Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 187 (2017) 425 – 434
10th International Scientific Conference Transbaltica 2017: Transportation Science and Technology
An Improvement in ant Algorithm Method for Optimizing a Transport Route with Regard to Traffic Flow Viktor Danchuka, Olena Bakulichb, Vitaliy Svatkoa,* a
Faculty of Transport and Information Technologies, National Transport University, Kyiv, Ukraine b Faculty of Management, Logistics and Tourism, National Transport University, Kyiv, Ukraine
Abstract The modification of ant algorithm method for optimizing the transportation route with regard to traffic flow in the street network has been developed in this paper. It was also made possible to confirm the results of optimization of partly covered distance for calculating a further route when changing the length of links while ant agents traveling on the links of a two-way graph. Besides, the procedure of ant agents’ traffic in the graph was improved so that ant agents can travel both synchronously and asynchronously. The proposed modification of ant algorithm for optimizing the goods delivery route when changing the speed of traffic flow in specific sections of the street network has been approbated, using the example of Kyiv’s specific street network within traveling salesman problem. We conducted the quantitative and comparative analysis of solving the problem of optimization of the goods delivery route in the street network, applying ant algorithm method and the respective findings of other existing classical methods. The obtained results of the study show the prospects of applying the proposed modification of ant algorithm for solving routing problems, particularly for transport networks which are characterized by high dimensionality and dynamism of functional parameters. © 2017 2017The TheAuthors. Authors. Published by Elsevier Ltd.is an open access article under the CC BY-NC-ND license © Published by Elsevier Ltd. This Peer-review under responsibility of the organizing committee of the 10th International Scientific Conference Transbaltica 2017: (http://creativecommons.org/licenses/by-nc-nd/4.0/). Transportation and Technology. Peer-review underScience responsibility of the organizing committee of the 10th International Scientific Conference Transbaltica 2017 Keywords: transport, methods for transport route optimization, ant algorithm, performance analysis
* Corresponding author. E-mail address:
[email protected]
1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of the 10th International Scientific Conference Transbaltica 2017
doi:10.1016/j.proeng.2017.04.396
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1. Introduction Nomenclature Ji,k ηij τij α, β Tk Lk Q tij lij υij
a list of cities which ant agent k has to visit and the city і is where ant agent k is located visibility is inversely proportional to the distance between the cities the amount of pheromone on the link (ij) at the instant in time t parameters which weigh with pheromone trail a way travelled by the ant k to the point of the time t the length of this way parameter which means the order of optimal way length the value of travel time length section ij average speed of a vehicle in section ij of street network
A great number of research papers are devoted to the problem of improving optimization processes when creating routes. Existing traditional methods for solving the problem of discrete optimization of the processes in logistics systems, which are considered in network representation, as a rule, are not perfect and do not provide unequivocal solutions (eg. [1]). What is more, proposed methods in [1] do not allow us to solve the problems of high dimensionality, and consider the actual state of transport network while creating a route. Knight [2] suggests a new algorithm, which, unlike [1], allows us to perform the procedure of discrete optimization when simultaneously considering all the ways of traffic between graph junctions, taking into account the capacity of each link. This method is based on presenting the graph in the form of an electric network, whose sections have a specific resistance, which characterizes the respective capacity. The proposed method, according to the author [2], also makes it possible to first of all focus on problem areas and structures of a high level instead of wasting a lot of time on finding insignificant solutions, which means it enabled us to effectively use the time of respective calculations. However, a deeper analysis of method [2] indicates that the solution to the problem of discrete optimization is to solve the problem of linear algorithm connected with solving respective system of linear equations of a specific dimensionality. Therefore, finding the ways of solving system equation of the maximum possible dimensionality still remains topical. It should be noted that method [2] in fact is analogous to the method of discrete optimization which is based on ant algorithm [3]. The classical method of ant colony self-organization provides the possibility for finding the optimal way for a static graph. Ant agents, which are located in the junctions of the graph in