**ALGORITHM FINDING MAXIMUM CONCURRENT MULTICOMMODITY LINEAR FLOW WITH LIMITED COST IN EXTENDED TRAFFIC NETWORK WITH SINGLE REGULATING COEFFICIENT ON TWO-SIDE LINES**

Tran Quoc Chien and Nguyen Dinh Lau

** ^{ }**The University of Education, University of Danang, Danang, Vietnam

**ABSTRACT**

Graphs and extended networks are is powerful mathematical tools applied in many fields as transportation, communication, informatics, economy, … Algorithms to find Maximum Concurrent Multicommodity Flow with Limited Cost on extended traffic networks are introduced in the works we did. However, with those algorithms, capacities of two-sided lines are shared fully for two directions. This work studies the more general and practical case, where flows are limited to use two-sided lines with a single parameter called regulating coefficient. The algorithm is presented in the programming language Java. The algorithm is coded in programming language Java with extended network database in database management system MySQL and offers exact results.

**KEYWORDS**

Graph, Network, Multicommodity Flow, Optimization, Approximation.

**1. INTRODUCTION**

Graphs are useful mathematical tools applied in various fields such as transportation, communication, information technology, economics and the others. So far in the graph there has been great and separate concern of the weight of the edges and vertices, in which the length of the road is simply the total of the weight of the edges and vertices. However, in many practical problems, weights at a vertice are different with various routes and depend on arrival edges and departure edge. For example, the period of time to go through the intersection of network traffic depends on the direction of movement of vehicles: turn right, go straight or turn left, even go to restricted direction.

The works [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25] built up maximum flow problem in the network without the capacity of vertex and cost at vertex. The above mentioned works don’t mention single regulating coefficient on two-side lines either.

Our article is far optimal than the above works because of two main reasons. First, we add the capacity of vertex in network to solve the problems in real situation, section 4 is devoted to illustrate the examples and our solutions. Secondly, we use single regulating coefficient on two-side lines (Y) to solve two direction traffic network problems in reality.

The work [4] constructs extended traffic network model and maximum concurrent multicommodity linear flow problem with limited cost to build model for the actual problems to have accurate and efficient solutions. Algorithms to find approximate maximum concurrent multicommodity linear flow with limited cost are developed in the works [4, 5]. Nevertheless, in this algorithm, the capacities of the routes are shared fully for both directions. In some cases, vehicle of one direction will occupy a two-way route, causing traffic digestion. In this work, capacity of traffic flow on the reserve direction route will be regulated by parameter Y. Y is so-called the regulating coefficient in two-way route. Regulating coefficient Y ranges from 0.5 to 1.

The main result of the article is the approximation algorithm to find maximum concurrent multicommodity linear flow with limited cost on two-side lines. The algorithm is coded in programming language Java with extended network database in database management system MySQL and offers exact results.

**2. MAXIMUM CONCURRENT MULTICOMMODITY LINEAR FLOW WITH LIMITED COST PROBLEM**

Given a mixed graph G = (V, E) with node set V and edge set E. The edges can be directed or undirected. There are many kinds of vehicles circulating in the graph. The undirected edges are assigned two-way route and commodities with the same line in the opposite direction shares the capacities. Let us consider the following functions [1], [6], [7], [8], [9], [10].

x ≥ 0, l ≥ 0

**3. ALGORITHM TO FIND MAXIMUM CONCURRENT MULTICOMMODITY LINEAR FLOW WITH LIMITED COST ON EXTENDED TRAFFIC NETWORK WITH SINGLE REGULATING COEFFICIENT ON TWO-SIDE LINES**

The algorithm is coded in programming language Java with extended network database in database management system MySQL.

Given an extended network as in figure 1 with 6 nodes, 6 directed edges và 3 undirected edges. The cost of edge b_{E} is shown in table 1 and the cost of node b_{V} in table 2.

The capacity of each edge is 10, the capacity for each node is là 20. There are 3 source-sink pairs (1,5), (2,4) and (3,6) within an amount of commodities.

d(1) = 10, d(2) = 10 và d(3) = 10.

Figure 1. Extended network G

Table 1. The cost of edge b_{E}

Table 2. The cost of node b_{V}

First, we consider the absence of regulatory constraints, ie regulating coefficient Y = 1.

Limited costs B = 600.

Choose approximation coefficient w = 0,024.

The results of maximum flow are demonstrated as following:

Maximum value l = 0,8968841421395084

Actual cost: 589,6342794580955

The number of commodities is directed or distributed for each source-sink pair is 9,0**.**

Flow distribution for commodities from source 1 to sink 5, source 2 to sink 4 và source 3 to sink 6 is illustrated in figure 2, figure 3 and figure 4.

Figure 2. Flow distribution from source 1 to sink 5

It can be seen that the total of flows from node 3 to node 5 is 2,8 + 6,2 = 9,0.

