1

Autonomous Vehicle Public Transportation System:Scheduling and Admission Control

Albert Y.S. Lam, Yiu-Wing Leung, and Xiaowen Chu

Abstract—Technology of autonomous vehicles (AVs) is gettingmature and many AVs will appear on the roads in the near future.AVs become connected with the support of various vehicularcommunication technologies and they possess high degree ofcontrol to respond to instantaneous situations cooperatively withhigh efficiency and flexibility. In this paper, we propose a newpublic transportation system based on AVs. It manages a fleetof AVs to accommodate transportation requests, offering point-to-point services with ride sharing. We focus on the two majorproblems of the system: scheduling and admission control. Theformer is to configure the most economical schedules and routesfor the AVs to satisfy the admissible requests while the latter isto determine the set of admissible requests among all requests toproduce maximum profit. The scheduling problem is formulatedas a mixed-integer linear program and the admission controlproblem is cast as a bilevel optimization, which embeds thescheduling problem as the major constraint. By utilizing theanalytical properties of the problem, we develop an effectivegenetic-algorithm-based method to tackle the admission controlproblem. We validate the performance of the algorithm withreal-world transportation service data.

Index Terms—Autonomous vehicle, admission control, bileveloptimization, smart city.

I. INTRODUCTION

HUMAN mobility is largely supported by public transport.Many people rely on public transport to move from

one place to another when the destinations of their journeysare not within walkable distances. To transform a city withlimited room for large-scale infrastructure into a smart city, itspublic transportation system may need to be further upgradedmainly from the existing road networks. Representatives ofroad-based public transport are buses and taxis, each typeof which has its pros and cons. In general, buses followfixed routes offering shared ride so that more passengers canbe served on each single journey. On the other hand, taxisoffer private services and run on flexible dedicated routesbased on the passengers’ requests. Nevertheless, no singleone type can support high throughput and flexibility at thesame time. The efficiency and capacity of the whole publictransportation system may be enhanced if there exists a newpublic transport which can accommodate many people in ashort period of time and concur high mobility. It may maintainflexibility by offering point-to-point services while enhancing

A preliminary version of this paper was presented in [1].A.Y.S. Lam is with the Department of Electrical and Electronic Engi-

neering, The University of Hong Kong, Pokfulam, Hong Kong (e-mail:ayslam@eee.hku.hk).

Y.-W Leung and X. Chu are with the Department of Computer Science,Hong Kong Baptist University, Kowloon Tong, Hong Kong (e-mail: {ywleung,chxw}@comp.hkbu.edu.hk).

efficiency by supporting shared ride. Such kind of publictransport requires several characteristics which may not bepossessed by a typical public transport. To develop such apublic transport, the vehicles need to cooperate to take upcustomers’ requests instead of cruising around the city forrandom offers. To enhance the efficiency and cooperativeness,a control center can be employed to coordinate all the vehicles,manage all the service requests, and assign the vehicles toserve the requests. Moreover, the vehicles should follow theroutes and carry out the travel plans instructed so as to achievesystem-wise objectives. Recently, autonomous vehicles (AVs)have been undergone active research and we can expectmany AVs running on the roads in the near future. TheAV is a good candidate possessing most of the requirementsmentioned above. Hence AVs can be adopted to construct anew smart public transportation system with high efficiencyand flexibility.

In this paper, we introduce an intelligent AV-based publictransportation system. It manages a fleet of AVs to accommo-date transportation requests, offering point-to-point serviceswith ride sharing. We focus on two important problems inthe system: scheduling and admission control. The former isabout how to assign the designated vehicles to the admissibletransportation requests, and when and where the vehiclesshould reach to provide services with the lowest cost. Thelatter is to determine the set of admissible requests amongall requests to achieve maximum revenue. As a whole, thecontributions of this paper include:

• proposing the AV public transportation system;• improving the model for scheduling proposed in [1], such

that the formulation developed in this paper can nowsupport both directed and undirected graphs;

• developing distributed scheduling;• formulating the admission control problem;• introducing the concept of admissibility and the related

analytical results;• designing an effective method to solve the admission

control problem; and• validating the performance of the solution method with

real-world transportation service data.

The rest of this paper is organized as follows. Related work isgiven in Section II and we present various system componentsand their operations in Section III. The scheduling problemis discussed in Section IV. In Section V, we formulate theadmission control problem and provide the related analyticalresults. We propose a genetic-algorithm-based solution methodfor admission control and develop distributed scheduling in

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Section VI. Section VII evaluates the system performance withreal-world transportation service data. Finally we conclude thispaper in Section VIII.

II. RELATED WORK

The concept of AVs was raised in the 1920’s and theresearch thereof has started for more than thirty years. AnAV is equipped with many sensors, which provide the vehiclewith full sensing ability so as to adapt to the neighborhoodenvironment and realize fully automated control. In 2007,the DARPA Urban Challenge boosted the awareness of AVscapable of being driven in traffic and performing complexmaneuvers [2]. In 2010, VisLab carried out the experimentthat several driverless vehicles successfully traveled 13,000 kmfrom Italy to China [3]. Google demonstrated an AV prototypein 2011 [4]. By the end of 2013, several states in the UnitedStates, including Nevada, Florida, California, and Michigan,had passed the law to allow AVs running on public roads[5]. The first self-driving shuttle on sale was from NAVIA[6]. Other automotive manufacturers, like Mercedes-Benz [7],BMW, and Audi [8], have invested in self-driving technologiesand include AVs in their production plans.

Most research work on AVs mainly focused on the con-trol and communication aspects. Mladenovic and Abbas [9]proposed a self-organizing and cooperative control frameworkfor distributed vehicle intelligence. Hu et al. [10] studied laneassignment strategies for connected AVs and proposed a lanechanging maneuver to balance the tradeoff between efficiencyand safety. Petrov and Nashashibi [11] developed a feedbackcontroller for autonomous overtaking without utilizing road-way marking and inter-vehicle communication. Li et al. [12]presented a multi-level fusion-based road detection system fordriverless vehicle navigation to ensure safety in various roadconditions. All these show that AV is a promising technologywith the support from governments, high-tech companies, andcar manufacturers.

Vehicles can communicate with each other and fixed infras-tructure via various vehicular wireless communication tech-niques [13]. Nowadays vehicular communications are mostlydeployed over satellite, cellular networks, and vehicular ad-hoc networks (VANETs) [13]. VANET is a mobile ad-hocnetwork where vehicles act as the mobile nodes [14] and itcan improve the communication capacity and organization ofAVs constituting an intelligent transportation system. Furda etal. [15] introduced a wireless communication framework fordriverless vehicles. It facilitated vehicle-to-vehicle and vehicle-to-infrastructure communications and improved the safety andefficiency of vehicles. Alsabaan et al. [16] made use of trafficlight signals and vehicle-to-vehicle (V2V) communicationsto help vehicles adapt their speeds and avoid unnecessarystop, acceleration, and excessive speed. Gomes et al. [17]designed a driver-assistance system which allowed a vehicleto collect real-time camera images from other vehicles in theneighborhood over V2V communications. In this way, AVsbecome connected and can communicate with the controlcenter.

Shareability of taxi services has been studied recently.Santi et al. [18] investigated the tradeoff between passenger

TABLE ICONTRIBUTIONS TO THE SYSTEM.

Technology/Example Contributions Ref.

feature

Hardware

VisLab Demonstrate the feasibilityof AVs

[3]

Google Show the confidence of theindustry in AVs

[4]

Mercedes-Benz, BMW,Audi, NAVIA

Guarantee supply of AVsfor the system

[7], [8], [6]

Software

Mladenovic &Abbas

Enhance self-organizing andcooperative control of AVs

[9]

Hu et al. Balance the efficiency andsafety of AVs

[10]

Petrov &Nashashibi

Enhance self-control of AVs [11]

Li et al. Improve safety of AVs [12]

LawNevada,Florida,California,and Michigan

Demonstrate the support ofgovernments

[5]

Cottingham Introduce the vehicularwireless communicationsavailable to be used in thesystem

[13]

Dahiya &Chauhan

Improve the communicationcapacity and organization ofAVs

[14]

CommunicationsFurda et al. Enhance the communica-tions between AVs and thecontrol center

[15]

Alsabaan et al. Improve the comfort of AVs [16]Gomes et al. Collect data for the system

to estimate traffic conditions[17]

RidesharingSanti et al. Confirm the ridesharing

functionality of the system[18]

Ma et al. [19]AV publictransporta-tionsystem

Lam et al. Provide a proof of concept [1]Lam et al. Investigate the scheduling

and admission control prob-lems

This work

inconvenience and collective benefits of sharing and concludedthat a small increase in discomfort could induce the significantbenefits of less congestion, less running costs, less split fares,less polluted, and cleaner environment. Ma et al. proposeda taxi ridesharing system called T-Share in [19], where thedynamic taxi ridesharing problem was studied. For a datasetof taxi services in Beijing, it showed that 25% additionaltaxi users could be served with saving of 13% of total traveldistance. These studies confirmed that ridesharing is beneficialbut they mostly focused on taxi services. In this paper, wefocus on AVs, which have a key intrinsic property hardlyfound in the standard taxis: the direct control of vehiclesdoes not involve any human factors. In other words, AVs cancompletely follow the instructions from the control center inthe sense that they neither undertake any unassigned requestsnor reject any assigned requests. We can see that AVs canfully cooperate to achieve the system objective but it may notbe the case for human-driving taxis.

