Bitwise Mutation Research Articles

Size Optimization of Truss Structures Based on Genetic Algorithm by Developing Single-point Crossover with Bit-wise Mutation Using FORTRAN Programming

A truss is a collection of axially loaded structural elements connected through pin joints. Optimization of truss structures indicates the determination of best possible conditions that are necessary for achieving the most economical design in terms of lowest possible weights of the truss elements. According to recent studies, Size, Topology and Shape based optimizations are the three possible independent optimization scopes for finding the optimum weight of a truss. The current study focuses on the size optimization of truss structure and aims to identify the most cost-effective sections of the truss using a powerful evolutionary optimization technique named as Genetic Algorithm (GA). Here single point cross over method and bit wise mutation operator are adopted for expanding the search space with the assigned displacement and stress constraints. Fortran based sub routines are developed for each individual steps of GA such as fitness evaluation, selection, cross-over and mutation. Finaly the complete program is run for validation of its outcome. While comparing with the solution of an existing literature regarding a minimization of objective function having two design variables the outcomes tend to show a good correlation. Further, the study is extended for the weight minimization of two widely used truss having 10 members and 17 members respectively with standard Finite Element Method of analysis. Minimum weight of each truss is obtained via convergence plot and compared with the existing literatures. With this simple but efficient global optimization technique optimum weight of truss structure can easily be achieved.

How to Use the Metropolis Algorithm for Multi-Objective Optimization?

The Metropolis algorithm can cope with local optima by accepting inferior solutions with suitably small probability. That this can work well was not only observed in empirical research, but also via mathematical runtime analyses on single-objective benchmarks. This paper takes several steps towards understanding, again via theoretical means, whether such advantages can also be obtained in multi-objective optimization. The original Metropolis algorithm has two components, one-bit mutation and the acceptance strategy, which allows accepting inferior solutions. When adjusting the acceptance strategy to multi-objective optimization in the way that an inferior solution that is accepted replaces its parent, then the Metropolis algorithm is not very efficient on our multi-objective version of the multimodal DLB benchmark called DLTB. With one-bit mutation, this multi-objective Metropolis algorithm cannot optimize the DLTB problem, with standard bit-wise mutation it needs at least Ω(n^5) time to cover the full Pareto front. In contrast, we show that many other multi-objective optimizers, namely the GSEMO, SMS-EMOA, and NSGA-II, only need time O(n^4). When keeping the parent when an inferior point is accepted, the multi-objective Metropolis algorithm both with one-bit or standard bit-wise mutation solves the DLTB problem efficiently, with one-bit mutation experimentally leading to better results than several other algorithms. Overall, our work suggests that the general mechanism of the Metropolis algorithm can be interesting in multi-objective optimization, but that the implementation details can have a huge impact on the performance.

A First Runtime Analysis of the NSGA-II on a Multimodal Problem

Very recently, the first mathematical runtime analyses of the multi-objective evolutionary optimizer NSGA-II have been conducted. We continue this line of research with a first runtime analysis of this algorithm on a benchmark problem consisting of multimodal objectives. We prove that if the population size N is at least four times the size of the Pareto front, then the NSGA-II with four standard ways to select parents, bit-wise mutation, and crossover with rate less than one, optimizes the OneJumpZeroJump benchmark with jump size 2≤k≤n/4 in time O(Nnk). When using fast mutation instead of bit-wise mutation this guarantee improves by a factor of kΩ(k). Overall, this work shows that the NSGA-II copes with the local optima of the OneJumpZeroJump problem at least as well as the global SEMO algorithm.

