Optimum solution search method and optimum solution search apparatus as well as storage medium in which optimum solution search program is stored

[0002] 1. Field of the Invention

[0003] This invention relates to an optimal solution search method and an optimal solution search apparatus as well as a storage medium in which an optimal solution search program is stored, suitable for use for optimization of a system using genetic algorithms (GAs), and contemplates provision of a technique for allowing genetic algorithms to be executed at a high speed to allow a high speed search for an optimal solution to a problem.

[0004] 2. Description of the Related Art

[0005] In recent years, optimal solution search apparatus which search for an optimal solution to a problem using genetic algorithms have been conceived from a principle of biological evolution (selection (reproduction), mutation or crossover).

[0006] An optimal solution search apparatus which employs genetic algorithms, represents candidates for solution to a problem as chromosomes. Each chromosome is an arrangement of genes. The apparatus performs various genetic operations for the chromosomes to search for one of the optimal solutions each of which is obtained as one of the chromosomes having the highest fitness value.

[0007] It is to be noted that the fitness value is calculated from a gene arrangement of a chromosome and represents a degree of fitness to a problem. That is, a degree of quality of a solution represented by a chromosome.

[0008] An optimal solution search apparatus of the type mentioned above is typically constructed such that, at the end (or start) of a generation, a fitness value of a gene arrangement obtained as a result of the performance of genetic operations is calculated.

[0009] The optimal solution search apparatus has a problem in that, when genetic algorithms are executed, for calculation processing of fitness values is time consuming.

[0010] Particularly, when fitness values are calculated for gene arrangements obtained as a result of the performance of genetic operations (as in the optimal solution search apparatus described above)) a large amount of time is required for calculation processing of fitness values.

[0011] The optimal solution search apparatus has another problem in that, as the number of chromosomes which make an object of processing increases, or as the length of a chromosome which makes an object of processing increases, or as the calculation processing for fitness values becomes complicated, the time required for execution of genetic algorithms increases.

[0012] It is an object of the present invention to provide an optimal solution search method and apparatus wherein genetic algorithms can be executed at a high speed to allow an optimal solution to a problem to be searched for at a high speed.

[0013] It is another object of the present invention to provide a storage medium in which an improved optimal solution search program for operating a computer to search for an optimal solution is stored.

[0014] In order to attain the objects described above, according to an aspect of the present invention, there is provided an optimal solution search method wherein candidates for solution to a problem are represented as chromosomes, which are arrangements of genes and genetic operations, and are performed for individual chromosomes for each generation to successively update the generation to search for an optimal solution to the problem based on fitness values calculated from gene arrangements of the chromosomes, the optimal solution search method comprising the fitness value updating step of updating, for each of those chromosomes for which a particular genetic operation which allows a fitness value after execution thereof to be calculated readily making use of a fitness value before the execution thereof, and the fitness value based on variation information of the fitness value by execution of the particular genetic operation.

[0015] The optimal solution search method according to the present invention is advantageous in that, since the time required for calculation processing of fitness values can be reduced, genetic algorithms can be executed at a high speed and an optimal solution to a problem can be searched at a high speed.

[0016] According to another aspect of the present invention, there is provided an optimal solution search method wherein candidates for solution to a problem are represented as chromosomes, which are arrangements of genes and genetic operations, and are performed for individual chromosomes for each generation to successively update the generation to search for an optimal solution to the problem based on fitness values calculated from gene arrangements of the chromosomes, the optimal solution search method comprising the schedule determination step of determining, prior to execution of those genetic operations which remain before the end of a generation, an execution schedule of the genetic operations to be performed for each of the chromosomes, the chromosome discrimination step of referring to the execution schedules determined by the schedule determination step to discriminate, for each of the chromosomes, whether or not the chromosome is a particular chromosome for which only a particular genetic operation which allows a fitness value after execution thereof to be calculated readily making use of a fitness value before the execution thereof, and the fitness value updating step of updating, for each of those chromosomes which have been determined to be the particular chromosomes by the chromosome discrimination step, and the fitness value based on variation information of the fitness value by execution of the particular genetic operation.

[0017] In another aspect of the present invention, the fitness value updating step, the fitness values may be updated in synchronism with execution of the particular genetic operation.

[0018] In another aspect of the present invention, the fitness value updating step, upon execution of the particular genetic operation, variation information of the fitness value may be calculated by execution of the particular genetic operation.

[0019] In another aspect of the present invention, the fitness value updating step, upon execution of the particular genetic operation from among those genetic operations which make an object of determination in the schedule determination step, variation information of the fitness value may be calculated by execution of the particular genetic operation independently of a result of the discrimination in the chromosome discrimination step.

[0020] In another aspect of the present invention, in the fitness value updating step, variation information of the fitness value may be calculated by execution of the particular genetic operation after execution of an arbitrary one of those genetic operations which make an object of determination in the schedule determination step.

[0021] Upon execution of the particular genetic operation from among those genetic operations executed prior to one of the genetic operations which has been determined as an object of determination in the schedule determination step, variation information of the fitness value by execution of the particular genetic operation may be calculated.

[0022] In the schedule determination step, it is only required that the execution schedules be determined before the particular genetic operation appearing last is executed.

[0023] In another aspect of the present invention, in the schedule determination step, the execution schedules may be determined before the first genetic operation is executed, or before the particular genetic operation appearing first is executed, or before an arbitrary genetic operation is executed.

[0024] In another aspect of the present invention, in the schedule determination step, the execution schedules of genetic operations may be determined for the individual chromosomes, or an execution schedule of genetic operations which is common to all of the chromosomes may be determined.

[0025] In another aspect of the present invention, in the chromosome discrimination step, the execution schedules determined by the schedule determination step and also execution conditions of those genetic operations which have been executed prior to a genetic operation determined as an object of determination by the schedule determination step may be referred to to discriminate, for each of the chromosomes, whether or not the chromosome is a chromosome for which only the particular genetic operation should be performed.

[0026] In another aspect of the present invention, in the chromosome discrimination step, for each of the chromosomes, whether or not the chromosome is a chromosome for which the particular genetic operation should be performed may be discriminated before the particular genetic operation which appears last is executed.

[0027] In this instance, in the chromosome discrimination step, for each of the chromosomes, whether or not the chromosome is a chromosome for which the particular genetic operation should be performed may be discriminated before the first genetic operation is executed, or before the particular genetic operation which appears first is executed, or after an arbitrary genetic operation is executed.

