An optimization approach for designing a transit service system is proposed. Its objective would be the maximization of total social welfare, by providing a profitable fare structure and tailoring operational strategies to passenger demand. These operational strategies include full route operation (FRO), limited stop, short turn, and a mix of the latter two strategies. The demand function is formulated to reflect the attributes of these strategies, invehicle crowding, and fare effects on demand variation. The fare is either a flat fare or a differential fare structure; the latter is based on trip distance and achieved service levels. This proposed methodology is applied to a case study of Dalian, China. The optimal results indicate that an optimal combination of operational strategies integrated with a differential fare structure results in the highest potential for increasing total social welfare, if the value of parameter
There has been a considerable increase in numbers for citizen travel, travel frequencies, and travel distances in the past few decades, which arose from the continuous increase in urban interspace and the development of city economies. Aside from the fact that the increased number of private cars could obviously not efficiently cover these increasing trips, they generate urban concerns such as traffic congestion and air pollution, as well as health and safety problems, and so on. Compared with the performances of private cars, public transit clearly can perform in an environmentally friendly, efficient, and sustainable manner [
Fundamentally, transit operational strategies include short turn, limited stop, deadheading, express, and zonal service all of which were initially illustrated by Furth and Day [
For limited stop strategy, based on the requirement of minimum level service, Fu et al. [
Most of the studies investigated operational strategies with different objective functions; it shows that constructing operational strategies is beneficial to improve the efficiency of the public transit operations system. However, these studies ignore the possible impact on passenger demand. The change of the demand is, therefore, important to consider in the optimization formulation. The use of strategies is linked to service levels changes for two groups of passengers: (a) improved service for passengers whose origins and destinations are served though some of their intermediate stops is skipped and (b) reduced service for those passengers who want to board or alight at skipped stops. Therefore, naturally, potential passengers on segments with high service levels are likely to find use of public transit attractive. Alternatively, it is likely that passengers who have used public transit will discontinue this practice due to the low levels of service offered. As a result, there is a fluctuation in demand and its distribution induced by the use of strategies. Even though Ulusoy et al. [
There are two common function forms used to describe variation of demand in the previous studies. The first form is exponential, presented in such studies as Evans [
Fare is found to be a key factor to capture in operational strategies planning context. There are four main elements related to fare: fare policy, strategies, structures, and enforcement technologies. An interaction among these four elements was illustrated in a report by Fleishman et al. [
The differences between flat fare and differential fare were discussed in detail by Fleishman et al. [
These previous studies mainly focused on the optimization of FRO strategy with a flat or a differential fare structure using distance based methods. This paper, as indicated above, aims to examine effects of both fare types on operational strategy optimization, for three basic strategies, FRO, limited stop, and short turn, and a mix of limited stop and short turn strategies. A fare table is constructed which can serve to present either a flat fare or a differential fare structure, in accordance with fare policy objectives. Considering the attributes of operational strategies, the differential fare structure is constructed based on travel distance and obtained service levels. Finally, an optimal combination of fares and strategies is determined by optimizing total benefits consisting of consumer surplus and operator profit, considering the variation of demand.
Following this introductory section, Section
Considering a single and bidirectional bus line, stop locations, route zones, and greatest potential passenger demand along this bus line are given. The passenger demand is assumed to be sensitive to fare, waiting time, crowding time, and invehicle time. It is also assumed that passenger boarding and alighting have a significant effect on bus dwell times at stops. Thus, invehicle time depends on buses passing through the number of bus stops and passenger demand at these bus stops, as well as bus speed. Users are assumed to arrive randomly at stops. The distribution of bus arrivals at each stop presents Poisson distribution. We assume that passengers would not transfer between vehicles operated by different strategies on the single bus line because of high transferring cost considered. This study period is assumed to be a single period of one hour, such as the morning peak hour.
Fare is of considerable importance to bus operation and management. A low fare may attract more passengers but reduce operator revenue. Alternatively, a high fare may increase the revenue but reduce demand. Therefore, it is necessary to determine an attractive and profitable fare that benefits both users and operators. This paper defines a fare structure for a bus line, as shown in Table
A fare structure table for a bus line.

