不确定随机规划及其应用

Uncertainty theory and probability theory are two different mathematical systems which are used to deal with belief degree and frequency, respectively. In many cases, uncertainty and randomness coexist in a complex system, and this phenomenon is called uncertain randomness. At present, mathematical programming models involving random variables or uncertain variables have been well studied, and some numerical algorithms have also been designed to solve those models. In order to deal with decision making problem involving not noly random variables but also uncertain variables, we will develop uncertain random programming models...This project is an interdisciplinary field among operations research, mathematical programming, information science and computer science. The research work consists of theory and application. In theoretical research, we will estiblish chance constrained programming models and dependent-chance programming models under an uncertain random environment, and extend them to multi-objective programming and goal programming cases. Application analysis includes establishing uncertain random programming models for project scheduling, machine scheduling, facility location-allocation, vehicle routing and some other problems; designing intelligent algorithms and analyzing the mathematical properties. Finally, we will develop a complete theoretical system of uncertain random programming.

不确定理论与概率理论是两套不同的理论体系，分别用来处理信度与频率。然而，不确定性与随机性往往同时出现在一个复杂系统中，这种现象称为不确定随机性。目前，仅含有随机变量和仅含有不确定变量的数学规划模型及其算法已得到了长足的发展。为了解决既有随机变量又有不确定变量的决策问题，本项目组将研究不确定随机规划。本项目为运筹学、数学规划、信息科学以及计算机科学的交叉项目，研究内容涉及理论和应用两个方面。理论研究包括建立不确定随机环境下的机会约束规划模型和相关机会规划模型，并把它们拓展到多目标规划和目标规划的情形下。应用分析包括如何在项目进度、机器排序、设备选址和车辆调度等问题中建立数学模型、设计智能算法、分析模型的数学性质。最终形成一套完整的不确定随机规划理论。

大量的研究都指出在实际应用里，我们所接触到的既不是单纯的随机变量，也不是单纯的不确定变量，而是一个同时包含随机性和不确定性的复杂系统。为了解决同时含有随机性和不确定性的决策难题，本项目组研究了不确定随机规划。本项目成功建立不确定随机环境下的机会约束规划模型并为其在项目进度、机器排序等实际问题里建模及设计算法，不仅如此，本项目还有广泛的拓展，包括不确定随机风险分析、不确定随机可靠度分析、不确定随机网络、不确定随机过程和不确定随机多目标优化。另外，本项目还通过公理化的数学方法解决了艾尔斯伯格罐子问题。文献表明，本项目的研究成果已经被应用到管理、信息、工程、物流等领域中。