[0028] In another aspect of the present invention, in the chromosome discrimination step, for each of the chromosomes, whether or not the chromosome is a chromosome for which only the particular genetic operation should be performed may be discriminated before the fitness value is updated by the fitness value updating step after the last genetic operation is executed.

[0029] With the optimal solution search method of the present invention, for each particular chromosome for which only the particular genetic operation which allows a fitness value after execution thereof to be calculated readily making use of a fitness value before the execution thereof, the fitness value is updated based on variation information of the fitness value by execution of the particular genetic operation. Consequently, the time required for calculation processing of fitness values can be reduced. Accordingly, the optimal solution search method is advantageous in that genetic algorithms can be executed at a high speed and an optimal solution to a problem can be searched at a high speed.

[0030] According to a further aspect of the present invention, there is provided an optimal solution search apparatus wherein candidates for solution to a problem are represented as chromosomes which are arrangements of genes and genetic operations are performed for individual chromosomes for each generation to successively update the generation to search for an optimal solution to the problem based on fitness values calculated from gene arrangements of the chromosomes, the optimal solution search apparatus comprising schedule determination means for determining, prior to execution of those genetic operations which remain before the end of a generation, an execution schedule of the genetic operations to be performed for each of the chromosomes, chromosome discrimination means for referring to the execution schedules determined by the schedule determination means to discriminate, for each of the chromosomes, whether or not the chromosome is a particular chromosome for which only a particular genetic operation which allows a fitness value after execution thereof to be calculated readily making use of a fitness value before the execution thereof, and fitness value updating means for updating, for each of those chromosomes which have been determined to be the particular chromosomes by the chromosome discrimination means, the fitness value based on variation information of the fitness value by execution of the particular genetic operation.

[0031] The schedule determination means may include a table for storing the execution schedules.

[0032] The chromosome discrimination means may include a table for storing information representing whether or not genetic operations to be performed for the individual chromosomes are the particular genetic operation, and another table for storing information representing whether or not the individual chromosomes are chromosomes for which only the particular genetic operation should be performed.

[0033] The optimal solution search apparatus of the present invention may further comprise updating pre-processing means for calculating, when the particular genetic operation from among those genetic operations executed prior to a genetic operation determined as an object of determination by the schedule determination means, variation information of the fitness value by execution of the particular genetic operation.

[0034] In another aspect of the present invention, the optimal solution search apparatus of the present invention may further comprise fitness value calculation means for calculating, for each of those chromosomes which have not been discriminated to be the particular chromosome by the chromosome discrimination means, a fitness value from a gene arrangement of the chromosome.

[0035] In another aspect of the present invention, the optimal solution search apparatus of the present invention may further comprise genetic operation execution means for performing the genetic operations for the individual chromosomes.

[0036] In another aspect of the present invention, the optimal solution search apparatus of the present invention may further comprise optimal solution discrimination means for discriminating an optimal solution to the problem based on the fitness value obtained by the fitness value updating means and the fitness value calculation means.

[0037] In another aspect of the present invention, the optimal solution search apparatus of the present invention may further comprise chromosome production means for representing candidates for solution to the problem as chromosomes which are arrangements of genes.

[0038] With the optimal solution search apparatus of the present invention, for each particular chromosome for which only the particular genetic operation which allows a fitness value after execution thereof to be calculated readily making use of a fitness value before the execution thereof, the fitness value is updated based on variation information of the fitness value by execution of the particular genetic operation. Consequently, the time required for calculation processing of fitness values can be reduced. Accordingly, the optimal solution search apparatus is advantageous in that genetic algorithms can be executed at a high speed and an optimal solution to a problem can be searched at a high speed.

[0039] According to a still further aspect of the present invention, there is provided a storage medium which stores an optimal solution search program for causing a computer to operate so that candidates for solution to a problem are represented as chromosomes which are arrangements of genes and genetic operations are performed for individual chromosomes for each generation to successively update the generation to search for an optimal solution to the problem based on fitness values calculated from gene arrangements of the chromosomes, wherein the optimal solution search program causes the computer to function as schedule determination means for determining, prior to execution of those genetic operations which remain before the end of a generation, an execution schedule of the genetic operations to be performed for each of the chromosomes, chromosome discrimination means for referring to the execution schedules determined by the schedule determination means to discriminate, for each of the chromosomes, whether or not the chromosome is a particular chromosome for which only a particular genetic operation which allows a fitness value after execution thereof to be calculated readily making use of a fitness value before the execution thereof, and fitness value updating means for updating, for each of those chromosomes which have been determined to be the particular chromosomes by the chromosome discrimination means, the fitness value based on variation information of the fitness value by execution of the particular genetic operation.

[0040] With the storage medium which stores the optimal solution search program of the present invention, for each particular chromosome for which the particular genetic operation which allows a fitness value after execution thereof to be calculated readily making use of a fitness value before the execution thereof, the fitness value is updated based on variation information of the fitness value by execution of the particular genetic operation. Consequently, the time required for calculation processing of fitness values can be reduced. Accordingly, the storage medium which stores the optimal solution search program is advantageous in that genetic algorithms can be executed at a high speed and an optimal solution to a problem can be searched at a high speed.

[0041] FIG. 1 is a block diagram showing a functional construction of an optimal solution search apparatus according to a preferred embodiment of the present invention.

[0042] FIG. 2 is a block diagram showing a detailed construction of the optimal solution search apparatus.

[0043] FIG. 3 is a schematic view showing a general construction of the optimal solution search apparatus of FIG. 1.

[0044] FIG. 4 is a diagrammatic view illustrating genetic algorithms.

[0045] FIG. 5 is a diagrammatic view illustrating execution timings of execution schedule determination processing and chromosome discrimination processing.

[0046] FIG. 6 is a table illustrating an example of an execution schedule produced by a schedule determination section.

[0047] FIG. 7 is a diagrammatic view illustrating the chromosome discrimination processing to be performed by a chromosome discrimination section.

[0048] FIG. 8 is a flow chart illustrating operation of the optimal solution search apparatus of FIG. 1.

[0049] FIGS. 9, 10 and 11 are diagrammatic views showing different an example of a result of a search for an optimal solution by the optimal solution search apparatus of FIG.

[0050] FIGS. 12 to 14 are diagrammatic views illustrating execution timings of various processes by a genetic algorithm execution section.

[0051] a. Embodiment of the Invention

[0052] An embodiment of the present invention will be described below with reference to the drawings.

