**4.2 CA-IM\_1**

398 Bio-Inspired Computational Algorithms and Their Applications

has a specific capacity as well each object has a specific weight. For example, Weing7 (105, 2)

Every experiment presented here has a similar capacity to the work described in DGA and DGA-SRM (Aguire et al., 2000) such as: population size, number of function evaluations in each run and a total of 100 independent runs. Each run uses a different seed for the random initial population. To improve understanding of DGA and DGA-SRM algorithms, some

• The parameter *K* represents the number of sub-populations (partitions). Hence,

• The parameter *M* is the number of generations between migration events (migration

• The symbol *N* represents the number of times the global optimum was found in the 100

• *Average* is the average of the best solutions and *Stdev* is the standard deviation around

In DGA and DGA-SRM, each sub-population broadcasts a copy of its *R* best individuals to all of its neighbor sub-populations. Hence, every sub-population in every migration event

and the sub-populations evolve in total isolation, the values corresponding to such a characteristic are denoted by *X* in the table. The results for knapsack problem Weing7 for

8 0.10 5 2 100 5 0 **1094423.4** 433.38 50 100 80 63 1095421.44 30.84 8 0.05 5 1 100 5 0 1093284.95 733.24 50 100 100 66 1095423.58 29.84 8 0.01 1 1 100 5 0 1089452.96 1082.41 50 100 80 77 **1095430.51** 26.51 8 X X 100 X 0 1087385.56 1729.4 50 100 X 60 1095419.80 30.86

According to Table 1 the best value found in *Average* is equal to 1094423.4, for DGA and 1095430.51 for DGA-SRM. Table 1 also indicates that the DGA-SRM improves the results in relation to DGA. Table 2 shows the results found for others knapsack problems by DGA and DGA-SRM. In order to simplify the results shown in Table 2, the following

configuration parameters should be considered: K = 16 sub-populations and

*<sup>m</sup>* / **DGA DGA-SRM** 

*<sup>m</sup>* = *L* x *R* migrants, where *L* is the number of links. When there is no migration

μ λ

**M N Average Stdev** 

*total* =800; T=8x105 ).

μ= 25

λ

• The symbol *T* represents the number of function evaluations in each run;

DGA and DGA-SRM is shown in the Table 1 (Aguirre et al., 2000).

**M N Average Stdev** 

Table 1.The best results for Weing7 (105, 2) by DGA and DGA–SRM (

represents a threshold (utilized for control of a normalized mutant's

λ

*total* (fixed in 800);

respectively;

μ and λ

represents a MKP with 105 objects and 2 knapsacks.

• The maximum size of the population is represented by

• The parent and offspring population sizes are represented by

parameters and symbols are presented:

τ

*Average*, respectively;

**L R** 

(Aguirre et al., 2000).

λ

*total* (maximum=800);

λ \**K*= λ

interval) ;

survival ratio).

λ

runs; • The symbol

receives

K λ λ For the algorithm proposed (CA-IM) various parameters and symbols are also considered such as:


For the tests carried out for CA-IM\_1, the selection chosen was tournament, whose value is 3, the mutation rate (*PM)* is 0.025 and recombination rate (*PR)* is 0.6. The situational knowledge configurations are: *SKP=0.2* and *SKM*=0.5. Table 3 shows the results found by CA-IM\_1, whose best value found in *Average* is 1095445 (the optimal value) and in the *Average of Generations* is 44.49. All values reached have optimum value. However, if *Average of Generations* is low in relation to total of generations, then this means that the optimum is found in few generations.

As it is shown in Table 3, it is possible to observe that CA-IM outperforms DGA-SRM for any configuration such as the number of sub-populations (islands) and size of subpopulation. Similarly, CA-IM also exhibits higher convergence reliability than DGA-SRM with higher values for *N* and *Average* with smaller *Stdev*. These results show that the CA-IM produces higher performance for all utilized parameters.

Performance Study of Cultural Algorithms Based on

Table 5. The best results for Weing7 (105,2) by CA-IM\_2 (

Table 6. The best results for other problems by CA-IM\_2 (

results for Weing7 as is shown in Table 3 and Table 5.

different seed for the random initial population.

α

knapsack problems.

