# A Multimetric Ant Colony Optimization Algorithm for Dynamic Path Planning in Vehicular Networks

Authors: Zhen Wang; Jianqing Li; Manlin Fang; Yang Li
Journal: International Journal of Distributed Sensor Networks
 (2015)

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## Abstract

With the rapid growth in the number of vehicles, energy consumption and environmental pollution in urban transportation have become a worldwide problem. Efforts to reduce urban congestion and provide green intelligent transport become a hot field of research. In this paper, a multimetric ant colony optimization algorithm is presented to achieve real-time dynamic path planning in complicated urban transportation. Firstly, four attributes are extracted from real urban traffic environment as the pheromone values of ant colony optimization algorithm, which could achieve real-time path planning. Then Technique for Order Preference by Similarity to Ideal Solution methods is adopted in forks to select the optimal road. Finally, a vehicular simulation network is set up and many experiments were taken. The results show that the proposed method can achieve the real-time planning path more accurately and quickly in vehicular networks with traffic congestion. At the same time it could effectively avoid local optimum compared with the traditional algorithms.


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## Body

## 1. Introduction

With the rapid development of green intelligent transportation, many intelligent transportation path planning algorithms were proposed, in order to achieve the optimal planning path with the least cost from a source to a destination within a reasonable time. These algorithms focused on the combination of swarm intelligent algorithm, such as artificial bee colony algorithm [1], genetic algorithm [2], and ant colony algorithm [3], to reduce energy consumption and environmental pollution in urban transportation with traffic congestion. Ant colony optimization algorithm (ACO) [4] is an abstract evolution based on the observation of ant colonies searching for food. Considering the similarity between a vehicle and an ant in searching for a path, ant colony optimization algorithm is widely used in the research and application of intelligent transportation. Many scholars have proposed different optimization models of the ant colony optimization algorithm, based on different research objects and in different application fields. Narasimha and kumar proposed an improved ant colony optimization algorithm based on solving the min-max Single Depot Vehicle Routing Problem [5] and min-max Multidepot Vehicle Routing Problem [6] but did not mention real-time path planning. Li et al. [7] studied the Travelling Salesman Problem (TSP) with the ant colony optimization algorithm and proposed a new multipath routing algorithm based on improved ant colony algorithm. In the paper the ACO was improved in three aspects: add the utilization ratio of router’s buffer queue into the criterion of selection; update the global pheromone with the utilization ratio of link; select multiple paths to transfer data. This algorithm can achieve network loading balance and reduce the likelihood of congestion, but unfortunately it did not consider real-time traffic and routing planning. Zeng et al. [8] proposed an improved ant colony optimization algorithm by dynamically adjusting the number of ants to solve the Chinese Traveling Salesmen Problem. The algorithm only considered how to get the global best result easily but failed to consider multifactors of road and real-time navigation. Moghaddam et al. [9] proposed an advanced particle swarm algorithm to solve uncertain vehicle routing problems in which the customers’ demands are supposed to be uncertain with unknown distributions. Lee et al. [10] investigated the vehicle routing problem with deadlines, to satisfy the requirements of a given number of customers with minimum travel distances, while respecting both uncertain customer demands and travel times. At the same time, Philip Chen et al. [11] used the algorithm of Multiple Attribute Decision Making and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method, to calculate the best path with the assumption that the road and vehicles are equipped with Internet of Things sensor device, to obtain instant traffic information. But the algorithm is similar to the Dijkstra algorithm [12] to a certain degree; it can only produce partial optimal results.The above-mentioned papers have explored intelligent path planning problems, but there are still some aspects that need to be tackled, particularly in view of the existing intelligent traffic navigation.The existing path planning algorithms mainly focus on the shortest path. Some of them can plan a better path in urban areas with the comprehensive factors, such as path length, driving speed, and road grade, and avoid the heavy traffic path in the beginning of path planning. But for urban traffic problems, especially during rush hours, it is obvious that the shortest path is not always the fastest one, as the traffic situation is a dynamic changing process. Once the result is produced by the existing algorithms, they will not be able to be changed. This means that they would fail to dynamically adjust path planning according to the real-time traffic situation.Based on the wireless sensor network in the application of intelligent transportation [13–15], A Multimetric Ant Colony Optimization (MACO) algorithm is presented to deal with intelligent traffic real-time path planning problems. The best path is calculated automatically according to dynamic real-time traffic using this algorithm, which can be the shortest distance, the shortest time, the best road condition, or the best path of comprehensive situations according to customers’ requirements. The MACO contributes to not only planning a best path with considering multiple factors of the road, but also adjusting the path planning dynamically with the real-time traffic situation. And, furthermore, the MACO can contribute to the best average velocity with the least energy consumption in green intelligent transportation. The rest of this paper is organized as follows. The system model of an intelligent transportation network is presented in Section 2. The path planning algorithm based on the ant colony optimization algorithm is discussed in Section 3. Section 4 shows the simulation results. Finally, the conclusions are given in Section 5.


