On the performance of MATLAB's inbuilt genetic algorithm on single and multi-objective unconstrained optimization problems

Multi-objective optimization problems (MOP) are frequently encountered in practice. In some cases, different computationally expensive analyses may be independently used for computing different objectives of the MOP. Additionally, the analyses may be executed to obtain estimates with different fidelity, typically higher fidelity requiring a longer run-time. For instance, in automotive design, the aerodynamic drag is computed using computational fluid dynamic (CFD) analysis and its crashworthiness/strength is assessed using finite element analysis (FEA). Both the objectives can be independently computed and the underlying fidelity of each analysis can also be controlled using different mesh sizes/thresholds on the residual errors. While there exist a number of generic MOP benchmark problems in the literature, there is scarce work on constructing MOPs with multi-fidelity (MF) analyses to support the development of multi-fidelity, multi-objective optimization algorithms. The existing MF benchmarks are limited to unconstrained, single-objective optimization problems only. Towards addressing this gap, in this paper, we introduce a test suite for multi-objective, multi-fidelity optimization (MOMF). The problems are derived by combining existing unconstrained, multi-objective design optimization problems with resolution/stochastic/instability errors that are common manifestations of MF simulations. The method allows for the construction of any number of low-fidelity functions with desired level of correlations for a given high-fidelity objective function. We hope that the test suite would motivate novel algorithmic developments to support optimization involving computationally expensive and independently evaluable objectives.

Although various optimization methods for solving single-objective problems have been developed in the last few decades, these methods have lost their eligibility due to the fact that today’s engineering problems are toward multiple objective optimization problems, in real applications. For single-objective optimization problems, for example, in case of a minimization problem, this value is the decision vector giving the smallest objective that can be achieved within the specified constraints. Hence the minimum decision vector within all possible (feasible) solution vectors is the so-called optimal solution and/or optimal design. However, in multi-objective optimization problems, since a different objective value is generated against each decision vector, the superiority of the solutions over each other is determined by considering the trade-off among the objective values. Therefore, the solution of multi-objective optimization problems, unlike single-objective problems, is a set of vectors rather than a single decision vector. In multi-objective optimization problems, especially if there are intricate objectives, the computational cost of the problem increases. In other words, while synchronously trying to maximize one of the objectives and to minimize another one makes it difficult to find the global optimum design. One of the important techniques used in multi-objective optimization problems is Pareto optimality which enables to select the global optimum solution taking into account the trade-off among all objectives. In this context, using of derivative-based methods has decreased, but the use of metaheuristic methods has increased due to the rapid availability of global optimum solution. This is because the improvements in the field of optimization are progressing in proportion to technology and varying according to the needs. In this chapter, one of the recent metaheuristic optimization methods based on swarm intelligence that is so-called a Pareto-based bat algorithm inspired by the behavior of determining the direction and distance of an object using the echo of the sound called the echolocation of bats is used in order to obtain optimum solutions for multi-objective engineering design problems. In this regard, a four-bar planar truss, a real-sized welded steel beam as well as a multi-layer radar absorber are selected as multi-objective engineering design optimization problems. In case the obtained results (optimal designs) are examined, the potency and the reliability of the proposed multi-objective Pareto-based bat algorithm are demonstrated.

Genetic algorithm based multi-objective reliability optimization in interval environment

Optimization problems in many industrial applications are very hard to solve. Many examples of them can be found in the design of aeronautical systems. In this field, the designer is frequently faced with the problem of considering not only a single design objective, but several of them, i.e., the designer needs to solve a Multi-Objective Optimization Problem (MOP). In aeronautical systems design, aerodynamics plays a key role in aircraft design, as well as in the design of propulsion system components, such as turbine engines. Thus, aerodynamic shape optimization is a crucial task, and has been extensively studied and developed. Multi-Objective Evolutionary Algorithms (MOEAs) have gained popularity in recent years as optimization methods in this area, mainly because of their simplicity, their ease of use and their suitability to be coupled to specialized numerical simulation tools. In this chapter, we will review some of the most relevant research on the use of MOEAs to solve multi-objective and/or multi-disciplinary aerodynamic shape optimization problems. In this review, we will highlight some of the benefits and drawbacks of the use of MOEAs, as compared to traditional design optimization methods. In the second part of the chapter, we will present a case study on the application of MOEAs for the solution of a multi-objective aerodynamic shape optimization problem.

