A framework to estimate dwell time of BRT systems using fuzzy regression

Abstract

This paper presents adjustments and validation of a model for the process of passenger boarding and alighting of BRT (Bus Rapid Transport) and express bus systems. Such models are of fundamental importance for the implementation of strategies to control the operation of those systems, which allow for quality and efficiency gains in the service. First, a generalized disjunctive programming model is presented in order to adapt a bus headway control formulation according to the number of onboard passengers. Later, as a way to improve the model, real data were collected from the boarding/alighting times of passengers in the Trunk line 10 system in the city of Blumenau, Brazil. These data were used in a two-stage fuzzy regression to take into account the uncertainties associated with the dwell time of the passengers. The actual data collected were used in simulations of the enhanced bus headway control system, showing that even in the presence of disturbances, the system can still operate properly without bunching.

Keywords

Fuzzy regression BRT systems headway control dwell time GDP

1 Introduction

Mass public transport, and more generally urban mobility, has been a topic of constant discussion and mobilization of society, which also draws attention in Brazil. Among the main topics discussed, besides the costs of transport tariffs, we can cite the quality and efficiency of the services. A negative factor observed in a large number of bus-based public transportation systems is the deviation of the schedule and/or bus headway (tendency of bunching). These deviations increase the waiting time of users and the costs of operating the transportation system [1 –3].

A solution to the problem of real time operating of bus public transport systems is the implementation of control strategies for their operation. Given the need for differentiated quality of express bus and BRT (Bus Rapid Transport) systems, the application of real-time operation strategies in these systems is of paramount importance, highlighting the bus holding strategies at stops [4 –7].

These real-time strategies make use of prediction models for boarding and alighting times of passengers at stops and/or terminals and stations. However, those commonly used models can be overly simplistic, implying a representation that does not match the actual observed behavior or is too complex for real-time applications. In a developed country scenario, with buses that do not operate crowded and with adequate road and station infrastructure, these simplifications typically used may not be significant. In the case of underdeveloped countries, in the everyday reality of many Brazilian cities, the real behavior of the boarding and alighting process is more significant and necessary.

The boarding and alighting process of passengers or “dwell time” is defined as the time spent by a bus at stops [8]. This time typically represents a considerable portion of the total travel time, which justifies the importance of a suitable model for representing this process. For example, [9] found that in American cities from 1957 to 1980 the average dwell time represented about 20% of total travel time within urban areas and increased to 26% in central areas. From data collected in Sydney, considering onboard billing, [10] concluded that the average “dwell time” is about 15% of total travel time.

As a way to improve the representation of the boarding/alighting process, this paper uses as a base the bus headway control strategy model presented by [5], which contemplates characteristics typically present in Brazil bus public transport systems. However, the model presented by [5] requires a more detailed study of the most appropriate value of some parameters, which reflect the behavior of the passengers during the boarding/alighting processes.

To improve the model, our approach uses a series of parameters that describe the behavior of the passengers, both in the boarding and alighting process, as well as parameters associated with the BRT system, like the number of passengers already boarded the bus. Thus, suitable values for these parameters are of fundamental importance for the model to adequately represent the real behavior of the system.

Also, in order to calibrate the model values, this paper makes use of real data collected from the passengers boarding/alighting process of the express bus Trunk line 10 located in the city of Blumenau, Brazil. These data were used in a two-stage fuzzy regression to develop a model that best represents the times associated with “dwell time”, taking into account the number of onboard passengers.

Everything considered, this paper contributes to the state-of-the-art by proposing a novel methodology for modeling and calibrating passenger boarding/alighting time of BRT systems. This methodology, when used in tandem with methods already established in the literature, is able to represent the behavior of a public transport system more accurately. As a consequence, the methodology can improve the performance of control methods at decreasing passenger waiting time.

Other proposed dwell time models make use of nonlinear functions [11, 12], details associated with the passenger’s location preference in the bus [13], door width [14], bus floor height [15], ticketing technique [16], passenger age [17], and grouping of people [18]. Some of these particularities are present directly and indirectly in the model to be used here; others imply in a model too complex, infeasible for real-time applications. Thus, it is considered that the model used here has the potential to improve the representation that describes the dwell time of the Trunk line 10 system, with applicability in other urban public transport systems found in Brazilian cities and similars.

Therefore, this paper is divided as follows. Section 2 presents the model used in the development of the work. Section 3 presents a review of fuzzy regression. The results are presented in Section 4. Finally, Section 5 presents the conclusion of the work.

2 Methodology

To understand the BRT control strategy employed in this paper, this section first presents the existing control strategy already available in the literature, and later proposes a modification of its boarding and alighting process.

2.1 Existing literature model

The original BRT and express bus model used in this paper for analysis and simulation is based on the one presented in [5], which was extended to multiple BRT lines in [6] and to bidirectional segments in [19]. The baseline model [5] allows the real-time control of BRT and express bus systems, more specifically, the control of bus headway through the holding of buses at stops. The model is deterministic and presented in the form of a frequently updated mathematical program with a rolling horizon. Before presenting the mathematical programming model, some preliminary definitions are necessary. First, the sets and indices used are presented in Table 1.

Table 1
Sets and indices

n _I number of buses of the BRT system

n _K number of bus stops of the BRT system

n _N number of stops in the prediction horizon of the buses

$I$ set of buses of the BRT system

$K$ set of stops of the BRT system

i bus index, $i \in I$

k stop index, $k \in K$

m index of the bus lap in the closed BRT circuit

$K_{i}$ set of stops belonging to the prediction horizon of the bus i

$K_{i}^{0}$ set of stops of bus i for which the initial values d_i,ω (departure times) and l_i,ω (onboard passengers) apply

Considering a circular itinerary, the following lap arithmetic is used: ω = (k, m); 〈i, ω^-〉 represents for bus i, the stop immediately preceding stop k, which can be stop (k - 1) on the same lap m, or else stop n_K of lap (m - 1) if the current stop is the first in the circuit (k = 1); 〈i^-, ω〉 corresponds to the previous bus that arrived at bus stop k before bus i in lap m, which can be a bus in the same lap or in the previous one. Given those considerations, the rest of the systems variables are presented in Table 2.