## 2. System Model

### 2.1. Intelligent Transportation Network Settings

A vehicular network is set up for intelligent transportation [15], which includes two subnetworks. The first one is the basic road network, which is deployed in the information node at the intersection and relay nodes along the roads. The second is the network of vehicle-mounted sensors. Figure 1 is a schematic view of transport at an intersection, in which the solid line arrows indicate the direction of traffic and the dotted line arrows indicate the direction in which data is transferred. The main functions of these nodes are described as follows.Figure 1
The model of road traffic network.(A) Information Nodes (as Solid Circles in Figure1). They are deployed at the intersections, to monitor and receive data (including traffic flow and average velocity) from the vehicles moving toward them in all directions, and transmit the comprehensive traffic data to every direction.(B) Relay Nodes (as Solid Squares in Figure1). They are deployed along both sides of the road and send the data received from the information node in the opposite direction of the traffic to subsequent vehicles to plan paths.(C) Vehicle-Mounted Sensor Nodes. They are installed in vehicles and can transfer data in two ways, that is, by gathering data from relay nodes or the information nodes along the road ahead. The optimal route can be identified through calculation; meanwhile, data of the vehicle itself, such as average velocity, average time, and path length, and other information will be sent to the node, allowing for the comprehensive integration of the information.The three kinds of nodes form a wireless Internet of Things for the transportation system. The vehicle-mounted nodes receive data on the traffic ahead from the information nodes and relay nodes to plan a path. Meanwhile, the vehicle can also send its own driving data to relay nodes and information node, for the traffic network to process data. The information nodes can simultaneously receive data from vehicles in all directions adjacent to the intersection and then send back processed comprehensive traffic data in the opposite direction and through relay nodes and spread such data to a further distance. Since the traffic conditions are constantly changing, it is necessary to put a timestamp on all the information transferred. When the data is received by the farther nodes overtime, the nodes will identify such data as non-real-time and invalid data and will simply discard the data.


### 2.2. Parameters

Assume there areK intersection nodes in total in the vehicle network. The road segment is either a one-way or two-way street. Based on the driving directions of vehicles, the road network can be understood as a directed graph G. For a vehicle on any segment, it must be driving away from an intersection and at the same time towards another intersection. For the road segment, the direction in which vehicles drive away from an intersection is defined as outdegree, and the direction in which the vehicles drive toward the intersection is defined as indegree. So the vehicle navigation problem can be the conversion of path planning problem from A to B on the directed graph.Suppose there aren outdegrees at an intersection, that is, n road segments leaving the intersection. For each outdegree, it is defined that length is L, width W, road grade C (based on speed limits), and average driving speed V. For the intersection node, comprehensive evaluation is applied to each intersection outdegree.DefinitionRoad Length
L
∗. Suppose, in a directed graph G abstracted from an urban road network, there are n outdegrees, with lengths of L1,L2,L3,…,Ln, respectively. L- is the road segment with the shortest length (outdegree length). The outdegree lengths are normalized as follows:(1)Li∗=L-Ln.Road Width
W. The width of the outdegree is based on traffic lane. Supposing only one lane, then Wi=1 and the number of i indicates the ith outdegree of the intersection.Road Grade
C
∗. The road grade is calculated according to the speed limits. To calculate simply, we suppose the maximum speed limit of the city road is 120 km/h; the road grade is normalized as follows:(2)Ci∗=Ci120.Average Velocity
V
∗. Suppose m vehicles pass a road segment in a particular period; the average velocity of each vehicle is (3)Vi=LiTend-Tstart,Vk∗=∑1iVi120∗n;i=1,2,…,n,where Vi denotes the average velocity of each vehicle, Li the length that the vehicle drives, Tstart the time the vehicle takes to drive in, and Tend the time the vehicle takes to drive out. Thus Vk∗ is the average speed after normalizing all the vehicles at a certain period.


## 2.1. Intelligent Transportation Network Settings

A vehicular network is set up for intelligent transportation [15], which includes two subnetworks. The first one is the basic road network, which is deployed in the information node at the intersection and relay nodes along the roads. The second is the network of vehicle-mounted sensors. Figure 1 is a schematic view of transport at an intersection, in which the solid line arrows indicate the direction of traffic and the dotted line arrows indicate the direction in which data is transferred. The main functions of these nodes are described as follows.Figure 1
The model of road traffic network.(A) Information Nodes (as Solid Circles in Figure1). They are deployed at the intersections, to monitor and receive data (including traffic flow and average velocity) from the vehicles moving toward them in all directions, and transmit the comprehensive traffic data to every direction.(B) Relay Nodes (as Solid Squares in Figure1). They are deployed along both sides of the road and send the data received from the information node in the opposite direction of the traffic to subsequent vehicles to plan paths.(C) Vehicle-Mounted Sensor Nodes. They are installed in vehicles and can transfer data in two ways, that is, by gathering data from relay nodes or the information nodes along the road ahead. The optimal route can be identified through calculation; meanwhile, data of the vehicle itself, such as average velocity, average time, and path length, and other information will be sent to the node, allowing for the comprehensive integration of the information.The three kinds of nodes form a wireless Internet of Things for the transportation system. The vehicle-mounted nodes receive data on the traffic ahead from the information nodes and relay nodes to plan a path. Meanwhile, the vehicle can also send its own driving data to relay nodes and information node, for the traffic network to process data. The information nodes can simultaneously receive data from vehicles in all directions adjacent to the intersection and then send back processed comprehensive traffic data in the opposite direction and through relay nodes and spread such data to a further distance. Since the traffic conditions are constantly changing, it is necessary to put a timestamp on all the information transferred. When the data is received by the farther nodes overtime, the nodes will identify such data as non-real-time and invalid data and will simply discard the data.


