg the followg codto For ay ε > 0,wecosder M = Nε cosecutvely served customers ad let y, y,, ym be a radom permutato of the locatos Cosder algorthms uder whch y, y,, ym are dstrbuted depedetly accordg to a dstrbuto f ( x ),where f ( x) s requred to satsfy the followg codto: there exsts α such that f x f y α x y,where x y s the dstace betwee x ad y Note that α does ot deped o ρ 6 The Fxed Partto Case Up to ths pot, we have examed algorthms for whch the umber ad sze of the parttos depeds upo ρ Now we cosder algorthms for whch the parttos are pre-determed Frst use a grd to dvde the ut square to fxed small parttos Let be the area of each partto Whe the server selects a partto to serve t wll ether fsh all customers watg the queue at the selected partto or fsh all the customers

14 arrvg durg some arbtrarly selected perod We show that as approaches zero, the average watg tme for servce wll be bouded from below by λβ mv(- ρ) Theorem For ay algorthm fallg to the category metoed above, whe s ρ small eough, lm W ( ρ ) the boud for the small partto case λβ > Note that ths exactly matches equato (6b), v m 7 symptotcally Optmal Geeral Partto lgorthm I ths secto we show the asymptotc optmalty of a specfc geeral partto algorthm Let P be the partto whch dvdes the area to * squares ad OPT deote the optmal algorthm For a specfc arrval sequece, let σ = r, r, deote the order whch the requestsare satsfed by OPT Gve a partto P of the L area, we wll devse a Geeral Partto lgorthm called P based o the behavor of OPT We take the sequece σ ad remove some of the requests to obta aother ' sequece σ as follows: Exame each r tur Suppose that request r s located rego R Suppose that at the tme that r s reached by OPT 's server, there are other outstadg requests R Remove these addtoal outstadg requests from σ ad cotue The requests that were removed from σ wll be called extra requests We ' deote the sequece σ = r, rk, The algorthm P wll work as follows: for each request ' r σ, vst rego R ad satsfy r ad ay extra requests whch are watg 3

15 R at the tme OPT serves r I other words, the perods are chose so that whe OPT serves a request r from σ ', the curret perod for R eds Proposto 4 (Bertsmas ad va Ryz) d * λ γ N + m/ where γ s a costat, γ, m s the umber of servers 3 π Proposto 5 For a fxed arrval rate λ,let * d λ deote the average dstace traveled per customer for the optmal algorthm For ay ε, for the partto P,whe s large eough, the average dstace traveled per customer served for the algorthm defed above s at most * d λ + ε N Proof: The maxmum dameter of a rego uder a gve partto P s,theeach extra request troduces a dstace of at most to P Whe N, we kow N Thus, as the parttos become more fe, the addtoal dstace troduced by a extra request decreases Furthermore, as the parttos become more fe, the probablty that ay gve request s a extra request also becomes smaller Usg these two facts, we coclude that for ay arrval rate λ,whe bg eough, the 4

16 average dstace traveled per customer served for the algorthm defed above s at most * d λ + ε N If we let d λ, p propostos 4 ad 5, we show that be the average dstace traveled per customer served, combg d λ, p = * d λ + ( * d λ ) ο 8 The Optmalty of BvR (, k ) * Lemma3(Bertsmas ad va Ryz) mog the Geeral Partto Class W * λβ +Ο,where m s 3/ mv ( ρ) ( ρ) the umber of servers, λs ρ = m Lemma 3 provdes a upper boud for the average watg tme uder the BvR (, k ) * algorthm Theorem provdes a lower boud for all geeral partto algorthms s ρ these bouds coverge Therefore, we show that uder hgh traffc testy, BvR (, k ) * s asymptotcally optmal amog the Geeral Partto class Sce we have show the asymptotc optmalty of oe algorthm ths class as we show that BvR (, k ) * s asymptotcally optmal amog the whole class, the we have also demostrated the asymptotc optmalty of BvR (, k ) * 5

