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2016
MCM/ICM
Summary Sheet
Water shortage, a universal issue existing all over the world, which may cause huge loss to human life can be
divided into two main aspects- Physical scarcity, caused by the finite amount of clean water and Economic
scarcity, caused by poor distribution and infrastructure limit. A method needs putting forward to measure the
capacity of a region to offer clean water to satisfy the demand of population and development with the
consideration of both natural constraints of water as well as artificial factors. What’s more, if the region is in
water shortage, how to give rational countermeasures and what impact the intervention plan will exert on the
water scarcity of this region and surrounding environments in the future. In order to address above problems, a
model is developed on the basis of building a relationship between parameters (the amount of supplying and
demanding water as well as water resources) and the water shortage problem so that we can not only judge
the water deficient type but also calculate the extent of each kind.
Since we think there are Physical scarcity (PS), Economic scarcity caused by distribution (ESD) and infrastructure
problem (ESI), we need to judge water shortage type of a region and how serious each type is with an abstract
function (Benefit Function). Under rational assumptions, Benefit Function can describe the positive impact
provided by water supply. If the water supply changes, which may be caused by water scarcity, the positive
impact will also change. And the differences between positive impacts can describe the type and extent of
water scarcity. Therefore, when we get the amount of supplying and demanding water predicted by
Exponential Model and Back Propagation, we can estimate the water shortage in the future. Furthermore,
numerical intervention plan can be formulated on the basis of the quantitative description of water scarcity.
Similar to Benefit Function, we construct a Damage Function to quantitatively depict relationship of negative
effect to surrounding aquatic ecosystems of our intervention.
We adopt our model to measure the water deficient condition of Shandong Province in 2013, and get the
conclusion that Shandong is free of PS, while ESD and ESI account for 97.09% and 2.91% of total water
shortage respectively. And we predict that in 2030, this region is free of ESI, while ESD and PS account for
47.55% and 52.45% of total water shortage respectively. Moreover, although our intervention plan can
mitigate the extent of water scarcity in local area, it cannot be eliminated in the future, and the plan has side
effect to the surroundings. Finally, we find that the change of the water resources will influent the local area
increasingly via the analysis of our model.
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CONTENT
1 INTRODUCTION ........................................................................................................... 2
1.1 BACKGROUND ........................................................................................................... 2
1.2 MODELING PURPOSE .................................................................................................. 2
2 TASK1: MODELING DEVELOPMENT............................................................................. 2
2.1 OVERVIEW OF THE MODEL ........................................................................................... 2
2.2 DECLARATION AND EXPLANATION OF SYMBOLS AND PARAMETERS ........................................ 4
2.3 ASSUMPTIONS........................................................................................................... 5
2.4 METHOD.................................................................................................................. 6
3 TASK 2: ANALYSIS OF WATER SCARCITY IN SHANDONG PROVINCE ............................ 9
4 TASK 3: PREDICTION OF WATER SCARCITY IN SHANDONG PROVINCE ..................... 11
5 TASK 4: INTERVENTION PLAN.................................................................................... 16
5.1 INTERVENTION PLAN:........................................................................................... 16
5.2 THE STRENGTH AND WEAKNESS OF THE INTERVENTION PLAN ............................................. 18
6 TASK 5: LONG TERM IMPACT OF INTERVENTION ON SHANDONG PROVINCE ......... 18
7 THE STRENGTH AND WEAKNESS OF THE MODEL ..................................................... 22
STRENGTH .................................................................................................................... 22
APPENDIX ..................................................................................................................... 24
APPENDIX A BP NET WORK ........................................................................................... 24
APPENDIX B F(X1,X2,…,XN) ........................................................................................... 26
APPENDIX C H(X) ...................................................................................................... 27
APPENDIX D SUSCEPTIBILITY ........................................................................................ 28
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1 Introduction
1.1 Background
Water is essential to all life on the Earth. Although the total amount of water resources in our
planet seemingly adequate, when taking the amount of total clean water into consideration,
there are still a lot of places is confronted with the water scarcity. There is no doubt that water is
a vital commodity without which the social and economic development in country or region
can’t be satisfied. And it is a universal phenomenon that this vital commodity is over exploited
in the areas with abundant natural water resources while there are numerous people suffering
from the water shortage. As was reported in the Vital Water Graphics，nearly one quarter of the
population on our planet are experiencing water scarcity.
Under this background, it is important to know whether our planet will run out of water, and
what are the true factors of water scarcity on earth. Despite of the fact that we can list as many
factors leading to water scarcity as we want, considering the fact that water is everywhere and
correlated with nearly all animate and non-living substances, we can still classify them into two
primary types——physical and economic water shortages. There are many models and
methods to measure and predict the extent of water deficiency, however, they are complicated
and limited to just one certain region. Thus we hope to get a simple method to describe and even
measure the content of water shortage quantitatively.
1.2 Modeling purpose
First of all, we develop this model to give a simple but practical enough method to give a
judgement of the water shortage types and measure the extent of each type. In our model, we
combine natural and artificial factors affect the change of water supply and demand.
What’s more, we want to predict the trend of water scarcity for one region and analyze the type
of water shortage.
Furthermore, we project the water availability in the future on the basis of our model, and hope
to provide practical reference to decision-makers how to take water shortage problem into
account when formulating policy.
2 Task1: Modeling development
2.1 Overview of the model
The factors deciding whether it is possible to provide clean water to all in a selected region can
be divided into two main aspects: physical scarcity, standing for the lack of available water in a
region and economic scarcity, which suggests the amount of water can’t satisfy the population
and the development of society even under the condition where there is adequate available
water. When it comes to economic scarcity, we can list numerous factors including pollution,
excessive personal consumption, industrial consumption and etc. However, we can classify
them into two categories—the limit of ability to get clean water and poor distribution
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management. The three main factors can be explained in Figure 1:
Clean water
Distribution problem
Not enough amount
Infrastructure
Physical
scarcity
Poor capacity
to
water
Poor capacity
to
water
Economic
scarcity
Water shortage
Figure 1: Relationship between the three water scarcity
According to Figure 1, we definite the two types of water shortage as Physical scarcity and
Economic scarcity. The effect of water shortage can be reflected by the conflict among water
resources, water supply and water demand. The most intuitive and useful method to judge if the
region is in short of water is to compare the water resources, the amount of demanding water
and supplying water. However, it is far from enough to give such a simple answer, we need to
definite the type of water scarcity. Furthermore, there is no doubt that a region can be Physical
scarcity and Economic scarcity at the same time, which requires us to analyze the propensity
between Physical scarcity and Economic scarcity. Therefore, we measure the capacity of a
region to offer the available water to satisfy the need of population and which kind of scarcity is
more serious by describing the relationship among water resources, supply and demand
quantitatively with our model. So the model achieves several important objectives:
1) Judge whether the ability of a region to provide clean water can meet the needs of its
population or not.