[0053] Referring first to FIG. 1, there is shown in block diagram a functional construction of an optimal solution search apparatus according to a preferred embodiment of the present invention.

[0054] The optimal solution search apparatus shown is generally denoted at 1 and includes a chromosome production section 2 and a genetic algorithm execution section 3. The optimal solution search apparatus 1 represents candidates for solution to a problem as chromosomes “j” (j=1 to n) which are arrangements of genes, and performs genetic operations OP_k(k=1 to m) for the chromosomes “j” for each generation to successively update the generation to search for an optimal solution to the problem based on fitness values f_j′ calculated from gene arrangements of chromosomes “j′”.

[0055] Here, prior to description of the optimal solution search apparatus 1 according to the present embodiment, several terms used in the description of the embodiment are explained.

[0056] The genetic algorithms (GAs) are algorithms conceived from a principle of biological evolution (selection (reproduction), mutation or crossover) and can be considered as a technique for probabilistic search, learning and optimization.

[0057] The chromosome exists as a substance which conveys genetic information. The chromosome is, in an actual organism, a physical substance composed of bases, but in genetic algorithms, a data area or a data array in the form of a code of candidates for solution to a problem. It is to be noted that a set of data represented by a data area or a data array is called individual, and components of a data area or a data array are called genes.

[0058] Further, the fitness values are values calculated from a gene arrangement of chromosomes and represent degrees of fitness to a problem, that is, degrees of quality of solutions represented by chromosomes as described hereinabove. It is to be noted that the fitness values are calculated using a fitness function. Further, since the fitness values are used when selection from among genetic operations which will be hereinafter described is performed, it must be calculated prior to such selection.

[0059] By the way, in the genetic algorithms, in order to vary a gene arrangement of chromosomes to produce chromosomes which may possibly make an optimal solution to a problem, various genetic operations (OP_k) are performed for individual chromosomes as seen in FIG. 4.

[0060] The genetic algorithms popularly use three different genetic operations including selection, crossover and mutation.

[0061] The selection (reproduction) is an operation of probabilistically determining a group of individuals to live in a next generation in accordance with a distribution of fitness values of a group, and basically acts so that individuals having higher fitness values leave larger numbers of descendants. Consequently, genes (or gene arrangements) which form better individuals are spread in the group. It is to be noted that the selection merely selects individuals from within a group, but does not produce new genes (or gene arrangements), and the crossover or mutation produces new genes on individuals.

[0062] The crossover is an operation of recombining chromosomes of two parents to produce chromosomes of a child. If good partial characters of parents can be successfully inherited in combination by a child, then remarkable augmentation in search can be anticipated.

[0063] The mutation is an operation of varying genes in a chromosome in a fixed probability to produce a new gene. The mutation in most cases destructs an effective gene arrangement if the mutation probability is set to a high value, but where no mutation is involved, it is generally considered that only chromosomes as possible combinations of genes of chromosomes in an initial stage can be obtained, and also the quality of a solution obtained is subject to some limitation. Therefore, the mutation is effective to maintain the multiplicity of a group (or to improve the solution).

[0064] Subsequently, the optimal solution search apparatus 1 according to the present embodiment will be described with reference to FIG. 1. The chromosome production section 2 represents candidates for solution to a problem as chromosomes “j” which are arrangements of genes to produce chromosomes “j”.

[0065] Meanwhile, the genetic algorithm execution section 3 executes genetic algorithms.

[0066] In particular, the genetic algorithm execution section 3 performs genetic operations OP_kfor chromosomes “j” produced by the chromosome production section 2 to successively update the generation and searches for an optimal solution to a problem based on fitness values f_j′ calculated from gene arrangements of chromosomes “j′”, and includes a schedule determination section 10, a chromosome discrimination section 11, a genetic operation execution section 12, a fitness value updating section 13, a fitness value calculation section 14, an optimal solution discrimination section 15 and an updating pre-processing section 19.

[0067] FIGS. 12 to 14 illustrate execution timings of various processes by the genetic algorithm execution section 3. It is to be noted that, in FIGS. 12 to 14, those execution timings adopted popularly are indicated by screen indications.

[0068] The schedule determination section 10 determines, for example, at the start of each generation, execution schedules of genetic operations OP_kto be performed for individual chromosomes “j” before the end of the generation in accordance with a probabilistic technique.

[0069] In particular, the schedule determination section 10 determines which ones of genetic operations OP_kincluding crossover, mutation and so forth should be performed for the individual chromosomes “j”, and further determines forms in which the genetic operations OP_kare performed (for example, for the crossover, a crossover probability and a crossover type (one-point crossover, two-point crossover or the like), or for the mutation, a mutation probability and a mutation type (exchange of two genes or the like)), and so forth.

[0070] The schedule determination section 10 includes an execution schedule storage table 16 into which execution schedules determined are stored.

[0071] An example of execution schedules determined by the schedule determination section 10 is shown in FIG. 6. According to the schedules shown in FIG. 6, for the chromosome “1”, a genetic operation OP₁is executed, a genetic operation OP₂is not executed, a genetic operation OP₃is executed, . . . , and a genetic operation OP_mis executed; for the chromosome “2”, the genetic operation OP₁is not executed, the genetic operation OP₂is not executed, the genetic operation OP₃is executed, . . . , and the genetic operation OP_mis executed; for the chromosome “n”, the genetic operation OP₁is not executed, the genetic operation OP₂is executed, the genetic operation OP₃is not executed, . . . , and the genetic operation OP_mis not executed.

[0072] It is to be noted that FIGS. 12 to 14 show another example of an execution schedule determined by the schedule determination section 10, wherein genetic operations capable of being performed for the individual chromosomes “j” are genetic operations OP₁to OP₁₂(genetic operations OP_k; k=1 to 12).

[0073] While, in the present embodiment, as seen in FIG. 6, the schedule determination section 10 determines execution schedules of the genetic operations OP_kfor the individual chromosomes “j”, it may otherwise determine an execution schedule of the genetic operations OP_kwhich is common to all of the chromosomes “j”.

[0074] According to an ordinary optimal solution search apparatus, since contents of each of the genetic operations OP_kto be performed for each of the chromosomes “j” before the end of the generation are determined in accordance with a probabilistic technique every time immediately before the genetic operation OP_kis executed, an execution schedule is not determined in advance.

[0075] However, in the optimal solution search apparatus of the present embodiment, if an execution schedule of the genetic operations OP_kis not determined in advance, then processing by the chromosome discrimination section 11 which will be hereinafter described cannot be executed. Therefore, an execution schedule of the genetic operations OP_kis produced by the schedule determination section 10.

