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A fuzzy clustering approach to real-time demand-responsive
bus dispatching control
Jiuh-Biing Sheu
∗Institute of Traffic and Transportation, National Chiao Tung University, 4F, 114 Chung Hsiao W. Rd., Sec. 1, Taipei 10012, Taiwan, R.O.C.
Received 20 August 2003; received in revised form 3 February 2004; accepted 7 May 2004 Available online 7 June 2004
Abstract
Quick response (QR) to passenger needs is a key objective for advanced public transportation systems (APTS), and it has become increasingly important for contemporary metropolitan bus operations to gain a competitive ad-vantage over private transportation. This paper presents a real-time control methodology for demand-responsive bus operations that respond quickly to passenger needs. The proposed method primarily involves two levels offunction-ality: (1) short-term forecasting of passenger demands using time-series prediction models, and (2) identification of service strategies coupled with the associated bus service segments using fuzzy clustering technologies in response to variances in passenger demand attributes and traffic conditions. The proposed bus operations method identifies the demand-responsive vehicle service strategies primarily according to the predicted up-to-date attributes ofpas-sengers’ demands, rather than deterministic passenger arrival rates, which were generally used in previous literature. In addition, the variation oftraffic conditions along bus lines is considered in the proposed method. Results from numerical studies using real data ofpassengers’ demands, including passenger volume at each bus stop and the passenger origin-destination (O-D) patterns, are presented to demonstrate the effectiveness of the proposed method for real-world applications.
© 2004 Elsevier B.V. All rights reserved.
Keywords: Forecasting; Fuzzy clustering; Advanced public transportation systems; Bus dispatching
1. Introduction
Correctly identifying passenger demands and quickly responding to those needs with advanced bus operations control strategies are vital to the development ofadvanced public transportation systems
∗Tel.: +886-2-2349-4963; fax: +886-2-2349-4953
E-mail address:[email protected](J.-B. Sheu).
0165-0114/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2004.05.006
(APTS) in urban areas. Generally, traditional public transportation systems are less convenient than pri-vate transportation, which can be quantified to a certain extent by passengers’ satisfaction with the total travel time, including walking time, waiting time and in-transit time. To improve the competitiveness of public transportation systems in urban areas, there have recently been increased applications ofadvanced technologies, such as automatic vehicle location (AVL) systems, wireless communication systems, and electronic fare payment systems for urban transit. However, APTS must also rely on advanced vehicle dispatching strategies. This is especially true since tremendous advances in information and communi-cation technologies have contributed to dramatic changes in passenger demands for public transportation services, and have exerted a profound influence on the operations of fleet management in public trans-portation systems.
Despite the importance ofadvanced demand-responsive bus transit operations control in APTS, there have been few related studies in previous literature, and this can be seen in the following two aspects.
First, most research in the field ofAPTS is devoted to analytical studies and related discussions for the application ofnew technologies in APTS, such as the assessment ofsystem performance based on only limited case studies. Adoption ofvarious intelligent transportation systems (ITS) technologies in public transportation systems and some local application cases are described in both Khattak et al. [18] and Hansen et al.[12]. Those studies indicate that the utilization ofadvanced ITS technologies such as global positioning systems (GPS), electronic fare payment systems, and automatic passenger counting systems appears to permit significant improvement in public transportation systems, such as metropolitan bus operations. Similar arguments can also be found in Hickman[13]. In addition, the idea ofusing historical AVL data to improve bus service, as proposed by Horbury[15], also helps to demonstrate the potential benefits ofthe emerging ITS technologies to the public transportation operations. Nevertheless, there is a lack offurther research to investigate the operational problems ofthese new technologies in APTS, using the analytical results from previous literature.
Second, although there is some literature on the operations ofpublic transportation, the validity of published methods seems to be subject to specific operational conditions, such as deterministic passen-ger arrival rates and fixed schedules, which may cause these previous methods to be problematic for real-time operations. In these previous studies, two types ofmethodology are noteworthy: simulation-based [1,23]and optimization-based [6,8,25,33], both ofwhich are broadly employed in the previous literature. FRABSIM, a fixed-route accessible bus service simulation model proposed by Adesanya[1]
was designed to evaluate the performance ofbus service under various scenarios ofpassenger demands. One notable application example ofFRABSIM is the estimation ofthe possible effects ofdifferent bus service alternatives on the performance of bus operations when serving persons in wheelchairs. Similarly, Santhakumar et al.[23]developed a simulation model to analyze the effects of diverse bus operational strategies on system performance, including express bus transit service. Given historical demand data collected from four bus lines, in order to improve the performance of local fixed-route bus operations, their numerical results highly recommended the strategy ofexpress bus transit operations coupled with re-configuration ofbus stops. In addition, Tsao et al.[33]proposed a nonlinear programming model to determine optimal solutions in terms ofthe number ofbus stops in each zone and the optimal dispatch-ing headway associated with each zone-based bus route, given passenger demand data. In addition to zonal service, short-turn service strategies have also attracted substantial research to address issues of heavy passenger demand, and considerable advances can be found in previous studies, including Furth
[8]and Site et al.[25]. One notable generalization obtained from recent related studies is that the effect of variation ofpassenger distribution on bus routing and scheduling has been given increasingly attention.
