For office use only
T1________________ T2________________ T3________________ T4________________
Team Control Number
55069
Problem Chosen
A
For office use only
F1________________
F2________________
F3________________
F4________________ 2017 MCM/ICM Summary Sheet
The Rehabilitation of the Kariba Dam Recently, the Institute of Risk Management of South Africa has just warned that the Kariba dam is in desperate need of rehabilitation, otherwise the whole dam would collapse, putting 3.5 million people at risk. Aimed to look for the best strategy with the three options listed to maintain the dam, we employ AHP model to filter factors and determine two most influential criteria, including potential costs and benefits. With the weight of each criterion worked out, our model demonstrates that option 3is the optimal choice.
According to our choice, we are required to offer the recommendation as to the number and placement of the new dams. Regarding it as a set covering problem, we develop a multi-objective optimization model to minimize the number of smaller dams while improving the water resources management capacity. Applying TOPSIS evaluation method to get the demand of the electricity and water, we solve this problem with genetic algorithm and get an approximate optimal solution with 12 smaller dams and determine the location of them.
Taking the strategy for modulating the water flow into account, we construct a joint operation of dam system to simulate the relationship among the smaller dams with genetic algorithm approach. We define four kinds of year based on the Kariba’s climate data of climate, namely, normal flow year, low flow year, high flow year and differential year. Finally, these statistics could help us simulate the water flow of each month in one year, then we obtain the water resources planning and modulating strategy.
The sensitivity analysis of our model has pointed out that small alteration in our constraints (including removing an important city of the countries and changing the measurement of the economic development index etc.) affects the location of some of our dams slightly while the number of dams remains the same. Also we find that the output coefficient is not an important factor for joint operation of the dam system, for the reason that the discharge index and the capacity index would not change a lot with the output coefficient changing.
Contents
1Overview (1)
1.1Background (1)
1.2 Restatement of the Problem (1)
1.3 Literature Review (2)
2Assumptions and Justifications (2)
3Notation (3)
4Model Overview (4)
5Model Theory (5)
5.1Determination of the Number and Location of the Dams (5)
5.2 Joint operation of dam system model (9)
6Model Implementation and Results (11)
6.1The Number and Location (11)
6.2The Strategy of Modulating Water Flow (14)
7 Sensitivity Analysis (15)
7.1 The Model of Determination of the Number and Location (15)
7.2 The Model of Modulating Water Flow (17)
8 Further discussion (17)
9 Strengths and Weaknesses (18)
9.1 Strengths (18)
9.2 Weaknesses (19)
10 Conclusion (19)
11 The Evaluation of Three Options (20)
11.1 Establish a Hierarchical Model (20)
11.2 Analysis and Results (21)
1 Overview
1.1 Background
A Zambezi River Authority conference was held in March 2014, engineers warned that the foundations of the dam had weakened and there was a possibility of dam failure unless repairs were made.
On 3 October 2014 the BBC reported t hat “The Kariba Dam is in a dangerous state. Opened in 1959, it was built on a seemingly solid bed of basalt. However, in the past 50 years, the torrents from the spillway have eroded that bedrock, carving a vast crater that has undercut the dam’s foundations, engineers are now warning that without urgent repairs, the whole dam will collapse. If that happened, a tsunami-like wall of water would rip through the Zambezi valley, reaching the Mozambique border within eight hours. The torrent would overwhelm Moz ambique’s Cahora Bassa Dam and knock out 40% of southern Africa’s hydroelectric capacity. Along with the devastation of wildlife in the valley, the Zambezi River Authority estimates that the live of 3.5 million people are at risk.”
On February 2015, Engineers have started on a R3.3bn rescue marathon to prevent the “catastrophic failure” of the Kariba Dam. According to a World Bank special repor t on the beleaguered structure-one of the biggest man-made dams in the world-a potential wall collapse threatens the lives of about 3-million people living on the Zambezi River floodplain between the hydro scheme on the Zambia-Zimbabwe border and the Mozambique coast. [1]
1.2 Restatement of the Problem
We are required to provide an overview of potential costs and benefits with the three options already listed. Then we need to establish a model to determine the number and placement of the new dams when removing the Kariba dam along the Zambezi River. The same overall water management capabilities are also needed. In addition, we should consider emergency water flow situations and restrictions regarding the locations and time, so that we could give out the strategy for modulating the water flow through our new multiple dam system.
In order to solve those problems, we will proceed as follows:
●Build a model to determine the number and location of the multiple dams.
●Give the corresponding strategy of modulating water flow in different conditions.
In our model, we first establish a multi-objective model and use genetic algorithm determine the number and location of the multiple dams. There are two goals improving the water resources management capacity and reducing the cost. Besides, we add some constraints such as water balance, water level, safety and water protection. We choose twenty suitable dam sites and employ the genetic algorithm to solve the optimal problem to determine the number and the location.
After determining the number and location of the dams, we construct our joint operation of dam
system model and employ the genetic algorithm to solve the problem based on the thought of dynamic programming. According to the Kariba’s climate data for about 30 years, we abstract normal flow year, low flow year, high flow year and differential year. We use them to work out the water resources planning and scheduling strategy. The construction of the discharge index and the capacity index benefits an analysis and evaluation for joint operation of the dam system’s performance in different month and year.
1.3 Literature Review
Dating back to 2004, the United States removed 72 dams in total, which created a historical record. Therefore, it is high time for us to focus on the construction of dams concerning their number and placement. Plenty of researchers have already made a number of notable papers to address these problems
Alfer Weber (1909) first proposed a framework for location problem, which is an allocation question with respect to space resource. Among the three classical location model, set covering problem is a significant branch of siting issues. They explored a multi-objective location model to tackle problems with siting optimal points. In their model, maximizing coverage rate in order to satis fy every place’s need is the target function, the concentric point and the capacity restrictions are constraint conditions. Thus, they could convert the optimization problem to the mixed integer linear programming question.
After the set covering model was established, we can optimize our choice of siting the dams. Then several scientists were devoted to building an optimal operation model to provide a reasonable balance between safety and balance. They begin to figure out how the multiple dam system would benefit or affect each other within its system. Masse(1940) first illustrated the concept of it; their main computing method was to optimize water modulating strategy during dispatching period.
Further studies are carried out about different methods to investigate the optimal operation model, including dynamic programming algorithm and neural network algorithm based on improved computer technique. Also, there is much theoretical analysis about location problem since 1990. John Current and Morton O'Kelly(1992) suggested using a modified version of the set covering location model, which still didn’t take the reality into account.
2 Assumptions and Justifications
To simplify our problems, we make the following basic assumptions, each of which is properly justified.
●The dam system is built downstream in the valley of the Kariba Dam. Because it’s
more convenient to build and also with less cost, which can be easily implemented.
