Optimal transit fare and service frequency of a nonlinear origin-destination based fare structure

Transportation Research Part E 96 (2016) 1–19

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Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

Optimal transit fare and service frequency of a nonlinear origindestination based fare structure Di Huang, Zhiyuan Liu ⇑, Pan Liu, Jun Chen Jiangsu Key Laboratory of Urban ITS, Jiangsu Province Collaborative Innovation Center of Modern Urban Traffic Technologies, Southeast University, Nanjing 210096, China

a r t i c l e

i n f o

Article history: Received 1 July 2016 Received in revised form 2 October 2016 Accepted 10 October 2016

Keywords: Transit fare structure MPEC model Principal–agent game Nonlinear fare structure Path-based stochastic transit assignment

a b s t r a c t This paper proposes a new nonlinear distance-based transit fare structure, which is measured by a function of the Euclidean distance between the origin and destination stations, termed as Origin-Destination (OD)-based fare. The novel fare structure encourages passengers to freely choose the most efficient trip plan. An optimization model is formulated based on a three-party game (involving the transport authority, transit company, and passenger) to determine the optimal fare function and frequency. An artificial bee colony algorithm is adopted to solve the model. Finally, a numerical example is provided to verify the proposed method. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Targeting at the sustainable development of urban transport systems, the urban transit system has been regarded as the backbone for a green transport network. Despite increasingly high investment in transit infrastructure, many of the current urban public transit systems still cannot satisfactorily accommodate the passenger demand. With increased transit operation costs, financial performance becomes one of the urgent issues for both transport authorities and enterprises. Ticket revenues in most cities hardly offset operation costs. Accordingly, government subsidies are necessary for a healthy transit system. However, it is difficult to propose a rational level of subsidy in the absence of reasonable assessments. This leads to most transit enterprises suffering deficits and facing a severe threat to their survival. Therefore, it is essential that the problem of how to regulate fair transit fares be studied to ensure that a financial balance can be achieved between the transit enterprises and transport authorities. For a given transport network, transit fares evidently have an impact on mode choice. They are also the key factor that affects the supply–demand equilibrium in the public transit system (Borndörfer et al., 2012). There are two types of transit fares: ‘flat’ and ‘differentiated’. In the early stages of urban transit systems, flat fare structures tend to be prevalent because of their simplicity and low fare-collection costs. However, such schemes oversimplify the fare charging system and do not consider the elasticity of passenger demand to the transit price. It gives rise to poor financial health of the public transit operators (Tsai et al., 2008). As smartcards become more widely adopted, it becomes more reasonable to upgrade to a differentiated fare structure. Differentiated fares can be classified into six categories: distance-based (including mileage-, station-, and origin–

⇑ Corresponding author. E-mail addresses: [email protected] (D. Huang), [email protected] (Z. Liu), [email protected] (P. Liu), [email protected] (J. Chen). http://dx.doi.org/10.1016/j.tre.2016.10.004 1366-5545/Ó 2016 Elsevier Ltd. All rights reserved.

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D. Huang et al. / Transportation Research Part E 96 (2016) 1–19

destination-based fares), time-based, quality-based, cost-based, route-based, and patron-based fares (Fleishman et al., 1996). For the ease of presentation, these different types of fare bases are referred to as ‘fare structures’. For a particular fare structure, the specific charging pattern is determined by a ‘fare function’. For instance, taking the distance-based fare as an example, a fare function such as the one shown in Fig. 1(a) might be adopted. With the advent of near-field communication (NFC) and global positioning system (GPS) technologies, distance-based fare structures (including station- and mileage-based) have already been adopted in several cities, e.g. Amsterdam, Singapore, and Melbourne (Pelletier et al., 2011). Several empirical studies have shown that distance-based fare structures have some advantages over flat fare structures in that they improve social equity and transit ridership (Daskin et al., 1988; Chien and Tsai, 2007; Tsai et al., 2008). This is because distance-based transit fares are proportional to the distances traveled by passengers and so they better reflect the ‘true cost’ of a passenger’s trip. However, Ling (1998) pointed out that a distance-based fare is only suitable under two conditions: (1) the elasticity of short trips is greater than that of long trips and (2) the number of long trips is greater than that of short trips. Thus, in an urban transit network (which usually involves a large proportion of short trips), the advantages of distance-based fare structures cannot be fully reflected. To overcome this drawback of distance-based fare structures, we propose that a nonlinear origin–destination (OD)-based fare structure should be used. We assume that the fare can be determined using a function of the Euclidean distance between origin and destination stations, regardless of the travel itinerary and number of transfers. This function will be referred to as the fare function, an example of which is given in Fig. 1(a). As shown in Fig. 1(b), the OD-based fare is only affected by the Euclidean distance between the origin and destination rather than the trajectory of the bus or the number of transfers involved. Thus, passengers are free to choose the most efficient travel plan. We note that special forms of an OD-based fare structure have already been adopted for the transit systems in Singapore (Meng et al., 2012), Hong Kong (Lo et al., 2003), and Taipei (Li et al., 2012). The passenger demand on a transit system is sensitive also to the level of service (e.g. the line frequency) as well as to the fare structure (Lo et al., 2003; Chien and Tsai, 2007; Tsai et al., 2013). Thus, the objective of this paper is to maximize the social welfare of a transit system. We do this by jointly optimizing the fare function for the OD-based fare structure as well as the service frequency of each bus line. 1.1. Literature review There are a number of studies on public transit pricing and subsidy strategy (Daskin et al., 1988; Ling, 1998; Lam and Zhou, 2000; Lo et al., 2003; Jørgensen and Preston, 2007; Farber et al., 2014; Liu et al., 2016, among many others). Differentiated fare structures were initially deemed to represent a better approximation to the ‘true cost’ of the passengers (Daskin et al., 1988). However, Ling (1998) pointed out that their advantages are conditional, and that the mathematical tools available could not properly model the passengers’ response to fare policies in practice yet. Following rapid expansion of transit networks, further studies on the relationship between fares and distance traveled were conducted. Jørgensen and Pedersen (2004) analyzed how the travel distance and operator’s objectives affected transit fares and generalized travel costs. A positive correlation between transit fare and travel distance was confirmed in the study based on realistic examples. Jørgensen and Preston (2007) found later that the relationship between fare and travel distance is dependent on the form of the demand function. It has also been proposed that in order to optimize the distance-based fare structure, the partition of the travel distance should be determined (Tsai et al., 2013). However, Lo et al. (2003) indicated that

Fig. 1. (a) A fare function for the OD-based fare structure. (b) The Euclidean distance involved in a bus trip directly linking an origin and a destination.

