JPH08194677A - Device and method for executing genetic algorithm - Google Patents

Abstract

PURPOSE: To shorten the time for getting an optimum solution by saving a useless calculation amount by narrowing down a solution space to be searched by providing an initializing processing means, chromosome selecting processing means, cross/variation processing means and genetic correcting processing means. CONSTITUTION: When expressing the chromosome of the candidate of the optimum solution, an initializing processing means 2 generates a chromosome group expressing chromosomes by using the rank of interobject cost based on a table prepared by a cost rank table preparing processing means 1 as a genetic factor. A chromosome selecting processing means 3 selects any chromosome out of the chromosome group corresponding to adaptability and a cross/variation processing means 4 performs the cross or mutation operation of the chromosome selected by the means 3. When the objects of the rank shown by a certain genetic factor are overlapped as a result of the cross or mutation operation of chromosomes due to the means 4, a genetic correcting processing means 5 changes that genetic expression into the rank of a minimum non-selected object in the rank of inter-object cost corresponding to that object since that genetic factor is a fatal genetic factor.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a genetic algorithm executing apparatus and an executing method for solving a combinatorial optimization problem using a genetic algorithm.

[0002] A genetic algorithm is a processing algorithm by a computer devised by mimicking the mechanism of natural selection, gene crossover, mutation, etc. in the evolutionary process of a living organism. For example, a method for simulating the evolution of a living organism. It is also used as a method for solving combinatorial optimization problems in engineering. There is a need for a technique for obtaining an optimum solution as fast as possible by this genetic algorithm.

[0003]

2. Description of the Related Art One of combinatorial optimization problems that is extremely difficult to solve is, for example, a traveling salesman problem. The traveling salesman problem is to find the least cost tour (route) for one salesman to travel through a designated city group without duplication and return to the first city. Here, the "cost" is a positive numerical value (for example, distance) given between any two points (city). "Tour (route)" means
It is a single stroke that passes through all cities, and the one with the minimum sum of costs on the tour (for example, the sum of distances) is the best cost and optimal solution.

As a method for rigorously solving the combinatorial optimization problem such as the traveling salesman problem, there are a dynamic programming method and a branch-and-bound method. Not a realistic solution. One of the various solution methods that have been proposed to obtain an approximate solution in a practical calculation time
One is using a genetic algorithm. For reference, see “DE Goldberg, Genetic Algorithms in Searc.
h, Optimization, and Machine Learning. Addison-Wesl
ey Publishing Company, Inc. (1989) ”.

Generally, in a genetic algorithm, a solution candidate of an optimization problem is converted into a sequence or a character string to represent a chromosome. A plurality of (n) chromosomes are randomly generated, and selection, crossover, and mutation operations are performed on these chromosomes.

In the selection, the chromosomes are selected according to the fitness value corresponding to the evaluation function of the solution represented by each chromosome. In crossover, a new chromosome (child) is generated from two chromosomes (parent).

Mutation causes mutations to occur randomly in the chromosome. These operations are repeated a fixed number of times or until the solution converges to some extent, and an optimal solution is obtained.

[0008]

In the coding in the conventional genetic algorithm for solving the traveling salesman problem, in general, alphabets representing city names are regarded as alleles, and alphabets arranged in a row in the order of city names are used. It was the chromosome of each individual.

FIG. 10 is an explanatory diagram of the traveling salesman problem. In FIG. 10, A to F are city names, and it is assumed that the tour is carried out in the order of arrow, A → B → C → D → E → F → A. The character string ABCDEF representing the tour (route) of A → B → C → D → E → F → A is a chromosome, and {A,
The elements of B, C, D, E, F} are alleles. Chromosomes in which alleles are duplicated or deleted carry lethal genes and cannot survive. Here, assuming that the total cost of traveling between cities, that is, the total distance is the fitness f, the fitness of the chromosome ABCDEF is, for example, f (ABCDEF) = distance (AB) + distance (BC) + distance (CD) + Distance (DE) + Distance (EF) + Distance (F
A).

