An Analysis of Dynamic Actions on the Big Long River

Control # 17126 Page 1 of 19 An Analysis of Dynamic Actions on the Big Long River MCM Team Control # 17126 February 13, 2012

Control # 17126 Page 2 of 19 Contents 1. Introduction... 3 1.1 Problem Background... 3 2. Problem Setup... 4 2.1 Information... 4 2.2 Simulator... 4 2.3 Outputs... 5 3. Simulator... 5 3.0 Problem s Original Conditions... 5 3.1 Mathematical Symbols... 6 3.2 Assumptions... 6 3.3 User s Input for Running of Simulator... 7 3.4 The Four most Important Ordered Conditions of Algorithm for Simulator... 8 3.5 The Basic Rules... 9 3.6 Recursive Formulas... 10 4. Analysis of Concrete Problem... 11 4.1 Application of The Example Case... 11 4.2 The Coordinate Data of the Example Case... 12 4.3 Applications and Descriptions with Diagram... 14 4.4 Sensitive Analysis... 16 5. Conclusions... 16 5.1 Strengths... 16 5.2 Weakness... 17 5.3 Alternative Model, Improvement and Future Work... 17 6. Reference... 19

Control # 17126 Page 3 of 19 1. Introduction Rafting is a modern outdoor recreational activity which became popular since 1970s. In the old time, rafting was just used as a method to meet the need of human s daily life. As the technology advances, rafting become a sport activity which provides people who busies themselves with study or work an opportunity to enjoy the nature and wild environment with an amusing and stimulating experience and refresh their minds and bodies. As the population of rafting increase rapidly in past few decades, there are many rafting tours built in U.S, such as smoky mountain rafting tour and Grand Canyon Colorado River rafting tour. In order to guarantee the quality of experience of each individual trip and also maximize the total capacity of each tour location during their rafting season. A simulation about the schedule for each tour location is significant and necessary. 1.1 Problem Background In this paper, we are going to build a model to simulate the trip schedule for a big long river which has 225 miles, one first launch and a final exit which means all trips will start at the same point and end at another same location. Visitors will have two options for the boats they will use, rubber rafts and motorized boats. Trips require camping during night and there are Y distributed fairly uniformly camping sites along the river which are not able to be occupied by different trip visitors at the same time. In order to provide the best experience of each trip, we will minimize the contact between different groups of people. Based on the current schedule of X trips which travel down the Big Long river each year during the rafting season, our model will add as many trips as we can to the current schedule so that we can get the an optimal schedule with mix of trips which range from 6-18 nights of camping. Based on the current situation of existing rafting tour operation, for example, Grand Canyon rafting tour on Corolla River 2, we can find some information about this kind problem. In 2000, in Joanna Bieri and Catherine Roberts s paper 1, they explained the main idea of using the Grand Canyon River Trip Simulator to test a new launch schedule and the simulator is designed by a group of faculty and students from the University of Arizona s School of Renewable Natural Resources and Northern Arizona University s Department of Mathematics and Statistics 3. Their simulator was based on data they collected and interview information collected from expert trip leaders, which included the average speed for different region along the river and how they decided where to set the camp. River trip situation is similar to estimate the largest traffic on a one-way high-way with finite rest areas which just allow one car to stay overnight.

Control # 17126 Page 4 of 19 However, the problem we are going to solve in our model is a little different. It is more similar to a train control simulator which is based on the old schedule and adding more trains into the base schedule, solving how to add in and how many to add in. Every trip on the river is similar to one train and each camping site is similar to a train station. 2. Problem Setup In this paper, we have two steps to approach the problem. Firstly, we need to find out all the information which is already given by the problem, identify the objects we are going to study the final goal for our model. Then we are going to analyze all the information and design our simulator. Finally, we will analyze the output of the model and give the final optimal solution. 2.1 Information Object: Every individual trip (or boat) is the object we are going to study. We will focus on how long (measured by nights) they will travel, which site they stop for each day, and the speed they will travel for each day. Constraint: There are finite camping sites (Y) on the river which are distributed fairly uniformly. Every trip have their own travel duration which require the boats speed must fall in a specific range and trips cannot travel in over speed or under speed for a long time. So occasionally speed is over or under the best speed range is acceptable. Under the condition of maximum condition, we need to minimize the contact between different trips. Goal: Given a current schedule of a rafting tour with X trips in it, our simulator will be able to add most amounts of trips into the schedule. And also, we need to make the number of group contact minimized. 2.2 Simulator In order to achieve our goal, maximum capacity and minimum contact, our simulator need to fulfill the following requirement: The simulator should be able to keep counting the passing date of rafting season and about to count the location of a specific trip at a specific date.

