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Berge's notion of equilibrium represents a complementary alternative to the Nash equilibrium when modeling socioeconomic behavior and human interactions. While the notion of Nash equilibrium is based on self-interest, as players seek to maximize their own payoffs given the action of the other players, the idea behind Berge equilibrium is mutual support, as given the action of one of the players, all others select their actions looking for her best interest. However, because of the demanding conditions involved, the existence of Berge equilibria is rarely guaranteed. In this paper, we propose vector-valued normal-form games as a unified framework in which to study and extend the concept of Berge equilibrium. Based on the equilibria of the so-called altruistic game, we introduce new equilibrium concepts which constitute different relaxations of Berge's notion, although they still retain the underlying idea of mutual support. We establish the links between these new equilibria, Nash equilibrium, Berge equilibrium, and other related concepts already existing in the literature. Our approach has the advantage that it permits the incorporation of preference information to identify the equilibria which are consistent with different altruistic attitudes of the players. [ABSTRACT FROM AUTHOR]

We develop a monetary model in which a private company issues digital currency and uses payment data to estimate consumers' preferences. Sellers purchase preference information to produce goods that better match consumers' preferences. A monopoly arises in the digital currency industry, and digital currency is not issued if the inflation rate is sufficiently high. Due to reinforcing interactions between the value of preference information and trade volume, multiple equilibria (with and without digital currency) can exist, depending on market structures for monetary exchanges. When left to market forces alone, socially efficient uses of payment data may not occur. [ABSTRACT FROM AUTHOR]

The transformation method between the linguistic judgement matrix and other four forms of preference information is researched in this paper. The four forms of preference information are preference ordering, utility value, reciprocal judgement matrices and complementary judgement matrices. First, based on the definition of preference information, the mutual transformation methods of linguistic judgement matrix and other four forms of preference information are given. Then new transformation equations are obtained. Next, it is proved that when the linguistic judgement matrix has complete consistency, the other four forms of preference information will have complete consistency too. Finally, a numerical analysis is offered to show that these methods are feasible and effective.

In this paper a ranking method using common weights methodology is presented. The goal of the method is enhancing the decision maker (DM)'s influence in the ranking procedure. Although DM's preference information is an important element in our method, the approach can also be modified to be used in the absence of it. Since we aim to implement the approach on an empirical instance, the model is modified to deal with the properties of the sample, so it is developed in the presence of the interval data and flexible measures. Finally, the results are discussed. [ABSTRACT FROM AUTHOR]

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We introduce novel concepts to solve multiobjective optimization problems involving (computationally) expensive function evaluations and propose a new interactive method called O-NAUTILUS. It combines ideas of trade-off free search and navigation (where a decision maker sees changes in objective function values in real time) and extends the NAUTILUS Navigator method to surrogate-assisted optimization. Importantly, it utilizes uncertainty quantification from surrogate models like Kriging or properties like Lipschitz continuity to approximate a so-called optimistic Pareto optimal set. This enables the decision maker to search in unexplored parts of the Pareto optimal set and requires a small amount of expensive function evaluations. We share the implementation of O-NAUTILUS as open source code. Thanks to its graphical user interface, a decision maker can see in real time how the preferences provided affect the direction of the search. We demonstrate the potential and benefits of O-NAUTILUS with a problem related to the design of vehicles. [ABSTRACT FROM AUTHOR]

With the continuous advancement of science and technology, the decision-making environment faced by managers is increasingly complex, which puts forward higher requirements for managers. Due to the complexity of the multi-attribute decision-making problem, it is difficult for the decision-maker to make the correct choice. In addition, due to the influence of the educational background and limited knowledge of the managers, it is impossible to evaluate with simple statistical data. In essence, the location problem of lowcarbon logistics parks is a MADM problem. Therefore, this paper establishes an optimization model to solve the multiple-attribute decision-making (MADM) problem with alternative preference and single-valued neutrosophic (SVN) information. Considering that the information of weight is unknown, a scientific model is built based on the minimum deviation method deriving the criterion weight. Furthermore, the above models and methods are extended to interval neutrosophic sets (INs). To verify the validity of the modified model, a numerical case for low-carbon logistics park site selection is taken as an example. Through the case study, we found that the method has strong operability and can make the most of the available information. [ABSTRACT FROM AUTHOR]

