Production, Manufacturing and LogisticsTraffic assignment model with fuzzy level of travel demand: An efficient algorithm based on quasi-Logit formulas
Introduction
Thank to traffic assignment (TA) models, network manager predicts the traffic policies influences, while traffic flows themselves are depended on independent drivers who only seek to optimize their own individual preference. Also, the urban drivers possibly use different preferable paths, without notifying or explaining their choice to the network manager (Henn, 2000; Hawas, 2004). Taking these concepts into account, some researchers utilized the preferred, absorbing or reasonable paths instead of shortest paths in TA models. Some path enumeration techniques to provide such reasonable paths are as follows: multi-objective shortest paths (Akiva et al., 1984), gateway paths (Lombard and Church, 1993), K-similar paths (Scott and Bernstein, 1997), dissimilar paths (Akgun et al., 2000, Kuby et al., 1997), Pareto dissimilar paths (Dell’Olmo et al., 2005), side-constrained paths (Larsson et al., 2004), constrained K-shortest paths (Zijpp and Catalano, 2005), shortest paths with forbidden paths (Villeneuve and Desaulniers, 2005), stochastic shortest paths (Miller-Hooks and Mahmassani, 2003), multicriteria stochastic time-varying shortest paths (Opasanon and Miller-Hooks, 2006) and finally fuzzy shortest paths (Okada and Soper, 2000), see e.g., Ghatee, 2005, Section 3 for a review.
Other researchers have introduced several functional forms to describe the relationship between travel times and flow rates (Clark and Watling, 2005, Watling, 2006). Nevertheless, Chen and Tzeng (2001), reported that considering a specific link travel time function in advance may not describe accurately the peak period traffic dynamics. Since, the travel times in a dynamic transportation network depend on both prevailing and future traffic conditions, they proposed the fuzzy integral for evaluating perceived travel costs in TA models.
The third group of researchers have generalized user equilibrium principle (Wardrop, 1952) to the multiple class (Van Vuren and Watling, 1991), stochastic (Daganzo and Sheffi, 1977) and fuzzy (Teodorovic and Kikuchi, 1990) variants. In stochastic user equilibrium samples, Logit model (Fisk, 1980) gives the error term as Gumbel distribution and can be implemented as a convex programming, see e.g., Damberg et al., 1996, for an efficient algorithm in order to obtain the Logit equilibrium flows.
Although, due to uncertainty on driver perceived information, fuzzy version of Logit model has been introduced by Henn (2000), but in established literature, there is no enough attention to uncertain level of travel demand between origin–destination (OD) pairs. The uncertain travel demands which are obtained based on incomplete statistics, can be represented as LR fuzzy numbers in which the average of data to be assumed as the most possible scenarios of LR fuzzy numbers (their centers) and the deviation of data to be assumed as the right and left spreads of those fuzzy numbers. For TA problem with fuzzy level of travel demand, in this paper, we propose a new variant of Logit model by combining the above three mentioned viewpoints. Here, we quantify the uncertain number of travelers with the simplest LR fuzzy numbers; triangular fuzzy numbers with linear shape functions. The proposed model is transformed into a mixed-integer problem with respect to path-flow variables. These paths can be generated by different path enumeration techniques. Since, in solving problems under uncertainty, it is necessary to exert maximal efforts in seeking the possibilities for overcoming the uncertainty, we maximize the level of certainty in our model. For this end, a branch and bound mechanism is used together with a bisection scheme. Then, by considering an optimal level of certainty, a simple model is obtained with travel cost in relating to congestion. The optimal solution of this problem, which we name as fuzzy equilibrium flow, is found. We show that such flow satisfies a quasi-Logit formula. The main contributions of this paper are as follows:
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Developing a framework of convex programming for TA model with the fuzzy level of demand.
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Transforming the original problem into a mixed-integer problem by maximizing the degree of certainty.
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Utilizing different path enumeration techniques for finding reasonable paths.
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Decreasing the computation time and increasing stability for finding Logit flows by utilizing a heuristic consisting of path generating before running Damberg et al., 1996.
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Obtaining the quasi-Logit formulas for equilibrium flow of the presented fuzzy TA model to construct efficient algorithm.
