Incentivizing the Use of Bike Trailers for Dynamic Repositioning in Bike Sharing Systems

Transcription

1 Incentivizing the Ue of Bike for Dynamic Repoitioning in Bike Sharing Sytem Supriyo Ghoh School of Information Sytem Singapore Management Univerity Pradeep Varakantham School of Information Sytem Singapore Management Univerity Abtract Bike Sharing Sytem (BSS) i a green mode of tranportation that i employed extenively for hort ditance travel in major citie of the world. Unfortunately, the uer behaviour driven by their peronal need can often reult in empty or full bae tation, thereby reulting in lo of cutomer demand. To counter thi lo in cutomer demand, BSS operator typically utilize a fleet of carrier vehicle for repoitioning the bike between tation. However, thi fuel burning mode of repoitioning incur a ignificant amount of routing, labor cot and further increae carbon emiion. Therefore, we propoe a potentially elf-utaining and environment friendly ytem of dynamic repoitioning, that move idle bike during the day with the help of bike trailer. A bike trailer i an add-on to a bike that can help with carrying 3-5 bike at once. Specifically, we make the following key contribution: (i) We provide an optimization formulation that generate repoitioning tak o a to minimize the expected lot demand over pat demand cenario; (ii) Within the budget contraint of the operator, we then deign a mechanim to crowdource the tak among potential uer who intend to execute repoitioning tak; (iii) Finally, we provide extenive reult on a wide range of demand cenario from a real-world data et to demontrate that our approach i highly competitive to the exiting fuel burning mode of repoitioning while being green. Introduction Due to it potential to mitigate the carbon emiion and traffic congetion, Bike Sharing Sytem (BSS) have been widely adopted in major citie acro the world. According to Meddin and DeMaio (2016), 1139 ytem with a fleet of over 1,445,000 bicycle are already intalled in major citie and additionally, 357 ytem are either in planning tage or under contruction. Popular example of BSS are Capital Bikehare in Wahington DC, Hubway in Boton, Bixi in Montreal, Vélib in Pari, Wuhan and Hangzhou Public Bicycle in Hangzhou etc. In a regular BSS, bae tation are cattered throughout a city and each tation i tocked with a pre-determined number of bike at the beginning of the day. According to peronal need, uer with memberhip card can pickup and drop-off bike at any bae tation, each of which ha a finite number of dock. At the end of the work Copyright c 2017, Aociation for the Advancement of Artificial Intelligence ( All right reerved. day, carrier vehicle (e.g., truck) are ued to rebalance the entire ytem o a to return to ome pre-determined configuration at the beginning of the day. BSS often experience a ignificant lo in cutomer demand during the day due to the uncoordinated movement of cutomer. Moreover, BSS companie (e.g., Vélib in Pari) are often penalized by local government for lo in cutomer demand (Schuijbroek, Hamphire, and Van Hoeve 2017), a it can reult in uage of fuel burning mode of private tranport. To addre thi problem, a wide variety of reearch paper and current ytem employ the idea of repoitioning idle bike with the help of vehicle during the day, by taking into account the movement of bike by cutomer. While uch a method of repoitioning can help reduce imbalance, there are multiple drawback: (a) Vehicle incur ubtantial routing and labor cot; (b) More importantly, the fuel burning model of repoitioning i at odd with the environment friendly nature of BSS; and (c) Finally, due to a limited number of thee vehicle, they are typically not ufficient to account for all the lot demand. A an alternative, ome BSS operator (e.g., CitiBike in NYC) have recently introduced the notion of bike trailer (O Mahony and Shmoy 2015). A bike trailer i an add-on to a bike that can carry a mall number of bike (e.g., each bike trailer can hold 3-5 bike) and i ueful to relocate bike to nearby tation. are an environment friendly mode of repoitioning the bike. Exiting reearch by O Mahony and Shmoy (2015) ha focued on computing the repoitioning tak for trailer with the aumption that dedicated taff can execute the repoitioning tak. However, given the limited budget availability, it i not economically viable to employ dedicated taff for each of the trailer. Thi paper introduce a potentially elf-utaining repoitioning ytem that addree thi Dynamic Repoitioning and Routing Problem with (DRRPT). We employ an unique combination of optimization and mechanim deign that crowdource the repoitioning tak to the potential uer while working within the budget contraint of the operator. Specifically, we provide a rolling horizon framework, where at each time tep we have two component executed one after another: 1. We firt employ mixed integer linear optimization to generate potential repoitioning tak along with their valuation at the next time tep.

