A technology selection and design model of a semi-rapid transit line

Abstract

We present a new optimization model for technology selection and design of a semi-rapid transit line. With respect to previous studies, we improve the synthetic representation of the temporal and spatial variability of demand, and of several operational and design aspects. We apply the model to two scenarios offering comparable performance by commercially available technologies in terms of service, rather than assuming that service quality is strongly associated with technology. The model is validated by comparing some computed performance indices with best practices. We show that planning for a faster technology can be more important than the choice between bus and rail per se, except at very low demand density, and that differences of total cost, sum of passengers’ time value and operator’s cost, between the technologies are smaller than commonly held across a wide range of higher demands. At high demand density multiple-unit rail offers the most cost-effective way to achieve high capacities under many conditions. A scenario variation analysis shows the relevance of differences between value of time components, the bias of averaging vehicle load ratios when assessing the crowding disutility, the usefulness of a demand index abstracting from some specific parameter choices, and the high impact of the project discount rate.

Appendices

Appendix 1: Demand profile

Figure 23 shows an example of a demand profile scaled to the peak maximum demand and subdivided in three periods. This demand profile is derived from 1 month of turnstile data of the Boston subway network. These data are extended to 1 year of service applying average month-specific corrective factors retrieved at MBTA (2017). In this type of chart, the average peak demand is equal to $1/\tau _{1}$, and the other parameters ($\gamma , \tau$, and $\chi$) can be derived by a graphical inspection as explained in the following. The vertical axis reports the demand levels as fractions with respect to the maximum demand in a year. The bottom horizontal axis shows the fraction of service hours in a year, and the top horizontal axis indicates the fraction of the period service hours with respect to the total service hours, i.e. the parameters $\chi$. The thick line indicates the demand fractions, and different shades of gray (different colors in the pdf version) represent the different periods. The remaining horizontal axes illustrate the average and maximum demand fractions for each period. The positions of these axes determine the parameters $\gamma$ and $\tau$.

Fig. 23

An example of the disaggregation of a demand profile in three periods

Appendix 2: Approximating the crowding penalty under non-uniform vehicle occupancy rate

As discussed in the literature review, there is a wide consensus that the value of the in-vehicle travel time should be multiplied by a crowding penalty function. Moccia and Laporte (2016), as well as previous studies, use a piecewise linear function $\delta$ with respect to the average vehicle occupancy rate $\bar{\theta }$. By such a function, there is no penalty up to an average occupancy rate $\theta _{min}$ that indicates the filling of available seats. For larger values of $\bar{\theta }$ the penalty increases linearly with a slope value $\rho$, see Fig. 24 for an example.

Fig. 24

Example of the crowding penalty function of Moccia and Laporte (2016)

Formally, this penalty function is

$$\begin{aligned} \delta (\bar{\theta }) = {\left\{ \begin{array}{ll} 1 + \rho \left( \bar{\theta }- \theta _{min}\right) &{} \bar{\theta }\ge \theta _{min}\\ 1 &{} \text {otherwise} \end{array}\right. }. \end{aligned}$$

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A caveat of this approach is that crowding is underestimated when the vehicle occupancy rate significantly varies along the cycle time. This underestimation derives from Jensen’s inequality. In the following we indicate by t the time step belonging to the cycle time, and we consider for notational brevity a cycle time of unitary length, i.e. $t \in [0,1]$. Assuming that the piecewise linear function $\delta$ accurately describes users’ preferences with respect to the instantaneous vehicle occupancy rate $\theta$, then, because of the convexity of $\delta$, we have the following Jensen’s inequality:

$$\begin{aligned} \delta (\bar{\theta }) = \delta \left( \int _{0}^{1}\theta (t) \mathrm {d} t\right)\le & {} \int _{0}^{1}\delta (\theta (t))\mathrm {d} t = \Delta , \end{aligned}$$

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where $\Delta$ is the penalty function that we want to estimate. Observe that $\delta (\bar{\theta }) = \Delta$ whenever $\theta (t)$ is a constant. In this case we have that $\delta (\bar{\theta })$ exactly gauges the users’ time at an occupancy rate larger than $\theta _{min}$, if any.

For the purpose of this paper, strategic appraisal, we want a synthetic representation of non-uniform occupancy rate $\theta (t)$. We assume that the occupancy rate can be approximated by a linear function. Because the users’ time in the unitary cycle time is an invariant, i.e. $\int _{0}^{1}\theta (t) \mathrm {d} t = \bar{\theta }$, such a linear function can be described by only one parameter, $\phi$, which indicates the ratio of the maximum to the average occupancy rate. Observe that for any practical use $\phi$ is constrained in the interval [1, 2], because values larger than two would imply that a fraction of the cycle time occurs with zero occupancy rate. Simple geometric analyses (see Figs. 25, 26 for the two relevant cases) allow to define the function $\Delta$ for the above defined occupancy rate:

$$\begin{aligned} \Delta (\bar{\theta }) = {\left\{ \begin{array}{ll} 1 + \rho \left( \bar{\theta }- \theta _{min}\right) &\quad \theta _{min} \le \bar{\theta }(2 - \phi )\\ 1 + \rho \dfrac{\left( \phi \bar{\theta }- \theta _{min}\right) ^{2}}{4\bar{\theta }(\phi - 1)} &\quad \bar{\theta }(2 - \phi )< \theta _{min} < \phi \bar{\theta }\\ 1 &\quad \theta _{min} \ge \phi \bar{\theta }\end{array}\right. }. \end{aligned}$$