the initial period, also travel on the links of the graph simultaneously. It eventually enables us to substantially reduce the time of calculations, taking this method’s features into consideration. In addition, various characteristics, which are attributed to ant agents, allow us to solve a wide range of discrete optimization problems, taking account of a large number of investigated system characteristics. Further numerous researches done through ant algorithm method showed the perspective of its application for solving the problems of discrete optimization of high dimensionality (see eg [4, 5]). Still, as a rule, existing methods, models of optimizing transportation route and corresponding software and hardware complexes of their realization mostly solve the optimization problems for stationary states of transport traffic. Meanwhile, the development and improvement of methods and models of controlling transport optimization processes, considering actual dynamics of traffic flows in the street network, are extremely topical in the modern conditions of transport functioning. First of all, it concerns large cities and functioning of street networks characterized by traffic capacity dynamism, substantial changes in traffic speed, congestions and etc. The problem of ant algorithm application for solving dynamic transport problems has drawn researchers’ attention recently. In particular, to employ ant colony algorithm while optimizing complex planning process was suggested in paper [6], with the possibility of taking dynamic emergency situations into account. But there are very few papers on such a topic in the literature nowadays. The purpose of this paper is to improve the existing ant algorithm method for optimizing transport route with regard to traffic flow dynamics in the street network. Within the developed modification of this algorithm, we made
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it possible to record the results of optimization of partly covered distance for calculating a further route while changing the procedure of ant agents’ traffic on the links of the two-way graph. Besides, the procedure of ant agents’ traffic (synchronous and asynchronous) in the graph was improved too. 2. An improvement in ant algorithm for solving the problems of optimizing a transport route 2.1. A Background: Ant algorithm method for optimizing a transport route The ant algorithm is one of the most effective polynomial algorithms for finding the solutions to traveling salesman problem as well as analogous problems of route optimization. The existing classical method of ant colony self-organization provides the possibility of finding the optimal way of a static graph [3, 7−8]. A static graph means a graph in which the position of links and distance between links do not change in the course of time. Thus, this algorithm enables us to find an optimal route for the problems, which describe the stationary state of a transport network with fixed parameters at a certain period of time, namely, static linear characteristics of transport network sections, their capacity and etc. The route optimization in this paper was carried out with the help of ant algorithm within traveling salesman problem. This problem is formulated as a task of the search of minimal route to all nodes which salesperson visits without the repetitions on the specific graph with the number of nodes m [3]. We consider that the place of vehicle’s departure and arrival can be from any node. For instance, the graph nodes can be delivery points from warehouse. In case of a static graph the price of travel between points (the weight of links) is determined only by the distance between them. The modeling of ants’ behavior is connected with pheromone which ants leave on the paths – graph links [3]. The likelihood of including the link in the route of a separate ant is proportional to the amount of pheromone on this link, and the amount of left pheromone is proportional to the route length. The shorter route is the more pheromone is left on the links, and more ants include it in the synthesis of their own routes. Modeling such an approach, which uses only a positive feedback, leads to premature convergence (most ants follow the optimal local route). It can be avoided through modeling a negative feedback in the form of pheromone evaporation. To obtain the stable optimal solution, it is necessary to select the sufficient time of evaporation. Taking into account the characteristics of traveling salesman problem, we are able to describe the local rules of ants’ behavior when choosing a route. According to [3], local rules of ants’ behavior when deciding on a route can be described as follows 1. Ants have their own “memory”. Since they can visit one city only once, each ant has a list of visited cities – a prohibition list. Let Jik be a list of cities which ant agent k has to visit and the city і is where ant agent k is located. 2. Ants have “eyesight” – visibility is a heuristic desire to visit the city j if an ant is located in the city і. Let us consider that visibility is inversely proportional to the distance between the cities
ηij =1 Dij .
