Continuous Approximation Model for Hybrid Flexible Transit Systems with Low Demand Density

Abstract

Flexible transit systems are a way to address challenges associated with conventional fixed route and fully demand responsive systems. Existing studies indicate that such systems are often planned and designed without established guidelines, and optimization techniques are rarely implemented on actual flexible systems. This study presents a hybrid transit system where the degree of flexibility can vary from a fixed route service (with no flexibility) to a fully flexible transit system. Such a system is expected to be beneficial in areas where the best transit solution lies between the fixed route and fully flexible systems. Continuous approximation techniques are implemented to model and optimize the stop spacing on a fixed route corridor, as well as the boundaries of the flexible region in a corridor. Both user and agency costs are considered in the optimization process. A numerical analysis compares various service areas and demand densities using input variables with magnitudes similar to those of real-world case studies. Sensitivity analysis is performed for service headway, percent of demand served curb-to-curb, and user and agency cost weights in the optimization process. The analytical models are evaluated through simulations. The hybrid system proposed here achieves estimated user benefits of up to 35% when compared with fixed route systems, under different case scenarios. Flexible systems are particularly beneficial for serving corridors with low or uncertain demand. This provides value for corridors with low demand density as well as communities in which transit ridership has dropped significantly because of the COVID-19 pandemic.

Current economic trends and population growth patterns pose challenges for the operation of fixed route systems, whereas demand responsive systems are often associated with high operating costs. There are several flexible transit services as an intermediate system between conventional fixed route and demand responsive transit services, which leads to the improved efficiency of transit systems. Flexible route systems are preferable in areas with demand density that is too low to support fixed route systems. The ability of flexible transit services to adapt to customer demands also makes it suitable for serving passengers with a disability. Changing demand for transit services, including disruptions caused by the COVID-19 pandemic, has created a need for alternative public transit systems that accommodate the need for user mobility and agency cost reduction associated with low transit demand.

Flexible transit systems can be designed under different service configurations according to service area characteristics and demand levels. It is thus important to properly identify the service areas where such systems may be effective, as well as the type of flexible transit that is most appropriate. There are many variations of flexible route services, and it is not uncommon for similar types of service to be referred to by different names, since individual transit agencies do not follow a standard naming practice. According to Koffman ( 1 ), there are four elements of service design that could assist in defining the type of flexible service: (a) where vehicles operate; (b) boarding and alighting locations; (c) schedule; and (d) advance notice requirements.

Existing literature includes flexible transit modeling approaches, such as analytical methods, simulation, empirical analysis, and stochastic processes. These are described in the following section. The study proposed here analyzes a hybrid fixed route transit system with elements of flexible services. More specifically, continuous approximation techniques are implemented to identify the optimal boundaries in a given corridor for providing flexible services in the form of route deviation. The proposed flexible hybrid service is compared with conventional fixed route service and fully flexible route deviation within the same corridor. The proposed model for flexible transit is expected to be beneficial in areas where the best transit solution lies between the fixed route and the fully flexible route systems.

Literature Review

An overview of flexible transit systems is provided by Mulley and Nelson ( 2 ), stating that the main goals of flexible systems are to improve the convenience of public transport and maintain a comparable price to existing public transit systems. A survey by Koffman ( 1 ) reveals that most flexible transit services are planned and designed without established guidelines. In Errico et al. ( 3 ) the authors classified existing studies on flexible transit into two categories. The first group includes studies that describe practical experiences, and the second refers to methodological contributions to assist planning processes. There are few cases of implementing optimization techniques for actual flexible systems ( 3 – 5 ). Many approaches are based on analytical models that consider rectilinear distances, because street networks often restrict vehicles to movements along a rectilinear grid ( 6 ).

Existing literature includes surveys related to flexible transit that aim to portray the current conditions under which flexible transit services operate. According to Koffman ( 1 ), at the time of the study development, flexible transit services were implemented in more than 50 transit agencies throughout North America. In Weiner ( 7 ), the authors complement Koffman ( 1 ) by focusing on integrated flexible transit services that were either designed according to the Americans with Disabilities Act (ADA) (1990) or have proven beneficial for riders with disabilities. In Potts et al. ( 4 ), the authors aim to provide a practical guide for implementing flexible transit services through the identification of best practices among 26 agencies, including Mason County Transportation Authority and Jacksonville Transportation Authority.

Most of the studies on public transit user preferences focus on the competition between fixed route and demand responsive services ( 8 , 9 ). Few studies have focused on flexible route transit more generally ( 10 ). In Broome et al. ( 11 ), the authors completed a study showing the public’s positive perception of flexible transit systems. In Chavis and Gayah ( 12 ), a stated preference survey is performed to develop a mode choice model that can be used to describe how transit users select among competitive transit options. Their study covered the entire public transit spectrum. The results indicated that there are statistically significant predictors of the flexible service type selected, such as monetary cost, expected in-vehicle time, waiting time, and walking time. According to Fu ( 13 ), the main beneﬁt of flexible services that involve vehicle route deviation is that they serve trips that would not be otherwise served or that would be served by a more expensive alternative.

Among the many types of flexible transit service, deviated fixed route services are the most widely used ( 14 ). In Zheng et al. ( 10 ), a methodology is proposed to support the decision-making process when choosing between a route deviation policy and a point deviation policy. In Nourbakhsh and Ouyang ( 15 ), the agency and user cost components of a flexible transit system are analyzed considering idealized square cities. In Kim et al. ( 16 ), a planning model is presented for optimizing a flexible system serving many-to-one and one-to-many demand patterns, identifying relations among optimal zone sizes, headways, and relevant exogenous factors. A study included in Pei et al. ( 17 ) summarizes valuable findings from the existing literature on modeling approaches for flexible transit systems.

