Robut Repoitioning to Counter Unpredictable Demand in Bike Sharing Sytem Supriyo Ghoh School of Info. Sytem Singapore Management Univ. upriyog.2013@phdi.mu.edu.g Michael Trick Tepper School of Buine Carnegie Mellon Univerity trick@cmu.edu Pradeep Varakantham School of Info. Sytem Singapore Management Univ. pradeepv@mu.edu.g Abtract Bike Sharing Sytem (BSS) experience a ignificant lo in cutomer demand due to tarvation (empty bae tation precluding bike pickup) or congetion (full bae tation precluding bike return). Therefore, BSS operator repoition bike between tation with the help of carrier vehicle. Due to unpredictable and dynamically changing nature of the demand, myopic reaoning typically provide a below par performance. We propoe an online and robut repoitioning approach to minimie the lo in cutomer demand while conidering the poible uncertainty in future demand. Specifically, we develop a cenario generation approach baed on an iterative two player game to compute a trategy of repoitioning by auming that the environment can generate a wore demand cenario (out of the feaible demand cenario) againt the current repoitioning olution. Extenive computational reult from a imulation built on real world data et of bike haring company demontrate that our approach can ignificantly reduce the expected lot demand over the exiting benchmark approache. 1 Introduction Bike Sharing Sytem (BSS) are widely intalled in major citie of the world to mitigate the concern aociated with extenive uage of private vehicle uch a increaed carbon emiion, traffic congetion and uage of non-renewable reource. Becaue of thi ability to provide healthier living and greener environment, bike haring ytem are widely adopted with 984 active ytem and 295 ytem under contruction [Meddin and DeMaio, 2016] in major citie of the world. Popular example of BSS include Capital Bikehare in Wahington DC, Hubway in Boton, Bixi in Montreal, and Velib in Pari. In a typical bike haring ytem, a et of bae tation are trategically placed throughout a city and each of the tation ha a finite number of dock, each holding one bike. At the beginning of the day, each tation i tocked with a pre-determined number of bike. Uer can hire bike from one tation and return them to a different tation. Due to the individualitic and uncoordinated movement of cutomer, there i often tarvation (fewer than required) or congetion (more than required) of bike at certain bae tation, which can reult in a ignificant lo of cutomer demand. Several bike haring operator employ carrier vehicle to repoition bike during the day uing myopic reaoning (e.g. tart filling when number of bike fall below 20% of the capacity) to better match the demand. Due to uncertainty in future demand, it i difficult to predict the ideal inventory level and therefore, myopic olution often fail to provide a good quality olution. While the exiting offline multi-tep algorithm [Ghoh et al., 2015] baed on expected future demand are uitable for ituation with table demand pattern, they perform poorly when demand varie throughout the day. While data driven olution approache that conider demand uncertainty have been propoed in everal application domain (ex: emergency medical ervice [Saiubramanian et al., 2015; Ghoh and Varakantham, 2016], taxi fleet optimization [Lowalekar et al., 2016]), progre remain low in handling the unpredictable demand in a robut manner, particularly in bike haring ytem. Thi erve a the motivation for thi paper. To addre uch cenario where demand ha high variance, we propoe an online and robut repoitioning approach to better match the demand and upply of bike and conequently to reduce the expected lot demand. We treat the problem of computing a robut olution a an iterative game between the deciion maker of the BSS and the environment acting a an adverary. In each iteration, the adverary identifie a feaible demand cenario that maximie the lot demand relative to the rebalancing trategy propoed by the deciion maker. From the deciion maker perpective, we olve thi game uing a cenario generation approach. That i to ay, the deciion maker take into account all the demand cenario generated by the adverary in previou iteration and compute a routing and repoitioning olution for the vehicle that minimie the wore cae lot demand over all the cenario. The proce continue until the objective of the adverary and the deciion maker converge. We develop an online approach where the robut trategy i generated at each time tep by conidering the current ditribution of bike acro the tation and the trategy i executed on a real world imulator to identify the ditribution of bike for the next time tep. Experimental reult on multiple
