a bi-level programming approach — optimal transit fare under line capacity constraints

Journal of Advanced Transportation, Vol. 34, No. 2, pp. I05- I24 ww w. advanced-transport corn

A Bi-level Programming Approach - Optimal Transit Fare Under Line Capacity Constraints

Jing Zhou William H. K. Lam

The fare of a transit line is one of the important decision variables for transit network design. It has been advocated as an efficient means of coordinating the transit passenger flows and of alleviating congestion in the transit network. This paper shows how transit fare can be optimized so as to balance the passenger flow on the transit network and to reduce the overload delays of passengers at transit stops. A bi-level programming method is developed to optimize the transit fare under line capacity constraints. The uppex- level problem seeks to minimize the total network travel time, while the lower-level problem is a stochastic user equilibrium transit assignment model with line capacity constraints. A heuristic solution algorithm based on sensitivity analysis is proposed. Numerical example is used to illustrate the application of the proposed model and solution algorithm.

Introduction

Transit systems are the lifelines of any urban area and are proving to be one of the viable solutions to the road traffic congestion and pollution problems in most of the urbanized cities with high density population. It is important that the transit systems should be designed efficiently to provide comfortable service and safety to passengers. Attention has been given to the transit network design problem in the past three decades. To develop the model and methodology in perspective, transit service planning for scheduled passenger carriers has been described by many scholars (e.g., Nihan and Morlok, 1976, Turnquist, 1985). The intent of their approaches is to optimize the system performance for various

Jing Zhou is in the Graduate School of Management Science and Engineering, Nanjing University, Nanjing, P.R. China. William H.K. Lam is in the Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, P.R. China. Received: October 2000; Accepted: April 2001

106 Jing Zhou and William H.K. Lam

objectives under specific constraints, and institutional environments. Focusing on urban transit networks, Ceder and Wilson (1986) described the transit system as a multi-stage process for solving the more detailed problems successively, beginning from network design and ending with crew scheduling. List (1990) presented a methodology and model for sketch-level planning that makes possible quick assessment of major changes in a transit network and service patterns. This model used trans- shipment equations to develop optimal transit vehicle flows in response to passenger flows, service frequency requirement, line segment capacities, system fleet size, and storage node (or station) capacities. In addition, there has been extensive work focusing on scheduling of urban transit lines (Ceder, 1986, Chakroborty and Subrahmanyam, 1995).

It is however noted that the aforementioned models ignored the response of passengers’ route choice behavior to the operators’ decisions, Most of the previous related studies considered the transit fare as fixed regardless of its effects on passenger flows. In fact, the fare of a transit line is one of the important decision variables for transit network design. It has been advocated as an efficient means of coordinating the transit passenger flows and of alleviating congestion in the transit network, particularly in Asian cities with high density population. In these cities such as Hong Kong and Singapore, the transit industry can make money without subsidy from the government.

This paper shows how a transit network fare can be determined so as to balance the passenger flow on the transit network and to reduce the overload delay of passengers. At the same time, the passenger’s route choice behavior is taken into account. A bi-level programming method is developed to determine the optimal transit fare under line capacity constraints. The upper level problem seeks to minimize the total network travel time, while the lower level problem is a stochastic user equilibrium (SUE) transit assignment model with line capacity constraints.

This paper is organized as follows. In the next section, the network representation is defined and some useful concepts for the transit network are introduced briefly. A bi-level programming formulation for the transit fare problem is presented in Section 3. In Section 4, a heuristic solution algorithm is proposed using the derivatives of equilibrium link flows and equilibrium passenger overload delays with respect to transit line fares. The derivative information is obtained by using sensitivity analysis for the congested transit network equilibrium problem with respect to line fare. A numerical example is given in Section 5 . Finally, conclusions and suggestions for further study are drawn in Section 6.

