历年美赛数学建模优秀论文大全

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 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

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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 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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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 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 ATS to 0.8823, compared to the original data 0.8116.

. We try to maximize the attainment (absolute total score) under the same total expenditure in2007.And we optimize the

Finally, we discuss strengths and weaknesses of our models and make necessary recommendations to the policy-makers。

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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 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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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 Table 1. Notation for goals and metrics Goals of Health System Health Responsiveness Fairness in Finance Notation U1 U2 U3 Metrics for Evaluation Overall level of health Distribution of health in the population Overall level of responsiveness Distribution of responsiveness Distribution of financial contribution Notation u1 u2 u3 u4 u5 4.1 Goals of Health Care System

According to the World Health Report in 2000, the WHO pointed out the three goal of health care system, each goal with different priority [WHO 2000].  Better Health

Better health is unquestionably the primary goal of a health system, with the highest priority.  Fairness in financial contribution.

Fairness in financial contribution is the second goal, with a relatively lower priority to health.  Responsiveness

Responsiveness to people’s expectations in regard to non-health matters reflects the importance of respecting people’s dignity, autonomy and the confidentiality of information, and is the third goal ,with the lowest priority.

4.2 Characteristics of a good health care system

Goodness&&Fairness [WHO 2000]

As the WHO defined what a good health care system was in its World Health Report in2000, a good health care system is a combination of Goodness and Fairness. A good health system, above all, contributes to good health. But it is not always satisfactory to protect or improve the average health of the population, if at the same time inequality worsens or remains high because the gain accrues disproportionately to those already enjoying better health. The health system also has the responsibility to try to reduce inequalities by preferentially improving the health of the worse-off, wherever these inequalities are caused by conditions amenable to intervention. The objective of good health is really twofold: the best attainable average level – goodness – and the smallest feasible differences among individuals and groups – fairness. A gain in either one of these, with no change in the other, constitutes an improvement, but the two may be in conflict.

4.3 Output metrics for measuring health care system

To assess a health care system, we must measure the following five output metrics：  Overall level of health

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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 We use the measure of disability-adjusted life expectancy – DALE to assess the overall level of population health. This measure converts the total life expectancy for a population to the equivalent number of years of ‘good health’.  Distribution of health in the population

We use the index of equality of child survival to assess distribution of health in the

population. It is based on the distribution of child survival across countries, and takes advantage of the widely available and extensive information on complete birth histories in the demographic and health surveys and small area vital registration data on child mortality. WHO defined it as follows[WHO 2000]：

nnxixji1j1 Equalityofchildsurvival120.52nx3 （1） Where x is the survival time of a given child and x is the mean survival time across children

 Overall level of responsiveness

Responsiveness includes two major components:

(1) Respect for people (including dignity, confidentiality and autonomy of individuals and

families to decide about their own health);

(2) Client orientation (including prompt attention, access to social support networks during

care, quality of basic amenities and choice of provider). The level of responsiveness was based on a survey of key informants in selected countries. And WHO defined the index of Overall level of responsiveness as weighted average of its seven components: [WHO 2000]

111LevelofResponsivenessDignitConfidentialityAutonomy33313 （2） PromptattentionQualityofamentities52011AccesstosocialSupportnetworkChoiceofprovider1020 Distribution of responsiveness

We use a simple approach to quantize the distribution of responsiveness. That is respondents in the key informant survey were asked to identify groups who were disadvantaged with regard to responsiveness. The number of times a particular group was identified as disadvantaged was used to calculate a key informant intensity score. Four groups had high key informant intensity scores: poor people, women, old people, and indigenous groups or racially disadvantaged groups (in most instances minorities). The key informant intensity scores for these four groups were multi- plied by the actual percentage of the population within these vulnerable groups in a country to calculate a simple measure of responsiveness inequality ranging from 0 to 1. The total score was calculated taking into account the fact that some individuals belong to more than one disadvantaged group.

 Distribution of financial contribution

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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 The fair financing measure estimates the degree to which health funding is raised according to the ability to pay for all members of the population. It captures concerns such as progressivity, and protection from catastrophic health costs. Fair financing is only concerned with distribution. In order that complete equality of household contributions is 1 and 0 is below the largest degree of inequality observed across countries, WHO defined the in fairness index. And the index is of the form:[WHO2000]

nHFCiHFCFairnessoffinancecontribution14i10.125n3(3)

Where HFC is the financial contribution of a given household and HFC is the average financial contribution across households.

