2008 国际大学生数学建模比赛参赛作品
---------WHO 所属成员国卫生系统绩效评估
作品名称：Less Resources, more outcomes
参赛单位： 重庆大学
参赛时间：2008 年 2 月 15 日至 19 日
指导老师： 何仁斌
参赛队员：舒强 机械工程学院 05 级
罗双才 自动化学院 05 级
黎璨 计算机学院 05 级
2008 年国际大学生数学建模比赛参赛作品 作者：舒强、罗双才、黎璨
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Content
Less Resources, More Outcomes ............................................................................................................................... 4
1. Summary ........................................................................................................................................................ 4
2. Introduction ................................................................................................................................................... 5
3. Key Terminology ............................................................................................................................................. 5
4. Choosing output metrics for measuring health care system.......................................................................... 5
4.1 Goals of Health Care System ................................................................................................................ 6
4.2 Characteristics of a good health care system ...................................................................................... 6
4.3 Output metrics for measuring health care system ............................................................................... 6
5. Determining the weight of the metrics and data processing ................................................................. 8
5.1 Weights from statistical data ............................................................................................................... 8
5.2 Data processing ................................................................................................................................... 9
6. Input and Output of Health Care System ....................................................................................................... 9
6.1 Aspects of Input ................................................................................................................................. 10
6.2 Aspects of Output .............................................................................................................................. 11
7. Evaluation System I : Absolute Effectiveness of HCS .................................................................................... 11
7.1Background ......................................................................................................................................... 11
7.2Assumptions ........................................................................................................................................ 11
7.3Two approaches for evaluation .......................................................................................................... 11
1. Approach A : Weighted Average Evaluation Based Model .................................................................. 11
2. Approach B: Fuzzy Comprehensive Evaluation Based Model [19][20] ................................................. 12
7.4 Applying the Evaluation of Absolute Effectiveness Method .............................................................. 14
8. Evaluation system II: Relative Effectiveness of HCS ..................................................................................... 16
8.1 Only output doesn’t work .................................................................................................................. 16
8.2 Assumptions ....................................................................................................................................... 16
8.3 Constructing the Model ..................................................................................................................... 16
8.4 Applying the Evaluation of Relative Effectiveness Method ................................................................ 17
9. EAE VS ERE: which is better? ........................................................................................................................ 17
9.1 USA VS Norway .................................................................................................................................. 18
9.2 USA VS Pakistan ................................................................................................................................. 18
10. Less Resources, more outcomes ................................................................................................................. 19
10.1Multiple Logistic Regression Model .................................................................................................. 19
10.1.1 Output as function of Input ........................................................................................................... 19
10.1.2Assumptions ................................................................................................................................... 19
10.1.3Constructing the model.................................................................................................................. 19
10.1.4. Estimation of parameters ............................................................................................................ 20
10.1.5How the six metrics influence the outcomes? ................................................................................ 20
10.2 Taking USA into consideration ......................................................................................................... 22
10.2.1Assumptions ................................................................................................................................... 22
10.2.2 Allocation Coefficient
............................................................................................................. 22
10.3 Scenario 1: Less expenditure to achieve the same goal ................................................................... 24
10.3.1 Objective function
：
..................................................................................................................... 24
10.3.2 Constraints .................................................................................................................................... 25
2008 年国际大学生数学建模比赛参赛作品 作者：舒强、罗双才、黎璨
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10.3.3 Optimization model 1 ................................................................................................................... 25
10.3.4 Solutions of the model .................................................................................................................. 25
10.4. Scenario2: More outcomes with the same expenditure ................................................................. 26
10.4.1Objective function .......................................................................................................................... 26
10.4.2Constraints ..................................................................................................................................... 26
10.4.3 Optimization model 2 ................................................................................................................... 26
10.4.4Solutions to the model ................................................................................................................... 27
15. Strengths and Weaknesses ........................................................................................................................ 27
Strengths .................................................................................................................................................. 27
Weaknesses ............................................................................................................................................. 27
16. References .................................................................................................................................................. 28
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Less Resources, More Outcomes
1. Summary
In this paper, we regard the health care system (HCS) as a system with input and output, representing total expenditure on
health and its goal attainment respectively. Our goal is to minimize the total expenditure on health to archive the same or maximize
the attainment under given expenditure.
