1 Introduction
The traffic assignment refers to a manner in which a given aggregate origindestination (OD) pair passenger traffic demand is assigned to the traffic routes of that OD pairBertsekas and Gafni (1982); Papageorgiou (1990); Ziliaskopoulos (2000); Liu et al. (2010); Du et al. (2016). As an important traffic assignment paradigm, the user equilibrium (UE) of travelers’ path choice in traffic networks was firstly conceptualised by Wardrop Wardrop (1952). The UE principle is based on the assuming that the traveller knows the precise route cost and will choose the route with the minimum cost. The user equilibrium is achieved when all travellers between the same OD pair have the same and minimum cost Beckmann et al. (1956); Chiou (2009). Additionally, the transportation cost of any traveller can’t be reduced by unilaterally changing routes.
In the existing literatures, many algorithms were designed to address the UE problem Florian et al. (2009); Chiou (2010); Lin and Leong (2014), which can be totally classed into three types: link based algorithms, path based algorithms and origin based algorithms. The linear approximation method of Frank and Wolfe (FW) Leblanc et al. (1975) has been the most popular algorithm in practice because of its simplicity. As a linear approximation method, the FW algorithm has a sublinear asymptotic convergence speed Florian et al. (2009). As a consequence, highly precise solutions can’t be achieved within reasonable computation time. In path based algorithms, the UE problem is solved by achieving the path flux directly Ryu et al. (2014). The traffic flux is assigned by fixing the flux from other OD pairs for each OD pair. In the exiting path based algorithms, the disaggregate simplicial decomposition algorithm (DSD) Larsson and Patriksson (1992) and the gradient projection algorithm (GP) Jayakrishnan et al. (1994); Cheng et al. (2003); Florian et al. (2009) are widely used. Both DSD and GP have shown excellent results when compared with FW algorithm Sun et al. (1996); Tatineni et al. (1998), but the memory requirement is usually regarded as impractical for largescale networks. The origin based algorithm proposed by BarGera BarGera (1999, 2002) doesn’t require as much memory as the path based algorithms. The algorithm computes sequentially for each origin subnetwork by using a quasiNewton approach. Later, Dial Dial (2006) developed a different origin based algorithm by shifting flows sequentially from the longest to the shortest path. Though the origin based algorithms can provide both link traffic flows and route traffic flows Xu et al. (2008), enumerating all route flows of each subnetwork is quite complicated.
Biological systems usually inspire computer scientists and engineers to process the information and make decisions. So far, some heuristic algorithms have been proposed to solve the traffic assignment problem, such as the ant colony algorithm
Matteucci and Mussone (2013); D Acierno et al. (2012)and the genetic algorithm
SanchezMedina et al. (2012). Recently, the slime mould Physarum polycephalum becomes a popular living computing substrate Adamatzky (2007); Adamatzky and Jones (2010). Physarum machines are proved to be the most successful biological substrates in solving problems of computation geometry, optimization, and logic because they are easy to realize Adamatzky and Jones (2015). In this article, a modified Physaruminspired model is proposed to solve the UE traffic assignment problem.Physarum polycephalum is a large amoeboid organism, which contains a great number of nuclei and tubular structures Stephenson and Stempen (1995). These tubular structures will distribute protoplasm as a transportation network. The experiment has shown that the Physarum has the capacity of finding the short path between two points in a given labyrinth Nakagaki et al. (2000). A mathematical model can capture the basic dynamics of network adaptability through iterations of local rules and produces solutions with properties comparable to or better than those of realworld infrastructure networks Tero et al. (2010). The convergence of Physarum finding the shortest path has been proved by Bonifaci et al. Bonifaci et al. (2012). Based on its foraging behavior, so far, the Physarum has been used to solve many problems, such as finding the shortest path in directed or undirected network Adamatzky (2012); Wang et al. (2014); Zhang et al. (2014); Wang et al. (2015), designing and simulating transport network Adamatzky (2012); Adamatzky et al. (2013); Tsompanas et al. (2014); Evangelidis et al. (2015), natural implementation of spatial logic Schumann and Adamatzky (2011a, b), and computer music Braund and Miranda (2013); Miranda (2014); Braund and Miranda (2015). In addition, the Physarum model can also find the shortest path under uncertain environment Zhang et al. (2013, 2014) in the real application Jiang et al. (2015).