A Bi-level Programming Apporach -

Paths rl r2

r3

r4 r5

107

Line sections (links) A-B(Li)

A-Y(L,), Y-B(L3) A-Y(LZ), Y-B(L4) A-X(LI), X-B(L3)

A-X(L2), X-Y(Ld, Y-B (L4)

Some Useful Concepts and Notations

Basic concepts and general assumptions

A transit network constitutes a set of stations (nodes) where passengers can board or alight and a set of transit lines. A transit line can be described by the frequency of the transit vehicles and by the vehicle type (e.g. bus or underground train). The section of a transit line between two consecutive stations is referred to as the line segment.

A transit route (or path) is a feasible path for transit passenger to travel between any given origindestination (OD) pair in the transit network. Generally it will be identified by a sequence of nodes, including origin node, destination node and all the intermediate nodes (transfer points). A line section is the portion of a transit route between two transfer points of its itinerary.

We can use the example network shown in Fig. 1 to elaborate the difference between line segments and line sections. In Fig. 1, L: , L$ are the line segments of L, from A to X and from X to Y respectively. Also, let L:, L; be the line segments of L, from X to Y and from Y to B respectively. In Table 1, the transit route rz is composed by two line sections (or links), A-Y(Lz) and Y-B(L,).

L;

G

Fig. 1. Example transit network

Table 1. Paths and line sections

108 Jing Zhou and William H. K. Lam

The different fares that are charged to the passengers and the levels at which they are set are integrated to form the transit fare problem. A wide range of fare structures can exist, and virtually all transit systems have some differential fares. Fares can be differentiated according to various factors, such as the type of passengers, the level of services, the time of travel and so on.

In this paper, only single-class passengers and single-service transit lines are considered as we have preferred a simpler and more intuitive approach to facilitate the presentation of the essential ideas. For simplicity, different flat fare is assumed for different transit line. In addition, the steady state assumption is adopted and the OD demand is fixed as the objective is to minimize the total network travel time in the transit system. These assumptions will however not limit the application of the proposed model for the purpose of strategic transit planning. Obviously, a temporary saturated steady-state may exist for a short duration during the peak period when passenger flows are constrained by limited actual capacities on a set of links (or line sections), and temporary steady-state passenger queuing holds. It is also assumed that passengers do not have perfect knowledge of the timetable for the transit lines and would select the transit route that minimizes their perceived total travel cost (including both travel time and transit fare).

Notations

We denote a transit network by G=(N, S) with node set N, representing transit stops and link set S, representing line sections of the transit network. The notations used throughout this paper are given as follows.

W: vr : k,: capacity on line segment e (i.e., capacity of the

corresponding line). R,: the set of feasible routes (or paths) associated with OD pair

i'(s) : the origin transfer node associated with link s (or line

c, :

the set of all OD pairs. passenger flow on line segment e.

W .

section). the travel time on link s (or line section).

A Bi-level Programming Apporach - . 109

g , : the passenger demand between OD pair w.

2 , :

u, :

the in-vehicle travel time on llnk s (or line section).

the passenger waiting time on link s (i.e., the waiting time at

node i + ( s ) ) . the equilibrium passenger overload delay, i.e. the time penalty that passengers will wait for the next coming vehicle or transfer to the alternative routes when they can't board the first coming vehicle because of insufficient capacity on link s (or line section). the total passenger flow on link s (or line section).

the flat fare of link s (or line section).

the passenger flow on path r E R, .

the vector of link flow. the vector of passenger overload. the vector of link fare. the vector of capacities of line segments. the vector of OD demand. the vector of path flow.

d, :

v, : p , : h," : V:

d: P: k: g : h:

A Bi-level Programming Formulation

Basic formulation

In principle, an optimal transit fare should balance the passenger flows and alleviate the congestion on the transit network. The transit fare problem concerned here can be represented as a leader-follower or Stackelberg game in which the transit operator is the leader, and the passengers are followers. It is assumed that the operator can influence, but cannot control the passengers' route choice behavior by choosing alternative fare policies. Under conditions of various transit fares for different lines, passengers choose their travel path in stochastic user equilibrium (SUE) manner (Lam et al., 1999). This interaction game can be described as the following bi-level programming problem (Yang and Lam, 1996).