17 3 The M-Server Case Now we assume that stead of a sgle server that there are m moble servers the Eucldea servce rego Comparg the sgle server ad multple server cases we fd the followg terestg result If we dvde the area to m sub-regos of equal sze ad assg each server to a sgle sub-rego, f each server works depedetly, the algorthm that s asymptotcally optmal for the sgle server case s also asymptotcally optmal for m server case 4 bbrevated Proof of the Theorems (detals the appedx) 4 Proof of the Small Partto Case Proof of Theorem From lemma, for ay ε > 0, ρ 0,whe ρ > ρ0, Qd f x dx β ε Γ ( ) β ε N f x dx Γ Nd Q From proposto 3, for the above fxed ε 0 Where ω = as ρ λr s + / v >,wehave, f ( + ε + ε ) N 3 Γ x dx, Qω Usg lemma, whe ρ ρ0 >, N d ( β ε ) ω 3 = ( β ε ) ( + ε + ε ) 3 ( + ε + ε ) Lettg ε 0,wehave lm ρ N d β 3 ( + ε ) Let ε 0,wehave(6a): lm ρ Nd β 6

18 Now we show (6b) based o (6a) Recall that λ s the average arrval rate for each server ad that m d s + s the actual v average servce tme for each demad I a stable system λ d s + m v must be less tha We kow that λ β λs s+ + o < Recallg that N = λw ad that ρ =, m v N N m λβ m v we have, lm ρ {(- ρ) W} 4 Proof of the Fxed Partto Case Proof of theorem 3: We observe that these systems, a server may arrve at a rego, ad provde cotuous servce to customers utl there are o customers the rego We call ths the tal busy perod The server may the rema dle at the curret rego utl a ew customer arrves t that tme t eters to what we refer to as a subsequet busy perod I prcple, a server may have may of these subsequet busy perods (later we show that oly poor algorthms wll allow the server to rema dle) Let p represet the fracto of customers that arrve durg the tal busy perods ad( p) represet the fracto that arrve durg the ether the dle perods or the subsequet busy perods Let Defe Z be the umber of customers served durg the tal busy perod for rego X to be the umber of customers served durg the subsequet busy perods for R rego R Let p represet the fracto of customers served at rego R durg the 7

19 tal busy perod ad ( p ) represet the fracto of customers served rego R durg subsequet busy perods t steady state we kow that [ ] [ ] Ε X p p = Ε = Ε Ε Z p p [ X ] [ Z ] From ow o, we oly cosder the case whch ρ s relatvely large ( 3 ρ > ) 4 Observato : (The costrats o Ε[ Z ] ad [ Z X ] Ε + ) We show the appedx that whe we have at least two parttos ( Λ< ), a ecessary codto for the system to be stable s p > If we oly cosder the algorthms whch satsfy the costrats that W x 0 < ϖ ϖ, W we kow that whe 3 4 ϖ NΛp 4 p p ρ >, Ε[ Z ] > ( ϖ NΛ ) + 3 Let g() be the rato of the average watg tme for partto total umber of customer served durg oe vst to rego R R to W Z X + s the [ X Z ] Ε + () Λ ( ρ ) g N p Observato : ( Lower boud for the average dstace traveled per customer served) 8

20 Gve that the locatos of demads ay gve area s uformly dstrbuted, from Beardwood, et al (96), we kow, for ay gve ε 0 >, there exsts 0 ε,suchthat whe > 0 ( ε ), L,, TSP X X Ε β ε K If the demad s from rego Ε ( + ) ( Z + X ) L Z X TSP R, The average travel dstace per demad served s Whe Z > 0, we kow ( + ) ( Z + X ) LTSP Z X Ε Z + X β ε Z + X β ε + = > 0 > 0 Λ > 0 Let ϖ = P{ Z X Z } P{ Z } lower boud for the average dstace traveled per customer served s ( + ) ( Z + X ) L Z X TSP Ε Λ ( + ) ( Z + X ) LTSP Z X = Ε Ε Z + X Λ ϖ g () NΛ 4 From observato, we kow Ε [ X + Z ] ad [ Z ] ( ϖ N ) From the defto of () g, we kow g() Λ= p Ε Λ 3 Mmzg ϖ uder above costrats leads to lm Nd p( ) Let 0 ε, we kow lm Nd p( ) ρ ρ β ρ ρ β ε (7a) 9

