A hybrid evolutionary approach based on the invasive weed optimization and estimation distribution algorithms

Abstract

Hybrid evolutionary methods combine approaches extracted from different evolutionary computation techniques to build a single optimization method. The design of such systems represents a current trend in the evolutionary optimization literature. In hybrid algorithms, the objective is to extend the potential advantages of the integrated approaches and eliminate their main drawbacks. In this work, a hybrid method for solving optimization problems is presented. The proposed approach combines (A) the explorative characteristics of the invasive weed optimization method, (B) the probabilistic models of the estimation distribution algorithms and (C) the dispersion capacities of a mixed Gaussian–Cauchy distribution to produce its own search strategy. With these mechanisms, the proposed method conducts an optimization strategy over search areas that deserve a special interest according to a probabilistic model and the fitness value of the existent solutions. In the proposed method, each individual of the population generates new elements around its own location, dispersed according to the mixed distribution. The number of new elements depends on the relative fitness value of the individual regarding the complete population. After this process, a group of promising solutions are selected from the set compound by the (a) new elements and the (b) original individuals. Based on the selected solutions, a probabilistic model is built from which a certain number of members (c) are sampled. Then, all the individuals of the sets (a), (b) and (c) are joined in a single group and ranked in terms of their fitness values. Finally, the best elements of the group are selected to replace the original population. This process is repeated until a termination criterion has been reached. To test the performance of our method, several comparisons to other well-known metaheuristic methods have been made. The comparison consists of analyzing the optimization results over different standard benchmark functions within a statistical framework. Conclusions based on the comparisons exhibit the accuracy, efficiency and robustness of the proposed approach.

Appendices

Appendix A: List of benchmark functions

In Tables 23, 24 and 25, $f\left( {{\mathbf{x}}^{*} } \right)$ is the optimum value of the function, ${\mathbf{x}}^{*}$ the optimum position and D the search space (subset of ${\mathbb{R}}^{n}$).

Table 23 Unimodal test functions considered in the experimental study

Table 24 Multimodal test functions considered in the experimental study

Table 25 Composite test functions considered in the experimental study

Appendix B: Engineering problems

2.1 B1. Gear train problem

In this problem (Sandgren 1990), it is required to minimize the squared difference between the teeth ratio of the gear and a given scalar value. The decision variables are the number of teeth corresponding to each gear. Labels $A$, $B$, $D$ and $F$ are used to identified the gears. The decision variables correspond to $x_{1} = A$, $x_{2} = B$, $x_{3} = D$ and $x_{4} = F$. The scalar value is $1/6.931$, see Fig. 8. The cost function and constraints are defined as follows:

Fig. 8

Gear train design

2.2 B2. Spring problem

In this formulation (Arora 2012), the objective is to minimize the tension or compression experienced by a spring when a load $P$ is applied. For optimization, the wire diameter $d$, the coil diameter $D$ and the number of active coils $n$ are considered. The decision variables are $x_{1} = d$, $x_{2} = D$ and $x_{3} = n$, see Fig. 9. The design problem is formulated as:

Fig. 9

Spring design

2.3 B3. Pressure vessel problem

In this problem (Sandgren 1990), the goal is to design a pressure vessel minimizing the required material used for its construction. Therefore, the optimization problem must consider the thickness of the shell $T_{\text{s}}$, the thickness of the head $T_{\text{h}}$, the internal radius of the vessel $R$ and the length of the vessel $L$, see Fig. 10. The decision variables are ${\mathbf{x}} = \left[ {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right]$ where $x_{1} = T_{\text{s}}$, $x_{2} = T_{\text{h}}$, $x_{3} = R$ and $x_{4} = L$. The cost function and constraints are defined as follows:

Fig. 10

Pressure vessel design

2.4 B4. FM synthesizer problem

An FM synthesizer generates a signal $y\left( {{\mathbf{x}},t} \right)$ similar to a target signal $y_{0} \left( t \right)$. To minimize the error between the signal and the target signal, a parameter estimator for the FM synthesizer is designed considering a finite wave amplitude $a_{i}$ and the frequency $\omega_{i}$ (Das and Suganthan 2011). The decision variables are ${\mathbf{x}} = \left[ {x_{1} = a_{1} ,x_{2} = \omega_{1} ,x_{3} = a_{2} ,x_{4} = \omega_{2} ,x_{5} = a_{3} ,x_{6} = \omega_{3} } \right]$. The cost function and constraints are defined as follows:

2.5 B5. Economic dispatch problem

Economic dispatch is a problem of allocating loads to $n$ plants (Park et al. 2005). The requirement is to minimize the fuel cost of all committed plants considering the total demand $D$ and losses $P_{\text{l}}$. In the following table, $G_{i}$ is the generation cost for the generator $i$ and $P_{i}$ is the electrical power generated by generator $i$. The cost coefficients for generator $i$ are $a_{i}$, $b_{i}$ and $c_{i}$. The transmission loss can be determined from $V_{nn}$ coefficients. The decision variables are ${\mathbf{x}} = \left[ {x_{1} , \ldots , x_{n} } \right]$. The objective function and constraints are defined as follows: