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
。
其中,
。
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
(其中
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。
![](Images/Table_Tmp.jpg)
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:
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:
设
。
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
理解为是
援助
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:
(其中
表示
援助
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:
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:
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:
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。
![](Images/Table_Tmp.jpg)
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。
![](Images/Table_Tmp.jpg)
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
f = [-100 -56.4 -51.35-55.81-62.54-57.24-57.24-51.52-67.93-61.7
-71.3 -44.36-52.86-44.78-61.87-54.46-53.28-56.06-54.8-57.41
-56.82-56.73-57.41-52.61-46.72-71.46-50.17-62.21-64.9-59.68];
intcon=[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30];
A=[1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00;
0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0;
0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1];
b=[4 1 5 3 3 2];
Aeq=[11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1;
00 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0;
00 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0];
beq=[16 0 0 0];
lb=zeros(30,1);
ub=[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1];
附录4求解模型2的matlab代码
f=[-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-35,-40,-45,-50,-55,-60,-65,-70,-75,-80,-85,-90,-95,-100,-100,-100,-25,-30,-35,-40,-45,-50,-55,-60,-65,-70,-75,-80,-85,-90,-95,-100,-75,-80,-85,-90,-95,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-50,-55,-60,-65,-70,-75,-80,-85,-90,-95,-100,-100,-100,-100,-100,-100,-85,-90,-95,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-65,-70,-75,-80,-85,-90,-95,-100,-100,-100,-100,-100,-100,-100,-100,-100,-90,-95,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-70,-75,-80,-85,-90,-95,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-30,-35,-40,-45,-50,-55,-60,-65,-70,-75,-80,-90,-95,-100,-65,-70,-75,-95,-100,-100,-100,-100,-100,-55,-70,-80,-85,-90,-95,-100,-100,-100,-100,-100,-40,-45,-50,-55,-60,-65,-70,-75,-80,-85,-90,-95,-100,-100,-100,-100,-55,-60,-65,-70,-75,-80,-85,-90,-95,-100,-100,-100,-100,-100,-100,-100,-95,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-80,-85,-90,-95,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-90,-95,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100,-100];
intcon=[12345678910111213141516
17181920212223242526272829303132
33343536373839404142434445464748
49505152535455565758596061626364
65666768697071727374757677787980
81828384858687888990919293949596
979899100101102103104105106107108109110111112
113114115116117118119120121122123124125126127128
129130131132133134135136137138139140141142143144
145146147148149150151152153154155156157158159160
161162163164165166167168169170171172173174175176
177178179180181182183184185186187188189190191192
193194195196197198199200201202203204205206207208
209210211212213214215216217218219220221222223224
225226227228229230231232233234235236237238239240
241242243244245246247248249250251252253254255256];
Aeq=[1111111111111111000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000;
0000000000000000111111111111111100000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000;
0000000000000000000000000000000011111111111111110000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000;
0000000000000000000000000000000000000000000000001111111111111111000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000;
0000000000000000000000000000000000000000000000000000000000000000111111111111111100000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000;
0000000000000000000000000000000000000000000000000000000000000000000000000000000011111111111111110 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000;
0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001 111111111111111000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000;
0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000111111111111111100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000;
0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000011111111111111110000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000;
0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000001111111111111111000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000;
0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000111111111111111100000000000000000000000000000000000000000000000000000000000000000000000000000000;
0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000011111111111111110000000000000000000000000000000000000000000000000000000000000000;
0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001111111111111111000000000000000000000000000000000000000000000000;
0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000111111111111111100000000000000000000000000000000;
0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011111111111111110000000000000000;
0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001111111111111111;
1000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001 000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000;
0100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000 100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000;
0010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000 010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000;
0001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000 001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000;
0000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000 000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000;
0000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000 000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000;
0000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000 000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000;
0000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000 000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000;
0000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000 000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000;
0000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000 000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000;
0000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000 000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000;
0000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000 000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000;
0000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000 000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000;
0000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000 000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100;
0000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100 000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010;
0000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010 000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001000000000000000100000000000000010000000000000001];
beq=[11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 111];
lb=zeros(256 , 1);
ub=[11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111];
NOTES
*共同第一作者。