An Enhanced Semi-Flexible Transit Service with Introducing Meeting Points

Abstract

Semi-flexible transit (SFT) has been widely investigated in recent years and is expected to improve the balance between accessibility and operational efficiency. However, a couple of critical issues remain, such as how to develop an optimization mechanism to achieve a win-win situation for both flexible-route and fixed-route service users simultaneously, how to identify a clear service boundary between flexible-route and fixed-route service areas, and how to improve the effectiveness and competitiveness of SFT. To address these issues, this study proposes an enhanced SFT system in which meeting points are introduced to establish one flexible stop that can serve multiple users. This system can directly contribute to reducing detours, which are the main problems hindering the promotion of flexible transit services. Analytical models are derived for this optimal design, and the artificial bee colony (ABC) algorithm is adopted to solve the proposed nonlinear problem. A series of numerical cases is designed to evaluate the performance of the proposed system, which is compared with the traditional fixed-route transit and conventional SFT without meeting points. Results demonstrate that the proposed SFT achieves a lower system cost and a shorter travel time in most cases where the observed benefits vary across different factors, including demand density, average travel distance, the wealth of population, and unit length of service segment. Moreover, the introduction of meeting points helps SFT significantly reduce the routing length.

Appendices

Appendix A: Notations

Table 15 Notation list

Appendix B: Conventional SFT System without Meeting Points

To be consistent with the proposed SFT system, a conventional SFT without meeting points is carefully derived to verify the performance of the proposed system as shown in Fig. 19.

Fig. 19

Conventional SFT system without meeting points

The generalized cost of conventional SFT can also be expressed as the sum of user cost $T$ and agency cost $AC$, which is in line with the proposed SFT system:

$$\underset{{l}_{a},{H}_{a}}{min} Z=AC+T$$

(33)

$${H}_{a}\ge {H}_{a}^{min}$$

(34)

$$w<{l}_{a}\le \mathrm{S}$$

(35)

$$\mathrm{C}\le {C}_{max}$$

(36)

$$O\le {O}_{max}$$

(37)

$${d}_{a}\le {d}_{m}$$

(38)

2.1 B.1 Average User Cost

The main difference between the conventional and proposed SFTs lies in the absence of meeting points in A2. In the conventional SFT, only one passenger can be picked at a flexible stop. By excluding the meeting points, the average user cost $T$ and the expected travel time between two adjacent stops $C$ in the conventional SFT can be expressed as

$$T=\frac{L}{S}C+\frac{{H}_{a}}{2}+\left(2-\frac{2{l}_{a}}{S}\right)\frac{{d}_{a}}{{v}_{w}}$$

(39)

$$C=\left(\frac{\rho {w}^{2}{H}_{a}}{3v}+\rho {H}_{a}w{\tau }_{1}\right){l}_{a}+\frac{w}{6v}+\frac{S}{v}+{\tau }_{1}+\rho {H}_{a}w{\tau }_{2}S$$

(40)

The user cost is then formulated as

$$\begin{aligned}T=&\;\frac{L}{S}\left[\left(\frac{\rho {w}^{2}{H}_{a}}{3v}+\rho {H}_{a}w{\tau }_{1}\right){l}_{a}\right.\\& \left.+\;\frac{w}{6v}+\frac{S}{v}+{\tau }_{1}+\rho {H}_{a}w{\tau }_{2}S\right]\\& +\;\frac{{H}_{a}}{2}+\left(2-\frac{2{l}_{a}}{S}\right)\frac{{d}_{a}}{{v}_{w}}\end{aligned}$$

(41)

where ${H}_{a}$ and ${l}_{a}$ are the decision variables, and ${d}_{a}$ is the expected walking distance to a fixed stop in the conventional SFT system, which can be computed as

$${d}_{a}=\frac{w+S-{l}_{a}}{4}$$

(42)

2.2 B.2 Agency Cost

Similarly, the capital cost $FC$ and operating cost $VC$ are included in the total agency cost $AC$. $VC$ includes those costs related to vehicle kilometers and hours traveled.

$$AC=VC+FC$$

(43)

$$VC=\frac{1}{\mu \rho }\left[{\pi }_{s}^{K}\frac{{L}_{a}+S-{l}_{a}}{{H}_{a}Sw}+{\pi }_{s}^{U}\left(\frac{{L}_{a}}{{H}_{a}{V}_{v}Sw}+\frac{S-{l}_{a}}{{H}_{a}{V}_{f}Sw}\right)\right]$$

(44)

where ${L}_{a}$ is the routing length in A2 when $k=1$, which can be expressed as

$${L}_{a}={l}_{a}+\frac{w}{2}+\frac{\left(w{l}_{a}\rho {H}_{a}-1\right)w}{3}$$

(45)

The average commercial speed of vehicles in both flexible- and fixed-route transit is given by

$$\frac{1}{{V}_{v}}=\frac{1}{v}+\frac{w{l}_{a}\rho {H}_{a}\left({\tau }_{1}+{\tau }_{2}\right)}{{L}_{a}}$$

(46)

$$\frac{1}{{V}_{f}}=\frac{1}{v}+\frac{{\tau }_{1}+w\left(S-{l}_{a}\right){\rho H}_{a}{\tau }_{2}}{S-{l}_{a}}$$

(47)

The above equation represents the time spent by a vehicle traveling a unit distance, which includes both traveling and dwelling times in line with the arguments of Daganzo (2010).

