Chapter 13: Data Mining > genetic_algorithm

genetic_algorithm

Optimizes a user defined fitness function using a tailored genetic algorithm.

Synopsis

#include <imsls.h>

Imsls_f_individual  *imsls_f_genetic_algorithm(float fitness(),
Imsls_f_population  *initial_population, ..., 0)

The type double function is imsls_d_genetic_algorithm.

Required Arguments

float fitness(Imsls_f_individual *individual)   (Input)
The fitness function.  Given the data structure for an individual within the population, fitness returns the fitness of that individual.  The fitness function must return non-negative values.

Imsls_f_population *initial_population (Input)
A pointer to the initial population.

Return Value

Function imsls_f_genetic_algorithm optimizes a user defined fitness function by evolving an initial population using a tailored genetic algorithm that searches for the fittest individual.  It returns a pointer to a clone of the fittest individual in the last generation.  Memory allocated to this data structure can be released using imsls_f_ga_free_individual.

Synopsis with Optional Arguments

#include <imsls.h>

Imsls_f_individual *imsls_f_genetic_algorithm (float fitness(),  
Imsls_f_population
*initial_population,
IMSLS_GRAY_ENCODING,
IMSLS_NO_ELITISM,
IMSLS_NO_DECODE,
IMSLS_PRINT_LEVEL, int level,
IMSLS_MAX_GENERATIONS, int max_generations,
IMSLS_MAX_FITNESS, float max_fitness,
IMSLS_LINEAR_SCALING, float c,
IMSLS_SIGMA_SCALING,
IMSLS_GENERATION_GAP, float p_gap,
IMSLS_MUTATION_PROB, float p_mutation,
IMSLS_CROSSOVER_PROB, float p_xover,
IMSLS_CROSSOVERS, int n_xover,
IMSLS_PMX_CROSSOVER,
IMSLS_INVERT_CROSSOVER,
IMSLS_SELECTION_MODEL, int selection_model,
IMSLS_FITNESS_FCN_WITH_PARMSfloat fitness(), void *parms,
IMSLS_N_GENERATIONS, int *n_generations,
IMSLS_ON_LINE_PERFORMANCE, float **on_line_performance,
IMSLS_OFF_LINE_PERFORMANCE, float **off_line_performance,
IMSLS_VELOCITY, float **velocity,
IMSLS_GENERATION_STATS, float **gen_statistics,
IMSLS_LAST_GENERATION, Imsls_f_population **last_generation,
0)

Optional Arguments

IMSLS_GRAY_ENCODING   (Input)
By default, alleles for integer and real phenotypes are encoded using Base-2 encoding.  This argument changes that default to Gray encoding for integer and real phenotypes. 

IMSLS_NO_ELITISM   (Input)
By default, elitism is used to preserve the fittest individual from one generation to the next.  This argument disables elitism.

IMSLS_NO_DECODE   (Input)
By default, chromosome information is decoded into the individual’s phenotypes before every call to the user’s fitness function.  This argument disables automatic decoding between generations. Decoding is only applied to the last generation, including the fittest individual.

IMSLS_PRINT_LEVEL, int level   (Input)
By default, no printing of intermediate and final results occur from this function.  The IMSLS_PRINT_LEVEL argument accepts the following values for level:

 

level

Enumeration

Description

0

IMSLS_NONE

Supresses printing of any results.

1

IMSLS_FINAL

Prints summary of final results.

2

IMSLS_TRACE_GEN

Prints summary of final results plus generation statistics.

3

IMSLS_TRACE_ALL

Prints summary of final results, generation statistics and individual crossovers.

 

IMSLS_MAX_GENERATIONS, int max_generations   (Input)
The maximum number of generations.  Optimization is halted when the number of generations exceeds max_generations
Default: max_generations=100.

IMSLS_MAX_FITNESS, float max_fitness   (Input)
The optimization is halted if the maximum fitness is greater than this value. 
Default: max_fitness=imsls_f_machine(7), i.e., optimization is not halted by large fitness values.  Optimization only stops when the number of generations exceeds max_generations.

IMSLS_LINEAR_SCALING, float c   (Input)
Specifies an upper limit for the linear fitness scaling constant.  Set c = 1 for no scaling.  A check is made to ensure that the minimum scaled fitness is non-negative.  If it falls below zero, then the scaling constant is automatically reduced to make the minimum scaled fitness equal to zero.  For linear scaling the scaling constant is typically between one and two. 
Default:  c =1, no linear fitness scaling.

IMSLS_SIGMA_SCALING  (Input)
By default, sigma scaling is not used for fitness scaling.  This argument enables sigma scaling.

IMSLS_GENERATION_GAP, float p_gap   (Input)
The proportion of weakest individuals replaced between generations.  If p_gap=1, all of the individuals are replaced. 
Default: p_gap=1.

IMSLS_MUTATION_PROB, float p_mutation   (Input)
The probability of mutation. Although most applications set this to a value between 0.005 and 0.1, any value between 0 and 1 is allowed.
Default: p_mutation=0.005.

IMSLS_CROSSOVER_PROB, float p_xover   (Input)
The probability of crossover.  p_xover can be any value between 0 and 1. Most genetic algorithms use a probability between 0.6 and 0.9.
Default: p_xover= 0.6.

IMSLS_CROSSOVERS, int n_xover   (Input)
The number of crossover points.  De Jong’s (1975) generalized crossover model R6 is implemented.  If n_xover is odd, then the chromosome is treated as a string with a default crossover at the beginning of the chromosome.  If n_xover is even, then the chromosome is treated as a ring with no beginning or end, and crossovers are selected using the uniform distribution on a circle.  Crossing points occur at the odd crossover points.  If the IMSLS_PMX_CROSSOVER optional argument is used, there are always two crossover points within the nominal portion of the chromosome.  For partially matched crossovers, this argument is only used to define the number of crossovers within the binary, integer and real portion of the chromosome. 
Default: n_xover=1.