5. Determining the weight of the metrics and data processing

5.1 Weights from statistical data

The key informant survey, consisting of 1791 interviews in 35 countries, yielded scores (from 0 to 10) on each element of responsiveness, as well as overall scores. A second, Internet-based survey of 1006 participants (half from within WHO) generated opinions about the relative importance of the elements, which were used to combine the element scores into an overall score instead of just taking the mean or using the key informants’ overall responses[World Health Report 2000].See figure 1 and 2:

25%50%HealthResponsivenessFairness inFinance25% Figure 1 Weights for the three goals of health system

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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 Overall level of health25%12.50%12.50%25%25%Distribution of health in the populationOverall level of responsivenessDistribution of responsiveness Distribution of financial contributionFigure 2 Weights of the five metrics

Figure1 and figure 2 illustrate the weights of three goals of health system and five metrics respectively.

5.2 Data processing

 Data Source

We get our data from WHO Statistical Information System on the official web site of WHO (http://www.who.int/whosis/en/index.html) And data in ‘THE WORLD HEALTH STATISTICS REPORT’ from 2005 to 2007 and ‘World Health Report ‘in 2000 is now accessible.  Normalization

To ensure comparability of effectiveness of health care system, metrics must be normalized by the following given formulation:

NormalizedDataRawDatamin(RawData) (4)

max(RawData)min(RawData)Where max greatest number of Raw Data and min is the least one.

6. Input and Output of Health Care System

In this paper, we consider Health Care System a system with both input and output (see fig.3). Five output metrics and six input metrics are specified in this paper.

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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 Input Health Care System Output

6.1 Aspects of Input

Figure3: How a health care system works?

Table 2 Notation for Input and Output Input Physician density per 1000 population Nurse density per 1000 population Social Security expenditure on health as % of government expenditure on health Private prepaid plans as % of Private expenditure on health External resources for health as % of total expenditure on health Out- of- Pocket expenditure as % of private expenditure on health Notation m1 m2 m3 m4 Output Overall level of health Distribution of health in the population Overall level of responsiveness Distribution of responsiveness Notation u1 u2 u3 u4 u5 m5 Distribution of financial contribution m6 We define Input Vector as a set of the four elements of input, that isInputVector{m1,m2,m3,m4,m5,m6}      

Physician density per 1000 population Nurse density per 1000 population

Social Security expenditure on health as % of government expenditure on health

Private prepaid plans as % of private expenditure on health Physician density per 1000 population External resources for health as % of total expenditure on health Out- of- Pocket expenditure as % of private expenditure on health

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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 6.2 Aspects of Output

Also, we define Output Vector as a set of the five elements of Output, that isOutputVector{u1u2u3,u4,u5}     

Overall level of health

Distribution of health in the population Overall level of responsiveness Distribution of responsiveness

Distribution of financial contribution

7. Evaluation System I : Absolute Effectiveness of HCS

7.1Background

In this part, we deal with the evaluation of health care system by the way of “absolute”, a way that only considers the output of the system. Then five typical metrics that can well represent the outcomes of the system are chosen for evaluation. Based on the five metrics, two empirical approaches are employed for evaluation. The former one is weighted average sum as a comprehensive indicator of the effectiveness, and the latter one is based on the theory of fuzzy mathematics.

7.2Assumptions

 We consider using output of the health system to evaluate the effectiveness acceptable here.  The five metrics can represent enough information for evaluation of the health care system,

thus we consider it reasonable and enough for us to use the metrics.  We don’t consider the interaction effect of metrics on the results.

 There is simply linear relationship between the metrics and the result of evaluation, thus

weighted average sum approach can reasonably reflect how the metrics influence the results.  As there is no specific definition on how well a health system is or the extent of “effectiveness”,

thus fuzzy comprehensive based approach employed here is acceptable.  Most the data collected is reliable, neglecting its accuracy.

7.3Two approaches for evaluation

1. Approach A : Weighted Average Evaluation Based Model

We define Absolute Total Score (ATS) as an indicator that can be used to describe how a heath system works. Based on the assumptions above, we can formulate the Absolute Total Score as follows:

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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 5AbsoluteTotalScoreiuii1 （5）Where ui represents the ith output metric and i is the weight corresponding to the metric.

