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International Journal of Digital Content Technology and its Applications Volume 4, Number 8, November 2010
Application Research on Optimal Path Based on Genetic Algorithm Lilin FAN*, Huili MENG College of Computer & Information technology, Henan Normal University, Xinxiang , China,453007 Email:[email protected] doi:10.4156/jdcta.vol4. issue8.22
Abstract At present, China's automobile logistics cost accounts for more than 20% of the production of motor vehicles. Therefore, lowering logistics cost by seeking its best path is an effective way to lower the cost of automobile manufacturing. This paper combines the Graph Theory and the genetic algorithm to research the optimizing problem of the lines of logistics distribution. Traffic flow data under normal conditions is obtained through field studies, based on which we can calculate the time of the network weight through appropriate method, find the best lines of distribution through the optimization theory of genetic algorithm, and show the feasibility and superiority of the method through examples.
Keywords: Genetic Algorithm, Automobile Logistics, Path Algorithm, Path Aptimization 1. Introduction Automobile logistics is a process that transferring the automobile parts, spare parts and whole vehicle from supply place to requirement place based on user’s requirement with the lowest cost [1]. The logistics cost is one of the main costs for automobile cost. The data shows that the logistics cost accounts for 8% of sales count in Europe and the United States’ automobile manufacturers, 5% in Japan, but 20% in China [2]. However, the automobile logistics is mainly taken by automobile manufacturer in China, which is a self-managed logistics combined supply, production and marketing. Therefore, how to provide a highly efficient, high quality and professional logistics service has been a core issue for the development of automobile industry in China. Genetic Algorithm(GA) is global heuristic optimization algorithm based on the biological theory of evolution and the thought of genetic, which has been widely used to solve complex optimization problems[3]. Its main advantages are simple, general, robust and suitable for parallel processing. It is a solving model which is not related to a problem. The development of GA is very fast in recent years. It is widely used to combinatorial optimization, adaptive control, machine learning and many other areas. It is used to solve the best logistics path. And the operation process and result are analyzed. Experimental data shows that this algorithm is effective and it is a superior performance heuristic algorithm for solving the vehicle path problem.
2. Genetic algorithms in Automotive Logistics path optimization The logistics network of automobile manufacturer, expects the whole vehicle distribution service to be a directed closure graph. With the constraint of accessory in every service station and the maximum load of distribution vehicle, the best path can be found based on the algorithm in computer graph theory [4]. Vehicle optimum route is changing with change of the time and road situations, so ought to calculate out a optimum route in current moment before the vehicle entering every road. We will weigh up value by the time used by passing the road section, and then can calculate each edge by using the directed graph theory, namely the weight of each road section.
2.1. The objective function of optimal path problem In directed graph G=(V,E), if the sequence（v1,v2,…，vj）satisfy (vi,vi+1) ∈E(i=1,2,…，j-1), then call
199
Application Research on Optimal Path Based on Genetic Algorithm Lilin FAN, Huili MENG j 1
the sequence is the route of node v1 to node vj, the length of the route is U= t (vi , vi 1 ) ,where i 1
t(vi,vi+1) is the time length of passing the road, namely the weight of route that is from vi to vi+1. The shortest route of vk to vj is minU, be that the sum of all section weight is minimum, also called optimum route. In city travel network, the weight of each edge t(vi,vi+1) is the function of average vehicle stream density on route vi to vi+1, namely t(vi,vi+1)=f( i ,i 1 ) ,among them, i ,i 1 denote that the average vehicle stream density from vi to vi+1 at time k. Therefore, we can get the target function of the optimum route at time k: k
k
j 1
minU= f ( k i ,i 1 )
(1)
i 1
For getting f( i ,i 1 (k )) ,we should get the average vehicle stream density, then getting the average time that passing the road section at time k by the distance of the two nodes and the average vehicle stream speed. First analyze the connection of the average vehicle stream speed and the vehicle stream density. While the vehicle stream is very rare, speed of a vehicle ought to comparatively highly. With the vehicle stream stopping increasing by comparatively, speed of a vehicle shouldn't coming down fast. The speed of a vehicle comes down rapidly when the vehicle stream increases to certain degree; Speed of a vehicle cuts to the minimum , is equal to having happened blocking up the car when the vehicle stream continues increasing by, henceforth, the vehicle stream is unable to increase by again , the vehicle stream speed has no changes also. Therefore the relation between the vehicle stream speed and the density is similar to the form of tangent function. The average vehicle stream speed between vertex vi to vi+1 is as follows:
v fi ,i 1
Vi,i+1=
1 e Among them,
is the constant,
(
0 k i ,i 1 0
(2)
)
v f i ,i1 is the free vehicle stream speed, 0 is the transition
vehicle stream speed when the road traffic stream rate of flow is maximal. Then, we can get from vi to vi+1 such road section right weight in k moment: ｆ(
k
L (1 e i ,i 1 ) = i
k i , j 1
00
)
(3)
v fi ,i 1 Among them Li denote the distance between the ith crossing and the i+1th crossing.