The specific real-time deadheading operational problem is elaborately investigated in Eberlein et al.[6], which assumes that such demand-related parameters as passenger arrival rates, the passenger alighting proportion at each transit station, and the boarding and alighting times per passenger are all constant. Nevertheless, the dynamics ofsuch demand-related parameters as the parameters mentioned above in Eberlein et al.[6], and the effect caused by time-varying traffic conditions along a bus route remain as critical issues in the existing approaches ofreal-time transit operations control.
Overall, it should be noted that the difficulties in formulating and solving real-time demand-responsive bus dispatching control problems are rooted in the uncertainty and vagueness ofshort-term changes in time-varying passenger origin-destination (O-D) flows and corresponding time-varying traffic flow conditions in bus routes. Correspondingly, it seems that any advanced bus control methodology must have the capability ofidentifying time-varying passenger O-D demand patterns coupled with corresponding traffic flow conditions so as to implement appropriate bus service strategies for quick response.
Considering the aforementioned uncertain effects on bus operations induced by variations in passenger O-D flow patterns as well as time-varying traffic conditions, we propose a real-time demand-responsive bus operations control method. This method utilizes fuzzy-clustering techniques and time-series predic-tion models for identificapredic-tion ofpassenger demand patterns and prior predicpredic-tion ofthe short-term changes in passenger volumes in the operations ofbus routing. In contrast with existing bus operations control approaches, the proposed method has several distinctive features, as follows.
(1) Conceptually, the proposed methodology stems from the concept of demand-oriented business op-erations, which advocates the primary goal ofsatisfying customer needs. Following this approach, we propose that the operations of bus routing should respond efficiently and effectively to short-term changes in passenger demand, thus satisfying passenger needs as much as possible.
(2) For real-time operations control, a two-stage QR demand-responsive bus operations control approach is proposed. Herein, we reiterate that the identification ofappropriate bus service strategies should be made before bus routing is established in order to respond to short-term changes in passenger demand patterns. Such a pre-trip measure is very important, especially in the APTS operational environment, when real-time information on passenger volumes and their O-D attributes is obtainable using advanced information and communication technologies. Therefore, two sequential procedures, including prior prediction oftime-varying passenger demand attributes and real-time identification ofbus service strategies, are used in the proposed method.
(3) For methodology, we integrate the fundamentals of fuzzy clustering and time-series prediction ap-proaches to address the real-time demand-responsive bus operations control problem. In the scenario ofprior prediction ofshort-term passenger demands, a time-series prediction model is developed to forecast the short-term changes in passenger demands in the process of bus routing. This is done using measured real-time passenger-related information together with historical data. Fuzzy clus-tering techniques are then utilized to identify appropriate bus service strategies in response to the variations in time-varying passenger demands and traffic conditions along bus lines. In contrast, pre-vious approaches, both optimization-based and simulation-based, are not used in this study due to their inadequacy in responding to the dynamics ofpassengers’ O-D flows and their inflexibility in dealing with uncertain and complicated time-varying traffic operational conditions for real-time bus operations control. At least, there is a lack ofevidence in the previous literature to show that the ex-isting approaches for real-time bus operations control can perform well in a dynamic bus operational environment with no ideal assumptions for either model parameters or input data.
It is worth mentioning that any advanced bus control strategy can be operated dynamically in the spatial (e.g., various bus stop strategies) and/or temporal (e.g., time-varying bus dispatching frequencies) domains to respond to the short-term changes oftime-varying passengers’ demand patterns. Nevertheless, the scope ofour current study is limited to dealing with the dynamic bus operations in the spatial domain. Correspondingly, appropriate bus service strategies are determined in response to the short-term changes ofpassenger O-D flows subject to constant bus dispatching frequency and given fleet size for bus operations.
Furthermore, it is worth noting that fuzzy clustering is a part of fuzzy data analysis, and can be viewed as an improved clustering technique. Compared to classical clustering techniques, the distinctive feature of fuzzy clustering is that fuzzy clustering utilizes fuzzy partitioning so that a given data point can belong to several groups with the degree ofbelongingness bounded within the range of0 and 1. However, utilizing classical clustering techniques, any given data point can be assigned to one and only one group among mutually exclusive data groups. In contrast with classical clustering techniques, this feature of fuzzy clustering techniques can make pattern recognition more flexible for real-time applications, especially in cases where patterns ofdata attributes change rapidly. For detailed descriptions ofproperties offuzzy clustering techniques and related applications, the reader is referred to the early literature[3,10,31] as well as our previous research[16,24].
The rest ofthe paper is organized as follows. The architecture ofthe proposed bus operations control method and the methodology development are presented in Section 2. Section 3 depicts numerical results obtained from cases studied. Finally, concluding remarks are made in Section 4.
2. Methodology development
Throughout this paper, the following three basic assumptions are postulated to facilitate development ofthe proposed methodology.
(1) Bus dispatching frequency is set to be time-invariant. Correspondingly, the prototype of the proposed method is limited to fixed-schedule operations, although it is claimed to be capable ofadjusting bus service strategies in real time in response to short-term changes in passenger demands. Such an assumption is postulated to facilitate modeling the proposed discrete-time time-series prediction model in the case offixed time intervals; however, it implies limitations ofthe proposed method for the implementation ofmore advanced operational strategies, such as dynamic headway control. (2) Real-time passenger demand data are assumed to be collectable via advanced ITS technologies such
as automatic passenger counting systems.
