The divergence of individual opinions in group decision making highlights the need to enhance consensus. Achieving this requires the cooperation of decision makers (DMs). However, such adjustments often conflict with personal interests, resulting in non-cooperative behavior of DMs. This paper leverages the concept of co-opetition, a blend of cooperation and competition, to explore effective strategies for consensus adjustment. By adopting a cooperative stance, we establish a comprehensive consensus adjustment strategy and allocation scheme while employing a competitive perspective to evaluate the payoffs of DM coalitions. Concretely, using Cournot and Stackelberg games, we develop multi-objective and bilevel programming models to determine the consensus adjustments of DM coalitions, which incorporate (optimal) counterstrategies for external DMs for any fixed coalition. The strength of new consensus adjustment schemes lies in their ability to balance collective and individual rationality, thereby ensuring that consensus distribution is equitable and effective for all involved parties. Additionally, we conduct a case study and comparison of Nanjing’s agricultural characteristic industries to validate our findings further.

Space-Air-Ground Integrated Networks (SAGIN) present stringent and complex requirements for network design and management due to their heterogeneity, self-organizing nature, dynamic characteristics and so on. Despite extensive research on deep learning in the context of SAGIN, challenges such as inflexibility, lack of generalization, limited robustness, and insufficient scalability persist. Large language models (LLMs), with their powerful natural language processing, understanding, and generation capabilities, show significant potential in meeting the complex demands of SAGIN. We thoroughly analyze the application of LLMs in optimizing channel models, improving communication algorithms, and enhancing network management and security performance to address the unique characteristics and requirements of SAGIN. Additionally, it identifies the challenges faced during the deployment of LLMs, including computational resource constraints, data transmission limitations, model updates, real-time processing, and system integration. Corresponding solutions are proposed to ensure the efficient operation and reliability of SAGIN. Through this research, we aim to provide innovative solutions for the intelligent management of SAGIN, promoting the continuous development and application of SAGIN.

In today's network era, people's decisions are susceptibly influenced by others, especially the ones they trust. This study confines to studying social network group decision making (SNGDM). Due to the mutual influence of consensus level and consensus adjustment among decision makers (DMs), this study utilizes biform game theory to propose an innovative consensus mechanism for facilitating group decision making with unconnected social networks. Specifically, in the context of a DM social network with multiple trust relationship-based connected components, we construct a multi-objective programming model to determine the consensus adjustment. Within each connected component, we employ the digraph game theory to study DMs' consensus adjustments, leveraging the directional and asymmetrical characteristics of trust-relationships. We then analyze the consensus adjustments of feasible DM coalitions using built optimization models and define the di-Myerson value. Additionally, we construct several axiomatic systems to show the rationality of consensus allocation results. We identify partial trust-relationships that increase the consensus adjustments of DMs as irrational, and design an algorithm to address them, thereby reducing the cost of consensus. Finally, we present a case study that showcases the real-world application of our new theoretical results. This is the first bi-form game consensus mechanism based on trust relationship for SNGDM.

In our daily life, players may reduce their cooperation levels for various reasons. This paper studies how to evaluate and allocate the cooperation payoff when the players cooperate partially. Different from previous research that directly uses the cooperation level to calculate the payoffs of fuzzy coalitions, we introduce the concept of dual fuzzy cooperative games, where the payoffs of fuzzy coalitions equal the difference between the original payoff and the lost payoff caused by the decline of the cooperation level. Meanwhile, the dual Shapley value is introduced, and its axiomatic systems are studied. Further, a special kind of dual fuzzy cooperative game with Choquet integral form is presented. Finally, we provide an application of dual fuzzy cooperative games in the manufacturer–retailer supply chain to allocate the payoff for vertical co-op advertising.

