Number Of Objective Functions Research Articles

Multi-objective search game: Long-term vs short-term

This study focuses on investigating a multi-objective search allocation game problem, specifically addressing the distribution of a search budget (player 1) across a searched area to detect an evading target (player 2). The game is defined as a two-person zero-sum game, as an optimal search strategy must consider the target’s optimal evasion strategy. Traditionally, the sole objective function in search theory literature has been to maximize the detection probability at the end of the time horizon. Our work extends the problem by proposing a multi-objective formulation that aims to strike a balance between long-term and short-term search objectives. In this context, the goal is to maximize the detection probability of the target at the end of each time period (e.g., hourly) throughout the time horizon. The main challenge in this problem lies in handling a multitude of objective functions, where the number of objective functions is an input data. To the best of our knowledge, no prior research has explored the multi-objective search allocation game. Therefore, we meticulously study the significance of this problem by highlighting the disparities and conflicts among these objectives. Subsequently, we propose a solution approach based on the epsilon-constraint algorithm, as well as a heuristic method that decomposes the problem into smaller multi-objective subproblems. We conduct a series of experiments to evaluate two key aspects of the methods: computation time and quality.

A reinforcement learning-based multi-objective optimization in an interval and dynamic environment

There are many fields involving multi-objective optimization problems in presence of dynamic and interval environments (DI-MOPs), in which the number of objective functions is greater than one, the objectives are conflicting with each other, the problem is varying with time, and the parameters are interval-valued. Conflicts between multiple objectives make interval problems more difficult to be optimized in the dynamic environment. Recent works suffer from the lack of accuracy in change severity detection, the lack of adaptability in change response, and insufficient consideration of reducing imprecision. To tackle these issues, a novel reinforcement learning-based algorithm is proposed in this study, which has three original contributions: (1) Internal interval similarity is specially designed for the interval detection of change severity. To be specific, this operator is proposed for higher accuracy, including the hybridization between the interval similarity and point similarity, and the decision of the detection object and strategy. (2) Q learning is embedded into the optimization algorithm to select the optimal change response after the change occurs. The benefit of this operator is that the response mechanism is dynamically changed in accordance to the environments. (3) To reduce the uncertainty of problems, a new crowding distance operator is presented to guide the search to simultaneously increase diversity, speed up convergence, and decrease imprecision. The computational results from the benchmark sets demonstrate that the proposed algorithm is more efficient than other state-of-the-art algorithms, generating Pareto sets with stronger convergence, wider distribution, and less uncertainty.

Multi-condition optimisation design of a hydrofoil based on deep belief network

A hydrofoil optimisation method based on the combination of deep belief network (DBN) and non-dominated sorting genetic algorithm II (NSGA-II) is proposed in this paper. Firstly, the number of objective functions in the multi-condition optimisation of hydrofoil is reduced by the performance distance formula. Then, Multi-DBN is adopted to replace computational fluid dynamics (CFD) simulation to establish the response relationship between geometric parameters and hydrodynamic performance of the hydrofoil. Moreover, the network parameters of Multi-DBN are optimised to improve prediction accuracy. Finally, the hydrofoil optimisation method is established by combining Multi-DBN and NSGA-II algorithms. Based on this method, the hydrodynamic performances of the hydrofoil NACA 63–815 at angles of attack (AoA) of 0°, 6° and 12° are optimised. Results indicate that the performances of the optimised hydrofoil are better than the initial at the three AoAs. Moreover, within the AoA range between 0° and 14°, the maximum improvements of the lift–drag ratio and the minimum pressure coefficients are 10.63% and 11.44%, respectively. Using Multi-DBN instead of CFD as the performance evaluation method in optimisation can greatly reduce the time of optimisation design.

