J. Indian Inst. Sci. A Multidisciplinary Reviews Journal ISSN: 0970-4140 Coden-JIISAD
© Indian Institute of Science 2019.
Modelling Methods for Planning and Operation of Bike‑Sharing Systems
Rito Brata Nath1 and Tarun Rambha1,2* REVIEW REVIEW ARTICLE
Abstract | Bike-sharing systems (BSSs) are emerging as a popular type of shared vehicle platform where users can rent bicycles without having to own and maintain them. BSSs are ideal for short trips and for connect‑ ing to public transit systems. Bicycle usage is associated with several unique characteristics which make planning and operation of BSSs very different from car sharing problems and other traditional transportation modelling approaches. In this paper, we summarize existing literature on strategic planning which involves selecting stations, designing bike paths, and fguring out station capacity. Research on operational meas‑ ures which include day-to-day and within-day repositioning activities are also collated. Additionally, models for understanding demand, pric‑ ing and incentives, maintenance, and other technological aspects are reviewed.
Keywords: Bike sharing, Strategic planning, Facility location, Operational planning, Repositioning
1 Introduction decisions on building dedicated bike lanes, setting Automobile usage is on the rise in many parts of up base stations,5 and choosing between pay-per- the world and cities are actively promoting eco- use and subscription-type services. Supply-side friendly transportation solutions to reduce traffc aspects can also in turn infuence demand. For congestion and emissions. Bike-sharing systems example, dedicated bike lanes make bike travel (BSSs) is one such alternative which can not only safer and has the potential to increase BSS serve short-distance trips, but can also enhance usage.6–8 connectivity to public transportation networks. For station-based BSSs, it is important to In a BSS, customers can pick up and drop off determine the capacity of each station and dis- cycles at specifc locations or anywhere in the tribute the feet across stations, although these city depending on the type of bikes in the system, decisions can also be made at an operational locking technology, and payment mechanisms. level.9–11 Within-day stochasticity in travel pat- Most of the current generation BSSs are either terns often leads to imbalances in the availability free-foating or station-based (Fig. 1). Station- of bikes and parking spots. Having stations that based BSSs may use both docked or geo-fenced are full or empty can affect ridership and render dockless bikes. A few examples of BSSs include the system ineffective. To address these situa- Capital Bikeshare (CaBi) in Washington, D.C., tions, cycles are often repositioned from one sta- 1 Department of Civil Citi Bike in New York, Blue Bikes in Boston, and tion to another using trucks12 or by providing Engineering, Indian Vélib’ in Paris. price incentives to users for dropping off bikes at Institute of Science, 13 Bangalore, India. Like any other transportation system, plan- nearby high-demand locations. 2 Center for infrastructure, ning and operation of BSSs require understand- When bikes are repositioned using motor Sustainable ing the spatio-temporal demand for cycles in a vehicles, one must decide how many cycles to Transportation and Urban Planning (CiSTUP), Indian city. Demand can either be inferred from exten- move between stations and also determine opti- Institute of Science, sive surveys or past data on traveller move- mal vehicle routes. Repositioning done dur- Bangalore, India. ments.3,4. This knowledge of demand can drive ing the day, in real-time, is classifed as dynamic *[email protected]
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Since then, BSSs have undergone many changes. An infographic of the historical development of BSSs through the years is shown in Fig. 2. The second generation of BSS saw the advent of coin deposit stations in which rides were free, but cus- tomers had to insert coins into a slot to unlock Figure 1: Docked BSS: Capital Bikeshare, US 1 bikes and could retrieve them once the bikes were (left) and Dockless BSS: Mobike, China2 (right). returned. The frst coin deposit bike program called Bycyklen started in Copenhagen in 1991.19 In 1995, it also became the frst large-scale BSS rebalancing,14,15 while that carried out at the end with around 1,100 bikes. This system was still vul- of a day, when the system is inactive, is called nerable to theft due to anonymity of users. The static rebalancing.12,16,17 Periodic maintenance of use of automated docked stations with registered bikes, vandalism, and theft are some other com- customers marked the beginning of the third gen- mon problems faced by a service provider of a eration of BSSs. This greatly reduced vandalism BSS. and theft issues associated with the previous gen- The rest of this review article is structured as erations of BSSs. Such a system frst appeared in follows. In Sect. 2, we discuss the history of BSSs Portsmouth University, England (1996) and stu- and motivate the need for developing decision dents had to pay for membership and bikes could support tools for studying planning and opera- be rented using a magnetic card. Other examples tional problems associated with BSSs. In Sect. 3, of third generation BSSs include LE Vélo STAR we discuss research on some of the strategic prob- in Rennes (1998), Bicing in Barcelona (2007), lems such as bike-lane design, station locations, Cycle Hire in London (2010), and Citi Bike in and dock size selection. Section 4 details various New York (2013). The fourth-generation bikes repositioning mechanisms that can be used when came into existence in 2005 with the Vélo’v pro- operating a BSS. Technological aspects and some gram in France. This system was operated by an emerging phenomena are addressed in Sect. 5 advertising frm JCDecaux and was equipped and the conclusions of this study are presented in with smart bikes that could be accessed using a Sect. 6. mobile app. The smart technology-based system provided real-time information on bike availabil- ity.19,25 Most BSSs in the recent past belong to the 2 Background and History ffth generation in which dockless bikes are used The frst BSS started in Amsterdam in 1965 in a free-foating or station-based set up. These (White bicycle plan) with just ffty bicycles.18 systems have lower setup costs and hence have However, a month later, all bikes were either sto- grown rapidly in many cities. len or dumped into canals. The white bicycle plan By December 2016, about a thousand cities in was a frst-generation BSS in which bikes were the world had a bike-sharing program.26 Mobike, free to use. Other frst-generation BSS examples a dockless BSS, is the world’s largest bike-sharing include Vélos Jaunes in La Rochelle, France (1974) operator. As of 2018, Mobike operated in over 19 and Green Bike Scheme in Cambridge, UK (1993). countries and 200 cities.27 One of the large-scale
Figure 2: Generations of BSSs. (Source: Midgley18, Chen et al.19; Picture source:20–24).
622 1 3 J. Indian Inst. Sci.| VOL 99:4 | 621–645 December 2019 | journal.iisc.ernet.in Modelling Methods for Planning and Operation station-based BSSs is the Hangzhou Public Bicycle 3 Strategic Planning System in China, which comprises of 2,965 sta- Strategic planning problems in the context of tions and approximately 69,750 bicycles,28, with a BSS typically involve designing the bike path plans to expand to 175,000 bicycles by 2020.29 network and determining the number and loca- Bike-sharing programs have grown exponen- tions of bike stations. These decisions must con- tially in the last decade, particularly in Asia. For sider construction costs, the effect of terrain, instance, thirteen of the world’s ffteen largest customer service level (which can be measured BSSs are in China.30 by the coverage level, bicycle availability, and user Although BSSs have been encouraged by pub- out-of-pocket costs), and the impact on existing lic agencies and users around the world, service automobile traffc. For instance, station location providers such as Mobike, Ofo, and Pedl had to decisions must make sure that cycles are at a con- shut down operations in many cities due to high venient walking distance (roughly 300–500 m) maintenance costs, low profts, theft, and vandal- from the actual trip origins and destinations.37 ism.31–33 Some of the new technologies like global Geographical factors are crucial not only for bike positioning system (GPS), anti-theft alerts, and lane design, but also for locating bike stations. For high-tech handlebars introduced in the fourth example, in Brisbane, it was observed that City- and ffth generation dockless bicycles have the Cycle users avoid returning bicycles to higher-ele- potential to address these issues to a certain vation stations.38 Stations must also be designed extent.34,35 such that there is enough curb-side space to Also, cycling is not perceived as a safe com- account for surges in pickups and dropoffs. mute mode, especially in mixed traffc, and the A key input to these decisions is the knowl- lack of dedicated bike lanes in most places proves edge of demand for bike sharing, which can be to be a major hurdle for the success of BSSs. Fur- estimated using census data,39 stated-preference ther, while BSSs work well in controlled environ- surveys, and by observing the travel patterns of ments such as offce and university campuses, commuters who might potentially shift from scaling them to a city level can be extremely other modes to cycling.40 BSS planners must challenging especially for dockless free-foating allocate bicycles at different stations in a man- systems. Often, bikes are left at remote locations ner that is consistent with the actual demand of where there is no demand, and this affects the uti- customers. Most studies in literature focus on lization rates of cycles. As bicycles are fairly inex- understanding demand patterns after a BSS sys- pensive, service providers tend to add more bikes tem is in place. For example, statistical regression- to the system as a knee-jerk reaction, but the based forecasting and time-series methods can oversupply of bikes has resulted in many aban- be used to predict the spatio-temporal activity doned and broken bicycles, especially in China of users. These have been successfully demon- (see Fig. 3). These observations strongly motivate strated using data from Bicing in Barcelona41,42 the need for planning and operating BSSs in an and Vélo’v in Lyon. 3,4 Others have used a data effcient manner. mining approach43 to cluster BSS stations accord- ing to the rate of bike pickups and dropoffs using Citybike Wien data from Vienna. Clustering methods were also used to identify ‘similar’ sta- tions for analysing the system before and after a policy change.44 Demand prediction for existing BSSs was also successfully done using machine learning and artifcial intelligence methods by learning customer behaviour from observed data and using it for prediction.45–49,50 However, these predictions are yet to be fully exploited in exist- ing research on operational planning that we will discuss in Sect. 4. Customer demand in existing BSSs is heavily infuenced by supply, and literature on demand forecasting before a BSS is planned remains sparse. Traditional demand models involving trip genera- tions and distribution were extended to bicycling Figure 3: Roadside dumping of bicycles in Xia‑ by Turner51 and Landis52. A few researchers have men, Fujian province, Chinan (Source: Alan36). proposed GIS-based methods that can provide