The AV public transportation system is uniquely designedand it can help improve the capacity and flexibility of the

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future transportation system. To further demonstrate its feasi-bility, we show how the existing work discussed above maycontribute to the system in Table I.

The scheduling problem has been introduced in [1] and itcan be considered as a variant of the Dial-A-Ride Problem(DARP) [20]. However, in our AV scheduling problem, weallow modifying the previously assigned but not yet servedrequests at desirable times to achieve system-wise performancegoal. When the system evolves, the AVs appear at differentlocations at different time instants. It may happen that aparticular request can be better served by a different AV atdifferent times. Consider an example with two AVs, I andII. At a paricular time, AV-I is in the neighborhood of alocation while AV-II is not. A request originated from thislocation may be better served by AV-I. After some time,AV-I may have gone away but AV-II may have come intothe neighborhood. Then the request may be better served byAV-II instead. As the AVs are connected through appropriatevehicular communication technologies, the schedules of AVscan be revised from time to time. We consider this in ourformulation making our scheduling problem different fromDARP. As the system involves a number of AVs, determiningtheir schedules in a distributed manner can undoubtedly speedup the process. Distributed scheduling has been advanced inmany engineering disciplines, e.g., communication networks[21], [22]. As a new system, we will dedicatedly design adistributed methodology for the scheduling thereof.

Admission control generally refers to a validation processin communication systems for quality-of-service assurance. Itdetermines which new connection or service request can begranted with resources for subsequent operations. For example,[23] designed an admission control mehanism to add or dropsession requests in 4G wireless networks and [24] discussedvarious admission control algorithms for multi-service IPnetworks. We adopt this idea in the transportation systemand design an admission control mechanism to differentiatethe transportation service requests for maximizing the totalprofit. There are many methods to facilitate admission control.Genetic Algorithm (GA) is one of them and it has beensuccessfully utilized to design admission control mechanisms,e.g., [25] and [26]. Based on the special formulation of theadmission control problem (to be discussed in Section V), wewill also adopt GA to solve the problem.

III. SYSTEM MODEL

In this section, we design the architecture for the systemwhich can manage a fleet of AVs to serve customers fortransportation services. In the following, we first introducethe system components and then describe the operationscharacterizing their interactions.

A. System Components

1) Network Structure: A graph is employed to modelthe region being served by the system. It characterizes thelocations and the road connections necessarily to describemovements of the AVs, origins and destinations of the servicerequests, and other required facilities. It is a directed graph

denoted by G(V, E), where V is a set of locations and Erefers to the road segments connecting the locations so thatwe can completely describe the routes of AVs with G. Fori, j ∈ V , each edge (i, j) ∈ E is associated with an operationalcost cij and a travel time tij , which is an estimation of timefor an AV to traverse from i to j based on historical data.Depended on the system objective, cij typically represents thedistance of the road segment (i, j) as the operational cost ofAVs is usually measured by the fuel consumption which is inturn characterized by the travel distance. If the system aimsto optimize the total service duration, we can set cij = tij forall (i, j)’s. We allow cij 6= cji and tij 6= tji to account forthe asymmetry of road segments. Moreover, refuel stations arelocated in some locations specified by V ⊂ V and each AVends its journey at any one of these refuel stations (reasonsexplained in Section III-B1). Based on the nature of the AVs,V ⊂ V will be the locations of charging (gas) stations ifthe AVs are electric (conventional) vehicles. For the case ofelectric vehicles, V ⊂ V can be determined based on thecharging demand and the connectivity of the charging stationnetwork according to [27].

2) Transportation Requests: Customers request services inthe form of transportation requests, which are collectivelydenoted by R. Each r ∈ R is represented by the 5-tuple〈sr, dr, Tr, [er, lr], qr〉. sr ∈ V and dr ∈ V represent thecustomer pickup and dropoff locations, respectively. Tr is themaximum ride time, an exceedance of which will lead tocustomer dissatisfaction. [er, lr] refers to the service startingtime window, where er and lr are the earliest and latest servicestarting times, respectively. qr stands for the number of seatsneeded in the request r.

3) Vehicles: The system coordinates a fleet of AVs de-noted by K. Each k ∈ K is represented by the 5-tuple〈ak, t0k, Tk, Qk,Rk〉. ak ∈ V is the first location where kwill visit from the current position of k while t0k is the timerequired to reach ak from its current position. It is possiblethat, at the time of scheduling, the AV is in the middleof a road segment heading to ak. ak and t0k can be easilyestimated by submitting its current position to the system. Tkdenotes the maximum remaining operation time that k cancontinue to provide services without refueling.1 Qk is thepassenger capacity that k can accommodate simultaneously.Rk = Rk ∪ Rk ∈ R is the set of requests previouslyassigned to k. Rk can be further categorized into two types;Rk contains those currently being served by k while Rk wasassigned to k at a previous schedule but the services havenot been implemented yet. For the former, some seats havealready been taken by the customers from Rk. On the contrary,seats have only been reserved but no actual seats have beentaken from Rk. We will handle Rk and Rk differently whenperforming scheduling in Section IV.

Without loss of generality, we assume that the number ofseats required in any request is no larger than the capacity ofany vehicle, i.e.,

qr ≤ Qk,∀r ∈ R, k ∈ K. (1)

1The maximum remaining operation time of k can be converted from itscorresponding remaining fuel level.

4

time

Data collection sub-interval

Duty assignment sub-interval

Interval i Interval i+1

Data collection sub-interval

(a few minutes) (a few seonds)

Fig. 1. Operating intervals in the system.

Receive transportation

requests

Merge requests

Providing services

Unserved requests

Acquire vehicle statuses

Admission control

Scheduling

Data collectionSub-interval

Duty assignmentSub-interval

Du

ty a

ssig

nm

ent

Fig. 2. Operation flow of the system.

We can always split those requests violating (1) into multiplerequests so that this condition always holds.

B. Operations

The system is managed and operated by a control centerwhose main duties are to collect all the required informationand assign the AVs to serve the transportation requests. Thesystem operates in a fixed time interval basis and each timeinterval is divided into data collection and duty assignmentsub-intervals (see Fig. 1). In each interval, the control centerfirst collects transportation requests and vehicle statuses inthe data collection sub-interval. Then the AVs are assigned toserve the transportation requests in the duty assignment sub-interval. On one hand, the duration of each interval shouldbe long enough such that the communication delays will notresult in any data missing from the customers and vehiclesfor scheduling. On the other hand, it should be short enoughsuch that the collected data can reflect the current situationhappening in that interval. In practice, the data collection sub-interval is longer than the duty assignment one. The formermay last for a few minutes while the latter may takes a fewseconds.

Fig. 2 illustrates the operation flow of the system with re-spect to an operating interval. As powered by various wirelessvehicular communication technologies, all AVs are connectedand can communicate with the control center instantaneously.In this way, the control center can collect the necessaryvehicle statuses, e.g., current locations of AVs, confirmationof serving requests, traffic congestion information, etc., in thedata collection sub-interval. Customers can also submit theirrequests to the control center by any appropriate means, e.g.,

phone calls, mobile apps, etc. After the data collection sub-interval, all the data required to perform duty assignment areready at the control center.

In the duty assignment sub-interval, the control centerprocesses the collected data and computes the duty assignment.There may exist some unattended requests incurred from someprevious intervals because of their unsuitability in the previoussystem conditions. They are merged with the newly submittedrequests and then all these requests are considered en masse.The duty assignment further consists of two processes: admis-sion control and scheduling. Admission control checks all theoutstanding requests and determines which requests are goingto be admitted in the current interval. The unadmitted requestswill be reserved for consideration in the next interval again.Any invalid or inappropriate requests are also permanentlyexcluded in the admission control process. We compute thetravel schedules of the AVs to serve the admitted requests inthe scheduling process. If a vehicle is assigned with a request,its schedule settled by the control center needs to satisfy thefollowing requirements:

1) Complete route specification: Since the vehicle is un-manned, we need to specify the exact route so that the vehiclecan follow the route to pick the passengers of the assignedrequests up and to drop them off at the required destinations.Moreover, the route should be short enough so that it hassufficient fuel to complete the route. The vehicle should endup at a refuel station to avoid breaking down in the middle ofany road segments. This can guarantee that the vehicle mustbe able to refuel after completing all the assigned services.