Continuous approximation methods are widely implemented in existing literature for transportation systems in general ( 18 ). An early study on this topic was conducted in Newell ( 19 ). The optimized coordination between rail and bus transit services through analysis of the user and agency benefits is presented in Wirasjnghe et al. ( 20 ). A recent study by Chen et al. ( 21 ) investigates the usage of local route and short-turn services to complement a regular fixed route transit service by implementing a continuous approximation to model the proposed hybrid system. A detailed review of continuous approximation techniques in existing literature for transportation systems is presented in Ansari et al. ( 22 ).

System Description

The service area considered in this study is rectangular with length $L$ and width $W$ . Vehicles are assumed to travel within the corridor on a rectilinear street network. The basic model of a fixed route transit service is a straight-line corridor in the middle of the service area, with one end being a major terminal station. A typical configuration for this network is given in Figure 1a. Demand in the corridor is assumed to follow a many-to-one pattern in which trips with uniformly distributed origins are all destined for the terminal and trips that originate at the terminal are destined for uniformly distributed points in the service area. The one-way demand in the corridor is the number of passenger trip origins per area per time, $Q$ , which is assumed to be uniformly distributed over space and time. The vehicle average speed, $V$ , accounts for stopping times and traffic delays. Vehicles operate on uniform headway, $H$ , and vehicles are assumed to be large enough that passenger capacity is not a binding constraint. The stop spacing, $S (x)$ , can vary across the fixed route corridor as a continuous function of the distance from the start of the route $x$ .

Figure 1.

Examples of system configuration for: (a) conventional fixed route; (b) flexible with route deviation; and (c) hybrid fixed with route deviation.

Users are assumed to travel from a location within the service area to a terminal station, or vice versa. The terminal station is assumed to connect the service area with a city center or other transportation hub. Thus, it is considered that passengers only board the vehicle as it moves toward the terminal station and they only alight in the opposite direction. Two types of users are analyzed:

Curb-to-curb users—system users that request curb-to-curb service either for their pick-up or drop-off and will be served by a vehicle that is routed to the requested stop.

Fixed stop users—system users that use only the fixed stops that are served by the flexible system.

Flexible services may involve only one or both the types of user presented above. Examples of curb-to-curb requests include users that are eligible for ADA paratransit or other passengers that want to avoid the efforts associated with accessing a fixed stop and waiting at a transit stop rather than their own private space. Such phenomena are expected to increase substantially during the ongoing COVID-19 pandemic, since public transit users aim to reduce their risk of infection to the extent possible. Alternatively, curb-to-curb requests could be assigned on a first-come-first-served basis to the first $a (%)$ of trips requested, based on the number of users that can be served curb-to-curb during a single trip time.

The modeling approach presented in this study assumes that all users are served as they desire, either curb-to-curb or at fixed stops. Thus, the factors that could lead to rejection of service (e.g., vehicle seating capacity) are considered negligible. Both types of demand are perfectly inelastic, which means that they are not affected by the quality of service. This study focuses specifically on a model of flexible service using route deviation, as described in the following section.

Modeling Route Deviation

A vehicle starts its trip from the terminal station and serves customers in a given corridor at fixed stops or by deviating to serve the curb-to-curb demand, which makes up a fraction $a \in [0, 1]$ of the total demand. The locations of fixed stops are defined in relation to the stop spacing at location $x$ , $S (x)$ . The curb-to-curb users are assumed to request their pick-ups or drop-offs with sufficient advanced notice that the vehicle routing can be scheduled and determined before dispatch. The route has a longitudinal length L, which is the length of the corridor. For each requested stop, the vehicle travels a lateral distance, $d$ , to pick up or drop off the curb-to-curb requests and then the same distance, $d$ , to return to the main route. The expected distance of a uniformly distributed requested stop from the main route is $W / 4$ . The vehicle does not backtrack to serve curb-to-curb demand. The remaining $(1 - a)$ portion of total demand is associated with passengers that walk to the nearest fixed stop and wait at that location for service. A typical configuration of a flexible system with route deviation is shown in Figure 1b.

The focus of this study is to optimize the operation of a transit system to identify when and where flexible service will be more beneficial for both agency and users. The resulting system is a hybrid system between a conventional fixed route and a flexible route deviation system. An example of such a system’s configuration is given in Figure 1c. The red dashed line indicates the flexible region where the vehicles may deviate from the fixed corridor to serve the curb-to-curb requested demand. The width of the flexible area around a point $x$ along the fixed corridor is $A (x)$ , where $A (x) \in [0, W]$ . The expected deviation is $A (x) / 4$ .

Here we present calculations for distributed demand and vehicle operations in a corridor heading toward the terminal. The reverse direction, with distributed destinations for passengers heading away from the terminal, is symmetric. The number of passengers boarding each vehicle per unit distance traveled in the corridor is the product of the demand rate, the headway since the last vehicle, and the corridor width, $QHW$ . Of this total demand, the number of passengers with request stop service is $aQHA (x)$ , where the width of the flexible service area can vary as a function of the location in the corridor, $x$ .

Vehicle distance and travel time can be calculated by integrating across the incremental vehicle distance and time required for the transit vehicle to traverse a distance $dx$ at any location $x$ . The total distance and time required to traverse the corridor is obtained by integrating the incremental values over the length $L$ . The one-directional value is then doubled to obtain the distance and travel time associated with a cycle of travel from the terminal back to the terminal.

The vehicle distance is the sum of longitudinal distance traveled along the corridor and the lateral distance traveled to serve each requested stop.