## 2.2. Parameters

Assume there areK intersection nodes in total in the vehicle network. The road segment is either a one-way or two-way street. Based on the driving directions of vehicles, the road network can be understood as a directed graph G. For a vehicle on any segment, it must be driving away from an intersection and at the same time towards another intersection. For the road segment, the direction in which vehicles drive away from an intersection is defined as outdegree, and the direction in which the vehicles drive toward the intersection is defined as indegree. So the vehicle navigation problem can be the conversion of path planning problem from A to B on the directed graph.Suppose there aren outdegrees at an intersection, that is, n road segments leaving the intersection. For each outdegree, it is defined that length is L, width W, road grade C (based on speed limits), and average driving speed V. For the intersection node, comprehensive evaluation is applied to each intersection outdegree.DefinitionRoad Length
L
∗. Suppose, in a directed graph G abstracted from an urban road network, there are n outdegrees, with lengths of L1,L2,L3,…,Ln, respectively. L- is the road segment with the shortest length (outdegree length). The outdegree lengths are normalized as follows:(1)Li∗=L-Ln.Road Width
W. The width of the outdegree is based on traffic lane. Supposing only one lane, then Wi=1 and the number of i indicates the ith outdegree of the intersection.Road Grade
C
∗. The road grade is calculated according to the speed limits. To calculate simply, we suppose the maximum speed limit of the city road is 120 km/h; the road grade is normalized as follows:(2)Ci∗=Ci120.Average Velocity
V
∗. Suppose m vehicles pass a road segment in a particular period; the average velocity of each vehicle is (3)Vi=LiTend-Tstart,Vk∗=∑1iVi120∗n;i=1,2,…,n,where Vi denotes the average velocity of each vehicle, Li the length that the vehicle drives, Tstart the time the vehicle takes to drive in, and Tend the time the vehicle takes to drive out. Thus Vk∗ is the average speed after normalizing all the vehicles at a certain period.


## 3. Path Planning Algorithm

### 3.1. The Ant Colony Optimization Algorithm

Based on the path planning problem in intelligent transportation and the similarity of ant colony foraging and pathfinding, we can use the ant colony optimization algorithm to implement the path planning in intelligent transportation [13]. As in the whole system, different vehicles have different destinations; however, it is assumed that the vehicles share the same route as the same group of ants in the same road segment. When vehicles pass through a road segment, they transmit integrated traffic data to the information node, which are similar to the pheromones of the ant colony optimization algorithm. When subsequent vehicles plan a path, they can use these data as an important reference.The parameters of ant colony optimization algorithm are defined as shown in the following.The Description of Parameters
m
: the number of ants (vehicles),
d
i
j: the distance between element (intersection) i and element (intersection) j,
τ
i
j
t: at t moment, the residual pheromone on the path between element (intersection) i and element (intersection) j, and at initial time, all the residual pheromones of paths which are precomputed,
p
i
j
k: the probability that ant k chooses element j from element i,
t
a
b
u
k: recording the element that the kth ant has gone through,
a
l
l
o
w
e
d
k
=
0,1
,
…
,
n
-
1
-
t
a
b
u
k: recording the allowed element that ant may choose next time,
η
i
j: the expectations that ants go through from element i to element j,
α: determining the relative influence of the pheromone trail,
β: determining the relative influence,
ρ: Pheromone evaporation rate,
Δ
τ
i
j
k
t: the amount of pheromone ant k deposits on the path form element i to element j visited at t time,
Q: the intensity coefficients of pheromone which are increased,
L
k: the length of the tour built by the kth ant.Suppose there arem ants (vehicles) in the system, every ant has the following characters:(A)
To select the next intersection according to the probability function with the value of the pheromones between intersections (supposeτijt as pheromone on edge ei,j at that moment t),(B)
To ensure walking along the legal path, the ants being forbidden to access roads which they have passed before except when there is no other route to the destination which is controlled by the tabu table (supposetabuk represents the kth ant’s tabu table and tabuks represents the sth element in tabu table),(C)
To update pheromones on every edge which it has walked along when the journey is completed.At beginning, the initial value of pheromone on every path is unequal. Supposeτij0=C0 (C0 as the precomputed constant). pijkt presents ant k’s probability to be transferred from i to j at t time:(4)pijk=τijαt·ηijβt∑s∈allowedτijα·τisβtifj∈allowedk0otherwise,where allowedk=0,1,…,n-1-tabuk denotes intersections permitted as the next step by ant k, tabukk=1,2,…,m records the intersections ant k walks through, and the set of tabuk is adjusted dynamically with evolution processing. The heuristic function of ηij denotes the visibility of edgei,j, calculated with some heuristic algorithms. Normally let ηij=1/dij, where dij denotes the distance between the nodes i and j. For ant k, the smaller the dij, the bigger the ηij and the greater the pijk. Obviously, the heuristic function denotes expectations of ants access node i to node j. α denotes the importance of trajectory and β denotes the importance of visibility. After n moments, the ant finishes one travel and the pheromones have to be adjusted on each path as follows:(5)τijt+n=1-ρ·τijt+Δτij,Δτij=∑k=1mΔτijk,where ρρ⊂0,1 is the pheromone decay factor; Δτijk describes pheromones of ant k left on the road segment eij of this travel; Δτij describes pheromones of all ants left on the road eij of this travel:(6)Δτijk=QLkifkth  ants  accessei,jin  this  travel0otherwise.