Table 2

Variables

Parameters and initial conditions
C ₀	time for the start of passengers boarding and alighting process from the arrival of the bus to the stop [s]
C _a	time spent for each passenger alighting the bus [s/passenger]
C _b	time spent for each passenger boarding the bus [s/passenger]
C _max	bus capacity [pax]
r _k	trip time from stop k - 1 to stop k
q _k	fraction of passengers alighting at stop k
λ _k	passenger arrival rate at stop k [pax/s]
State and control variables
d _i,ω	departure time of bus i from stop k in lap m [s], with ω = (k, m)
h _i,ω	holding control time for bus i at stop k in lap m [s]
l _i,ω	onboard passengers of bus i when it departs from stop k in lap m
Auxiliary variables
a _i,ω	arrival time of bus i at stop k in lap m [s]
p _i,ω	number of passengers to board the buses at the stops [pax]
re _i,ω	passenger residue at stops [pax]
s _i,ω	dwell time of bus i at stop k in lap m [s]
$t_{i, ω}^{a}$	bus passenger alighting time [s]
$t_{i, ω}^{b}$	bus passenger boarding time [s]
v _i,ω	bus remaining capacity [pax]

The total passengers’ waiting time for the horizon considered is given by: $f = \sum_{i \in I} \sum_{ω \in K_{i}} [\begin{matrix} \frac{λ_{k}}{2} {(d_{i, ω} - d_{〈 i^{-}, ω 〉})}^{2} \\ + r e_{〈 i^{-}, ω 〉} (d_{i, ω} - d_{〈 i^{-}, ω 〉}) \\ + (1 - q_{k}) l_{〈 i, ω^{-} 〉} (h_{i, ω} + s_{i, ω}) \end{matrix}],$ (1) subject to the following constraints: $d_{i, ω} = d_{i, ω}^{0}, \forall ω \in K_{i}^{0}$ (2) $l_{i, ω} = l_{i, ω}^{0}, \forall ω \in K_{i}^{0}$ (3) $r e_{i, ω} = {re}_{i, ω}^{0}, \forall ω \in K_{i} \cup K_{i}^{0}$ (4) $a_{i, ω} = d_{〈 i, ω^{-} 〉} + r_{k}$ (5) $v_{i, ω} = C_{max} - l_{〈 i, ω^{-} 〉} (1 - q_{k})$ (6) $p_{i, ω} = r e_{〈 i^{-}, ω 〉} + (d_{i, ω} - d_{〈 i^{-}, ω 〉}) λ_{k}$ (7) $d_{i, ω} = a_{i, ω} + s_{i, ω} + h_{i, ω}$ (8) $d_{i, ω} \geq a_{〈 i^{-}, ω 〉}$ (9) $l_{i, ω} = min {(1 - q_{k}) l_{〈 i, ω^{-} 〉} + p_{i, ω}, C_{max}}$ (10) ${re}_{i, ω} = max {0, p_{i, ω} - v_{i, ω}} .$ (11)

Constraint (2) gives initial conditions of the bus departure time from a stop, bus onboard passengers when departing from a stop by (3), and initial residue of passengers at a stop by (4). The constraint (5) represents the bus arrival time to the stop k; (6) the bus remaining capacity and (7) the number of passengers to board the bus. Departure time is given by (8), and (9) guarantees that bus overtaking is not allowed. The number of onboard passengers after bus departure is given by (10), and the residue of passengers by (11).

The original formulation presented in [5], the boarding and alighting processes are defined as follows: $t_{i, ω}^{a} = C_{a} q_{k} l_{〈 i, ω^{-} 〉}$ (12) $t_{i, ω}^{b} = min {C_{b} v_{i, ω}; C_{b} (p_{i, ω} - h_{i, ω} λ_{k})}$ (13) $s_{i, ω} = C_{0} + t_{i, ω}^{a} + t_{i, ω}^{b}$ (14)

Constraint (12) gives the bus passengers alighting time. The boarding time is given by the time spent to board the passengers who occupy the remaining capacity of the bus or the time spent for boarding the accumulated passenger queue at the stop. This is given by means of a minimum operator as (13). Considering terminals and stations as bus stops of traditional BRT systems, the passenger alighting process occurs first. The boarding process, with pre-billing, proceeds through the same bus rear doors used for alighting. Thus, dwell time is calculated as the sum of the boarding and alighting processes as (14), Fig. 1 illustrates the process when the number of boarding passengers is smaller than the bus remaining capacity, and Fig. 2 illustrates the process when the number of boarding passengers is bigger than the bus capacity.

Fig. 1

Boarding and alighting processes for terminal and stations with the number of boarding passengers smaller than the bus remaining capacity. The queue boarding rate is μ_k.

Fig. 2

Boarding and alighting processes for terminal and stations with the number of boarding passengers bigger than the bus remaining capacity. The queue boarding rate is μ_k.

As can be seen in (12), (13) and (14), boarding and alighting times depend on the parameters C_a and C_b which are typically considered constant. However, what is observed in real systems is that both parameters, besides the typical variation around the average value, vary as a function of other parameters of the BRT system such as bus capacity. More insights on how to solve this model can be seen in [5, 6].

2.2 Proposed model enhancement

Given the characteristics and peculiarities of urban transport systems by buses in Brazil and other countries, this paper proposes an enhanced estimation of boarding and alighting of passengers that takes into account a more realistic behavior, with characteristics compatible with real-time applications.

In the new proposed boarding/alighting model, the time spent for each passenger boarding the bus (C_b) and the time spent for each passenger alighting the bus (C_a) can vary depending on the number of onboard passengers. Consider the following set containing the limits of the categorical variable “quantity of onboard passengers”:

$U = {low, medium, large},$ (15) i.e.:

$U = {(0, 40), (41, 50), (51, 60)} .$ (16)

The decision between the categorical variables for the alighting time can be represented as a Generalized Disjunctive Program (GDP): $⋁_{(\underline{u}, \bar{u}) \in U} [\begin{matrix} g_{i, ω, (\underline{u}, \bar{u})}^{a} \\ \underline{u} \leq l_{〈 i, ω^{-} 〉} \leq \bar{u} \\ t_{i, ω}^{a} = C_{a, (\underline{u}, \bar{u})} q_{k} l_{〈 i, ω^{-} 〉} \end{matrix}]$ (17a) $g_{i, ω, (\underline{u}, \bar{u})}^{a} \in {True, False}, \forall (\underline{u}, \bar{u}) \in U$ (17b)

Notice that the alighting time depends on the passengers that boarded the bus on the last stop (l_〈i,ω^-〉). Big-M and convex hull reformulations are the most common methodologies for transforming a GDP into a mixed-integer linear programming (MILP) problem. The proposed GDP can be reformulated as a MILP, using Big-M notation, in the following form ( $\forall (\underline{u}, \bar{u}) \in U$ ):