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macro-level bicycle demand using socio-demo- Their formulation included fow-balance graphic and geographical attributes.53–55 Using constraints, connectivity constraints for every daily trips by different modes and stated prefer- OD pair, a constraint that limits the path length ence surveys that provided mode shift propensi- beyond which users will not cycle, constraints ties,40 forecasted the bike trips for Philadelphia, US that ensure a suitable level of service, and con- assuming three different levels of system usage. In straints that select intersections belonging to the absence of elaborate travel demand models or a chosen path. Extensions in which these con- surveys, studies that understand factors infuenc- straints are reformulated to speed up computa- ing bike trips can be transferred to other cities for tion were also proposed. The model was solved Branch-and-bound: is an predicting demand.56 For example, Faghih-Imani using a branch-and-bound method for small enumeration technique for in- et al.57 analyses the role of factors such as popu- problem instances and the authors studied the teger optimization problems in which the feasible region is lation density, accessibility, points of interest, and effect of the level of service and the number of iteratively decomposed into supply-side features such as the number of stations OD pairs on the total cost. smaller sets and bounds are per unit area and capacity on bicylce trip gen- Others have formulated bi-level programs for estimated to prune certain 62 search directions. eration and attractions. Data from Barcelona and the bike route design problem. At the upper Seville, Spain were used in this work to estimate level, benefts to cars and cyclists were considered, Bi-level Programs: are optimization models in which model parameters using a restricted maximum and at the lower level, an assignment model for two objective functions are likelihood approach. In another work, Singhvi bikes and automobile traffc was optimized. A optimized: one at the upper et al.58 uses taxi data from New York City, US along genetic algorithm was used to solve the bi-level and another at the lower level. Upper level decisions affect with population information to build regression formulation on medium-sized examples using a the lower level constraints or models that predict bicycle usage. Other predictors special crossover and mutation technique. objective. that have been found to signifcantly infuence bike Another optimization framework was proposed 59 60 63 Maximum likelihood: is demand are weather and seasons . by Mauttone et al. in which the roadway net- a statistical procedure for Bicycle trips may also be used for frst- and work could have sections with no cycling infra- estimating a distribution’s pa- rameters in order to maximize last-mile access to transit systems. In such cases, structure and the total number of discontinuities the probability of observing transit ridership and accessibility must be fac- in bike paths was minimized. A mixed-integer the data. tored in when estimating the demand for a BSS. multi-commodity fow problem was proposed, In the next subsections, we discuss a few math- and a metaheuristic was used to handle large ematical models that have tried to incorporate problem instances, including a test case from the the integrated effects of various input parameters city of Montevideo, Uruguay. The optimal bike in the design of a BSS. For better readability, we path design was also addressed in64 to separate have altered the notation from the original papers bicyclists from motorized vehicles for an existing at several places to describe similar variables and transportation network. The objective was to parameters wherever possible. maximize the cyclists’ utilities assuming that their route choices could be modelled using a path-size logit framework. The problem was formulated as 3.1 Route Design Logit models: are a class of a mixed-integer linear program (MILP) and Researchers have addressed the bike network random-utility econometric tested on the Sioux Falls and Anaheim, US net- models in which decision design problem in multiple ways using different works using a global solver and a metaheuristic. makers’ utilities are charac- objectives and assumptions. For example,61 for- terized by a deterministic While previously mentioned studies consid- mulated models that connect origin-destination component and an error term ered a single objective function, Zhu and Zhu which is Gumbel distributed. (OD) pairs with bike paths while minimizing 65 formulated a multi-objective function that total cost and meeting a specifed bicycling level comprised of accessibility, bicycle level of ser- of service. The framework considers a network MILP: Mixed integer linear vice, number of intersections, and the construc- G = (R, S) , where R is the set of intersections programs are optimization tion cost. (Since intersections pose safety risk for models in which the objective and S is the set of roads. Each roadway segment bicyclists, they are assumed to prefer connected function and constraints are has an associated cost c to make it cyclable. The affne functions and some or ij bikeway networks over fragmented ones.66,67). cost associated with improving each intersection all variables are integral. Accessibility was measured by not only consider- is d . Decision variable δ is 1 if roadway segment i ij ing the connectivity between the points of inter- (i, j) ∈ S is improved and is 0 otherwise. Similarly, est, but by also considering the travel budget of γ is 1 if intersection i ∈ R is improved and is 0 i commuters on the road. The problem was solved otherwise. The objective can thus be mathemati- by an augmented ǫ-constraint method using cally expressed as follows. hypothetical data from Jurong Lake District in Singapore. A few other route design models have min cijδij + diγi (1) been summarized in Table 1. ∈ (i ,j)∈S i R
624 1 3 J. Indian Inst. Sci.| VOL 99:4 | 621–645 December 2019 | journal.iisc.ernet.in Modelling Methods for Planning and Operation
Table 1: Summary of route design models.
References Description
Su et al.68 Developed a GIS-based route planner considering user preferences and the data was used to identify and improve disconnected segments in the network 69 Černá et al. Integer linear programming model for tourists which maximizes the attractiveness of paths. Constraints include fow-balance, maximum riding time, and budget limits Teschke et al.70 Statistical analysis was carried out to infer the effect of locations of streets or sidewalks, characteristics of trips, personal characteristics, and temporary features like construction sites on the risks due to cycling. These results were used to make decisions on improving existing infrastructure Winters and Teschke71 A population-based survey was used in multiple linear regression models to show the need for having dedicated lanes. The likelihood of choosing routes with attributes such as paved/unpaved, residential/arterial, and the presence of on-street parking were estimated, and route design recommendations were provided Putta and Furth72 Proposed methods to detect barriers in low-stress bike networks that comprise of links belonging to dedicated bike lanes and shared lanes with low automobile traffc. Their meth- ods were demonstrated on real-world networks of Boston and Arlington, US
n 3.2 Facility Location cost would be Kn(n − 1)/2 , since 2 arcs have to Research on station location selection is heavily be built. 73 infuenced by the hub location and maximal The bike station design problem is not exactly covering problem.74 In the basic version of the similar to this model since it involves a pick up single hub location problem, it is assumed that and a drop off. Such scenarios resemble a two- there are n nodes which act as both origins and hub facility location problem.73 Suppose 1 and destinations. The objective is to fnd the optimal 2 represent two hub locations and let ui be 1 if hub location such that the cost of transporting an origin or a destination i is serviced by hub 1 demand between nodes via the hub is minimized. and be set to 0 otherwise. Likewise, let vi be 1 if That is, the hub acts as a switch for all interactions an origin or a destination i is serviced by hub 2 in the network. Suppose that the fow between and is 0 otherwise. Note that when a node can be OD pair (i, j) is denoted by wij and cij represents served by both hubs, the one nearest to the node the distance between nodes i and j. The optimal is assumed to serve the node and the binary vari- location of the hub q can be obtained by solving able corresponding to the other hub is set to 0. The goal is to send the OD fows passing through min wij ciq + cqj q (2) both hubs. The hub locations are chosen such i j that the overall transportation cost is minimized. Note that in the absence of set up costs, there is no requirement of a hub since, min wij uivj(ci1 + c12 + cj2) i j w c ≤ w c + c (5) ij ij ij iq qj (3) i j i j + ujvi(ci2 + c21 + cj1) is satisfed if triangle inequality is assumed. How- The problem of locating multiple facilities ever, if K is the cost associated with setting up is also widely addressed in the literature using a each inter-city route, a hub is needed if maximum covering model or a p-median prob- lem.74–77 In the maximum covering model, the wij ciq + cqj + Kn objective is to locate a fxed number of facilities i j to maximize the total demand that can be cov- n(n − 1) (4) ered assuming that demand located farther than < w c + K ij ij 2 S units from a hub cannot be served. Mathemati- i j cally, it can be expressed as The Kn term on the left-hand side of inequality max aiyi (4) corresponds to the cost of connecting each of J (6) ∈ the n stations to the hub. On the other hand, if i I routes were to be built between each pair of sta- where I and J are the set of demand nodes and tions without creating a hub, the construction facility sites respectively, ai is the demand at node i, decision variable xj is 1 if a facility is opened at