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transit fares are, in practice, not always directly proportional to the travel distance. That is, there is a nonlinear fare structure, which imposes further modeling difficulties because of the non-additive nature of the travel costs. Chin et al. (2016) were the first to propose a model to deal with nonlinear fare pricing by extending the stochastic equilibrium transit assignment model proposed by Lam et al. (1999). As a reflection of the level of service, the line frequency is sometimes optimized jointly with the fare structure. Chien and Tsai (2007), for example, developed an efficient model to optimize the transit service frequency and fare structure that maximized the total profit subject to service capacity constraints. Tsai et al. (2013) also presented an approach to jointly optimize frequency and zone-based fare structure that yielded the maximum operating profit. To elucidate the optimal fare pattern for a specific fare structure, however, a system-wide objective should be taken into consideration. Thus, exactly how to obtain the optimal fare pattern is still an open question, especially for nonlinear fare structures. This is the issue addressed in the current paper. The determination of transit fares and line frequency is a three-party game involving three stakeholders: transport authorities, transit enterprises, and, passengers, each individually pursuing their various objectives (Schmöcker et al., 2009). The game between the transport authority and transit enterprise is usually described as a principal–agent game, wherein the transport authority is the principal and the transit enterprise is the agent (Laffont and Martimort, 2009). The principal–agent game has been previously studied from a macroscopic point of view. Gagnepain and Ivaldi (2002a) assessed the regulatory schemes in France in relation to the principal–agent game, and gave an estimate of the inefficiency of and effort expended by the transit operators. Empirical studies have shown that the behavior of passengers with respect to route choice is governed by many factors, including the level of service, travel time, and fares, etc. (Borndörfer et al., 2012). Many of these factors depend on the transit network structure and can be subsumed into the public transit network design problem (PTNDP). However, the studies in the literature which consider passengers’ route choice in the principal–agent game are relatively scarce. This has resulted in an ambiguous relationship between fare policies and passengers’ travel behavior. The PTNDP is usually formulated as a leader–follower game (Lam and Zhou, 2000; Li et al., 2009; Farahani et al., 2013; Farber et al., 2014). The upper-level problem depicts the system-wide objectives of the leaders (who plan/regulate or manage/operate the transit network), while the lower-level problem reflects the user reaction in terms of passenger flow pattern, which can be formulated using mathematical programs. Lam and Zhou (2000) introduced such a model to optimize a fare structure with elastic demand in a monopolistic transit market. They adopted the total revenue as the objective function of the upper-level in which the fare structure was the decision variable. This work was later extended by Li et al. (2009) and applied to three market regimes: a monopoly market, an oligopoly market, and a social optimum market with various objectives. In most of the previous studies, the objectives of the public transit regulator and operator are considered separately. In this case, the game between the three stakeholders cannot be reflected properly. In a transit network, once the fare function of the fare structure and line frequency have been determined, the passengers make their route choices in response, which may be considered a transit assignment problem. As mentioned before, nonlinear fare structures result in travel costs being non-additive. Hence, travel costs cannot be calculated via link-based formulation because the total path travel cost is not equal to the sum of the link travel costs on that path (Szeto et al., 2011). Therefore, a link-based transit assignment model cannot be used in conjunction with nonlinear fare structure scenarios (Spiess and Florian, 1989; De Cea and Fernández, 1993; Lo et al., 2004). Wu et al. (1994) proposed a path-based solution method for transit assignment based on the assumption of fixed demand and known path set. Szeto et al. (2011) modified the path generation procedure proposed by Chen et al. (2001) by calculating the path travel cost margins and effective travel costs in each iteration of path generation. To the best of our knowledge, the problem of finding a path-based transit assignment considering the non-additivity of a nonlinear fare structure is still unresolved. To sum up, two gaps may be identified in the extant literature. Although the pros and cons of flat and nonlinear fare structures have been pointed out in several studies, a fare structure (OD-based) which combines the advantages of both kinds of fare structures (flat and nonlinear) is yet to be proposed. Secondly, the games between transit planner, operator, and users are usually modeled separately when determining transit pricing and operation. And yet, in practice, the determination of transit fares and frequency is a joint decision made by all three stakeholders. That is, the objectives of all the stakeholders should be considered together. How to comprehensively depict this game is still an open question. This paper thus aims to fill these two gaps.

1.2. Objectives and contributions As decisions on fare policy and service level involve a three-party game, the objectives of all the involved stakeholders should be taken into consideration. The principal–agent game is suitable to depict the relationship between the transport authority and transit company on a macroscopic level. Considering the principal role of the transport authority in this game, a model employing mathematical programming with equilibrium constraints (MPEC) can be developed (with an objective based on a system-wide index describing the pursuit of the authority) to determine the optimal fare function and frequency. As the agent, the transit company is entrusted by the authority to provide transit service. Subsequently, the transit company responds to achieve its own benefits according to the fare policy and service level. Thus, the objective of the company can be described by equilibrium constraints in the proposed MPEC model.

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As discussed above, by considering the passengers’ response to the fare policy and service level, the leader–follower game can be adopted to describe the game between transit planners and users. Due to the non-additivity of travel costs, it is challenging to find an efficient transit assignment model that considers the nonlinearity of the fare structure. Hence, for this study, we also adopt a path generation subroutine before performing transit assignment. At the same time, the transit assignment procedure is treated via constraints on the MPEC model. The remainder of this paper is organized as follows. In the next section, some basic concepts and assumptions are briefly introduced. Then, a mathematical programming model with equilibrium constraints is introduced in Section 3 to determine the optimal fare structure and subsidy required. Subsequently, a solution algorithm is introduced in Section 4 based on an artificial bee colony. In Section 5, a sample transit network is used to test the proposed model and verify the solution algorithm employed. Our conclusions and recommendations for future research are given in the final section. 2. Problem description For the ease of presentation, the list of notation used in this paper is first provided as follows. Sets D K L N O S W Ls Mk Parameters Ch cs hk m q0w qw RT l ts ws b k h s


scr ss;k ss;k u

Decision variables a f

s s0m sm s0 ; . . . ; sn

set set set set set set set set set

of of of of of of of of of

destinations transit paths lines in the transit network nodes (stations) in the transit network origins sections OD pairs lines in section s, Ls 2 S paths in section-based paths k

unit cost factor depending on the roundtrip travel time, in $/h total travel cost for transit users on route section s passenger flow on path k cost related parameter the maximum demand between OD pair w total resultant passenger demand between OD pair w, w e W roundtrip travel time of line l, l 2 L in-vehicle travel cost on route section s waiting time associated with route section s the sensitivity to the expected travel cost value-of-time (VOT) the degree of passengers’ perception of the path travel time average fare in the transit network regulation cost total fare of path k transit fare on section s of path k shadow cost caused by taxation effort level defined by the transport authority frequency vector for all bus lines l, l 2 L. That is, f ¼ ðf l ; l 2 LÞ the vector made from the variables in the fare function of the fare structure fixed components of the mileage-based fare function variable components of the mileage-based fare function variables for the OD-based fare function

