# The Sensitivity Analysis for Supply and Demand Dynamics System of Aged Services Resource Allocation in China

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## Abstract

**:**

## 1. Introduction

## 2. Causality Analysis of Aged Service Resources Allocation System

## 3. Materials and Methods

**A**is the $m\times m$ order parameter matrix,

**B**is the $m\times p$ order parameter matrix,

**C**is the $q\times m$ order parameter matrix,

**D**is the $q\times p$ order parameter matrix, $x$ is the $m$ dimension state variable, $u$ is the $p$ dimension input variable, $\alpha $ is the r dimension parameter variable, y is the $q$ dimension output variable, and t represents time.

- (1)
- Sensitivity index 1

- (2)
- Sensitivity index 2

## 4. Result

#### 4.1. Simulation Analysis

#### 4.2. Dynamic Analysis of Parameter Sensitivity in Supply and Demand System of Aged Service Resources

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Number | State Variables | Parameter Variables | Out Variables | ||
---|---|---|---|---|---|

1 | Elderly Population (EP) | GDP Growth Rate (GDPGR) | Average Number of Beds in Pension Institutions (ANPI) | Initial Value of Market Engagement of Pension Communities (IVPC) | Total Ratio between Supply and Demand (TRSD) |

2 | Total Population (TP) | Factor of Per Capita Disposable Income (FPCDI) | Average Amount of Elderly Population Served by Pension Communities (ANPC) | Regulation Threshold of Market Engagement of Pension Communities (RTPC) | Ratio between the Supply and Demand of Pension Institutions (RSDPI) |

3 | Gross Domestic Product (GDP) | Average of CPI (ACPI) | Maximum Values of WDEPP (MaxWD) | Regulation Coefficient of Market Engagement of Pension Institution (RCMEPC) | Ratio between the Supply and Demand of Pension Communities (RSDPC) |

4 | Total National Pension Expenditure (TNPE) | Birth Rate (BR) | Minimum Values of WDEPP (MinWD) | Regulation Threshold of Total Ratio between Supply and Demand (RTT) | |

5 | Per Capita Pension Expenditure (PCPE) | Death Rate (DR) | Maximum Values of WNAPP (MaxWN) | Regulation Coefficient of Pension Expenditure (RCPE) | |

6 | Operating Subsidies of Pension Institutions (OSPI) | Disability Rate of Elderly Population (DREP) | Minimum Values of WNAPP (MinWN) | Initial Value of Weight Factor of Pension Expenditure in Institutions (IVWFPI) | |

7 | Operating Subsidies of Pension Communities (OSPC) | Rate of Increase in Elderly Population (RIEP) | Gain of Willingness of Disabled Elderly Population to Pay ${K}_{A1}$ | Balance Coefficient of the Supply and Demand Ratio of Institutional Community Pension (BCRIC) | |

8 | Factor of OSPI and OSPC (FOO) | Gain of Willingness of Non-disabled Elderly Population to Pay ${K}_{A2}$ | Regulation Threshold of Weight of Pension Expenditure in Communities (RTPE) | ||

9 | Willingness of Elderly Population to Participate in Pension Institutions (WEPPI) | Initial value of market engagement of pension institutions (IVPI) | Gain of Weight Regulation of Pension Expenditure (GPE) | ||

10 | Willingness of Elderly Population to Participate in Pension Communities (WEPPC) | Regulation Threshold of market engagement of Pension institutions (RTPI) | |||

11 | Weight Factor of Beds’ Construction Subsidies of Pension Institutions (WBSPI) | Regulation Coefficient of Market Engagement of Pension Institution (RCMEPI) |

Number | Variable Name | Number | Variable Name |
---|---|---|---|

1 | GDP Growth (GDPG) | 15 | Total Amount of Demand of Elderly Population to Participate in Both Pension Community and Pension Institutions (TND) |

2 | Per Capita GDP (PCGDP) | 16 | Increase in National Pension Expenditure (INPE) |