ynthetic data et and a real world data et demontrate that our approach ignificantly reduce the expected lot demand over the exiting benchmark approache and i robut to the uncertainty in demand. 2 Related Work Given the practical importance of bike haring ytem, they have been tudied extenively in the literature. We broadly categorize the repoitioning problem into three thread of reearch. The firt thread of reearch focue on tatic repoitioning [Chemla et al., 2013] where the goal i to find the route for a fixed et of vehicle for achieving the deired configuration of bike acro the bae tation at the beginning of the day. [Raviv and Kolka, 2013; Raviv et al., 2013; Rainer-Harbach et al., 2013] propoe calable exact and approximate algorithm to olve the tatic repoitioning problem by employing contraint from inventory management literature or by uing variable neighborhood earch heuritic. [Di Gapero et al., 2013; 2015] employ contraint programming (CP) and efficiently olve the problem uing large neighbourhood earch. Thee notably calable tatic repoitioning olution are fruitful if the demand pattern i table and predictable. However, if the demand change over time, the tation get imbalanced during the day and tatic repoitioning i not ufficient in thoe ituation. Therefore, our approach focue on repoitioning during the day. The econd thread of reearch focue on performing dynamic repoitioning of bike during the day. [Shu et al., 2013] provide an optimiation model for dynamic repoitioning to minimie the number of unatified cutomer. [Ghoh et al., 2015] conider the dynamic repoitioning of bike in conjunction with the routing problem for vehicle. Due to the inherent complexity of the joint problem, they employ decompoition and abtraction baed heuritic to olve the real world large cale problem. [Contardo et al., 2012] develop a myopic repoitioning approach by conidering the recently oberved demand to reduce the unmet demand in ruh hour. [Pfrommer et al., 2014] provide myopic online deciion baed on aement of near future demand. [Schuijbroek et al., 2013] propoe a calable approximate olution for thi problem by abtracting bae tation and olving it uing a clutered vehicle routing [Battarra et al., 2014] approach. All the paper in thi thread aume a known ditribution of demand and they are not enitive to the fluctuating or unpredictable demand cenario. In contrat, we propoe a robut olution approach for the dynamic repoitioning problem by conidering the poible uncertainly in future demand. The lat thread of reearch focue on the prediction and analyi of demand in BSS. [Nair and Miller-Hook, 2011; Nair et al., 2013] provide ervice level analyi of BSS uing dual-bounded chance contraint. [Leurent, 2012] repreent the BSS a a dual markovian waiting ytem. [Borgnat et al., 2009; 2011] propoe the idea of predicting temporal uer demand and forecating that information to uer. [George and Xia, 2011; Shu et al., 2013; Kabra et al., 2015] repreent the cutomer arrival proce at bae tation uing Poion ditribution. Due to it implicity and accuracy in repreenting random arrival procee, we evaluate the performance of our robut trategie on the demand cenario generated uing Poion ditribution. 3 Model: Bike Sharing Sytem The generic model for Dynamic Repoitioning and Routing Problem with demand Uncertainty (DRRPU) in BSS i formally defined uing the following tuple: S, V, C #, C, d #,0, d,0, {σv}, 0 P, F S repreent the et of bae tation, where each tation S ha a fixed capacity (number of dock) denoted by C #. V repreent the et of vehicle and each vehicle v V ha a fixed capacity (number of lot for bike) denoted by Cv. The number of bike tocked at a bae tation, at the beginning of the day, i given by d #,0. d,0 v denote the number of bike preent initially in a vehicle v. σv() 0 i et to 1 if vehicle v i preent at tation initially. For eae of notation, we