2 2. We employ an incentive compatible mechanim to crowdource (uing payment/trip baed incentive) the repoitioning tak to the uer who are intereted in executing thoe tak within the budget contraint of the operator. There ha been exiting reearch (Singla et al. 2015; Pfrommer et al. 2014) that ha focued on providing incentive to uer for aiting with repoitioning. However, thi line of work ha primarily focued on individual bike and ha taken a myopic (individual tation) view on whether a bike i required at a tation. In thi work, we provide an end to end ytem that take the global view (all tation) of the repoitioning requirement and incentive their execution within the budget contraint. We evaluate our ytem uing a imulation model which i built on the realized demand cenario from a real-world data et. At each time tep the two component of the rolling horizon framework are executed on thi imulator to identify the ditribution of bike for the next time tep. Thi iterative proce continue until we reach the lat time tep. Experimental reult on multiple ynthetic and a real-world data et of Hubway (Boton) BSS demontrate that our approach i highly competitive in term of reducing the expected lot demand, over the fuel burning model of repoitioning. Related Work Given the practical importance of BSS, repoitioning problem ha been tudied extenively in the literature. We categorize exiting reearch into three thread: (a) Static and (b) Dynamic repoitioning uing carrier vehicle; (c) Incentivizing cutomer and utilizing trailer for repoitioning. Static repoitioning i the problem of finding route for a fleet of vehicle to repoition bike at the end of the day when the movement of bike by cutomer are negligible, to achieve a pre-determined inventory level at the bae tation. Chemla, Meunier, and Wolfler Calvo (2013) employ branch and cut algorithm to olve the tatic repoitioning problem with more than a hundred tation. Benchimol et al. (2011) propoe an approximate olution for the tatic rebalancing and routing problem with a ingle vehicle uing inight from the olution of C-delivery TSP (Chalaani and Motwani 1999). Raviv, Tzur, and Forma (2013) and Raviv and Kolka (2013) provide calable approximate olution for multiple vehicle uing mathematical optimization model where they deign an objective function that penalize unavailability of bike or empty dock. Di Gapero, Rendl, and Urli (2013; 2015) employ contraint programming (CP) to efficiently olve the tatic repoitioning problem uing large neighbourhood earch. A the uncertainty and change in demand alter the tation inventory level, thee tatic approache are not uitable to olve our problem during the day. Dynamic repoitioning i referred to a the cae when the movement of cutomer during the day are conidered in the planning period. Nair and Miller-Hook (2011) and Nair et al. (2013) provide a dynamic repoitioning approach by employing dual-bounded joint-chance contraint to enure that the predicted near future demand i erved with a certain probability. Schuijbroek, Hamphire, and Van Hoeve (2017) develop a calable approximate olution by clutering the bae tation uing maximum panning tar (MAXSPS) and allocate one vehicle in each cluter o a to meet the ervice level requirement. Furthermore, they repreent the problem a a clutered vehicle routing problem [CluVRP] (Battarra, Erdogan, and Vigo 2014) and olve it in an online fahion. Contardo, Morency, and Roueau (2012) preent a calable myopic repoitioning olution by conidering the current demand that wa recently oberved and olve it uing Dantzig-Wolfe and Bender decompoition technique. Recently, Lowalekar et al. (2017) propoe a calable online repoitioning olution uing multitage tochatic optimization and online anticipatory algorithm. Ghoh, Trick, and Varakantham (2016) propoe a robut and online repoitioning approach for the vehicle to counter the uncertainty in future demand. In contrat to the rolling horizon baed online olution, Shu et al. (2010; 2013) conider the future expected demand for a long period to deal with the future demand urge and propoe an optimization model for dynamic repoitioning to minimize the number of unatified cutomer. However, they did not conider the pecific routing contraint and the phyical limitation of the vehicle in their model. Ghoh et al. (2015; 2017) overcome thi concern by jointly conidering the dynamic repoitioning of bike in conjunction with the routing problem for vehicle. Our approach differ from thi thread of reearch a we are utilizing the bike trailer for repoitioning and crowdourcing the tak to cutomer in contrat to uing vehicle with dedicated taff for repoitioning. The lat thread of reearch focue on incentivizing cutomer and utilizing trailer for rebalancing the ytem. There ha been exiting reearch in bike haring (Singla et al. 2015; Pfrommer et al. 2014) and car haring (Chow and Yu 2015; Mareček, Shorten, and Yu 2016) that preent pricing mechanim to provide incentive to uer for aiting with repoitioning. However, thi line of work ha taken a myopic (individual tation) view on whether a bike or car i required at a tation. Furthermore, unlike car haring (Chow and Yu 2015), the BSS operator cannot order uer baed on their utility and operate within tight budget contraint. In thi work, we provide an end to end ytem that take the global view (all tation) of the repoitioning requirement and incentive their execution within the budget contraint. On the other hand, O Mahony and Shmoy (2015) predict the ervice level requirement for bae tation in ruh hour and introduce the notion of repoitioning with bike trailer, by matching each trailer to it uitable producer and conumer tation, baed on the aement of inventory tate of the bae tation. However, they aume that all the tak for the trailer can be achieved with dedicated taff which i not an economically viable option. In contrat, we propoe an optimization model to generate the repoitioning tak for trailer and deign a mechanim to crowdource thoe tak to the uer while enuring the given budget contraint. Model: DRRPT In thi ection, we formally decribe the generic model of Dynamic Repoitioning and Routing Problem uing (DRRPT) extending from the DRRP model introduced