As a process for ensuring the agreeable degree of individual opinions, consensus analysis is crucial for GDM. This paper focuses on the adaptive consensus mechanism. That's, different adjustment strategies are employed for various consensus levels. Unlike the feedback iteration method, this paper introduces an optimization model-based consensus-reaching procedure. To do this, optimal models are built to determine the minimum consensus adjustment at different levels. Then, the individual minimum consensus adjustment is analyzed, and the inconsistency between individual and group minimum consensus adjustments is concluded. After that, consensus adjustment cooperative games at three levels are proposed to allocate the total minimum consensus adjustment in view of the comprehensive evaluation. We can obtain the coalitional stability allocation scheme using the core of constructed cooperative games. Additionally, core-Nash bargaining games at three levels are proposed to ensure the fairness and coalitional stability of allocation results. Finally, a numerical example is offered to indicate the application of the new theoretical developments.

This paper focuses on the cardinality constrained mean-variance portfolio optimization, in which only a small number of assets are invested. We first treat the covariance matrix of asset returns as a diagonal matrix with a special matrix processing technique. Using the dual theory, we formulate the lower bound problem of the original problem as a max-min optimization. For the inner minimization problem with the cardinality constraint, we obtain its analytical solution for the portfolio weights. Then, the lower bound problem turns out to be a simple concave optimization with respect to the Lagrangian multipliers. Thus, the interval split method and the supergradient method are developed to solve it. Based on the precise lower bound, the depth-first branch and bound method are designed to find the global optimal investment selection strategy. Compared with other lower bounds and the current popular mixed integer programming solvers, such as CPLEX and SCIP, the numerical experiments show that our method has a high searching efficiency. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by the National Natural Science Foundation of China [Grants 12101317, 12271071, and 11991024], the Natural Science Foundation of Jiangsu Province [Grant BK20200819], the Team Project of Innovation Leading Talent in Chongqing [Grant CQYC20210309536], the Contract System Project of Chongqing Talent Plan [Grant cstc2022ycjh-bgzxm0147], and the Philosophy and Social Science Fund of Education Department of Jiangsu Province [Grant 2020SJA0168]. K.F.C. Yiu is supported in part by the Research Grants Council of Hong Hong [Grant PolyU 15223419], the Hong Kong Polytechnic University [Grants 4-ZZPT and 1-WZ0E], and the Research Centre for Quantitative Finance [Grant 1-CE03]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0344 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0344 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Consensus reaching process (CRP) is a key topic in the area of group decision making (GDM). When the consensus level is not high enough, it becomes necessary to adjust the original opinions of decision makers (DMs). To offer the adjustment reference for DMs, we build the programming models to determine the minimum modification to be carried out from the individual and global perspectives. Meanwhile, all DMs are divided into two subgroups: DMs with acceptable and unacceptable consensus levels. If some DMs with unacceptable consensus level do not accept the relevant modifications, the Nash bargaining game-based programming model is built for the fairness and efficiency of modifications. When some DMs refuse to make any modifications or tend to modify the opinions in their way, with respect to different group consensus situations, we make the minimum hybrid penalty mechanism by the Nash bargaining game-based programming models. For each case, we determine the corresponding optimal modification mechanism in view of the fixed individual total modification and the maximum consensus level. Furthermore, we study the arrangements of weights of DMs according to their cardinal and ordinal consensus contributions. Based on these results, we present a new algorithm and illustrate its application by a numerical example. Moreover, we carry out the sensitivity and comparison analysis. We summarize the conclusions and future research directions in the end. The main originality of the new method includes: the fairness and efficiency of modifications, and the determination of the hybrid penalty mechanism.

This paper focuses on games on augmenting systems with a coalition structure that can be seen as an extension of games with a coalition structure and games on augmenting systems. Considering the player payoffs, the quasi-Owen value is defined. To show the rationality of this payoff index, five representative axiomatic systems are established. The population monotonic allocation scheme (PMAS) and the core are introduced. Moreover, the relationships between the PMAS and quasi-Owen value as well as the core and quasi-Owen value are discussed. Finally, an illustrative example is given to show the concrete application of the new payoff indices.

