[数] 随机规划
The convexity of the models for large scale transportation networks is analyzed. It is proved that this type of models are convex stochastic programming problems.
分析了大规模运输网络随机规划模型的凸性，证明了该类模型为一随机凸规划问题。
In addition, the efficiency solution scheme of this problem is given, after the multiple objective stochastic programming with probabilistic constrained is discussed.
此外，提出了这种具有概率约束多目标随机规划问题的一种有效解模型。
Present two classes of multivalued stochastic programming models, i. e. , multivalued random chance-constrained programming model and set-valued random expected-value programming model;
建立了两类多值随机规划模型：多值随机机会约束规划模型和多值随机期望值规划模型；
And then, we present an approximation method for solving this probabilistic constrained stochastic programming, and prove certain convergence of the method under some conditions.
随后我们提出了求解这类概率约束随机规划的一种近似算法，并在一定的条件下证明了算法的收敛性。
In this paper we give a survey of several stochastic programming models and algorithms for unit commitment problem under uncertainty load demands.
文章给出在不确定的荷载需求下机组组合问题的一些常见的随机规划模型和算法的综述。
In this paper we propose a stochastic programming model to determine the location and capacity of centralized return centers and distribution centers in reverse logistics network.
为此，本文提出了一个随机规划模型用于确定逆向物流网络中集中式回收中心和分销中心的位置和容量。
As using stochastic simulation and fuzzy simulation to resolve the problem in complex stochastic programming and fuzzy programming, the concept and method of grey simulation was advanced in .
正如解决复杂随机规划和模糊规划使用随机模拟和模糊模拟的手段一样，本文提出了灰色模拟的概念和方法。
Transform the bilevel stochastic programming model into single-level stochastic programming model under certain confidence, and a corresponding mixed method is designed.
在一定的置信度下，将双层规划模型转化为单层随机规划模型，并设计了相应的混合智能算法。
A stochastic programming model was established for selection of container shipping routes, and in the model uncertainty in container demands was taken into consideration.
基于海运集装箱运输问题特性的分析，建立了需求不确定的海运集装箱路径随机规划模型。
The other one is approaching method, i. e. getting the approximate optimal value and solution of stochastic programming through genetic algorithm based on stochastic simulation.
另一种是逼近方法，利用随机模拟技术，通过一定的遗传算法程序，得到随机规划问题的近似最优解和目标函数的近似最优值。
According to the optimal investment model with minimum-risky, we give a united stochastic programming with probabilistic constrained.
根据最小风险的投资最优问题，我们给出了一个统一的概率约束随机规划模型。
A genetic algorithm is used to solve this stochastic programming model.
采用遗传算法对该随机规划模型进行求解。
In the light of fuzzy theory, fuzzy optimization model is changed into the stochastic programming.
利用模糊随机理论转化模糊优化模型，使其成为一种易求解的随机规划模型。
This thesis presents the combination of the stochastic programming and generalized goal programming.
在本文中 ，我们提出了双凹规划问题和更一般的广义凹规划问题 。
Uncertainty environmental; inventory control; inventory coordination; remanufacturing logistics system; stochastic programming; fuzzy programming.
不确定性；库存控制；库存协调；再制造物流系统；随机规划；模糊规划。
Traditional short-term line maintenance scheduling is always formulated as a stochastic programming problem.
传统方法将短期线路检修计划作为单重不确定性优化问题进行建模和求解。
The convergence of the optimal value and optimal solution of the stochastic programming are given as the function sequence is epi-convergence and the random variable sequence is square convergence.
主要讨论了一类随机规划在函数序列上图收敛和随机变量序列均方收敛意义下，该类随机规划的最优解和最优值的收敛情况。
A stochastic programming model with fuzzy chance constraint is presented which has stochastic and fuzzy parameters.
提出一类模糊机会约束的随机期望值规划模型，该模型同时含有随机和模糊参数。
Based on stochastic programming, a multi objective chance constrained optimization model is developed for an illustrative coal blending problem.