### 3.2. The Pheromone Computation of MACO

The key part of MACO is to determine parameterpijk, namely, the probability of ant k transfer from node i to node j at time t, and firstly we determine the initial value of pheromone τij0.Every intersection has some attribute values and every attribute needs a different weight. This shows the important grade, to calculate the pheromones in ant colony optimization algorithm. So set a weight value for each attribute of each outdegree of intersection asω1,ω2,…,ωn, and calculate the weight with the maximizing deviation method [16] and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method [17]. The maximum deviation method was presented in [18] to solve the problem of Multiple Attribute Decision Making (MADM). TOPSIS method is often used for solving MADM problems [19, 20]. MADM problems can have several solutions [21], in which the most ideal solution is named the positive ideal solution and the worst solution named the negative ideal solution. In the TOPSIS method, the solution which is closest to the positive ideal solution and farthest from the negative ideal solution is the optimal solution [22].


#### 3.2.1. Normalized Road Decision Matrix

Suppose each intersection containsm outdegrees; each of outdegree contains n attribute value in the road network; then the matrix below is formed:(7)A=a1,a2,a3,…,am=a11a12⋯a1na21a21⋯a2n⋮⋮⋯⋮am1am1⋯amn,where a1i,a2i,…,ami represent different outdegrees section of the same intersection; ai1,ai2,…,ain represent different parameters of the same section.


#### 3.2.2. Calculating the Weight Vectors

Letω′=ω1′,ω2′,…,ωn′ be the initialized weight value of attribute vector. It needs to calculate weight vector ω′ maximizing all deviation values for all the vectors. Formulate a nonlinear programming model:(8)maxFω′=∑i=1m∑j=1n∑l=1nωj′drij,rlj,Subject  to∑j=1mωj′2=1,0≤ωj′≤1,where drij,rkj is the distance between different varieties which belong to the same attribute. Set that (9)drij,rkj=rij-rlj2.To solve formula (9), let(10)Hω′,ξ=∑j=1m∑i=1n∑l=1nωj′drij,rlj-12ξ∑j=1mωj′-1.The partial derivatives of formula (10) are computed as(11)∂Hω′∂ωj′=∑i=1n∑l=1ndrij,ril-ξωj′=0,1≤j≤m,∂Hω′∂ξ=-12∑j=1mξj′2-1=0,1≤j≤m.From formula (11), get(12)ωj′=∑i=1n∑l=1ndrij,rlj∑j=1m∑i=1n∑l=1ndrij,rlj2,1≤j≤m.Normalizing the final weight fromωj′ to ωj, get(13)ωj=ωj′∑j=1mωj′=∑i=1n∑l=1ndrij,rlj∑j=1m∑i=1n∑l=1ndrij,rlj,1≤j≤m.So we have the weights of different attributes, and then we can obtain the data of every intersection.


#### 3.2.3. Calculating the Positive and Negative Ideal Solutions

According to the weight value determined from (13), we can formulate the normalized decision matrix and obtain the positive and the negative ideal solutions as follows:(14)χij=ωjrij,χ∗=χ1∗,χ2∗,…,χn∗,χ-=χ1-,χ2-,…,χn-,subject toχ∗=1, χ-=0.


#### 3.2.4. Calculating the Pheromone Values of MACO

Euler distance can be achieved withm degrees on each intersection node as follows:(15)pi∗=∑j=1mdχij,χj∗2,1≤i≤n,pi-=∑j=1mdχij,χj-2,1≤i≤n.According to the definition of  TOPSIS method, the nearer it is to the positive ideal solution the farther it is from the negative one, the better the solution will be obtained. The pheromones of each outdegree can be achieved as follows:(16)pi=pi∗pi-+pi∗,1≤i≤n.Combining the experiment with the above algorithm, several other parameters in the ant colony optimization algorithm are determined as follows:(17)Δτij0=pi,Δτijk=QLijif  thekth  ant  passed  thiseijin  this  travel0else,where Lij is the length of eij.


### 3.3. The Algorithm Complexity of MACO

The algorithm complexity of MACO is similar to ACO according to MACO based on ACO. We assume thatm ants have to traverse n elements (intersections in this paper) after N circulations to get a solution of MACO. The effects of low power can be neglected if n is lager enough when calculating the time complexity, and the time complexity of MACO is Tn=ON·n2·m. And the real-time complexity of MACO is lower than ACO because of distributed computing adopted in MACO, which means the pheromones are computed in information nodes and the MACO run in every vehicle to get a very path for the vehicle itself. And next step we will study the algorithm complexity furthermore.