IMSLS_PMX_CROSSOVER, (Input)
By default this optional argument applies partially matched crossover to the nominal portion of the chromosome.  Crossovers for other phenotypes are still applied using standard crossover and inversion crossover if requested.  With partially matched crossover, the number of crossovers for nominal phenotypes is set to 2, and partially matched crossover is applied only to the nominal phenotype.  The number of crossovers for non-nominal phenotypes is still controlled by the value of n_xover.  However, if this optional argument is used, crossover points are randomly selected separately for nominal and non-nominal alleles.

IMSLS_INVERT_CROSSOVER, (Input)
This option augments standard or partially matched crossover with inversion.  Inversion crossover inverts the values of the alleles in every other crossover segment.  If this is applied with partially matched crossover, inversion is applied within the matched segment of the alleles for the nominal phenotypes and then within every other segment of any non-nominal phenotype.

IMSLS_SELECTION_MODEL, int selection_model,   (Input)
The model used for selecting individuals for reproduction.  Selection models are described in the following table:

 

selection_model

Description

IMSLS_DETERMINISTIC

Individuals with highest fitness are selected for reproduction using their expected sampling frequency.  See Goldberg (1989)

IMSLS_ROULETTE_WITH

Original fitness-proportional selected described by Holland(1975).  Sampling is done with replacement.

IMSLS_ROULETTE_WITHOUT

The original fitness-proportional selected except that sampling is done without replacement.  This is also referred to as De Jong’s (1975) R3 model.

IMSLS_REMAINDER_WITH

Remainder selection with replacement.

IMSLS_REMAINDER_WITH

Remainder selection with replacement.

IMSLS_REMAINDER_WITHOUT

Remainder selection without replacement

IMSLS_SUS_SELECTION

Stochastic Universal Sampling as described by Baker (1987).

IMSLS_RANK_SELECTION

Rank selection.  The individuals with the highest fitness are selected once for reproduction. 

IMSLS_TOURNAMENT_1

Tournament selection as described by Wetzel (1983).  Only the fittest individual in a pair is selected.

IMSLS_TOURNAMENT_2

Tournament selection as described by Goldberg and Deb (1991).  The fittest individual in a pair is selected with probability 0.75.  Otherwise the weaker individual is selected.

            Default:  The original selection method described by Holland (1975), selection_model=IMSLS_ROULETTE_WITH.

IMSLS_FITNESS_FCN_WITH_PARMS, float fitness(Imsls_f_individual *individual, void *parms), void *parms,   (Input)
The fitness function calculated for individual.  If this is supplied, fitness values are calculated for each individual and included in the expanded population data structure.  The parameters in parms are passed to the function. 

IMSLS_N_GENERATIONS, int *n_generations   (Output)
The number of generations used to find the fittest individual.

IMSLS_ON_LINE_PERFORMANCE, float **on_line_performance   (Output)
An array of length max_generations containing on-line performance statistics for each generation.

IMSLS_OFF_LINE_PERFORMANCE, float **off_line_performance (Output)
An array of length max_generations containing off-line performance statistics for each generation.

IMSLS_VELOCITY, float **velocity (Output)
An array of length max_generations containing velocity statistics for each generation. The velocity for the i-th generation is equal to  where  is the maximum fitness for the i-th generation.

IMSLS_GENERATION_STATSfloat **gen_statistics (Output)
An array of size max_generations*4 containing the maximum fitness, minimum fitness, average fitness and standard deviation of the fitness for each generation.  The
i-th row of gen_statistics contains the statistics for the i-th generation.  When n_generations<max_generations, rows greater than n_generations-1 are filled with NaN values.  The four columns contain the following statistics calculated for each generation:

Column

Description

1

Maximum Fitness

2

Minimum Fitness

3

Fitness Average

4

Fitness Standard Deviation

IMSLS_LAST_GENERATION, Imsls_f_population **last_generation (Output)
The last generation produced by the genetic algorithm.  Memory allocated to this data structure can be released using imsls_f_ga_free_population.

Description

Genetic algorithms search for the optimum individual in a population. This is defined as the individual with the highest fitness. Function imsls_f_genetic_algorithm returns the fittest individual in the last generation.  Mathematically, this is equivalent to finding the values of the phenotypes that maximize a user provided fitness function.  Although there are no requirements that the fitness function be non-negative, in general, convergence to optimum fitness is faster when values of the fitness function are non-negative.  Constraints can be applied by incorporating a penalty function within the fitness calculation.  Phenotypes can consist of any combination of nominal, binary, integer and real values.  Integer and real values must be encoded into a binary representation.  This procedure provides for either Base-2 or Gray encoding.  However, users can supply other encodings within the fitness function.

The function imsls_f_genetic_algorithm uses the population data structure and fitness with simulated genetic processes of reproduction to search for the optimum individual, i.e. settings of phenotype values.  Genetic algorithms have been successfully applied to a wide variety of optimization and search problems, see Holland (1975) and Goldberg (1985).

There are many refinements to the basic genetic algorithm originally described by Holland (1975).   His basic algorithm begins with an initial population of n individuals, a fitness function, and probabilities for crossover and mutation of p_xover and p_mutation respectively.  The initial population is transformed from one generation to the next using the following steps:

1.     Select n individuals from the current population to generate a mating pool.

2.     Apply crossover with probability p_xover to pairs of the selected individuals within the mating pool to produce two offspring.

3.     Apply mutation with probability p_mutation to the offspring to generate the next generation.

4.     Check stopping criteria.  If they are met, stop and report the fittest individual within the last generation.

By default Holland’s approach to these steps are used.  However, many variations of these can be selected using optional arguments.

