针对合作对策中联盟值（或特征函数）常表示为区间值而非实数的现象，提出两种新的二次规划求解方法，该方法能快速、有效的获得n人区间值合作对策的区间值最小二乘预核仁解和核仁解。首先，利用反映联盟不满意度的平方超量e（S，x）=（υL（S）-xL（S））2+（υR（S）-xR（S））2，构建求解区间值最小二乘预核仁解的二次规划模型，据此确定每个局中人的区间值分配xiE*=[xLiE*，xRiE*]（i∈N）。其中，xLiE*=（υL（N））/n+1/（n2n-2）（naLi（υ）-???20171209???aLj（υ）），xRiE*=（υR（N））/n+1/（n2n-2）（naRi（υ）-???20171209???aRj（υ））（i∈N）。接着，考虑个体合理性，拓展所导出的数学优化模型，获得区间值最小二乘核仁解。然后，讨论最小二乘预核仁解和核仁解的一些重要性质，如，存在性和唯一性、有效性、可加性、对称性、匿名性，等。最后，利用供应链合作利益分配的数学算例验证所提出的二次规划模型和方法的合理性、有效性和优越性。
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)=(υL(S)-xL(S))2+(υR(S)-xR(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 xiE*=[xLiE*,xRiE*](i∈N), where xLiE*=(υL(N))/n+1/(n2n-2)(naLi(υ)-???20171209???aLj(υ)),xRiE*=(υR(N))/n+1/(n2n-2)(naRi(υ)-???20171209???aRj(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.
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