By comparing the Absolute Total Score of a system, we can compare systems among

countries. Meanwhile, by calculating the value of five metrics, we can also get the rank of systems with respect to each metric.

2. Approach B: Fuzzy Comprehensive Evaluation Based Model [19][20]

As there is no specific definition on how well a health system is or the extent of “effectiveness”, we employ the theory of fuzzy mathematics to assess it.

 Combination of factors

To assess the absolute effectiveness of health care system, we focus on three aspects of health care system that is health, responsiveness and fair financial contribution. Health can be divided into two major parts, the overall level of health; the distribution of health in the population. Responsiveness can be divided into two major part, the overall level of responsiveness; the distribution of responsiveness.

The following figure illustrates the relationships and levels of those indicators.

the overall level of healthHealthAbsolute effectiveness ofhealth care systemResponsivenessthe distribution of responsivenessthe distribution of health in populationthe overall level of responsivenessFairness in financialthe distribution of responsiveness

Figure 5: Hierarchy structure of factors

We use fuzzy set U{u1u2u3....u5}’

Where u1u2u3....u5 is the indication for the five basic metrics respectively,

To include all the five basic metrics, and divided it into three groups, we have

U{U1U2U3} ,

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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 Where fuzzy subset U1 U2 U3 represents health, responsiveness and fair financial contribution respectively.

Then we have U1{u1u2} , U2{u3u4} , and U3{u5}. The weight set for U is W(12)3,

Where

123 is the weight of U1, U2and U3 respectively.

And the weight set for U1is indicated by W1(1,11，2), where w11w12is weight that metrics u1 and u2 account for respectively. The weight set for U2is indicated by W2(2,12，2), where

2，12，2is weight that metrics u3 and u4 account for respectively.

 Determine membership degree for each metric

Assume that there are n countries of to be compared in terms of absolute effectiveness of their health care system. We take normalized form membership functions for each metric so that values of all the metrics of different levels can be constrained between 0 and 1. By the membership degree function

（uU~i,k）ui,jmin(ui,k)1kn1knmax(ui,k)min(ui,k)1knai,j, (6)

Where ui,kindicates the ith metric of the kth country.

 Deducing of model

For the fuzzy setU1, the single factor judgment matrix

a1，a1，12...a1,nR1

aa...a12，22,n2，By weighted average method, we can easily have matrix B1

B1[b1,1b1,2...b1,n] , where b1,j1,i.ai,ji12(j1,2...n)

For the levelU1, the single factor judgment matrix

a3，a3，12...a3,nR2

aa...a14，24,n4，Similarly, we have matrix B2

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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 4B2[b2,1b2,2...b2,n] ，where b2,ji32,i2.ai,j(j1,2...n)

And B3[b3,1b3,2n.b..3]u,[u3,1u..,n.3.

]Finally, we perform comprehensive evaluation on the top level. Then the R is

bn1,B1b1,1b1,...2 RB2b2,1b2...b,2n2,Bbbn233,1b3,...3,By weighted average method, we have overall synthetic judge matrix

(7)

B[b1b2...bn] , where bji.bi,ji13(j1,2...n)

The value of each element in B can be looked on as the absolute effectiveness of health care system for each country. So the larger the value of element in matrix B is, more effective the health care system of the country to which this value is corresponding is.

7.4 Applying the Evaluation of Absolute Effectiveness Method

 Applying Approach A

Apply approach A to 34 countries (USA included), and the rank is given in the following table. We focus on the three goals of health system, the five output metrics as well as the overall rank.

Table3 Absolute Effectiveness of 34 countries, rank by 5 output metrics , estimates for 2007

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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

From table 3, we can see:

 With respect to overall health, Japan ranks the first and Rwanda the lowest, while the USA

ranks in the lower level.

 With respect to Responsiveness, the USA is leading in the 23 developed countries, while

Uganda ranks last.

 With respect to Absolute Effectiveness, Japan leads first, while the USA ranks 3, a relative

lower level.



Comparison between Approach A and Approach B

By the Evaluation of Absolute Effectiveness(EAE) method, the policy makers and other related department can judge whether the current system approaches its goal, in other words , we can identify whether the system can satisfy residents’ requirement of health. And the Evaluation of Relative Effectiveness (ERE) method can evaluate the efficiency of usage of resources, which can give guidance for adjusting and improving health care system.