The initial population can be generated automatically using imsls_f_ga_random_population or it can be created by first creating individuals using imsls_f_ga_individual and then a population for those individuals using imsls_f_ga_population.

By default Holland’s roulette wheel with replacement is used for selecting the mating pool.  The optional argument IMSLS_SELECTION_MODEL allows users to select alternate selection methods including remainder, tournament and stochastic universal selection.  Default crossover and mutation probabilities are p_xover= 0.8 and p_mutation= 0.005.  These defaults can be changed using the optional arguments IMSLS_CROSSOVER_PROB and IMSLS_MUTATION_PROB.

In the original algorithm only a single crossover point was randomly selected.  The optional argument IMSLS_CROSSOVERS allows users to designate any number of crossover points.

Standard crossover proceeds by combining the genes from both parents in the order found in those parents.  Inversion crossover inverts this order for one of the parents.  Inversion crossover is selected using the optional argument IMSLS_INVERT_CROSSOVER.

For certain problems, such as the traveling salesman problem, standard crossover can produce infeasible individuals.  One approach is to assign zero fitness to those solutions, but this can be very inefficient.  Partially matched crossover is an approach that ensures individuals are feasible for a certain class of problems.  If the problem is best represented using nominal phenotypes with values 0, 1, …, n_nominal-1 where all values must appear once and only once in the chromosome, then partially matched crossover preserves that condition.  Partially matched crossover is selected using the optional argument IMSLS_PMX_CROSSOVER

One issue with some applications of genetic algorithms is premature convergence or convergence to false local solutions.  This can occur when dominant individuals within early generations take over the population prematurely reducing population diversity.  One approach to this problem is fitness scaling.  This implementation allows users to use either linear or sigma fitness scaling. By default, no scaling is used.  However, the optional arguments IMSLS_LINEAR_SCALING and IMSLS_SIGMA_SCALING allow users to have fitness values automatically scaled before selection.

The genetic algorithm is stopped when any one of the stopping criteria is met.  The algorithm is stopped when the number of generations exceeds max_generations or when the maximum fitness exceeds max_fitness.  By default max_generations= 100; this can be changed using IMSLS_MAX_GENERATIONS.  By default max_fitness is imsls_f_machine(7); this can be changed using IMSLS_MAX_FITNESS

On some platforms, imsls_f_genetic_algorithm can evaluate the user-supplied function fitness in parallel. This is done only if the function imsls_omp_options is called to flag user-defined functions as thread-safe. A function is thread-safe if there are no dependencies between calls. Such dependencies are usually the result of writing to global or static variables.

Examples

Example 1

De Jong (1975) examined the performance of a genetic algorithm for finding the maximum of a multivariate function.  This is an example of optimizing a variation of De Jong’s R2 function:

 where

and  .

Since there were only two real phenotypes and the function is easily calculated, the phenotypes were encoded using discretization with 65,536 values over the interval [-2.048, 2.048].  By default, encoding and decoding is done within imsls_f_genetic_algorithm.  This allows the fitness function to calculate individual fitness using the real phenotypes instead of the chromosome.  Both the chromosome and its phenotype representation are available within the Imsls_f_individual data structure argument.

The default selection algorithm IMSLS_ROULETTE_WITH was used, but the number of crossover probability was set to 0.6.  The genetic algorithm was more efficient using a lower crossover probability and Gray encoding instead of the defaults 0.7 and Base-2 encoding.  Each generation consisted of 40 individuals.

 

#include <imsls.h>

#include <stdio.h>

 

int main(){

   int i, j;                           /* index variables            */

   int n = 40;                         /* population size            */

   int n_generations = 0;              /* final number of generations*/

   int n_real = 2;                     /* number of real phenotypes  */

   int   r_intervals[2] = {65536, 65536};

   float r_bounds[4]    = {-2.048, 2.048, -2.048, 2.048};

   float* genStats;                    /* generation statistics      */

   static float deJongR2(Imsls_f_individual* individual);

   Imsls_f_chromosome* chromosome;     /* chromosome data structure  */

   Imsls_f_individual* best_individual;/* optimum                    */

   Imsls_f_population* population;     /* population data structure  */

   Imsls_f_population* last_generation;/* last generation            */

   char *dashes=

      "**************************************************************";

   /******************************************************************/

   /* In this example the user function is thread safe.  Let CNL     */

   /* know it is safe, which allows genetic algorithm to run in      */

   /* parallel, if that capability exists on the user computer.      */

   /******************************************************************/

   imsls_omp_options(IMSLS_SET_FUNCTIONS_THREAD_SAFE, 1, 0);

   imsls_random_seed_set(12345);

   chromosome = imsls_f_ga_chromosome(

      IMSLS_REAL, n_real, r_intervals, r_bounds, 0);

   population = imsls_f_ga_random_population(n, chromosome,

      IMSLS_GRAY_ENCODING,

      IMSLS_FITNESS_FCN, deJongR2, 0);

 

   best_individual = imsls_f_genetic_algorithm(deJongR2, population,

      IMSLS_PRINT_LEVEL, IMSLS_FINAL,

      IMSLS_MAX_FITNESS, 3999.999,

      IMSLS_CROSSOVER_PROB, 0.6,

      IMSLS_GRAY_ENCODING,

      IMSLS_MAX_GENERATIONS, 1000,

      IMSLS_N_GENERATIONS, &n_generations,

      IMSLS_GENERATION_STATS, &genStats,

      IMSLS_LAST_GENERATION,&last_generation, 0);

   printf(

      "\n*****************GENERATION STATISTICS*****************\n");

   printf("Generation  Max. Fit.   Avg. Fit.  Min. Fit.     CV\n");

   printf("*******************************************************\n");

   for(i=0; i<=n_generations; i++){

      printf("Gen. %3d: %11.5f %10.2f %10.2f %9.2f \n",

         i, genStats[4*i], genStats[4*i+2], genStats[4*i+1],

         100*genStats[4*i+3]/genStats[4*i+2]);