$\underline{u} (g_{i, ω, (\underline{u}, \bar{u})}^{a}) \leq l_{〈 i, ω^{-} 〉, (\underline{u}, \bar{u})} \leq \bar{u} (g_{i, ω, (\underline{u}, \bar{u})}^{a})$ (18a) $t_{i, ω}^{a} \leq C_{a, (\underline{u}, \bar{u})} q_{k} l_{〈 i, ω^{-} 〉} + M (1 - g_{i, ω, (\underline{u}, \bar{u})}^{a})$ (18b) $t_{i, ω}^{a} \geq C_{a, (\underline{u}, \bar{u})} q_{k} l_{〈 i, ω^{-} 〉} - M (1 - g_{i, ω, (\underline{u}, \bar{u})}^{a})$ (18c) $l_{〈 i, ω^{-} 〉} = \sum_{(\underline{u}, \bar{u}) \in U} l_{〈 i, ω^{-} 〉, (\underline{u}, \bar{u})}$ (18d) $\sum_{(\underline{u}, \bar{u}) \in U} g_{i, ω, (\underline{u}, \bar{u})}^{a} = 1$ (18e) $g_{i, ω, (\underline{u}, \bar{u})}^{a} \in {0, 1}$ (18f)

The decision between the categorical variables for the boarding time can also be represented as a GDP, but the decision variables depends on the number of passengers that didn’t alight the bus on the present stop (l_〈i,ω^-〉 (1 - q_k)), since the alighting occurs first:

$⋁_{(\underline{u}, \bar{u}) \in U} [\begin{matrix} g_{i, ω, (\underline{u}, \bar{u})}^{a} \\ \underline{u} \leq l_{〈 i, ω^{-} 〉} (1 - q_{k}) \leq \bar{u} \\ t_{i, ω}^{b 0} = C_{b, (\underline{u}, \bar{u})} v_{i, ω} \\ t_{i, ω}^{b 1} = C_{b, (\underline{u}, \bar{u})} (p_{i, ω} - h_{i, ω} λ_{k}) \end{matrix}]$ (19a) $g_{i, ω, (\underline{u}, \bar{u})}^{b} \in {True, False}, \forall (\underline{u}, \bar{u}) \in U$ (19b) which can be reformulated as a MILP using Big-M notation similarly to the last one. So that the dwell time can now be calculated as: $s_{i, ω} = C_{0} + t_{i, ω}^{a} + min {t_{i, ω}^{b 0}; t_{i, ω}^{b 1}}$ (20)

With those propositions, there is no need to depend on a single deterministic value during the control estimation. The control system can automatically adapt to the number of passengers boarded. Note that although the example presented here has only three categories (namely low, medium, and large), it becomes simple to extrapolate the model to as many categories as necessary to better represent the system.

2.3 Data acquisition

To populate C_a and C_b with more realistic data, the approach of this paper uses two-stage fuzzy regression coefficients (discussed in the next section) of real collected data as deterministic values in the control model. To do so, the measurements were made considering the number of passengers to board/alight, the total “dwell time”, and the number of passengers already boarded. During the holding control holding calculation stage, the equations (17) and (19) are responsible for identifying which categorical variable to be used in the process.

2.4 Test setup

In order to test the model, a stochastic scenario is used in a simulation (transit model) in which boarding/alighting times are given randomly. Therefore, two-stage fuzzy regression data were used to collect samples from a triangular distribution over the interval. The triangular distribution is a continuous probability distribution with lower limit left, peak at mode, and upper limit right. Unlike other distributions, these parameters directly define the shape of the probability density function (pdf). The pdf for the triangular distribution is: $P (x; l, m, r) = {\begin{matrix} \frac{2 (x - l)}{(r - l) (m - l)} & for l \leq x \leq m, \\ \frac{2 (r - x)}{(r - l) (r - m)} & for m \leq x \leq r, \\ 0 & otherwise . \end{matrix}$ (21)

For the stochastic scenario simulation, the setup presented in Fig. 3 will be used. Further details of the simulation parameters can be found in [5].

Fig. 3

Test setup for the headway control, consisting of a deterministic predictive control and a stochastic transit scenario.

3 Fuzzy regression

Fuzzy regression was proposed in the literature as a way to consider the uncertainties of the coefficients of linear regression, mostly used when the model is indefinite, the relationships between model parameters are vague, the sample size is small, or the data is hierarchically structured [21 –24]. This kind of analysis was initially proposed by [21] based on the concepts of fuzzy functions presented by [25], which aimed to take into account the natural uncertainties present in a certain phenomenon, case of the Trunk line 10 scenario analyzed here.

While for traditional linear regression the deviation between the observed and estimated values is assumed to be due to observation errors, [21] assumed that these differences occur because of the vagueness of the system structure. The fuzzy parameters of a linear system measured by the model correspond to the system’s possibility distribution, being responsible for measuring its uncertainty (fuzziness).

3.1 Possibilistic regression

By generating parameters corresponding to the possibility distribution of a system, the model presented by [21] is known in the literature as possibilistic regression and with a general form given by the equation below: $\tilde{y} (k) = {\tilde{A}}_{0} + {\tilde{A}}_{1} x_{1} (k) + \dots + {\tilde{A}}_{N} x_{N} (k)$ (22) with k = 1, 2, …, K and where x = [x₀, x₁, …, x_N] ^T is a vector of independent variables, $\tilde{A}$ = $[{\tilde{A}}_{0}, {\tilde{A}}_{1}, . . ., {\tilde{A}}_{N}]^{T}$ a vector of fuzzy coefficients denoted in the form of a triangle by ${\tilde{A}}_{j}$ = $(a_{j}^{c}, a_{j}^{w})$ , having its pertinence function described by: $μ_{{\tilde{A}}_{j}} (a_{j}) = {\begin{matrix} 1 - \frac{| a_{j}^{c} - a_{j} |}{a_{j}^{w}}, \\ 0, \end{matrix} \begin{matrix} a_{j}^{c} - a_{j}^{w} \leq a_{j} \leq a_{j}^{c} + a_{j}^{w} \\ otherwise \end{matrix}$ (23) where $a_{j}^{c}$ and $a_{j}^{w}$ are respectively the center point and the maximum dispersion of the function, as shown in Fig. 4. Thus, the regression presented in (22) can be rewritten as: $\begin{matrix} \tilde{y} (k) = & (a_{0}^{c}, a_{0}^{w}) + (a_{1}^{c}, a_{1}^{w}) x_{1} (k) \\ + \dots + (a_{n}^{c}, a_{n}^{w}) x_{n} (k) \end{matrix}$ (24) or in the general formulation: $\tilde{y} (k) = (a_{0}^{c}, a_{0}^{w}) + \sum_{n = 1}^{N} (a_{n}^{c}, a_{n}^{w}) x_{n} (k)$ (25)

Fig. 4

Symmetric triangular membership function.