ue the generic σv() t and et it to 0, if t > 0. P, repreent the ditance between tation and. F repreent the et of demand bound that i computed from the hitorical trip data. We compute three type of bound on the arrival cutomer demand: (a) ˇF t, ˆF t denote the lower and upper bound on the ytem wide demand acro all the tation at time tep t; (b) ˇF, t ˆF t denote the bound on the demand in tation at time tep t; (c) ˇF, t, ˆF, t denote the bound on the demand that arie in tation at time tep t and reach tation at time tep t+1. Thee demand bound are ued in the olution approach to generate the trategy. On the other hand, the trategie are evaluated on a wide range of teting demand cenario that are created uing Poion ditribution and thee cenario are not forced to follow the bound ued in the planning proce. Given the DRRPU model, our goal i to provide a repoitioning and routing trategy for the vehicle at each time tep that minimie the wore cae lot demand. We are primarily intereted in minimiing lot demand that arie becaue of the tarvation of bike at tation. A we compute the trategy for one time tep, we have no control over the lot demand that arie due to the congetion of bike at the detination tation (which depend on the unknown demand) in the next time tep. However, experimental reult on the real world data et demontrate that repoitioning bike to reduce the lot demand at the time of hiring, determine the inventory level efficiently and furthermore, reduce the number of unatified cutomer at the return time. 4 Solution approach We compute a robut repoitioning and routing trategy uing rolling horizon framework. In each deciion epoch, for a given ditribution of bike at tation, we compute a robut trategy by auming that the arrival demand in each tation and in aggregate follow the input bound. Once we obtain the repoitioning trategy for a deciion epoch, we imulate the cutomer flow for the given demand cenario along with the repoitioning number to achieve the ditribution of bike acro tation for the next deciion epoch. Thi iterative proce continue until we reach the lat deciion epoch.
For the eae of repreentation, we made three key aumption: (a) Cutomer complete their trip in one deciion epoch. That i to ay, cutomer who hire bike at deciion epoch t hould return their bike to the detination tation 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; (c) The event at each time tep follow a particular equence. Firt, the cutomer return their bike which wa hired in the previou time tep, then the repoitioning event by the vehicle are done and latly, the arrival cutomer hire bike. Variable Definition y,v +,t Number of bike picked up from tation by vehicle v at time index t y,v,t Number of bike dropped off at tation by vehicle v at time index t z, t Set to 1 if vehicle v ha to move from tation,v to at time index t d,t v Number of bike in vehicle v at time index t F, k Arrival cutomer demand from tation to for k th demand cenario Table 1: Definition of the variable To compute a robut trategy in each deciion epoch, we propoe an iterative two player game approach between the repoitioning planner and an adverary. We provide two novel Mixed Integer Linear Programming (MILP) formulation to repreent the planning problem for the adverary and the repoitioning planner. For eae of undertanding, the deciion variable employed in the MILP are provided in Table (1). 4.1 The Adverarial Planner Once the intention of the repoitioning planner are revealed, the adverary aim at providing the wort poible demand cenario that reult in lowet bike uage during the planning period. More pecifically, the goal i to find a demand cenario that maximie the amount of lot demand, while enuring contraint related to demand feaibility. In the firt iteration the adverary find a wore demand cenario with the aumption of no repoitioning in the ytem. In the ubequent iteration, the adverary plan againt a particular repoitioning trategy that i propoed by the repoitioning planner. The MILP for the demand election proce by the adverary i hown compactly in Table (2). The input for the MILP are the current repoitioning trategy, i.e., the number of bike to pickup, Y + and drop-off, Y at tation. The ditribution of bike, d #,t at tation and in the deciion epoch t i alo