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With the changing business environment and the active participation of various stakeholders in the decision making process, it plays an increasingly important role to the weight of decision makers and the preference information given by decision makers. This paper presents a novel approach for group decision making under uncertainty with the involvement of the third-party evaluator in the decision making process. Recognizing the challenge in adequately determining the weight of decision makers in group decision making, the evidence theory is appropriately used with the involvement of the third-party evaluator. To effectively model the uncertainty and imprecision in the decision making process, fuzzy preference relations are used for better representing the subjective assessment of individual decision makers. To adequately determine the ranking of available alternatives, the logarithmic least square method is applied for appropriately aggregating the fuzzy preference relation of individual decision makers. A group consensus index is developed for facilitating consensus building in group decision making. This leads to better group decisions being made. A real-world application is presented that shows the proposed approach is effective in solving group decision making problems under uncertainty.

The challenge of complex multi-attribute large group decision-making (CMALGDM) is reflected from three perspectives: interrelated attributes, large group decision makers (DMs) and dynamic decision environments. However, there are few decision techniques that can address the three perspectives simultaneously. This paper proposes a random intuitionistic fuzzy factor analysis model, aiming to address the challenge of CMALGDM from the three perspectives. The proposed method effectively reduces the dimensionality of the original data and takes into account the underlying random environmental factors which may affect the performances of alternatives. The development of this method follows three steps. First, the random intuitionistic fuzzy variables are developed to deal with a hybrid uncertain situation where fuzziness and randomness co-exist. Second, a novel factor analysis model for random intuitionistic fuzzy variables is proposed. This model uses specific mappings or functions to define the way in which evaluations are affected by the dynamic environment vector through data learning or prior distributions. Third, multiple correlated attribute variables and DM variables are transformed into fewer independent factors by a two-step procedure using the proposed model. In addition, the objective classifications and weights for attributes and DMs are obtained from the results of orthogonal rotated factor loading. An illustrative case and detailed comparisons of decision results in different environmental conditions are demonstrated to test the feasibility and validity of the proposed method.

In real bargaining problems, players care not only about their own shares, but also about others’ shares. In addition, a player’ attitude toward others depends on how this player feels he is being treated. To model such preferences, the Rubinstein bargaining game is reconsidered, where players’ preferences is characterized as altruism and spite. First, a subgame perfect equilibrium (SPE) is constructed, where player’s strategy depends on the opponent’s share through altruistic and spiteful preferences. The uniqueness of SPE is shown if it shares with SPE in the classical Rubinstein case: no delay and stationarity. Then, a comparative statics analysis with respect to players’ altruism and spite is performed. It is shown that the equilibrium share of a player is negatively related to the opponent’s global spite and his own global altruism, and positively to global altruism of the other one and the global spite of himself. It is also found that the impact of the intrinsically altruistic and spiteful levels of a player on equilibrium share depends on this player’s attitudes towards the opponent. Furthermore, it is found that a more positive attitude towards the opponent leads to the increase (decrease) of this player’s share if this player is more (less) intrinsically altruistic than the opponent. Finally, we establish a relationship with asymmetric two-person Nash bargaining game. It is found that bargaining power of a player decreases with the globally altruistic and spiteful preferences of himself, and increases with the opponent’s. It is further found that the effects of players’ attitudes towards the opponent on their own bargaining power depend on the gap between the intrinsic altruistic and spiteful levels of players.

Bimatrix games are reconsidered under assumption that players are loss averse and payoffs are fuzzy, where reference points are given exogenously. The impact of loss aversion on bimatrix game is investigated by applying credibility theory. Three solution concepts of credibilistic loss aversion Nash equilibria (i.e., expected loss aversion Nash equilibrium, optimistic loss aversion Nash equilibrium and pessimistic loss aversion Nash equilibrium) and their existence theorems are proposed. The sufficient and necessary conditions are presented to find three equilibria. It is found that three credibilistic loss aversion Nash equilibria are equivalent if confidence level is equal to 0.5. Furthermore, an analysis on credibilistic loss aversion equilibria with respect to loss aversion is performed in a 2 × 2 bimatrix game with triangular fuzzy payoffs. It is found that with a higher probability a more loss-averse player receives a preferred payoff in the mixed strategy Nash equilibrium in some situations, but receives the second highest payoff in other situations. Finally, a case study is shown to illustrate the validity of the bimatrix game with triangular fuzzy payoffs developed in this paper.