[0076] It is to be noted that the determination processing for an execution schedule by the schedule determination section 10 may be performed before a particular genetic operation which appears last is executed. The particular gene operation will be hereinafter described.

[0077] In the present embodiment, the determination processing for an execution schedule by the schedule determination section 10 will be described by way of an example wherein it is performed at the start of each generation (before the first genetic operation OP₁is executed) (refer to “CASE 1” of “EXECUTION SCHEDULE DETERMINATION PROCESSING” in FIG. 12).

[0078] The genetic operation execution section 12 actually performs the genetic operations OP_kfor the individual chromosomes “j” in accordance with execution schedules determined by the schedule determination section 10 to produce chromosomes “j” as descendants of the chromosomes “j”.

[0079] Meanwhile, the chromosome discrimination section 11 refers to execution schedules determined by the schedule determination section 10 to discriminate, for each of the chromosomes “j”, whether or not it is a chromosome for which only a particular gene operation by which a fitness value f_jk′ after execution can be calculated readily making use of a fitness value f_jkprior to the execution should be performed.

[0080] Here, the particular genetic operation signifies a genetic operation of a type by which a variation amount Δf_jkof the fitness value by execution of the particular genetic operation can be calculated readily by a difference calculation or the like (including some other calculation processing such as a multiplication calculation). For example, from among the three kinds of genetic operations OP_k(selection, crossover and mutation), the mutation corresponds to this.

[0081] It is to be noted that, if a location of a chromosome “j” at which the gene arrangements changes, a manner in which the gene arrangement has changed and so forth become known, then the variation amount Δf_jkof the fitness value can be calculated by performing a difference calculation or the like taking a relationship of the chromosome “j” to gene arrangements positioned forwardly and rearwardly of the location.

[0082] In particular, the particular genetic operation signifies a genetic operation by which, for each chromosome “j”, a fitness value f_jk′ after execution of the particular genetic operation can be calculated readily by reflecting a variation amount Δf_jkof the fitness value by execution of the particular genetic operation upon the fitness value f_jkbefore the execution of the particular genetic operation. It is to be noted that, in the figures of the accompanying drawings, the particular genetic operation is indicated as “EASY OP”.

[0083] More particularly, the chromosome discrimination section 11 includes an attribute information storage table 17 (refer to FIG. 7) for the genetic operations OP_kand a chromosome discrimination information storage table 18 (refer to FIG. 7). It is to be noted that an example of the attribute information storage table 17 and the chromosome discrimination information storage table 18 is shown in FIG. 7.

[0084] The attribute information storage table 17 stores attribute information of the genetic operations OP_kwhich represents whether or not the genetic operations OP_kwhich can be performed for the individual chromosomes “j” are the particular genetic operation. The chromosome discrimination information storage table 18 stores chromosome discrimination information which represents whether or not the chromosomes “j” are chromosomes for which only the particular genetic operation should be permitted.

[0085] The chromosome discrimination section 11 performs such processing as will be described below to discriminate, each of the chromosomes “j”, whether or not it is a chromosome to which only the particular genetic operation should be performed (refer to FIG. 7).

[0086] In particular, the chromosome discrimination section 11 first refers to the execution schedule storage table 16 to recognize an execution schedule, for example, for the chromosome “1” and then refers to the attribute information storage table 17 to discriminate whether or not all of the genetic operations OP₁, OP₃, . . . and OP_mwhich should be performed for the chromosome “1” are the particular genetic operation (EASY OP).

[0087] In this instance, since all of the genetic operations OP₁, OP₃, . . . and OP_mare the particular genetic operation, Y (Yes) is set into a place of the chromosome discrimination information storage table 18 corresponding to the chromosome “1” in order to indicate that the chromosome “1” is a chromosome for which only the particular genetic operation should be performed.

[0088] Then, such processing is performed for all of the chromosomes “j”. It is to be noted that, in order to indicate that a chromosome “j” which makes an object of processing is not a chromosome for which only the particular genetic operation should performed, N (No) is set into a place of the chromosome discrimination information storage table 18 which corresponds to the chromosome “j”.

[0089] In particular, the chromosome discrimination section 11 performs such processing as to refer to the execution schedule storage table 16 and the attribute information storage table 17 of the schedule determination section 10 to produce the chromosome discrimination information storage table 18 as seen in FIG. 7.

[0090] It is to be noted that, while, in FIGS. 12 to 14, information of the particular genetic operation (EAST OP) from among the attribute information of the genetic operations OP_k(in FIGS. 12 to 14, k=1 to 12) is described simultaneously in locations for the genetic operations OP_k, the attribute information of the genetic operations OP_kis stored in a similar manner as described above in the attribute information storage table 17.

[0091] Further, the discrimination processing for a chromosome by the chromosome discrimination section 11 need not be performed simultaneously with the determination processing for an execution schedule by the schedule determination section 10.

[0092] In the present embodiment, the discrimination processing for a chromosome by the chromosome discrimination section 11 will be described by way of an example wherein it is performed at the start of each generation (before execution of the first genetic operation OP₁) but immediately after the determination processing for an execution schedule by the schedule determination section 10 is completed.

[0093] Further, the fitness value updating section 13 updates, for each chromosome “j” which has been discriminated by the chromosome discrimination section 11 to be a chromosome for which only the particular genetic operation should be performed (each chromosome for which Y is placed in the chromosome discrimination information storage table 18 of FIG. 7), the fitness value f_jbased on a variation amount Δf_jkof the fitness value by execution of the particular genetic operation to calculate a fitness value f_j′ of the chromosome “J” when the generation comes to an end (a fitness value of a chromosome “j′” which is a descendent of the chromosome “j” ).

[0094] In particular, the fitness value updating section 13 performs such processing as described below to calculate a fitness value f_j′ at the end of the generation of each chromosome “j” which has been discriminated to be a chromosome for which only the particular genetic operation should be performed.

[0095] First, every time the particular genetic operation is executed, the fitness value updating section 13 refers to the chromosome discrimination information storage table 18 (refer to FIG. 7) produced by the chromosome discrimination section 11 to discriminate whether or not a chromosome “j” which makes an object of processing is a chromosome for which only the particular genetic operation should be performed, that is, whether or not a variation amount Δf_jkof the fitness value of the chromosome “j” which is an object of processing by execution of the particular genetic operation should be calculated (refer to “CASE 1” of “DISCRIMINATION PROCESSING REGARDING CALCULATION OF Δf_jk” in FIG. 12).