Differently from the ordinary least squares regression, the deviation between the data and the estimated model depends on the imprecision of the parameters and not on the measurement errors. In this case the model of [21] proposes that, in order to minimize the uncertainty of the estimated model, one can minimize the total spread of the system’s fuzzy coefficients [26].

This spread may also depend on a degree of pertinence known as the “h value”, defined by the operator, which specifies the degree of feasibility of the system conditions [27]. The higher the degree of feasibility, the higher the spreading of the system, i.e., the h value expands the confidence interval of the model. Fig. 5 illustrates the change in the lower and upper limits for h of 0.5 and 0.7.

Fig. 5

Increase of fuzzy interval with a higher h value.

Fig. 6 illustrates the behavior of the spreading of the system with the increase of the h factor, both to the right and to the left of the central part of the system. Its increase is directly proportional to the expansion of the limits, allowing the inclusion of more or fewer data between the limits of the model. While the typical values are contained within the limits, even with a reduced h factor, the outliers are included with the factor expansion.

Fig. 6

Triangular membership function with h value interval.

From these considerations, [21] show that the possibilistic regression is reduced to a linear programming problem whose objective is to minimize the spread of the system, or its uncertainties, as presented in (26) subject to the constraints given in (27). $min_{a^{c}, a^{w}} J = \sum_{n = 1}^{N} \sum_{k = 1}^{K} a_{n}^{w} x_{n} (k)$ (26) $\begin{array}{l} s . t . (\sum_{n = 1}^{N} a_{n}^{c} x_{n} (k)) \\ - (1 - h) (\sum_{n = 1}^{N} a_{n}^{w} x_{n} (k)) \leq y (k), \forall k \in K \end{array}$ (27a) $\begin{matrix} (\sum_{n = 1}^{N} a_{n}^{c} x_{n} (k)) \\ + (1 - h) (\sum_{n = 1}^{N} a_{n}^{w} x_{n} (k)) \geq y (k), \forall k \in K \end{matrix}$ (27b)

$a^{w} \geq 0$ (27c) where $a^{c} = [a_{1}^{c}, \dots, a_{n}^{c}]$ , $a^{w} = [a_{1}^{w}, \dots, a_{n}^{w}]$ , y is the dependent variable to be predicted by the regression, and $K = {1, \dots, K}$ is the set of points.

However, although still present in the literature due to its ease of implementation and lower computational cost when compared to improved models [26 –28], the possibilistic regression using linear programming has received several criticisms, some of which are remarked by [22]:

The model proposed by Tanaka resembles linear regression, but there is no evidence of how it relates to the classical concept of least squares [29];

High sensitivity to outliers;

Tendency to become multicolinear as more independent variables are collected;

In this matter, the literature has presented alternatives for the possibilistic regression using linear programming, such as fuzzy regression using least squares [29], rigid fuzzy regression [30], piecewise fuzzy linear regression [31, 32], fuzzy regression with linear neural networks [33] and nonlinear neural networks [34].

The authors of [21] themselves continued to improve the model employing quadratic programming, which according to the authors, increases the tendency of the coefficients to become diffuse when compared to the initial model [23]. The possibilistic regression using quadratic programming starts from the quadratic minimization of the spreads of the function, as presented in (28). $min_{a^{c}, a^{w}} J = \sum_{n = 1}^{N} \sum_{k = 1}^{K} (a_{n}^{w} | x_{n} (k) |)^{2}$ (28)

The function can be rewritten as presented in (29), with the addition of a new term, where ξ is a sufficiently small positive number added to the objective function so that (29) is strictly convex. The constraints of the function remain the same, as shown in (30). $min_{a^{c}, a^{w}} J = {a^{w}}^{T} (\sum_{n = 1}^{N} | x_{n} | {| x_{n} |}^{T}) a^{w} + ξ {a^{c}}^{T} a^{c}$ (29) $\begin{array}{l} s . t . (\sum_{n = 1}^{N} a_{n}^{c} x_{n} (k)) \\ - (1 - h) (\sum_{n = 1}^{N} a_{n}^{w} x_{n} (k)) \leq y (k) \end{array}$ (30a) $\begin{matrix} (\sum_{n = 1}^{N} a_{n}^{c} x_{n} (k)) \\ + (1 - h) (\sum_{n = 1}^{N} a_{n}^{w} x_{n} (k)) \geq y (k) \end{matrix}$ (30b)

$a^{w} > 0$ (30c)

To allow an optimization capable of differentiating the possibility of left propagation of the possibility of right propagation, one can rewrite the quadratic possibilistic regression as in (31), subject to the restrictions presented in (32). $\begin{array}{l} \min_{a^{c}, a^{w L}, a^{w R}} J = (a^{w L} + a^{w R})^{T} \\ \times (\sum_{n = 1}^{N} |x_{n}| {|x_{n}|}^{T}) (a^{w}^{L} + a^{w}^{R}) \\ + ξ a^{c}^{T} a^{c} \end{array}$ (31) $\begin{array}{l} s . t . (\sum_{n = 1}^{N} a_{n}^{c} x_{n} (k)) \\ - (1 - h) (\sum_{n = 1}^{N} a_{n}^{w} L x_{n} (k)) \leq y (k) \end{array}$ (32a) $\begin{array}{l} (\sum_{n = 1}^{N} a_{n}^{c} x_{n} (k)) \\ + (1 - h) (\sum_{n = 1}^{N} a_{n}^{w} R x_{n} (k)) \geq y (k) \end{array}$ (32b) $a_{n}^{wL} > 0$ (32c) $a_{n}^{wR} > 0$ (32d)

From this modification the optimization starts to consider a non-symmetric triangular membership function, as shown in Fig. 7.

Fig. 7

Non-symmetric fuzzy triangular membership.

3.2 Hybrid regression

Criticism of the initial model presented by [21], including the uncertainty of how it relates to the traditional model of least squares regression [29], has led to the creation of hybrid models.

One of these hybrid models seeks to maintain the central tendency of the model; for instance, [35] introduced a two-stage fuzzy regression considered important by its results and low computational cost [28]. In the two-stage fuzzy regression, the central trends of the model are defined, typically by ordinary least squares, which are then used as known variables in the optimization.