2) Provide a principle to calculate the extent of different kinds of scarcity
3) Judge which kind of scarcity is dominant.
4) Judge the influence on surroundings by the usage of water.
5) Analyze the susceptibility to water scarcity of a region
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2.2 Declaration and explanation of symbols and parameters
1) Water resources
According to most document literatures, the impact of environmental constraints on water
resources can be reflected by the amount of water provided by surface water and underground
water. Hence, when calculate the total available water resources, we take surface water,
underground water as well as water offered via other methods such as water transfer and
sewage reuse into consideration. And we can give the declaration of the following parameters:
W = 𝑊𝑆 + 𝑊𝑈 + 𝑊𝑂
𝑊: 𝑡𝑜𝑡𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑟𝑒𝑠𝑜𝑢𝑟𝑐𝑒𝑠
𝑊𝑆 : 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑟𝑒𝑠𝑜𝑢𝑟𝑐𝑒𝑠 𝑓𝑟𝑜𝑚 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑤𝑎𝑡𝑒𝑟
𝑊𝑈 : 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑟𝑒𝑠𝑜𝑢𝑟𝑐𝑒𝑠 𝑓𝑟𝑜𝑚 𝑢𝑛𝑑𝑒𝑟𝑔𝑟𝑜𝑢𝑛𝑑 𝑤𝑎𝑡𝑒𝑟
𝑊𝑂 : 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑜𝑓𝑓𝑒𝑟𝑒𝑑 𝑏𝑦 𝑜𝑡𝑕𝑒𝑟 𝑤𝑎𝑦𝑠(𝑤𝑎𝑡𝑒𝑟 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟, 𝑠𝑒𝑤𝑎𝑔𝑒 𝑟𝑒𝑢𝑠𝑒 )
2) Water supply and demand
Given that usages of water vary in different parts of world and country, we use set to
representative the amount of water supply and water demand.
D = *𝑑1 , 𝑑2 , ⋯ , 𝑑𝑛 +
S = *𝑠1 , 𝑠2 , ⋯ , 𝑠𝑛 +
D refers to the set of demand water while S refers to the set of supply of water. D means how
much water the region really need to satisfy the population and social development, it can be
obtained from the prediction of the Regional Development Standard. What’s more ∑ 𝑑𝑖 and
∑ 𝑠𝑖 represents the total amount of demanding water and supplying water respectively.
S can be got from Regional Water Management Report given by the local governments. Table 1
shows the relationship between water usages, water demand and water supply.
Usages of water
D(Demand)
S(Supply)
Usage 1
𝑑1
𝑠1
Usage 2
𝑑2
𝑠2
⋮
⋮
⋮
Usage i
𝑑𝑖
𝑠𝑖
⋮
⋮
⋮
Usage n
𝑑𝑛
𝑠𝑛
(Usages are like agriculture use, industrial use and so on.)
Table 1: Relationship between water usages, water demand and water supply
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3) Benefit Function
Water brings benefit when consumed. A function 𝑓𝑖 (𝑥𝑖 ) 𝑖 = 1,2, … , 𝑛 can be used to describe
the relationship between water supply and benefit in usage i. 𝑥𝑖 refers to the water supply to
usage i. Different usage has different benefit, therefore has different function.
4) Benefit reduction by different kinds of scarcity
𝑅𝑃 ,𝑅𝐸 , 𝑅𝐷 and 𝑅𝐼 are definited as the reduction respectively resulting from Physical scarcity,
Economic scarcity, the distribution problem and infrastructure problem. Obviously, 𝑅𝐸 =
𝑅𝐷 + 𝑅𝐼 and the more the value of the reduction, the more serious this type of shortage will be.
5) Damage Function
When water is consumed by people, waste water produces and this will cause environment
disruption. So Damage Function 𝑝𝑖 (𝑥𝑖 ) 𝑖 = 1,2,3 … , 𝑛 is used to describe the damage caused
by usage i. 𝑥𝑖 refers to the water supply to usage i.
6) susceptibility to water scarcity
The impact caused by the change of water resource varies in different place. That is to say
different region has different susceptibility to water scarcity. S is used to stand susceptibility
2.3 Assumptions
In order to streamline our model, we have following key assumptions:
1) When the infrastructure limit does not exist, 𝑅𝐼 = 0 which is the same as physical scarcity,
economic scarcity and distribution limit.
2) When the physical scarcity doesn’t exist, 𝑊 ≥ ∑ 𝑑𝑖 . because the amount of available water
is enough to meet the requirement of total amount of water supply.
3) Infrastructure limit doesn’t exist, which means all the demands are gratified when there is
no physical scarcity or the total water supply is no less than water resources when physical
scarcity exists. That is to say when infrastructure limit doesn’t exist, if 𝑊 ≥ ∑ 𝑑𝑖 then
∑ 𝑠𝑖 ≥ ∑ 𝑑𝑖 or if 𝑊 < ∑ 𝑑𝑖 then ∑ 𝑠𝑖 ≥ 𝑊.
4) The extent of distribution problem is determined by the benefit. When the distribution
limitation doesn’t exist, the distribution of water is optimal given that the total amount of
supply water is certain. And the benefit reach its maximum value under this condition. (The
standard of optimal varies with the region because the situation in different places is
different. This can be affected by the climate, the topography and the policy.)
5) To a certain usage, when the water supply reaches the demand, the benefit will not increase
with the increase of supply water.
6) To a certain usage, when water supply increases, the benefit will dose not decreases.
7) Although the Physical scarcity and infrastructure limitation can be directly calculated, they
are also judged by the benefit in order to compared with distribution limitation.
8) The extent of environment damage is only related to water supply. The larger the water
supply is the deeper extent of environment damage will be. Although water transfer,
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overexploitation and desalinate seawater can also disrupt environment, all of them cause
the increase of water supply. So we only take water supply into account.
9) We use benefit to calculate the susceptibility. That is to say susceptibility means the
decrease of benefit due to unit water resources decrease.