α

**Problem (n, m) P K**

**4.4 CA-S (Standard CA)** 

*P* **K** 

Genetic Algorithm with Single and Multi Population for the MKP 401

Table 5 that shows the results for Weing7 and in Table 6 that shows the results for others

400 8 0.125 50 20 100 **70.48** 1095445 0.0 400 8 0,125 50 05 100 72.72 1095445 0.0 100 7 1.0 100 05 100 107.11 1095445 0.0

Petersen6 (39,5) 400 8 0.125 50 20 100 37.89 10618.0 0.0 Petersen6 (39,5) 400 4 0,25 100 05 100 **33.39** 10618.0 0.0 Petersen7(50,5) 400 8 0.125 50 20 100 81.46 16537.0 0.0 Petersen7(50,5) 400 4 0,25 100 05 100 **74.38** 16537.0 0.0 Sento1(60,30) 400 8 0,25 50 20 98 **112.55** 7771.75 1.7717 Sento1(60,30) 400 4 0,25 100 05 100 126.46 7772.0 0.0 Sento2(60,30) 400 8 0.125 50 20 71 183.35 8720.0 3.7199 Sento2(60,30) 400 4 0,25 100 05 88 **173.53** 8721.38 2.1732

The implementation of random rate for mutation and recombination in CA-IM\_2 doesn't produce satisfactory results in comparison to CA-IM\_1, as it is shown in Table 6. In addition, the *Average of Generations* from algorithm CA-IM\_2 is greater than CA-IM\_1 for all knapsack problems. However, in comparison to CA-IM\_1, there are few differences in

For CA-S we also utilized the same configuration such as: tournament value=3, *PM*= 0.025 and *PR* = 0.6. The situational knowledge configuration is equal to 0.2 (*SKP=0.2).* Every experiment presented here also consists of 100 independent runs and each run uses a

> Problem (, m) P N Average Stdev. Petersen6 (39,5) 800 97 10617.58 2.4002 Petersen7 (50,5) 800 81 16533.7 6.8703 Sento1 (60,30) 800 100 7772.0 0.0 Sento2 (60,30) 800 82 8721.14 2.4495 Weing7 (105,2) 800 100 1095445.0 0.0

Table 7. The best results for all knapsack problems by CA-S (T=4x105).

**SI M N Average of Generations Average Stdev** 

**SI M N Average of** 

λ

λ

*total* =800, T=8x105).

*total* = 800, T=4x105).

**Generations Average Stdev.** 


Table 3. The best results for **Weing7** (105, 2) by CA-IM\_1 ( λ*total* =800 and T=8x105).

A new result "Average of Generations" was introduced so as to evaluate other type other type of performance whose value represents the average of generations that the optimum value was found for 100 independent runs for each problem presented. Particularly, it occurs when *M* is low and *K* is high (see result for Average of Generations). This means that a larger number of islands with small populations produce better convergence.

According to Table 3 the best value found in *Average* is 1095445 (the optimal value) while the *Average of generations* is 44.49 that means a low value, considering that 500 generations was utilized in each run which T=4x105. This represents 500 generations with a population size equal to 800 (including all subpopulations). Table 4 shows the results for others MKPs found by algorithm CA-IM\_1.


Table 4. The best results for other problems by CA-IM\_1 ( λ*total* = 800, T=4x105).

Thereby, it is possible to observe that CA-IM\_1 outperforms DGA-SRM. Similarly, CA-IM\_1 also exhibits higher convergence reliability (higher values of *N* and *Average* with smaller *Stdev*) than DGA-SRM. These results show that the CA-IM\_1 is able to find global optimal for MKP, taking into consideration the tests results with 100% success.

The problem that presented greater difficulty was Sento2, that presented in some cases optimal values near to 100% such as N=98 and N=99. Even with results of *N* < 100 they are still better than the results obtained in the chosen benchmarks. In the meantime, the implementation of some adjustments allows CA-IM\_1 to reach N=100 for Sento2.

### **4.3 CA-IM\_2**

For the tests carried out for CA-IM\_2 the selection chosen was tournament whose value is 3. The mutation rate (*PM)* is a random value in a specific interval: *PM=* [0.01, 0.5]. The Recombination rate (*PR)* is also a random value in an interval: *PR= [0.1,* 0.99]. The situational knowledge configurations are: *SKP=0.2* and *SKM*=0.5. The CA-IM\_2 results are presented in 400 Bio-Inspired Computational Algorithms and Their Applications

400 8 0.125 50 20 100 52.9 1095445 0.0 400 8 0,125 50 05 100 **44.49 1095445** 0.0 100 7 1.0 100 05 100 68.87 1095445 0.0

