Abstract: Due to the character that values (or characteristic functions) of coalitions of players are usually expressed with intervals rather than real numbers in real situations, two quadratic programming methods are proposed, which can quickly and effectively compute n-person interval-valued least square prenucleolus and nucleolus of interval-valued cooperative games. In this methodology, based on the square excess e(S,x)=(υ_L(S)-x_L(S))²+(υ_R(S)-x_R(S))² which can be interpreted as a measure of the dissatisfaction of the coalitions, the quadratic programming model is firstly constructed for interval-valued least square prenucleolus of interval-valued cooperative games and obtain players' interval-valued imputations x_i^E*=[x_Li^E*,x_Ri^E*](i∈N), where x_Li^E*=(υ_L(N))/n+1/(n2^n-2)(na_Li(υ)-???20171209???a_Lj(υ)),x_Ri^E*=(υ_R(N))/n+1/(n2^n-2)(na_Ri(υ)-???20171209???a_Rj(υ))(i∈N). Then, taking into account the individual rationality, the aforementioned optimization mathematical model is extended and all players' interval-valued least square nucleolus is solued. Moreover, some useful properties of the interval-valued least square prenucleolus and nucleolus are discussed,such as existence and uniqueness, efficiency, additivity, symmetry, and anonymity. Finally, the quadratic programming models and methods are illustrated with a numerical example about the optimal profit allocation of supply chain and the computational result is analyzed to show the validity, applicability, and advantages.

Key words: fuzzy game, cooperative game, interval, square Excess, nucleolus