《数学建模商人过河问题论文》Word文件下载.docx

《数学建模商人过河问题论文》Word文件下载.docx

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王鹏道:

选择论文题目、设计论文版面字体、分配成员任务、总结任利伟:

一、问题提出、关键、分析。

二、模型假设、三、模型建立孙祎:

四、模型求解、五、模型的检验、拓展及延伸

2014年11月24日

摘要

为了求解3个商人和3个随从的过河问题，用数学分析方法，建立数学模型，并且加以求解，展示动态规划思想的应用步骤。

最后利用计算机蝙程进行求解，获得过河问题的完整求解过程;

有效地求解类似多步决策问题的作用。

关键词:

多步决策计算机求解状态转移律图解法

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一、问题的提出

随从们密约,在河的任一岸,一旦随从的人数比商人多,就杀人越货，但是乘船渡河的方案由商人决定.商人们怎样才能安全过河?

二、问题的关键

解决的关键集中在商人和随从的数量上，以及小船的容量上，该问题就是考虑过河步骤的安排和数量上。

各个步骤对应的状态及决策的表示法也是关键。

三、问题的分析

在安全的前提下（两岸的随从数不比商人多）,经有限步使全体人员过河。

由于船上人数限制，这需要多步决策过程，必须考虑每一步船上的人员。

动态规划法正是求解多步决策的有效方法。

它要求把解的问题一层一层地分解成一级一级、规模逐步缩小的子问题。

直到可以直接求出其解的子问题为止。

分解成所有子问题按层次关系构成一棵子问题树（树根是原问题。

原问题的解依赖于子问题树中所有子问题的解。

四、模型假设

记第k次过河前A岸的商人数为X，K

随从数为Yk=1，2，?

K

X,Y=0，1，2，3，将二维向量S=（X，Y）定义为状态（把满足安全渡KKKKK

河条件下的状态集合称为允许状态集合。

记作S。

则

S={（X,Y）|（X=0,Y=0,1,2,3）,（X=3,Y=0,1,2,3）,（X=Y=1）（X=Y=2）}KKKKKKKKKK

记第k次过河船上的商人数为UK

随从数为VK

将二维向量D=（U,V）定义为决策（由小船的容量可知允许决策集合（记作KKK

D）为

D={（U,V）|U+V=l,2}={（O,1）;

（O,2）;

（1,O）;

（1,1）;

（2,O）}KKKK

五、模型建立:

它要求把解的问题一层一

层地分解成一级一级、规模逐步缩小的子问题。

直到可以直接求出其解

的子问题为止。

分解成所有子问题按层次关系构成一棵子问题树（树根

是原问题。

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