基于多目标非线性规划的公平分级供需模型

Fair Hierarchical Supply and Demand Model Based on Multi-Objective Nonlinear Programming

本文为对2022年美国大学生数学建模竞赛B题的后续研究.首先,为确定模型更新时间,我们利用马尔科夫链模型分析历史气候数据,得到降雨的概率.再通过蒙特卡洛模拟仿真一个月内的天气情况,汇总分析,得到连续t天干旱的概率最大,以此作为模型的更新时间.其次,为简化计算,先在一年的长度上构建水资源的公平分级供需模型,查阅资料得到水库水位与容量的函数关系,以此构建供给函数;设计公平-效益系数、价格系数、成本系数、满意度来确定需求函数.接着,构建多目标规划方程,利用NSGA2遗传算法求解,得经济效益可达287.01亿美元.对于一年的供给量,鲍威尔湖需要10个月可以满足需求,米德湖需要12个月,最后剩余20亿立方米左右的水流入加利福尼亚湾.最后,当水资源紧缺,无法满足水电需求时,通过控制供给函数的范围,实现节流.我们还对模型进行了灵敏度分析,对于需求侧的变化引起供给侧的变动进行全面的分析.

In this paper,a further study is carried out on the Problem B of The 2022 Mathematical Contest in Modeling.First of all,the updating time of the model from the consecutive drought days of one month in five states is considered,and the continuous drought days with the highest probability as the updating time are selected.In this way,considering the worst case,our reservoir allocation model can be more robust and moderate.The specific method is to use Markov chain model to analyze the historical climate data and get the probability of rainfall.Then through Monte Carlo to simulate the weather in a month,summary analysis,the probability of continuous T days of drought is the largest.Secondly,through the analysis of the topic,it is found that if water resources are allocated in units of month or day,it will bring huge complexity to the model,and the solution will be tedious and time-consuming.In order to allocate water resources fairly and effectively,a fair hierarchical supply and demand model of water resources on the length of a year is built,and the functional relationship between reservoir water level and capacity by consulting data is obtained,so as to build the supply function.Design the equity-benefit coefficient,price coefficient,cost coefficient and satisfaction to determine the demand function.Then,the multi-objective programming equation is constructed and solved by NSGA2 genetic algorithm.Firstly,the annual water resources amount is allocated to determine the annual supply amount.The economic benefits can reach us$28.701 billion;For a year's supply,Powell needed 10 months to meet demand and Mead needed 12 months,leaving about 2 billion cubic meters of water flowing into the Gulf of California.Moreover,when water resources are scarce and the demand for hydropower cannot be met,the range of supply function is controlled to achieve throttling.Sensitivity analysis on the model is also constructed,and the changes of supply side caused by the changes of demand side are comprehensively analyzed.