Final%20Paper

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Optimized River Trip Model

Introduction

River rafting is an outdoor activity that has received rising popularity across America. Rafting America, for instance, is one of the largest rafting companies that has attracted over 4.1 millions of visitors since 1970.  More than 75 rivers across the U.S., Canada and South America are made available for rafting. Trips of multiple durations, from half day to multi-day, aim at both beginner rafters and experts who love the adventure of flowing in nature. A shared characteristic among all river trips are that visitors can enjoy locations and destinations near the river some of which are not available in any other way. River trips are designed to have varying levels of adventuress with different class ratings for each river. International Scale of River Difficulty classifies rivers according to their levels of challenge changes with fluctuations in water levels, using roman numerals from I to VI. Class I is characterized by rivers with small waves and no serious obstacles whereas class VI is characterized by rivers with waterfalls and death-traps. Rivers from class  I to V are chosen as appropriate for rafting, ranging from easy to extremely difficult.

Visitors who choose river trips are enchanted with the desire to “getting away from it all”, where “it all” mainly refers to the stress and unhappiness from daily life (Arnould & Price, 1993). In other words, visitors expect to have fun on river trips. Previous researches have found that the level of communication with nature is positively proportional with visitor’s level of satisfaction of river trip experience (Arnould & Price, 1993). Communication with nature is affected by the traffic condition on the river (Biere & Roberts, 2000). Specifically, visitors’ sense of involvement with nature is lower when the river is more crowded, that is, when there are more trips on the river. Therefore, in order to provide the visitors with more satisfaction of the trip, there should be as minimal chance of seeing others on the river as possible. More trips certainly generate more profit but lead to low satisfaction for visitors in each trip: the trade-off is obvious. However, the responsibility to ensure satisfaction for visitors is not contradictory with the maximized usage of campsites available, which is another way to maximize profit while providing the best satisfaction simultaneously.

Problem and ApproachProblem B poses the question of how to launch as more trips as possible to fill every

campsite while visitors in these trips do not see other groups on the river, which would interfere with their enjoyment of the river sight-seeing and their close involvement with nature. From mathematically macro perspective, Problem B requires the designed model to ensemble an optimal mix of trips with varying durations, under the condition that the probability of cross-trip contact is minimized and the usage of campsites available is maximized.

Optimized River Trip ModelA mathematical model is developed to evaluate the adding-trip problem. It is an abstract

model initially, but only after the macro-scope model has been built can we plug in numbers and solve the real life problem. There are two crucial ideas in our model: First, minimizing contact among groups of trips on the river, including keeping each campsite with only one group of visitors(one trip), which is required by the government agency to give the travelers wilderness experiences. Second, maximizing the river capacity so as to allow more river trips.

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Figure 1.Overall view of two models.

Basic Assumptions and Design:

As it is shown in the flow chart above, this model consists of Model A and Model B. Starting from Model A, one assumption is made that all boats are uniformly distributed on the river, regardless of mix of trips.

Under this assumption, we first divide the Big Long River into m parts, naming each part from A1to Am, that is

Ai ϵ {A1 , A2 , … Am}.

Given the length of the river is 225 miles, we define the length of each part to be l, such that:

l ≡225m

Given Y campsites in total on the Big Long River, define the number of campsites in each Ai as b i such that:

Model

model A (regardless of mix of trips)P(minimize contact among groups on the river)

R(maximize usage of sites)Q=P*R

Model B (take mix of trips into consideration)H1 (minimize contact between different speed boats)

H2 (maxmize mix of trips)H=H1/H2

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b i≡Ym

Define the number of total boats, including both oar boats and motorboats, to be “n”; thus, the number of boats in Ai can be defined as ni, such that:

ni ≡nm

(Assuming that: b1=b2==b i=Y/m; n1=n2==ni=n/m)The graph below shows the partition of the Big Long River:

Figure 2. Basic Design and Approach

.

Model A: Model A is a simplified model of the real situation; we did not take the mix of trips into

consideration in Model A.

Probability of minimal contact To achieve the minimal contact of boats on the river, the concept of probability is used to

describe the situation of contact and to minimize the contact. We take an arbitrary interval Ai,

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and divide it into a parts; the length of each part is the eyesight, E, of a normal person. Thus, number of parts, a is given by: a=l / E.

By our assumption, there are ni boats in the interval Ai. To compute the probability of minimal contact, p, define the term “ minimal contact” in this case to be the condition that there are no two boats squeezed in one same part; in another words, no two boats in Ai are within each other’s eyesight.

Now, compute the number of all possible ways that ni boats randomly flow in a intervals:

N1=a¿

Then compute the number of ways that no two boats flow in the same interval:

N2=a!