ЗАГАЛЬНІ ВИМОГИ ДО ПОБУДОВИ ТА ВИКОРИСТАННЯ ГІБРИДНИХ МІКРОСИСТЕМ В РОЗПОДІЛЬНИХ МЕРЕЖАХ

The article substantiates the expediency of forming hybrid microgrids that can be integrated into a centralized power supply system or operate autonomously in order to expand the use of renewable energy sources and prevent their possible negative impact on electrical network modes. A two-level approach to managing the operation of a hybrid microgrid is proposed, which involves a preliminary planning stage based on forecasted loads, output power of renewable generation sources, market price of electric energy, and a stage of operational control based on data obtained in real time. A number of objective functions and constraints are proposed to solve the task of optimal planning of the operation of a hybrid microgrid as a multicriteria problem. At the same time, the possibility of transforming this problem into a monocriteria one is substantiated. An analysis of the advantages and disadvantages of each of the proposed approaches was carried out. Ref. 12.

Synthesis of WPTS compensation networks considering multiple criteria

The paper deals with an evolutionary method for solving many-objective optimization problems exhibiting a high-dimensionality objective space, which is a challenging problem. An application in the optimal synthesis of Compensation Networks (CNs) of wireless power transfer systems for charging the batteries of electric vehicles is developed. This design problem is characterized by a set of multiple objectives in mutual conflict, which should be simultaneously considered. The optimization aims to the maximization of both the efficiency and the transferred power; a further criterion selects the networks with a suitable profile of impedance vs. frequency. Moreover, the minimization of current and voltage values relevant to inductors and capacitors in the networks, respectively, is pursued. These five design criteria are optimized exploiting the concept of the degree of conflict, which is the core of the proposed method, named “EStra-many”. The method is applied by considering two approaches: the single-objective one, based on the degree of conflict function only, and the bi-objective approach in which the tradeoff between the degree of conflict function itself and another objective function (in turn, the efficiency, the transferred power, the distance of the resonance frequency from the supply frequency, the maximum value of the inductance current, the maximum value of the capacitor voltage, the distance from the Utopia point, and the number of inductors in the CN), is searched for. This way, all in one, seven different optimization problems are solved. The main element of novelty of the paper is a method to solve an optimization problem characterized by a high number of objective functions. In view of this, instead of considering a weighted sum of the objectives, a preference function inspired by the concept of least-conflict solution is formulated accordingly, the preference function is minimized by a cost-effective evolutionary algorithm of lowest order.

A literature review of the perishable inventory routing problem

The inventory routing problem arises when inventory management and vehicle routing decisions are integrated instead of treated as separate problems. The technique of combining such decisions could lead to solutions that are better than merging the optimal solutions of the smaller subproblems. Hence, the problem is prominent and has been the focus of extensive research in recent years. When the products considered in such a problem are perishable, the importance is intensified; the current paper presents a literature review of the inventory routing problem for perishable products. This review classifies papers according to five attributes, namely, the number of products, the type of product (including the types of product perishability and types of perishable products), the type of demand, the number of objective functions, and the solution approach. A comprehensive analysis is performed based on these five attributes. Finally, based on 89 relevant reviewed papers, directions for future research on the perishable inventory routing problem are presented.

Enhancement of the Improved Recursive Method for Multi-objective Integer Programming Problem

In this paper, we developed a new algorithm to find the set of a non-dominated points for a multi-objective integer programming problem. The algorithm is an enhancement on the improved recursive method where the authors have used a lexicographic method for analysis. In this approach a sum of two objectives is considered as one weighted sum objective for each iteration. Computational results show that the proposed approach outperforms the currently available results obtained by the improved recursive method with respect to CPU time and the number of integer problems solved to identify all non-dominated points. Many problems such as assignment, knapsack and travelling salesman have been investigated on different sized problems. The benefit of this approach becomes more visible with the increase in the number of objective functions.

Identification of elastic properties utilizing non-destructive vibrational evaluation methods with emphasis on definition of objective functions: a review

Defining a suitable objective function is important as it affects the solution search process and the solution quality. In vibrational material identification, a number of objective functions have been employed since the last decades. It is the intent of the present study to provide a comprehensive review of the previous and current trends of non-destructive vibrational material identification with emphasis on the definition of objective function. Prior to defining an objective function, several aspects should be given utmost attention, including selection of responses, design variables and goals. Along with the critical review of past research, potential areas that could be worth for future investigation are as well explicated.

