数学建模美赛2012MCM B论文

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2012数学建模优秀论文 葡萄酒

2012数学建模优秀论文 葡萄酒

江苏师范大学第五届(2011)数学建模竞赛我们选择的题号是: B我们的参赛队号为:2012江苏师范大学数学建模竞赛题目B题研究生录取问题摘要:根据问题的背景和题目要求,研究在不同条件的研究生录取问题,在对笔试,面试以及导师信息量化,加权平均求解的基础来解决研究生录取的问题。

通过构造选择矩阵和满意度矩阵建立导师和学生之间的双向选择矩阵的0-1规划模型。

利用测发编程计算求出最优解,从而求得问题的最优方案,同时采用降阶技巧和创建定理,快速的求解出实用的最优解,得到对应的最优方案!一问题重述某学校M系计划招收10名计划内研究生,依照有关规定由初试上线的前15名学生参加复试,专家组由8位专家组成。

在复试过程中,要求每位专家对每个参加复试学生的以上5个方面都给出一个等级评分,从高到低共分为A,B,C,D四个等级,并将其填入面试表内。

所有参加复试学生的初试成绩、各位专家对学生的5个方面专长的评分。

该系现有10名导师拟招收研究生,分为四个研究方向。

导师的研究方向、专业学术水平(发表论文数、论文检索数、编(译)著作数、科研项目数),以及对学生的期望要求。

在这里导师和学生的基本情况都是公开的。

要解决的问题是:(1) 首先,请你综合考虑学生的初试成绩、复试成绩等因素,帮助主管部门确定10名研究生的录取名单。

然后,要求被录取的10名研究生与10名导师之间做双向选择,即学生可根据自己的专业发展意愿(依次申报2个专业志愿)、导师的基本情况和导师对学生的期望要求来选择导师;导师根据学生所报专业志愿、专家组对学生专长的评价和自己对学生的期望要求等来选择学生。

请你给出一种10名研究生和导师之间的最佳双向选择方案(并不要求一名导师只带一名研究生),使师生双方的满意度最大。

(2) 根据上面已录取的10名研究生的专业志愿,如果每一位导师只能带一名研究生,请你给出一种10名导师与10名研究生双向选择的最佳方案,使得师生双方尽量都满意。

(3) 如果由十位导师根据初试的成绩及专家组的面试评价和他们自己对学生的要求条件录取研究生,那么,10名研究生的新录取方案是什么?为简化问题,假设没有申报专业志愿,请你给出这10名研究生各申报一名导师的策略和导师各选择一名研究生的策略。