21 Observato 3: (the cost of remag dle whe the system s ot empty) Now we make the followg observato: the average dle tme per demad served durg λλ ρ each subsequet busy perod(s) s bouded from below by ( ) Ths comes from dvdg the average terarrval tme by the average umber of customers served durg a sgle busy perod a M/G/ queue So the average extra-cost due to dle perods per customer served for customers rego R s equal to ( ρ )( p ) λλ ad the average extra-cost due to dle perods per overall demad served s bouded by p ( p ) ( ρ ) Λ = ( ρ ) λλ λλ (7b) From (7a) ad (7b), we kow that the average extra-cost due to swtchg ad dlg s at least β ( ρ ) p + ο N N p + ( ρ ) λλ Because our system s steady-state λ β p( ρ ) m p s + + ο + ( ρ ) < lgebrac mapulato leads v N N λλ p ρ λβ p ρ tolm ρ W ρ m > Λ vm We ca show that whe p =, w( p) = vm λβ p ρ m ( ρ ) ( p)( ρ ) Λ s mmzed Fally, we kow lm W ( ρ ) ( ρ ) λβ > v m ρ 0

22 Because ρ ρ Λ,whe Λ s small eough, we have lm W ( ρ ) λβ v m 4 Cocluso We costruct a class of algorthms ad demostrate that BvR (, k ) * s ths class ad that t s optmal amog algorthms ths class The we show that a algorthm ths class s asymptotcally optmal Therefore BvR (, k ) * s asymptotcally optmal If t s BvR (, k ) * s asymptotcally optmal for the sgle vehcle case, t s also asymptotcally optmal for the multple vehcle case Our results demostrate the robustess of partto algorthms for routg ad schedulg problems These results mrror those developed earler for the travelg salesma problem (Karp, 985) ckowledgemets Ths research was partally supported by a grat from the Uversty of Calfora Trasportato Ceter (UCTC) ad by the US Natoal Scece Foudato (NSF) uder Grats No CMS ad No CCR The authors gratefully ackowledge ths geerous support y opos, fdgs ad coclusos or recommedatos expressed ths paper are those of the authors ad do ot ecessarly reflect the vews of the UCTC or the NSF

23 Refereces Beardwood, J, J Halto ad J M Hammersley, The shortest path through may pots, Proceedgs of the Cambrdge Phlosophcal Socety, 55, 9-37, 96 Bertsmas, D, ad D Smch-Lev, ew geerato of vehcle routg research: Robust algorthms, addressg ucertaty, Operatos Research, 44, , 995 Bertsmas, D, ad G va Ryz, Stochastc ad dyamc vehcle routg problem the Eucldea Plae, Operatos Research, 39, 60-65, 99 Bertsmas, D, ad G va Ryz, Stochastc ad dyamc vehcle routg problem wth geeral demad ad terarrval tme dstrbutos, dvaces ppled Probablty, 5, , 993 Bertsmas, D, ad G va Ryz, Stochastc ad dyamc vehcle routg problem the Eucldea Plae wth multple capactated vehcles, Operatos Research, 4,60-76, 993 Bramel, J ad D Smch-Lev, The Logc of Logstcs, Sprger-Verlag, New York, 997 Jallet, P, ad Odo, The probablstc vehcle routg problem, I Vehcle Routg: Methods ad Studes, BL Golde ad ssad(eds), North-Hollad, msterdam, 988 Karp, RM ad Steele JM, Probablstc aalyss of heurstcs, The Travelg Salesma Problem, Lawler EL, Lestra JK, Rooy Ka HG ad Shmoy (eds) Joh Wley & Sos Ltd, 985 Powell, W, P, Jallet ad Odo, Stochastc ad dyamc etworks ad routg, Network Routg, MBall,TMagat,CMomaadGNemhauser(eds) North-Hollad, msterdam, 995 Psarafts, H, Dyamc Vehcle Routg Problems, I Vehcle Routg: Methods ad Studes, BL Golde ad ssad (eds), North-Hollad, msterdam, 988

