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The mixed logit model is a flexible discrete choice model that can approximate any random utility model (McFadden and Train, 2000; Hensher, 2001). Recent advances in discrete choice modelling, have promoted the treatment of attitudes and perceptions affecting decision-making to get a more realistic representation of choice behaviour.

The ML model generalizes the MNL model by allowing the coefficients of observed variables to vary randomly between people rather than being fixed. Additionally, it partitions the stochastic

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component of the random utility equation into two parts: correlated and uncorrelated components. This allows for the possibility that the information relevant to making a choice that is unobserved may indeed be sufficiently rich in reality to induce correlation across the transport mode alternatives in each choice situation. One part of the random component is allowed to be correlated over alternatives and is heteroscedastic, and the other part is IID over alternatives and individuals. That is,

𝑈𝑖𝑗 = (𝛽 + 𝜗𝑖𝑗)𝑋𝑖𝑗+ 𝜀𝑖𝑗 (3.9)

where 𝑋_𝑖𝑗 is the observed variables and is related to shipper 𝑖 and alternative 𝑗, 𝛽 is a vector of coefficients, 𝜀_𝑖𝑗 is once again a random term (with zero mean) that is independently and identically distributed over alternatives and individuals, and 𝜗𝑖𝑗 is an error component that can be correlated among alternatives and heteroscedastic for each individual. The mixed logit model assumes a general distribution for 𝜗_𝑖𝑗 (e.g. normal, log-normal, triangular, uniform, etc.) and an IID Gumbel distribution for 𝜀_𝑖𝑗 (Hensher and Greene, 2002). The density function of the error component 𝜗_𝑖𝑗 is denoted as 𝑓(𝜗_𝑖𝑗|𝜏) , where 𝜏 is a parameter vector of the distribution of 𝜗𝑖𝑗. The conditional probability of choosing option j given the value of component 𝜗_𝑖𝑗, is

𝑄_𝑖(𝑗|𝜗_𝑖𝑗) = 𝑃_𝑖𝑗 = exp (𝑥𝑖𝑗𝛽+𝜗𝑖𝑗)

∑𝐽_{𝑘∈𝑍𝑖}exp (𝑥𝑖𝑘𝛽+𝜗𝑖𝑗) (3.10)

Since 𝜗𝑖𝑗 is not given, the unconditional choice probability, 𝑃𝑖(𝑗), is the integral of the conditional choice probability, 𝑄_𝑖(𝑗|𝜗_𝑖𝑗), over the distribution of 𝜗_𝑖𝑗 . This model is called the mixed logit (ML) model since the choice probability is a mixture of logits with 𝑓(𝜗_𝑖𝑗|𝜏) as the mixing distribution (Hensher et al., 2005; Rose et al., 2005). In general, the ML model does not have an exact

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likelihood function because the probability 𝑃𝑖(𝑗) does not always have a closed form solution. Therefore, the ML model uses simulated maximum likelihood estimation for computing the approximate probability (McFadden and Train, 2000).

For simulation purposes, it is assumed that the error component has a specific structure. The ML specification is formed by allowing the individual parameter estimates 𝜗_𝑖𝑗 in the vector 𝜗 to be defined as follows,

𝜗_𝑖𝑗 = 𝜗_𝑗+ 𝜎_𝑗 𝜑_𝑖𝑗 (3.11)

where 𝜑𝑖𝑗 is the individual specific heterogeneity with mean zero and standard deviation equal to one, 𝜎_𝑗 is standard deviation of the distribution of 𝜗_𝑖𝑗 around 𝜗_𝑗, and 𝜗_𝑗is the population mean.

One can observe 𝑥 and the choices, and estimate the random parameters 𝜗_𝑗 and 𝜎_𝑗. The random parameters allow heterogeneity across individuals in their sensitivity to observed exogenous variables. There are many distributions that can be used for the random parameters. The following four types of distributions are commonly used for the random parameters: normal, uniform, lognormal, and triangular distribution (Hensher and Greene, 2003; Hensher et al., 2005; Rose et al., 2005). It is assumed that the 𝑚 th _{element of 𝜎}

𝑚 is denoted as 𝜎𝑚∗. Under the initial assumption, the coefficients are independently distributed with mean 𝜎_𝑚 and spread 𝑠_𝑚 being estimated 𝜎_𝑚∗ _{= 𝑐}

𝑚+ 𝑠𝑚𝜀𝛽 in the population. That is 𝜀𝛽 ~ 𝑁(0,1) for a normal distribution and 𝜀_𝛽 ~ 𝑈(−𝑠, +𝑠) for a uniform distribution and 𝑈(𝜎 − 𝑠, 𝜎 + 𝑠) for a triangular distribution with mean 𝜎 and spread 𝑠. The coefficient of a lognormal distribution can be estimated as 𝜎_𝑚∗ ₌ exp (𝑐𝑚+ 𝑠𝑚𝜀𝛽) and 𝜀𝛽 ~ 𝑁(0,1) (Train, 2003).

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The most widely used distribution is the normal, mainly for its simplicity. The normal and lognormal distributions have an infinite range. For coefficients that take the same sign for all people (e.g. the coefficient of transport cost in the utility function is usually negative), such as a price coefficient that is necessarily negative or the coefficient of a desirable attribute, distributions with support on only one side of zero, like the lognormal, are commonly used (Hensher and Greene, 2005).When coefficients cannot logically be unboundedly large or small, or one wishes to restrict the range of variation of a parameter, then bounded distributions are often used, such as uniform or triangular distributions.

Several studies have used mixed logit models to analyse heterogeneity of preference in freight transport mode choice (Abate and Jong, 2014; Bergantino et al., 2013; O’Malley et al., 2013; Mitra, 2013; Hensher et al., 2013; Brooks et al., 2012; Samimi et al., 2011; Arunotayanun and Polak, 2011; Feo-Valero et al., 2011; Masiero and Hensher, 2010; Beuthe and Bouffioux, 2008; Bolis and Maggi, 2002; Kang-Soo, 2002). Bergantino et al., (2013) analysed the determining choice behaviour of Sicilian road carriers when faced with transhipment-related modal alternatives, using a RP/SP data set of 632 choice observations. The study revealed that attributes of road carriers' attitudes towards time, punctuality and risk of loss/damage can significantly enhance the explanatory power of the choice model in determining the attractiveness of two alternatives: logistics terminals and road-sea intermodal services.

Arunotayanun and Polak (2011) dealt with shippers' mode choice behaviour and, through ML and latent class (LC) model, showed that the conventional practice of using commodity type as the only segmenting variable is not adequate to account for taste heterogeneity. Their study found that the accommodation of taste heterogeneity within commodity segments leads to significant improvements in model fit in all segments. It also affects the estimates of the mean

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effects of cost and time attributes and service attributes, leading to an increase in the estimated parameters.

Feo-Valero et al. (2011) analysed the viability of a maritime logistics chain in the motorway of the Sea of South-West Europe and carried out a detailed evaluation of the performance and the potential of using cost-oriented measures to support traffic reallocation toward sea transport using SP survey data. Kang-Soo (2002) estimated two versions of the ML models, an error component and random coefficient logit, for the freight mode choice across the Channel Tunnel using SP data. The results showed the superiority of both models over traditional logit and showed the relevance of taste variations.