## 3.1. The Ant Colony Optimization Algorithm

Based on the path planning problem in intelligent transportation and the similarity of ant colony foraging and pathfinding, we can use the ant colony optimization algorithm to implement the path planning in intelligent transportation [13]. As in the whole system, different vehicles have different destinations; however, it is assumed that the vehicles share the same route as the same group of ants in the same road segment. When vehicles pass through a road segment, they transmit integrated traffic data to the information node, which are similar to the pheromones of the ant colony optimization algorithm. When subsequent vehicles plan a path, they can use these data as an important reference.The parameters of ant colony optimization algorithm are defined as shown in the following.The Description of Parameters
m
: the number of ants (vehicles),
d
i
j: the distance between element (intersection) i and element (intersection) j,
τ
i
j
t: at t moment, the residual pheromone on the path between element (intersection) i and element (intersection) j, and at initial time, all the residual pheromones of paths which are precomputed,
p
i
j
k: the probability that ant k chooses element j from element i,
t
a
b
u
k: recording the element that the kth ant has gone through,
a
l
l
o
w
e
d
k
=
0,1
,
…
,
n
-
1
-
t
a
b
u
k: recording the allowed element that ant may choose next time,
η
i
j: the expectations that ants go through from element i to element j,
α: determining the relative influence of the pheromone trail,
β: determining the relative influence,
ρ: Pheromone evaporation rate,
Δ
τ
i
j
k
t: the amount of pheromone ant k deposits on the path form element i to element j visited at t time,
Q: the intensity coefficients of pheromone which are increased,
L
k: the length of the tour built by the kth ant.Suppose there arem ants (vehicles) in the system, every ant has the following characters:(A)
To select the next intersection according to the probability function with the value of the pheromones between intersections (supposeτijt as pheromone on edge ei,j at that moment t),(B)
To ensure walking along the legal path, the ants being forbidden to access roads which they have passed before except when there is no other route to the destination which is controlled by the tabu table (supposetabuk represents the kth ant’s tabu table and tabuks represents the sth element in tabu table),(C)
To update pheromones on every edge which it has walked along when the journey is completed.At beginning, the initial value of pheromone on every path is unequal. Supposeτij0=C0 (C0 as the precomputed constant). pijkt presents ant k’s probability to be transferred from i to j at t time:(4)pijk=τijαt·ηijβt∑s∈allowedτijα·τisβtifj∈allowedk0otherwise,where allowedk=0,1,…,n-1-tabuk denotes intersections permitted as the next step by ant k, tabukk=1,2,…,m records the intersections ant k walks through, and the set of tabuk is adjusted dynamically with evolution processing. The heuristic function of ηij denotes the visibility of edgei,j, calculated with some heuristic algorithms. Normally let ηij=1/dij, where dij denotes the distance between the nodes i and j. For ant k, the smaller the dij, the bigger the ηij and the greater the pijk. Obviously, the heuristic function denotes expectations of ants access node i to node j. α denotes the importance of trajectory and β denotes the importance of visibility. After n moments, the ant finishes one travel and the pheromones have to be adjusted on each path as follows:(5)τijt+n=1-ρ·τijt+Δτij,Δτij=∑k=1mΔτijk,where ρρ⊂0,1 is the pheromone decay factor; Δτijk describes pheromones of ant k left on the road segment eij of this travel; Δτij describes pheromones of all ants left on the road eij of this travel:(6)Δτijk=QLkifkth  ants  accessei,jin  this  travel0otherwise.


## 3.2. The Pheromone Computation of MACO

The key part of MACO is to determine parameterpijk, namely, the probability of ant k transfer from node i to node j at time t, and firstly we determine the initial value of pheromone τij0.Every intersection has some attribute values and every attribute needs a different weight. This shows the important grade, to calculate the pheromones in ant colony optimization algorithm. So set a weight value for each attribute of each outdegree of intersection asω1,ω2,…,ωn, and calculate the weight with the maximizing deviation method [16] and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method [17]. The maximum deviation method was presented in [18] to solve the problem of Multiple Attribute Decision Making (MADM). TOPSIS method is often used for solving MADM problems [19, 20]. MADM problems can have several solutions [21], in which the most ideal solution is named the positive ideal solution and the worst solution named the negative ideal solution. In the TOPSIS method, the solution which is closest to the positive ideal solution and farthest from the negative ideal solution is the optimal solution [22].


### 3.2.1. Normalized Road Decision Matrix

Suppose each intersection containsm outdegrees; each of outdegree contains n attribute value in the road network; then the matrix below is formed:(7)A=a1,a2,a3,…,am=a11a12⋯a1na21a21⋯a2n⋮⋮⋯⋮am1am1⋯amn,where a1i,a2i,…,ami represent different outdegrees section of the same intersection; ai1,ai2,…,ain represent different parameters of the same section.


### 3.2.2. Calculating the Weight Vectors

Letω′=ω1′,ω2′,…,ωn′ be the initialized weight value of attribute vector. It needs to calculate weight vector ω′ maximizing all deviation values for all the vectors. Formulate a nonlinear programming model:(8)maxFω′=∑i=1m∑j=1n∑l=1nωj′drij,rlj,Subject  to∑j=1mωj′2=1,0≤ωj′≤1,where drij,rkj is the distance between different varieties which belong to the same attribute. Set that (9)drij,rkj=rij-rlj2.To solve formula (9), let(10)Hω′,ξ=∑j=1m∑i=1n∑l=1nωj′drij,rlj-12ξ∑j=1mωj′-1.The partial derivatives of formula (10) are computed as(11)∂Hω′∂ωj′=∑i=1n∑l=1ndrij,ril-ξωj′=0,1≤j≤m,∂Hω′∂ξ=-12∑j=1mξj′2-1=0,1≤j≤m.From formula (11), get(12)ωj′=∑i=1n∑l=1ndrij,rlj∑j=1m∑i=1n∑l=1ndrij,rlj2,1≤j≤m.Normalizing the final weight fromωj′ to ωj, get(13)ωj=ωj′∑j=1mωj′=∑i=1n∑l=1ndrij,rlj∑j=1m∑i=1n∑l=1ndrij,rlj,1≤j≤m.So we have the weights of different attributes, and then we can obtain the data of every intersection.


### 3.2.3. Calculating the Positive and Negative Ideal Solutions

According to the weight value determined from (13), we can formulate the normalized decision matrix and obtain the positive and the negative ideal solutions as follows:(14)χij=ωjrij,χ∗=χ1∗,χ2∗,…,χn∗,χ-=χ1-,χ2-,…,χn-,subject toχ∗=1, χ-=0.