3 by Ghoh et al. (2015; 2017). DRRPT i compactly repreented uing the following tuple: < S, V, C #, C, d #,0, {σ 0 v}, H, F, B > S denote the et of bae tation where C # repreent the capacity of the tation S. We have a et of bike trailer V where Cv denote the number of bike lot in the trailer v V. d #,0 repreent the initial ditribution of bike in the tation. σv 0 ymbolie the initial location of the trailer, i.e., σv() 0 i fixed to 1 if trailer v i tationed at initially and 0 otherwie. H denote a two dimenional matrix that tore the relative ditance between each pair of tation. F repreent a et of K dicrete training demand cenario. Specifically, F, k denote the demand for the planning period for cenario k that arie at tation and reache tation in the next time tep. Finally, B denote the amount of budget allocated for the repoitioning tak for a given planning period. We make the following aumption for the eae of explanation and repreentation. However, thee aumption can eaily be relaxed with minor modification to our method; (a) In the imilar direction of Ghoh, Trick, and Varakantham (2016), we aume that uer who carry bike and trailer at deciion epoch t alway return their bike at the beginning of the deciion epoch t + 1; (b) Cutomer are impatient in nature and leave the ytem if they encounter an empty tation. On the other hand, they return their bike to the nearet available tation if the detination tation i full. Solving DRRPT We propoe a rolling horizon framework for olving DR- RPT, where the following two component are run continuouly at each time tep: Generate repoitioning tak for the next time tep; Mechanim to incentivize execution of tak (within the central budget contraint) by intereted uer. Generating Repoitioning Tak In thi ection, we decribe the method for computing repoitioning tak for the trailer and alo etimate the valuation of thoe tak from center perpective. A a trailer can travel at mot to one tation in each time tep (equivalent to bike), the repoitioning tak for a trailer i to pickup a certain number of bike from the neighbourhood of it origin tation and drop them to another tation. To formally repreent the repoitioning tak, we introduce the following deciion variable: y +,v denote the number of picked up bike by trailer v from tation ; y,v denote the number of bike dropped off by trailer v at tation ; b +,v i a binary deciion variable which i et to 1 if trailer v pick up bike from tation and 0 otherwie; b,v repreent a binary deciion variable which i et to 1 if trailer v return bike at tation and 0 otherwie. In addition, we ue the ymbol G v to denote the et of neighbouring tation from where vehicle v i allowed to min y L k (1),k.t. L k F k, ( d #,t + (y,v y,v) + ), k, (2) v y,v + b +,v min(d #,t, Cv ),, v (3) y,v + d #,t, (4) v y,v C # v d #,t, (5) y,v = b,v y,v, +, v (6) (b +,v + b,v 1) H, Hmax,,, v (7) b +,v = 1, v (8) b +,v = 0, v (9) / G v b,v = 1, v (10) b +,v, b,v {0, 1}; y +,v, y,v C v ; L k 0 (11) Table 1: TASKGENERATION(F,t,d #,drrpt) pick up bike. A tation i included in G v if it i ituated within a threhold ditance from the origin tation of trailer v. Our goal i to compute the bet routing and repoitioning trategy for each of the bike trailer o a to minimize the total number of expected lot demand over K training demand cenario. Let L k denote the lot demand at tation for demand cenario k, after the repoitioning tak are achieved. We repreent the problem of minimizing expected lot demand a a Mixed Integer Linear Programme (MILP). The MILP for the tak generation i compactly repreented in Table (1). Our objective (delineated in expreion 1) i to minimize the expected lot demand (equivalent to total lot demand, a each cenario ha equal probability) over all the K training cenario. The contraint aociated with thi repoitioning tak generation are decribed a follow: 1. Compute the lot demand a the deficiency in upply of bike: The number of bike preent in a tation after accomplihing the repoitioning tak i etimated a + v (y,v y,v). + Therefore, contraint (2) enure that the number of lot demand at tation for cenario k i lower bounded by the difference between demand and upply of bike at that tation. Note that, a we are minimizing the um of lot demand over all the cenario, thee contraint are ufficient alone to compute the exact number of lo in cutomer demand. d #,t 2. capacity i not exceeded while picking up bike: Contraint (3) enure that the number of bike picked up by trailer v from tation i bounded by the minimum value between the number of bike preent in the tation and the capacity of the trailer.

