疫情援助中的分配问题

Distribution Issues in Epidemic Aid

摘要: 自2019年底以来，新冠肺炎疫情肆虐，其中武汉及整个湖北省疫情严峻。为践行一方有难八方支援的理念，我国给出了一个省援助湖北省一个市的援助思路，即“一省包一市”。针对这个问题，我们依据运筹学中线性规划以及整数规划的思想，运用分配问题的模型以及算法，最后利用Matlab进行求解。在进行推导和运算之后，我们得到了这个问题的优化后的可行解，以解决相应问题。

Abstract: Since the end of 2019, the COVID-19 epidemic has been raging, with severe epidemics in Wuhan and the entire Hubei Province. In order to implement the concept of trouble in one side, support in all directions, China puts forward the idea of “one province covers one city”. To solve this problem, we use the model and algorithm of the allocation problem based on the ideas of linear programming and integer programming in operations research, and finally use Matlab to solve it. After derivation and calculation, an optimized feasible solution is put forward to solve this problem, helping solve the corresponding problem.

1. 引言

自2019年底以来，新冠肺炎疫情肆虐，尤其在2020年初时，武汉市及整个湖北省全境范围疫情异常严峻。国家卫生健康委对武汉市的援驰力度不断加大，全国医疗卫生系统共有4.2万名医务人员驰援武汉。同时，除武汉以外的一些地市，医疗资源和病人需求之间也存在矛盾。因此，国家卫生健康委建立了16个省支援武汉以外地市的一一对口支援关系，以“一省包一市”的方式，全力支持湖北省加强病人的救治工作，维护好人民群众的生命安全和身体健康 [1]。中共中央政治局常委会召开会议强调，“要分类指导各地做好疫情防控工作”。针对湖北除武汉外地区的医疗资源缺口，“一省包一市”的做法正是“对症下药”。这既是全国一盘棋的防控思路，也是一方有难、八方支援的强大合力。湖北各地区疫情的态势不尽相同，“一省包一市”的援助模式在实践中需要根据具体情况尽快适应、迅速到位。本文旨在运用整数规划模型探求一种解决此问题的有效方案。

2. 模型构建与求解

2.1. 问题描述

我们将30个可供参考的援助省划分为六个片区(华北、华东、东北、中南、西北、西南)。经过分析发现，此类问题是典型的运筹整数规划问题 [2]。而且由于是1省对1市，此类问题又可以作为分配问题进行求解。在规划的过程中有多种因素需要考虑：

1) 如何从30个省份中选出16个省份对武汉进行援助。

2) 这16个省份要分别对接哪一个被援助城市。

3) 在援助后，6个片区一定要留守足够的医疗资源，以应对当地的疫情。

4) 制定援助计划时，要合理分析当地的GDP水平以及医疗水平对决策的影响。

2.2. 模型建立

2.2.1. 模型1——援助省份的选择

1) 问题分析：

从30个省，自治区，直辖市中选择16个，考虑GDP，医疗水平等因素，以及华北，东北，华东，中南，西南，西北六个地区要留守足够的医疗资源。因而可以采用0-1规划进行求解。

2) 变量设定：

设北京，天津，河北，山西，内蒙古，辽宁，吉林，黑龙江，上海，江苏，浙江，安徽，福建，江西，山东，河南，湖南，广东，广西，海南，重庆，四川，贵州，云南，西藏，陕西，甘肃，青海，宁夏，新疆分别为 $X_i\left(i=1,2,\cdots ,30\right)$。

其中， $X_i=\{\begin{array}{c}1,当选中i时\\ 0,当不选i时\end{array}$。

3) 目标函数设立：

考虑到选择地区是去援助湖北，因此将各地医疗水平分数作为价值系数，目标函数为使得医疗水平最大化。

即 $\max z_1={\displaystyle {\sum }_{i=1}^{30}{a}_i{x}_i}$ (其中 $a_i$ 为各地医疗水平分数)。

4) 约束条件确立：

约束条件主要从保证各个片区分别留守足够的医疗资源着手。医疗、经济实力的强弱与的外派医护力量的强弱大致是正相关的，因而可以从此着手确立基本约束。在这里假定以50%为界，根据经济与医疗水平的高低确定从每个片区选中的地方数量上限。

同时，考虑到一些地方自身实力薄弱，难以胜任此工作，因此结合GDP，人均GDP和医疗实力三个数据，将3个数据同时位于倒数前10位置的地区排除，见表1。

Table 1. Region Medical level, DGP, GDP per capita

表1. 各地区医疗实力，DGP，人均GDP

从中可以看出黑龙江，西藏，甘肃符合条件。

具体约束条件如下：

$s.t.\{\begin{array}{l}{x}_1+x_2+x_3+x_4+x_5\le 4\hfill \\ x_6+x_7+x_8\le 1\hfill \\ x_9+x_{10}+x_{11}+x_{12}+x_{13}+x_{14}+x_{15}\le 5\hfill \\ x_{16}+x_{17}+x_{18}+x_{19}+x_{20}\le 3\hfill \\ x_{21}+x_{22}+x_{23}+x_{24}+x_{25}\le 3\hfill \\ x_{26}+x_{27}+x_{28}+x_{29}+x_{30}\le 2\hfill \\ {\displaystyle {\sum }_{i=1}^{30}{x}_i=16}\hfill \\ x_8=0\hfill \\ x_{25}=0\hfill \\ x_{27}=0\hfill \\ x_i\in \left\{0,1\right\}\left(i=1,2,\cdots ,30\right)\hfill \end{array}$