2.4 Method
According to the assumption, our model should meet the request show in the following tables:
W ≥ ∑ 𝑑𝑖
∑ 𝑑𝑖 > ∑ 𝑠𝑖
∑ 𝑑𝑖 > ∑ 𝑠𝑖
∑ 𝑑𝑖 ≤ ∑ 𝑠𝑖
∑ 𝑑𝑖 ≤ ∑ 𝑠𝑖
∀𝑑𝑖 ≤ 𝑠𝑖
∃𝑑𝑖 > 𝑠𝑖
∃𝑑𝑖 > 𝑠𝑖
∀𝑑𝑖 ≤ 𝑠𝑖
ESI
ESD
No water shortage
ESD may exists
W < ∑ 𝑑𝑖
W ≤ ∑ 𝑠𝑖
W ≤ ∑ 𝑠𝑖
W > ∑ 𝑠𝑖
W > ∑ 𝑠𝑖
∀𝑑𝑖 ≤ 𝑠𝑖
∃𝑑𝑖 > 𝑠𝑖
∃𝑑𝑖 > 𝑠𝑖
∀𝑑𝑖 ≤ 𝑠𝑖
PS
PS and ESI
PS
ESD may exists
(PS represents the Physical scarcity, ES represents the Economic scarcity, ESD represents the
economic scarcity caused by distribution problem and ESI represents the economic scarcity
caused by infrastructure problem).
Table 2: 8 permutations of three types of relationships
As discussed above, we need to describe how much the extent of each type: PS, ESD ESI,
therefore, we adopt a Benefit Function 𝑓𝑖 (𝑥𝑖 ), and according to assumption, 𝑓𝑖 (𝑥𝑖 ) can be
defined as:
𝑓𝑖 (𝑥𝑖 ) = {
𝑓𝑖 (𝑥𝑖 ),
𝑓𝑖 (𝑑𝑖 ),
𝑥𝑖 < 𝑑𝑖
𝑥𝑖 ≥ 𝑑𝑖
which means the beneficial affect brought about by clean water in usage i. Table 3 details the
Benefit Function :
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Usages of water
D(Demand)
S(Supply)
Benefit Function
Usage 1
𝑑1
𝑠1
𝑓1 (𝑠1 )
Usage 2
𝑑2
𝑠2
𝑓𝑖 (𝑠2 )
⋮
⋮
⋮
Usage i
𝑑𝑖
𝑠𝑖
⋮
⋮
⋮
Usage n
𝑑𝑛
𝑠𝑛
𝑓𝑖 (𝑠𝑖 )
𝑓𝑖 (𝑠𝑛 )
(Usages are like agriculture use, industrial use and so on.)
Table 3: Relationship between water usages, water demand , water supply and Benefit Function
According to Table 3, the total beneficial affect brought about by all the usages in a selected
region can be written as :
𝑛
F(𝑥1 , 𝑥2 , ⋯ 𝑥𝑛 ) = ∑ 𝑓𝑖 (𝑥𝑖 )
𝑖<1
A new function 𝑕(𝑥) represent the maximum value of benefit when the total water supply is x
that is to say:
𝑛
𝑕(𝑥) = max({𝐹(𝑥1 , 𝑥2 , … , 𝑥𝑛 )| ∑ = 𝑥 })
1
𝑖
the realistic beneficial affect which can be expressed as:
𝑛
F(𝑠1 , 𝑠, ⋯ 𝑠𝑛 ) = ∑ 𝑓𝑖 (𝑠𝑖 )
𝑖<1
Whichever type of water shortage leads to the reduction of the value of the Benefit Function.
Therefore, 𝑅𝑃 , 𝑅𝐸 , 𝑅𝐷 and 𝑅𝐼 represent the reduction respectively resulting from Physical
scarcity, Economic scarcity, the distribution problem and infrastructure problem. We definite R
as the reduction respectively resulting from water scarcity. Obviously, 𝑅𝐸 = 𝑅𝐷 + 𝑅𝐼 and the
more the value of the reduction, the more serious this type of shortage will be. 𝑅𝑃 , 𝑅𝐸 , 𝑅𝐷 ,𝑅𝐼
and R can be computed as follows:
𝑛
𝑅𝑃 = 𝑕(∑ 𝑑𝑖 ) − 𝑕(𝑤)
𝑖<1
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𝑛
𝑛
𝑕(𝑤) − 𝑕(∑ 𝑠𝑖 ) , 𝑕(𝑤) > 𝑕(∑ 𝑠𝑖 )
𝑖<1
𝑅𝐼 =
0
{
𝑛
𝑖<1
, 𝑕(𝑤) ≤ 𝑕(∑ 𝑠𝑖 )
𝑖<1
𝑛
𝑅𝐷 = 𝑕(∑ 𝑠𝑖 ) − 𝐹(𝑠1 , 𝑠2 , … , 𝑠𝑛 )
𝑖<1
𝑅𝐸 = 𝑅𝐼 + 𝑅𝐷
R = 𝑅𝑃 + 𝑅𝐼 + 𝑅𝐷
𝑕(∑𝑛𝑖<1 𝑑𝑖 ) − 𝑕(𝑤) can represent the reduction of benefit due to physical scarcity, when
𝑤 ≥ ∑𝑛𝑖<1 𝑑𝑖 , 𝑕(𝑤) = 𝑕(∑𝑛𝑖<1 𝑑𝑖 ), 𝑅𝑃 = 0, which is corresponded to the assumption.
𝑕(𝑤) − 𝑕(∑𝑛𝑖<1 𝑠𝑖 ) can represent the reduction of benefit due to device problem because not all
the water resources is utilized. When there is no device problem, ∑𝑛𝑖<1 𝑠𝑖 can equal w even
larger (means overexploitation), so 𝑅𝐼 = 0. if 𝑤 ≥ ∑𝑛𝑖<1 𝑑𝑖 , in reality it is not necessary to
make use of all water resources, which means ∑𝑛𝑖<1 𝑠𝑖 ≥ ∑𝑛𝑖<1 𝑑𝑖 (when equal, this is the best
situation). Under this circumstance, 𝑕(𝑤) = 𝑕(∑𝑛𝑖<1 𝑑𝑖 ) = 𝑕(∑𝑛𝑖<1 𝑠𝑖 ) and 𝑅𝐼 = 0, which is
corresponded to the assumption, so we can use this formula to calculate 𝑅𝐼 .
𝑕(∑𝑛𝑖<1 𝑠𝑖 ) − 𝐹(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) can represent the reduction of benefit due to distribution, this is
obvious and corresponded to the assumption.
𝑅𝑃 + 𝑅𝐼 + 𝑅𝐷 can represent the reduction of benefit due to all factors, so we use it to present R.
So the model can cover all the circumstances shown in Table 2 and meet their requirement.
