2023年12月11日发(作者：中考数学试卷真题吕梁)
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 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 . 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 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]：
nnxixji1j1
Equalityofchildsurvival120.52nx3 （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]
111LevelofResponsivenessDignitConfidentialityAutonomy33313 （2）
PromptattentionQualityofamentities52011AccesstosocialSupportnetworkChoiceofprovider1020 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]
nHFCiHFCFairnessoffinancecontribution14i10.125n3(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
(/whosis/en/)
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:
NormalizedDataRawDatamin(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
5AbsoluteTotalScoreiuii1
（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{u5}’
Where
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(12)3,
Where
123 is the weight of
U1,
U2and
U3 respectively.
And the weight set for
U1is indicated by
W1(1,11，2), where
w11w12is weight that metrics
u1
and
u2 account for
respectively. The weight set for
U2is indicated by
W2(2,12，2), where
2，12，2is 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,jmin(ui,k)1kn1knmax(ui,k)min(ui,k)1knai,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,nR1
B1
B1[b1,b1,n] , where
b1,j1,,ji12(j)
For the levelU1, the single factor judgment matrix
a3，a3，12...a3,nR2
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,b2,n] ，where
b2,ji32,i,j(j)
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
bn1,B1b1,1b1,...2
RB2b,2n2,Bbbn233,1b3,...3,By weighted average method, we have overall synthetic judge matrix
(7)
B[bn] , where
bj,ji13(j)
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 ScoreInput(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 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 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:
dATSATSRATS(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:
dATSaATSbATS2 (10)
dMWith the initial condition
ATS(M0)ATS0,the equation has closed-form solutions:
ATS(M)aeaMATSabATSbeaMATS(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
Mimi0 (12)
i16i is the weight and
mi is the
ith input metric
19 / 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
Then from (11) and (12), we can get
a(ATS(M)aeimi0)i166ATSATSa(abATSbeimi0)i1(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)1.10321.0958eM（14）
WithM29.98220.498m1593923m257.78m38.4232m418.556m59.9023m6(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
ATSf(M)f(m1,m2,...,m6) (16)
Let us consider
And (11) and (12) can get
ATSATSMMi (17)
miMmmiiATSaATSbATS2 (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
Mi (19)
miThen from (17),(18) and (19), we can get
ATSATSM(aATSbATS2)i (20)
Mmi
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
get all concerned data for USA in 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
90.544
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)
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(Mimi)
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)1.10321.0958eM
ATS=0.8116, this guarantees the absolute total score doesn’t change.
Mi>0, this means no metrics is negative.
m3,4,6100 ,this is the definition of metricsm1,2,51000,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）=0.8116AST
m0im3,4,6100m1,2,51000i10.3.4 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
0.4000
0.1648
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=$388.18, so the total money saved in America is 107.8 billion. We
reduce the cost mainly through decreasing the number of physicians, nurses, and increasing hospital beds.
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2008年国际大学生数学建模比赛参赛作品 作者：舒强、罗双才、黎璨
10.4. Scenario2: More outcomes with the same expenditure
In 2007, USA ranks 3rd by the evaluation of absolute effectiveness and gets an absolute total
score (ATS) try to maximize absolute total score with the same expenditure on health of
1146 billion dollars.
10.4.1Objective function
Our goal is to maximize the absolute total score ATS and it is given as the function of total expenditure on health
in the form of logistic equation:
Thus our objective function is
Max AST
（M）10.4.2Constraints
m2109.810, this guarantees total expenditure on health is 2109.8 billion dollars.
9ii=16 Mi>0, this means no metrics is negative.
m3,4,6100 ,this is the definition of metricsm1,2,51000,this is the definition of metrics
10.4.3 Optimization model 2
From the analysis above, we can get an optimization model, shown as follows:
Max AST（M）69im2109.810i=1mi0m13,4,6
st： 26 / 29
2008年国际大学生数学建模比赛参赛作品 作者：舒强、罗双才、黎璨
10.4.4Solutions to the model
Table8 Solution to optimization to model2 Unit: million$
USA Physician Nurse
Social
security
0
0.4000
Private Hospitaprepaid
l Bed
0.4000
0.1000
３３
６５
Out-of
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 189, Issue 1, 16 August 2008, Pages 132-145
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