First, five output metrics and six input metrics are specified. Output metrics are overall level of health, distribution of health
in the population,etc. Input metrics are physician density per 1000 population, private prepaid plans as % private expenditure on
health, etc.
Second, to evaluate the effectiveness of HCS, two evaluation systems are employed in this paper:
Evaluation of Absolute Effectiveness(EAE)
This evaluation system only deals with the output of HCS，and we define Absolute Total Score (ATS) to quantify the
effectiveness. During the evaluation process, weighted average sum of the five output metrics is defined as ATS, and the
fuzzy theory is also employed to help assess HCS.
Evaluation of Relative Effectiveness(ERE)
This evaluation system deals with the output as well as its input, and also we define Relative Total Score (RTS) to
quantify the effectiveness. The measurement to ATS is units of output produced by unit of input.
Applying the two kinds of evaluation system to evaluate HCS of 34 countries (USA included), we can find some countries which
rank in a higher position in EAE get a relatively lower rank in ERE, such as Norway and USA, indicating that their HCS should have
been able to archive more under their current resources .
Therefore, taking USA into consideration, we try to explore how the input influences the output and archive the goal: less input,
more output. Then three models are constructed to our goal:
Multiple Logistic Regression
We model the output as function of input by the logistic equation. In more detains, we model ATS (output) as the
function of total expenditure on health system. By curve fitting, we estimate the parameters in logistic equation, and
statistical test presents us a satisfactory result.
Linear Optimization Model on minimizing the total expenditure on health
We try to minimize the total expenditure and at the same time archive the same, that is to get a ATS of 0.8116. We
employ software to solve the model, and by the analysis of the results. We cut it to 2023.2 billion dollars, compared to
the original data 2109.8 billion dollars.
Linear Optimization Model on maximizing the attainment
. We try to maximize the attainment (absolute total score) under the same total expenditure in2007.And we optimize the
ATS to 0.8823, compared to the original data 0.8116.
Finally, we discuss strengths and weaknesses of our models and make necessary recommendations to the
policy-makers。
2008 年国际大学生数学建模比赛参赛作品 作者：舒强、罗双才、黎璨
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2. Introduction
Today and every day, the lives of vast numbers of people lie in the hands of health systems.
From the safe delivery of a healthy baby to the care with dignity of the frail elderly, health systems
have a vital and continuing responsibility to people throughout the lifespan. They are crucial to the
healthy development of individuals, families and societies everywhere. Due to the irreplaceable
role that the health care systems play in residents’ life, better health care system is needed.
“Improving performance” is therefore the key words today.
However, nowadays health care systems in many countries do not exhibit enough high
effectiveness in guaranteeing residents’ good health and a long life expectancy. In some countries,
their government invests large amount of money on the health care systems, however, they didn’t
archive what they should have been to archive. We try to explore an optimized system in this
paper.
3. Key Terminology
Health Care System (HCS)
Health Care System is such a system that has its input and output, representing total
expenditure on health and its goal attainment respectively.
Evaluation of Absolute Effectiveness of Health Care System (EAE)
It is a kind of evaluation system that only considers the outcomes of the health care system,
saying nothing to do with the input (resources), and adapts the outcomes as measurement to
effectiveness.
Evaluation of Relative Effectiveness of Health Care System (ERE)
It is a kind of evaluation system that considers the outcomes of the health care system as well
its inputs, and adapts units of output produced by unit of input as measurement to
effectiveness.
Absolute Total Score (ATS)
Overall score for the evaluation of absolute effectiveness of health care systems
Relative Total Score (RTS)
Overall score for the evaluation of relative effectiveness of health care systems
Input Metrics (IM)
Metrics that are specified to assess input of HCS
Output Metrics (OM)
Metrics that are specified to assess output of HCS
4. Choosing output metrics for measuring health care system
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