When comparing different kinds of scarcity, we can calculate as follows:
𝑅𝑃 % =
𝑅𝑃
𝑅𝑃 + 𝑅𝐷 + 𝑅𝐼
𝑅𝐷 % =
𝑅𝐷
𝑅𝑃 + 𝑅𝐷 + 𝑅𝐼
𝑅𝐼 % =
𝑅𝐼
𝑅𝑃 + 𝑅𝐷 + 𝑅𝐼
𝑅𝐸 % =
𝑅𝐸
𝑅𝐼 + 𝑅𝐷
=
𝑅𝑃 + 𝑅𝐸 𝑅𝑃 + 𝑅𝐷 + 𝑅𝐼
We define 𝑝𝑖 (𝑥𝑖 ) as environment damage. And according to assumption 𝑝𝑖 (𝑥𝑖 ) is a monotone
increasing function. The total environment P disruption is defined as follows:
𝑛
𝑃(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) = ∑ 𝑝𝑖 (𝑥𝑖 )
𝑖<1
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According to assumption, the susceptibility to water scarcity is determined by benefit, so S can
be calculated as follows:
𝑕(𝑤) − 𝑕(𝑤 − ∆𝑤)
∆𝑤→0
∆𝑤
𝑆 = lim
3 Task 2: Analysis of water scarcity in Shandong Province
In the UN water scarcity map, we circled a heavily overloaded place with green line which is
located in Shandong Province in China as the following picture. To measure its degree of water
scarcity, we use the model from task 1 to calculate all of the indexes which are relevant to water
scarcity. Since most of the indexes are related to time, we allocated the time to be the year of
2013.
Figure 2: The location of Shandong Province in UN water scarcity map
From the National Bureau of Statistics of the People’s Republic of China and the Statistical
Yearbook of Shandong Province, we can get the relevant statistics and the relationship between
water usages, water demand, water supply and Benefit Functions is detailed in the
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W(108m3)
D(108m3)
Usages of water
Benefit Function
𝑑1 =144.1
𝑠1=129.73
𝑓1 (𝑠1 )
Industry water
𝑑2 = 24.25
𝑠2 = 28.86
𝑓2 (𝑠2 )
Domestic water
𝑑3 = 21.32
𝑠3 = 25.97
𝑓3 (𝑠3 )
Urban public water
𝑑4 = 7.45
𝑠4 = 7.34
𝑓4 (𝑠4 )
Ecology use
𝑑5 = 9.69
𝑠5 = 6.06
𝑓5 (𝑠5 )
Forestry and Breeding
water
𝑑6 = 20.00
𝑠6< 19.99
𝑓6 (𝑠6 )
Farmland irrigation
291.70
S(108m3)
Table 4: Data of W, D and S[1-4]
6
From these data, we can get ∑6𝑖<1 𝑑𝑖 = 226.80𝑚3 and ∑𝑖<1
𝑠𝑖 = 217.95𝑚3 , and we find
that W > ∑6𝑖<1 𝑑𝑖 > ∑6𝑖<1 𝑠𝑖 and ∃𝑑𝑖 > 𝑠𝑖 . Therefore, we can make a qualitatively judgment
that Shandong Province has water shortage caused by distribution problem.
For the sake of city’s development and according to the Law of Water in China, we give every
independent variable 𝑘1 , 𝑘2 , 𝑘3 , 𝑘4 , 𝑘5 , 𝑘6 a coefficient 1, 2, 6, 6, 5, 1 and definite the Benefit
Function as:
𝑘𝑖
𝑥,
𝑓𝑖 (𝑥𝑖 ) = {𝑑𝑖 𝑖
𝑘𝑖 ,
𝑥𝑖 < 𝑑𝑖
𝑥𝑖 ≥ 𝑑𝑖
If the ∑ 𝑆 ≥ ∑ 𝐷, then the best distribution could be ∀𝑥𝑟 = 𝐷𝑟 , and the function can reach its
largest value: 𝑕(∑𝑛𝑖<1 𝑠𝑖 ) = 1 + 2 + 6 + 6 + 5 + 1 = 21.
If the ∑ 𝑆 < ∑ 𝐷, then the best distribution could be calculated by linear programming, while
the sum of every independent variable is a constant value ∑ 𝑆.
Therefore, the realistic benefit affect can be calculated as:
129.73
+2
144.10
= 18.94
F(𝑠1 , 𝑠2 , ⋯ 𝑠6 ) =
6
𝑕(∑ 𝑠𝑖 ) =
𝑖<1
135.24
+2
144.10
24.25
+6
24.25
24.25
+6
24.25
21.32
+6
21.32
21.32
+6
21.32
𝑕(𝑊) = 21
6
𝑕 (∑ 𝑑𝑖 ) = 21
𝑖<1
7.34
+5
7.45
7.45
+5
7.45
6.06 19.99
+
9.69 20.00
9.69 20.00
+
= 20.94
9.69 20.00
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𝑛
𝑅𝑃 = 𝑕(∑ 𝑑𝑖 ) − 𝑕(𝑤) = 21 − 21 = 0
𝑖<1
𝑛
𝑅𝐷 = 𝑕(∑ 𝑠𝑖 ) − 𝐹(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) = 20.94 − 18.94 = 2.00
𝑖<1
𝑛
𝑅𝐼 = 𝑕(𝑤) − 𝑕(∑ 𝑠𝑖 ) = 21 − 20.94 = 0.06
𝑖<1
𝑅𝐷 % =
𝑅𝐷
= 97.09%
𝑅𝐷 + 𝑅𝑝 +𝑅𝐼
𝑅𝐼 % =
𝑅𝐼
= 2.91%
𝑅𝐷 + 𝑅𝑝 +𝑅𝐼
𝑅𝑃 % = 0
Taking all the account into consideration, we can draw a conclusion that in 2013, our chosen
region is free of physical scarcity. The water scarcity in this region is economic scarcity which
is mainly caused by poor management.
The total degree of water scarcity in Shandong Province in 2013 can be measured as:
𝑅 = 𝑅𝐷 + 𝑅𝐼 + 𝑅𝑃 = 2.00 + 0.06 = 2.06
4 Task 3: Prediction of water scarcity in Shandong Province
All the parameters will change due to population increase, environment change, pollution and
so on. So it is necessary to make predictions of all the parameters in 15 years. Different
parameters have different ways to predict. The water demand of industry, Farmland irrigation
and Forestry and Breeding can be predicted by exponential model which can be described by
the following function:
𝐵
𝑤(𝑡) = 𝐴𝑒 𝑡;𝐶
Where A, B, C are constant, t refers to time
The demand of Domestic water can be regarded as being proportioned to the population. That is
to say
𝑤𝑑𝑜𝑚𝑒𝑠𝑡𝑖𝑐 (𝑡) = 𝑘𝑃
Where k is the coefficient and P means population.