2.1. Network representation Consider a strongly connected network, denoted by G ¼ ðN; LÞ, where N is the set of all nodes (transit stations) and L is the set of transit lines. A helpful illustration of such a network is shown in Fig. 2. In this network, a transit line is a fixed path connecting two terminals, e.g. transit line L1 can be seen to run between terminals N 1 and N 4 . The same sequence of nodes is followed by all the vehicles serving one transit line, which can be defined as an ‘itinerary’, such as L1 : N 1 ; N 2 ; N 4 . A transit L1

link is any link portion of a transit line connecting two consecutive stations, for instance, N 1 ! N 2 . A transit path is defined as

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D. Huang et al. / Transportation Research Part E 96 (2016) 1–19

(a)

(b)

Fig. 2. Transit network representation using: (a) lines and (b) route sections.

a transit travel strategy between any OD pairs, which can be represented by the transit stations (nodes) and transit links used L1

L3

L1

on the trip, e.g. N 1 ! N 2 ! N 4 . Note that if two stations are served by different transit links, e.g. N 1 ! N 4 , they are regarded as L3

being on different transit paths. Between any OD pair N 1 and N 4 , the passengers may choose to transfer along the route. For instance, passengers taking L1 in Fig. 2(a) to node N 2 may transfer to L3 . For the ease of presentation, any section between two transfer stations (or origin/ L1

L3

L1

destination) on a transit path k, is termed a route section (denoted by S). For example, on the path N 1 ! N 2 ! N 4 , N 1 ! N 2 and L3

N 2 ! N 4 are two different route sections. It should be pointed out herein that a ‘transfer station’ on path k means that all the passengers on this path will transfer at this station. 2.2. Nonlinear fare structures As discussed in the Introduction, this paper deals with three representative transit charging structures involving: flat (see Fig. 3(a)), mileage-based (see Fig. 3(b)), and OD-based fares (see Fig. 1). Only in the first type of structure is the transit fare additive, i.e. the total transit fare of path k, sk , is the sum of all the section fares in the path. That is,

sk ¼

X

ss;k

ð1Þ

s2Sk

where

ss;k , is the fare on section s of path k. f ss;k , on each section. We use ss;k and sm s;k to denote flat and mileage-based fares,

We proceed to discuss the transit fare,

respectively. As shown in Fig. 4, for the flat fare case: f ss;k ¼ sf

ð2Þ

where sf is a fixed value. For a mileage-based fare structure, the transit fare s

m s;k

is determined by the in-vehicle distance trav-

eled after boarding at station i to alighting at station j. Thus:

(

sms;k ¼

s0m lði; jÞ 6 l0 s0m þ sm  ðlði; jÞ  l0 Þ lði; jÞ > l0

ð3Þ

where lði; jÞ represents the distance traveled on path k between nodes i and j. l0 is a predetermined threshold, within which the passengers only have to pay a flat fare s0m . If lði; jÞ > l0 , the passengers have to pay an additional per-mileage price sm calculated with respect to the difference between the actual travel distance and l0 . In this paper, the nonlinear OD-based fare structure introduced is based on the Euclidean distance between the boarding and alighting stations. This approach has the combined advantages of both flat fare and traditional distance-based fare approaches. The transit fare is assumed to be positively proportional to the Euclidean distance, following a nonlinear func-

Fig. 3. Illustration of two different fare structures: (a) flat fare structures and (b) mileage-based fare structures (Vuchic, 2005).

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Fig. 4. Fare structures for flat and mileage-based fare schemes.

tion /ðgÞ like that shown in Fig. 5(a). It is unreasonable to restrict this nonlinear function to any particular functional form. Also, there is a lack of practical data to calibrate the functional form required. Thus, a piecewise linear function is used here to approximate the general nonlinear function (Meng et al., 2012; Liu et al., 2014). As illustrated in Fig. 5, /ðgÞ is divided into n equal intervals, with g0 and gn corresponding to the minimal and maximal Euclidean distances, respectively. That is, the fare function is divided into n sections, each of which can be sufficiently accurately approximated by a linear function. The linear function can, in fact, be defined by its vertexes s ¼ ðs0 ; s1 ; . . . ; sk ; . . . ; sn Þ, as shown in Fig. 5(b). The piecewise linear approximation function can thus be expressed as follows:


n ðgÞ ¼ si þ siþ1  si ðg  g Þ; / i giþ1  gi

gi 6 g < giþ1 ; i ¼ 1; 2; . . . ; n

ð4Þ

where g1 ¼ gmin and gnþ1 ¼ gmax . It should be mentioned that the nonlinear transit fare /ðgÞ used in this study is general and so includes the flat fare scenario and any sort of differentiated fare function (see Eq. (1)). However, due to the high nonlinearity of the mileage-based P and OD-based fare structures (see Eqs. (3) and (4)), the path fare is non-additive, that is, sk – s2Sk ss;k . Therefore, the path fare cannot be determined by simply adding up the link fares due to the intercept in Eq. (3) and the nonlinearity of Eq. (4). In this regard, a path-based transit assignment model is introduced in the following subsection to estimate the passenger flows in a network with an OD-based fare /ðgÞ. 2.3. Path-based stochastic transit assignment model with elastic demand The generalized travel cost for each section, s, of path k can be considered to consist of an in-vehicle travel time, a waiting time, and a transit fare. In addition to these three costs, the passengers’ path travel cost, cw k , is also assumed to have a random term, nw s;k , which represents their perception error in the travel time. Thus,

cw k ¼

X XX w w w ðtw s;k þ ws;k þ kss;k þ ns;k Þ

w2W k2K s2S

Fig. 5. A comparison of (a) a general nonlinear fare function, / and (b) the piecewise linear fare function, /n.