2) Time constraints: The vehicle should be able to pick thepassengers up at a time within the service starting time windowspecified in the request. Moreover, the actual ride time shouldbe no longer than the maximum value stated in the request.

3) Capacity constraints: When the vehicle arrives at thepickup location, there should always be enough free seatsavailable to accommodate all the passengers of the request.

Admission control and scheduling are inter-related and wewill discuss their details in the subsequent sections. Afterdetermining the result, the control center then distributes theassignments to the corresponding AVs, which provide servicesto the customers.

IV. SCHEDULING

Scheduling involves determining the following:• the assignment of AVs to the requests;• the routes of AVs to accomplish the assigned requests;

and• the times by which the AVs should reach particular

locations.Here we assume that all requests being scheduled are ad-missible, where the admittability of a request is handled byadmission control. Thus all requests will be served by appro-priate vehicles after scheduling. When discussing admissioncontrol in Section V, we will explain the relationship betweenadmission control and scheduling.

To facilitate scheduling, we assume that all vehicles areconnected and can communicate with the control center with

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reasonably short delays. This ensures that no apparent changesin positions happen to the AVs in each interval given inFig. 1. With the support of modern advanced communicationtechnologies, this assumption can go through. In our model,we require that the computation of scheduling can be done in ashort period of time. This ensures the validity of the traffic datawhen the vehicles traverse along their assigned routes. Thereare basically two types of traffic data: the distances and traveltimes of road segments. The former is time-invariant whilethe latter usually changes gradually. In other words, significantchanges in travel times only take place in a timespan muchlonger than the time interval.

A. Preprocessing

We schedule the AVs to accomplish the transportationrequests to achieve the minimum total operational cost interms of fuel costs, which are in turn measured by the totaldistance traveled. The distance between any pair of locationsis invariant and we transform G(V, E) to G′(V ′, E ′) with anyshortest path algorithm, e.g., Dijkstra’s algorithm [28], whereV ′ ⊂ V is the set of locations at which we need to determinethe arrival times of the assigned AVs in order to configuretheir travel schedules. V ′ includes the first locations visitedby all the vehicles (i.e., ak’s), the sources and destinationsof the requests (i.e., sr’s and dr’s, respectively), and thelocations of the refuel stations (i.e., i ∈ V). E ′ is definedas {(i, j)|i, j ∈ V ′} such that there exists a shortest path fromi ∈ V to j ∈ V in G. For (i, j) ∈ E ′, the associated cij andtij are the sums of costs and times, respectively, of all theedges constituting the corresponding shortest path in G. Inthe subsequent computation, we focus on G′(V ′, E ′) insteadof G(V, E). The reasons why we adopt this transformationare two-fold: First, the number of variables needed in theformulation can be dramatically reduced. The set V\V ′ are notimportant as all conditions confining to the locations specifiedby the vehicles and requests are restricted to V ′ only. In thisway, the efficiency of solving the scheduling problem can beimproved significantly. Second, this can improve the flexibilityof the schedules. Consider that AV k goes from vertices 1to 4 and there exist two paths connecting them as, Path 1:1→ 2→ 4, and Path 2: 1→ 3→ 4. Suppose that vertices 1and 4 belong to V ′ but vertices 2 and 3 do not. To satisfy therequirements imposed on k, we need to determine the times bywhich k should arrive at vertices 1 and 4 only, i.e., tk1 and tk4 . Ifvertices 2 and 3 are also included in the formulation and Path 1is finally chosen, tk2 will be specified by solving the schedulingproblem and thus k needs to arrive at the vertices by tk1 , tk2 , andtk4 , respectively. If not, only tk1 and tk4 are specified and we cangive flexibility to k of arriving at vertex 2. tk2 can be any timebetween tk1 and tk4 as long as the required travel times spent on(1, 2) and (2, 4) have been considered. This flexibility givesroom for k to respond to any instantaneous traffic incidentswhich may disturb its original travel plan. This also allows kto change to Path 2, if needed, without altering the originaltravel plan.

Note that the preprocessing step can be skipped if thescheduling problem constructed directly from G(V, E) can be

solved efficiently. However, if the preprocessing is requiredto simplifiy the scheduling problem, it can be considered asa number of result lookups. As cij’s generally refer to thetravel distances which are invariant, the results of the shortestpath computations are also invariant. In fact, before the systemoperates, we can first compute the shortest path for every pairof locations in V . When the preprocessing is triggered in aninterval, we just need to look up the pre-computed shortestpath results. Hence, the time cost of preprocessing can beconsidered negligibly small.

B. Problem Formulation

We formulate the scheduling problem based on G′(V ′, E ′).The given data for the problem parameters include the graphG′(V ′, E ′) with costs cij’s and travel times tij’s, the set oftransportation requests R, and the set of AVs K. We defineseveral variables for the problem. Binary variables xkij’s areused to indicate which connections will be traversed by thevehicles, as

xkij =

{1 if vehicle k traverses (i, j),0 otherwise.

We define binary variables ykr ’s for the assignment of thevehicles to the requests, as

ykr =

{1 if vehicle k is assigned to request r,0 otherwise.

For i ∈ V , binary variables gki ’s are utilized to indicate therefuel stations at which the vehicles end their routes, as

gki =

{1 if vehicle k ends its route at vertex i ∈ V ,0 otherwise.

We need to specify the times and occupancy conditions atvarious locations along the routes. Let tki be the time by whichk should arrive at vertex i and fki be the number of passengersin k right before it leaves i.

We aim to construct economical schedules for the AVs andthus we minimize the total operational cost with the objectivefunction as ∑

i,j∈V,k∈K

cijxkij . (2)

We define a set of constraints to confine the scope of thevariables so that the requirements discussed in Section III-Bare satisfied. Each transportation request can only be servedonce and thus we have∑

k∈K

ykr = 1,∀r ∈ R. (3)

Each AV will end at one of the refuel stations if it is assignedto a request. This is specified by∑

i∈V

gki ≤ 1,∀k ∈ K. (4)

If AV k is not assigned to any request, we do not need todetermine a path for k so as the final stopping refuel stationfor k. Thus it is possible to have

∑i∈V g

ki = 0 for some k.

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Let N+(i) and N−(i) be the sets of incoming and outgoingneighbors of vertex i, i.e., N+(i) = {j ∈ V ′|(j, i) ∈ E ′}and N−(i) = {j ∈ V ′|(i, j) ∈ E ′}. We model a path with anetwork flow model. A path starting at ak and ending at i ∈ Vcan be defined with the following:

0 ≤∑

i∈N−(ak)

xkaki −∑

i∈N+(ak)

xkiak ≤∑r

ykr ,∀k ∈ K, (5)

0 ≤∑

j∈N+(i)

xkji −∑

j∈N−(i)

xkij ≤ gki ,∀i ∈ V, k ∈ K, (6)

∑j∈N+(i)

xkji =∑

j∈N−(i)

xkij ,∀i ∈ V ′ \ V ∪ {ak|k ∈ K}. (7)

Eq. (5) defines for the starting vertex of k, where a startingvertex has one unit of net outgoing flow.

∑r y

kr specifies if a

path needs to be defined for k. If there are no requests assignedto k,

∑r y

kr becomes zero and ak is not the starting vertex of

any paths for k. Similarly, (6) defines for the destination vertexof k and the exact vertex i ended by k is indicated by gki . If kends at i ∈ V , (6) will allow i to have one unit of net incomingflow for k. For other vertices, (7) sets the conversation of flowby equalizing the corresponding incoming and outgoing flows.

If request r is assigned to vehicle k, k needs to pass throughthe pickup location sr of r. It is equivalent to having positiveoutgoing flow for k at sr as∑

i∈N−(sr)

xksri ≥ ykr ,∀r ∈ R, k ∈ K. (8)

Similarly, k needs to pass through the dropoff point dr ofrequest r when r is served by k. This requires positiveincoming flow for k at dr as∑

i∈N+(dr)

xkidr ≥ ykr ,∀r ∈ R, k ∈ K. (9)

Note that specifying incoming flow for sr is not sufficient as itis possible to have zero incoming flow when k begins its pathat sr exactly. Similarly, it is not sufficient to specify outgoingflow for dr as it is possible to have zero outgoing flow whenk ends its path at dr.