Large-scale group decision making (LSGDM) is a powerful technology to deal with complex decision problems. As decision makers (DMs) in LSGDM come from different fields with various backgrounds and expertise, their perception of the same decision problem is largely divergent. The measure and improvement of consensus can obtain widely accepted decision results. In this process, the behaviors of DMs are interactive. The amount and direction of adjustments influence not only their consensus level but also that of other DMs. Therefore, the fairness and rationality of the consensus adjustment mechanism are fundamental. Considering the characteristics of LSGDM, this paper introduces two-stage consensus adjustment mechanisms from the perspective of cooperative game theory. In the first stage, players are DMs within each subgroup, and subgroups are regarded as players in the second stage. Further, the coalition payoffs are defined as the minimum consensus adjustment determined by the built models. To show the designated consensus adjustment cooperative games are essential, we prove that they are supperadditive. Considering the marginal contribution of DMs to consensus adjustment, a two-stage Shapley consensus adjustment mechanism is offered. Namely, the Shapley function is adopted to allocate the consensus adjustment to DMs and subgroups. When the Shapley consensus adjustment does not satisfy coalitional stability, a two-stage core-Nash bargaining consensus adjustment mechanism is provided, which can overcome the drawback of the two-stage Shapley consensus adjustment mechanism. Moreover, two new algorithms for LSGDM are proposed. Finally, a numerical example is provided to show the feasibility of the new method, and a comparison analysis is made. (c) 2023 Elsevier Ltd. All rights reserved.

Linguistic variables are flexible and intuitive attraction for expressing the wording of decision makers. This paper introduces a new type of linguistic fuzzy sets called linguistic dual hesitant fuzzy sets to express the hesitancy of decision makers' qualitative preferences and non-preferences. Considering the application in decision making, linguistic dual hesitant fuzzy preference relations (LDHFPRs) are introduced that permit the decision makers to apply several linguistic variables to indicate a qualitative preferred judgment and a qualitative non-preferred judgment, respectively. To rank objects from LDHFPRs rationally, a consistency concept is first presented. Then, two optimal models are built to judge the consistency of LDHFPRs. When LDHFPRs are inconsistent, an optimal model-based iteration algorithm for obtaining consistent LDHFPRs is offered. Based on consistent linguistic intuitionistic fuzzy preference relations, a method for calculating the weighted linguistic intuitionistic fuzzy priority vector is introduced. In the setting of group decision making (GDM), a consensus measure based on individually weighted consistent reverse complementary linguistic intuitionistic fuzzy preference relations is defined. When the consensus does not satisfy the requirement, a two-step optimal model-based method for increasing the consensus level is offered. Furthermore, an approach for GDM with LDHFPRs is developed. Finally, an illustrative example concerning the evaluation of basic services internet enterprise websites is provided to show the efficiency of the new method.

Probabilistic linguistic variable is a kind of powerful qualitative fuzzy sets, which permits the decision makers (DMs) to apply several linguistic variables with probabilities to denote a judgment. This paper studies group decision making (GDM) with normalized probability linguistic preference relations (NPLPRs). To achieve this goal, an acceptably multiplicative consistency based interactive algorithm is provided to derive common probability linguistic preference relations (CPLPRs) from PLPRs, by which a new acceptably multiplicative consistency concept for NPLPRs is defined. When the multiplicative consistency of NPLPRs is unacceptable, models for deriving acceptably multiplicatively consistent NPLPRs are constructed. Then, it studies incomplete NPLPRs (InNPLPRs) and offers a common probability and acceptably multiplicative consistency based interactive algorithm to determine missing judgments. Furthermore, a correlation coefficient between CPLPRs is provided, by which the weights of the DMs are ascertained. Meanwhile, a consensus index based on CPLPRs is defined. When the consensus does not reach the requirement, a model to increase the level of consensus is built that can ensure the adjusted LPRs to meet the multiplicative consistency and consensus requirement. Moreover, an interactive algorithm for GDM with NPLPRs is provided, which can address unacceptably multiplicatively consistent InNPLPRs. Finally, an example about the evaluation of green design schemes for new energy vehicles is provided to indicate the application of the new algorithm and comparative analysis is conducted.