(1)
3. Ants are able to distinguish smells. They can feel pheromone trail, which confirms their desire to visit the city j from the city і on the basis of other ants’ experience. We regard τij(t) as the amount of pheromone on the link (ij) at the instant in time t. 4. Based on this, probability-proportional rule can be formulated, which determines the probability of transition of the ant k from the city i to the city j:
[
] [ ]
⎧ τ ij (t ) α ⋅ ηij β , j ∈ J i, k ; ⎪ Pij , k (t ) = α β ⎪ τ ⋅ η ( t ) [ ] [ ] il il ⎨ l ∈ J i .k ⎪ ⎪ P (t ) = 0, j ∉ J , i, k ⎩ ij , k
∑
(2)
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where α, β − parameters which weigh with pheromone trail; when α = 0 algorithm degenerates to greedy algorithm (the nearest city is chosen). While carrying out the algorithm, which is described by conditions 1−4, the rule (2) does not change, but two different ants have different probability of transition as they have a different list of allowed cities. 1. Passing the link (i, j), an ant leaves some amount of pheromone which should be associated with an optimum choice. Let Tk(t) be a way travelled by the ant k to the point of the time t, Lk(t) − the length of this way, and Q – parameter which means the order of optimal way length. In this case the amount of left pheromone can be determined as
⎧ Q ,(i, j )∈Tk (t ); ⎪ Δτ ij ,k (t ) = ⎨ Lk (t ) ⎪ ⎩0,(i, j )∉Tk (t ).
(3)
The environment rules in the first place determine pheromone evaporation. p ∈ [0.1] is the coefficient of evaporation, then the evaporation rule is as follows m
τ ij (t +1) = (1− p ) ⋅ τ ij (t ) + Δτij (t ); Δτij (t ) = ∑ Δτij ,k (t ),
(4)
k =1
where m is the number of ants in a colony. At the beginning of solving, the amount of pheromone is taken equally and it is small. The total number of ants remains constant and equals the number of graph nodes. Each ant starts the route from its node. The choice of the first node for each ant is determined by the rule “go to the nearest”. Each subsequent step of an ant is determined by probabilistic equation [3]. Interpreting the presented algorithm to the real circumstances within salesman problem, it can be said following. The objective function here is the shortest found route. The number of graph nods is determined by the number of goods delivery points performed by a vehicle. The place of vehicle’s departure and arrival can be from the point. Therefore, the number of graph nodes is determined by the necessary number of points to which goods has to be delivered, and the links between these nodes are distances between points. So, the distance matrix of specified dimensionality is formed this way. The procedure of creating the minimal route of a vehicle is described in detail in a large number of papers (see eg [3, 7−8]). 2.2. The modification of ant algorithm method for optimizing the transportation route with regard to traffic flow in the street network In order to solve the problem of optimization, considering dynamics of traffic flow (speed change, traffic congestions, accidents, road repairs and etc.), the street network is presented in the form of two-way dynamic graph with its nodes in which there are goods delivery points (warehouses, supermarkets and etc.), and links correspond to either average speed (Fig. 1b) or the time (Fig. 1c) of the vehicle traveling in the traffic flow between two specific nodes of the graph. Each link of the graph of length is the full section of the road between its two nodes. So, each link j of the graph correspond with total nj of street network sections, each i is characterized by length lij, average speed υij, and the time of travel tij. For convenience’s sake, we assume that the vehicle’s departure and return can occur from any node of the graph, and the optimization is carried out within traveling salesman problem. Unlike static graph, length between objects (Fig. 1a) in dynamic graphs of average speed (Fig. 1b) and the time of travel (Fig. 1c) lengths of links are variables which are determined by the character of traffic flow dynamics in specific sections of the street network. In case of dynamic transport problem, the route optimization was made through minimization of objective function for the time of travel to the nodes of the graph.
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Fig. 1. Two-way oriented graphs: (a) lengths between objects; (b) average speed between objects; (c) time of travel between objects.