Recently, quite a few researchers try to apply the Physarum model to solve the traffic assignment problem Zhang et al. (2015); Zhang (2015). These Physaruminspired models can affect well in specific conditions where networks are unilaterally connecting. But the exiting models can’t be realised in traffic networks with twoway traffic characteristics. The Physarum model for UE traffic assignment problem Zhang (2015) can’t distinguish the flux from different OD pairs, which means the model isn’t reasonable in traffic networks with multiple OD pairs. In this paper, a modified Physaruminspired model for the UE traffic assignment problem is proposed for traffic networks with twoway traffic characteristics. In the proposed model, the flows are decomposed by origin node as the origin based algorithms. For each subnetwork, the flows are assigned by the Physarum model based on its protoplasmic network adaptivity and continuity.
This paper is structured as follows. In Section 2, the user equilibrium traffic assignment principle is reviewed. The original Physarum polycephalum model and Physaruminspired model for UE traffic assignment problem proposed by Zhang Zhang (2015) are briefly introduced. In Section 3, a modified Physaruminspired model for the UE traffic assignment problem is presented. In Section 4, numerical examples are given to demonstrate the rationality and convergence properties of the proposed model. Finally, the paper ends with conclusions and suggestions for further researches in Section 5.
2 Preliminaries
In this section, some preliminaries are briefly introduced, including the traffic assignment model Beckmann et al. (1956), the original Physarum polycephalum model Tero et al. (2007) and the Physaruminspired model for UE problem proposed by Zhang Zhang (2015).
2.1 User Equilibrium Traffic Assignment Model
The transportation network is a strongly connected directed graph , where is the set of nodes and is the set of directed links. Assume that there is no links from a node to itself and only one link, if any, between two different nodes. Suppose that and denote the set of origin nodes and the set of destination nodes, and , , . Let and denote the set of all the paths and the travel demand between OD pair , than we have:
(1) 
where is the path traffic flow along path between OD pair . Let denote the traffic flow along the link . Then all nodes, except source nodes and destination nodes, satisfy the flow conservation law Zhang (2015), shown as follows:
(2) 
Let denote the traffic time experienced by each user among the link when units of vehicles flux along the link. is a monotonously non decreasing and continuously differentiable traffic time functions for the flux on link due to the effects of congestion on the travel time Beckmann et al. (1956), which can be expressed as follows:
(3) 
(4) 
Let represent the path traffic time along the path between OD pair and denote the correlation coefficient between traffic link and traffic path, if the path between OD pair traverses link , otherwise . The path traffic time and link traffic flow can be expressed as Beckmann et al. (1956):
(5) 
(6) 
The Wardrop’s user equilibrium principle Wardrop (1952) is that travelers seek to minimize the cost associated with their chosen routes. Travelers are assumed to have perfect information about actual travel conditions, and they can be identical in the sense that they valued time, monetary cost, and other route attributes in the same way. The Wardrop’s user equilibrium is obtained when no traveller s traffic time can be reduced by unilaterally changing routes, which can be expressed as follows:
(7) 
where is the shortest traffic time between OD pair under traffic equilibrium.
Under the assumptions above all, the traffic assignment problem can be mathematically formulated as the following convex optimization problem Beckmann et al. (1956):
(8) 
2.2 The Origin Physarum Polycephalum Model
Physarum polycephalum is a singlecelled amoeboid organism, which is also called as plasmodium in the vegetative phase Tero et al. (2007). It has the ability to solve the shortest path selection, based on its special foraging mechanism: the transformations of tubular structures and a positive feedback from flow rates. The high rates of the flow motivate tubes to thicken and the diameter of the tube minishes at a low flow rate Tero et al. (2007). A total introduction for the physarum polycephalum path finding model is given below.
