Mountain railway alignment optimization using stepwise & hybrid particle swarm optimization incorporating genetic operators

Mountain railway alignment optimization considering geological impacts: A cost‐hazard bi‐objective model

“…The objective of the least‐cost alignment optimization problem is to find an alignment with the lowest comprehensive cost between start and end points (Jong & Schonfeld, 2003). The cost function used in this study is the same as in our previous publications (Li et al., 2016; Pu, Song, Schonfeld, Li, Zhang, Hu et al., 2019). It contains the following components: Construction cost, including costs spent on bridge construction ( C B ), tunnel construction ( C T ), earthwork construction ( C E ), length‐dependent construction ( C L ), and right‐of‐way expenses ( C R ). Operation cost, which is the sum of annual operating costs related to the deflection angles at HPI's ( C A ), to alignment length ( C M ) and to gradients ( C G ). …”

“…The detailed mathematical formulations for computing the cost components in Equation () can be found in our earlier publications (Li et al., 2016; Pu, Song, Schonfeld, Li, Zhang, Hu et al., 2019) and are not duplicated here.…”

“…Meanwhile, alignment cost is another deciding factor in mountain railway alignment design. That occurs because mountain railway alignments are largely comprised of expensive bridge and tunnel sections (Pu, Song, Schonfeld, Li, Zhang, Hu et al., 2019). It is quite difficult to properly design these expensive constructions without excessive alignment cost.…”

“…Then, these paths were used as centerlines to generate a set of railway corridors (Area‐corridor stage). In the next stage (Li et al., 2017; Pu, Song, Schonfeld, Li, Zhang, Hu et al., 2019; H. Zhang et al., 2020), we used a stepwise & hybrid particle swarm optimization‐genetic algorithm (PSO‐GA), which is an evolutionary‐based method for engineering problems (e.g., Adeli & Cheng, 1993, 1994; Adeli & Kumar, 1995a, 1995b; Kociecki & Adeli, 2015; Sarma & Adeli, 2001), to obtain final alignment solutions in specific railway corridors (Corridor‐alignment stage). Although these methods have been verified to be successful for optimizing mountain railway alignments, the aims of both of these optimization processes were to find least‐cost solutions in the search spaces, without accounting for geological hazard factors.…”

A three‐dimensional distance transform for optimizing constrained mountain railway alignments

“…The usual goal of alignment optimization is to find an alignment with the lowest comprehensive cost between two given endpoints. In the literature, representative methods for optimizing alignments include particle swarm optimization (Shafahi & Bagherian, 2013;Babapour, Naghdi, Ghajar, & Mortazavi, 2018;Pu et al, 2019), two-stage method that combines global optimization methods with a gradient type algorithm (Vázquez-Méndez, Casal, Santamarina, & Castro, 2018), derivative-free algorithms (Mondal, Lucet, & Hare, 2015), discrete algorithms (Hirpa, Hare, Lucet, Pushak, & Tesfamariam, 2016;, dynamic programming (Hogan, 1973;Li, Pu, Zhao, & Liu, 2013), mixed integer programming (Easa & Mehmood, 2008), linear programming (Revelle, Whitlatch, & Wright, 1996;Chapra & Canale, 2006), network optimization (Trietsch, 1987a(Trietsch, , 1987b), heuristic neighborhood search with mixed integer programming (Cheng & Lee, 2006;Lee, Tsou, & Liu, 2009), calculus of variations (Howard, Bramnick, & Shaw, 1968), numerical search (Robinson, 1973), enumeration (Easa, 1988), average-end-area method for improving earthwork calculation accuracy from 2D to 3D (Cheng & Jiang, 2013), genetic algorithms (Maji & Jha, 2009, and distance transforms (DTs) (de Smith, 2006;Li et al, 2016;Li et al, 2017;Pu et al, 2018).…”