24 ppedx Frst, secto, we troduce a smoothg techque used to prove lemma I secto, we troduce the lemmas eeded to proof the lemma We use lemma ad to prove lemma Usg lemma 3 to prove lemma I secto 3, we provde the proof for the propostos ad the last secto, we provde proof of the lemmas Smoothg techque: To proof lemma, we are terested a smoothed verso of f varety of smoothg techques wll work We wll choose oe to make our dscusso precse We select a parameter η, for each area o whch f ( x) s ot zero, we defe a ew area whch cotas the orgal oe wth the same ceter, the rato of the parameter of ew area to the orgal oe s η Frstwedefea g( x) based o f ( x) ad η : g ( x) ( x) f = η d( x, ) whe x η We defe fη ( x) to be a smoothed verso of f ( x) as followg: f ( x) g x η = g x dx Note that as η gets small, f η approaches f the lmt Furthermore, for ay fxed value ofη, we ca fd a costat α such that for ay two pots x ad y the area, f x f y x y, σ σ α 3

25 where x y s the Eucldea dstace from x to y ad α = f Γ ( x) { } max x f x η Note that s bouded ad so after applyg ths smoothg techque, the smoothed verso of f ( x) Γ wll satsfy the Lpschtz α -codto Lemmas Lemma Let y,, y,,, y be the d radom varables wth dstrbuto f µ µ, µ µ Σ α, where Σ = α { f f x f y α x y } Let LTSP ( y,, y,,, y, ) µ µ µ be the legth of optmal TSP tour over y,, y,,, y,forayε > 0, we ca fd µ µ, µ N 0,whe > N0,we have for ay µ, (, µ,, µ,,, µ ) LTSP y y y Ε β f ( x) dx µ ε,where β s the TSP costat defed Beardwood et al (96) K Lemma ssume y,, ym are d radom varables wth commo dstrbuto f ad z,, zm K are d radom varables wth dstrbuto f η Let L ( y y ) TSP K,, L (,, TSP z K zm ) be the legth of the optmal TSP tour over y,, K ym ad z, K, zm respectvely For ay ε,wheη s suffcet small ad M s suffcet large, we have, Ε K (,, ) (,, ) L y y L z z TSP M TSP M < M K ε M ad 0 Lemma 3 Let { B } = = be a grd partto over such that each B has the same area Let f ( x) be costat wth B ad X, X, X be d radom varables dstrbuted 4

26 accordg to f ( x ) Foray ε 3 >0, we ca fd N ( ε ),whe > 3 N ε,wehave 3 Ε L ( X, X, X ) TSP β f( X) dx -ε 3 holds for ay f x To prove lemma ad lemma 3, we borrow the method from the classcal paper by Beardwood, Halto ad Hammersley (959) To prove lemma, we rely o optmzato methods ad algebrac techques Frst we state the ma result from the paper of Beardwood, Halto ad Hammersley (959) that we wll use later our proof L Theorem BHM: ssume that X, X,, X are d radom varables wth dstrbuto f ( x) ad let L ( X, X,, X ) TSP L be the legth of TSP tour over X, X, L, X (,,, ) L X X L X f X dx TSP lm β as = (a) 3 Proof of propostos Proof of proposto We have Q customers, the locatos of these customers are d radom varables wth dstrbuto f ( x ) We fx a rego frst, let t be R, ssume f Let of be the umber of customers located at s Bomal wth parameter radom selected customer located at x c R Let W( x) x = c whe x R R It s easy to kow that the dstrbuto be the expected watg tme for a 5

27 ssume from the momet that the server eters a rego ad begs to provde the servce ad keeps workg utl there s o other customer the curret rego, there are r customers served totally We estmate the average watg tme the followg way: whe the server fshes the prevous customer, there are r customers the rego, the average watg tme s bgger tha of the legth of the total sum of ( r ) terarrval tme mus the legth of the total stay perod, whch s less tha the sum of r o ste servce tme ad travel tme (whch s less tha Therefore, v ) to provde servce =Ε Ε [ = ] W x W r Qf Ε s+ λr v = x R Qf ( x) R s λr + v = Qf ( x) R s + λ v λr Note that whe ρ, N mples R 0 as ρ For ay NR such that > R s+ 4λ v ad NR θ > 0, we kow, whe ρ s bg eough, we have W( x) ( x) Qf 4λ λr Proof of proposto From proposto, we kow whe ρ s bg eough, W x ( x) Qf 4λ W ( x ) Qf ( x ) λr W 4N NR f ( x) 4NW x 4N WQ + Γ NRQ 6

