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A Method of Allocating Assistance Resources Based on Single-Objective Fractional Model
DOI: 10.12677/MSE.2022.114086, PDF , HTML, XML, 下载: 73  浏览: 128

Abstract: Over the past few decades, China’s economy has grown rapidly and it has gotten rid of absolute poverty in 2020. With the overall economic growth slowing down, promoting balanced development among regions has become an important issue. Under this background, this paper establishes a single-objective fractional programming mathematical model to minimize the regional development difference. Based on this model, a series of quantitative allocation schemes of assistance resources can be obtained. By using the actual economic data of counties and cities in Gansu province to carry out numerical simulation, it is proved that if we distribute the assistance resources based on an allocation scheme in this paper, it can promote the balanced development of the region, and at the same time promote overall development. Therefore, the method of this paper can be applied to the actual non-profit organizations to provide substantive management suggestions, so as to promote balanced and coordinated development of regions.

1. 引言

2. 文献综述

3. 建立模型

${\alpha }_{2}b\le {\sum }_{j\in J}{f}_{j}{p}_{j}\le b$

${t}_{j}={g}_{j}+{g}_{j}{f}_{j}{c}_{j},j\in |J|$

$\left({\sum }_{j\in J}{t}_{j}-{\sum }_{j\in J}{g}_{j}\right)/{\sum }_{j\in J}{t}_{j}\ge lb$

$CV=\frac{\sqrt{{\sum }_{j\in J}{\left({t}_{j}-\mu \right)}^{2}/|J|}}{{\sum }_{j\in J}{t}_{j}/|J|}=\frac{\sqrt{{\sum }_{j\in J}{\left({t}_{j}-\mu \right)}^{2}|J|}}{{\sum }_{j\in J}{t}_{j}}$

$\mu =\frac{{\sum }_{j\in J}{t}_{j}}{|J|}$

$\mathrm{min}\frac{\sqrt{{\sum }_{j\in J}{\left({t}_{j}-\mu \right)}^{2}|J|}}{{\sum }_{j\in J}{t}_{j}}$

s.t

${t}_{j}={g}_{j}+{g}_{j}{f}_{j}{c}_{j},j\in |J|$ (1)

$\left({\sum }_{j\in J}{t}_{j}-{\sum }_{j\in J}{g}_{j}\right)/{\sum }_{j\in J}{g}_{j}\ge lb$ (2)

${\alpha }_{2}b\le {\sum }_{j\in J}{f}_{j}{p}_{j}\le b$ (3)

$0<{\alpha }_{2}\le 1$ (4)

${t}_{j}\ge 0,j\in |J|$ (5)

${f}_{j}\in \left\{0,1\right\},j\in |J|$ (6)

4. 求解方法

$\mathrm{min}\frac{{f}_{1}\left(x\right)}{{f}_{2}\left( x \right)}$

s.t.

$x\in Χ$

$\mathrm{min}{f}_{1}\left(x\right)-q{f}_{2}\left(x\right)$

s.t.

$x\in Χ$

$F\left(q\right)={\mathrm{min}}_{x\in Χ}\left\{{f}_{1}\left(x\right)-q{f}_{2}\left(x\right)\right\}$。当且仅当 ${x}^{\ast }$$P\left({q}^{\ast }\right)$ 的最优解时(此时需满足 $F\left({q}^{\ast }\right)=0$ )， ${x}^{\ast }$ 为问题P的最优解 [20]。因此为了能够找到 ${x}^{\ast }$，首先需要找到 ${q}^{\ast }$，现在引入定理1 [21]，如下：

1) 当且仅当 $q<{q}^{\ast }$ 时，有 $F\left(q\right)>0$

2) 当且仅当 $q={q}^{\ast }$ 时，有 $F\left(q\right)=0$

3) 当且仅当 $q>{q}^{\ast }$ 时，有 $F\left(q\right)<0$

5. 数值实验

$b={\alpha }_{3}\underset{j\in J}{\sum }{p}_{j}={\alpha }_{3}\underset{j\in J}{\sum }0.1\ast GD{P}_{j}$

Table 1. Poverty classifications of counties and cities in Gansu province

Table 2. The effects of different assistance schemes

· 甘肃各县级市经济发展水平的原始方差为0.4413。在本次数值实验中，得到的4个不同的资源分配方案，均可以不同程度地促进地区发展均衡。其中方案1可以将子区域的经济水平方差降低至0.4073，降低幅度达8.35%；

· 通过设置增长速率约束中的下限，可以发现扶助资金分配方案能帮助地区在保持一定整体发展均速的同时，促进区域发展均衡性。但两者为反相关关系，如图1，即一个资源分配方案最终能够实现的子区域间经济水平差异越小，其对子区域的整体发展增速的提升作用也越小；

Figure 1. Trade-off curve between coefficient of variance and overall development rate

· 基于第2点的结论，可以总结到若子区域间均衡发展视为最重要的目标时，得到方案的资金利用率也会较低，因为此时会通过放弃资助部分经济发展水平较高的子区域的方式实现均衡。此时针对未被利用的部分资金，可以通过本模型确定进一步的分配方案，将该部分的剩余资源优先分配给经济发展较差的地区，进而进一步缩小各区域间的经济发展水平。

6. 总结

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