Planning Approximations to the average length of vehicle routing problems with time window constraints

Transcription

1 Planning Aroximations to the average length of vehile routing rolems with time window onstraints Miguel Andres Figliozzi ABSTRACT This aer studies aroximations to the average length of Vehile Routing Prolems (VRP) with time window, route duration, and aaity onstraints The aroximations are valuale for the strategi and lanning analysis of transortation and logistis rolems Using asymtoti roerties of vehile routing rolems and the average roaility of suessfully sequening a ustomer with time windows a new exression to estimate VRP distanes is develoed The inrease in the numer of routes when time onstraints are added is modeled roailistially This aer introdues the onet of the average roaility of suessfully sequening a ustomer with time windows It is roven that this average roaility is a unique harateristi of a vehile routing rolem The aroximation roosed is tested in instanes with different ustomer satial distriutions, deot loations and numer of ustomers Regression results indiate that the roosed aroximation is not only intuitive ut also redits the average length of VRP rolems with a high level of auray KEYWORDS: Vehile Routing Prolem, Distane Estimation, Time Window Constraints Assistant Professor, College of Engineering and Comuter Siene, Deartment of Civil Engineering, Portland State University, Portland, USA, figliozzi@dxedu

2 INTRODUCTION In many logistis rolems it is neessary to estimate the distane that a fleet of vehiles travel to meet a set of ustomer demands Traveled distane is not only an imortant element of arriers variale osts ut it is also a key inut in tatial and strategi models to solve rolems suh as faility loation, fleet sizing, and network design The transortation deisions assoiated with high value - high time sensitive roduts, are the most demanding ativities in terms of transort servie requirements and usually require servie within a hard time windows (Figliozzi, 006) Time windows are a key onstraint also for make to order-jit rodution systems as well as emergeny reair work and exress (ourier) delivery servies Time windows have a signifiant imat on dereasing the effiieny of routes, reduing servie areas, and signifiantly inreasing distane travelled (Figliozzi, 007) Desite the growing imlementation of ustomer-resonsive and made-to-order suly hains, the imat of time window onstraints and ustomer demand levels on average VRP distane traveled has not yet een studied in the literature The existing ody of literature has mostly foused on the estimation of distanes for either the Traveling Salesman Prolems (TSP) or the aaitated vehile routing rolems (CVRP) This researh rovides an intuitive and arsimonious mathematial framework to estimate average distanes in VRP rolems The aer is organized as follows: Setion two rovides a literature review Setion three resents asymtoti results for the VRP and exressions to estimate the additional numer of routes due to time window onstraints Setion four resents a new exression to estimate distane traveled Setion five desries the exerimental design and test results Setion six ends with onlusions LITERATURE REVIEW A seminal ontriution to the estimation of the length of a shortest losed ath or tour through a set of oints was estalished y Beardwood et al (959) These authors demonstrated that for a set of n oints distriuted in an area A, the length of the TSP tour through the whole set asymtotially onverges, with a roaility of one, to the rodut of a onstant k y the square root of the numer of oints and the area, ie k na when n The asymtoti validity of this formula for TSP rolems was exerimentally tested y Ong and Huang (989) using a nearest neighor and exhange imrovement heuristis With Hereafter, VRP denotes vehile routing rolems with time window, route duration, and aaity onstraints

