Inventory Rebalancing and Vehicle Routing in Bike Sharing Systems

Transcription

1 Inventory Rebalancng and Vehcle Routng n Bke Sharng Systems Jasper Schujbroek School of Industral Engneerng, Endhoven Unversty of Technology, The Netherlands, j.m.a.schujbroek@student.tue.nl Robert Hampshre* H John Henz III College, Carnege Mellon Unversty, Pttsburgh, PA 15213, hamp@cmu.edu Wllem-Jan van Hoeve Tepper School of Busness, Carnege Mellon Unversty, Pttsburgh, PA 15213, vanhoeve@andrew.cmu.edu Bke sharng systems have been nstalled n many ctes around the world and are ncreasng n popularty. A major operatonal cost drver n these systems s rebalancng the bkes over tme such that the approprate number of bkes and open docks are avalable to users. We combne two aspects that have prevously been handled separately n the lterature: determnng servce level requrements at each bke sharng staton, and desgnng (near-)optmal vehcle routes to rebalance the nventory. Snce fndng provably optmal solutons s practcally ntractable, we propose a new cluster-frst route-second heurstc, n whch the polynomalsze Clusterng Problem smultaneously consders the servce level feasblty constrants and approxmate routng costs. Extensve computatonal results on real-world data from Hubway (Boston, MA) and Captal Bkeshare (Washngton, DC) are provded, whch show that our heurstc outperforms a pure mxed nteger programmng formulaton and a constrant programmng approach. Key words : vehcle routng and schedulng, nventory, queues: applcatons, programmng: nteger, programmng: constrants, heurstcs 1. Introducton Bke sharng systems are experencng wde-spread adopton n major ctes around the world, wth over 300 actve systems and more than 200 n plannng (Meddn and DeMao 2012). In these systems, users can pckup and return bkes at desgnated bke sharng statons wth a fnte number of docks. Unfortunately, user behavor results n spatal mbalance of the bke nventory over tme. The system equlbrum s often characterzed by unacceptably low avalablty of bkes or open docks, for pckups or returns respectvely (Frcker et al. 2012, p. 375). Therefore, operators deploy a fleet of trucks to rebalance the bke nventory. We focus on the effcency of these rebalancng operatons, a major cost drver for operators (DeMao 2009, p. 50). Ths problem conssts of two man components. Frst, determnng the desred nventory level at each bke staton, whch s typcally done by an analyss of hstorc user data. Second, desgnng truck routes that wll perform the necessary pckups and delveres n order to reach the target * Ths materal s based upon work partally supported by the Natonal Scence Foundaton under Grant No. CMMI #

2 2 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng nventory levels. In the lterature, these two aspects have been manly consdered separately (see Secton 2 for a lterature revew). However, n practce operators often face a tradeoff between user satsfacton and the cost of rebalancng. Moreover, as wll be shown n ths paper, t may be advantageous to nclude statons n a route even though ther nventory s already satsfactory, n order to satsfy the servce level requrements at a nearby locaton. For these reasons, we propose to consder the nventory balancng problem and the routng problem smultaneously. We model the stochastc demand by vewng the nventory at each staton as a queung system wth fnte capacty and derve closed-form servce level requrements on the transent dstrbuton of the avalablty of bkes and docks (Secton 3). We recognze that servce level requrements can be met when the nventory s between a lower and upper bound. We then ntroduce the noton of self-suffcent statons. Self-suffcent statons meet the servce level requrements wth ther startng nventory and hence should not necessarly be vsted by a vehcle. Next, we consder the routng of vehcles to acheve the servce level requrements. Observe that the pckup and delvery of bkes s unpared,.e., our stuaton corresponds to an extended One- Commodty Pckup-and-Delvery Vehcle Routng Problem (1-PDVRP, see Hernández-Pérez and Salazar-González 2003). However, whle preventng unnecessary vehcle vsts, the nventory flexblty at each staton adds another layer of complexty to the classcal Routng Problem (Secton 4). Snce exact mxed nteger programs (MIPs) for vehcle routng problems become ntractable for realstc nstances, we present a new cluster-frst route-second heurstc n Secton 5. We ncorporate the servce level feasblty constrants n a polynomal-sze Clusterng Problem (modeled as a MIP). The routng costs are ncorporated nto the clusterng problem va a new approxmaton based on maxmum spannng stars. Lastly, we present an mprovement scheme based on elmnaton cuts to mtgate the approxmaton error. In addton, we present a Constrant Programmng (CP) formulaton n Secton 6, actng both as an effectve (exact) soluton method for smaller nstances and a benchmark for our dedcated clustered routng heurstcs. Computatonal experments (Secton 7) usng data from Hubway (Boston, MA) and Captal Bkeshare (Washngton, DC) show that our heurstcs strongly outperform the classcal MIP model. Wthn seconds, we dentfy a feasble soluton wth a reasonable optmalty gap. In a mnute, we fnd better solutons for almost all nstances than the best MIP soluton after 2 hours. Moreover, our dedcated Clustered MIP heurstc outperforms Constrant Programmng on larger nstances. We summarze the man contrbutons of ths paper as follows. Frst, we present a novel formulaton of the rebalancng problem usng dual-bounded servce level constrants. Second, we present a new polynomal-sze Clusterng Problem wth servce level feasblty constrants. Thrd, we develop heurstcs that are fast and accurate for large nstances, and outperform state-of-the-art exstng methods.

