﻿ 基于部分变需求的系统最优疏散路径模型
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 山东大学学报(工学版)  2016, Vol. 46 Issue (3): 87-92  DOI: 10.6040/j.issn.1672-3961.0.2015.243 0

### 引用本文

GAO Jianjie, SHI Chenpeng, CAO Jianjun, OU Jushang. Department of Transportation Management, Sichuan Police College, Luzhou 646000, Sichuan, China[J]. Journal of Shandong University(Engineering Science), 2016, 46(3): 87-92. DOI: 10.6040/j.issn.1672-3961.0.2015.243.

### 文章历史

Department of Transportation Management, Sichuan Police College, Luzhou 646000, Sichuan, China
GAO Jianjie, SHI Chenpeng, CAO Jianjun, OU Jushang
System optimal model of evacuation route selection based on partial variable demand
Abstract: The traffic demand after effects of the emergency was divided into two parts: fixed demand and variable demand. One was the evacuation demand and the other was the original traffic demand in non hazardous areas. A system optimal model of evacuation route selection based on partial variable demand was proposed aimed at minimizing the total travel time of traffic network system. The equivalence between the solution of the model and the system optimal condition of varying demand was proved. And the existence and uniqueness of the solution of the model was analyzed. At last an algorithm of the model was designed from the convex combination method.
Key words: traffic engineering    evacuation route    partial variable demand    system optimal model    distribution of traffic volumes
0 引言

1 突发事件对交通需求的影响

2 系统最优疏散路径分配模型 2.1 模型

 $\min :Z\left( X \right)=\sum\limits_{a}{{{x}_{a}}{{t}_{a}}\left( {{x}_{a}} \right)}-\sum\limits_{rs}{\int_{0}^{{{D}_{rs}}}{{{F}_{rs}}\left( v \right)\text{d}v,}}$ (1)
 $\text{s}\text{.t}\text{.:}\sum\limits_{k}{f_{k}^{rs}}={{D}_{rs}},\forall r,s,k,$ (2)
 ${{D}_{rs}}={{D}_{rs0}}+{{D}_{rs1}},\forall r,s,$ (3)
 ${{D}_{rs1}}\ge 0,f_{k}^{rs}\ge 0,\forall r,s,k,$ (4)
 ${{x}_{a}}=\sum\limits_{r,s}{\sum\limits_{k}{f_{k}^{rs}\cdot x_{a,k}^{r,s}}},\forall a$。 (5)

2.2 模型的解与部分变需求的系统最优条件之间的等价性

 $L\left( f,D,\gamma \right)=Z\left[ X\left( f \right),D \right]+\sum\limits_{rs}{{{\gamma }_{rs}}\left( {{D}_{rs}}-\sum\limits_{rs}{f_{k}^{rs}} \right)},$ (6)

 \left\{ \begin{align} & f_{k}^{rs}\frac{\partial L}{\partial f}=0,\forall r,s,k, \\ & \frac{\partial L}{\partial f}\ge 0,\forall r,s,k, \\ & f_{k}^{rs}\ge 0,\forall r,s,k, \\ \end{align} \right. (7)

 \left\{ \begin{align} & {{D}^{rs}}\frac{\partial L}{\partial D}=0,\forall r,s, \\ & \frac{\partial L}{\partial D}\ge 0,\forall r,s, \\ & {{D}_{rs}}\ge {{D}_{rs0}},\forall r,s, \\ \end{align} \right. (8)
 $\frac{\partial L}{\partial {{\gamma }_{rs}}}=0,\forall r,s$。 (9)

 ${{t}_{a}}\left( {{x}_{a}} \right)={{t}_{a}}\left( {{x}_{a}} \right)+{{x}_{a}}\frac{\text{d}{{t}_{a}}\left( {{x}_{a}} \right)}{\text{d}{{x}_{a}}},\forall a,$