●The cost of the dam is mostly the same. Owing to the fact that the length of the canyon
is not large（24km）,geological conditions and climate conditions are mostly the same.
● Each dam’s water supply is mostly the same. Taking into account of safe operation of
the entire multi-dam system, we should make the burden of each dam to be the same as much as possible. ● The water quality of the dam system is the average of the water quality between the
two reservoirs. The river is flowing, so the water quality is mostly similar.
● Water of the dams downstream only comes from dams upstream and natural
precipitation. According to Google Maps, there are no tributaries near the canyon. Also with the principle of conservation of water, the formula should be maintained.
3 Notation
Abbreviation Description
{}12,,m Y y y y = The set of cities
{}12=,,n X x x x
The set of dams
(,)i n d y X The distance from the th i city to the nearest smaller dam
i Ele The electricity demand of th i city
i Wat The water demand of th i city
(,)W t i
The discharge amount of the th i dam at the end of period t,
(,)Z t i The total amount of water released during period t
(,)T t i The amount of natural water in period t
ij V The volume in the th j period of the th i dam
ij Q The inflow in the th j period of the th i dam
ij q The discharge volume in the th j period of the th i dam
ij QJ The runoff volume between the th i dam and the (1)th i -dam
ij S The actual water supply th i dam in the th j period
ij D The planned water supply of the th i dam in the th j period
ij H The hydraulic head of th i dam in the th j period
i K The fore-voltage factor of the th i dam
t λ The discharge flow indicators in the th t period
t μ The storage capacity indicators in the th t period
4 Model Overview
To provide a detailed analysis of the option (3)-removing the Kariba dam and newly building ten to twenty smaller dams. We need to determine the number and location of the multiple dams first. And on that basis we must establish a model to modulate the water flow through the multiple dam system to adapt to different situations. A reasonable balance between safety and cost of our strategy is also needed.
Our first model allows us to determine the number and the location of the multiple dams. We regard the optimal problem as a set covering problem and establish a multi-objective model to solve the problem. There are two goals, namely improving water resources management capacity and reducing the cost. And there are also some constraints including water balance, water level, safety, water resources protection and number constraints. On the account that the optimal problem is difficult to solve in polynomial time, so we use genetic algorithm to get the solution.
After determining the numbers and the location, we establish a joint operation of dam system model to gain a strategy about modulating water flow in different condition. Though it’s also a multi-objective problem, it is different from the previous model. We use the maximization of economic and social benefits as the objective and set some constraints, such as water balance, reservoir capacity and discharge flow constraints. We use genetic algorithm to get the
modulating strategy in different conditions.
In conclusion, we use programming and heuristic algorithm to solve the problem of building dams and the modulating strategy. It’s relatively easy to achieve and it has a significant guidance for the reality.
5 Model Theory
5.1 Determination of the Number and Location of the Dams 5.1.1 Establishment of the model
The construction of dams needs to consider many aspects, while at the same time it is subject to economic, social, environmental constraints and other factors. In order to obtain the proper number of dams and their location, we establish a multi-objective model. The Objective
● Improve Water Resources Management Capacity
The purpose of building smaller dams instead of larger dams is to manage water resources better, mainly to satisfy dweller’s demand for the electricity consumption and water consumption (Including agricultural, industrial and domestic water) of the neighboring cities. Demand may vary between cities, but it is clear that for cities with greater demand, the dam should be built closer to them. so we get that
1
min (,)m
i i n i Ele d y X =∑
1
min (,)m
i i n i Wat d y X =∑
● Reduce the Cost
On the basis of ensuring the water supply and power supply, we should minimize the cost of our plan. The whole cost consists of the removal of the Kariba dam and building new smaller dams. Since the cost of removal is fixed, so we only consider the variable cost(building cost), which is only related to the number of dams. So we should minimize the quantity of the smaller dams.
min n
The Constraints ● Water Balance
()
(,)(1,)(,)(,)k i W t i W t i Z t k T t i π
∈=-+
+∑
(,)W t i ，(,)Z t i and (,)T t i represent the discharge amount of the th i dam at the end of period
t, the total amount of water released during period t and the amount of natural water in period t respectively.()i π denotes the set of all higher dams of th i dam. If ()i π is empty, the corresponding summation is zero.[2] ● Water Level
The water level in the dam area should be kept between the dead water level and the limited water level in the flood season. Dead Water Level, namely the lowest water level that allows a reservoir to fall off under normal operating conditions. Flood limited water level is the requirements of control over flood to limit reservoirs’ water storage.
DWL WL FWL ≤≤
● Safety
The construction of multiple smaller dams is at least as safe as the original larger dams. The safety considerations for dam construction mainly include reducing the probability of dam failure, thereby reducing damage to dams downstream and enhancing resistance ability to extreme weather.
sd Bd Safety Safety ≥
sd Safety represents the safety of multiple dams system, while Bd Safety denotes the safety of
the existing Kariba dam. ● Water Resources Protection
The construction of smaller dams should require a higher degree of protection for the water resources as the replacement of the existing dam.
sd Bd WRP WRP ≥
sd WRP represents the environment protection of the multiple dams system, while Bd WRP
denotes the environment protection of the existing Kariba dam. ● Number
The number of small dams should be greater than ten and less than twenty. Option 3 is replacing the Kariba dam with a series of ten to twenty smaller dams.
To ensure the continuity of the model establishment, the parameters of the constraints will be described in the next part. And in the sixth section, we will clarify how to calculate the demand of electricity(i Ele ),the demand of the water(i Wat ),the safety of different dams and the
environment protection of different dams.
To sum up, we model the problem about smaller dams’ number and location decision with multi-objective optimization. The formulas of this model is
1min (,)m
i i n i Ele d y X =∑ 1min (,)m
i i n i Wat d y X =∑
min n
..
S t ()
(,)(1,)(,)(,)k i W t i W t i Z t k T t i π
∈=-++∑ DWL WL FWL ≤≤ sd Bd Safety Safety ≥ sd Bd WRP WRP ≥
1020n ≤≤(n=10, 11, 12…..20)
5.1.2 The Parameter of the Constraints
● The Demand of Electricity
The power demand of a city is mainly related to its economic and demographic factors. Our
paper uses the city's latest available GDP data as the representative of its economic factors, and the total urban population as the representative of its demographic factor. The GDP of the th i city is i GDP .The population of the th i city is i pop .
● The Demand of Water
The water consumption of residents is mainly affected by residents' income, water price and other factors. Since within a country, the price difference is relatively small, so here we ignore the price difference.. Therefore, the water consumption of urban residents is mainly affected by the income of residents, We use the city's GNI to represent its urban residents. The GNI of the th i city is i GNI .
● The Safety of Dams and the Kariba Dam [3]
For the safety of dam construction, our paper mainly consider the index of risk loss as the agent of its safety indicators.
For a single Big dam, its Risk Loss Index(RLI) Bd R equals to P multiply N . P is the failure probability of the dam beyond the current code standard. For example, the current code specifies that if a dam's seismic safety standard is 50 years beyond 10%, the probability of
failure exceeding the current code specification is 0.1 / 50 = 0.002.N is the life loss of dam failure (we can convert economic loss into life loss in a certain percentage). We use the Dekay & McClelland method to calculate the loss. Thus, the RLI of the Kariba dam is
()()Bd R P K N K =?.
For multiple dam system, the RLI of the th j dam is
1
()()(|)()()n
j l j R P j N j P l j P j N l =+=?+
??∑
and the RLI of the multiple dam system is
1
n
sd
sd j j R R ==∑
The safety of the multiple dam system should be greater than or at least equal to the safety of the existing Kariba dam. So we get that sd
Bd
R R ≤, that is to say 1()()n
sd i i R P K N K =≤?∑.And n
is an integer greater than ten and less than twenty ● The water resources protection of dams
According to the International Commission On Large Dams(ICOLD), there are three goals of
water resources management [4]
? Improved management of the water supply ? Improved water quality in our rivers
? Improved environmental conditions in the watershed
Since there are few branches in the Kariba Gorge, we mainly consider the management of the water supply and the water quality.
According to the constraints, the water supply of the multiple-dam system should be greater than the big dam. That is to say
sd Bd WS WS ≥
sd WS equals to the number of the smaller dams multiply the water supply of a smaller dam,
while Bd WS represents the water supply of the Kariba Dam.
Besides, the water quality of the multiple-dam system should be greater than the Kariba dam.
sd Bd WQ WQ ≥,
sd WQ represents the expected average water quality of the Lake Kariba if the authority adopt
the option three, while Bd WQ represents the present water quality of the Lake Kariba.
5.2 Joint Operation of Dam System Model
Joint operation of dam system can utilize the differences between dam capacity and hydrologic condition, so that we develop a joint compensation effect for the dam group and make full use of water resources. Compared with the traditional single large dam, the dam group establish on the basis of multi –dam system, which could help us take advantage of water resources for our social benefits, including economic and social benefits. Therefore, it is necessary to establish an effective water resources dispatching scheme.
In order to better express our model ，we let 1...i n =indicates the number of dams ， 1...j T =denotes the cycle of water dispatching ，ij V illustrates the th i dam ’s water storage during the th j period.[5]
The Objective
The hydropower station mainly bring about economic and social benefits. But obviously, hydropower station can bring magnificent economic benefits. Among them, the most direct indication is the dam power generation. Therefore, taking the reservoir power generation as the revelation of economic benefit.
11
***n T
e i ij ij i j I K q H t ===?∑∑
ij q is the discharge flow of the th i dam in the th j period ，ij H is the hydraulic head of th i dam
in the th j period ，i K is the force-voltage factor of the th i dam ，t ?is the time period. In addition to creating enormous economic benefits, a large dam can also bring huge social benefits. Through the construction of dams, it can meet the need of a large number of industrial and agricultural water, also the domestic water.
11
()n T
s ij ij i i I S D ===-∑∑
ij S is the actual water supply th i dam in the th t stage ，ij D is water demand or planned water supply of the th i dam in the th t stage.
Therefore, we should make full use of water resources so as to make the economic and social benefits as large as possible.[6]
max()e s I I I =+
The Constraints ● Water Balance
,(1)()i t it it it it V V Q q t L +=+-??-
,(1)(1),i t i t it Q q QJ +-=+
it Q 、ij q are the average inflow into the dam and discharge volume in the th t period of the th i
dam,it L is the evaporation and leakage loss.it QJ is the runoff volume between the th i dam and the th j dam. ● Capacity of the Dam
To prevent floods and droughts ，The capacity of each dam must be limited.
min max i it i V V V ≤≤
min i V 、max i V are the upper and lower limit of the th i dam’s water storage ，namely the dead capacity and the flood control capacity. ● Discharge Flow Constraint
In order to keep the normal life standard and to maintain the downstream ecological water use, we must ensure the certain discharge of water, we employ min i q to indicate the minimal discharge of water. Because of the limitation of the dam itself and the purpose of flood control, there need to be a maximal discharge capacity constraint max i q
m max i it i q in q q ≤≤
In conclusion, in order to make economic benefit and social benefit as large as possible while satisfying the constraint conditions, we need to develop water resources optimized dispatching scheme.
11
11
max ***()n T n T
i ij ij ij ij i j i i I K q H t S D =====?+-∑∑∑∑
,(1),(1)(1),()*..min max m max i t it it it it i t i t it i it i
i it i V V Q q t L Q q QJ s t V V V q in q q ++-=+-?-??=+?
?
≤≤??≤≤?
it V 、it q are the interval-type variable ，
min i V 、min i q correspond to meet the basic industrial and agricultural water 、domestic water and ecological water ，they are the dead water level. For safety considerations such as flood control ，max i V 、max i q are the maximal water level. We let [,]i i Vf VF to be the of it V ,[,]i i qf qF is the optimum interval of it q .[7]
1()/(min ),min 1,1()/(max ),max 0,i it i i i it i
i it i it it i i i i it i Vf V Vf V V V Vf Vf V VF V VF V VF VF V V else λ---≤≤??≤≤?
=?
---≤≤???
1
1N
t it i N λλ==∑
1()/(min ),min 1,1()/(max ),max 0,i it i i i it i i it i it it i i i i it i qf q qf q q q qf qf q qF q qF q qF qF q q else μ---≤≤??≤≤?
=?
---≤≤???
1
1N
t it i N μμ==∑
6 Model Implementation and Results 6.1 The Number and Location
Measuring the demand of the nearby city
TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) is a commonly used and effective method in multi-objective decision analysis. Hence ,we employ this model and then investigate extended TOPSIS to evaluate the demand of electricity and water of each city.
Then we use the data to calculate the demand of the electricity and water with TOPSIS, and part of the results is shown in Table 1.
Table 1 The Demand of Electricity and Water
Calculate the shortest distance from each dam to each city
The dam should be built in the valley of the river fully, because the project is small with low cost. Moreover, the construction of small dams is mainly to replace the larger dam, so the small dam should be constructed mainly in the downstream of the Kariba dam in the canyon. Figure 1 is the suitable site for building dams.
Figure 1 The Suitable Gorge for Building Dams
With respect to the option 3 that we need to construct 10 to 20 small dams, considering the geological factors around the dam, we first select the suitable location of the 20 dams.