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Fig. 25

Example of an occupancy rate function such that $\theta _{min} \ge \bar{\theta }(2-\phi )$. The area corresponding to the users’ time under crowding is depicted as hatched and can be computed by the formula reported in the upper left corner of the figure

Fig. 26

Example of an occupancy rate function such that $\theta _{min} \le \bar{\theta }(2-\phi )$. The area corresponding to the users’ time under crowding is depicted as hatched and can be computed by the formula reported in the upper left corner of the figure. This formula is the same as in the case of a constant occupancy rate

Appendix 3: Lower convex approximation scheme

We construct a separable lower convex envelope of the objective function $C_{tot}$ in the feasible frequency range as follows. First, we use the minimum feasible value of the crowding penalty (13). Second, we assume a discounted waiting time by the factor $\mu$ whenever the minimum frequency is smaller than $f_{l}$. Third, we compute the intersection delay in the cycle time at the minimum frequency. Fourth, we remove non-convex terms by fixing some variables to proper bounds.

Thus, we can define a separable lower convex envelope of the objective function for a given vector of consists $\mathbf {\bar{n}}$ as

$$\begin{aligned} \tilde{C}_{tot}(\mathbf {f}, d, \mathbf {\bar{n}}) = a_{0} + a_{1}d + \dfrac{a_{2}}{d} + \sum _{p}\left( a_{3,p} + \dfrac{a_{5,p}}{d_{max}}\right) f_{p} + \sum _{p}\dfrac{a_{4,p}}{f_{p}}. \end{aligned}$$

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The coefficients of the previous equation are defined as follows

$$\begin{aligned} a_{0}&= {} c_{0l} + c_{1v}\beta \zeta \dfrac{t_{b}q}{3600} + c_{1t}\beta \dfrac{t_{b}q}{3600}\sum _{p}\dfrac{\gamma _{p} \chi _{p}}{\bar{n}_{p}} \nonumber \\&\quad +\, V_{v}\dfrac{\lambda }{2}q\sum _{p}\chi _{p}\gamma _{p}\bar{\Delta }_{p}(R + t_{\omega ,p}) + c_{1v}\beta \zeta f_{1,min}\bar{n}_{1}t_{\omega ,1} \nonumber \\&\quad+ \,c_{1t}\beta \sum _{p}\chi _{p}f_{min,p}t_{\omega ,p}\end{aligned}$$

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$$\begin{aligned} a_{1}= & {} \dfrac{q V_{a}}{2 s}\sum _{p } \gamma _{p}\chi _{p}\end{aligned}$$

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$$\begin{aligned} a_{2}= & {} 2L(c_{0s} + c_{0sv}(\bar{n}_{1} - 1) ) + T_{l}V_{v}\dfrac{\lambda }{2}q\sum _{p}\chi _{p}\gamma _{p}\bar{\Delta }_{p}\end{aligned}$$

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$$\begin{aligned} a_{3,p}= & {} c_{1t}\chi _{p}R^{'}_{p} + 2 c_{2v}L\chi _{p}\bar{n}_{p} +_{\text {if} \, p = 1} c_{1v}\beta \zeta \bar{n}_{1}R^{'}_{1}\end{aligned}$$

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$$\begin{aligned} a_{4,p}= & {} V_{w}q \gamma _{p}\chi _{p}\epsilon \mu _{p} + V_{v}l(q\gamma _{p})^{2}\chi _{p}\bar{\Delta }_{p}\dfrac{t_{b}}{3600 \bar{n}_{p}}\end{aligned}$$

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$$\begin{aligned} a_{5,p}= & {} 2 c_{1t}\beta L T_{l}\chi _{p} +_{\text {if} \, p = 1} 2 c_{1v}\beta \zeta \bar{n}_{1}L T_{l}, \end{aligned}$$

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where, for notational compactness, we have used the following notation:

$$\begin{aligned} R^{'}_{p}= \beta R + \dfrac{t_{tf} + t_{tv}\bar{n}_{p}}{3600}\end{aligned}$$

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$$\begin{aligned} \mu _{p}= {\left\{ \begin{array}{ll} \mu &{} \text {if} \,\,\,\, f_{min,p} < f_{l}\\ 1 &{}\text {if} \,\,\,\, f_{min,p} \ge f_{l} \end{array}\right. } \end{aligned}$$

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$$\begin{aligned} \bar{\Delta }_{p}= \Delta (f_{max,p}, \bar{n}_{p}) \end{aligned}$$

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$$\begin{aligned} t_{\omega ,p}= \omega _1 \frac{2t_u L}{60}\left( \frac{f_{min,p}}{\dot{f}}\right) ^{\omega _2}. \end{aligned}$$

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By calculus, as done in Moccia and Laporte (2016), the unconstrained optimal stop spacing of the approximation $\tilde{d}_{unc}$ is

$$\begin{aligned} \tilde{d}_{unc} = \sqrt{\dfrac{ 2s(2L(c_{0s} + c_{0sv}(\bar{n}_{1} - 1) ) + T_{l}V_{v}\dfrac{\lambda }{2}q\sum _{p}\chi _{p}\gamma _{p}\bar{\Delta }_{p})}{q V_{a}\sum _{p } \gamma _{p}\chi _{p}}}. \end{aligned}$$

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The unconstrained optimal frequencies $\tilde{f}_{unc, p}$ are

$$\begin{aligned} \tilde{f}_{unc, p}= \sqrt{\dfrac{a_{4,p}}{a_{3,p} + \dfrac{a_{5,p}}{d_{max}}}}. \end{aligned}$$

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