Improvement of the method of calculating a group of sound­insulating panels

Original single-layer and multilayer structures of sound-insulating panels are considered, as well as their advantages compared to traditional analogs. The shortcomings of the method of optimization calculation of a group of sound-insulating panels designed to reduce noise in several production rooms are analyzed. This method has limited functionality due to a relatively small number of objective functions and the corresponding conditions for their use. Given this, an improved method of optimization calculation of a group of sound-insulating panels is proposed. The improvement of the method consists in increasing the number of objective functions intended for multipurpose optimization taking into account real production conditions. The refinement of the algorithm consists in the preliminary selection of a subgroup of panels with additional requirements for operating conditions (increased strength, fire safety, etc.). Under these conditions, the operator directionally distributes structures and materials among the selected panels. The statement of the optimization problem of group calculation with an extended list of objective functions and restrictions is given. Recommendations on selecting an objective function in specific production conditions are given. The regulatory requirements for noise reduction in production rooms and spectral characteristics of sound insulation of panels made of various materials are given. The spectral characteristics of noise in the room before and after application of the sound-insulating panel are also given. The effectiveness of the method is confirmed by the steady reduction of the mathematical expectation and variance of the total noise load on people in production premises with an increase in the number of iterations. The calculation method demonstrated the reduction of excess noise load in comparison with standard methods

Improved Fuzzy-Based Taguchi Method for Multi-Objective Optimization of Direct-Drive Permanent Magnet Synchronous Motors

The original fuzzy-based Taguchi method is restricted with the number of objective functions. Aiming to solve this problem, this paper describes an improved fuzzy-based Taguchi method (IFBTM) coupled with finite-element analysis (FEA) for the multi-objective optimization of direct-drive permanent magnet synchronous motors (DDPMSMs). To reduce the number of dimensions in the fuzzy rule base, the IFBTM uses a factor stratification method to establish multilayer fuzzy rule base. The fuzzy inference system with multilayer fuzzy rule base is introduced to translate the multiple optimization objectives to a single objective. After optimal selection analysis, the best combination of motor structure factors is obtained with small FEA computation. Finally, the performances of DDPMSM are analyzed and compared with those of IFBTM. The results certify the superiority and validity of IFBTM.

BÜTÜNLEŞİK TEDARİK ZİNCİRİ ÇİZELGELEME MODELLERİ: BİR LİTERATÜR TARAMASI

Research on integration of supply chain and scheduling is relatively recent, and number of studies on this topic is increasing. This study provides a comprehensive literature survey about Integrated Supply Chain Scheduling (ISCS) models to help identify deficiencies in this area. For this purpose, it is thought that this study will contribute in terms of guiding researchers working in this field. In this study, existing literature on ISCS problems are reviewed and summarized by introducing the new classification scheme. The studies were categorized by considering the features such as the number of customers (single or multiple), product lifespan (limited or unlimited), order sizes (equal or general), vehicle characteristics (limited/sufficient and homogeneous/heterogeneous), machine configurations and number of objective function (single or multi objective). In addition, properties of mathematical models applied for problems and solution approaches are also discussed.

Solving many-objective optimisation problems by an improved particle swarm optimisation approach and a normalised penalty method

In this paper, a novel modified particle swarm optimisation (NMPSO) approach is presented to handle the many-objective knapsack (MOK) problem. NMPSO relies on the global best particle to guide the search of all particles in each generation. Furthermore, a randomisation-based mutation is adopted to overcome the premature convergence. A normalised penalty method (NPM) is devised to reach a compromise between objective functions and inequality constraints, which enables particles to explore solution space more precisely. In summary, the contribution of our work can be summarised in two aspects: 1) a more powerful approach called NMPSO is proposed; 2) a reasonable NPM is devised. Five improved PSOs are used to handle the MOKs with different number of objective functions and dimensions. Experimental results show that NMPSO has higher efficiency than the other four approaches. It uses the lowest computational cost and achieves the smallest penalty function values for most MOKs.