(3) The changes ofpassenger O-D patterns in the time interval between bus dispatching from the terminal and arriving at the origin bus stop are not considered. Correspondingly, the passenger O-D patterns are assumed to be known in a given time interval, and measurable directly from related advanced technologies such as electronic fare payment systems.
Accordingly, a real-time demand-responsive bus operations control methodology is proposed, and
Fig. 1illustrates the framework of the proposed control method. As shown inFig. 1, there are two primary mechanisms, (1) short-term passenger demand forecasting and (2) identification of bus service strategies coupled with the served bus stops in given bus routes. Here the real-time control ofdemand-responsive
time interval k=k+1 Real-time passenger O-D flows Historical data of passenger O-D patterns
Short-term passenger demand forecasting
Identification of passenger demand patterns
Determination of bus service strategies and segments Time-varying
traffic conditions
Fig. 1. Framework ofthe proposed bus operations control method.
bus operations refers to the identification of appropriate bus service strategies each time when a bus is dispatched to serve passengers in a given route. Therefore, such proposed real-time bus dispatching control methodology relies mainly on both the mechanisms ofreal-time passenger demand data collection and quick response to the changes ofcorresponding passenger demand patterns for determination ofsuitable bus service strategies at that moment. However, the highlight ofthe proposed method is on the latter, i.e., the capability ofresponding to the changes ofpassenger demand patterns, which can be found in the corresponding model formulation in the following two subsections. As to the real-time data collection, it is assumed to be available via other ITS-related technologies, as mentioned in the second assumption. The details in the fundamentals of the aforementioned two major functions are described in the following two subsections.
2.1. Short-term passenger demand forecasting
This function serves to forecast the time-varying passenger volume at each bus stop in a given time interval k of bus headway. To execute the aforementioned short-term passenger demand forecasting, a time-varying passenger demand variablei(k) is introduced, and derived using the fundamentals of exponential smoothing-based prediction approaches, wherei(k) is defined as the time-varying passenger volume at a given bus stop i in the time interval k.
It is noteworthy that although there have been a certain amount ofshort-term forecasting techniques, e.g., basic and extended time-series prediction methods, filtering techniques, neural network based ap-proaches and nonparametric regression models, published previously[2,5,7,11,21,22,27–30,36], most ofthem aim at traffic state prediction (e.g., speed, volume and occupancy) rather than passenger trip generation, as conducted in this study. In addition, the analytical results obtained by comparing the
Fig. 2. Relationship ofbus dispatching and time-varying passenger demands.
existing typical techniques may vary in different studies due to different study purposes and evaluation criteria[20,26,28,34]. Considering the need ofthis study and the corresponding requirement in terms ofthe functionality of passenger demand forecasting performed in the proposed method, a simple exponential smoothing technique is employed for convenience.
In the proposed demand-responsive bus operations control method,i(k) should be predicted at the beginning ofthe time interval k, before bus dispatching, and is derived as follows. As shown inFig. 2, a bus is dispatched at the start ofa given time interval k to serve the passengers at a given bus stop i at the beginning ofthe next time interval k+1. Herei(k − 1) is the real increased demand volume of passengers measured from passenger counting systems in a given time interval k−1; andi(k−1|k−2) is the increased demand volume ofpassengers predicted at a given bus stop i in the given time interval k−1. Utilizing the theories ofexponential smoothing methodology, the prior prediction ofshort-term changes in the passenger volume at the given bus stop i in the time interval k (i(k|k − 1)) can be formulated as
i(k|k − 1) =×i(k − 1) + (1 −)
×i(k − 1|k − 2), (1)
wherecorresponds to a user-specific weight, which is set to be 0.63 in this study. Conveniently,is here determined utilizing the SAS statistical analysis package, which searches for the optimal value ofwith the objective ofminimizing the sum ofsquared prediction errors. First, we sum up the real passenger volume remaining at a given bus stop i at the end ofthe time interval k−1i(k −1|k −1) and the predicted valuei(k|k − 1), both measured at the beginning ofthe time interval k. From this, we then have the prior prediction ofi(k) at the beginning ofthe given time interval k (i(k|k − 1)) as
i(k|k − 1) =i(k − 1|k − 1) +×i(k − 1)
+ (1 −) ×i(k − 1|k − 2), (2)
Herein, the estimate of i(k|k − 1) is used as one of determinants in the following fuzzy clustering mechanism to identify appropriate bus dispatching strategies in response to the time-varying passenger demands in the time interval k. In addition, the passenger volume remaining at the given bus stop i at the
end ofthe time interval k (i(k|k)) should be updated as i(k|k) = [i(k − 1|k − 1) +i(k)] − ∀ f i(k) , (3)
where i(k) is the real increased demand volume ofpassengers, as measured by passenger counting systems at the given bus stop i in a given time interval k; and fi(k) represents the real volume of passengers which are served by a given bus using a given bus dispatching strategyat the given bus stop
i, as measured by on-board electronic fare payment systems in the given time interval k. It is noteworthy
that the updated value ofi(k|k) is used for recursive estimation in the next time interval k+1. In addition, further details referring to testing the model’s validity can be found elsewhere[32].