Control # 17126 Page 5 of 19 The simulator should be able to count the duration of each trip on the river and measure the speed of each trip (the suitable speed or speed range it will travel for next day). The simulator should be able to let user input their current schedule (type of trip and the sites the trip supposed to stop at) and the preference about the added trips. The simulator should be able to count the times of contact and minimize it. 2.3 Outputs Based on the goals and requirement we set up for our simulator, the simulator we give up an output which included the following point It will provide the trips which finish the travel, and these trips are the ones we will add into the base schedule. It will show the user the trips which fail to finish the travel, and these trips are the ones which are ruled out by conditions It will also provide the minimum number of contact which is based on the algorithm of the simulator. 3. Simulator: In this section, we describe the recursive formula from two sides, dynamic programming and net-work. In addition, we discuss why we use the 4 conditions to constrain the recursive formula. In all, our assumptions, user s inputs, the 4 conditions, and recursive formula simultaneously make an algorithm, which let the simulator work. 3.0. Problem s Original Conditions: The entire period includes 6 months, which is also 180 night-times and 181 day-times. There are 13 types of trips, 6 night-trip, 7 night-trip,, and 18 night-trip. X 1 trips travel down the river and each of them already on base schedule.

Control # 17126 Page 6 of 19 Y sites on the river, and with uniform distribution. Two types of boats, rubber rafts with average speed 4 mph, and motorized boats with average speed 8 mph. 3.1. Mathematical Symbols (those symbols occupy in paper with italics): s i : the i th controlling trip, which is in the base schedule x i : the i th normal trip which is added into the base schedule f(n, s i ): the camping site that the i th controlling trip moved in on the n th night Y is the total number of sites X1 and X2 are the number of trips which are already on the base schedule and added into the schedule X is the total number of trips, and y i : the i th site on the river (counted downstream) A j : there are j possible choices for in following action. lns : least nights for trips nc : no contacting N: the total number of days in this period. In this case, N is 180. ar: action range, the domain of distance for one day s travel n: the n th night in the entire period, or we also call it the n th stage in our dynamic network. CHO{}: the decision that manager made for a trip s camping site 3.2 Assumptions: Every trip is rational, there is no any trip wants to keep adventure overnight. So, in the night-time, every trip on the river must go to site, and no site has two or more than two trips.

Control # 17126 Page 7 of 19 If the managers want to control the schedule, and with the controlling power on hand; then there is always at least one controlling trip on the river. Otherwise, the schedule will only have one type of trips and one type of boats. Treat every trip has a same unique action range. After getting the outcomes from simulator, if some trips highest actual action range limit for each day which are lower than rubber s highest action range, then let managers make those decisions, which is whether or not let those trips use lower action range boats. For instance, in this case, if some trips can use rubber rafts or motorized boats, then we let managers to make those decisions. The managers cannot put multiple controlling trips in a same site, which means multiple controlling trips must have different schedules. Meanwhile, normal trips must give the way for controlling trips; if they may both go to a same site in the next action. For instance, in the next action if a controlling trip s i wants to go to site y j in and a normal trip x i has choices to go to site y j-1, y j or y j+1. Then the normal trip x i will have two choices left, site y j-1 or y j+1. Normal trips cannot pass another normal trip but they may be able to pass a control trip. We rule out the unexpected situations and unexpected accidents in the entire period. For example, there is no flood happen and no water animals kill tourists. 3.3 User s Input for the Running of Simulator: The managers need to give the exact number of sites to the simulator, which is the value of Y. And, the number of sites must be an integer ( ).