   }

   printf("\nLAST GENERATION\n");

   printf("%s\n",dashes);

   printf("\nIndv  Fitness              ");

   printf("Chromosome              X1     X2\n");

   for(i=0; i<last_generation->n; i++){

      printf(" %2d   %6.2f  ", i, last_generation->fitness[i]);

      for(j=0; j<last_generation->chromosome->c_length; j++)

         printf("%d",

         last_generation->individual[i]->chromosome->allele[j]);

      printf("%7.3f %6.3f\n",

         last_generation->individual[i]->realPhenotype[0],

         last_generation->individual[i]->realPhenotype[1]);

   }

   printf("\nMaximum:    %6.2f for Individual %d\n",

      last_generation->maxFitness,

      last_generation->indexFittest);

   printf("Minimum:    %6.2f for Individual %d\n",

      last_generation->minFitness,

      last_generation->indexWeakest);

   printf("Average:    %6.2f\n", last_generation->avgFitness);

   printf("Std. Dev:    %6.2f\n\n", last_generation->stdFitness);

   printf("%s\n", dashes);

}

/*********************************************************************/

/* De Jong's R2 Function                                             */

/*********************************************************************/

static float deJongR2(Imsls_f_individual* individual)

{

   float f, x1, x2;

   x1 = individual->realPhenotype[0];

   x2 = individual->realPhenotype[1];

   f = 100*(x1*x1-x2)*(x1*x1-x2) + (1.0-x1)*(1.0-x1);

   f = 4000 - f;

   return f;

}

Output

In this example, the print level is set to IMSLS_FINAL in order to print the optimum solution.  The generation statistics are requested using the IMSLS_GENERATIONS_STATS option, and the last population is requested using the IMSLS_LAST_GENERATION option.

Although the maximum number of generations is set to 100 using the IMSLS_GENERATIONS option,  the genetic algorithm halted after 26 generations when the maximum population fitness exceeded 3999.999. 

OPTIMUM SOLUTION

 

   Fitness: 3999.999512

   Phenotypes:

      Real:     2

   Function Calculations: 1080

   Population Size:       40

   Number of Generations: 26

 

   Real Phenotype(s):

 

      1.023594 1.047844

 

   Chromosome (Gray Encoded):

 

      11100000000001011010000111000011

 

 

*****************GENERATION STATISTICS*****************

Generation  Max. Fit.   Avg. Fit.  Min. Fit.     CV

*******************************************************

Gen.   0:  3996.12915    3244.16     578.91     28.62

Gen.   1:  3996.25269    3725.90    2770.41      8.24

Gen.   2:  3999.22974    3699.14    1917.26     10.28

Gen.   3:  3999.22974    3683.41    2551.70      9.87

Gen.   4:  3999.73779    3778.83    2551.70      8.35

Gen.   5:  3999.73779    3823.50    3187.72      5.22

Gen.   6:  3999.73779    3796.59    3187.72      5.76

Gen.   7:  3999.73779    3850.94    3302.26      4.28

Gen.   8:  3999.73779    3860.17    3358.92      3.93

Gen.   9:  3999.73779    3886.33    3138.96      4.23

Gen.  10:  3999.74683    3896.13    3292.64      3.85

Gen.  11:  3999.74683    3900.24    3638.17      2.93

Gen.  12:  3999.74683    3899.95    3376.35      3.17

Gen.  13:  3999.74683    3900.57    3476.12      3.19

Gen.  14:  3999.74683    3897.88    3408.28      3.36

Gen.  15:  3999.74683    3908.28    2331.26      3.36

Gen.  16:  3999.99585    3897.28    3301.18      3.94

Gen.  17:  3999.99585    3910.99    3236.92      3.31

Gen.  18:  3999.99585    3953.46    3429.17      1.31

Gen.  19:  3999.99585    3944.98    3764.08      1.41

Gen.  20:  3999.99585    3945.62    3751.01      1.41

Gen.  21:  3999.99585    3934.07    3751.10      1.81

Gen.  22:  3999.99585    3947.08    3739.52      1.62

Gen.  23:  3999.99609    3943.99    3652.86      1.95

Gen.  24:  3999.99609    3942.42    3652.86      1.99

Gen.  25:  3999.99927    3970.69    3845.42      0.92

Gen.  26:  3999.99951    3970.61    3845.42      0.90

 

LAST GENERATION

***************************************************************

 

