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An ant colony system for transportation user equilibrium analysis in congested networks

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Abstract

In this paper we present Ant Colony System for Traffic Assignment (ACS-TA) for the solution of deterministic and stochastic user equilibria (DUE and SUE, respectively) problems. DUE and SUE are two well known transportation problems where the transportation demand has to be assigned to an underlying network (supply in transportation terminology) according to single user satisfaction rather than aiming at some global optimum. ACS-TA turns the classic ACS meta-heuristic for discrete optimization into a technique for equilibrium computation. ACS-TA can be easily adapted to take into account all aspects characterizing the traffic assignment problem: multiple origin-destination pairs, link congestion, non-separable cost link functions, elasticity of demand, multiple classes of demand and different user cost models including stochastic cost perception. Applications to different networks, including a non-separable costs case study and the standard Sioux Falls benchmark, are reported. Results show good performance and wider applicability with respect to conventional approaches especially for stochastic user equilibrium computation.

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Notes

  1. It must be stressed that we are not providing here the description of how the network reaches the equilibrium condition (if any), but of how to calculate the equilibrium solution using a so called assignment model.

  2. For every od pair there is an ant colony, with its own origin (centroid o) and a destination (centroid d); in the case of the multi-class approach a colony is characterized not only by origin and destination, but also by the user category.

  3. The generality of the approach in terms of dealing with elastic demand can be seen from

    $$ f^c = P\bigl(\tau^c\bigr),\qquad \tau^c = T\bigl(d^c\bigr),\qquad d^c = D\bigl(c^c\bigr),\qquad c^c = \mathcal{H}\bigl(f^c\bigr), $$
    (21)

    the demand d c being some function D of the path costs. If the function D is continuous and monotone we have guaranteed again the existence and uniqueness of the equilibrium.

  4. Other conditions are required for convergence, we point the interested reader to Robbins and Monro (1951) and Blum (1954).

  5. What we want to demonstrate by citing (24) is that the expected value of the left member is equal to the expected value of \(T(\mathcal{H}(f_{t}^{c} ))\) (which is the value of pheromone update at each iteration); that is, \(E(\tau^{c})=E(T(\mathcal{H}(f_{t}^{c})))\), where E is the expectation operator. This can be easily demonstrated by noting that \(E(\tau^{c})=E(\tau_{(t-1)}^{c} )\). Taking into consideration this simple demonstration (which indeed reflects the property of the fixed point approach) no bias is introduced through the right member of (24).

  6. In type 1 function the travel time on link l is \(\operatorname{tr}_{l}=c_{0}\times f + c_{1} /2 \times f^{2}\), with c 0 and c 1 real valued coefficients, and f the flow on the link l; this function was not used in these experiments.

  7. With calibration we mean the tuning of the model with respect to the observed phenomenon through the exploitation of observed data in order to adapt the cost function to the real scenario.

  8. Transportation Research Board, Washington, D.C., USA.

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Authors and Affiliations

  1. Dipartimento di Elettronica, Informazione e Bioingegneria, Politecnico di Milano, p.zza Leonardo da Vinci 32, Milan, Italy

    Matteo Matteucci

  2. Department of Architecture, Built Environment, Construction Engineering, Politecnico di Milano, p.zza Leonardo da Vinci 32, Milan, Italy

    Lorenzo Mussone

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  1. Matteo Matteucci
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Matteucci, M., Mussone, L. An ant colony system for transportation user equilibrium analysis in congested networks. Swarm Intell 7, 255–277 (2013). https://doi.org/10.1007/s11721-013-0083-x

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