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j ∈ J and is 0 otherwise, Ni ={j ∈ J |dij ≤ S} is demand between i and j passes through bike sta- a subset of facility sites which can serve demand tions k and l and is 0 otherwise. Denoting the set from i, and yi is 1 if the demand at i can be served of origins, destinations, and bike stations as I, J, and is 0 otherwise. and K, respectively, the transportation cost com- The x and y variables are connected using ponent of the objective was formulated as constraints xj ≥ yi ∀ i ∈ I and xj = p , j∈Ni j∈J where p is the total number of facilities. α dik yiklj ij ∈ ∈ The p-median problem on the contrary mini- i I k ∈K l∈K j J mizes the total cost of serving the demand and + β d y kl iklj ij (8) can be expressed as ∈ ∈ k ∈K l∈K i I j J min a c w + γ d y i ij ij (7) lj iklj ij i∈I j∈J ∈ ∈ l∈K j J i I k ∈K where cij represents the unit cost of serving To address the issue of unmet demand, the demand at i using a facility at j and wij is the frac- authors introduce a penalty term tion of the total demand ai served by the facility at j. (Hence, it must satisfy j∈J wij = 1 ∀ i ∈ I .) δ qik yiklj ij As before, a binary variable x is used to repre- i∈I j∈J j k ∈K l∈K sent facility location decisions and x = p (9) j∈J j + q y ensures that p such locations are opened. Finally, jl iklj ij ∈ ∈ the x and the w variables are connected using an j J l∈K k ∈K i I additional constraint wij ≤ xj ∀ i ∈ I, j ∈ J. where δ is the additional unit cost of uncovered There are a few key differences in bicycle net- demand and qrs is 1 if a bike station located at s works that prohibit the direct use of standard cannot cover demand starting or ending at r. In facility location models. For instance, the hub addition, setup costs location model implicitly assumes that the fow f x + c z from a certain node can frst be sent to hub 1 or k k kl kl (10) 2 (whichever is closer) and it can be redirected k ∈K k ∈K l∈K to the destination. However, in a BSS, some trips are introduced to model the cost of constructing may not be feasible if the stations are far from stations and bike lanes. Here, the binary decision the actual origins and destinations. Second, there variable xk is 1 if a station is opened at k and zkl is are more than two hubs in a bike network. On set to 1 if a bike lane is needed between stations k the other hand, the maximum covering and the and l. The associated costs are fk and ckl respec- p-median problems can be used to model unmet tively. Finally, the authors also include a couple of demand, but they are applicable to single com- extra terms in the objective that refect the aver- modity, single source/destination-type fows age holding costs and the cost of replenishing whereas locating bike stations involves a multi- bicycles assuming some stochasticity in demands. commodity, multiple OD pair problem. Consistency between the decision variables is One of the frst models to tackle these issues achieved using constraints. For example, if bike was proposed by Lin and Yang37 using multiple stations are opened at two nodes, a bike lane objectives and found the optimal bicycle loca- could be built between them using tions along with the paths needed for connectiv- 2z ≤ x + x ∀ k ∈ K, l ∈ K\{k} ity. The formulation, explained below, balances kl k l (11) the cost incurred and the level of service provided Similarly, demand can be routed between two sta- to customers. tions only if a bike lane connects them and this is Let drs denote the distance between nodes r modelled using and s (which could be trip origins or destinations y ≤ z ∀ i ∈ I, k ∈ K, l ∈ K\{k}, j ∈ J or bike stations). Different components of the iklj kl (12) objective are weighted by parameters to convert it Finally, constraint (13) is used to route the to cost units. For example, α , β , and γ represent demand between each OD pair along some path the unit travelling cost from the trip origin to the connecting the OD pair. pickup bike station, between the pickup and the y = 1 ∀ i ∈ I, j ∈ J dropoff bike station, and the dropoff bike sta- iklj (13) tion to the trip destination respectively. Assume k ∈K l∈ K\{k} that the yearly mean travel demand between OD Some researchers have also proposed tools pair (i, j) is ij and decision variable yiklj is 1 if the to locate bike stations while simultaneously
626 1 3 J. Indian Inst. Sci.| VOL 99:4 | 621–645 December 2019 | journal.iisc.ernet.in Modelling Methods for Planning and Operation modelling the interactions with other modes. For hence the daily demand at station k was com- example, Romero et al.9 capture the mode choices puted using between cars and a BSS using a multinomial logit framework within a bi-level optimization pro- 1 k = yiklj ij ∀ k ∈ K gram that determines the optimal bike station T (14) i∈I l∈K\{k} j∈J locations. Using data from Santander City, Spain, a genetic algorithm was used to demonstrate the where T is the number of days in a year. Assuming applicability of their model. Results indicate that that the lead time for replenishing bikes at a sta- optimally located bikes can induce a signifcant tion k is τk , and a desired level of service is set by mode shift from cars to cycles. In another line the probability of running out of stock (1 − γk ) , of related, but tangential, work on car sharing, the inventory level required sk can be expressed as Kumar and Bierlaire78 developed regression mod- s−1 els that predict the demand for shared services as e−�kτk (� τ )q s = min s : k k ≥ γ a function of transit ridership, personal car usage, k q! k q=0 (15) and other land use attributes and integrated the outputs with an optimization model to select car ∀k ∈ K stations. A few other facility location models have Constraints (14) and (15) are both non-linear been summarized in Table 2. and make the problem highly intractable. The authors proposed an iterative greedy heuristic in which for a given set of bike stations, lanes 3.3 Capacity Allocation and inventory levels are chosen one at a time to After deciding the locations of the bike stations reduce the overall costs. Their method was dem- and paths, another key strategic decision that is onstrated on a hypothetical test network and sen- crucial to a station-based BSS is the capacity allo- sitivity of optimum inventory levels with respect cation of bikes at each station. Many studies have to the frequency of replenishment and network modelled this jointly with the location decisions design was studied. of bicycles.81,83 In this section, we will discuss one Some researchers have proposed MILPs to model proposed by Lin et al.6 that builds on the address the capacity allocation problem. For formulation by Lin and Yang37 discussed earlier. instance, Sayarshad et al.84 formulated a multi- In addition to (8)–(10),6 introduce a term period optimization model in which the demand h s that refects the overall holding costs, k∈K k was known, and the objective function included where h is the inventory holding cost of a bicycle revenue from trips, relocation costs, capital and and s is a non-negative decision variable repre- k maintenance costs, and a penalty for unmet senting the inventory level at station k. The yearly demand. A similar multi-period MILP was sug- travel demand between OD pair (i, j) is assumed gested by Martinez et al.5 and it also included to follow a Poisson distribution with rate and ij relocation decisions. Heuristics that decompose
Table 2: Summary of facility location models.
References Description
García-Palomares et al.53 A GIS-based method was used to study bike location for two objectives: p-median and maximum coverage models. Quantitative accessibility analysis to identify the stations that are relatively isolated was carried out using data from Madrid, Spain Yan et al.79 Mixed-integer programming models for deterministic and stochastic demand instances where the goal was to minimize the cost of routing demand as well as fxed costs of locating bike stations Frade and Ribeiro80 A maximum coverage model that captures relocations over time using constraints. Budget constraints that feature inventory, maintenance, and relocation costs are also modelled Park and Sohn81 Maximum coverage and p-median model were solved using taxi data from Seoul, South Korea. Their model also suggested station capacity using the frequency of bike trips and a clustering technique Zhang et al.82 Analysed re-design strategies for an existing BSS using historic demand usage and crowd suggestions. Objectives included increasing convenience at a minimum cost 83 Dobešová and Hỳbner Used ArcGIS to locate a minimum number of bike stations (and determined their capacities) while maximizing coverage. An existing bike network and the number of inhabitants in different regions were taken as inputs
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the problem by time periods were proposed and overall performance of a BSS. Figure 4 shows a tested on a network from Lisbon, Portugal. A few snapshot of the inventory levels for a portion of other capacity allocation models have been sum- Citybike Wien in Vienna, Austria and CaBi, Wash- marized in Table 3. ington D.C., US and one can notice bicycle sta- tions which are nearly full or empty. To address these situations, day-to-day and 4 Operational Planning within-day operational measures such as relocat- As discussed in Sect. 3, strategic planning can be ing bicycles from one place to another is a must. used to locate stations and allocate an optimum These repositioning tasks are usually carried out number of bicycles at those locations. However, using trucks or bike-trailers (see Fig. 5). In addi- at an operational level, uncertainty in demands tion, one can provide incentives that might nudge and maintenance requirements create supply customers to pick up (or drop off) their bicycles imbalances rendering re-optimization necessary. at nearby stations that are close to capacity (or For station-based systems, these types of stochas- short of bicycles). Repositioning strategies are tic events might make some stations go empty, mainly classifed as static and dynamic depending preventing customers to rent a bicycle. It may on the timing of repositioning. Some authors also also happen that some stations become full and classify it as online and offine methods and the force customers to wait or return their cycles at subtle distinction in the nomenclature will be dis- another station. Supply imbalances in free-foat- cussed in Sect. 4.3. ing systems do not affect dropoffs but demand fuctuations can make it diffcult to fnd a bike in the frst place. Such departures from strategic plans can lead to loss of customers and affect the
Table 3: Summary of capacity allocation models.
References Description
Caggiani et al.85 A bi-level optimization model which uses data from an existing BSS was proposed to create spatio- temporal clusters. The model optimizes the number of times out-of-stock events occur subject to a budget constraint Çelebi et al.86 An optimization method that determines station locations and capacity using a set covering method. A queuing model is used to estimate service levels and unmet demand is minimized using a dynamic program Freund et al.87 Optimization formulation to minimize out-of-stock events under budget constraints by re-allocat- ing dock capacity. A polynomial-time allocation algorithm was also proposed Cavagnini et al.88 Two-stage stochastic programming formulation in which capacity allocation is made in the frst stage and relocation decisions are made in the second stage. Demand scenarios and associated probabilities are assumed, and the objective minimizes the total expected penalty for re-balanc- ing and stock-out Lu89 A robust optimization approach is used for multi-period feet allocation to minimize the total system cost that includes holding and redistribution costs and penalties for lost customers
Figure 4: Station inventory levels of Citybike Wien (left) and CaBi (right). (Source: Citybike Wien System Map90, Capital Bikeshare System Map91).