The ecological use water can be simply seen as being proportional to the green area:
𝑤𝑒𝑛𝑣 (𝑡) = 𝑘𝐺
Where k is the coefficient and G is the green area.
But parameters like water supply, population, GDP, green area are strongly affected by the
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policy, the environment and so on. There is no formula to predict these parameters. So Back
Propagation is used. Back Propagation is a common method of training artificial neural
networks used in conjunction with an optimization method. The method calculates the gradient
of a loss function with respect to all the weights in the network. The gradient is fed to the
optimization method which in turn uses it to update the weights, in an attempt to minimize the
loss function. When it is used to predict, the future value of xi (i=1, 2, …, n) is related to former
m numbers which can be defined as
𝑥𝑛:1 = 𝐹(𝑥𝑛 , 𝑥𝑛;1 , 𝑥𝑛;2 , … 𝑥𝑛;𝑚:1 )
We can use 𝑥𝑖 (𝑖 = 1,2 … 𝑛) as input number, 𝑥𝑛:1 as expectation to find out the relation
𝐹(𝑥𝑛 , 𝑥𝑛;1 , 𝑥𝑛;2 , … 𝑥𝑛;𝑚:1 ) and use this relation to predict the future parameter. The
following tables show all the historical parameters and the parameters in 2030 we take into
account: (D1,D2,D3,D4,D5,D6 respectively stand for water demand of farmland irrigation,
industry, domestic, urban public, ecological use, forestry and breeding water. S1, S2, S3, S4,
S5, S6 respectively stand for water supply to farmland irrigation, industry, domestic, urban
public, ecological use, forestry and breeding water.)
Year
Water
resource
8
3
(10 m )
Population
GDP
(106)
(108 yuan)
Agriculture
GDP
Industry
GDP
Livestock
GDP
(108 yuan)
(108 yuan)
(108 yuan)
2000
252.09
89.98
8337.47
1268.57
4164.45
993.91
2001
238.81
90.41
9195.04
1359.49
4556.01
1052.62
2002
98.14
90.82
10275.50
1390.00
5184.98
1105.17
2003
489.69
91.25
12078.15
1480.67
6485.05
1303.13
2004
349.46
91.80
15021.84
1778.45
8478.69
1562.18
2005
415.86
92.48
18366.87
1963.51
10478.62
1963.51
2006
199.32
93.09
21900.19
2138.90
12574.03
2138.90
2007
387.11
93.67
25776.91
2509.14
14647.53
2509.14
2008
328.71
94.17
30933.28
3002.65
17571.98
3002.65
2009
284.95
94.70
33896.65
3226.64
18901.83
3226.64
2010
309.12
95.88
39169.92
3588.28
21238.49
3588.28
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2011
347.61
96.37
45361.85
3973.85
24017.11
3973.85
2012
274.08
96.85
50013.24
4281.70
25735.73
4281.70
2013
291.70
97.33
55230.32
4565.97
27442.85
4742.63
2014
148.44
97.89
59426.59
4798.36
28788.11
4992.88
2030
59.65
108.34
126758.88
8755.97
61988.80
10206.59
Table 5: Part of parameters in a period of time (2000) [1-4]
years
S1
S2
S3
S4
S5
S6
Total
(108m3)
(108m3)
(108m3)
(108m3)
(108m3)
(108m3)
(108m3)
2000
43.65
0.00
2001
167.32
41.92
0.34
15.59
2002
170.98
36.59
0.29
17.29
2003
142.87
31.62
20.22
3.59
1.38
19.67
219.35
2004
138.78
28.39
20.70
3.97
1.68
21.36
214.88
2005
141.49
21.76
21.39
3.78
2.37
20.24
211.03
2006
175.14
25.53
24.41
4.60
2.97
23.18
255.83
2007
144.53
24.12
23.02
4.40
3.20
20.28
219.55
2008
142.33
24.69
23.76
4.95
3.73
20.43
219.89
2009
141.11
24.70
24.12
5.65
3.94
20.49
220.01
2010
139.01
26.84
25.03
6.31
4.64
20.64
222.47
2011
131.91
29.72
25.80
7.09
7.17
22.35
224.04
2012
133.29
28.10
25.50
7.31
6.66
20.93
221.79
2013
129.73
28.86
25.97
7.34
6.06
19.99
217.95
2014
127.54
28.64
26.07
7.32
5.78
19.18
214.53
2030
127.56
28.63
26.06
7.34
14.25
19.18
223.02
Table6: Data of supplying water [1-4]
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years
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D1
D2
D3
D4
D5
D6
Total
(108m3)
(108m3)
(108m3)
(108m3)
(108m3)
(108m3)
(108m3)
39.93
19.71
2.50
2000
2001
203.03
33.95
19.80
2.78
18.07
203.03
2002
174.66
30.48
19.89
3.26
16.88
174.66
2003
158.14
30.51
19.98
3.54
4.11
17.71
158.14
2004
162.95
32.33
20.10
3.81
4.39
18.88
162.95
2005
155.66
32.76
20.25
4.11
4.76
21.11
155.66
2006
147.87
32.58
20.39
4.43
4.98
20.46
147.87
2007
152.36
31.76
20.51
4.77
5.89
21.35
152.36
2008
161.22
32.19
20.62
5.13
6.43
22.73
161.22
2009
154.13
29.49
20.74
5.52
7.09
21.73
154.13
2010
153.36
28.45
21.00
5.98
7.60
21.49
153.36
2011
152.78
27.82
21.11
6.44
8.12
21.18
152.78
2012
148.80
25.96
21.21
6.93
8.72
20.30
148.80
2013
144.10
24.25
21.32
7.45
9.69
20.00
144.10
2014
138.11
22.42
21.44
8.02
10.34
18.73
138.11
2030
87.04
11.23
23.73
26.40
19.22
5.89
173.52
Table 7: Data of demanding water(2000)
Figure 3shows the water resources, total water supply and total water demand in the past and in
the future we predict.
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600.00
water supply/(108m3)
500.00
400.00
water demand
300.00
water resources
200.00
water supply
100.00
0.00
1995
2000
2005
2010
2015
2020
2025
2030
2035
year
Figure 3: Trend of water demand, water resources and water supply
From the prediction, the water resources decreases. This may be caused by water pollution,
climate change and so on. Although the population and green area increases, the water demand
decreases, which mainly due to the improvement of technology that reduce the water demand in
agriculture and industry. The water supply is almost stable and exceeds the water demand and
water resources which shows the existence of water waste and water overexploitation
All the parameters are showed in the tables. Use the model to show the impact on citizens of
this region. The following table shows the demand and supply of all usage in 2030.
usage
demand
supply
Supply/demand
Farmland irrigation
87.05
127.56
1.47
industry
11.23
28.63
2.55
Domestic water
23.73
26.06
1.10
Urban public water
26.40
7.34
0.28
Ecological use
19.22
14.25
0.74
Forestry and breeding
5.89
19.18
3.26
total
173.52
223.02
1.29
Table 8: Part of data of supplying and demanding water
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The situation in 2030 can be computed as follows.