No matter where vehicle k goes, it cannot travel continu-ously longer than its operational time limit specified by Tk.Moreover, it needs to take at least t0k in order to reach theinitial vertex of its path. Hence we have

t0k ≤ tki ≤ Tk,∀i ∈ V ′, k ∈ K. (10)

Let M be a sufficiently large positive number. When vehiclek traverses edge (i, j), the time at j should be larger than orequal to the time at i together with the travel time on (i, j),i.e., tij . This can be specified by

tkj ≥ tki + tij −M(1− xkij),∀k ∈ K, i, j ∈ V ′. (11)

When vehicle k is assigned to request r, the actual ride timeto reach dr from sr should be no larger than the maximumride time Tr specified by r, i.e.,

tkdr − tksr ≤ Tr +M(1− ykr ),∀r ∈ R, k ∈ K. (12)

If request r is served by vehicle k, k should arrive at srwithin the service starting time window [er, lr] specified by r.This can be expressed as

er −M(1− ykr ) ≤ tksr ≤ lr +M(1− ykr ),∀r ∈ R, k ∈ K.(13)

Passengers being served occupy seats and the capacity limitsof all vehicles should be satisfied at all times. So we have

0 ≤ fki ≤ Qk,∀i ∈ V ′, k ∈ K. (14)

At ak, some passengers induced from Rk may get off kand new passengers may get on k from other requests. Theoccupancy conditions of the AVs at their initial vertices ak’sare given by

fkak ≥∑

r|sr=ak

qrykr −

∑r|dr=ak

qrykr ,∀k ∈ K. (15)

When k traverses from i to j along (i, j), vertex j may bethe pickup locations of some requests and dropoff locations ofsome other requests. The relationship between the occupancyconditions of AV k at i and j can be specified as

fkj ≥ fki −M(1− xkij) +∑

r|sr=ak

qrykr −

∑r|dr=ak

qrykr , (16)

∀i, j ∈ V ′, k ∈ K.

When an AV reaches a refuel station, all requests assignedto it should have been settled and no passenger should beaccompanied to the end of the route. This is described by

fki ≤M(1− gki ),∀i ∈ V , k ∈ K. (17)

Recall that there are two kinds of requests which havealready been assigned to the AVs before the current schedulinginterval, i.e., Rk = Rk ∪ Rk. As a (nearly) real-time appli-cation, with updated information, we may further improve thesystem performance by revising the already assigned requests.For those requests currently being served, e.g., r ∈ Rk withthe passengers sitting in k, we can consider those r’s as“new” requests starting the service at the the starting nodeak by setting sr = ak and affirming ykr = 1. As k has beenserving r by following a previously determined schedule, wecan update its Tr by shortening the elapsed time. The servicestarting time window is no longer important and thus we seter = −∞ and lr = +∞. There is no change to qr. Forthose requests Rk’s which have been previously assigned tok but not yet been served, we may reschedule r ∈ Rk withother AVs if it can result in lower cost. As the passengers donot concern about which vehicle would eventually provide theservice, it may be more efficient to re-allocate those r’s inRk to other more appropriate vehicles with lower operationalcost. This enhances the flexibility of the system. As a whole,the scheduling problem is defined as

Problem 1 (Scheduling):

minimize (2)subject to (3)− (17)

over xkij ∈ {0, 1}, ykr ∈ {0, 1}, gkl ∈ {0, 1}, tki ∈ R+,

fki ∈ Z+,∀i, j ∈ V ′, l ∈ V, r ∈ R, k ∈ K.

7

Problem 1 has a linear objective function and linear equalityand inequality constraints. Some of its variables are binarywhile the rest are real. Thus the scheduling problem is amixed-integer linear program (MILP). Although the prepro-cessing step discussed in Section IV-A helps simplify theproblem, the numbers of variables and constraints also growwith the sizes of R and K. As those invalid requests have beenremoved by admission control (discussed in Section V), thisMILP is always feasible and all requests must be served. Aslong as all cij’s are positive, the solution of Problem 1 doesnot result in zero cost and the schedule without serving anyrequests will never be a solution.

C. Complete Schedule Construction

Since the vehicles are unmanned, we need to providecomplete instructions about the paths and schedules so thatthey know when and where they should go in order toprovide services to the customers. Solving the MILP gives thesolutions for xkij’s, ykr ’s, bk’s , tki ’s, and fki ’s. As being binaryvariables, the results of xkij’s and ykr ’s are unambiguous. Thelatter tells which vehicles are assigned to the requests. Theformer explains the route of each k in G′ starting at ak andending at one of the refuel stations. The paths determinedin G′ in turn infer the corresponding complete routes in G.Recall that we have determined the shortest path from i toj in G corresponding to the edge (i, j) ∈ E ′ . By insertingthe shortest paths for every pair of adjacent vertices along thepaths based on G′, the complete routes in G can be derivedaccordingly.

Note that (10)–(13) define the scope of tki ’s in the formof inequality. The resulting tki ’s make feasible time schedulesbut may not be specific enough leading to ambiguity. Forexample, if the arrival of k at location i at any moment in[t′, t′′] is feasible, a reasonable way is to set tki = t′ and thisenhances the flexibility for the later scheduling intervals. Toconstruct the schedule of k, we examine the path computedfrom xkij’s. For the first vertex, we set tkak = t0k. For anysubsequent vertices, says from i to j, we can add the traveltime on edge (i, j) to the settled time at i to obtain the settledtime at j, i.e., tkj = tki + tij . If vertex j induces a request, weneed to fulfill its service starting time window and thus wehave tkj = max{tki + tij , er}.

Similarly, (14)–(17) also confine the occupancies of thevehicles at various locations with inequalities. The exact seatconditions cannot be told from the resulting fki ’s. Usually, weonly concern about the seat conditions at the customer pickupand dropoff points, i.e., sr’s and dr’s. We can examine theroute computed from xkij’s again and determine the occupancyconditions. For example, k goes from i to j on (i, j). If j isthe service starting location of request r, we add the numberof seats required for r to the occupancy of k at i to get itsoccupancy at j, i.e., fkj = fki +qr. If j is a service destinationlocation instead, we subtract the seats taken by r from theoccupancy of k at i to get its occupancy at j, as fkj = fki −qr.In this way, the complete schedules of the vehicles with dutyassigned can be determined and the vehicles just need to followthe schedules to accomplish the services.

V. ADMISSION CONTROL

Recall that, in Section IV, all requests submitted forscheduling are assumed to be admissible and need to be served.In this section, we investigate the admission control problem.We first formulate the problem and then study the variations inthe presence of traffic congestion and no-show of passengers.

A. Problem Formulation

Admission control is responsible for determining a set ofrequests suitable for scheduling. In other words, after admis-sion control, we will produce a subset R ⊂ R for subsequentscheduling, where R is the set of all available requests and Rwill be settled by appropriate AVs in scheduling. However, tojudge if a particular request r is admissible, we need to checknot only its feasibility but also its profitability, i.e., whetherserving r will induce a positive net profit. Determining thenet profit from r involves its induced cost, which is regulatedthrough scheduling. Hence there is no clear precedence rela-tionship between scheduling and admission control and thesetwo processes should be considered simultaneously.

We can interpret the requests and AVs as the demand andsupply of transportation services, respectively, and then theconstraints of Problem 1 define the scope of matching betweenthe demand and supply. The constraints can be satisfied moreeasily with larger K and smaller R. Practically, the size of Kis generally fixed as the system would not suddenly employmore AVs into the fleet or many AVs become out of service allof a sudden. However, the requests submitted are absolutelyexternal from the system; the system can neither forbid thecustomers from submitting requests nor modify the attributesin the requests to match the conditions of AVs. In fact, justa single inappropriate request (e.g., a request with very shorttolerable ride time) can make Problem 1 infeasible and thescheduling collapse. To avoid this, the system should performadmission control by screening out any inappropriate requestsbefore undergoing the scheduling (see Fig. 2). Consider thatentertaining a request results in revenue. Although the sys-tem cannot modify the submitted requests, it has the rightto dismissing any requests by sacrificing the correspondingrevenue. Admission control manipulates R with the followingobjectives: 1) Produce a subset of requests R ⊂ R so that thescheduling process can be performed, i.e., Problem 1 is madefeasible with R; 2) Maximize the profit incurred.

Consider that we admit R for scheduling with Problem 1,which can be re-written as

minimize φ(α) (18a)

subject to α ∈ Z(R), (18b)

where α , {xkij} ∪ {ykr } ∪ {gki } ∪ {tki } ∪ {fki }, φ(α) ,∑i,j∈V,k∈K cijx

kij , and let Z(R) be the feasible region of

Problem 1 with respect to R. Let ρr be the revenue madewhen admitting r ∈ R and define

zr =

{1 if we admit r ∈ R for scheduling,0 otherwise.