 

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The elements of a differential fare structure table vary in accord with differential fare strategies, such as distance based, time based, quality based, cost based, route based, and patron based strategies, as well as market based fare strategies. They depend on fare policy objectives. For instance, in order to attract more passengers to use public transit in offpeak periods, the transit service planner may consider varied discounts to lower fares during offpeaks, which could cause some users to shift their traveling times from peak to offpeak times. If the objectives of a fare policy are to increase the operator’s revenue and achieve social equity, other differential fare strategies such as cost based, quality based, and distance based strategies are considered when setting bus fares. Accordingly, a differential fare structure table will display hundreds of thousands of possible bus fares for varied purposes of public transit service optimization. However, in consideration of transit system operation characteristics, this work constructs a differential fare structure based on travel distance and obtained service levels.
In order to formulate the methodology as a programming model, the parameters and variables are defined in Notation.
In this study, operational strategies are constructed based on passenger demand. Meanwhile, strategy application generates fluctuations in passenger demand because of the beneficial attributes of these strategies, like high speed between successive stops, greater comfort, and so on. Passenger demand on a single bus line is considered variations with respect to waiting time, invehicle time, fare, and invehicle crowding time, when designing strategies to further improve transit system efficiency.
The product term between two binary variables,
In addition, it is also assumed in this paper that the processes of boarding and alighting are simultaneous (different doors for boarding and alighting) and that boarding and alighting flows are independent of each other. Therefore, the larger one between passenger boarding and alighting times at a stop is dwell time of strategy serving this stop. Accordingly, dwell time of strategy
The weight factor of fare per unit distance for pair
Thus, an integrated differential fare for pair
When the number of onboard passengers
Consequently, average crowding time for pair
An efficient transit system matches the complex relationship between supply and demand from the users’ and operators’ perspectives [
Consumer surplus can be defined as follows [
Substituting (
In this formulation, Constraint (
The above problem is formulated as a mixed integer nonlinear programming (MINLP) problem. It can be handled by the outer approximation with both equality relaxation and augmented penalty (OA/ER/AP) algorithm of Viswanathan and Grossmann [
In this section, a real bus line in the city of Dalian, China, Line 26, is taken as a numerical example for applying the proposed models. Line 26, as shown in Figure
A real life transit route, Line 26, in Dalian.
In this case, a survey was conducted to collect route and travel demand data of Line 26 in the morning peak hour. The running time and distance between successive stops are shown in Table
Running time and distance between successive stops.
Segment  12  23  34  45  56  67  78  89  910 

Running time (min)  2  1.5  0.5  1.5  0.5  1.5  2.5  2  2 
Distance (km)  0.7  0.4  0.5  0.5  0.5  0.6  0.7  0.4  0.5 


Segment  1011  1112  1213  1314  1415  1516  1617  1718  1819 


Running time (min)  5.5  1.5  1.5  3.0  5.5  3.0  3.0  3.0  3.0 
Distance (km)  0.5  0.6  0.5  0.6  0.7  1.1  0.6  0.72  0.78 
Potential passenger