Control # 17126 Page 8 of 19 The managers need to give the independent schedule of each controlling trip. Suppose the managers think the schedule of last period was excellent, and what they want for the schedule of this period is based on last period; so there is no any change to last period schedule. Thus, we would make all trips from last period to be the controlling trips on this period s schedule. While keeping the same schedule of last period, we maximize the number of normal trips and minimize the contacting times. But the number of normal trips and number of contacting time are depended on manager s decision. For example, if managers have different input option of camping site of a normal trip and there is a control trip stay in the normal trip s action range, then manger choose low speed for normal trip will lead to no contact and highest speed will cause a contact. While the simulator is doing the optimization, it would produce some action choices from normal trips. So, the managers also can decide that how to allocate those choices. If the managers have no idea how to do this, we suggest those normal trips with multiple action choices to be as slow as possible. Hence, those trips can experience more adventures and scene. 3.4 The 3 Most Important Ordered Conditions of Algorithm of the Simulator: We call them 4 conditions. So, the simulator not only uses our recursive formulas to do the dynamic optimization, but also follows the 4 conditions. Typically, how the recursive formulas work is based on the 4 conditions. Meanwhile, the recursive formulas orderly follow the 4 conditions. Every controlling trip has the right of priority to decide when they want to go to sites and what sites to go. So, if a site may include a normal trip and controlling trip, it would

Control # 17126 Page 9 of 19 only let the controlling trip stay. And in the simulator, we always let controlling trips make the next action first. The order of priority of normal trip is depended on their traveled distance. The further a trip has traveled the higher priority it has. When the every normal trip gets the chance to move, the user will get several options for its travel speed. Based on the decision of travel speed user makes for the normal trips in a day, the simulator will add maximum number of normal trips into the river and output the contract times. 3.5 The Basic Rules: Contacting rule: We divide the entire period that is 180 nights into 180 stages. If trips launch at the same stage, there are no contacting. For instance, there are 3 trips launch at the first stage, x 1 goes to y 2, x 2 goes to y 3, and s 1 goes to y 1. Thus there is no contacting; even the sub-number of x 2 is less than the sub-number of x 1 and the distance of x 2 is greater than x 1. The moves of priority: To make our net-work works reasonable, we always let far trips whose locations close to the end make actions first. Since, we can let more new trips on the river from the launch location. The moves of controlling trips: To make the all actions reasonable, we always let controlling trips move first. If what sites they go to have trips, then we emerge those trips to their next possible sites. If there are no possible sites for those emerged normal trips, then we erase them. So which means those trips are not reasonable in the river

Control # 17126 Page 10 of 19 trip at that time. But, we will keep their routes in the final analysis diagram, to show how they traveled and why we erased them. The simple rule for all trips: No trip can go backward, and no trip can stay at one site two nights or more than two nights. The rule of lease erase: If the number of nights left is less than lns, all trips get onto river would be erased. The rule of action range: No trip can make one action more over the action range. Suppose every trip every day at most can go 5 sites, thus ar=5, and no trip can go over 5 sites at any stages. 3.6 Recursive Formulas: 1) Recursive formula for normal trips: { [ ] Explanation: If A j = A 1, which means the normal trip x i only has one option of action at stage n. If A j = A 2, which means the normal trip x i has two options of action at stage n; meanwhile, if j >1, we need user s decision to decide which action the trip will make. If A j = A 0, which means the normal trip x i has no site to go at this stage, then it is a failed trip. So, we need to erase it from the river and its trout that is made in previous stages. If, the normal trip x i is also need to be erased. For instance, in a period with 180 nights and 181 days, if a trip launches at 175 th night and lns is 6, then the trip need to be erased from the period s schedule.