Solving many-objective optimisation problems by an improved particle swarm optimisation approach and a normalised penalty method

In this paper, a novel modified particle swarm optimisation (NMPSO) approach is presented to handle the many-objective knapsack (MOK) problem. NMPSO relies on the global best particle to guide the search of all particles in each generation. Furthermore, a randomisation-based mutation is adopted to overcome the premature convergence. A normalised penalty method (NPM) is devised to reach a compromise between objective functions and inequality constraints, which enables particles to explore solution space more precisely. In summary, the contribution of our work can be summarised in two aspects: 1) a more powerful approach called NMPSO is proposed; 2) a reasonable NPM is devised. Five improved PSOs are used to handle the MOKs with different number of objective functions and dimensions. Experimental results show that NMPSO has higher efficiency than the other four approaches. It uses the lowest computational cost and achieves the smallest penalty function values for most MOKs.

Unified Planning of Wind Generators and Switched Capacitor Banks: A Multiagent Clustering-Based Distributed Approach

In this paper, a multiagent clustering-based distributed approach for the optimal planning of wind-distributed generators (DGs) and switched capacitor banks (SCBs) is proposed. First, electrical distance matrices for the power systems are constructed. Additionally, a constrained optimization problem, which includes several indices and a few constraints, for the optimal clustering of distribution networks is formulated and solved. After obtaining optimal clusters, agents are assigned to the clusters, and a second multiobjective optimization problem (MOOP) for the distributed planning of wind DGs and SCBs is formulated and assigned to a head agent. The number of objective functions in the MOOP is equal to the number of agents. The objective function of an agent consists of three indices: annual energy losses, investment costs, and voltage enhancement. Moreover, a deep neural network architecture is designed, and four independent networks are trained with six years of wind speed data for the seasonal wind speed forecasting. Two IEEE unbalanced test feeders, one with 37 nodes and the other with 123 nodes, and eight test cases are considered for simulations. The results show that losses and costs are optimized, and the voltage unbalance of the system is reduced.

A novel mixed Spider’s web initial solution and data envelopment analysis for solving multi-objective optimization problems

When it comes to solving engineering design problems, the existing techniques require a significant amount of time. In the following case the objective function value has been obtained using optimization methods which has been the subject of many research efforts, recently accepted methods possess advantages however the biggest challenge is the number of objective functions and constraints required. In this paper, we proposed a novel combined approach involving the method of Data Envelopment Analysis and the Spider’s web initial solution. It turns out that the selected method requires no additional algorithm and can be used to solve most multi-objective problems, in contrast to previous methods in which the number of objective functions increase due to a rising level of complexity in such algorithms thereby leading to a reduction in effectiveness. Furthermore the proposed method is capable of yielding a more efficient result in non-convex problems as well as typical convex problems.

Smart imposing of operational limits in optimising the technical and economic indexes of a distribution network in the presence of distributed generation sources

ABSTRACTOptimal placement of distributed generation sources in order to enhance the technical and economic indexes of the distribution network maximally is one of the problems for which very diverse solutions are competitively developing. Each of these methods tries to reach a more optimum point by choosing a number of objective functions from economic and technical indexes and applying a number of common limits of the electricity grid. Little attention has been paid to the extent of practicality of implementation of these optimum points on a real network with limits imposed by set up condition of distributed generation sources and the common electrical limits. In this paper, depending on the grid's environmental conditions, a method is proposed to smartly impose limits regarding operational conditions in maximally optimising the technical and economic indexes of a distribution network in the presence of distributed generation sources. The firefly nonlinear smart algorithm is utilized for smart optimisation and the impact of operational limits in optimal placement of a wind turbine and the extent of enhancement in economic and technical indexes of the studied grid is investigated graphically.

Multi-objective colliding bodies optimization algorithm for design of trusses

Abstract This article presents a new population-based optimization algorithm to solve the multi-objective optimization problems of truss structures. This method is based on the recently developed single-solution algorithm proposed by the present authors, so called colliding bodies optimization (CBO), with each agent solution being considered as an object or body with mass. In the proposed multi-objective colliding bodies optimization (MOCBO) algorithm, the collision theory strategy as the search process is utilized and the Maximin fitness procedure is incorporated to the CBO for sorting the agents. A series of well-known test functions with different characteristics and number of objective functions are studied. In order to measure the accuracy and efficiency of the proposed algorithm, its results are compared to those of the previous methods available in the literature, such as SPEA2, NSGA-II and MOPSO algorithms. Thereafter, two truss structural examples considering bi-objective functions are optimized. The performance of the proposed algorithm is more accurate and requires a lower computational cost than the other considered algorithms. In addition, the present methodology uses simple formulation and does not require internal parameter tuning. Highlights A new population-based algorithm is presented for multi-objective optimization. The algorithm is based on the recently developed single-solution colliding bodies optimization (CBO). The proposed multi-objective colliding bodies optimization is abbreviated as MOCBO. MOCBO utilizes the maximin fitness procedure for sorting the agents. A series of well-known test functions and number of objective functions are studied. The MOCBO is more accurate and requires lower computational cost. The MOCBO method uses simple formulation and requires no internal parameter tuning.