[0096] Then, when the chromosome “j” which is an object of processing is a chromosome for which only the particular genetic operation should be performed, a variation amount Δf_jkof the fitness value by execution of the particular genetic operation is calculated after the particular genetic operation is executed, and the fitness value fj is updated based on the calculated variation amount Δf_jkof the fitness value, whereafter the fitness value updating section 13 waits until the next particular genetic operation is executed (refer to “CASE 1” of “FITNESS VALUE UPDATING PROCESSING” in FIG. 13).

[0097] This processing is repeated until execution of the last particular genetic operation is completed to calculate fitness values f_j′ at the end of the generation of those chromosomes “j” which have been determined to be chromosomes for which only the particular genetic operation should be performed. It is to be noted that whether or not an genetic operation OP_kexecuted is the particular genetic operation can be recognized by referring to the attribute information storage table 17 shown in FIG. 7.

[0098] The updating pre-processing section 19 calculates, when there is a genetic operation which has been executed prior to a genetic operation determined to be an object of determination by the schedule determination section 10, the variation amount Δf_jkof the fitness value by execution of the particular genetic operation after the particular genetic operation is executed.

[0099] This is because, although it cannot be discriminated before an execution schedule is determined whether or not a chromosome “j” which makes an object of processing when the particular genetic operation is to be executed prior to determination of the execution schedule is a chromosome for which only the particular genetic operation should be performed, if contents of the genetic operation or operations OP_kexecuted already are referred to after the execution schedule is determined, then it can be discriminated whether or not the chromosome “j” which is an object of processing is a chromosome for which only the particular genetic operation should be performed, and in this instance, if the variation amount Δf_jkof the fitness value by execution of the particular genetic operation executed already is not known, then the fitness values f_j′ of the chromosomes “j” at the end of the generation cannot be calculated.

[0100] It is to be noted that, where the determination processing for an execution schedule by the schedule determination section 10 is performed at the start of each generation, the updating pre-processing section 19 does not function since no generic operation has been executed prior to the genetic operation which is made an object of determination by the schedule determination section 10.

[0101] In the present embodiment, the determination processing for an execution schedule by the schedule determination section 10 is described by way of an example wherein it is performed at the start of each generation, and another example wherein the updating pre-processing section 19 functions will be hereinafter described (refer to “b. Others”).

[0102] Meanwhile, the fitness value calculation section 14 calculates, for each chromosome “j” which has been determined by the chromosome discrimination section 11 not to be a chromosome for which only the particular genetic processing should be performed, that is, for each chromosome for which N is set in the chromosome discrimination information storage table 18 of FIG. 7, from the gene arrangement of a chromosome “j′” which is a descendant of the chromosome “j”, a fitness value f_j′ at the end of the generation of the chromosome “j”, that is, a fitness value of the chromosome “j” (refer to “FITNESS VALUE CALCULATION PROCESSING” in FIG. 14).

[0103] More particularly, the fitness value calculation section 14 performs such processing as described below to calculate the fitness values f_j′ at the end of the generation of those chromosomes “j” which have been determined not to be chromosomes for which only the particular genetic operation should be performed.

[0104] First, after execution of all of the genetic operations OP_kis completed, the fitness value calculation section 14 refers to the chromosome discrimination information storage table 18 (refer to FIG. 7) to discriminate whether or not a chromosome “j′” which makes an object of processing is a chromosome for which only the particular genetic operation should be performed, that is, whether or not the chromosome “j′” is a chromosome for which the fitness value f_j′ after execution of the genetic operations OP_khas been calculated already by the fitness value updating section 13 described above.

[0105] Then, if the chromosome “j′” which is an object of processing is not a chromosome for which only the particular genetic operation should be performed, the fitness value calculation section 14 calculates the fitness value f_j′ at the end of the generation of the chromosome “j” from the gene arrangement of the chromosome “j′” similarly as in an ordinary optimal solution search apparatus.

[0106] Further, the optimal solution discrimination section 15 discriminates, based on the fitness values f_j′ obtained by the fitness value updating section 13 and the fitness value calculation section 14 described above, whether or not any of the chromosomes “j′” obtained as a result of the performance of all of the genetic operations OP_kto be performed has reached an allowable solution which is regarded as an optimal solution (including a solution which seems to be an optimal solution), or in other words, whether or not the genetic algorithms by the genetic algorithm execution section 3 should be ended.

[0107] Then, if the optimal solution discrimination section 15 discriminates that any of the chromosomes “j′” has reached an allowable solution which is regarded as an optimal solution or the genetic algorithms should be ended, it outputs one of the chromosomes “j′” which has the highest fitness value f_j′ as an optimal solution. However, if the optimal solution discrimination section 15 discriminates that none of the chromosomes “j′” has reached an allowable solution which is regarded as an optimal solution or the genetic algorithms should not be ended, it updates the generation and activates the schedule determination section 10 again.

[0108] It is to be noted that, for example, the following methods (1) to (3) are available as a method of discriminating whether or not the genetic algorithms should be ended.

[0109] (1) It is discriminated whether or not the highest fitness value f_j′ of chromosomes obtained from each generation has reached a desired value set in advance (including a fitness value corresponding to an allowable solution which is regarded as an optimal solution).

[0110] (2) For example, the increasing rate of the highest fitness value f_j′ of chromosomes when the generation is updated is referred to to discriminate whether or not the increasing rate has reduced to a value equal to or lower than a predetermined value.

[0111] (3) The number of generations for which the genetic algorithms should be performed is determined in advance, and it is discriminated whether or not execution for the number of generations has been completed.

[0112] The functions of the chromosome production section 2 and the genetic algorithm execution section 3, that is, the schedule determination section 10, chromosome discrimination section 11, genetic operation execution section 12, fitness value updating section 13, fitness value calculation section 14 and optimal solution discrimination section 15, described above are actually realized as operations of a CPU (central processing unit, computer) which executes an optimal solution search program stored in and read out from a storage medium such as a hard disk, an optical disk or a CD-ROM (compact disk read only memory) and stored in a memory such as a read only memory (RAM).