However, when the above expression in the conventional genetic algorithm is used, a chromosome having a gene sequence that actually connects distant cities is generated in the operation of initialization, crossover or mutation. ， A lot of useless calculations are required to find the optimal solution. Also,
A complicated crossover method was required to prevent the chromosome containing the lethal gene generated by the crossover operation.

An object of the present invention is not to use a conventional city name directly as a chromosome to express a solution of a problem, but to reflect a degree of proximity between cities (degree of closeness of distance) to a chromosome. By using the expression, it is possible to prevent the generation of a chromosome having a gene sequence that expresses unnecessary connection between cities,
It is possible to narrow the solution space to be searched, reduce the amount of calculation, and shorten the time to reach the optimal solution.

[0012]

In order to achieve the above object, the genetic algorithm execution apparatus of the present invention comprises means as shown in FIG. 1, for example. In FIG. 1, 1 is a cost ranking table preparation processing means, 2 is an initialization processing means, 3 is a chromosome selection processing means, 4 is a crossover / mutation processing means, and 5 is a gene correction processing means.

The cost ranking table creation processing unit 1 pays attention to the costs between the objects, and creates the cost ranking table in which the objects are ranked so that the costs from the target to other objects are arranged in ascending order. Is a means to do.

The initialization processing means 2 expresses a chromosome by using the rank of the inter-target costs in the table prepared by the cost ranking table preparation processing means 1 as a gene when representing the chromosome of the optimal solution candidate. It is a means of creating a group.

The chromosome selection processing means 3 is means for selecting chromosomes from the chromosome group according to the fitness.
The crossover / mutation processing means 4 is means for performing crossover or mutation operations on the chromosomes selected by the chromosome selection processing means 3. Either one or both of the crossover or mutation operations may be performed.

The gene correction processing means 5 is a lethal gene if the objects of the order indicated by a certain gene overlap as a result of the crossover or mutation operation of the chromosomes by the crossover / mutation processing means 4, the gene is a lethal gene. The gene expression,
It is a means for changing the rank of an unselected target in the rank of inter-target costs for that target.

In the present invention, when the candidates for the optimal solution are expressed as chromosomes, the cost between objects is focused on instead of the letters or numbers representing the objects themselves, and the cost between objects is ascended in order from the smallest. A cost ranking table is created by ranking the objects in a line, and the rankings in this cost ranking table are used as genes.

In addition, regarding genes (lethal genes) whose targets are duplicated in the gene expression in the initialization process or the operation of crossover or mutation, the gene expression is most ranked in the order of cost between targets for the target. Change to a small unselected target rank. As a result, it is possible to prevent the generation of gene sequences that actually connect high-cost objects and speed up the convergence to the optimal solution.

[0019]

[Operation] Examining the optimal solutions for some examples of the traveling salesman problem reveals that the optimal tours (routes) connect cities with short distances. For example, 239, which is well known as a library for the traveling Thessalman problem
In the case of the problem of two cities, 99.2% of the passes are connected between cities up to the ninth place. 21st even in the farthest cities
Rank. Further, in the case of the problem of 30 cities, all the paths are up to the 9th place (see FIG. 3, which will be described later. The underlined city is the city selected by the optimal solution).

From these things, in order to obtain the optimum solution,
It turns out that it is not necessary to consider the paths between all cities (objects). By avoiding passing between distant cities, it is possible to narrow the solution space of the optimization problem to be searched and reduce the amount of calculation. In order to take advantage of this, the present invention uses a chromosomal expression that reflects the degree of closeness of the distances between cities, rather than expressing the solution of the problem in a sequence of cities. That is, each chromosome is represented by a number that represents the rank with the lowest cost among the objects. This allows for efficient and effective modification of the lethal gene when it occurs.

[0021]

EXAMPLE An example of the present invention will be described below. First, the expression method of chromosomes will be explained. In this embodiment,
Given n cities, number i (i = 1, 2,
..., n) are attached. It is assumed that the city to which the tour travels next to the city with the number i is the city closest to the city _i . Chromosomes that are candidate solutions to the problem
These are represented by the sequence {a _i , i = 1, 2, ..., N}.