2.2. Chromosome coding and function definition of fitness Design of chromosome structure. It's natural ordinal number encoder and its length is k+n+1. Among them, k is the count of service stations that they need accessories, n is the count of vehicle that they take distribution task. For example, chromosome 01203450 refers to one road from service station of manufacturer back to departure point by service place 1 and service place 2 and another road back to service station by service place 3, 4, 5 orderly. The interior of the chromosome structure sub-path is orderly. Exchanging position in the interior of sub-path will lead the change to the value of objective function. However, there is no order between sub-paths. Exchanging position will not lead to the change to the value of objective function. Certainly, the count of chromosome gene must exceed nine in practical application. In this situation, one gene can be represented with the combination of two or
200
International Journal of Digital Content Technology and its Applications Volume 4, Number 8, November 2010
three bits. Function definition of fitness by (4): m
F min U i i 1
(4)
Where minU is the objective function defined by (1). It is restricted by the requirement of accessories in every station and the maximum load of distribution vehicle. In this situation, the smaller the F value of the object is, the higher its fitness is. Generation method of the initial group Generate a full permutation contained k service stations. Insert 0 at begin and the end of the permutation firstly, select m-1 count insert points and insert 0, which construct a initial chromosome.
2.3. The design of genetic operator First, the paper constructs a directed close graph whose nodes are automobile service station, then get the shortest path by use of Dijkstra algorithm, at last apply GA to optimize this path, by which to get the shortest path in whole. Genetic operator is one of the key factors to determine the capability of GA. In the application of path layout , it can’t reflect the special knowledge of the problem that need to be solved if using normal genetic operator , the evolution of algorithm belongs to self-searching states to great degree , which leads the efficiency is low . So it is necessary to design genetic operator especially on the problem of path layout. Operator selection. According to the fitness function selected above and fitness size, it is selected
Pc. max Pct Pc. min P 1 t / t max P Pct Pc. min c. min 1 0t Pmt Pm. min Pm. max e tmax t Pm Pmt Pm. min Pm. min t c
(5)
(6)
with roulette strategy. In order to avoid destruction from cross or mutation operation, the best saving policy is adopted. That is recording the best object of every generation, which is used to substitute the worst. That will not only improve the average fitness continuously, but also ensure to reduce the fitness of the best. Crossover operator. Adopt the sequence cross method proposed by Davis [5]. Improvement is made to cross algorithm for the diversity of species. If the two genes are equal to zero in the cross point of chromosome, make sequence cross operation directly. If not all of the genes in the cross point of chromosome are zero, move the cross point to right or left until the genes in two cross points are zero, and take sequence cross operation. Mutation operator. Mutation operator is referred to replace some genes of the individual code by other genes ,a accessorial method to form a new individual , which is a absolutely necessary calculate process..Mutation itself is a local search. Reversal mutation method is proposed in this paper [6]. For example, Ybefore=a1a2a3a4a5a6, Assumption facture happens between a3 and a5, Fracture fragment a3a4a5 were reversed and then inserted into fractured chromosome again: Yafter= a1a2a5a4a3a6. It’s more important that only when the fitness was improved the reversion is valid, otherwise invalid. Crossover rate and mutation rate. The crossover operator of genetic algorithm is based on crossover rate. Theory investigation and application practice shows that crossover rate and mutation rate are associated, and if the data are improper, not only the efficiency of this algorithms will be reduced, but also bringing can be divergence, which may lead to failed result. In the paper, function used to calculate the Self-adaptive crossover rate is defined in (5)， the function used to calculate the Self-adaptive mutation rate is defined in (6) [7]. formula (5)、(6) , Pct is rossing-over rate of the t generation, Ptm is mutation rate of the t generation; Pc..min, Pc..max, Pc..max is the scope of rossing-over rate changes, P m..min, Pm..max is the scope of mutation
201
Application Research on Optimal Path Based on Genetic Algorithm Lilin FAN, Huili MENG
rate changes; t,tmax is genetic algebra and most genetic algebra.