ð5Þ

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D. Huang et al. / Transportation Research Part E 96 (2016) 1–19

w where t w s;k is the in-vehicle travel time relating to section s of path k between the OD pair w and ws;k is the combined waiting time of section s at the initial node of this section, named transfers. k is the passenger’s VOT. This variable can be obtained using the expression

0 @ ww s;k ¼

X

11 f lA

ð6Þ

l2Lw s;k

where f l is the frequency of line l and Lw s;k is the set of all the transit lines in the section s of path k between OD pair w. Note that due to the non-additivity of the nonlinear fare structure, the transit fare calculation is path-based. Eq. (5) should be then rewritten as

cw k ¼

XX X w w w ðtw s;k þ ws;k þ ns;k Þ þ ksk w2W k2K

! ð7Þ

s2S

The random term nw s;k is assumed to follow an identically and independently distributed Gumbel distribution (Sheffi, 1985), which gives rise to a logit model:

P Pw k ¼

expðhcw kÞ w k2K expðhc k Þ

ð8Þ

where Pw k is the probability that path k is chosen between OD pair w 2 W. h is the degree of passengers’ perception of the w path travel time. Thus, the passenger flow on path k, hk , can be calculated using the expression w

hk ¼ qw Pw k

ð9Þ

Subsequently, the expected minimum disutility Sw (also termed the ‘satisfaction’) can be measured via the expression

  1 X Sw ¼ E minðcw Þ ¼  ln expðhcw k kÞ k2K h k2K

ð10Þ

In addition to the stochasticity of the path travel costs, this paper further considers the elasticity of the passenger demand. It is necessary to include demand elasticity in transit fare studies because higher fares may affect the mode choice decision of many travelers and thus lead to elastic demand. In this regard, the total passenger demand, qw , between OD pair w is determined by a demand function:

qw ¼ Dw ðSw Þ

ð11Þ

The demand function Dw ðÞ is assumed to be continuous and monotonically decreasing. Previous studies have commonly adopted a linear form of demand function (Lam and Zhou, 2000):

qw ¼ q0w  bSw

ð12Þ

where q0w is the maximum demand between OD pair w and b is the ‘sensitivity’ to the expected minimum disutility. 3. Mathematical model Three stakeholders are involved in the PTNDP (the transport authority, transit enterprise, and passenger). The transport authority regulates the fare function and subcontracts the transit operation to bus companies who maintain and provide transit services to passengers. Therefore, there are three games in this PTNDP. One game is between the authority and transit companies (for pricing and subsidy) and is described as a principal–agent game. The other two are leader–follower games relating to the interaction between the passenger and authority, as well as between the passenger and transit company (see Fig. 6). 3.1. The principal–agent game In the principal–agent game, the urban transport authority acts as the principal and the transit enterprise as the agent (Laffont and Martimort, 2009). This scenario is a suitable description of the game used to decide fare and subsidy determination for two reasons. First, the transport authority entrusts the transit companies and subcontracts the operation of transit services to them. During this contracting process, the authority and companies have different objectives that are both conflicting and incompatible. The transport authority, which considers public transit to be a ‘public good’, needs to take full consideration of the social equity issues arising, including fare levels and the amount of subsidy imposed on the transit enterprises (Drevs et al., 2014). Thus, the transport authority attempts to maximize the social welfare. As for the transit enterprises, they care more about their total profit, which ensures their continued operation and sustainable development.

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D. Huang et al. / Transportation Research Part E 96 (2016) 1–19

Principal Transport Authority Objective : Max of Total Social Welfare

Principal-agent Game

Agent Transit Enterprise Objective : Max of Profit

Leader

Leader Fare Policy and Level of Service

Leader-follower Game

Leader-follower Game

Passenger Objective : Min of Total Travel Cost Follower Fig. 6. The three-party game in the proposed PTNDP.

Second, another characteristic of this game is that the authority cannot observe the actual action that the company chooses, namely the effort level, and so it lacks a complete set of information about the operation. However, it can obtain information about variables that are determined by the company’s actions and other exogenous stochastic factors, e.g. the operation cost, total passenger demand, etc. In other words, the information obtained by the authority is incomplete, but based on this information the authority has to decide how to incite the companies to choose their effort levels to comply with the authority’s wishes. Accordingly, the core aspect of the principal–agent game is how the principal decides the amount of subsidy required to incite the agents and induce them to make decisions that comply with the principal’s desired benefit. However, the decisions made by the agents cannot be completely observed by the principal because of the asymmetric nature of the information flow between these two participators. This results in problems relating to morality and adverse selection (Perrigne and Vuong, 2011). Put more bluntly, the agents tend to take risks to maximize their own benefit at the expense of society’s and so the maximization of social welfare is difficult to be realized in practice. In this paper, an incentive mechanism is considered in order to overcome these two potential conflicts in three ways (Tscharaktschiew and Hirte, 2012): (i) If the benefit resulting from a higher effort level outweighs that obtained in other ways, the transit enterprise may choose a higher level in order to obtain additional subsidy. This improves the level of service in the whole transit system and, at the same time, eliminates the effects of adverse selection. (ii) The binding enforced on the transit enterprise is strengthened by the introduction of an incentive mechanism, and so the risk of immoral behavior will be reduced. (iii) This mechanism should promote financial efficiency from the perspective of the balance between the transit companies’ incomes and expenditures. Mathematically, we assume that a transit enterprise makes its decision from a finite set of actions A = {a1, a2, . . . , an} e [0, 1], which may be viewed as effort levels (Gagnepain and Ivaldi, 2002b). For ease of assessment, we assume that a is a one-dimensional, continuous variable in the range from 0 to 1. The subsidy function, S, is defined as:

Sða; sÞ ¼

X

½scr ð1 þ rÞ  AðsÞ  qw ða; sÞ

ð13Þ

w2W

where scr is the regulation operation cost per passenger which is determined by the transport authority based on the financial performance of the previous year, r is the regulation profit rate for the transit enterprises (to make sure they have sustainable development), and AðsÞ is a function yielding the average fare in the transit network. It should be mentioned that if the average trip fare is larger than the regulation cost, i.e. AðsÞ > scr ð1 þ rÞ, the subsidy is treated as zero. As mentioned before, the principal cannot observe the actual effort level a of the agent, only other exogenous factors, e.g. the total passenger demand or the result of the transit assignment (Tscharaktschiew and Hirte, 2012). Therefore, the transit enterprise’s profit (u) can be defined as the sum of the net profit represented by total revenue (TR) minus the operation cost (OC) plus the subsidy (S):

uða; f; sÞ ¼ TR  OC þ u  S ¼ RðsÞ  Cða; fÞ þ u  Sða; sÞ

ð14Þ

D. Huang et al. / Transportation Research Part E 96 (2016) 1–19

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where RðsÞ is the function giving the total revenue, and Cða; fÞ is the cost function of the transit operator in terms of effort level a and line frequency f, f ¼ ðf l ; l 2 LÞ. A cost function with quadratic form has been widely adopted in the Economics of Information literature (Socorro and de Rus, 2010), thus,

Cða; fÞ ¼

X a2 RT l  f l  Ch  2 l2L

ð15Þ

B

To ensure it is advantageous for an agent to participate in the principal–agent game, individual-rationality (IR) and incentive-compatibility (IC) constraints associated with the transit enterprises’ expected utility need to be satisfied simultaneously. The IR, or participation, constraint indicates that the agent’s expected utility under the principal–agent contract should be not less than the maximum expected utility if the agent does not take this contract. In other words, the expected utility of the effort level a chosen by the transit enterprise should be larger than the average utility of the transit market:


uða; f; sÞ P u

ð16Þ

The IC constraint is intended to persuade the agent to choose the action that the principal expects the agent to do under the incentive mechanism. Therefore, the utility of effort level a, as chosen by the transit enterprise, should be larger than that of the other effort levels a0 , a; a0 2 A. That is,

uða; f; sÞ P uða0 ; f; sÞ

ð17Þ

The IC constraint reflects the aim of the transit enterprise (i.e. the maximization of its own profit). The transit enterprises choose their effort levels according to the incentive contract, and then try to reduce their operation costs and achieve more subsidy at the same time. Consequently, the IC constraint in Eq. (17) coincides with the maximization of uða; f; sÞ. Additionally, the following feasibility constraints need to be satisfied that stipulate both minimum and maximal values for the effort levels, line frequencies, and fares,

06a61

ð18Þ

f min 6 f l 6 f max

ð19Þ

smin 6 s 6 smax

ð20Þ

3.2. The leader–follower game As shown in Fig. 6, there are two leader–follower games (between the transport authority and passengers, and between the transit company and passengers) in this PTNDP. As mentioned in Section 2.3, the passengers’ behavior with respect to route choice is denoted by the path flow, which is modeled by the path-based stochastic transit assignment (see Eqs. (5)– (9)). Therefore, the transit assignment results could be affected in two ways: (i) a change in the fare level, which results in a perturbation to the passengers’ travel costs and will ultimately directly lead to redistribution of the passenger flow and (ii) a higher effort level being chosen by a transit company, which results in a higher level of service being provided on the transit network (Hensher and Stanley, 2008). The level of service is an essential factor that influences and changes passenger demand between any OD pair (Fan and Machemehl, 2008), leading to a variety of transit assignments. Accordingly, the demand function adopted in this paper (see Eq. (12)) can be written in the form:

qw ða; f; sÞ ¼ a  q0w  b  Sw ðf; sÞ

ð21Þ

where the first term represents the maximum potential passenger demand which is influenced by the effort level of the transit company. Perturbation of the passenger flow will subsequently lead to changes in the objectives of both the authority and transit company. On the one hand, the path flow and total passenger demand are inputs required to calculate the transit company’s total revenue. On the other hand, the authority cannot observe the actual effort level chosen by the transit company (due to the asymmetric nature of the information flow in the principal–agent game), but it can monitor some exogenous factors. Therefore, the results for the total passenger demand and transit assignment are the decision parameters in the objective function of the authority. 3.3. The MPEC model From the discussion above, it should be clear that the MPEC model proposed in this paper involves three stakeholders connected by three games (one principal–agent game and two leader–follower games). As the authority plays a leading role in this problem, maximization of the social welfare (SW) is adopted as the objective function in the MPEC model. Here, the social welfare (or net social benefit) is taken to be the sum of the consumer surplus (CS) and the producer benefit (Borndörfer et al., 2012). Furthermore, the producer benefit is considered to the operator’s profit (u). Hence,

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SW ¼ CS þ u

ð22Þ

The CS is derived from the total social benefit (TSB) minus the total cost (TC) (Chien and Spasovic, 2002):

CS ¼ TSB  TC

ð23Þ

where the TSB is obtained by inverting the demand function (i.e. the transit fare is cast as a function of the demand) and then integrating the inverted function with respect to the demand:

TSB ¼

XZ

w2W

qw

0

D1 w ðxÞdx

ð24Þ

where D1 w ðÞ is the inverse of the demand function between OD pair w. The TC is the generalized cost that the passengers actually pay which can be found using:

TC ¼

XX w hk  c w k ðf; sÞ:

ð25Þ

w2W k2K

In the principal–agent game, the behavior of the transit company is influenced by the IR (Eq. (16)) and feasibility (Eqs. (18)– (20)) constraints to ensure its benefit. Once the fare policy and the level of service are decided by the principal–agent game, the passengers will make their response via their route choice (Eqs. (5)–(9), and (21)). Therefore, the MPEC model can be expressed mathematically as:

max SW ¼

XZ

qw

0

w2W

þu

D1 w ðxÞdx 

X

XX w XX X a2 hk ða; f; sÞ  cw swk hwk ða; f; sÞ   C h  RT l  f l k ðf; sÞ þ 2 w2W k2K w2W k2K l2LB !