数学建模美赛2012MCM B论文

数学建模美赛2012MCM B论文

Camping along the Big Long RiverSummaryIn 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 Equation1 IntroductionThe number of visits to outdoor recreation areas has increased dramatically in last three decades. Among all those outdoor activities, rafting is often chose as a family get-together during May to September. Rafting or white water rafting is a kind of interesting and challenging recreational outdoor activity, which uses an inflatable raft to navigate a river or sea [1]. It is very popular in the world, especially in occidental countries. This activity is commonly considered an extreme sport that usually done to thrill and excite the raft passengers on white water or different degrees of rough water. It can be dangerous.During the peak period, there are many tourists coming to experience rafting. In order to satisfy tourists to the maximum, we must make full use of our facilities in hand, which means we must do the utmost to utilize the campsites in the best way possible. What's more, to make more people feel the wildness life, we should minimize the encounters to the best extent; meanwhile no two sets of parties can occupy the same campsite at the same time. It is naturally coming into mind that we should consider where to stop, and when to stop of a party [2].In previous studies [3-5],many researchers have simulated the outdoor creation based on real-life data, because the approach is dynamic, stochastic, and discrete-event, and most recreation systems share these traits. But there exists little research aiming at describing the way that visitors travel and distribute themselves within a recreation system [6]. Hence, in our paper, we consider the whole process of parties in detail and simulate every party ’s behavior, including the location of their campsites, and how long it will last for them to stay in a campsite to finish their itineraries. Meanwhile minimize the numbers of encounters.Aiming at showing the whole process of rafting, we firstly focus on analyzing the situation s of a single-party's rafting by using status transfer equation, then consider the problems of two parties' encounters on the river. Finally, after several times of simulation on the whole process of rafting, we obtain the optimal value of X . 2 Symbols and DefinitionsIn this section, we will give some basic symbols and definitions in the following for the convenience.Table 1. Variable Definition SymbolsDefinition i vi pj i q , S dThe velocity of oar or motor 0-1 variables on choosing rafting transportation 0-1 variables on the occupation of campsitesLength of the riverAverage distance between two campsites3 General AssumptionsIn order to have a better study on this paper, we simplify our model by the following assumptions:1) 19 : 00 to 07: 00 is people's sleeping time, during this time, people are stationedin the campsite. The total time of sleeping is 12 hours, as rafting is an exiting sport game, after a day's entertainment, people have cost a lot of energy, and nearly tired out. So in order to have a better recreation for the next day, we set that people begin their trip at 07:00, and end at 19:00 for a day's schedule.2) Oar- powered rubber rafts and motorized parties can successfully raft from FirstLaunch to Final Exit, there exist no accident over the whole trips.3) All the rubber rafts and motorized boats have the same exterior except velocities;we regard a rubber raft or a motorized boat as a party and don't consider the tourists individuals on the parties.4) There is only one entrance for parties to enter the recreation system.5) Regardless of the effects that the physical features of the river brings to oar andmotorized parties, that is to say we ignore the stream ’s propulsion and resistance to both kinds of rafting boats. Oar and motorized parties can keep the average velocity of 4 mph and 8mph.6) Divide the whole river into N segments.4 Analysis of This Rafting ProblemRafting is a very popular spots game world-wide. In the peak period of rafting, there are more people choosing to raft, it often causes congestion that not all people can raft at any time they want. Hence, it is important for managers to set an optimal schedule for every party (from our assumptions, we regard a rafting boat as a party) in advance. Meanwhile, the parties need to experience wildness life, so the managers should arrange the schedules which minimize the encounters' time between parties to the best extent. What's more, no two sets of parties can occupy the same site at the same time.Our aim is to determine an optimal mix of trips over varying duration (measured YXNj i t ,j TtK Numbers of campsites Numbers of parties Numbers of attraction sites Time of the i th party finishing the whole trip ranges from6 days to 18 days Random staying time at each campsite Delay time of rafting from beginning Threshold value of encounterin nights on the river. That is to say,we must obtain an optimal value of X through lots of trails. This optimal value represents that the campsites have a high usage while more people are available to raft.The Long Big River is 225 miles long, if we discuss the river as a whole and consider all the parties together, it will be difficult for us to have a clear recognition on parties' behaviors. Hence, we divide the river into N attraction sites. Each of the attraction sites has Y/N campsites since the campsites are uniformly distributed throughout the river corridor. So build up a model based on single-party ’s behavior of rafting in small distance. At last, we can use computer simulation to imitate more complex situations with various rafting boats and large quantities of parties. 5 Mathematic Models5.1 Rafting of the Single-party Model (Status Transfer Equation [7])From the previous analysis, in order to have a clear recognition of the whole rafting process, we must analyze every single-party's state at any time.In this model, we consider the situation that a single-party rafts from the First Launch to the Final Exit. So we formulate a model that focus on the behavior of one single-party.For a single-party, it must satisfy the following equation: status transfer equation. it represents the relationships between its former state and the latter state. State here means: when the i th party arrives at the j th campsites, the party may occupy the j th campsite or not.As a party can choose two kinds of transportation to raft: oar- powered rubber rafts(i v =4mph) and motorized rafts(i v =8mph). i v is the velocity of the rafting boats,and i p is the 0-1 variables of the selecting for boats. Therefore, we can obtainthe following equation:)1(84i i i p p v -+= (i=1,2,…,X ). (1) where i p =0 if the i th party uses motorized boat as their rafting tool, at thistime i v =8mph ; while i p =1 when , the i th party rafts with oar- powered rubber raft with i v =4mph. In fact, Eq.(1) denotes which kind of rafting boat a party can choose.A party not only has choice on rafting boats, but also can select where to camp based on whether the campsites are occupied or not. The following formulation shows the situation whether this party chooses this campsite or not:⎩⎨⎧=p a r t y p r e v i o u s a b y o c c u pi e d is campsite the 0,party previous a by occupied not is campsite the q ij ,1 (2) where i =1,2,…,X ; j =1,2,…,Y .Where the next one can’t set their camp at this place anymore, that is to say thelatter party’s behavior is determined by the former one.As campsites are fairly uniformly distributed throughout the river corridor, hence, we discrete the whole river into segments, and regard Y campsites as Y nodes which leaves out (Y +1) intervals. Finally we get the average distance between th e j th campsite and (j+1)th campsite:1+=Y S d (3) where is the length of the river, and its value is 225 miles.What’s more, the trip -days for a party is not infinite, it has fluctuating intervals: h t h j i 432144,≤≤ (4) where is the t i ,j itinerary time for a party ranges from 144 hours to 432 hours (6 to 18 nights).From Eq.(1), (2) and (3), the status transfer equation is given as follows: ),...2,1,,...2,1(11,1,,Y j X i T q v d t t j j i i j i j i ==⨯++=--- (5)The i th party’s arriving time at the j th campsite is determined by the time when the i th arrived at (j-1)campsite, the time interval i v d , and the time T j-1 random generated by computer shown in Eq.(5). It is a dynamic process and determined by its previous behavior.5.2 The Analysis of Two Parties’ Encounter on the RiverOur goal is to making full use of the campsites. Hence, the objective of all the formulation is to maximize the quantities of trips (parties )X while consider getting rid of the congestion. If we reduce the numbers of the encounters among parties, there will be no congestion. In order to achieve this goal, we analysis the situations of when two parties’ to encounter, and where they will enco unter.In order to create a wildness environment for parties to experience wildness life, managers arrange a schedule that can make any two parties have minimal encounters with each other. Encounter is that parties meet at the same place and at the same time. Regarding the river as a whole is not convenient to study, hence, our discussion is based on a small distance where distance=d (Eq.3), between the j th and ( j+1)th campsites. Finally the encounter problem of the whole river is transferred into small fractions. On analyzing encounter problem in d and count numbers of each encounter in d together, we get a clear recognition of the whole process and the total numbers of encounter of two parties.The following Figure 1 represents random two parties rafting in d :Figure 1. Random two parties' encounter or not on the riverThe i th party arrives at j th campsite (t j k ,-t j i ,) time earlier than the k th party reaches the j th campsite. After t time, interval distance between the i th party and the k th party can be denoted by the following function:)()(t t t v t v t S ij kj i k j +-⨯-⨯=∆ (6) Where k,i =1,2,…,X , j =1,2,…,Y . k i ≠.Whether the two parties stationed on the j th campsite and(j +1)th campsite are based on the state of the campsites’occupation, yields we obtain:⎩⎨⎧=⨯01,,j k j i q q ( i,k =1,2,…,X ; j =1,2,…,Y ; k ≠i ) (7)Note that Eq.6 is constrained by Eq.7, for different value of )(t S J ∆ andj k j i q q ,,⨯we can obtain the different cases as follows:Case 1:⎩⎨⎧=⨯=∆10)(,,j k ji j q q t S (8) Which means both the i th and k th party don’t choose the j th campsite, they arerafting on the river. Hence, when the interval distance between the two parties is 0, that is )(t S J ∆=0,they encounter at a certain place in d on the river.Cases2:⎩⎨⎧=⨯=∆00)(,,j k ji j q q t S (9) Although the interval distance between the two parties is 0, the j th campsite is occupied by the i th party or the k th party. That is one of them stop to camp at a certain place throughout the river corridor. Hence, there is no possibility for them to encounter on the river.Cases 3:⎩⎨⎧=⨯≠∆10)(,,j k j i j q q t S⎩⎨⎧=⨯≠∆00)(,,j k j i j q q t S (10) No matter the j th campsite is occupied or not for )(t S J ∆≠0 , that is at the same time, they are not at the same place. Hence, they will not encounter at any place in d .5. 3 Overview of Computer Simulation Modeling to Rafting5.3.1 Computer SimulationSimulation modeling is a kind of method to imitate the real-word process or a system. This approach is especially suited to those tasks which are too complex for direct observation, manipulation, or even analytical mathematical analysis (Banks and Carson 1984, Law and Kelton 1991, Pidd 1992).The most appropriate approach for simulating out-door recreation is dynamic, stochastic, and discrete-event model, since most recreation systems share these traits. In all, simulation models can reflect the real-world accurately.5.3.2 Simulation for the Whole Process of Parties on Rafting [8]This simulation can approximate show a party’s behavior on the river under a wide rang of conditions. From the analysis of the previous study, we have known that the next party’s behavior is affected by the former one. Hence, when the first party enters the rafting system, there is no encounter, and it can choose every campsite. then the second party comes into the rafting system , at this time, we must consider the encounter between them, and the limit on choosing the campsite. As time goes by, more and more parties enter this system to raft which lead to a more complex situation. A party who satisfies the following two conditions will be removed from the current order to the next order. So he can’t “finish his trip” right away. The two conditions are as follows:(1) He chooses a campsite where has been occupied by other parties.(2) He has two many encounters with other parties.So in order to determine typical trip itineraries for various types of rafting boat ,campsite, and time intervals (See Trip Schedule Sheet 1),we need to perform a series of trails run that can represent the real-life process of rafting based on these considerations,. A main flowchart of the program is shown in Figure 2.Figure 2. Main simulation flowchartAfter several times of simulation, we obtain the optimal X (the numbers of campsites), minimal E(Encounter) and TP (Trip Time).Followed by Figure 2, we simulate the behavior of a party whether it can enterthe rafting system or not in Figure 3.Figure 3.Sub flowchart5.3.3 The Results of SimulationAfter simulating the whole process of parties rafting on the river, we get three figures (Figure 4, Figure 5 and Figure 6) to present the results.In order to simulate the rafting process more conveniently, we divide the whole river into 31 segments (31 attraction sites), and input an initial value of Y=155(numbers of campsites), where there are 5 campsites in every attraction sites.We represent the times of campsites occupied by various parties on Figure 2 by coordinates( x, y), where x is the order of the campsites from 0 to 155(these campsites are all uniformly distributed thorough the corridor), and y is the numbers of each campsite occupied by different parties. For example, (140,1100)represents that at the campsite, there exists nearly 1100 times of occupation in total by parties over 180days. Hence, the following Figure 4 shows the times of campsites’ usage from March to September.Figure 4.Numbers of campsites' usage during six-month period from March to SeptemberFrom Figure 4, The numbers of campsites’ usage can be identified the efficiency of every campsites’ usage. The higher usage of the campsites, the higher efficiency they are. Based on these, we give a simple suggestion to managers (see in Memo to Managers).Figure 5. the ratio of usage on campsites with time going byFigure 5 shows the changes of the ratio on campsites. when t =0, the campsites are not used , but with time going by, the ratio of the usage of campsites becomes higher and higher.We can also obtain that when t >20, the ratio keeps on a steady level of 65% ; but when t >176 , the ratio comes down, that is, there are little parties entering the recreation system. In all, these changes are rational very much, and have high coincidence with real-world.Then we obtain 1599 parties arranged into recreation system after inputting the initial value Y=155, and set orders to every party from number 0 to number 1599. Plotting every party's travelling time of the whole process on a map by simulating, as follows:Figure 6. Every party’s travelling timeFigure 6 shows the itinerary of the travelling time, most of the travelling time is fluctuating between 13days and 15.3 days, and most of travelling time are concentrated around 14 days.In order to create an outdoor life for all parties, we should minimize the numbers of encounter among different parties based on equations (6) and (7):So we get every party’s numbers of encounter by coordinates ( x, y), where x is the order of the parties from 0 to 1600, y is the numbers of encounters. Shown in Figure 7, as follows:Figure 7. Every party’s numbers of encounterFigure 7 shows every party’s numbers of encounter at each campsite. From this figure, we can know that the numbers of their encounter are relatively less, the highest one is 8 times, and most of the parties don’t encounter during their trips, which is coincident with the real-world data.Finally, according to the travelling time of a party from March to September, we set a plan for river managers to arrange the number of parties. Hence, by simulating the model, we obtain the results by coordinate ( x, y), where y is the days of travelling time, x is the numbers of parties on every day. The figure is shown as follows:Figure 8. Simulation on travelling days versus the numbers of parties From Figure 8.we set a suitable plan for river manager, which also provide reference on his managements.6 Sensitive AnalyzeSensitive analysis is very critical in mathematical modeling, it is a way to gauge the robustness of a model with respect to assumptions about the data and parameters. We try several times of simulation to get different numbers of parties on changing the numbers of campsites ceaselessly. Thus using the simulative data, we get the relationship between the numbers of campsites and parties by fitting. On the basis of this fitting, we revise the maximal encounter times (Threshold value) continually, and can also get the results of the relationships between the numbers of campsites and parties by fitting. Finally, we obtain a Figure 9denoting the relations of Y (numbers of campsites)and X(numbers of parties), as follows:Figure 9 .Sensitive analysis under different threshold values Given the permitted maximal numbers of encounters (threshold value= K), we obtain the relationships between Y(numbers of campsites) and X( numbers of trips). For example, when K=1 , it means no encounters are allowed on the river when rafting; when K=2, there is less than 2 chances for the boats to meet. So we can define the K=4,6,8 to describe the sensitivity of our model.From Figure 9, we get the information that with the increase of K , the numbers of boats available to rafts till increase. But when K>6 , the change of the numbers of boats is inconspicuous, which is not the main factor having appreciable impact on theIn all, when the numbers of campsites ( Y ) are less than 250, they would have a great effect on the numbers of boats. But in the diverse situations, like when Y>250 , the effect caused by adding the numbers of campsites to hold more boats is not notable.When K<6 , the numbers of boats available increases with the ascending of K , While K>6 , the numbers of boats don’t have g reat change .Take all these factors into consideration, it reflects that the numbers of the boats can’t exceed its upper limit. Increase the numbers of campsites and numbers of encounter blindly can’t bring back more profits.7 Strengths and WeaknessesStrengthsOur model has achieved all of the goals we set initially effectively. It is not only fast and could handle large quantities of data, but also has the flexibility we desire.Though we don’t test all possibilities, if we had chosen to inpu t the numbers of campsites data into our program, we could have produced high-quality results with virtually no added difficulty. Aswell, our method was robust.Based on general assumptions we have made in previous task, we consider a party’s state in the first place, then simulate the whole process of rafting. It is an exact reflection of the real-world. Hence, our main model's strength is its enormousedibility and stability and there are some key strengths:(1) The flowchart represents the whole process of rafting by given different initialvalues. It not only makes it possible to develop trip itineraries that are statistically more representatives of the total population of river trips, but also eliminates the tedious task of manual writing .(2)Our m odel focuses on parties’ behavior and interactions between each other, notthe managers on the arrangement of rafting, which can also get satisfactory and high-quality results.(3) Our model makes full use of campsites, while avoid too many encounters, whichleads to rational arrangements.WeaknessesOn the one hand, although we list the model's comprehensive simulation as a strength, it is paradoxically also the most notable weakness since we don’t take into account the carrying capacity of the water when simulates, and suppose that a river can bear as much weight as possible. But in reality, that is impossible. On the other hand, our results are not optimal, but relative optimal.8 ConclusionsAfter a serial of trials, we get different values of X based on the general assumptions we make. By comparing them, we choose a relative better one. From this problem, it verifies the important use of simulation especially in complex situations. Here we consider if we change some of the assumptions, it may lead to various results. For example,(a) Let the velocity of this two kinds of boats submit to normal distribution.In this paper, the average velocity of oar- powered rubber rafts and motorized boats are 4mphand 8mph, respectively. But in real-world, the speed of the boats can’t get rid of the impacts from external force like stream’s propulsion and resistance. Hence, they keep on changing all the time.(b) Add and reduce campsites to improve the ratio of usage on campsites. By analyzing and simulating, the usage of each campsite is different which may lead to waste or congestion at a campsite. Hence, we can adjust the distribution of campsites to arrive the best use.A Memo to River ManagersOur simulation model is with high edibility and stability in many occasions. It can imitate every party’s behavior when rafting so as to make a clear recognition of the process.Internal Workings of The ModelInputsOur model needs to input initial value of Y , as well as the numbers of attraction sites. Algorithm ( Figure 2, and Figure 3)Our algorithm represents the whole process of rafting, so we can use it to simulate the process of rafting by inputting various initial values.OutputsBased on the algorithm in our paper, our model will output the relative optimalnumbers of parties X. Furthermore, we can also get other information, such as the interval time between two parties at First Launch, a detailed schedule for each party of rafting , the relationship between X and Y and so on.Summary and RecommendationsAfter 100 times of simulating, we come to two conclusions:(a) The numbers of parties (X) have relations with the numbers of campsites(Y), that is to say, with the increasing of Y , the increasing speed of X goes fast at the first place and then goes down, finally it tends to be steady. Hence, we advice river managers to adjust the numbers of campsites properly to get the optimal numbers of parties.(b) Add campsites to the high usage of the former campsites and deduce campsites at the low usage of the former campsites. From Figure 4, we know that the ratios of every campsites are different, some campsites are frequently used, but some are not. Thus we can infer that the scenic views are attractive, and have attracted lots of parties camping at the campsite. so we can add campsites to this nodes. Else the campsites with low usage have lost attractions which we should reduce the numbers of campsites at those nodes.References[1] Karlo Šimović,Wikipedia,Rafting,/wiki/Rafting.[2] C. A. Roberts and R. Gimblett,Computer Simulation for Rafting Traffic on theColorado River,COMPUTER SIMULATION FOR RAFTING TRAFFIC, 2001, 19-30.[3] C.A. Roberts, D. Stallman,J. A. Bieri. Modeling complex human-environmentinteractions: the Grand Canyon river trip simulator,Ecological Modeling153(2002) 181-196.[4] J. A. Bieri and C. A. Roberts,Using the Grand Canyon River Trip Simulator toTest New Launch Scheduleson the Colorado River,Washington DC, AWISMagazine, V ol. 29, No. 3, 2000, 6-10.[5] A.H. Underhill and A. B. Xaba,The Wilderness Simulation Model as aManagement Tool for the Colorado River in Grand Canyon National Park,NATIONAL PARK SERVICE/UNIVERSITY OF ARIZONA Unit SupportProject CONTRIBUTION NO. 034/03.[6] B. Wang and R. E. Manning,Computer Simulation Modeling for RecreationManagement: A Study on Carriage Road Use in Acadia National Park, Maine, USA, USA, Vermont 05405, 1999.[7] M. M. Meerschaert,Mathematical Modeling(Third Edition). China MachinePress publishing,2009.[8] A.H. Underhill,The Wilderness Use Simulation Model Applied to Colorado RiverBoating in Grand Canyon National Park, USA,Environmental Management V ol.10, No. 3, 1986, 367-374.。