3 an Eulidian metri and a uniform distriution of ustomers the onstant term has een estimated at k = 0765 (Stein, 978) For reasonaly omat and onvex areas, the limit rovided y Beardwood et al onverges raidly (Larson and Odoni, 98) Jaillet (988) estimated the onstant k = 097 for a Manhattan metri Aroximations to the length of aaitated vehile routing rolems were first ulished in the late 960 s and early 970 s (We, 968, Christofides and Eilon, 969, Eilon et al, 97) We (968) studied the orrelation etween route distane and ustomer-deot distanes Eilon et al (97) roosed several aroximations to the length of CVRP ased on the shae and area of delivery, the average distane etween ustomers and the deot, the aaity of the vehile in terms of the numer of ustomers that an e served er vehile, and the area of a retangular delivery region Daganzo (984) roosed a simle and intuitive formula for the CVRP length when the deot is not neessarily loated in the area that ontains the ustomers CVRP(n) rn/ Q+ 057 na In this exression CVRP( n) is the total distane of the CVRP rolem serving n ustomers, the average distane etween the ustomers and the deot is r, and the maximum numer of ustomers that an e served er vehile is Q Hene, the numer of routes m is a riori known and an e alulated as n/ Q Daganzo s aroximation an e interreted as having: (a) a term related to the distane etween the deot and ustomers and () a term related to the distane etween ustomers The oeffiients of Daganzo s aroximation were derived assuming Q > 6 and n > 4Q Daganzo s aroximation works etter in elongated areas as the routes were formed following the stri strategy Rouste et al (004) tested Daganzo s aroximation using simulations and ellitial areas; they roose adjustments ased on area shae, vehile aaity, and numer of ustomers A dissertation rodued y Erera (000) extended the usage of ontinuous aroximations to estimate the distane of detours and routes in a stohasti version of the CVRP Chien (99) arried out simulations and linear regressions to test the auray of different models to estimate the length of TSP Chien tested retangular areas with 8 different length/width ratios ranging from to 8 and irular setors with 8 different entral angles ranging from 45 to 360 degrees Exat solutions to solve the TSP rolems were used and the size of the rolems ranged from 5 to 30 ustomers The deot was always loated at the origin, ie the left-lower orner of a retangular area Chien randomly generated test rolems and using linear regressions found the est fitting arameters The mean asolute erentage 3

4 error (MAPE) was the enhmark to omare seifiations Chien finds that the lowest MAPE for the est model is equal to 69% TSP(n) r + nr R = MAPE = In this exression, TSP( n) is the total distane of the TSP rolem serving n ustomers The area of the smallest retangle that overs the ustomers is denoted R The usage of R instead of the total area, A, ontaining all ustomers may not e onvenient for lanning uroses when there may e many ossile susets of ustomers that are not known a riori Chien also estimated the revious models for eah of the 6 different regions; linear regression and MAPE are reorted for eah tye of region and model The estimated arameters hange aording to the shae of the region Kwon et al (995) also arried out simulations and linear regressions ut in addition to Chien s work also used neural networks to identify etter aroximations To test the auray of different models they tested TSP rolems in retangular areas with 8 length/width ratios ranging from to 8 Models were estimated with the deot eing loated at the origin and at the middle of the retangle The sizes of the rolems range from 0 to 80 ustomers Kwon et al roosed aroximations that make use of the geometri information roortioned y the ratio length/width of the retangle and a shae fator S The results otained for the deot loated at the origin are as follows: TSP(n) n+ + S n+ na R = MAPE = [ ( ) /( )] TSP(n) r n S n na R MAPE 04 + [ ( + ) /( + )] = 099 = 36 By aounting for the shae of the area, Kwon et al imroved the auray of the d estimations although this ame at the exense of adding two extra terms R is defined as the area of the smallest retangle that overs the ustomers and the deot; with the deot loated at the enter of the retangle the results otained y Kwon et al are as follows: TSP(n TSP(n d ) [ ( n+ ) + 34 S/( n+ )] nr R = 099 MAPE = 388 d ) 5 r+ [ ( n+ ) S/( n+ )] nr R = 099 MAPE = 370 It an e oserved that MAPE slightly inreases when the deot is loated at the enter of the retangle Kwon et al also used neural networks to find a model that etter redits TSP length They onluded that the aaility of neural networks to find hidden relationshis rovides a slight advantage against regression models However, the models are less arsimonious and the aroximations harder to interret in geometri terms R 4