The palm oil industry faces significant pressure because of the environmental problems from the plantation process and the palm oil milling process in their supply chain, which is due to the constraints of natural resources and the growing ecological awareness among customers. Reverse logistics (RL) has been considered as a systematic approach for the palm oil industry to improve the environmental impact. However, there are more barriers which cannot be detected in the RL. Therefore, this study proposes robust and flexible strategies for overcoming such barriers by focusing on identifying and ranking the solutions on RL adoption in Thailand’s palm oil industry. We use a fuzzy multi-criteria decision-making (MCDM) methods based on fuzzy analytical network process (ANP) and Vise Kriterijumska Optimizacija I Kompromisno Resenje (VIKOR) to prioritize related solutions and strategies. Firstly, fuzzy ANP is applied to identify the weighting of each barrier using pairwise comparison, while the final ranking of the solution on RL adoption is obtained through VIKOR. This model will help business analysis and supply chain managers formulate both short-term and long-term, flexible decision strategies for successfully managing and implementing RL adoption in the supply chain scenarios. In addition, sensitivity analysis is used to illustrate the method, and comparisons with other existing methods is also presented in this study.

By using Moore&rsquo, s subtraction operator and a total order on the set of closed intervals, we introduce a new variation of the Banzhaf value for cooperative interval games called the interval Banzhaf-like value which may accommodate the shortcomings of the interval Banzhaf value. We first reveal the relation between this introduced value and the interval Banzhaf value. Then, we present two sets of properties that may be used to determine whether an interval value is median-indifferent to the interval Banzhaf-like value. Finally, in order to overcome the disadvantages of the interval Banzhaf-like value, we propose the contracted interval Banzhaf-like value and give an axiomatization of this proposed value.

In real multi-attribute decision-making problems, attributes are usually correlated and performances of alternatives are affected by random environmental factors. This paper proposes an intuitionis...

As an extension of the classical averaging operators, Choquet integral has been shown a powerful tool for decision theory. In this paper, a method based on the generalized interval-valued intuitionistic fuzzy Choquet integrals w.r.t. the generalized interaction indices is proposed for multiattribute group decision making problems, where the importance of the elements is considered, and their interactions are reflected. Based on the given operational laws on interval-valued intuitionistic fuzzy sets, the interval-valued intuitionistic fuzzy Choquet integrals with respect to the generalized Shapley and Banzhaf indices are defined. Moreover, some of their properties are studied, such as idempotency, boundary, comonotonic linearity and μ–linearity. Furthermore, a decision procedure based on the proposed operators is developed for solving multi-attribute group decision making under interval-valued intuitionistic fuzzy environment. Finally, a numerical example is provided to illustrate the developed procedure.

Triangular fuzzy reciprocal preference relations (TFRPRs) are good choices to denote the decision makers' fuzzy preferences. Consistency analysis is a very important topic to preference relations that can avoid the illogical ranking order. However, all previous multiplicative consistency concepts for TFRPRs cannot well address this type of preference relations. To address this issue, this paper introduces a new multiplicative consistency concept for TFRPRs, which can be seen as a natural extension of Tanino's multiplicative consistency concept for reciprocal preference relations. Based on the new consistency concept, 0–1 mixed programming models for judging the multiplicative consistency of TFRPRs are constructed, and an approach for deriving multiplicative consistent TFRPRs is provided. Considering the incomplete case, programming models to determine missing values are constructed. Subsequently, a multiplicative consistency based method for decision making with TFRPRs is presented that can address the inconsistent and incomplete cases. Meanwhile, illustrative examples are provided to show the concrete application of the proposed procedure, and comparison analysis is also made.