Suppose the shape of the network formed by the Physarum is represented by a graph , where the edge of the graph denotes the plasmodial tube and the node denotes the junction between tubes. And is a set of nodes, where and are signed as the source and destination nodes, any others are labeled as , , , etc. The edge connecting node and node are remarked as and the flux from node to node through edge is remarked as , shown as follows Tero et al. (2007):
(9) 
where is the viscosity of the fluid and is the measure of the conductivity of the edge tube . And is the measure of the pressure at the node , is the length of the edge . According to the conservation law of flow, the inflow and outflow must be balanced, namely:
(10) 
especially for the source nodes and , the flux equations can be expressed as:
(11) 
(12) 
where is the flux from the source node to the destination node, which is assumed as a constant in the model. According to the Eqs.(912), the network Poisson equation for the pressure is derived as following:
(13) 
by further setting as the basic pressure level, the pressure of all nodes can be determined, the pressure of all nodes can be determined according to Eq.(13) and all can be determined by solving Eq.(9).
To accommodate the adaptive behavior of the plasmodium, the conductivity is assumed to change when adapting to the flux . And tubes with zero conductivity will die out. The conductivity of each tube is described as follows: Tero et al. (2010):
(14) 
where is the decay rate of the tube and is monotonously increasing continuous function which satisfies . Obviously, the positive feedback exists in the model.
2.3 The Physaruminspired Model for UE Problem
According to the feature of Physarum foraging behavior system, the optimum problem and the user equilibrium problem in directed traffic networks are solved by modified Physarum models Zhang et al. (2015); Zhang (2015) . There are mainly two points different from from the original Physarum model in Zhang’s method Zhang (2015).
First, the original Physarum model can only find the shortest path in undirected networks shown in Figure 3(a). The modified method was proposed to extend the original Physarum model to directed networks. A total introduction for the modified model is given below.
In the modified model, each edge is regard as two tubes with opposite directions and equal weight, which is shown in Figure 3(b). And there is only one direction between two nodes in the network, which means that flux can only flow from node to node and it can’t flow in the opposite direction as shown in Figure 3(b). And the network Posson equation defined in Eq.(13) was modified as follows Zhang (2015):
(15) 
where denotes the travel time of from node to node and denotes the travel time of from node to node . Similarly, and have different meanings. In the initialization, if , then . Otherwise, assign a value between and to . To guarantee the inconsistency, should be kept during iterations if .
Second, there is only one source and one destination in networks. However, usually there are multiple OD pairs in traffic networks. To handel the network with multiple origins and destinations, Eq.(15) was modified as follows Zhang (2015):
(16) 
where denotes the origin node and , denotes the destination node and . and represent the units of flow supplied by the origin node and consumed by the destination node .
3 Proposed Method
In this section, we will discuss the shortcomings of the original Physaruminspired method for UE problem and propose the modified Physaruminspired method to solve the UE problem.
3.1 Shortcomings of the Origin Physaruminspired Method
Compared with the previous slime mould models, the Physaruminspired method for UE problem is superior when characterizing its foraging activity. However, two shortcomings of the original Physaruminspired method for UE problem are founded as follows:

Note that there is only one direction between two nodes in the network shown in Figure 3(b), which means the flux can only flow from one node to another but never in the opposite direction. However, in real traffic networks, most roads have the properties of twoway traffic characteristics as shown in Figure 4. Clearly, opposite directions are separated with each other, where flows don’t interfere in two opposite directions. Obviously, the original Physaruminspired method can’t be implemented in the traffic network shown in Figure 4.

