美模B题最终答案.docx

20 / 20Team#14727The Simulation Model applied to the Big Long River1. Restatement of the ProblemConsidering more people want to go camping on the Big Long River, the managers of the Big Long River need to develop better solutions to meet the needs of the passengers. The government responsible for the river asks that passengers can enjoy wild experience during the trip. Since the quality of the wilderness experience is inversely proportional to the number of encounters with other parties, so the number of encounters should be as few as possible. Here we consider one trip meets with another as the condition that the motorized boats catch up oar-powered rubber rafts, if two trips will meet in a camp site but one of them is going to the camp site to rest, this case is not the meet we define.Through the analysis of the subject, the time spent in camping is far outweigh the time of drifting, and groups can spent more than one time in camp site every day. We hypothesize that each team shall spend the night on camp site. Each camp site is for a travel team at any time.As for this subject, we need to provide guidance for managers in charge of the Big Long River by making the best trip itineraries to increase the number of tourists under the condition that a camp site cannot be simultaneously occupied by two trips. With comprehensive consideration of the travel time length and drifting tools choice, we concluded the way to maximize the use of the campsites.Our aim is to:l Maximize the use of the campsites, thus making the Big Long River accept more tourists.l Try to reduce the encounter between trips, namely reduced the catch up phenomenon. l Tourists can go to more different camps to enjoy different entertainment and appreciate the scenery.We will use three models to achieve the above objects.2. Further ConsiderationsThe problem is a typical dynamic planning issue, according to the total days of the travel we divide the river into even sections, and the trips need a day to travel through each section. Since camping point is evenly distributed, so if each campsite is best used, all the campsites will be of utmost used. So now, we need to make maximum use of campsite in one section, then using recursion relation we can make all campsites be of utmost used.If assume interval tending to zero here, just dont consider the gap between the trips. According to limit principle, we can issue a continuous of boats in one type. Because a single type of boats operation at the same rate, so they will never meet. Theoretically there is enough space to camping, river can accommodate great number of ships, but it doesnt fit with the practical situation. In practice, the number of campsites is limited, thus we should consider the gap between the trips, it is not only related to whether passengers can enjoy wild experience, but more important is related to the safety of the passengers.3. Assumptions and HypothesesThe following assumptions were made when tackling the problem.l The number of visitors uniform distribution in six months.l The speed of the boats is not influenced by the flow of water.l The movement direction of trips can only from upstream to downstream, and wont retrograde.l Y campsites are fixed, here we assume that there are 224 campsites, and isometric evenly distributed. Thus there is a campsite every 1 mile. Each camp site is a sight port, and we want to maximize the use of resources.l Each group has a lot of boats, and is one type of two varieties boats, oar-powered boats or motorized boats.l Oar-powered boats group and motorized boats ratio of group m: n.l All trips must have a rest on camp site in the evening, one camp site can only allow a trip to rest, we regulate 8 pm and 6 am as the rest time during which all trips must be in the camp site. l During the day, each trip can decide to rest in camp site according to their demands.l The distribution of 6-18 nights camping arrangement is decided by passengers demands, and it is not mandatory.l If two trips will meet with each other in a camp site, and one can enter the camp site for a rest, in this case it is not the meet we define.4. Analysis of the ProblemThe main problem now is how we can make the Bid Long River hold more of the trips. According to the actual situation of historical data, the number of camping are 200 or so, here we define Y = 200. We think, the limit to all the trips in a time of the tour should be mainly for the following two constraints: l The average travel time of the project. This time, in this problem is 6 to 18 nights (to spend 6-18 nights). The shorter the time, the more boats through, thus absorbing more tourists.l The number of the campsites .Because every trip have to rest on shore in the camp site in the evening, and each camp site can only hold one trip, so the number of camp site should be the upper limit of all group number.We are requested to design the travel plan for each trip, including each place to camp and the rest time. However there is no need to design one by one for each trip since a time schedule is enough, which allows them to play 6-18 nights, and the trips have a place to rest every night. Another goal is to make the trip as less as possible contact with other trips. The speed of the motorized boats is eight miles/hour, and the speed of the oar-powered boats is 4 miles/hour. The demands of tourists lead to there must be accompanied motorized boats and oar-powered boats. We temporarily hypothesize that the ratio of oar-powered boat and motorized boat is 1:1. As we all know, the motorized boats behind is easy to catch up oar-powered boats.We think that visitors has the specific demand for the play time, so we wont reduce their play time to increase the number of