2012年美国大学生数学建模竞赛B题特等奖文章翻译

2012年美国大学生数学建模竞赛B题特等奖文章翻译

We develop a model to schedule trips down the Big Long River. The goalComputing Along the Big Long RiverChip JacksonLucas BourneTravis PetersWesternWashington UniversityBellingham,WAAdvisor: Edoh Y. AmiranAbstractis to optimally plan boat trips of varying duration and propulsion so as tomaximize the number of trips over the six-month season.We model the process by which groups travel from campsite to campsite.Subject to the given constraints, our algorithm outputs the optimal dailyschedule for each group on the river. By studying the algorithm’s long-termbehavior, we can compute a maximum number of trips, which we define asthe river’s carrying capacity.We apply our algorithm to a case study of the Grand Canyon, which hasmany attributes in common with the Big Long River.Finally, we examine the carrying capacity’s sensitivity to changes in thedistribution of propulsion methods, distribution of trip duration, and thenumber of campsites on the river.IntroductionWe address scheduling recreational trips down the Big Long River so asto maximize the number of trips. From First Launch to Final Exit (225 mi),participants take either an oar-powered rubber raft or a motorized boat.Trips last between 6 and 18 nights, with participants camping at designatedcampsites along the river. To ensure an authentic wilderness experience,at most one group at a time may occupy a campsite. This constraint limitsthe number of possible trips during the park’s six-month season.We model the situation and then compare our results to rivers withsimilar attributes, thus verifying that our approach yields desirable results.Our model is easily adaptable to find optimal trip schedules for riversof varying length, numbers of campsites, trip durations, and boat speeds.No two groups can occupy the same campsite at the same time.Campsites are distributed uniformly along the river.Trips are scheduled during a six-month period of the year.Group trips range from 6 to 18 nights.Motorized boats travel 8 mph on average.Oar-powered rubber rafts travel 4 mph on average.There are only two types of boats: oar-powered rubber rafts and motorizedTrips begin at First Launch and end at Final Exit, 225 miles downstream.*simulates river-trip scheduling as a function of a distribution of trip*can be applied to real-world rivers with similar attributes (i.e., the Grand*is flexible enough to simulate a wide range of feasible inputs; andWhat is the carrying capacity of the riverÿhe maximum number ofHow many new groups can start a river trip on any given day?How should trips of varying length and propulsion be scheduled toDefining the Problemmaximize the number of trips possible over a six-month season?groups that can be sent down the river during its six-month season?Model OverviewWe design a model thatCanyon);lengths (either 6, 12, or 18 days), a varying distribution of propulsionspeeds, and a varying number of campsites.The model predicts the number of trips over a six-month season. It alsoanswers questions about the carrying capacity of the river, advantageousdistributions of propulsion speeds and trip lengths, how many groups canstart a river trip each day, and how to schedule trips.ConstraintsThe problem specifies the following constraints:boats.AssumptionsWe can prescribe the ratio of oar-powered river rafts to motorized boats that go onto the river each day.There can be problems if too many oar-powered boats are launched with short trip lengths.The duration of a trip is either 12 days or 18 days for oar-powered rafts, and either 6 days or 12 days for motorized boats.This simplification still allows our model to produce meaningful results while letting us compare the effect of varying trip lengths.There can only be one group per campsite per night.This agrees with the desires of the river manager.Each day, a group can only move downstream or remain in its current campsiteÿt cannot move back upstream.This restricts the flow of groups to a single direction, greatly simplifying how we can move groups from campsite to campsite.Groups can travel only between 8 a.m. and 6 p.m., a maximum of 9hours of travel per day (one hour is subtracted for breaks/lunch/etc.).This implies that per day, oar-powered rafts can travel at most 36 miles, and motorized boats at most 72 miles. This assumption allows us to determine which groups can reasonably reach a given campsite.Groups never travel farther than the distance that they can feasibly travelin a single day: 36 miles per day for oar-powered rafts and 72 miles per day for motorized boats.We ignore variables that could influence maximum daily travel distance, such as weather and river conditions.There is no way of accurately including these in the model.Campsites are distributed uniformly so that the distance between campsites is the length of the river divided by the number of campsites.We can thus represent the river as an array of equally-spaced campsites.A group must reach the end of the river on the final day of its trip:A group will not leave the river early even if able to.A group will not have a finish date past the desired trip length.This assumption fits what we believe is an important standard for theriver manager and for the quality of the trips.MethodsWe define some terms and phrases:Open campsite: Acampsite is open if there is no groupcurrently occupying it: Campsite cn is open if no group gi is assigned to cn.Moving to an open campsite: For a group gi, its campsite cn, moving to some other open campsite cm ÿ= cn is equivalent to assigning gi to the new campsite. Since a group can move only downstream, or remain at their current campsite, we must have m ÿ n.Waitlist: The waitlist for a given day is composed of the groups that are not yet on the river but will start their trip on the day when their ranking onthe waitlist and their ability to reach a campsite c includes them in theset Gc of groups that can reach campsite c, and the groups are deemed “the highest priority.” Waitlisted groups are initialized with a current campsite value of c0 (the zeroth campsite), and are assumed to have priority P = 1 until they are moved from the waitlist onto the river.Off the River: We consider the first space off of the river to be the “final campsite” cfinal, and it is always an open campsite (so that any number of groups can be assigned to it. This is consistent with the understanding that any number of groups can move off of the river in a single day.The Farthest Empty CampsiteOurscheduling algorithm uses an array as the data structure to represent the river, with each element of the array being a campsite. The algorithm begins each day by finding the open campsite c that is farthest down the river, then generates a set Gc of all groups that could potentially reach c that night. Thus,Gc = {gi | li +mi . c},where li is the groupÿs current location and mi is the maximum distance that the group can travel in one day.. The requirement that mi + li . c specifies that group gi must be able to reach campsite c in one day.. Gc can consist of groups on the river and groups on the waitlist.. If Gc = ., then we move to the next farthest empty campsite.located upstream, closer to the start of the river. The algorithm always runs from the end of the river up towards the start of the river.. IfGc ÿ= ., then the algorithm attempts tomovethe groupwith the highest priority to campsite c.The scheduling algorithm continues in this fashion until the farthestempty campsite is the zeroth campsite c0. At this point, every group that was able to move on the river that day has been moved to a campsite, and we start the algorithm again to simulate the next day.PriorityOnce a set Gc has been formed for a specific campsite c, the algorithm must decide which group to move to that campsite. The priority Pi is a measure of how far ahead or behind schedule group gi is:. Pi > 1: group gi is behind schedule;. Pi < 1: group gi is ahead of schedule;. Pi = 1: group gi is precisely on schedule.We attempt to move the group with the highest priority into c.Some examples of situations that arise, and how priority is used to resolve them, are outlined in Figures 1 and 2.Priorities and Other ConsiderationsOur algorithm always tries to move the group that is the most behind schedule, to try to ensure that each group is camped on the river for aFigure 1. The scheduling algorithm has found that the farthest open campsite is Campsite 6 and Groups A, B, and C can feasibly reach it. Group B has the highest priority, so we move Group B to Campsite 6.Figure 2. As the scheduling algorithm progresses past Campsite 6, it finds that the next farthest open campsite is Campsite 5. The algorithm has calculated that Groups A and C can feasibly reach it; since PA > PC, Group A is moved to Campsite 5.number of nights equal to its predetermined trip length. However, in someinstances it may not be ideal to move the group with highest priority tothe farthest feasible open campsite. Such is the case if the group with thehighest priority is ahead of schedule (P <1).We provide the following rules for handling group priorities:?If gi is behind schedule, i.e. Pi > 1, then move gi to c, its farthest reachableopen campsite.?If gi is ahead of schedule, i.e. Pi < 1, then calculate diai, the number ofnights that the group has already been on the river times the averagedistance per day that the group should travel to be on schedule. If theresult is greater than or equal (in miles) to the location of campsite c, thenmove gi to c. Doing so amounts to moving gi only in such a way that itis no longer ahead of schedule.?Regardless of Pi, if the chosen c = cfinal, then do not move gi unless ti =di. This feature ensures that giÿ trip will not end before its designatedend date.Theonecasewhere a groupÿ priority is disregardedisshownin Figure 3.Scheduling SimulationWe now demonstrate how our model could be used to schedule rivertrips.In the following example, we assume 50 campsites along the 225-mileriver, and we introduce 4 groups to the river each day. We project the tripFigure 3. The farthest open campsite is the campsite off the river. The algorithm finds that GroupD could move there, but GroupD has tD > dD.that is, GroupD is supposed to be on the river for12 nights but so far has spent only 11.so Group D remains on the river, at some campsite between 171 and 224 inclusive.schedules of the four specific groups that we introduce to the river on day25. We choose a midseason day to demonstrate our modelÿs stability overtime. The characteristics of the four groups are:. g1: motorized, t1 = 6;. g2: oar-powered, t2 = 18;. g3: motorized, t3 = 12;. g4: oar-powered, t4 = 12.Figure 5 shows each groupÿs campsite number and priority value foreach night spent on the river. For instance, the column labeled g2 givescampsite numbers for each of the nights of g2ÿs trip. We find that each giis off the river after spending exactly ti nights camping, and that P ÿ 1as di ÿ ti, showing that as time passes our algorithm attempts to get (andkeep) groups on schedule. Figures 6 and 7 display our results graphically.These findings are consistent with the intention of our method; we see inthis small-scale simulation that our algorithm produces desirable results.Case StudyThe Grand CanyonThe Grand Canyon is an ideal case study for our model, since it sharesmany characteristics with the Big Long River. The Canyonÿs primary riverrafting stretch is 226 miles, it has 235 campsites, and it is open approximatelysix months of the year. It allows tourists to travel by motorized boat or byoar-powered river raft for a maximum of 12 or 18 days, respectively [Jalbertet al. 2006].Using the parameters of the Grand Canyon, we test our model by runninga number of simulations. We alter the number of groups placed on thewater each day, attempting to find the carrying capacity for the river.theFigure 7. Priority values of groups over the course of each trip. Values converge to P = 1 due to the algorithm’s attempt to keep groups on schedule.maximumnumber of possible trips over a six-month season. The main constraintis that each trip must last the group’s planned trip duration. Duringits summer season, the Grand Canyon typically places six new groups onthe water each day [Jalbert et al. 2006], so we use this value for our first simulation.In each simulation, we use an equal number of motorized boatsand oar-powered rafts, along with an equal distribution of trip lengths.Our model predicts the number of groups that make it off the river(completed trips), how many trips arrive past their desired end date (latetrips), and the number of groups that did not make it off the waitlist (totalleft on waitlist). These values change as we vary the number of new groupsplaced on the water each day (groups/day).Table 1 indicates that a maximum of 18 groups can be sent down theriver each day. Over the course of the six-month season, this amounts to nearly 3,000 trips. Increasing groups/day above 18 is likely to cause latetrips (some groups are still on the river when our simulation ends) and long waitlists. In Simulation 1, we send 1,080 groups down river (6 groups/day?80 days) but only 996 groups make it off; the other groups began near the end of the six-month period and did not reach the end of their trip beforethe end of the season. These groups have negligible impact on our results and we ignore them.Sensitivity Analysis of Carrying CapacityManagers of the Big Long River are faced with a similar task to that of the managers of the Grand Canyon. Therefore, by finding an optimal solutionfor the Grand Canyon, we may also have found an optimal solution forthe Big Long River. However, this optimal solution is based on two key assumptions:?Each day, we put approximately the same number of groups onto theriver; and?the river has about one campsite per mile.We can make these assumptions for the Grand Canyon because they are true for the Grand Canyon, but we do not know if they are true for the Big Long River.To deal with these unknowns,wecreate Table 3. Its values are generatedby fixing the number Y of campsites on the river and the ratio R of oarpowered rafts to motorized boats launched each day, and then increasingthe number of trips added to the river each day until the river reaches peak carrying capacity.The peak carrying capacities in Table 3 can be visualized as points ina three-dimensional space, and we can find a best-fit surface that passes (nearly) through the data points. This best-fit surface allows us to estimatethe peak carrying capacity M of the river for interpolated values. Essentially, it givesM as a function of Y and R and shows how sensitiveM is tochanges in Y and/or R. Figure 7 is a contour diagram of this surface.The ridge along the vertical line R = 1 : 1 predicts that for any givenvalue of Y between 100 and 300, the river will have an optimal value ofM when R = 1 : 1. Unfortunately, the formula for this best-fit surface is rather complex, and it doesn’t do an accurate job of extrapolating beyond the data of Table 3; so it is not a particularly useful tool for the peak carrying capacity for other values ofR. The best method to predict the peak carrying capacity is just to use our scheduling algorithm.Sensitivity Analysis of Carrying Capacity re R and DWe have treatedM as a function ofR and Y , but it is still unknown to us how M is affected by the mix of trip durations of groups on the river (D).For example, if we scheduled trips of either 6 or 12 days, how would this affect M? The river managers want to know what mix of trips of varying duration and speed will utilize the river in the best way possible.We use our scheduling algorithm to attempt to answer this question.We fix the number of campsites at 200 and determine the peak carrying capacity for values of R andD. The results of this simulation are displayed in Table 4.Table 4 is intended to address the question of what mix of trip durations and speeds will yield a maximum carrying capacity. For example: If the river managers are currently scheduling trips of length?6, 12, or 18: Capacity could be increased either by increasing R to be closer to 1:1 or by decreasing D to be closer to ? or 12.?12 or 18: Decrease D to be closer to ? or 12.?6 or 12: Increase R to be closer to 4:1.ConclusionThe river managers have asked how many more trips can be added tothe Big Long Riverÿ season. Without knowing the specifics ofhowthe river is currently being managed, we cannot give an exact answer. However, by applying our modelto a study of the GrandCanyon,wefound results which could be extrapolated to the context of the Big Long River. Specifically, the managers of the Big Long River could add approximately (3,000 - X) groups to the rafting season, where X is the current number of trips and 3,000 is the capacity predicted by our scheduling algorithm. Additionally, we modeled how certain variables are related to each other; M, D, R, and Y . River managers could refer to our figures and tables to see how they could change their current values of D, R, and Y to achieve a greater carrying capacity for the Big Long River.We also addressed scheduling campsite placement for groups moving down the Big Long River through an algorithm which uses priority values to move groups downstream in an orderly manner.Limitations and Error AnalysisCarrying Capacity OverestimationOur model has several limitations. It assumes that the capacity of theriver is constrained only by the number of campsites, the trip durations,and the transportation methods. We maximize the river’s carrying capacity, even if this means that nearly every campsite is occupied each night.This may not be ideal, potentially leading to congestion or environmental degradation of the river. Because of this, our model may overestimate the maximum number of trips possible over long periods of time. Environmental ConcernsOur case study of the Grand Canyon is evidence that our model omits variables. We are confident that the Grand Canyon could provide enough campsites for 3,000 trips over a six-month period, as predicted by our algorithm. However, since the actual figure is around 1,000 trips [Jalbert et al.2006], the error is likely due to factors outside of campsite capacity, perhaps environmental concerns.Neglect of River SpeedAnother variable that our model ignores is the speed of the river. Riverspeed increases with the depth and slope of the river channel, makingour assumption of constant maximum daily travel distance impossible [Wikipedia 2012]. When a river experiences high flow, river speeds can double, and entire campsites can end up under water [National Park Service 2008]. Again, the results of our model don’t reflect these issues. ReferencesC.U. Boulder Dept. of Applied Mathematics. n.d. Fitting a surface to scatteredx-y-z data points. /computing/Mathematica/Fit/ .Jalbert, Linda, Lenore Grover-Bullington, and Lori Crystal, et al. 2006. Colorado River management plan. 2006./grca/parkmgmt/upload/CRMPIF_s.pdf .National Park Service. 2008. Grand Canyon National Park. High flowriver permit information. /grca/naturescience/high_flow2008-permit.htm .Sullivan, Steve. 2011. Grand Canyon River Statistics Calendar Year 2010./grca/planyourvisit/upload/Calendar_Year_2010_River_Statistics.pdf .Wikipedia. 2012. River. /wiki/River .Memo to Managers of the Big Long RiverIn response to your questions regarding trip scheduling and river capacity,we are writing to inform you of our findings.Our primary accomplishment is the development of a scheduling algorithm.If implemented at Big Long River, it could advise park rangerson how to optimally schedule trips of varying length and propulsion. Theoptimal schedule will maximize the number of trips possible over the sixmonth season.Our algorithm is flexible, taking a variety of different inputs. Theseinclude the number and availability of campsites, and parameters associatedwith each tour group. Given the necessary inputs, we can output adaily schedule. In essence, our algorithm does this by using the state of theriver from the previous day. Schedules consist of campsite assignments foreach group on the river, as well those waiting to begin their trip. Given knowledge of future waitlists, our algorithm can output schedules monthsin advance, allowing managementto schedule the precise campsite locationof any group on any future date.Sparing you the mathematical details, allow us to say simply that ouralgorithm uses a priority system. It prioritizes groups who are behindschedule by allowing them to move to further campsites, and holds backgroups who are ahead of schedule. In this way, it ensures that all trips willbe completed in precisely the length of time the passenger had planned for.But scheduling is only part of what our algorithm can do. It can alsocompute a maximum number of possible trips over the six-month season.We call this the carrying capacity of the river. If we find we are below ourcarrying capacity, our algorithm can tell us how many more groups wecould be adding to the water each day. Conversely, if we are experiencingriver congestion, we can determine how many fewer groups we should beadding each day to get things running smoothly again.An interesting finding of our algorithm is how the ratio of oar-poweredriver rafts to motorized boats affects the number of trips we can send downstream. When dealing with an even distribution of trip durations (from 6 to18 days), we recommend a 1:1 ratio to maximize the river’s carrying capacity.If the distribution is skewed towards shorter trip durations, then ourmodel predicts that increasing towards a 4:1 ratio will cause the carryingcapacity to increase. If the distribution is skewed the opposite way, towards longer trip durations, then the carrying capacity of the river will always beless than in the previous two cases—so this is not recommended.Our algorithm has been thoroughly tested, and we believe that it isa powerful tool for determining the river’s carrying capacity, optimizing daily schedules, and ensuring that people will be able to complete their trip as planned while enjoying a true wilderness experience.Sincerely yours,Team 13955。