VMT = 2 \int_{0}^{L} [1 + aQHA (x) \frac{A (x)}{2}] dx

(1)

The first term is the longitudinal distance traveled per unit length of the corridor; the total longitudinal distance is $2 L$ per cycle. The second term is the product of the expected number of passengers with request stop service per unit length of the corridor and the expected lateral distance per request stop, which is twice $A (x) / 4$ .

The cycle time, $C$ , includes the travel times for the longitudinal and lateral travel at speed V. It also includes dwell time for three kinds of stop: the dwell time at fixed stops, $τ^{f}$ ; the dwell time at requested stops, $τ^{r}$ ; and the dwell time at the terminal station, $τ^{t}$ . Fixed stops have spacing $S (x)$ , as defined above, so the expected number of fixed stops per unit length of corridor is $1 / S (x)$ . The number of requested stops per unit length of the corridor is the same as the expected number of passengers with request stop service, because each request trip is served individually. The vehicle stops once at the terminal. As a result, the cycle time is given by:

C = 2 \int_{0}^{L} [\frac{1}{V} + aQHA (x) \frac{A (x)}{2 V} + τ^{f} \frac{1}{S (x)} + aQHA (x) τ^{r}] dx + τ^{t}

(2)

Modeling Costs

The continuous approximation approach is adopted here to determine the optimal width of the flexible service area, $A (x)$ , as well as the optimal spacing between fixed stops, $S (x)$ . Both characteristics are treated as continuous functions of the distance, $x$ , from the edge of the corridor and serve as decision variables in the optimization process presented in this paper. Specifically, $A (x)$ can be implemented as a continuous function, and $S (x)$ is approximated by a continuous function. Like the formulation for $VMT$ and $C$ , the analysis is focused on the costs associated with the cycle of vehicle traversing the corridor from the terminal to the end and back.

Agency Costs

The agency cost per vehicle cycle, $AC$ , consists of three parts: costs attributed to vehicle distance traveled; costs attributed to vehicle hours of operation; and costs associated with the fleet size. Each of these costs is calculated by multiplying a cost coefficient by the corresponding value,

AC = a_{VMT} VMT + a_{VHT} C + a_{M} M \frac{H}{O}

(3)

where

$a_{VMT}$ is the cost per vehicle distance traveled,

$a_{VHT}$ is the cost per vehicle time operated,

$a_{M}$ is the daily capital cost per vehicle, and

$M$ is the number of vehicles in the fleet.

The vehicle distance traveled per cycle, $VMT$ , and the cycle time, $C$ , are given by Equations 1 and 2. The fleet size is considered to be constant for this analysis, so its cost must be spread over the number of vehicle cycles operated within a daily period of operations. Details on properly selecting $M$ are given in the following section. If the daily operating hours are denoted by $O$ and the service headway is $H$ , then there are $O / H$ vehicle cycles operated per day.

User Costs

User costs include costs associated with walking, waiting, and riding as experienced by the users. Like the analysis of vehicle operations and agency costs, the user costs can be calculated by integrating the incremental user cost associated with each unit length across the corridor. As a result, the total daily user cost is the sum of these components, weighted by corresponding user cost coefficients: $a_{WK}$ for time spent walking per vehicle cycle, $WK$ ; $a_{WT}$ for time spent waiting per vehicle cycle, $WT$ ; and $a_{R}$ for time spent riding per vehicle cycle, $R$ .

UC = a_{WK} WK + a_{WT} WT + a_{R} R

(4)

The models for each of these components of time spent by users are presented in the subsections below.

Walking

Passengers that receive request stop service do not experience walking time, so the remaining demand $QH (W - aA (x))$ per unit length of the corridor must walk to the nearest fixed transit stop. On average, this is $W / 4$ in the direction perpendicular to the main corridor and $S (x) / 4$ along the corridor. The walking speed is assumed to be $V_{WK}$ . The total walking time for all users served in a vehicle cycle is thus given by

WK = 2 \int_{0}^{L} QH (W - aA (x)) \frac{W + S (x)}{4 V_{WK}} dx

(5)

Waiting

All transit users, either served curb-to-curb or at fixed stops, are expected to experience waiting time equal to half the headway. User choices, such as planning trips around the timetable, are not considered. The total waiting time for all passengers served in a vehicle cycle is simply the product of the demand, $2 QHW$ , and the average waiting time, $H / 2$ .

WT = Q H^{2} W

(6)

Riding

The expected riding time is calculated based on the incremental riding time experienced by all passengers on board a vehicle as it traverses a unit length of the corridor at location $x$ . The number of passengers onboard the vehicles is the cumulative number of passengers that have boarded since the beginning of the line. It is useful to think of this in relation to a vehicle trip from the edge of the corridor that starts empty and picks up passengers en route to the terminal. By the time the vehicle reaches location $x$ , there are $QHWx$ passengers onboard. Each of these passengers experiences travel time associated with longitudinal and lateral vehicle distance as well as loss time per fixed and requested stop. The incremental travel time per unit length of the corridor for all passengers is the product of $QHWx$ and the incremental vehicle travel time, which is the integrand of Equation 2. Therefore, total riding time for a vehicle cycle, $R$ , has a similar structure to the expression for cycle time, $C$ .

R = 2 \int_{0}^{L} QHWx [\frac{1}{V} + aQHA (x) \frac{A (x)}{2 V} + τ^{f} \frac{1}{S (x)} + aQHA (x) τ^{r}] dx + QHWL τ^{t}

(7)

To estimate the total riding costs, the dwell time at the terminal should also be considered. For a fixed corridor of length $L$ , there are $2 QHWL$ passengers that each experience half of the dwell time at the terminal, $τ^{t} / 2$ .