2.2. Identification of bus service strategies
This mechanism serves to identify appropriate bus service strategies which are implemented to satisfy both time-varying passenger demands in a given time interval k. To perform this function in real time, we propose a fuzzy clustering-based algorithm, which analyzes multiple state variables in relation to both time-varying passenger demand attributes and traffic conditions. This algorithm then determines a suitable bus service strategy together with its service segment in the given bus route, in quick response to the time-varying passenger demands under the present traffic conditions along the given bus route. The entire architecture of fuzzy-clustering functionality built in this mechanism is shown inFig. 3.
Herein, four well-known bus service strategies are considered as candidates for demand-responsive bus service strategies, responding to various conditions ofpassenger demand patterns, as well as traffic conditions. They are: (1) all-stop service, (2) express service, (3) short-turn service, and (4) zonal service.
Fig. 4graphically characterizes the specific en-route stopping operations ofthese bus service strategies. Details relevant to their advantages and limitations can be found in previous studies[9,33,35,37]. In the proposed method, all-stop service is regarded as a basic bus service strategy which aims for the case that passengers’ O-D trip flows disperse at bus stops on a given bus route, and the proportion ofshort-distance passenger trips is relatively high in a given time interval. In contrast, the other strategies differ from the all-stop service strategy in terms ofthe specific patterns ofpassenger trip flows that they mainly serve. Note that there are no limitations on the number ofbus service strategies potentially used in the proposed method. Other sophisticated bus service strategies such as deadheading service and skip-stop service can also be involved in further development of a comprehensive demand-responsive bus operations control system. Nevertheless, this study aims to investigate the feasibility of the proposed approach as well as the primary procedures ofmodel formulation, and thus only certain bus service strategies are illustrated. To perform real-time bus service strategy identification, four groups of state variables are specified in this study, and are defined as follows.
(1) s∗(k) represents the time-varying degree ofpassenger trip centralization in a specific short-turn bus service regions∗in a given time interval k, and is denoted by
s∗(k) = ∀os∗ ∀ds∗ vos∗,ds∗(k)/los∗,ds∗ ∀oT ∀dT voT,dT(k)/loT,dT , (4)
passengers' demand
data traffic data
demand-related state
variables traffic condition index
initialization of clustering-center vectors associated with bus service
strategies
Determination of multi-variable datum of passenger demand
Computation of the fuzzy correlation matrix
identification of bus service strategies and the associated service
segments all-stop service group of express service group of short-turn service group of zonal service
Fig. 3. Functional architecture for the identification of bus service strategies.
whereos∗ andds∗ represent a given pair oforigin and destination bus stops located in the specific short-turn service regions∗, andlos∗,ds∗ is the geographic distance betweenos∗ andds∗; similarly,oT
anddT correspond to a given pair ofthe origin and destination bus stops served by a given all-stop bus coded with T in a given bus route, and loT,dT is the geographic distance betweenoT anddT; vos∗,ds∗(k) represents the time-varying passengers’ O-D trip volume between os∗ andds∗, estimated
in a given time interval k; and similarly,voT,dT(k) represents the time-varying passengers’ O-D trip volume between a given pair ofthe originoT and the destination dTin the given bus route T in a given
time interval k. Herein, the time-varying passengers’ O-D trip volumesvos∗,ds∗(k) and voT,dT(k) are estimated in real time using the previous mechanism ofthe proposed system, denoted respectively by vos∗,ds∗(k) =os∗(k|k − 1) × pos∗,ds∗(k − 1|k − 1), (5) voT,dT(k) =oT(k|k − 1) × poT,dT(k − 1|k − 1), (6)
where the prior predictions ofpassenger demands (os∗(k|k − 1) andoT(k|k − 1)) at given bus stops os∗ andoT are predicted using Eq. (2);pos∗,ds∗(k − 1|k − 1) represents the time-varying percentage
ofpassengers originating at a given bus stopos∗ and getting off at a given bus stopds∗ measured at time step k−1; and similarly, poT,dT(k − 1|k − 1) refers to the time-varying percentage of passen-gers originating atoT and alighting atdT. Note that as described previously in assumption (3), both
Fig. 4. Illustration ofen-route stopping operations for the specified bus service strategies.
pos∗,ds∗(k − 1|k − 1) and poT,dT(k − 1|k − 1) are given, and can be determined employing advanced
APTS-related technologies.
(2) e∗(k) denotes the time-varying degree ofpassenger trip centralization in a specific express service regione∗in a given time interval k, and is given by
e∗(k) =
∀oe∗
∀de∗voe∗,de∗(k)
ne∗ ∀i i(k|k − 1) N , (7)
whereoe∗ andde∗represent a given pair ofthe origin and destination bus stops located in the specific express service region e*;ne∗represents the total number ofbus stops in the specific express service
regione∗; N is the total number ofbus stops in a given bus route; and similar to the definition of vos∗,ds∗(k), voe∗,de∗(k) is referred to as the time-varying passengers’ O-D trip volume between oe∗
andde∗, estimated in a given time interval k.
(3) z∗(k) represents the time-varying degree ofpassenger trip centralization in a specific zonal service regionz∗in a given time interval k, and is denoted by
z∗(k) = ∀oz∗ ∀dz∗ voz∗,dz∗(k) ∀i i(k|k − 1) , (8)
whereoz∗ anddz∗ represent a given pair oforigin and destination bus stops located in the specific zonal service regionz∗; andvoz∗,dz∗(k) is referred to as time-varying passengers’ O-D trip volume betweenoz∗anddz∗, as predicted in a given time interval k.