3 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng 3 2. Related Work The study of bke sharng systems s ncreasng n popularty. DeMao (2009) and Shaheen et al. (2010) provde a hstory of bcycle sharng, startng wth the frst generaton whte bkes n Amsterdam as early as From 1995 onwards, the thrd generaton IT-based systems ncorporate advanced technologes for bcycle reservatons, pckup, drop-off, and nformaton trackng (Shaheen et al. 2010, p. 7). We dentfy four research substreams n bke sharng lterature: strategc desgn, demand analyss, servce level analyss, and rebalancng operatons. Strategc desgn. Snce most major ctes have planned or consdered mplementaton of bke sharng systems, and exstng systems are often expanded, several studes present models dedcated to strategc desgn. Dell Olo et al. (2011) develop a comprehensve methodology for mplementaton, from estmatng potental demand to optmzng locatons. Martnez et al. (2012) and Prem Kumar and Berlare (2012) present MIP models for the locaton problem. Ln and Yang (2011) create a model tradng off the nterests of both users and nvestors. We note that none of the studes nclude a noton of expected nventory mbalance costs. Demand analyss. The purpose of demand analyss s twofold: forecastng future demand (e.g. for servce level requrements) and understandng the explanng factors for manageral decson makng. Kaltenbrunner et al. (2010) predct the system nventory state and suggest makng such nformaton avalable to users. Froehlch and Olver (2008), Borgnat and Abry (2009), Borgnat et al. (2011), and Latha et al. (2012) dentfy a temporal demand pattern and forecast the number of rentals. Hampshre and Marla (2012) seek land use and soco-economc factors that explan the use of bke sharng systems. Vogel et al. (2011) construct clusters of statons wth smlar demand patterns. These studes could provde nsghts that are helpful n mprovng the servce level requrements developed n ths paper. Servce level analyss. Several studes focus on servce levels n bke sharng systems. We recall that servce level requrements are typcally two-sded: for avalable bkes and docks. Most notably, Nar and Mller-Hooks (2011) and Nar et al. (2013) decompose system-wde relablty nto a set of dual-bounded chance constrants for each staton. We adapt ther dual-bounded servce level constrants, but use a more realstc Markov chan to model the staton nventory over tme, as opposed to observng the total net demand (see Secton 3). Ravv et al. (2011a) present a queung system wth fnte capacty to model expected user dssatsfacton at each staton, smlar to our approach. However, we assume tme-ndependent user arrval rates to derve closed-form solutons. Leurent (2012) models bke sharng statons as a dual Markovan watng system, but contrary to ther assumpton of watng customers, we assume mmedate lost sales.

4 4 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng Rebalancng operatons. Usng mean feld analyss, Frcker et al. (2012) conclude that the equlbrum system performance collapses under heterogenety of user behavor and that a pressng need for rebalancng operatons exsts. Vogel and Mattfeld (2010) motvate rebalancng actvtes usng a system dynamcs model. Shu et al. (2010) estmate rebalancng operatons can lead to an addtonal 15-20% of trps supported system-wde. We dentfy two modes of rebalancng: provdng user ncentves and deployng a truck fleet. Whle Frcker and Gast (2012) and Waserhole and Jost (2012) develop a prcng strategy, and some actve systems have already mplemented ncentve schemes (e.g. V+ for Vélb, see Frcker and Gast 2012, p. 20), we observe that every bke sharng system stll operates a vehcle fleet for rebalancng. Therefore, the underlyng vehcle routng problem has receved most attenton. Most studes use an exact target nventory for each staton, whch mples ther routng problem s more closely related to the 1-PDVRP than ours, whch adds nventory flexblty. Routng costs to attan exact target nventores wll always be hgher than strctly necessary to mantan approprate servce levels (see Secton 3). We note that our models can be parameterzed to solve the target nventory problem. Caggan and Ottomanell (2012) construct a decson support system for routng. Chemla et al. (2012) present a branch-and-cut algorthm for the sngle-vehcle problem, wth results on nstances of up to 100 statons. Approxmaton algorthms for the same problem are gven by Benchmol et al. (2011). The work of Contardo et al. (2012) can be consdered state of the art for the mult-vehcle target nventory problem, usng Dantzg-Wolfe and Benders decomposton to derve lower bounds and feasble solutons wth low computng tmes (approxmately 5 mnutes) for nstances of 5 vehcles and up to 100 statons. Ravv et al. (2011a) present an alternatve approach: the expected system-wde user dssatsfacton s mnmzed subject to a tme lmt. Ther arc-, tme-, and sequence-ndexed MIP models are ntractable for systems of reasonable sze, however we rely strongly on these models n Secton 4 to present the Bke Sharng Rebalancng Problem n ts pure MIP form. As mentoned, the dualbounded chance constrants from Nar et al. (2013) nspre our servce level requrements, but ther paper presents a swappng-based rebalancng model and no explct vehcle routng. Ln and Chou (2012) propose an optmzaton method takng road condtons, traffc regulatons, and geographcal factors nto account, rather than smply usng Eucldan dstance. Ther actual dstance path calculaton s used to mplement a heurstc for the VRP. Naturally, usng actual dstances would lead to decreased costs n practce. 3. Servce Level Requrements In general, nventory rebalancng efforts are made n order to mprove customer servce. Contrary to tradtonal nventory theory, where the servce level ncreases as nventory ncreases, bke sharng

5 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng 5 systems are subject to a net demand process, wth an empty staton preventng users from pckng up bkes and a full staton preventng returns. For the remander of the paper, we let S represent the set of bke sharng statons. Let C denote the capacty (.e. number of docks) of staton S. Net demand mples that after a pckup and a return occur, the nventory s unchanged. Prevous studes (Nar et al. 2013) observe the total net demand (pckups mnus returns durng observaton perod) at each staton S. However, we note that whle observng a net demand of zero, we often stll need bkes and docks. We vew the net demand as a stochastc process whch needs to be satsfed durng the entre observaton perod, not only at the end. In Appendx A we motvate why observng the total net demand s nsuffcent Servce Level Defnton Operators can beneft greatly from measurng ther servce level (Gunasekaran et al. 2001). But due to censorng, t s nontrval to observe lost sales n bke sharng systems. Therefore, operators commonly measure the (fracton of) tme that ther statons are full or empty. Some operators are even penalzed by the local government n proporton to such a tme measure (e.g. Vélb n Pars). In the next secton, we show that the number of dssatsfed customers s proportonal to ths tme measure under the assumpton of Posson demand. Therefore, we can mplement a measurable type 2 servce level: the fracton of demands satsfed drectly should be larger than β and larger than β + for pckups for returns. We assume no backorders,.e., the effect of watng customers s neglgble. Customers often choose alternatve statons n case of stock outs, we assume ths behavor s mplct n the (ndependent) demand processes. Defnton 1. The servce level requrements at staton S are for gven β, β + [0, 1]. E[Satsfed bke pckup demands] E[Total bke pckup demands] E[Satsfed bke return demands] E[Total bke return demands] β β Markov Chan Formulaton The nventory at staton S can be modeled as an M/M/1/K queung system (Kendall 1953), wth the number of customers n the queue representng the nventory. Ths mples that the customer nter-arrval tmes (for bke returns) and servce tmes (.e. nter-arrval tmes for bke pckups) are exponentally dstrbuted wth rates λ and µ, respectvely, at each staton S. We mplctly assume that user behavor durng the observaton perod s statonary. Whle some users (e.g. toursts) arrve smultaneously (compound Posson process), we assume ths effect s