Total Weighted Generalized Costs

The generalized cost for a day of flexible transit operations, $GC$ , is the sum of agency costs, $AC$ , and user costs, $UC$ , weighted by $w_{AC}$ and $w_{UC}$ , respectively.

GC = w_{AC} AC + w_{UC} UC

(8)

This cost depends on the size of the flexible region, $A (x)$ , and the fixed stop spacing, $S (x)$ , which can be designed as functions of $x$ .

The total daily generalized cost, $TGC$ , is then calculated by multiplying the cost per cycle by the number of vehicle cycles that are operated in a day, $O / H$ .

TGC = GC \frac{O}{H}

(9)

The objective in this study is to minimize $TGC$ with respect to $A (x)$ and $S (x)$ for given $O$ and $H$ to achieve the optimal performance for the hybrid transit system studied here. The respective analysis is presented in the following section.

Optimal Spacing and Flexible Region

Given that the duration and daily operations, $O$ , and the service headway, $H$ , are treated as exogenous values in this analysis, the minimization of $TGC$ is equivalent to minimizing $GC$ . Thus, the optimization problem that this paper addresses if the following:

min_{A (x), S (x)} GC

(10)

s.t.

0 \leq A (x) \leq W \forall x \in (0, L)

(11)

0 \leq S (x) \leq 2 L \forall x \in (0, L)

(12)

The constraints on $A (x)$ ensure that the flexible region is always a subset of the corridor. The stop spacing is constrained to $2 L$ , which would be the extreme case with one stop at the terminal, thereby forcing any customer that does not receive request stop service to walk all the way to their destination.

To facilitate the optimization, it is useful to note that in Equations 1, 2, 5, 6, and 7, which are the inputs to Equation 8, the decision variables, $A (x)$ and $S (x)$ , only appear within the integrand. This integrand containing the terms with decision variables can be rewritten as:

\begin{matrix} w_{AC} [a_{VMT} (\frac{aQH}{2} A {(x)}^{2}) + a_{VHT} (\frac{aQH}{2 V} A {(x)}^{2} + \frac{τ^{f}}{S (x)} + aQH τ^{r} A (x))] \\ + w_{UC} [a_{WK} (\frac{QH}{4 V_{WK}} (W - aA (x)) (W + S (x))) + a_{R} QHWx (\frac{aQH}{2 V} A {(x)}^{2} + \frac{τ^{f}}{S (x)} + aQH τ^{r} A (x))] \end{matrix}

(13)

The value of the continuous approximation formulation is that we can now focus on identifying the values of $A (x)$ and $S (x)$ that minimize the integrand at any $x$ , and the results are functions that minimize the integral, and thus the generalized cost.

Expression 13 is not quite separable with respect to $S (x)$ and $A (x)$ , because the term associated with walking cost includes $(W - aA (x)) (W + S (x))$ , which combines both decision variables. This combined term prevents us from being able to derive a closed form analytical solution for the optimal values for $S (x)$ and $A (x)$ at any location $x$ , $S^{*} (x)$ and $A^{*} (x)$ . We note that expression 13 is convex in S(x) if A(x) is treated as given, and it is convex in A(x) if S(x) is treated as given. Therefore, a closed form for the optimal stop spacing at each location, $S^{*} (x)$ , can be expressed in relation to $A (x)$ by solving the first order conditions for expression 13 with respect to $S (x)$ ; that is, setting the first derivative equal to zero and solving for $S (x)$ .

S^{*} (x) = 2 {[\frac{v_{WK} τ^{f} (w_{AC} a_{VHT} + w_{UC} a_{R} QHWx)}{w_{UC} a_{WK} QH (W - aA (x))}]}^{0.5}

(14)

Likewise, a closed form for the optimal size of the flexible service area at each location, $A^{*} (x)$ , can be expressed in relation to $S (x)$ by solving the first order conditions with respect to $A (x)$ .

A^{*} (x) = \frac{V}{4 V_{WK}} \cdot \frac{w_{UC} a_{WK} (W + S (x)) - 4 w_{UC} a_{R} QHWx τ^{r} V_{WK} - 4 w_{AC} a_{VHT} τ^{r} V_{WK}}{w_{UC} a_{R} QHWx + w_{AC} a_{VMT} V + w_{AC} a_{VHT}}

(15)

Equations 14 and 15 are directly applicable to cases where one of the two decision variables is exogenous. For example, Equation 14 provides the optimal fixed stop spacing for a system in which an agency has already decided how big the flexible service area should be (e.g., $A (x) = 1.5$ mi to satisfy minimum ADA requirements). Similarly, Equation 15 defines the optimal size of the flexible service area for a transit agency that may not want to move the stop locations of a fixed route service that has already been designed.

The more complex case is to optimize both decision variables, $S (x)$ and $A (x)$ , simultaneously, because each depends on the other. A computational approach can be implemented to identify the fixed-point solution satisfying Equations 14 and 15. This can be solved substituting the expression for $S^{*} (x)$ in Equation 14 into Equation 15 to obtain an expression with only $A (x)$ terms. The optimal value, $A^{*} (x)$ , that satisfies the equation can be identified by iterating through potential values of $A (x) \in (0, W)$ for increments of $x$ . A numerical solution can be obtained quickly with a computer. Once $A^{*} (x)$ has been identified, $S^{*} (x)$ is given by Equation 14.

Finally, it is necessary to confirm that the available fleet size, $M$ , is sufficient for the designed service operation. Although it is theoretically possible to make $M$ a variable that depends on design variables, the reality is that a flexible transit service in low density corridors typically operates at such long headways that only a small number of vehicles are ever needed. Therefore, $M$ is treated as an input parameter. The fleet size must be at least large enough to sustain the headway, $H$ , with the cycle time, $C$ .