628 1 3 J. Indian Inst. Sci.| VOL 99:4 | 621–645 December 2019 | journal.iisc.ernet.in Modelling Methods for Planning and Operation
Figure 7: Markov chain for station inventory. (Source: Schuijbroek et al.17).
with different objectives by forecasting the opera- tions on the subsequent day. 17 Figure 5: Rebalancing using a trailer. (Source: For example, Schuijbroek et al. suppose that Panhard92). C is the capacity of a station and the state transi- tions for the number of bicycles at the station occur according to a non-stationary Mt /Mt /1/C (in Kendall notation) process. That is, the inter- Kendall notation: A/S/c/K arrival times for returns and pickups at time t are represents queuing processes using arrival process (A), distributed exponentially with rates (t) and µ(t) service time (S), number respectively (see Fig. 7). These transition rates are of servers (c), and capacity estimated using maximum likelihood methods. limit (K). Thus, M/M/1/C indicates that the arrivals are Additional assumptions are often needed when Poisson and service times are Figure 6: Static bicycle repositioning. (Source: developing a demand forecasting tool since lost exponential in a single-server Zhang et al.15). demand due to empty or full stations is censored queue with capacity C. and is not a part of the observed data. Assuming that a station starts with s cycles 4.1 Static Repositioning after static repositioning, let p(s, s′, t) be the In static repositioning, bicycles are rebalanced probability of fnding s′ bikes at time t on the during the night when customer movements next day. These transition rates satisfy Chapman- are minimal. Past data may be used to forecast Kolmogorov equations. To calculate the expected demand for bikes at different stations and guide fraction of successful pickups and dropoffs, the the repositioning operation. The repositioning authors defne and forecasting periods do not overlap as shown T in Fig. 6 and hence real-time demand variations µ(t)(1 − p(s, 0, t))dt g(s, 0, T) = 0 (16) are not addressed. Nevertheless, moving bicycles T 0 µ(t)dt during the night is convenient from the opera- tor’s perspective since there are no parking and T congestion issues. Modelling within-day demand (t)(1 − p(s, C, t))dt g(s, C, T) = 0 (17) fuctuations requires a more dynamic approach T 0 (t)dt and will be discussed in Sect. 4.2. Most research on static repositioning is geared where T is the time limit for the next day’s opera- towards addressing the following key questions. tions. The bounds for the desired inventory level First, how many cycles should be moved between at the end of the static rebalancing procedure is different pairs of stations. (This problem is also defned as referred to as the inventory balancing problem.) smin = min s : g(s, 0, T) ≥ β− (18) Second, what is the most effcient way to route vehicles which move these bicycles (which con- + smax = max s : g(s, C, T) ≥ β (19) stitutes the routing problem). These two problems are often jointly solved using optimization mod- where β− and β+ are desired service levels for the els with integer decision variables. next day. A commonly used target stock level in the Many studies also allow deviations from the inventory balancing problem is the number desired inventory levels but penalize them in determined from the capacity allocation problem. objective functions.94,95 The penalty could just be Alternately, researchers have also proposed mod- an absolute value of the difference between the els in which the inventory level at the end of the desired and achievable inventory level or could rebalancing procedure falls within an ideal pre- factor in the next day’s operations. For instance, determined interval.93 The limits of such intervals Raviv et al.95 assume a penalty for out-of-stock can be obtained from Markovian queuing models pickup and dropoff events as φ and ψ respectively
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and defne a function to describe the expected excess when compared to the desired inventory si . shortage using If qi > 0 , station i is a pickup node and if qi < 0 , T it is a dropoff node. A second decision variable F(s) = p(s, 0, t)φ + p(s, C, t)ψ dt (20) yij represents the total number of cycles that are 0 carried by the vehicle on arc (i, j). Supposing that An approximation of this function was used as the total defcit equals the total excess (this can be a penalty term in the objective function of an easily relaxed assuming that the depot has extra MILP. The authors used data from Tel-O-Fun in inventory or space for extra cycles), the 1-PDTSP Tel Aviv, Israel to estimate the model parameters. can be formulated as follows. Inventory levels after rebalancing have also 96 min c x been set using a chance constraint approach . ij ij (22) + (i,j)∈A In this method, the number of pickups ( ξi ) and dropoffs ( ξ − ) at a station i ∈ N are assumed i s.t. x = x = 1 ∀ i ∈ N random and one of the constraints in the ij hi (23) ∈ model ensures that the probability of success- j N h ∈N ful pickups and dropoffs are greater than a pre- xij ≥ 1 ∀ S ⊂ N, S �= ∅ specifed parameter p. Specifcally, let ri and Ci (24) ∈ ∈ denote the current inventory level and capac- i S j /S ity of station i respectively. If uij indicates the y − y = q ∀ i ∈ N number of bicycles moved from station i to sta- ij hi i (25) ∈ tion j, then the number of available bikes at a j N h ∈N station i is r + ξ − + (u − u ) . Likewise, i i j ji ij 0 ≤ y ≤ Qx ∀ (i, j) ∈ A the number of available spaces at station i is ij ij (26) C − r + ξ + + (u − u ) . The chance con- i i i j ij ji x ∈{0, 1}∀(i, j) ∈ A straint can thus be written as ij (27) Constraint (23) ensures that each station is vis- ited exactly once and (24) eliminates subtours. P − + ri + ξ + uji − uij ≥ ξ , Flow conservation of the cycles is guaranteed i i j � � � using (25) and inequality (26) forces the fow C − r + ξ + + u − u variables to be zero on links that are not traversed i i i ij ji (21) j by the vehicle. This formulation was extended � � � by Raviv et al.95 to the multiple vehicle scenario using a three-index formulation with less restric- ≥ ξ − ∀ i ∈ N ≥ p i tive assumptions. Suppose that previous notation is modifed such that xijv is a decision variable After deciding the target inventory levels or their which is 1 if vehicle v ∈ V traverses arc (i, j) and intervals, the routing problem needs to be solved is 0 otherwise. Similar to (23), fow conservation to fgure out how a single or multiple vehicles can of vehicles can be expressed as redistribute cycles in an optimal manner. The sin- x = x = 1 ∀ i ∈ N, v ∈ V gle vehicle routing problem can be formulated as ijv hiv (28) ∈ a one-commodity pickup and delivery travelling j N h ∈N salesman problem (1-PDTSP).97 To mathemati- x ≤ 1 ∀ i ⊂ N\{0}, v ∈ V cally model this problem, consider a complete ijv (29) ∈ graph (without self-loops) G = (N, A) where j N N ={0, 1, ..., n} represents the set of bike sta- Note that (29) makes sure that each vehicle can tions and A is the set of arcs. The assumption visit a station at most once. It is also possible that that the graph is complete is not necessary but a station is visited by more than one vehicle. Just is made only to simplify the notation. Suppose like the 1-PDTSP, Raviv et al.95 defne another node 0 represents the depot where the vehicle variable yijv which indicates the number of cycles (with capacity Q) that is used to move bicycles carried by vehicle v while traversing arc (i, j). begins its trip and suppose nodes 1, ..., n denote These are linked to the xijv variables in a manner the other stations in the network. Let cij be the similar to (26) as shown below travel costs between i and j and binary decision 0 ≤ yijv ≤ Qvxijv ∀ (i, j) ∈ A, v ∈ V (30) variable xij be 1 if the vehicle takes arc (i, j) and is 0 otherwise. Each station i is assumed to have a where Qv is the capacity of vehicle v. Two new + − demand/supply qi = ri − si which is the defcit or decision variables ziv and ziv are introduced which
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represent the number of bikes added and removed to revisit stations. Models in literature also allow by vehicle v at station i respectively. Hence, we exchanging bikes between vehicles. − + 95 may write qi = ri − si = v∈V (ziv − ziv ) and The optimization program by Raviv et al. fow conservation of bicycles (25) can be recast as has been a starting point for many MILP formu- lations in static repositioning research. For y − y = z− − z+ ∀ i ∈ N, v ∈ V ijv hiv iv iv instance, instead of penalty functions, Erdoğan ∈ 93 j N h ∈N et al. use pre-determined inventory levels and (31) Schuijbroek et al.17 use the bounds obtained from Assuming that we need not meet the desired equations (18) and (19) as extra constraints. Oth- inventory level exactly (i.e., we need not clear ers have included service times and unloading the excess or defcits), the following sets of con- and loading costs as part of the objective.93,94 straints on the z variables follow naturally. Most MILP models, however, tend to be compu- tationally intractable for large problem instances. z− ≤ r ∀ i ∈ N iv i (32) To address these issues, solution methods such as ∈ 93,100 v V branch-and-cut; heuristics such as cluster- Branch-and-cut: is solution frst-route-second which solves the multiple vehi- method which combines z+ ≤ C − r ∀ i ∈ N 17,101 branch-and-bound with a iv i i (33) cle problem using single vehicle problems; cutting plane method for im- ∈ 102 v V and metaheuristics such as tabu search have proving the linear program- been proposed. A summary of the papers that ming relaxation solutions at + − − = ∀ ∈ the nodes of the search tree. ziv ziv 0 v V (34) address static rebalancing is presented in Table 4. i∈N Almost all of them use integer programming Metaheuristics: are generic higher-level heuristic proce- Subtour elimination