28 If θ s NR θ ε ad 0 W( x) < ϖ ϖ, we kow f ( x) W 4ϖ Γ ε + εθ s Proof of proposto 3 Frst we try to gve a lower boud o the expected total watg tme Let Z deote the umber of requests R Let T be the total watg tme for the customers R Let be the total umber of customers served R Usg the fact that the dstrbuto of the Q requests s f ( x ), the dstrbuto of these radom varables of Z s s gve by the followg multomal dstrbuto: Q! { =, r = r} = Π( ) P Z z Z z c R Π( z ) z Ε T = Ε Ε T ( ) Ε ( ) s+ λr v Q Q c R = λ ( ) c R Q Q s + v We have Ε T ( Q ) = f dx λr s Q λ + v Because Ε[ W ] W Ε + Q W W as + Soforayε > 0 =, there exsts T > 0,whet > T,wehave W Ε + + Q ε = W 7

29 N + ε So whe t > T, fγ ( x) dx Qω 4 Proof of lemmas Proof of lemma Observato There exsts For ay f ( x) Σ α,let f ( y 0 ) = m x { f ( x) } C > 0, such that f ( x) dx C sup f x Σα Because f ( x ) dx=, we kow: f ( y ) 0 D f ( y 0 ) D { } f ( y0 ) +α D + α D,Where D max x f x D s the dameter of Let be the area of adc = C D + α D, therefore, sup f x f Σα x dx Observato Foraygve> ε3 > 0,letδ =, for ay for ay x ad y satsfyg α x y δ, f( x)- f( y) ε3 The reaso for t s as followg: for ay xysatsfyg, x y δ,assume f ( y ) =τ ad f ( x ) =τ +ξ where 0 ε 3 τ, ξ 0, ote that ξ ε3, therefore, f( x)- f( y) ξ = max τ τ + ξ + τ 0 ξ τ + ξ + τ = ξ ε3 8

30 We dvde to grd parttos of detcal sze wth dameter δ s before, let B be the Let th partto f ( x) = m f( y), x B δ { } { } yy B Observato leads us to observato 3 Observato 3 f ( x ) dx - δ ε3 ad fδ ( x) dx f ( x) dx -ε3 We place Y to oe of the two sets ( Ω, Ω) as follows: If fδ ( Y ) =0, let Y Ω wth probablty oe; otherwse, let Y Ω wth probablty ( Y ) fδ ad Y Ω f Y wth probablty f δ ( Y ) f Y Let be the umber of requests belogg to Ω ad LTSP( Ω) be the legth of optmal TSP tour over all the odes belogg to Ω We ca show that (I) f x dx as as δ (II) The radom varables the set of Ω are d radom varables wth the probablty desty fucto f f δ δ ( x) x dx Usg lemma 3 ad (II), we kow that for ay 4 whe > N ε, the followg holds, 4 4 ε >0, there exsts N ε,suchthat 4 4 9

31 ( Ω ) δ β L f x TSP Ε dx fδ x dx From (I), we kow there exsts 0 > 0,whe > 0, ε 4 P f δ ( x ) dx > ε 3 > ε3 Whe N ( ε ) 4 4 > max 0, ε3, L ( Ω ) ( ) ( ) TSP Ε β ε3 ε 3 fδ x dx ε4 β f x dx δ ε3 f x dx δ ε4 From observato, observato 3, we kow L ( Ω ) TSP Ε β f x dx βε3 ε 3C ε 4 Let ε 5 = ε 3 ε 3 C ε 4, because C ad β are costat ad ε 3, ε4 are arbtrary small umber, ε5 s arbtrary t last, we fd that ( Ω) L ( Ω ) L Ε Ε β TSP TSP Ths cocludes the proof of lemma f x dx ε 5 Proof of lemma Step We place Y to oe of the two sets ( Ω, Ω) as follows: 30