Considering the advantages of dual hesitant fuzzy elements (DHFEs) for describing the hesitant and intuitionistic judgments of experts and identifying the limitations of previous research about dual hesitant fuzzy decision making, this paper studies decision making with dual hesitant fuzzy preference relations (DHFPRs) and provides a new group decision making (GDM) method based on a series of built optimization models. A multiplicative consistency concept for DHFPRs is first introduced and analyzed. On the basis of this concept, optimization describing the multiplicative consistency of DHFPRs is realized. Meanwhile, stepwise optimization models for obtaining complete multiplicatively consistent DHFPRs are constructed, and a method for calculating the intuitionistic fuzzy priority vector is offered. Then, optimization models for determining unknown values in incomplete DHFPRs are formed in view of the multiplicative consistency. An index for measuring the consensus is defined and optimization models for reaching the consensus requirement of a group are established. Furthermore, consensus-based optimization models for determining the weights of experts are established. In view of the multiplicative consistency and consensus analysis, a detailed procedure for GDM with incomplete and inconsistent DHFPRs is provided. Finally, the new method is illustrated by a case study concerning an evaluation of the environmental performance of enterprises and associated comparative analyses are reported.

The concept of sponge city receives more and more attention by Chinese government, and the evaluation of sponge city construction is an important aspect. To cope with the complexity and uncertainty of the evaluation process, this paper adopts interval type-2 trapezoidal fuzzy numbers (IT2TFNs) to express decision-making information and develops an approach for evaluating sponge city construction. To do these, two prioritized-guided interval type-2 trapezoidal fuzzy Hamacher operators are first defined to infuse IT2TFNs offered by experts, which can cope with the situation where there is prioritization among experts/attributes. In order to further consider the interactions among experts/attributes, two generalized-Shapley interval type-2 trapezoidal fuzzy prioritized Hamacher Choquet integral operators are presented. To measure the discrimination degree between IT2TFNs, a new interval type-2 trapezoidal fuzzy cross-entropy is defined. After that, cross-entropy based models for obtaining the optimal fuzzy measure on the expert/attribute set are constructed to handle the situation where the weighting information is interactive and partly known. Furthermore, an interval type-2 trapezoidal fuzzy multi-attribute group decision-making approach is developed. Finally, a practical example about the evaluation of residential land design plans in sponge city is provided to illustrate the utilization of the new method, and comparison analysis is provided.

This paper investigates decision making with normalized probabilistic linguistic preference relations (NPLPRs). Consistency analysis is indispensable for deriving the reasonable ranking. After recalling previous research, we find that all previous concepts cannot fully define consistent NPLPRs. As a fundamental topic of decision making with preference relations, it is necessary to further study the consistency of NPLPRs. For this purpose, an interactive algorithm for deriving disjunctive probabilistic additive linguistic preference relations (DPALPRs) is provided, by which an additive consistency concept for NPLPRs is defined. When NPLPRs are unacceptably consistent, models for obtaining acceptably additively consistent NPLPRs are built. Considering the situation where only incomplete NPLPRs are obtained, a disjunctive probability and additive consistency based interactive algorithm for ascertaining missing judgments is provided. Meanwhile, we discuss group decision making (GDM) with NPLPRs and offer a distance measure based formula to determine the weights of the decision makers. In addition, the method defines a consensus index and builds models for improving the consensus level. Under the additive consistency and consensus discussions, an interactive algorithm for GDM with NPLPRs is proposed. Finally, the new method is applied to select green raw material suppliers to illustrate the application and compared with several previous ones.