Mixed-model sequencing problem under capacity and machine idle time constraints in JIT production systems

Abstract Sequencing problems in mixed-model production systems assume a negligible set-up time that permits unconstrained product interspersing. This assumption offers better production sequences in terms of just-in-time objectives such as minimizing parts usage rate variation and minimizing Production Rate Variation (PRV). However, set-up times turn out to be critical in many realistic situations. The present paper addresses the sequencing problem in mixed-model production systems, where internal and external set-up times are considerable measures. Internal set-up time limits free mixing of products, resulting in batching of products; while external set-up time causes machine idle time. The authors present a mathematical model for the problem to find sequences that minimize PRV, satisfying capacity and zero machine idle time constraints. LINGO 15.0 solver could solve the model for small-size problems in a reasonable computational time. For large-size problems, a modified bit-wise mutation algorithm that uses a single unary genetic operator has been developed. The authors have applied the algorithm for diverse problem instances at different production capacities. The same problems are also analyzed by relaxing the zero machine idle time constraint. Results demonstrate how the constraints on capacity and machine idle time disturb the leveling of sequences in mixed-model production systems. The model provides decision makers with the opportunity to select smooth sequences based on the available production time and acceptable machine idle time.

A note on ‘A bit-wise mutation algorithm for mixed-model sequencing in JIT production systems’ (Nazar & Pillai, 2015, IJPR, 53:19, 5931-5947)

This note discusses three points about the problem of sequencing units on a mixed-model assembly line in ‘A bit-wise mutation algorithm for mixed-model sequencing in JIT production systems’ (Nazar & Pillai, 2015, IJPR, 53:19, 5931-5947). Specifically, the mixed-integer quadratic model, the bit-wise mutation algorithm and the bi-objective problem to optimise both product rate variation and makespan. The conclusion of the discussion is that the three alleged contributions are not valid or are outperformed by those presented in some previously published papers.

A bit-wise mutation algorithm for mixed-model sequencing in JIT production systems

Product sequencing in a mixed-model production system is an operational level problem and is computationally hard. This necessitates the use of non-traditional optimisation methods to solve it in a short time. In the first phase, this study addresses the mixed-model sequencing problem with the objective of minimising production rates variation (PRV) which is a just-in-time objective. We formulated a mixed integer quadratic assignment model for this problem. LINGO 14.0 solver could solve this model for small- and medium-size problems. The study presents a bit-wise mutation algorithm to solve the sequencing problem with the same objective. This algorithm uses a single unary operator which makes it simpler and faster. Computational results show that the algorithm succeeds in solving large-size PRV problems in a reasonable time, for which the algorithms reported in the literature could only get near-optimal solutions. In the second phase, we consider both PRV and one of the system performance measures, makespan as objectives for the mixed-model sequencing problem. A mathematical description of the underlying model is provided. The bit-wise mutation algorithm is modified to generate a set of non-dominated solutions which provide the decision-maker with the opportunity to trade-off between the two objectives.

On the Impact of Mutation-Selection Balance on the Runtime of Evolutionary Algorithms

The interplay between mutation and selection plays a fundamental role in the behavior of evolutionary algorithms (EAs). However, this interplay is still not completely understood. This paper presents a rigorous runtime analysis of a non-elitist population-based EA that uses the linear ranking selection mechanism. The analysis focuses on how the balance between parameter η, controlling the selection pressure in linear ranking, and parameter χ controlling the bit-wise mutation rate, impacts the runtime of the algorithm. The results point out situations where a correct balance between selection pressure and mutation rate is essential for finding the optimal solution in polynomial time. In particular, it is shown that there exist fitness functions which can only be solved in polynomial time if the ratio between parameters η and χ is within a narrow critical interval, and where a small change in this ratio can increase the runtime exponentially. Furthermore, it is shown quantitatively how the appropriate parameter choice depends on the characteristics of the fitness function. In addition to the original results on the runtime of EAs, this paper also introduces a very useful analytical tool, i.e., multi-type branching processes, to the runtime analysis of non-elitist population-based EAs.