2012年全国大学生数学建模大赛B题--论文

2012年全国大学生数学建模大赛B题--论文

4
开始
山西大同市地理 参数(纬度,地 形高度)
某一时刻的太阳辐射量W总
中间参数(日 序、倾角、方 位角、时刻)
Wd(N,β)=∫WtdT
中间变量 (日出、日 落时刻)
Wy(β)=Wt
结束
图 4 倾斜放置的光伏板表面太阳辐射量数学模型建立 已知山西大同市的地理参数(纬度、地形高度等)以及中间参数(日序、光 伏板倾角、方位角和时刻) ,可以得到逐时太阳能光伏板表面的辐射量和中间参 数的关系。 将逐时太阳能光伏板表面辐射量关于时间积分得到某一天的日辐射总 量������������ (������, ������),再将������������ (������, ������)关于 N 累加得到太阳能光伏板表面的年累计辐 射量������ 。 ������ (������ ) 计算地球表面任一点的太阳辐射量,首先确定一些基本的天文参数,主要包 括地球表层大气外界上空的垂直太阳辐射强度、赤纬角、太阳高度角、太阳方位 角和日出日落时刻等。
cos A
sin sin sin , cos cos
其中 A 为太阳的方位角, 为太阳高度角, 为时角, 为当时的太阳赤纬, 为当地的地理纬度。 (该定义摘自维基百科) 1.2 太阳能光伏板上太阳能总辐射量的计算 光伏板的放置方式可分为朝向赤道和任意方向两种,在相同倾角的情况下, 前者斜面接收的辐射能量要大于后者,所以在此仅讨论第一种情况。
2012 高教社杯全国大学生数学建模竞赛 B题 太阳能小屋的设计
摘要:
在太阳能小屋的设计中为实现太阳能光伏板最佳朝向、 倾角及排布阵列设计 及优化, 通过建立倾斜放置的光伏板表面接收太阳辐射能模型,计算到达光伏板 上的太阳辐射能量, 推导出光伏板的最佳朝向及倾角。为使光伏板最大限度地接 收太阳辐射的能量,在选择合适的朝向及倾角的基础上,对光伏电池排布阵列, 建立目标规划,并通过与实际逆变器的相互匹配,不断对目标进行优化,最终得 到一组最优解。通过上述研究,结合山西大同市本地情况,重新设计出一个更加 适合当地地理及气象条件的太阳能光能房屋并为其选择最优的阵列排布方案。 针对问题一: 电池板只是铺设房屋的表面, 没有涉及到电池板放的角度问题, 先求算出房屋的角度为 10.62 度,再根据角度,建立模型算出光伏板上太阳能辐 射量。 并用目标规划阵列排列方案计算出电池的排布。再通过排布计算出经济效 益,最后得出 35 年之内无法收回成本。 针对问题二:通过对角度建立模型,计算得出最佳角度 44.66 度,通过排布 计算出电池板排布最佳方案,建立模型计算出经济效益,在 28.5 年收回成本。 如考虑货币时间价值,35 年的经济效益是亏损的。 针对问题三: 要通过目标构建一个产电量尽量大, 而成本尽量小的理想模型。 假设小屋无挑檐、挑雨棚(即房顶的边投影与房体的长宽投影相等) ,建立模型 计算出最佳的图形,并画出模型图。