constraints for each vehicle methods and hence integrality constraints have dures that can be applied to were described in the form proposed by Miller not been explicitly mentioned in the table. a wide range of optimiza- et al.98 as shown in (35) using an additional con- tion problems. They have been successfully applied in tinuous decision variable wiv and a suffcient large 4.2 Dynamic Repositioning transportation logistics to number M. fnd approximate solutions. While static repositioning helps reset a BSS to a Examples include genetic wjv − wiv + M(1 − xijv) ≥ 1 state with ideal inventory levels, it can perform algorithms, simulated anneal- (35) ing, and tabu search. ∀ i ∈ N, j ∈ N\{0}, v ∈ V poorly when the spatio-temporal demand pat- terns exhibit high variance. It also cannot han- The complete formulation is shown below. dle non-recurring forms of demand fuctuations min f (s ) + α c x such as those due to weather, special events, etc. i ij ijv (36) i∈N (i,j)∈A v∈V In such situations, a BSS operator must reposi- tion bicycles during the day and in real-time to s.t. (28) − (35) match supply and demand. Hence, this opera- xijv ∈{0, 1}, yijv ≥ 0 ∀ (i, j) ∈ A, v ∈ V tion is more challenging to carry out than its (37) static counterpart. Unlike in Fig. 6, repositioning and forecasting periods of dynamic repositioning − + Z procedures overlap. ziv , ziv ∈ + ∀ i ∈ N, v ∈ V (38) Two approaches are popular in literature on wiv ≥ 0 ∀i ∈ N, v ∈ V (39) dynamic repositioning. The frst divides the oper- ating period into a fnite number of time steps ( ) where f si is a penalty function for reaching and assumes perfect knowledge of time-varying an inventory level si at station i. In addition, the demand. This allows us to extend the static repo- authors also impose a constraint on the maxi- sitioning formulations to determine the number mum duration of operations assuming a fxed of cycles to be moved between stations and the loading and unloading time per bike. Note that vehicle routes at each time step. For example, the formulation assumes that bikes can be picked Ghosh et al.103 formulated a dynamic reposition- up at stations with excess and dropped off at ing model in which the goal was to reduce the places where there is a defcit, but stations can- lost customer demand. To understand their for- not be used as buffers. This assumption is also mulation, assume that N, A, and T are the set of referred to as the monotonicity condition for fll nodes, arcs, and time steps respectively and let xt levels99. The time-indexed and sequence-indexed ijv 95 be a binary variable which is 1 if a vehicle v starts formulations in further relaxed some of these to move between stations i and j at time step t. model assumptions by dividing the time avail- t Defne another binary variable χiv which captures able into smaller intervals and allowed vehicles initial conditions by taking a value 1 if vehicle v
631 J. Indian Inst. Sci.| VOL 99:4 | 621–645 December 2019 | journal.iisc.ernet.in 1 3 R. B. Nath, T. Rambha bounds elementary paths without any cycles, capacity constraints ables and fows, fow conservation, capacity constraints visiting certain ments, constraints which force stations, fow conservation, connectivity, capacity constraints total operation time, subtour elimination time limits, subtour elimination repositioning load, capacity constraints diameter of clusters time limit, subtour elimination, loading and capacity constraints unloading quantity, budget of bikes, repositioning - time limit, subtour elimination, rout capacity, ing constraints Eulerian condition, subtour elimination, capacity Constraints ensuring the existence of connected Dependency between loading instruction vari - - demand interval require Constraints to ensure vehicle routing, Flow conservation, capacity, Routing, balancing, time limits, cost constraints Flow conservation, vehicle load, station capacity, Flow conservation, subtour elimination, vehicle Inventory level constraints, constraint on the single-visit, subtour elimination Capacity, Unmet demand conditions, fow conservation, types substitution of different Loading, routing, station Flow conservation, vehicle capacity, Constraints 2-approximation algorithm 2-approximation tabu search vari - procedure, randomized adaptive search able neighbourhood search demand intervals, minimum cost network branch-and-cut, Benders decomposition fow, approach search techniques 9.5-approximation algorithm and a greedy algorithm and a greedy 9.5-approximation branch-and-cut algorithm, Integer program, iterative lookahead technique, greedy Preferred with problem static bicycle relocation 1-PDTSP, heuristic Iterated tabu search large neighbourhood Constraint programming, iterated local search Bilevel Programming, branch-and-cut MILP, clustering-based heuristic MILP, metaheuristic, local search Destroy-and-repair optimization Enhanced chemical reaction a combined hybrid genetic algorithm MILP, tabu search Hybrid large neighbourhood search, Model/solution technique tory, total travel cost tory, inventory level of bicycles after repositioning total travel time, loading and unloading tory, service times inventory and lower level minimizes routing cost inventory substitution costs depend on the fnal inventory level Total travel distance Total travel cost Total and target inven - between proposed Difference travel cost, handling costs of bicycles Total Quadratic penalty cost which is a function of the and target inven - between proposed Difference Upper level minimizes penalty function at fnal travel cost Total travel distance, penalty function at fnal Total travel cost Total unmet demand, total travel time Total travel cost including penalty and Total travel time and penalty costs which Total Objective function S M M S S S/M S S S S S S S # visits S S M S S M S S M M S S M # veh. 99 94 100 105 16 93 102 108 12 101 Summary of static rebalancing models. 106 104 107 4: Table Chemla et al. et al. Rainer-Harbach et al. Erdogan Ho and Szeto et al. Di Gaspero et al. Tang Dell’Amico et al. Forma et al. Dell’Amico et al. Szeto et al. Li et al. Ho and Szeto Benchimol et al. References
632 1 3 J. Indian Inst. Sci.| VOL 99:4 | 621–645 December 2019 | journal.iisc.ernet.in Modelling Methods for Planning and Operation - - vation constraints tion, travel time limit, excess total demand dissatisfaction constraints its, subtour elimination, vehicle capacity bounds, vehicle capacity follow inhomogeneous Poisson process dropoffs bike fow conservation capacity, vehicle capacity operational time limits vation, vehicle capacity, the depot, minimal time for moving a from vehicle between stations, fow conser vehicle and bike fow conserva - cle capacity, tion, subtour elimination stations trip length limit vehicle capacity, quantities, broken bikes, vehicle capacity quantities, broken Capacity, fow conservation, subtour elimina - Capacity, Flow conservation, inventory bounds, time lim - Time limits, subtour elimination, inventory bike demand, pickups and level, broken Target Demand level, fow balance, capacity constraints Level-of-service chance constraints, station Subtour elimination, bicycle fow conservation, bike fow conser Inventory level, vehicle routes, time of vehicles Inventory intervals, Departure tour starts and ends at the depot, vehi - Vehicle Flow conservation of bikes, capacity limits and bicycle fow balance constraints, Vehicle Routing, inventory level, loading and unloading Constraints enhanced artifcial bee colony algorithm insertion algorithm neighbourhood descent with variable neighbourhood descent search algorithm loading and unloading quantity adjustment procedure heuristic approach greedy Many-to-many pickup and delivery problem, Many-to-many pickup and delivery problem, Branch-and-cut algorithm, iterated local search and assignment model routing mTSP, minimum cost fow problem, Integer program, and randomized variable Iterated local search Stochastic optimization model hybrid nested large neighbourhood MILP, optimization, a Enhanced chemical reaction Branch-and-bound algorithm, genetic algorithm Branch-and-cut algorithm Branch-and-bound Service network design IP, set covering formulation, Integer program, MILP, geographical clustering MILP, Model/solution technique emitted from all the emitted from 2 CO travel time, loading and unloading service times time on all routes vehicles repositioning system), inaccessible locations (free-foating total unmet demand, operational time not balanced which are total (driving, parking, handling) time tory, anced vehicles Total unmet demand and maximum of total Total Minimizing the total travel cost Total travel cost Total travel cost, coeffcient of variation the Total travel cost Total cost, penalty costs for lost demand Travel time of Minimize the maximum rebalancing Inconvenience level of picking up bikes from sum of waiting times at stations Weighted and target inven - between proposed Difference fows Minimize the total cost of relocation - rebal Maximize the number of stations that are Objective function S M M S M M S/M M M S M M S # visits M M M M M S NA S/M M S M NA M # veh. 11 113 112 110 109 119 116 111 118 114 continued 115 117 120 Espegren et al. Espegren Wang and Szeto Wang Angeloudis et al. et al. Alvarez-Valdes Cruz et al. Nair et al. Pal and Zhang Liu et al. Kadri et al. Vogel O’Mahony and Shmoys Bulhoes et al. Szeto and Shui Table 4: References a station not explicitly modelled). Multiple visits imply that each vehicle is allowed to revisit S single, M multiple, NA not applicable (i.e., vehicle movements are
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is at station i at time t = 0 and is 0 for all other are assumed to be proportional to the demand to times. That is, vehicles are not required to be pre- those stations. sent at the depot at the start of the rebalancing Mathematically, this is modelled using (44).