Table 4 Horizontal and vertical comparison of HCS by EAE, estimates for 2006 and 2007 Approach A 2007 1 2 4 5 6 7 3 9 11 8 12 13 10 14 16 15 18 Approach B 2006 1 2 3 8 4 6 4 7 9 12 13 10 1 14 16 15 19 2007 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 country Portugal Poland Hungary Mexico Turkey Approach A 2006 2007 18 18 17 20 20 21 21 23 25 24 22 26 27 28 29 30 32 31 33 34 22 19 26 23 24 25 27 30 29 28 32 33 31 34 Approach B 2006 18 17 20 21 23 25 24 22 26 28 27 29 30 32 31 33 34 2007 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 country 2006 Japan 1 Norway 2 Iceland 3 Australia Canada Austria USA Finland Denmark France UK Belgium Italy New Zealand Spain Israel Ireland 5 4 6 8 7 9 12 11 10 13 14 16 15 19 Republic of Korea Uzbekistan India Mongolia Turkmenistan Pakistan China Uganda Sudan Rwanda Zambia Nepal From table 4, we can see

 Through comparing the ranks of countries using the two approaches respectively in the

same year, we find that the results of two different approaches to determine Evaluation of

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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 Absolute Effectiveness (EAE) do not change significantly, with ranks of most countries interested in having not big change. The comparing between the two approaches proves correctness and rationality of each other.

 Through comparing the ranks of countries using the two approaches respectively in the

different year, we find the ranks of countries are nearly stable.

 Comparing to Japan which has a quite good health system, the USA’s absolute

effectiveness of health care system is not as high as Jａpan.

8. Evaluation system II: Relative Effectiveness of HCS

8.1 Only output doesn’t work

The overall indicator of attainment, like the five specific metrics which compose it, is an absolute measure. It says how well a country has done in reaching the different goals, but it says nothing about how that outcome compares to what might have been achieved with the resources available in the country. It is achievement relative to resource that is the critical measure of a health system’s performance.

For example, if Sweden enjoys better health than Uganda – life expectancy is almost exactly twice as long – that is in large part because it spends exactly 35 times as much per capita on its health system. But Pakistan spends almost precisely the same amount per person as Uganda, out of an income per person that is close to Uganda’s, and yet it has a life expectancy almost 25 years higher. This is the crucial comparison: why are health outcomes in Pakistan so much better, for the same expenditure? And it is health expenditure that matters, not the country’s total income, because one society may choose to spend less of a given income on health than another. Therefore, each health system should be judged according to the resources actually at its disposal, not according to other resources which in principle could have been devoted to health but were used for something else. Therefore, corresponding to the Evaluation of Absolute Effectiveness, we introduce another evaluation system, the Evaluation of Relative Effectiveness (ERE).

8.2 Assumptions

 We can assess the input of health care system by the total money it needs to operate.

 Total expenditure on health as % of GDP alone can be used to quantify the input of health

care system.

8.3 Constructing the Model

The concept of Value Engineering was introduced to describe the relationship between costs, function and value [L· D· Miles 1943]. It defines value as function of costs and function in the form:

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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 Value=Similarly, we define Relative Total Score as:

FunctionCostsAbsolute Total Score

Input

(7)

Relative Total Score = (8)

Where Relative Total Score is defined to assess relative effectiveness of health care system

By comparing the Relative Total Score, we can assess how a health care system works according to what it should have been archived. Here, to be simplified, we use Total expenditure on health as % of GDP to quantify the input.

8.4 Applying the Evaluation of Relative Effectiveness Method

Table 5 Relative Effectiveness of HCS, ranked by the Relative Evaluation system, estimates for 2007

Country Pakistan Poland Iceland Ireland Finland Japan Korea Uzbekistan

U K Spain China Mexico Denmark Turkmenistan

Italy India Israel

Total expenditure on health as % of GDP

2.2 6.2 7.2 7.1 7.4 7.8 5.5 5.1 8.1 8.1 4.7 6.5 8.6 4.8 8.7 5 8.7

R-Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Country Netherlands Hungary Norway Australia Canada Belgium Austria Mongolia France Portugal Turkey Sudan U S A Uganda Nepal Zambia Rwanda

Total expenditure on health as % of GDP

9.2 7.9 9.7 9.6 9.8 9.7 10.3 6 10.5 9.8 7.7 4.1 15.4 7.8 5.6 6.3 7.5

R-Rank 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

From the table (5), we can find that

 Pakistan ranks the first, and Rwanda is last. Especially some developed ones, such as America,

ranks in the lower level.

