Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 4527402, 10 pages

http://dx.doi.org/10.1155/2016/4527402

## Simulation Experiment Exploration of Genetic Algorithm’s Convergence over the Relationship Advantage Problem

Department of Industrial Engineering, School of Mechanical and Electronic Engineering, Wuhan University of Technology, Wuhan 430070, China

Received 12 April 2016; Revised 14 June 2016; Accepted 15 June 2016

Academic Editor: László T. Kóczy

Copyright © 2016 Yabo Luo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Concentrating on the convergence analysis of Genetic Algorithm (GA), this study originally distinguishes two types of advantage sources: value advantage and relationship advantage. Accordingly, the quantitative feature, complete quantization feature, and the partial quantization feature in the fitness evaluation are proposed. Seven simulation experiments show that these two types of advantages have different convergence properties. For value advantage problems, GA has a good convergence. However, for a relationship advantage problem, only from the practical point of view, it is possible to get a feasible and even satisfactory solution through large-scale searching, but, in theory, however, the searching process is not convergent. Therefore, GA is not reliable to solve relationship advantage problems, to which most engineering problems involving combinatorial optimization belong. This study systematically shows convergence properties of “relationship advantage” through simulation experiments, which will be a new area for the further study on GA.

#### 1. Introduction

There are now many operation research tools for solving combinatorial optimization problems, such as the Ant Colony Optimization (ACO) algorithm, Particle Swarm Optimization (PSO) algorithm, and Artificial Neural Network (ANN). However, no method has been proven to be better than some other methods, since each method has its own advantages and disadvantages. GA, as one of the oldest and the most used heuristic algorithms for NP-hard problems, has been attracting a large number of scholars to study its properties and applications. This research is motivated by a flexible job shop reconstruction employing GA [1], whose objective is to lay out machines to get a high logistic efficiency. However, the search is not stable and it is difficult to get a satisfactory solution. By a great deal of literature review [2–5], the same problem has been found in other engineering cases, such as the path planning problem and the bin packing problem, among other problems, in which a feasible solution may be gotten by great size searching, but the convergence cannot be guaranteed and the efficiency is really low. To explore the reasons for low search efficiency and poor convergence, this research carries out a series of simulation experiments, since simulation technology is a good approach to finding the root cause.

Since the GA is a kind of simulation method which simulates the process of biological evolution [6], for a long time, its effectiveness has been proved through experiments and application results based on the experience design theory [7], and so far there is no strict mathematical proof to demonstrate its convergence. Even in* Adaptation in Natural and Artificial Systems* [8], originally published in 1975, in which the GA was originally formally proposed by Michigan University’s Professor Holland, and republished in 1992, there was no strict mathematical proof given. GA is one of heuristic optimization algorithms, which simulates Darwin’s natural selection and genetic mechanism in the evolution [9]. From a practical perspective, the researches of GA in recent years focus on how to improve it to solve actual industrial problems such as the traffic scheduling, packing and layout optimization, and job shop scheduling problems.

For example, in the field of vehicle and path planning, Vidal et al. [10] present an efficient hybrid genetic search with advanced diversity control, which is proven feasible by outperforming all current state-of-the-art approaches on classical literature benchmark instances for any combination of periodic, multi-depot, site-dependent, and duration-constrained vehicle routing problem with time windows. Tasan and Gen [11] propose a GA based approach to vehicle routing problem with simultaneous pickup and deliveries, which is evaluated by solving several test problems.

In the field of packing and layout optimization, Gonçalves and Resende [12] present a multipopulation biased random-key GA for the single container loading problem, and the proposed algorithm is extensively tested on the complete set of benchmark instances and is compared with 13 other approaches to demonstrate its better performance. Moradi and Abedini [13] present a novel combined GA and Particle Swarm Optimization algorithm model for optimal location and sizing of distributed generation on distribution systems, the effectiveness of which is demonstrated by a detailed performance analysis carried out on bus systems.