Most studies in dynamic multi-objective optimization have predominantly focused on rapidly and accurately tracking changes in the Pareto optimal front (POF) and Pareto optimal set (POS) when the environment undergoes changes. However, there are real-world scenarios where it is necessary to simultaneously solve changing objective functions and satisfy the preference of Decision Makers (DMs). In particular, the DMs may be only interested in a partial region of the POF, known as the region of interest (ROI), rather than requiring the entire POF. To meet the challenge of simultaneously predicting a changing POF and/or POS and dynamic ROI, this paper proposes a new dynamic multi-objective evolutionary algorithm (DMOEAs) based on the preference. The proposed algorithm consists of three key components: an evolutionary direction adjustment strategy based on changing reference points to accommodate shifts in preferences, an angle-based search strategy for tracking the varying ROI, and a hybrid prediction strategy that combines linear prediction models and population manifold estimation within the ROI to ensure convergence and distribution in scenarios where preferences remain unchanged. Experimental studies conducted on 30 widely used benchmark problems in which it outperforms contrasting algorithms on 71% of test suits. Empirical results demonstrate the significant advantages of the proposed algorithm over existing state-of-the-art DMOEAs. [ABSTRACT FROM AUTHOR]

Interactive methods are useful decision-making tools for multiobjective optimization problems, because they allow a decision-maker to provide her/his preference information iteratively in a comfortable way at the same time as (s)he learns about all different aspects of the problem. A wide variety of interactive methods is nowadays available, and they differ from each other in both technical aspects and type of preference information employed. Therefore, assessing the performance of interactive methods can help users to choose the most appropriate one for a given problem. This is a challenging task, which has been tackled from different perspectives in the published literature. We present a bibliographic survey of papers where interactive multiobjective optimization methods have been assessed (either individually or compared to other methods). Besides other features, we collect information about the type of decision-maker involved (utility or value functions, artificial or human decision-maker), the type of preference information provided, and aspects of interactive methods that were somehow measured. Based on the survey and on our own experiences, we identify a series of desirable properties of interactive methods that we believe should be assessed. [ABSTRACT FROM AUTHOR]

The data are not always real numbers due to the complexity and uncertainty of the real-world decision environment. This requires us to use probability-based expressions to describe the complex decision-making data in reality. In order to understand the probability-based decision-making information more comprehensively, this paper reviews and summarizes the current development of probability-based decision-making information. First, the number and the distribution of publications related to the probability-based decision-making information are analyzed. Then, based on presenting literature related to different concepts, all the existing different concepts are reviewed and discussed. Considering different forms of decision-making information, this paper summarizes decision-making methods from two aspects: the decision-making methods based on decision matrix and the decision-making methods based on preference information. We also summarize the existing literature from the perspective of application. In addition, based on current developments and the discussions above, some challenges concerning probability-based expressions are listed, which can be regarded as future research directions. [ABSTRACT FROM AUTHOR]

In real life, some complex problems are multi-objective optimization problems. Most of the existing studies have focused on how to obtain the optimal solutions distributed on the whole Pareto-optimal frontier. However, in some fields such as industrial production, the decision-makers of enterprises usually care about what specific measures can maximize the comprehensive benefits of enterprises. Due to this kind of realistic demands, we prefer to find a small part of the optimal solutions according to the preference information suggested by the decision-makers rather than obtain all of the Pareto-optimal solutions. However, almost all of the existing methods only repeat calculation when they meet the scenario where the user-preferences change over time. To address the multi-objective optimization problem under the scenario with dynamic user-preferences information, we propose a MOEA/D-DPRE (multi-objective optimization algorithm based on dynamic preference information) algorithm in this paper, and its framework is inspired by the MOEA/D-PRE (decomposition user-preference multi-objective evolutionary) algorithm. We analyze the four position relations between the distribution region of the old preference weight vectors (old preference region), and we also present the distribution region of the new preference weight vectors (new preference region) and propose the different strategy to the different case respectively. When the preference information changes, the MOEA/D-DPRE can converge to the region of new interest by responding to the change of preference and the historical information. Experimental results show that the proposed method is better than the compared method in convergence speed and distribution under the scenario with dynamic user-preferences information. [ABSTRACT FROM AUTHOR]