M \geq \frac{C}{H}

(16)

Numerical Analysis

We now present a numerical analysis to illustrate application of the model to realistic corridors. Optimal values of $A^{*} (x)$ and $S^{*} (x)$ are calculated every 0.001 mi to provide a high-resolution representation of optimized functions. The input values for the numerical examples presented here are summarized in Table 1.

Table 1.

Input Values

Parameter	Value	Units
Percent of curb-to-curb demand, $α$	0.50	unitless
Fleet cost coefficient, $a_{M}$	100	$$ / veh$
Riding cost coefficient, $a_{R}$	10	$$ / h$
Vehicle hours traveled (VHT) cost coefficient, $a_{VHT}$	20	$$ / veh . h$
Vehicle miles traveled (VMT) cost coefficient, $a_{VMT}$	0.5	$$ / veh . mi$
Walking cost coefficient, $a_{WK}$	20	$$ / h$
Waiting cost coefficient, $a_{WT}$	10	$$ / h$
Vehicle headway, $H$	1	$h / veh$
Operational hours, O	18	$h / day$
Cruising speed, $V$	25	$mph$
Walking speed, $V_{wk}$	3	$mph$
Weighting factor for AC, $w_{AC}$	1	unitless
Weighting factor for UC, $w_{UC}$	1	unitless
Dwell time at fixed stops, $τ^{f}$	0.008	$h / stop$
Dwell time at curb-to-curb stops, $τ^{r}$	0.005	$h / stop$
Dwell time at terminal stop, $τ^{t}$	0.010	$h / stop$

Note: AC = agency costs; UC = user costs.

The fundamental assumption for user costs is that walking should have higher cost coefficients than waiting and riding and the latter two are considered equal. Insights on the transit user cost coefficients can be found in Wardman ( 23 ). The magnitudes considered here for agency costs are derived from existing literature for the paratransit services in New Jersey and the Greater Boston area, which are considered the worst case scenario, since demand responsive operations in large cities tend to be made more expensive by the high costs of labor. For more details on the agency cost coefficients, readers are referred to Rahimi et al. ( 24 ) and Turmo et al. ( 25 ).

Real-world flexible service areas where vehicles deviate from their routes to serve customers as needed can be identified in existing literature. In Zheng et al. ( 10 ), Route 289 in a suburban area of Zhengzhou City, China, is evaluated for an implementation of point and route deviation services. A single service vehicle is considered for a service area of $W = 1$ mi and $L = 3$ mi. The demand density ranges from 4 to 17 (pax/mi²/h). The MTA Line 646 in Los Angeles County is used in several case studies of flexible system ( 14 , 26 ). The service area has a width of $W = 1$ mi and length of $L = 10$ mi, with one operating service vehicle. In Zheng et al. ( 26 ) demand ranging from 0.8 to 2.8 (pax/mi²/h) is considered. In Qiu et al. ( 14 ), slightly higher demand levels are considered for the same service area and a corridor of size $W = 2$ and $L = 5$ mi is evaluated. A third real-world case study for flexible systems is the Plymouth Area Link in Greater Attleboro Taunton Regional Transit Authority in Massachusetts, U.S.A., which operates the Manomet/Cedarville Deviated Link, where two vehicles operate on a fixed corridor of $L \approx 8$ mi, with a headway $H = 1$ h, which is a common headway for such services. The vehicles are allowed to deviate to serve passengers within $3 / 4$ mi of the fixed route, indicating a service area of width $W = 1.5$ mi.

In the remaining analyses, the magnitudes of $W$ , $L$ , and $Q$ are based on values in existing literature to investigate the implementation of the proposed method under different service area scenarios. For input values with no clear indications from existing literature, a sensitivity analysis is performed to investigate their impact on the proposed flexible transit service.

Optimal Decision Variables

Figure 2 shows the flexible region boundaries for $W \in {1, 2, 3}$ mi for a service area of length $L = 10$ mi. Since $A^{*} (x)$ and $S^{*} (x)$ depend only on cumulative demand up to $x$ , shorter corridors are represented by the same figures, just truncated to $L < 10$ . The horizontal line in the middle of each service area represents the fixed route corridor. Three cases of demand density per direction are investigated in this figure, $Q \in {2.5, 5, 7.5} (pax / m i^{2} / h)$ . The three shaded areas represent the flexible regions in each case, colored with gray, blue, and red, respectively. For all $W$ , the lower value of $Q$ leads to a greater flexible service region and the flexible region gets smaller as $W$ increases. Station locations are also shown for each demand density by black, blue, $ and red dots, respectively. The station spacing increases with $x$ , because greater vehicle occupancy increases the generalized cost of stopping. For more details on determining the station location from a continuous function of spacing between stations, the readers are referred to Wirasjnghe et al. ( 20 ).

Figure 2.

Service area configuration for: (a) W = 1; (b) W = 2; and (c) W = 3.

Optimized System Cost Components

Figure 3 shows that increasing $W$ , $L$ , and $Q$ increases all of the cost components. The costs associated with walking have the greatest impact, and the costs associated with $VMT$ have the least impact. Fleet size costs shown in Figure 3d present a step increase of one vehicle after $x = 4$ . The maximum fleet size for all scenarios investigated here is equal to two vehicles. Although fleet size costs and waiting costs are independent of the optimization process, their values offer insights into the relative magnitudes of the components of the generalized costs. In the case of $VHT$ and $VMT$ shown in Figure 3, e and f, it is apparent that the increase of $Q$ has a lower effect on costs, compared with the increase of $W$ .