On the approximation ability of evolutionary optimization with application to minimum set cover

Evolutionary algorithms (EAs) are heuristic algorithms inspired by natural evolution. They are often used to obtain satisficing solutions in practice. In this paper, we investigate a largely underexplored issue: the approximation performance of EAs in terms of how close the solution obtained is to an optimal solution. We study an EA framework named simple EA with isolated population (SEIP) that can be implemented as a single- or multi-objective EA. We analyze the approximation performance of SEIP using the partial ratio, which characterizes the approximation ratio that can be guaranteed. Specifically, we analyze SEIP using a set cover problem that is NP-hard. We find that in a simple configuration, SEIP efficiently achieves an Hn-approximation ratio, the asymptotic lower bound, for the unbounded set cover problem. We also find that SEIP efficiently achieves an (Hk−k−18k9)-approximation ratio, the currently best-achievable result, for the k-set cover problem. Moreover, for an instance class of the k-set cover problem, we disclose how SEIP, using either one-bit or bit-wise mutation, can overcome the difficulty that limits the greedy algorithm.

Self-Adaptation of Mutation Operator and Probability for Permutation Representations in Genetic Algorithms

The choice of mutation rate is a vital factor in the success of any genetic algorithm (GA), and for permutation representations this is compounded by the availability of several alternative mutation operators. It is now well understood that there is no one "optimal choice"; rather, the situation changes per problem instance and during evolution. This paper examines whether this choice can be left to the processes of evolution via self-adaptation, thus removing this nontrivial task from the GA user and reducing the risk of poor performance arising from (inadvertent) inappropriate decisions. Self-adaptation has been proven successful for mutation step sizes in the continuous domain, and for the probability of applying bitwise mutation to binary encodings; here we examine whether this can translate to the choice and parameterisation of mutation operators for permutation encodings. We examine one method for adapting the choice of operator during runtime, and several different methods for adapting the rate at which the chosen operator is applied. In order to evaluate these algorithms, we have used a range of benchmark TSP problems. Of course this paper is not intended to present a state of the art in TSP solvers; rather, we use this well known problem as typical of many that require a permutation encoding, where our results indicate that self-adaptation can prove beneficial. The results show that GAs using appropriate methods to self-adapt their mutation operator and mutation rate find solutions of comparable or lower cost than algorithms with "static" operators, even when the latter have been extensively pretuned. Although the adaptive GAs tend to need longer to run, we show that is a price well worth paying as the time spent finding the optimal mutation operator and rate for the nonadaptive versions can be considerable. Finally, we evaluate the sensitivity of the self-adaptive methods to changes in the implementation, and to the choice of other genetic operators and population models. The results show that the methods presented are robust, in the sense that the performance benefits can be obtained in a wide range of host algorithms.

A New Approach for Analyzing Average Time Complexity of Population-Based Evolutionary Algorithms on Unimodal Problems

In the past decades, many theoretical results related to the time complexity of evolutionary algorithms (EAs) on different problems are obtained. However, there is not any general and easy-to-apply approach designed particularly for population-based EAs on unimodal problems. In this paper, we first generalize the concept of the takeover time to EAs with mutation, then we utilize the generalized takeover time to obtain the mean first hitting time of EAs and, thus, propose a general approach for analyzing EAs on unimodal problems. As examples, we consider the so-called (N + N) EAs and we show that, on two well-known unimodal problems, leadingones and onemax , the EAs with the bitwise mutation and two commonly used selection schemes both need O(n ln n + n(2)/N) and O(n ln ln n + n ln n/N) generations to find the global optimum, respectively. Except for the new results above, our approach can also be applied directly for obtaining results for some population-based EAs on some other unimodal problems. Moreover, we also discuss when the general approach is valid to provide us tight bounds of the mean first hitting times and when our approach should be combined with problem-specific knowledge to get the tight bounds. It is the first time a general idea for analyzing population-based EAs on unimodal problems is discussed theoretically.

