There are some mistakes in the computation results of the real example in the article by Li, “TOPSIS-based nonlinear-programming methodology for multi-attribute decision making with interval-valued intuitionistic fuzzy sets” [ IEEE Trans. Fuzzy Syst. , vol. 18, no. 2, pp. 299–311, 2010], and this article provides corrections to that paper.

The primary aim of this paper is to develop one kind of easy and effective method to solve fuzzy cooperative games with coalition values expressed by triangular fuzzy numbers (TFNs). This method ensures that each player should receive a TFN-typed fuzzy pay-off from the grand coalition because each coalition value is expressed by a TFN. Using the concept of Alpha-cut sets, an arbitrary TFN’s Alpha-cut set can be shown as an interval. If the 1-cut sets and 0-cut sets of the TFN-typed coalition values are known, we can easily gain some important values, such as the means, the lower limits, and the upper limits of the TFN-typed payoffs via the proposed quadratic programming models and method. Furthermore, it is also easy for us to compute the lower and upper limits of Alpha-cut sets at any confidence levels of the TFN-typed payoffs for any TFN-typed cooperative game through solving the constructed quadratic programming models. Hereby the players’ TFN-typed payoffs for the TFN-typed cooperative game can be explicitly solved via the representation theorem for fuzzy sets. It is easy to prove that the proposed solutions of the fuzzy cooperative games with coalition values expressed by TFNs satisfy some useful and important properties, such as symmetry, additivity, and anonymity. Finally, the validity, applicability and advantages of the proposed method is proved and discussed through a numerical example.

There exist some fuzziness and uncertainty in the process of ecological construction. The aim of this paper is to develop a direct and an effective simplified method for obtaining the cost-sharing scheme when some interested parties form a cooperative coalition to improve the ecological environment of Min River together. Firstly, we propose the solution concept of the least square prenucleolus of cooperative games with coalition values expressed by trapezoidal intuitionistic fuzzy numbers. Then, based on the square of the distance in the numerical value between two trapezoidal intuitionistic fuzzy numbers, we establish a corresponding quadratic programming model to obtain the least square prenucleolus, which can effectively avoid the information distortion and uncertainty enlargement brought about by the subtraction of trapezoidal intuitionistic fuzzy numbers. Finally, we give a numerical example about the cost-sharing of ecological construction in Fujian Province in China to show the validity, applicability, and advantages of the proposed model and method.

Intense market competition has driven small- and medium-sized enterprises (SMEs) in the manufacturing sector to collaborate and form supply chain coalitions, which can improve the information flow and resource sharing and significantly enhance supply chain management efficiency. However, the distribution of cooperative benefits poses a core challenge for the long-term stability of coalitions. This paper addresses the impact of dynamic changes in complex business environments by utilizing triangular fuzzy numbers to represent the value of coalition, effectively depicting the uncertainty and ambiguity in the cooperation process. Compared to traditional models (which do not use triangular fuzzy numbers), this model is better suited to dynamic changes, offering flexible response mechanisms that ensure adaptability and fairness in the decision-making process. In addition, considering the influence of each member’s weight in the coalition, the fuzzy comprehensive evaluation method is used to determine the weights. With the goal of minimizing the dissatisfaction of enterprises in benefit distribution, a least square contribution with triangular fuzzy numbers is constructed to replace the marginal contribution of the classical Banzhaf value, and an improved Banzhaf value based on the player’s triangular fuzzy number-weighted excess contribution is proposed to arrive at a fair and reasonable benefit allocation strategy in order to enhance the long-term stability and cooperative benefits of coalition. By analyzing an example of the supply chain coalition, the effectiveness of the proposed improved Banzhaf value is verified, which satisfies the uniqueness, the individual rationality, and the group rationality. It not only promotes the level of risk management and decision making under the uncertainty conditions of complex business, but also deepens the theoretical foundation of cooperative game theory and expands its possibilities in practical applications and future development.