We also define the admission function σ(R, [zr]r∈R) whichreturns R ⊂ R based on zr such that r ∈ R if zr = 1. The

8

total profit is the difference between the total revenue and totalcost, i.e.,

∑r∈R ρrzr−φ(α). Then we formulate the admission

control problem asProblem 2 (Admission Control):

maximize Φ(R, [zr]r∈R) =∑r∈R

ρrzr − φ(α) (19a)

subject to R = σ(R, [zr]r∈R), (19b)zr = 1,∀r ∈ Rk, k ∈ K, (19c)

α ∈ arg min{φ(α) : α ∈ Z(R)}, (19d)

over α, R ∈ R, zr ∈ {0, 1},∀r ∈ R, (19e)

where (19c) ensures that those requests admitted in the previ-ous operating intervals will still be admitted in the currentinterval. We cast admission control as a bilevel optimiza-tion problem, which consists of an upper- and a lower-leveloptimization. Φ is the upper-level objective function withupper-level variables R and zr’s. φ represents the lower-levelobjective function with lower-level variable α. Eq. (19d) isin fact (18), and thus, we cast Problem 1 as a constraintof Problem 2. The upper-level optimization is to manipulatethe whole set of requests R and determine R such that Rcan maximize the total profit. The lower-level optimization isto schedule the AVs to serve the set of admissable requestsR so that the retained cost is the lowest. The two levelsof optimization are inter-related; the upper level requires theresult of the lower level, i.e., α, in order to get R, while thelower level needs the result from the upper level, i.e., R, inorder to output α. Note that if the upper level produces Rwhich makes Z infeasible, the resulting α will return +∞ forthe objective function of (19d), which will in turn make theobjective function (19a) retain −∞.

Bilevel optimization is in general difficult to solve. A bilevelproblem with a linear objective function and linear constraintsis NP-hard [29]. As seen from (19), we are manipulatingdiscrete variables in the problem. As classical methods forbilevel optimization usually assume smoothness or convexity[30], those classical methods are not applicable to Problem 2.As inspired by [31], [32], we decide to tackle the problem withan evolutionary heuristic approach. Evolutionary approachesare commonly applied to bilevel optimization problems intransport science. For example, in [33], Differential Evolution(DE) is employed to address the optimal toll problem, which isabout setting polls to control congestion, and the road networkdesign problem, which determines the capacity enhancementsof network facilities. In [34], GA is applied to the transitroad space priority problem, which optimizes the system byreallocating the road space between private car and transitmodes. We will design a GA-based algorithm to solve Problem2. Before discussing the details of the algorithm, we defineadmissibility and give some analytical results for Problem 2,which can help design the algorithm in the next section.

Definition 1 (Admissibility): A set of requests R is admis-sible if [zr]r∈R produces R, which results in finite profit, i.e.,Φ(R, [zr]r∈R) > −∞.

Theorem 1: We have the following results for admissibility:1) Consider that a subset of requests R ⊂ R are admissible.

Let P(R) be the power set of R. Any R′ ∈ P(R) is also

admissible.2) For any singleton {r} ⊂ R, if {r} is not admissible, any

superset R ⊃ {r} are also non-admissible.3) Consider subsets of requests, R1, R2 ⊂ R, and subsets

of vehicles K1, K2 ⊂ K. Suppose K1 ∩ K2 = ∅. If R1

and R2 are admissible by K1 and K2, respectively, thenR1 ∪ R2 are also admissible.Proof: For Statement 1, Constraint (19b) defines R,

which is an input of Constraint (19d). It is sufficient to showthat the removal of any r ∈ R will not make (19d) infeasible ifthe participating AVs can serve all the requests in R. Supposethat r is removed from R and AV k would have assigned toserve r if r had been admitted. k can still follow the path asif r is present. Hence (19d) is still feasible for R \ r.

For Statement 2, a non-admissible r means that it isimpossible to arrange an AV to entertain r. We will neverbe able to provide services to a set of requests containing ras its component r can never be served.

For Statement 3, we can represent R1 ∪ R2 by three non-overlapping sets R1\(R1∩R2), R2\(R1∩R2), and R1∩R2.Since K1 and K2 are mutually exclusive, R1 \ (R1∩R2) andR2 \ (R1 ∩ R2) can be served by K1 and K2 simultaneously.Each r ∈ R1 ∩ R2 can be admitted by either k ∈ K1 ork ∈ K2.

Lemma 1: The system will not make negative profit. Thatis, for any R, Problem 2 must have at least one feasiblesolution whose objective function value is non-negative.

Proof: We separate R into the previously admitted andnewly received requests, i.e., {Rk} and R \ {Rk}.

For the newly received requests, we can always set zr =0,∀r ∈ R\{Rk}. Then (19b) gives R = ∅. Eq. (19d) returnsα with φ(α) = 0 as no requests need to be served and thusno AVs have been used to provide service. Hence we have∑r∈R\{Rk} ρrzr − φ(α) = 0.For the previously admitted requests, since they are admitted

in some previous admission control processes, they must incurnon-negative profit when they were admitted as new requestsbefore. Otherwise, we would not have admitted them at thefirst place.Lemma 1 implies that Problem 2 must be feasible.

Theorem 2: Consider two subsets of requests R and R′with R ⊂ R′ ⊂ R. If both R and R′ are admissible, thenR′ will not be less profitable than R, i.e., sup Φ(R, {zr|r ∈R}) ≤ sup Φ(R′, {zr|r ∈ R′}).

Proof: Suppose sup Φ(R, {zr|r ∈ R}) >sup Φ(R′, {zr|r ∈ R′}). We write R′ = R ∪ (R′ \ R). Thenwe have

sup Φ(R′, {zr|r ∈ R′})= sup Φ(R, {zr|r ∈ R}) + sup Φ(R′ \ R, {zr|r ∈ R′ \ R}).

By Lemma 1, sup Φ(R′ \ R, {zr|r ∈ R′ \ R}) has a valuelarger than or equal to zero. This induces a contradiction.Theorem 2 implies that entertaining more requests will notreduce the amount of profit made.

B. VariationsHere we investigate how traffic congestion and no-show

of paasengers impact on admission control (and scheduling).

9

Basically, we will see that under these circumstances, theproposed admission control and scheduling mechansims canstill be applied but we may need some additional minorarrangements to handle various situations.

1) Traffic Congestion: Traffic congestion has direct impacton the travel time tij for some (i, j) ∈ E and subsequentlyaffects the admissibility of requests. Recall that the systemoperates in a fixed-interval basis and each interval generallylasts for a few minutes (see Section III-B). We basicallyassume that, within an interval, the parameters, including thetravel times, are constant or with very small changes such thatthe results of admission control completed for that intervalare still valid. If the travel times are relatively fast changing,we need to shorten the duration of the intervals to makethe assumption valid. On the other hand, if the travel timesare slowly varying, we may lengthen the durations to reducethe computation burden. Hence, the duration of the operatingintervals depends on the traffic conditions of the deployedservice area.

Now consider that tij in the current interval has beenupdated such that its value is different from that used inthe previous interval. There are three cases for the possibleinfluence: (i) tij does not involve in Rk for all k; (ii) tijinvolves in Rk for some k; and (iii) tij involves in Rk forsome k. For Case (i), since tij has not been used to serve anyrequests, its change does not affect the schedules of any AVs.Hence, nothing needs to be done solely based on tij . For Case(ii), although tij has been used to determine the schedules ofsome AVs, the involved requests have not been served yet.We can simply consider these requests as newly submittedrequests and perform admission control and scheduling withthem again. For Case (iii), tij affects those schedules which arebeing implemented by some AVs. In the subsequent intervals,the scheduling process will see if the road segment (i, j) canbe avoided by determining other shortest paths. If not, asthe passengers are being served, it may not be appropriateto ask them to shift to other vehicles for their journeys andnothing can be done further operationally. However, we maycompensate the passengers in the marketing perspective, e.g.,by issuing cash coupons for future rides.

2) No-show of Passengers: No-show refers to the situationthat some or all passengers of a paricular request are absentat the scheduled pickup time. If a passenger cannot arrive atthe pickup location on time, this will be considered as no-show. If some but not all passengers are absent, the schduleof the designated AV is unaffected but fewer seats are required.These unused seats can be released to serve other appropriaterequests in the later intervals. If all passengers are absent, the“resources” allocated to the request can be released in thesubsequent intervals right after its original pickup time. Thisgives the AV more flexibility in time and occupancy to servefuture requests. In the business perspective, there may existsome penalty policies to discourage such activities.

VI. GENETIC-ALGORITHM-BASED SOLUTION METHOD

In this section, we propose a solution method to tackleProblem 2. We adopt a GA-based framework to structure the

method. Some of its components are designed based on theanalytical results discussed in Section V.

A. Working Principle of Evolutionary Algorithms

Evolutionary Algorithms (EAs) refer to a class of opti-mization algorithms, whose designs are inspired by variousnatural phenomena. Examples include GA [35], DE [36], andChemical Reaction Optimization (CRO) [37]. Different EAsgenerally have similar working principles: An EA samples thesolution space of the problem iteratively and tries to locate aglobal optimum after examining a limited number of candidatesolutions in the solution space. In each iteration, with someoperators, it generates a population of candidate solutionsbased on those obtained from the previous iterations and theircorresponding objective function values. It tends to converge tothe global optimum along the iterations and it terminates whena stopping criterion is matched. Different EAs have differentdesigns of their operators. For example, GA is designed basedon the ideas of natural selection in genetics while CRO mimicsthe nature of chemical reaction processes. Unlike most of thetraditional optimization approaches, EAs require the problemto be neither convex nor differentiable. In each algorithm run,they only need to sample a number of candidate solutions andevaluate their solution qualities with the objective function.Hence, a search with an EA usually incurs many objectivefunction calls. As discussed, EAs have been shown effectivein solving bilevel optimization problems in transport science.We are going to adopt the well-established GA frameworkto facilitate the design of a method which can return goodsolutions for Problem 2 in a practical sense.