According to our model, we use the distance from the dam to the center of the city as a representative of the distance to the city. A total of 25 cities are selected close to the Kariba gorge, including 15 cities in Zambia, 10 cities in Zimbabwe. We search and calculate the shortest distance and path of each dam to the city through Google maps, and the results of some small dams to the cities are shown in Table 2.
Table 2 The Distance Between the Dams and the City(Part)
● The Data of Other Factors
At the same time, we collect the data of other related factors through the Zambia National Bureau of Statistics, the World Bank, the World Commission on Dams and other institutions to gain the relevant statistics.
Table 3 The Data of Other Factors
DWL
NWL
FWL
BFWL
RLI
WS
WQ
Kariba 475.5 484.6 490 476.7 9000 608.5 19
Multiple 382 408 412 385
1
n
sd i
i R
=∑
e WS n ?
e WQ
Under normal circumstances the water level should be higher than the dead water level, and lower than the normal position. But before the flood season, the water level is lower than the limit of water level.
● The numbers and the location
This optimal problem belongs to the NP-hard problem. We can’t solve it in polynomial time. So we can only use heuristic algorithms such as artificial neural networks, genetic algorithm, simulated annealing to obtain the approximate solution. In this paper, we select the genetic algorithm to solve the problem.
We run the program repeatedly, and finally get a relatively good result and determine our number and location of the dams. There should be 12 smaller dams in the Kariba Gorge. Besides, we select the optimal 12 dam location to maximize our objective function from 20 appropriate
dam sites. The sixth and twelfth site locations are listed in the Table 4
Table 4 The Location of the Dam
At the same time, we will also mark the location of the small dam on the map, we put the location of the sixth and twelfth dam on the map. We can see it on the Figure 2 and
Figure 3
Figure 2 The Location of the 6th Dam Figure 3 The Location of the 12th Dam 6.2 The Strategy of Modulating Water Flow
Traditional optimization algorithm mostly belongs to the convex optimization category. It demands a unique global optimal solution, which is often used in practical applications. A number of heuristic intelligent algorithm, for example genetic algorithm etc.These methods show that the algorithm can work out satisfactory results for solving the optimization problem. Employing joint operation of dam system model, we use genetic algorithm and Matlab2014 for the solution.[8]
By running the genetic algorithm and adjusting the parameters, we obtain a relatively satisfactory result, and then develop a joint operation of dam system model.(The picture of the results is shown in the appendix part four.)
Normal Flow Year：Take the climate and hydrological conditions of the average of nearly
5 years to simulate normal climatic conditions.
In normal years, the water resources dispatching of the dam system shows obvious seasonal difference. In the dry season from June to September, the discharge of each reservoir reaches the bottom line. Compared with the middle and lower reaches of the river, upper and middle
reaches of water resources and reserves are relatively abundant, therefore, still a lot of water could be scheduled downstream. But until the rainy season from November to March, the precipitation intensity and the discharge of each dam are very large. The results show that the discharge index and storage capacity index perform better in May and October, which are in the optimum range, while the other month are all above 0.7, this is a satisfactory result.
●Low Flow Year: The monthly precipitation is taken as the highest value in the past 30
years, simulating the more serious flood threat..
When the annual precipitation decreases, each dam discharge is significantly lower, but the discharge index and the capacity index is still above 0.7, which reflects the joint operation of dam system still has obvious effect to resist abnormal weather like the drought condition.
●High Flow Year：The monthly precipitation is taken as the highest value inthe past 30
years, simulating the more serious flood threat.
When the annual precipitation has increased significantly, then the reservoir’s discharge would increase. While the discharge of the rainy season increases a lot, but in dry season the increase range is smaller, which makes the discharge capacity index and volume index increase significantly from May to September. Although the index decreases significantly in November and February the figure are more than 0.7, is still relatively satisfactory.
●Differentiation Year：The precipitation from May to September is taken as the lowest
value in the past 30 years, the precipitation from October to April was taken as the highest value in the past 30 years. So that we could simulate drought, flood significantly differentiated condition..
When confronted with the differentiation year where there comes rainy season when the precipitation increases and the dry season when the precipitation decreases all come about in one year. The discharge flow of each reservoir expands seasonally, which makes the discharge index and storage capacity index of each month decrease, and it is more obvious in the rainy season. Which can not be ignored is that the discharge flow indicators and storage capacity indicators remained at 0.7 or more.[9]
7. Sensitivity Analysis
7.1 The Model of Determination of the Number and Location
●Remove an important city of both countries
In the questions we discussed above，we have selected 15 cities in Zambia，10 cities in Zimbabwe to specify the number and placement of the newly built smaller dams. Among the sensitivity analysis, we remove important cities of the two countries, thus, we could see whether this movement would influence the number and placement of constructing dams.
In our analysis, we leave out Lusaka (The capital city of Zambia) and Harare(The capital city of Zimbabwe) from our original cities, yet, we still use the multi objective optimization model and genetic algorithm we already established to calculate the number and placement of smaller dams. We discover that the optimal number of constructing dams is also 12, but the location of each dam is slightly changed. Just the fourth and the eleventh dam’s place changes a lot, while other dam’s place just changes a little. Therefore, we list the location of the fourth and the eleventh dam’s site in Table 5.
To sum up, the influence of removing bigger cities is not that much. Still in accordance with our former model, we could also draw the conclusion that large cities exert magnificent impact on the selection of the dam’s location. Hence, putting larger cities into our model while constructing dams is an appropriate choice.
Chang e the measurement of the economic development index
In the model we have discussed before, we mainly employ the current GDP of each city as a measure of the economic development, which is also used as one of the indicators to measure the water and electricity demand. Here, we will use its GDP growth rate as an agent indicator for the city's economic development, and change its GDP growth rate to observe whether it has the impact on the number and location of the construction of dams.
Our paper first use economic growth data of the city to measure its water and electricity demand, and then float up and down of the basis, the maximum change is up to 8%. The impact of economic growth on the number of dams is shown in Table 6.
Table 6 The Sensitivity of the GDP growth rate
From the table above we can see that the changing of the economic growth rate for the number of the building small dam is relatively small. Even reaching to 8% changes, building two more small dams from the original can meet the demand of electricity and water. While Zambia and Zimbabwe are African countries, so that they are in a relatively poor condition. Due to the lack of economic development momentum, the change in its economic growth in the future is relatively small. Therefore, it is a reasonable choice of constructing 12 dams.
Table 5 The Location of 4th Dam and 11th Dam
Alteration in the economic growth rate has little impact on the position of dams. Basing on the fact that Kariba gorge is about 34.9 kilometers, and we only need to build 10 to 20 small dam in the canyon, also small dams need to maintain a certain distance, so that the small dam’s location is relatively stable and the change is not great.