(a−¿ ) !Therefore,

P=N2

N1

= a!(a¿¿¿(a−¿))!¿

P= a!a¿∗(a−¿)!

The Campsites Usage Index (CUI)

Besides minimizing minimal contact on the river, define another R, which is the Campsites Usage Index (CUI). And it is given by R is defined to illustrate the relationship between bi and ni (x =100-ni). To maximize the usage of campsites, it is important to make sure that in each Ai, bi campsites can be used as much as possible by niboats; however, it is also crucial to avoid the overuse of each campsite, namely that each campsite can only be occupied by one boat. More specifically, we need R to describe a situation such that: the value of R keeps increasing before ni= b i, illustrating the fact that CUI would get higher as the number of boats in Ai gets closer to the number of campsites in Ai; and afterni = b i, any ni value bigger than b i would bring a sharp drop in R, describing the fact that the government agency strongly avoid the overuse of each campsite, any more boats after the saturation of campsite use would cause R to be very low. Thus, numerically, the bigger R is, the better CUI, R, is.

That being said, one can imagine a graph interpreting relationship between ni and R, for a certain amount of b i, in some Ai:

(In plotting Figure 3 below, b i is chosen to be 12 and it turns out when ni = 12, we will get the peak value of R.)

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Figure 3. Graph of CUI Index.To get the shape of R look like the shape drawn in the graph above, we inferred a

concept in Physics: Plank’s Law in blackbody radiation:

The graph interpreting Plank’s Law always looks like:

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Apparently, the shape of the curve looks very similar to our expectation to the shape of R curve. Thus, from Plank’s Law, we abstract a simpler expression for R that has the similar shape:

R=

a

kxb∗c

ed

hx

, where a ,b , c , d , k , h are constants ;

x is the variable .We plug in negative signs in front of x to flip the curve around y-axis and the graph we get looks more precisely to our expectation. Then, we also try out different ways of managing constant in R. And we finally got the expression below:

R=25∗{

1

[−( x−100 )2∗bi ]

2∗1

e[ 2∗bi−( x−100 ) ]

},

Where x=100- ni

The Q IndexIn model A, we define two important terms P and R, which measure the possibility

of minimal contact and the usage of camping site respectively. The major idea here is to maximize the probability of minimal contact and maximize the usage of the camping sites simultaneously. To achieve this goal, we define a new term Q, where

Q Ξ P*ROnce we have the Q index, we can maximize both P and R simultaneously by

maximizing Q, which measures the overall maximization of river trips. By plotting the graph of Q with certain value of bi, we can see the location of the peak of Q; and the ni value with respect to the peak value is the quantity of boats that we should put in some Ai.

There is no certain unit of Q, since it is merely considered as an index of maximization. Given that Q is the multiplication of P and R, Q is proportional to R. According to the figure of R that we obtained earlier, we can plot the figure of Q, which indicates a similar pattern as the figure of ROne of the graphs of Q looks like:

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Figure 4. Graph of Q index.

Model B:In Model B, we start to take mix of trips into our consideration. Recall the Q index in Model A, it describes how we should maximize the river

usage while keep minimal contact in a general sense. By assuming certain number of campsites Y and certain number of m, we can plot graphs of Q with respect to ni. Such graphs tells us: generally, with given amount of campsites Y in total, and with the river divided into m parts, how many boats in total, namely n, we should put into the river to keep minimal contact and maximal river usage. Details are provided in the later section.)

Now, with number of total boats n, we can start to achieve the maximal goal of river usage while keep minimal contact by mixing the trips in appropriate proportion.

The H Index Our plan to minimize contact among motor boats and oar boats is to try to put motor boats (which has a higher speed) into the intervals nearer Final Exit  and oar boats(which has the lower speed) into the intervals nearer First Launch. To achieve this goal, we define a new index H 1. H 1 measures the abstract portion of the low-speed boat in the intervals near the First Launch and high-speed boat in the intervals near the Exit. Thus, define:

H 1≡∑i=1

m

nmb∗i+∑i=1

m

nob∗(m−i)

where the letter m refers to the number of total intervals, i is the numeral of an arbitrary interval, and nmb is the number of motor boats in Ai and nob is the number of oar boats in Ai. By the formula of H 1,the value of H 1 increases as i becomes larger, since the closer and the more motor boats are to Final Exit and the closer and the more oar boats are to First Launch, the larger value each sum will have. Therefore, by maximizing H 1, which will maximize the number of motorboats near to the exit and the number of oar boats near to the First Launch, the minimal contact is achieved. However, the maximum of H 1is obtained by having either only motor boats and have

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them all at the last interval, or only oar-powered boats and have them all at the first interval. This extreme situation is unrealistic. Therefore, H2is developed to ensure the mix of both types of boats. Define:

H 2 ≡nmb

n0

+n0

nmb

The value of H 2s minimized when there’s an equal amount of each types of boats. Therefore, by defining

H ≡H 1

H 2

and maximizing the value of H, which maximizes H 1and minimizes H 2at the same time, the mix of different boats are ensured to achieve maximal river usage while minimal contact is also maintained.  

Results and Discussion

Model A Results And Discussion: In Model A, the major formula we use to maximize river usage and minimize

contact is Q. Plotting graphs of Q has two parts:1. Assume several different values of total campsites Y with the same partitioning

number m, and then plot Q with respect to ni. 2. Assume several different values of m with the same value of total campsites Y,

and then plot Q with respect to ni. Under different circumstances, namely changing total campsites Y or changing

partitioning number m, Q would respond differently to ni; thus, we can find the best value ni in different situations which maximize value of Q.

Before plotting Q, we used excel to general two groups of basic values:

Constant M and Changing Y

Eyesight(miles) l (miles) Y per mile Y m b i (Y/m)a= l

E0.50 5.00 3.00 675 45 15 100.50 5.00 2.75 619 45 13.75 100.50 5.00 2.50 563 45 12.5 100.50 5.00 2.25 506 45 11.25 100.50 5.00 2.00 450 45 10 100.50 5.00 1.75 394 45 8.75 100.50 5.00 1.50 338 45 7.5 100.50 5.00 1.25 281 45 6.25 100.50 5.00 1.00 225 45 5 100.50 5.00 0.75 169 45 3.75 100.50 5.00 0.50 113 45 2.5 100.50 5.00 0.25 56 45 1.25 10

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Constant Y and Changing M

Eyesight (miles) l (miles) Y per mile Y M b i (Y/M)a= l

E0.50 2.50 1.00 225 90 2.5 50.50 5.00 1.00 225 45 5 100.50 7.50 1.00 225 30 7.5 150.50 10.00 1.00 225 22.5 10 200.50 12.50 1.00 225 18 12.5 250.50 15.00 1.00 225 15 15 300.50 17.50 1.00 225 12.8571

428617.5 35

0.50 20.00 1.00 225 11.25 20 400.50 22.50 1.00 225 10 22.5 450.50 25.00 1.00 225 9 25 500.50 27.50 1.00 225 8.18181

818227.5 55

0.50 30.00 1.00 225 7.5 30 60

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Table 1. Values of bi and a resulting from Y and M.

As shown above, we generated 2 groups of basic data. In both two groups, we assigned the same Eyesight distance to both cases. In the first group, the partitioning method is constant, only total campsites are changing. This results in changing values in b iand constant values ina. Similarly, in the second group, the total campsites number Y is constant, and only partitioning number m is changing. This results in changing values in both b i anda.

Then, we plug different values of b iand a into Matlab to plot the graphs of Q, with respect to ni .

For the first group, with constant m and changing Y, we got following graphs: (here, we only show 5 out 12 of them)

Graph1: m=45, Y=675, Peak at ni=5

Graph2: m=45, Y=619, Peak at ni=5

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Graph3: m=45, Y=563, Peak at ni=3

………………

Group10: M=45, Y=169, Peak at ni=3

Graph11: m=45, Y=113, Peak at ni=2

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Comparing the shift of peak with respect to ni, one might find as the total campsite number Y decreases, the number of boats ni, in any Ai also decreases. This means, if the total number of campsites Y increases, ni has to increase too so as to maximize Q.

In the second group, we get following graphs, here we show 5 out of 12:

Graph1: Y=225, m=90, Peak at ni =2

Graph2: Y=225, m=45, Peak at ni=3

Graph 3, Y=225, m=30, Peak at ni=4

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………………

Graph11: Y=225, m=8.18, Peak at ni=12

Graph12: Y=225, m=7.5, Peak at ni=13

Analyzing these graphs, one finds a fact that: when total camping site is constant, changing the number of intervals, m, will also affect the peak of Q with respect to ni. Sparser intervals will make the peak appears at bigger ni and denser intervals will make the peak appears at smaller ni. This difference makes sense; intuitively, how we partition the river should not affect the maximization of Q. Different results from the second group exactly proves this point: more intervals, less ni in each interval; less intervals, more ni in each

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interval.

Partial Conclusion of Model A:

Comparing two groups of data and graphs, we have following conclusions about Model A:

1. How we partition the river does not necessarily influence the maximization of Q. 2. Increasing total campsites number Y would allow more ni in each Ai, for a constant

m intervals on the river. 3. For each constant total campsites number Y and constant number of intervals, m, Q

will have a unique peak at some ni.