6 6 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng neglgble. There s 1 server and there are K = C watng spaces n the system for staton S. The M/M/1/K queue s well-studed (Morse 1958) and closed-form expressons for the transent probabltes gven a startng state are avalable. Fgure 1 Markov chan for the nventory S (t) at staton S. λ λ λ λ 0 1 C 1 C µ µ µ µ Note. Users arrve wth rate λ to return bkes and wth rate µ to pckup bkes. Denote by {S (t) : t 0} the stochastc process on state space {0,..., C } representng the nventory of staton S at tme t 0. Defne p (s, σ, t) Pr(S (t) = σ S (0) = s), the transent probablty that the nventory at staton S equals σ {0,..., C } at tme t 0 gven startng nventory s {0,..., C }. Defne g (s, σ) 1 T p T 0 (s, σ, t) dt as the expected fracton of the observaton perod [0, T ] for whch the nventory at staton S equals σ gven startng nventory s. Then, n order to meet both the pckup and return servce level from Defnton 1 durng the observaton perod [0, T ], we calculate the expected values: E[Satsfed bke pckup demands] E[Total bke pckup demands] T 0 = µ (1 p (s, 0, t)) dt µ T T 0 = 1 p (s, 0, t) dt T = 1 g (s, 0) and smlarly E[Satsfed bke return demands] E[Total bke return demands] = 1 g (s, C ). Thus, nventory level s satsfes the servce level requrements at staton S when 1g (s, 0) β and 1 g (s, C ) β +. Lemma 1. A closed-form expresson for g (s, σ) exsts. Proof of Lemma 1. (note that N = C ): Morse (1958, p. 64) presents a transent soluton for the M/M/1/N queue p (s, σ, t) = π (σ) + 2ρ 1 2 (σs) C + 1 C m=1 K,m e k,mt

7 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng 7 wth ρ = λ π (σ) = µ { 1 ( µ K,m = C f ρ +1 = 1 1ρ ρ σ 1ρ C +1 otherwse ) ( sn msπ k,m C + 1 m(s + 1)π ρ sn C + 1 k,m = λ + µ 2 ( ) mπ λ µ cos C + 1 The antdervatve P s,σ (t) follows naturally: Thus, g (s, σ) = 1 T P (s, σ, t) = p (s, σ, t) dt = π (σ)t 2ρ 2 1 (σs) C + 1 ) ( sn mσπ C + 1 ρ sn C m=1 K,m e k,mt k,m. T 0 p (s, σ, t) dt = 1 T (P (s, σ, T ) P (s, σ, 0)). ) m(σ + 1)π C + 1 Intutvely, as the startng nventory ncreases, the bke pckup servce level ncreases and the bke return servce level decreases. We note that ndeed g (s + 1, 0) g (s, 0) 0 and g (s + 1, C ) g (s, C ) 0 for s {0,..., C 1}, whch gves us Lemma 2. Lemma 2. To meet the servce level requrements from Defnton 1 at staton S, t must hold that s s mn and g (s, σ) = 1 T and s s max s mn s max wth T 0 p (s, σ, t) dt. = mn { s {0,..., C } : 1 g (s, 0) β = max { s {0,..., C } : 1 g (s, C ) β + Hence, vehcles should rebalance the startng nventory s 0 } } such that s mn s s max staton S, n order to meet the servce levels β, β +. Denote by S 0 = { S : s mn for each s 0 s max } the set of self-suffcent statons whch satsfy the servce level requrements wth ther startng nventory. Note that the servce level requrements could theoretcally be nfeasble, e.g. durng peak hours. We dentfy three types of nfeasblty:

8 8 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng Fgure 2 Example of Servce Level Requrements for Hubway (Boston, MA). 50 Inventory bounds (bkes) Statons Startng nventory Insuffcent Selfsuffcent Note. Observaton perod 8 9AM on weekdays wth β = β + = 95%. The range [s mn, s max ] s dsplayed sold wth [0, C ] dsplayed dotted. For reference, we show the startng nventory s 0 for each staton as a sold marker, based on a snapshot of the system taken at 8AM on Frday June 1st s max < s mn requres the operator to prortze the servce level requrement for ether pckups or returns at staton S, or choose a weghted average. We observe that most operators prortze returns, to prevent users ncurrng unntended late-return fees. 1g (C, 0) < β or 1g (0, C ) < β + mples the nventory always volates (one of) the servce level requrements. Ths requres the operator to choose the best possble nventory bounds. S s0 < S smn or S s0 > S smax vehcle nventory and capacty for the sake of smplcty). mples a system-wde shortage or excess (gnorng All types of nfeasblty would requre the operator to take alternatve measures n case servce level problems persst, e.g., ncrease staton capacty, nfluence user behavor, or ntroduce or remove bkes from the system. However, we encountered no nfeasbltes n processng any of our data. Usng Lemma s 1 and 2, we are able to effcently calculate the servce level requrements at each staton usng closed-form expressons. Example 1. For ths example, we use trp data provded by Hubway (Boston, MA) for the 60 statons that were actve between November 1st 2011 and May 31st In Appendx B, we gve a detaled overvew of the data sources used for our examples and computatonal results. The observaton perod s 8 9AM (T = 1) on weekdays, for whch we have 82 observatons. We estmate λ and µ by the mean (Maxmum Lkelhood Estmator for Posson varables) number of returns and pckups, respectvely, per observaton perod. By requrng a β = β + = 95% servce level at each staton S, we calculate s mn and s max usng Lemma 2. The servce level requrements for each staton are depcted as bounds n Fgure 2. We