(4) (k) denotes the time-varying congestion index, which is used to quantify the effect of road congestion on the performance of diverse bus service strategies in a given time interval k. In addition to the variety ofpassenger trip patterns, the severity ofroad traffic congestion along a given bus route is also consid-ered as a critical factor in determining the performance of real-time bus operations control in the study. Correspondingly, the bus service strategies determined in real time by the proposed method may vary with present traffic conditions using the specified traffic state variable. Conveniently employing the advantages and characteristics of S-shaped fuzzy membership functions suggested by Zimmermann
[38] and Kandel [17], (k) is herein formulated as an S-shaped fuzzy membership function with respect to the aggregate road traffic occupancy in the given bus route T (¯uT(k)), and is denoted by
(k) = F ( ¯uT(k); a, b, c) = 0, for ¯uT(k)a
=2 ¯uT(k)−a c−a 2 , for a¯uT(k) < b =1 − 2 ¯uT(k)−c c−a 2 , for b¯uT(k) < c =1 f or ¯uT(k)c (9)
where ¯uT(k) corresponds to the route-based occupancy value which is determined by averaging the measured occupancies on the links located in the given bus route T in the time interval k; the parameters a, b, and c (termed the primary values in fuzzy theories), represent the pre-set thresholds to characterize the degree ofroad traffic congestion indicated by the linguistic terms light, medium, and heavy congestion, respectively.
Given the real-time passenger demand data as well as the measured raw traffic data in a given time interval k, we then have an (ns+ ne+ nz+ 1)×1 time-varying state variable vector ((k)) as:
(k) = Col s(k),e(k),z(k),(k), (10)
where ns, ne, andnz are defined as the numbers ofbus stopping strategies associated with the types ofshort-turning (s for short), express (e for short), and zonal services (z for short), respectively;s(k), e(k), andz(k) are the (ns × 1), (ne× 1) and (nz× 1) state vectors which involve the groups ofstate variables associated with the types ofshort-turning, express, and zonal services, respectively, and can be further expressed as s(k) = [s1(k),s2(k), . . . ,sns(k)] T, (11) e(k) = [e1(k),e2(k), . . . ,ene(k)] T, (12) z(k) = [z1(k),z2(k), . . . ,znz(k)] T. (13)
Herein, the elements present ins(k),e(k), andz(k) represent the state variables estimated, respec-tively, under the bus operational conditions ofshort-turning (s for short), express (e for short), and zonal services (z for short) associated with corresponding service regions. Accordingly, all the state variables involved ins(k),e(k), andz(k) are estimated using Eqs. (4), (7) and (8), respectively.
Utilizing the time-varying state variables, a fuzzy clustering-based algorithm is proposed to determine ifit is necessary to replace the basic all-stop service strategy with another appropriate bus service strat-egy coupled with specific service regions in response to the time-varying patterns ofpassengers’ trip demands and the present traffic conditions along a given bus route in a given time interval. Otherwise, the
conventional all-stop bus service strategy remains in the given time interval. The following summarizes the primary steps executed in this algorithm.
Step 0: Initialize the vector ofclustering centers associated with the specified bus service strategies,
where each clustering center represents the typical measurements ofstate variables associated with a given bus service strategy and a specific service region∗. Conveniently, the mean vector of (k), denoted as follows, is employed as the(ns+ ne+ nz+ 1) × 1 clustering-center vector ().
= E[(m)] = [E[s(m)], E[e(m)], E[z(m)], E[(m)]]T, (14) where m denotes any given time interval; andE[s(m)], E[e(m)], and E[z(m)] are given by
E[s(m)] = ¯us1, ¯us2, . . . , ¯usns
T
, (15)
E[e(m)] = ¯ue1, ¯ue2, . . . , ¯uene
T
, (16)
E[z(m)] = ¯uz1, ¯uz2, . . . , ¯uznz
T
. (17)
The elements ofare time-invariant, and pre-determined using the historical data, which were collected under the implementation ofspecific bus service strategies.
Step 1: Calculate the time-varying state variable vector(k) in a given time interval k. As mentioned
previously, these time-varying state variables can be readily estimated in real time using the collected passengers’ trip patterns and traffic data, according to the aforementioned definitions of state variables.
Step 2: Compute the fuzzy correlation matrix. In this stage, we attempt to estimate the time-varying
(ns+ ne+ nz+ 1) × (ns+ ne+ nz+ 1) fuzzy correlation matrix (W(k)), which is given by
W(k) (ns+ne+nz+1)×(ns+ne+nz+1) = w,(k) W,s(k) (1×ns) W,e(k) (1×ne) W,z(k) (1×nz) Ws,(k) (ns×1) Ws,s(k) (ns×ns) Ws,e(k) (ns×ne) Ws,z(k) (ns×nz) We,(k) (ne×1) We,s(k) (ne×ns) We,e(k) (ne×ne) We,z(k) (ne×nz) Wz,(k) (nz×1) Wz,s(k) (nz×ns) Wz,e(k) (nz×ne) Wz,z(k) (nz×nz) . (18)
Herein, each given element of W(k) (wg,h(k)) represents the degree ofsimilarity between a given datum pair g and h, as estimated by the following rules:
IFg = h, THEN wg,h(k) = 1, (19)
ELSE IFwg,h(k) ∈ Row(W(k)) ∨ Col(W(k)), THEN wg,h(k) = wh,g(k) = 1 − 1 3 j=0 {g(k − j) − ¯uh+(k − j) − E[(k − j)]}2, (20) ELSE wg,h(k) = wh,g(k) = 1 − 2¯ug− ¯uh
, (21)
where is a pre-determined value set for the upper and lower boundaries ofwg,h(k), namely 1 and 0, respectively, and here it is set to be 15 in the numerical study. It is also worth noting that according to Eqs. (19)–(21), W(k) turns out to be a symmetrical matrix.
