1、 Print This Page Close This Window Team Control NumberFor office use only For office use onlyT1 _ F1 _T2 _ 8631 F2 _T3 _ F3 _T4 _ Problem Chosen F4 _145BSummaryIn order to determine the mix of trips, from the point of view of the managers and the passengers, depend on using the dynamic programming,
2、Monte Carlo methods and genetic algorithm optimization to set up two models, we determine the optimal mix of trips. For Model 1: from the point of view of the managers, , due to the large number of camps along the River is unknown, and in order to ensure the utilization rate of camp at 100% We deter
3、mine the 26 kinds of trips as 26 trip packages to be used by analyzing varying duration and propulsion in a certain condition.For Model 2: from the point of view of the managers, to begin with, by considering how to choose two vessels in A trip. Therefore, the average speeds of the vessels, as well
4、as whether campsites are occupied, the drifting time between the two campsites are determined. Then, we take the any two vessels encounter problem into account. We divide the whole problem into several phases, and simulate the whole trip by using Monte Carlo methods. After the trip simulation, And t
5、hen using dynamic programming and genetic algorithms to solve the problem. We conclude that carrying capacity of the river is 1854 vessels and allow for eleven vessels to be launched each day of the season.Contents1. Introduction 11.1 Problem Background . 1 1.2 Our Work. 12. Symbol Definitions . .33
6、. Model Assumptions.34. Problem Analysis55. Model 1 54.1 The Foundation of Model.54.2 Solution and Result.54.3 Analysis of the Result.56. Model 2. 55. 1 The Foundation of Model.55. 2 Solution and Result.65. 3 Analysis of the Result.77. Discussion and Conclusion. 77.1 Strength and Weakness.77.2 Concl
7、usions88. References239. Appendix.23Team # 8631 Page 1 of 101 Introduction1.1 Problem Background The Big Long River (225 miles downstream) , there are two kinds of vessels could be taken by the passengers: oar- powered rubber rafts, which travel on average 4 mph, or motorized boats, which travel on
8、average 8 mph. The park managers want to determine how they might schedule an optimal mix of trips, and how much more boat trips could be added to the Big Long Rivers rafting season.We are now faced with the following: How to optimize the mix of trips and the schedule? By considering the carrying ca
9、pacity of the river, how to determine the times boat trips could be added to the Big Long Rivers rafting season ?1.2 Our work As we know, X trips travel down the Big Long River each year during a six month period. There are Y camp sites on the Big Long River, distributed fairly uniformly throughout
10、the river corridor. The trips range from 6 to 18 nights of camping on the river, start to finish. Thus we will consider as following: Based on the Y camp sites certain conditions, determine varying duration (measured in nights on the river) and propulsion (motor or oar) to utilize the campsites in t
11、he best way possible, with minimal contact with other groups of boats on the river. After mathematically analyzing the Camping schedule, our modeling group would like to describe our key findings to the managers of the river.Team # 8631 Page 2 of 102. Symbol DefinitionsVariable DefinitionivThe speed
12、 of the i-th driftingipi-th drifting push mode ( motor or oar) Variable ( 0 or 1)jiq,Camp site occupied variable ( 0 or 1)S The length of the riverd The average distance between the two campsitesY The number of campsitesX The number of packagesjit,The time of starting at i-th drifting to the camp si
13、te j-thjTRandom residence time in the campsitexx-th days in the six-month open period（ 180x）y y-th motorized boat in a dayx x-th oar- powered rubber rafts in a day0tThe first motorized boat departure time everydayTThe oar- powered rubber rafts departure time everydayytMotorized boat drifting space（
14、01t）Yoar- powered rubber rafts drifting space（ T）M a total of vessels in a dayaThe number of motorized boat every dayb The drifting time of motorized boat every day .B The drifting time of oar- powered rubber rafts every day .xydDrift days of yth Motorized boat in x-th dayYDDrift days of yth oar- po
15、wered rubber rafts in x-th dayj j-th Camp sitexykjyth motorized boat in x-th day to the j-th camp site at k-th dayYjyth oar- powered rubber rafts in x-th day to the j-th camp site at k-th dayTeam # 8631 Page 3 of 10xykjpProbability of yth oar- powered rubber rafts in x-th day to the j-th camp site a
16、t k-th dayYjProbability of yth motorized boat in x-th day to the j-th camp site at k-th day3. Model AssumptionsWe assume that : For camping 6 to 12 nights tourists, we only provide oar - powered rubber rafts, called package 1-7; For camping 13 to 18 nights tourists, we only provide oar powered motor
17、ized boats, called package 8-13; For the same kind of travel package, every day in the big river rafting distance is the same, start to finish; The oar- powered rubber rafts which travel on constant average 4 mph. The motorized boats which travel on constant average 8 mph. Changing vessels in a trip
18、 can be neglected; The traveling distance of each vessel in a day must be positive. The impact of natural factors of the trip can be temporarily ignored.Team # 8631 Page 4 of 104. Problem AnalysisTo the question 1: The question 1 belongs to a kind of optimization problem in mathematics, which is, na
19、mely, with the different time and the different promotion methods (motor or plasma), arrange an optimal mixed travel plan, make maximum use of camp, and to make ships as little access to other ships on the river. In this paper we use the computer simulation, the Monte Carlo method, integer programmi
20、ng, genetic algorithm and other related theory to establish and solution model.To the question 2: We must calculate how many times of trips can be scheduled per day on the basis of the question 1. Only in this way can we make the maximum use of camp, thus further find out how many times of trips can
21、 be scheduled in the six months. Then, the carrying capacity of the river is concluded. To realize the bearing capacity of the river to maximize, we must both consider every hair ship time, and to consider the maximization of unreasonable ship time minimization. Then, we analyze the bearing capacity
