Abstract
Diversity of the population in a genetic algorithm plays an important role in impeding premature convergence. This paper proposes an adaptive neurofuzzy inference system genetic algorithm based on sexual selection. In this technique, for choosing the female chromosome during sexual selection, a bilinear allocation lifetime approach is used to label the chromosomes based on their fitness value which will then be used to characterize the diversity of the population. The motivation of this algorithm is to maintain the population diversity throughout the search procedure. To promote diversity, the proposed algorithm combines the concept of gender and age of individuals and the fuzzy logic during the selection of parents. In order to appraise the performance of the techniques used in this study, one of the chemistry problems and some nonlinear functions available in literature is used.
1. Introduction
A large scale of design, control, scheduling, or other engineering problems results in solution of optimization problems. Genetic algorithms (GAs) were first considered by Holland [1]. A genetic algorithm is a numerical optimization procedure that is based on evolutionary principles such as selection, recombination, and mutation. In many areas of chemistry, there are problems to which GAs can be used. For example, one of the principal subfield of analytical chemistry is the qualitative and quantitative identification of the main components of unknown mixtures by means of spectroscopic methods that investigate the molecules utilizing electromagnetic radiation. Genetic algorithms have been used here as they are effective at finding patterns in data even when the data contains a large amount of extraneous information. Genetic algorithms have also been used for the generation of regression curves, protein folding, and structure elucidation and for system optimization [2–4]. Yang et al. [5–8] utilized GA in real applications of various soft-computing techniques in different fields.
Premature convergence is a classical problem in finding optimal solution in GA. The population diversity is a way of avoiding the premature convergence in a GA. In a traditional GA, chromosomes reproduce asexually where any two chromosomes may be parents during crossover. Gender division and sexual selection here inspire a model of gendered GA in which crossover takes place only between chromosomes of opposite sex. The sex of chromosomes is not only accountable for preserving diversity in population and maintaining a victorious genetic pool by means of selection, crossover, and mutation, but it is also accountable for the optimization of the different tasks which are very important for survival.
Jalali Varnamkhasti and Lee [9] introduced a new sexual selection. In their technique, the population is divided into two groups of males and females. During the sexual selection, the female chromosome is selected by the tournament selection while the male chromosome is selected based on the hamming distance from the selected female chromosome, fitness value, or active genes. In another study conducted by Jalali Varnamkhasti and Lee [10], a fuzzy genetic algorithm based on this technique for selection mechanism was suggested. They used some nonlinear numerical functions, and by considering the results from each test function, they showed that the proposed technique of grouping the male and female chromosomes alternately outperforms other grouping techniques of sexual selection mechanisms. Also, the application of this technique for sexual selection is given by Jafari et al. [11] for committee neural networks.
In this study, an obvious characteristic between the two gender groups, with the possibility of embedding different tasks for each one, is considered such as the determination of which partners are suitable for mating and crossover. We believe the relations between age, effectiveness and fitness as in biological systems will affect the selection procedure. A bilinear allocation lifetime approach is used to label the chromosomes based on their fitness value [12]. The obtained chromosomes labels are used to characterize the diversity of the population. The population is then divided into two groups: male and female, so that they are selected in an alternate way. In each generation, the layout of selection for male and female is different.
In short, the aim of this paper is to keep the diversity of population by female preference. The selection of the female chromosome is done through a set of fuzzy rules and a newly developed genetic-neurofuzzy algorithm.
2. Neurofuzzy Inference Systems Genetic Algorithm
In this section, we concentrate on the discussion of the proposed neurofuzzy inference systems genetic algorithm for solving combinatorial optimization problems. In the remainder of this section, we explained the framework of the proposed algorithm as illustrated in Figure 1.
2.1. Initial Population
For the nonlinear functions, a solution can be simply encoded by a string of 0’s and 1’s. A good initial population makes it easy for a GA to converge to good solutions while a poor initial population can prolong a GA convergence. There are different approaches in generating initial population for a GA. The most common method is by random generation [10]. In this paper, we use the random generation method to achieve better diversity in the population.
2.2. Lifetime
Three methods are presented for the calculation of chromosomes lifetime in genetic algorithms with varying population size, chromosomes are categorized according to their fitness. The purpose is to use the allowed range of lifetime values in a way which is more suitable to search the optimum than proportional, linear, and bilinear strategies [12]. In this study, a bilinear allocation lifetime approach proposed in [12] is used to label the chromosomes based on their fitness value which will then be used to characterize the diversity of the population. In this case, a linguistic variable “age” is utilized for chromosomes. Figure 2 describes the linguistic variable age where infant, adult and old are the linguistic values. The membership functions for the linguistic terms are called semantic rules.