5 A simle and intuitive analytial model of VRP with time window onstraints is rovided y Daganzo (987a, 987) Daganzo divides a day into time eriods or ins of equal length and then lusters ustomers in retangles Eah ustomer is then laed in a alaned time eriod or in, onsistent with his or her time window; this allows a simlifiation of the rolem as ustomer individual time window harateristis are now assoiated with a time eriod Using this time in-luster first-route seond aroah, Daganzo analyzes main routing tradeoffs and determines that distane traveled is a funtion of the square root of the numer of time eriods and that lower distanes are ossile when routes are allowed to overla Different aroximations are rovided if the dominant onstraint is either vehile aaity or route duration Although Daganzo s formulas are useful and intuitive they are not easily alied to estimate VRP distane sine his aroah does not guarantee feasiility Unfortunately, no systemati method or general exression for lustering and determining the numer of eriods that guarantees alaned eriods and feasile routes is rovided Aroximations to the average length of vehile routing rolems have reently een ontriuted y Figliozzi (008) to estimate VRP distane when the numer of ustomers served ( n) and the numer of routes ( m) are given The formula roosed aounts for the tradeoffs etween onneting distane and loal tour distane as the numer of routes inreases: n m VRP( n) kl An + m r n The term ( n m)/ n is shown to imrove MAPE values in rolems with aaity onstraints, time windows onstraints, and a varying numer of ustomers served ( n ) 3 CHARACTERIZATING THE IMPACT OF TIME WINDOW CONSTRAINTS This setion introdues a roailisti aroah to ature the imat of time windows on distane traveled for VRP instanes that serve N = {,,, n} ustomers Assoiated with eah ustomer i N there is a quintulet ( x, q, s, e, l ) that reresents, resetively, the i i i i i oordinates, demand, servie time, earliest servie starting time, and latest servie ending time The deot quintulet is denoted ( x0, q0, s0, e0, l 0) with q 0 = 0, s 0 = 0 and e 0 = 0 The distane etween eah ustomer i N and the deot is denoted d( x ); feasiility onditions inlude d( x i ) l i, d( x i ) + s i l0, and q i Q Customers with time windows are drawn from a roaility measure ν with ounded suort Without loss of generality, attriutes of the quintulet are saled and shifted so they elong to the real interval [0,] The oordinates i x i 5

6 are indeendently and identially distriuted aording to a distriution with omat suort in R, [0,] [0,] ; the ustomer arameters ( q, s, e, l ) are drawn from a joint roaility i i i i distriution Φ with a ontinuous density funtion φ The suort of φ is the feasile suset 4 of ( x, x, x, x ) [0,] It is also assumed that ostumer loations and their arameters are 3 4 indeendent of eah other Customers without time windows are drawn from the same roaility measure ut their time windows are relaxed, ie ( ei, li) is relaed y ( e0, l 0) The relaxed roaility measure is denoted, whose suort is the feasile suset of (, ) [0,] x x with 3 4 x = 0, x = The exeted numer of routes needed to serve n ustomers with and without time windows is denoted mν ( n) and m ( n) resetively Known results for the aaitated vehile routing rolem (Bramel et al, 99) indiate that: CVRP * ( n, ) limn = γ E( d ) n where γ > 0 is a onstant that deend only on, E( d ) is the exeted distane etween the deot and ustomers, and CVRP * ( n) is the est VRP solution for travel distane The ratio / γ is the average numer of ustomers er route Similar results an e derived for the vehile routing rolem with time windows (Bramel and Simhi-Levi, 996, Federgruen and Van Ryzin, 997): VRP * ( n, ν ) limn = γν E( d) n The next lemma rovides a useful ound for the additional numer of routes due to time window onstraints Lemma The ontriution of time windows to the distane traveled is ounded Asymtotially, the numer of additional routes due to time window onstraints an e exressed as kn, eing k a onstant suh that 0 k Proof Asymtotially, the ontriution of time windows to the distane travelled er ustomer an e exressed as ne( d)( γν γ ) The inrease in the numer of routes due to time windows, denoted m ν, an e aroximated y m ( n) = n( γ γ ) = m ( n) m ( n) ν ν ν There annot e more routes than ustomers, hene γ Time windows, additional onstraints, annot redue the VRP distane; hene ( γ γ ) 0 ν ν 6