9 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng 9 observe that there s a lot of flexblty n the target nventory, wth most statons havng relatvely wde servce level bounds. For one of the statons, s mn s relatvely hgh, but ths s pared wth a hgher than average s max, n antcpaton of bke pckups clearng suffcent docks. 4. Routng Problem Recall that vehcles should rebalance the startng nventory s 0 such that s mn s s max for each staton S, n order to meet the servce level requrements. We defne the Routng Problem, a pure MIP approach to the Bke Sharng Rebalancng Problem, nspred by Ravv et al. (2011a). The bke sharng system s represented as a complete drected graph wth vertex set S and dstances d,j for all, j S. We observe that usng a model that s both arc- and sequence-ndexed ncreases the sze, but yelds much stronger relaxatons than the sequence-ndexed model. Several objectves could be appled to ths routng problem, e.g. mnmzng the total dstance. However, to maxmze user satsfacton t s usually desred to fnsh the rebalancng operatons as soon as possble. For ths reason, we mnmze the maxmum tour length of the vehcles,.e. the makespan of the schedule. Denote by h v the routng costs (dstance traveled) of vehcle v V. Then, our objectve s to mnmze the makespan H = max v V h v of rebalancng the bke nventory such that servce level requrements at each staton are met. We assume user actvty durng the rebalancng operatons s neglgble. We allow arbtrary route start and end ponts (no closed tour) to mplement the model on a rollng horzon. We use a set T = {1,..., T } for sequence ndexng. We ntroduce bnary decson varables x,j,t,v to ndcate whether vehcle v V traverses arc (, j) n tme step t T. Decson varables y,t,v, y +,t,v ndcate bke pckup or delvery, respectvely, by vehcle v V. We use q 0 v nventory and capacty of vehcle v V, respectvely. The formulaton then becomes: and Q v for the ntal mnmze H (P1) s.t. s 0 + (y +,t,v y,t,v) s mn S (1) t T v V s 0 + (y +,t,v y,t,v) s max S (2) t T v V x,j,1,v 1 v V (3) S j N x j,,t1,v S, t T \ {1}, v V (4) x,j,t,v j N j S x,,t,v = 0 (5) S t T v V

10 10 q 0 v + S q 0 v + S t T : t t t T : t t Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng y,t,v Q v x,j,t,v S, t T, v V (6) j N y +,t,v Q v x j,,t,v S, t T, v V (7) j S y,t,v s 0 S (8) t T v V y +,t,v C s 0 S (9) t T v V (y,t,v y +,t,v) 0 t T, v V (10) (y,t,v y +,t,v) Q v t T, v V (11) h v = d,j x,j,t,v v V (12),j S t T H h v v V (13) x,j,t,v {0, 1} y,t,v, y +,t,v N 0 S, j N, t T, v V S, t T, v V H, h v 0 v V Constrants (1)-(2) mpose the servce level requrements from Lemma 2. Constrants (3)-(7) take care of vehcle routng. Constrants (3) mply that each vehcle starts at most one route. Constrants (4) take care of flow conservaton. Constrants (5) prevent dwellng. Constrants (6)-(7) ensure pckup or delvery can only take place when leavng from or arrvng at a staton, respectvely. Note that N S {0} extends the staton set wth an artfcal vertex 0 to allow pckng up bkes at the fnal stop wthout ncurrng routng costs. Constrants (8)-(9) lmt the amount of transshpments, because of the model s nablty to track staton nventory over tme (cf. Ravv et al. 2011a, p. 10). Constrants (10)-(11) ensure that the vehcle nventory remans non-negatve and wthn vehcle capacty at all tmes. Constrants (12) defne the routng costs for each vehcle. Fnally, constrants (13) lnearze the objectve H = max v V h v. Note that the ntegralty constrants on y,t,v, y +,t,v can be relaxed wthout mplcatons. Denote by H (S, V) the optmal soluton obtaned by solvng (P1) for staton set S and vehcle set V. Solvng (P1) gves an optmal soluton for the Bke Sharng Rebalancng Problem (apart from the lmted transshpments). However, we observe that(p1) s practcally ntractable for realstc staton sets wth S 50 and vehcle fleets wth V 3. Therefore, we ntroduce Clustered Routng heurstcs n Secton 5 and a Constrant Programmng heurstc n Secton 6 as alternatves to fnd hgh qualty solutons n a short amount of tme.

11 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng Clustered Routng We formulate a Clusterng Problem to decompose the Routng Problem (P1) nto separate snglevehcle Routng Problems, thereby reducng combnatoral complexty. A feasble clusterng soluton assgns dsjont clusters of statons S v S to vehcles v V such that the servce level requrements can be satsfed usng only wthn-cluster vehcle routng. Implementng these feasblty constrants n exstng clusterng algorthms, e.g. the Fsher-Jakumar algorthm (Fsher and Jakumar 1981), s non-trval (see Secton 5.3). Therefore, we propose to formulate the Clusterng Problem as an extended Set Parttonng Problem. The core of the model s a set of bnary decson varables: { 1 f staton S s assgned to vehcle v V, z,v = 0 otherwse. These varables assgn a cluster of statons S v = { S : z,v = 1} to each vehcle v V. The objectve of the Clusterng Problem s to fnd a feasble soluton whle mnmzng makespan H = max v V h v,.e., rebalance the system as soon as possble and dvde the workload between vehcles. Optmally, statons are clustered wth known exact routng costs for any (feasble) combnaton of statons S v. However, the computatonal complexty (see Secton 4) requres us to use approxmatons to estmate the routng costs. Therefore, we are nterested n non-algorthmc routng costs approxmatons that correlate hghly wth the exact vehcle routng costs wthn a cluster, wth a consstent over- or underestmaton Routng Costs Approxmaton It s not straghtforward to approxmate the optmal routng costs for a cluster, because feasblty constrants on the vehcle route may requre (many) revsts. However, assumng that all statons n S v need to be vsted, the wthn-cluster routng costs H (S v, {v}) are bounded from below by the optmal soluton of a Travelng Salesman Problem wth an added artfcal zero-dstance depot (to model the open tour for rollng horzon mplementaton). If we mpose the trangle nequalty, then ths lower bound equals the length of the shortest Hamltonan path over S v. TSP approxmatons are wdely studed, see e.g. Laporte (1992). However, exponental approxmatons do not scale well n MIPs and, as mentoned, we refran from algorthmc approxmatons because of the feasblty constrants. For example, the MIP formulaton of the Held-Karp relaxaton (Held and Karp 1970, Charkar et al. 2004) would requre constrants for each non-empty subset of S. Furthermore, the polynomal-sze Assgnment Problem relaxaton (Dantzg et al. 1954) has lmted applcablty, because t does not satsfy monotoncty,.e., addng statons to a cluster may