procedure. Additionally, it is assumed that vehi- tt′ ′ d cles can travel between a pair of stations within tt ≤ t ij ∀ ∈ , ∈ , ∈ , ′ ∈ uij ri tt′ i N j N t T t T one-time step. This assumption is reasonable if dik the duration of each time step is large. If not, the k∈N (44) underlying graph can be modifed by creating dummy nodes and arcs. Just like the static case, The actual fow of bicycles between the stations fow conservation constraints (28) and (29) can must also be less than or equal to the demand. equivalently be written as Further, for each station i, the inventory level must not exceed the station capacity Ci . These t − t−1 = χ t ∀ ∈ ∈ ∈ xijv xhiv iv i N, v V , t T conditions are implied in constraints (45) and ∈ j N h ∈N (46). (40) ′ ′ 0 ≤ utt ≤ dtt ∀ i ∈ N, j ∈ N, t ∈ T, t′ ∈ T xt ≤ 1 ∀ i ∈ N, t ∈ T ij ij ijv (41) ∈ ∈ (45) j N v V Constraint (40) equates the number of vehi- 0 ≤ rt ≤ C ∀i ∈ N, t ∈ T cles coming into station i to the number going i i (46) out of i. Inequality (41) restricts the number of As before, let Qv denote the capacity of vehicle v. vehicles that can be present at a station to avoid A vehicle can be loaded or unloaded at a station overcrowding. only when present at that station. These observa- Extending other notation in a similar man- tions are ensured using (47) and (48). t ner, let ri be the inventory level of bikes at station +t −t t +t −t ziv + ziv ≤ Qv xijv ∀ i ∈ N, t ∈ T, v ∈ V i at time t and let ziv and ziv be the number of ∈ bicycles added and removed by a vehicle v at sta- j N tt′ tion i at time t respectively. Denote using uij , the (47) number of bicycles trips made by customers from 0 ≤ z+t , z−t , yt ≤ Q ∀ i ∈ N, t ∈ T, v ∈ V station i at time step t to station j at time t + t′ . iv iv v v (Customers take different times to travel between (48) tt′ stations, but vehicles are assumed to take one Let bij be the revenue generated from one bicy- time step.) Flow conservation of bicycles can thus cle trip that departs from station i at t and reaches ′ be expressed as station j at time t + t and cij be the vehicle cost of traversing (i, j). With these constraints, objective t t−t′,t′ tt′ ri + uhi − uij (49) is maximized to improve the overall proft ′ ′ ∈ t
t+1 t −t +t yv = yv + ziv − ziv ∀ t ∈ T, v ∈ V ∈ i N (43) Let the demand at time t for travelling between ′ tt′ station i and station j in t time steps be dij . The actual number of customer trips starting from a station at each time step should not exceed the number of bicycles present in the station at that time. When the demand at a station is greater Figure 8: Rolling horizon method for dynamic bicycle repositioning. (Source: Zhang et al.15). than its supply, bounds on rentals to destinations
634 1 3 J. Indian Inst. Sci.| VOL 99:4 | 621–645 December 2019 | journal.iisc.ernet.in Modelling Methods for Planning and Operation unloading constraints constraints tion capacity constraints fow balance, state equation describing incentives and customer behaviour completed in a single time period, proportionality constraints limiting the number of bikes that can one station to another be moved from defcit and excess stations but not both; aggregate or a donor, a receiver fow balance constraints cluster-level tory level, vehicle and bicycle fow conservation, vehicle capacity bounds, travel time limits, vehicle capacity vehicle, subtour elimination Route starts and ends at the depot, loading Constraints Vehicle capacity, fow conservation, vehicle usage capacity, Vehicle and bicycle fow balance, vehicle sta - Vehicle in time-expanded graph, bicycle Feasible routes time periods, rides are Flow conservation across Dock capacity and vehicle constraints, Distance limits for feasible matching between Constraints ensuring that each cluster is either Unsatisfed demand of bikes and docks, inven - Time-varying demand functions, vehicle capacity and bicycle fow balance, level of service Vehicle stations for each Bicycle and fow balance, clustered - colony metaheuristic, route truncation heuristic, colony metaheuristic, route genetic algorithm Benders decomposition non-linear multi-commodity time-space network horizon heuristic rolling fow, dynamic price incentives using model routing, control predictive bounds linear programming using artifcial neural network and fuzzy logic bound algorithm neural network, two TSP model autoregressive frst collected and then distributed) (bikes are horizon framework, Benders algorithm, rolling decomposition randomized adaptive search greedy hood search, machines, chance constrained model for inven - model tory balancing, vehicle routing TSP heuristic nearest-neighbour Rolling horizon approach, enhanced artifcial bee Rolling horizon approach, Model/solution technique Dantzig-Wolfe decomposition, column generation, Dantzig-Wolfe Non-homogeneous Poisson pickups and dropoffs, Receding horizon, mixed integer quadratic vehicle deterministic Stochastic network fow problem, demand forecasting Non-linear integer program, branch-and- k -center problem, Integer program, foating system, clustering methods, non-linear Free greedy Deterministic time-varying demand, MILP, construction heuristic, variable neighbour Greedy using gradient boosting Demand forecasting to model station occupancy, Birth-death processes emis - 2 CO sion cost docks, total cost operating costs based on availability of bikes and docks stations across lost demand, travel cost of a given threshold, vehicles inventory level of visiting station(s) with defcits cost rebalancing Objective function Total unmet demand Total User dissatisfaction due to absence of bikes or unmet demand, fuel consumption, Total Number of extra trips possible due to redistribution, Maximize expected number of successful trips Redistribution costs, lost user cost, satisfaction Maximize a weighted distance matching objective Duration for which bike inventory level falls below travel cost, total unmet demand Total the desired from unmet demand, difference Total utility Minimize number of bikes to be repositioned, Minimize the duration of out-of-stock events, # veh. M M S M NA S NA S S M M M 11 130 129 14 123 122 131 127 15 Summary of dynamic rebalancing models. 125 126 128 manelli 5: Table Pfrommer et al. Pfrommer Contrado et al. References not explicitly modelled) S single, M multiple, NA not applicable (i.e., vehicle movements are Zhang et al. Shui and Szeto Shu et al. Caggiani and Otto - O’Mahony and Shmoys Caggiani et al. Wang Kloimüllner et al. Regue and Recker Chiariotti et al.
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The MILP model is NP-hard and hence does obtained from updated demand forecasts for the not scale well with the problem size. To tackle interval 12:00 to 15:00 and the process continues this issue, Ghosh et al.103 proposed a Lagrangian till the end of the time horizon. This method has Dual Decomposition (LDD) approach in which a greater practical applicability since it can react the original problem is decomposed into a mas- to current conditions by adjusting the initial con- ter problem and two slaves (one for repositioning ditions for each roll period. and the other for routing). A few other models which addresses the Since the repositioning variables z and the dynamic rebalancing problem are summarized in routing variables x in constraint (47) are coupled, Table 5. t it is relaxed by introducing dual variables αiv . The Lagrangian function L(α) can thus be written as 4.3 Offine and Online Repositioning Some authors have also classifed repositioning min αt z+t + z−t z iv iv iv activities as offine and online methods. Offine i∈N v∈V t∈T algorithms assume perfect knowledge of input ′ ′ − btt utt data and do not react to changing system states. ij ij (50) Hence, they can be both static and dynamic. (i,j)∈A t∈T t′>t When applied in a dynamic setting (see 123–125,103 t t for example), one can view offine methods as + min (cij − Qvα )x x iv ijv (i,j)∈A v∈V t∈T open-loop control measures. They are suitable in situations with stable demand patterns. How- The frst component of (50) only involves repo- ever, if the demand exhibits high variance or if sitioning and the second component is related to there is supply-side uncertainty due to traffc, vehicle routing. For a given α , these slaves are sep- weather, broken bikes, etc., the recommended arately solved and the α vector is updated using solutions may be infeasible since re-optimiza- a sub-gradient descent method for the master tion is not done in such methods. It can, how- problem max L(α) . To speed up computation, α≥0 ever, be used to compute value-of-information an additional clustering approach was used to benchmarks by determining how well the system create abstract stations and the proposed method could be operated in retrospect, using data on the was tested on CaBi and Hubway data sets. Com- events that occurred. In that way, dynamic offine parison with other benchmark solutions showed algorithms are still useful compared to static a reduction in lost demand. repositioning methods. The formulation discussed so far was extended Online methods on the other hand can react to stochastic demand settings using a robust opti- to the current inventory level and potentially mization approach.121 In this framework, the BSS other external factors such as the day of the week, operator and the users/environment were viewed temperature, and precipitation.132 Most online as players in a two-player game. At each iteration, methods in the literature are posed using a roll- the environment generates a demand scenario ing horizon14,122 or a Markov decision process which maximizes the lost demand considering (MDP) and reinforcement learning (RL) frame- the repositioning strategy of the operator. The work.133,134 MDPs prescribe the sequence of operator reacts by proposing a new repositioning actions to be taken at different system states by strategy that minimizes the lost demand consid- considering the rewards/costs incurred for vari- ering the worst demand scenario presented by the ous state-action pairs and the stochastic nature of environment and the process is continued until transitions between states after an action is taken. both objectives converge. In the context of bike repositioning, states typi- The second popular approach for dynamic cally comprise of inventory levels and locations repositioning is to use rolling horizon models in of repositioning vehicles and their contents. State which the overall problem is broken down into transitions may occur when customers pick up or multiple dynamic rebalancing problems. The drop off bicycles or when vehicles remove or add observed demand in each time interval is used to cycles to stations. update forecasts for the next interval and rebal- Transition probabilities depend on the arrival ancing decisions are recomputed.15,122 For exam- processes of customers and the time it takes for ple, in the set up shown in Fig. 8, using forecasts vehicles and cycles to move between stations. of demand between 10:00 and 13:00, a reposition- Owing to large state and action spaces, the opti- ing strategy is frst constructed for the roll period mal policies to these problems are solved using which, for time period 1, begins at 10:00 and ends RL, particularly off-policy RL methods. In these at 12:00. At 12:00, a new repositioning strategy is