𝐹(𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 , 𝑥5 , 𝑥6 ) = 𝑓1 (𝑥1 ) + 𝑓2 (𝑥2 ) + 𝑓3 (𝑥3 ) + 𝑓4 (𝑥4 ) + 𝑓5 (𝑥5 ) + 𝑓6 (𝑥6 )
𝐹(𝑠1 , 𝑠2 , 𝑠3 , 𝑠4 , 𝑠5 , 𝑠6 ) = 15.3749
6
𝑕 (∑ 𝑠𝑖 ) = 𝑕(223.02) = 𝑕(173.52) = 21
𝑖<1
𝑕(𝑤) = 𝑕(59.6503) = 14.7962
6
𝑕 (∑ 𝑑𝑖 ) = 𝑕(173.52) = 21
𝑖<1
𝑅𝑃 = 21 − 14.7962 = 6.2038
𝑅𝐼 = 0
𝑅𝐷 = 21 − 15.3749 = 5.6251
𝑅𝑃
= 52.45%
𝑅𝑃 + 𝑅𝐷 + 𝑅𝐼
𝑅𝐷
= 47.55%
𝑅𝑃 + 𝑅𝐷 + 𝑅𝐼
𝑅𝐼
=0
𝑅𝑃 + 𝑅𝐷 + 𝑅𝐼
From the consequence, we can know that there is no device problem (even overexploitation).
The benefit reduction caused by physical scarcity is a little more than caused by distribution,
but both of them are much serious than the situation nowadays. Also, the water supply to urban
public and ecological use can meet the demand while in farmland irrigation, industry and
forestry and breeding, but the waste of water is serious. So an intervention plan is necessary to
change the situation.
5 Task 4: Intervention plan
To mitigate water scarcity, we designed an intervention plan which can solve the water scarcity
problem.
5.1 Intervention plan:
Plan1: To decrease the total demand:
1) By increasing the reuse rate of water for industry;
2) By increasing the use ratio of water for farm irrigation and forestry and breeding;
3) By decreasing the amount of water usage of domestic and urban public;
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Plan2: To increase the total supply:
1) By invoking clean water from surrounding areas;
2) By sea water desalination;
3) By improving the infrastructure;
Plan3: Redistribute the water supply on the basis of Plan 1 and Plan 2.
Plan4: Improve the production equipment to reduce the pollution brought by supply water.
1
2
3
4
5
6
Farm
irrigation
Industry
Domestic
Urban
Public
Ecological use
Forestry and
breeding
Under the intervention plan, we can get some new parameters:
1;𝜂
𝑑2′ = 𝑋𝑖 𝑞2 , 𝑞2 = 𝑞1 (1 − 𝛼) 1;𝜂2
1
182.38
𝑑1′ = 𝛾𝑎 𝑑1 = 𝛾𝑎 𝑋𝑎 ∙ 0.005𝑒 𝑡;1969
𝑑6′ = 𝛾𝑙 𝑑6 = 𝛾𝑙 𝑋𝑙 ∙ 0.0193𝑒 ;0.017(𝑡;2000)
𝑑3′ = 𝛾𝑝 𝑑3 = 0.6𝑃 ∙ 𝛾𝑝 𝐾
𝑑4′ = 𝛾𝑐 𝑑4 = 𝑃 ∙ 𝛾𝑐 ∙ 0.0362𝑒 0.0681(𝑡;2002)
Where 𝑋𝑖 , 𝑋𝑎 , 𝑋𝑙 represent the GDP brought by industry, farm irrigation and forestry and
breeding; 𝛾𝑎 , 𝛾𝑙 , 𝛾𝑝 , 𝛾𝑐 represent the progressing coefficient of every trade which value from
zero to one; 𝑞1 , 𝑞2 represent the amount of water demanded by unit GDP of the first and the
last year; 𝜂1 , 𝜂2 represent the repeating utilization factor of water in the first and the last year; 𝛼
represents the industrial progressing coefficient; t represents the year; P represents the local
population; K represents the per capita water consumption.
Assuming that the amount of clean water invoked from surrounding areas, desalinated with sea
water and improved by better infrastructure can be respectively valued as a, b, c.
With the new distribution, we can calculate its degree of water scarcity (R’) as:
𝑅 ′ = 𝑕 (∑ 𝑑′ ) − 𝑕 (∑ 𝑠 ′ )
If we regard the degree of water scarcity before use the intervention plan as R, we can use the
value of 𝛥𝑅 to measure the effect of the plan to the water availability of local area.
𝛥𝑅 = 𝑅 ′ − 𝑅
𝛥𝑅 = 𝑕 (∑ 𝑑′ ) − 𝑕 (∑ 𝑠 ′ ) − (𝑅𝑃 + 𝑅𝐼 + 𝑅𝐷 )
To measure the impact of the local productivity on surrounding areas, we use the Damage
Function P(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) which can be used to descript the total impact result from n factors.
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We assume that this impact is only influenced by the water supply of every trade (𝑠𝑖 ), so the
function can be described as:
𝑃(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) = 𝑃(𝑠1 , 𝑠2 , … , 𝑠𝑛 )
𝑃(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) = 𝜆1 𝑠1 + 𝜆2 𝑠2 + ⋯ + 𝜆𝑛 𝑠𝑛
While 𝜆1 , 𝜆2 , … , 𝜆𝑛 represent a series of weighted values of water supply of every trade.