12 12 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng actually decrease the routng costs approxmaton. Snce sub-tours are not elmnated, our expermentaton wth the Assgnment Problem resulted n the undesrable assgnment of geographcally separated groups of statons to the same cluster. Instead, we ntroduce the Maxmum Spannng Star approxmaton (we note that Wu et al. (1998, p. 2) ntroduce the algorthmc mnmum k-star approxmaton). Denote by SPS (S v ) = j S v d,j the cost of the spannng star (spannng tree wth depth one) of S v rooted at staton S v. The routng costs are approxmated by the maxmum-cost spannng star max Sv SPS (S v ). In Secton 5.3 we show that the Maxmum Spannng Star can be mplemented usng a polynomal number of bnary assgnment varables and constrants. Moreover, the Maxmum Spannng Star s an upper bound on the shortest Hamltonan path over the cluster. Most mportantly, the Maxmum Spannng Star satsfes monotoncty, such that statons are only assgned to a cluster f ths s necessary for feasblty of the servce level requrements Propertes Maxmum Spannng Star Next, we prove the monotoncty and upper bound of the Maxmum Spannng Star approxmaton, to motvate our ntuton that MAXSPS(S v ) and H (S v, {v}) correlate. Namely, both H (S v, {v}) and MAXSPS(S v ) are bounded from below by the shortest Hamltonan path over S v, gven that the Maxmum Spannng Star satsfes monotoncty (whch ensures all statons are vsted). Lemma 3. The Maxmum Spannng Star approxmaton satsfes monotoncty: Proof of Lemma 3. MAXSPS(S v {}) MAXSPS(S v ). Assume wthout loss of generalty that MAXSPS(S v ) s rooted at staton j S v. Then, MAXSPS(S v ) = SPS j (S v ) SPS j (S v ) + d j, MAXSPS(S v {}). Denote by TSP 0(S v ) the length of the shortest Hamltonan path over S v, whch, under the trangle nequalty, equals the optmal soluton of a Travelng Salesman Problem wth an artfcal zero-dstance depot. Lemma 4. The Maxmum Spannng Star approxmaton s an upper bound on the length of the shortest Hamltonan path: Proof of Lemma 4. MAXSPS(S v ) TSP 0(S v ). Denote by C(S v ) =,j S v d,j the cost of the drected clque on S v. Denote by n = S v the number of statons n a canddate cluster. Then: S v SPS (S v ) = C(S v ), whch yelds MAXSPS(S v ) C(S v) n.

13 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng 13 Akyama et al. (2004, p. 40) present the Waleck decomposton of a complete undrected graph K n wth odd n 3 nto (n 1)/2 Hamltonan cycles. It follows that K n wth even n 2 can be decomposed nto n/2 Hamltonan paths. Ths result gves us: Case 1: n 2 s even. The complete drected graph on S v Hamltonan paths. Hence, TSP 0(S v ) C(S v) n. can be decomposed nto n drected Case 2: n 3 s odd. The complete drected graph on S v can be decomposed nto n 1 drected Hamltonan cycles. We can remove one of the n edges n any of these n 1 Hamltonan cycles to obtan a Hamltonan path. Thereby, Thus, for both even and odd n 2 we have TSP 0(S v ) n 1 C(S v ) n n 1 = C(S v) n. TSP 0(S v ) C(S v) n MAXSPS(S v ). We report on the approxmaton performance of the Maxmum Spannng Star n Secton 7.1. In partcular, we show that the Maxmum Spannng Star approxmaton s hghly correlated wth the actual routng dstance for our data sets (correlaton of more than 85%) Clusterng Problem Next, we mplement the Maxmum Spannng Star approxmaton n our Clusterng Problem. The objectve s to mnmze the estmated makespan Ĥ such that servce level requrements can be satsfed usng only wthn-cluster vehcle routng. mnmze Ĥ (P2) s.t. q 0 v + S z,v = 1 S \ S 0 (14) v V z,v 1 S 0 (15) v V s 0 z,v S s mn z,v v V (16) (Q v q 0 v) + S s 0 z,v S s max z,v v V (17) ĥ v j S d,j (z,v + z j,v 1) S, v V (18) Ĥ ĥv v V (19) z,v {0, 1} S, v V Ĥ, ĥv 0 v V

14 14 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng Fgure 3 Example of a soluton of the Clusterng Problem (P2) for Hubway (Boston, MA). Startng nventory Insuffcent Selfsuffcent Constrants (14)-(17) ensure feasblty of the clusterng soluton. Insuffcent statons must be vsted by a vehcle (14) and self-suffcent statons can be vsted (15). A vehcle cluster must contan enough bkes, possbly usng the vehcle startng nventory qv, 0 such that servce level requrements can be met through wthn-cluster repostonng (16). Constrants (17) are smlar but then for the maxmum nventory n the cluster, possbly usng vehcle surplus capacty Q v qv. 0 Constrants (18) mpose SPS (S v ) as a lower bound on the estmated routng costs ĥv f staton S s assgned to vehcle v V. Snce ĥv s ndrectly mnmzed, ĥv MAXSPS(S v ). The estmated makespan Ĥ = max v V ĥv s lnearzed through constrants (19). Note that we have formulated a compact clusterng model wth routng costs approxmaton, whch guarantees that the servce level requrements can be satsfed at each staton whle usng only wthn-cluster vehcle routng. Example 2. Fgure 3 shows the assgnment of statons to the two Hubway vehcles obtaned by solvng the Clusterng Problem (P2) for the servce level requrements from Example 1. We use real nventory data, based on a snapshot of the system taken at 8AM on Frday June 1st Heurstcs Havng presented MIP formulatons for the Clusterng Problem and the Routng Problem, we now formally ntroduce our Clustered Routng heurstcs. Heurstc 1 (Clustered MIP). Subsequently: 1. Solve the Clusterng Problem (P2).