Indv  Fitness              Chromosome              X1     X2

  0   3916.58  11100000000011001100110001010011  1.023  0.134

  1   3996.09  01010000000001011100110001010011 -0.512  0.134

  2   3965.22  11010000000011101110111110011011  0.511  0.849

  3   3993.47  01100101000001011010000001100011 -0.928  1.028

  4   3941.02  11010000000001011010000001100011  0.512  1.028

  5   3998.11  11110000000001011100110001110011  0.512  0.134

  6   3951.04  01100100000101011100110001100011 -0.898  0.132

  7   3919.76  01000000000001011110110001110011 -0.000  0.890

  8   3965.35  11110000000001011110111110011011  0.512  0.849

  9   3913.56  11010000000011101010111101110011  0.511  1.190

 10   3948.39  11010000000011101110001110110011  0.511  0.978

 11   3996.04  01010000000001011100110001100011 -0.512  0.132

 12   3997.44  11100000100111001110111101101001  1.009  0.859

 13   3992.45  11110000000001110100000001000011  0.512 -0.008

 14   3958.07  01010000000101011110110001110011 -0.510  0.890

 15   3951.33  01100100000001011100110001100011 -0.896  0.132

 16   3948.39  11010000000011101110001110110011  0.511  0.978

 17   4000.00  11100000000001011010000111010001  1.024  1.046

 18   3999.96  11100000000001011010000001100011  1.024  1.028

 19   3925.31  01010110010001011010000111010001 -0.440  1.046

 20   3999.37  11010000000001011101111001101111  0.512  0.325

 21   3845.42  01100101010110110101110001100011 -0.921 -0.380

 22   3999.64  11010101000011101100111001000011  0.415  0.184

 23   3992.37  01100000000011101110111101101001 -1.023  0.859

 24   3990.55  01010000000001110100000001110011 -0.512 -0.006

 25   3993.26  11110000000011101100000001000011  0.513  0.008

 26   3964.15  11010000000011101110111101110011  0.511  0.858

 27   3998.36  11010000000011101100110010011011  0.511  0.143

 28   3950.63  01100101000011101100111001000011 -0.927  0.184

 29   3996.08  01010000000001011100110001110011 -0.512  0.134

 30   3997.89  01010110000011101100111001100111 -0.447  0.188

 31   3998.11  11110000000001011100110001110011  0.512  0.134

 32   3993.47  01100101000001011010000001100011 -0.928  1.028

 33   3992.45  11110000000001110100000001000011  0.512 -0.008

 34   3916.61  11100000000011011100110001010011  1.023  0.134

 35   3929.38  01010110000001011010000001100011 -0.448  1.028

 36   3997.89  01010110000011101100111001100011 -0.447  0.188

 37   4000.00  11100000000001011010000111010001  1.024  1.046

 38   4000.00  11100000000001011010000111000011  1.024  1.048

 39   3991.28  01100100000001011010000001110001 -0.896  1.030

 

Maximum:    4000.00 for Individual 38

Minimum:    3845.42 for Individual 21

Average:    3970.61

Std. Dev:     35.71

Example 2

The traveling saleman problem creates a problem for traditional crossover.  In this problem, the objective is to find the shortest route while traveling to each city once.  In this example, there are eight cities, labeled using the letters a-h, with distances ranging from 17 to 113 miles.

Traditional crossover would create unfeasible routes; that is some routes after crossover would not visit every city once.  Some would not be visited and others would be visited more than once.

Partially matched crossover (PMX) preserves the feasibility of a route.  In the general sense, PMX assumes that the nominal phenotypes consists of a string of numbers from zero to n_nominal-1, with each number appearing once and only once in that string.  Partially matched crossover uses two crossover points within the nominal portion of the chromosome and swaps the middle segment between the parents.  The first and third segments are manipulated to ensure the resulting offspring is feasible.

 

#include <imsls.h>

#include <stdio.h>

#include <stdlib.h>

 

int main(){

    int i, j, k;                        /* index variables             */

    int n = 50;                         /* population size             */

    int n_generations;                  /* number of generations       */

    int n_nominal = 8;                  /* number of nominal phenotypes*/

    int n_categories[8] = {8, 8, 8, 8,

        8, 8, 8, 8}; /* nominal category limits     */

    float x1;                           /* temporary storage           */

    float avg;                          /* average fitness             */

    float* genStats;                    /* generation statistics       */

    static float pmxFitness(Imsls_f_individual* individual);

    Imsls_f_chromosome* chromosome;     /* chromosome data structure   */

    Imsls_f_individual* best_individual;/* optimum                     */

    Imsls_f_population* population;     /* population data structure   */

    Imsls_f_population* last_generation;/* last generation             */

    char *cities[8] = {"a", "b", "c", "d",  /* Cities Label a-h        */

        "e", "f", "g", "h"};

 

    /*******************************************************************/

    /* In this example the user function is thread safe.  Let CNL      */

    /* know it is safe, which allows genetic algorithm to run in       */

    /* parallel, if that capability exists on the user computer.       */

    imsls_omp_options(

        IMSLS_SET_FUNCTIONS_THREAD_SAFE, 1,

        0);

    imsls_random_seed_set(12345);

    chromosome      = imsls_f_ga_chromosome(

        IMSLS_NOMINAL, n_nominal, n_categories,

        0);

 

    population      = imsls_f_ga_random_population(n, chromosome,

        IMSLS_PMX_CROSSOVER,

        IMSLS_FITNESS_FCN, pmxFitness,

        0);

    best_individual = imsls_f_genetic_algorithm(pmxFitness, population,

        IMSLS_PRINT_LEVEL, IMSLS_FINAL,

        IMSLS_PMX_CROSSOVER,

        IMSLS_INVERT_CROSSOVER,

        IMSLS_CROSSOVER_PROB, 0.8,

        IMSLS_MAX_GENERATIONS, 10,

        IMSLS_GENERATION_STATS, &genStats,

        IMSLS_N_GENERATIONS, &n_generations,

        IMSLS_LAST_GENERATION, &last_generation,

        0);

 

    printf("GENERATION STATISTICS\n");

    printf("Total Number of Generations: %d\n\n", n_generations);

    printf("Generation  Max. Fit. Min. Fit.  Avg. Fit.    CV\n");

 

    for(i=0; i<=n_generations; i++){

        printf("Gen. %3d: %8.0f %8.0f %12.2f %8.2f\n",

            i, genStats[4*i], genStats[4*i+1], genStats[4*i+2],

            100*genStats[4*i+3]/genStats[4*i+2]);

    }

 

    printf("\n\n            LAST GENERATION\n");

    printf("*************************************\n");

    printf("Individual  Fitness  Phenotype Values \n");

 

    avg = last_generation->avgFitness;

    for(i=0; i<last_generation->n; i++){

        x1 = last_generation->fitness[i];

        printf("    %2d     %6.0f    ", i, x1);

        for(j=0; j<last_generation->chromosome->c_length; j++) {

            k = last_generation->individual[i]->nominalPhenotype[j];

            printf("%s ", cities[k]);

        }

        if(x1 == last_generation->maxFitness){

            printf("***\n");