4 3. Total number of bike picked up from a tation i le than the available bike: A multiple trailer can pick up bike from the ame tation, contraint (4) enforce that the total number of picked up bike by all the trailer from tation during the planning period t i bounded by the number of bike preent in the tation, d #,t. 4. Station capacity i not exceeded while dropping off bike: Contraint (5) enure that the total number of dropped off bike at tation i bounded by the number of available lot for bike at that tation. 5. A trailer hould return the exact number of bike it ha picked up: Note that b,v i the binary deciion variable that control the drop-off location for the trailer v. Therefore, contraint (6) enforce that the number of bike dropped off by a trailer in a tation i exactly equal to the number of picked up bike if the tation i viited. 6. Total traveling ditance for a trailer i bounded by a threhold value: To repreent the phyical limitation of route, we need to enure that the total ditance travelled by a trailer in a given planning period i within a few kilometer. Contraint (7) enforce thi condition by enuring that the ditance between the pick up and the drop-off tation for a trailer i bounded by a threhold value, H max. 7. A trailer hould pick up bike from one tation only: Contraint (8) enforce thi condition by allowing only one pick up deciion variable to be et to 1 for each trailer. 8. The pick up location for a trailer hould be within the geographical proximity of it origin tation: Contraint (9) aure thi requirement by fixing all the pick up deciion variable for trailer v to 0 for the tation which doe not belong to it nearby tation et, G v. 9. A trailer hould return bike to one tation only: Contraint (10) enure thi condition by allowing only one drop-off deciion variable to be et to 1 for each trailer. Note that, contraint (6) are non-linear in nature. However, one component in the right hand ide i a binary variable. Therefore, we can eaily linearize them uing the following contraint (12)-(14). y,v C v b,v, v (12) y,v y,v y +,v, v (13) y +,v (1 b,v) C v, v (14) Although we are uing big-m method for the linearization, the upper bound for the pickup or drop-off variable (or alternatively the value of M) i the capacity of the trailer which i relatively mall and therefore, thee contraint are computationally inexpenive. Mechanim to Incentivize Tak Execution within Budget Contraint Once we determine the tak, our goal i to deign a mechanim which allocate the tak among the uer who are intereted in executing thee tak and generate a payment method to enure that the uer bid for the tak truthfully. If the payment method doe not enure truthful behaviour, then either the bike haring operator are unhappy (a they pay more money to uer than required) or uer are unhappy (a they get paid le) while repoitioning bike through trailer. To deign a mechanim for crowdourcing the repoitioning tak, the firt tep i to compute the value of the tak from center perpective. A our goal i to minimize the expected lot demand, the valuation of the tak i proportional to the expected lot demand reduced by the trailer job over all the training demand cenario. Specifically, the value of tak for trailer v i defined a follow (ξ repreent unit value of lot demand to compute overall value): U(v)= ξ K k, ( min [ min ( max( F k, d#,t, 0), y +,v) max ( y,v (d #,t F k, ), 0), y,v )] (15) Intuitively thi value i the weighted difference in reduced lot demand uing the trailer minu increae in lot demand due to moving bike uing trailer. The firt term in equation (15) compute the expected lot demand reduced by trailer v in it detination tation over K cenario. The econd term compute the expected lot demand ariing becaue of the pickup deciion by trailer v at it origin tation. We aume that the et of intereted uer for each pair of tak are dijoint. One uer can execute a ingle tak in any given deciion epoch, o thi aumption can be eaily enforced. To enure thi aumption i atified, we can either aociate a huge penalty for bidding on multiple tak or dicard all bid of a uer except the firt one. Once all the bid arrive, the goal of the center i two-fold: (a) Deign an incentive compatible mechanim to enure that uer bid truthfully on every tak; (b) Allocate the tak in a fahion that maximize the efficiency of the entire ytem while atifying the budget feaibility. Obervation 1 A the et of bidder for different tak are pairwie dijoint and the mechanim initiate once all the bid information i available, the tak are primarily independent but coupled by the central budget contraint. Therefore, the mechanim or payment method can be deigned for each of the tak eparately. However, the final allocation of tak hould be accomplihed in a fahion o that the budget feaibility i enured. By exploiting obervation (1), we deign a mechanim for each of the tak eparately. Let the et of repoitioning tak be T = {1,..., V }. We begin the dicuion with the mechanim deign for a ingle tak for trailer v. Let, N v = {1,..., n v } repreent the et of rational uer who are bidding privately to the center for the tak of trailer v. Each uer i N v privately reveal their type θ i =< C i (v) >, where C i (v) denote their private cot for executing the tak of trailer v. The center profit for the bid of uer i i defined a W i (v) = U(v) C i (v). We reject a bid from a uer i if C i (v) > U(v), which enure that W i (v) i alway poitive. Our goal i to aign the tak to a bidder o that the center profit i maximized and deign a payment method to