The critical occupancy for conventional SFT systems, $O$ is formulated as:

$$O=\frac{{H}_{a}Sw}{2\left({L}_{a}+S-{l}_{a}\right)}\rho L$$

(48)

Appendix C: Traditional FRT (Fixed-Route Transit)

In FRT service, the distance between two adjacent stops is decision variable S, which is demonstrated in a system layout (Fig. 20). Note that, the FRT can be treated as a simplified case of proposed SFT, as vehicles are not allowed to deviate but have to follow a predetermined route, as:

$${l}_{s}=0$$

(49)

where ${l}_{s}$ is the length of the flexible segment.

Fig. 20

Traditional FRT system between each two adjacent fixed stops

Therefore, the generalized cost for the FRT system can also be expressed as the sum of user cost ${T}_{f}$ and agency cost ${AC}_{f}$, with headway to be optimized:

$$\underset{{H}_{f}}{\mathit{min}}{Z}_{f}={AC}_{f}+{T}_{f}$$

(50)

$${H}_{f}\ge {H}_{f}^{min}$$

(51)

$${O}_{f}\le {O}_{max}$$

(52)

3.1 C.1 Average User Cost

Passenger travel time has three parts, namely, waiting time $\frac{{H}_{f}}{2}$, in-vehicle time $\frac{L}{S}{C}_{f}$, and access and egress time $\frac{{2d}_{t}}{{v}_{w}}$.

$${T}_{f}=\frac{{H}_{f}}{2}+\frac{L}{S}{C}_{f}+\frac{{2d}_{t}}{{v}_{w}}$$

(53)

where ${C}_{f}$ is the travel time of vehicles between any pair of fixed stops, which can be computed as

$${C}_{f}=\frac{S}{v}+{\tau }_{1}+wS\rho {H}_{f}{\tau }_{2}$$

(54)

${d}_{t}$ represents the average walking distance to a fixed stop, which is modeled as the sum of expected longitudinal and lateral routing distances:

$${d}_{t}=\frac{S+w}{4}$$

(55)

3.2 C.2 Agency Cost

${VC}_{f}$ Follows the same structure in the proposed and conventional SFTs and is related to both vehicle-operated mileage and vehicle hours traveled.

$${AC}_{f}={VC}_{f}+FC$$

(56)

$${VC}_{f}=\frac{1}{\mu \rho }\left({\pi }_{s}^{K}\frac{S}{{H}_{f}wS}+{\pi }_{s}^{U}\frac{\mathrm{S}}{{H}_{f}wS{V}_{fr}}\right)$$

(57)

The average travel speed of vehicles for fixed-route transit is given by

$$\frac{1}{{V}_{fr}}=\frac{1}{v}+\frac{{\tau }_{1}+wS\rho {H}_{f}{\tau }_{2} }{S}$$

(58)

The critical occupancy of FRT systems, ${O}_{f}$, is formulated as:

$${O}_{f}=\frac{{H}_{f}w}{2}\rho L$$

(59)

Appendix D: The Demand Threshold

To derive the critical demand density, the following simplifications are made:

$${H}_{s}={H}_{a}={H}_{f}=\widetilde{H}={H}_{min}$$

(60)

$${l}_{s}={l}_{a}=\widetilde{l}$$

(61)

$$\widetilde{{d}_{w}}=0.508*\sqrt{U}=0.508*\sqrt{k/\widetilde{H}/\rho }$$

(62)

where $k$ is the expected number of served passengers at one meeting point. Equation (62) is to simplify two decision variables $r$ and $d$ into one variable $k$. The difference between $\widetilde{{d}_{w}}$ (Daganzo 1984) and ${d}_{w}$ calculated in Eq. (24) is about 7%.