½scr ð1 þ rÞ  s  qw ða; f; sÞ

ð26Þ

w2W

subject to Eqs. (5)–(9), (16), and (18)–(21). 4. Solution algorithm As depicted in Fig. 6, the MPEC optimization model consists of a principal–agent game and two leader–follower games, which, intrinsically, is a three-level programming problem. In optimal transportation network design and scheme setting problems, the bi-level model is widely adopted and is acknowledged to be an NP-hard problem (Ben-Ayed et al., 1988; Yang and Bell, 2001; Gao et al., 2005). In nature, bi-level models can be reduced to a three-level model, and thus the problem addressed here is also NP-hard. Hence, in order to efficiently solve this problem in practice, a heuristic approach is a more suitable one than the use of an exact algorithm. In this section, a hybrid artificial bee colony (ABC) algorithm (Szeto and Jiang, 2012; Chen et al., 2015) is employed in the solution algorithm. The hybrid method used relies on the ABC algorithm to solve the proposed MPEC model by incorporating a path-generation subroutine based on a node deletion algorithm to generate the path set (Ramming, 2001). Details of the steps of the algorithm are described below. 4.1. Two-phase path generation subroutine To enable path-based transit assignment, it is necessary to first generate a path set. In previous work, the k-shortest path algorithm has generally been adopted for path generation (Fan and Machemehl, 2006, 2008; Szeto et al., 2013). However, because of the overlapping and the variance in the travel cost between transit links in a specific route section, the kshortest path algorithm cannot be directly adopted in our problem. Thus, in this section, a two-phase procedure is proposed for path generation. The first stage aims to generate the shortest path connected by route sections, i.e. the section-based shortest path, and in the second stage one obtains the shortest path in terms of transit lines, i.e. the line-based shortest path. Due to the consolidation of transit lines in a route section, the combined route section cost should be defined first. At this stage, only the in-vehicle travel time and waiting time are considered to be the components of the route section cost due to
s . Let tls be the in-vehicle travel time for line l on route section s. Then, the absence of specific fare functions, that is,
cs ¼
t s þ w the average of the in-vehicle travel times on section s,
t s , is given by:


ts ¼

X l f s t ls ;

s2S

ð27Þ

l2Ls l

l

l

where Ls is the set of lines in section s, and f s is the relative frequency of line l defined as f s ¼ f =f s , where f s is the effective P l frequency, and f s ¼ l2Ls f . In previous work, the waiting time on route section s is assumed to be inversely related to the effective frequency (De Cea and Fernández, 1993; Szeto et al., 2011, 2013), so that,

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s ¼ w

a fs

;

8s 2 S

ð28Þ

where a is a calibration parameter for the conversion. The specific steps of the two-stage path generation subroutine are as follows: 4.1.1. Stage 1: Generation of the section-based shortest path Step 1. Initialization. Set m = 0. For simplicity, let M k be the set of all line-based paths in the section-based path k. Step 2. Convert the transit network representation from transit lines into route sections — as in the example shown in
s , where
t s and w
s are calFig. 2 (from (a) to (b)). Calculate the combined travel cost,
cs , of each section s, and
cs ¼
ts þ w culated via Eqs. (27) and (28). Step 3. Solve the shortest-path problem based on the combined route section cost. Obtain a feasible path connected by route sections (e.g. N 1

S1 ðL1 ;L3 Þ

!

N 4 , in Fig. 2), that is, the shortest section-based path k.

4.1.2. Stage 2: Generation of the line-based shortest path Step 4. Extend path k into transit line representation. For example, in Fig. 2, the section-based path N 1 L1

S1 ðL1 ;L3 Þ

!

N 4 can be

L3

extended to two line-based paths: N 1 ! N 4 and N 1 ! N 4 . Step 5. If Mk is empty, go to Step 6; otherwise, solve the shortest-path problem based on the in-vehicle travel time and waiting time of each line individually. Obtain the shortest line-based path m. Eliminate path m from M k . Set m = m + 1. Go to Step 7. Step 6. Delete the middle transfer node of the section-based path k. If the node sequence of the path k is 1 ! 2 ! . . . ! i ! . . . ! N, the sequence of the node to be deleted should be the integer part of ðN  1Þ=2. Go to Step 3. Step 7. If m < mmax , go to Step 3; otherwise, stop. The number mmax is a predetermined constant specifying the maximal number of candidate paths. 4.2. The ABC algorithm The ABC algorithm is a metaheuristic approach inspired by the food scavenging behavior of honey bees. It is superior to other often-used evolutionary algorithms, e.g. genetic algorithms, because of its inherent local search mechanism (Chen et al., 2015). In the algorithm, the colony of bees is deemed to consist of three categories: employed bees, onlookers, and scouts. Employed bees are in charge of exploring for food sources (‘solutions’) until the food source is exhausted. During this time, a local search procedure is carried out. The onlookers and scouts are responsible for evaluating and searching for new food sources, respectively. It should be pointed out that, in this study, each food source represents a feasible solution, as defined by the decision variables in the fare functions of Eqs. (2)–(4). Each set of these variables can be referred to as a ‘fare pattern’. As shown in Table 1, the fare function for a flat fare (Eq. (2)) is determined by only one variable sf . For a mileage-based fare structure (Eq. (3)), two variables are needed, one for the fixed and one for the variable component. The nonlinear OD based fare structure is represented by a piecewise linear function determined by the end points at the boundaries of each interval (see Eq. (4)). In this case, the number of variables, s ¼ ðs0 ; s1 ; . . . ; sk ; . . . ; sn Þ, is determined by the length of the intervals and the maximum Euclidean distance in the transit network. Step 1. Initialization of input parameters. Set the colony size N c , the number of employed bees N e , the number of onlooker bees N 0 , the number of scout bees N s , and the limit counter L. Set the iteration counter I to 0, and set the maximum number of iterations, Imax . Step 2. Initialize employed bees. Generate the initial set of food sources and set the limit counter of each food source to zero. Step 3. Employed bee phase. In each of the food sources that have been chosen by the employed bees, a neighborhood search is executed. If the fitness of the new neighbor solution is better than the previous one, replace the previous solution by this neighboring solution and set its limit counter to zero. Otherwise, the previous solution is retained and the limit counter is increased by 1.

Table 1 Comparison of the variables used in different fare structures. Fare structure

Fare function

Fare function variables

Frequency variables

Flat Mileage-based OD-based

sf s0m þ sm  lði; jÞ

sf s0m and sm s0 ; s1 ; . . . ; sk ; . . . ; sn

f 1; f 2; . . . ; f n

–

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D. Huang et al. / Transportation Research Part E 96 (2016) 1–19