        }else{

            printf("\n");

        }

    }

 

    printf("Average Fitness: %6.1f\n\n", avg);

    printf("*************************************\n");

    printf("OPTIMUM SOLUTION:\n\nFitness:%4.0f\n\nChromosome:        ",

        pmxFitness(best_individual));

    for(i=0; i<best_individual->chromosome->c_length; i++)

        printf("%3d", best_individual->chromosome->allele[i]);

    printf("\n\nPhenotype Values:  ");

    for(i=0; i<n_nominal; i++) {

        k = best_individual->nominalPhenotype[i];

        printf("->%s", cities[k]);

    }

 

    printf("\n\n");

    printf("freeing best individual\n");

    imsls_f_ga_free_individual(best_individual);

    printf("freeing last generation\n");

    imsls_f_ga_free_population(last_generation);

    printf("freeing chromosome\n");

    free(chromosome);

    printf("freeing population\n");

    imsls_f_ga_free_population(population);

    system("pause");

}

/***********************************************************************/

/* FITNESS FUNCTION                                                    */

/********************************************************************* */

static float pmxFitness(Imsls_f_individual* individual)

{

    int i=0, k=0, i1, i2; /* Index variables                           */

    int n_nominal = 8;    /* number of nominal phenotypes              */

    float f = 0.0;        /* fitness value                             */

    /* cities:  a    b    c    d    e    f    g   h       */

    float distances[64] = { 0,  17,  27,  73,  61,  57,  51, 23, /* a  */

        17,   0,  37,  73,  72,  74,  66, 40, /* b  */

        27,  37,   0,  48,  35,  49,  65, 50, /* c  */

        73,  73,  48,   0,  47,  82, 113, 95, /* d  */

        61,  72,  35,  47,   0,  38,  80, 78, /* e  */

        57,  74,  49,  82,  38,   0,  48, 65, /* f  */

        51,  66,  65, 113,  80,  48,   0, 40, /* g  */

        23,  40,  50,  95,  78,  65,  40,  0};/* h  */

 

    n_nominal = individual->chromosome->n_nominal;

    k = individual->chromosome->nominalIndex+1;

    f = 0.0;

    for(i=k; i<k+n_nominal-1; i++){

        i1 = individual->nominalPhenotype[i-1];

        i2 = individual->nominalPhenotype[i  ];

        f += distances[i1*n_nominal + i2];

    }

    return 516-f;

}

Output

This program produced the following output.  Since the print level was set to IMSLS_FINAL, the optimum solution was printed.  The generation statistics were requested using the IMSLS_GENERATIONS_STATS option, and the last population was requested using the IMSLS_LAST_GENERATION option.

The maximum number of generations was set to 10.  The genetic algorithm found the optimum route after evaluating the fitness of 550 routes in 10 generations.  Generation zero is the initial generation provided to the algorithm and is not counted towards the maximum generation count.

 

OPTIMUM SOLUTION

 

   Fitness: 269.000000

   Phenotypes:

      Nominal:  8

   Function Calculations: 550

   Population Size:       50

   Number of Generations: 10

 

   Nominal Phenotype(s):

 

        3   4   2   1   0   7   6   5

 

   Chromosome (Base-2 Encoded):

 

      3 4 2 1 0 7 6 5

 

GENERATION STATISTICS

Total Number of Generations: 10

 

Generation  Max. Fit. Min. Fit.  Avg. Fit.    CV

Gen.   0:      194       29       113.64    33.55

Gen.   1:      251       24       119.10    35.08

Gen.   2:      251       37       131.16    35.99

Gen.   3:      251       38       128.96    33.68

Gen.   4:      251       29       135.08    33.25

Gen.   5:      255       38       136.18    33.44

Gen.   6:      255       28       142.38    33.23

Gen.   7:      255       56       150.56    29.31

Gen.   8:      269       55       155.06    28.88

Gen.   9:      269       56       148.62    26.68

Gen.  10:      269       48       146.32    29.59

 

 

            LAST GENERATION

*************************************

Individual  Fitness  Phenotype Values

     0        106    a c d g b h e f

     1        269    d e c b a h g f ***

     2        222    d e c h g f a b

     3        154    e a g h b d c f

     4        215    e d c a b f g h

     5        215    e d c a b f g h

     6        158    a h b c d f g e

     7        113    c a d b f e h g

     8         78    f a d b h c g e

     9        125    e d h g b c f a

    10        118    c b e g f h a d

    11        209    d e c b h g a f

    12        161    h b a d c f g e

    13        167    g d b a h c e f

    14        169    a f h g b c e d

    15        137    f g a e d h b c

    16        127    h d a b c f e g

    17         90    b d h g c e a f

    18        216    d h b a c e f g

    19        167    g d e c f h a b

    20        120    e h f c d a b g

    21         86    f a h d b c g e

    22        171    a b d f g h c e

    23        156    d h g a c e f b

    24        146    c e a d b h g f

    25        116    f d g e c h a b

    26        118    b c a h e f d g

    27         95    d g a b f e h c

    28        143    d h b a f c e g

    29        157    b f c g h a e d

    30        175    b e a h g f c d

    31        146    c e a d b h g f

    32        111    c b e h g f a d

    33        140    b e a f g h c d

    34        164    b c e d g h a f

    35        166    f e b c a h g d

    36        199    b a g h f c d e

    37        171    e d f a b c g h

    38         97    f a c d h b e g

    39        222    d e c h g f a b

    40        128    d h c g a b e f

    41         99    e g d c b a f h

    42        112    d h f c b a e g

    43        107    h d a b e f c g

    44        143    c h a b f d e g

    45        144    g a c e d b f h

    46         64    a e c g h d f b

    47        103    b d f h g c a e

    48        129    d a g b f e c h

    49        172    f h b g a c d e

Average Fitness:  146.3

 

*************************************

OPTIMUM SOLUTION:

 

Fitness: 269

 

Chromosome:          3  4  2  1  0  7  6  5

 

Phenotype Values:  ->d->e->c->b->a->h->g->f

 

freeing best individual

freeing last generation

freeing chromosome

freeing population

Example 3

This example uses the N-Queens problem to illustrate the use of a fitness function with parameters in implementing a genetic algorithm.  The N-Queens problem is derived from chess. The genetic algorithm provides an efficient search for a valid solution.  For this problem the chess board consists of N rows and N columns.  The objective is to place N queens on this board with no conflicts.  A conflict occurs when one queen can move and capture another.  Since queens can move diagonally, vertically and horizontally this problem is challenging when N becomes large.