B. Distributed Scheduling

When an EA is employed to address Problem 2, manycandidate solutions will be generated. To evaluate the qualityof a particular candidate solution, we need to compute (19a)once, which also needs to examine (19d) one time. In otherwords, a single run of EA requires to solve Problem 1many times. When the lower-level optimization is simple,the computational burden of solving it many times may stillbe acceptable. However, this is not the case for Problem 1,where the required numbers of variables and constraints growexponentially with the quantities of transportation requests andserving AVs. This implies that we need to a more effective wayto solve Problem 1, in order to tackle Problem 2.

Consider that Rk ⊂ R is the subset of requests assignedto vehicle k. Suppose that we know the distribution of therequests to the vehicles, i.e., Rk for all k. Since each requestis only served by one vehicle, we have Rk ∩ Rl = ∅, for anyk, l ∈ K, k 6= l, and

⋃k∈K Rk = R. When given Rk, we

consider the following problem:Problem 3 (Scheduling Subproblem for vehicle k):

maximize∑i,j∈V′

cij xkij (20a)

subject to∑i∈V

gki ≤ 1, (20b)

10

0 ≤∑

i∈N−(ak)

xkaki −∑

i∈N+(ak)

xkiak ≤∑r

ykr ,

(20c)

0 ≤∑

j∈N+(i)

xkji −∑

j∈N−(i)

xkij ≤ gki ,∀i ∈ V,

(20d)∑j∈N+(i)

xkji =∑

j∈N−(i)

xkij ,∀i ∈ V ′ \ V ∪ {ak}

(20e)∑i∈N−(sr)

xksri ≥ ykr ,∀r ∈ Rk, (20f)

∑i∈N+(dr)

xkidr ≥ ykr ,∀r ∈ Rk, (20g)

t0k ≤ tki ≤ Tk,∀i ∈ V ′, (20h)

tkj ≥ tki + tij −M(1− xkij),∀i, j ∈ V ′ (20i)

tkdr − tksr ≤ Tr +M(1− ykr ),∀r ∈ Rk, (20j)

er −M(1− ykr ) ≤ tksr ≤ lr +M(1− ykr ),∀r ∈ Rk,(20k)

0 ≤ fki ≤ Qk,∀i ∈ V ′, (20l)

fkak ≥∑

r|sr=ak

qrykr −

∑r|dr=ak

qrykr , (20m)

fkj ≥ fki −M(1− xkij) +∑

r|sr=ak,r∈Rk

qrykr

−∑

r|dr=ak,r∈Rk

qrykr ,∀i, j ∈ V ′, (20n)

fki ≤M(1− gki ),∀i ∈ V , (20o)

over xkij ∈ {0, 1}, ykr ∈ {0, 1}, gkl ∈ {0, 1}, tki ∈ R+,

fki ∈ Z+,∀i, j ∈ V ′, l ∈ V, r ∈ Rk. (20p)

Solving Problem 3 only allows us to obtain the serving path,the schedule to reach various locations along the path, andthe capacity conditions of vehicle k for serving the requestsindicated by Rk. Problem 3 looks similar to Problem 1 butindeed much simpler. It does not contain (3) and it manipulatesfewer variables as those related to vehicles other than k arenot included. It also possesses fewer constraints because offewer variables.

For simplicity, similar to (18), we also write the solution,objective function, and the solution space of Problem 3 as αk,φk(αk) and Zk, respectively.

Theorem 3: When given Rk ⊂ R,∀k ∈ K, such that Rk ∩Rl = ∅, for any k, l ∈ K, k 6= l, and

⋃k∈K Rk = R, solving

Problem 3 for all k ∈ K is equivalent to solving Problem 1,i.e.,

infα∈Z

φ(α) =∑k∈K

infαk∈Zk

φk(αk),

and xkij = xkij , ykr = ykr , gkl = gkl , tki = tki , and fki = fki ,

∀i, j ∈ V ′, l ∈ V, r ∈ Rk, k ∈ K.Proof: When given such Rk ⊂ R,∀k ∈ K, we can

construct ykr ,∀r ∈ R, k ∈ K, such that (3) holds. In thisway, we can remove (3) from Problem 1. Without Constraint(3), the objective function and the rest of the constraints of

Problem 1 become separable in terms of k: (2) gives the sumof costs spent on the vehicles; Eqs. (4)–(9) specify the pathstraversed by the vehicles, each of which are independent; Eqs.(10)–(13) confine the time requirements at various locationsalong the vehicular paths; Eqs. (14)–(17) limit the passengercapacity conditions along the vehicular paths. If we group theterms of (2) and the constraints (4)–(17) for each k, we willhave |K| problems, each of which is given by (20).

Theorem 3 states that when the assignment of requests tothe vehicles is known, solving the |K| individual schedul-ing subproblems distributedly can retain the solution of theoriginal scheduling problem. Note that this result is dedicat-edly developed based on some characteristics of the problemformulations and it generally cannot be applied to the otherscheduling problems. Unlike general distributed optimization[38], [39], our result here does not require techniques likemessage-passing. As a result, the |K| subproblems can besolved by |K| computing units distributedly. Assuming thatthe vehicles are connected through advanced vehicular com-munication technologies at all times, an obvious option ofthe computing unit is the AV. Thus, by Theorem 3, if wecan assign each vehicle with the requests it needs to serve,each vehicle can determine a feasible path per se to servethe assigned requests with the lowest cost by solving Problem3 concurrently. However, when the communications betweena particular AV and the control center are interrupted, thecorresponding subproblem can be delegated to an unoccupiedcomputing unit at the control center or even to the cloudinstead. The computed scheduling result can be returned tothe AV when its communications have been resumed.

C. Algorithmic ComponentsSince GA is one of the most popular EAs, we adopt a

GA-based design to address the admission control problem.GA generates a sequence of candidate solutions using opera-tions inspired by natural evolution, e.g., inheritance, selection,crossover, and mutation. Here we introduce various algo-rithmic components before discussing the overall algorithmicdesign:

1) Chromosome: A chromosome specifies a candidate so-lution of Problem 2. While the lower-level optimization ishandled by a standard MILP method, our GA approachis mainly used to handle the upper-level optimization. Achromosome is represented by a 1 × |R| binary vectorz = [z1, . . . , zr, . . . , z|R|], together with a vehicle assignmentvector κ = [κ1, . . . , κr, . . . , κ|R|], where κr represents thevehicle assigned to r if zr is of unity. Note that the introductionof κ is the trick to carry out distributed scheduling discussedin Section VI-B. Although κ can be determined in (19d) if weapply the original formulation of scheduling (18), there is noharm in manipulating κ together with z in the chromosomelevel. This makes distributed scheduling feasible and thebenefit of computation time saving will be clear in SectionVII-A. During the course of search, we maintain a populationof Npop chromosomes.

2) Fitness Evaluation: We evaluate the fitness of eachchromosome in a distributed manner. The fitness evaluationprocess is illustrated in Fig. 3 and it consists of five steps:

11

(1) Grouping requests in Rk: Each chromosome i containszi and κi. For those r’s with zir = 1, based on κi, atthe control center, we can divide R into |K| groups, i.e.,Rk,∀k ∈ K.

(2) Request information distribution: For each k, the controlcenter transmits Rk to AV k, e.g., via VANET.

(3) Distributed scheduling: Modern vehicles are generallyequipped with computers and thus each k can solve theindividual Problem 3 simultaneously with other vehicles.Those AVs with empty Rk assigned can skip the com-putation.

(4) Individual cost return: The individual vehicles transmitthe computed costs of scheduling to the control center,e.g., via VANET.

(5) Fitness computation: Based on Theorem 3, the costassociated to the chromosome is the sum of the ob-jective function values of Problem 3 determined by theindividual AVs, i.e., φ(α) =

∑k∈K φk(αk|κr = k).

Then the fitness of the chromosome can be computedas Φ(z, κ) =

∑r∈R ρrzr −

∑k∈K φk(αk|κr = k).2

Note that Φ(z, κ) becomes −∞ if and only if any request rwith zr = 1 is non-admissible. The advantages of undergoingthe above process are three-fold:

(i) The computation time can be dramatically reduced.Among all the computation components in the algorithm,scheduling is the most computationally demanding. Ifeach vehicle can compute their own schedules, all theindividual scheduling subproblems can be solved simul-taneously.