The Physaruminspired model for UE problem isn’t reasonable in traffic networks with multiple OD pairs. The modified equation Eq.(16) only satisfies the flow conservation law of the traffic network. However, it can’t distinguish the flux in each OD pair, which means that for a given OD pair , the output flux at node doesn’t equal the input flow at node . In fact, the Physaruminspired model for UE problem is only suitable for traffic networks with one source and multiple sinks or multiple sources and one sink. Here, we illustrate this problem with the following example.
Example A small network with multiple sources and multiple sinks
Here, we use a small traffic network with 4 nodes, 4 links and 2 OD pairs which are shown in Figure 5. And the origindestination demands, in vehicles per hour, are =100 and . For simplicity, the link travel time is calculated by the US Bureau of Public Roads(BPR) function Hansen (1959), which is expressed as follows:
(17) 
where , , , denote the travel time, free flow travel time flow and the capacity on link , respectively.
By using the Physaruminspired model, the path flows are shown in Table 1. It’s clear that traffic assignment doesn’t satisfy the traffic demands, which the flux along path and path should equal 100. Note that the free flow travel time of and is much less than that of and . Clearly the model doesn’t distinguish the flux from different OD pairs. As a result, the Physaruminspired model assigns more flux to path and .
Path(node sequence)  Path flow  Path cost 

1 2  19.5177  10.0348 
1 3  80.4823  10.0348 
4 2  80.4823  10.0348 
4 3  19.5177  10.0348 
3.2 The Modified Physaruminspired Model for UE Problem
Here, we proposed a modified physaruminspired model for UE problem to overcome these abovementioned shortcomings.
3.2.1 Modified Physarum model for the shortest path in directed networks
To satisfy the characteristic of the real traffic network shown in Figure 4, each edge is regarded as two tubes with opposite directions and different weight. And flux can flow in opposite directions without interfering with each other. The network Posson equation is same as Eq.(15). But to maintain the validity of conductivity, the conductivity equation defined in Eq.(9) should be improved as follows:
(18) 
when the flux in each tube is negative, the flux will be assigned as 0. That’s because the flux can’t get through the given tube in the opposite direction. In the initialization, if , we assign a value between 0 and 1 to . Otherwise, conductivity is assigned as .
3.2.2 Modified Physarum model for directed networks with sources and multiple sinks
To overcome the defect that the Physaruminspired model can’t distinguish the flux in each OD pair, we modify the Physaruminspired model as below. Let denote the set of destination nodes which are originated from node . Clearly, we have:
(19) 
where denotes the input flow at the origin node and denotes the output flow at the destination node originated from node . Here, we regard the multiple sources and multiple sinks network as the superposition of one source and multiple sinks networks. For each one source and multiple sinks network, we modify the original network Poisson equation Eq.(13) as follows:
(20) 
Naturally, the flux of each tube in all one source and multiple sinks subnetworks can be calculated by Eq.(18). According to the superposition principle, the flux of each tube in the original multiple sources and sinks network can be expressed as follows:
(21) 
where denotes the flux from node to node in the original multiple sources and multiple sinks network and represents the flux from node to node in the subnetwork originated from node .
3.2.3 Modified Physarum model for UE problem
Note that in the process of Phyasrun approaching the shortest path, the flow and the conductivity along each link are continuous. Further consideration of the continuity and dynamic reconfiguration of Physarum model, we can update the link travel time within each iteration. The flux will be redistributed by the modified Physarum model when the link travel time is updated during iterations. The length of link is updated as follows:
(22) 
where denotes the total flow on link at the iteration, and represent the length of link at the and iteration. And the search direction of link length is guided by . Note that in equilibrium, there will be , which means the length of link equals the travel time along link .
Here, the main steps of the modified Physaruminspired model for user equilibrium problem is presented in Algorithm 1.