5 enure that uer alway bid truthfully. We ue the tandard Vickrey-Clarke-Grove [VCG] mechanim (Vickrey 1961; Clarke 1971; Grove 1973) to olve thi problem. According to thi mechanim, the tak i alway allocated to the lowet bidder, but the lowet bidder get paid the bid of the econd lowet bidder. For intance, if there are 3 bid of 10$, 12$ and 14$ to perform a repoitioning tak, then the tak i allocated to bid 1 and the peron putting in bid 1 get paid 12$. More formally, let λ = {0, 1} Nv denote the allocation of the tak o that the center profit i maximized. { } λ 1 if i = argmaxj Nv W i (v) = j (v) 0 Otherwie Then the payment to the uer i for tak v i computed uing the following expreion: P i (v) =λ i (v) [ U(v) max W j(v) ] = λ i (v) [ min C j(v) ] j i j i (16) Equation (16) indicate that the payment for uer i i the econd lowet cot revealed in the bid proce if the tak i allocated to him, otherwie the payment i et to 0. Since, we directly adapt the tandard VCG mechanim, the mechanim for ingle tak i truthful or incentive compatible. However, thi doe not enure incentive compatibility over all tak, a there i a budget contraint. We now provide a method that will enure incentive compatibility over all tak without violating the budget feaibility. Enuring the Budget Feaibility: A mentioned previouly, the BSS operator work within a fix budget B. We have a et of tak T = {1,..., V }, where each tak v T ha a valuation, U(v) (computed uing equation 15) and the payment for the tak i denoted by P (v) (computed uing equation 16). Our goal i to allocate the tak in a fahion that maximize the overall valuation of the center while the total payment i bounded by the given budget, B. Let x(v) denote a binary deciion variable which i et to 1 if tak v i allocated and 0 otherwie. We compactly repreent the problem a an Integer Program (IP) in table (2). max x x(v) U(v) (17) v T.t. x(v) P (v) B v T (18) x(v) {0, 1} v T (19) Table 2: TASKALLOCATIONIP(U, P, T, B) Our objective in expreion (17) aim to find an allocation of the tak o that the cumulative valuation from the center perpective i maximized. Contrain (18) enforce that the total payment made to the uer due to the reulting allocation hould repect the given budget B. The IP in Table (2) i exactly equivalent to the 0/1 knapack problem which i a known NP-Hard problem. However, we can employ the well-known dynamic programming (DP) approach (refer to chapter 6 of Dagupta, Papadimitriou, and Vazirani, 6) to peed up the olution proce. The complexity of uch a DP approach i O( T B) in comparion to the brute force method that ha the complexity of O(2 T ). Propoition 1 The mechanim for tak allocation for the trailer in bike haring ytem i incentive compatible (IC), individually rational (IR) and economically efficient (EE). Proof: The mechanim for ingle tak atifie the IC and IR property a it follow the tandard VCG mechanim. A all the tak are independent and payment are made for a ubet of tak to enure the budget feaibility, all the allocated tak atify the IC and IR property. Hence, the budget feaible mechanim for the entire BSS meet the requirement to atify the IC and IR property. Finally, the mechanim maximize the difference between center valuation and the cot for executing the tak which i equivalent to expected total welfare, hence, our mechanim atifie the EE property. Overall Flow of Our Approach To better undertand the overall flow of our approach, we provide Algorithm (1). We begin by olving the MILP of Table (1) to generate the repoitioning tak for each of the trailer to better atify cutomer demand over a et of training demand cenario. Then the value of the tak from center perpective are computed uing equation (15) and broadcated to the uer. Next, a et of rational uer bid for the tak privately. Once all the bid are ubmitted, we employ the tandard VCG mechanim to generate the payment (refer to equation 16) for each tak. Finally, we allocate budget to tak (and make payment only if the tak can be allocated money) by olving a 0/1 knapack problem that maximize the global welfare of the ytem without violating the budget contraint of the operator. Algorithm 1: olverepoitioning(drrpt, t, d #, F t, B) Initialize: Y +, Y 0 ; Y +, Y TASKGENERATION(F t, t, d #, drrpt); for each v V do U(v) COMPUTETASKVALUE(Y v +, Yv ); for each v V do C(v) COLLECTBIDS(Y v +, Yv, U(v)); for each v V do P (v) GENERATEPAYMENT(U(v), C(v)); X TASKALLOCATION(U, P, V, B); for each v V do Y + v Y + v X(v); Y v X(v); Y v return Y +, Y ; Empirical Evaluation In thi ection, we explain the imulation model ued to execute the tak, the benchmark approache implemented for the computational comparion and the experimental reult.