636 1 3 J. Indian Inst. Sci.| VOL 99:4 | 621–645 December 2019 | journal.iisc.ernet.in Modelling Methods for Planning and Operation methods, the optimal policy is learnt using a sim- 4.4 Incentivizing Users ulator which generates demand data after train- Apart from using vehicles and bike-trailers to ing it on a real-world dataset. This procedure is rebalance a BSS, operators can provide incentives done offine (not to be confused with the earlier to customers and infuence them to pick up or description of offine repositioning methods) and drop off bikes at desired stations to avoid stations a near-optimal policy is generated in the form of from becoming empty or full. Incentive design a look-up table that prescribes the action to take may be ideal if it is cheaper than deploying repo- in each state. Using this policy, one could imple- sitioning vehicles but is relatively diffcult since ment the suggested actions, in the feld, in an user behaviour can be unpredictable. Researchers online manner. have presented different models to address this An MDP model proposed by Legros135 problem. attempted to minimize the long-run rate of A deep RL framework was proposed in137 to 1 2 unmet demand. Suppose that Tit and Tit represent rebalance dockless BSSs. The problem was mod- the expected arrival rate of customers who are elled as an MDP in which the actions at each not able to rent and return a bike at station i up time step are the prices for renting bikes in dif- 1 2 to time t respectively. Also let ci and ci be the unit ferent regions of the network. A policy gradient costs incurred by the operator due to the non- approach was used to develop a novel hierarchi- availability of bikes and docks at station i respec- cal reinforcement pricing (HRP) algorithm, the tively. Then, the objective was written as objective of which was to maximize the total number of satisfed customers with a limited rebalancing budget. Experiments for HRP were min lim c1T 1 + c2T 2 (51) t→∞ i it i it conducted based on datasets from Mobike. i∈N A two-choice model and a mean-feld approx- The time-varying nature of arrival processes was imation was proposed in138 for incentivizing modelled by dividing the time horizon into inter- users to rebalance a homogeneous BSS in which vals within which the parameters of the random unmet pickup demands are assumed lost. Users processes could be assumed constant. Next, a are requested to provide two nearest destination rebalancing problem for a single station was cast stations and they are incentivized to return their as an average cost MDP and was extended to the bicycles to the station with lower inventory. It was multi-station case using approximate relative found that this incentivizing scheme improved value functions and policy improvement steps. the redistribution rate signifcantly, even when a A spatio-temporal RL approach was used in133 small fraction of users complied. for online repositioning of bikes in a BSS with an Another approach was developed by Pfrom- objective to minimize lost demand. To reduce the mer et al.14 to rebalance a BSS using vehicle-based problem complexity, a clustering algorithm was redistribution and user-based price incentives. used to group stations and multiple trikes (repo- Their model predictive control algorithm com- sitioning tricycles which typically carry 3–4 bikes) puted dynamic rewards depending on the current were used within each cluster. A deep neural net- and predicted future system states to optimize work was used to learn the optimal value func- the operating costs while ensuring a desired ser- tions and the model was tested on real-world Citi vice quality. A Monte Carlo model was formu- Bike data. Another MDP model was proposed lated using historical data from the London Cycle in134 to solve the dynamic repositioning problem Hiring and results showed that on weekends, the with a similar objective. A coordinated lookahead incentive scheme alone could improve the service policy heuristic was used to address the curse of level. On weekdays, however, price incentives were Curse of dimensionality: is a dimensionality. The resulting policy was tested on found to be insuffcient for achieving the desired term coined by Richard Bell- man to describe the complex- data sets from BSSs in Minneapolis and San Fran- service level, especially during rush-hours. ity of dynamic programs that cisco, US and was shown to perform better than A bilevel optimization formulation was pre- result from high-dimensional benchmark policies in reducing lost demand. sented in139 where link-level incentives/prices state, action, and disturbance spaces. Online problems have also been formulated are decided at the upper level to minimize the as multi-stage stochastic programs.136 This model number of imbalanced stations. The lower level extends103 by considering demand scenarios assigns users to routes and destinations assum- drawn from known distributions that are con- ing that they take the minimum cost paths. The structed from data. They proposed a sample aver- proposed method attempted to create hubs in age approximation which was solved using a LDD the system through which most of the demand is method and a greedy online anticipatory heuris- routed and ensured that only a small number of tic on CaBi and Hubway problem instances. vehicles are deployed for further repositioning. A
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heuristic approach named iterative price adjust- Use of E-bikes in a BSS brings with it a new set ment scheme was used to solve the problem. of problems. First, the demand for E-bikes may be A different kind of incentive mechanism very different from that of traditional BSSs since design problem was proposed in140 where reposi- factors such as age, gender, trip purposes, and tioning activity was crowd sourced. Their model destinations infuence the adoption of these sys- frst determines all repositioning tasks and inter- tems. When compared with regular bikes, E-bikes ested customers could bid for carrying out these may also attract a signifcant portion of travel- tasks using bike trailers. Instead of bids,13 pro- lers using other motorized forms of transport. posed a dynamic incentive scheme in which the Demand forecasts for E-bikes can be obtained system offers its users incentives to change their using stated preference surveys143 and other pickup or dropoff location using a fnite set of methods as explained in Sect. 3. For instance,144 possible prices (subject to an overall budget con- use a multinomial logit model on data collected straint) and observes binary acceptance/rejection from a survey in Beijing to analyse the impact of decisions. An online learning mechanism varies socio-demographic factors, environmental condi- these prices across time and customers and using tions, and transit supply on E-bike usage. their acceptance/rejection decisions, a cost curve Second, E-bike batteries need to be recharged F(p) representing the probability of accepting an which, depending on the vehicle design, can be alternate station when offered a price incentive done at the stations or using solar energy.145 p is discovered. The proposed mechanism was Hence, there are other dimensions to station loca- deployed for one month on a real-world BSS, tion such as connection with the grid and the MVGmeinRad, in Europe. Rental requests were amount of daylight received. Stochastic demand made on a smart phone app with information on results in fuctuations in charging patterns and intended pickup and dropoff stations. About 60 this needs to be considered when designing a percent of the offered incentives were accepted by low voltage grid network of bike stations with users during the pilot implementation. recharging capabilities.146 E-bikes can also be recharged by swapping batteries.147 The move- ment of charged batteries and the swapping 5 Technological Aspects activities can also be modelled as a logistical BSSs are going through a transformative phase optimization problem. For instance,148 generated in which technological advancements to improve different demand scenarios using Poisson distri- existing systems are constantly being tested and butions and determined the number of E-bikes deployed. For instance, a study by Woodcock and batteries to be placed at different stations et al.141 uses secondary data sources to estimate using Monte Carlo simulations. A pilot experi- the disabled-life adjusted years (DALY) of BSS ment was also run at the University of Tennessee, users in London by considering levels of air pol- Knoxville campus, where E-bikes powered by Li- lution and traffc injuries. With modern day tech- ion batteries were deployed. nology, it is possible to track bicycle activity of registered members and the health impacts can be more accurately captured and relayed to users 5.2 Locking Mechanisms via their apps in real-time. In this section, we Cycles in a BSS are prone to theft which neces- examine potential future improvements to BSSs sitates the use of foolproof locking mechanisms. and discuss related research issues. Most BSS operators provide keyless locks on their bicycles. For example, Ofo used a number lock system in the early stages and later adopted a bar/ 5.1 Electric Bikes QR code. Mobike also uses a smart lock mecha- Electric bicycles that use rechargeable batteries nism which can be unlocked using a mobile and a motor to assist pedalling have the potential app.149 Even though GPRS-based smart locks to replace traditional bikes of a BSS. These offer are in use, network connectivity issues are not greater ease of cycling especially in cities with uncommon. To address this problem, a narrow uneven terrains and can support long-distance band Internet of Things (NBIoT)-based smart trips. In January 2018, Limebike (currently known bike locks are being developed.150 Other options as Lime) unveiled a pedal-assisted electric bicy- include bluetooth low energy (BLE) powered cle142 which was made operational in cities like smart locks151 and electromechanical locking sys- Seattle, Miami, and San Francisco in the US. Cur- tem for E-bikes (which can also verify if the bicy- rently, E-bikes of Lime and JumpBike are opera- cle was returned to a dock).145 tional in many cities across the globe.