provided a input. Let L denote the number of lot demand occurred at tation during the planning period. F, denote the number of cutomer arrived in tation at the current deciion epoch and reach tation at the beginning of the next deciion epoch. The objective delineated in expreion (1) i to generate a demand cenario, F that maximie the total amount of lot demand over all the tation. The number of bike preent at tation after the repoitioning event can be computed a max F L (1).t. L = max(0, F, (d #,t + Y Y + )), (2) ˇF t, F, ˆF t (3) ˇF t F, ˆF t, (4) ˇF t, F, ˆF t,,, (5) Table 2: ADVERSARY(Y +, Y, t, d #,drrpu) (d # +Y Y + ). Therefore, contraint (2) compute the lot demand at tation a the deficiency between the demand for bike (i.e., F, ) and the upply of bike. Thee contraint are non-linear in nature and we linearie them with a et of inequality contraint uing the well known Big-M method. Contraint (3-5) enure that the generated demand follow the given input bound. Specifically, contraint (3) enure that the aggregated ytem wide demand at the deciion epoch t i bounded by ˇF t and ˆF t. Contraint (4) enforce that the arrival demand in tation at deciion epoch t i bounded by ˇF t and ˆF. t Contraint (5) enforce that the demand arie in tation at deciion epoch t and reach tation in the next deciion epoch i bounded by ˇF, t and ˆF t,. 4.2 The Repoitioning Planner Given a et of K demand cenario (computed by the adverary in K iteration), the goal of the repoitioning planner i to find the bet routing and repoitioning trategy for the vehicle that maximie the bike uage or alternatively, minimie the wore cae lot demand. Let F, k denote the arrival demand from tation to for cenario k. L k denote the lot demand at tation for cenario k. The outcome of the repoitioning planner i two-fold: (a) A et of deciion z for the vehicle route; (b) The repoitioning trategy y + and y. The MILP for olving the joint problem of routing and repoitioning i repreented compactly in Table (3). The objective function delineated in expreion (8) i to minimie the maximum lot demand over all the cenario. We further implify the objective function by introducing an additional et of contraint (7) to enure that the total lot demand for cenario k i bounded by the variable λ and we minimie λ in objective function (6). min y,z.t. λ (6) λ L k, k (7) Note that a vehicle can viit multiple tation in one deciion epoch. Let a vehicle viit a maximum of ˆT number of tation within one deciion epoch. To repreent the equence of move, we ue a time index ˆt [0, ˆT ]. After repoitioning, the number of bike preent at tation in the deciion
ˆt,v (y,ˆt epoch t can be computed a (d #,t +,v y,v +,ˆt )). Therefore, contraint (9) enure that the lot demand at tation for cenario k i equal to the difference between the total arrival demand (i.e., F k, ) and the upply of bike. Contraint (10) enure that the total number of bike picked up from a tation during the planning period i le than the available bike, d #,t. Contraint (11) enforce that the total number of bike dropped off at tation i le than the number of available dock, C # d #,t. min y,z max k.t. L k ˆt,v ˆt,v d,ˆt v y +,ˆt,v L k (8) F k, (d#,t + ˆt,v (y,v,ˆt y,v +,ˆt )),, k (9) d #,t, (10) y,ˆt,v C # d #,t, (11) + [(y,v +,ˆt y,v,ˆt )] = d,ˆt+1 v, ˆt, v (12) S zˆt,k,v zˆt 1 k,,v = σˆt v (), ˆt,, v (13) k S k S y,v +,ˆt + y,v,ˆt Cv zˆt,i,v, ˆt,, v (14) i S α ˆt,, P, zˆt,,v +M ˆt, (y,v +,ˆt + y,v,ˆt ) Q, v (15) L k 0, y,v +,ˆt, y,v,ˆt Cv, d,ˆt v Cv, zˆt i,j,v {0, 1} (16) Table 3: REDEPLOYMENT(F,k,t,d #,drrpu) The initial ditribution of bike in vehicle, d,0 and the initial ditribution of vehicle at tation, σ 0 are computed from the tate of the ytem at the end of previou deciion epoch. Contraint (12) enure the flow conervation of bike in the vehicle. The number of bike preent in vehicle v at ) i equivalent to the number of bike preent in the vehicle at time index ˆt (i.e., d,ˆt v ) plu time index ˆt + 1 (i.e., d,ˆt+1 v the net incoming bike at time index ˆt. Contraint (13) enforce the flow conervation of vehicle at tation by enuring the equivalence between the inflow and outflow of vehicle in each tation. For ˆt = 0, depending on the initial location of vehicle, σ 0 v thee contraint enure that vehicle move appropriately out of