FIG. 2 is a diagram for explaining a method of expressing a chromosome in one embodiment. As shown in FIG. 2A, in the four cities A, B, C, and D, the distance relationship between the cities is as follows: distance (AD) ≦ distance (AB) ≦ distance (BD) ≦ distance (C
D) ≦ distance (BC).

From the relationship of distances shown in FIG.
A cost ranking table for ranking proximity relationships between cities as shown in (B) is created. This cost ranking table indicates that the distance from the city A is closer to the city A in the order of city D, city B, and city C from the distance relationship between the cities. Similarly, when viewed from city B, it is shown that city A, city D, and city C are closer in this order. Here, "City A → City B → City C → City D →
The expression of the chromosome representing the tour that goes around in the order of “city A” is as follows from the cost ranking table shown in FIG.

First, regarding city A → city B,
Next, city B where the tour goes is the second closest to city A
Because it is a city, a in city A_A= 2. Next, the capital
About city B → city C, city C is the third closest to city B
It is a big city, so in city B a_B= 3. Further
For city C → city D, city D is the first from city C
Since it is a city close to the eyes, in city C a_C= 1,
Regarding city D → city A, city A is the first from city D
Since it is a near city, in city D a_D= 1. Follow
"City A-> City B-> City C-> City D-> City A"
Representing chromosome [a _Aa_Ba_Ca_D] Is [2 ・ 3 ・ 1 ・
1] (see FIG. 2C).

Similarly, considering a chromosome that expresses a tour that goes around in the order of "city A → city D → city C → city B → city A", the next to city A is the city D closest to city A. a
_A = 1, next to city B is city A closest to city B, and a _B =
1, next to city C is city B, which is the second closest to city _C , and a _C =
2, next to city D is city C, which is the third closest to city _D , and a _D =
Therefore, the chromosome [a _A a _B a
_C a _D ] becomes [1, 1, 2, 3] (see FIG. 2D).

Next, the case of the problem of 30 cities will be described. FIG. 3 is a diagram showing an example of a cost ranking table showing the ranking of the degree of proximity between cities in the traveling salesman problem in 30 cities. A cost ranking table showing such relationships between cities is created in advance. This cost rank table is usually expressed as an array on a computer, but the expression format is arbitrary as long as it has similar information about the rank.

For each city, the nearest city is ranked first,
It represents the rank and the cost to reach that city. In FIG. 3, 1 to 30 are numbers given to city names.
The cost is shown in parentheses, and here the value of the distance between cities is expressed as a positive number. For example, in city 1, the closest city is city 30, and the distance value between city 1 and city 30 is 4. Second place is city 2, city 1 and city 2
Has a distance value of 5.

The first city of the tour is city 1. For example, if the k-th closest city to the city i is represented by n (i, k), the number of the second city to follow in the tour is n (1,
a ₁ ), the number of the third city to follow is n (n (1, a
₁ ), a ₂ ), etc.

The correspondence between the chromosome and the solution will be described. For example, chromosome _{[a 1 · a 2 · a} 3 · ... · a 30] = [1 · 3 · 1 · 8 ·
9 ・ 2 ・ 8 ・ 1 ・ 1 ・ 2 ・ 4 ・ 1 ・ 2 ・ 1 ・ 2 ・ 3 ・ 1
・ 9 ・ 3 ・ 13 ・ 5 ・ 1 ・ 3 ・ 3 ・ 5 ・ 1 ・ 4 ・ 2 ・ 1
・ 2]

Since a ₁ = 1, the city which the tour follows second after city 1 is n from the cost ranking table of FIG.
(1,1) = city 30. Also, the city that the tour follows the third is a ₃₀ = 2, so n (30,2) =
It is city 2. The city the tour will follow fourth is a ₂ =
Since it is 3, n (2,3) = city 25. By repeating these, the order of cities traced by this chromosomal expression is as follows.

