1 Introduction
The traffic assignment or network equilibrium problem is to predict the steadystate flow of a transportation network Friesz (1985); Wang and Liao (1999); Ghatee and Hashemi (2009). Based on this equilibrium, a traffic network can be designed or managed more effectively Farahani et al. (2013); George and Kim (2013). There are many literatures for the traffic assignment problem such as multiple route assignment Burrell (1968), probabilistic multipath traffic assignment Dial (1971) and paired alternative segments for traffic assignment BarGera (2010). However, in reality, the state of a transportation network is always determined by independent travelers who only seek to choose the minimumcost path between every origindestination pair. According to this phenomenon, in 1952, Wardrop et al. Wardrop (1952) proposed the user equilibrium principle for the traffic network equilibrium problem. Then, a mathematical model of the user equilibrium is developed by Beckmann et al. Beckmann et al. (1956) in 1956. Based on this model, the dynamic user equilibrium and the stochastic user equilibrium are presented and considered by a lot of researchers Friesz et al. (2011); Chen et al. (2014); Zhou et al. (2012).
However, in real world, travelers always can not obtain global information of the traffic network, in which many uncertain events may occur at any moment such as traffic accident. Usually these uncertain events can be taken by fuzzy viewpoints
Ghatee and Hashemi (2009). Akiyama et al. Akiyama and Yamanishi (1993) develop a model for route choice behavior due to the fuzzy reasoning approach. Henn Henn (2000) proposes a fuzzy route choice model by representing travelers with various indices such as risktaking travelers or riskingaverting travelers. Consider the spatial knowledge of individual travelers, Ridwan Ridwan (2004) suggest a fuzzy preference for travel decisions because some travelers do not follow maximizing principles in route choice. Ghatee et al. Ghatee and Hashemi (2009)propose a method based on quasilogit formulas to obtain a fuzzy equilibrium flow assuming a fuzzy level of travel demand. Many researchers have focused on the fuzzy traffic assignment problem, but, there is not a method dominating the others. Therefore, it is meaningful for us to investigate new method to focus on the fuzzy network equilibrium problem.
Recently, a large, singlecelled amoebalike organism, Physarum polycephalum, was found to be adaptively capable of solving many graph theoretical problems such as the shortest path found through a maze Nakagaki et al. (2000, 2001); Adamatzky (2012), path selection in networks Nakagaki et al. (2007); Zhang et al. (2013a, b), network design Tero et al. (2010); Adamatzky and Prokopenko (2012). In this paper, for taking full of advantage of Physarum Polycephalum’s adaptivity in network design, we modify the Physarumtype algorithm to unity the Physarum network and the traffic network so that they can propagate mutually. By this way, the fuzzy user equilibrium can be approached by Physarum Polycephalum. Comparing with other existing algorithms, the main advantage of this algorithm is its adaptivity. To test the performance of this method, some experiments are developed and the results demonstrate that our approach is efficient.
The rest of the paper is organized as follows: in section 2, some preliminaries are presented. In section 3, the proposed method is described. In section 4, experimental results are evaluated. In the final, a brief conclusion is given.
2 Preliminaries
Some basic theories are shown in this section, including Physarumtype algorithm for shortest path selection, user equilibrium in the traffic network and basic concepts of fuzzy set.
2.1 Physarumtype algorithm for shortest path selection
The shortest pathselection process of Physarum polycephalum is based on the morphogenesis of the tubular structure Nakagaki et al. (2001); Tero et al. (2007): on the one hand, high rate of protoplasmic flow stimulates an increase in tubes diameter, whereas tubes tend to decline at low flow rate. Tube thickness therefore adapts to the flow rate. On the other hand, the decrease of tube thickness is accelerated in the illuminated part of the organism. Thus, the tube structure evolves according to a balance of these mutually antagonistic processes. Based on the observed phenomena of the tube structure s evolution, a simple Physarum polycephalum model, which takes a mathematically simplified and tractable form, is proposed by Tero et al. Tero et al. (2007).
Using the graphic illustrated in Tero et al. (2007), the model can be described as follows. Each segment in the diagram represents a section of tube. Two special nodes, which are also called food source nodes, are named and , and the other nodes are denoted as , , , and so on. The section of tube between and is denoted as . If several tubes connect the same pair of nodes, intermediate nodes will be placed in the center of the tubes to guarantee the uniqueness of the connecting segments. The variable is used to express the flux through tube from to . Assuming the flow along the tube as an approximately Poiseuille flow, the flux can be expressed as:
(1) 
where is the viscosity coefficient of the sol. is a measure of the tube conductivity. is the pressure at the node . is the length of the edge .
Assume zero capacity at each node, can be obtained according to the conservation law of sol. For the source node and the sink node , and , respectively. is the flux flowing from the source node (or into the sink node). Then the network Poisson equation for the pressure can be obtained as follows:
(2) 
By setting as the basic pressure level, all ’s can be determined by solving above equation system, and each is also obtained.