3. Application This paper takes the 20 service stations in southwest area and a automobile manufacturer as an example. The weight values are computed out by the computation method in section 2.1 and coordinate value in table 1. Chromosome gene is represented with 00-20 two digits. The maximum iterative number of the algorithm is 50. The count of distribution vehicle is 3. The maximum load is 10 tons. The data is stored with SQL Server2005 for the extension of the program. This program is implemented with C# language. The shortest path is 1218 after 25 times iterative. The corresponding chromosome coder as follows: 00 19 15 16 13 06 11 00 05 02 12 09 10 01 07 08 20 00 14 04 03 17 18 00 Table 1. coordinate and parts requirement in every service station 01 128 75 0.12 08 95 74 0.56 15 64 20 1.72
02 174 44 0.43 09 145 31 1.02 16 12 27 1.01
03 162 154 1.22 10 158 62 0.04 17 22 182 1.06
04 178 163 1.51 11 57 178 0.89 18 142 165 1.62
05 145 126 0.84 12 138 36 1.08 19 74 67 1.73
06 49 96 1.32 13 28 77 1.32 20 86 136 1.52
07 97 75 0.63 14 181 120 1.91
4. Conclusion Arranging the vehicle path reasonably is the important ways to reduce the waste and enhance economic efficiency in logistics distribution system. However, due to the NP complete of the problem, it is very difficult to resolve accurately. It is a feasible direct to research heuristic algorithm. The vehicle path heuristic algorithm based on genetic algorithm is raised and implemented in this paper. Experiments show that this algorithm is a feasible way to resolve the vehicle path question.
5. References [1] HUANG Kunbing, “Domestic Automobile Logistics Development Strategy.” Application of
Logistics Technology, No.3, pp.101~104, 2008 [2] WANG Lei, “The Development Strategy of China Automotive Logistics.” Journal of Logistics SciTech, No.7, pp.4~6, 2008 [3] Ali Hadian, "Clustering Based Multi-Objective Rule Mining using Genetic Algorithm", Internatio nal Journal of Digital Content Technology and its Applications, Vol.4, No.1, pp.37~42, 2010 [4] WANG Xiadong, FAN, Lilin, “Road Network Optimization Scheme.” International Conference on Transportation Engineering, pp.1997 ~2002, 2007. [5] TANG Guoxin, CHEN Xiong, YUAN Yang,“Improved Genetic Algorithm for Robotic Path Plan ning”,Journal of Computer Engineering and Applications, 2007,Vol.43,No.22,pp.67 ~70,2007 [6] WU Qinghong，ZHANG Jihui，XU Xinhe,“An Ant Colony Algorithm With Mutation Features.” Journal of Computer Research & Development, Vol. 36, No. 10, pp.1242 ~1243, 1999 [7] YUAN Huimei, “Adaptive Crossover Rate and Mutation Rate in Genetic Algorithms”, Journal of aptial Normal University , Vol. 21, No. 3, pp.14 ~19,2000 [8] The Scientific and Technological Project of Henan Province under Grant No.092102210316, 1021022107
202
Application Research on Optimal Path Based on Genetic Algorithm Lilin FAN*, Huili MENG College of Computer & Information technology, Henan Normal University, Xinxiang , China,453007 Email:[email protected] doi:10.4156/jdcta.vol4. issue8.22
Abstract At present, China's automobile logistics cost accounts for more than 20% of the production of motor vehicles. Therefore, lowering logistics cost by seeking its best path is an effective way to lower the cost of automobile manufacturing. This paper combines the Graph Theory and the genetic algorithm to research the optimizing problem of the lines of logistics distribution. Traffic flow data under normal conditions is obtained through field studies, based on which we can calculate the time of the network weight through appropriate method, find the best lines of distribution through the optimization theory of genetic algorithm, and show the feasibility and superiority of the method through examples.