Thus, the total of flows from node 3 to node 5 is nearly 10. It means that it almost encroaches upon the other side of the route from node 5 to node 3.

Now for regulating coefficient Y = 0.6.

This means that the total of flows of one direction in two-way route do not exceed 60% of the capacity.

Figure 3. Flow distribution from source 2 to sink 4

Figure 4. Flow distribution from source 3 to sink 6

Let limited cost B = 600.

Choose approximation coefficient w = 0,024.

The results of maximum flow are demonstrated as following:

Maximum value l = 0.798637155967107

Actual cost: 546,3976778135944

The number of commodities is directed or distributed for each source-sink pair is 8,0.

Flow distribution for commodities from source 1 to sink 5, source 2 to sink 4 and source 3 to sink 6 is illustrated in figure 5, figure 6 and figure 7.

Figure 5. Flow distribution from source 1 to sink 5

Figure 6. Flow distribution from source 2 to sink 4

Figure 7. Flow distribution from source 3 to sink 6

It can be seen that the total of flows 8,0 is smaller than 9,0. In this case, it is impossible to regulate two-way route.

In contrast, the total of flows from node 3 to node 5 is 2,5 + 3,5 = 6,0

It is clear that the regulated flows do not exceed 60% of capacity.

**5. CONCLUSION**

The main result of the article is the approximation algorithm to find maximum concurrent multicommodity linear flow with limited cost on two-side lines. The algorithm is presented in the programming .

The algorithm applied to solve many practical problems can be the form of the optimization problem through extented multicommodities network, such as the traffic management problem, freight, flow distribution in the Internet, means circulation and others.

The algorithm is coded in programming language Java with extended network database in database management system MySQL and offers exact demonstrations and reliable results.

**REFERENCES**

[1] Tran Quoc Chien, “Extended traffic network problem and applied in Danang city”, (2011) No 507/QĐ-SKHCN date 28/12/2011

[2] Tran Quoc Chien, (2012) “Algorithms for finding the shortest paths in a general network”, journal of science and technology – university of danang, No 12(61)/2012, pp 16-21.

[3] Tran Quoc Chien, Nguyen Mau Tue, Tran Ngoc Viet, “Algorithms for finding the shortest paths in a extended network”, proceeding of the 6^{th} national conference on fundamental and applied information technology research (FAIR), Hue, 20-21/6/2013, pp 522-527.

[4] Tran Ngoc Viet, Tran Quoc Chien, Nguyen Mau Tue, (2014) “Extended traffic network and the traffic multicommodity linear assignment problems”, journal of science and technology – university of danang, No 1(74)2014, pp 136-139.

[5] Tran Quoc Chien, (2013) “Application of the fastest path algorithm to finding maximum concurrent multicommodity linear flow with minimal cost on extended traffic network”, journal of science and technology – university of danang, No 10(71)2013, pp 85-91.

[6] Chien Tran Quoc, Lau Nguyen Dinh, Trinh Nguyen Thi Tu,(2013) “Sequential and Parallel Algorithm by Postflow-Pull Methods to Find Maximum Flow”, Proceedings 2013 13th International Conference on Computational Science and Its Applications, ISBN:978-0-7695-5045-9/13 $26.00 © 2013 IEEE, DOI 10.1109/ICCSA.2013.36, published by IEEE- CPS pp 178-181.

[7] Lau Nguyen Dinh, Chien Tran Quoc and Manh Le Thanh, (2014) “Parallel algorithm to divide optimal linear flow on extended traffic network”, Research, Development and Application on Information & Communication Technology, Ministry of Information & Communication of Vietnam, No 3, V-1, pp 14-29.

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**Authors**

Ass. Prof. DrSc. Tran Quoc Chien (http://scv.ued.udn.vn/ly_lich/chi_tiet/275).He **has 14 papers in SCIE of Journal (http://www.kybernetika.cz/contact.html). Born in 1953 in Dien Ban, Quang Nam, Vietnam. He graduated from Maths_IT faculty. He got Ph.D Degree of maths in 1985 in Charles university of Prague, Czech Republic and hold Doctor of Science in Charles university of Prague, Czech Republic in 1991. He received the tittle of Ass. Pro in 1992. He work for university of Danang, Vietnam. His main major: Maths and computing, applicable mathematics in transport, maximum flow, parallel and distributed process, discrete mathemetics, graph theory, grid Computing, distributed programming.**

Dr. NGUYEN DINH LAU (http://scv.udn.vn/ndlau).Born in 1978 in Dien Ban, **Quang Nam, Vietnam. He graduated from Maths_IT faculty of Hue university of science in 2000. He got master of science (IT) at Danang university of technology and hold Ph.D Degree in 2015 at Danang university of technology. His main major: Applicable mathematics in transport, parallel and distributed process, discrete mathemetics, graph theory, grid Computing and distributed programming.**

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