In conclusion, we can see that the representative index of its economic development is reasonable. Despite the fact that the figure floats up and down, its influence on the number and location of small dams is relatively small. It also shows that the model we established in the previous paper about the location and quantity of small dams is robust and has good adaptability to the reality.
7.2 The Model of Modulating Water Flow
Output coefficient is a comprehensive benefit coefficient which reflects water-turbine generator set’s overall efficiency. It equals to turbine efficiency, generator efficiency, transfer efficiency multiply invariant constant 9.81. Its value is up to the scale of Hydropower Station Dam. For Large hydropower station the value is about 8.5.For media hydropower station the value is about 8-8.5. While for small hydropower station the value is about 6.0-8.0. Multiply the coefficient, water-head and water-flow and the answer will be the unit’s output.
We analyze the change of output coefficient’s influence on the joint operation of dam system in the Table 7.
Table 7 The Sensitivity of the Output Coefficient
±would bring We could see from the Table 7 that,output coefficient K change about 15%
±. This indicates that output coefficient K is not aboutλandμchange for about only 5%
an important factor for joint operation of the dam system.
8. Further Discussion
Take the future benefits into account. Reservoir operation not only has the current economic and social benefits, but also has the economic and social benefits in the future The future economic benefits can be expressed by the total energy storage at the end of the
dispatching cycle, the greater the total energy storage, the more the potential for future development. The social benefits in the future can be illustrated as the total capacity in the last scheduling period. Generally speaking, the more water storage in the reservoir, the more able to meet the needs of future generations.
(1),,11
[***()()]n
T
e i ij ij i j i j ij ij i j I K q H t h h Q q =-==?+--∑∑
11
()n T
s T ij ij i i I V S D ===+-∑∑
ij h illustrates the water level of th i dam in the th j period,T V indicates the dam’s water storage
in the end of the dispatching period.
● Increase ecological benefits ：Most people pay more attention to the traditional functions
of power generation, flood control, water supply and so on. We will further consider the ecological benefits of the reservoir. We do not take the ecological flow as a constraint, but consider it as one of the scheduling objectives, but there are many difficulties.
11
(,,,)n
T
en ij ij ij ij i j I f V Q q de ===∑∑
ij de is the ecological water requirement of the th i dam in the th j period.
● In order to avoid the large change of the reservoir water level, which could lead to the
change of the geological condition, and even cause the reservoir earthquake. So we will
increase the constraint condition.,(1)it i t i h h δ+-≤,in which i δ is the allowable variation of reservoir water level.
9 Strengths and Weaknesses 9.1 Strengths
● Our models are robust while the parameters changes. That is to say, a slight change of
parameter will not cause a significant change of the results. ● Our models can be well applied in other places and we just need to change a little the
specific conditions. ● Based on different situation, we develop the corresponding modulating strategy and make
问题A：除非超车否则靠右行驶的交通规则 在一些汽车靠右行驶的国家（比如美国，中国等等），多车道的高速公路常常遵循以下原则：司机必须在最右侧驾驶，除非他们正在超车，超车时必须先移到左侧车道在超车后再返回。建立数学模型来分析这条规则在低负荷和高负荷状态下的交通路况的表现。你不妨考察一下流量和安全的权衡问题,车速过高过低的限制，或者这个问题陈述中可能出现的其他因素。这条规则在提升车流量的方面是否有效？如果不是，提出能够提升车流量、安全系数或其他因素的替代品（包括完全没有这种规律）并加以分析。在一些国家，汽车靠左形式是常态，探讨你的解决方案是否稍作修改即可适用，或者需要一些额外的需要。最后，以上规则依赖于人的判断，如果相同规则的交通运输完全在智能系统的控制下，无论是部分网络还是嵌入使用的车辆的设计，在何种程度上会修改你前面的结果？ 问题B：大学传奇教练 体育画报是一个为运动爱好者服务的杂志，正在寻找在整个上个世纪的“史上最好的大学教练”。建立数学模型选择大学中在一下体育项目中最好的教练：曲棍球或场地曲棍球，足球，棒球或垒球，篮球，足球。 时间轴在你的分析中是否会有影响？比如1913年的教练和2013年的教练是否会有所不同？清晰的对你的指标进行评估，讨论一下你的模型应用在跨越性别和所有可能对的体育项目中的效果。展示你的模型中的在三种不同体育项目中的前五名教练。 除了传统的MCM格式，准备一个1到2页的文章给体育画报，解释你的结果和包括一个体育迷都明白的数学模型的非技术性解释。 使用网络测量的影响和冲击 学术研究的技术来确定影响之一是构建和引文或合著网络的度量属性。与人合写一手稿通常意味着一个强大的影响力的研究人员之间的联系。最著名的学术合作者是20世纪的数学家保罗鄂尔多斯曾超过500的合作者和超过1400个技术研究论文发表。讽刺的是,或者不是,鄂尔多斯也是影响者在构建网络的新兴交叉学科的基础科学,尤其是，尽管他与Alfred Rényi的出版物“随即图标”在1959年。鄂尔多斯作为合作者的角色非常重要领域的数学,数学家通常衡量他们亲近鄂尔多斯通过分析鄂尔多斯的令人惊讶的是大型和健壮的合著网络网站(见http:// https://www.doczj.com/doc/047275249.html,/enp/)。保罗的与众不同、引人入胜的故事鄂尔多斯作为一个天才的数学家,才华横溢的problemsolver,掌握合作者提供了许多书籍和在线网站(如。,https://www.doczj.com/doc/047275249.html,/Biographies/Erdos.html)。也许他流动的生活方式,经常住在带着合作者或居住,并给他的钱来解决问题学生奖,使他co-authorships蓬勃发展并帮助构建了惊人的网络在几个数学领域的影响力。为了衡量这种影响asErdos生产,有基于网络的评价工具,使用作者和引文数据来确定影响因素的研究,出版物和期刊。一些科学引文索引,Hfactor、影响因素,特征因子等。谷歌学术搜索也是一个好的数据工具用于网络数据收集和分析影响或影响。ICM 2014你的团队的目标是分析研究网络和其他地区的影响力和影响社会。你这样做的任务包括: 1)构建networkof Erdos1作者合著者(你可以使用我们网站https://files.oak https://www.doczj.com/doc/047275249.html,/users/grossman/enp/Erdos1.htmlor的文件包括Erdos1.htm)。你应该建立一个合作者网络Erdos1大约有510名研究人员的文件,与鄂尔多斯的一篇论文的合著者,他但不包括鄂尔多斯。这将需要一些技术数据提取和建模工作获
For office use only T1________________ T2________________ T3________________ T4________________ Team Control Number 55069 Problem Chosen A For office use only F1________________ F2________________ F3________________ F4________________ 2017 MCM/ICM Summary Sheet The Rehabilitation of the Kariba Dam Recently, the Institute of Risk Management of South Africa has just warned that the Kariba dam is in desperate need of rehabilitation, otherwise the whole dam would collapse, putting 3.5 million people at risk. Aimed to look for the best strategy with the three options listed to maintain the dam, we employ AHP model to filter factors and determine two most influential criteria, including potential costs and benefits. With the weight of each criterion worked out, our model demonstrates that option 3is the optimal choice. According to our choice, we are required to offer the recommendation as to the number and placement of the new dams. Regarding it as a set covering problem, we develop a multi-objective optimization model to minimize the number of smaller dams while improving the water resources management capacity. Applying TOPSIS evaluation method to get the demand of the electricity and water, we solve this problem with genetic algorithm and get an approximate optimal solution with 12 smaller dams and determine the location of them. Taking the strategy for modulating the water flow into account, we construct a joint operation of dam system to simulate the relationship among the smaller dams with genetic algorithm approach. We define four kinds of year based on the Kariba’s climate data of climate, namely, normal flow year, low flow year, high flow year and differential year. Finally, these statistics could help us simulate the water flow of each month in one year, then we obtain the water resources planning and modulating strategy. The sensitivity analysis of our model has pointed out that small alteration in our constraints (including removing an important city of the countries and changing the measurement of the economic development index etc.) affects the location of some of our dams slightly while the number of dams remains the same. Also we find that the output coefficient is not an important factor for joint operation of the dam system, for the reason that the discharge index and the capacity index would not change a lot with the output coefficient changing.