Control # 17126 Page 11 of 19 2) Recursive formula for controlling trips: Explanation: In the n th stage, the controlling trip s i only has one option of action, and it was scheduled before the running of the simulator. 3) The principle of dynamic actions: The two recursive formulas above with the conditions and rules can make the simulator works. The principle of dynamic action is to let one trip move, then affect others movement, until the first a few sites empty, since the first a few sites relate to the launch point. In the next, the simulator starts to add new trips to the empty sites, so we can always reach the maximal capacity. Anyway, the recursive formulas are always follow the 3 conditions when start to calculate stage by stage. 4. Analysis of Concrete Problem: In this section, we give an example with 18 sites (Y=18), and first 13 nights to run the simulator. Then we cut partial data from the beginning of period and put those data into excel to make the linear relation functions diagram. 4.1 Applications of the Example Case: We want to the, which means the maximal distance for any trip at any day is about 35.53miles. Y=18 makes there are 19 blocks on the river. (Block = the space between sites or launch point or end point) We write a concrete partial data in the 4.2 section,

Control # 17126 Page 12 of 19 So, s 1 is a 6 nights controlling trip, s 2 is an 18 nights controlling trip, s 3 is a 16 nights controlling trip, and s 4 is a 15 nights controlling trip.. We also apply those partial data into diagram in which multiple linear relation functions participate. The diagram is in the 4.3 section. In this example, we suppose we are the managers and prefer the normal trips with fastest actions. 4.2 The Coordinate Data of the Example Case: 1 st stage: 2 nd stage: 3 rd stage: 4 th stage: =4 5 th stage: 6 th stage: 7 th stage:

Control # 17126 Page 13 of 19 8 th stage: For the further stages, please see Diagram #1 Diagram #1 4.3 Applications and Descriptions with Diagram: 1) First example related to the data from previous parts of section # 4: The partial diagram below is consisted with 1 st stage to 13 th stage and sites y 1 to y 18. Typically, there are 28 linear relation functions, 4 of them are represent controlling trips and 24 normal trips. Obviously, the diagram is made by the data from 4.1 and 4.2

Control # 17126 Page 14 of 19 sections. What s more, the intersections from the diagram below represent the contacting information of the partial actions from an entire period with 180 nights, which includes the time of contacting, the place of contacting and the other trips of contacting. The contacting times already are optimized as well as the maximal capacity. Definitely, the maximal value of vertical line is 18; because we setup the example with 18 sites. On the other hand, the managers also can change the total number of sites to some greater integer. 2) Analysis of the first example: Actually, the diagram is influenced by the route of controlling trips and the managers willing decisions. In the beginning of the simulator, we as the managers prefer to faster normal trips; so, the normal trips between s 1 and s 2 are relative fast trips. Meantime, on the right-bottom corner, there are some intersections. Those happen because of the routes of the new controlling trips. In our basic rules, we let some trips were scheduled first, and nothing can change their routes. So, those controlling trips can be the VIP trips which reserved by people. 3) Another partial example with different schedule: The diagram below is another new example with different data. In this example, as the managers, we prefer the normal trips move slowly and with less contacting. Obviously, in the entire diagram, there is no intersection, which means we have a favorite schedule without any contacting. So, sometimes, the outcomes of the simulator or the model are not independent. Since many things are based on the users or the managers.

Control # 17126 Page 15 of 19 Diagram #2 4) The comparison between diagram #1 and diagram #2: The differences between two examples are the users willing speed or schedules from the normal trips and the schedules of the controlling trips. In the first example, the users prefer to use faster speed normal trips rather than lower speed normal trips. However, in the second example, the users prefer to use lower speed normal trips rather than faster speed normal trips. Actually, whatever how fast the faster normal trips move every day, no one can move over the action range. Also, whatever how low the lower normal trips move every day, no one go backward or stay at a same site two days or more than two days. Moreover, the schedules of the controlling trips are made before the running of simulator. So, those schedules are independent. Nevertheless, the schedules of the normal trips are not independent, and this is the one of most interesting part.