 America has the largest percentage of GDP spent on health care, while Pakistan has only 2.2%.

9. EAE VS ERE: which is better?

Apply the two evaluation system to 34 countries, we focus on the different ranks from the two evaluation

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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 systems.

Table 6 EAE VS ERE, rank comparison

From table 6, we can see: Comparing to ranks in terms of Absolutely Evaluation of Effectiveness, the new ranks of these countries change significantly.

 Ranks of countries having large percent of GDP spent on health care such as USA, Norway, Australia, Canada,

Austria, France decrease by more then 15, especially for USA of which rank declines from 7 to30.This means that these countries do not make the most of their inputs.

Ranks of countries having small percent of GDP spent on health care such as Pakistan increases from 28 to 1.This means that this country makes the most of its inputs. This may be a good example that those developed countries like the USA should learn from. But for developing countries, especially those having poor health care system, no matter how efficient their health care system is, they still can not supply good enough health service, simply because they have not enough resources to input into health care system. 9.1 USA VS Norway From the aspect of Evaluation of Absolute Effectiveness, we can see that USA ranks 7th, while Norway ranks 2, while from the aspect of Evaluation of Relative Effectiveness, the USA ranks 30th, and Norway 20th.

9.2 USA VS Pakistan Health care system of the USA is better than Pakistan from the aspect of Evaluation of Absolute Effectiveness obviously. However, Pakistan ranks first from aspect of Evaluation of Relative Effectiveness, while America ranks only 30th, a quite low rank.

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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 10. Less Resources, more outcomes

10.1Multiple Logistic Regression Model

10.1.1 Output as function of Input

We need to determine whether various changes can improve the overall quality of a country’s health care system. Thus, we focus on how the output of a system changes due to variation of input. We employ the logistic equation to model the relationship between output and input [Gotelli 1998]. By the equation, we can clearly see how input influences the output.

10.1.2Assumptions

 Input can be qualified by weighted average sum of the six input metrics, and the weight

reflects how the metric contributes to the input.

 Output can be qualified by weighted average sum of the five output metrics (ATS), and the

weight reflects how the metric contributes to the input,

 Relationship between input and output of health system can be quantized as logistic equation,

that is the output grows as the inputs growth, and the growth rate is rising at first, but as the output approaches a certain value, its growth rate will gradually decrease to zero.

10.1.3Constructing the model

Here we set the Absolute Total Score as the quantification of output, and the logistical equation is given as:

dATSATSRATS(1) (9) dMKWhere R is the growth rate, K is the upper bound of output and M is the quantification of input. For simplicity, we let a=R and b=R/K, so that:

dATSaATSbATS2 (10) dMWith the initial condition ATS(M0)ATS0,the equation has closed-form solutions:

ATS(M)aeaMATSabATSbeaMATS(11)

According to the assumption that input can be quantified by the linear weighed average sum of input metrics, we can quantify input as:

Where

Mimi0 (12)

i16i is the weight and mi is the ith input metric

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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 Then from (11) and (12), we can get

a(ATS(M)aeimi0)i166ATSATSa(abATSbeimi0)i1(13)

The figure below illustrates how output changes as input varies:

Figure 6 Solution to the logistic equation, with output plotted as a function a input

10.1.4. Estimation of parameters

We estimate the parameters for (13) by curve fit, statistical data collected from the 34 countries mentioned above is employed to help the curve fit, and we get

AST(M)1.10321.0958eM（14）

WithM29.98220.498m1593923m257.78m38.4232m418.556m59.9023m6(15)

Also, we do statistical tests for our model, and it presents us a satisfactory result:

Residual=0.051, and Confidence Degree=1-Residual=0.949, indicating that it passes the statistical test.

10.1.5How the six metrics influence the outcomes?