Online tourist reviews are the real feeling of tourists after the journey, which have a strong reference value for potential tourists to make travel decisions. However, it is almost impossible for a potential tourist to look through the massive online reviews related to tourist attractions (TAs) so that he/she can make the most appropriate decision. To this end, this paper proposes a recommender system to rank the alternative TAs through online reviews based on aspect-level sentiment analysis and multi-criteria decision-making (MCDM) with intuitionistic and hesitant fuzzy information. In this methodology, the aspects that the experienced tourists concern are extracted from online reviews to construct a three-level evaluation system (including target layer, criteria layer and sub-criteria layer), which not only ensures the comprehensive evaluation of TAs as much as possible, but also reduces the complexity of the decision-making process. Then, the online reviews related to these sub-criteria are transformed into the corresponding intuitionistic and hesitant fuzzy performance scores through aspect-level sentiment analysis. Furthermore, in order to obtain the final ranking result that more in line with the expectations of the potential tourist, the preference information from the potential tourist and experienced tourists is integrated to determine the weights of criteria. Subsequently, the intuitionistic and hesitant fuzzy TOPSIS (IHF-TOPSIS) method is proposed to rank the alternative TAs. Finally, a case study is provided to verify the validity and applicability of the methodology. [ABSTRACT FROM AUTHOR]

We study how visual interaction techniques considered in visual analytics can be utilized when implementing interactive multiobjective optimization methods, where a decision maker iteratively participates in the solution process. We want to benefit from previous research and avoid re-inventing ideas. Our aim is to widen awareness and increase the applicability of interactive methods for solving real-world problems. As a concrete approach, we introduce seven high-level tasks that are relevant for interactive methods. These high-level tasks are based on low-level tasks proposed in the visual analytics literature. In addition, we give an example on how the high-level tasks can be implemented and demonstrate this in the context of a real-world multiobjective optimization problem related to wastewater treatment plant operation. Finally, we make recommendations for implementations of interactive methods. We conclude that task-based visual analytics can help in implementing interaction between human decision makers and interactive multiobjective optimization methods. [ABSTRACT FROM AUTHOR]

Interactive multiobjective optimization (IMO) aims at finding the most preferred solution of a decision maker with the guidance of his/her preferences which are provided progressively. During the process, the decision maker can adjust his/her preferences and explore only interested regions of the search space. In recent decades, IMO has gradually become a common interest of two distinct communities, namely, the multiple criteria decision making (MCDM) and the evolutionary multiobjective optimization (EMO). The IMO methods developed by the MCDM community usually use the mathematical programming methodology to search for a single preferred Pareto optimal solution, while those which are rooted in EMO often employ evolutionary algorithms to generate a representative set of solutions in the decision maker's preferred region. This paper aims to give a review of IMO research from both MCDM and EMO perspectives. Taking into account four classification criteria including the interaction pattern, preference information, preference model, and search engine (i.e., optimization algorithm), a taxonomy is established to identify important IMO factors and differentiate various IMO methods. According to the taxonomy, state-of-the-art IMO methods are categorized and reviewed and the design ideas behind them are summarized. A collection of important issues, e.g., the burdens, cognitive biases and preference inconsistency of decision makers, and the performance measures and metrics for evaluating IMO methods, are highlighted and discussed. Several promising directions worthy of future research are also presented.

Finding the overall Pareto optimal front while addressing the effect of an increasing number of objectives has become an essential and challenging issue for multiobjective optimization in real-world applications. Preference information provided by a decision maker can guide the search for preferred regions of the Pareto front and accelerate the convergence of the population. In this paper, a new variant of the Pareto dominance relation, called preference angle and reference information-based dominance, is proposed to create a stricter partial order among nondominated solutions. In the proposed method, the Euclidean distance and angle information between candidate solutions and reference points are calculated to evaluate the degree of convergence and population diversity, respectively. In addition, an adaptive threshold is designed to adjust the judgment condition of ar-dominance using an iterative process in a prespecified interval. The proposed algorithm increases the convergence speed of the population and reduces the number of solutions in the nonpreferred region. Comparative evaluation experiments are presented with respect to two performance metrics for a variety of benchmark test problems and real-world aluminum electrolytic production cases. The results demonstrate that the proposed approach is effective for highly complex, multiobjective optimization problems when compared with five state-of-the-art evolutionary algorithms. [ABSTRACT FROM AUTHOR]