Quadratic approximation based hybrid genetic algorithm for function optimization

Probably the popular form of binary genetic algorithms for function optimization use tournament selection (TS) or roulette wheel selection (RS) for function optimization. Also single point crossover (SC) and uniform crossover (UC) are most popular and effective crossover operators. In an earlier paper we had considered all four combinations of these crossover and mutation operators along with bit-wise mutation, called GA1 (TS + SC), GA2 (TS + UC), GA3 (RS + SC) and GA4 (RS + UC). In this paper, an attempt is made to hybridize these four GAs by incorporating the quadratic approximation (QA) operator into them. The four resultant hybrid GAs, called HGA1, HGA2, HGA3 and HGA4, are compared with the four simple GAs on a set of 22 test problems taken from literature. Based on the extensive numerical and graphical analysis of results it is concluded that the HGA3 outperforms all rest 7 versions. Further, we study the depth and frequency of the QA should be applied for better performance for the particular problem suite.

The effectiveness of mutation operation in the case of Estimation of Distribution Algorithms

The Estimation of Distribution Algorithms are a class of evolutionary algorithms which adopt probabilistic models to reproduce individuals in the next generation, instead of conventional crossover and mutation operators. In this paper, mutation operators are incorporated into Estimation of Distribution Algorithms in order to maintain the diversities in EDA populations. Two kinds of mutation operators are examined: a bitwise mutation operator and a mutation operator taking account into the probabilistic model. In experiments, we do not only compare the proposed methods with conventional EDAs on a few fitness functions but also analyze sampled probabilistic models by using KL-divergence. The experimental results shown in this paper elucidate that the mutation operator taking account into the probabilistic model improve the search ability of EDAs.

A genetic algorithm approach to the simultaneous scheduling of machines and automated guided vehicles

This article addresses the problem of simultaneous scheduling of machines and a number of identical automated guided vehicles (AGVs) in a flexible manufacturing system (FMS) so as t minimize the makespan. For solving this problem, a genetic algorithm (GA) is proposed. Here, chromosomes represent both operation sequencing and AGV assignment dimensions of the search space. A third dimension, time, is implicitly given by the ordering of operations of the chromosomes. A special uniform crossover operator is developed which produces one offspring from two parent chromosomes. It transfers any patterns of operation sequences and/or AGV assignments that are present in both parents to the child. Two mutation operators are introduced; a bitwise mutation for AGV assignments and a swap mutation for operations. Any precedence infeasibility resulting from the operation swap mutation is removed by a repair function. The schedule associated with a given chromosome is determined by a simple schedule builder. After a number of problems are solved to evaluate various search strategies and to tune the parameters of the proposed GA, 180 test problems are solved to evaluate various search lower bound is introduced and compared with the results of GA. In 60% of the problems GA reaches the lower bound indicating optimality. The average deviation from the lower bound over all problems is found to be 2.53%. Additional comparison is made with the time window approach suggested for this same problem using 82 test problems from the literature. In 59% of the problems GA outperforms the time window approach where the reverse is true only in 6% of the problems.

An Improved Genetic Algorithm for Pipe Network Optimization

An improved genetic algorithm (GA) formulation for pipe network optimization has been developed. The new GA uses variable power scaling of the fitness function. The exponent introduced into the fitness function is increased in magnitude as the GA computer run proceeds. In addition to the more commonly used bitwise mutation operator, an adjacency or creeping mutation operator is introduced. Finally, Gray codes rather than binary codes are used to represent the set of decision variables which make up the pipe network design. Results are presented comparing the performance of the traditional or simple GA formulation and the improved GA formulation for the New York City tunnels problem. The case study results indicate the improved GA performs significantly better than the simple GA. In addition, the improved GA performs better than previously used traditional optimization methods such as linear, dynamic, and nonlinear programming methods and an enumerative search method. The improved GA found a solution for the New York tunnels problem which is the lowest‐cost feasible discrete size solution yet presented in the literature.

Abranching process model for genetic algorithms

A new stochastic analysis of schema processing in genetic algorithms is presented in this paper. We have developed a branching process model for the haploid, binary, fixed-length genetic algorithm with fitness-proportionate selection, single-point crossover and bitwise mutation. We provide empirical evidence on a ten-problem test suite.