在随机规划学的基础上，建立了电厂优化配煤的多目标机会约束数学模型。
A stochastic programming model was established for selection of container shipping routes, Author put forward an effective way of handling uncertain data in the models.
建立需求不确定的海运集装箱路径随机规划模型，在模型中提出了处理不确定数据的有效方法。
Combined with the historical data, the model predicts the stochastic parameter using Monte-Carlo simulation method, which improved accuracy and feasibility of stochastic programming model.
模型结合历史数据，用蒙特卡罗模拟方法来预测参数，提高了随机规划模型的准确性和可行性。
This paper discusses a method for two-stage stochastic programming with recourse in which the objective function is replaced by its empirical mean.
探讨了以随机变量的子样为条件，使用目标函数的经验均值逼近法来求解有补偿二阶段问题，并分析了相关的收敛性。
In order to generate scenarios, this article applies the method of autoregression and Monte-Carlo simulation in stochastic programming models.
本文将自回归和蒙特卡罗模拟方法运用到随机规划模型中，生成情景元素。
Stochastic programming is recognized as a powerful modeling paradigm for several areas of application, include DALM in financial institutions.
随机规划就是带不确定性的数学规划，能应用到很多领域，包括金融机构动态资产负债管理。
In this paper, the multiple vehicles coordinated Stochastic Vehicle Routing problem with time-constrained is (analysed. ) A kind of stochastic programming model for this problem is established.
分析了有时间约束的基于多车辆协作的随机路径问题。 提出了问题的随机规划期望值模型。
We formulated scenario based multi-objective stochastic programming model to describe the problem of capacity planning under uncertainty and applied improved Multi-Objective PSO (MOPSO) to solve it.
提出了基于场景的多目标随机规划模型来构建不确定市场需求环境下的能力计划问题模型，并用改进的多目标粒子群优化算法求解。
In this paper, maximum entropy theorem is used to give an approximate computational method for the single stage stochastic programming.
基于双随机理论，本文提出了一类两阶段双随机规划模型，并对这类模型的数学性质进行了研究。
We propose a two-stage stochastic programming model with uncertain demand.
建立了消费者需求不确定状态下的两阶段随机规划模型。
随机规划（Stochastic Programming）是运筹学与数学优化领域的重要分支，主要用于解决包含不确定参数的决策问题。其核心思想是在建模过程中将不确定性以概率分布的形式纳入目标函数或约束条件，从而设计出对随机因素具有鲁棒性的最优策略。
随机规划通过数学规划方法处理多阶段决策问题，其中部分参数（如需求、价格、资源供给等）具有随机性。例如，在电力系统调度中，发电量可能受天气影响，需预先建立不同场景下的概率模型，并优化期望成本或风险值（来源：Springer《Stochastic Programming》专著）。
典型的随机规划模型分为两阶段或多阶段：
常用算法包括随机分解法、Benders分解和蒙特卡洛采样法。现代求解工具（如GAMS、PySP）结合场景生成技术，可高效处理大规模问题（来源：MathWorks优化工具箱文档）。
Stochastic programming（随机规划）是一种数学优化方法，用于处理包含不确定性参数的决策问题。它结合了概率论和优化理论，通过建立多阶段模型来寻找在不确定环境下最优的决策策略。以下是详细解释：
不确定性建模
系统参数（如需求、价格、资源量等）不是固定的，而是服从某种概率分布或可能出现的多种场景。例如，电力公司需在未知未来电价的情况下规划发电量。
多阶段决策
决策分阶段进行：
目标函数
通常优化期望值，例如最小化总成本的期望，或结合风险度量（如条件风险价值CVaR）以控制极端情况。
两阶段随机线性规划
公式示例：
$$
begin{aligned}
min_{x} quad & c^T x + mathbb{E}[Q(x, xi)]
text{s.t.} quad & Ax leq b
& x geq 0
end{aligned}
$$
其中，$Q(x, xi)$是第二阶段应对随机变量$xi$的补救成本。
场景树方法
将不确定性离散化为有限场景（如“高需求”“中需求”“低需求”），求解大规模线性规划问题。
如果需要更具体的数学形式或应用案例，可以进一步探讨！
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