(ii) All entities need to manage the necessary data only. ρris the result of the deal between the customer and thecontrol center. With distributed scheduling, the usage ofρr is restricted to the control center and no vehicles areinvolved. Moreover, after a vehicle solves its schedulingsubproblem, its computed schedule is stored in thatvehicle only, but not the control center nor any othervehicles.

(iii) The amount of communications keeps minimal. In eachevaluation, the only data needed to be communicatedbetween the control center and the vehicles are the re-quests assigned to the individual vehicles (in Step 2) andthe computed scheduling costs (in Step 4). The systemdoes not require a sophisticated communication systemto satisfy the communication requirements.

3) Tabu List: We construct a tabu list τr for each requestr to reduce the size of the search space. τr contains thosevehicles k which cannot serve r. As implied by Theorem 1, if arequest r is not admissible by k, any set of requests containingr will also not be admissible by k. In other words, we willnever need to consider those k in τr when configuring κr.Unlike Tabu Search [40], we do not need to update the tabu

2By abuse of notation, we write Φ(z, κ) = Φ(R, [zr]) to emphasize thestructure of the chromosome.

... ...

1

2

3

4

5

Fig. 3. Fitness evaluation process.

lists during the course of search.3 τr’s are only constructed inthe initialization phase of the algorithm and utilized in bothinitial population generation and mutation.

4) Selection: In each generation, a fraction Xrate of Npopsurvives and the rest of (1 − Xrate) will be replaced by thechildren bled in the processes of crossover. We apply weightedrandom pairing [41] to select the survived chromosomes toperform crossover.

5) Crossover: Crossover is an operator in GA to achieveintensification. In each operation, it manipulates two parentchromosomes to breed two offspring. The offspring inheritthe merits from their parents and thus they tend to have betterfitness values, i.e., higher objective function values of (19a).By Theorem 2, a larger set of requests will improve the fitness.Also based on Statement 3 of Theorem 1, we manipulatethe chromosomes with crossover as follows. Parents i andj reproduce offspring i′ and j′. i′ admits all those r’s as idoes with the same set of vehicles. If there is any k whichis adopted in j but not in i, we randomly adopt one such kin i on those r’s which are not admitted in its parent i. Weproduce an offspring j′ dominantly inherited by the parent jsimilarly. In this way, the offspring are likely to admit morerequests resulting in higher fitness.

6) Mutation: Mutation exhibits diversification to preventthe algorithm from getting stuck in local optimums and webasically follow [41] to design mutation. We control theamount of mutation with a mutation rate µ ∈ [0, 1]. Weapply elitism to the chromosome with highest fitness in thepopulation and only the rest undergo mutation. A mutationoccurs on bit zir of chromosome i and the number of mutations

3As discussed in Section III-B, admission control is completed in the dutyassignment sub-interval once in each operating interval. Such sub-interval isshort so that it is unlikely to have great changes to the positions of the AVs.Thus the tabu lists can be assumed to be static throughout the admissioncontrol process happened in each interval. However, the tabu lists may needto be updated in the next interval as the vehicles may have moved to otherpositions.

12

Define system parameters

Generate initial population

Fitness evaluation

Selection

Crossover

MutationStopping criteria check

Output the best solution found

Fitness evaluation

Init

ializ

atio

n

Iter

atio

n

Fin

al s

tage

Solution backupConstruct tabu lists

Fig. 4. Flow chart of the algorithm.

taken place in each generation is µ× (Npop− 1)× |R|. If weperform mutation on zir, we toggle zir. If zir is changed from0 to 1, we randomly assign κr a k which is not in the tabulist τr. If zir is changed from 1 to 0, we set κr = 0. Tofurther enhance diversification, besides the elite chromosome,each chromosome has a probability of γ to be replaced by arandom chromosome.

D. Algorithmic Design

We basically follow [41] to design the algorithm, whichconsists of three stages: initialization, iterations, and the finalstage. The flow chart of the algorithm is given in Fig. 4. Wemaintain the chromosomes with feasible candidate solutionsduring the whole course of search.

1) Initialization: In initiation, we define all the systemparameters, e.g., Npop and Xrate, and construct the tabulist τr for each r. Then we create the initial population ofchromosomes, each of which is assigned with one randomrequest r associated with a vehicle not in its tabu list τr. Thiscan ensure all chromosomes are initially feasible. We evaluatethe fitness of the initial chromosomes before the iterationsstart.

2) Iterations: In each iteration (or called generation), wemanipulate the candidate solutions held by the chromosomes.Before any modification, we back up the feasible candidatesolutions stemmed from the previous generation. Then weperform selection, crossover, and mutation to manipulate thechromosomes, followed by fitness evaluations. If any chromo-some possesses an infeasible solution, we retain its originalfeasible one from the backup. We check the stopping criteria tosee if we continue with the next iteration or proceed to the finalstage. One commonly used stopping criterion is terminationafter undergoing a certain number of generations.

3) Final Stage: We output the best solution found in thisstage.

In general, the solution method is implemented in a centralmanner at the control center. When evaluating the fitness of

the chromosomes, the scheduling tasks are distributed to thevehicles based on distributed scheduling.

VII. PERFORMANCE EVALUATION

We perform a series of simulations to evaluate differentaspects of the algorithm. We consider a set of real taxi servicedata from [42], containing the pickup and dropoff times, andpickup and dropoff locations of a number of taxi trips servedin the City of Boston. We sample 100 trip data whose pickuptimes happened within a period of 30 minutes in a day of2012 as the transportation request pool. Since no existingtransport can offer flexible shared-ride services as our systemdoes, we adopt the data for our system as follows: the earliestservice starting time as the pickup time of the data, the latestservice starting time as the pickup time plus 15 minutes, themaximum ride time as the actual trip time times 1.5, randomseat occupancy in the range of [1, 5], and 50% of the actualtaxi fare as the charges. The driving distance and travel timebetween any two locations are determined through the GoogleMaps API. Based on [43], we assume that the fuel cost is16 cents per mile. We select five gas stations in Boston asthe refuel stations for AVs. Each vehicle is assumed to beequipped with five seats and we randomly place the vehiclesin the city.

We perform the simulations on a computer with Intel Corei7-2600 CPU at 3.40 GHz and 32 GB of RAM. They areconducted in the MATLAB environment, where the schedulingproblem is addressed with YALMIP [44] and CPLEX [45].We follow [41] to set the GA parameters: Npop = 16,Xrate = 0.5, and µ = 0.15, and we set γ = 0.5. Recallthat, to operate the system for a period of time, we needto do admission control for each operating interval withinthe period. To perform admission control for an interval, weneed to undergo a number of scheduling processes. We try toevaluate the performance of the algorithm incrementally fromthe smallest module. First we evaluate the computation timefor scheduling. In the second test, we evaluate the performanceof the algorithm on solving the admission control problem. Atlast, we examine the profits made when the system operatescontinuously for a period of time.

A. Computation Time for Scheduling

As Problem 1 is an MILP, we assume that CPLEX canreturn the optimal solution if the problem is tractable. So wefocus on the computation time. When we look at Problem1, the numbers of variables and constraints grow exponen-tially with the problem size in terms of the quantities oftransportation requests and vehicles. Hence the computationtime for scheduling grows very fast with the problem size. Fordemonstrative purposes, we focus on small problem instances.We randomly generate 9 cases from the Boston dataset: threecases with three requests, three with four requests, and threewith five requests. All the cases are served with five vehicles.Recall that we have two main ways to address the schedulingproblem: (1) by solving Problem 1 as a whole and (2) bysolving a number of Problem 3 collectively. For the latter,we can further arrange the subproblems to be solved (2.1)

13

Centralized approach Cumulative approach Distributed approach

(Problem solving)

(Problem solving)

(Problem solving)

Request and vehicle data

Schedule

Request and vehicle data

Schedule

Request data

Request data Cost

Fig. 5. Data processing, communications, and computation of the threeapproaches in scheduling.

en masse at the control center or (2.2) separately at theindividual vehicles. Thus, there are three approaches in totaland we call (1), (2.1), and (2.2) the centralized, cumulative,and distributed approaches, respectively. The data processing,communications, and computation of the three approachesare depicted in Fig. 5. For the centralized and cumulativeapproaches, all data need to be collected and gathered at thecontrol center from the passengers and vehicles for processing.After scheduling, the computed schedules will be distributedto the corresponding vehicles. For the distributed approach,the vehicular data are only maintained at the problem solv-ing agents, i.e., that vehicles per se, before and after thecorresponding subproblems being solved. After scheduling,the resulting costs are transmitted back to the control centerfor the subsequent scheduling. When different numbers ofvehicles are involved, the computation time can be noticeablydifferent. To see this, for each of Cases I-IX, we examine allpossible combinations of z and κ (i.e., candidate solutionsfor chromosomes) and check their computation times forscheduling. We consider the time spent on communicationsnegligible as it is usually much smaller when compared withthe computation time. Fig. 6 shows the average computationtimes for feasible schedules with different numbers of vehiclesinvolved in each case. Since the computation time of thecentralized approach grows too fast (e.g., 8.30 s, 69.25 s,and 6.72 × 103 s for 3–5 requests, respectively), the timechanges for the cumulative and distributed approaches wouldhave become indistinguishable if the centralized data had alsobeen displayed. For clearer representation, we skip the resultsfor the centralized approach in Fig. 6. In Fig. 6, some barsare missing because no feasible schedule can be computedwith particular numbers of vehicles involved. For example,one request in Case III cannot be scheduled with any vehicle,and thus, no results are shown for three vehicles for CaseIII. Generally, for the cumulative approach, the computationtime grows linearly with the number of vehicles involvedas more subproblems with similar size need to be solved.