However, according to the fundamentals of fuzzy clustering techniques [19], the estimated fuzzy correlation matrix W(k) should be processed through the composition operation such that the following condition, Eq. (22), holds for further use in sample clustering.
˜
W(k) ◦ ˜W(k) = ˜W(k), (22)
where ˜W(k) represents the processed fuzzy correlation matrix of W(k). To generate ˜W(k), we conduct a
routine ofthe max-min composition operation with respect to each given element ofW(k) (e.g., wg,h(k)) as:
W(k) ◦ W(k) =ns+nmaxe+nz+1
t=1 {min[wg,t(k), wt,h(k)]} (23)
until the condition shown in Eq. (22) is satisfied. Then, we have the processed fuzzy correlation matrix
˜
W(k), which is represented mathematically by
˜ W(k) (ns+ne+nz+1)×(ns+ne+nz+1) = ˜w,(k) W˜ ,s(k) (1×ns) ˜ W,e(k) (1×ne) ˜ W,z(k) (1×nz) ˜ Ws,(k) (ns×1) ˜ Ws,s(k) (ns×ns) ˜ Ws,e(k) (ns×ne) ˜ Ws,z(k) (ns×nz) ˜ We,(k) (ne×1) ˜ We,s(k) (ne×ns) ˜ We,e(k) (ne×ne) ˜ We,z(k) (ne×nz) ˜ Wz,(k) (nz×1) ˜ Wz,s(k) (nz×ns) ˜ Wz,e(k) (nz×ne) ˜ Wz,z(k) (nz×nz) . (24)
Step 3: Identification ofappropriate bus service strategies by the following truncation rules. Let ˜wg,h(k) be a given element ofthe processed fuzzy correlation matrix ˜W(k), and be truncated as
˜wg,h(k) = 1, if ˜wg,h(k)1,
= 0, otherwise, (25)
where 1 is a pre-determined threshold, which is bounded by the range of0 and 1. Then, comparing
the first column vector (i.e., Col1( ˜W(k))) with the other column vectors of ˜W(k), a given bus service
strategyassociated with a specific service region∗is identified for serving passengers in the given time interval k in case of Col1( ˜W(k)) = Col∗( ˜W(k)); otherwise, an all-stop bus is dispatched in the given
time interval k.
Note that the aforementioned threshold 1 appears to influence the result offuzzy clustering in the
proposed method, and the effect caused herein by1 may be similar to that caused by a fuzzifier, as
discussed in Hoppner et al. [14]. The corresponding effect of 1 on the identification ofbus service strategies is illustrated inFig. 5, which implies that different values of1may result in different sets of bus service strategies. Theoretically, the greater1is, the faster ˜wg,h(k) becomes 0, which may result in more groups. One extreme case is that the all-stop bus service strategy may remain to be implemented in any time interval ifwe choose1to be 1. It is also noteworthy that the identification ofmultiple bus service strategies is allowed in the proposed method when more than one strategy is identified simultaneously in a given time interval, and thus multiple buses associated with different service strategies and service regions can be dispatched in the given time interval. Nevertheless, such a multi-bus dispatching strategy is not absolutely needed in all practical operational cases because the limited operational resources (e.g., available bus drivers and bus fleet size) and induced operational costs should also be considered, as in
value
demand
data sample strategy-1 strategy-2 strategy-3 strategy-4... strategy-n
one group two groups "n+1" groups "n" group 1 0.85 0.75 0.2 0 λ1
Fig. 5. Illustration ofthe effect of1on the determination ofbus service strategies.
the following numerical study. It is therefore suggested that 1 should be carefully determined using
historical data before being used for real-world applications, and here is preset to be 0.96, according to our previous calibration results [32]. Related discussions on the specification offuzzifiers can also be found in the previous literature[14].
3. Illustrative application
To demonstrate the potential advantages ofthe proposed methodology for the operations ofdemand-responsive bus operations control, the model was applied to a hypothetical but realistic case in which the real passenger demand data collected from the Taipei City bus route 255 was used to evaluate the performance of the proposed approach in comparison with the existing all-stop bus service strategy implemented in the given bus route.
Bus route 255 primarily serves the community trips along one radial corridor ofTaipei, with one end in the CBD during peak hours, and with a total route length of30.5 km. The total number ofstops on the route is 50, as illustrated inFig. 6, where stop 25 serves as a transfer stop for the connection with the metropolitan mass transit system. The existing dispatching frequency is one bus per 30 min. To generate a real database used for the numerical study, the 30-min passenger volume profiles in morning peak hours were collected from 06:00 am to 08:00 am over one month at the study site.