15 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng For each v V solve the Routng Problem (P1) wth S = S v and V = {v} to obtan H (S v, {v}). 3. H = max v V H (S v, {v}). Snce the routng costs approxmaton s mperfect and may therefore lead to sub-optmal clusters, the Clustered MIP heurstc has an optmalty loss. However, we can use the nformaton obtaned from routng the ndvdual clusters to add cuts to the Clusterng Problem. Assume that an unknown optmal soluton exsts wth makespan H OPT strctly less than our best found soluton H. Ths mples that all clusters S v wth known optmal routng costs H (S v, {v}) H are not part of the optmal soluton. Thereby, these clusters can be elmnated from the soluton space of the Clusterng Problem. Note that, contrary to problems n whch total routng costs are mnmzed, these cuts do not need to be removed later. Heurstc 2 (Clustered MIP wth Cuts). Subsequently: 1. Defne a cut set C 2 S of subsets of S. Intalze C =. 2. Solve the Clusterng Problem (P2) wth addtonal constrants: z,v z,v c 1 c C, v V (20) c S\c 3. For each v V solve the Routng Problem (P1) wth S = S v and V = {v} to obtan H (S v, {v}). (a) If max v V H (S v, {v}) < H or C = then (re)defne H and store the routng soluton. (b) For each v V wth H (S v, {v}) H redefne C = C {S v } 4. Go to step 2. Note that H s only redefned f an mprovement s found (3a), n whch case at least one cut s added per teraton (we have H (S v, v) = H for at least one v V). If no mprovement s found, then H s not redefned and at least one cluster has actual costs strctly larger than the best found soluton H. We add a cut for all non-mprovng clusters (3b), and contnue. Note that any strct subset or superset of a cut c C s not elmnated by constrants (20). The cuts force the Clusterng Problem to teratvely adjust the assgnment of statons to vehcles, thereby mtgatng the approxmaton error. After fntely, but possbly exponentally many steps, the heurstc dentfes the optmal soluton. Unfortunately, to our knowledge no stronger termnatng condton for Clustered MIP wth Cuts heurstc than an exhaustve search of the Clusterng Problem soluton space exsts. However, Table 1 n Secton 7.2 shows how quckly mprovements over the Clustered MIP heurstc are attaned.

16 16 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng 6. Constrant Programmng In ths secton we present our Constrant Programmng (CP) model for the Bke Sharng Rebalancng Problem. Constrant Programmng s among the state of the art for solvng complex routng and schedulng problems, even though t apples a generc modelng and solvng approach. In partcular, CP has been appled before to constraned routng problems (Klby and Shaw 2006). Most ndustral CP solvers combne constrant propagaton wth large neghborhood search for solvng routng problems (Shaw 1998). In order to take advantage of the strengths of CP, t s common to represent routng problems as schedulng problems, by representng the vst of a locaton as an actvty. An actvty s a hgh-level CP modelng structure that mplctly defnes nteger varables for ts start tme, duraton, and end tme, and a Boolean varable for ts presence. Actvtes for whch the presence s not fxed to true are called optonal actvtes. Travelng the dstance between two locatons s represented by sequence-dependent setup tmes between the respectve actvtes (for each par of locatons),.e., f staton j s vsted drectly after staton, we need to respect the dstance d,j as setup tme. In CP, actvtes mpact resources whch, n case of routng problems, correspond to the vehcles. For example, for each vehcle, we must ensure that no two actvtes overlap. We next specfy the detals of our CP model, followng the AIMMS notaton for actvtes and resources (Roelofs and Bsschop 2012). In partcular, each actvty A nduces the varables A.Start, end tme A.End, and presence A.Present, as explaned above. For each staton S and vehcle v V, we defne optonal actvtes Pckup[,v] and Delvery[,v] wth duraton 0. That s, each staton may be vsted by any of the vehcles. Varables y +,v and y,v represent the pckup, respectvely delvery, amount for vehcle v at staton, as n our models above. Our CP model then becomes: s.t. mnmze max { max S v V {Pckup[, v].end}, max {Delvery[, v].end} S v V s 0 + v V } (P3) y +,v y,v s mn S (21) s 0 + y +,v y,v s max S (22) v V Pckup[, v].present + Delvery[, v].present 1 S (23) v V Pckup[, v].present = 1 y,v 1 S, v V (24) Pckup[, v].present = 0 y,v = 0 S, v V (25) Delvery[, v].present = 1 y +,v 1 S, v V (26)

17 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng 17 Delvery[, v].present = 0 y +,v = 0 S, v V (27) y,v, y +,v {0,..., Q v } S, v V wth resources Sequental resource VehcleTme[v]( Schedule doman: {0,..., MaxTme} Actvtes: Pckup[,v], Delvery[,v] Transton: d,j ) Parallel resource VehcleInventory[v]( Actvtes: Pckup[,v], Delvery[,v] Level range: {0,..., Q v } Intal value: qv 0 Begn change: Delvery[,v]: y +,v End change: Pckup[,v]: y,v ) Parallel resource StatonInventory[]( Actvtes: Pckup[,v], Delvery[,v] Level range: {0,..., C } Intal value: s 0 Begn change: Delvery[,v]: y +,v End change: Pckup[,v]: y,v ) As before, constrants (21)-(22) mpose the servce level requrements from Lemma 2. Constrants (23) are so-called alternatve resource constrants, whch lmt the number of vsts to one per staton. Addng these constrants can greatly mprove the performance of the constrant propagaton, but the optmal soluton may be elmnated. Note that t may not always be possble to mpose the alternatve resource constrants, for example due to lmted vehcle capacty. In such cases, the model can trvally be extended to allow multple vsts by ncreasng the rght-hand sde. We can ndex the actvtes and varables y,v, y +,v correspondngly. However, our prelmnary expermentaton showed strongly decreasng computatonal performance f we allowed multple vsts per locaton. Therefore, we mposed the alternatve resource constrants n our experments. Constrants (24)-(27) lnk the vehcle presence constrants wth performng a pckup or delvery. Note that the f and only f constrants enhance propagaton.