It should be noted that the calculation of fuel consumption for vehicles when delivering goods is generally included a large number of parameters. The length of way of delivery plays the key role in the calculations of fuel consumption. However, the deliver time is more important for some groups of goods. Besides, the reduction of work time of vehicles decreases fuel consumption. What is also significant is the real traffic situation (accidents, congestions and etc.), which can substantially impact on the vehicle’s speed, and the delivery time. Because time is variable value, which depends on traffic flow dynamics in the street networks, we determine the objective function. The application of ant algorithm method modification developed in this paper for the route optimization of goods delivery with allowance for transport flow was performed under the following assumptions: • in all the sections of street network, the vehicle travels along two-lane two-way traffic flow; • in each totality of sections nj of street network which corresponds to j graph link, there are always alternative ways of travel; • the change of travel average time tij and the average speed υij mainly depend on traffic flow dynamics, which does not comprise stops and delays caused by traffic lights, traffic and etc. The value of travel time in one i section from the totality of sections nj of the street network which corresponds with j link of the graph (Fig. 1c), tij was determined as follow
tij =
lij
,
(5)
υij
lij length section ij, m; υij average speed of a vehicle in section ij of street network, m/s. Here the speed is variable and it depends on the intensity I and density ρ of traffic. For example, according to Kerner’s theory [10], the phase transition between traffic dynamics appear when density of traffic flow increases, and the rise in vehicles’ interaction in the flow grows. In other words, traffic flow dynamics changes from free to sync, then wide mobile clusters, and in the end to congestion. Such phase transitions come together with significant abrupt decrease in the speed of traffic flow as well as the decrease in average speed of vehicles. Thus, according to (5), time tij is also variable. As an example, Fig. 2 displays the process of optimization of vehicle’s route, when changing speed υij to the specific critical value in section ij of totality nj of street network sections, which corresponds link j of the graph of travel time from k and m. Also, we assume that the alternative variants of travel can be found between these nodes.
a)
Ŭ
Ŭ
b)
l1 j ,ν 1 j l1 j ,ν 1 j
lij ,ν ij
l2 j ,ν 2 j ͘͘͘ ͘͘͘
l2 j ,ν 2 j
lnj ,ν nj ͘͘͘ ͘͘͘
lij ,ν ij ͘͘͘ ͘͘͘
ν 2 j ≤ν k , t2 jk >> t2 j
lnj ,ν nj ͘͘͘ ͘͘͘
ŵ
ŵ
Fig. 2. Scheme of optimization of vehicle in the totality nj of street network sections which correspond with link j of the graph of travel time between nodes k and m: (a) before the reduction of speed in section ij; (b) after the reduction of speed in section ij (υij ≤ υk).
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As can be seen from Fig. 2b, in case there is decrease in speed in a specific road section to the specific critical value, which corresponds with, for example, congestion, accident road repair and etc., this section ij is excluded from consideration and optimization is realized according to a different route. According to the model presented above, the modification of classical ant algorithm was developed in this paper. 1. The possibility of recording the results of optimization of partly covered distance was realized for calculating a further route when changing the length of links while driving. It enables us to recreate the part of the route, which was not traveled, considering the real conditions of traffic of the street network. 2. The procedure of ant colony travel in the graph, in which cyclical travel of ant colony was changed to asynchronous travel of each ant agent at a specific speed, was improved for reducing time consumption on finding an optimal route and increase of accuracy of found solutions. 