 \begin{align} & \frac{\partial L}{\partial f}=\frac{\partial }{\partial f}\left( \sum\limits_{a}{{{x}_{a}}{{t}_{a}}\left( {{x}_{a}} \right)} \right)-\frac{\partial }{\partial f}\left( \sum\limits_{rs}{\int_{0}^{{{D}_{rs}}}{{{F}_{rs}}\left( \omega \right)\text{d}\omega }} \right)-{{\gamma }_{rs}}= \\ & \sum\limits_{a}{\frac{\partial {{x}_{a}}{{t}_{a}}\left( {{x}_{a}} \right)}{\partial {{x}_{a}}}\frac{\partial {{x}_{a}}}{\partial f}-\frac{\partial }{\partial f}\left( \sum\limits_{rs}{\int_{0}^{{{D}_{rs}}}{{{F}_{rs}}\left( \omega \right)\text{d}\omega }} \right)}-{{\gamma }_{rs}}= \\ & \sum\limits_{a}{\chi _{a,k}^{r,s}\frac{\partial }{\partial {{x}_{a}}}\sum\limits_{a}{{{x}_{a}}{{t}_{a}}\left( {{x}_{a}} \right)}}-\frac{\partial }{\partial f}\left( \sum\limits_{rs}{\int_{0}^{{{D}_{rs}}}{{{F}_{rs}}\left( \omega \right)\text{d}\omega }} \right)-{{\gamma }_{rs}}= \\ & \sum\limits_{a}{\chi _{a,k}^{r,s}\left[ {{t}_{a}}\left( {{x}_{a}} \right)+{{x}_{a}}\frac{\text{d}{{t}_{a}}\left( {{x}_{a}} \right)}{\text{d}{{x}_{a}}} \right]}-\frac{\partial }{\partial f}\left( \sum\limits_{rs}{\int_{0}^{{{D}_{rs}}}{{{F}_{rs}}\left( \omega \right)\text{d}\omega }} \right)-{{\gamma }_{rs}}=\tilde{c}_{k}^{rs}-{{\gamma }_{rs}}, \\ \end{align} (10)
 $\frac{\partial L}{\partial D}=-{{F}_{rs}}\left( {{D}_{rs}} \right)+{{\gamma }_{rs}},$ (11)

 \left\{ \begin{align} & f_{k}^{rs}\left( \tilde{c}_{k}^{rs}-{{\gamma }_{rs}} \right)=0,\forall k,r,s, \\ & \tilde{c}_{k}^{rs}-{{\gamma }_{rs}}\ge 0,\forall k,r,s, \\ & f_{k}^{rs}\ge 0,\forall k,r,s, \\ \end{align} \right. (12)
 \left\{ \begin{align} & \left( {{D}_{rs}}-{{D}_{rs0}}\left[ {{\gamma }_{rs}}-{{F}_{rs}}\left( {{D}_{rs}} \right) \right]=0,\forall r,s, \right. \\ & {{\gamma }_{rs}}-{{F}_{rs}}\left( {{D}_{rs}} \right)\ge 0,\forall r,s, \\ & {{D}_{rs}}\ge {{D}_{rs0}},\forall r,s, \\ \end{align} \right. (13)
 $\sum\limits_{k}{f_{k}^{rs}={{D}_{rs}},{{D}_{rs}}={{D}_{rs0}}+{{D}_{rs1}},}\forall r,s$。 (14)

2.3 模型解的存在性和唯一性

 $\frac{\partial }{\partial {{x}_{b}}}\sum\limits_{a}{{{x}_{a}}{{t}_{a}}\left( {{x}_{a}} \right)}={{t}_{b}}\left( {{x}_{b}} \right)+{{x}_{b}}\frac{\text{d}{{t}_{b}}\left( {{x}_{b}} \right)}{\text{d}{{x}_{b}}}$。 (15)

 $\frac{{{\partial }^{2}}}{\partial {{x}_{b}}\partial {{x}_{a}}}\sum\limits_{a}{{{x}_{a}}{{t}_{a}}\left( {{x}_{a}} \right)}=\frac{\partial }{\partial {{x}_{a}}}\left[ {{t}_{b}}\left( {{x}_{b}} \right)+{{x}_{b}}\frac{\text{d}{{t}_{b}}\left( {{x}_{b}} \right)}{\text{d}{{x}_{b}}} \right]=\left\{ \begin{matrix} 2\frac{\text{d}{{t}_{a}}\left( {{x}_{a}} \right)}{\text{d}{{x}_{a}}}+{{x}_{a}}\frac{{{\text{d}}^{2}}{{t}_{a}}\left( {{x}_{a}} \right)}{\text{d}x_{a}^{2}}, & a=b, \\ 0, & a\ne b \\ \end{matrix} \right.$。 (16)

2.4 模型的算法思路及步骤

$\sum\limits_{rs}{\frac{\left| {{F}_{rs}}\left( D_{rs}^{n} \right)-\gamma _{rs}^{n} \right|}{\gamma _{rs}^{n}}+\sum\limits_{rs}{\frac{\left| \gamma _{rs}^{n}-\gamma _{rs}^{n-1} \right|}{\gamma _{rs}^{n}}\le \varepsilon }}$,(ε是预先确定的小正数),停止计算;否则,令n=n+1,返回至第2步。

3 实例验证

 图 1 试验网络 Figure 1 The traffic network

4 结语

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