6 Simulation Model Once the repoitioning tak for the trailer and their allocation are determined for a time tep, we execute them on a imulator (adapted from Ghoh, Trick, and Varakantham 2016) for evaluating their performance on the realized demand cenario for that particular time tep. Let, f, t denote the number of cutomer who arrive in tation at time tep t and plan to reach tation at the beginning of time tep t+1. Let, d #,t repreent the number of bike preent in tation at time tep t after the repoitioning tak are completed. Due to the repoitioning, the number of bike available in tation change and therefore, the flow of bike by the cutomer alo change in comparion to the oberved data denoted by f. However, a reaonable aumption employed in previou work (Shu et al. 2013; Ghoh et al. 2017; Ghoh, Trick, and Varakantham 2016) for any configuration i that the aggregated tranition probabilitie between tation that i oberved in the data remain the ame during execution. Therefore, the flow of bike between the tation are determined baed on the following two cae: (a) If the arrival demand at a tation i le than the number of bike preent in that tation, then all the cutomer are able to hire bike; (b) If the arrival demand at a tation i higher than the number of bike preent in that tation, then the actual flow of bike (denoted a x t, ) i computed uing the relative ratio f t, f a hown in equation (20). t, { f t x t, =, if f, t d#,t f t, d #,t Otherwie f t, } (20) Once the actual flow of bike by the cutomer at time tep t i determined, we calculate the ditribution of bike in tation at time tep t + 1 a the um of un-hired bike at time tep t, the net incoming bike in tation at the beginning of time tep t + 1 and the net drop-off bike at tation by the trailer at time tep t + 1 (i.e., Y,t+1 Y +,t+1 ). d #,t+1 =d #,t + [ x t, [ x t, ]+ Y,t+1 Y +,t+1 ] (21) Equation (21) for computing the number of bike in tation at time tep t+1 (i.e., d #,t+1 ) doe not take into account the tation capacity contraint. To handle uch boundary condition and to enure the capacity contraint are conidered for the tation, we tranfer extra bike (i.e., d #,t+1 C # ) to the nearet available tation if d #,t+1 exceed the tation capacity, C #. In our experimental reult, we how thee extra number a the lot demand at the time of return. Once the ditribution of bike acro the tation for time tep t + 1 i obtained, we utilize thi information to compute the repoitioning trategy for trailer for time tep t + 1. Thi iterative proce continue until we reach the lat deciion epoch. Experimental Setup We conducted our experiment on a real-world data et 1 of Hubway BSS. The Hubway data et contain the following 1 Data i taken from Hubway bike haring company of Boton [ detail: (1) Cutomer trip record, from which we compute demand for each origin detination pair at each time tep; (2) The number of tation, their capacity and initial ditribution of bike at the tation; (3) Geographical location of bae tation to calculate the ditance between two tation; (4) The number of vehicle ued for repoitioning and their capacity. Furthermore, we generate two et of ynthetic demand cenario uing Poion ditribution with the mean computed from real-world data et. More pecific detail about thee ynthetic data et are mentioned later. We evaluate our approach with repect to the key performance metric of lo in cutomer demand. We compare the utility of our approach with two exiting benchmark approache. Benchmark-1: Static Repoitioning implie the practice of no repoitioning during the day. The vehicle are only ued at the end of the day to rebalance tation to achieve a predefined inventory level. We ue thi a a baeline approach where no repoitioning i done during the planning period. Benchmark-2: Dynamic Repoitioning implie the practice of repoitioning idle bike uing vehicle during the day to meet the expected future demand. We adapted a recently developed cenario baed approach from Ghoh, Trick, and Varakantham (2016). In their iterative approach, a wore demand cenario i generated in each iteration to counter the repoitioning trategy of the current iteration and then they produce a robut repoitioning olution by conidering all the previouly generated cenario. However, for a fair comparion with our approach (a hown in Table 1), we make the following modification in their optimization model: (1) We take the exact et of training demand cenario ued in our approach rather than generating the wore cae cenario; (2) We minimize the total number of lot demand over all the demand cenario (equivalent to our objective function of Table 1) in contrat to optimizing for the wort cae. To enure a fair comparion, all the benchmark approache and our approach are evaluated on the imulation model decribed in the previou ubection. Empirical Reult We now how the performance 2 of our approach on Hubway data et. The Hubway BSS conit of 95 bae tation and 3 vehicle. We tudy with 10 trailer and their capacity i aumed to be three in our default etting of experiment. We take 6 hour of planning horizon in the morning peak (6AM-12PM) and the duration of each deciion epoch i conidered a 30 minute. The demand cenario are generated from three month of hitorical trip data. A the trip data only contain ucceful booking and doe not capture the unoberved lot demand, we employ a micro-imulation model (courtey: Ghoh, Trick, and Varakantham 2016) with 1 minute of time tep to determine the time lot when a tation wa empty and introduce artificial demand at the empty tation baed on the oberved demand at that tation in previou time tep. We demontrate three et of reult on the Hubway data et: 2 All the linear optimization model were olved uing IBM ILOG CPLEX Optimization Studio V12.5. incorporated within python code.