[1 → 30 → 2 → 25 → 29 → 28 → 26
→ 27 → 24 → 23 → 29 → 15 → 17 → 16 → 18 →
22 → 21 → 20 → 14 → 13 → 12 → 11 → 10 → 6
→ 9 → 8 → 7 → 3 → 4 → 5 → 1] In other words, in the present invention, the tour that follows in such an order is represented by a chromosome that reflects the degree of closeness between cities as described above. To do.

Next, the flow of processing in the embodiment will be described. FIG. 4 is a diagram showing an outline of the processing flow in the embodiment. The flow of processing in the problem of 30 cities will be described with reference to FIGS. 4 to 8.

Cost ranking table creation (FIG. 4) The cost ranking table creation processing unit 1 is shown in FIG. 3 in which, based on the distance information between the given cities, other cities are arranged in the order close to the city. Create a ranking table as shown.

Initial population generation The initialization processing unit 2 generates an initial population of chromosomes that represent solution candidates. Here, the sequence {a _i , i = 1, 2, ..., N} based on the cost ranking table created in the process of
Is generated by a random number. If a city that appears once is selected again, select a city that is not yet selected. From experience, it is closer to the optimal solution that a _i is selected to be a value closer to 1 than n.

Chromosome selection The chromosome selection processing means 3 selects a chromosome according to the fitness of each chromosome. Selection is performed using ordinary roulette selection or rank selection.

Crossover operation and correction processing Crossover operation and correction processing of the selected chromosome are performed. For the crossover, an ordinary method such as one-point crossover or multipoint crossover is used. However, after the crossover, a lethal gene occurs that the same city appears twice, so it is necessary to correct it. Therefore, as in the case of the initialization process, when a city that once appeared is selected again, it is sufficient to select a city that has not been selected. An example of correction when the same city is selected twice as at the one-point crossover will be described according to a specific example.

As shown in FIG. 5A, it is assumed that chromosome A and chromosome B are selected as parents. Chromosome A (parent A) is [2 ・ 2 ・ 1 ・ 2 ・ 2 ・ 9 ・ 3 ・ 1 ・ 1 ・ 3 ・ 4 ・ 1 ・
2 ・ 1 ・ 1 ・ 1 ・ 5 ・ 3 ・ 1 ・ 1 ・ 5 ・ 1 ・ 9 ・ 2 ・ 1
・ 1 ・ 6 ・ 2 ・ 1 ・ 3], chromosome B (parent B) is [1 ・ 10 ・ 1 ・ 18 ・ 2 ・ 9 ・ 1 ・ 2 ・ 3 ・ 2 ・ 4 ・
1, 2, 5, 1, 1, 3, 1, 2, 15, 5, 1, 1,
1, 2, 1, 2, 5, 2, 4, 4]. FIGS. 7A and 7B are diagrams showing tours represented by the chromosome A and the chromosome B shown in FIG. 5A.

FIGS. 5B and 5C are diagrams showing an example of correction when the same city is selected twice by performing one-point crossover. Here, it is assumed that the intersection point is given as 15 using a random number.

As shown in FIG. 5B, the first chromosome A
~ 15th gene [2 ・ 2 ・ 1 ・ 2 ・ 2 ・ 9 ・ 3 ・ 1 ・ 1 ・ 3 ・ 4 ・ 1 ・
2.1.1] and the 16th to 30th genes of chromosome B [1.3.1 / 21.5.5.15.11.11.21.2]
・ 5 ・ 2 ・ 4] and the chromosome A'of the child [2 ・ 2 ・ 1 ・ 2 ・ 2 ・ 9 ・ 3 ・ 1 ・ 1 ・ 3 ・ 4 ・ 1 ・
2 ・ 1 ・ 1 ・ 1 ・ 3 ・ 1 ・ 2 ・ 15 ・ 5 ・ 5.1.1 ・ 1 ・ 1 ・
2, 1, 2, 5, 2, 4, 4] are made. FIG. 8A is a diagram showing the tour represented by the chromosome A ′.