Keywords: Genetic Algorithm, Automobile Logistics, Path Algorithm, Path Aptimization 1. Introduction Automobile logistics is a process that transferring the automobile parts, spare parts and whole vehicle from supply place to requirement place based on user’s requirement with the lowest cost [1]. The logistics cost is one of the main costs for automobile cost. The data shows that the logistics cost accounts for 8% of sales count in Europe and the United States’ automobile manufacturers, 5% in Japan, but 20% in China [2]. However, the automobile logistics is mainly taken by automobile manufacturer in China, which is a self-managed logistics combined supply, production and marketing. Therefore, how to provide a highly efficient, high quality and professional logistics service has been a core issue for the development of automobile industry in China. Genetic Algorithm(GA) is global heuristic optimization algorithm based on the biological theory of evolution and the thought of genetic, which has been widely used to solve complex optimization problems[3]. Its main advantages are simple, general, robust and suitable for parallel processing. It is a solving model which is not related to a problem. The development of GA is very fast in recent years. It is widely used to combinatorial optimization, adaptive control, machine learning and many other areas. It is used to solve the best logistics path. And the operation process and result are analyzed. Experimental data shows that this algorithm is effective and it is a superior performance heuristic algorithm for solving the vehicle path problem.
2. Genetic algorithms in Automotive Logistics path optimization The logistics network of automobile manufacturer, expects the whole vehicle distribution service to be a directed closure graph. With the constraint of accessory in every service station and the maximum load of distribution vehicle, the best path can be found based on the algorithm in computer graph theory [4]. Vehicle optimum route is changing with change of the time and road situations, so ought to calculate out a optimum route in current moment before the vehicle entering every road. We will weigh up value by the time used by passing the road section, and then can calculate each edge by using the directed graph theory, namely the weight of each road section.
2.1. The objective function of optimal path problem In directed graph G=(V,E), if the sequence（v1,v2,…，vj）satisfy (vi,vi+1) ∈E(i=1,2,…，j-1), then call
199
Application Research on Optimal Path Based on Genetic Algorithm Lilin FAN, Huili MENG j 1
the sequence is the route of node v1 to node vj, the length of the route is U= t (vi , vi 1 ) ,where i 1
t(vi,vi+1) is the time length of passing the road, namely the weight of route that is from vi to vi+1. The shortest route of vk to vj is minU, be that the sum of all section weight is minimum, also called optimum route. In city travel network, the weight of each edge t(vi,vi+1) is the function of average vehicle stream density on route vi to vi+1, namely t(vi,vi+1)=f( i ,i 1 ) ,among them, i ,i 1 denote that the average vehicle stream density from vi to vi+1 at time k. Therefore, we can get the target function of the optimum route at time k: k
k
j 1
minU= f ( k i ,i 1 )
(1)
i 1
For getting f( i ,i 1 (k )) ,we should get the average vehicle stream density, then getting the average time that passing the road section at time k by the distance of the two nodes and the average vehicle stream speed. First analyze the connection of the average vehicle stream speed and the vehicle stream density. While the vehicle stream is very rare, speed of a vehicle ought to comparatively highly. With the vehicle stream stopping increasing by comparatively, speed of a vehicle shouldn't coming down fast. The speed of a vehicle comes down rapidly when the vehicle stream increases to certain degree; Speed of a vehicle cuts to the minimum , is equal to having happened blocking up the car when the vehicle stream continues increasing by, henceforth, the vehicle stream is unable to increase by again , the vehicle stream speed has no changes also. Therefore the relation between the vehicle stream speed and the density is similar to the form of tangent function. The average vehicle stream speed between vertex vi to vi+1 is as follows:
v fi ,i 1
Vi,i+1=
1 e Among them,
is the constant,
(
0 k i ,i 1 0
(2)
)
v f i ,i1 is the free vehicle stream speed, 0 is the transition
vehicle stream speed when the road traffic stream rate of flow is maximal. Then, we can get from vi to vi+1 such road section right weight in k moment: ｆ(
k
L (1 e i ,i 1 ) = i
k i , j 1
00
)
(3)
v fi ,i 1 Among them Li denote the distance between the ith crossing and the i+1th crossing.