3. Increasing the effectiveness of applying modified ant algorithm for solving routing optimization problem of control parameters (α, β, p), in (2), (4) is performed with the help of local search method [3]. So, the optimal values of these parameters (α = 1; β = 5; p = 0.67) were determined, which allows us to considerably reduce the time of finding an optimal solution and the accuracy of a found result. All other conditions and stages of the procedure of creating the vehicle’s minimal route coincide with classical ant algorithm, which are given in 2.1 (see also, for example, [3]). Thus, the proposed modification of ant algorithm considers the additional elements of intellectualization, which, for example, are connected with excluding those sections in which the time of vehicle’s travel is excessively long. Besides, there exists the possibility of finding an optimal route in the real conditions of traffic flow (change of speed, congestions and so on). 3. Results and discussion 3.1. The results of applying ant algorithm for solving a routing static problem The test studies of ant algorithm, exhaustive search method, and branch and bound method application were made to verify the accuracy of ant algorithm application for solving the problems of finding an optimal route. The test computing was performed on the workstation with processor Intel® Celeron® DCPU 3.06 ГГц and 2048 MB of RAM. While testing, the dimensionality of distance matrix increased from 3 to 75 nodes, and the obtained results were compared with the best known value. The obtained results of made test studies are given in Table 1. Table 1. Comparative analysis of different methods for solving the routing problem. Number of nodes
The best known result of given length, 103 m
Ant algorithm length, 10 m
divergence
length, 10 m
divergence
length, 103 m
divergence
3
48
48
0.00%
48
0.00%
48
0.00%
5
74
74
0.00%
74
0.00%
74
0.00%
3
Branch and bound method 3
Exhaustive search
7
80
80
0.00%
85
6.25%
80
0.00%
10
82
82
0.00%
99
20.73%
82
0.00%
12
86
86
0.00%
133
54.65%
86
0.00%
13
94
95
1.06%
120
27.66%
94
0.00%
14
98
98
0.00%
−*
−*
−*
−*
15
101
106
4.95%
−*
−*
−*
−*
50
425
433
1.85%
−*
−*
−*
−*
75
535
564
5.14%
−*
−*
−*
−*
* − means the absence of data on the obtained solution or the impossibility of finding the solution for the given number of nodes.
The obtained results show that the number of graph nodes is not large (not more than 13) so the optimal solution can be obtained through exhaustive search method and ant algorithm method. The solution can be only obtained
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through ant algorithm method if the dimensionality of the problem increases (more than 13 nodes). Employing the other methods in this problem for the graph of higher dimensionality than 13 nodes essentially deteriorates the possibility of finding optimal solutions, and it even leads to the impossibility of their finding. Thus, we can infer that a considerable drawback of existing classical methods is the impossibility of applying them to the problems with a large number of parameters. 3.2. The results of applying modified ant algorithm for solving a dynamic routing problem. To conduct the testing, we applied the real data received in the company which is engaged in producing and delivering its own production to most (sales points) supermarkets in Kiev. As an example, we take the vehicle traveling to 15 sales points. We assume that the sales points are connected by one section road of street network. Then we assume that each link of two way graph corresponds to the specific section of street network. Calculations of distances and average speed of goods delivery in each section of route to each sales point within Kyiv’s street network were taken from [9]. These values are given in Table 2. Table 2. The values of distances (×103 m) and average speed of travel (m/s) between sales points.
Values of distances between points (х103m)
Points between objects
Values of average speed of traffic flow (× 103 / 3.6 × 103 m/s) 1
2
3
4
5
6
7
8
9
10
11
12
13
14
1
0
54
49
44
47
44
45
25
15
32
45
44
36
25
15 30
2
17
0
38
55
51
52
53
48
46
55
58
58
51
44
40
3
16
15
0
49
56
38
45
58
38
36
39
42
45
56
45
4
28
20
16
0
52
49
53
55
43
57
46
51
55
58
48 58
5
17
5
10
25
0
55
58
50
41
43
43
51
47
59
6
23
6
21
21
11
0
42
49
47
57
45
52
56
45
51
7
19
2
17
18
7
3
0
55
59
54
54
55
55
43
51
8
25
23
14
2
28
24
21
0
44
49
45
50
40
41
45
9
15
8
13
28
3
14
10
31
0
53
54
43
51
56
57 53
10
32
15
22
6
20
16
13
7
23
0
45
45
47
52
11
16
15
1
16
10
21
17
14
13
22
0
56
54
43
45
12
34
16
24
8
21
17
14
9
24
2
24
0
56
59
60
13
15
10
15
30
5
16
12
32
8
25
10
25
0
33
45
14
25
9
17
13
14
10
7
15
10
9
16
7
18
0
57
15
30
15
20
8
20
16
13
9
15
4
21
3
24
6
0
*Below the main diagonal, there are distances between any two sales points, and above the main diagonal of the matrix, there is average speed between them.