Figure 3.

Daily costs of: (a) walking; (b) waiting; (c) riding; (d) fleet size; (e) VHT; and (f) VMT.

Optimal Percent Flexibility

The percentage of the service area that is covered by the flexible region, $f (%)$ , can be calculated considering the results of implementing Equation 15 and the dimensions of the service area, as shown in Equation 17.

f (%) = \frac{\int_{0}^{L} A (x) dx}{WL} 100 %

(17)

Figure 4 shows $f (%)$ for different service areas using inputs of Table 1. Transit agencies that do not implement hybrid services could also use such a graph as a guide for choosing either a fixed route or flexible system.

Figure 4.

Optimal percent flexibility of a service area with length L = 10 and headway equal to: (a) 0.5; (b) 1; and (c) 1.5 h/veh.

Comparison between Fixed Route, Hybrid Transit, and Route Deviation System Costs

Table 2 compares the benefit of the optimized hybrid system with a fixed route and the fully flexible service. The agency cost components considered in optimizing the hybrid service are the $VHT$ and $VMT$ costs. Three corridor lengths are considered, $L \in {3, 5, 10}$ mi. The percent benefit, $B^{S} (%)$ , from implementing hybrid transit $(HT)$ is,

Table 2.

Percent Agency Benefit from Implementing the Optimized Hybrid Transit Instead of Fixed Route (FR) and Route Deviation (RD)

Q = 2.5 (pax/mi²/h)
Benefits (%)	W = 1 mi			W = 2 mi			W = 3 mi
	L = 3	L = 5	L = 10	L = 3	L = 5	L = 10	L = 3	L = 5	L = 10
FR	−26.3	−19.6	−12.0	−52.1	−35.9	−20.4	−77.7	−52.4	−29.3
RD	11.4	18.0	25.5	44.3	52.3	60.0	63.2	70.1	76.3
Q = 5 (pax/mi²/h)
Benefits (%)	W = 1 mi			W = 2 mi			W = 3 mi
	L = 3	L = 5	L = 10	L = 3	L = 5	L = 10	L = 3	L = 5	L = 10
FR	−28.8	−19.8	−11.2	−53.5	−35.5	−19.5	−78.7	−51.9	−28.4
RD	31.7	38.7	45.6	65.3	71.1	76.2	79.0	83.3	87.0
Q = 7.5 (pax/mi²/h)
Benefits (%)	W = 1 mi			W = 2 mi			W = 3 mi
	L = 3	L = 5	L = 10	L = 3	L = 5	L = 10	L = 3	L = 5	L = 10
FR	−28.8	−19.2	−10.5	−53.2	−34.8	−19.0	−78.2	−51.1	−28.0
RD	45.0	51.3	57.3	74.9	79.3	83.1	85.4	88.5	91.0

B^{S} (%) = \frac{C^{S} - C^{HT}}{C^{S}} 100 %

(18)

where $C^{S}$ represents the cost of system $S$ , with $S \in (FR, RD)$ for fixed route and route deviation, respectively. Fixed route service has the lower agency costs among all three systems, so the benefit of hybrid service is negative. Route deviation has the highest agency costs, so the benefit of hybrid service is positive.

The user benefits associated with the hybrid system compared with the fixed route are shown in Figure 5. The user costs of walking and riding affect the optimization process and are considered here. The user benefits range from 0% to 35 % for all combinations of service areas and demand densities. Smaller service areas and lower demand densities lead to greater user benefits from the implementation of hybrid systems compared with fixed routes. Comparing with full route deviation systems, the implementation of the hybrid transit has a user benefit of up to $~ 80 %$ , with some cases having a small loss (i.e., $\leq 5 %$ for small areas and low demand densities). This loss is caused by agency costs in the optimization process for the hybrid service.

Figure 5.

Percent user benefits from implementing hybrid transit instead of fixed route for: (a) Q = 2.5; (b) Q = 5; (c) Q = 7.5; and route deviation for: (d) Q = 2.5; (e) Q = 5; and (f) Q = 7.5.

Given the COVID-19 pandemic and the resulting decrease in transit ridership, it is noteworthy that for any one of the service areas studied here, there is a significant increase in user benefits with a hybrid system as the demand density decreases. The hybrid system is also more beneficial for users than full route deviation systems, especially for $W > 1$ . Finally, the agency loss associated with the hybrid system compared with conventional fixed route is slightly affected by falling demand. For these reasons, the proposed hybrid system has the potential for many beneficial applications in low density communities or in areas where demand has dropped significantly because of the pandemic.

Optimization of Station Spacing Based on Fixed Route and Route Deviation Systems

We now consider the effect of the the size of the flexible region, $A (x)$ on the optimal fixed stop spacing and the costs of the system. Specifically, we are interested in the two extreme cases: $A (x) = 0$ , which is a fixed route system, and $A (x) = W$ , which is a route deviation system. Although $S^{*} (x)$ , as calculated in Equation 14 is sensitive to the value of $A (x)$ used, $A^{*} (x)$ from Equation 15 is not greatly affected whether $S^{*} (x, A (x) = 0)$ or $S^{*} (x, A (x) = W)$ is used. Figure 6a shows that the optimized fixed stop spacing for the hybrid transit, $S_{HT}^{*} (x)$ lies between the optimized station spacings for fixed route, $S_{FR}^{*} (x)$ , and route deviation, $S_{RD}^{*} (x)$ . In this case, $S_{HT}^{*} (x)$ overlaps $S_{RD}^{*} (x)$ for locations $x$ where $A^{*} (x) = W$ and then moves toward $S_{FR}^{*} (x)$ . Although the optimized spacings differ depending on what type of service is considered for their optimization, the optimized flexible regions that result from implementing each of the three optimal spacings are very similar, as shown in Figure 6b.