FP-ABC: Fuzzy-Pareto dominance driven artificial bee colony algorithm for many-objective software module clustering

Recently, multi-objective evolutionary algorithms (MOEAs) gained wide attention to solve various search-based software engineering (SBSE) problems. The formulation of software module clustering problem (SMCP) as SBSE problem facilitates the application of many MOEAs. Even though MOEAs are being used successfully to solve the SMCPs, the performance of Pareto-dominance based MOEAs degrade if the number of objective functions is greater than three. To address the large and complex many-objective SMCPs (MaSMCPs), changes in MOEAs are essential. This paper proposes a Fuzzy-Pareto dominance driven Artificial bee colony (FP-ABC) to solve the MaSMCPs effectively and efficiently. In this contribution, fuzzy-Pareto dominance and two external archive concepts have been integrated into artificial bee colony (ABC) algorithm. The fuzzy-Pareto dominance improves the selection process of candidate solution and two external archives concept helps in balancing the convergence and diversity. To validate the supremacy of the proposed approach, a comparative study is performed with the existing many-objective optimization algorithms such as Two-Arch2, NSGA-III, MOEA/D, and IBEA, over seven real-world problems. The statistical analysis of the results indicates that the proposed approach outperforms existing approaches in terms of modularization quality (MQ), coupling, cohesion, and Inverted Generational Distance (IGD).

A redundancy detection algorithm for fuzzy stochastic multi-objective linear fractional programming problems

ABSTRACTThe computational complexity of linear and nonlinear programming problems depends on the number of objective functions and constraints involved and solving a large problem often becomes a difficult task. Redundancy detection and elimination provides a suitable tool for reducing this complexity and simplifying a linear or nonlinear programming problem while maintaining the essential properties of the original system. Although a large number of redundancy detection methods have been proposed to simplify linear and nonlinear stochastic programming problems, very little research has been developed for fuzzy stochastic (FS) fractional programming problems. We propose an algorithm that allows to simultaneously detect both redundant objective function(s) and redundant constraint(s) in FS multi-objective linear fractional programming problems. More precisely, our algorithm reduces the number of linear fuzzy fractional objective functions by transforming them in probabilistic–possibilistic constraints characterized by predetermined confidence levels. We present two numerical examples to demonstrate the applicability of the proposed algorithm and exhibit its efficacy.

A branch-and-bound based heuristic algorithm for convex multi-objective MINLPs

We study convex multi-objective Mixed Integer Non-Linear Programming problems (MINLPs), which are characterized by multiple objective functions and non linearities, features that appear in real-world applications. To derive a good approximated set of non-dominated points for convex multi-objective MINLPs, we propose a heuristic based on a branch-and-bound algorithm. It starts with a set of feasible points, obtained, at the root node of the enumeration tree, by iteratively solving, with an ε-constraint method, a single objective model that incorporates the other objective functions as constraints. Lower bounds are derived by optimally solving Non-Linear Programming problems (NLPs). Each leaf node of the enumeration tree corresponds to a convex multi-objective NLP, which is solved heuristically by varying the weights in a weighted sum approach. In order to improve the obtained points and remove dominated ones, a tailored refinement procedure is designed. Although the proposed method makes no assumptions on the number of objective functions nor on the type of the variables, we test it on bi-objective mixed binary problems. In particular, as a proof-of-concept, we tested the proposed heuristic algorithm on instances of a real-world application concerning power generation, and instances of the convex biobjective Non-Linear Knapsack Problem. We compared the obtained results with those derived by well-known scalarization methods, showing the effectiveness of the proposed method.