[0113] Here, the optimal solution search program in the present embodiment causes the CPU to function as chromosome production means (to which the chromosome production section 2 corresponds) which represents candidates for solution to a problem as chromosomes “j” which are arrangements of genes, schedule determination means (to which the schedule determination section 10 corresponds) which determines, before those genetic operations OP_kwhich remain by the end of the generation are executed, execution schedules of the genetic operations OP_kto be performed for the individual chromosomes “j”, chromosome discrimination means (to which the chromosome discrimination section 11 corresponds) which refers to the execution schedules determined by the schedule determination means to discriminate, for each of the chromosomes “j”, whether or not it is a particular chromosome for which such a particular genetic operation that can calculate a fitness value f_jk′ after execution readily making use of the fitness value f_jkbefore the execution, fitness value updating means (to which the fitness value updating section 13 corresponds) which updates, for each chromosome “j” which has been discriminated to be such a particular chromosome as mentioned above by the chromosome discrimination means, the fitness value f_jbased on variation information Δf_jkof the fitness value by execution of the particular genetic operation to obtain a fitness value f_j′, genetic operation execution means (to which the genetic operation execution section 12 corresponds) which performs the genetic operations OP_kfor the individual chromosomes “j”, fitness value calculation means (to which the fitness value calculation section 14 corresponds) which calculates, for each chromosome “j” which has not been discriminated to be such a particular chromosome as described above by the chromosome discrimination means, the fitness value f_j′ from the gene arrangement of the chromosome “j′”, and optimal solution discrimination means (to which the optimal solution discrimination section 15 corresponds) which discriminates an allowable solution which is regarded as an optimal solution to the problem based on the fitness values f_j′ obtained by the fitness value updating means and the fitness value calculation means.

[0114] A detailed construction of the optimal solution search apparatus 1 according to the present embodiment is shown in FIGS. 2 and 3.

[0115] Referring to FIG. 2, the optimal solution search apparatus 1 shown includes a display unit 5A for displaying various setting screens, a result of search for an optimal solution and so forth, a display control section 5 for controlling the display condition of the display unit 5A, an inputting section 7B such as a keyboard or a mouse operable by an operator who refers to data displayed on the display unit 5A for inputting response information to the displayed data, and an input control section 7D for controlling the inputting section 7B.

[0116] The optimal solution search apparatus 1 further includes a hard disk 8 which stores information such as an operating system (OS) for controlling operation of the optimal solution search apparatus 1 and stores the optimal solution search program in the present embodiment.

[0117] The optimal solution search apparatus 1 further includes an external file reading/writing section 6, an external file 6A, a printing section 7A and a printing control section 7C. The external file reading/writing section 6 or the printing section 7A records a result of search for an optimal solution or the like displayed on the display unit 5A into the external file 6A or on predetermined printing paper in response to an instruction from the inputting section 7B.

[0118] The optimal solution search apparatus 1 further includes a CPU 4 for collectively controlling the various components of the optimal solution search apparatus 1.

[0119] In particular, the optimal solution search apparatus 1 in the present embodiment can be realized using an ordinary computer system (computer; refer to FIG. 3) which includes the CPU 4, display unit 5A, display control section 5, external file reading/writing section 6, printing section 7A, printing control section 7C, inputting section 7B, input control section 7D, hard disk 8 and so forth described above. It is to be noted that reference numeral 9 in FIG. 3 denotes a computer main frame.

[0120] In the optimal solution search apparatus 1 of embodiment of the present invention having the construction described above, the chromosome production section 2 first represents candidates for solution to a problem as chromosomes which are arrangements of genes to produce chromosomes “j”.

[0121] Then, the genetic algorithm execution section 3 successively performs genetic operations OP_kfor the chromosomes “j” produced by the chromosome production section 2 to successively update the generation to search for an optimal solution to the problem based on fitness values f_j′ calculated from gene arrangements of chromosomes “j′”.

[0122] Here, the processing by the genetic algorithm execution section 3 will be described with reference to the flow chart shown in FIG. 8.

[0123] In the genetic algorithm execution section 3, after the generation i (i is an arbitrary natural number) is started, the schedule determination section 10 determines execution schedules of the genetic operations OP_kto be performed for the individual chromosomes “j” (j=1 to n) in accordance with a probabilistic technique (step A1).

[0124] Then, the chromosome discrimination section 11 refers to the execution schedule storage table 16 and the attribute information storage table 17 illustrated in FIG. 7 to discriminate whether or not those genetic operations to be performed for each of the chromosomes “j” are a combination only of particular genetic operations (EASY OP), that is, whether or not each of the chromosomes “j” is a chromosome for which only the particular genetic operation should be performed, and produces the chromosome discrimination information storage table 18 (step A2).

[0125] In this manner, after the determination processing for an execution schedule by the schedule determination section 10 (step A1) and the discrimination processing for a chromosome by the chromosome discrimination section 11 (step A2) are completed, the genetic operations OP_kare executed by the genetic operation execution section 12.

[0126] In particular, the genetic operation execution section 12 sets the variable “k” to “1” to select the genetic operation OP₁from among the genetic operations OP_k(step A3). Then, the genetic operation execution section 12 checks it based on the execution schedules determined by the schedule determination section 10 whether or not the genetic operation OP₁should be performed for each of the chromosomes “j”, and executes the genetic operation OP₁for each of those chromosomes “j” for which it is indicated that the genetic operation OP₁should be performed (step A4).

[0127] Further, the fitness value updating section 13 recognizes, based on the attribute information storage table 17 shown in FIG. 7, that the genetic operation OP₁executed is the particular genetic operation. Then, for each of those chromosomes “j” for which Y has been set in the chromosome discrimination information storage table 18 of FIG. 7, that is, for each of those chromosomes to which only the particular genetic operation should be performed, from among the chromosomes “j” for which the genetic operation OP₁has been performed, the fitness value updating section 13 updates the fitness value f_j1(whose initial value is a fitness value f_jat the start of the generation i) based on a variation amount Δf_j1of the fitness value by execution of the genetic operation OP₁(step A5).

[0128] In particular, for a chromosome “j”, where the fitness value prior to execution of a genetic operation OP_kas the particular genetic operation is represented by f_jk, the fitness value after execution of the genetic operation OP_kis represented by f_jk′, and the variation amount of the fitness value by the genetic operation OP_kis represented by Δf_jk, the fitness value f_jk′ after execution of the genetic operation OP_kis calculated (updated) in accordance with the following expression for calculation:

[0129] f_jk′=f_jk+Δf_jk

[0130] Then, at a point of time at which the execution of all of the genetic operations OP_kis completed, a fitness value f_j′ of the generation i after execution is calculated.