1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  
1  0  20  7  3  6  11  15  23  2  7  0  6  12  7  5  9  5  5  37 
2  8  0  13  7  12  25  30  45  3  14  0  13  24  13  10  19  9  9  474 
3  3  4  0  10  18  39  48  272  5  22  0  20  37  21  15  29  14  14  616 
4  222  233  5  0  4  9  10  15  1  5  0  4  8  4  3  6  3  3  25 
5  16  25  5  34  0  16  19  28  2  9  0  8  50  8  6  12  6  5  46 
6  6  8  1  12  13  0  18  29  2  9  0  8  14  8  6  11  5  5  45 
7  2  3  0  4  5  26  0  28  2  8  0  8  14  8  6  11  5  5  45 
8  10  15  2  21  24  325  43  0  3  15  0  214  126  14  11  20  10  9  78 
9  1  1  0  1  1  9  2  5  0  11  0  10  17  10  8  14  7  7  56 
10  0  0  0  0  0  1  1  1  2  0  0  5  8  5  4  7  3  3  25 
11  0  1  0  1  1  4  1  4  8  0  0  4  6  3  2  4  2  2  17 
12  0  0  0  0  0  3  1  2  4  0  0  0  39  22  17  31  15  15  721 
13  0  1  0  1  1  3  1  4  8  0  0  1  0  17  12  22  11  11  187 
14  1  3  0  4  4  21  7  18  37  1  1  2  1  0  15  28  13  13  611 
15  1  2  0  2  2  12  4  11  21  1  1  1  1  1  0  50  8  8  67 
16  3  5  1  7  8  38  13  33  167  2  2  4  2  2  13  0  4  4  32 
17  2  2  0  3  4  19  7  17  33  1  1  2  1  1  6  1  0  2  18 
18  6  9  1  12  13  168  23  59  317  3  3  7  3  4  23  4  4  0  20 
19  19  28  5  38  44  421  177  296  383  12  12  23  12  12  178  15  15  10  0 
Passenger load profiles on bus Line 26.
Passenger load profile in Direction 1
Passenger load profile in Direction 2
Boarding time and alighting time per passenger are, respectively,
Figure
Start stops and end stops of feasible strategies.
Strategies  Direction 1  Direction 2  

Start stop  End stop  Start stop  End stop  
FRO  Stop 1  Stop 19  Stop 19  Stop 1 
Limited stop  Stop 1  Stop 19  Stop 19  Stop 1 
Short turn  Stop 3  Stop 19  Stop 19  Stop 9 
Mixed strategy  Stop 3  Stop 19  Stop 19  Stop 9 
The optimal results obtained by the proposed methodology are shown in Table
Results of two objectives with and without optimal strategies based on a differential fare table.
Type  Frequency 
Round trip time 
Demand 
Unit fare 
Fleet size 
Operator profit 
Consumer surplus 
Objective function value  

Optimal strategies  FRO  13.0  137.4  6439  0.50  60 

23957.6  20823.1 
Limited stop  14.8  122.8  
Without strategies  FRO  24.1  137.4  6315  0.51  55 