The proposed algorithm was then utilized to determine the bus service strategies implemented in the specific four sequential test intervals, namely 06:00–06:30, 06:30–07:00, 07:00–07:30, and 07:30–08:00, in response to the time-varying passenger demand patterns. Note that herein, the processed time-varying passenger volume at each bus stop in a given test time interval is directly used as the data basis for the determination ofboth bus service strategies and the associated bus stops served. Corresponding model testing for the short-term passenger demand forecasting executed in the first mechanism of the proposed algorithm was completed previous to this study[32], and thus is not presented here.
To quantitatively assess the performance of the proposed method with respect to the improvement of bus operations for a given bus service route, we compared the results obtained from the proposed approach and from the original all-stop service strategy utilizing three proposed measures, defined as follows:
44 45 46 47 48 49 50 43 42 41 40 38 37 36 39 10 34 33 32 31 30 29 9 12 21
Fig. 6. Illustration ofthe Taipei City bus route 255 in the study.
(1) The time-varying transportation cost for routing with the bus service strategyin a specific service region∗in a given time interval k (T C∗(k)), which is denoted by
T C∗(k) = t1× L∗(k), (26)
whereL∗(k) is the total distance for a given bus routing with a given bus service strategy, in a specific service region∗and in a given time interval k; andt1 is the unit cost for transportation, as determined
according to statistics provided by the municipal bus administration ofTaipei.
(2) The total monetary value ofpassengers’ in-vehicle travel time under the service associated with a given bus routing with service strategy and a specific service region ∗ in a given time interval k (P C∗(k)), as given by P C∗(k) = t2× ∀o∗ ∀d∗ vo∗,d∗(k) × lo∗,d∗ ∗(k) , (27)
wherevo∗,d∗(k) represents the time-varying passenger trip volume traveling from a given bus stop o∗to a given bus stopd∗served by a bus associated with a given bus service strategyin a give time interval k; lo∗,d∗(k) is the geographical distance between bus stops o∗andd∗; ∗(k) is the average running speed
ofthe bus associated with the given bus service strategyin the specific service region∗; andt2is herein
defined as the average monetary value ofpassengers’ in-vehicle travel time in the unit ofUS dollars per hour. Conveniently, t2is pre-determined using the analytical results ofa related study[4]for the case of
Taiwan. It is worth noting thatt2 may vary with the study site, and is adjustable in consideration ofthe distinction ofpassengers’ perception oftime value in different countries.
(3) The total monetary value ofpassengers’ waiting time under the service provided by a given bus with service strategyin the specific service region∗in time interval k (WC∗(k)), which is given by
WC∗(k) = t3× ∀o ×o∗(k|k) + l ,o∗ ∗(k) ×f ∗ o∗(k) + ∀o∗ ε × f∗ o∗(k) , (28)
whereo∗represents a given bus stop associated with a given bus service strategyin the specific service region∗; similarly,o∗also belongs to the set ofbus stops associated with the given bus service strategy
and the specific service region∗, but is located beforeo∗;o∗(k|k) is the time-varying passenger volume at the given bus stopo∗ in a given time interval k; l ,o∗ represents the geographical distance from the beginning of the given bus route ( ) to the bus stopo∗;fo∗∗(k) and fo∗
∗(k) represent the real
passenger volumes which are served by a given bus using a given service strategyat bus stopso∗ and
o∗ in a given time interval k, respectively; is the length ofa given time interval;ε is defined as the
average riding time;t3is the average monetary value ofpassengers’ waiting time in the unit ofUS dollars
per hour, and for the case of Taiwan, is herein set to be the triple of t2, according to the above-cited
study[4]. Moreover, the numerical results yielded by other strategies are also used to demonstrate the comparative advantages ofthe bus service strategies determined by the proposed algorithm.
To illustrate the applicability ofthe proposed method in real-time demand-responsive bus dispatching control, we summarized the main numerical results yielded from the proposed fuzzy clustering-based algorithm in the process of identifying an appropriate bus service strategy for the first test interval (i.e., 06:00–06:30). According to the proposed fuzzy clustering algorithm mentioned above, the vector of clustering centers associated with the specified three bus service strategies, including short-turn, express and zonal service strategies, was generated in the initialization step, as can be seen inTable 1. Then, in Step 1, the time-varying state variable vector(k) in the given time interval was estimated using the collected passengers’ trip patterns and traffic data. Using the results obtained in the previous steps, the estimated fuzzy correlation matrix ˜W(k), as presented inTable 2, was measured in Step 2.Table 3
summarizes the output ofthe strategy identification in the current test interval using the truncation rule mentioned in Step 3 ofthe proposed algorithm. Accordingly, the short-turn strategy is identified as the suitable bus service strategy for the first test interval.
The comparison results according to the aforementioned criteria are summarized inTable 4, where the strategies identified by the proposed method are highlighted in bold. The generalizations obtained from the numerical results are itemized as follows.