To examine the multiplicative consistency of interval fuzzy preference relations (IFPRs), this paper first analyzes the limitations associated with the previous consistency concepts. Accordingly, a new consistency concept is defined that is an extension of the crisp case and overcomes limitations in the previous concepts. Next, a linear programming model to judge the consistency of IFPRs is constructed, and an approach to derive multiplicative consistent IFPRs is introduced. Furthermore, goal-programming models to determine missing values in an incomplete IFPR are constructed that have the highest consistent level with respect to known values. Moreover, a multiplicative consistency and consensus based method for group decision making with IFPRs is developed that can address incomplete and inconsistent cases. Finally, two practical decision-making problems are offered to demonstrate the feasibility and efficiency of the new method, and an analysis of a numerical and theoretical comparison with several related methods is performed.

Display Omitted A new consistency concept for IMPRs is defined.It is a natural extension of Saatys consistency concept for crisp case.It is robust to the permutation of the object labels.Models to address inconsistent and incomplete IMPRs are built.An algorithm to IMPRs is presented. Consistency analysis is very important to ensure the reasonable ranking order. However, all previous consistency concepts for interval multiplicative preference relations (IMPRs) are insufficient to address this type of preference relations. This paper introduces a new consistency concept for IMPRs that is a natural extension of the Saatys consistency concept for multiplicative preference relations. Several desirable properties are discussed, and the relationship between the new concept and two previous ones is studied. Then, a method to judge the consistency of IMPRs is proposed. Considering the inconsistent case, a 01 mixed programming model to derive consistent IMPRs from inconsistent ones is established. To determine missing values in incomplete HFPRs, 01 mixed programming models are constructed that can address the situation where ignore objects exist. Meanwhile, illustrative examples are offered to show the feasibility and efficiency of the developed theoretical results, and comparison analysis is provided. Finally, a consistency analysis based algorithm to decision making with IMPRs is developed that can address inconsistent and incomplete cases.

The increase of environmental pollution has led to the rise of sustainable awareness in recent years. This trend has motivated various industries to recognize the importance of implementing sustainable supply chain practices to seek economic, environmental and social advantages. From a sustainability perspective, selecting a suitable supplier is the main component of modern enterprises. It is also a challenging problem since several criteria concerning supplier selection are interdependent with a complex character. Therefore, the contribution of this paper is a new extension to multi-criteria decision-making model (MCDM) under an intuitionistic fuzzy environment for sustainable supplier selection (SSS) based on sustainable supply chain management SSCM practices. It consists of intuitionistic fuzzy set theory (IFS) with a decision making trial and evaluation laboratory (DEMATEL) combined with an analytic network process (ANP) to identify uncertainties and interdependencies among criteria as well as analyzing the criteria weights. We modified Vise Kriterijumska Optimizacija I Kompromisno Resenje (VIKOR) to evaluate and rank the desired level of sustainable supplier performance. The suggested approach is conducted by a case study from the Thailand palm oil industry. Results show that the proposed model not only can find the most suitable sustainable supplier, but also the enterprises can aid their suppliers in improving sustainability by using the proposed method and can improve enterprises&rsquo, socio-environmental performance, which is key to achieving sustainable development.

The consumer environmental awareness promotes green manufacturing and the behavioral preferences of members become prevailing in supply chain management. To promote further development of green supply chains, a two-echelon green supply chain with a manufacturer and a retailer is considered, where the manufacturer is loss-averse and the retailer is risk-neutral. We use a Stackelberg game to investigate the impacts of loss aversion and green efficiency coefficient on retail price, wholesale price, green degree, profits of members, and profit of the green supply chain under the assumption that manufacturer&rsquo, s reference point of loss aversion is equal to the subgame perfect equilibrium partition. It is shown that, in the centralized decision-making setting (CDS), green degree and profit of the green supply chain are higher than those in the decentralized decision-making setting (DDS), while in the decentralized decision-making setting with a loss-averse manufacturer (DDS-LAM) loss aversion of manufacturer further decreases green degree and profit of green supply chain. It is also found that profits of the manufacturer and the retailer decrease with levels of loss aversion of manufacturer. Furthermore, it is also shown that wholesale price and retail price in the three decision-making settings depend on the green efficiency coefficient. Wholesale price and retail price in DDS-LAM are always the lowest (highest) if the green efficiency coefficient is sufficiently high (low). Finally, executing a greening cost-sharing contract can improve chain members&rsquo, profits if the retailer shares an appropriate proportion with the loss-averse manufacturer.