Recently, multi-objective evolutionary algorithms (MOEAs) have been widely explored and applied to many real-world problems. Particularly, preference-based MOEAs are among the mostly studied. Several preference-based optimization algorithms have already been proposed in literature. However, most existing studies focus on how to locate the region of interest (ROI) and how to control the size of ROI, and overlook the relationship between preference information and distribution of the final solutions. Given that the distribution of the final solutions is also an important factor, in this paper, we propose a new preference-based MOEAs called MOEA/D-AWV using an adaptive weight vector generation strategy (AWV). The weight vectors are generated adaptively by the decision maker's preference, and finally guide the solutions to converge to a preferred distribution. Solutions will converge to the reference point as close as possible with the AWV strategy, which will lead to the loss of diversity. In order to prevent the search process from being trapped at local optima, we propose an adaptive parameter tuning scheme (APT) to maintain diversity during the search process. In addition, since the distribution of weight vectors should adapt to the changes of decision makers' preference, the APT scheme can help algorithm find desired results in different scenarios. Compared with five state-of-the-art preference-based MOEAs on 23 test instances, MOEA/D-AWV achieves the best performance. Especially, in many-objective optimization problems with high-dimensional objective space, our proposed MOEA/D-AWV still shows a competitive performance. [ABSTRACT FROM AUTHOR]

To estimate life-cycle cost and future seismic capacity, the life-cycle analysis in seismic design of bridges is formulated as an interactive multiobjective decision-making and optimization problem. Specifically, preference information based on engineering judgement and theoretical analysis is incorporated in the optimization procedure. The posteriori evolutionary multiobjective optimization (EMO) algorithm (nonpreference) and the preference-based interactive EMO algorithm are both applied to the seismic design of an reinforced concrete (RC) pier with two, three, and four objectives, namely, flexural strength coefficient, shear strength coefficient, reliability index of drift, and life-cycle cost coefficient. In terms of a rational displacement ductility, the safety preference information is applied to establish a value function after every few optimization runs, progressively directing the search of the EMO algorithm to more preferred solutions. By comparing the results of both algorithms, it is found that when there are more than three objectives, it is difficult for the nonpreference optimization to converge with the global Pareto frontier, resulting in a local optimal solution as the final design. The combination of preference structure and multiobjective optimization effectively avoids the challenges of global search and purposefully converges with the most preferred design. [ABSTRACT FROM AUTHOR]

Abstract The measure called partial area under the curve (pAUC) has attracted increasing interest in recent years due to the wide range of real-world applications it can be used for, such as biomarker selection and pedestrian detection. Compared with AUC, pAUC only acts upon a partial (instead of a full) area under the ROC curve. In this paper, a multi-objective evolutionary algorithm termed MOPA is proposed to optimize the pAUC in an arbitrary false positive range. Within this range, two important components are proposed to focus on a particular region within the AUC. First, a new metric (K-FPR), is proposed that is created by considering the partial range of the false positive rate (FPR). This is combined with the true positive rate (TPR) to provide the two optimized objectives of MOPA. Second, a preference-based multi-objective evolutionary algorithm is developed within the framework of a recently proposed algorithm (AR-MOEA), by which the search is concentrated on a partial area under the ROC curve. Numerical experiments on different data sets demonstrate the competitiveness of the proposed method in comparison with state-of-the-art algorithms. [ABSTRACT FROM AUTHOR]

To reduce the dimensionality of high-dimensional data and enhance its classification accuracy, feature selection can be regarded as a multiobjective optimization problem that can be addressed by evolutionary computation algorithms with promising results. However, balancing the convergence and diversity of nondominated solutions remains challenging. To address these issues, this paper proposes a multiobjective sparrow search feature selection approach with sparrow ranking and preference information and its applications for high-dimensional data. First, during the population updating process, the updating formula of observers is combined with the mutualism phase of the symbiotic organisms search algorithm to design a location updating formula of observers in the sparrow search algorithm, balancing the local development ability and global search ability of sparrow search algorithm and increasing the possibility of moving closer to the optimal solution. Second, based on the dominant and nondominated sparrow individuals of each sparrow, the dominant and nondominated distances are defined, and the sparrow ranking is proposed. The feature ranking is then designed by using the definition of sparrow ranking. Following the definition of sparrow ranking and feature ranking, the selection strategy and location updating formula of the producers in the sparrow search algorithm are proposed. Finally, a preference information-based mutation algorithm is designed to augment the diversity of the nondominated solutions and more effectively guide the whole sparrow population to a better solution. The experimental results on 14 high-dimensional datasets show that the proposed algorithm outperforms different single-objective and multiobjective optimization feature selection algorithms in terms of classification efficiency and diversity of solutions. • A new location updating formula of observers is designed for the mutualism phase of the symbiotic organisms search algorithm. • The dominant and nondominated distances are defined and the feature ranking is designed with the sparrow ranking. • The novel selection strategy and location updating formula of the producers in the sparrow search algorithm are improved. • A preference information-based mutation algorithm is designed to augment the diversity of the nondominated solutions. [ABSTRACT FROM AUTHOR]