In many recent works, the potential of Exploratory Landscape Analysis (ELA) features to numerically characterize single-objective continuous optimization problems has been demonstrated. These numerical features provide the input for all kinds of machine learning tasks in the domain of continuous optimization problems, ranging, for example, from High-level Property Prediction to Automated Algorithm Selection and Automated Algorithm Configuration. Without ELA features, analyzing and understanding the characteristics of single-objective continuous optimization problems is-to the best of our knowledge-very limited. Yet, despite their usefulness, as demonstrated in several past works, ELA features suffer from several drawbacks. These include, in particular, (1) a strong correlation between multiple features, as well as (2) its very limited applicability to multiobjective continuous optimization problems. As a remedy, recent works proposed deep learning-based approaches as alternatives to ELA. In these works, among others, point-cloud transformers were used to characterize an optimization problem's fitness landscape. However, these approaches require a large amount of labeled training data. Within this work, we propose a hybrid approach, Deep-ELA, which combines (the benefits of) deep learning and ELA features. We pre-trained four transformers on millions of randomly generated optimization problems to learn deep representations of the landscapes of continuous single- and multiobjective optimization problems. Our proposed framework can either be used out of the box for analyzing single- and multiobjective continuous optimization problems, or subsequently fine-tuned to various tasks focusing on algorithm behavior and problem understanding.

The well-known NP-hard traveling salesman problem (TSP) primarily considers distance as its single objective. However, applications modeled from real world systems repeatedly involve more than one objective giving rise to multi-objective optimization. Fusing ideas of dimension reduction, decomposition approaches, and genetic algorithms, this paper presents a multiobjective minimum matrix search algorithm (MOMMS) for the heuristic resolution of the bi-objective TSP (bTSP). The MOMMS uses dimension reduction to obtain a reduce matrix network that is used to obtain or to approximate the set of efficient solutions. The reduce matrix network aids in the decomposition of a multiobjective combinatorial optimization (MOCO) problem into a single objective combinatorial optimization problem. Moreover, using the reduce matrix network MOMMS introduces a population generator that creates an initial population composed of an approximation to the extreme supported efficient solutions. The MOMMS does not use any numerical parameter. Also, MOMMS uses family competitive metamorphosis and short-term memory selection to maintain population diversity in MOCO problems. The proposed algorithm showed respectable results in testing on well-known benchmark problems of the bTSP. Comparisons are performed with the results of state-of-the-art algorithms from the literature. Moreover, the MOMMS is tested on largescale instances of the bTSP. The computational study shows that the proposed algorithm is able to solve large-scale instances in reasonable time. Therefore, the MOMMS is a competitive tool for solving the bTSP.

This paper deals with the multi-objective optimization of tire reinforcement structures such as the tread belt and the carcass path. The multi-objective functions are defined in terms of the discrete-type design variables and approximated by artificial neutral network, and the sensitivity analyses of these functions are replaced with the iterative genetic evolution. The multi-objective optimization algorithm introduced in this paper is not only highly CPU-time-efficient but it can also be applicable to other multi-objective optimization problems in which the objective function, the design variables and the constraints are not continuous but discrete. Through the illustrative numerical experiments, the fiber-reinforced tire belt structure is optimally tailored. The proposed multi-objective optimization algorithm is not limited to the tire reinforcement structure, but it can be applicable to the generalized multi-objective structural optimization problems in various engineering applications.

Decision makers tend to define their optimization problems as multi-objective optimization problems. Generating the whole nondominated set is often unrealistic due to its size but also because most of these points are irrelevant to the decision maker. Another approach consists in obtaining preference information and integrating it a priori, in order to reduce the size of the nondominated set and have a gain in computation time. In this work we focus on a partial preference relation based on requirement and tolerance thresholds that translate the Pareto dominance cone. After introducing this preference relation, we present adaptations to use it in existing discrete multi-objective optimization algorithms. Numerical experiments on multi-objective combinatorial optimization problems show the applicabiliy of our approach.

In a real life situation, due to the complexity of the industrial systems and non-linearity of their behavior, it is very difficult to achieve optimum performance of system for desired industrial goals using uncertain, vague and imprecise data. Herein, an approach has been proposed through which the behavior of the system is analyzed in the form of well- known six reliability indices by using triangular fuzzy numbers which allow the consideration of expert opinions, linguistic variables, operating conditions in reliability information. Using their behavior analysis a fuzzy multi-objective optimization problem (FMOOP) has been formulated. Due to the conflicting nature of the multiple objectives, the decision making is difficult and it leads to the Pareto optimal solutions instead of single optimal solutions. Many evolutionary algorithms (EAs) already exist in the literature for solving a multi-objective optimization problem (MOOP), and are termed as multi-objective evolutionary algorithms (MOEAs). Particle swarm optimization (PSO) is one of such MOEA which demonstrates the ability to identity a Pareto-optimal front efficiently. Here, a crisp optimization problem is reformulated from FMOOP by taking into account the preference of decision maker (DM) and then PSO is applied to solve the resulting fuzzified MOOP. The presented approach is applied in order to solve the multi-objective series-parallel system reliability optimization problem for a crystallization unit of a urea plant.