The classical Bertrand game is assumed that players are perfectly rational. However, many empirical researches indicate that people have bounded rational behavior with fairness concern, which is important in the two-person game and has attracted much attention. In this paper, fairness concern is incorporated into the Bertrand game with two homogeneous products and the effect of fairness concern on this extended Bertrand game is explored. Nash bargaining solution of player is applied to be his own fairness reference point. Then, a Bertrand game model with fairness concern is established, and its equilibrium price is also derived and analyzed. It is shown from some numerical examples that fairness concern and bargaining power of players have a significant influence on their equilibrium price, expected profits, and utilities. As a player gets more fair-minded, if the other player has a less focus on fairness, the price competition between them will be intensified and both of them suffer loss. Thus, fairness concern may be advantageous or disadvantageous for players. In most situations, the fairness concern behavior is not beneficial for players. Additionally, the effect of bargaining power is relative to fairness concern. A player who manufactures a low-cost product should have a weak bargaining power if he terribly focuses on fairness and should have a strong bargaining power if he pays little attention to fairness. However, a player who manufactures a high-cost product should have a weak bargaining power if he is rarely concerned about fairness. Anyway, the same bargaining power is the best for two players.

Rubinstein bargaining game is extended to incorporate loss aversion, where the initial reference points are not zero. Under the assumption that the highest rejected proposal of the opponent last periods is regarded as the associated reference point, we investigate the effect of loss aversion and initial reference points on subgame perfect equilibrium. Firstly, a subgame perfect equilibrium is constructed. And its uniqueness is shown. Furthermore, we analyze this equilibrium with respect to initial reference points, loss aversion coefficients, and discount factor. It is shown that one benefits from his opponent’s loss aversion coefficient and his own initial reference point and is hurt by loss aversion coefficient of himself and the opponent’s initial reference point. Moreover, it is found that, for a player who has a higher level of loss aversion than the other, although this player has a higher initial reference point than the opponent, this player can(not) obtain a high share of the pie if the level of loss aversion of this player is sufficiently low (high). Finally, a relation with asymmetric Nash bargaining is established, where player’s bargaining power is negatively related to his own loss aversion and the initial reference point of the other and positively related to loss aversion of the opponent and his own initial reference point.

Aggregation operators play a key role in multiple criteria decision-making (MCDM). Extensions of aggregation operators to intuitionistic fuzzy sets (IFSs) usually involve replacing the standard arithmetic operations with those defined over the membership and nonmembership of IFS, which is essentially a pair of special Archimedean triangular norm (t-norm) and triangular conorm (t-conorm), called probabilistic sum t-conorm and product t-norm, on the membership and nonmembership of IFS, respectively. In this paper, we first introduce some operations on IFSs by means of Archimedean t-norm and t-conorm. Then some generalized Archimedean intuitionistic fuzzy aggregation operators are proposed, such as generalized Archimedean intuitionistic fuzzy weighted averaging operator, generalized Archimedean intuitionistic fuzzy ordered weighted averaging (GAIFOWA) operator, and generalized Archimedean intuitionistic fuzzy hybird averaging operator. Some desirable properties of these operators are investigated. The relations between these operators and the existing intuitionistic fuzzy aggregation operators are discussed. Finally, applying these proposed operators, we develop an approach for multi-criteria decision-making with intuitionistic fuzzy information, an illustrative example is used to verify the developed approach and to demonstrate its practicality and effectiveness.

In most existing literatures on Cournot game, game behaviour between players is based on the hypothesis that people are complete rationality. However, players’ decisions are often affected by their behavioural characteristic and psychology preference. Traditional Cournot model also doesn’t deal with ambiguous information. Based on rank-dependent utility theory, this paper develops an incomplete information Cournot game in an ambiguous decision environment, where the form of ambiguity is described by a set of fuzzy parameters, and behavioural pattern is reflected by means of emotional function in rank-dependent utility. Further, we investigate the Nash equilibrium quantity of each manufacture in this kind of fuzzy Cournot game with incomplete information. Finally, the proposed Cournot model is applied to a case study, and dynamic variation and sensitivity analysis of optimal quantity with respect to decision-maker’s behaviour pattern are discussed in detailed, which illustrates that the proposed Cournot mo...