32 If f ( Y ) f ( Y ) η,lety Ω wth probablty f ( Y ) η f( Y ) ad Y Ω wth probablty - fη ( Y ) f( Y ) ( η ) f x f x dx Now we dspatch the elemets the set Ω Step Let τ = > f x fη x accordg to f x f x η τ fter these two steps, from y, K, ym, we get z,, K zm,where z, K, zm are d radom varables wth dstrbuto f η Now we exame the umber of elemets set Ω Let be the umber of requests belogg to Ω ad LTSP( Ω) be the legth of optmal TSP tour over all the odes belogg to Ω To calculate the dfferece betwee Ε L ( Z, Z, K, Z ) ad Ε L ( Y, Y,, Y ) we ote the followg fact, (I) L ( Z, Z, K, Z ) TSP M TSP TSP L Ω ad L ( Y, Y,, Y ) TSP M M TSP K LTSP( Ω ) K, (II) From Karp ad Steel (985), lemma, we kow there exsts a tour whose legth s M less tha ( M ) + + over M odes Combg these two facts we kow that Ε K K ( M ) (,,, ) Ε (,,, ) L Z Z Z L Y Y Y TSP M TSP M [ M ] Ε + + Ε + + It s easy to see that whe η goes to zero, Ε[ M ] goes to zero too 3

33 So for ay ε, whe η s suffcet small, M s bg eough, we have Ε K (,, ) (,, ) L y y L z z TSP M TSP M < M K Ths cocludes the proof of lemma ε Proof of lemma 3 We prove lemma 3 by ducto based o the dfferet values f ( x) ca have Frst we assume that f ( x) s equal to zero or ay costat over all the B e B, { f x B B ( x) 0 otherwse f( x) = cι ( x) Ι = ad c equals to zero or c Let m be the umber of the parttos o whch f ( x) s c The probablty that ay demad falls to ay specfc partto o whch f ( x) 0 s m Let LTSP be the legth of optmal TSP subtour over all the demads belogg to the th pece of the partto o whch f ( x) s c Let LTSP demads Let x s the legth of the crcumferece of the partto be the optmal TSP tour over all the B For the optmal TSP tour over all the odes, we are terested the pots that le the optmal TSP tour ad the permeter of the partto For these pots, we costruct a tour through all the pots ad select each fter these steps, for each partto, we have a coected graph whch each ode has eve degree We kow there exsts oe Euler tour that traverses all the lks exactly oce Rememberg that 0 s the total umber of 3

34 parttos, The legth of ths tour s less tha LTSP + 30 x ad s as least the same legth of the L Fally, we have TSP TSP 3 TSP 0 L L x LTSP ( X, X, X ) Now we try to estmate x 0 Because 0 as, we focus o L TSP To express the dea clearly, from ow o, we focus oly o the parttos whch f ( x) s a costat, c ssume these parttos are B, {,,, m} L Let be the umber of odes B,let D = > k m m Usg Chebychev s Iequalty, PD { } k (a) By De Morga s rule, P{ D} =- P{ D } - [- P{ D }] c m Fally, P{ D} - [- P{ D }] as k 4 k = (a3) m Let B be the sze of the area of ay partto ad betheszeofthewholearea,we use the result of Beardwood et al (959), please refer the (a) ad combg wth (a), (a3), for ay 0 ε >, there exsts ( ε ) >,whe ( ε ) 0 >,wehave P L TSP B > β ε > ε (a4) 33

35 L > β ε > ε TSP e P B 4/3 { } fter some calculato, we kow whe max 6 m, m ( ε ) >,wehave (a5) k m m > - > ( ε )Sowhe max 6 m, m ( ε ) 4/3 { } >, wth at least P{ D } probablty that all satsfes ( ε ) > Because whe ( ε ) >,wehave k From (a5), (a6) ad P( B) P + P( B), we kow, L m P B m = m = m = = k TSP > ( β ε) > ε L P m B k m > (a6) m m = m TSP > ( β ε) - > ε = m m k L m m P m B k m = m > β ε - > ε = k TSP e m Because k = m /4,soatlastwehave = m = /4 3/ L m m P m B m TSP > ( β ε) > ε Ε = m 3/ TSP mε ( β ε ) /4 L m m m B = 34