Experimental observation shows that tubes with larger fluxes are reinforced, while those with smaller fluxes degenerate. To accommodate the adaptive behavior of the tubes, all corresponding conductivities ’s change in time according to the following equation:
(3) 
where is a decay rate of the tube. The functional form is generally given by for the sake of simplicity Tero et al. (2007). To solve the adaption of Eq. (3), a semiimplicit scheme is used as follows:
(4) 
where is a time mesh size and the upper index indicates a time step. Hence, the timevarying conductivity generates the state of the system to make the shortest path emerge.
2.2 User equilibrium in the traffic network
Nomenclature
the set of network nodes
the set of network links
the set of origindestination () nodes
the set of paths from to for each
: the demand of trips through
the total flow through link
the capacity of link
the freeflow cost of link
the cost on link ,
the traffic flow of path
the cost of path ,
In 1952, Wardrop et al. Wardrop (1952) proposed the User equilibrium principle: any travelers can not decrease themselves’ cost by changing travel route when the traffic system is equilibrium. According to this principle, a flowcost formula was developed by Beckmann et al. Beckmann et al. (1956):
(5) 
where is the minimum cost of each in the network equilibrium. Assuming that the cost of each link is only associated with the flow of that link and the cost is strictly increasing with the flow increasing, Eq. 5 can be transformed into the mathematical model shown as follows:
(6) 
(7) 
It can be proved that there exists one unique solution for Eq. 6 Smith (1979)
. Hence, The user equilibrium problem can be solved by solving a mathematical optimization problem as Eq.
6 and Eq. 7.2.3 Fuzzy set
Fuzzy set proposed by Zadeh Zadeh (1965) in 1965 is widely used in many fields such as statistics Yang et al. (2010); Giri et al. (2014), computer programming Azadeh et al. (2012); Hu et al. (2011), engineering and experimental science Nguyen et al. (2011); Deng et al. (2011). Based on this theory, The concept of fuzzy number was first used by Nahmias in the United States and by Dubois and Prade in France in the late 1970s. In this paper, the triangular fuzzy number will be used. Therefore, according to Kauffman and Gupta (1991); Ezzati and Saneifard (2010),some basic definitions of fuzzy set and fuzzy number are given as follows.
Definition 1
A fuzzy set defined on a universe X may be expressed as:
(8) 
where is the membership function of . The membership value describes the degree of in .
Definition 2
A fuzzy set of X is normal iff .
Definition 3
A fuzzy set of X is convex iff , where denotes the minimum operator.
Definition 4
A fuzzy set is a fuzzy number iff is normal and convex on X.
Definition 5
A triangular fuzzy number is a fuzzy number with a piecewise linear membership function defined by:
(9) 
which can be denoted as a triplet . A triangular fuzzy number in the universe set that conforms to this definition shown in Fig. 1.
Based on Giachetti and Young (1997); Chen (1994), fuzzy arithmetic on triangular is shown as follows.
Definition 6
Assuming that both and are triangular numbers, then the basic fuzzy operations are:
(10)  
(11)  
(12)  
(13) 
Definition 7
Assuming that both and are trapezoidal numbers, then the basic fuzzy operations are:
(14)  
(15)  
(16)  
(17) 
For example, let and be two triangular fuzzy numbers. Based on Eqs. 10, four basic operations can be derived as:
The results of the above operations are depicted in Fig. 2.
Recently, fuzzy distance, as a measure of distance between two fuzzy numbers, has gained much attention from researchers and been widely applied in data analysis, classification, and so on Guha and Chakraborty (2010); SadiNezhad and Khalili Damghani (2010). In this paper, the distance proposed in Gildeh and Gien (2001) is adopted to measure the difference between two fuzzy numbers.
Definition 8
The analytical properties of depend on the first parameter , while the second parameter of characterizes the subjective weight attributed to the end points of the support. Having close to 1 results in considering the right side of the support of the fuzzy numbers more favorably. Since the significance of the end points of the support of the fuzzy numbers is assumed to be same, the is adopted in this paper.
According to studies by Mahdavi et al. Mahdavi et al. (2009); Hassanzadeh et al. (2011), with and , the general form of fuzzy distance can be converted into different forms, as two fuzzy numbers and take different types.