the initial location. Contraint (14) enforce that the number of bike picked up or dropped off i conditional to the tation being viited at that time index. Let α denote the unit for converting ditance to time, M denote the time required to pickup/drop-off one bike and Q denote the duration of planning period. Then, contraint (15) enforce the phyical limitation of the carrier route. That i to ay, total time pent by the vehicle for traveling between the tation plu the time pend on picking up or dropping off the bike, i bounded by the duration of the planning period. Finally, contraint (16) enforce that the number of bike picked Algorithm 1: olvedrrpu(drrpu, t, d # ) 1 Initialize: F {}, Y + 0, Y 0 0, i 0 ; 2 repeat 3 i i + 1; 4 O a, F i ADVERSARY(Y + i 1, Y i 1, t, d#, drrpu) 5 F F F i 6 O r, z i, Y i, Y+ i REDEPLOYMENT(F, i, t, d#, drrpu) 7 until Converge; 8 return y + i, y i, z i up or dropped off i bounded by the capacity of vehicle. To better undertand the robut optimiation approach, we provide the key iterative tep in Algorithm (1). The repoitioning trategie are initialied a 0, therefore, in the firt iteration adverary compute a demand cenario againt the no repoitioning trategy. From the ubequent iteration, the adverary generate a wore demand cenario againt the repoitioning trategy revealed by the repoitioning planner. At iteration k, the repoitioning planner ha k demand cenario (communicated by the adverary) and it compute a repoitioning trategy that minimie the wore cae lot demand over all the cenario. The proce top when the objective of the repoitioning planner, O r and the adverary, O a converge. Therefore, at the convergence, the olution guarantee to provide an upper bound on the lot demand for any poible demand cenario that follow the given bound. 4.3 Simulation Model We employ the olvedrrpu procedure from Algorithm (1) to compute a repoitioning trategy at each time tep and execute the trategy on a imulator for the evaluation. Let f, t denote the number of arrival cutomer in tation at time tep t and want to reach tation at the beginning of time tep t+1. d #,t denote the number of bike preent in tation at time tep t after the repoitioning i done. The flow of bike i determined baed on the following two cae: (a) If the arrival demand at a tation i le than the number of bike preent in the tation, then all the cutomer are erved. (b) If the arrival demand at a tation i higher than the number of bike preent in the tation, then actual flow (denoted a x t, ) depend on the relative ratio f t, f t,. { f t x t, =, if f, t d#,t f t, d #,t Otherwie f t, Once we determine the flow of bike between tation at time tep t, we can compute the ditribution of bike at a tation at time tep t + 1 a the um of un-hired bike at time tep t, net incoming bike at the beginning of time tep t + 1 and the net drop-off bike by vehicle at time tep t + 1. d #,t+1 = d #,t + [ x t, x t, ]+ [ Y,t+1 } ] Y +,t+1 If the number of bike in tation at time tep t+1, d #,t+1 exceed the tation capacity, C # then we tranfer the extra bike
(i.e., d #,t+1 C # ) to the nearet tation to enure the capacity contraint of the tation. Thee extra number are hown a the lot demand at the time of return in the experimental reult. Once we obtain the ditribution of bike acro the tation for time tep t + 1, thi information can readily be utilied to compute the repoitioning trategy for time tep t + 1. Thi iterative proce continue until we reach the lat time tep. 5 Experimental Setting We evaluate our approach with repect to key performance metric of lo in demand, on real world 1 and ynthetic data et. The real world data et contain the following data: (1) Cutomer trip record, from which we etimate the bound on demand; (2) The number of tation, their capacity and initial ditribution of bike in each of the tation; (3) Geographical location of bae tation, from which we calculate the ditance between two tation; (4) The number of vehicle and their capacity. We generate the ynthetic data et a follow: (a) We take a ubet of the tation from the real world data et (b) Station capacity, their geographical location and initial ditribution of bike are drawn from the real world data for thoe pecific tation. (c) We generate the demand bound manually. More pecific detail on the