To find the membership function, we use the fitness value of each chromosome and the minimum, maximum, and average fitness values of the population in each generation. Each chromosome has its own label determined by the age function.
Let , , and , where is fitness value of chromosome ; is average fitness value; is minimum fitness value; and is maximum fitness value of population.
The age function can be written as where is chromosome ; is population size; and ; and are minimum and maximum age respectively.
Equation (2.1) is suited for maximization problems which relate to higher fitness values while (2.2) is more suitable for minimization problems which relate to lower fitness values. The fuzzification interface defines for each chromosome the possibilities of the three linguistic values. These values determine the applicability degree for each rule premise. The computation takes into account all chromosomes in each generation and relies on the triangular membership functions shown in Figure 3 with and .
On the other hand, we can consider linguistic rules and membership function for each rule as follows: A bilinear allocation lifetime approach proposed in [12] is used to label the chromosomes based on their fitness value which will then be used to characterize the diversity of the population: Let be the label of half of the population, and then the population can be divided into three levels, low, medium and high diversity as follows: where is a parameter that has correlation with the domain of labels in the population and ( means nearest integer number to x, e.g., and ). This computation is performed in every generation and relies on the triangular membership functions shown in Figure 4.
We can consider linguistic rules and membership function for each rule as follows:
2.3. Sexual Selection
Darwin differentiated sexual and natural selections as different types of processes operating on different sorts of traits each in accordance with varying evolutionary dynamics types. From Darwin’s perspectives, natural selection enhances the abilities of an organism to survive in a competitory environment whereas sexual selection (SX) pertains more to the capabilities of attracting and selecting mates in an effort to generate an offspring that is viable and attractive.
Indeed, organisms reproducing sexually should refrain from random mating since the genetic quality of a mate will dictate half of the genetic quality of the respective offspring. The clue to successful selection of mates is evolving a mechanism for mate selection that internalizes the long-term fitness reproducing results with various types of possible mates. The gain of mate selection is that passive fitness features which in the past posed threat to survival can be escaped while in the meantime the positive ones can be utilized. Accordingly, choice of mate is well calibrated if the recognized sexual attractiveness of the likely mates is strongly linked with the factual attractiveness, viability, and fertility of those offspring they may produce.
In numerous species, females choose the males for mating and producing offspring. This, on one hand, implies that the males must compete with one and another to make sure they will be selected as mates. On the other hand, females will be concerned in finding those males having high fitness and attractiveness as by mating with them they will guarantee production of offspring having high survival potentials and being attractive well enough to be later selected as mates by females in the succeeding generations.
Inspired by the nongenetic sex determination system prevalent in some reptile species where sex is determined by the temperature at which the egg is incubated, the population is divided such that the male and female would be selected in an alternate way. The layout of the male and female chromosomes in each generation is different [10].
During the sexual selection, the male chromosome is selected randomly from the male category and the selection of the female chromosome is done through a set of fuzzy rules and a newly developed genetic-neurofuzzy algorithm.
2.3.1. Adaptive Neurofuzzy Inference Systems Genetic Algorithm
The Sugeno fuzzy model [13] was proposed for a systematic approach to generate fuzzy rules from a given input-output dataset. A typical Sugeno fuzzy model given in Figure 5 shows the ANFIS architecture that corresponds to the first-order Sugeno fuzzy model. For simplicity, we assume that the ANFIS has two inputs and but only one output .
(a)
(b)
In adaptive neurofuzzy inference systems genetic algorithm (ANFISGA), there are two inputs: male’s age and population diversity and one output: female’s age, .
In ANFISGA, we have five layers. Layer 1 is the input layer. Neurons in this layer simply pass external crisp signal to Layer 2. Layer 2 is the fuzzification layer. Neurons in this layer perform fuzzification. Layer 3 is the rule layer. Each neuron in this layer corresponds to signal Sugeno-type fuzzy rule. Layer 4 is the normalization layer. Each neuron in this layer receives inputs from all neurons in the rule layer and calculates the normalized firing strength of given rule. Layer 5 is the defuzzification layer. Each neuron in this layer is connected to the respective normalization neuron, and also receives initial inputs, and .