One solution for N = 4 is displayed in the following table.  A valid solution must place every queen in a different row and column.  The problem is to ensure the queens are not in conflict because of lying on the same diagonal.

 

Row/Col

0

1

2

3

0

 

 

Q

 

1

Q

 

 

 

2

 

 

 

Q

3

 

Q

 

 

Similar to the traveling salesman problem, the N-Queens problem can be expressed using N nominal phenotypes with values 0, 1, …, N-1.  The value of the i-th phenotype represents the row number for the queen in the i-th column.  This ensures that any arrangement of the phenotype values represents a board with N queens, each placed in a different row and column. 

The solution for N queens displayed above can be represented by the phenotype values {1, 3, 0, 2}.  The search looks for arrangements that also do not place queens on the same diagonal.  Two queens fall on the same diagonal if the absolute value of  the difference between their row numbers equals the absolute value of the difference between their column numbers.

This example uses this representation with a fitness function equal to (N – C) where C equals the number of conflict among the queens.  With this fitness, a solution to the N-Queens has a fitness of N.

 

#include <imsls.h>

#include <imsls.h>

#include <stdlib.h>

#include <stdio.h>

#include <math.h>

 

typedef struct

{

   int n_queens;

} inputArgs;

int main(){

   int i, j, k;        /* index variables             */

   int n = 500;        /* population size             */

   int n_generations;  /* number of generations       */

   int n_queens = 25;  /* number of nominal phenotypes*/

   int n_categories[25];

   float maxFit;       /* maximum fitness hurdle      */

   float* genStats;    /* generation statistics       */

   static float queensFitness(

      Imsls_f_individual* individual, inputArgs* input);

   inputArgs* parameters;

   Imsls_f_chromosome* chromosome;     /* chromosome */

   Imsls_f_individual* best_individual;/* optimum    */

   Imsls_f_population* population;     /* population */

   /**************************************************/

   /* In this example the user function is thread    */

   /* safe.  Let CNL know it is safe, which allows   *

   /* genetic algorithm to run in parallel, if that  */

   /* capability exists on the user computer.        */

   imsls_omp_options(IMSLS_SET_FUNCTIONS_THREAD_SAFE, 1, 0);

   imsls_random_seed_set(12345);

   maxFit = n_queens - 0.5;

   for(i=0; i<n_queens; i++) n_categories[i] = n_queens;

   chromosome      = imsls_f_ga_chromosome(

            IMSLS_NOMINAL, n_queens, n_categories, 0);

  

   parameters = (inputArgs*) malloc(sizeof(inputArgs));

   parameters->n_queens = n_queens;

   population = imsls_f_ga_random_population(n,

            chromosome,

            IMSLS_PMX_CROSSOVER,

            IMSLS_FITNESS_FCN_WITH_PARMS,

            queensFitness, parameters, 0);

   best_individual = imsls_f_genetic_algorithm(

         NULL, population,

         IMSLS_FITNESS_FCN_WITH_PARMS, queensFitness, parameters,

         IMSLS_PRINT_LEVEL, IMSLS_FINAL,

         IMSLS_PMX_CROSSOVER,

         IMSLS_LINEAR_SCALING, 2.0,

         IMSLS_CROSSOVER_PROB, 0.7,

         IMSLS_MUTATION_PROB, 0.01,

         IMSLS_MAX_GENERATIONS, 10000,

         IMSLS_MAX_FITNESS, maxFit,

         IMSLS_GENERATION_STATS, &genStats,

         IMSLS_N_GENERATIONS, &n_generations, 0);

 

   printf("GENERATION STATISTICS\n");

   printf("Total Number of Generations: %d\n\n", n_generations);

   printf("Generation  Max. Fit. Min. Fit.  Avg. Fit.    CV\n");

   printf("*************************************************\n");

   for(i=0; i<=n_generations; i+=25){

      printf("Gen. %3d: %8.0f %8.0f %12.2f %8.2f\n",

         i, genStats[4*i], genStats[4*i+1], genStats[4*i+2],

         100*genStats[4*i+3]/genStats[4*i+2]);

   }

   i = n_generations;

   printf("Gen. %3d: %8.0f %8.0f %12.2f %8.2f\n",

         i, genStats[4*i], genStats[4*i+1], genStats[4*i+2],

         100*genStats[4*i+3]/genStats[4*i+2]);

   printf("*************************************************\n");

   printf("OPTIMUM SOLUTION:\n\nFitness:%4.0f\n\nChromosome:        ",

      queensFitness(best_individual, parameters));

   for(i=0; i<best_individual->chromosome->c_length; i++)

      printf("%3d", best_individual->chromosome->allele[i]);

   if(n_queens<100){

      printf("\n\nBoard Positions:  \n\n");

      for(i=0; i<n_queens; i++) printf("--");

      printf("-\n");

      for(i=0; i<n_queens; i++) {

         for(j=0; j<n_queens; j++){

            k = best_individual->nominalPhenotype[j];

            if(i==k) printf("|Q");

            else     printf("| ");

         }

         printf("|\n");

         for(k=0; k<n_queens; k++) printf("--");

         printf("-\n");

      }

      printf("\n\n");

   }

}

static float queensFitness(Imsls_f_individual* individual, inputArgs* input)

{

   int i=0, j=0, k=0; /* Index variables */

   int n_queens;

   float f = 0.0;     /* Fitness value   */

   n_queens = input->n_queens;

   f = 0;

   for(i=0; i<n_queens-1; i++){

      for(j=i+1; j<n_queens; j++){

         k = individual->chromosome->allele[i] -

            individual->chromosome->allele[j] ;

         k = abs(k);

         if (abs(i-j) == k) f++;

      }

   }

   f = n_queens - f;

   return f;

}

Output

This program produced the following solution to the N-Queens problem with N=50 queens.  Notice that some of the minimum fitness values are negative.  This alters the random selection of the fittest parents, but if these values are few and small, then the effect is not enough to halt the genetic algorithm.

For 50 queens there are over 1064 ways of placing queens ensuring each is in its own row and column.  An exhaustive search of these possible solutions to find a arrangement without diagonal conflicts would be time consuming.  The genetic algorithm search found a solution in only 568 generations, requiring 284,500 function evaluations.

OPTIMUM SOLUTION

 

   Fitness: 25.000000

   Phenotypes:

      Nominal:  25

   Function Calculations: 284500

   Population Size:       500

   Number of Generations: 568

   Nominal Phenotype(s):

 

   23  12  16   9  13   2  18   1  21  10   6  19

    3  20   0   7  15   4   8  17  22  14   5  11

   24

   Chromosome (Base-2 Encoded):

 

   23 12 16 9 13 2 18 1 21 10 6 19 3 20 0 7 15 4

    8 17 22 14 5 11 24

 

GENERATION STATISTICS

Total Number of Generations: 568

 

Generation  Max. Fit. Min. Fit.  Avg. Fit.    CV

*************************************************

Gen.   0:       19      -10         8.11    52.98

Gen.  25:       21       -2        11.02    30.33

Gen.  50:       21       -4        11.49    29.48

Gen.  75:       21       -1        11.15    29.51

Gen. 100:       21       -3        11.42    33.10

Gen. 125:       21       -3        11.17    31.50

Gen. 150:       21       -4        10.64    34.01

Gen. 175:       21       -8        10.60    34.64

Gen. 200:       21       -4        11.16    32.47

Gen. 225:       21       -2        10.76    34.18

Gen. 250:       22       -1        11.08    32.50

Gen. 275:       22       -4        10.98    34.14

Gen. 300:       23       -3        10.97    33.84

Gen. 325:       23       -1        11.75    29.37

Gen. 350:       23      -10        11.08    33.76

Gen. 375:       23       -4        11.54    31.17

Gen. 400:       23        1        11.55    30.43

Gen. 425:       23       -8        12.76    26.36

Gen. 450:       23       -2        12.01    27.39

Gen. 475:       23        0        12.94    27.34

Gen. 500:       23        3        13.78    22.91

Gen. 525:       23        4        15.05    17.90

Gen. 550:       24        4        16.47    18.41

Gen. 568:       25       12        19.48    11.63

*************************************************

OPTIMUM SOLUTION:

 

Fitness:  25

 

Chromosome: 23 12 16  9 13  2 18  1 21 10  6 19  3 20

             0  7 15  4  8 17 22 14  5 11 24

Board Positions:

 

---------------------------------------------------

| | | | | | | | | | | | | | |Q| | | | | | | | | | |

---------------------------------------------------

| | | | | | | |Q| | | | | | | | | | | | | | | | | |

---------------------------------------------------

| | | | | |Q| | | | | | | | | | | | | | | | | | | |

---------------------------------------------------

| | | | | | | | | | | | |Q| | | | | | | | | | | | |

---------------------------------------------------

| | | | | | | | | | | | | | | | | |Q| | | | | | | |

---------------------------------------------------

| | | | | | | | | | | | | | | | | | | | | | |Q| | |

---------------------------------------------------

| | | | | | | | | | |Q| | | | | | | | | | | | | | |

---------------------------------------------------

| | | | | | | | | | | | | | | |Q| | | | | | | | | |

---------------------------------------------------

| | | | | | | | | | | | | | | | | | |Q| | | | | | |

---------------------------------------------------

| | | |Q| | | | | | | | | | | | | | | | | | | | | |

---------------------------------------------------

| | | | | | | | | |Q| | | | | | | | | | | | | | | |

---------------------------------------------------

| | | | | | | | | | | | | | | | | | | | | | | |Q| |

---------------------------------------------------

| |Q| | | | | | | | | | | | | | | | | | | | | | | |

---------------------------------------------------

| | | | |Q| | | | | | | | | | | | | | | | | | | | |

---------------------------------------------------

| | | | | | | | | | | | | | | | | | | | | |Q| | | |

---------------------------------------------------

| | | | | | | | | | | | | | | | |Q| | | | | | | | |

---------------------------------------------------

| | |Q| | | | | | | | | | | | | | | | | | | | | | |

---------------------------------------------------

| | | | | | | | | | | | | | | | | | | |Q| | | | | |

---------------------------------------------------

| | | | | | |Q| | | | | | | | | | | | | | | | | | |

---------------------------------------------------

| | | | | | | | | | | |Q| | | | | | | | | | | | | |

---------------------------------------------------

| | | | | | | | | | | | | |Q| | | | | | | | | | | |

---------------------------------------------------

| | | | | | | | |Q| | | | | | | | | | | | | | | | |

---------------------------------------------------

| | | | | | | | | | | | | | | | | | | | |Q| | | | |

---------------------------------------------------

|Q| | | | | | | | | | | | | | | | | | | | | | | | |

---------------------------------------------------

| | | | | | | | | | | | | | | | | | | | | | | | |Q|

----------------------------------------------------


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