23185.8  20174.9 


124 

5 

771.8  648.2 
Resulting strategies’ topologies under scenario using optimal strategies.
FRO strategy
Limited strategy
In Section
Optimal fares for each pair along a bidirectional bus line with and without use of strategies.
Optimal fare for each pair with strategies
Optimal fare for each pair with FRO strategy and no other strategies added
Interestingly, Figure
Optimal fares using strategies minus optimal fares without strategies.
Figure
Optimal results of two models using flat and differential fare structures.
Objective function values ($/h)
Consumer surpluses ($/h)
Demands (passengers/h)
Average optimal fares ($)
Operator profits ($/h)
In this study, a differential fare function is developed based on trip distance and obtained service levels. This section will discuss the degree of how much passengers are willing to pay for service levels provided. Weight factor of fare per unit distance for each pair calculated by (
The second part on the right hand side of (
Figure
The effects of parameter
Objective function value ($/h)
Operator profit ($/h)
Consumer surplus ($/h)
Demand (passengers/h)
Fare per unit distance
Frequency of operational strategy (vehicles/h)
It is obvious that the effect of an additional service fee on a trip fare will become greater, as the value of
This work focuses on operational strategies and fare problems on a bus line. These operational strategies consist of FRO, limited stop, short turn, and mixed strategies, constructed on the basis of passenger demand. According to characteristics of this transit operation system, a fare table is proposed. This fare table can be presented for a flat fare and a differential fare structure, depending on fare policy objectives. Both types of fare structure are considered in this study, in order to seek a more suitable, profitable fare structure for the application of strategies. The differential fare structure proposed is constructed on the basis of trip distance and obtained service levels. Moreover, passenger demand is sensitive with respect to waiting time, invehicle time, fare, and crowding time in vehicles. This allows for investigating the effects of using operational strategies and two types of fare structures on passengers and the operators, as well as the transit system as a whole. The strategies and fare problems, therefore, are formulated as optimization models, with the objective of maximizing a sum of benefits of users and operators.
This model has been applied to a real life example in Dalian, China. It shows that using strategies can improve results by gains of $648.2/h for objective function value and consumer surplus gains of $771.8/h and attracting more than 124 passengers, despite a small loss in operating profit of $123.6/h, compared with applying FRO strategy exclusively, when a differential fare structure is applied for both optimization models. Clearly, passenger gains can compensate for this loss of operator profit.
In addition, in comparison with optimizing strategies considering a flat fare, a differential structure proves more profitable in terms of objective function value, passenger surplus, and demand, if the value of parameter
Moreover, sensitivity analysis is conducted to examine the effect of an additional service fee on optimal results. It is found that operational strategies using a flat fare provide greater objective function value than those using a differential fare, when the value of parameter
Future research could extend the proposed methodology to include provision of information on bus arrival/departure and running, passengers transferring between vehicle trips associated with different operational strategies, and transport emissions considerations. In the modern, multimodel, urban transportation system, the application of an optimal combination of operational strategies integrated into a differential fare structure, may drive more travelers to leave their cars and turn to public transit services in deference to resulting high service levels of the transit service. The online information can be provided via smartphone and Internet and thus allows for reducing waiting time by timing arrival at stops. We also note that some passengers may transfer between vehicle trips associated with different operational strategies when transferring costs are less than their obtained travel savings. Moreover, operational strategies’ attributes can reduce transit emissions by reducing the number of “stops and goes,” though this is not explicitly discussed in this study. This would be inserted into the objective function.
The maximum number of passengers inside a vehicle with an accepted comfortable level (passengers/vehicle)
Passenger alighting times for a bus using strategy
Passenger boarding times for a bus using strategy
Operating cost per bus hour ($/vehicle h)
Average crowding time for pair
Consumer surplus for trips ($/h)
Operator cost ($/h)
Dwell time of a bus using strategy
Trip distance for pair
Set of end stop of strategies
Demand elasticity parameter for waiting time
Demand elasticity parameter for invehicle time
Demand elasticity parameter for fare
Demand elasticity parameter for crowding time
Bus fare for pair
Frequency of strategy
Operator revenues ($/h)
Potential demand for pair
Layover time of strategy
A set of bus strategies serving a bus line, including full route operation (FRO), short turn (ST), limited stop (LS), and a mix of limited stop and short turn (MLS)
Set of stops on a bus line
Set of feasible stops on strategy
The skipped number of stops that a bus using strategy
The number of stops from stop
Operator profit ($/h)
Fare per unit distance ($/km)
Capacity of vehicle (passengers/vehicle)
An demand variable of the inverse function of elastic demand function with the upper boundary
Running time of a bus using strategy
Running time of strategy
Running time for a bus from stop
Set of start stop of strategies
Average invehicle time for pair
Waiting time for pair
Average boarding time per passenger (h/passenger)
Average alighting time per passenger (h/passenger)
Flat fare for all
The resulting number of passengers for pair
The number of onboard passengers for a bus using strategy
The inverse function of elastic demand function for pair
Crowding time for pair
The sum of consumer surplus and operator profit ($/h)
Equals 1 if passengers of pair
Equals 1 if passengers of pair
Equals 1 if passengers of pair
A constant consisting of deceleration/acceleration as well as doors opening and closing time at each stop (h/vehicle)
A parameter depending on the distribution of bus arrivals at each stop; when bus arrival is Poissondistributed,
A parameter used to adjust the proportion of obtained additional service fee in fare per unit distance
Weight factor of unit fare for pair
Binary variable; taking the value of 1 if strategy
The authors declare that they have no conflicts of interest.
The first author would like to acknowledge the support of the China Scholarship Council (CSC) for this study.