tourists, after all, the long time visitors spent on camp site also means visitors need higher consumption, wont cause investors income decrease.5. Model OneConsidering the preference life style and travel style of American people, visitors hope to have more autonomy space to spend their travel times, so in most cases the administrators shall not interfere in their specific travel which lasts for 6 to 18 days. In view of this, the schedule prohibited by the investor is only the departure time. The obligation of visitors is to leave the Big Long River from the end exit after spending 6-18 nights on the river.Since the managers do not arrange specific schedules of the play time for the tourists, the play times of each trip is random while the randomness of each trips play time can cause the phenomenon that some camp sites are left at night while some groups need to spend a lot of time looking for campsites to rest, some even cant find a campsite which leading to the great decrease of tourists pleasure.Goalsl The big long river can accommodate more tourists to meet the needs of the tourists, thus improving the income of the manager.l Visitors can free to look for camp site to rest easily (the satisfaction of the tourist satisfaction decrease with the increase time looking for camp sites, until the tourists cannot tolerate); Tourists can visit the place they want, that is when they want to visit a place, the probability of achieving this goal is high;l Camp sites can be occupied as many as possible at night, but this goal is conflict to the first goal; l The number of contacts between trips should be as few as possible.Consideration of the Modell The preferences to a camp site when tourists visit the camp site are influenced by two factors. One is the popularity of campsite which is determined by the number of being visited and the rank in the past. Here, we assume that it subjects to uniform distribution, randomly generated by the computer. , is on behalf of the likelihood of this attraction be accepted by tourists;The second factor is the will tourists want to land on the campsite, the will of landing changes with time. For example, if a group just starts its travel, tourists tend to drifting a period of time before landing, especially after a night resting, tourists would more likely to continue drifting a longer time before landing; after drifting a long time, their drifting will would be reduced. Overall, the probabilities of visitors landing in a subsequent distance are normal distribution. In consideration of the attraction degrees of scenic spots, we normalized the attraction degrees of some campsites which are likely to be visited, and make the sum of them equal 100%.We can get a new distribution by normalizing this distribution with the normal distribution. The landing probabilities of tourists obey this new distribution. Under the new distribution, the comprehensive probability for visitors landing on campsite x is: （Note: the normal distribution is continuous distribution, but the distribution here is discrete distribution, so it needs necessary processing.）(1) The probabilities for a group to land on a campsite are the integrals of the interval it is in(2) In the normal distribution, 95% probabilities fall in the interval, so the part outside can be neglected which can be added to the middle campsite(3) Represents the comprehensive will for tourists to land on campsite x, s represents half of the distance between two adjacent campsites. stands for the campsite point which tourists are most likely to land, the point is among the several camp site points after point x. l The times tourists staying on campsites are normal distribution. Tourists wont rush to leave a campsite, the longer time tourists stay on the camp site, and the greater tourists want to leave. l When the camp site tourists want to land is occupied, tourists would have to continue flow down the river. In this case the wills of landing for tourists obey the index distribution, because tourists have a strong will to land. When this case happens many times, the wills of landing for tourists obey more gradient index distribution.l In a certain algorithm, the average play time of tourists will be certain with the increasing of the number of simulation, so the travel time and the time staying on the campsite in a day is certain. So the factor that the stay times in every campsite are different has the similar influence to other groups.l The number of camp sites is fixed in the reality. Under the request Y should be utmost used, if the number of visitors enter the big long river are certain, the longer the time visitors stay on the river, the more campsites will be utilized , and this can allow visitors more time to enjoy the sights. This is consistent with the goals to maximize the river capacity and maximize profit of the managers. The opportunity to go to the want campsite will become difficult with the increasing number of the tourists; meanwhile the opportunity of contact will increase, thus reducing the satisfaction of tourists. We think the main trouble is the difficulty to find the preference camp site, according to the literature. We think that the influence of the increasing contact is little.l We divide each hour into eight sections, each section is 7.5 minutes, and in this case motorized boats can reach a campsite every 7.5 minutes.l Now we only consider motorized boats, the average of the 6 to 18 nights is 12 nights which has 13 days. Totally need 28.125 (225/8 = 28.125) hours drifting on the river, the average time for drifting each day is 2.163 hours (28.125/13 = 2.163 h), the average rest time each day is 11.837 hours, except the provisions of 10 hours to rest at evening. Suppose there