22、 of the river and state our own opinions.After solving and analyzing the problem, we set up two models.5. Model 1Model 1: From the view of tourism managerBecause the number of camps along the coast is unknown, and for the convenience of tourism managers to manage, the tourism strategy we have formul
23、ated is based on the number of camps. Under the premise of the camp utilization rate of 100%, we make sure the days of trips we can provide for visitors.Taking both travel time and travel tool into consideration, we divide travel mode into 13 * 2 = 26 kinds of travel package .But because the number
24、of camps along the river is unknown, and we have to make assumptions: the distance of traveling every day in each package is same and drift constant progress. So according to the number of unknown camps, we need to provide the combination of traveling days so as to make the sum of all combinations o
25、f the traveling days equal the sum of the number of the camps. And then put all the camp on distribution, make all kinds of camping days drift has a definite camp with the corresponding, i.e., Before the tourists set off, the given camp visitors should stay in has been made sure. So the utilization
26、rate of camp can reach 100%. And then, the tourists choose from by oar - powered drift, or by rubber rafts drift by themselves. Finally which kinds of the package types should be provided for passengers is determined.Team # 8631 Page 5 of 10During 6 months of the tourist season, in addition to the l
27、ast 17 days, the remaining 163 days of every day are the same, so for the first 163 days, we only need to determine the tour arrangement of the first day, and then the travel arrangements of the rest of the 163 days are the same with the first day.Figure.1 6. Model 2Model 2：From the view of the pass
28、engersThere are Y camp sites on the Big Long River, distributed fairly uniformly throughout the river corridor. If two adjacent camp sites apart d miles, we will get the relation between d and y: 1YSEach team has to reach the a camping site to have a rest every day, Supply energy for the next day ra
29、fting , so we can know the vessel traveling distance in one day for an integer multiple of d, Thus dnSkSince the Each vessel traveling distance on a day trip is less than the maximum traveling distance in a day trip, and milesSck251Team # 8631 Page 6 of 10We have the table 1 as follows:,Table 1: Dri
30、fting time of different type vessels Types of 6 7 8 9 10 11 12 13 14 15 16 17 18Packages oar- powered 9.4 8 7.03 6.3 5.63 5.11 4.7 4.3 4 3.75 3.5 3.3 3.13Rubber rafts Motorized 4.7 4 3.52 3.1 2.81 2.56 2.3 2.2 2 1.88 1.8 1.7 1.56Boats Using the model 2, we compute Drifting time of different type ves
31、sels. The results of this method are presented in Table1. Therefore, we conclude that A total of 25 kinds of packages could meet the conditions. Monte Carlo was used to simulate the approximation of the probability distribution of totally drifting time of different travel package.Figure 3. Totally d
32、rifting time of different travel package.Monte Carlo was used to simulate the approximation of the probability distribution of everyday drifting time of different travel package.Team # 8631 Page 7 of 1000.050.10.150.20.250.31 2 3 4 5 6 7 8 9 10摩 托 艇皮 划 艇Figure 3. Drifting time ofthe different travel
33、 packagee in each day.7. Discussion and Conclusion.7.1 Strength and WeaknessThe mathematical model has both the advantages and disadvantages. We have completed the two main problems, putting forward our views on carrying capacity of the river .Advantages and disadvantages are shown as follows:Streng
34、thThe mathematical model is based on general assumptions. In the previous task, considering a rafting trip, slowly for two trips driving situation, we finally simulate the whole process of drifting step by step. The results are obvious. Then we establish another model, the model of the problem is di
35、vided into six different factors, and then a thorough analysis of the six factors was established. So it is reliable.WeaknessTeam # 8631 Page 8 of 10In reality packages costs could potentially have an effect, especially in the short term. However, there is the vital drawback. Our algorithms do not t
36、ake the natural factors into account.7.2 ConclusionsPerhaps the most striking and valuable result is that all three models propose a maximum seasonal throughput for the Big Long River as some value between 1150 and 1250 boats, with up to 1400 allowed if the itinerary model is used and there is a hig
37、h tolerance for daytime crowding on the river. Despite this common conclusion, the differing motivations behind the models allow each to provide unique insight into the behavior of boats on the river. The agent-based model shows the importance of starting boats early in the day and restricting the d
38、uration of trips by slower boats to twelve or more days. The itinerary-based model suggests that placement preference should go to the slower boats, which are more restricted in their range of potential campsites each night. The probabilistic model advocates for keeping the relative proportions of e
39、ach type of boat constant over the course of the season and adjusting this proportion depending on whether the rafting management wishes to favor oar or motor boats. Taken together, these results provide a good picture as to how the river should be managed. A manager who is confident that he or she
40、can trust boats to travel to predetermine campsites and is comfortable with turning down some potential customers should pick the itinerary-based approach, thus preventing any one boat from having to travel in potentially unsafe conditions. A manager who believes that the boaters cannot be trusted t
41、o travel to a pre-specified campsite each evening should rely on the agent-based and probabilistic models, sending out a wave of boats each morning and keeping the relative proportion of each type of boat on the river constant, knowing that each boat runs some small chance of having to travel an exceedingly long distance to find an open campsite. Regardless of the approach chosen, the manager can be confident that their choice will not negatively affect the total number of people they serve in a season.Team # 8631 Page 9 of 10Appendix