2012高教社杯全国大学生建模竞赛B题论文

2012高教社杯全国大学生建模竞赛B题论文

5、建模与求解
5.1 问题 1:请根据山西省大同市的气象数据,仅考虑贴附安装方式,选定 光伏电池组件,对小屋(见附件 2)的部分外表面进行铺设,并根据电池组件分 组数量和容量,选配相应的逆变器的容量和数量。并计算出小屋光伏电池 35 年 寿命期内的发电总量、经济效益(当前民用电价按 0.5 元/kWh 计算)及投资的 回收年限。 对于问题 1 模型的建立,讨论如何选择 PV 电池类型和选择组件的连接方式 使转换的能量最大。从成本方面考虑,由于各个电池的转换效率不同,所以应该 考虑各个电池的综合性能。具体分析如下:从光伏电持组件中选取组件对小屋铺 设,尽可能使铺设组件面积最大,组建的最大和外表的面积相等;三种类型的光 伏电池( A 单晶硅、 B 多晶硅、 C 非晶薄膜)表示如下:
5.2.1 本参数的确定 计算地球表面任一点的太阳辐射量,需要确定一些基本的天文参数,主要包 括地球表层大气外界上空的垂直太阳辐射强度,赤纬,太阳高度角,太阳方位角 和日出日落时刻等。 1.1.1 层外的太阳辐射强度(I。) 当太阳光垂直入射在大气上界时,其太阳辐射强度 I。=S。(1+0.033cos(2π *N/365)) (1)
太阳能小屋设计的优化模型
摘要: 随着社会越来越发达,工业、农业、制造业的现代化程度越来越高,人类对能 源的需求就越来越大,为了保证不对生存环境造成严重的污染,对于能源质量的 要求也越来越高。其中太阳能是各种可再生能源中最重要的基本能源,也是人类 可利用的最丰富的能源。太阳能虽然具有含量多、分布广、无污染等诸多优势, 但由于其密度低、不稳定、成本高、效率低等缺点,使得太阳能的利用受到很大 的限制。在太阳能小屋日渐进入人们的视野的同时,人们也意识到提高光伏模块 太阳能转换效率是提高太阳能的利用率的最有效方法。 这对于加快太阳能光伏产 业的发展具有重要意义。 对于第一问根据题目给出的数据, 运用多目标线性规划和统计学相结合求出 最优解。并根据电池组件分组数量和容量,选配出相应的逆变器的容量和数量。 并给出小屋各个外表面电池组件铺设分组阵列图形及组件连接方式。 然后小屋的 部分外表进行分析,最后进行铺设。并计算出小屋光伏电池 35 年寿命期内的发 电总量、最大经济效益及投资的回收年限。 对于第二问根据题意可得,运用定量分析法,分析了光伏模块的倾角和方位 角与接收太阳辐射量之间的关系, 利用 LabVIEW 软件对光伏模块全年平均辐射量 进行计算,分析各种因素对全年平均辐射量的影响,得到光伏模块全年最佳倾角 和方位角,为正确安装光伏模块及提高太阳能转换效率提供理论依据。 对于第三问由题意可得,根据第二问求得的最佳倾角设计出新的房屋构造, 根据题目中附件 7 的设计要求,完成设计,并给出小屋各个外表面电池组件铺设 分组阵列图形及组件连接方式,选配逆变器,计算相应结果。 关键词:光伏发电 太阳能转换效率 光伏电池铺设

2012年数学建模美赛B题

2012年数学建模美赛B题
5 Model Optimization ……………………………………………………………...4 Reference………………………………………………………………………........5
Team#12591
Page 3 of 18
1 Introduction & Backgrounds
Tourism is the sum of relationships and the phenomena which happens in the process when people make the non resident trip in order to seek the spirit of the pleasant feeling. it is a kind of spiritual pursuit when people have met the material life. Every time when we plan to select a tourist destination, we are always considering many issues, such as: the safety, the quality, the time and so on. As the administrator of the tourism area, we hold the tourism resources, At the same time, in order to improve the safety of tourist and the quality of tourism, we must do our best to avoid too many tourists at the same time or in a given period of time to enter the scenic area. For hot spots and spots of limited opening hours, if conditions permit, improving the system to achieve the maximum utilization of resource is the problem to be solved urgently, in other words, as the managers, we must give access to more tourists in the case of quality assurance.

2012年南京理工大学数学建模竞赛论文

2012年南京理工大学数学建模竞赛论文

承诺书我们仔细阅读了全国大学生数学建模的竞赛规()。

我们完全明白,在竞赛开始后参赛队员不能以任何方式(包括电话、电子邮件、网上咨询等)与本队以外的任何人(包括指导教师)研究、讨论与赛题有关的问题。

我们知道,抄袭别人的成果是违反竞赛规则的, 如果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明确列出。

我们郑重承诺,严格遵守竞赛规则,以保证竞赛的公正、公平性。

如有违反竞赛规则的行为,我们愿意承担由此引起的一切后果。

我们的参赛(报名)队号为: 6我们选择的题号为(A或B):B参赛组别(研究生或本科):本科参赛队员 (先打印,后签名,并留联系电话) :可持续利用森林资源的策略摘要森林资源是人类赖以生存的重要资源,保护森林资源并对其进行可持续性利用是当代科学的重要课题,本文研究了森林的可持续利用问题,并对于森林砍伐、种植策略等问题进行了较深入的分析与讨论。

为了解决问题,我们首先建立了描述一棵树生长过程的含材体积模型。

通过查找资料确定树木生长速度的二次曲线模型bt at dtdv+-=2,一棵树木的含材体积⎰+-==232131bt at dt dt dv v (假设t =0时,含材体积为0)通过观察树木含材体积的曲线确立成材年限m 与成长停滞年限max 的关系。

a b m 43=,ab=max 接着,我们建立了单维离散动态模型,先考虑稳定状态(即时间充分长)下森林中只有一种树的各年龄段树数量变化情况,引入砍伐强度变量,认为木材的需求按年计算,即树木每年按需求砍伐一次。

我们认为种植策略是将砍伐过后的空地上种上幼苗,并及时将死掉的幼苗清除并重栽幼苗。

为了保持系统稳定,这里认为树木自我繁殖率与死亡率大致相等。

随后给出了该树种各年龄段的动态差分方程组并构造Leslie 矩阵:)()1(t X L t X *=+,计算矩阵最大特征根为1,表明该树种各年龄段的在确定的砍伐强度下分布情况),...,,...,,(max 21*X X X X X m =最终趋于稳定(即每年砍伐前*X 稳定)以3.0,2.01==k k 为例(种植面积100万公顷) 树木年龄(年)12345678稳定值(810⨯棵)2.460 2.463 2.462 2.458 2.454 2.452 1.473 1.773 这样,对于该树种的利用做到了可持续发展。

2012年美国数学建模MCM的B题翻译

2012年美国数学建模MCM的B题翻译

PROBLEM B: Camping along the Big Long RiverVisitors to the Big Long River (225 miles) can enjoy scenic views and exciting white water rapids. The river is inaccessible to hikers, so the only way to enjoy it is to take a river trip that requires several days of camping. River trips all start at First Launch and exit the river at Final Exit, 225 miles downstream. Passengers take either oar- powered rubber rafts, which travel on average 4 mph or motorized boats, which travel on average 8 mph. The trips range from 6 to 18 nights of camping on the river, start to finish.. The government agency responsible for managing this river wants every trip to enjoy a wilderness experience, with minimal contact with other groups of boats on the river. Currently, X trips travel down the Big Long River each year during a six month period (the rest of the year it is too cold for river trips). There are Y camp sites on the Big Long River, distributed fairly uniformly throughout the river corridor. Given the rise in popularity of river rafting, the park managers have been asked to allow more trips to travel down the river. They want to determine how they might schedule an optimal mix of trips, of varying duration (measured in nights on the river) and propulsion (motor or oar) that will utilize the campsites in the best way possible. In other words, how many more boat trips could be added to the Big Long River’s rafting season? The river managers have hired you to advise them on ways in which to develop the best schedule and on ways in which to determine the carrying capacity of the river, remembering that no two sets of campers can occupy the same site at the same time. In addition to your one page summary sheet, prepare a one page memo to the managers of the river describing your key findings.游客在“大长河”(225英里)可以享受到秀丽的风光和令人兴奋的白色湍流。

2012年美国数学建模MCM题目(中英对照版)

2012年美国数学建模MCM题目(中英对照版)

2012 Contest ProblemsMCM PROBLEMSPROBLEM A: The Leaves of a Tree"How much do the leaves on a tree weigh?" How might one estimate the actual weight of the leaves (or for that matter any other parts of the tree)? How might one classify leaves? Build a mathematical model to describe and classify leaves. Consider and answer the following:• Why do leaves have the various shapes that they have?• Do the shapes “minimize” overlapping individual shadows that are cast, so as to maximize exposure? Does the distribution of leaves within the “volume” of the tree and its branches effect the shape?• Speaking of profiles, is leaf shape (general characteristics) related to tree profile/branching structure?• How would you estimate the leaf mass of a tree? Is there a correlation between the leaf mass and the size characteristics of the tree (height, mass, volume defined by the profile)?In addition to your one page summary sheet prepare a one page letter to an editor of a scientific journal outlining your key findings.“一棵树的叶子有多重?”怎么能估计树的叶子(或者树的任何其它部分)的实际重量?怎样对叶子进行分类?建立一个数学模型来对叶子进行描述和分类。

2012高教社杯全国大学生数学建模竞赛论文模板(1)

2012高教社杯全国大学生数学建模竞赛论文模板(1)
[编号]作者,论文名,杂志名,卷期号:起止页码,出版年。
[编号]作者,资源标题,网址,访问时间(年月日)。
附录一
正文用小四号宋体书写,……
注:打印前请加上页码,从承诺书开始为第一页。页码在页面底端居中,用阿拉伯数字。
2012高教社杯全国大学生数学建模竞赛
承诺书
我们仔细阅读了中国大学生数学建模竞赛的竞赛规则.
我们完全明白,在竞赛开始后参赛队员不能以任何方式(包括电话、电子邮件、网上咨询等)与队外的任何人(包括指导教师)研究、讨论与赛题有关的问题。
我们知道,抄袭别人的成果是违反竞赛规则的,如果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明确列出。
2.问题的具体内容。……
问题二
1.问题的具体内容。……
2.问题的具体内容。……
二、问题分析
分析问题一
对问题一进行分析。正文小四宋体,……
1.……
2.……
分析问题二
对问题二进行分析,……
1.……
2.……
三、模型假设
1.正文小四宋体,……
2.……
四、符号说明
M 质量
V速度
T时间
……
……
……
i对j的力
……
赛区评阅记录(可供赛区评阅时使用):







全国统一编号(由赛区组委会送交全国前编号):
全国评阅编号(由全国组委会评阅前进行编号):
XXXX(论文题目)
摘要
本部分书写论文摘要及关键词,摘要要用简洁的文字把意思表达清楚,篇幅限定在本页之中。格式要求是用小四号宋体书写。最好不要有图表,如避免不了并且能够满足在本页之中的要求,则采用三线式图表,在图的正上方用小五号宋体标明题目及图序号(这是刊物发表的图表格式要求)。例:

2012全国数学建模竞赛B题参赛论文PDF版

2012全国数学建模竞赛B题参赛论文PDF版

4.1.3求小屋各表面的年发电总量及经济效益计算 小屋某一表面的年发电总量与该表面年太阳辐射单位面积强度的关系式如下:
M n ( H n Si ni i i )
i 1 k