In the field of job shop scheduling, Yusof et al. [14] propose a new hybrid parallel GA based on a combination of asynchronous colony and autonomous immigration GAs to solve benchmark job shop scheduling problem, whose better performance is shown by decreasing the makespan considerably as compared to the conventional GA. As to the hybrid flow shop scheduling with multiprocessor task problem, Engin et al. [15] propose a new mutation operator and a full factorial experimental design was determined by using the best values of the control parameters and the operators. Zhang et al. [16] study a job shop scheduling problem with two new objective functions based on the setup and synergy costs besides the traditional total weighted tardiness criterion and present a Pareto-based GA incorporating a local search module, whose effectiveness is verified by the computational experiments on both real-world and randomly generated scheduling instances.

So far, it can be seen that for the application of GA it fails to carry out the strict mathematical proof of convergence, and its validity and practicability are proved mainly on the following four aspects:(1)Taking a standard test case as the instance to test its validity.(2)Comparing with standard GA to illustrate the superiority of the improved algorithm.(3)Comparing with the relative research to illustrate the superiority of the proposed method.(4)Using a large number of simulation experiments to show the reliability of the improved algorithm.

This study is motivated by a flexible machine layout problem [1, 17]. There are some researches which apply GA to the flexible machine layout optimization [18–20]. However, in a flexible workshop layout optimization study involving 6 machines [1], we found that although there could be feasible solutions or even satisfactory solutions appearing in the process of search, better solutions show no sign of spreading in the population, and the search space also does not appear to narrow; in other words, the search process is not convergent. After a detailed investigation of individuals in each generation, we found that values of design variables may be completely different between those better solutions, while there are similar relative position relations between those variables. These better position relations cannot be kept and spread in the subsequent iterations using GA.

In this paper, the advantage derived from relationship between variables is defined as relationship advantage, and the advantage derived from values of design variables is defined as value advantage. This study discusses the convergence property of the value and the relationship advantage by designing a series of simulation experiments.

#### 2. Value Advantage and Relationship Advantage

In the biological world, two different types of advantages can generate higher fitness: value advantage and relationship advantage. Value advantage comes from a single trait’s value. For example, in Darwin’s theory of evolution, a giraffe with a longer neck has the stronger ability for reaching food on the tree; this is a value advantage produced by “neck length.” The other instance of value advantage in Darwin’s theory of evolution is the moth: the darker the color of the moths, the stronger the survival ability; this is a typical value advantage too. The degree of superiority depends on the value of a single trait.

Most advantages in the biological world, however, are not from the value of a single trait, but from the relationship between multiple traits, which is defined as relationship advantage in this paper. For example, cheetah is the running champion of animal world, but the cheetah’s leg is not longer than that of the horse, and its body is also not stronger than that of the tiger; cheetah’s superiority of running ability is not from longer legs or a more robust body, but from the high degree of coordination among locomotive organs; this is the typical relationship advantage. Again, the dragonfly’s superiority of balance ability does not come from the size of head or body, but from the perfect proportion between head size and body length, which is a typical relationship advantage. In fact, in the world of biology, most advantages belong to the type of relationship advantage; even in the giraffe example referred to in Darwin’s theory of evolution, if there is no coordination with long neck muscles, heart function, and other traits, the only characteristic of long neck is unable to form a survival advantage.

In the field of engineering, there are value and relationship advantages as yet. The cause of carrying out this study is the following: in the process of using a GA for machine layout optimization considering the workshop logistics characteristics, we found that the standard GA for this problem is not convergent. The reason is as follows: the optimization degree of machine layout depends on the relative position relationship among machines and the compatibility with the workshop logistics characteristics and does not depend on a specific place of a single machine. That is to say, even if the locations of multiple machines change, its logistics characteristic value still does not change as long as the relative position relationship among machines keeps changeless. Machine layout problem is a typical relationship advantage problem. The following two examples can more clearly show the difference between value advantage and relationship advantage.

Figure 1 is a circuit including a slide rheostat, a light bulb, and a set of batteries. Taking the slide rheostat’s value as the design variables and the battery voltage and the maximum resistance of the slide rheostat as constraint conditions and maximizing the brightness of the light bulb as the optimization objective, an optimization model can be constructed. The optimization model is as follows.