Subject to G(x, y(x)) I 0

Where y(x) is implicitly defined by

Min f(x,y)

Subject to g(x, y) I 0 Y

where: F is the objective fbnction of upper level decision-maker (transit operator); x is the decision vector (transit fare) of upper level decision- maker; G is the constraint set of upper level decision vector; f is the objective function of lower level decision-maker (passengers); y is the decision vectors (passenger flows) of lower level decision-maker; and g is the constraint set of lower level decision vector.

It is also assumed that the transit operator would have a feasible range of his decision variables (i.e. fares of transit lines) so as to optimize his objective. The passengers, in view of-the transit operator’s decision, would make route choice decisions in an attempt to minimize their perceived total travel cost, resulting in a SUE network flow pattern.

The lower level problem

For a given transit fare, passengers will make their route choice in the SUE manner. In view of the unreliable transit services during peak hour periods, it is reasonable to consider that passengers perceive with less than perfect information. The behavior assumption of stochastic user equilibrium assignment is then more realistic than that of user equilibrium (UE) assignment.

Passenger flows on transit links (or line sections) that satisfy the following constraints are defined to be feasible. For each OD pair w E W ,

Chrw = g , w E W rsR,

For each link s E S ,

A Bi-level Programming Apporach - ... 111

Let A be the link-path incidence matrix with elements a,, equal 1 if link s lies on path r and equals 0 otherwise. The relationships between the passenger path flows and link flows, shown in Eqn. (2), may then be expressed as

v=Ah (3)

Furthermore, let B be the OD-path incidence matrix with elements bWr being equal to 1 if path r connects OD pair w and 0 otherwise. The relationship between the OD flows and passenger path flows, shown in Eqn. (l), may then be expressed as

g=Bh (4)

On the other hand, the capacity on each line should not be exceeded. In fact, different line sections (or links) may share the same line segment of a transit line. It is obvious that a line section (or link) is not overloaded if and only if the first line segment on it will not be overloaded. We should therefore impose the line segment capacity constraints so that the resultant line flow will not exceed its capacity. It is assumed that each line segment on a line has the same capacity as the line. Let r = (ye,) denote the first line segment-line section incidence matrix, which equals 1 if line segment e is the first part of line section s, otherwise 0. Therefore, passenger flows on line segment e of line 1 can be subject to the following capacity constraint:

It is known that a feasible path for passengers travelling from an origin to a destination consists of a series of line sections (or links) belonging to different transit lines. Passengers will incur the waiting time and the overload delay when they change lines at the transfer nodes. Therefore, the link travel time c, consists of three elements; namely, the

passenger waiting time us and overload delay d, before boarding the


link, and the in-vehicle time t , on the link. The passenger waiting time

us at the transfer node associated with link s is the average time that the passengers wait before the arrival of the first coming vehicle on the link s and is related to the frequency f' . The method of estimating passenger

waiting time us has been described by Lam et al. (1999).

The passenger overload delay d, at transfer node i' (s) associated with the link s would be present when some passengers cannot get on the first arrival vehicle due to insufficient capacity of the arrival vehicle. The overload delay d, is actually equivalent to the passenger overload delay

d, on the first line segment e which connects to station i' (s) . It would

be present only if v, >k, and equals zero if V, c k, . It is because there

will be a passenger overload delay (i.e. d,>O) when some passengers cannot get on the first arrival vehicle on the line segment e connecting to station i ' ( s ) due to insufficient capacity of the arrival vehicle. The relationships can be expressed as:

d, =0, if V , < k , .

d, L O , if V, = k , .

In a transit network with bottlenecks, only a proportion of passengers may get on the first arriving vehicle at some stations and therefore the passenger overload delay needs to be modeled carefully because of insufficient capacity on the in-vehicle links. Lam e f al. (1999) determined endogenously the passenger overload delay according to the characteristics of the congested transit network.

In view of the above discussion, Lam et al.'s model (1999) can be modified to describe passengers' route choice behavior in congested transit network. Consider the following problem:

s.t. g, = ChrW , w E w rcR,

A Bi-level Programming Apporach - ... 113

vS = c Casrhrw s E S

v, = C C Cyesasrhrw 5 k, e E E UEW r€R,

KW rsRw s d

where 8 is the dispersion parameter and is used to measure the different degree of passengers' perception on the path travel time. As 8 + 0 0 , the results of SUE approximate to that of deterministic user equilibrium (UE). /2 is the factor which converts the transit fare into equivalent travel time. E is the set of all the first line segments of the line sections in the transit network.