18 18 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng For each vehcle we ntroduce two types of resources. The frst represent the no-overlap condtons wth respect to the vehcle tme, usng a Sequental resource named VehcleTme[v]. For each such resource, we dentfy the dscrete tme horzon as ts Schedule doman, whle the keyword Actvtes specfes whch actvtes mpact the resource. The arc-dependent transton tmes model the travel dstances va Transton. The second resource assocated wth a vehcle s ts nventory, modeled as a Parallel resource named VehcleInventory[v]. In addton to specfyng the set of actvtes n ts scope, we defne ts Level range to be {0,..., Q v }, whch s ntalzed at qv. 0 Furthermore, we specfy for each actvty n ts scope how t mpacts the level. For Delvery[,v], the level s changed at the start of the actvty, wth amount y,v. + Lkewse, for Pckup[,v], the level s changed at the end of the actvty wth amount y,v. Lastly, for each staton we defne a Parallel resource representng the staton nventory, named StatonInventory[]. The range of ths nventory s {0,..., C } wth ntal value s 0. Level changes for pckups and delveres are exactly opposte to the vehcle nventory changes. 7. Computatonal Results In ths secton we report on the performance of our routng costs approxmaton and heurstcs. Recall that we gve a detaled overvew of our data sources n Appendx B Approxmaton performance In Secton 5.2 we proved the monotoncty and upper bound MAXSPS(S v ) TSP 0(S v ) of the Maxmum Spannng Star to motvate our ntuton that MAXSPS(S v ) approxmates H (S v, {v}) TSP 0(S v ). Fgure 4 vsualzes ths relatonshp wth a scatterplot, n whch the lne H (S v, {v}) = MAXSPS(S v ) s shown for reference. Whle we establshed that H (S v, {v}) TSP 0(S v ), we note that the addtonal feasblty constrants on the vehcle route mposed n (P1) do not lead to routng costs hgher than the MAXSPS approxmaton for any nstance. Rather, the Maxmum Spannng Star consstently overestmates the routng costs, whch s reflected n the hgh correlaton of 87.99%. A lnear regresson on H (S v, {v}) wth an mposed zero ntercept estmates a.4238 coeffcent for MAXSPS(S v ) yeldng R 2 = 94.28%. Nonetheless, a small approxmaton error s present. Next, we show how the mprovement scheme wth elmnaton cuts overcomes ths lmtaton of the Clustered MIP heurstc Heurstcs performance We report computatonal results comparng the exact MIP (P1), Clustered MIP (Heurstc 1), Clustered MIP wth Cuts (Heurstc 2), and CP (P3) approaches to the Bke Sharng Rebalancng

19 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng 19 Fgure 4 Computatonal results for the Maxmum Spannng Star routng costs approxmaton. Routng costs (m) Approxmaton (m) Number of statons Note. Routng costs H (S, {v}) are plotted aganst the approxmaton MAXSPS(S v) for all 82 two-vehcle nstances of Hubway (Boston, MA). Each data pont corresponds to a cluster. The correlaton equals 87.99%. Problem. We consder multple famles of nstances based on real trp and nventory data provded by Hubway (Boston, MA) and Captal Bkeshare (Washngton, DC). All experments were performed on an Intel Xeon 2.50GHz wth 4GB of memory, usng AIMMS 3.13 FR1 modelng software wth MIP solver GUROBI 5.0 (whch outperformed CPLEX 12.4 on test nstances) and CP solver IBM ILOG CP Optmzer Parameters. A famly of nstances s defned by a market (Hubway or Captal Bkeshare), observaton perod (8 9AM or 4 5PM), servce level (consstently 95%) and number of vehcles. We observe that the mornng and afternoon commute are the most challengng rebalancng problems. Furthermore, dfferent servce levels (90%, 99%) dd not substantally mpact our fndngs. Subsequently, for each famly we generate multple nstances by usng dfferent nventory snapshots contanng the startng nventory s 0 for each staton S at the begnnng of the observaton perod. We calculate Eucldan dstances d,j = d j, n meters based on the lattude and longtude of statons. We assume q v = 0 to refran from random data generaton. For MIP we set T = S \ S 0 (ths ensures feasblty for our nstances) and for the Clustered MIP heurstcs we set T = S v +1 (ths allows a revst). Hubway (Boston, MA). We restrct to S = 60 statons to obtan suffcent trp observatons to calbrate the servce level requrements on 82 weekdays between November 1st 2011 and May 31st We use 41 nventory snapshots on weekdays between June 1st and July 29th Hubway currently operates V = 3 vehcles wth capacty Q v = 22 snce openng an addtonal 40 statons n Summer However, durng our observatons Hubway only operated two vehcles. We nvestgate both truck fleet szes to reveal the possble mplcatons for performance.

20 20 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng Table 1 Computatonal results for Hubway (Boston, MA). MIP Clustered MIP Clustered MIP wth Cuts CP Famly V LP bound Best found Tme Soluton Tme Soluton Iteratons Tme Soluton Tme 8 9AM AM PM PM These results are averaged over the 41 nstances of each nstance famly. The mean number of nsuffcent statons was equal to S \ S 0 = 10. The MIP solver was unable to fnd a soluton of the full model for three nstances. These are excluded from ths summary and shown n Appendx C. Fgure 5 Computatonal results for Hubway (Boston, MA). Soluton / MIP best found 125% 100% 75% 50% 25% 0% 125% 100% 75% 50% 25% 0% 89AM Complexty 45PM 2 vehcles 3 vehcles Soluton Clustered MIP wth Cuts CP MIP LP bound Note. Instances are sorted wth decreasng MIP LP bound to MIP best found soluton rato, to approxmate complexty. Here, the MIP best found soluton s shown as 100%. If the LP bound equals the best found soluton (100%), then the solver was able to solve the exact MIP to optmalty. After extensve experments we set the computatonal cut-offs (f unsolved): MIP after 7200 seconds; Clustered MIP after 20 seconds for both (P2) and (P1); Clustered MIP wth Cuts after 60 seconds n total wth 20 seconds for (P2) and (P1); CP after 60 seconds. For CP we set the schedule doman wth MaxTme = 50000, whch proved necessary to quckly dentfy a feasble (but low-qualty) soluton. Table 1 summarzes our computatonal results for Hubway. We show the average solutons and computaton tmes per nstance famly for the exact MIP and our heurstcs. For our mprovement scheme (Clustered MIP wth Cuts), we show the number of teratons. We observe that n 1 mnute, our Clustered MIP wth Cuts and CP heurstc outperform the best found soluton of the MIP after 2 hours wth 5 10% on average for the two-vehcle famles. For the three-vehcle famles, the mprovement s 15 25%. Note that our Clustered MIP heurstc can, on average, even generate better solutons than the MIP model wthn 1 second. Ths allows more