Let a^c be a central trend vector previously defined. The two-stage fuzzy regression can determine the spreads as in (33) subject to the conditions presented in (34). Notice that in order to allow an optimization capable of differentiating between the possibility of left and right propagation, one can rewrite the quadratic possibilistic regression decomposing a^w in a^wL and a^wR. $min_{a^{wL}, a^{wR}} J = \sum_{n = 1}^{N} \sum_{k = 1}^{K} (a_{n}^{wL} + a_{n}^{wR}) x_{n} (k)$ (33) $\begin{array}{l} s . t . (\sum_{n = 1}^{N} a_{n}^{c} x_{n} (k)) \\ - (1 - h) (\sum_{n = 1}^{N} a_{n}^{w} L x_{n} (k)) \leq y (k), \forall k \in K \end{array}$ (34a) $\begin{array}{l} (\sum_{n = 1}^{N} a_{n}^{c} x_{n} (k)) \\ + (1 - h) (\sum_{n = 1}^{N} a_{n}^{w} R x_{n} (k)) \geq y (k), \forall k \in K \end{array}$ (34b) $a^{wL} \geq 0$ (34c) $a^{wR} \geq 0$ (34d)

This model can be augmented to use quadratic programming in order to increase the diffusion of the coefficients. Rewriting (33) in its quadratic form it is possible to arrive at (35), very similar to the model of [22]. Since the vector a^c is already defined in the first stage of the regression and does not need to be optimized, the function is naturally strictly convex. $J = \sum_{n = 1}^{N} (a_{n}^{wL} + a_{n}^{wR})^{2} x_{n}^{T} x_{n}$ (35)

With two-stage fuzzy regression it is possible to maintain the original central tendency of the model, which can be replaced by other types of estimation (ridge, minimax) and minimize spreads based on these values to include the human uncertainties in the model.

3.3 Solving the fuzzy regression

All the problems presented in this section are linear programming (LP) problems or quadratic programming (QP) problems. The first can be written in the following canonical form: $min c^{T} x$ (36a) $s . t . A x \leq b$ (36b) $x \geq 0,$ (36c) while the canonical form of the second is given by: $min \frac{1}{2} x^{T} Q x + c^{T} x$ (37a) $s . t . A x \leq b$ (37b) $x \geq 0,$ (37c) where x is a vector defined by stacking variables (the variables to the minimized); and parameter vectors b and c and matrices A and Q are easily obtained from the coefficients of the previously presented formulations. Notice that both problems are classic in the optimization literature, and the process to convert any linear or quadratic problem into their corresponding standard form is a straightforward process, resulting in no loss of generality.

Linear problems given by (36) can be solved for the global optimum by the use of efficient algorithms such as the simplex method [36], while the global optimum of a quadratic problem (37) can be reached by employing algorithms such the interior-point method [37]. Both algorithms are available in commercial and non-commercial solvers, like Gurobi [38] and IPOPT [39].

3.4 Goodness-of-fit

3.4.1 Index of confidence (IC)

As a way of interpreting the results of a fuzzy regression and contributting to the interpretation regarding the regression interval, [40] have proposed an index of confidence (IC) that measures the variation between the independent upper and lower limits of the variable, which is similar to the goodness-of-fitness R² in statistical regression. Thus, we consider that y^L (k) and y^R (k) are respectively the lower and upper bounds of a fuzzy estimate $\tilde{y} (k)$ , which has as a central estimation y^c (k).

The index of confidence (IC) is calculated as (38), and the larger the value, the greater the system’s representation by the regression performed. $IC = \frac{SSR}{SST}$ (38)

The SST (Total Sum of Squares) measures the total variation of y (k) to the left spread (y^L (k)) and to the right spread (y^R (k)), calculated as: $SST = \sum_{k = 1}^{K} {(y (k) - y^{L} (k))}^{2} + \sum_{k = 1}^{K} {(y^{R} (k) - y (k))}^{2}$ (39)

The SSR (Regression Sum of Squares) measures the variation of the center estimate (y^c (k)) to both spreads (y^L (k) and y^R (k)), calculated as: $SSR = \sum_{k = 1}^{K} {(y^{c} (k) - y^{L} (k))}^{2} + \sum_{k = 1}^{K} {(y^{R} (k) - y^{c} (k))}^{2}$ (40) The h value is directly related to the index of confidence, being directly proportional to its increase [40].

3.4.2 Average fuzzy spread (AFS)

Since fuzzy regression aims to find a minimal range that covers all collected samples, the Average Fuzzy Spread (AFS), which considers the absolute difference between the longest and shortest interval of the model and the h value, can be used to estimate the range size and general diffusion of the model. The smaller the AFS, the better the model. So the AFS equation is given by [41]: $AFS = h \cdot \sum_{n = 1}^{N} Δ {\tilde{y}}_{n}^{*} = h \cdot \sum_{n = 1}^{N} | y_{n}^{R} - y_{n}^{L} |$ (41)

However, AFS considers only the model range and may not take into account the central approach, which differs from the previous IC and the following metric, the Mean Fuzzy Credibility (MFC).

3.4.3 Mean fuzzy credibility (MFC)

The fuzzy regression can generate models with low membership and high diffusion, resulting in a low credibility model. In order to measure the model’s credibility, the MFC assesses the relevance of the collected data to the fuzzy regression model and the general diffusion of the model. Thus, it measures the credibility of ${\tilde{y}}_{n}^{*}$ in representing y_n, so the higher the MFC, the better the model. The MFC is calculated as [42]: $MFC = \sum_{n = 1}^{N} \frac{u_{{\tilde{y}}^{*}} (y_{n})}{Δ {\tilde{y}}_{n}^{*}} = \sum_{n = 1}^{N} \frac{u_{{\tilde{y}}^{*}} (y_{n})}{| y_{n}^{R} - y_{n}^{L} |}$ (42)

3.5 An optimized h value

Since h directly influences the relevance of the model members, so the model’s credibility is also affected by its value. In order to obtain maximum credibility based on the h factor value, [43] proposed a systematic approach to obtain an optimized h value, as: $max_{h} z^{h} = - p^{0} h^{2} + (p^{0} - z^{0}) h + z^{0}$ (43a) $s . t . 0 \leq h < 1$ (43b) where z⁰ is the credibility (MFC) obtained when h = 0 and p⁰ is a parameter defined in [42], as follows: $p^{0} = \sum_{n = 1}^{N} \frac{1 - u_{{\tilde{y}}^{*}} (y_{n})}{| y_{n}^{R} - y_{n}^{L} |}$ (44) where $u_{{\tilde{y}}^{*}} (\cdot)$ is the fuzzy triangular membership function.