7 The inrease in the numer of routes when time onstraints are added is modeled roailistially Given any two ustomers i, j N there is a roaility ij that a vehile an suessfully visit ustomer j after visiting ustomer i without violating js ' time window In general, ij is a random variale that will deend on the roaility measure ν The goal is to find an exression that rovides the average numer of additional vehiles needed due to time window onstraints, ie mν ( n) An exat solution using ij is likely to e intratale and to the est of the author s knowledge there is no general analytial exression that an e used to estimate the imat of time window onstraints on VRP distanes To model m ( n), the onet of an average roaility of suessfully sequening any ν given ustomer with time window onstraints is introdued; let s denote this average suess roaility ν Let s denote = / γ as the average numer of ustomers er route or in without time window onstraints The roaility assoiated to finding a feasile route with ustomers, eah with time window onstraints, an e exressed as: () ( ) = ν By definition () = eause it is assumed that all ustomers an feasily e served from the deot When = the numer of routes is simly m= n When =, the numer of exeted routes needed to serve n ustomers an e exressed as the weighted sum of routes with one and two ustomers: n () + n ( ()) and generalizing for any : n () Em [ ν ( n)] = ( ( j)) j= + A similar exression an e found in the work of Diana et al (006) whih estimated demand resonsive transit fleet sizes The exeted numer of additional routes due to time window onstraints, E[ m ν ], an e exressed as: n () n Em [ ν ( n)] = [ ( j ( ))] () j= + Lemma The exeted numer of additional routes due to time window onstraints, E[ mν ( n)], is a ontinuously dereasing funtion of ν 7

8 Proof The omlete roof is resented in Aendix A; a sketh of the roof is resented in this setion The sum of weight fators w () w () = () ( ()) j adds u to one j= + w () = () ( ()) j = j= + As ν the sum nw inreases from zero to one, the weight fators are shifted from = to, hene, () dereases as s inreases Lemma 3 The exeted numer of additional routes due to time window onstraints E[ mν ( n)] is ounded etween (0, n n/ ) The value of E[ mν ( n)] numer of ustomers is a fration of the Proof By sustitution, it an e shown that E[ mν ( n)] = n n/ when ν = 0 and Em [ ν ( n)] = 0 when ν = Sine E[ mν ( n)] is a dereasing funtion it is ounded etween (0, n n/ ) Theorem A routing rolem with ustomers drawn from a roaility measure ν has a unique ν suh that Em [ ( n )] = n ( γ γ ) as n ν ν Proof Asymtotially, the additional numer of routes is m ( n) = n( γ γ ) with ν ν 0 ( γ γ ) Due to Lemma and 3, E[ m ( n)] is a ontinuously dereasing funtion ν Hene, there is a unique ν suh that Em [ ( n)] = n( γ γ ) Corollary The value of ν ν ν ν, the average roaility of failing to sequene a ustomer with time window onstraints rovides a measure, in a sale (0,), of the imat of time window onstrains on VRP distane As inreases the relative imat of time windows onstraints on the numer of routes and the distane traveled inreases ν 8

9 4 APPROXIMATING VRP DISTANCES This setion rovides an aroximation to VRP distane assuming a distriution enter that serves a set of N = {,,, n} ustomers on any given day or time eriod The numer of daily requests may vary ut it never exeeds n, ie n n The total numer of ustomers with time windows is denoted n t, nt n, and the total demand is denoted q N = q The i fous of this researh is the derivation of general aroximations to the average distane traveled to serve a total of n ustomers with i N n t time windows, n n and 0 nt n This average distane is denoted VRP( n, nt, ν ) Instanes of daily demands are formed y joining nt ustomers, drawn aording to a roaility measure ν, and n nt ustomers drawn aording to roaility measure A ustomer has a time window if either ei > e0 or li < l0 The value of ν is aroximated as the value that minimizes the asolute value of the differene: n () min mν ( n) ( ( j)) () j= + st: =,0 ν () ( ) ν, and = n/ m ( n) From Theorem, it is guaranteed that there is only one ν that minimizes the asolute value of () The value of m ( n) and mν ( n) an e estimated y samling from the resetive distriutions and determining the numer of routes need To estimate the numer of additional routes due to time windows when 0 < n < n, it is neessary to model how time windows are distriuted among routes Assuming a inomial distriution, the roaility of having a route with k time windows out of ustomers is: ( ) inomial( k;, ) = ( ) where:! ( k ) = k!( k)! k k nt k t t k = numer of suesses in trials, = numer of indeendent trials, and = n / n= the roaility of suess on eah trial t t Then, the numer of additional routes to serve a total of n ustomers with an e aroximated as follows: t n t time windows 9