Inspired by Shalev’s model of loss aversion, we investigate the effect of loss aversion on a bimatrix game where the payoffs in the bimatrix game are characterized by triangular fuzzy variables. First, we define three solution concepts of credibilistic loss aversion Nash equilibria, and their existence theorems are presented. Then, three sufficient and necessary conditions are given to find the credibilistic loss aversion Nash equilibria. Moreover, the relationship among the three credibilistic loss aversion Nash equilibria is discussed in detail. Finally, for2×2bimatix game with triangular fuzzy payoffs, we investigate the effect of loss aversion coefficients and confidence levels on the three credibilistic loss aversion Nash equilibria. It is found that an increase of loss aversion levels of a player leads to a decrease of his/her own payoff. We also find that the equilibrium utilities of players are decreasing (increasing) as their own confidence levels when players employ the optimistic (pessimistic) value criterion.

In this paper, we extend the generalized weighted geometric and generalized ordered weighted geometric operators to intuitionistic fuzzy environments, that is, we develop a series of generalized intuitionistic fuzzy geometric operators to aggregate input arguments that are expressed by intuitionistic fuzzy values based on Archimedean t-conorm and t-norm. Then some desired properties of these aggregation operators are investigated. The relations between these operators and some existing intuitionistic fuzzy geometric aggregation operators are discussed in detail. Furthermore, applying these proposed operators, we develop an approach for multi-criteria decision making with intuitionistic fuzzy information. Finally, a practical example is given to verify the developed approach and to demonstrate its practicality and effectiveness.

In this paper, a generalized form of fuzzy games with interval characteristic functions is proposed, which can be seen as an extension of games with crisp characteristic functions. Based on the extended Hukuhara difference, the interval Shapley function for interval fuzzy games is studied. Then, the concept of interval population monotonic allocation function (IPMAF) is defined. When interval fuzzy games are convex, we prove that the interval Shapley function is an IPMAF. Furthermore, two special types of interval fuzzy games are researched, and the associated interval Shapley function is studied.

Environmental concerns have globally driven the encouragement of green supply chain management. Accordingly, business and industrial organizations try to seek green supply chain strategies to respond to market pressure regarding corporate social responsibility. Green supplier selection is one of the practical strategies for modern enterprises. With the large-scale development of the palm oil products industry, green supplier selection technique is the key for decision making when dealing with mass information and possible risks of biased data. For instance, the preference of decision makers possibly causes a misleading decision, thus leading to unnecessary waste of resources. Therefore, the contribution of this paper is to apply the integrated multi-criteria decision method using the &lsquo, fuzzy decision-making trial and evaluation laboratory&rsquo, (fuzzy DEMATEL) method to consider the cause and effect relationship and then using fuzzy analytic network process (fuzzy ANP) to assign the weight of each relevant criteria. The initial results are useful for strategic procurement planning. In the final step, we adopt the prospect theory to synthesize procurement&rsquo, s psychological and behavioral factors when selecting green suppliers. The final result refers to the comprehensive prospect value to rank the eligible suppliers into orders. Moreover, the results of both sensitivity analysis and comparison method confirm that the proposed model is adequately realistic and robust.

In this paper, a novel discrete differential evolution (DDE) algorithm is proposed to solve the vehicle routing problems (VRP) in B2C e-commerce, in which VRP is modeled by the incomplete graph based on the actual urban road system. First, a variant of classical VRP is described and a mathematical programming model for the variant is given. Second, the DDE is presented, where individuals are represented as the sequential encoding scheme, and a novel reparation operator is employed to repair the infeasible solutions. Furthermore, a FLOYD operator for dealing with the shortest route is embedded in the proposed DDE. Finally, an extensive computational study is carried out in comparison with the predatory search algorithm and genetic algorithm, and the results show that the proposed DDE is an effective algorithm for VRP in B2C e-commerce.