With this function, we can measure the effect of the plan on water availability of surrounding
areas:
Before the intervention plan is carried out, the effect of local water supply on the surrounding
areas can be calculated as:
𝑃(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) = 𝑃(𝑠1 , 𝑠2 , 𝑠3 , 𝑠4 , 𝑠5 , 𝑠6 )
𝑃(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) = 𝜆1 𝑠1 + 𝜆2 𝑠2 + 𝜆3 𝑠3 + 𝜆4 𝑠4 + 𝜆5 𝑠5 + 𝜆6 𝑠6
After the intervention plan is carried out, the effect of local water supply on the surrounding
areas can be calculated as:
𝑃′(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) = 𝑃′(𝑠1′ , 𝑠2′ , 𝑠3′ , 𝑠4′ , 𝑠5′ , 𝑠6, )
𝑃′(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) = 𝜆1′ 𝑠1′ + 𝜆′2 𝑠2′ + 𝜆′3 𝑠3′ + 𝜆′4 𝑠4′ + 𝜆′5 𝑠5′ + 𝜆′6 𝑠6,
Thus, the impact on surrounding areas brought by the intervention plan can be calculated as:
𝛥𝑃 = 𝑃′ (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) − 𝑃(𝑥1 , 𝑥2 , … , 𝑥𝑛 )
𝛥𝑃 = 𝑃′ (𝑠1′ , 𝑠2′ , 𝑠3′ , 𝑠4′ , 𝑠5′ , 𝑠6, ) − 𝑃(𝑠1 , 𝑠2 , 𝑠3 , 𝑠4 , 𝑠5 , 𝑠6 )
5.2 The strength and weakness of the intervention plan
As for water scarcity, our intervention plan decreases the total amount of water demand,
increases the total amount of water supply and solved the distribution problem. That is to say,
the value of 𝛥𝑅 will be negative. Thus, this plan will surely mitigate the water scarcity of local
area.
As for environmental impact,
Plan 1 decreases the total amount of water demand which might also decreases the improved
water supply of every trade (𝑠𝑖 ′). Since the P’ changes with 𝑠𝑖 ′ , P’ might also be decreased.
Plan 2 increases the total amount of water supply which might also increases the improved
water supply of every trade (𝑠𝑖 ′). Since the P’ changes with 𝑠𝑖 ′ , P’ might also be increased.
Plan 3 makes the improved water supply of every trade (𝑠𝑖 ′) changes with the improved water
demand of every trade (𝑑𝑖 ′), so P’ might also be changed.
Plan 4 changes the weighted values of water supply of every trade from 𝜆𝑖 to 𝜆′𝑖 . If 𝜆′𝑖 < 𝜆𝑖 ,
the P’ will be decreased.
6 Task 5: Long term impact of intervention on Shandong Province
To calculate the environmental impact of our intervention plan, we regulate computational
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formula for all the improved water demand.
182.38
𝑑1′ = 𝛾𝑎 𝑑1 = 𝛾𝑎 𝑋𝑎 ∙ 0.005𝑒 𝑡;1969
1;𝜂
𝑑2′ = 𝑋𝑖 𝑞2 , 𝑞2 = 𝑞1 (1 − 𝛼) 1;𝜂2
1
𝑑3′ = 𝛾𝑝 𝑑3 = 0.6𝑃 ∙ 𝛾𝑝 𝐾 = 0.6𝑃 ∙ 𝛾𝑝 ∙ 0.365
𝑑4′ = 𝛾𝑐 𝑑4 = 𝑃 ∙ 𝛾𝑐 ∙ 0.0362𝑒 0.0681(𝑡;2002)
𝑑5′ = 𝑑5 = 0.5588 ∙ G − 1.1288 (The G represents the green area of our region.)
𝑑6′ = 𝛾𝑙 𝑑6 = 𝛾𝑙 𝑋𝑙 ∙ 0.0193𝑒 ;0.017(𝑡;2000)
And we give all the parameters a certain value:
𝛾𝑎 = 𝛾𝑝 = 𝛾𝑐 = 𝛾𝑙 = 0.9
𝛼 = 0.025, 𝜂1 = 0, 𝜂2 = 0.15, 𝑞0 =
𝑑2,0
𝑋𝑖,0
= 0.009588(𝑚3 /yuan)
𝑋𝑖 , 𝑋𝑎 , 𝑋𝑙 , 𝑃 and G can be calculated by the BP model.
So, all of the computational formula will be:
182.38
𝑑1′ = 0.9𝑋𝑎 ∙ 0.005𝑒 𝑡;1969
𝑞2 = 𝑞1 (1 − 0.025)
1 − 0.15
= 0.8288𝑞1 = 0.8288𝑡;2000 ∙ 𝑞0 = 0.009588 ∙ 0.8288𝑡;2000
1
𝑑2′ = 𝑋𝑖 𝑞2 = 𝑋𝑖 ∙ 0.009588 ∙ 0.8288𝑡;2000
𝑑3′ = 0.6𝑃 ∙ 0.9 ∙ 0.365 = 0.1971𝑃
𝑑4′ = 𝑃 ∙ 0.9 ∙ 0.0362𝑒 0.0681(𝑡;2002)
𝑑5′ = 𝑑5 = 0.5588 ∙ Ga − 1.1288
𝑑6′ = 0.9𝑋𝑙 ∙ 0.0193𝑒 ;0.017(𝑡;2000)
According to Plan 2, we allocate that the total amount of improved water supply could be:
∑ 𝑠𝑖′ = 1.05 ∙ ∑ 𝑠𝑖
As all of these parameters were settled, we can predict the degree of water scarcity under the
intervention plan in the future by calculating 𝑅 ′ :
𝑅 ′ = 𝑕 (∑ 𝑑′ ) − 𝑕 (∑ 𝑠 ′ )
Also, by using the former parameters, we can predict the degree of water scarcity in the future
before the intervention plan is adopted:
𝑅 = 𝑅𝑃 + 𝑅𝐼 + 𝑅𝐷
Now, we use a new function S to describe the susceptible degree of our chosen region to water
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scarcity:
𝑕(𝑊) − 𝑕(𝑊 − ∆𝑊)
∆𝑊→0
∆𝑊
𝑆 = 𝑙𝑖𝑚
Here comes the calculated outcomes:
Year
R
R'
∆𝑹
S
2015
2.6764
0.0000
-2.6764
0.0000
2016
2.9883
0.0000
-2.9883
0.0000
2017
3.2776
0.0000
-3.2776
0.0000
2018
3.5496
0.0000
-3.5496
0.0000
2019
3.8000
0.0000
-3.8000
0.0000
2020
4.0354
0.0000
-4.0354
0.0000
2021
4.2518
0.0000
-4.2518
0.0000
2022
4.4550
0.0000
-4.4550
0.0000
2023
4.7204
0.0000
-4.7204
0.0000
2024
5.0435
0.0000
-5.0435
0.0000
2025
5.3631
0.1532
-5.2099
0.4415
2026
5.6779
0.3322
-5.3457
0.5128
2027
5.9934
0.5219
-5.4715
0.5952
2028
6.3058
0.7223
-5.5835
0.6920
2029
8.0079
0.9332
-7.0747
0.8065
2030
11.8289
3.7196
-8.1093
0.9390
Table 9: The trend of R, R’ and S
The trend of R, R’ and S are as Figure 4 and Figure 5:
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Figure 4: Trend of R and R’
Figure 5: Trend of S
From these outcomes, we can find that with time going on, the degree of water scarcity under
the intervention plan (𝑅 ′ ) in the future will increase, which means water scarcity in this region
will become more serious. Meanwhile, the degree of water scarcity in the future before the
intervention plan (R) will also increase, and will make the value of 𝛥𝑅 become a constant
negative value. That is to say, our intervention plan effectively mitigate the water scarcity in
this region.