are two groups of normal distributions, the decision on whether land or not made by tourists in drifting meet the normal distributions in group 1, and the decision on whether leave or not made by tourists on the camp site is accord with the normal distributions in group 2.Distribution 1 meets the normal distribution N (4, 2) based on the distance. The landing will of motorized boats always reach the peak at the campsites whose distance is the multiples of four, it is idealized here, and the camp sites whose distance is the multiples of four are set as rest sites during the day.Distribution 2 meets the normal distribution N（3，1） based on the time, The drifting will of tourists always reach the peak after three hours staying on the campsites.In this way, we can find that the total number of landing are four, the average time for camping each day is three hours, the total camping time are 14 hours （4*0.5+4*3）. The simplified model of oar-powered rubber rafts can be got in the similar way.Methods of the ModelThe schedule we designed considering the first launch time. From the angle of reference, we design 14 groups which are mutually contrast in the total number of parties、the proportions between motorized boats and oar-powered rubber rafts、weekly unevenly distributed groups (various distributions formed by different distributions everyday)、daily distribution of groups to find the characteristics an excellent schedule should have. Based on this, we design schedules meet the four goals. Through the computer simulation, we compare the results of each schedule, and select the optimal solution.The schedule is as follows:Sun. Sat.: The corresponding number of these lines indicates boats launched every day.%Motors%Oars: Indicates the proportion of motorized boats and oar-powered rubber rafts.10:302:30: Indicates the launch time of the boats.BaseSim1Sim2Sim3Sim4Sim5Sim6Sim7Sim8Sim9Sim10Sun565765786510Mon45465567642Tues45465567642Wed45465567642Thurs565865810651Fri33343545638Sat454555556410Total/week2935294235354249422935%Motors70005060601005060050%Oars30100100504040050401005010:3012122222210211:004547547853611:30101210141251416611012:005657657861612:30454650676051:00333335336131:30111114135422:00111214225412:3000000600150Sim1, Sim2, Sim3 Sim13, are generated by the control variable. For example, the difference between Sim2 and Base is the proportion of motorized boats and oar-powered rubber rafts; Sim5，the number of daily hair is uniformly distributed, the proportion of motorized boats and oar-powered rubber rafts is equal to Sim4;On the basis of Sim5，linearly increase the daily number of issued boats, keep the proportion of motorized boats and oar-powered rubber rafts unchanged, get Sim8；On the basis of Sim2，keep the number of trips and the proportion unchanged，change the number of boats at the corresponding time，the number of daily hair is distributed at both ends.In the expansion of our model, we randomly generated daily schedules and the number of motorized boats and oar-powered rubber rafts to look for conditions meeting necessary restrictions and achieving the goals. Genetic algorithm can be considered to solve this problem.We change some goals into restrictions for multi-objective programming. We no longer consider goal 3 and goal 1, because they are homogeneous. We change the target in goal 2 that visitors can find campsites to rest at night as easily as possible to a restriction. We set that the tourists at least in 99% of cases can rest in a campsite before ten o clock at night for the reason that the safety of the night drifting is very low and the satisfaction will fell sharply if visitors cant find campsites to rest at night for many times. This will also affect the reputation of tourism project. In a certain schedule, if more than 99% of the cases meet the constraints in the hundreds of times trials through the computer simulation, the schedule is feasible; otherwise, the schedule will be abandoned no matter how many visitors he can contain. The influence of the contact between tourists is not big and the happened opportunity is little, so it will not be considered until the first two goals are realized.Supplementary Assumptionsl In this model, we assume that the visitors travel along the river down to search for camp site for the night, but cant upstream. Tourists cant know the situation of other trips, so they dont know whether the ahead camp site will have been occupied or not.l Visitors flow downstream, at the random moment of a period of time they prepare to entry into the camp site, when they find a camp site they immediately check in, if they found camp site is occupied, they will continue to downstream until finding an available camp site to rest.l The preferences of visitors to camp site during the day obey a probability distribution. That is the preference enter probability of a trip enter into to a camp site is, the preference not enter the camp site probability is .If the camp site that the trip prefer to into has been occupied, they cannot enter.l Since tourists dont know whether there are other trips at front or at last, so at any time, they wont enter a camp site for the reason of avoiding other groups while they enter a camp site depending only on their preference and whether the campsite has been occupied.l The preference probability of every group to a camp site is same, but the preference degree among campsites is different.Our ModelNote: G represents the largest number of tourists of a week on the big long river;U is for the number of tourists daily into the campsite; Represents the proportion of groups who entering into the wanted campsite, Represents the probabilities of realizes when can realize.(, represent time-distance function of a group