2.2. Chromosome coding and function definition of fitness Design of chromosome structure. It's natural ordinal number encoder and its length is k+n+1. Among them, k is the count of service stations that they need accessories, n is the count of vehicle that they take distribution task. For example, chromosome 01203450 refers to one road from service station of manufacturer back to departure point by service place 1 and service place 2 and another road back to service station by service place 3, 4, 5 orderly. The interior of the chromosome structure sub-path is orderly. Exchanging position in the interior of sub-path will lead the change to the value of objective function. However, there is no order between sub-paths. Exchanging position will not lead to the change to the value of objective function. Certainly, the count of chromosome gene must exceed nine in practical application. In this situation, one gene can be represented with the combination of two or
200
International Journal of Digital Content Technology and its Applications Volume 4, Number 8, November 2010
three bits. Function definition of fitness by (4): m
F min U i i 1
(4)
Where minU is the objective function defined by (1). It is restricted by the requirement of accessories in every station and the maximum load of distribution vehicle. In this situation, the smaller the F value of the object is, the higher its fitness is. Generation method of the initial group Generate a full permutation contained k service stations. Insert 0 at begin and the end of the permutation firstly, select m-1 count insert points and insert 0, which construct a initial chromosome.
2.3. The design of genetic operator First, the paper constructs a directed close graph whose nodes are automobile service station, then get the shortest path by use of Dijkstra algorithm, at last apply GA to optimize this path, by which to get the shortest path in whole. Genetic operator is one of the key factors to determine the capability of GA. In the application of path layout , it can’t reflect the special knowledge of the problem that need to be solved if using normal genetic operator , the evolution of algorithm belongs to self-searching states to great degree , which leads the efficiency is low . So it is necessary to design genetic operator especially on the problem of path layout. Operator selection. According to the fitness function selected above and fitness size, it is selected
Pc. max Pct Pc. min P 1 t / t max P Pct Pc. min c. min 1 0t Pmt Pm. min Pm. max e tmax t Pm Pmt Pm. min Pm. min t c
(5)
(6)
with roulette strategy. In order to avoid destruction from cross or mutation operation, the best saving policy is adopted. That is recording the best object of every generation, which is used to substitute the worst. That will not only improve the average fitness continuously, but also ensure to reduce the fitness of the best. Crossover operator. Adopt the sequence cross method proposed by Davis [5]. Improvement is made to cross algorithm for the diversity of species. If the two genes are equal to zero in the cross point of chromosome, make sequence cross operation directly. If not all of the genes in the cross point of chromosome are zero, move the cross point to right or left until the genes in two cross points are zero, and take sequence cross operation. Mutation operator. Mutation operator is referred to replace some genes of the individual code by other genes ,a accessorial method to form a new individual , which is a absolutely necessary calculate process..Mutation itself is a local search. Reversal mutation method is proposed in this paper [6]. For example, Ybefore=a1a2a3a4a5a6, Assumption facture happens between a3 and a5, Fracture fragment a3a4a5 were reversed and then inserted into fractured chromosome again: Yafter= a1a2a5a4a3a6. It’s more important that only when the fitness was improved the reversion is valid, otherwise invalid. Crossover rate and mutation rate. The crossover operator of genetic algorithm is based on crossover rate. Theory investigation and application practice shows that crossover rate and mutation rate are associated, and if the data are improper, not only the efficiency of this algorithms will be reduced, but also bringing can be divergence, which may lead to failed result. In the paper, function used to calculate the Self-adaptive crossover rate is defined in (5)， the function used to calculate the Self-adaptive mutation rate is defined in (6) [7]. formula (5)、(6) , Pct is rossing-over rate of the t generation, Ptm is mutation rate of the t generation; Pc..min, Pc..max, Pc..max is the scope of rossing-over rate changes, P m..min, Pm..max is the scope of mutation
201
Application Research on Optimal Path Based on Genetic Algorithm Lilin FAN, Huili MENG
rate changes; t,tmax is genetic algebra and most genetic algebra.