[0131] Further, the genetic operation execution section 12 increments the value of the variable “k” by one (step A6) and discriminates whether or not the value of the variable “k” is “m+1”, that is, whether or not execution of the last genetic operation OP_mhas been completed (step A7; refer to “DISCRIMINATION PROCESSING OF COMPLETION OF EXECUTION OF ALL OP OF GENERATION i” in FIG. 14).

[0132] In this instance, since execution only of the genetic operation OP₁has been completed, the processing in steps A4 to A7 described above is executed again until execution of the last genetic operation OP_mis completed (from the NO route of step A7 to step A4).

[0133] On the other hand, if execution of the last genetic operation OP_mis completed and it is discriminated in step A7 that the value of the variable “k” is “m+1”, then the fitness value calculation section 14 calculates, for each of the chromosomes “j” for which N is set in the chromosome discrimination information storage table 18 of FIG. 7, that is, for each chromosome for which not only the particular genetic operation should be performed, a fitness value f_j′ at the end of the generation from the gene arrangement of a chromosome “j′” which is a descendent of the chromosome “j” (step A8).

[0134] It is to be noted that, in the genetic algorithm execution section 3, the optimal solution discrimination section 15 discriminates, based on the fitness values f_j′ obtained by the fitness value updating section 13 and the fitness value calculation section 14, whether or not a chromosome which corresponds to an optimal solution to the problem is included in the chromosomes “j′” obtained as a result of the performance of all of the genetic operations OP_kto be performed, that is, whether or not any of the chromosomes “j′” produced has reached an allowable solution which is regarded as an optimal solution. Then, if it is discriminated that a chromosome corresponding to an optimal solution to the problem is not included, then the genetic algorithm execution section 3 updates the generation and repeats the processing in steps A1 to A8 described above. Then, if it is discriminated that a chromosome corresponding to an optimal solution is included, then the genetic algorithm execution section 3 outputs the chromosome “j′” as an optimal solution.

[0135] Here, an example of a result when an optimal solution is searched for actually by the optimal solution search apparatus 1 according to the present embodiment is illustrated in FIGS. 9 to 11.

[0136] FIGS. 9 to 11 illustrates a result of solution to a traveling salesman (salesperson) problem by the optimal solution search apparatus 1. It is to be noted that the traveling salesperson problem signifies a problem of finding out the shortest route (a route of the shortest distance) when a sales person visits all of several predetermined towns (in FIGS. 9 to 11, 100 towns) each by one time.

[0137] FIG. 9 is a diagram obtained by plotting a distance corresponding to a solution (route) calculated from a chromosome having the highest fitness value in each generation with respect to time. Meanwhile, FIGS. 10 and 11 are diagrams both showing part of the diagram of FIG. 9 enlarged in the direction of the axis of ordinate, and FIG. 10 is a diagram when the lower limit value of the axis of ordinate is determined as a correct answer to the problem while FIG. 11 is a diagram when the lower limit value of the axis of ordinate is determined as d (allowable solution reached in the process of search for an optimal solution) indicated in FIG. 9.

[0138] In FIGS. 9 to 11, each curve C indicates a result when it is attempted to search for an optimal solution using the optimal solution search method according to the present embodiment, and each curve D indicates another result when it is attempted to search for an optimal solution using an ordinary method. It is to be noted that the optimal solution search method corresponding to the curve C uses a certain non-particular genetic operation from among genetic operations used in the optimal solution search method corresponding to the curve D while lowering the frequency of execution of the non-particular genetic operation.

[0139] It can be seen from FIGS. 9 to 11 that the optimal solution search method according to the present embodiment provides a higher processing speed than the ordinary method.

[0140] It is to be noted that, more strictly, it may be considered otherwise that there is the possibility that, with the optimal solution search method according to the present embodiment corresponding to the curve C, since the execution frequency of some genetic operation is reduced, for example, the other genetic operations exhibit higher contributions to an increase in fitness value than the genetic operation whose execution frequency has been decreased and the influence of such contribution contributes much to a difference in performance as a whole while the contribution of the functions based on the optimal solution search method according to the present embodiment is low. However, it has been observed by an actual measurement that the contribution of the functions based on the optimal solution search method according to the present embodiment is sufficiently high.

[0141] As described above, with the optimal solution search apparatus 1 according to the embodiment of the present invention, the time required for calculation processing of fitness values can be reduced by determining execution schedules of genetic operations OP_kto be performed for individual chromosomes “j”, discriminating those chromosomes “j” for which only a particular genetic operation of such a form with which a variation amount Δf_jkof the fitness value can be calculated readily by a difference calculation or the like in accordance with the execution schedules and successively updating, for each of those chromosomes “j”, the fitness value f_jat the start of each generation after execution of the particular genetic operation. Consequently, the genetic algorithms can be executed at a high speed, and an optimal solution to the problem can be searched at a high speed.

[0142] Further, with the optimal solution search apparatus 1 according to the present embodiment, the effect described above is particularly high where the number of chromosomes which make an object of processing increases or when the length of a chromosome which makes an object of processing increases or otherwise when the calculation processing for fitness values is complicated, that is, where an evaluation function is complicated.

[0143] It is to be noted that the optimal solution search apparatus 1 according to the present embodiment can be utilized to solve various optimization problems, that is, for optimization of a delivery route where a truck is used, optimization of an operation schedule of a factory, optimization of a diagram of a traffic facility, optimization of a working schedule of a nurse or the like and so forth.

[0144] b. Others

[0145] While the foregoing description is given by way of an example wherein the schedule determination section 10 performs the determination processing for an execution schedule at the start of each generation (prior to execution of the first genetic operation OP₁), only it is required that the determination processing for an execution schedule be performed before the particular genetic operation which appears last is executed as described hereinabove.

[0146] In particular, the schedule determination section 10 may perform the determination processing for an execution schedule before the particular genetic operation which appears first is performed (refer to FIG. 5) or before an arbitrary one of genetic operations OP_kis executed (refer to “CASE 2” of “EXECUTION SCHEDULE DETERMINATION PROCESSING” in FIG. 12).

[0147] When the schedule determination section 10 performs the determination processing for an execution schedule other than at the start of each generation in this manner, there is a genetic operation OP_kwhich has been executed already, that is, there is a genetic operation which has been executed prior to a genetic operation which is determined as an object of determination by the schedule determination section 10.

[0148] Further, whether or not each of those genetic operations OP_kshould be executed is determined for each chromosome “j” using a probabilistic technique at the point of time of the execution.