Substituting the constraint (7c) directly into the objective hnction (7a) and the capacity constraint (7d), the Lagrangian function for problem (Pl) can be formulated as below:

L= c Zhrw(lnhrw - I )+% C(f, + u s +@,) C C asrhrw UEW 6 R W S€S KW rcR,,

+ c l w ( g w - Chrw)+ C meFe - C C Y e s a s r h r " ) (8) WEW rcR,, e c E K W rER,, s d

The Kuhn-Tucker conditions of problem (Pl) can be given as follows:

W € W ( 9 4

ChrW = g , W EW rERw

114

hrw 2 0

Jing Zhou and William H.K. Lam

r e R,, W E W (90

Let

m, is actually the multiplier associated with the capacity on line section s, Z denotes the first line segment of the links. The form of Eqn.(9a) ensures that hrw >O. Eqn. (9a) can be rewritten as

lnhrw=-$(trw +urw +A.prW)+mrw+Zw r E R, W E W (11)

where I, is the corresponding Lagrangian multiplier, ( trw + urw) is the sum of the actual in-vehicle time and the passenger waiting time on path Y E R, , prw is the sum of the actual travel fare on path r E R,. m y is

the sum of Lagrangian multipliers m, along path Y E R,. With Eqn. (Sc), Eqn. (1 1) can easily be transformed to the following logit -type choice model:

exp( - 6ytrw+urw + ilprW)+mrw) Cexp( - e(t;+u; + i lp;)+m;)

h," = g, r E R , W E W (12) keR,

Let d, be the passenger overload delay on line segment e. Clearly, if me = -0 d , , according to Eqn. (1 0), we have

Consequently, mrw = -Bd,", it implies that Eqn. (12) is compatible with the SUE transit assignment problem with capacity constraints. Similar to Bell's (1995) proof, me = -d, is a necessary and sufficient condition for SUE transit assignment with capacity constraints. The role of me is to ensure that the capacity on line segment e is not exceeded.

A Bi-level ProgrammingApporach - ... 115

It should be noted that the objective fbnction (Pl) could be shown to be strictly convex to the passenger path flows. The passenger path flow variable is uniquely defined and hence the passenger link flows are also uniquely determined by Eqn. (7c). However, the Lagrange multipliers or passenger overload delay may not be determined uniquely. On the basis of the Proposition 4 in Bell (1999, it can be deduced that the linear independence of the capacity constraints is a necessary and sufficient condition for the equilibrium passenger overload delay in congested transit network.

The upper level problem

Generally, the transit network design problem can be formulated with different forms of decision variables and objective functions that are dependent on the characteristics of the particular problem of interest. As the transit fares are considered as decision variables and OD demand is fixed, the meaningful objective is thus naturally to minimize the total network travel time instead of maximizing the total revenue raised from the fare charges. In general, the latter could be adopted particularly for the interest of transit operators.

On the other hand, the transit fare or flat h r e of each line should be confined to a reasonable and acceptable range. The upper level problem can therefore be formulated as:

Subject to p,"" 5 P, 5 P,"" (14b)

where p,"'" is the lower bound of fare charges and might be set to be zero or predetermined values for covering part or total operating cost of the transit line concerned; p,"" is the upper bound of fare charges which would be accepted by passengers. It should be noted that different line sections (or links) of a transit line are charged identically as the flat fare is considered here. Moreover, passengers will be charged only once when they get on the transit line at a particular stop. Since the transit fare influences the passenger's route choice behavior, the link flows and passenger overload delays are therefore dependent on the fare of each transit h e . The objective function of the upper problem, which is to


minimize the total network travel time, is implicitly related to the transit fare.