10 nnn (,, t, ν ) mν ( n, nt ) = ( ( j)) ] m ( n) where j= + k t ν = nt s k = 0 (, n, n, ) inomial(;, k )( ) Aroximating the numer of routes related to in-aking onstraints suh as vehile aaity or tour duration is relatively straightforward: q nτ m ( n) max(, ) N Q l0 e0 where τ is the sum of estimated travel time lus servie time er ustomer Although asymtoti results indiate that numer of routes is the only essential fator to estimate VRP distanes, the literature review has shown that the est aroximations to VRP distane aount for (a) a term related to the distane traveled etween the deot and ustomers and () a term related to the distane traveled etween ustomers The roosed aroximation (3) also aounts for oth tyes of distanes ut adding terms to estimate the additional imat of time windows m m VRP( n, n, ν) k na + k n A + k γ r m ( n) + k r m ( n, n ) (3) t λ t ν ν t m m The vetor of oeffiients ( k, k, k, k ) is estimated y linear regression The oeffiients λ ν m k and k m ν are related to the distane generated y the numer of routes needed; the oeffiients k and k λ are related to the interustomer distane, as in Beardwood et al (959) If k ν nt A and k na aroximate the interustomer distane with and without time windows resetively, then, k ν reresents the hange in interustomer distane when time window onstraints are added: k n A = k na+ k n A where k = k k and nt ν t ν t ν ν = n The other two remaining oeffiients relate to the numer of routes as follows: γ m ( n) nγ as n, and γ m ( n, n ) n( γ γ ) as n ν ν t ν The next setion desries the exerimental setting where aroximation (3) is tested 0

11 5 EXPERIMENTAL SETTING AND RESULTS The exerimental setting is ased on the lassial instanes of the VRP with time windows roosed y Solomon (987) The Solomon instanes inlude distint satial ustomer distriutions, vehiles aaities, ustomer demands, and ustomer time windows These rolems have not only een widely studied in the oerations researh literature ut the datasets are readily availale The well-known 56 Solomon enhmark rolems for vehile routing rolems with hard time windows are ased on six grous of rolem instanes with 00 ustomers The six rolem lasses are named C, C, R, R, RC, and RC Customer loations were randomly generated (rolem sets R and R), lustered (rolem sets C and C), or mixed with randomly generated and lustered ustomer loations (rolem sets RC and RC) Prolem sets R, C, and RC have a shorter sheduling horizon, tighter time windows, and fewer ustomers er route than rolem sets R, C, and RC resetively Random samles of the Solomon rolems are used to examine the auray of models Out of N =00 ossile ustomers in a servie area A, a rolem or instane is formed y a suset of n randomly seleted ustomers Using the first instane of the six rolem tyes roosed y Solomon, 5 susets of ustomers of sizes 70, 60, 50, 40, 30, 0, and 0 were randomly seleted from the original 00 ustomers; n = 70 All rolem instanes in this researh were solved with a VRP imrovements heuristi that has otained the est ulished solution in terms of numer of vehiles (Figliozzi, 008a) Real-world routes have a relatively small numer of ustomers er route due to aaity, time windows, or tour length onstraints (Figliozzi et al, 007) For examle, in Denver over 50% of single and omination truk routes inlude less than 6 stos (Holguin-Veras and Patil, 005) and 95% of the truk routes inlude less than 0 stos This researh tests the roosed VRP distane aroximation in instanes that range from to over 35 ustomers er route To otain this range of ustomers er route, new instanes were systematially reated varying the levels of ustomer demand and the erentages of ustomers with time windows To test different levels of ustomer demand, new instanes were reated alying the demand fators resented in Tale to eah suset of ustomers Alying the fators in the seond row of demand fators in Tale, the ustomers have similar demands as in the original Solomon rolems (the row haraterized y all ones []) The resulting rolems using the highest demand multiliers (last row of Tale ) are suh that some ustomers are trukload (TL) or almost TL ustomers Inreasing some ustomer demands to or lose to the