0  0.2  0.4  0.6  0.8  1  

1.2  1.4  1.6  1.8  2  

1   2   3  

Compu

ta(o

n  (m

e  (s)  

No.  of  vehicles  involved  

I-­‐cum   I-­‐dist   II-­‐cum   II-­‐dist   III-­‐cum   III-­‐dist  

(a) 3 requests

0  0.5  1  

1.5  2  

2.5  3  

3.5  4  

4.5  5  

1   2   3   4  

Compu

ta(o

n  (m

e  (s)  

No.  of  vehicles  involved  

IV-­‐cum   IV-­‐dist   V-­‐cum   V-­‐dist   VI-­‐cum   VI-­‐dist  

(b) 4 requests

0  

1  

2  

3  

4  

5  

6  

7  

8  

1   2   3   4   5  

Compu

ta(o

n  (m

e  (s)  

No.  of  vehicles  involved  

VII-­‐cum   VII-­‐dist   VIII-­‐cum   VIII-­‐dist   IX-­‐cum   IX-­‐dist  

(c) 5 requests

Fig. 6. Computation times for scheduling.

For the distributed approach, the computation times withdifferent vehicle sizes are more or less similar because theinvolved subproblems can be handled at different vehiclessimultaneously. While the computation time of the centralizedapproach grows exponentially with the number of requests,that of the cumulative approach increases at a much slower rateand that of the distributed approach is approximately steady.Hence, it is not feasible to adopt the centralized approach. If

14

95%  96%  97%  98%  99%  100%  101%  102%  103%  

5   10   15   20  

Percen

tage  cha

nge  

No.  of  vehicles  

Case  I   Case  IV   Case  VII  

Fig. 8. Profits made with different numbers of vehicles.

the vehicles have sufficient communication and computationcapabilities, we suggest the distributed approach. Otherwise,we can only endorse the cumulative approach for scheduling.

B. Admission Control in an Operating Interval

Next we investigate the performance of the algorithm toaddress admission control for an operating interval. Eachfitness evaluation involves solving the scheduling problemonce and the computation time for each fitness evaluation isdominated by that for scheduling. Moreover, the computationtime of the algorithm depends on the number of fitnessevaluations needed. Since the population size is fixed in everygeneration, the run time of the algorithm can be estimatedfrom the number of generations taken place and the resultsdetermined in Section VII-A. Hence here we focus on thesolution quality instead.

We run the algorithm for Cases I-IX. As we have examinedall candidate solutions, we can acquire the optimal solutionsof these cases. We repeat running the algorithm 20 times foreach case. Fig. 7 shows the average objective function valuecomputed during the course of search for 40 generations. Asabsolute values do not help reveal the performance of thealgorithm, the objective function values are instead normalizedwith the corresponding optimal values to standardize thepresentation.4 For each data point, we also provide the errorbars for the maximum and minimum values computed in the20 repeats. The performance of the algorithm in each caseis similar. The algorithm starts with relatively low qualitysolutions and then converges rapidly to the global optimal ina few generations. The gap between the error bars diminishesafter more generations have been taken place and this furtherconfirms the convergence of the algorithm. When the problemsize increases, it takes slightly more generations to have thealgorithm converged. We can conclude that our algorithm isvery effective in solving the admission control problem.

We further investigate the total profits gained for the testcases with different AV population sizes. We perform thesimulations with the same settings and repeat each test 20times. Fig. 8 shows the average results with respect to 5, 10,15, and 20 vehicles. Since the resultant profit highly depends

4An optimal solution has the normalized objective function value equal toone.

Fig. 9. Cumulative total profits.

0  

10  

20  

30  

40  

50  

60  

70  

80  

90  

100  

1   2   3   4   5   6   7   8   9   10   11   12   13   14   15   16   17   18   19   20  

Cumula&

ve  re

quests  adm

i0ed

 

Interval  

Case  1   Case  2  

Fig. 10. Cumulative numbers of admitted requests.

on the parameters of the respective requests and vehicles,the total profits gained from different cases are not directlycomparable. Instead for each case, we show the percentagechange of profit by normalizing the results with the profit madewith 5 AVs. Since all cases show similar trends, for clearerpresentation, we give the results for Cases I, IV, and VII inFig. 8 only. In general, the more vehicles available, the higherprofit can be made. However, the increase of profit is marginal;when compared with 5 AVs, the increase is just 1−2% in thepresence of 20 AVs. The reason is that more available vehiclesmay result in more economical routes but the total distancetravelled would not be shortened significantly.

C. Admission Control in Consecutive Operating Intervals

Here we consider operating the system consecutively for aperiod of time to entertain the 100 requests in the transporta-tion request pool. We consider two cases of different operatinginterval durations. In Case 1, there are 10 intervals, in each ofwhich 10 random requests from the pool are to be scheduled.If a request is successfully admitted in an interval, it will beeliminated from the pool. Otherwise, it will be consideredagain in the subsequent intervals. The setting for Case 2 issimilar but we consider total 20 intervals with 5 requests beingprocessed in each interval. Five vehicles are arranged to servethe requests in both cases and we apply our algorithm to eachinterval for admission control. In other words, we perform10 and 20 admission control processes in Cases 1 and 2,respectively.

15

Case I Case II Case III

Case IV Case V Case VI

Case VII Case VIII Case IX

Fig. 7. Evolutions of the algorithm in solving admission control.

Fig. 9 shows the profit accumulated along the intervals, inwhich we consider the duration of one interval for Case 1is that of two intervals for Case 2. Note that the cost is theactual expense on gas based on the traversed distance and therevenue gained from serving each request is the discountedresult of having 50% off from the real fare as if the requestwould be served by a normal taxi in Boston. The discountis used to compensate for the inconvenience of ride sharingand possibly longer ride time. This discount rate may bealready attractive to many people to adopt our system insteadof the normal taxi service. Hence the profit shown can beprojected to a real business running in a similar scale. Fig. 10provides the numbers of successfully admitted requests alongthe same interval horizon as in Fig. 9. We can see that Case2 can produce more profit by successfully admitting moretransportation requests. With the same number of vehicles inservice, the smaller the number of requests to be scheduled inan interval, the higher the success rate of admission control is.In real situation, we normally cannot dramatically increase thesize of the AV fleet and we would not intentionally reduce thenumber of AVs in service. On the other hand, it is much easierto adjust the number of requests to be scheduled each time bycontrolling the duration of each operating interval. In general,the shorter the interval, the smaller number of requests thereare. Therefore, we would suggest to set the operating intervalshorter, resulting in fewer requests to be scheduled each timeand higher profits. Moreover, this will make the schedulingproblem smaller by requiring shorter computation time to runthe algorithm.

VIII. CONCLUSION

With advancements in technologies, AVs become feasibleand can run on the roads. Various vehicular wireless communi-cation technologies allow AVs to be connected and respond co-operatively to instantaneous situations. This constitutes a newform of public transport with high efficiency and flexibility.In this paper, we propose the AV public transportation systemsupporting point-to-point services with ride sharing capability.The system manages a fleet of AVs and accommodates anumber of transportation requests. We focus on two majorproblems in the system: scheduling and admission control.The former is to configure the most economical schedules androutes for the AVs in order to satisfy the admissible requests.The latter is to determine the set of admissible requestsamong all requests so as to produce maximum profit. Weformulate the scheduling problem as an MILP. The admissioncontrol problem is cast as a bilevel optimization problem,in which the scheduling problem is set as a constraint. Wepropose a GA-based solution method to address admissioncontrol. We perform a series of simulations with a real taxiservice dataset recorded in Boston and the simulation resultsshow that our solution method is effective in solving theproblem. By shortening the operating intervals, the systemcan curtail the computation time required to solve the problemby limiting the quantity of the submitted requests and it canalso produce higher profit cumulatively. To summarize, ourcontributions in this paper include: (i) designing the AV publictransportation system, (ii) formulating the scheduling problem,(iii) developing distributed scheduling, (iv) formulating theadmission control problem, (v) introducing the concept of

16

admissibility and deriving the related analytical results, (vi)proposing an effective method to solve the admission controlproblem, and (vii) validating the performance of the solutionmethod with real-world transportation service data.

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