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Route choice modelling is a key topic in transport engineering, with a well-established research stream spanning over more than thirty years of relevant literature, summarized in various state- of-the-art reviews, including Ramming (2001), Prato and Bekhor (2007), Prato (2009), Papola and Marzano (2013). Researchers and analysts acknowledge unanimously some distinct, unique challenging features characterizing route choice contexts, primarily along three main viewpoints.

The first deals with the behavioural framework underlying route choices by decision-makers. In general, the concept of route itself as elemental choice alternative can be questioned. Various alternative paradigms have been formulated in the attempt to explain how decision-makers actually perceive a route, for instance based on sequences of waypoints, or on destination- oriented macro-directions. In terms of behavioural choice mechanism, many approaches have been explored with success, for instance the application of prospect theory to model risk- seeking and risk-averse behaviour (e.g. Katsikopoulos et al. 2000; Avinieri and Prashker, 2004; de Palma et al., 2008; Gao et al., 2010; de Luca and Di Pace, 2015). Notwithstanding, the classical definition of route as an ordered sequence of links connecting an origin-destination o- d pair and the Random Utility Models (RUMs) framework still represent the most effective assumptions to operationalize a route choice model for large-scale transport applications. The second deals with the considerable number of alternative routes usually available for each o-d pair. As a first consequence, assuming full knowledge/perception of the choice set by decision-makers is unrealistic. This problem is circumvented often by applying a route choice set generation method prior to the route choice model, that is selecting a subset of routes to choose from based on heuristic rules (Ben-Akiva et al., 1984; De La Barra et al., 1993; Azevedo

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et al., 1993; Bekhor et al., 2001; Van der Zijpp and Catalano, 2005; Prato and Bekhor, 2006; Bekhor et al., 2006). In addition, the implementation of RUM-based route choice models on real-size network requires developing algorithms and/or procedures capable to specify effectively and with limited computational burden the route choice model directly from the graph representing the underlying transport network.

The third deals with the presence of a complex correlation structure amongst perceived utilities of route choice alternatives, structurally determined by the topological overlapping of alternative routes in a transport network. Under the usual definition of route as an ordered sequence of links in the network, there is consensus in the literature towards considering the Daganzo and Sheffi (1977) assumption – that is, a correlation between pair of routes proportional to their topological overlapping, measured using a given link impedance – as a key reference. Along this line, Frejinger and Bierlaire (2007) proposed the so-called subnetwork approach, that is an application of the assumption by Daganzo and Sheffi (1977) only to a portion of the network given by primary, most likely perceived, roads.

Summarizing, notwithstanding many variations on the theme, a prevailing and widely adopted research track in route choice modelling is to specify RUM-based route choice models with routes defined as ordered sequences of links and underlying correlation consistent with the Daganzo and Sheffi (1977) assumption.

Within this track, many relevant contributions are available in the concerned literature. The simplest model was proposed by Dial (1971), who applied the Multinomial Logit model (MNL) to route choice with an elegant and computationally very effective algorithm to calculate route choice probabilities without explicit route enumeration. Unfortunately, the MNL model hypothesizes null correlation amongst perceived utilities of alternatives, because of its underlying assumptions. Thus, Daganzo and Sheffi (1977) operationalized the Multinomial Probit model (MNP) as a natural for embedding their assumption in a route choice model, thanks to the possibility offered by the MNP model to specify directly its correlation matrix. However, the MNP suffers from the absence of a closed-form probability statement, leading to computational issues related to the need to simulate choice probabilities, see e.g. Horowitz (1982), McFadden (1989), Bunch (1991), Geweeke (1991), Train (2009), Connors et al. (2014). The same also apply to Mixed Logit applications to route choice, e.g. Bekhor et al. (2002), Frejinger and Bierlaire (2007).

A natural alternative research direction aimed at developing closed-form route choice models leveraging the class of Generalised Extreme Value (GEV) models proposed by McFadden (1978). Many models have been proposed so far in this context, including the Link-Nested Logit (LNL) model by Vovsha and Bekhor (1998); the Paired-Combinatorial Logit (PCL) model by Prashker and Bekhor (1998); the Path Multilevel PML model by Papola and Marzano (2013). A noticeable variation on the theme is represented by the so-called recursive models (Fosgerau et al., 2013; Mai et al., 2015; Mai, 2016). However, there is no evidence in the literature on their capability to target Daganzo and Sheffi (1977) correlations. Alternatively, several researchers tried to introduce correction/penalty factors in the systematic utility of a MNL model to mimic the effect of correlations on route choice probabilities, for instance the C-Logit model by Cascetta et al (1996), and Russo and Vitetta (2003), and the Path-size model by Ben-Akiva and Ramming (1998), Ben-Akiva and Bierlaire (1999), Ramming (2001), Hoogendoorn-Lanser et al. (2005). These models exhibit limitations in capturing the proper effect of route correlations on choice probabilities, as addressed amongst others by Prashker and Bekhor (1998), Prashker and Bekhor (2004), Marzano (2005), Papola and Marzano (2013). In the light of the above, an interesting opportunity for route choice modelling is offered by the Combination of RUMs (CoRUM) model proposed by Papola (2016), particularly its Nested

117 Logit component-based form, termed Combination of Nested Logit (CoNL). For the purposes of this chapter, the key feature of the CoRUM, and thus of the CoNL, is the availability of a closed-form statement for both choice probabilities and correlations. This allows handling effectively the relationship between the CoNL specification (model structure, parameters) and its underlying correlations, thus enabling the possibility of specifying a CoNL capable to target Daganzo and Sheffi (1977) correlations.

In a nutshell, the cumulative distribution function (cdf) of a CoRUM is defined as a convex combination of cdf’s of other RUMs, termed mixing components of the CoRUM. All mixing components should embed by definition the same alternatives in the choice set, that will be the choice set also of the resulting CoRUM. Such specification, resembling a latent class model (Gopinath, 1995; Bhat, 1997; Swait, 1994; Greene, 2001; Greene and Hensher, 2003; Walker and Li, 2007; Vij et al., 2011), allows expressing probability statements, correlations, and elasticities of a CoRUM as a convex combination of the corresponding expressions of the mixing components. As a matter of fact, choosing Nested Logit (NL) models as mixing components, i.e. components with closed-form probability statements and correlations, yields a CoNL with corresponding closed-form expressions. In addition, as addressed by Papola (2016), the NL components can be specified so as to obtain a CoNL correlation matrix with maximal flexibility.

Primary target of this chapter is to explore the applicability of the CoNL model to the route choice context, proposing a new route choice model: (a) capable to target Daganzo and Sheffi (1977) correlations, (b) characterized by a closed-form expression of choice probabilities, and (c) operationalized by means of an algorithm providing automatically the specification and the route choice probabilities of a CoNL route choice model on a transport network. The structure of the chapter is the following: Section 5.2 recalls key features of the CoRUM and of the CoNL models, Section 5.3 describes the specification of the CoNL model for route choice modelling, Section 5.4 introduces a methodology and an algorithm for its operationalization on real-size networks, Section 5.5 provides tests on synthetic and real-size networks, Section 5.6 draws conclusions and research prospects.