则可得到该表面的单位发电量
mn = a Mn

其中,成本 某电池的发电总量
a = (P bi) + (j Ci)
2012 高教社杯全国大学生数学建模竞赛



我们仔细阅读了中国大学生数学建模竞赛的竞赛规则. 我们完全明白,在竞赛开始后参赛队员不能以任何方式(包括电话、电子邮件、网 上咨询等)与队外的任何人(包括指导教师)研究、讨论与赛题有关的问题。 我们知道,抄袭别人的成果是违反竞赛规则的, 如果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 考文献中明确列出。 我们郑重承诺,严格遵守竞赛规则,以保证竞赛的公正、公平性。如有违反竞赛规 则的行为,我们将受到严肃处理。 我们授权全国大学生数学建模竞赛组委会,可将我们的论文以任何形式进行公开展 示(包括进行网上公示,在书籍、期刊和其他媒体进行正式或非正式发表等) 。
关键词:光伏电池组件;光伏阵列;太阳辐射总量;优化铺设;
一、 问题重述
当前,能源与环境问题已经成为制约人类社会可持续发展的首要问题。随着社会的 发展,太阳能作为一种天然无污染能源,无疑具有很大的利用价值。作为世界上最大的 太阳能热水器产销大国, 中国能源领域正孕育着新的革命。 尽可能节约资源, 降低能耗, 已成为住宅建设发展要考虑的首要问题。太阳能作为一种清洁能源,其开发和利用日益 广泛。如今,太阳能与建筑一体化是太阳能应用领域的新趋势。 现需为山西省大同市设计一种太阳能小屋,研究光伏电池在小屋外表面的优化铺设 方案,使小屋的全年太阳能光伏发电总量尽可能大,而单位发电量的费用尽可能小,并 计算出小屋光伏电池 35 年寿命期内的发电总量、经济效益及投资的回收年限。根据题 意,本文需要解决以下三个问题: 问题一: 根据山西省大同市的气象数据, 仅考虑贴附安装方式, 选定光伏电池组件, 对小屋的部分外表面进行铺设,并根据电池组件分组数量和容量,选配相应的逆变器的 容量和数量。 问题二:电池板的朝向与倾角均会影响到光伏电池的工作效率,选择架空方式安装 光伏电池,重新考虑问题 1。 问题三:根据附件 7 给出的小屋建筑要求为大同市重新设计一个小屋,要求画出小 屋的外形图,并对所设计小屋的外表面优化铺设光伏电池,给出铺设及分组连接方式, 选配逆变器,计算相应结果。

2012年国际数模竞赛B题分析及优秀论文讲评

2012年国际数模竞赛B题分析及优秀论文讲评

读书报告2012年国际数模竞赛C题陈润泽李思瑾颜颖摘要本题是要我们从八十二名成员中根据给出的业务信息(Message)找出犯罪团伙的同谋和领导人。

这是一道典型的图论题,其信息量之大、成员间关系之复杂着实让人感觉毫无头绪。

本着简化问题的原则,我们组在阅读论文之前进行了深入思考,并建立了自己的模型。

首先,我们运用了布尔代数(Boolean Algebra),将每个话题被谈论的与否表示为1 和0,即如果某个话题被某人谈论,则其相应位置的值为1 ,反之为0 。

最后得到每个成员后都跟上一个15位只有0 和1 的数(其中的每一位都代表一个话题)。

然后设定一个15位的布尔数,其三个可疑话题的位置为1 其余位置为0 。

再同每个成员对应的布尔数做and(与)运算,可选出存在可疑话题的成员,即我们需要研究的对象。

在对选出对象按优先次序排列的过程中,我们主要进行了以下两个步骤。

1.我们给出三个可以话题中每个话题被同谋者谈论的概率(如:若可疑话题一的概率为0.5,则谈论这个话题的人有50%的可能性为同谋者)。

然后对研究对象进行加权求和,根据所得进行排序。

2.我们对每个人与已知同谋者的相关性进行了分析。

以每个话题在每个人的业务信息出现的概率为维,对于每个研究对象建立一个15维的向量。

然后利用余弦定理,将每个研究对象的向量同已知同谋者的向量的夹角余弦值求出,再取平均数。

在既得排序的基础上,按降序对夹角进行排序,最后剔除已知非同谋者,即可按照排序结果确定犯罪团伙的领导人以及每个成员是同谋者的可能性。

一、问题重述与理解1.1 问题重述题目的背景是ICM组织在进行对一项密谋犯罪的调查。

已知罪犯和嫌疑人都在一家大公司的一个综合办公室里工作,公司里有82名成员,其中有7名已知同谋和8名已知非同谋。

ICM最近掌握了82名员工的一部分信息,并且想通过对信息的分析找出同谋以及犯罪组织的领导。

所有信息中包含15个话题,其中有3个可疑话题。

而且,只要成员的交流信息中包含可疑话题的,其可疑性便增加一些。

2012年美国大学生数学建模竞赛国际一等奖(Meritorious Winner)获奖论文

2012年美国大学生数学建模竞赛国际一等奖(Meritorious Winner)获奖论文

AbstractFirstly, we analyze the reasons why leaves have various shapes from the perspective of Genetics and Heredity.Secondly, we take shape and phyllotaxy as standards to classify leaves and then innovatively build the Profile-quantitative Model based on five parameters of leaves and Phyllotaxy-quantitative Model based on three types of Phyllotaxy which make the classification standard precise.Thirdly, to find out whether the shape ‘minimize’ the overlapping area, we build the model based on photosynthesis and come to the conclusion that the leaf shape have relation with the overlapping area. Then we use the Profile-quantitative Model to describe the leaf shape and Phyllotaxy-quantitative Model to describe the ‘distribution of leaves’, and use B-P Neural Network to solve the relation. Finally, we find that, when Phyllotaxy is determined, the leaf shape has certain choices.Fourthly, based on Fractal Geometry, we assume that the profile of a leaf is similar to the profile of the tree. Then we build the tree-Profile-quantitative Model, and use SPSS to analyze the parameters between Profile-quantitative Model and tree-Profile-quantitative Model, and finally come to the conclusion that the profile of leaves has strong correlation to that of trees at certain general characteristics.Fifthly, to calculate the total mass of leaves, the key problem is to find a reasonable geometry model through the complex structure of trees. According to the reference, the Fractal theory could be used to find out the relationship between the branches. So we build the Fractal Model and get the relational expression between the mass leaves of a branch and that of the total leaves. To get the relational expression between leaf mass and the size characteristics, the Fractal Model is again used to analyze the relation between branches and trunk. Finally get the relational expression between leaf mass and the size characteristics.Key words:Leaf shape, Profile-quantitative Model, Phyllotaxy-quantitative Model, B-P Neural Network , Fractal,ContentThe Leaves of a Tree ........................................................ 错误!未定义书签。

2012高教社杯全国大学生数学建模竞赛论文

2012高教社杯全国大学生数学建模竞赛论文

承诺书我们仔细阅读了中国大学生数学建模竞赛的竞赛规则.我们完全明白,在竞赛开始后参赛队员不能以任何方式(包括电话、电子邮件、网上咨询等)与队外的任何人(包括指导教师)研究、讨论与赛题有关的问题。

我们知道,抄袭别人的成果是违反竞赛规则的, 如果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明确列出。

我们郑重承诺,严格遵守竞赛规则,以保证竞赛的公正、公平性。

如有违反竞赛规则的行为,我们将受到严肃处理。

我们授权全国大学生数学建模竞赛组委会,可将我们的论文以任何形式进行公开展示(包括进行网上公示,在书籍、期刊和其他媒体进行正式或非正式发表等)。

我们参赛选择的题号是(从A/B/C/D中选择一项填写): A我们的参赛报名号为(如果赛区设置报名号的话):所属学校(请填写完整的全名):兰州理工大学技术工程学院参赛队员(打印并签名) :1. 种王涛2. 王世刚3. 邹永海指导教师或指导教师组负责人(打印并签名):窦祖芳蒙頔日期: 2012 年 9 月 10 日赛区评阅编号(由赛区组委会评阅前进行编号):编号专用页赛区评阅编号(由赛区组委会评阅前进行编号):全国统一编号(由赛区组委会送交全国前编号):全国评阅编号(由全国组委会评阅前进行编号):葡萄酒的评价摘要本文通过合理的假设,重点分析重要指标,忽略次要、不相关的指标,从而得到合理可靠的结论。

用单因素试验的方差分析、多项式拟合、函数适线、三维立体效果图等方法,反映出了酿酒葡萄和葡萄酒的理化指标和质量。

在问题一中,采用方差分析,用EXCEL软件,对大量数据的处理绘制出波动图,由图分析了评价酒员结果有显著差异,第二组的评价结果更可靠。

在问题二中,查阅大量资料分析了酿酒葡萄的理化指标的主次,然后对酿酒葡萄的主要理化指标和葡萄酒质量迭多次筛选,结合葡萄理化指标的含量和葡萄酒的质量,对酿酒葡萄做出了超特级、特选级、精选级、优选级四个等级。

2012年MCM

2012年MCM

2013 Contest ProblemsMCM PROBLEMSPROBLEM A: The Ultimate Brownie PanWhen baking in a rectangular pan heat is concentrated in the 4 corners and the product gets overcooked at the corners (and to a lesser extent at the edges). In a round pan the heat is distributed evenly over the entire outer edge and the product is not overcooked at the edges. However, since most ovens are rectangular in shape using round pans is not efficient with respect to using the space in an oven.Develop a model to show the distribution of heat across the outer edge of a pan for pans of different shapes - rectangular to circular and other shapes in between.Assume1. A width to length ratio of W/L for the oven which is rectangular in shape.2. Each pan must have an area of A.3. Initially two racks in the oven, evenly spaced.Develop a model that can be used to select the best type of pan (shape) under the following conditions:1. Maximize number of pans that can fit in the oven (N)2. Maximize even distribution of heat (H) for the pan3. Optimize a combination of conditions (1) and (2) where weights p and (1- p) are assigned to illustrate how the results vary with different valuesof W/L and p.In addition to your MCM formatted solution, prepare a one to two page advertising sheet for the new Brownie Gourmet Magazine highlighting your design and results.PROBLEM B: Water, Water, EverywhereFresh water is the limiting constraint for development in much of the world. Build a mathematical model for determining an effective, feasible, and cost-efficient water strategy for 2013 to meet the projected water needs of [pick one country from the list below] in 2025, and identify the best water strategy. In particular, your mathematical model must address storage andmovement; de-salinization; and conservation. If possible, use your model to discuss the economic, physical, and environmental implications of your strategy. Provide a non-technical position paper to governmental leadership outlining your approach, its feasibility and costs, and why it is the “best water strategy choice.”Countries: United States, China, Russia, Egypt, or Saudi Arabia。

2012年美国国际大学生数学建模竞赛(MCM+ICM)题目+翻译

2012年美国国际大学生数学建模竞赛(MCM+ICM)题目+翻译

2012 Contest ProblemsPROBLEM A: The Leaves of a Tree"How much do the leaves on a tree weigh?" How might one estimate the actual weight of the leaves (or for that matter any other parts of the tree)? How might one classify leaves? Build a mathematical model to describe and classify leaves. Consider and answer the following:• Why do leaves have the various shapes that they have?• Do the shapes “minimize” overlapping individual shadows that are cast, so as to maximize exposure? Does the distribution of leaves within the “volume” of the tree and its branches effect the shape?• Speaking of profiles, is leaf shape (general characteristics) related to tree profile/branching structure?• How would you estimate the leaf mass of a tree? Is there a correlation between the leaf mass and the size characteristics of the tree (height, mass, volume defined by the profile)?In addition to your one page summary sheet prepare a one page letter to an editor of a scientific journal outlining your key findings.2012美赛A题:一棵树的叶子(数学中国翻译)“一棵树的叶子有多重?”怎么能估计树的叶子(或者树的任何其它部分)的实际重量?怎样对叶子进行分类?建立一个数学模型来对叶子进行描述和分类。

2012数学建模优秀论文..