摘 认真书写摘要（注意篇幅不能超过一页，但要充分利用本页），勿庸置疑，摘要 在整个数模论文中占有及其重要的地位，它是评委对你所写论文的第一印象，因此在这一部分的写作上一定要花大功夫， 千万不能马虎。摘要是论文是否取得好名次的决定性因素，评委们通过你的摘要就决定是否继续阅读你的论文。换句话说，就算你的论文其他方面写得再好，摘要不行，你的论文也不会得到重视。我认为在写摘要时应包括6个方面：对问题稍做描述（问题的研究有什么意义），用了什么方法，建立了什么样的模型（线性规化模形），针对所建立的模型用什么算法、软件解的，得到什么结论，模型、结论有什么特色。 简而言之，摘要应该体现你用什么方法，解决了什么问题，得出了什么结论。另外，好的摘要都包含了两个共同的特点：简要simple 和明确clear 。 学术论文要求：括地陈述论文研究的目的、方法、结果、结论，要求200～300字．应排除本学科领域已成为常识的内容；不要把应在引言中出现的内容写入摘要，不引用参考文献；不要对论文内容作诠释和评论．不得简单重复题名中已有的信息．用第三人称，不使用“本文”、“作者”等作为主语．使用规范化的名词术语，新术语或尚无合适的汉文术语的，可用原文或译出后加括号注明．除了无法变通之外，一般不用数学公式和化学结构式，不出现插图、表格．缩略语、略称、代号，除了相邻专业的读者也能清楚理解的以外，在首次出现时必须加括号说明．结构严谨，表达简明，语义确切。 摘要是论文的门面，摘要写的不好评委后面就不会去看了，自然只能给个成功参赛奖。摘要首先不要写废话，也不要照抄题目的一些话，直奔主题，要写明自己怎样分析问题，用什么方法解决问题，最重要的是结论是什么要说清楚，在中国的竞赛中结论如果正确一般得奖是必然的，如果不正确的话评委可能会继续往下看，也可能会扔在一边，但不写结论的话就一定不会得奖了，所以要认真写。摘要至少需要琢磨两个小时，不要轻视了它的重要性。很有必要多看看优秀论文的摘要是如何写的，并要作为赛前准备的内容之一。 关键词：关键词1；关键词2；关键词3用的方法中的重要术语） 其它汉字 小四号宋字，行距用单倍行距（由于数学论文中通常有汉字和公式，建议行距用固定行距22磅。）
Camping along the Big Long River Summary In this paper, the problem that allows more parties entering recreation system is investigated. In order to let park managers have better arrangements on camping for parties, the problem is divided into four sections to consider. The first section is the description of the process for single-party's rafting. That is, formulating a Status Transfer Equation of a party based on the state of the arriving time at any campsite. Furthermore, we analyze the encounter situations between two parties. Next we build up a simulation model according to the analysis above. Setting that there are recreation sites though the river, count the encounter times when a new party enters this recreation system, and judge whether there exists campsites available for them to station. If the times of encounter between parties are small and the campsite is available, the managers give them a good schedule and permit their rafting, or else, putting off the small interval time t until the party satisfies the conditions. Then solve the problem by the method of computer simulation. We imitate the whole process of rafting for every party, and obtain different numbers of parties, every party's schedule arrangement, travelling time, numbers of every campsite's usage, ratio of these two kinds of rafting boats, and time intervals between two parties' starting time under various numbers of campsites after several times of simulation. Hence, explore the changing law between the numbers of parties (X) and the numbers of campsites (Y) that X ascends rapidly in the first period followed by Y's increasing and the curve tends to be steady and finally looks like a S curve. In the end of our paper, we make sensitive analysis by changing parameters of simulation and evaluate the strengths and weaknesses of our model, and write a memo to river managers on the arrangements of rafting. Key words: Camping；Computer Simulation; Status Transfer Equation
全国大学生数学建模竞赛论文格式规范 ●本科组参赛队从A、B题中任选一题，专科组参赛队从C、D题中任选一题。（全国评奖时，每个 组别一、二等奖的总名额按每道题参赛队数的比例分配；但全国一等奖名额的一半将平均分配给本组别的每道题，另一半按每道题参赛队比例分配。） ●论文用白色A4纸单面打印；上下左右各留出至少2.5厘米的页边距；从左侧装订。 ●论文第一页为承诺书，具体内容和格式见本规范第二页。 ●论文第二页为编号专用页，用于赛区和全国评阅前后对论文进行编号，具体内容和格式见本规 范第三页。 ●论文题目、摘要和关键词写在论文第三页上，从第四页开始是论文正文，不要目录。 ●论文从第三页开始编写页码，页码必须位于每页页脚中部，用阿拉伯数字从“1”开始连续编号。 ●论文不能有页眉，论文中不能有任何可能显示答题人身份的标志。 ●论文题目用三号黑体字、一级标题用四号黑体字，并居中；二级、三级标题用小四号黑体字， 左端对齐（不居中）。论文中其他汉字一律采用小四号宋体字，行距用单倍行距。打印文字内容时，应尽量避免彩色打印（必要的彩色图形、图表除外）。 ●提请大家注意：摘要应该是一份简明扼要的详细摘要（包括关键词），在整篇论文评阅中占有重 要权重，请认真书写（注意篇幅不能超过一页，且无需译成英文）。全国评阅时将首先根据摘要和论文整体结构及概貌对论文优劣进行初步筛选。 ●论文应该思路清晰，表达简洁（正文尽量控制在20页以内，附录页数不限）。 ●在论文纸质版附录中，应给出参赛者实际使用的软件名称、命令和编写的全部计算机源程序（若 有的话）。同时，所有源程序文件必须放入论文电子版中备查。论文及程序电子版压缩在一个文件中，一般不要超过20MB，且应与纸质版同时提交。 ●引用别人的成果或其他公开的资料(包括网上查到的资料) 必须按照规定的参考文献的表述方 式在正文引用处和参考文献中均明确列出。正文引用处用方括号标示参考文献的编号，如[1][3]等；引用书籍还必须指出页码。参考文献按正文中的引用次序列出，其中书籍的表述方式为： ●[编号] 作者，书名，出版地：出版社，出版年。 ●参考文献中期刊杂志论文的表述方式为： ●[编号] 作者，论文名，杂志名，卷期号：起止页码，出版年。 ●参考文献中网上资源的表述方式为： ●[编号] 作者，资源标题，网址，访问时间（年月日）。 ●在不违反本规范的前提下，各赛区可以对论文增加其他要求（如在本规范要求的第一页前增加 其他页和其他信息，或在论文的最后增加空白页等）；从承诺书开始到论文正文结束前，各赛区不得有本规范外的其他要求（否则一律无效）。 ●本规范的解释权属于全国大学生数学建模竞赛组委会。 ●[注] 赛区评阅前将论文第一页取下保存，同时在第一页和第二页建立“赛区评阅编号”（由各 赛区规定编号方式），“赛区评阅纪录”表格可供赛区评阅时使用（各赛区自行决定是否在评阅时使用该表格）。评阅后，赛区对送全国评阅的论文在第二页建立“全国统一编号”（编号方式由全国组委会规定，与去年格式相同），然后送全国评阅。论文第二页（编号页）由全国组委会评阅前取下保存，同时在第二页建立“全国评阅编号”。 全国大学生数学建模竞赛组委会 2017年修订