A new result "Average of Generations" was introduced so as to evaluate other type other type of performance whose value represents the average of generations that the optimum value was found for 100 independent runs for each problem presented. Particularly, it occurs when *M* is low and *K* is high (see result for Average of Generations). This means that

According to Table 3 the best value found in *Average* is 1095445 (the optimal value) while the *Average of generations* is 44.49 that means a low value, considering that 500 generations was utilized in each run which T=4x105. This represents 500 generations with a population size equal to 800 (including all subpopulations). Table 4 shows the results for others MKPs

Petersen6 (39,5) 400 8 0.125 50 20 100 30.22 10618.0 0.0 Petersen6 (39,5) 400 4 0,25 100 05 100 **26.29** 10618.0 0.0 Petersen7 (50,5) 400 8 0.125 50 20 100 78.49 16537.0 0.0 Petersen7 (50,5) 400 4 0,25 100 05 100 **71.51** 16537.0 0.0 Sento1 (60,30) 400 8 0.125 50 20 100 100.21 7772.0 0.0 Sento1 (60,30) 400 4 0,25 100 05 100 **87.44** 7772.0 0.0 Sento2 (60,30) 400 8 0.125 50 20 99 185.19 8721.81 0.099 Sento2 (60,30) 400 4 0,25 100 05 100 **166.12** 87722.0 0.0

Thereby, it is possible to observe that CA-IM\_1 outperforms DGA-SRM. Similarly, CA-IM\_1 also exhibits higher convergence reliability (higher values of *N* and *Average* with smaller *Stdev*) than DGA-SRM. These results show that the CA-IM\_1 is able to find global optimal

The problem that presented greater difficulty was Sento2, that presented in some cases optimal values near to 100% such as N=98 and N=99. Even with results of *N* < 100 they are still better than the results obtained in the chosen benchmarks. In the meantime, the

For the tests carried out for CA-IM\_2 the selection chosen was tournament whose value is 3. The mutation rate (*PM)* is a random value in a specific interval: *PM=* [0.01, 0.5]. The Recombination rate (*PR)* is also a random value in an interval: *PR= [0.1,* 0.99]. The situational knowledge configurations are: *SKP=0.2* and *SKM*=0.5. The CA-IM\_2 results are presented in

implementation of some adjustments allows CA-IM\_1 to reach N=100 for Sento2.

**SI M N Average of** 

λ

**Generations**

*total* = 800, T=4x105).

**Average Stdev.** 

a larger number of islands with small populations produce better convergence.

α

**SI M N Average of Generations Average Stdev** 

λ

*total* =800 and T=8x105).

*P* **K** 

α

found by algorithm CA-IM\_1.

**Problem (n, m) P K** 

**4.3 CA-IM\_2** 

Table 3. The best results for **Weing7** (105, 2) by CA-IM\_1 (

Table 4. The best results for other problems by CA-IM\_1 (

for MKP, taking into consideration the tests results with 100% success.

Table 5 that shows the results for Weing7 and in Table 6 that shows the results for others knapsack problems.



Table 5. The best results for Weing7 (105,2) by CA-IM\_2 ( λ*total* =800, T=8x105).

Table 6. The best results for other problems by CA-IM\_2 ( λ*total* = 800, T=4x105).

The implementation of random rate for mutation and recombination in CA-IM\_2 doesn't produce satisfactory results in comparison to CA-IM\_1, as it is shown in Table 6. In addition, the *Average of Generations* from algorithm CA-IM\_2 is greater than CA-IM\_1 for all knapsack problems. However, in comparison to CA-IM\_1, there are few differences in results for Weing7 as is shown in Table 3 and Table 5.

## **4.4 CA-S (Standard CA)**

For CA-S we also utilized the same configuration such as: tournament value=3, *PM*= 0.025 and *PR* = 0.6. The situational knowledge configuration is equal to 0.2 (*SKP=0.2).* Every experiment presented here also consists of 100 independent runs and each run uses a different seed for the random initial population.


Table 7. The best results for all knapsack problems by CA-S (T=4x105).

Performance Study of Cultural Algorithms Based on

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Table 7 shows the results from standard Cultural Algorithm (CA-S) that utilizes single population. According to results, the CA-S reaches optimum average for 100 runs only for Sento1 and Weing7. However, the results from CA-S for Petersen6, Pertersen7 and Sento2 outperform the results presented by DGA-SRM.