According to [42], we can directly obtain the optimized h value with the following formulation: $h^{*} = {\begin{matrix} \frac{1}{2} (1 - \frac{z^{0}}{p^{0}}) \\ 0, \end{matrix} \begin{matrix} z^{0} \leq p^{0} \\ otherwise \end{matrix}$ (45) where h^* is the optimized value of h. It is possible to define the lower and upper fuzzy coefficients as follows: $a_{w γ} = \frac{1}{1 - h^{*}} a_{w γ}^{0}$ (46) with $a_{w γ}^{0}$ the fuzzy value obtained with h = 0 and γ = [L, R].

4 Case study

The BRT baseline model and the two-stage fuzzy regressions formulations were expressed in Python, using the Pyomo library [44]. They were solved with the MIQP solver Gurobi 8.1.1 in a Windows workstation, with an Intel Core i7-7820HK 2.9 GHz processor and 16 GB of RAM.

4.1 Studied scenario

For comparative purposes, the studied scenario is the same presented in [5], a high-frequency segregated transit system, more specifically the transit system Trunk line 10 of the city of Blumenau, Santa Catarina, Brazil. The Trunk line 10 system consists of 2 bus terminals, 32 bus stations, and 17 buses in a closed circuit of 17.7 km. Other data of the Trunk line 10 systems and simulation parameters are: n_N = 5, C₀ = 3 and C_max = 120. The scenario is shown in Fig. 8.

Fig. 8

Trunk line 10 scenario.

During the data collection, filming was carried out at bus stops in the city of Blumenau/SC for later determination of boarding/alighting times, taking into account the number of passengers to be processed, as well as the number of passengers already onboard.

4.2 Two-stage hybrid fuzzy regression results

With the collected data, the two-stage hybrid fuzzy regression presented in the previous section was used on each of the categorical variables (related to the number of passengers standing on the buses), considering the collected data.

First, the regressions were performed without optimizing the h value, i.e. with the parameter set to 0. The quality indicators for this case are presented in Table 3. Subsequently, h values were optimized as suggested by [42], which improved the quality indicators, as presented in Table 4.

Table 3
Fuzzy regression quality indicators without h optimization

h IC AFS MFC

$C_{lo}^{a}$ 0 0.944 0.0 5.607

$C_{me}^{a}$ 0 0.865 0.0 0.355

$C_{hi}^{a}$ 0 0.895 0.0 0.992

$C_{lo}^{b}$ 0 0.816 0.0 4.411

$C_{me}^{b}$ 0 0.758 0.0 1.894

$C_{hi}^{b}$ 0 0.836 0.0 1.965

	IC	MFC
$C_{lo}^{a}$	0.944	5.607
$C_{me}^{a}$	0.865	0.355
$C_{hi}^{a}$	0.895	0.992
$C_{lo}^{b}$	0.816	4.411
$C_{me}^{b}$	0.758	1.894
$C_{hi}^{b}$	0.836	1.965

Table 4

Fuzzy regression quality indicators with h optimized

	h	IC	AFS	MFC
$C_{lo}^{a}$	0	0.944	0.0	5.607
$C_{me}^{a}$	0.341	0.937	18.997	0.487
$C_{hi}^{a}$	0.145	0.921	9.047	1.022
$C_{lo}^{b}$	0.362	0.916	25.984	6.512
$C_{me}^{b}$	0.031	0.770	1.890	1.896
$C_{hi}^{b}$	0	0.836	0.0	1.965

To illustrate the behavior or the IC variation depending on the h value, Figure 9(a) shows IC changes for the lower limit, while Figure 9(b) shows the same change, now related to the upper index. Notice that both plots also have a a^L or a^R axis, which grows according to h.

Fig. 9

IC variation relative to h value for the case of bus boarding with low occupancy and its influence to the lateral spreads.

Considering the optimized h values, the regression parameters are presented in Table 5 as [a^wL; a^c; a^wR].

To illustrate the results of the fuzzy regression, Fig. 10(a) shows the regression obtained for $C_{a}^{lo}$ (alighting time from a bus with a low number of passengers), while Fig. 10(b) shows the results obtained for $C_{b}^{lo}$ (boarding time into a bus with a low number of passengers).

Table 5

Fuzzy regression coefficients

Item	Bus occupancy	Fuzzy regression coefficients
$C_{a}^{lo}$	low	[0.156; 0.368; 2]
$C_{a}^{me}$	medium	[0.07; 0.367; 2.848]
$C_{a}^{hi}$	high	[0.134; 0.388; 2.275]
$C_{b}^{lo}$	low	[0.145; 0.433; 1.32]
$C_{b}^{me}$	medium	[0.129; 0.489; 1.532]
$C_{b}^{hi}$	high	[0.140; 0.565; 2]

Fig. 10

Fuzzy regression for alighting/boarding time for passengers onboard of a bus with low occupancy.

4.3 Scenario application

In [5] and [6] the authors used a deterministic scenario in order to test holding control strategy for BRT systems. Although testing deterministic models is important to demonstrate the features of control algorithms, it is necessary to apply disturbances to the scenario in order to bring the model closer to reality. Also, it is vital that the quantities involved in the perturbations, as well as the deterministic data used in the model, approach the system operation.

4.3.1 Model robustness

The setup of Fig. 3 was used to inject system disturbances in the system. Thus, the system is composed of a deterministic control and a stochastic traffic model. The stochastic traffic model here presented uses the limit values found by fuzzy regression to populate a triangular distribution from which stochastic data will be sampled.

First, we analyzed the deterministic behavior of the control system, considering an ideal deterministic scenario. Second, we run simulations for the stochastic scenarios considering different random generator seeds. The bus trajectories for the simulations are shown below, namely Fig. 11 presents the ideal scenario, while Fig. 12 presents the results for the stochastic scenario. All simulation run for 15 bus stops ahead of the beginning of the process.

Fig. 11

Bus trajectories for the deterministic scenario.

Fig. 12

Bus trajectories for the stochastic scenario with different seeds.

Considering each case, the results of the passenger waiting time appear in Table 6.

Table 6

Passengers waiting time after 15 simulation steps.

Scenario	Passenger’s waiting time
Deterministic	205,234.96
Stochastic (seed = 300)	218,202.47
Stochastic (seed = 800)	213,034.14
Stochastic (seed = 1400)	226,356.78

Although the waiting time of passengers is obviously longer in the stochastic scenario, it is possible to notice that the control managed to ensure the correct system operation (bus headway control) even in this kind of situation, keeping the headway between the buses and avoiding bunching situations. This is possible due to the predictive nature of the control proposed in [5], which is extended in this paper in order to enhance the boarding/alighting process, whereby real system disturbances are compensated by the feedback nature of the control strategy.