3 | Per Capita Disposable Income (PCDI) | 17 | Increase Rate of National Pension Expenditure (IRNPE) |

4 | Elderly Population Increase (EPI) | 18 | Total Pension Expenditure of Pension Institutions (TPEPI) |

5 | Born Population (BP) | 19 | Total Pension Expenditure of Pension Community (TPEPC) |

6 | Death Population (DP) | 20 | Numbers of Pension Institutions (NPI) |

7 | Per Capita Pension Expenditure Increase (PCPEI) | 21 | Amount of Supply of Elderly Population Provided by Pension Institutions (NSPI) |

8 | Willingness of Disabled Elderly Population to Pay (WDEPP) | 22 | Amount of Pension Community (NPC) |

9 | Number of Disabled Elderly Pension Needs (NDDEP) | 23 | Amount of Supply of Elderly Population Provided by Pension Community (NSPC) |

10 | Willingness of Non-Disabled Elderly Population to Pay (WNEPP) | 24 | Total Amount of Supply of Elderly Population Provided by Both Pension Institutions and Pension Community (TNS) |

11 | Numbers of Demand of theNon-disabled Elderly Population (NDNEP) | 25 | Market Engagement of Pension Institutions (MEPI) |

12 | Ratio of Pension Expenditure to Per Capita Disposable Income (RPETP) | 26 | Market Engagement of Pension Community (MEPC) |

13 | Amount of Demand of the Elderly Population to Participate in Pension Institutions (NDPI) | 27 | Weight Factor of Pension Expenditure for Pension Institutions (WFPEPI) |

14 | Amount of Demand of Elderly Population to Participate in Pension Community (NDPC) | 28 | Weight Factor of Pension Expenditure for Pension Community (WFPEPC) |

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**Figure 6.**Causal closed-loop logic diagram for supply and demand system of aged service resources (the blue variable is the variable related to the total supply and demand ratio, the green variable is the variable related to institutional supply and demand ratio, and the orange variable is the variable related to community supply and demand ratio).