### 3.2.4. Calculating the Pheromone Values of MACO

Euler distance can be achieved withm degrees on each intersection node as follows:(15)pi∗=∑j=1mdχij,χj∗2,1≤i≤n,pi-=∑j=1mdχij,χj-2,1≤i≤n.According to the definition of  TOPSIS method, the nearer it is to the positive ideal solution the farther it is from the negative one, the better the solution will be obtained. The pheromones of each outdegree can be achieved as follows:(16)pi=pi∗pi-+pi∗,1≤i≤n.Combining the experiment with the above algorithm, several other parameters in the ant colony optimization algorithm are determined as follows:(17)Δτij0=pi,Δτijk=QLijif  thekth  ant  passed  thiseijin  this  travel0else,where Lij is the length of eij.


## 3.2.1. Normalized Road Decision Matrix

Suppose each intersection containsm outdegrees; each of outdegree contains n attribute value in the road network; then the matrix below is formed:(7)A=a1,a2,a3,…,am=a11a12⋯a1na21a21⋯a2n⋮⋮⋯⋮am1am1⋯amn,where a1i,a2i,…,ami represent different outdegrees section of the same intersection; ai1,ai2,…,ain represent different parameters of the same section.


## 3.2.2. Calculating the Weight Vectors

Letω′=ω1′,ω2′,…,ωn′ be the initialized weight value of attribute vector. It needs to calculate weight vector ω′ maximizing all deviation values for all the vectors. Formulate a nonlinear programming model:(8)maxFω′=∑i=1m∑j=1n∑l=1nωj′drij,rlj,Subject  to∑j=1mωj′2=1,0≤ωj′≤1,where drij,rkj is the distance between different varieties which belong to the same attribute. Set that (9)drij,rkj=rij-rlj2.To solve formula (9), let(10)Hω′,ξ=∑j=1m∑i=1n∑l=1nωj′drij,rlj-12ξ∑j=1mωj′-1.The partial derivatives of formula (10) are computed as(11)∂Hω′∂ωj′=∑i=1n∑l=1ndrij,ril-ξωj′=0,1≤j≤m,∂Hω′∂ξ=-12∑j=1mξj′2-1=0,1≤j≤m.From formula (11), get(12)ωj′=∑i=1n∑l=1ndrij,rlj∑j=1m∑i=1n∑l=1ndrij,rlj2,1≤j≤m.Normalizing the final weight fromωj′ to ωj, get(13)ωj=ωj′∑j=1mωj′=∑i=1n∑l=1ndrij,rlj∑j=1m∑i=1n∑l=1ndrij,rlj,1≤j≤m.So we have the weights of different attributes, and then we can obtain the data of every intersection.


## 3.2.3. Calculating the Positive and Negative Ideal Solutions

According to the weight value determined from (13), we can formulate the normalized decision matrix and obtain the positive and the negative ideal solutions as follows:(14)χij=ωjrij,χ∗=χ1∗,χ2∗,…,χn∗,χ-=χ1-,χ2-,…,χn-,subject toχ∗=1, χ-=0.


## 3.2.4. Calculating the Pheromone Values of MACO

Euler distance can be achieved withm degrees on each intersection node as follows:(15)pi∗=∑j=1mdχij,χj∗2,1≤i≤n,pi-=∑j=1mdχij,χj-2,1≤i≤n.According to the definition of  TOPSIS method, the nearer it is to the positive ideal solution the farther it is from the negative one, the better the solution will be obtained. The pheromones of each outdegree can be achieved as follows:(16)pi=pi∗pi-+pi∗,1≤i≤n.Combining the experiment with the above algorithm, several other parameters in the ant colony optimization algorithm are determined as follows:(17)Δτij0=pi,Δτijk=QLijif  thekth  ant  passed  thiseijin  this  travel0else,where Lij is the length of eij.


## 3.3. The Algorithm Complexity of MACO

The algorithm complexity of MACO is similar to ACO according to MACO based on ACO. We assume thatm ants have to traverse n elements (intersections in this paper) after N circulations to get a solution of MACO. The effects of low power can be neglected if n is lager enough when calculating the time complexity, and the time complexity of MACO is Tn=ON·n2·m. And the real-time complexity of MACO is lower than ACO because of distributed computing adopted in MACO, which means the pheromones are computed in information nodes and the MACO run in every vehicle to get a very path for the vehicle itself. And next step we will study the algorithm complexity furthermore.