7 450 Lot Demand (Realied Demand) 250 Lot Demand (Poion at Origin) 300 Lot Demand (Poion on OD pair) # Lot Demand # Lot Demand # Lot Demand Static Repoitioning Vehicle Repoitioning (Capacity=3) (Capacity=5) 0 Static Repoitioning Vehicle Repoitioning (Capacity=3) (Capacity=5) 0 Static Repoitioning Vehicle Repoitioning (Capacity=3) (Capacity=5) Figure 1: Lot demand tatitic for (a) Demand cenario from real-world data; (b) Demand cenario follow Poion ditribution at origin tation; (c) Demand cenario follow Poion ditribution for each OD pair. 320 Effect of β 245 Effect of γ 280 Effect of α # Lot Demand # Lot Demand # Lot Demand Budget/hr. (β) % of Intereted Uer (γ) α Figure 2: (Average) Lot demand tatitic for varying (a) Allocated budget (β) [α = 0.3, γ = 0.3]; (b) Percentage of uer intereted in trailer tak (γ) [α = 0.3, β = 50]; (c) Ratio of lower and upper bound of bid (α) [β = 50, γ = 0.3]. The performance comparion between our approach and the benchmark in term of reducing the lot demand; The effect of key tunable input parameter on the mechanim deign over a wide range of demand cenario; Runtime performance of our approach. Performance comparion: To evaluate the performance of our approach, we produce three type of demand cenario: (1) We took the real demand data for 60 weekday. The actual demand i etimated by introducing artificial demand at empty tation uing a imilar heuritic a dicued earlier. We ue 20 day of demand cenario for training purpoe and other 40 day of demand for teting; (2) We generate 100 demand cenario, where the arrival demand at each tation i generated uing Poion ditribution with the mean computed from hitorical data. Similar to Shu et al. (2013), we aume that cutomer reach their detination tation with a fixed probability; (3) We generate 100 demand cenario, where demand for each origin detination [OD] pair at each time tep i computed uing Poion ditribution. For the demand cenario type 2 and 3, we ue 30 demand cenario for training and 70 demand cenario for teting. For all the three type of demand cenario, we compute the cumulative lot demand at the time of bike pickup and return for the following four approache: (a) Static repoitioning, i.e., no rebalancing i done during the planning period; (b) Dynamic repoitioning uing 3 exiting vehicle; (c) Dynamic repoitioning uing 10 trailer, each having a capacity of 3; (d) Dynamic repoitioning uing 10 trailer, each having a capacity of 5. For thi et of experiment, we aume that there i ufficient budget available to allocate all the tak. Therefore, we directly took the repoitioning olution from Table (1) for evaluation. Figure (1) depict the average number of lot demand along with tandard deviation for all the three type of demand cenario. Figure 1(a) how the lot demand tatitic on the real-world demand cenario. By utilizing trailer with capacity 3, the average lot demand over 40 teting cenario reduce by 41% over the no repoitioning approach. The repoitioning olution for the trailer with capacity 3 are alo proven to be highly competitive to the olution achieved by vehicle. A expected, the repoitioning olution for the trailer with capacity 5 produce better reult and outperform the lot demand obtained by 3 carrier vehicle. Similar performance tatitic are hown in Figure 1(b) and 1(c) for the demand cenario generated uing Poion ditribution at origin tation and for each OD pair repectively. In both the etting, we oberve a conitent pattern that the repoitioning olution uing trailer with capacity 3 reduce the average lot demand over 70 tet cenario by 69% and 63% in comparion to the baeline approach. Moreover, both the figure clearly demontrate that the olution for trailer with capacity 5 are better than the fuel burning mode of repoitioning by the vehicle. Effect of tunable parameter: In the next et of reult we demontrate the performance of our approach by varying the different input parameter of the mechanim. We employ the real-world demand cenario (demand cenario type 1) for thee experiment, where 20 demand cenario are ued for training and the evaluation i done on other 40 cenario. The outcome of the mechanim depend on the following three input parameter: Hourly budget allocated by the operator (β): Ideally the BSS operator allocate a fixed amount beforehand for