12 TL level was done in order to test the aroximation when rolems are highly onstrained and have a large numer of routes In addition, for eah samle, out of the n ustomers a random suset of time windows is turned off; the erentage of ustomers with time windows ranges from 0 to 00%, in inrements of 0% In all ases, route durations were limited y the deot time window For eah Solomon rolem lass, variaility is introdued in three distint ways: a) different susets of ustomer loations, ) different levels of ustomer demands, and ) different erentages of time window onstraints Analysis of Exerimental Results All the regression results resented in this setion are otained foring the interet or onstant term to e zero; this is onsistent with revious studies y Chien (99) and Kwon et al (995) The average roailities ( ν ) of failing to onnet any two ustomers due to time window onstraints are shown in Tale The values of ( ν ) do reflet the harateristis of the underlying rolem tyes Tye rolems where time windows are tight result in higher ( ν ) values Tale also rovides an understanding of the relative imat of time window onstraints on distane traveled As the level of demand inreases, the relative size of the in or vehile aaity is redued and there is a onsequent redution in the feasile numer of ustomers er route Hene, the imat of time window onstraints is redued as aaity onstraints eome more inding The estimated regression arameters disaggregated y rolem tye are shown in Tale 3 These arameters are otained y ooling the data of all different demand levels er rolem m m tye, ie using one set of arameters ( k, k, k, k ) for all instanes It is reassuring that the λ ν regression arameters are not only statistially signifiant ut also reflet the harateristis of the underlying rolem tyes The values of k are lowest and highest for lustered and random rolems resetively In all ases the oeffiients k λ are signifiant and ositive whih suggests that time window onstraints inrease the distane traveled etween ustomers The oeffiients k λ follow a similar trend as the k oeffiients; lowest and highest values for lustered and random rolems resetively As exeted, the values of are slightly less than one ut signifiantly different than zero The tye C oeffiients demonstrate that although k m ν is zero, k λ an e ositive and signifiant; ie time window m k

13 onstraints inrease the interustomer distane ut do not affet the numer of routes that is determined y aaity onstraints To evaluate the redition auray, the MPE (Mean Perentage Error) and the MAPE (Mean Asolute Perentage Error) are used and alulated as follows: Di Ei MPE = *00% D i= Di Ei MAPE = *00% D i= i i Where the atual distane for instane i is denoted D i and the estimated distane is denoted E i For a given set of instanes it is always the ase that MPE MAPE The MPE indiates whether the estimation, on average, overestimates or underestimates the atual distane The MAPE rovides the average deviation etween atual and estimated distane as a erentage of the atual distane Model fit R, MAPE, and MPE are dislayed for eah rolem lass and ooled data in Tale 4 The aroximation quality is high, artiularly for random and random lustered rolems The values of MAPE range from 34 to 56% with an average of 45% for the ooled data As exeted, a etter fit an e otained if a regression is run for eah demand level Aroximation quality, as evaluated y MAPE, imroves signifiantly as shown in Tale 5 6 CONCLUSIONS This is the first researh effort to study and test aroximations to the average length of vehile routing rolems when there is variaility in the numer of ustomers, time window onstraints, and demand levels Based on asymtoti roerties of vehile routing rolems, a roailisti modeling aroah was develoed to aroximate the average distane traveled An exression to estimate the numer of additional routes needed for a varying numer of time windows onstraints is derived using the average roaility of suessfully sequening a ustomer with time windows It is roven that this average roaility is a unique harateristi of a vehile routing rolem This roaility also indiates the relative imortane of time windows onstraints on VRP distanes The exerimental results demonstrate that the quality of the aroximation is roust in terms of MAP and MAPE In addition, the estimated regression arameters are intuitive and reflet the harateristis of the underlying routing rolems 3