Trapezoidal fuzzy numbers offer a suitable and flexible way to express vague judgments of decision makers. This paper studies decision making with additive trapezoidal fuzzy preference relations. It first analyzes the issues encountered in previous consistency concepts for additive trapezoidal fuzzy preference relations. Then, two new consistency concepts, namely, an additive consistency concept and a multiplicative consistency concept, are introduced. To verify the rationality, several of their properties are discussed. To efficiently assess the consistency of additive trapezoidal fuzzy preference relations, several corresponding optimization models are formed. When the objective function values of the built models are zero, the associated additive trapezoidal fuzzy preference relations are consistent. Considering the case where additive trapezoidal fuzzy preference relations are incomplete, optimization models based on the consistency analysis are built, by which identified unknown judgments have the highest consistency level with the known ones. In general, additive trapezoidal fuzzy preference relations are unacceptably consistent. To rank objects from additive trapezoidal fuzzy preference relations, several optimization models for improving the consistency level are built, which consider the self-confidence of the decision makers, the total adjustment and the number of adjusted elements. According to the above discussion, two frameworks for ranking objects from additive trapezoidal fuzzy preference relations are provided that are based on the additive and multiplicative consistency analysis, respectively. Finally, numerical examples are provided to highlight the concrete application and to deliver the comparative analysis.

To address the situation where the complete consistency is unnecessary, a stepwise optimization model-based method for testing the acceptably additive consistency (AAC) of hesitant fuzzy preference relations (HFPRs) is introduced. Then, an AAC concept for HFPRs is defined. Meanwhile, incomplete HFPRs (iHFPRs) are discussed and a series of optimization models to acquire complete HFPRs is constructed. If the consistency is unacceptable, an optimization model for revising unacceptably consistent HFPRs under the conditions of the AAC and maximizing the ordinal consistency (OC) is offered. Subsequently, a model for minimizing the number of adjusted variables is presented. Considering the weighting information and the consensus for group decision making (GDM), the weights of fuzzy preference relations (FPRs) obtained from each individual HFPR and the decision makers (DMs) are determined using the distance measure. With regard to the consensus, two models for reaching the consensus requirement and minimizing the amount of revised variables are separately constructed, which are both based on the analysis of maximizing the OC. Furthermore, the thresholds of the additive consistency and the consensus are studied using the Monte Carlo simulation method. A GDM algorithm with HFPRs is offered. Finally, an example and comparison are provided to show the efficiency of the new procedure.

Dual hesitant fuzzy elements (DHFEs) are suitable to express hesitant possible preferred and nonpreferred judgments of decision makers. Preference relation is an important tool in decision making that only needs the decision makers to compare a pair of objects at one time. This study focuses on decision making with dual hesitant fuzzy preference relations (DHFPRs). Considering the consistency, an additive consistency concept is defined. Meanwhile, the property of the new concept is studied. Using this consistency concept, a method for assessing the additive consistency of DHFPRs is offered. To extend the application of DHFPRs, a programming model to determine the missing DHFEs in incomplete DHFPRs is built, which have the highest additive consistency level for the known ones. Two equivalent methods to calculate the priority vector are offered. One method obtains the probabilistic dual hesitant fuzzy priority vector, and the other derives the intuitionistic fuzzy priority vector. Furthermore, a consensus index is defined to measure the consensus of individual opinions in group decision making (GDM), and an interactive method for increasing the consensus level is offered. On the basis of the additive consistency and consensus, an algorithm to GDM with DHFPRs is offered that can address inconsistent and incomplete cases. Finally, a practical example about evaluating color TV is provided to demonstrate the usefulness of the new procedure.

This paper studies the reverse channel in dynamic closed-loop supply chain (CLSC) system which consists of manufacturers and retailers. Based on the dynamic CLSC model, we research the decisions and profits of CLSC members in different reverse channels that consider the quantitative characteristic of products. The results show that the optimal collection decisions for the given quantitative characteristic of products are diverse in different development levels of CLSC. Furthermore, we propose a transfer payment coordination mechanism based on Nash bargaining model to address the objective inconsistency between CLSC and its members. Moreover, we carry out the case study using the statistical data of Chinese new energy vehicles (NEVs). We show the robustness of results through the sensitivity analysis and offer some suggestions for the government and NEV managers in China.