21 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng 21 Table 2 Computatonal results for Captal (Washngton, DC). Clustered MIP Clustered MIP wth Cuts CP Famly V S \ S 0 Soluton Tme Soluton Iteratons Tme Soluton Tme 8 9AM PM The S \ S 0 column shows the mean number of nsuffcent statons per nstance famly. than 60 teratons of the Clustered MIP wth Cuts heurstc n 1 mnute, yeldng an mprovement of approxmately 10% over the Clustered MIP heurstc. In Fgure 5 we present a more detaled comparson of our dedcated Clustered MIP wth Cuts heurstc wth the CP heurstc for the Hubway nstance famles. We show how they perform n comparson to the best found MIP soluton after two hours (depcted as 100%). We note that the performance of the full MIP model decreases strongly for the three vehcle famles, wth our 1 mnute heurstc solutons up to 75% better than the best found solutons after two hours. CP performs very well for the Hubway nstances. We beleve that due to the low number of nsuffcent statons, on average 10 out of 60, CP s able to quckly dentfy a feasble soluton whch s subsequently mproved. We observe that the Routng Problem (P1), when appled to the ndvdual clusters, becomes ntractable when there are more than 15 statons n a cluster. For nstances on whch CP outperformed the Clustered MIP heurstcs, ndeed one vehcle was assgned more than 15 statons. In these stuatons, we suggest a hybrd cluster-frst, CP-second approach would leverage both strengths, because CP performed well on sngle-vehcle test nstances. Captal Bkeshare (Washngton, DC). We restrct to S = 135 statons to obtan 130 trp observatons on weekdays between January 1st and June 30th We use 27 8AM and 25 4PM nventory snapshots of weekdays obtaned between December 17th 2012 and January 25th Captal Bkeshare currently operates V = 5 vehcles wth Q v = 25. We ncrease the computatonal cut-offs of Clustered MIP wth Cuts and CP to 120 seconds to accommodate the ncreased complexty. For CP we ncrease MaxTme to Other cut-offs are dentcal to those used for Hubway. Wth 135 statons and 5 vehcles, the Captal Bkeshare nstances were too complex to derve feasble solutons or even useful LP bounds from the full MIP (P1) wthn a reasonable amount of tme. Therefore, we report only on the performance of our heurstcs. Table 2 summarzes our computatonal results for Captal. The 8 9AM nstances are more complex than the 4 5PM nstances, wth on average 25 nsuffcent statons per nstance nstead of 11 (see column S \ S 0. The Clustered MIP wth Cuts heurstc s 45% better than CP for the 8 9AM famly of nstances. Ths hghlghts exactly where we beleve our cluster-frst route-second

22 22 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng Fgure 6 Computatonal results for Captal Bkeshare (Washngton, DC). Soluton / Clustered MIP soluton 300% 250% 200% 150% 100% 50% 0% 89AM Complexty 45PM 5 vehcles Soluton Clustered MIP wth Cuts CP Note. Instances are sorted wth ncreasng CP soluton to Clustered MIP soluton rato, to approxmate complexty. The Clustered MIP soluton s shown as 100%. heurstc excels. The polynomal-sze Clusterng Problem (P2) allows rapd decomposton of the mult-vehcle problem nto reasonably good sngle-vehcle clusters. Then, the Routng Problem (P1) can be solved to optmalty for ndvdual clusters n under a second. The mprovement cuts mtgate both the approxmaton error and (possbly) sub-optmalty ncurred from cuttng off the Clusterng Problem before the solver s fnshed. In Fgure 6 we present a comparson of the heurstcs, wth the Clustered MIP heurstc shown as 100%. For some nstances, even the smple Clustered MIP heurstc s up to 65% better than CP. Fgure 6 also shows clearly how addng cuts n the Clustered MIP wth Cuts heurstc can yeld mprovements of up to 40% over the smple Clustered MIP heurstc. The results for Captal Bkeshare show that our dedcated Clustered MIP (wth Cuts) heurstc performs better than CP, especally for nstances wth a large vehcle fleet and a low number of statons per vehcle. Ths mples the polynomal-sze Clusterng Problem can handle large sets of nsuffcent statons, gven that enough vehcles are avalable to dvde the workload. When an nstance s more smlar to a schedulng problem (.e., a lower number of vehcles and longer routes, lke for some Hubway nstances), the technques embedded n CP show ther strength. 8. Conclusons Ths paper s the frst to unfy dual-bounded servce level constrants, whch add nventory flexblty, and vehcle routng n bke sharng systems. We represent the nventory at each staton as a fnte-buffer sngle-server queung system and use closed-form analyss of the transent probabltes to calculate servce level requrements. We ntroduce the noton of self-suffcent statons, whch fulfll these requrements wth ther startng nventory. Hence, self-suffcent statons do not necessarly need to be vsted by a vehcle, but may act as source or snk nodes.

23 Schujbroek, Hampshre, and Van Hoeve: Inventory Rebalancng and Vehcle Routng 23 We present a mxed nteger programmng based Clusterng Problem that decomposes the multvehcle rebalancng problem nto separate sngle-vehcle problems, whle takng nto account servce level feasblty constrants (a cluster-frst route-second approach). We ntroduce a novel polynomal-sze Maxmum Spannng Star routng costs approxmaton for the Clusterng Problem to acheve hgh computatonal performance. We develop an mprovement scheme based on elmnaton cuts to mtgate the approxmaton error. Furthermore, we provde the frst constrant programmng formulaton of the bke sharng rebalancng problem. Usng emprcal data from two bke sharng systems, we extensvely test the Clustered MIP heurstcs aganst the classcal full MIP model and the constrant programmng approach. Our Clustered MIP wth Cuts heurstc outperforms the constrant programmng formulaton as the number of vehcles (and correspondngly, the number of statons) grows. Constrant programmng performs well when the number of vehcles s low and the number of statons per vehcle s hgh. Both the Clustered MIP and constrant programmng approaches dentfy better solutons wthn one or two mnutes, than the often-used full MIP after two hours. We thus beleve that our approach s sutable for practcal mplementaton n bke sharng systems. The novel heurstcs and our approxmaton may be applcable to other constraned routng problems, specfcally to (extended) One-Commodty Pckup-and-Delvery VRPs lke the empty freght contaner rebalancng problem, as well as to other sharng systems. Appendx A: Net demand process Fgure 7 motvates why we adapt a process vew to net demand for bkes, nstead of observng total net demand. Appendx B: Data sources In order to produce the examples and computatonal results, we processed two data sets from Hubway (Boston, MA) and Captal Bkeshare (Washngton, DC), whch have dentcal formattng. These data sets were made avalable through ther webstes thehubway.com and captalbkeshare.com. Each data set conssts of three tables: Statons, Trps and Snapshots. The Statons table contans the followng felds for each staton: d, name, lat, lng, nstalled, locked and temporary. We use the d feld to create the relatonshp wth the Trps and Inventory tables. We use the lat and lng felds to calculate the Eucldean dstance matrx d n meters wth the spdsts functon of the sp package n R. The Trps table contans the followng felds for each trp: d, start date, end date, start staton, end staton, bke name, bke and member type. We calculate the number of pckups at a staton durng the observaton perod usng start date and start staton, and the number of returns usng end date and end staton, to synchronze these events. As mentoned