**4.1 DGA and DGA-SRM**

The DGA works with various 0/1 multiple knapsack problems (NP hard combinatorial) which from previous efforts seem to be fairly difficult for GAs (Aguirre et al., 2000). Those algorithms were evaluated on test problems which are taken from the literature. The problem sizes range from 15 objects to 105 and from 2 to 30 knapsacks and can be found in OR-Library (Beasley, 1990). The knapsack problems are defined by: problem *(n, m)* where *n* represents the number of objects and *m* represents the number of knapsacks. Each knapsack

Performance Study of Cultural Algorithms Based on

**LR**

• The parameter *P* is the size of main population;

α

through the Situational Knowledge.

island at function of *P*.

• The number of islands is *K* (number of sub-populations);

• The sub-population size in each island is *SI*, since *SI* **=**

multi population space is represented by *SKM*.

produces higher performance for all utilized parameters.

λ

Table 2. The best results for other problems by DGA and DGA-SRM (

λ λ

**Problem (n, m)** 

**4.2 CA-IM\_1** 

• The parameter

runs.

*Average*;

each run.

found in few generations.

such as:

Genetic Algorithm with Single and Multi Population for the MKP 399

**M N Average Stdev** 

Petersen6 (39,5) 0.01 1 1 50 5 0 **10506.90** 26.11 50 140 77 10614.82 5.82 Petersen7 (50,5) 0.10 5 1 100 5 0 1093284.95 733.24 50 40 89 16535 5.94 Sento1 (60, 30) 0.10 5 1 100 5 0 1089452.96 1082.41 50 40 98 7771.78 1.54 Sento2 (60, 30) 0.10 5 1 100 5 0 1087385.56 1729.4 50 40 84 8721.32 2.11

For the algorithm proposed (CA-IM) various parameters and symbols are also considered

• The parameter *PM* is the probability of mutation and *PR* probability of recombination.

• The percentage of best individuals in Situational Knowledge on population space is represented by *SKP* and the percentage of best individuals in Situational Knowledge on

• The parameter *M* is the number of generations between migration events (migration interval). Here *M* determines the interval of influence from the islands population

• 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

• *Average of generations* is the average of the generations whose best solution was found in

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

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

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

is the percentage which defines the size of the population of each

α**\*P**.

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

λ

λ

τ**=0.35)** 

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

**M N Average Stdev** 

has a specific capacity as well each object has a specific weight. For example, Weing7 (105, 2) represents a MKP with 105 objects and 2 knapsacks.

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 parameters and symbols are presented:


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 receives λ*<sup>m</sup>* = *L* x *R* migrants, where *L* is the number of links. When there is no migration 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 DGA and DGA-SRM is shown in the Table 1 (Aguirre et al., 2000).


Table 1.The best results for Weing7 (105, 2) by DGA and DGA–SRM ( λ*total* =800; T=8x105 ).

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 μ = 25 (Aguirre et al., 2000).


Table 2. The best results for other problems by DGA and DGA-SRM ( λ*total* = 800; T=4x105).
