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. 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