The bi-level programming model

By combining the problems (P l ) and (P2), we obtain the following bi-level programming problem:

Subject to p$'" I p , I p,"" (14b)

where v(p) and d(p) are obtained by solving:

s.t. g , = Ch,, , w E W rER,,

This bi-level problem is to determine an optimal transit fare p* so as to minimize the total travel time on the transit network. It should be noted that the above problem, like any other form of bi-level mathematical problems, is essentially non-convex. Hence it might be difficult to solve this type of problem exactly.

A Bi-level Programming Apporach - 117

Sensitivity Analysis Based Algorithm

The above bi-level programming problem is a Stackelberg game, where the leader (transit operator) determines the fare of each line and the follower (passengers) will make route choice decisions according to the leader's strategy. The transit operator would predict the response of passengers before making his decision. The evaluation of the upper level objective hnction thus requires solving the lower level problem. Recently, sensitivity analysis has become an effective approach to solve this type of bi-level programming problem. Based on the earlier method of Tobin and Friesz (1988), Yang (1995a) adopted the sensitivity analysis approach for road network assignment problem with considering both traffic queue and congestion. One of the important applications of these sensitivity analysis methods is the development of a solution algorithm for some bi-level transportation optimization problems in which the traffic equilibrium problem is taken as the lower level problem. Such applications include the network design (Friesz et al., 1990), OD demand matrix estimation (Yang, 1995b) as well as the congestion pricing problem (Yang and Lam, 1996) and so on.

The aforementioned sensitivity analysis methods are designed to calculate the derivatives of decision variables and constraint multipliers with respect to a variety of perturbation parameters. This derivative information could be used to determine a nearby equilibrium solution resulting from a variety of parameter perturbations once an equilibrium solution has been obtained. In the bi-level problem, the decision variables of the upper level problem are considered as the perturbation parameters of the lower level problem.

Let us recall the Kuhn-Tucker conditions (9a)-(9f) of problem (P 1). Giving a solution po of the upper level problem (P2), then the corresponding passenger flows and multipliers of the lower level

problem (P l ) are denoted by (h*T, m , I ) . Note that the Lagrange multipliers associated with the non-binding constraints equal zero and remain zero near po . The corresponding derivatives of the multipliers

with respect to po thus equal zero. Let us consider only the non- degenerate extreme point of positive path flows and deleting the non- binding constraints, the Kuhn-Tucker conditions can then be reduced to:

*T *T

In h," +O(trw +urw + Aprw)- zasrrns - Z w = 0 r E R,,w E W (16a) SE s


C C asrhrw* - k, = 0 S € S WEW rcRW

where equations can be rewritten in vector form as follows:

is the set of saturated links. For convenience, the above

$(h*,po)-ATm* -BTI* = O

A h * - k = O (17) B h * - g = 0

where the element of @(h',po) is In h:' + O (2: + u: + @:). Let

then,

@(h*, P o ) - ATp * = 0

A h * - z = O

The Jacobian matrix of the equations (18) with respect to (hT ,p ) and evaluated at po is

Let

A Bi-level ProgranimingApporach - ... 119

Therefore, the Jacobain matrix of (18) with respect to p and evaluated at po is

According to Tobin and Friesz 's (1988) theorem, the following equation is obtained

Thus,

With Eqn. (3),

V,V = AV,h (25)

On the other hand, the passenger overload delays are equivalent to the correspondent multipliers associated with line segment's capacity constraints, i.e.,

m = - d

Thus,

It can be obtained by Eqn. (24).


Therefore, a heuristic solution algorithm based on sensitivity analysis is proposed to solve the bi-level programming problem (P12) as follows.

Step 1 Step 2 Step 3

Step 4

Step 5

Step 6

Set an initial transit fare p''), n=O. Solve the problem (P 1) for given p W , get h (") and v (n) . Calculate the sensitivity information VPv(") and Vpd'"'. Formulate local linear approximation of the upper level objective function (P2) using the derivative information and solve the resulting linear programming problem to obtain an auxiliary solution y.

I Compute P ( ~ + ' ) = p'") + -(y - p'")) . n + l

If 1 p("+') - p(") 1s E , E is the tolerated error. Otherwise return to Step 2.