## 4. Simulation

We mainly pay attention to several different types in the process of navigation algorithm: time priority, distance priority, road condition priority, and integrated priority. Time priority refers to arriving at the destination with minimum time, distance priority refers to arriving at the destination with the shortest path, road condition priority refers to arriving at the destination with the best intersection line, and integrated priority preference refers to combining different road properties and then arriving at the destination with the least cost.To illustrate conveniently, a road map measured with 8 by 8 squares is adopted to simulate road navigation. In the designed grid map (Figure2), each intersection represents a road traffic intersection. Each node can adjoin the adjacent nodes bidirectionally. To stimulate path planning under different conditions, it is assumed that the path(49-56) and path(2-58) have two-way four-lane widths and other roads are of two-way two-lane width; here (i-j) represents the linear connection from node i to node j. Each of the horizontal distances between the intersections is set at 200 meters; the vertical distance between each intersection is set at 100 meters. At the same time, the average speed of road(57-64)is set at 20 km/h, the average speed of road(49-56) is 30 km/h, the average speed of roads(41-48) and(33-40) is 40 km/h, and the average speed of the other roads is 60 km/h.Figure 2
The navigation path planning under different conditions.In the following, different navigation algorithms of MACO and two famous algorithms (the Dijkstra algorithm and theA∗ algorithm) are adopted to analyze the network. First, the Dijkstra algorithm and MACO are adopted to plan the shortest path and the result show that the planned paths are the same one (57-64), with the length of 1400 meters and time of 252 seconds. If MACO is only concerned with the path length, the result obtained is consistent with the Dijkstra algorithm. When only road width is considered, the planned path by MACO is (57-49-56-64), with the length of 1600 meters and the time of 178 seconds. The path planned by MACO is (57-25-32-64) with the length of 2200 meters and 132 seconds if only the average velocity be cared about, that is the same as the A∗ algorithm planned. When MACO integrates road conditions such as path length, road width, road grade, and average velocity for analysis, the optimal planning path is (57-41-48-64), with the length of 1800 meters and the time of 150 seconds. The simulation results of several different ways of navigation are shown in Table 1.Table 1
Different planning path ON 8× 8 grid chart.
Navigation mode
Path length (m)
Time (s)
Dijkstra navigation algorithm
1400
252
A
∗ algorithm
2200
132
MACO
Distance priority
1400
252
Road condition priority
1600
178
Time priority
2200
132
Integrated priority
1800
150Now we simulate with Zhuhai City map; as shown in Figure3, we choose the area with 60 road intersections as coordinates. For each section between two adjacent intersections, we stimulate the path length, road width, road grade, and average velocity and then simulate MACO in SUMO environment and then carry on the simulation under velocity and time preference.Figure 3
Normal traffic map in Zhuhai City.As shown in Figure3, the Dijkstra algorithm, A∗ algorithm, and MACO apply to plan a path from point A to point C and path from point B to point C separately; the results are shown in Table 2. First of all, the algorithms are adopted and make a comparison from point A to point C. The Dijkstra algorithm chooses a path with distance of 8870 meters and time 539 seconds; MACO chooses a path with distance of 9090 meters and time of 467 seconds, which is the same as the A∗ algorithm. Then the algorithms are adopted to compare from point B to point C, respectively. Path distance of all three algorithms is 10300 meters, with the time of 589 seconds. The results show that the path planned by the Dijkstra algorithm is the shortest path and would not consider velocity and other factors.Table 2
Statistics of three algorithms in normal situation.
Algorithm
Starting point → destination
Distance (meter)
Time (second)
Dijkstra algorithm
A
→
C
8870
539
B
→
C
10300
589
A
∗ algorithm
A
→
C
9090
467
B
→
C
10300
589
MACO
A
→
C
9090
467
B
→
C
10300
589MACO can consider the comprehensive factors, so, from point A to point C, although the path planned by MACO is longer than the path planned by the Dijkstra algorithm, it still saves time since the average velocity is higher, which is the same as theA∗ algorithm planned. For the path whose average path length and velocity are all optimal, MACO is consistent with the result of the Dijkstra algorithm and the A∗ algorithm, such as from point B to point C, passes through the distance of 10300 meters, using identical times of 589 seconds, respectively.Among the above, the left convergence curves of Figure4 are based on time (velocity) priority; the right convergence curves of Figure 4 are based on distance priority. On the left of the Figure 4 there are two convergence curves and we get the below convergence curve of travel time when we try to get a planning path only considering the high velocity or short travel time with MACO, and at same time we get the convergence curve of travel distance accordingly as the above one. And on the right of Figure 4 there are two convergence curves, too. We get the above convergence curve of travel distance when we try to get a planning path only considering the shortest travel distance with MACO, and at the same time we get the convergence curve of travel time accordingly as the below one. We can see that all the curves can be converged normally under two situations.Figure 4
Convergence curves of MACO.
(a)
Time (velocity) priority convergence curves
(b)
Distance priority convergence curvesIn order to further illustrate the advantages of MACO, simulation is carried out with this map and path, but set limited velocity in some sections as shown in the shadow part of Figure5. Assuming that it is facing congestion, the simulation results are shown in Table 3.Table 3
Statistics of three algorithms in congestion situation.
Algorithm
Starting point → destination
Distance (meter)
Time (second)
Dijkstra algorithm
A
→
C
8870
803
B
→
C
10300
872
A
∗ algorithm
A
→
C
10380
591
B
→
C
11290
663
MACO
A
→
C
10380
591
B
→
C
11290
663Figure 5
Traffic map of congestion situation in Zhuhai.From Table3 we can see that if we set the shadow part as a traffic jam, the Dijkstra algorithm would not consider special cases and the path planned is still consistent with the normal circumstances, path length from point A to point C and point B to point C is 8870 meters and 10300 meters, but the time it takes increases to 803 and 872 seconds, respectively. And MACO fully considers the congestion; the paths planned are longer than normal and the path planned by the Dijkstra algorithm increased to 10380 meters and 11290 meters from point A to point C and point B to point C, but the time is less than the Dijkstra obviously, as 591 seconds and 663 seconds, which are consistent with the A∗ algorithm planned.MACO not only considers the integrated traffic information according to the requirement of path planning, but also can make path plans according to users’ requirements. Below we still use Figure5 traffic map and make analysis with only focusing on the path length, road width, average velocity, and comprehensive situation separately, as shown in Table 4.Table 4