Table 1
Estimated clustering centers for identification of bus service strategies
Type ofstrategy state variable Short-turn Express Zonal
s∗(k) 0.67 0.25 0.28
e∗(k) 0.17 0.89 0.36
z∗(k) 0.23 0.42 0.73
Table 2
Estimated time-varying fuzzy correlation matrix ˜W(k)
Data sample Short-turn Express Zonal
Data sample 1 0.968 0.653 0.274
Short-turn 0.968 1 0.217 0.468
Express 0.653 0.217 1 0.639
Zonal 0.274 0.468 0.639 1
Table 3
Identification results for the first test interval 06:00-06:30
Data sample Short-turn Express Zonal
Data sample 1 1 0 0
Short-turn 1 1 0 0
Express 0 0 1 0
Zonal 0 0 0 1
1. Overall, the numerical results shown inTable 4reveal that there would be significant improvement in the performance of bus routing by implementing the proposed real-time demand-responsive bus dispatching approach in comparison with the system performance of the existing all-stop bus service strategy. According to the results with respect to the average performance during the four-interval test period, the existing bus operations can be improved by 16.65% by implementing the proposed bus service strategies, and herein, the average passengers’ waiting time is particularly improved up to 23.65%, which is a significant improvement for customers. In addition, the supply-oriented trans-portation cost is reduced by 19.45%, which extends the potential benefit ofthe proposed method to the supply side, in addition to the demand side.
2. The identified strategies appear superior to the other un-identified bus service strategies in their capa-bility ofresponding to diverse passenger demand patterns during the test period. This can be seen in that the aggregate cost-based measure associated with the identified strategy is less than that associated with any other strategies in any given time interval. Such a generalization also implies the validity ofthe proposed method in the identification ofappropriate bus strategies in response to time-varying patterns ofpassenger demands.
3. The measurements shown inTable 4may also be helpful to analyze the relative performance of the existing bus dispatching operations. For example, there is an interesting finding which can be seen in the results ofTable 4that, under certain patterns ofpassengers’ trip volumes, the existing all-stop bus service strategy proves to be more suitable than some specific strategies, although its overall performance may not satisfy the peak-hour passenger demands for the criterion of waiting time.
4. Concluding remarks
This paper has presented an advanced demand-responsive bus operations control approach in response to the variety ofboth passenger demands and road traffic conditions. Using the proposed short-term
Table 4
Comparison ofsystem performance
Criteria strategies T C∗(k) P C∗(k) WC∗(k) Aggregate
k = 1 (06:00 am–06:30 am) All-stop 49.41 4.25 355.88 409.54 Express 36.60 3.14 344.09 383.83 Short-turn 40.26 3.36 287.15 330.77 Zonal 42.09 3.55 346.94 392.58 k = 2 (06:30 am–07:00 am) All-stop 49.41 4.92 402.62 456.95 Express 36.60 2.83 291.89 331.32 Short-turn 40.26 3.55 281.71 325.52 Zonal 42.09 3.78 323.09 368.96 k = 3 (07:00 am–07:30 am) All-stop 49.41 3.97 365.06 418.44 Express 36.60 3.27 362.04 401.91 Short-turn 40.26 3.17 444.11 487.54 Zonal 42.09 3.36 355.49 400.94 k = 4 (07:30 am–08:00 am) All-stop 49.41 4.61 389.84 443.86 Express 36.60 3.30 344.01 383.91 Short-turn 40.26 4.49 389.21 433.96 Zonal 42.09 3.33 347.19 392.61
Average performance (identified strategy vs. all-stop strategy)
Identified strategy 39.80 3.39 317.09 360.28
All-stop strategy 49.41 4.44 378.35 432.20
Relative improvement (%) 19.45 23.65 16.19 16.65
Note: the regions highlighted in bold represent the strategies suggested in given time intervals.
passenger demand forecasting model coupled with advanced APTS-related technologies, the time-varying patterns ofpassengers’ O-D trip volumes are recognized, and then appropriate bus service strategies associated with specific service regions are identified using the proposed fuzzy clustering based algorithm in response to the time-varying passenger demands.
Our numerical study, using the processed data ofreal passenger demands collected from one city bus route in Taipei, demonstrates the potential advantages ofthe proposed method over both the existing all-stop bus service strategy and the other specific bus service strategies. Utilizing three specified criteria measures, including one supply-based and two demand-based criteria, the comparison results reveal the applicability ofthe proposed method for the use ofreal-time demand-responsive bus operations control. Nevertheless, there is potential for improving the performance of the proposed method by adding more elaborate bus service strategies such as skip-stop and limited-stop service strategies to the possible bus service strategies in response to the variety ofpassenger demands. Moreover, more elaborate strategies, such as the adjustment ofdispatching headways, can also be implemented for quick response to a growing variety ofpassengers’ trip volumes under conditions ofcomplicated urban transportation networks. Based on the present results, our further research will explore the possibility of integrating the time-varying
passengers’ demands on multiple bus routes in a given urban area to satisfy the systematic optimization ofnetwork-wide bus routing by extending the proposed methodology. Clearly, the acquisition ofvalid data for model testing and evaluation will be significant for our further research.
More importantly, it is expected that this study can help to demonstrate the applicability ofthe real-time data collected from advanced APTS-related technologies, and stimulate more research on operational models in the APTS environment. Furthermore, the integration ofthe proposed method with other public transportation systems, including mass transit systems, also warrants further research in order to improve the competitiveness ofurban public transportation systems.
Acknowledgements
The author would like to thank the referees for their constructive comments. Any errors or omissions remain the sole responsibility ofthe author.
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