Numerical Example

To illustrate the performance of the proposed solution algorithm, the example transit network shown in Fig. 1 is adopted. The example network consists of one OD pair (i.e. A-B) and four nodes where X and Y are transfer nodes only. There are four transit lines serving the network.

The line data of the transit network in Fig. 1 is given in Table 2.

A Bi-level PrograntmingApporach - ... 121

Obviously, the line L, provides an express service and line L, is faster

than lines L, and L, . There are five feasible paths from A to B. The link composition (or line sections) of each path is shown in Table 1. Assuming that OD demand is 350 passfhr, and the dispersion parameter 9 is given as 0.5. We also assume that the lower bound and the upper bound of fare charges are $10 and $50 respectively.

The conversion factor A converts transit fares to equivalent travel times. The inverse of A is actually equivalent to the value of unit time, which affects the result of the optimal fare. The optimal fares associated with different values of time are shown in Table 3. It can be seen that the fares of the express line L, and faster line L, are markedly sensitive to

the value of time. The optimal fare charges of lines L, and L, become

higher as the value of XI increases. This is because the larger value of A-' implies that passengers weigh the travel time and the overload delay more important. In comparison, due to the longer in-vehicle travel times and the excess capacities on lines L, and L,, their fares always maintain at the lowest value and are insensitive to the change of value of time. As a result, the total revenue from the fare charges increase accordingly as the value of time increases.

Table 3. The resultant oDtimal fares. total travel times and total revenues


On the other hand, it can be found in Table 3 that the total minimal travel times maintain unchanged and the link flows with the optimal transit fares are identical although the values of time are vaned. This is because that the optimal link flows lead to the unchanged minimal total travel time. This result implies that the optimal link flow can always be reached by determining the optimal fare so as to minimize the total travel time on the transit network.

In order to illustrate how the fare on a transit line affects the passenger overload delay on that line, the relationship between the fare and the passenger overload delay on line L, is shown in Fig. 2. In this sensitivity test, the value of time is assumed as $4/min and only the fare of line L, is changed, while the fares of the other lines are remained at the optimal values (i.e., $10 for line L, and L,, and $22.2 for line L3) . It can be seen in Fig.2 that the passenger overload delay on line L, decreases as the fare on line L, increases until the optimal fare of $44.3.

This numerical example illustrates that the optimal fare of transit lines can always be determined by the proposed model so as to balance the passenger flows and alleviate congestion on the transit network.

2 t 4

.09 0 0

10 15 20 25 30 35 40 45 50

Fare on line L 1 ($)

Fig. 2. The relationship between the passenger overload delay and the fare on line L,

A Bi-level Programming Apporach - 123

Conclusions

In this paper, the model formulation and solution algorithm for the design problem of transit fare have been presented. It is well known that an efficient transit fare should balance the passenger flows and alleviate congestion on the transit network.

The transit fare problem discussed here can be represented as a leader-follower or Stackelberg game where the transit operator is the leader, and the passengers are the followers. It is assumed that the operator can influence, but cannot control the passengers’ route choice behavior by choosing alternative fare policies. In view of various transit fares, passengers choose their travel path in the SUE manner. A bi-level programming approach is developed to determine the optimal transit fare in congested transit network. The upper level problem seeks to minimize the total network travel time, while the lower level problem is a stochastic user equilibrium transit assignment model with congestion.

Furthermore, the sensitivity analysis approach has been employed to solve the bi-level programming problem. A numerical example is given to illustrate the relationship between the optimal fare and the value of time. It is noted that the fare charges of lines with shorter travel time are more sensitive to the value of time. As a result, the total revenue from fare charges increases as the value of time increases.

It should be pointed out that the proposed model has recently been extended to elastic transit demand. At the lower level problem, the elasticity of passenger demands has been explicitly incorporated by employing an elastic demand SUE transit assignment model (including overload delays). On the other hand, different transit fare structures (such as distance-based, OD-based and multi-class fares) might however have significant effects on the passenger demands. This issue leads to a topic for our hrther research.

Acknowledgment

Th~s work described in this paper was substantially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region (Project No. Polyu5046/00E).


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a bi-level programming approach — optimal transit fare under line capacity constraints

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