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.
LIU Jia-cai, LI Deng-feng, HU Xun-feng
. Interval-Valued Least Square Nucleolus and Its Application in Cooperative Profit Allocation of Supply Chain[J]. Chinese Journal of Management Science, 2017
, 25(12)
: 78
-87
.
DOI: 10.16381/j.cnki.issn1003-207x.2017.12.009
[1] 李登峰.模糊多目标多人决策与对策[M].北京:国防工业出版社,2003.
[2] Branzei R, Dimitrov D, Tijs S. Models in cooperative game theory:Crisp, fuzzy and multichoice games[M]. Berlin,Heidelberg:Springer,Verlag,2005.
[3] Butnariu D. Stability and shapley value for an n-person fuzzy game[J]. Fuzzy Sets and Systems, 1980, 4(1):63-72.
[4] Mares M. Fuzzy cooperative games:Co-operation with vague expectations[M]. Heidelberg:Physica-Verlag, 2001.
[5] Li Dengfeng. Decision and game theory in management with intuitionistic fuzzy sets[M]. Heidelberg:Springer-Verlag, 2012.
[6] 李登峰.直觉模糊集决策与对策分析方法[M].北京:国防工业出版社,2012.
[7] Aubin J P. Coeur et valeur des jeux flous à paiements latéraux[J]. C. R. Acad. Sci. Paris Sér. A, 1974, 279(A):891-894.
[8] Hinojosa M A, Mármol A M, Fernández F R. Set-valued cooperative games with fuzzy payoffs. The fuzzy assignment game[J]. European Journal of Operational Research, 2013, 225(1):85-90.
[9] Borkotokey S. Cooperative games with fuzzy coalitions and fuzzy characteristic functions[J]. Fuzzy Sets and Systems, 2008, 159(2):138-151.
[10] Aubin J P. Cooperative fuzzy games[J]. Mathematics of Operations Research, 1981, 6(1):1-13.
[11] Aubin J P. Locally lipschitz cooperative games[J]. Journal of Mathematical Economics, 1981, 8(3):241-262.
[12] Branzei R, Dimitrov D, Tijs S. Convex fuzzy games and participation monotonic allocation schemes[J]. Fuzzy Sets and Systems, 2003, 139(2):267-281.
[13] Branzei R, Branzei O, Alparslan Gök S Z, et al. Cooperative interval games:A survey[J]. Central European Journal of Operations Research, 2010, 18(3):397-411.
[14] Branzei R, Alparslan Gök S Z, Branzei O. Cooperative games under interval uncertainty:On the convexity of the interval undominated cores[J]. Central European Journal of Operations Research, 2011, 19(4):523-532.
[15] Alparslan Gök S Z, Branzei O, Branzei R, et al. Set-valued solution concepts using intervaltype payoffs for interval games[J]. Journal of Mathematical Economics, 2011, 47(4-5), 621-626.
[16] Alparslan Gök S Z, Branzei R, Tijs S. The interval Shapley value:An axiomatization[J]. Central European Journal of Operations Research, 2010, 18(2), 131-140.
[17] Alparslan Gök S Z, Miquel S, Tijs S. Cooperation under interval uncertainty[J]. Mathematical Methods of Operational Research, 2009, 69(1), 99-109.
[18] Mallozzi L, Scalzo V, Tijs S. Fuzzy interval cooperative games[J]. Fuzzy sets and systems, 2011, 165(1), 98-105.
[19] Li Dengfeng. Models and methods for interval-valued cooperative games in economic management[M]. Cham, Switzerland:Springer, 2016.
[20] 于晓辉,张强.具有区间支付的合作对策的区间Shapley值[J].模糊系统与数学,2008,22(5):151-156.
[21] 高作峰,邹正兴,马栋,等.区间合作对策在增广系统上的区间Shapley值[J].模糊系统与数学,2013,27(4):148-156.
[22] Gillies D B. Some theorems on n-person games[M]. Princeton:Princeton University Press, 1953.
[23] Shapley L S, Shubik M. On the core of an economic system with externalities[J]. American Economic Review, 1969, 59(4):678-684.
[24] Schmeidler D. The nucleolus of a characteristic function game[J]. SIAM Journal on Applied Mathematics, 1969, 17(6):1163-1170.
[25] 徐伟,郑石桥,陈丹萍.闭环供应链中制造商与销售商合作创新模型研究[J].中国管理科学,2014,22(7):116-123.
[26] 程永波,陈洪转,何利芳,等.复杂装备主制造商-供应商主从合作激励协调Stackelberg模型[J].中国管理科学,2016,24(1):91-96.
[27] Moore R E. Methods and applications of interval analysis[M]. Philadelphia:Studies in Applied Mathematics, 1979.
[28] Ruiz L M, Valenciano F, Zarzuelo J M. The least square prenucleolus and the least square nucleolus. Two values for TU games based on the excess vector[J]. International Journal of Game Theory, 1996, 25(1):113-134.
[29] 钟昌宝,魏晓平,聂茂林,等.一种考虑风险的供应链利益两阶段分配法-正交投影熵值法[J].中国管理科学,2010,18(2):68-74.
[30] 赵晓丽,乞建勋.供应链不同合作模式下合作利益分配机制研究-以煤电企业供应链为例[J].中国管理科学,2007,15(4):70-76.