The main difference between ANFISGA and ANFIS is the adaptation. The ANFIS relies on the weights, during adaptation. But in the ANFISGA, the weights are all constant. For adaptation, we use sexual selection based on female choice and population diversity. Figure 6 shows the ANFISGA architecture that corresponds to the first-order Sugeno fuzzy model. The rules for ANFISGA are given in Table 1, where , , and are defined in (2.1), is given in (2.3), is taken from (2.6), and the output of ANFISGA is which refers to the lower bound of the during the selection of the female chromosome.
After finding , if we are not able to find a chromosome that has a value at least equal to , then we select a chromosome having the nearest fitness value to . On the other hand, if we are able to find more than one chromosome which satisfies having condition, then we choose a chromosome having the highest fitness value among them. This technique is called complement method.
3. Computational Experiments
The experiments are conducted to compare the performance of our proposed ANFISGA to other GAs found in [14] (see Table 2) for solving the well-known generalised Rosenbrock’s Function introduced in [15] and one of the chemistry problems, namely, multieffect systems.
3.1. Generalized Rosenbrock’s Function ()
The generalized Rosenbrock’s function is a continuous and unimodal function, with the optimum located in a steep parabolic valley with a flat bottom. This feature will probably cause slow progress in many algorithms since they must continually change their search direction to reach the optimum: where .
We also included a binary-coded GA (BGA) with 30 genes per variable, multiple crossovers with two points and proportional selection probability into the experiments. Each algorithm is tested for 30 times with a maximum of 5000 generations per each run. The results of the test function are listed in Table 3. The entries reported the average over 30 runs of the best fitness value found at the end of each run.
3.2. Multieffect Systems
Multieffect systems have two sequences of integration: forward integration sequence, where the heat integration is in the direction of the mass flow and backward or reverse integration sequence, where the integration is in the opposite direction of the mass flow as shown in Figure 7.
(a) Direct
(b) Indirect
In this study, we considered multieffect distillation that is used in [16]. In this multieffect distillation of methanol/water system comprised of two 16 staged columns. The feed stream contains 0.73 mole percentage of methanol is fed to the column at the rate of 4320 kmol/h.
The objectives are to determine the appropriate configuration, feed locations, reflux, and boilup flows in both columns that minimize the operating cost.
For formulation, Preechakul and Kheawhom [16] used the common assumptions of equi-molar overflow, total condenser, and partial reboiler. This method creates acostly product by using as little energy as possible. The product from both columns must contain at least 99% methanol, and the methanol concentration in bottom stream from the second column must not be larger than 1%. The temperature difference between the condenser of high-pressure column and the reboiler of low-pressure column must be larger than 10°C.
The following equations describe the model: where : activity coefficient; : binary parameter; : bottom stream flow rate; : overhead stream flow rate; : feed stream flow rate; : enthalpy of vapor; : enthalpy of liquid; : liquid flow rate at stage ; : number of equilibrium stages; : feed location; : pressure; : absolute pressure, : heat flow rate at condenser; : heat flow rate at reboiler; : ideal gas constant; : temperature; : normal boiling point temperature; : molar volume; : vapor flow rate at stage ; : mole fraction of liquid phase at stage .
The objective function is where and are the products (methanol) of the high-pressure and low-pressure column, respectively. is the heat load to the column, and mol/MJh is the relative cost of energy [16].
This fuzzy genetic algorithm is used in order to find the suitable configuration and to seek other operating situation.
The performance of our algorithms was compared with the presentation of the classical genetic algorithm and Hammersley sequence sampling (HSS) in [16] these results are shown in Table 4.
For traditional GA and genetic algorithm based on HSS, the population size 500 is considered and probability of crossover and mutation used are 0.8 and 0.1 respectively. In ANFISGA, probability of crossover is considered 1 and probability mutation and population size are the same other algorithms.
4. Conclusion
The principle conclusions derived from the results of experiments carried out are the following.(i)The procedure presented is the most successful one for controlling diversity as compared with other methods proposed in the GA literature that have been considered for the experiments.(ii)The adaptation capability of this procedure allows suitable parent to be used for producing a robust operation for test function with different difficulties.Therefore, we may conclude that the female choice by ANFISGA is a suitable way for improving the performance of GAs in keeping the diversity of the population.
Acknowledgment
The authors are indebted to Universiti Kebangsaan Malaysia for funding this research under the Grant OUP-UKM-FST-2012.