Camping along the Big Long River
Summary
In this paper, the problem that allows more parties entering recreation system is investigated. In order to let park managers have better arrangements on camping for parties, the problem is divided into four sections to consider.
The first section is the description of the process for single-party's rafting. That is, formulating a Status Transfer Equation of a party based on the state of the arriving time at any campsite. Furthermore, we analyze the encounter situations between two parties.
Next we build up a simulation model according to the analysis above. Setting that there are recreation sites though the river, count the encounter times when a new party enters this recreation system, and judge whether there exists campsites available for them to station. If the times of encounter between parties are small and the campsite is available, the managers give them a good schedule and permit their rafting, or else, putting off the small interval time ?tuntil the party satisfies the conditions. Then solve the problem by the method of computer simulation. We imitate the whole process of rafting for every party, and obtain different numbers of parties, every party's schedule arrangement, travelling time, numbers of every campsite's usage, ratio of these two kinds of rafting boats, and time intervals between two parties' starting time under various numbers of campsites after several times of simulation. Hence, explore the changing law between the numbers of parties (X) and the numbers of campsites (Y) that X ascends rapidly in the first period followed by Y's increasing and the curve tends to be steady and finally looks like a S curve.
In the end of our paper, we make sensitive analysis by changing parameters of simulation and evaluate the strengths and weaknesses of our model, and write a memo to river managers on the arrangements of rafting.
Key words: Camping；Computer Simulation; Status Transfer Equation
1 Introduction
The 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 Definitions
In this section, we will give some basic symbols and definitions in the following for the convenience.
Table 1. Variable Definition Symbols vi pi qi,j Definition The velocity of oar or motor 0-1 variables on choosing rafting transportation 0-1 variables on the occupation of campsites Length of the river Average distance between two campsites S d Y X N ti,j Tj ?t K Numbers of campsites Numbers of parties Numbers of attraction sites Time of the ith 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 encounter 3 General Assumptions
In 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 stationed
in 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 First
Launch 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 and
motorized 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 Problem
Rafting 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
in 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 Models
5.1 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 ith party arrives at the jth campsites, the party may occupy the jth campsite or not.
As a party can choose two kinds of transportation to raft: oar- powered rubber rafts(vi=4mph) and motorized rafts(vi=8mph). viis the velocity of the rafting boats,and pi is the 0-1 variables of the selecting for boats. Therefore, we can obtain the following equation:
vi?4pi?8(1?pi) (i=1,2,…,X). (1) where pi=0 if the ith party uses motorized boat as their rafting tool, at this timevi=8mph ; whilepi=1 when , the ith party rafts with oar- powered rubber raft with vi=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:
(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 the
?1,the campsite is not occupied by a previous partyqij?? biyea dprevi poaursty?0,the campsite is occuplatter 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 the jth campsite and (j+1)th campsite:
Sd? (3)
Y?1where 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:
144h?ti,j?432h (4)
where is the ti，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:
ti,j?ti,j?1?d?qi,j?1?Tj?1vi(i?1,2,...X,j?1,2,...Y) （5）
The ith party’s arriving time at the jth campsite is determined by the time when the i arrived at（j-1）campsite, the time interval
th
dvi, and the time Tj-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 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 jth 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:
数学建模美赛2012MCM B论文