When the tour city represented by the chromosome A'is traced in order from the city 1 to the tour city with reference to the cost ranking table of FIG. 3, city 1 → city 2 → city 30 → city 29 → city 26 → city 27 → city 28 → city 24 → city 25 → city 2 ...

Here, the city 2 represented by the 25th gene on the chromosome A'has already been selected in the second tour (FIG. 8 (A): dotted arrow). Similarly, in Figure 5 (B) *
As shown by the mark, the cities indicated by the 17th to 20th genes are also already selected.

Therefore, the gene (lethal gene) expressing the already selected city is corrected to the rank of the city closest to that city among the cities not yet selected.
For example, regarding the 25th gene of chromosome A ′, a city closest to the city 25 among the cities not yet selected is searched for by referring to the cost ranking table shown in FIG. Is applicable. Therefore, the 25th gene is changed from "2" to "6". As a result, the tour is conducted from city 25 to city 23 as shown by the solid line in FIG.
Will be modified to proceed to.

Similarly, since the 17th to 20th chromosomes A'are also lethal genes expressing already selected cities, similar corrections are made to these genes. As a result of the above correction processing, the chromosome A ′ is, as shown in FIG. 5 (C), [2,2,1,2,2,9,3,1,1,3,4,1.
2 ・ 1 ・ 1 ・ 1 ・ 5 ・ 2 ・ 1 ・ 1 ・ 5 ・ 1 ・ 1.1 ・ 6
・ 1, 2, 5, 2, 4, 4].

On the other hand, as shown in FIG.
16th to 30th genes of [1/5 ・ 3 ・ 1 ・ 1 ・ 5 ・ 1 ・ 9 ・ 2 ・ 1 ・ 1 ・ 6 ・
2 ・ 1 ・ 3] and the 1st to 15th genes on chromosome B [1/10 ・ 1 ・ 18 ・ 2 ・ 9 ・ 1 ・ 2 ・ 3 ・ 2 ・ 4 ・
1 ・ 2 ・ 5 ・ 1] and the chromosome B'of the child [1 ・ 10 ・ 1 ・ 18 ・ 2 ・ 9 ・ 1 ・ 2 ・ 3 ・ 2 ・ 4 ・
1, 2, 5, 1, 1, 5, 3, 1, 1, 5, 1, 9, 2
・ 1 ・ 1 ・ 6 ・ 2 ・ 1 ・ 3] is generated.

4th, 9th, 14th of chromosome B '
Since the second gene indicates the city already selected (Fig. 6
(B): *), these genes are modified by the same method as above. As a result, as shown in FIG. 6 (C), the chromosome B ′ is [1 · 10 · 1 · 4 · 2 · 9 · 1 · 2 · 13 · 2 · 2 · 4.
1 ・ 2 ・ 1 ・ 1 ・ 1 ・ 5 ・ 3 ・ 1 ・ 1 ・ 5 ・ 1 ・ 9 ・ 2
・ 1 ・ 1 ・ 6 ・ 2 ・ 1 ・ 3] FIG. 8B is a diagram showing the tour represented by the chromosome B ′.

Mutation Operation and Correction Processing Next, mutation operation and correction processing are performed. Mutation operation is performed on an arbitrary gene by a method as used in a general genetic algorithm. When the lethal gene is generated by the mutation operation, the correction process described in the process of is performed.

Generation Number Judgment It is judged whether or not the processing from the above processing is repeated for a predetermined number of generations, and if the predetermined number of generations has been reached, the processing ends, and if not, Return to the processing, and repeat the processings of to. Alternatively, instead of determining the number of generations, the degree of convergence of the solution indicated by the chromosome group may be determined and the processing may be terminated.

Here, the correction process in the crossover operation has been described in detail. However, even when the initialization or mutation operation is performed, by selecting the proximity order to be selected within a certain range, it is possible to unnecessarily move between cities far away from each other. The connection can be avoided.