Control # 17126 Page 16 of 19 4.4 Sensitive Analysis In the previous parts of section #4, we use concrete data to make examples. However, the length of period, the number of sites, the type of boats, the schedules of the controlling trips and the types of the normal trips (long trip or short trip) all are possible to be changed. In the beginning of making this model, we used some simple cases, such as 6 sites with 18 nights (ar =3), 5 sites with 10 nights (ar = 2), and so on. So, we develop this model not only by the dynamic net-work, but also the method of induction. So, typically, the model and the simulator can be developed to large number of sites, the nights in the periods, or even make the period longer. We consider that if some trips want to have lunch at noon, we can divide an entire day into two parts: one is the first half day-time, another one is the second half day-time; thus we also keep the rule of being in site in the night-time. So, the problem of food or restroom time or others can also be solved. Meanwhile, if trips want to get into sites multiple times during day-time, we also can divide the day-time into multiple parts. 5. Conclusions 5.1. Strengths Our simulator has the following characteristics. It considers the behavior of each trips and the mutual effect between different trips. As the closer to the exit a trip is, the higher priority this trip will have and control trips have the highest priority, the trips which have moved will leave as many available sites behind as possible so that it will guarantee that we can launch most amount of new trips and maximize the capacity of the river based on the current schedule provided by the manager. Our simulator will also minimize the contact between different groups as lower priority trip is able to avoid passing the higher priority (the trip launched before it and the control trip) by the user s decisions. Our simulator will launch as many trips as possible and then erase the trips which are suitable (the ones cannot move in a day as there are no available site in front path under the condition of action range). In another word, the trip will be canceled if it cannot move anywhere in a day during its travel.

Control # 17126 Page 17 of 19 As these characteristics we has explained above, given a specific original schedule of different types of rafting trips, we will be able to reform the schedule by adding more optional trips in order to maximize the capacity of the river. It also helps to minimize the contact frequency and make it as low as possible and also gives greatest capacity of the river under the user s choice (pass or not pass front trip). Besides that, if we are not given a very detailed schedule but just the demand of different types of trips or the speed of trips (fast or slow) the user would like to have during some periods, we can also provide an optimal schedule for this river. Making an extremely random situation become controllable and reasonable is the most important idea and function of our simulator. Based on a schedule of launch time and sites of each site supposed to go, the managers can use the model to control all the trips and select the trips they prefer during some period. 5.2. Weakness While our simulator try to simulate every trip s position and each day s situation, it require large amount of calculation, especially when the number of sites is really large and number of control boat on the river is large. Also because of the control power, the ability of visitors self-decision will be weakened, which means, visitor have to obey their schedules to go to the camping sites where they are supposed to go, and cannot decide the camping place themselves. When visitor spent too much time during the day time to enjoy a specific sight, it will become impossible for them to get to their camping site, which is regarded as an unexpected accident in our model. Once accident happens, the control ability will be limited but managers are still able to control them as they know which site is available. So we need experienced group leader to control the speed and prevent accidents happening. Also, the new trips are kind of similar to each other in selected period, so it just can satisfy the most demand of a specific type of trip. If users need to add some particular types of trips, they have to make them become control trip. 5.3. Alternative Model, Improvement and Future Work Under the limit time constraint, our simulator is not impossible to be perfect in considering of all kinds of situation. In the future, we change our model about user s preference effect so that they can give their expected trips duration range and our simulator will prove as many expected types of trips as possible, which may lower the capacity of the river but the result will more suit the demand of market. Besides that,

Control # 17126 Page 18 of 19 we can perfect our simulator to consider the effect of different situations from different regions on the river, which means that, for example, in a specific region, the water velocity may be extremely high and it is impossible to slow down. We need to consider the specialization of different area and set the speed range of each region.

Control # 17126 Page 19 of 19 6. Reference: 1. Joanna A. Bieri and Catherine A. Roberts, Using the Grand Canyon River Trip Simulator to Test New Launch Schedules on the Colorado River AWIS Magazine, Vol 29, No.3 2000 2. http://www.raftarizona.com/rates-dates/default.asp 3. http://mathcs.holycross.edu/~croberts/research/gcrtsim/