**Figure 7.**Comparison of simulated values of output variables. (

**a**) The model before optimization. (

**b**) The model after optimization.

**Figure 8.**Comparison for dynamic sensitivity of each parameter change to each output variable. (

**a**) Dynamic sensitivity comparison for the variation of the FPCDI to each output variable. (

**b**) Dynamic sensitivity comparison for the variation of the ACPI to each output variable. (

**c**) Dynamic sensitivity comparison for the variation of DRAP to each output variable. (

**d**) Dynamic sensitivity comparison for the variation of the RIAP to each output variable. (

**e**) Dynamic sensitivity comparison for the variation of the FOO to each output variable. (

**f**) Dynamic sensitivity comparison for the variation of the MEPI to each output variable. (

**g**) Dynamic sensitivity comparison for the variation of the WFPEPI to each output variable. (

**h**) Dynamic sensitivity comparison for the variation of the MEPC to each output variable. (

**i**) Dynamic sensitivity comparison for the variation of the WAPPI to each output variable. (

**j**) Dynamic sensitivity comparison for the variation of the WBSPI to each output variable. (

**k**) Dynamic sensitivity comparison for the variation of the ANPI to each output variable. (

**l**) Dynamic sensitivity comparison for the variation of the ANPC to each output variable. (

**m**) Dynamic sensitivity comparison for the variation of the MaxWD to each output variable. (

**n**) Dynamic sensitivity comparison for the variation of the MinWD to each output variable. (

**o**) Dynamic sensitivity comparison for the variation of the MaxWN to each output variable. (

**p**) Dynamic sensitivity comparison for the variation of the MinWN to each output variable. (

**q**) Dynamic sensitivity comparison for the variation of the GWD to each output variable. (

**r**) Dynamic sensitivity comparison for the variation of the GWN to each output variable.

**Figure 9.**Three-dimensional comparison diagram of dynamic sensitivity of 10% increase on each parameter. (

**a**) The influence of 10% increase on each parameter on the TRSD. (

**b**) The influence of 10% increase on each parameter on the RSDPI. (

**c**) The influence of 10% increase on each parameter on the RSDPC. (

**d**) The influence of 10% increase on each parameter on the TRSD. (

**e**) The influence of 10% increase on each parameter on the RSDPI. (

**f**) The influence of 10% increase on each parameter on the RSDPC.

**Figure 10.**Histogram of influence degree for 10% change of each parameter on the TRSD. (

**a**) Sensitivity index 1. (

**b**) Sensitivity index 2.

**Figure 11.**Histogram of influence degree for 10% change of each parameter on the RSDPI. (

**a**) Sensitivity index 1. (

**b**) Sensitivity index 2.

**Figure 12.**Histogram of influence degree for 10% change of each parameter on the RSDPC. (

**a**) Sensitivity index 1. (

**b**) Sensitivity index 2.

No. | Parameter Variables | No. | Parameter Variables | No. | Parameter Variables |
---|---|---|---|---|---|

1 | Factor of Per Capita Disposable Income (FPCDI) | 7 | Weight Factor of Pension Expenditure for Pension Institutions (WFPEPI) | 13 | Maximum Values of WDAPP (MaxWD) |

2 | Average of CPI (ACPI) | 8 | Market Engagement of Pension Community (MEPC) | 14 | Minimum Values of WDAPP (MinWD) |

3 | Disability Rate of Aging Population (DRAP) | 9 | Willingness of Aging Population to participate in Pension Institutions (WAPPI) | 15 | Maximum Values of WNAPP (MaxWN) |

4 | Rate of Increase in Aging Population (RIAP) | 10 | Weight factor of Beds’ construction Subsidies of Pension Institutions (WBSPI) | 16 | Minimum Values of WNAPP (MinWN) |

5 | Factor of OSPI and OSPC (FOO) | 11 | Average Numbers of beds in Pension Institutions (ANPI) | 17 | GWD ${K}_{A1}$ |

6 | Market Engagement of Pension Institutions (MEPI) | 12 | Average Amount of aging population served by Pension Communities (ANPC) | 18 | GWN ${K}_{A2}$ |

No. | Parameter Variables | No. | Parameter Variables | No. | Parameter Variables |
---|---|---|---|---|---|

19 | GDPGR | 23 | Regulation Coefficient of Market Engagement of Pension Institution (GMEPI) | 27 | Regulation Coefficient of Pension Expenditure (GNPE) |

20 | BR | 24 | Regulation Threshold of Market Engagement of Pension Communities (RTPC) | 28 | Balance Coefficient of Pension Community Supply and Demand Ratio (BGRR) |

21 | DR | 25 | Regulation Coefficient of Market Engagement of Pension Institution (GMEPC) | 29 | Regulation Threshold of Weight of Pension Expenditure in Communities (RTPE) |

22 | Regulation Threshold of Market Engagement of Pension institutions (RTPI) | 26 | Regulation Threshold of Total Ratio between Supply and Demand (RTT) | 30 | Gain of Weight Regulation of Pension Expenditure (GPE) |

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## Share and Cite

**MDPI and ACS Style**

Zhang, Y.; Zhang, M.; Hu, H.; He, X.
The Sensitivity Analysis for Supply and Demand Dynamics System of Aged Services Resource Allocation in China. *Systems* **2022**, *10*, 147.
https://doi.org/10.3390/systems10050147

**AMA Style**

Zhang Y, Zhang M, Hu H, He X.
The Sensitivity Analysis for Supply and Demand Dynamics System of Aged Services Resource Allocation in China. *Systems*. 2022; 10(5):147.
https://doi.org/10.3390/systems10050147

**Chicago/Turabian Style**

Zhang, Yijie, Mingli Zhang, Haiju Hu, and Xiaolong He.
2022. "The Sensitivity Analysis for Supply and Demand Dynamics System of Aged Services Resource Allocation in China" *Systems* 10, no. 5: 147.
https://doi.org/10.3390/systems10050147