Model A allows us to determine the number of boats, ni in each Ai, if we were given a constant Y and m and try to minimize contact while maximize river usage.

Model B Results And Discussion:

From Model A, one can estimate the total number of boats that can be put into the river, for a given Y and m, and the total number of boats, n (n=ni*m) also satisfies the maximization of Q and therefore minimizes contact and maximizes the river usage in a general sense.

In this part, we tested a specific set of data, which came from Model B. And then, investigate, for this certain amount of intervals, and total boats n, how should we maximize the river usage and minimize contact by mixing trips.

The data we picked is: Number of total boats: n=130. Number of Intervals: m= 20.

Recall in Model B, we have

H 1≡∑i=1

m

nmb∗i+∑i=1

m

nob∗(m−i)

H 2 ≡nmb

n0

+n0

nmb

H ≡H 1

H 2

Thus, Excel is used to calculate H 1 and H 2 for each interval of increasing fast boats percentage. The full EXCEL is attached at Appendix.

Screenshot 1: In this screenshot, you will see how I compute each component part in H 1with respect to different percentage of fast boats in Ai. The formula for cell C4 is

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f ( x )=( D 31 )∗( 2

21 )∗2

20in this formula, 2/20 is our function of

f ( x )= iz

This function is not mentioned in the previous part, because it is a minor part of the model and it is part of computing H 1.

In the formula, 2/21 is a factor to normalize the function of f ( x )= iz

.

After normalizing this function, we will have 2

21∗i

z=¿of total boats∈Ai∗i

. And recall the

first part in H 1is the sum of the number of motor boats in Ai * i; therefore, for each cell, the formula is:

f ( x )=%of fast

boats∗221

∗i

z

In this way, H 1 ,H 2 and H are calculated for each different percentage of fast boats on the river.

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Then, we plotted the graph of H with respect to percentage of fast boats in Kaleidagraph. (Shown on the right.)

As the graph shows, the peak of H is at 55%of fast boat out of total 130 boats in the river. Thus, if we were to maximize river usage and minimize contact, we can mix 55% percent of fast boats with 45% of slow boats.

Here, the figure “55%”, is specifically for this case, in which m=20 and n=130. But, this process of calculating H and plotting H with respect to percentage of fast boats illustrates a general process of determining the exact ratio of mix of trips.

Partial Conclusion of Model B:

Model B provides us a way to calculate the exact proportion of mix of two trips, and by this proportion to maximize river usage and minimize contact in a more specific way.

One thing notable is that, for other circumstances, where m and n is different from the specific case I mentioned just now, the best proportion of mix of trips is always a little bit more than 50% of fast boats.

Final Conclusion:

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Model A and Model B are closely connected to each other, rather than Model A is an estimate in a general sense and the Model B is more specific.

Two models together constitute a complete and systematic way to estimate how to manage total number of boats and how to mix them in the right proportion so as to maximize the river usage and minimize the contact. Model A allows us to estimate the biggest value of total boats that a river can hold while keep the contact minimal; from Model A, one can move on to Model B and analyze more specifically, in terms of how many fast boats should be on the river and ho many slow boats should be on the river so as to maximize the river usage and also keep the minimal contact.

LimitationThis model is deviated from reality in the following three ways. First, in real life situation, motor trips normally have more visitors in each trip and stop at campsites with greater capacities, whereas oar boats have fewer visitors in each trip and stays at campsites with smaller capacities. The established model fails to consider the capacity of campsites and the size of the trip for different boats. Second, one of the assumptions is that the number of boats in each interval Ai is the same. This assumption is adopted for convenience in calculation but fails to reflect the reality. In other words, at a given time, the probability that the number of boats is the same in each interval is considerably smaller than the probability that it is not. Third, one of the assumptions is that, Y, the number of campsites, is large enough such that every campsite is in the reachable distance of either oar boat or motor boat, at a given time.

Reference

http://en.wikipedia.org/wiki/File:Blackbody-lg.pn.

http://en.wikipedia.org/wiki/Plank%27s_law.

Arnould, E., & Price, L. (1993). River magic: extraordinary experience the extended service encounter. Journal of Consumer Research: Vol.20, 24-45.

Bieri, J., Roberts, C. (2000). Using the grand canyon river trip simulator to test new launch schedules on the Colorado River. AWIS Magazine Vol.29 No.3.

Rafting America is a site that presents the commercial river rafting service of this company, including history, data, and typical river trips it provide(http://www.raftingamerica.com/).

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