For triangular fuzzy numbers and , the fuzzy distance between them can be represented as:
(19) 
3 Proposed method
3.1 Fuzzy user equilibrium
In the model of traditional user equilibrium, a basic assumption is that all travelers know the global information of the traffic network. Based on this, every one can make the decision having no conflict with each other. In reality, however, because of some uncertain events or local information, travelers always need to make the decision according to the fuzzy information of the traffic network. In general, the fuzzy information can be divided into three types Ghatee and Hashemi (2009): inexact travel cost, unsure network topology and imprecise travel demand. In this paper, we consider the user equilibrium problem in traffic assignment with fuzzy travel cost or fuzzy user equilibrium problem. According to section 2.2, the fuzzy user equilibrium can be stated as follows:
(20) 
(21) 
where is the fuzzy cost function. It denotes the inaccuracies of perceived time of travelers. For example, one traveler may have a larger perceived travel time of one path than its real travel time while some travelers increase their speed through this paths for some special reasons and a lower if a car accident occurs on this path. For the sake of simplicity, we assign with a triangular fuzzy number. To associate with , we use and to extend . Therefore, can be calculated as follows:
(22) 
where is the left limit and is the right limit of , respectively. The advantage of using such a strategy is it considers link capacity as an effective factor in traffic assignment Ramazani et al. (2011).
3.2 Physarumtype algorithm for fuzzy user equilibrium algorithm
According to the network structure, Physarum Polycephalum is able to make full use of its protoplasm (flow) for building a new network (Physarum network) based on its adaptivity Tero et al. (2010). To take full of advantage of this feature for fuzzy user equilibrium problem, it is necessary for us to find out the similarities and differences between the Physarum network and the traffic network, and then to find a way to unify the Physarum network and the traffic network. Therefore, some modifications of Physarumtype algorithm should be carried out.
There are many similar properties between the Physarum network and the traffic network. As a result, we can treat the links in the traffic network as the tubes in Physarum network, the traffic flow as the protoplasm, the traffic nodes as the food sources and the cost as the distance. Meanwhile, there are some differences between them. For example, in reality, the traffic nodes have many their own features such as education center, political centers and transportation hubs. Besides, the traffic flow is always determined by many factors such as the choices of travelers, the accidents and the transport facilities. While in the Physarum network, the protoplasm in the tubes only flow from the highpressure node to the lowpressure node.
Based on these similarities and differences, we rewrite some formulas of Physarumtype algorithm to unify the Physarum network and the traffic network. Firstly, consider the distance of tubes and the cost of links:
(23) 
where arc from node to node is equal to link . Like Zhang et al. (2013a), fuzzy cost can be denoted as . Then, Eq. 2 can be rewrite as follows:
(24) 
where is associated with the fuzzy cost . According to , the can be obtained for each pair. Then, the globe pressure can be calculated as follows:
(25) 
Next, can be calculated as follows:
(26) 
where is the function of fuzzy distance measure. Finally, according to Eq. 4 and , is obtain as follows:
(27) 
where is a time mesh size and the upper index indicates a time step.
4 Experimental results
In this section, we briefly illustrate the efficiency of the proposed algorithm by studying on some sample networks.
4.1 A test problem of Ramazani
The first experiment involves a test problem introduced by Ramazani et al. Ramazani et al. (2011). It is based on a network with 4 nodes, 6 links and one origindestination travel demands shown as Fig. 3. According to the well known cost function represented by the US Bureau of Public Roads Manual (1964), the cost of link can be calculated as follows:
(28) 
where and are the freeflow cost and the capacity of link depicted in Table 1. Parameters and are fixed values (usual values are and ).
The demand for origindestination is 700, namely . Parameter and are assumed to be . The results of assignment after Physarumtype algorithm (PA) are shown as Fig. 4 and Table 2. Consider this equilibrium state, the fuzzy cost of each path can be calculated according to Eq. 22. For comparing these results, we use the method in Deng et al. (2012); Hassanzadeh et al. (2013) to transform the fuzzy cost to crisp number shown as Table 3. Therefore, PA is more efficient than FITA.
4.2 A test problem of Ghatee
The second experiment involves a test problem introduced by Ramazani et al. Ghatee and Hashemi (2009). It is based on a network with 13 nodes, 15 links and five junctions depicted in Fig. 5. The capacity and freeflow cost of network links are presented in Table 4. Assume six pairs with demand , the results of the equilibrium flows and the equilibrium path data of each pair are presents in Table 5 and Table 6, respectively. Table 5
5 Conclusion
In this paper, we propose a Physarum network mode to address the fuzzy user equilibrium problem. We modify the Physarumtype algorithm to build a relationship between the Physarum network and the traffic network so that they can propagate mutually. By this way, an equilibrium state occurs approaching the solution of the fuzzy user equilibrium problem. To test the performance of the proposed method, some experiments are developed. The results demonstrate the feasibility and effectiveness of the proposed algorithm.
6 Acknowledgments
The author greatly appreciate the reviews’ suggestions. The work is partially supported by National Natural Science Foundation of China (Grant No. 61174022), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20131102130002), R&D Program of China (2012BAH07B01), National High Technology Research and Development Program of China (863 Program) (Grant No. 2013AA013801), the open funding project of State Key Laboratory of Virtual Reality Technology and Systems, Beihang University (Grant No.BUAAVR14KF02).
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