Figure 6.

Optimized decision variable of: (a) station spacing; and (b) flexible region for a corridor with $W = 2, L = 10, and Q = 5$ .

The difference in cost is more important than that difference in the design variables, because it is the generalized cost of the system that we seek to minimize. The percent change in cost for implementating either the fixed route or full route deviation system relative to the optimized hybrid system is given by

Δ (%) = \frac{C (S_{T}^{*}) - C (S_{HT}^{*})}{C (S_{HT}^{*})} 100 %

(19)

where

$C (S_{T}^{*})$ is the cost of implementing the optimal station spacing for system $T$ , $T \in (FR, RD)$ , and

$C (S_{HT}^{*})$ is the cost of implementing the optimal spacing for the hybrid service.

The costs considered in this analysis are the sum of costs that participate in the optimization process, namely walking, riding, VHT, and VMT costs.

This analysis shows that the effect of different optimized station spacings on the user and agency costs is always small, less than 2% for the cases presented in Figure 7. As a result, it is acceptable to approximate the joint optimization of $A (x)$ and $S (x)$ by implementing Equations 14 and 15 independently. Although the optimized station spacing might differ based on what system is considered in its optimization, the optimal flexible region and the resulting operating costs are not severely impacted.

Figure 7.

Percent difference between costs for: (a) Q = 2.5; (b) Q = 5; and (c) Q = 7.5.

Sensitivity Analysis

Effect of Headway, H

The headway of service has a significant effect on the system design and cost, because it determines the number of passengers served by each vehicle and the number of vehicles needed in the fleet. To facilitate the analysis in this study, $H = 1$ h was used as an exogenous value in accordance with many real-world flexible systems. We now consider the effect of varying $H \in (0.1, 2)$ h on the optimized design variables and the resulting costs. Figure 8, a and b, show the effect of $H$ on $S^{*} (x)$ and $A^{*} (x)$ for a service area with $W = 2$ mi, $L$ up to 10 mi, and $Q = 5 (pax / m i^{2} / h)$ . Greater $H$ leads to smaller flexible regions and shorter stop spacing as the system more closely resembles fixed route.

Figure 8.

Optimal decision variable of: (a) S* (x) for various headways, H; (b) A* (x) for various headways, H; (c) S* (x) for various percentages, a; (d) A* (x) for various percentages, a; (e) S* (x) for various weights w_uc; and (f) A* (x) for various weights, w_uc.

Table 3 shows that, as $H$ increases, daily user costs increase significantly for any percentage of demand served curb-to-curb, $a$ . Lower $H$ is associated with greater impact of $a$ on user costs. The costs included in Table 3, refer to all types of user and agency costs to offer an overview of the overall cost magnitudes.

Table 3.

User and Agency Costs per Day for Different Headways and Percent of Demand Served Curb-To-Curb

H = 0.5 h/veh	User costs ($/day)			Agency costs ($/day)
H = 0.5 h/veh	$a$ = 0.25	$a$ = 0.50	$a$ = 0.75	$a$ = 0.25	$a$ = 0.50	$a$ = 0.75
L = 3 mi	7,599.9	7,260.9	6,908.1	722.3	832.0	947.8
L = 5 mi	13,984.5	13,591.2	13,183.4	968.8	1,087.6	1,212.7
L = 10 mi	33,509.7	33,066.5	32,608.2	1,738.5	1,864.4	1,996.6
H = 1.0 h/veh	User costs ($/day)			Agency costs ($/day)
	$a$ = 0.25	$a$ = 0.50	$a$ = 0.75	$a$ = 0.25	$a$ = 0.50	$a$ = 0.75
L = 3 mi	10,435.2	10,246.7	10,053.2	471.5	529.4	589.5
L = 5 mi	18,659.78	18,458.4	18,252.0	792.9	852.8	914.9
L = 10 mi	42,730.3	42,525.6	42,315.9	1,375.7	1,436.4	1,499.3
H = 1.5 h/veh	User costs ($/day)			Agency costs ($/day)
	$a$ = 0.25	$a$ = 0.50	$a$ = 0.75	$a$ = 0.25	$a$ = 0.50	$a$ = 0.75
L = 3 mi	13,192.0	13,064.4	12,934.2	483.6	522.5	562.6
L = 5 mi	23,226.6	23,096.3	22,963.4	764.0	803.4	844.1
L = 10 mi	51,800.2	51,669.9	51,537.1	1,452.2	1491.7	1,532.4

Effect of Flexible Service Demand

The percentage of demand receiving request stop service within the flexible region, $a$ , affects the distance and time traveled to serve the requested stops. Figure 8c shows that $S^{*} (x)$ increases with a. For the extreme case of $a = 1.00$ , the fixed spacing tends to infinity for $x \leq 0.22$ mi, $A^{*} (x) = W$ in this range so no pasesngers use fixed stops. Further along the corridor, $A^{*} (x)$ drops, increasing the number of passengers using fixed stops. Gray lines in Figure 8, c and d, are associated with increments of $a$ from 0 to 1, with a step of 0.1. Figure 8d shows that the optimal flexible region is very insensitive to $a$ . Only when $a = 0.00$ does it have no impact on costs. Therefore, advanced knowledge of the percentage of users served with request stops is not necessary for identifying that $A^{*} (x)$ is a result of $S^{*} (x)$ .