FIG. 9 is a diagram showing an example of a tour in the problem of 30 cities of this embodiment. FIG. 9A shows a tour represented by the optimal chromosome in the early generation chromosome group, and the cost (distance value) is 473. Figure 9
(B) shows an optimal solution in the 358th generation, and its cost (distance value) is 420.

In the above description of the embodiment, the case where the distance between cities is used as the cost between objects has been described, but other evaluation factors may be used instead of the distance. Moreover, the present invention can be similarly applied to the case where various traveling conditions exist, such as the arrival time restriction.

[0051]

As described above, according to the present invention,
Since it is possible to avoid the connection between cities that are far apart,
The solution space of the optimization problem to be searched can be narrowed and the calculation speed can be improved. In addition, no special crossover method is required to prevent the occurrence of lethal genes.

[Brief description of drawings]

FIG. 1 is a principle configuration diagram of the present invention.

FIG. 2 is a diagram illustrating chromosomal expression in an example.

FIG. 3 is a diagram showing a cost ranking table in which other cities are arranged in the order of being closer to each city.

FIG. 4 is a diagram illustrating a processing flow according to the embodiment.

FIG. 5 is a diagram illustrating a crossover operation and a correction process in the embodiment.

FIG. 6 is a diagram illustrating a crossover operation and a correction process in the embodiment.

FIG. 7 is a diagram showing a tour expressed by a chromosome.

FIG. 8 is a diagram showing a tour represented by a chromosome.

FIG. 9 is a diagram showing a tour represented by a chromosome of an optimal solution in an example.

FIG. 10 is an explanatory diagram of a traveling salesman problem.

[Explanation of symbols]

 1 cost ranking table preparation processing means 2 initialization processing means 3 chromosome selection processing means 4 crossover / mutation processing means 5 gene correction processing means

 ─────────────────────────────────────────────────── --- Continuation of the front page (72) Kazuhiro Matsuo, Kazuhiro Matsuo, Kanagawa Prefecture Nakazaki-ku, Nakahara-ku, 1015 Kamiodanaka, Fujitsu Limited (72) Inventor, Yukiko Yoshida 1015, Kamiodanaka, Nakahara-ku, Kawasaki, Kanagawa Prefecture, Fujitsu Limited

Claims (3)

[Claims] 1. When n objects and costs between the objects are given, a tour path that minimizes the sum of costs of paths that go through all of the objects without overlapping, In a genetic algorithm execution device that calculates using a genetic algorithm, a cost ranking table creation process that creates a table in which other objects are ranked such that the costs from that object to other objects are arranged in ascending order for each object Means, and an initialization processing means for generating a chromosome group expressing a chromosome by using the ranks of the inter-target costs in the table created by the cost order table creating processing means as genes when executing the genetic algorithm; Of the chromosomes selected according to the fitness and the chromosomes selected by the chromosome selection processing means. When a crossover / mutation processing means for performing mutation or both of them and an object of the order indicated by a gene overlap as a result of the operation of crossover or mutation of chromosomes, the gene expression of that gene is given to that object. A gene algorithm execution device, comprising: a gene correction processing unit that changes the rank of an unselected target in the rank of inter-target costs. 2. When n objects and costs between the objects are given, a traveling path that minimizes the total cost of paths that go through all of these objects without overlapping is calculated. In the method for executing a genetic algorithm, the cost for connecting a target chromosome, which is the target of selection, crossover, or mutation in the execution of the genetic algorithm, between the target and the next target is Expressed in the sequence of the number that is the second smallest, when a crossover or mutation operation causes a lethal gene in which the object appears in duplicate in the expression of the chromosome, for one gene indicating the overlapping object, Genetically characterized by eliminating lethal genes by selecting a selectable target in the order of lower cost and replacing one gene of the duplicated target Algorithm execution method. 3. The method for executing a genetic algorithm according to claim 2, wherein for each of the n objects to be cycled in advance, the other objects are ranked so that they are arranged in ascending order of cost from the object to the other object. A method for executing a genetic algorithm, characterized in that when a cost ranking table is created and the lethal genes are eliminated, the cost ranking table is used to select a target with a lower cost.