| Virtual enterprises (VE) can be viewed as a temporary alliance of member companies. It is rapidly created for the purposes of pursuing specific market opportunities and disbanded when the market opportunity has passed. It is a kind of cooperation relation for pursuing the further profit based on the contract in nature. The essential reason of the formation of the VE is member companies all expect to seek more profit than they do singly.The profit allocation is the problem of easily engendering the most prominent conflict in the cooperation of VE. The profit allocation strategy among partners in VE is a rational selective result in the process of cooperation. This problem can be viewed as a cooperative game. Lots of methods in classical cooperative games theory are used to allocate profit among partners. However, the classical cooperative games are based on two assumptions: (1) The players fully participate in a specific coalition. It means that the rate of participation of each player in a coalition is either 1 or 0; (2) The players, at the very beginning of the games, know exactly the expected profit. It means that they know the total payoffs of coalitions that will be distributed in the fist case, and the vectors of individual profit of the cooperation in the second case. It can be observed that, in most real-world situations, the players may—more or less—participate in several coalitions, and the expected profit of the different cooperation strategies are often only imprecisely or ambiguously known to the players.With the observation mentioned above in mind, the paper introduced fuzzy cooperatibe games theory to solve the problem of profit allocation among partners in VE. We used two routes—the cooperative games with fuzzy coalitions and the cooperatve games with fuzzy payoffs—to research the profit allocation schemes among partners in VE.Firstly, we studied the profit allocation problem in the situation that the participation rates of the partners in VE are fuzzy. This kind of problem can be viewed as the cooperative games with fuuzy coalitions in deed. In the kind of cooperative games, a membership degree represented by a fuzzy number shows to what extent a player transfers his representability and is called a rate of participation, and the collection of players who transfer fractions of their representability to each coalition is called a fuzzy coalition. Based on the former research results about the cooperative games with real number valued fuzzy coalitions, we extended the cooperative games with real number valued fuzzy coalitions to the cooperative games with interval number valued fuzzy coaltions by using the fuzzy sets theory, fuzzy measures and Choquet integrals. Further, we extended the cooperative games with interval number valued fuzzy coalitions to the cooperative games with general fuzzy number valued fuzzy coaltions. Thus, we introduced the profit allocation strategies among partners in VE based on the cooperative games with real number valued, interval number valued and general fuzzy number valued fuzzy coalitions.In the studying of the cooperative games with real number valued fuzzy coalitions, the players'participation rates in the coalitions were denoted by the membership degrees represented by a real number in the interval of [0, 1]. The outcomes of Butnariu and Tisurumi were introduced in the part. The shotages of the method brought by Butnariu were discussed. On the base of these, we put forword the prfit allocation stategy among partners in VE based on the cooperative games with real number valued fuzzy coaltions.In the studying of the cooperative games with interval number valued fuzzy coalitions, the players'participation rates in the coalitions were denoted by the interval numbers in the interval of [0, 1]. Holding the character that the payoff in classical cooperative games is a fuzzy measure, using the Choquet integrals with interval number valued integrand, we defined the payoff functions and Shapley functions on the cooperative games with interval number valued fuzzy coalitions. These functions have the Choquet integral expressions. Some properties of the payoffs and Shapley values defined by us were discussed as well. The profit allocation strategy among partners in VE based on the cooperative games with interval number valued fuzzy coalitions was introduced as well.In the studying of the cooperative games with general fuzzy number valued fuzzy coalitions, the players'participation rates in the coalitions were denoted by the membership functions represented by a general fuzzy numbers in the interval of [0, 1]. Based on the outcome of the cooperative games with interval number valued fuzzy coalitions studied above, we extended the cooperative games with interval number valued fuzzy coaltions to the cooperative games with general fuzzy number valued fuzzy coalitions by using the extension principle. According to the Choquet integral with general fuzzy number integrand, we defined the payff functions and Shapley functions on the cooperative games with general fuzzy number valued fuzzy coalitions. These functions have the Choquet integral expressions. On the base of these, we discussed some properties of the payoffs and Shapley values defined by us, and also put forword the profit allocation stategy among partners in VE based on the cooperative games with general fuzzy number valued fuzzy coaltions.Secondly, we studied the profit allocation problem in the situation that the expected profits of VE are fuzzy. This kind of problem can be viewed as the cooperative games with fuzzy payoffs in deed. In the kind of cooperative games, the payoffs are represented by a fuzzy number. Using the fuzzy sets theory such as extension principle, decomposition theorem and representation theorem, we firstly denfined the cooperative games with interval number valued fuzzy payoffs and their Shapley values. Secondly, we discussed the cooperative games with general fuzzy number valued fuzzy payoffs and their Shapley values defined by Mares, and pointed that the kind of cooperative games with fuzzy payoffs defined by Mares is an extension of the cooperative games with interval number valued fuzzy payoffs. Considering the two kinds of cooperative games with fuzzy payoffs mentioned above can not satisfy the validity axiom for Shapley values, we defined the cooperative games with triangular fuzzy number valued fuzzy payoffs and their Shapley values. The Shapley values meet the validity axiom, and is an extension of the cooperative games with interval number valued fuzzy payoffs. Thus, we introduced the profit allocation strategies among partners in VE baed on the cooperative games with interval number valued, general fuzzy number valued and triangular fuzzy number valued fuzzy coalitions.In the studying of the cooperative games with interval number valued fuzzy payoffs, the payoffs in the coalitions were denoted by the interval numbers. Using the calculating principle, we defined the payoff functions and Shapley functions on the cooperative games with interval number valued fuuzy payoffs. Because the Shapley functions defined by us in the part have not the validity similar with the classical Shapley values, we introduced a concept of the relative validity. The Shapley functions defined by us satisfied the relative validity. Then, the profit allocation strategy among partners in VE based on the cooperative games with interval number valued fuzzy payoffs was introduced.In the studying of the cooperative games with general fuzzy number valued fuuzy payoffs, the payoffs in the coalitions were denoted by the general fuzzy numbers. This kind of cooperative games with fuzzy payoffs and their Shapley values was discussed by Mares. We introduced the Mares dedinition method and pointed that it does not satisfy the validity similar with the classical Shapley values but meet the relative validity. Then, the profit allocation strategy among partners in VE based on the cooperative games with general fuzzy number valued fuzzy payoffs was introduced.In the studying of the cooperative games with triangular fuzzy number valued fuzzy payoffs, the payoffs in the coalitions were denoted by the triangular fuzzy numbers. For the two coopearative games with fuzzy payoffs mentioned above, their Shapley values all do not satisfy the validity axioms. In order to avoid the shortage, we selected a sepecial fuzzy number—triangular fuzzy number—to define the cooperative games with fuzzy payoffs. We also proved the Shapley values defined on the kind of cooperative games meet the validity axioms. On the base of thses, we put forword the profit allocation stategy among partners in VE based on the cooperative games with general fuzzy number valued fuzzy coaltions.At last, on the base of studying the profit allocation problem, we discussed the method and process of the partner management in VE simply. The method for partner selection in VE under the resource restriction was introduced. We firstly discovered Bernardo group decision making method is the extension of the linear distribution method, analyzed the disfigurements of Bernardo algorithm and then ameliorated it. Secondly, we set up an ameliorated Bernardo model for partner selection in virtual enterprises under the multiple resource restrictions. This model not only gets the best partner amount, but also gains the best combination scheme for partner under the multiple resource restrictions.
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