3.3 Discussion
Usually, there are three convergence measures for the traffic assignment BarGera (2002); Boyce et al. (2004), which are briefly introduced as follows:

According to the error of traffic flows or travel time calculated in two adjacent times to decide whether the iteration stops Leblanc et al. (1975). The computing will stop only when the flows become stable in two adjacent iterations. The measure is simple and usually effective, which can get the flows satisfying the UE principle. Naturally the convergence principle can be expressed as follows:
(23) 
According to the error between the travel time pathbased and linkbased network, the relative gap (RGAP) and normalized gap (or excess average cost) are taken into consideration to measure the convergence Florian et al. (2009). The measure of RGAP can be expressed as:
(24) 
The error between the maximum path travel time and the minimum path travel time between the OD pair is also an evaluation index of convergence BarGera (2002):
(25) by giving a suitable value to , the UE principle can be archived.
Usually, Principle 1 is used in the link based method, such as the FW method. Principle 2 is usually used in the pathbased method and Principle 3 is generally used in the originbased algorithm. Note that the modified Physaruminspired model for UE problem is actually basing on the link travel time. During iterations, the shortest travel paths and the maximum path travel time aren’t available. In order to reduce computing time, we choose the Principle 1 as the convergence measure in this article, which can be expressed as:
(26) 
In the proposed Physaruminspired algorithm, solving a linear system of equations is necessary for each origin based subnetwork. Note that many investigations about paralleled Physarum model have been achieved Adamatzky and Jones (2008), it’s clear that the paralleled computing can be implemented for each origin based subnetwork. Besides, the linear system of equations for each origin based subnetwork is very special and can be formulated as Laplacian system, which is solvable in Koutis et al. (2010). For the sake of simplicity, paralleled computing isn’t implemented in the following. And the linear system of equations for each origin based subnetwork is solved in a general method with .
4 Numerical Examples
In this section, numberical examples are illustrated to demonstrate the rationality and convergence properties of the modified method. And the effect of the stopping criterion is further discussed.
Computational examples reported in this article are using Matlab on Intel(R) Core(TM) i55200U processor (2.2Ghz) with 8.00 GB of RAM under Windows Eight.
4.1 Example 1
Here, we use the simple network shown in Figure 5 and the OD demands are same as those showed in Sec. 3.1. By utilizing the modified Physaruminspired model, the path flows are shown in Table 2. It’s clear that the modified Physaruminspired model has the ability of distinguishing the flux in different OD pairs.
stopping criterion()  Path(node sequence)  Path flow  Iteration 

0.01  1 2  100  17 
4 3  100 
4.2 Example 2
In this example, we test the modified Physaruminspired model on the Sioux Falls network Leblanc et al. (1975) which is used in many publications for the traffic assignment problem.
OD flows are given in thousands of vehicles per day, with integer values up to 44 Leblanc et al. (1975). OD flows here are the values from the table multiplied by 100. They are therefore 0.1 of the original daily flows, and in that sense might be viewed as approximate hourly flows. The parameters in Leblanc et al. (1975) are given as:
(27) 
where denotes the free flow travel time given here. The original parameter can be expressed as:
(28) 
where and denote the free flow traffic time and the capacity flow, respectively. is set as and assume the traditional BPR value of , so we can get the same travel time equation as Eq.(17). And the free flow travel time() and the capacity flow() are shown in Table 3.