14 REFERENCES BEARWOOD, J, HALTON, H & HAMMERSLEY, J (959) The Shortest Path Through Many Points Proeedings of the Camridge Philosohial Soiety 55, BRAMEL, J, COFFMAN, E G, SHOR, P W & SIMCHI-LEVI, D (99) Proailisti analysis of the aaitated vehile routing rolem with unslit demands Oerations Researh, 40(6), BRAMEL, J & SIMCHI-LEVI, D (996) Proailisti analyses and ratial algorithms for the vehile routing rolem with time windows Oerations Researh, 44(3), CHIEN, T W (99) Oerational Estimators For The Length Of A Traveling Salesman Tour Comuters & Oerations Researh, 9(6), CHRISTOFIDES, N & EILON, S (969) Exeted Distanes In Distriution Prolems Oerational Researh Quarterly, 0(4), DAGANZO, C F (984) The Distane Traveled To Visit N-Points With A Maximum Of C- Stos Per Vehile - An Analyti Model And An Aliation Transortation Siene, 8(4), DAGANZO, C F (987a) Modeling Distriution Prolems With Time Windows Transortation Siene, (3), 7-79 DAGANZO, C F (987) Modeling Distriution Prolems With Time Windows - Customer Tyes Transortation Siene, (3), DIANA, M, DESSOUKY, M M & XIA, N (006) A model for the fleet sizing of demand resonsive transortation servies with time windows Transortation Researh Part B, 40(8), EILON, S, WATSON-GANDY, D & CHRISTOFIDES, N (97) Distriution Management: Mathematial Modelling and Pratial Analysis, New York, Hafner ERERA, A (000) Design of Large-Sale Logistis Systems for Unertain Environments Ph D dissertation, University of California-Berkeley FEDERGRUEN, A & VAN RYZIN, G (997) Proailisti analysis of a generalized in aking rolem and aliations Oerations Researh, 45(4), FIGLIOZZI, M (008a) An Iterative Route Constrution and Imrovement Algorithm for the Vehile Routing Prolem with Soft and Hard Time Windows Aliations of Advaned Tehnologies in Transortation (AATT) 008 Conferene Proeedings Athens, Greee, May 008 FIGLIOZZI, M A (006) Modeling the Imat of Tehnologial Changes on Uran Commerial Tris y Commerial Ativity Routing Tye Transortation Researh Reord 964, 8-6 FIGLIOZZI, M A (007) Analysis of the effiieny of uran ommerial vehile tours: Data olletion, methodology, and oliy imliations Transortation Researh Part B, 4(9), FIGLIOZZI, M A (008) Planning Aroximations to the Average Length of Vehile Routing Prolems with Varying Customer Demands and Routing Constraints Transortation Researh Reord, Forthoming 008 FIGLIOZZI, M A, KINGDON, L & WILKITZKI, A (007) Analysis of Freight Tours in a Congested Uran Area Using Disaggregated Data: Charateristis and Data Colletion Challenges Proeedings nd Annual National Uran Freight Conferene, Long Beah, CA Deemer HOLGUIN-VERAS, J & PATIL, G (005) Oserved Tri Chain Behavior of Commerial Vehiles Transortation Researh Reord 906, JAILLET, P (988) Ariori Solution Of A Traveling Salesman Prolem In Whih A Random Suset Of The Customers Are Visited Oerations Researh, 36(6),