[0149] Therefore, the chromosome discrimination section 11 must refer not only to execution schedules determined by the schedule determination section 10, but also to an execution condition of the genetic operation OP_kwhich has been executed already, such as whether or not all of the executed genetic operations OP_kare the particular genetic operation (EASY OP) or the like, to discriminate for each of the chromosomes “j” whether or not it is a chromosome for which only the particular genetic operation should be performed.

[0150] Further, only it is required that the determination processing for a chromosome by the chromosome discrimination section 11 be performed before the particular genetic operation which appears last is executed only if the determination processing for an execution schedule by the schedule determination section 10 has been completed.

[0151] In particular, only if the determination processing for an execution schedule by the schedule determination section 10 has been completed, the chromosome discrimination section 11 may perform, for each of the chromosomes “j”, the determination processing for a chromosome before the particular genetic operation which appears first is executed (refer to FIG. 5) or after an arbitrary genetic operation is executed before the particular genetic operation which appears last is executed.

[0152] It is to be noted that, only if the determination processing for an execution schedule by the schedule determination section 10 has been completed, the chromosome discrimination section 11 may perform the determination processing for a chromosome for each of the chromosomes “j” before updating of the fitness value by the fitness value updating section 13 after execution of the last genetic operation OP_m.

[0153] Further, where the schedule determination section 10 performs the determination processing for an execution schedule other than at the start of each generation as described above, upon execution of the particular genetic operation from among genetic operations OP_kwhich have been executed already, the variation amount Δf_jkof the fitness value by execution of the particular genetic operation must be calculated unconditionally (refer to the “CASE 2” of “DISCRIMINATION PROCESSING REGARDING CALCULATION OF Δf_jk” in FIG. 12).

[0154] It is to be noted that the variation amount Δf_jkof the fitness value calculated here is used for updating of the fitness value f_j.

[0155] Further, while the calculation of the variation amount Δf_jkof the fitness value in this instance is performed by the updating pre-processing section 19 as described hereinabove, the fitness value updating section 13 may otherwise have the function as the updating pre-processing section 19.

[0156] Furthermore, the fitness value updating section 13 may calculate, upon execution of the particular genetic operation from among genetic operations OP_kwhich make an object of determination for a schedule by the schedule determination section 10, the variation amount Δf_jkof the fitness value by execution of the particular genetic operation independently of a result of the determination by the chromosome discrimination section 11 (refer to “CASE 3” of “DISCRIMINATION PROCESSING REGARDING CALCULATION OF Δf_jk” in FIG. 12).

[0157] It is to be noted that the fitness value updating section 13 may otherwise calculate, after execution of an arbitrary one of genetic operations OP_kwhich make an object of determination for a schedule by the schedule determination section 10, the variation amount Δf_jkof the fitness value by execution of the particular genetic operation.

[0158] On the other hand, while, in the embodiment described above, the fitness value updating section 13 updates the fitness values f_jbased on the variation amounts Δf_jkof the fitness values calculated every time upon execution of the particular genetic operation (refer to “CASE 1” of “FITNESS VALUE UPDATING PROCESSING” in FIG. 13), only the variation amounts Δf_jkof the fitness values may alternatively be calculated upon execution of the particular genetic operation whereas updating of the fitness values f_jis performed collectively at the end of each generation, or otherwise, calculation of the variation amounts Δf_jkof the fitness values and updating of the fitness values f_jmay be performed collectively at the end of each generation (for both, refer to “CASE 2” of “FITNESS VALUE UPDATING PROCESSING” in FIG. 13).

[0159] Further, where the fitness value updating section 13 cannot execute a certain particular genetic operation unless the fitness value immediately prior to that point of time is known, final updating of the fitness values f_jmay be performed based on the fitness values f_jwhich are updated once before execution of the directly preceding particular genetic operation is completed (refer to “CASE 3” of “FITNESS VALUE UPDATING PROCESSING” in FIG. 13).

[0160] In the embodiment described above, the fitness value updating section 13 updates the fitness value f_jevery time when a chromosome “j” which makes an object of processing upon execution of the particular genetic operation is a chromosome for which only the particular genetic operation should be performed, the updating manner of the fitness value updating section 13 is not limited to this, and the fitness value updating section 13 may calculate, even when a chromosome “j” which-makes an object of processing is not a chromosome to which only the particular genetic operation should be performed, the variation amount Δf_jkof the fitness value every time and check the validity of a fitness value f_j′ calculated using the variation amount Δf_jkof the fitness value at the end of the generation.

[0161] It is to be noted that the checking of the validity of the fitness value f_j′ then may be performed at the end of the generation, that is, after execution of all of the genetic operations OP_kis completed, referring to a result of confirmation of execution conditions of the genetic operations OP_k.

[0162] Further, depending upon a manner in which a genetic operation OP_kis performed, even if the genetic operations OP_kis performed, the gene arrangement may possibly exhibit no variation by chance.

[0163] One of examples of this may be a case wherein the position at which a chromosome is to be cut for crossover is any of the opposite ends of the chromosome. In this instance, even if crossover is performed, the gene arrangement of the chromosome of the child does not exhibit a variation from the gene arrangement of the parent.

[0164] Thus, upon determination of execution schedules of genetic operations OP_k, also the position at which a chromosome should be cut for crossover may be determined so that, when the cutting position of a chromosome comes to any of the opposite ends of the chromosome, the genetic operation (crossover) may not be executed regarding the genetic operation (crossover) as a particular genetic operation so as to be reflected on the execution schedule.

[0165] Another example of a case wherein the gene arrangement exhibits no variation by chance even if a genetic operation OP_kis performed may be a case wherein crossover does not occur depending upon setting of the crossover probability for crossover. For example, where the crossover probability is 50%, crossover is performed only by a frequency of once for two times.

[0166] Or, if crossover is not performed actually although crossover should be performed depending upon an execution schedule, a fitness value may be passed as it is.

[0167] It is to be noted that, while the schedule determination section 10 in the optimal solution search apparatus 1 according to the embodiment described above determines execution schedules of genetic operations OP_kfor each generation, the schedule determination section 10 may alternatively determine execution schedules of genetic operations OP_kfor a plurality of generations collectively or may otherwise perform determination processing for an execution schedule by a plurality of times for each generation.

[0168] Further, while the optimal solution search apparatus 1 according to the embodiment described above performs only genetic operations for individual chromosomes to update the chromosomes “j”, an engineering operation such as, for example, a hill-climbing method may be performed in addition to the genetic operations to update the chromosomes “j”.