From Figure 4, the green line means that if the value of R exceeds 1(the minimum value of
critical water scarcity), then the situation of water scarcity in our chosen region is considered as
a critical issue. In 2029, the value of R’(0.9332) nearly equals 1, and the R’ far over 1 in 2030.
So, the critical scarcity will occur in 2030.
What is more, the susceptible degree of our chosen region to water scarcity stays invariant in
the first few years and then increases. Thus, our chosen region will become more susceptible to
water scarcity.
To measure the effect of our intervention on the surrounding areas water availability with
function P, we give every independent variable 𝑠1 , 𝑠2 , 𝑠3 , 𝑠4 , 𝑠5 , 𝑠6 a coefficient 5, 7, 2, 2, 0, 5,
according to the negative effect of every usage.
𝑃(𝑥1 , 𝑥2 , … , 𝑥𝑛 ) = 5𝑠1 + 7𝑠2 + 2𝑠3 + 2𝑠4 + 0 + 5𝑠6
The trend of P and P’ are as Figure 6:
Figure 6: Trend of P and P’
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From this picture, the predict P’ values over P, which means 𝛥𝑃 > 0. Thus, we can draw a
conclusion that the invention plan has side effects on water availability of surrounding areas.
7 The strength and weakness of the model
Strength:
1)
The model is not complicated and can easily give result.
2) The model is flexible, all the Benefit Functions can be determined by the environment
situation and policy in a certain region or country. So, the model can be used in any country and
region.
3) The model can judge what kind of water scarcity causes the water shortage and how much
a certain kind of water scarcity causes water shortage.
4) We can analyze which parameters result in the water scarcity via this model and then we
can analyze how social and environmental factors affect the parameters in the model. Then an
optimal scheme can be made to solve the water scarcity by referring this model. So this model
has practical value.
Weakness:
1) This model only take three kinds of water scarcity in to account. The water scarcity can be
described more in details and specific.
2) Some coefficients are determined subjectively. Although the environment, social and
policy factors can qualitative determined which coefficient is large which coefficient is small,
different people have different idea when quantitative determine the coefficient, and this may
result in the various of outcome.
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Reference
[1] China statistical yearbook. 2000-2014.
[2] Statistical yearbook of Shandong province. 2000-2014.
[3] Water resources statistical yearbook of Shandong province.
[4] Living consumption water quantity standard in Shandong province.
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Appendix
Appendix A
BP net work
function [epoch Ok pre wij wki]=BP2(X,T,X1)
lr=0.1;err_goal=0.000001;max_epoch=100000;a=0.9;
Oi=0;Ok=0;[M
N]=size(X);q=3;[L,N]=size(T);wij=rand(q,M);wki=rand(L,q);wij0=zeros(s
ize(wij));wki0=zeros(size(wki));
for epoch=1:max_epoch
NETi=wij*X;
for j=1:N
for i=1:q
Oi(i,j)=2/(1+exp(-NETi(i,j)))-1;
end
end
NETk=wki*Oi;
for i=1:N
for k=1:L
Ok(k,i)=2/(1+exp(-NETk(k,i)))-1;
end
end
E=((T-Ok)'*(T-Ok))/2;
if (E<err_goal)
break;
end
deltak=Ok.*(1-Ok).*(T-Ok);
w=wki;
wki=wki+lr*deltak*Oi'+a*(wki-wki0);
wki0=w;
deltai=Oi.*(1-Oi).*(deltak'*wki)';
w=wij;
wij=wij+lr*deltai*X'+a*(wij-wij0);
wij0=w;
end
NETi=wij*X;
for j=1:N
for i=1:q
Oi(i,j)=2/(1+exp(-NETi(i,j)))-1;
end
end
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NETk=wki*Oi;
for i=1:N
for k=1:L
Ok(k,i)=2/(1+exp(-NETk(k,i)))-1;
end
end
NETi=wij*X1;
for j=1:N
for i=1:q
Oi(i,j)=2/(1+exp(-NETi(i,j)))-1;
end
end
NETk=wki*Oi;
for i=1:N
for k=1:L
pre(k,i)=2/(1+exp(-NETk(k,i)))-1;
end
end
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Appendix B
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F(x1,x2,…,xn)
function rben=F(S,D)
rben=zeros(1,16);
Q=ones(6,16);;
Q(2,:)=Q(2,:)*2;
Q(3,:)=Q(3,:)*6;
Q(4,:)=Q(4,:)*6;
Q(5,:)=Q(5,:)*5;
T=Q./D;
for i=1:16
for j=1:6
if S(j,i)>D(j,i)
rben(i)=rben(i)+T(j,i)*D(j,i);
else
rben(i)=rben(i)+T(j,i)*S(j,i);
end
end
end
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Appendix C
h(x)
function ben=h(X,D)
S=zeros(1,16);D1=zeros(6,16);dw=X; ben=zeros(1,16);
Q=ones(6,16);
Q(2,:)=Q(2,:)*2;
Q(3,:)=Q(3,:)*6;
Q(4,:)=Q(4,:)*6;
Q(5,:)=Q(5,:)*5;
T=Q./D;
for i=1:16
[s(:,i),l]=sort(T(:,i));
for j=1:6
D1(j,i)=D(l(j,1),i);
end
end
for i=1:16
for j=6:-1:1
if dw(i)>D1(j,i)
ben(i)=ben(i)+D1(j,i)*s(j,i);
dw(i)=dw(i)-D1(j,i);
else
ben(i)=ben(i)+dw(i)*s(j,i);
break
end
end
end
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Appendix D
susceptibility
function S=susceptibility(D,w)
S=zeros(1,16);D1=zeros(6,16);dw=w;
Q=ones(6,16);
Q(2,:)=Q(2,:)*2;
Q(3,:)=Q(3,:)*6;
Q(4,:)=Q(4,:)*6;
Q(5,:)=Q(5,:)*5;
T=Q./D;
for i=1:16
[s(:,i),l]=sort(T(:,i));
for j=1:6
D1(j,i)=D(l(j,1),i);
end
end
for i=1:16
for j=1:6
dw(i)=dw(i)-D1(j,i);
if dw(i)<=0
S(i)=s(j,i);
end
end
end