The supply chain operating model is constructed as a multi - objective programming problem to satisfy several conflict objectives , such as coordination , of supply chain , the maximum profit of all participants , and robustness of decision to uncertain product demands and raw material supplies 供应链的运作模型为一个多目标规划问题，满足诸如供应链协调运作、所有供应链成员的目标利润尽可能最大、对应于不确定供求的决策的鲁棒性等多个相互冲突的目标。
Furthermore , a mixed type dual for the primal problem is established and duality results are obtained . after that , the optimality sufficiency conditions and duality results for the nonlinear fractional programming problem are presented under generalized ( f , or , p , d ) - convexity assumptions 接着建立了含有等式约束和不等式约束的非线性多目标规划问题的wolfe型对偶和mond - weir型对偶及原问题的混合类型对偶，并获得了相应类型下的弱对偶定理和强对偶定理。
Involve mathematical economics , finance , control theory , mechanics , physics and so on . they become an important foundation and tool for studying multiobjective and multilevel programs and one of focal point problems paid close attention by scholars in the field of applied mathematics 集值变分包含问题涉及数理经济学、金融学、控制论、机械学、物理学等学科，是研究多目标规划和多层规划的重要基础和工具，也是目前应用数学领域中备受关注的热点之一。
In all , from above research , this paper concluded as follow : ( 1 ) making out the four kinds solution ’ s definition , and in the basis putting out the existence theorem of pareto efficient solution on blmop found . ( 2 ) giving out the algorithm of blmop based on the genetic algorithm 总体说来,通过以上研究,本文主要得出以下结论: ( 1 )给出了二层多目标规划四种类型的解的定义,并在此基础上提出并证明了二层多目标规划问题的pareto有效解存在的充分性和必要性条件。
In this thesis , the optimality sufficiency conditions and duality theory are discussed in multiobjective nonlinear programming involving ( f , a , p , d ) - convexity and generalized ( f , a , p , d ) - convexity . at that time , an algorithm is discussed for nonlinear multiobjective problem 本文主要讨论了( f , , , d ) -凸及广义( f , , , d ) -凸条件下非线性多目标规划问题的最优性充分条件和对偶理论，同时，也探讨了求解具有线性等式约束的非线性多目标规划问题的一种新算法。