The rest of the paper is organized as follows. In the next section, Logit TA model and a modification of Damberg et al. algorithm are presented. In Section 3, we first review on affects of uncertainty in urban networks. Also, we study a convex programming to exhibit the fuzzy travel demands in TA model. Furthermore, some properties of the equilibrium solution of this problem are studied which are the base for constructing an algorithm in order to maximize the level of certainty. In Section 4, a quasi-Logit formula for fuzzy equilibrium flows is introduced as a part of an algorithm for assigning flows with respect to equilibrium status. Implementation concepts and numerical examples are illustrated in Section 5. Final section ends this paper with conclusions and future research directions.
Section snippets
Logit TA
The mathematical model of TA problem has been introduced by researchers including Fisk, 1980, Sheffi, 1985. Also, some drawbacks in Wardrop’s user equilibrium principle (1952), were reported by Daganzo and Sheffi (1977). They extended stochastic user equilibrium model to tackle variations in the travelers’ perception of travel cost. In this category, the Logit model assumes Gumbel distribution as random component in perceived travel times. This model can be stated as the following convex
Constructing fuzzy TA model
The aim of TA problem is to predict the amount of flows on network links, while it depends on the route choice models. The vast majority of classic TA models are deterministic as they assume that the travel demand between the pairs of OD nodes to be known a priori for the entire planning. These hypotheses aid the solution process and highlight a fundamentally complex characteristic of this problem. As it reflects the different provision of traffic information, the different user equilibrium
Finding equilibrium flows in fuzzy TA model
In this section, we show that how it is possible to combine the modified Damberg Algorithm 2.1 and branch and bound scheme to solve efficiently the subproblem (29). Furthermore, we construct some formulas representing equilibrium flows of fuzzy TA model. Hereafter, since, is assumed to be a fixed number, for simplicity without any ambiguity, we eliminate from the formulation of models. Thus, we can rewrite our model as follows:
Implementation and numerical results
In this section, we briefly illustrate the efficiency of the proposed algorithm by studying on a sample network with 13 nodes, 30 links and five junctions (depicted in Fig. 2. In Table 1, the capacity and free-flow cost of network links are represented.
According to the well known cost function presented by the US Bureau of Public Roads, 1964, we consider the link costs as follows:where and are the capacity and free-flow cost of link and parameters α and β are
Conclusion and future directions
With respect to the uncertainty as a crucial issue in TA models and insufficiency of traditional random utility to take into account the uncertainty due to randomness of traffic patterns and its important role in ATIS and ATMS, a TA model with fuzzy level of travel demand is introduced in this paper. We utilize triangular fuzzy numbers to exhibit the imprecise number of traveler whom want travel between the OD pairs. Such numbers reflects the perception of individuals to travel. Then we
Acknowledgement
The authors would like to express particular thanks to the anonymous referees and honorable editors for their valuable comments, which led us to improvements in this paper. Also, the support from Tehran Control Traffic Organization is gratefully acknowledged.
References (53)
- et al.
On finding dissimilar paths
European Journal of Operational Research
(2000) - et al.
Traffic signal timing optimisation based on genetic algorithm approach, including drivers’ routing
Transportation Research B
(2004) - et al.
A fuzzy user-optimal route choice problem using a link-based fuzzy variational inequality formulation
Fuzzy Sets and Systems
(2000) - et al.
Using fuzzy integral for evaluating subjectively perceived travel costs in a traffic assignment model
European Journal of Operational Research
(2001) - et al.
Modelling network travel time reliability under stochastic demand
Transportation Research B
(2005) - et al.
An algorithm for the stochastic user equilibrium problem
Transportation Research B
(1996) - et al.
On finding dissimilar Pareto-optimal path
European Journal of Operational Research
(2005) - et al.
An inventory model for single-period products with reordering opportunities under fuzzy demand
Computers & Mathematics with Applications
(2007) Some developments in equilibrium traffic assignment
Transportation Research B
(1980)A non-cooperative neuro-fuzzy system for integrating ATIS and ATMS decisions
Fuzzy Sets and Systems
(2004)