Abstract

Virtual Enterprises（VE）can be seen as a kind of cooperation relationship pursuing economic profit based on the contract.The motivation for the partners to take part in VE is to obtain the maximum return under given conditions.The un⁃fair allocation scheme which is the usual reason for the conflict between the play⁃ers in VE is also one of the important reasons for making VE fall by the way⁃side.Compared with game theory，the cooperation relationship of VE is a typical cooperative game problem，and the profit allocation problem in VE is equal to the solution searching problem in the cooperative game.However，most of learn⁃ers use cooperative crisp game to undertake research on the profit allocation prob⁃lem of VE under certainty，while few learners consider the profit allocation prob⁃lem of VE under uncertainty.

As an excellent combination of cooperative crisp game theory and fuzzy sets，the cooperative fuzzy game provides an efficient method for profit allocation prob⁃lem in VE.In recent years，cooperative fuzzy game has been developed to be an important branch of game theory，and it has been focused on by more and more researchers.However，it is still a difficult issue to seek a reasonable and effective solution of cooperative fuzzy game.

In view of the observations above，we make a systematical and intensive re⁃search on cooperative fuzzy game through three routes，i.e.，cooperative game with fuzzy coalitions，cooperative game with a fuzzy characteristic function，co⁃operative game with fuzzy coalitions and a fuzzy characteristic function simultane⁃ously.The main work in this book can be concluded as follow.

（a）The cooperative game with fuzzy coalitions and its application in the profit allocation problem of VE.Firstly，Considering the fact that the cooperative crisp game can not solve the profit allocation problem on condition that the partic⁃ipation rates of the partners in VE are fuzzy information，we propose the general expression of cooperative game with fuzzy coalitions extended by cooperative crisp game.Secondly，we point out that the Shapley value axioms defined by Tsurumi can be adopted for any class of cooperative games with fuzzy coalitions.Two ex⁃plicit Shapley values，which are applicable to the fuzzy game defined by Butnariu and Tsurumi respectively，are used for the profit allocation problem of VE.Thirdly，the explicit Shapley value of Owen's multilinear extension is given，and the Shapley value is proved to satisfy the Shapley axioms defined by Tsurumi.Then，the fuzzy core of cooperative game with fuzzy coalitions is proposed，and the explicit cores of cooperative fuzzy game defined by Owen，Butnariu and Tsu⁃rumi andtherelationshipbetweenfuzzycoreandShapleyvalueare studied.Moreover，the allocation schemes in VE based on cooperative game with fuzzy coalitions are proposed.Finally，we take research on multiobjective linear production programming game with fuzzy coalition and its solution.

（b）The cooperative game with a fuzzy characteristic function and its appli⁃cation in the profit allocation problem of VE.For the reason that the cooperative crisp game can not solve the profit allocation problem on condition that the expec⁃ted profits of VE are fuzzy information，we introduce the fuzzy Shapley value giv⁃en by Mareš and the fuzzy Shapley value based on Hukuhara⁃difference.By a spe⁃cial kind of fuzzy Shapley，i.e.，interval Shapley value，we discuss the proper⁃ties of two fuzzy Shapley values as well as the relationship between them.Sec⁃ond⁃ly，the fuzzy cores proposed by Mareš，Nishizaki and Sakawa are discussed，and the deficiencies of the fuzzy cores and the relationship between them are pointed out.Finally，considering the deficiencies of two kinds of fuzzy cores，we define two fuzzy max order cores.Also，the properties of fuzzy max order cores，the relationship between the fuzzy max order cores and fuzzy Shapley value，and the necessary and sufficient conditions that triangular fuzzy number vector belongs to fuzzy max order cores are studied.Besides，the allocation schemes in VE based on fuzzy max order cores and fuzzy Shapley value are proposed.

（c）The cooperative game with fuzzy coalitions and a fuzzy characteristic function and its application in the profit allocation problem of VE.In view of that neither the cooperative game with fuzzy coalitions nor the cooperative game with a fuzzy characteristic function can solve the profit allocation problem on condition that the participation rate of partner and the expected profits of VE are both fuzzy information，we extend the cooperative crisp game to cooperative game with fuzzy coalitions and a fuzzy characteristic function by Borkotokey's method.We define the general fuzzy Shapley value in order to solve cooperative game with fuzzy coa⁃litions and a fuzzy characteristic function.The explicit form of general fuzzy Shap⁃ley value for the general cooperative game with Choquet form is presented，and it is shown that the general fuzzy Shapley value is uniquely defined for such games.Finally，based on the general fuzzy Shapley value，the profit allocation schemes in VE are proposed.