The Keep-Right-Except-To-Pass Rule Summary As for the first question, it provides a traffic rule of keep right except to pass, requiring us to verify its effectiveness. Firstly, we define one kind of traffic rule different from the rule of the keep right in order to solve the problem clearly; then, we build a Cellular automaton model and a Nasch model by collecting massive data; next, we make full use of the numerical simulation according to several influence factors of traffic flow; At last, by lots of analysis of graph we obtain, we indicate a conclusion as follow: when vehicle density is lower than 0.15, the rule of lane speed control is more effective in terms of the factor of safe in the light traffic; when vehicle density is greater than 0.15, so the rule of keep right except passing is more effective In the heavy traffic. As for the second question, it requires us to testify that whether the conclusion we obtain in the first question is the same apply to the keep left rule. First of all, we build a stochastic multi-lane traffic model; from the view of the vehicle flow stress, we propose that the probability of moving to the right is 0.7and to the left otherwise by making full use of the Bernoulli process from the view of the ping-pong effect, the conclusion is that the choice of the changing lane is random. On the whole, the fundamental reason is the formation of the driving habit, so the conclusion is effective under the rule of keep left. As for the third question, it requires us to demonstrate the effectiveness of the result advised in the first question under the intelligent vehicle control system. Firstly, taking the speed limits into consideration, we build a microscopic traffic simulator model for traffic simulation purposes. Then, we implement a METANET model for prediction state with the use of the MPC traffic controller. Afterwards, we certify that the dynamic speed control measure can improve the traffic flow . Lastly neglecting the safe factor, combining the rule of keep right with the rule of dynamical speed control is the best solution to accelerate the traffic flow overall. Key words：Cellular automaton model Bernoulli process Microscopic traffic simulator model The MPC traffic control
Team Control Number For office use only 13215 For office use only T1 ________________ F1 ________________ T2 ________________ F2 ________________ T3 ________________ Problem Chosen F3 ________________ T4 ________________ F4 ________________ C 2012 Mathematical Contest in Modeling (MCM) Summary Sheet (Attach a copy of this page to each copy of your solution paper.) Type a summary of your results on this page. Do not include the name of your school, advisor, or team members on this page. Message Network Modeling for Crime Busting Abstract A particularly popular and challenging problem in crime analysis is to identify the conspirators through analysis of message networks. In this paper, using the data of message traffic, we model to prioritize the likelihood of one’s being conspirator, and nominate the probable conspiracy leaders. We note a fact that any conspirator has at least one message communication with other conspirators, and assume that sending or receiving a message has the same effect, and then develop Model 1, 2 and 3 to make a priority list respectively and Model 4 to nominate the conspiracy leader. In Model 1, we take the amount of one’s suspicious messages and one’s all messages with known conspirators into account, and define a simple composite index to measure the likelihood of one’s being conspirator. Then, considering probability relevance of all nodes, we develop Model 2 based on Law of Total Probability . In this model, probability of one’s being conspirator is the weight sum of probabilities of others directly linking to it. And we develop Algorithm 1 to calculate probabilities of all the network nodes as direct calculation is infeasible. Besides, in order to better quantify one’s relationship to the known conspirators, we develop Model 3, which brings in the concept “shortest path” of graph theory to create an indicator evaluating the likelihood of one’s being conspirator which can be calculated through Algorithm 2. As a result, we compare three priority lists and conclude that the overall rankings are similar but quite changes appear in some nodes. Additionally, when altering the given information, we find that the priority list just changes slightly except for a few nodes, so that we validate the models’ stability. Afterwards, by using Freeman’s centrality method, we develop Model 4 to nominate three most probable leaders: Paul, Elsie, Dolores (senior manager). What’s more, we make some remarks about the models and discuss what could be done to enhance them in the future work. In addition, we further explain Investigation EZ through text and semantic network analysis, so to illustrate the models’ capacity of applying to more complicated cases. Finally, we briefly state the application of our models in other disciplines.