demand bound are mentioned later. We compare the utility of our approache with three exiting benchmark approache a mentioned bellow. Benchmark-1: Static Repoitioning implie the practice of no repoitioning during the day. The tation are rebalanced at the end of the day to achieve a predefined inventory level. We ue thi a a baeline approach where no repoitioning i done during the planning period. Benchmark-2: Myopic Repoitioning entail that bike are repoitioned at each time tep to reach a certain inventory level. Through the peronal communication with bike haring operator, we infer that 50% of the tation capacity i the ideal inventory level and ome operator rebalance the tation in a myopic fahion (without conidering the demand pattern) to reach that pecific inventory level. Benchmark-3: Online Heuritic i adapted from [Schuijbroek et al., 2013]. Thi tatic repoitioning approach can be executed online due to it aumption of negligible cutomer movement during the rebalancing period. A we evaluate the trategie on the demand cenario generated uing Poion ditribution with a known mean, the goal of the online heuritic i to bound the inventory level within 10% of the Poion mean, while enuring the phyical limitation of the vehicle route. To enure a fair comparion, all the three benchmark approache and our robut trategy are evaluated by employing a imulation model a decribed in ection (4.3). Furthermore, we compute an upper bound on the optimal olution for the ynthetic data et where exact future demand for the entire horizon i aumed to be known. We employ an MILP formulation propoed by [Ghoh et al., 2015] to compute the optimal olution. 1 Data i taken from Hubway bike haring company of Boton [http://hubwaydatachallenge.org/trip-hitory-data] Static Myopic Online Robut Offline MEAN 822 758 641 638 451 STDEV 37 47 38 38 38 MAX 938 908 713 730 521 (a) Scenario for Uniform Data (data et: 1) MEAN 956 769 734 704 491 STDEV 48 62 45 48 39 MAX 1069 974 825 826 568 (b) Scenario for Two-Peaked Data (data et: 2) Table 4: Lot demand tatitic on ynthetic data et 6 Empirical Evaluation We report 2 reult on two ynthetic data et. Both the data et conit of 20 tation and 1 vehicle. We generate demand for 14 time tep. Figure 1(a) how the demand pattern for both the ynthetic data et. We generate the aggregated mean demand at each time tep for firt data et randomly, while the aggregated mean demand for econd data et follow a realitic pattern with two peak hour. For both the data et, we compute the lower bound on the arrival demand a (1-ɛ) of the mean demand and upper bound a (1+ɛ) of the mean demand. To compute the bound on arrival demand for each tation and for each origin detination pair we et ɛ a 100%, while for the bound on the ytem wide demand at each time tep, ɛ i et a 10%. Demand 260 240 220 200 180 160 140 120 100 80 Demand-tructure (Synthetic Data et) Uniform Two-Peaked 0 2 4 6 8 10 12 14 Time Step Lot Demand 160 150 140 130 120 110 100 90 Convergence Plot (Synthetic Data et) Repoitioning Adverary 0 5 10 15 20 25 30 35 40 #Iteration Figure 1: (a) Demand pattern for ynthetic data et; (b) Convergence of cenario generation approach on ynthetic data et Figure 1(b) how the convergence of our cenario generation approach on ynthetic data. A expected, the gap between the objective of the adverary and the repoitioning planner reduce monotonically and converge after 40 iteration. A both the objective converge to 112, we can claim that the wore cae lot demand i bounded by 112 if the robut trategy i adopted. To compare the utility of our policy with the exiting benchmark approache, we generate 100 teting demand cenario, where demand from tation to at time tep t, f t, i generated uing Poion ditribution with known mean parameter. We report average performance tatitic in term 2 All the linear optimiation model were olved uing IBM ILOG CPLEX Optimiation Studio V12.5 on a 3.2 GHz Intel Core i5 machine with 8GB DDR3 RAM