ARC  ARCS  ARCS  ARCS  

2.5900  6  2.3403  4  2.5900  6  0.4958  5  
2.3403  4  1.7111  4  2.3403  4  1.7111  4  
1.7783  2  0.4909  6  1.7783  2  0.4948  4  
1.0000  5  0.4958  5  0.4948  4  0.4899  2  
0.7842  3  2.3403  2  0.4899  2  0.7842  3  
0.5050  10  0.5046  5  1.0000  5  0.5050  10  
1.3916  3  1.3916  3  1.0000  5  1.3512  6  
0.4855  4  0.4994  8  0.4909  6  1.0000  5  
0.4909  6  0.4877  4  2.3403  4  0.4909  6  
2.5900  3  2.5900  3  0.5091  4  0.4877  4  
0.5128  5  0.4925  4  1.3512  6  0.5128  5  
1.4565  3  0.9599  3  0.5046  5  0.4855  4  
0.5230  2  1.9680  3  0.4994  8  0.5230  2  
0.4824  2  2.3403  2  1.9680  3  2.3403  4  
1.4565  3  0.4824  2  0.5003  4  2.3403  4  
0.5003  4  0.5060  6  0.5076  5  0.5060  6  
0.5230  2  0.4885  3  0.9599  3  0.5076  5  
0.5230  2  0.5000  4  0.4925  4  0.5000  4  
0.5079  2  0.5091  4  0.4885  3  0.5079  2 
There are 76 arcs, 24 nodes, 552 conservations of flow constraints and 1824 nonnegativity constrains in the network. Here, we set as the stopping criterion of the modified Physaruminspired algorithm for the Sioux Falls network. And traffic flows of each link calculated by the proposed algorithm and FW algorithm are shown in Table 4 and Figure 6, respectively. Compared with the FW algorithm, the proposed Physaruminspired algorithm obtains the same traffic flows.
ARC  ARCS  ARCS  ARCS  

4.4945  8.1189  4.5189  5.9674  
8.0945  14.0068  10.0226  14.0307  
18.0068  5.2000  18.0307  8.7983  
15.7812  5.9919  8.8065  12.4928  
12.1012  15.7966  12.5254  12.0405  
6.8824  8.3886  15.7969  6.8363  
21.7448  21.8145  17.7266  23.1267  
11.0469  8.1000  5.3000  17.6041  
8.3654  9.7764  9.9742  8.4052  
12.2881  12.3794  11.1209  9.8142  
9.0363  8.4002  23.1929  9.0798  
19.0836  18.4094  8.4066  11.0728  
11.6939  15.2805  8.1000  11.6829  
9.9528  15.8573  15.3354  18.9793  
19.1171  9.9417  8.6874  18.9950  
8.7098  6.3023  7.0000  6.2404  
8.6188  10.3095  18.3857  7.0000  
8.6069  9.6618  8.3945  9.6261  
7.9028  11.1122  10.2595  7.8613 
Let and denote the summation of the error and the max relative error between each equilibrium link flow and the assignment link flow at iteration, namely:
(29) 
where denotes the equilibrium flow along link , denotes the error between equilibrium flow and the assignment flow at the iteration along link . The change of and during iterations are indicated in Figure 7. The summation of the error and the maximum relative error are monotone decreasing during iterations. The maximum relative error of each link flow is no more than at the iteration. After 100 iterations, the maximum relative error of each link flow is no more than and the summation of error between each equilibrium link flow and the assignment link flow equals 54.2587.
5 CONClUSIONS
To address UE traffic assignment problem, a modified Physaruminspired model for UE traffic assignment is proposed in this paper. Considering the foraging behavior of Physarum, the equilibrium flows can be obtained when the Physarum can’t find a shorter travel time between each OD pair. The modified algorithm is more efficient in real traffic networks with twoway traffic characteristics and multiple OD pairs . By decomposing flows according to origin node, the flows from different OD pairs can be distinguished. Numerical examples are illustrated to show the rationality and convergence properties of the proposed algorithm.
In the future, one of our works is to use paralleled computing and the optimal model for the linear system of equations Koutis et al. (2010) to reduce computing time. Besides, the theoretical analysis of convergence is also our research topic.
Acknowledgment
The work is partially supported by National Natural Science Foundation of China (Grant No. 61671384), Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2016JM6018), the Fund of SAST (Program No. SAST2016083), the Seed Foundation of Innovation and Creation for Graduate Students in Northwestern Polytechnical University (Program No. Z2016122).
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