The distances between any two sales points are given below the main diagonal while average speed between these objects are given above the main diagonal. The graphs of distance between objects, average speed and the travel time between the objects are two-way, respectively lij = lji, υij = υji, tij = tji. We find the corresponding values of travel between all the points through formula (5). The route optimization is performed within traveling salesman problem. Before the travel, the optimal route is created. The values of travel average speed as well as the time calculated through (5) are given in Table 2. When creating this route, the objective functions are the distance and time of cargo delivery. The minimization of these objective functions is carried out through the algorithm given in 2.2. It is known that street network in Kyiv is characterized by very dynamic traffic flow. To test the proposed modification of ant algorithm method route optimization, we consider the dynamic transition of traffic flow from free to nearly congested state in some sections of street network. In this regard, there appears the necessity to take into account the change in speed and time of travel in such sections, and accordingly reconstruct the previous
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optimal route. In case the speed of traffic flow changes to critical value υK, which is nearly congested in specific sections of created route, the route is changed and these sections are excluded from the calculations. Besides, it is necessary to consider that the part of the route is covered and it cannot be changed. The variant of creating a route for dynamic salesman problem with 15 nodes is displayed for illustrative purposes (see Fig. 3a). The stationary problem of optimizing the route of salesman was first solved for this graph. The vehicle’s departure is from the first node, and its speed in all the links was the same all the way, equaling m/c. The found optimal route for this problem is shown in Fig. 3a, and its length is 106·103 m, the time of the trip is 8280 sec.
a)
b)
c)
Fig. 3. Results of route optimization before (3a, solid line) and after (3b, red dashed line) change in the traffic speed in the section 11−8, and after (3c, blue dashed line) change in the traffic speed in the section 15−14.
Let the traffic situation in some sections of the created route change after covering some part of the route, namely, two nodes 11 and 8 from node 1. For instance, the speed of traffic flow decreased from 12.5 m/s to 1.4 m/s in the road section between node 11 and 8, which is less than critical value υK for this section. The value of decrease of average traffic speed, according to (5), causes the increase in the time of covering this section, which eventually can lead to changing the previously optimal route of a salesman. Hence, the procedure of route optimization was repeated for the way which was left to cover all the graph nodes, taking into account the covered distance. Also, the road section between points 11 and 8 was removed from the calculation. As can be seen from the created route (Fig. 3b, solid line), the sequence of traveling through the nodes has changed in the part of the uncovered route at the time of its recreating. The time of the new route increased by 1620 sec and it made up 9900 sec (initial time – 8280 sec). The length of the new route grows by 20·103 m in comparison with the initial length, and it is 126·103 m. After traveling the part of the new route (nodes 1, 3, 11, 13, 5, 7, 6 and 15) in the road section between point 15 and 14, the speed of traffic flow decreased from 15.8 m/s to 1.4 m/s, which is less than critical value υK for this section. The speed decrease led to the considerable time increase of travel of the respective section (from 360 sec to 4320 sec), which results in changing the previously found route. Also, the road section between points 15 and 14 is removed from the calculation, and the route is recreated. In case of continuing the travel on the initial route, the time of its covering increases from 9900 sec to 13860 sec, the length of the route stays the same 126·103 m (Table 3). The sequence of traveling through the nodes changed in the section of uncovered route after recreating the route (Fig. 3c), at the time of its recreating. The time of covering the new route increased by 2880 sec and it is 12780 sec. The length of the new route increased by 31·103 m, and it makes up 157·103 m. On the one hand, the obtained calculations show that the route length increased by 51·103 m, and its time by 3690 sec, taking into consideration the real traffic conditions. On the other hand, if the route is not changed in the