The generalized cost of proposed SFT, ${Z}^{s}$, can be expressed as $\widetilde{{Z}^{s}}\left(k\right)$ after simplification

$$\begin{aligned}\widetilde{{Z}^{s}}\left(k\right)=&\;\frac{L}{S}\left(\frac{{w}^{2}}{3v}+w{\tau }_{1}\right)\frac{\widetilde{l}\rho \widetilde{H}}{k}P+\frac{1}{\mu \rho }\\& \left[{\pi }_{s}^{K}\frac{w\widetilde{l}\rho \widetilde{H}P}{3k\widetilde{H}S}+\frac{{\pi }_{s}^{U}}{Sw\widetilde{H}}\left(\frac{{w}^{2}\widetilde{l}\rho \widetilde{H}P}{3vk}+\frac{\widetilde{l}w{\tau }_{1}\rho \widetilde{H}}{k}P\right)\right]\\& +\frac{2\widetilde{l}}{S{v}_{w}}*0.508*\sqrt{\frac{k}{\rho \widetilde{H}}}+A\end{aligned}$$

(63)

where A is the cost unrelated to k.  

Firstly, we can derive the optimal solution for $k$. Since Eq. (63) is convex in $k$, and when the constraints are restricted the first-order condition with $k$ is obtained.

$${\widetilde{k}}^\frac{3}{2}=\frac{{v}_{w}\sqrt{\rho \widetilde{H}}}{0.508}\left[L\rho \widetilde{H}P\left(\frac{{w}^{2}}{3v}+w{\tau }_{1}\right)+\frac{1}{\mu }\left[{\pi }_{s}^{K}\frac{wP}{3}+{\pi }_{s}^{U}\left(\frac{wP}{3v}+{\tau }_{1}P\right)\right]\right]$$

(64)

$k$ is bounded from below and above by 1 and ${k}_{max}$, respectively. Then the optimal solution for $k$ can be written as a function of demand density $\rho$ as follows:

$$\widetilde{{k}^{*}}=mid\left\{1,\widetilde{k},{k}_{max}\right\}$$

(65)

The generalized cost of Conventional SFT, Z, can be expressed as $\widetilde{Z}$ after simplification. The difference in generalized cost between conventional SFT and proposed SFT is calculated as: $G\left(\rho \right)=\widetilde{{Z}^{s}}\left(\rho \right)-\widetilde{Z}(\rho )$.

$$\begin{aligned}G\left({\rho }_{1}\right)=&\;L{\rho }_{1}\widetilde{H}\left(\frac{{w}^{2}}{3v}+w{\tau }_{1}\right)\left(\frac{P}{\widetilde{{k}^{*}}}-1\right)\\& +\frac{2*0.508*}{{v}_{w}}*\sqrt{\frac{\widetilde{{k}^{*}}}{\widetilde{H}{\rho }_{1}}}+\frac{1}{\mu }\\& \left[{\pi }_{s}^{K}\frac{w}{3}+{\pi }_{s}^{U}\left(\frac{w}{3v}+{\tau }_{1}\right)\right]\left(\frac{P}{\widetilde{{k}^{*}}}-1\right)=0\end{aligned}$$

(66)

where ${\rho }_{1}$ is the critical demand density that represents the switching point between conventional SFT and proposed SFT $(P\approx 1)$.

The generalized cost of FRT, ${Z}_{f}$, can be expressed as $\widetilde{{{Z}_{f}}}$ after simplification. The difference in generalized cost between FRT and proposed SFT is calculated as $F\left(\rho \right)=\widetilde{{Z}^{s}}\left(\rho \right)-\widetilde{{Z}_{f}}(\rho )$.

$$\begin{aligned}F\left({\rho }_{2}\right)=&\;\frac{1}{\mu S}\left[{\pi }_{s}^{K}\left(\frac{1}{6\widetilde{I}\widetilde{H}{\rho }_{2}}+\frac{w}{3\widetilde{k*}}\right)\right.\\& \left.+\;{\pi }_{s}^{U}\left(\frac{\frac{1}{6\widetilde{I}\widetilde{H}{\rho }_{2}}+\frac{w}{3\widetilde{k*}}}{v}+{\tau }_{1}\frac{1}{\widetilde{k*}}\right)\right]\\& +\;\frac{L}{S}\left(\frac{{w}^{2}}{3v}+w{\tau }_{1}\right)\frac{\widetilde{H}{\rho }_{2}}{\widetilde{{k}^{*}}}+\frac{L}{S}\frac{w}{6v\widetilde{l}}\\& +\;\frac{2}{S}\frac{0.508}{{v}_{w}}\sqrt{\frac{\widetilde{{k}^{*}}}{\widetilde{H}{\rho }_{2}}}-\frac{1}{{2v}_{w}}\\& -\;\frac{\left(S+w\right)}{2S{v}_{w}}+\frac{\widetilde{l}}{2S{v}_{w}}=0\end{aligned}$$

(67)

where ${\rho }_{2}$ is the critical demand density that represents the switching point between FRT and proposed SFT $(P\approx 1)$.