Effect of Cost Weights

These weights can control the relative effect that agency costs and user costs have on the optimal values for the two decision variables. Figure 8, e and f, show the effects of changing user cost weights from 0.1 to 1 with a step of 0.1 and 1 to 10 with a step of 1. When a cost weight is examined the other is considered equal to one.

Figure 8e shows that station distance is decreased as user costs are taken on higher consideration. Intuitively, this could be attributed to walking costs, which are reduced as user costs have a higher impact on the total generalized costs. The change in $S^{*} (x)$ is greater for $0.1 < w_{UC} < 1$ and much lower for $1 < w_{UC} < 10$ . At $x ~ 0.20$ mi, the lines that correspond to $w_{uc} > 1$ overlap, indicating that station spacing is independent of the user costs weight. This location is the point where the optimal flexible region boundaries reach their maximum value (i.e., $W = 2$ in this case). At some locations $x$ , the optimal value for $A^{*} (x)$ is bounded by the feasibility condition that $A^{*} (x) \leq W$ and the optimal spacing is estimated based on this bounded value of $A^{*} (x)$ . These points are at $x = 0.12$ for $w_{UC} = 1$ and at $x = 0.41$ for $w_{UC} = 10$ . For $w_{UC} = 0$ , the optimal value for station spacing goes to infinity. Similarly, for $w_{uC} = 0$ , the optimal flexible region boundaries go to 0. Figure 8f shows the increase of flexible region boundaries as the weight of user costs increase. Again, for $0.1 < w_{UC} < 1$ , the boundaries present a greater rate of increase compared with the respective changes in $1 < w_{UC} < 10$ .

In relation to the effect of the agency cost weight, $w_{AC}$ , it is observed that an increase in $w_{AC}$ leads to an increase in the station spacing and a decrease in the flexible region boundaries. At the same locations x as in the case of $w_{UC}$ , there are overlaps between the lines of station spacing when $A^{*} (x)$ is bounded by $W$ . The overlap in this case occurs when the agency costs are undervalued. Overall, undervaluing the agency costs has a smaller effect on the two optimized decision variables than overvaluing it. Undervaluing the weight of agency costs has a greater effect on the two decision variables than overvaluing them, even if the agency costs are overvalued by ten times (i.e., $w_{AC} = 10$ ).

Simulation

The simulation process is developed using the R programming language and aims to evaluate the assumptions made during the analytical model development. The output of the simulation algorithm is the scheduling of vehicles in relation to times of arrival at the fixed and curb-to-curb stops, as well as the costs that result from their operation. The demand is generated considering a Poisson distribution. The generated values include location coordinates and requested time. The algorithm serves curb-to-curb passengers following a first-come-first-served pattern and the vehicles do not backtrack.

Figure 9 presents costs resulting from analytical models accompanied by error bars based on the confidence interval from running $N = 50$ simulations. The t-value considered is $t = 2.01$ and the confidence intervals, $CI$ , are calculated as:

CI = (\bar{X} - t \frac{s . d .}{\sqrt{N}}, \bar{X} + t \frac{s . d .}{\sqrt{N}})

(20)

where $\bar{X}$ represents the mean value and $s . d .$ the standard deviation of each cost component for 50 simulation runs. The case study considered here is $W = 2$ mi, $L = 10$ mi and $Q = 5 (pax / m i^{2} / h)$ . Input values are as in Table 1, with $O = 8$ h. The system’s flexible region and station spacing in each run are optimized for the expected demand density per direction, $Q = 5 (pax / m i^{2} / h)$ using Equations 14 and 15. The simulation, however, allows in each run the demand density per direction to be determined by the randomly generated demand. Finally, for this case study it is confirmed that the analytical model’s results are always statistically equivalent to the simulation since the analytical values are always within the simulation confidence intervals. In the case of agency costs, the error bars are not visible since the confidence intervals are narrow.

Figure 9.

Validation of analytical costs of: (a) users; and (b) agency.

Conclusions

Flexible transit systems are widely implemented in real-world service areas, but according to existing literature there is still area for improvement in current services both in relation to operation and design. This study focuses on a hybrid transit system, with elements of both fixed route and route deviation systems. The main outputs of this study include two formulas for optimizing the flexible region boundaries and the station spacing for a hybrid transit service in any given service area. It is highlighted that if agencies prefer to use either fixed route or fully flexible route deviation services, the proposed formulas can serve as a guidance in deciding which service to choose. The numerical analysis performed here adopts input values based on existing flexible service areas and reveals the behavior of the modeling approach under various case scenarios. The analytical method’s performance is evaluated considering a simulation approach developed in R programming language. The hybrid transit has significant user benefits over full route deviation services. In relation to agency costs for the three systems considered here, fixed route is associated with the lowest agency costs, followed by the optimized hybrid system and the fully flexible route deviation system. An important finding is that a service area could switch from fixed route or full deviation to hybrid service within a day, adjusting to any level of demand and maintaining the same station spacing and infrastructure without negative impacts on the operational costs. The benefits from the analyzed hybrid system as transit demand decreases is promising for the implementation of such systems during and after the COVID-19 pandemic. Future work includes the application of these models in case studies and, if possible, their calibration to make them reflect actual operational characteristics to the greatest possible extent.

Footnotes

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: E.J. Gonzales; data collection: C. Sipetas; analysis and interpretation of results: E.J. Gonzales, C. Sipetas; draft manuscript preparation: E.J. Gonzales, C. Sipetas. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was undertaken as part of the Massachusetts Department of Transportation Research Program. This program is funded with Federal Highway Administration (FHWA) and State Planning and Research (SPR) funds. Through this program, applied research is conducted on topics of importance to the Commonwealth of Massachusetts transportation agencies.

Data Accessibility

Data not available.

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