2012数学建模优秀论文..

基于系统综合评价的城市表层土壤重金属污染分析摘要本文针对城市表层土壤重金属污染问题,首先对各重金属元素进行分析,然后对各种重金属元素的基本数据进行统计分析及无量纲化处理,再对各金属元素进行相关性分析,最后针对各个问题建立模型并求解。

针对问题一,我们首先利用EXCEL 和 SPSS 统计软件对各金属元素的数据进行处理,再利用Matlab 软件绘制出该城区内8种重金属元素的空间分布图最后通过内梅罗污染模型:2/12max22⎪⎪⎭⎫ ⎝⎛+=P P P 平均综,其中平均P 为所有单项污染指数的平均值,max P 为土壤环境中针对问题二,我们首先利用EXCELL 软件画出8种元素在各个区内相对含量的柱状图,由图可以明显地看出各个区内各种元素的污染情况,然后再根据重金属元素污染来源及传播特征进行分析,可以得出工业区及生活区重金属的堆积和迁移是造成污染的主要原因,Cu 、Hg 、Zn 主要在工业区和交通区如公路、铁路等交通设施的两侧富集,随时间的推移,工业区、交通区的土壤重金属具有很强的叠加性,受人类活动的影响较大。

同时城市人口密度,土地利用率,机动车密度也是造成重金属污染的原因。

针对问题三,我们从两个方面考虑建模即以点为传染源和以线为传染源。

针对以点为传染源我们建立了两个模型:无约束优化模型()[]()[]()22y i y x i x m D -+-=,得到污染源的位置坐标()6782,5567;有衰减的扩散过程模型得位置坐标(8500,5500),模型为:u k zu c y u b x u a h u 2222222222-∂∂+∂∂+∂∂=∂∂, 针对以线为传染源我们建立了l c be u Y ∆-+=0模型,并通过线性拟合分析线性污染源的位置。