Step 4. Onlooker bee phase. The probability that the onlooker bees will choose a certain food source is determined by conducting a roulette-wheel selection based on the fitness of each food source. Once the onlooker bees have found their food sources, they execute a neighborhood search. If the fitness of the new solution is better than the former one, replace it with the new one and set the relevant limit counter to zero; otherwise, keep the former solution and increase the limit counter by 1. Step 5. Scout bee phase. If a food source cannot be improved within the maximal number of allowed trials, it is treated as a poor one. Then, it is discarded and the corresponding employed bee becomes a scout bee. The latter will randomly search for a new food source location and the limit counter of the new food source is set to zero. Step 6. Stop test. Increase the iteration number, I, by one: I ¼ I þ 1. If I < Imax , return to Step 3; otherwise, stop. Flow charts illustrating the proposed path-generation subroutine and ABC algorithm are shown in Fig. 7. 5. Numerical example A numerical example is presented to illustrate the properties of the proposed MPEC and effectiveness of the algorithm used. The programs used to perform the calculations involved in this section were all coded in MATLAB (ver. R2015b) and implemented on a personal computer (Intel Core i5-5200 CPU @ 2.20 GHz and 4 GB RAM). A 9-node example (Fig. 8) is adopted in this section to assess the proposed methodology. The example involves a single OD pair (N1 – N9), consisting of nine transit stops (N1, N 2 , N 3 , N 4 , N 5 , N 6 , N 7 , N 8 , and N 9 ), and five bus lines (L1, L2 , L3 , L4 , and L5 ). The route section representation of the transit network is shown in Fig. 8(b) and the basic input data for the transit network is presented in Table 2. For the sake of simplicity, it is assumed that all of these lines have fixed frequencies, constant operation costs, and travel times. The potential demand q0w for the OD pair N 1 – N9 is taken to be 1000 passenger/h. The total passenger demand is determined using Eq. (21) with b equal to 3. The parameters used in the ABC algorithm were set to: N c ¼ 40, N e ¼ 20, N 0 ¼ 20, L ¼ 5, and the maximum number of iterations Imax is 50. The other model parameters used are: scr ¼ $7, r ¼ 0:6, k ¼ 0:3, and h ¼ 0:9. 5.1. Optimization results The computational results obtained using the proposed model are presented in Table 3 for three scenarios. The results show that the flat fare scenario produces the worst financial performance — it has a negative profit associated with it and requires a sizeable subsidy. On the other hand, the mileage-based fare structure (Scenario 2) achieves the maximum social welfare and attracts more passengers than the other two. This confirms the conclusions made in previous studies that a distance-based fare is preferable for transit operators as it reflects the ‘true cost’ of the passengers’ trips and performs better in social equity and welfare (Daskin et al., 1988; Ling, 1998). However, as a result of this true reflection of the cost, passenger

Fig. 7. Flow charts of: (a) the path generation subroutine and (b) the ABC algorithm.

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D. Huang et al. / Transportation Research Part E 96 (2016) 1–19

Fig. 8. An example of a small transit network presented in: (a) bus line and (b) route section representation.

Table 2 Basic input data used in the transit network example. Line no.

Itinerary

L1 L2 L3 L4 L5

N1 , N1 , N1 , N1 , N1 ,

N2 , N2 , N5 , N4 , N4 ,

N3 , N5 , N8 , N5 , N7 ,

Roundtrip travel time (h) N6 , N9 N6 , N9 N9 N9 N8 , N9

0.83 0.74 0.65 0.67 0.85

Table 3 Optimization results. Parameter

Fare function Average fare ($/passenger) Demand (passenger/h) Social welfare Revenue ($/h) Profit ($/h) Subsidy ($/h) Effort level

Fare structure Scenario 1: flat

Scenario 2: mileage-based

Scenario 3: OD-based

8.45 8.87 935 174894.65 5008.8 158.51 679.45 0.93

3:32 þ 2:1  lði; jÞ 9.62 704 388216.5 5534.77 2603.83 0 0.78

See Fig. 9 13.12 661 222339.12 5377.63 2446.68 0 0.71

flow becomes more concentrated in the most attractive paths (this issue is discussed further in the next section). The optimal OD-based fare structure is shown in Fig. 9, which clearly reveals its nonlinear nature. Table 4 presents the optimal line service frequencies for each scenario. Lines 1 and 5 have the highest service frequencies in both scenarios 1 and 3, and these two lines have relatively large roundtrip travel times compared to the others as well (see Table 2). Line 3, which has the lowest roundtrip travel time, has the highest service frequency in Scenario 2. In scenarios 1 and 3, the transit fares relate to the passengers’ travel trajectories to a lesser extent, and so the passengers tend to choose paths that have lower in-vehicle travel times or waiting times. This results in the transit lines with larger roundtrip travel times having higher service frequencies in order to accommodate passenger demand. The distribution of passenger demand will be further discussed in the following sections. The social welfare values associated with the three scenarios with basic inputs are provided in Fig. 10, which also shows the convergence behavior of the proposed model and solution algorithm in the three cases considered. To further illustrate the new methodology used, the demand distributions associated with the different scenarios, and a sensitivity analysis of the effort level and passenger value-of-time (VOT), are presented in the next two sections.

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D. Huang et al. / Transportation Research Part E 96 (2016) 1–19

Fig. 9. The optimized OD-based fare structure.

Table 4 Optimized service frequencies (vehicle/h).

L1 L2 L3 L4 L5

Scenario 1

Scenario 2

Scenario 3

12 9 6 9 12

8 5 13 6 7

14 7 6 5 10

Fig. 10. Convergence trend of the social welfare values in three scenarios.

5.2. Distribution of passenger demand To illustrate the various passenger distributions, Table 5 shows the passenger flows along the ten paths as derived from the path-generation subroutine. (The percentage contribution made by each path with respect to the total passenger flow is presented to the right of each flow.) Some specific path flow distributions are also shown in Fig. 11 (note that only the five most common paths are presented for each scenario). The figure clearly shows that, for a given transit network, different fare structures lead to different passenger distributions. In Scenario 1, the five most common paths constitute nearly 93% of the total demand, all of which are direct paths from the origin node N 1 to the destination node N 9 . The passenger distribution is relatively even but fewer passengers select paths with a greater number of transfers. For the mileage-based fare structure (Scenario 2), the top three paths account for 64% of the total demand, and the paths with transfers are starting to be chosen more often. This reflects the fact that, if the transit fare has a positive correlation with the distance traveled, passengers will tend to choose the paths with the lowest costs (in terms of both time and monetary cost). This consolidates demand into the ‘attractive’ transit lines. Moreover, passengers start to choose paths with transfers because this allows them to make trade-offs between time and monetary concerns.

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D. Huang et al. / Transportation Research Part E 96 (2016) 1–19 Table 5 Path flow results. No.