Since we have equation (14) and (15), we can consider

ATSf(M)f(m1,m2,...,m6) (16)

Let us consider And (11) and (12) can get

ATSATSMMi (17) miMmmiiATSaATSbATS2 (18) M 20 / 29

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

Mi (19) miThen from (17),(18) and (19), we can get

ATSATSM(aATSbATS2)i (20) Mmi miAnd the value of partial differential

ATS show how metric mi influences the output. mi

Also, by controlling variable m2,m3,m4,m5, m6, and vary variable m1, we can see how m1 influences the output; similarly we can get how m2,m3,m4,m5,and m6 influences the output respectively.

Figure7 How input metric influences the output

As figure 7 illustrates：

 With respect to private prepaid plans

It is negatively correlated to AST. That is as the increase of private prepaid plans, AST decreases. The reason for this is mainly due to people of their own country do not trust the health care system, they store a large amount of money to spend by the sick and hospitalized, which reflects the health care system is far from perfect, so lower scores.  With respect to the other five metrics

We can see that the AST increases as the other five input metrics increase, only that the increasing rate is different.

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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 10.2 Taking USA into consideration

As we have analyzed above, USA ranks 3rd by the evaluation of absolute effectiveness while ranks 7th by the evaluation of relative effectiveness. The difference between the ranks indicates that health system of USA should have archived more under the current total expenditure on health. In this part, we try to explore an optimized combination of input metrics to minimize the input or maximize the output. Thus we focus the USA in 2007,trying to minimize the total expenditure on health and at the same time archive the same attainment, or to maximize the attainment under the same expenditure.

In 2007, by the evaluation of absolute effectiveness, USA gets an absolute total score 0.8116.We get all concerned data for USA in 2007.See the following table:

10.2.1Assumptions

 The total expenditure on health is employed to as the quantification of the input of health care system.

 Total expenditure on health every year is divided into six parts: expenditure on physician wage, expenditure

on nurse wage, expenditure on social security, and expenditure on private prepaid plans, expenditure on

Out-of-pocket expenditure compensation and expenditure on the purchase of hospital beds.

Total expenditure on Physician Total expenditure on Nurse Total expenditure on Social security Total expenditure on health

Total expenditure on Private prepaid plans Total expenditure on Hospital beds Total expenditure on Out-of-pocket compensation 10.2.2 Allocation Coefficient 

To transit the six input metrics into expenditure, we define the “Allocation Coefficient” as the coefficient that Transits the six input metrics to expenditure:

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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

Table 5 Data collected to determine Allocation Coefficient Items we concern Total population Gross domestic product per capita Total expenditure on health as % of GDP General government expenditure on health as % of Total expenditure on health Private expenditure on health as % of Total expenditure on health Social security expenditure on health as % of general government expenditure on health Private prepaid plans as % private expenditure on health Hospital bed on per 1000 capital Out-of-poke expenditure as % of private expenditure on health The average income of Physician per year The average income of Nurse per year Physician Density per 1000 population Nurse Density per 1000 population Average fees for bed

Value 298,213,000 46950 15.40% 44.70% 55.30% 0.1% 0.40% 33 84.90% 181850 70000 2.3 12.12 300  GDP=Total population×GDP per capita

 Total expenditure on health=(GDP×Total expenditure on health as % of GDP)/100

 General government expenditure on health= (General government expenditure on health as % of Total expenditure on health×Total expenditure on health)/100

 Social security expenditure on health= (Social security expenditure on health as % of general

government expenditure on health×general government expenditure on health)/100

 Private expenditure on health = (Private expenditure on health as % of Total expenditure on health×

Total expenditure on health)/100

 Private prepaid plans expenditure= (Private prepaid plans as % private expenditure on health×Private

expenditure on health)/100

 Out-of-pocket expenditure= (Out-of-pocket expenditure as % of private expenditure on health×

Private expenditure on health)/100

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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  Expenditure on payment of Physician= (Total Population×Physician Density per 1000 population×The

average income of Physician per year)/1000  Expenditure on payment of Nurse= (Total Population×Nurse Density per 1000 population×The

average income of Nurse per year)/1000

 Expenditure on beds= (Hospital bed on per 1000 capital×Total Population ×Average fees for

bed)/1000

Table 6 Values for allocation coefficient Input metrics Physician Density per1000population Nurse Density per 1000 population Social security expenditure on health as% of general government expenditure on health Private prepaid plans as % private expenditure on health Hospital bed on per 1000 capital Out-of-poke expenditure as % of private expenditure on health Allocation Coefficient 163.38 90.4 3231.9 3998.5 0.4 39.984 Notation 1 2 3 4 5 6 10.3 Scenario 1: Less expenditure to achieve the same goal

In 2007, USA ranks 3rd by the evaluation of absolute effectiveness and gets a absolute total score (ATS) 0.8116.We try to minimize the total expenditure on health and at the same time get a ATS of 0.8116.

