networks is crucial for accommodating patrons’ first- and last-mile trips, ensuring seamless connectivity
and enhancing the overall effectiveness of mass transit systems.
This paper focuses on the optimal design of fixed-route feeder services under spatially
heterogeneous demand. Compared to emerging feeder modes, such as ride sourcing, flex-route transit,
and shared bikes, fixed-route feeders offer greater reliability thanks to their fixed service schedules and
routes, as well as the benefit of remaining impervious to inclement weather conditions. Furthermore,
fixed-route feeders are more affordable and energy-efficient, particularly in operating scenarios with
relatively high demand densities (e.g., Chang and Schonfeld, 1991).
Two types of models have been commonly employed to address the feeder service optimization
problem: discrete models and continuous models. Discrete models incorporate numerous details
representing specific physical and geographical constraints in the operating environment, such as the
exact locations of candidate stops (e.g., Martins and Pato, 1998; Shrivastava and O’Mahony, 2006; Ciaffi
et al., 2012; Lee et al., 2021). These models are often solved using heuristic methods (e.g., Fan and Ran,
2021), making it challenging to gauge the quality of the generated solutions. In contrast, continuous
models are more parsimonious, utilizing only a few parameters and design variables to represent the
feeder system. Specifically, continuous approximation (CA) models approximate heterogeneous discrete
details with continuous functions for demand density, line and stop spacings, and so on. This makes
them particularly well-suited for developing optimal transit network layouts under heterogeneous
demand (Ouyang et al., 2014; Chen et al., 2015; Chen et al., 2018; Luo et al., 2021). Continuous models
are often solved to global optimality or near optimality using efficient analytical or numerical methods.
Notable studies in this field are summarized in Table 1 and compared with our research.
Most works in this field assumed spatially uniform demands (Chang and Schonfeld, 1991; Kim
and Schonfeld, 2012, 2013, 2014; Sivakumaran et al., 2014; Kim and Schonfeld, 2015, Guo et al., 2018;
Su and Fan, 2019; Badia and Jenelius, 2020, 2021). Thus, they cannot be directly applied to real-world
feeder system designs under spatially heterogeneous demand. Spatial heterogeneity in demand cannot
be overlooked when the feeder’s service region is large, as exemplified by the satellite cities mentioned
earlier. Most of the cited works also assumed uniform line spacing and headway, while largely neglecting
stop spacing optimization. Moreover, they typically assumed fixed dwell time and transfer time between
trunk and feeder services, disregarding the relationship between dwell and transfer times and the number
of boarding and alighting patrons. Notably, Chang and Schonfeld (1991) and Kim and Schonfeld (2012,
2013, 2014, 2015) considered vehicle size optimization.
Only a few studies have developed feeder network design models for the more realistic, spatially
heterogeneous demand. Among them, Wirasinghe (1980) and Sivakumaran et al. (2012) assumed
demand is heterogeneous in only one direction (either along the rail line or perpendicular to it), while
Kuah and Perl (1988) and Yang et al. (2020) considered demand heterogeneity in both directions of the
service region. Most of these studies did not optimize stop spacings, likely due to mathematical
complexity. For instance, Quadrifoglio and Li (2009) stated that deriving the optimal stop spacing was
“often quite hard.” To the best of our knowledge, Kuah and Perl (1988) is the only study that optimized
stop spacings under heterogeneous demand. However, their model relied on oversimplified and
somewhat unrealistic assumptions (probably necessary to make the problem mathematically tractable),
such as very small rail station spacing and ignoring the operating cost associated with bus dwell times.
In addition, nearly all these works assumed fixed dwell and transfer times and unlimited feeder bus
capacity.
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