Lot demand at Iue Time Lot demand at Return Time Static Myopic Online Robut Static Myopic Online Robut MEAN 267 269 257 197 61 138 46 50 STDEV 79 77 82 68 19 32 18 15 MAX 460 471 453 393 103 196 91 96 (a) Demand cenario from real-world data MEAN 145 155 146 100 53 126 37 33 STDEV 18 19 19 18 9 21 14 10 MAX 193 202 192 137 83 165 68 74 (b) Demand cenario follow Poion ditribution at each tation MEAN 163 171 154 113 69 143 50 54 STDEV 24 22 17 19 13 28 16 14 MAX 206 220 204 158 103 208 86 85 (c) Demand cenario follow Poion ditribution for each OD pair Table 5: Lot demand tatitic on the Hubway data et of mean, tandard deviation and the wore cae lot demand over 100 demand cenario. The performance tatitic for the ynthetic data et with uniform pattern are demontrated in Table 4(a). Our approach reduce the average lot demand by 22% over the tatic approach and by 15% over the myopic approach and i highly competitive with the online approach. Similar performance tatitic for the ynthetic data et with 2 peak hour are hown in Table 4(b). The average performance of our approach i ignificantly better than all the three benchmark approache, which verify the fact that our approach i able to better handle the lot demand at ruh hour. More interetingly, the competitive ratio for our olution i approximately 70% of the optimal olution for both the data et. Reult on the Hubway data et: The next thread of reult demontrate the performance tatitic on the Hubway data et. The Hubway BSS conit of 95 bae tation and 3 vehicle. We conider a planning horizon of 6 hour in the morning peak (6AM-12PM) and the duration of each deciion epoch i 30 minute. We compute the bound on demand from three month of hitorical trip data. A the hitorical trip data only contain ucceful booking and doe not capture the unoberved lot demand, we employ a microimulation model with 1 minute of time tep to identify the duration when a tation got empty and introduce artificial demand at the empty tation baed on the oberved demand at that tation in previou time tep. We produce three thread of demand cenario (1) We took the real demand data for 60 weekday. We etimate the actual demand by introducing artificial demand at empty tation uing a imilar heuritic mechanim dicued earlier; (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 the demand for each origin detination pair at each time tep i computed uing Poion ditribution. For all the three etting of demand cenario, we ummarie the key performance tatitic for all the approache in Table (5). A the planning period for one deciion epoch i 30 minute, we et a time threhold of 3 minute a a convergence criterion for our cenario generation approach. We provide tatitic for two type of lot demand: (a) Lot demand occurred at the time of hiring the bike due to tarvation of bike at tation; (b) Lot demand occurred at the time of returning the bike due to the congetion of bike at tation. The performance tatitic for real demand cenario are demontrated in Table 5(a). On an average our approach reduce the overall lot demand by atleat 18% over all the benchmark approache. Moreover, our approach reduce the wore cae lot demand by atleat 10%, hence, i robut to the uncertainty in demand. Similar performance tatitic for other two etting of demand cenario are hown in Table 5(b) and 5(c). For both the etting, the average and wore cae performance of our approach i noticeably better than all the three benchmark approache. The average lot demand i reduced by atleat 27% and 18%, while the wore cae lot demand i decreaed by atleat 19% and 16%, over all the three benchmark approache. 7 Concluion We develop a robut optimiation approach to olve the dynamic repoitioning problem in bike haring ytem. We propoe an iterative cenario generation approach where an adverary identifie the wore demand cenario for a given repoitioning trategy and the deciion maker compute a repoitioning trategy by conidering a et of demand cenario propoed by the adverary. The empirical reult on a real world and multiple ynthetic data et hown that our approach outperform the exiting benchmark approache in term of reducing the expected and wore cae lot demand and therefore, improve the operational efficiency of the bike haring company. In future, thi work can be extended with multi-tep planning by conidering the expected future demand bound for multiple epoch to better account for the future demand urge. Furthermore, a decompoition technique can be employed for the repoitioning planner to cale up the olution proce for problem with thouand of tation.
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