‎The wide variety of available interactive methods brings the need for creating general‎ ‎interactive algorithms enabling the decision maker (DM) to apply freely several convenient methods which best fit his/her preferences‎. ‎To this end‎, ‎in this paper‎, ‎we propose a general scalarizing problem for multiobjective programming problems‎. ‎The relation between optimal solutions of the introduced scalarizing problem and (weakly) efficient as well as properly efficient solutions of the main multiobjective optimization problem (MOP) is discussed‎. ‎It is shown that some of the scalarizing problems used in different interactive methods can be obtained from proposed formulation by selecting suitable transformations‎. ‎Based on the suggested scalarizing problem‎, ‎we propose a general interactive algorithm (GIA) that enables the DM to specify his/her preferences in six different ways with capability to change his/her preferences any time during the iterations of the algorithm‎. ‎Finally‎, ‎a numerical example demonstrating the applicability of the algorithm is provided‎.

Considering future seismic risk and life-cycle cost, the life-cycle seismic design of bridge is formulated as a preference-based multi-objective optimization and decision-making problem, in which the conflicting design criteria that minimize life-cycle cost and maximize seismic capacity are treated simultaneously. Specifically, the preference information based on theoretical analysis and engineering judgment is embedded in the optimization procedure. Based on reasonable displacement ductility, the cost preference and safety preference information are used to progressively construct value function, directing the evolutionary multi-objective optimization algorithm’s search to more preferred solutions. The seismic design of a reinforced concrete pier is presented as an application example using the proposed procedure for the global Pareto front corresponding with engineering designers’ preference. The results indicate that the proposed model is available to find the global Pareto front satisfying the corresponding preference and overcoming the difficulties of the traditional multi-objective optimization algorithm in obtaining a full approximation of the entire Pareto optimal front for large-dimensional problems as well as cognitive difficulty in selecting one preferred solution from all these solutions. [ABSTRACT FROM AUTHOR]

Proposing a suitable skin care regimen based on tbe skin condition of human body is an urgent problem to be disposed of. This paper adopted the expert grading method .standard deviation method and entropy weight method. aimed at optimizing the position weight value of the facial skin. and acquired skin attribute values more comprehensively. It established goal-project-attribute hierarchical model,which combining the preference of tested sample and project-attribute characteristic by expert, to realize multiple attribute weights adjustment under individual preferences and rational recommendation. By analyzing measured data.it calculated the weighted distance between the schemes and the positive and negative ideal schemes utilizing optimized TOPSIS method dependent upon multi attribute weights, to realize the ranking of skin care programs. Finally.it verifies the feasibility and effectiveness of proposed method through an application example and comparison analysis with two existing methods. [ABSTRACT FROM AUTHOR]

Many evolutionary multi-objective optimization (EMO) methodologies have been proposed and performed very well on finding a representative set of Pareto-optimal solutions, but this advantage will be weakened with the increasing number of objectives. And in real applications, what decision makers (DMs) want is a unique solution or a set of solutions rather than the overall Pareto-optimal front. It is a difficult task to solve many-objective problems by using preference information provided by decision maker (DM) during optimization process. In this paper, a new angle-based preference selection mechanism is proposed, which replaces the traditional crowding distance with the aid of preference information provided by the DMs. Particularly, we combine it with a multi-objective immune algorithm with non-dominated neighbor-based selection. The proposed method has been extensively compared with other recently proposed preference-based EMO approaches over DTLZ1, DTLZ2, and DTLZ3 test problems with 4-100 objectives. The results of the experiment indicate that the proposed algorithm can achieve competitive and better results. [ABSTRACT FROM AUTHOR]