15 KWON, O, GOLDEN, B & WASIL, E (995) Estimating The Length Of The Otimal Ts Tour - An Emirial-Study Using Regression And Neural Networks Comuters & Oerations Researh, (0), LARSON, R C & ODONI, A R (98) Uran Oerations Researh, Prentie-Hall, In ONG, H L & HUANG, H C (989) Asymtoti Exeted Performane Of Some Ts Heuristis - An Emirial-Evaluation Euroean Journal Of Oerational Researh, 43(), 3-38 ROBUSTE, F, ESTRADA, M & LOPEZ-PITA, A (004) Formulas for Estimating Average Distane Traveled in Vehile Routing Prolems in Elliti Zones Transortation Researh Reord 873, SOLOMON, M M (987) Algorithms For The Vehile-Routing And Sheduling Prolems With Time Window Constraints Oerations Researh, 35(), STEIN, D (978) An asymtoti roailisti analysis of a routing rolem Mathematis Of Oerations Researh, 3(), 89-0 WEBB, M (968) Cost funtions in the loation of deots for multile-delivery journeys Oerational Researh Quarterly, 9(3),

16 TABLES Tale Demand Fators Prolem C R CR C R RC Vehile Caaity Max Demand Demand Level Tale Average Proaility ( ( ν )) C R RC C R RC 0 36% 54% 05% 00% 9% 06% 36% 54% 05% 00% 9% 06% s Demand Level % 3% 70% 00% 0% 0% 3 00% 98% 4% 00% 00% 00% 4 00% 6% 06% 00% 00% 00% 5 00% 8% 00% 00% 00% 00% 6 00% 00% 00% 00% 00% 00% 6

17 Tale 3 Estimated Regression Coeffiients y Prolem Class Prolem k t stat k λ t stat m k t stat m k ν t stat C R RC C R RC Tale 4 Aroximation Quality y Prolem Class (Pooled data) Prolem R MAPE MAP C % 06% R % 03% RC % 05% C % 8% R % 0% RC % 07% Tale 5 Average Aroximation Quality y Prolem Class (By Distriution) Prolem R MAPE MAP C % 07% R % 04% RC % 04% C % 03% R % 04% RC % 03% 7

18 APPENDIX A Lemma The exeted numer of additional routes due to time window onstraints, E( m ν ), is a ontinuously dereasing funtion of ν n () n Em ( ν ) = [ ( ( j))] j= + Proof This is a ontinuous funtion eause it is a linear omination of ontinuous funtions of the variale ν The weight fators: () ( ()) j j= + are alied to eah feasile route with ustomers er route Develoing the sum of weight fators and denoting ( ) = for the sake of revity: j= + ( ) = j = + ( ) + + ( )( )( ) + ( )( )( ) = 3 making a ommon fator: = [ + ( + + ( )( ) + ( )( )] = 3 making a ommon fator: = [ + [ + + ( )( ) + ( )( )]] = 3 The term [ + ] =, then: ( ) = + + ( )( ) + ( )( ) j 3 j= + The same roess an e ontinued until: ( j) = + ( ) = j= + 8

19 Sine = as all ustomers an e served from the deot without violating time window onstraints, this roves that: j= + ( ) = j Develoing the sum and relaing ( ) = = for the sake of revity, for any sum u to i = i: n () j= + i ( ( j) ) = i i i 3 i i i n[ + ( ) + ( )( ) + i i i 0 i i i i + ( )( )( ) + ( )( )( )( )] = j Making the terms, j =,, i ommon fators: i i i i n[ + [( )[ + [( )[ + [ + [( )[ + [( ) ]]]] i i 3 ν j For any inrease in ν, any, j =,, i will have an inrease However, any inrease in j will redue the term j that multilies the sum of the j, j,,, terms Sine the sum of the weight fators remains onstant and equal to one, as ν inreases the weight alied to the terms: j j 3 n, n,, n, n j j dereases whereas the term j n j inreases for any j =,, i Hene, as ν inreases E( m ν ) dereases In artiular, as ν inreases the term with the largest index always inreases whereas the term with index one always dereases 9