针对问题四,我们在已有信息的基础上,还应收集不同时间内的样点对应的浓度以及各污染源重金属的产生率。

根据高斯浓度模型建立高斯修正模型,得到浓度关于时间和空间的表达式ut e C C -⋅=0。

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Camping along the Big Long RiverSummaryIn 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 Equation1IntroductionThe number of visits to outdoor recreation areas has increased dramatically in last three decades.Among all those outdoor activities,rafting is often chose as a family get-together during May to September.Rafting or white water rafting is a kind of interesting and challenging recreational outdoor activity,which uses an inflatable raft to navigate a river or sea [1].It is very popular in the world,especially in occidental countries.This activity is commonly considered an extreme sport that usually done to thrill and excite the raft passengers on white water or different degrees of rough water.It can be dangerous.During the peak period,there are many tourists coming to experience rafting.In order to satisfy tourists to the maximum,we must make full use of our facilities in hand,which means we must do the utmost to utilize the campsites in the best way possible.What's more,to make more people feel the wildness life,we should minimize the encounters to the best extent;meanwhile no two sets of parties can occupy the same campsite at the same time.It is naturally coming into mind that we should consider where to stop,and when to stop of a party [2].In previous studies [3-5],many researchers have simulated the outdoor creation based on real-life data,because the approach is dynamic,stochastic,and discrete-event,and most recreation systems share these traits.But there exists little research aiming at describing the way that visitors travel and distribute themselves within a recreation system [6].Hence,in our paper,we consider the whole process of parties in detail and simulate every party ’s behavior,including the location of their campsites,and how long it will last for them to stay in a campsite to finish their itineraries.Meanwhile minimize the numbers of encounters.Aiming at showing the whole process of rafting,we firstly focus on analyzing the situation s of a single-party's rafting by using status transfer equation,then consider the problems of two parties'encounters on the river.Finally,after several times of simulation on the whole process of rafting,we obtain the optimal value of X .2Symbols and DefinitionsIn this section,we will give some basic symbols and definitions in the following for the convenience.Table 1.Variable Definition Symbols Definitioni v i p j i q ,S dThe velocity of oar or motor0-1variables on choosing rafting transportation0-1variables on the occupation of campsitesLength of the riverAverage distance between two campsites3General AssumptionsIn order to have a better study on this paper,we simplify our model by thefollowing assumptions:1)19:00to 07:00is people's sleeping time,during this time,people are stationedin the campsite.The total time of sleeping is 12hours,as rafting is an exiting sport game,after a day's entertainment,people have cost a lot of energy,and nearly tired out.So in order to have a better recreation for the next day,we set that people begin their trip at 07:00,and end at 19:00for a day's schedule.2)Oar-powered rubber rafts and motorized parties can successfully raft from FirstLaunch to Final Exit,there exist no accident over the whole trips.3)All the rubber rafts and motorized boats have the same exterior except velocities;we regard a rubber raft or a motorized boat as a party and don't consider the tourists individuals on the parties.4)There is only one entrance for parties to enter the recreation system.5)Regardless of the effects that the physical features of the river brings to oar andmotorized parties,that is to say we ignore the stream ’s propulsion and resistance to both kinds of rafting boats.Oar and motorized parties can keep the average velocity of 4mph and 8mph.6)Divide the whole river into N segments.4Analysis of This Rafting ProblemRafting is a very popular spots game world-wide.In the peak period of rafting,there are more people choosing to raft,it often causes congestion that not all people can raft at any time they want.Hence,it is important for managers to set an optimal schedule for every party (from our assumptions,we regard a rafting boat as a party)in advance.Meanwhile,the parties need to experience wildness life,so the managers should arrange the schedules which minimize the encounters'time between parties to the best extent.What's more,no two sets of parties can occupy the same site at the same time.Our aim is to determine an optimal mix of trips over varying duration (measured YXNj i t ,jT t∆KNumbers of campsites Numbers of parties Numbers of attraction sites Time of the i th party finishing the whole trip ranges from6days to 18daysRandom staying time at each campsiteDelay time of rafting from beginning Threshold value of encounterin nights on the river.That is to say,we must obtain an optimal value of X through lots of trails.This optimal value represents that the campsites have a high usage while more people are available to raft.The Long Big River is 225miles long,if we discuss the river as a whole and consider all the parties together,it will be difficult for us to have a clear recognition on parties'behaviors.Hence,we divide the river into N attraction sites.Each of the attraction sites has Y/N campsites since the campsites are uniformly distributed throughout the river corridor.So build up a model based on single-party ’s behavior of rafting in small distance.At last,we can use computer simulation to imitate more complex situations with various rafting boats and large quantities of parties.5Mathematic Models5.1Rafting of the Single-party Model (Status Transfer Equation [7])From the previous analysis,in order to have a clear recognition of the whole rafting process,we must analyze every single-party's state at any time.In this model,we consider the situation that a single-party rafts from the First Launch to the Final Exit.So we formulate a model that focus on the behavior of one single-party.For a single-party,it must satisfy the following equation:status transfer equation.it represents the relationships between its former state and the latter state.State here means:when the i th party arrives at the j th campsites,the party may occupy the j th campsite or not.As a party can choose two kinds of transportation to raft:oar-powered rubber rafts(i v =4mph)and motorized rafts(i v =8mph).i v is the velocity of the rafting boats,and i p is the 0-1variables of the selecting for boats.Therefore,we can obtainthe following equation:)1(84i i i p p v −+=(i=1,2,…,X ).(1)where i p =0if the i th party uses motorized boat as their rafting tool,at thistime i v =8mph ;while i p =1when ,the i th party rafts with oar-powered rubber raft with i v =4mph.In fact,Eq.(1)denotes which kind of rafting boat a party can choose.A party not only has choice on rafting boats,but also can select where to camp based on whether the campsites are occupied or not.The following formulation shows the situation whether this party chooses this campsite or not:⎩⎨⎧=party previous a by occupied is campsite the 0,party previous a by occupied not is campsite the q ij ,1(2)where i =1,2,…,X ;j =1,2,…,Y .Where the next one can’t set their camp at this place anymore,that is to say thelatter party’s behavior is determined by the former one.As campsites are fairly uniformly distributed throughout the river corridor,hence,we discrete the whole river into segments,and regard Y campsites as Y nodes which leaves out (Y +1)intervals.Finally we get the average distance between th e j th campsite and (j+1)th campsite:1+=Y Sd (3)where is the length of the river,and its value is 225miles.What’s more,the trip-days for a party is not infinite,it has fluctuating intervals:h t h j i 432144,≤≤(4)where is the t i ,j itinerary time for a party ranges from 144hours to 432hours (6to 18nights).From Eq.(1),(2)and (3),the status transfer equation is given as follows:),...2,1,,...2,1(11,1,,Y j X i T q v d t t j j i i j i j i ==×++=−−−(5)The i th party’s arriving time at the j th campsite is determined by the time when the i th arrived at (j-1)campsite,the time interval i v d ,and the time T j-1random generated by computer shown in Eq.(5).It is a dynamic process and determined by its previous behavior.5.2The Analysis of Two Parties Parties’’Encounter on the River Our goal is to making full use of the campsites.Hence,the objective of all the formulation is to maximize the quantities of trips (parties )X while consider getting rid of the congestion.If we reduce the numbers of the encounters among parties,there will be no congestion.In order to achieve this goal,we analysis the situations of when two parties’to encounter,and where they will encounter.In order to create a wildness environment for parties to experience wildness life,managers arrange a schedule that can make any two parties have minimal encounters with each other.Encounter is that parties meet at the same place and at the same time.Regarding the river as a whole is not convenient to study,hence,our discussion is based on a small distance where distance=d (Eq.3),between the j th and (j+1)th campsites.Finally the encounter problem of the whole river is transferred into small fractions.On analyzing encounter problem in d and count numbers of each encounter in d together,we get a clear recognition of the whole process and the total numbers of encounter of two parties.The following Figure 1represents random two parties rafting in d :Figure 1.Random two parties'encounter or not on the riverThe i th party arrives at j th campsite (t j k ,-t j i ,)time earlier than the k th party reaches the j th campsite.After t time,interval distance between the i th party and the k th party can be denoted by the following function:)()(t t t v t v t S ij kj i k j +−×−×=∆(6)Where k,i =1,2,…,X ,j =1,2,…,Y .k i ≠.Whether the two parties stationed on the j th campsite and(j +1)th campsite are based on the state of the campsites’occupation,yields we obtain:⎩⎨⎧=×01,,j k j i q q (i,k =1,2,…,X ;j =1,2,…,Y ;k ≠i )(7)Note that Eq.6is constrained by Eq.7,for different value of )(t S J ∆andj k j i q q ,,×we can obtain the different cases as follows:Case 1:⎩⎨⎧=×=∆10)(,,j k j i j q q t S (8)Which means both the i th and k th party don’t choose the j th campsite,they are rafting on the river.Hence,when the interval distance between the two parties is 0,that is )(t S J ∆=0,they encounter at a certain place in d on the river.Cases2:⎩⎨⎧=×=∆00)(,,j k j i j q q t S (9)Although the interval distance between the two parties is 0,the j th campsite is occupied by the i th party or the k th party.That is one of them stop to camp at a certain place throughout the river corridor.Hence,there is no possibility for them to encounter on the river.Cases 3:⎩⎨⎧=×≠∆10)(,,j k j i j q q t S ⎩⎨⎧=×≠∆00)(,,j k j i j q q t S (10)No matter the j th campsite is occupied or not for )(t S J ∆≠0,that is at the same time,they are not at the same place.Hence,they will not encounter at any place in d .5.3Overview of Computer Simulation Modeling to Rafting5.3.1Computer SimulationSimulation modeling is a kind of method to imitate the real-word process or a system.This approach is especially suited to those tasks which are too complex for direct observation,manipulation,or even analytical mathematical analysis (Banks and Carson 1984,Law and Kelton 1991,Pidd 1992).The most appropriate approach for simulating out-door recreation is dynamic,stochastic,and discrete-event model,since most recreation systems share these traits.In all,simulation models can reflect the real-world accurately.5.3.2Simulation for the Whole Process of Parties on Rafting [8]This simulation can approximate show a party’s behavior on the river under a wide rang of conditions.From the analysis of the previous study,we have known that the next party’s behavior is affected by the former one.Hence,when the first party enters the rafting system,there is no encounter,and it can choose every campsite.then the second party comes into the rafting system ,at this time,we must consider the encounter between them,and the limit on choosing the campsite.As time goes by,more and more parties enter this system to raft which lead to a more complex situation.A party who satisfies the following two conditions will be removed from the current order to the next order.So he can’t “finish his trip”right away.The two conditions are as follows:(1)He chooses a campsite where has been occupied by other parties.(2)He has two many encounters with other parties.So in order to determine typical trip itineraries for various types of rafting boat ,campsite,and time intervals (See Trip Schedule Sheet 1),we need to perform a series of trails run that can represent the real-life process of rafting based on these considerations,.A main flowchart of the program is shown in Figure 2.Figure2.Main simulation flowchartAfter several times of simulation,we obtain the optimal X(the numbers of campsites),minimal E(Encounter)and TP(Trip Time).Followed by Figure2,we simulate the behavior of a party whether it can enterthe rafting system or not in Figure3.Figure3.Sub flowchart5.3.3The Results of SimulationAfter simulating the whole process of parties rafting on the river,we get three figures(Figure4,Figure5and Figure6)to present the results.In order to simulate the rafting process more conveniently,we divide the whole river into31segments(31attraction sites),and input an initial value of Y=155(numbers of campsites),where there are5campsites in every attraction sites.We represent the times of campsites occupied by various parties on Figure2by coordinates(x,y),where x is the order of the campsites from0to155(these campsites are all uniformly distributed thorough the corridor),and y is the numbers of each campsite occupied by different parties.For example,(140,1100)represents that at the campsite,there exists nearly1100times of occupation in total by parties over180days. Hence,the following Figure4shows the times of campsites’usage from March to September.Figure4.Numbers of campsites'usage during six-month period from March to SeptemberFrom Figure4,The numbers of campsites’usage can be identified the efficiency of every campsites’usage.The higher usage of the campsites,the higher efficiency they are.Based on these,we give a simple suggestion to managers(see in Memo to Managers).Figure5.the ratio of usage on campsites with time going byFigure5shows the changes of the ratio on campsites.when t=0,the campsites are not used,but with time going by,the ratio of the usage of campsites becomes higher and higher.We can also obtain that when t>20,the ratio keeps on a steady level of65%;but when t >176,the ratio comes down,that is,there are little parties entering the recreation system.In all,these changes are rational very much,and have high coincidence with real-world.Then we obtain1599parties arranged into recreation system after inputting the initial value Y=155,and set orders to every party from number0to number1599.Plotting every party's travelling time of the whole process on a map by simulating,as follows:Figure6.Every party’s travelling timeFigure6shows the itinerary of the travelling time,most of the travelling time is fluctuating between13days and15.3days,and most of travelling time are concentrated around14days.In order to create an outdoor life for all parties,we should minimize the numbers of encounter among different parties based on equations(6)and(7):So we get every party’s numbers of encounter by coordinates(x,y),where x is the order of the parties from0to1600,y is the numbers of encounters.Shown in Figure7,as follows:Figure7.Every party’s numbers of encounterFigure7shows every party’s numbers of encounter at each campsite.From this figure,we can know that the numbers of their encounter are relatively less,the highest one is8times,and most of the parties don’t encounter during their trips,which is coincident with the real-world data.Finally,according to the travelling time of a party from March to September,we set a plan for river managers to arrange the number of parties.Hence,by simulating the model,we obtain the results by coordinate(x,y),where y is the days of travelling time,x is the numbers of parties on every day.The figure is shown as follows:Figure8.Simulation on travelling days versus the numbers of parties From Figure8.we set a suitable plan for river manager,which also provide reference on his managements.6Sensitive AnalyzeSensitive analysis is very critical in mathematical modeling,it is a way to gauge the robustness of a model with respect to assumptions about the data and parameters. We try several times of simulation to get different numbers of parties on changing the numbers of campsites ceaselessly.Thus using the simulative data,we get the relationship between the numbers of campsites and parties by fitting.On the basis of this fitting,we revise the maximal encounter times(Threshold value)continually,and can also get the results of the relationships between the numbers of campsites and parties by fitting.Finally,we obtain a Figure9denoting the relations of Y(numbers of campsites)and X(numbers of parties),as follows:Figure9.Sensitive analysis under different threshold values Given the permitted maximal numbers of encounters(threshold value=K),we obtain the relationships between Y(numbers of campsites)and X(numbers of trips). For example,when K=1,it means no encounters are allowed on the river when rafting;when K=2,there is less than2chances for the boats to meet.So we can define the K=4,6,8to describe the sensitivity of our model.From Figure9,we get the information that with the increase of K,the numbers of boats available to rafts till increase.But when K>6,the change of the numbers of boats is inconspicuous,which is not the main factor having appreciable impact on the numbers of boats.>In all,when the numbers of campsites(Y)are less than250,they would have a great effect on the numbers of boats.But in the diverse situations,like when Y>250, the effect caused by adding the numbers of campsites to hold more boats is not notable.When K<6,the numbers of boats available increases with the ascending of K, While K>6,the numbers of boats don’t have great change.Take all these factors into consideration,it reflects that the numbers of the boats can’t exceed its upper limit.Increase the numbers of campsites and numbers of encounter blindly can’t bring back more profits.7Strengths and WeaknessesStrengthsOur model has achieved all of the goals we set initially effectively.It is not only fast and could handle large quantities of data,but also has the flexibility we desire.Though we don’t test all possibilities,if we had chosen to input the numbers of campsites data into our program,we could have produced high-quality results with virtually no added difficulty.Aswell,our method was robust.Based on general assumptions we have made in previous task,we consider a party’s state in the first place,then simulate the whole process of rafting.It is an exact reflection of the real-world.Hence,our main model's strength is its enormousedibility and stability and there are some key strengths:(1)The flowchart represents the whole process of rafting by given different initialvalues.It not only makes it possible to develop trip itineraries that are statistically more representatives of the total population of river trips,but also eliminates the tedious task of manual writing.(2)Our model focuses on parties’behavior and interactions between each other,notthe managers on the arrangement of rafting,which can also get satisfactory and high-quality results.(3)Our model makes full use of campsites,while avoid too many encounters,whichleads to rational arrangements.WeaknessesOn the one hand,although we list the model's comprehensive simulation as a strength,it is paradoxically also the most notable weakness since we don’t take into account the carrying capacity of the water when simulates,and suppose that a river can bear as much weight as possible.But in reality,that is impossible.On the other hand,our results are not optimal,but relative optimal.8ConclusionsAfter a serial of trials,we get different values of X based on the general assumptions we make.By comparing them,we choose a relative better one.From this problem,it verifies the important use of simulation especially in complex situations. Here we consider if we change some of the assumptions,it may lead to various results. For example,(a)Let the velocity of this two kinds of boats submit to normal distribution.In this paper,the average velocity of oar-powered rubber rafts and motorized boats are 4mphand8mph,respectively.But in real-world,the speed of the boats can’t get rid of the impacts from external force like stream’s propulsion and resistance.Hence,they keep on changing all the time.(b)Add and reduce campsites to improve the ratio of usage on campsites.By analyzing and simulating,the usage of each campsite is different which may lead to waste or congestion at a campsite.Hence,we can adjust the distribution of campsites to arrive the best use.A Memo to River ManagersOur simulation model is with high edibility and stability in many occasions.It can imitate every party’s behavior when rafting so as to make a clear recognition of the process.Internal Workings of The ModelInputsOur model needs to input initial value of Y,as well as the numbers of attraction sites. Algorithm(Figure2,and Figure3)Our algorithm represents the whole process of rafting,so we can use it to simulate the process of rafting by inputting various initial values.OutputsBased on the algorithm in our paper,our model will output the relative optimalnumbers of parties X.Furthermore,we can also get other information,such as the interval time between two parties at First Launch,a detailed schedule for each party of rafting,the relationship between X and Y and so on.Summary and RecommendationsAfter100times of simulating,we come to two conclusions:(a)The numbers of parties(X)have relations with the numbers of campsites(Y), that is to say,with the increasing of Y,the increasing speed of X goes fast at the first place and then goes down,finally it tends to be steady.Hence,we advice river managers to adjust the numbers of campsites properly to get the optimal numbers of parties.(b)Add campsites to the high usage of the former campsites and deduce campsites at the low usage of the former campsites.From Figure4,we know that the ratios of every campsites are different,some campsites are frequently used,but some are not.Thus we can infer that the scenic views are attractive,and have attracted lots of parties camping at the campsite.so we can add campsites to this nodes.Else the campsites with low usage have lost attractions which we should reduce the numbers of campsites at those nodes.References[1]KarloŠimović,Wikipedia,Rafting,/wiki/Rafting.[2] C.A.Roberts and R.Gimblett,Computer Simulation for Rafting Traffic on theColorado River,COMPUTER SIMULATION FOR RAFTING TRAFFIC,2001, 19-30.[3] C.A.Roberts,D.Stallman,J.A.Bieri.Modeling complex human-environmentinteractions:the Grand Canyon river trip simulator,Ecological Modeling153(2002)181-196.[4]J.A.Bieri and C.A.Roberts,Using the Grand Canyon River Trip Simulator toTest New Launch Scheduleson the Colorado River,Washington DC,AWISMagazine,Vol.29,No.3,2000,6-10.[5] A.H.Underhill and A.B.Xaba,The Wilderness Simulation Model as aManagement Tool for the Colorado River in Grand Canyon National Park,NATIONAL PARK SERVICE/UNIVERSITY OF ARIZONA Unit SupportProject CONTRIBUTION NO.034/03.[6] B.Wang and R.E.Manning,Computer Simulation Modeling for RecreationManagement:A Study on Carriage Road Use in Acadia National Park,Maine,USA,USA,Vermont05405,1999.[7]M.M.Meerschaert,Mathematical Modeling(Third Edition).China MachinePress publishing,2009.[8]A.H.Underhill,The Wilderness Use Simulation Model Applied to Colorado RiverBoating in Grand Canyon National Park,USA,Environmental Management Vol.10,No.3,1986,367-374.。

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