Path

Passenger flow (passenger/h) Scenario 1

1 2 3 4 5 6 7 8 9 10

L1

1!9 L2 1!9 L3 1!9 L4 1!9 L5 1!9 L2 L4 1!5!9 L4 L3 1!5!9 L3 L4 1!5!9 L3 L5 1!8!9 L1 L2 1!6!9

Scenario 2

Scenario 3

Flow

%

Flow

%

Flow

%

141.74 150.54 196.98 197.20 146.09 21.41 21.73 24.87 17.88 16.99

15.15 16.09 21.06 21.08 15.62 2.29 2.32 2.66 1.91 1.82

18.29 32.88 168.01 220.49 27.32 46.57 47.42 122.00 12.12 8.77

2.33 4.18 21.37 28.05 3.48 5.92 6.03 15.52 1.54 1.12

55.61 58.62 73.16 75.58 58.34 71.31 71.23 81.53 59.31 57.01

8.40 8.86 11.06 11.42 8.82 10.78 10.77 12.32 8.96 8.62

Fig. 11. Passenger distributions for different fare structures.

In Scenario 3, passenger demand is more evenly distributed over all the paths (compared to the other two scenarios) and the number of passengers choosing paths with transfers is increased further. The inherent cause of this situation is that the fares in the OD-based fare structure are only related to the Euclidean distance between boarding and alighting stations — they are in no way related to the number of transfers or the distance actually traveled by the passengers. 5.3. Sensitivity analyses 5.3.1. Effort level and passenger rate A sensitivity analysis was conducted aimed at the relationship between effort level and passenger rate. As discussed in Section 3.1, effort level is adopted as a decision variable of the transit company and reflects the level of service it provides (a higher effort level indicates a higher service level). When the effort level is varied from 0 to 1.0, the fluctuation in the passenger flow in paths 1–5 in Scenario 1 is as shown in Fig. 12. The figure shows that, as the effort level increases, the passenger flow rates along the five most common paths in Scenario 1 tend to decrease. As mentioned earlier, the effort level reflects the level of service that the transit company provides, and this includes the line frequency. The optimal frequencies of each line when a is 0.1 and 1.0 are presented in Table 6. The table clearly shows that the service frequency is larger if the transit com-

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D. Huang et al. / Transportation Research Part E 96 (2016) 1–19

Fig. 12. Sensitivity analysis results showing the passenger flow rate as a function of the effort level for Scenario 1.

Table 6 Line frequencies (vehicle/h). Transit line

L1 L2 L3 L4 L5

Optimal frequency a = 0.1

a = 1.0

8 8 9 7 13

10 10 7 13 14

pany chooses a higher effort level. In addition, such a change decreases the average passenger waiting time from 0.21 to 0.13 h. Thus, a higher effort level will also result in a passenger flow that is more evenly distributed as well. 5.3.2. VOT and passenger rate In this section, value-of-time is examined in more detail to show the relationship between the fare structure and passenger demand distribution. VOT is an important parameter in the calculation of passenger costs. In this particular section, we assume that all passengers are homogeneous in the sense that they all have identical VOT values. In Fig. 13, we show the results of a sensitivity analysis which looks at the change in the passenger flow rates for certain paths as a function of a given, fixed VOT value. The results allow easy comparison to be made of the situation in which the time value is low (e.g. k ¼ 0:1), to one in which it is high (e.g. k ¼ 1:0). For each fare structure, the results for the five most common paths are presented in Fig. 13. It can be readily observed that Scenario 2 is the most sensitive one with respect to VOT. This reflects the fact that in a mileage-based fare structure a bigger VOT value encourages passengers to choose paths 3 and 4 instead of the other paths. As shown in Tables 2 and 4, these two paths have lower total travel times and itinerary mileages, which results in lower total travel costs (taking into account both time and monetary components). This is especially so in Scenario 2. Flat and OD-based fare structures are not as sensitive to travel distance as a mileage-based fare. As VOT increases, passengers will tend to choose the more attractive paths that have lower travel costs. It is thus not surprising in Fig. 13 that the results for Scenario 3 are not very sensitive to the VOT value. This is because the fares for all the paths associated with the same OD pair, when calculated using the OD-based fare structure, are only dependent on the Euclidean distance between the origin and destination nodes, which is, of course, identical for the same OD pair. This reflects one of the advantages of the proposed OD-based fare structure: even if the VOT value is very high, passenger demand can still be evenly distributed across the transit network. 6. Conclusions In this paper, a new nonlinear distance-based fare structure has been proposed, described, and investigated in detail. The fare structure is based on the Euclidean distance between origin and destination stations (which we refer to as an OD-based fare structure).

D. Huang et al. / Transportation Research Part E 96 (2016) 1–19

17

Fig. 13. Sensitivity analysis results showing the passenger flow rate with respect to VOT value.

The OD-based fare structure has two advantages over existing fare structures. First, as a distance-based fare, it reflects the ‘true cost’ of a passenger’s trip. This is an important issue for social equity. Secondly, the OD-based fare is based on the Euclidean distance between the origin and destination stations rather than the trajectory of the bus. Thus, passengers can freely choose the travel plans that are the most efficient for them. This results in a more even passenger distribution over the various transit lines. To evaluate the new fare structure, and compare it with other fare structures, an MPEC model was adopted to describe the games between the transport authority (planner), transit enterprise (operator), and passenger (user). In comparison to previous studies, the model more comprehensively considers the three stakeholders in the urban transit network. It also describes a complete picture of the decision process used for the transit fare and line frequency, which makes it a more convincing optimization model. A numerical example has been presented to further verify the feasibility of the proposed fare structure and model. The results obtained show that by adopting the new OD-based fare one can efficiently distribute the passenger demand to include paths with transfers or relatively high in-vehicle travel costs. It can thus relieve the concentration of passenger flow into a few specific paths. The model developed here can be used to optimize the fare function and service frequency of existing transit services, as well as for planning new transit systems. Moreover, the model was formulated in a completely general manner, which

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means it can accommodate flat fares and other nonlinear fare structures. Thus, the frequencies and fares can easily be determined for a variety of different scenarios. In future work, more attention should be given to solving the problem of optimizing the fare policy in a multimodal transit framework (i.e. including combined use of bus, metro, and light rail) in order to model the coordination of different fare structures. In the meantime, the current work also needs to be extended by accounting for other practical considerations, e.g. vehicle capacities, congestion on the transit network, etc., as well as extending the optimization strategies used in the lowerlevel problems. These improvements should allow a more effective model to be developed to describe the passengers’ route choice behavior. In addition, several parameters are also in need of further calibration to improve the model’s accuracy. 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