36 Note that fdx= mb,wehave = m 3/ /4 L m m Ε TSP mε ( β ε) f dx = 3/ m Because 0, m x 0 ad 0 0 as, LTSP L - 3 TSP 0x ad ε s arbtrary, so for ay 3 0 ε >, we ca fd N ( ε ),whe > 3 N ε,wehave 3 Ε L ( X, X, X ) TSP β f( X) dx -ε 3 The above argumet holds for ay fxed m,whe m goes from to 0,wefx = max m, we kow whe > *,wehave * * m Ε L ( X, X, X ) TSP β f( X) dx -ε 3 The last step ths ducto proof s to assume the lemma s true whe f ( x) has k dfferet values ad to show that the lemma holds whe f ( x) has k + dfferet values The proof s based o smlar deas ad s rather tedous, so we omt the rest of proof here Ths cocludes the proof of lemma 3 35

37 Proof of lemma We kow that,,, M y y y are d radom varables wth dstrbuto of f ( x) applyg the smoothg techque, we kow that the for ay f ( x) codto of f ( x) Γ Γ that satsfes the 4 ϖ + + ε, we ca fd commo α such that f Γ η ( x),the ε εθ ε smoothed verso of f ( x) has the followg two propertes: fter Property I: Its satsfes the Lpschtz α -codto; Property II: Let z, z,, zq we kow that for ay 0 L be d rvs wth dstrbuto of f ( x) ε >, ρ ( ε ) >,whe ρ > 0 ρ ε, (,, L, ) Ε (,, L, ) Ε L y y y L z z z Q Q From lemma ad property I, we kow for ay ε 3 > 0, ρ TSP Q TSP Q (,, L, Q ) Ε LTSP z z z fη ( x) dx Because f ( x) dx ( 4 η + 4 η ) f ( x) dx,adε η Γ, 3 sup f x ε Q η From lemma, ε,whe ρ max { ρ, ρ } ε ε are arbtrary, 3 >, 4ϖ f ( x) dx Γ Γ +, by the smlar argumet we made the proof of ε εθ lemma, we ca show that for ay ε > 0 ρ 0,whe ρ > ρ0, Qd β ε fγ x dx Ths cocludes the proof of lemma 36

38 Proof of lemma By addg some addtoal o-bdg costrats we ca show that the orgal problem ca be traslated to the followg problem: m = = x * subect to: = = x * =, = * x + ε, x ε, = for some small value ε Wheε the costrats { } = x ε = are o-bdg ad hece, do ot affect the solutotoourproblem For ths ew problem, we use a stadard optmzato techques as follows: Let = = = = * * * LX (, λµγ,, ) = x + λ x - + µ + ε - x + uur uur γ ( x - ε) = = = = By cosderg the Kuh-Tucker codtos we kow that the optmal soluto must L L L satsfy the followg: = 0, = 0, = 0, γ = 0 x λ µ Therefore, for the optmal 3 soluto + λ x - 4µ x = 0, must hold 3 Cosderg the set of equatos + λ x - 4µ x = 0, Fromthetheoryof algebrac equatos, we kow the followg facts, Case Oe: The equatos have the same oegatve soluto e x = x for ay, Case Two: There are at most two dfferet posto solutos ssume that the postve solutos are a ad b respectvely ad that a b Let Z = * = m x = subect to: 37

39 = = x * = = = * x + ε x {,} ab Z = m subect to * * a + b { x = a} { x = b} * * a b x = a x = b { } + = { } a + b + ε * * x = a x = b { } { } So f we let { x= a} * x = ad y { x= b} * =, we obta, Z m{ ax+ by} subect to: ax+ by = ax+ by + ε a > 0, b> 0, b a Therefore, Z + ε Ths cocludes the proof of lemma 38