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road section between point 11 and 8, 15 and 14 is substantially longer and it is 10080 sec and 4320 sec respectively, which results in the general increase in the time of the trip by 21180 sec. In its turn it would lead to the considerable growth in transport spending on goods delivery if the driver didn’t change the route. Table 3. The result of conducted calculations of reconstructing the optimal delivery route. The route of sales representative’s travel
Characteristics of sales representative’s travel Length, 103m
The time, sec
Optimal route by the time 1-3-11-8-4-12-10-15-14-6-7-2-9-13-5-1 when there are unchanged initial values of average speed and average time of travel between sales points
106
8280
Route 1-3-11-8-4-12-10-15-14-6-7-2-9-13-5-1 when there is decrease in speed between sales points 11-8 from 12.5 m/s to 1.4 m/s
106
17220
Optimal route by the time 1-3-11-13-5-7-6-15-14-8-4-10-12-9-2-1 when there is decrease in speed between sales points from 11-8 12.5 m/s to 1.4 m/s
126
9900
Optimal route 1-3-11-13-5-7-6-15-14-8-4-10-12-9-2-1 when there is decrease in speed between sales points 15-14 from 15.8 m/s to 1.4 m/s
126
13860
Optimal route 3 by the time 1-3-11-13-5-7-6-15-12-8-4-2-9-10-14-1 when there is decrease in speed between sales 15-14 від from 15.8 m/s to 1.4 m/s
157
12780
Thus, the obtained results of route optimization in the street network in Kyiv point out the perspective of applying the proposed ant algorithm modification for solving transport problems considering traffic flow dynamics. The approbation of the method was performed only for the case of phase transition from free traffic flow to nearly congested state. Meanwhile, it is known that large cities’ street network is characterized by a large number of macro, microscopic static and dynamic parameters, which describe different states of traffic flow dynamics that affects the average speed and time of travel of street network sections. It in its turn influences the price of goods delivery when performing the rout optimization. So, we can ascribe traffic capacity, intensity, density, characteristics of traffic flow content and traffic lights to the aforementioned parameters. 4. Conclusions In this paper, the quantitative and comparative analysis for solving the stationary problem of optimizing the goods delivery route in the street network was made through ant algorithm method and other classical methods. The obtained results points at the effectiveness of applying the proposed method for solving routing problems, especially those, which are characterized by high dimensionality. The modification of ant algorithm for optimizing the transport route was developed with regard to traffic flow dynamics in the street network. It was made possible to record the results of optimization of partly covered distance for calculating a further route when changing the length of links while ant agents traveling on the links of a two-way graph. Besides, the procedure of ant agents’ traffic in the graph was improved so that ant agents can travel both synchronously and asynchronously. The implemented additional elements of intellectualization enabled us to exclude those road sections in which the speed reaches the critical value, and the time of covering the respective section substantially rises. The proposed modification of ant algorithm for optimizing the goods delivery route when changing the speed of traffic flow in specific sections of the street network has been approbated, using the example of Kyiv’s specific street network within traveling salesman problem. The route optimization was performed for the case of change of traffic flow dynamic state when there was a phase transition from free traffic flow to nearly congested state. The obtained results of study show the prospects of the proposed method application, especially for traffic networks which are characterized be high dimensionality and dynamics of parameters.
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The further strategy of study has to be directed at improving the proposed modification of ant algorithm method for solving routing problems in real conditions of traffic in street network, considering factual traffic capacity in certain sections of street network, various kinds of traffic flow dynamics and traffic lights. Thus, the obtained results can be used to improve intellectual transport systems and traffic control technologies.
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