Statistics under congestion situation using MACO.
Algorithm
Priority
Distance (meter)
Time (second)
A
→
C
Distance priority (as the Dijkstra algorithm)
8870
803
Road condition priority
11600
655
Time priority (as theA∗ algorithm)
10380
591
Integrated priority
10380
591
B
→
C
Distance priority (as the Dijkstra algorithm)
10300
872
Road condition priority
12510
726
Time priority (as theA∗ algorithm)
11530
647
Integrated priority
11290
663According to Table4, taking congestion into consideration, from point A to point C, the distance priority takes the distance of 8870 meters but the longest time of 803 seconds; the road condition priority only considers the road width which chooses the longest distance of 11600 meters and takes 655 seconds; it is similar between time priority and the integrated priority, the distance of 10380 meters, and takes 591 seconds. From point B to point C, the distance priority has the shortest distance of 10300 meters but takes the longest time of 872 seconds; the road condition priority planned the longest distance path for 12510 meters and takes 726 seconds in total; compared to the comprehensive case, time priority planned a longer distance path of 11530 meters and takes 647 seconds; integrated priority has a distance of 11290 meters and takes 663 seconds. From the above it can be concluded that the comprehensive situation, namely, integrated priority, takes account of path length, road width, road grade, and average velocity, so the distance and the time of the path planned cannot be the best, but it provides the optimal comprehensive solution.The MACO has an important feature that it can plan path in real time with the dynamic traffic situation. Using ring map of Areia Preta in Macau, for example, these further illustrate the simulation results obtained by the algorithm as shown in Figure6.Figure 6
Map of Areia Preta in Macau.
(a)
In the normal traffic situation
(b)
In the heavy traffic situationFigure6 shows two different situations in the urban traffic of Macau. The left of the maps shows normal traffic situation and that means there are no traffic jams. The right one of the maps shows the dynamic traffic situation, meaning that the traffic congestion is dynamically changing. The shadow part of Figure 6 is the traffic jam with an average speed of 20 km/h; other sections are normal traffic roads. Bold line shows two-way four-lane road with the average speed of 60 km/h and other roads indicate two-way two-lane roads with the average speed of 50 km/h. The unidirectional road segments are omitted for simplification. Simulation is carried out with the path planning frompoint A topoint B. Also the Dijkstra algorithm, A∗ algorithm, and MACO are adopted to make contrastive analysis, as shown in Table 5.Table 5
Statistics of Macau under the real-time traffic situation.
Traffic situation
Priority method
Distance (meter)
Time (second)
Normal traffic situation
Dijkstra algorithm
1290
104
A
∗ algorithm
1406
94
MACO algorithm
1406
94
Dynamic traffic situation
Dijkstra algorithm
1290
142
A
∗ algorithm
1406
216
MACO algorithm
1464
120The obvious differences are shown among three algorithms from the planned path. In the normal traffic situation, the Dijkstra algorithm plans a path whose travel distance is 1290 meters and the travel time is 104 seconds; it has the shortest path but the longest travel time. MACO chooses a path as good as theA∗ algorithm whose travel distance is 1406 meters and the travel time is 94 seconds. The path which MACO chose is longer than that of the Dijkstra algorithm’s but the time is shorter.In order to describe the algorithms in detail, suppose there are three cars travelling along the different paths planned above. When they travel to point C, the congestion happens in front of the path. Because they cannot plan dynamically the path according to the traffic situation, the cars adopt the planned path by the Dijkstra algorithm and theA∗ algorithm toward destination with no change of the route. They take 1290-meter travel distance with 142 seconds and 1406-meter travel distance with 216 seconds finally, respectively. At the same time, the car which takes the path planned (shown as red dotted line) by MACO changes to a new route (shown as red solid line) which is planned dynamically and has a travel distance of 1464 meters with travel time of 120 seconds. The real paths they traveled are shown in the right map of Figure 6.From Figure7, which shows the distance cost comparison of normal traffic situation and dynamic traffic situation, we can see that in two situations the Dijkstra algorithm and A∗ algorithm have no change of planning path. But for MACO algorithm, the length of the planned path in dynamic traffic situation is longer than that in normal traffic situation. Then from Figure 8, which shows the time cost comparison of three algorithms in normal traffic situation and dynamic traffic situation, we can see that all the time costs of three algorithms have increased, and the MACO algorithm has the minor increase of time cost comparing with other two algorithms. To further compare the three algorithms, we construct comprehensive cost data using time cost to multiply distance cost as shown in Figure 9. From normal traffic situation to dynamic traffic situation, the comprehensive costs of three algorithms are all increased. The cost of A∗ algorithm has increased drastically, the Dijkstra algorithm is the second, and the MACO algorithm stands with least increase. The results show that though it can give the same planned path in normal traffic situation, MACO is better than other algorithms, that is, the Dijkstra algorithm and A∗ algorithm, because MACO can adjust dynamically planning path according to dynamic traffic situation and represent an optimal path with integrating the path length, road width, road grade, and average velocity with the least cost.Figure 7
The distance cost comparison of two situations.Figure 8
The time cost comparison of two situations.Figure 9
The comprehensive cost comparison of two situations.


## 5. Conclusion

This paper has developed a novel multiple-metric ant colony optimal algorithm for vehicle path planning, in which four kinds of traffic attributes are considered, and some phases of the algorithm are discussed. A new general metric has been introduced as the pheromone value of ant colony optimization algorithm. Based on multiple traffic information and planning requirements, the proposed algorithm selects the most effective and suitable planning path. Simulation results demonstrate that the presented algorithm could easily achieve the suitable path according to different requirements, including the shortest path planning and the shortest time planning, in the real-time vehicular traffic networks. This algorithm provides a potential solution for energy consumption and environmental pollution in increasingly complex urban traffic environment, which could be used in intelligent transportation system.



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*Source: 271067-2015-10-25.xml*