Multi-objective Optimization of Engineering Design Problems Through Pareto-Based Bat Algorithm

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.

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.

This paper presents optimality criteria for fuzzy-valued fractional multi-objective optimization problem. There are numerous optimality criteria which have been established for the deterministic fractional multi-objective optimization problems. Very few studies are available on the establishment of optimality criteria for fuzzy-valued multi-objective optimization problem. So, Karush–Kuhn–Tucker optimality criteria for fuzzy-valued fractional multi-objective problem are established by using Lagrange multipliers. First, the original problem is modified using the parametric approach of Dinkelbach into multi-objective non-fractional optimization problem, and then, the optimality conditions are established for the modified problem using the Hukuhara derivative. The established optimality criteria are verified by two numerical examples.

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.

A multi-objective particle swarm optimization (MOPSO) approach for multi-objective economic load dispatch problem in power system is presented in this paper. The economic load dispatch problem is a non-linear constrained multi-objective optimization problem. The proposed MOPSO approach handles the problem as a multi-objective problem with competing and non-commensurable fuel cost, emission and system loss objectives and has a diversity-preserving mechanism using an external memory (call “repository”) and a geographically-based approach to find widely different Pareto-optimal solutions. In addition, fuzzy set theory is employed to extract the best compromise solution. Several optimization runs of the proposed MOPSO approach were carried out on the standard IEEE 30-bus test system. The results revealed the capabilities of the proposed MOPSO approach to generate well-distributed Pareto-optimal non-dominated solutions of multi-objective economic load dispatch. Comparison with Multi-objective Evolutionary Algorithm (MOEA) showed the superiority of the proposed MOPSO approach and confirmed its potential for solving multi-objective economic load dispatch.

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.

In many optimization problems, the main goal is to improve a single performance index in which a minimum or maximum value of this index fully reflects the quality of the response obtained from a system. However, in some cases, it is impossible to rely solely on a single index, so a multi-objective optimization problem with multiple performance indicators is considered where the values of all of them should be optimized simultaneously. The mentioned process requires a multi-objective optimization algorithm that can deal with the complexity of problems with simultaneous indexes. This paper presents the multi-objective version of a recently proposed metaheuristic algorithm called Crystal Structure Algorithm (CryStAl) which was inspired by the principles underlying the formation crystal structures. For the performance evaluation of this algorithm which is called MOCryStAl, the benchmark problems of the Completions on Evolutionary Computation (CEC) on multi-objective optimization, called CEC-09, are utilized. Some real-world engineering design problems are used to evaluate the efficiency of the proposed approach. The results demonstrate that the proposed methods can provide excellent results in dealing with the considered multi-objective problems.

The multi-objective unconstrained combinatorial optimization problem (MUCO) can be considered as an archetype of a discrete linear multi-objective optimization problem. It can be interpreted as a specific relaxation of any multi-objective combinatorial optimization problem with linear sum objective function. While its single criteria analogon is analytically solvable, MUCO shares the computational complexity issues of most multi-objective combinatorial optimization problems: intractability and NP-hardness of the $$\varepsilon $$ -constraint scalarizations. In this article interrelations between the supported points of a MUCO problem, arrangements of hyperplanes and a weight space decomposition, and zonotopes are presented. Based on these interrelations and a result by Zaslavsky on the number of faces in an arrangement of hyperplanes, a polynomial bound on the number of extreme supported solutions can be derived, leading to an exact polynomial time algorithm to find all extreme supported solutions. It is shown how this algorithm can be incorporated into a solution approach for multi-objective knapsack problems.

The supply chain nowadays is more dynamic than before partly due to the information techniques and changes of consumer behaviors. This proposal analyzes the multiple objective optimization problems existing in the supply chain management, and the stochastic and dynamic circumstance of current supply chain. It presents a general overview of multi-objective optimization techniques, stochastic and dynamic programming. We attempt to apply these techniques to solve some optimization problems in supply chain, under changing environment. It can be a guideline of future research. There are mainly three parts of the proposal; the first is multiple objective optimization problems in the real supply chain management, and the stochastic and dynamic environment of the supply chain. The second part is the suggesting approaches dealing with multi-objective optimization problems and the stochastic, dynamic circumstance. The last part is attempting to apply these methods into a real case existing in the supply chain.

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.

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.

Many real-world design problems involve simultaneous optimisation of multiple objectives. The considered grillage design is a multi-objective optimisation problem since there are two objective functions, namely: the volume design and the cost design. Thus, the solution of this optimisation problem requires specialised method suitable for multi-objective problems. In this article, a multi-objective design optimisation using real-coded genetic algorithm is proposed for the Pareto-optimality of grillage system. The objective functions in the optimisation problem measure the design sensitivities in grillage system. The non-linear constrained multi-objective optimisation is very important from the point of view of practical problem solving. Therefore, the real-coded genetic algorithm with multiple genetic operators is proposed to find the optimum grillage system without handling any of the penalty functions. Direct strength calculation defined from the class rules of DNV was applied for structural design of grillage system. The hybrid method (real-coded genetic algorithm including the non-dominated sorting and sharing approaches) performs a marvellous explorability in finding a diverse set of solutions and in converging near the true Pareto-optimal set. The results obtained are very encouraging since they show that we can produce an important portion of the Pareto-front at a very low computational time frame.

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.