美国数学建模论文格式翻译 你的论文需要从此开始 请居中 使用Arial14字体 第一作者，第二作者和其他(使用Arial14字体) 1.第一作者的详细地址，包括国籍和email(使用Arial11) 2.第二作者的详细地址，包括国籍和email(使用Arial11) 3.将所有的详细信息标记为相同格式 Keywords: List the keywords covered in your paper. These keywords will also be used by the publisher to produce a keyword index. 关键词 列出文章的关键词。这些关键词会被出版方用作关键词索引(使用Arial11字体) 论文正文使用Times New Roman12字体 Abstract. This document explains and demonstrates how to prepare your camera-ready manuscript for TransTechPublications. The best is to read these instructions and follow the outline of this text. The text area for your manuscript must be 17 cm wide and 25 cm high (6.7 and 9.8 inches, resp.). Do not place any text outside this area. Use good quality, white paper of approximately 21 x 29 cm or 8 x 11 inches. Your manuscript will be reduced by approximately 20% by the publisher. Please keep this in mind when designing your figures and tables etc. 摘要 这一部分阐述说明了如何为TransTechPublications.准备手稿。最好阅读这些用法说明并且整篇论文都是遵照这个提纲。手稿的正文部分应该是 17cm*25cm(宽*高)的格式(或者是6.7*9.8英尺)。请不要在这个区域以外书写。
2019 MCM/ICM Summary Sheet (Your team's summary should be included as the first page of your electronic submission.) Type a summary of your results on this page. Do not include the name of your school, advisor , or team members on this page. Ecosystems provide many natural processes to maintain a healthy and sustainable environment after human life. However, over the past decades, rapid industrial development and other anthropogenic activities have been limiting or removing ecosystem services. It is necessary to access the impact of human activities on biodiversity and environmental degradation. The main purpose of this work is to understand the true economic costs of land use projects when ecosystem services are considered. To this end, we propose an ecological service assessment model to perform a cost benefit analysis of land use development projects of varying sites, from small-scale community projects to large national projects. We mainly focus on the treatment cost of environmental pollution in land use from three aspects: air pollution, solid waste and water pollution. We collect pollution data nationwide from 2010 to 2015 to estimate economic costs. We visually analyze the change in economic costs over time via some charts. We also analyze how the economic cost changes with time by using linear regression method. We divide the data into small community projects data (living pollution data) and large natural data (industrial pollution data). Our results indicate that the economic costs of restoring economical services for different scales of land use are different. For small-scale land, according to our analysis, the treatment cost of living pollution is about 30 million every year in China. With the rapid development of technology, the cost is lower than past years. For large-scale land, according to our analysis, the treatment cost of industrial pollution is about 8 million, which is lower than cost of living pollution. Meanwhile the cost is trending down due to technology development. The theory developed here provides a sound foundation for effective decision making policies on land use projects. Key words: economic cost , ecosystem service, ecological service assesment model, pollution. Team Control Number For office use only For office use only T1 ________________ F1 ________________ T2 ________________ F2 ________________ T3 ________________ Problem Chosen F3 ________________ T4 ________________ F4 ________________ E
点评：航空货运问题 一、基本参数 1、货机：假设均匀分布 每天三架货机。 2、工作时间5：00—20：00设置为 t ：[0，15]？ 每天货机到达时间：5：00—20：00； 一工作组装满装卸场：6小时；一货机装满：3小时； 装卸台的容量：1.5货机； 3、费用系数： 停机费（等待装货）：15000元/小时架 一工作组：每小时9000元；二工作组：每小时12000元 4、服务原则：假设先来先服务 二、模型建立：概率计算模型 （一）概率分布 1、三架货机到达的时刻3,2,1,=i t i 服从[0，15]上的均匀分布，则： 密度函数：()1 ,01515 f t t = ≤≤ 分布函数：(),01515 t F t t = ≤≤ 2、设τ,δ,ε分别是首架货机到达时刻、第一架与第二架间隔、第二架与第三架间隔，
（1）τ的分布函数 3 31321321321321321))(1(1))(1(1)()()(1),,(1) ()()},,(min{) ()(1t F t t P t t P t t P t t P t t t t t t P t t t t t t P t t t t t t P t t t t P t P t F t --=≤--=>>>-=>>>-=≤?≤?≤-Ω=≤?≤?≤=≤=≤=ττ τ 的密度函数： ()()()1125 15151)151(3]1[3)(')(2 22 11-=-=-==t t t f t F t F t f t t ττ ]15,0[∈t （2）其余两货机到达与第一个到达的货机的间隔21,t t ??在0到15-τ之间是均匀分布的 于是： τ -=?151)(t f i t , τ-≤≤150t ；τ-=?15)(t t F i t , τ-≤≤150t ，i =1,2 δ 的密度函数 /121212()()()1() 1()() F t P t P t t t t P t t t t P t t P t t δτδ=≤=?≤?≤=-?>?>=-?>?> 221)](1[1)](1[11t F t t P t ?--=≤?--= ()()2//15152)()](1[2)(')(11---= -==??τττδτδt t f t F t F t f t t （3）第三架货机到达与第二个到达的货机的间隔ε在0和15-δ-τ之间是均匀分布的， 于是： ε 的密度函数
你的论文需要从此开始 请居中 使用Arial14字体 第一作者，第二作者和其他（使用Arial14字体） 1.第一作者的详细地址，包括国籍和email(使用Arial11) 2.第二作者的详细地址，包括国籍和email(使用Arial11) 3.将所有的详细信息标记为相同格式 关键词 列出文章的关键词。这些关键词会被出版方用作关键词索引(使用Arial11字体） 论文正文使用Times New Roman12字体 摘要 这一部分阐述说明了如何为TransTechPublications.准备手稿。最好阅读这些用法说明并且整篇论文都是遵照这个提纲。手稿的正文部分应该是17cm*25cm（宽*高）的格式（或者是6.7*9.8英尺）。请不要在这个区域以外书写。请使用21*29厘米或8*11英尺的质量较好的白纸。你的手稿可能会被出版商缩减20%。在制图和绘表格时候请特别注意这些准则。 引言 所有的语言都应该是英语。请备份你的手稿（以防在邮寄过程中丢失）我们收到手稿即默认为原作者允许我们在期刊和书报出版。如果作者在论文中使用了其他刊物中的图表，他们需要联系原作者，获取使用权。将单词或词组倾斜以示强调。除了每一部分的标题（标记部分的标题），不要加粗正文或大写首字母。使用激光打印机，而不是点阵打印机 正文的组织： 小标题 小标题应该加粗并注意字母的大小写。第二等级的小标题被视为后面段落的一部分（就像这一大段的一小部分的开头） 页码 不要打印页码。请用淡蓝色铅笔在每一张纸的左下角（在打印区域以外）标注数字。 脚注 脚注应该单独放置并且和正文分开理想地情况下，脚注应该出现在参考文献页，并且放在文章的末尾，和正文用分割线分开。 表格 表格（如表一，表二，...）应该放在正文当中，是正文的一部分，但是，要避免文本混乱。一个描述性的表格标题要放在图表的下方。标题应该独立的放在表格的下方或旁边。 表中的单位应放在中括号中[兆伏]如果中括号不可用，需使用大括号{兆}或小括号（兆)。1.这就是脚注