4.3.2 Model comparison

In order to test the quality of the proposed enhancement, a comparison was made with the baseline model from [5], and also with two traditional controls, namely Forward Headway Control (FH) and Two-Way Headway Control (TWH), as shown in [7]. The same setup of Fig. 3 was adopted in this analysis whose results are presented in Table 7.

Table 7
Model Comparison.

Scenario Passenger’s waiting time Delay Increase

Proposed 213,034.14 -

[5] 224,094.66 +5.2%

TWH 238,950.05 +12,16%

FH 278,335.72 +30.65%

Scenario	Passenger’s waiting time	Delay Increase
Proposed	213,034.14	-
[5]	224,094.66	+5.2%
TWH	238,950.05	+12,16%
FH	278,335.72	+30.65%

As presented by [7], optimal and predictive control methods are expected to improve passenger waiting times. Owing to the nature of the predictive models embedded in such control systems, which rely on historical data, the model parameters need to be fined adjusted to better represent the dynamics of the BRT system of interest.

Notice that the baseline model [5] uses fixed values for C_a and C_b based only on historical records, without taking into account the number of boarded passengers. Thus, it is not surprising that, by adjusting the number of boarded passengers, the proposed methodology leads to a shorter waiting time compared to the original model. This is due to the fact that poorly adjusted parameters can lead, for example, to an overestimation of the waiting time of passengers, generating control actions (holding signals) above what is necessary. The opposite is also true.

Despite the original model from [5] bringing about significant gains when compared to traditional control methods (FH and TWH), it can be seen that the adoption of the methodology proposed here can further benefit the model, and consequently the users of the public transportation system.

The advance of computational intelligence technologies, together with infrastructures for system interconnection, can facilitate the tuning of the control parameters of the BRT lines, which must be carried out individually for each line. Despite the potentially high deployment and maintenance cost of the technologies and resources needed to implement an optimal control, such as APC (Automatic Passenger Counter), such cost can be diluted over time in the case of a well-adjusted control, reducing bottlenecks of the system.

5 Conclusion

This paper presented a methodology to take into account the number of passengers already boarded in the modeling of a BRT system. In order to populate and test this formulation, an approach using two-stage fuzzy regression to estimate the time of boarding/alighting of passengers was employed. This approach used data from the real Trunk line 10 scenario, which was collected in the city of Blumenau, Brazil, measuring the number of passengers to be processed, the time of boarding/alighting by passengers, and the number of passengers already boarded.

The fuzzy regression results were also used in the stochastic simulation model to represent the random behavior present in the transit system better. In this way, the methodology presented here can be used as a framework to test other forms of control in a most accurate scenario.

Simulation results suggest that, for the studied scenario, the predictive nature of the control is capable of maintaining the bus headways and avoiding bunching even in the presence of disturbances.

For future works, it is suggested to estimate other stochastic parameters of the model, bringing the estimated model closer to the system operation. One could, for example, consider investigating possible disturbances in travel times, and passengers boarding and alighting rates.

Footnotes

Acknowledgments

This research was funded in part by CAPES under Program PrInt #698503P.

References

Eberlein

X.J.

, Wilson

N.H.M.

and Bernstein

, The holding problem with real-time information available, Transportation Science 35(1) (2001), 1–18.

Hickman

, An analytic stochastic model for the transit vehicle holding problem, Transportation Science 35(3) (2001), 215–237.

Zolfaghari

, Azizi

and Jaber

M.Y.

, A model for holding strategy in public transit systems with real- time information, International Journal of Transport Management 2 (2004), 99–110.

Koehler

L.A.

, Kraus

and Camponogara

, Iterative quadratic optimization for the bus holding control problem, IEEE Transactions on Intelligent Transportation Systems 12(4) (2011), 1568–1575.

Koehler

L.A.

, Seman

L.O.

, Kraus

and Camponogara

, Real-time integrated holding and priority control strategy for transit systems, IEEE Transactions on Intelligent Transportation Systems 20 (2019), 3459–3469.

Seman

L.O.

, Koehler

L.A.

, Camponogara

, Zimmermann

, Kraus

, Headway control in bus transit corridors served by multiple lines, IEEE Transactions on Intelligent Transportation Systems (2019), 1–13.

Zimmermann

, Kraus

, Koehler

L.A.

and Camponogara

, Holding control of bus bunching without explicit service headways, IFAC-PapersOnLine 49(3) (2016), 209–214.

Abkowitz

M.D.

and Engelstein

, Factors affecting running time on transit routes, Transportation Research Part A: General 17 (1983), 107–113.

Levinson

H.S.

, Analyzing transit travel time performance, Transportation Research Record: Journal of the Transportation Research Board 915 (1983), 1–6.

10.

Tirachini

, Estimation of travel time and the benefits of upgrading the fare payment technology in urban bus services, Transportation Research Part C: Emerging Technologies 30 (2013), 239–256.

11.

Guenthner

R.P.

and Sinha

K.C.

, Modeling bus delay due to passenger boardings and alightings, Transportation Research Record 7(13) (1983) 915.

12.

Lin

T.-M.

, Wilson

N.H.M.

, Dwell time relationships for light rail systems, Transportation Research Record (1361) (1992), 287–295.

13.

M.T.

, Zhao

, Chow

L.-F.

, Zhang

and Li

S.-C.

, Simulation model for estimating bus dwell time by simultaneously considering numbers of disembarking and boarding passengers, Transportation Research Record: Journal of the Transportation Research Board 1971 (2006), 59–65.

14.

Fernández

, Zegers

, Weber

and Tyler

, Influence of platform height, door width, and fare collection on bus dwell time, Transportation Research Record: Journal of the Transportation Research Board 2143 (2010), 59–66.

15.

Dueker

, Kimpel

, Strathman

and Callas

, Determinants of bus dwell time, Journal of Public Transportation 7 (2004), 21–40.

16.

Dorbritz

, Luthi

, Nash

, Weidmann

, Effects of on-board ticket sales on public transport reliability, Institue for Transport Planning and Systems (IVT) Swiss Federal Institute of Technology, ETH Zurich, Switzerland, 2008.

17.

Tirachini

, Bus dwell time: The effect of different fare collection systems, bus floor level and age of passengers, Transportmetrica A: Transport Science 9(1) (2013), 28–49.

18.

Currie

, Delbosc

, Harrison

and Sarvi

, Impact of crowding on streetcar dwell time, Transportation Research search Record: Journal of the Transportation Research Board 2353 (2013), 100–106.

19.

Seman

L.O.

, Koehler

L.A.