2.2.2. 模型2——援助省份与被援助地区的匹配

1) 问题分析：

建立选出的16个援助地与被援助地区的的帮扶关系，可以利用分配模型求解该问题。

2) 变量设定：

见表2：

Table 2. Assistance mark sheet

表2. 援助评分表

设 $X_{ij}=\{\begin{array}{l}1,当A_i援助B_j时\hfill \\ 0,当A_i不援助B_j时\hfill \end{array}\left(i,j=1,2,\cdots ,16\right)$。

3) 目标函数设立：

由于是医疗援助分配，所以此处主要考虑援助地的医疗水平以及被援助地的疫情严重程度。因此可以将表2中的数值 $a_{ij}$ 理解为是 $A_i$ 援助 $B_j$ 的效果，满分100。假设理想状态为：医疗水平最高的援助地援助任何地区效果都为满分，医疗水平第二的援助地则援助除最严重地区外效果都为满分，援助最严重地区效果则为95分。以此类推，根据附录1、2计较分析得到表2的数据。

故目标函数为：

${\displaystyle \underset{j=1}{\overset{16}{\sum }}{a}_{ij}{x}_{ij}}$ (其中 $a_{ij}$ 表示 $A_i$ 援助 $B_j$ 的效果)

4) 约束条件确立

正如一般运输问题一样，此问题的约束条件如下：

$s.t.\{\begin{array}{l}{\displaystyle {\sum }_{i=1}^{16}{x}_{ij}=1}(i=1,2,\cdots ,16)\hfill \\ {\displaystyle {\sum }_{j=1}^{16}{x}_{ij}=1}(j=1,2,\cdots ,16)\hfill \\ x_{ij}\in \left\{0,1\right\}\left(i,j=1,2,\cdots ,16\right)\hfill \end{array}$

2.3. 模型求解

2.3.1. 模型1求解

利用matlab求解 [3]，结果整理见表3：

Table 3. Regional distribution

表3. 地区分配情况

最终结果为1代表该地区被选中，为0则代表该地区不会作为援助地区。所以选择的16个地方为北京，天津，山西，内蒙古，辽宁，上海，江苏，浙江，山东，广东，海南，重庆，四川，贵州，陕西，宁夏。

2.3.2. 模型2求解

利用Matlab求解该分配问题 [4]，结果整理见表4：

Table 4. Result of Matlab

表4. Matlab求解情况

标黄色的数据代表援助省份以及被援助地区的具体情况。例如，北京在此次援助过程中要援助的地区为黄冈，天津要援助的地区是潜江……

该模型的求解代表着从经济实力、医疗水平、疫情严重程度等多角度出发，各被援助地区都得到了良好的援助，并且各援助地也能在本地区留有余力的情况下献出最大的贡献。此模型可以在更多的分配问题、更广泛的领域应用。

2.4. 进一步分析

模型建立在一定的假设基础之上，所求得的最优解具有一定的局限性。换言之，在现实的决策考量中，往往还会受到其他因素的影响。比如：北京作为首都，具有稳定器的作用，所以在北京一定要留有足够的医疗资源。再比如援助过程中还要考虑到地形、运输距离、运输方式、天气等等这些因素的变化。但此模型考虑利用主成分分析法的思想，放大关键条件作用，忽略其他细碎条件的影响，具有一定的合理性与可行性。

3. 总结与展望

本文利用0-1规划模型与分配问题模型，为“一省包一市”政策提供了一种可行的方法。此模型经过改进可以推广到很多类似的现实问题，具有广阔的应用前景。但同时，这个模型有一定的主观因素在里面，具有一定的缺陷。该模型不仅局限于援助领域，具有一定的延展性，在诸多的有限资源分配问题上都能得到一定应用，例如，物流合理调配这类微观问题，解决区域发展不平衡所面对的经济分配问题。

附录

附录1我国目前的省份区域分布、医疗水平、GDP、人均GDP情况 [5]，见表5。

Table 5. Our country present province area distribution, medical treatment level, GDP, percapita GDP situation

表5. 我国目前的省份区域分布、医疗水平、GDP、人均GDP情况

附录2湖北省除武汉市以外的16个地市及其疫情情况，见表6。

Table 6. Sixteen cities and their epidemic situation in Hubei Province except Wuhan

表6. 湖北省除武汉市以外的16个地市及其疫情情况

附录3求解模型1的matlab代码

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-56.82-56.73-57.41-52.61-46.72-71.46-50.17-62.21-64.9-59.68];

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b=[4 1 5 3 3 2];

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lb=zeros(30,1);

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附录4求解模型2的matlab代码

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NOTES

^*共同第一作者。