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5.3 Impact of Pricing Schemes slots at stations could also malfunction. These One of the major challenges of BSS operators is to problems warrant periodic maintenance of bicy- draw more customers to use their services. User cles and the BSS infrastructure. Operators often satisfaction is not only dependent on the spatial allow users to report issues with their rented bicy- location of stations and the availability of bikes cles using mobile apps. This information can be or empty docks, but also on the pricing scheme. used to deploy maintenance vehicles and crew to Low rental fares can increase ridership but also repair faulty cycles at bike stations or to take bro- reduces revenue. Revenue also decreases when the ken bikes to dedicated repair stations. Decisions cost of rentals is high since the demand for bike- support tools for locating repair stations and sharing will drop in such situations. In this con- routing of maintenance crew can be built using text, some studies have focused on understanding optimization models and these operations can be the optimal pricing policy and the sensitivity of combined with repositioning.156 BSS operators users to BSS pricing. must, from time to time, perform a cost beneft In June 2016, CaBi introduced a single-trip analysis to decide if bicycles must be discarded or fare (STF) scheme to allow customers to take repaired and reintroduced into the feet.155 Some a trip up to 30 min for $2. The timing of this of the latest bicycles are equipped with tyreless scheme coincided with a SafeTrack metro rail tubes, disk brakes, and chainless drive shaft all of maintenance program because of which an which can drastically reduce the frequency and increase in bike rentals was anticipated and the extent of maintenance required for the upkeep of number of docks was increased by about 23%. a BSS. Before STF, CaBi also had a 24-h pass and a 3-day pass, priced at $8 and $17 respectively, for unlim- 6 Conclusions ited trips less than 30 min. It also had monthly The growth of bike sharing systems has spurred and annual subscription passes that cost $28 and considerable research, especially in the last dec- $85. The presence of different options allowed152 ade, on problems related to its planning and to study the impact of the price differences on operations. BSSs have the potential to transform the ridership across price classes. They found into a competitive transportation mode in many that rentals by casual users rose to about 79% per cities around the world. It has a positive effect on dock but there was not much change in the rid- the environment and the health of individuals ership of those with monthly and annual passes. and can also serve as a cost-effective intermediate It was also found that there was a decrease in the public transportation mode to address last- and revenue generated from users having a 24-h pass frst-mile issues that plague most transit systems. and a 3-day pass, indicating that some of them In this paper, we reviewed the history of BSSs and started to shift to the STF scheme. literature on various mathematical models that The price sensitivities were further analysed can help planners and operators design, improve, in153 where STF was decreased from $2 to $1.50 and optimize new or existing BSSs. We also briefy and annual membership changed from $85 to examined the effects of pricing schemes, techno- $73. This new pricing scheme improved both rev- logical aspects such as E-bikes, and the mainte- enue and ridership. Further analysis was made nance challenges posed by the broken bicycles. using an ordered logit regression model which Specifcally, we examined literature on stra- suggested that low-income groups were relatively tegic and operational planning models. Strate- more sensitive to price changes and women were gic planning involves forecasting the demand about 30% more price sensitive than men in the for BSSs, designing stations and bike paths, and case of the STF pricing scheme. determining dock capacity. Potential directions Although changes to pricing structure in BSSs for future research on route design must consider are usually infrequent, such opportunities can the effects of elastic demand induced by supply- be put to good use to infer the effect of prices on side changes, automobile congestion on route revenue and ridership. choices of travellers, multi-modal trip making behaviour and transit connectivity, and socio- 5.4 Periodic Bike Maintenance economic characteristics of demand between dif- Another crucial problem in the operation of a ferent zones. BSS is to identify broken or faulty bicycles that When designing station locations and capaci- need repair.154 Bicycles typically face issues with ties, most studies assume knowledge of demand tyre punctures, broken chains, and braking sys- which lacks spatio-temporal complexity. Diurnal tems.155 In addition, GPS devices, locks, and dock patterns of travel are known to cause a reversal
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of origins and destinations between the morning 3. Borgnat P, Abry P, Flandrin P, Rouquier J-B (2009) and evening peak periods. While these effects have Studying Lyon’s Vélo’v: a statistical cyclic model. In: been widely studied in the context of rebalancing, Proceedings of the European conference on complex they do play a major role in station location and systems (ECCS). Complex System Society. [Online]. capacity allocation as well. Stylized versions of spa- http://prunel.ccsd.cnrs.fr/ensl-00408 147/en/ tio-temporal variation in demand, time to travel 4. Borgnat P, Abry P, Flandrin P, Robardet C, Rouquier between stations, and relocation strategies must be J-B, Fleury E (2011) Shared bicycles in a city: a signal used to make these decisions at a strategic level. processing and data analysis perspective. Adv Complex Our synthesis of literature on operational Syst 14(03):415–438 aspects of BSSs predominantly included static and 5. Martinez LM, Caetano L, Eiró T, Cruz F (2012) An dynamic repositioning. Static repositioning mod- optimisation algorithm to establish the location of sta- els assume that bikes are redistributed at night tions of a mixed feet biking system: an application to when bike usage is negligible. Dynamic reposi- the city of Lisbon. Procedia-Soc Behav Sci 54:513–524 tioning operations on the other hand are carried 6. Lin J-R, Yang T-H, Chang Y-C (2013) A hub location out during the day when the system is in use. inventory model for bicycle sharing system design: for- Models for rebalancing consider single or multi- mulation and solution. Comput Ind Eng 65(1):77–86 ple vehicles/bike-trailers which can make single or 7. Vogel P, Mattfeld DC (2011) Strategic and operational multiple visits to stations and can also feature user planning of bike-sharing systems by data mining—a incentives. Many of these formulations were tested case study. In: International conference on computa- on real-world data sets and were found to improve tional logistics. Springer, pp 127–141 the operational effciency when compared to a 8. Saharidis G, Fragkogios A, Zygouri E (2014) A multi- do-nothing policy. However, supply-demand periodic optimization modeling approach for the estab- interactions are modelled only to a limited extent lishment of a bike sharing network: a case study of the in existing literature. Although, econometric and city of Athens. Proc Int MultiConf Eng Comput Sci machine learning models have been found to 2(2210):1226–1231 uncover infuential factors and have good pre- 9. Romero JP, Ibeas A, Moura JL, Benavente J, Alonso B dictive power for short- and long-term demand, (2012) A simulation-optimization approach to design embedding them within optimization frameworks effcient systems of bike-sharing. Procedia-Soc Behav for managing supply is an uphill task. A right bal- Sci 54:646–655 ance between predictive demand models and sup- 10. Garcia-Gutierrez J, Romero-Torres J, Gaytan-Iniestra J ply optimization is much needed for data-driven (2014) Dimensioning of a bike sharing system (BSS): tools that can practically be used for BSSs. a study case in Nezahualcoyotl, Mexico. Procedia-Soc Behav Sci 162:253–262 11. O’Mahony E, Shmoys DB (2015) Data analysis and Publisher’s Note optimization for (Citi) bike sharing. In: Twenty-Ninth Springer Nature remains neutral with regard to AAAI conference on artifcial intelligence jurisdictional claims in published maps and insti- 12. Chemla D, Meunier F, Calvo RW (2013) Bike sharing tutional affliations. systems: solving the static rebalancing problem. Dis- crete Optim 10(2):120–146 Acknowledgements 13. Singla A, Santoni M, Bartók G, Mukerji P, Meenen M, The authors thank IMPacting Research, INnova- Krause A (2015) Incentivizing users for balancing bike tion and Technology (IMPRINT), Department sharing systems. In: Twenty-Ninth AAAI conference on of Science and Technology, India (Project no. artifcial intelligence IMP/2018/001850) for supporting this study. 14. Pfrommer J, Warrington J, Schildbach G, Morari M (2014) Dynamic vehicle redistribution and online price Received: 21 September 2019 Accepted: 9 October 2019 incentives in shared mobility systems. IEEE Trans Intell Published online: 30 November 2019 Transp Syst 15(4):1567–1578 15. Zhang D, Yu C, Desai J, Lau H, Srivathsan S (2017) A time-space network fow approach to dynamic reposi- References tioning in bicycle sharing systems. Transp Res Part B: 1. Webster T (2014) Capital Bikeshare DC. [Online]. Methodol 103:188–207 https://www.fickr.com/photos/diversey/14326445924. 16. Benchimol M, Benchimol P, Chappert B, De La Taille Accessed 16 July 2019 A, Laroche F, Meunier F, Robinet L (2011) Balancing 2. Shankar S (2017) Environmentally friendly public the stations of a self service “bike hire” system. RAIRO- transport. [Online]. https://www.fick r.com/photo s/ Oper Res 45(1):37–61 shankaronl ine/35607 26063 4 . Accessed 17 July 2019
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Rito Brata Nath is currently an M.Tech. (CiSTUP). He received his Ph.D. from the University of (Research) student in the Transportation Texas at Austin where he worked on network equilibrium, Systems Engineering division of the congestion pricing, and adaptive routing in stochastic tran- Department of Civil Engineering, Indian sit and traffc networks. Prior to joining IISc, he was a post- Institute of Science (IISc), Bangalore. He doctoral researcher at Cornell University where he studied received his bachelor’s degree in Civil Engi- hospital evacuations and demand estimation during hurri- neering from Jadavpur University, Kolkata. His current canes. His research interests include network optimization, research is on bike-sharing systems with a focus on static shared ride and transit systems, and real-time control of and dynamic repositioning and vehicle routing. traffc.
Tarun Rambha is an assistant professor in Civil Engineering at the Indian Institute of Science (IISc) Bangalore and an affliate fac- ulty at the Center for infrastructure, Trans- portation and Sustainable Urban Planning
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