, Camponogara

and Kraus

, Junior, Integrated headway and bus priority control in transit corridors with bidirectional lane segments, Transportation Research Part C: Emerging Technologies 111 (2020), 114–134.

20.

Castro

P.M.

, Tightening piecewise McCormick relaxations for bilinear problems, Computers and Chemical Engineering 72 (2015), 300–311.

21.

Tanaka

, Uejima

and Asai

, Linear regression analysis with fuzzy model, IEEE Transactions on Systems, Man, and Cybernetics 12(6) (1982), 903–907.

22.

Tanaka

, Hayashi

and Watada

, Possibilistic linear regression analysis for fuzzy data, European Journal of Operational Research 40(3) (1989), 389–396.

23.

Tanaka

and Lee

, Interval regression analysis by quadratic programming approach, IEEE Transactions on Fuzzy Systems 6(4) (1998), 473–481.

24.

Martinkova

, Skrabanek

, fuzzyreg: An r package for fuzzy linear regression, in ENBIK2018 Conference Proceedings (P. Cech and D. Svozil, eds.), (Prague), (2018), pp. 7, University of Chemistry and Technology.

25.

Zadeh

L.A.

, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Transactions on Systems, Man and Cybernetics SMC-3 (1) (1973), 28–44.

26.

Romano

, Fuzzy Regression and PLS Path Modeling: A combined two-stage approach formulti-block analysis. PhD thesis, Università degli Studi di Napoli, 2006.

27.

Shapiro

, Fuzzy regression models, Article of Penn State University, (2005), 1–17.

28.

Chan

, Wuz

, Fuzzy regression: Methods and evaluations, Journal of Macau University of Science and Technology, 2007.

29.

Diamond

, Fuzzy least squares, Information Sciences 46(3) (1988), 141–157.

30.

Donoso

, Quadratic programming models for fuzzy regression, International Conference on Mathematical and Statistical Modeling in Honor of Enrique Castillo, (2006), 1–12.

31.

J.R.

, Tzeng

G.H.

and Li

H.L.

, A general piecewise necessity regression analysis based on linear programming, Fuzzy Sets and Systems 105(3) (1999), 429–436.

32.

Bisserier

, Galichet

, Boukezzoula

, Fuzzy piecewise linear regression, IEEE International Conference on Fuzzy Systems, (2008), pp. 2089–2094.

33.

Ishibuchi

and Tanaka

, Fuzzy regression analysis using neural networks, Fuzzy Sets and Systems 50 (1992), 257–265.

34.

Pandit

, Fuzzy nonlinear regression using artificial neural networks, Int J Adv Appl Math and Mech, 2014.

35.

Savic

D.A.

and Pedrycz

, Evaluation of fuzzy linear regression models, Fuzzy Sets and Systems 39(1) (1991), 51–63.

36.

Vanderbei

R.J.

, Linear Programming 37 of International Series in Operations Research & Management Science. Boston, MA: Springer US, 2001.

37.

Boyd

, Vandenberghe

, Convex Optimization. New York, NY, USA: Cambridge University Press, 2004.

38.

I. Gurobi Optimization, Gurobi optimizer reference manual, 2016.

39.

Wächter

and T.

, Biegler, On the implementation of aprimal-dual interior point filter line search algorithm for largescalenonlinear programming, Mathematical Programming 106(1) (2006), 25–57.

40.

Wang

H.F.

and Tsaur

R.C.

, Insight of a fuzzy regressionmodel, Fuzzy Sets and Systems 112(3) (2000), 355–369.

41.

Chan

K.Y.

and Engelke

, Varying spread fuzzy regressionfor affective quality estimation, IEEE Transactions on FuzzySystems 25 (2017), 594–613.

42.

Liu

, Chen

, Zhou

and Zhong

, Fuzzy linear regressionmodels for qfd using optimized h values, EngineeringApplications of Artificial Intelligence 39 (2015), 45–54.

43.

Liu

and Chen

, A systematic approach to optimizingvalue for fuzzy linear regression with symmetric triangularfuzzy numbers, Mathematical Problems in Engineering 2013 (2013), 1–9, 12.

44.

Hart

W.E.

, Laird

C.D.

, Watson

J.-P.

, Woodruff

D.L.

, Hackebeil

G.A.

, Nicholson

B.L.

and Siirola

J.D.

, Springer Science &Business Media, Pyomo-Optimization Modeling in Python, second ed., 67, 2017.

n _I	number of buses of the BRT system
n _K	number of bus stops of the BRT system
n _N	number of stops in the prediction horizon of the buses
$I$	set of buses of the BRT system
$K$	set of stops of the BRT system
i	bus index, $i \in I$
k	stop index, $k \in K$
m	index of the bus lap in the closed BRT circuit
$K_{i}$	set of stops belonging to the prediction horizon of the bus i
$K_{i}^{0}$	set of stops of bus i for which the initial values d_i,ω (departure times) and l_i,ω (onboard passengers) apply

A framework to estimate dwell time of BRT systems using fuzzy regression

Abstract

Keywords

1 Introduction

2 Methodology

2.1 Existing literature model

2.4 Test setup

3.1 Possibilistic regression

3.4.1 Index of confidence (IC)

4.1 Studied scenario

Table 3 Fuzzy regression quality indicators without h optimization h IC AFS MFC C lo a 0 0.944 0.0 5.607 C me a 0 0.865 0.0 0.355 C hi a 0 0.895 0.0 0.992 C lo b 0 0.816 0.0 4.411 C me b 0 0.758 0.0 1.894 C hi b 0 0.836 0.0 1.965

4.3.1 Model robustness

Table 7 Model Comparison. Scenario Passenger’s waiting time Delay Increase Proposed 213,034.14 - [5] 224,094.66 +5.2% TWH 238,950.05 +12,16% FH 278,335.72 +30.65%

Footnotes

Acknowledgments

References

Table 3
Fuzzy regression quality indicators without h optimization

h IC AFS MFC

$C_{lo}^{a}$ 0 0.944 0.0 5.607

$C_{me}^{a}$ 0 0.865 0.0 0.355

$C_{hi}^{a}$ 0 0.895 0.0 0.992

$C_{lo}^{b}$ 0 0.816 0.0 4.411

$C_{me}^{b}$ 0 0.758 0.0 1.894

$C_{hi}^{b}$ 0 0.836 0.0 1.965

Table 7
Model Comparison.

Scenario Passenger’s waiting time Delay Increase

Proposed 213,034.14 -

[5] 224,094.66 +5.2%

TWH 238,950.05 +12,16%

FH 278,335.72 +30.65%