Construction site layout is multi-objective optimization (MOO) problem. However, the previous researchers treated site layout as a single objective optimization problem, which is far away from the reality. In this study, site layout was treated as MOO problem. The two objective functions are minimizing the likelihood of accidents happened and total handling cost of interaction flows between the facilities. The modified Pareto-based ant colony optimization (ACO) algorithm was proposed to solve site layout problem. Moreover, fuzzy rule-based system was used to quantify the interaction flows. Finally, a case of office building was used to evaluate the modified Pareto-based ACO algorithm for solving MOO site layout problem. The results showed the feasibility and effectiveness of the modified Pareto-based ACO algorithm to solve MOO site layout problem.

In order to solve the problem of the multi-objective spectrum allocation on the joint optimization of maximal network utility and fairness of users in cognitive radio network, based on quantum bee colony theory and membrane computing, a novel multi-objective discrete combinatorial optimization algorithm, named membrane-inspired quantum bee optimization, is proposed. The global optimal solution of single objective can be searched in the elementary membranes, and Pareto front solutions which take account of network utility and fairness, can be obtained from skin membrane with the proposed method. The multi-objective optimization algorithm, which can solve both single objective and multi-objective optimization problems at the same time, is designed by the communication rules between membranes, the cooperative evolution of foraging behavior based on quantum state, and non-dominated sorting. Compared with classical color-sensitive graph coloring algorithm, genetic algorithm, quantum genetic algorithm, and particle swarm optimization under different objective functions, the proposed spectrum allocation method can search the global optimal solution of single objective as shown by the simulation results, and it is superior to classical spectrum allocation algorithms and existing intelligence spectrum allocation methods. The optimal Pareto front solutions of multi-objective spectrum allocation are also obtained.

It is shown that finding the equivalence set for solving multiobjective discrete optimization problems is advantageous over finding the set of Pareto optimal decisions. An example of a set of key parameters characterizing the economic efficiency of a commercial firm is proposed, and a mathematical model of its activities is constructed. In contrast to the classical problem of finding the maximum profit for any business, this study deals with a multiobjective optimization problem. A method for solving inverse multiobjective problems in a multidimensional pseudometric space is proposed for finding the best project of firm’s activities. The solution of a particular problem of this type is presented.

The search region in multiobjective optimization problems is a part of the objective space where nondominated points could lie. It plays an important role in the generation of the nondominated set of multiobjective combinatorial optimization (MOCO) problems. In this paper, we establish the representation of the search region by half-open polyblocks (a variant concept of “polyblock” in monotonic optimization) and propose a new procedure for updating the search region. We also study the impact of stack policies to the new procedure and the existing methods that update the search region. Stack policies are then analyzed, pointing out their performance effectiveness by means of the results of rich computational experiments on finding the whole set of nondominated points of MOCO problems.

The present work focuses on obtaining optimal operational policies of an industrial scale penicillin fed-batch fermentation process using dynamic optimization (DO). The three process objectives considered in this work include, maximization of total penicillin formed (J 1), minimization of total biomass formed (J 2), and minimization of fed-batch operation time (J 3). The control variables for the optimization study include, the feed flowrates of sugar, soyabean oil, and phenylacetic. We perform a single-objective optimization study with J 1 as the objective function and showcased the importance of DO. Consequently, we solve three multi-objective optimization (MOO) problems corresponding to all the pairwise combinations of the aforementioned three objectives. Outcomes of MOO problems are presented in terms of several non-dominated Pareto-optimal solutions. Furthermore, descriptive analysis of three representative Pareto points for all the MOO problems is presented, which highlights the importance of MOO study of the penicillin bio-reactor.

Multiobjective Optimization Problems (MOP) can be found in many issues of scientific research, engineering, and in everyday social life. A MOP problem has several objectives that conflict with one another which must be optimized simultaneously. This paper presents a quantum-inspired evolutionary algorithm (QEA) to solve continuous multiobjective optimization problem (MOP). The proposed method employs Fast Nondominated Sorting and Crowding Distance from NSGA-II and implements all common operators of genetic algorithms (GA), such as crossover and mutations with additional Quantum Gate quantum operators. The proposed method is then run in a distributed manner and is proven to be able to significantly outperform the hypervolume and MOEA/D metrics and have hypervolumes that are comparable to NSGA-II while maintaining a better average Δ’ in all testing problems. From this result, it is concluded that using quantum-inspired individual genetic algorithms to solve continuous MOP can produce hypervolume and Δ’ metrics that are good in all specified test problems.