10.3.1 Objective function：

Our goal is to minimize the total expenditure on health by the optimized combination of the six

input metrics. Thus we can get the total expenditure M by Objective function Min(Mimi)

i=16

Where i is the allocation coefficient, see table 6.

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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 10.3.2 Constraints

From the logistic model, we get the relationship between total expenditure and the absolute total score ATS by the logistic equation:

AST(M)1.10321.0958eM

 ATS=0.8116, this guarantees the absolute total score doesn’t change.  Mi>0, this means no metrics is negative.  

m3,4,6100 ,this is the definition of metricsm1,2,51000,this is the definition of metrics

10.3.3 Optimization model 1

From the analysis above, we can get an optimization model, shown as follows:

Min ( M=imi)i=16st：（M）=0.8116AST m0im3,4,6100m1,2,51000i10.3.4 Solutions of the model

Table 7 Solution to optimization to model1 Unit: million$ USA Current Solution Physician 2.5600 1.8755 Nurse 9.370 6.510 Social security 0 0 Private Hospital Out-of prepaid bed -pocket 0.4000 0.18 ３３ 50.3 AST 0.8116 0.8116 Money Per capita 6263.8 5992.2 84.9 82.9656 For the condition of America, we recommend the USA spend as less expenditure as possible and at the same time maintains its current AST. Through adjusting each kind of resource according to table (7), we can make each American save 6263.8-5992.2=$388.18, so the total money saved in America is 107.8 billion. We reduce the cost mainly through decreasing the number of physicians, nurses, and increasing hospital beds.

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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 10.4. Scenario2: More outcomes with the same expenditure

In 2007, USA ranks 3rd by the evaluation of absolute effectiveness and gets an absolute total score (ATS) 0.8116.We try to maximize absolute total score with the same expenditure on health of 1146 billion dollars.

10.4.1Objective function

Our goal is to maximize the absolute total score ATS and it is given as the function of total expenditure on health in the form of logistic equation:

Thus our objective function is Max AST （M）10.4.2Constraints



m2109.810, this guarantees total expenditure on health is 2109.8 billion dollars.

9ii=16 Mi>0, this means no metrics is negative.  

m3,4,6100 ,this is the definition of metricsm1,2,51000,this is the definition of metrics

10.4.3 Optimization model 2

From the analysis above, we can get an optimization model, shown as follows:

Max AST（M）69im2109.810i=1mi0m13,4,6

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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 10.4.4Solutions to the model

Table8 Solution to optimization to model2 Unit: million$ USA Physician Nurse Social security 0 0.4000 Private Hospitaprepaid l Bed 0.4000 0.1000 ３３ ６５ Out-of -pocket 84.9 50.0000 AST Money Per capita 6263.8 6263.8 Current Solution 2.5600 5.7000 9.37 17.2 0.8116 0.8823

We recommend the USA improve AST as much as possible and at the same time maintain its current expenditure. Through adjusting each kind of resource according to table (8), we can make each AST of America up to 0.8823, increasing by 0.0707. We can improve AST mainly through increasing number of physicians and nurses and hospital beds, at the same time reducing Out-of-pocket expenditure as % of private expenditure on health.

15. Strengths and Weaknesses Strengths

 By the Evaluation of Absolute Effectiveness(EAE) method, the policy makes and other

related department can judge whether the current system approaches its goal, in other words , we can identify whether the system can satisfy residents’ requirement of health. And the Evaluation of Relative Effectiveness (ERE) method can evaluate the efficiency of usage of resources, which can give guidance for adjusting and improving health care system.

Weaknesses

 When applying the ERE method, we only choose Total expenditure on health of GDP as

metrics, which can evaluate the efficiency of usage of expenditure but can not help to find the concrete reasons for low output of health care system.

 Inefficiency of metrics of some countries limits our choosing of metrics; as a result,

sometimes we have to exclude some metrics that may have big influence on the assessment of health care systems.

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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 16. References

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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 European Journal of Operational Research Volume 1, Issue 1, 16 August 2008, Pages 132-145

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