3. Application This paper takes the 20 service stations in southwest area and a automobile manufacturer as an example. The weight values are computed out by the computation method in section 2.1 and coordinate value in table 1. Chromosome gene is represented with 00-20 two digits. The maximum iterative number of the algorithm is 50. The count of distribution vehicle is 3. The maximum load is 10 tons. The data is stored with SQL Server2005 for the extension of the program. This program is implemented with C# language. The shortest path is 1218 after 25 times iterative. The corresponding chromosome coder as follows: 00 19 15 16 13 06 11 00 05 02 12 09 10 01 07 08 20 00 14 04 03 17 18 00 Table 1. coordinate and parts requirement in every service station 01 128 75 0.12 08 95 74 0.56 15 64 20 1.72
02 174 44 0.43 09 145 31 1.02 16 12 27 1.01
03 162 154 1.22 10 158 62 0.04 17 22 182 1.06
04 178 163 1.51 11 57 178 0.89 18 142 165 1.62
05 145 126 0.84 12 138 36 1.08 19 74 67 1.73
06 49 96 1.32 13 28 77 1.32 20 86 136 1.52
07 97 75 0.63 14 181 120 1.91
4. Conclusion Arranging the vehicle path reasonably is the important ways to reduce the waste and enhance economic efficiency in logistics distribution system. However, due to the NP complete of the problem, it is very difficult to resolve accurately. It is a feasible direct to research heuristic algorithm. The vehicle path heuristic algorithm based on genetic algorithm is raised and implemented in this paper. Experiments show that this algorithm is a feasible way to resolve the vehicle path question.
5. References [1] HUANG Kunbing, “Domestic Automobile Logistics Development Strategy.” Application of
Logistics Technology, No.3, pp.101~104, 2008 [2] WANG Lei, “The Development Strategy of China Automotive Logistics.” Journal of Logistics SciTech, No.7, pp.4~6, 2008 [3] Ali Hadian, "Clustering Based Multi-Objective Rule Mining using Genetic Algorithm", Internatio nal Journal of Digital Content Technology and its Applications, Vol.4, No.1, pp.37~42, 2010 [4] WANG Xiadong, FAN, Lilin, “Road Network Optimization Scheme.” International Conference on Transportation Engineering, pp.1997 ~2002, 2007. [5] TANG Guoxin, CHEN Xiong, YUAN Yang,“Improved Genetic Algorithm for Robotic Path Plan ning”,Journal of Computer Engineering and Applications, 2007,Vol.43,No.22,pp.67 ~70,2007 [6] WU Qinghong，ZHANG Jihui，XU Xinhe,“An Ant Colony Algorithm With Mutation Features.” Journal of Computer Research & Development, Vol. 36, No. 10, pp.1242 ~1243, 1999 [7] YUAN Huimei, “Adaptive Crossover Rate and Mutation Rate in Genetic Algorithms”, Journal of aptial Normal University , Vol. 21, No. 3, pp.14 ~19,2000 [8] The Scientific and Technological Project of Henan Province under Grant No.092102210316, 1021022107
202