8 the repoitioning tak. In our default etting of experiment we have fixed the hourly budget to 50 dollar 3 ; Percentage of uer intereted in trailer tak (γ): To execute a mechanim, it i important to compute the number of uer bidding for each trailer tak. Typically, a certain percentage of uer whoe origin and detination location i within kilometer of the pickup and drop-off location of the trailer, are the potential uer intereted in executing the tak and bid for it. In our experiment we et the value of to 1 kilometer 4. We ue the default value of γ a 30%; Ratio of lower and upper bound of bid (α): The third and mot important parameter for the mechanim i the bid value ubmitted privately by the uer. We generate the bid value uing Gauian ditribution 5 from the range [C min, C max ]. A the upper limit of the bid value for tak v i bounded by it valuation U(v), we et the C max for the tak of trailer v to U(v). The value of C min i et to αc max. A the bid are generated from a ditribution, we run the mechanim 100 time for every tak and ue the expectation over 100 run a the payment. The default value of α i et to 30%. Figure (2) depict the effect of the tunable parameter on the performance of our approach. Figure 2(a) plot the average lot demand over 40 day of teting demand cenario, where we vary the hourly budget (β) in the X-axi from 10 dollar to 80 dollar. A expected, the average number of lot demand reduce monotonically a we increae the hourly budget. Due to the randomne in bid value in different run, the lot demand might increae for ome cenario, even after increaing the hourly budget. We oberve that for more than 78% of the cae, lot demand decreae if we increae the hourly budget by 10 dollar. Figure 2(b) plot the average lot demand over 40 teting demand cenario, when we vary the interet rate of the uer (γ) in the X-axi from 10% to 70%. The average number of lot demand reduce monotonically a we increae the interet rate of uer, becaue increaing the interet rate implie that additional bid are ubmitted to the center and therefore, the likelihood of the payment value reduce which in turn enable u to execute extra trailer tak within the given budget, hence, the number of expected lot demand reduce. We oberve that the lot demand decreae for around 60% of the cae, if we increae γ by 10%. Figure 2(c) plot the average lot demand over 40 teting demand cenario, where we vary the ratio of the lower and upper bound of the bid (α) in the X-axi from 20% to 90%. 3 The link: from the US Department of Labor provide hourly alarie for driver operating light truck that are ued for repoitioning the bike. It how that the median hourly cot for a hired driver i 14 dollar. Therefore, the hourly budget for 3 exiting vehicle including the fuel cot for routing would be around 50 dollar. 4 We experimented with a 0.5 kilometer and oberve imilar reult a hown in Figure 2. 5 We chooe Gauian bid, becaue exiting work (Singla et al. 2015) in bike haring that experimented with real human employed Gauian ditribution for repreenting uer bid. Increaing the value of α indicate that the lower bound of the bid increae, o the expected bid value alo increae. Now, increae in the bid value implie that the expected payment for the tak alo increae and the number of tak that can be executed within a fixed budget decreae, hence, the number of expected lot demand alo increae. A expected, Figure 2(c) clearly depict that the average number of lot demand increae monotonically a we increae the value of α. For around 74% of the cae, the lot demand increae if we increae α by 10%. Runtime performance: In the lat et of reult, we how the runtime performance of our approach in comparion to the repoitioning olution of the vehicle on the real-world demand cenario. The time to find a repoitioning olution i a crucial factor in our etting, a we are generating the trategy after every 30 minute of interval. Figure (3) depict the runtime performance where in the X-axi we vary the number of deciion epoch and the Y-axi denote the cumulative runtime in logarithmic cale. For every value of deciion epoch, our approach wa able to olve the problem within a couple of econd with 95 tation and 20 training cenario. On the other hand, it took more than 15 minute for each deciion epoch to generate the olution for the vehicle with the ame number of training cenario. Cumulative Runtime (Sec) (Cumulative) Runtime Comparion #Time-tep Vehicle Figure 3: (Cumulative) Runtime comparion between the repoitioning olution of vehicle and trailer. Concluion In thi paper we explore the dynamic repoitioning problem in bike haring ytem with the help of bike trailer. We propoe a novel optimization model to generate the repoitioning tak for the trailer to better meet the cutomer demand. Additionally, we deign a budget feaible incentive compatible (incentivize truth telling) mechanim to crowdource the tak among the uer who are intereted in executing thoe tak. The empirical reult on a real-world data et how that our green mode of repoitioning i economically viable and highly efficient in term of reducing the lot demand. In future thi work can be extended in the following direction: (a) Developing a budget feaible mechanim by conidering the uncertaintie in completion time of the trailer tak; (b) Developing a model that jointly conider the dynamic repoitioning problem for vehicle and trailer and dicover an efficient olution while enuring the central budget contraint.

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