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For office use onlyT1________________ T2________________ T3________________ T4________________Team Control Number 46639Problem ChosenCFor office use onlyF1________________F2________________F3________________F4________________2016 MCM/ICM Summary SheetAn Optimal Investment Strategy ModelSummaryWe develop an optimal investment strategy model that appears to hold promise for providing insight into not only how to sort the schools according to investment priority, but also identify optimal investment amount of a specific school. This model considers a large number of parameters thought to be important to investment in the given College Scorecard Data Set.In order to develop the required model, two sub-models are constructed as follows: 1.For Analytic Hierarchy Process (AHP) Model, we identify the prioritizedcandidate list of schools by synthesizing the elements which have an influence on investment. First we define the specific value of any two elements’ effect on investment. And then the weight of each element’s influence on investment can be identified. Ultimately, we take the relevant parameters into the calculated weight, and then we get any school’s recommended value of investment.2.For Return On Investment M odel, it’s constructed on the basis of AHP Model.Let us suppose that all the investment is used to help the students to pay tuition fee.Then we can see optimal investment as that we help more students to the universities of higher return rate. However, because of dropout rate, there will be an optimization investment amount in each university. Therefore, we can change the problem into a nonlinear programming problem. We identify the optimal investment amount by maximizing return-on-investment.Specific attention is given to the stability and error analysis of our model. The influence of the model is discussed when several fundamental parameters vary. We attempt to use our model to prioritize the schools and identify investment amount of the candidate schools, and then an optimal investment strategy is generated. Ultimately, to demonstrate how our model works, we apply it to the given College Scorecard Data Set. For various situations, we propose an optimal solution. And we also analyze the strengths and weaknesses of our model. We believe that we can make our model more precise if more information are provided.Contents1.Introduction 21.1Restatement of the Problem (2)1.2Our Approach (2)2.Assumptions 23.Notations 34.The Optimal Investment Model 44.1Analytic Hierarchy Process Model (4)4.1.1Constructing the Hierarchy (4)4.1.2Constructing the Judgement Matrix (5)4.1.3Hierarchical Ranking (7)4.2Return On Investment Model (8)4.2.1Overview of the investment strategy (8)4.2.2Analysis of net income and investment cost (9)4.2.3Calculate Return On Investment (11)4.2.4Maximize the Total Net Income (11)5.Test the Model125.1Error Analysis (12)5.2Stability Analysis (13)6.Results136.1Results of Analytic Hierarchy Process (13)6.2Results of Return On Investment Model (14)7.Strengths and Weaknesses157.1Strengths (15)7.2Weaknesses (16)References16 Appendix A Letter to the Chief Financial Officer, Mr. Alpha Chiang.171.Introduction1.1Restatement of the ProblemIn order to help improve educational performance of undergraduates attending colleges and universities in the US, the Goodgrant Foundation intends to donate a total of $100,000,000 to an appropriate group of schools per year, for five years, starting July 2016. We are to develop a model to determine an optimal investment strategy that identifies the school, the investment amount per school, the return on that investment, and the time duration that the organization’s money should be provided to have the highest likelihood of producing a strong positive effect on student performance. Considering that they don’t want to duplicate the investments and focus of other large grant organizations, we interpret optimal investment as a strategy that maximizes the ROI on the premise that we help more students attend better colleges. So the problems to be solved are as follows:1.How to prioritize the schools by optimization level.2.How to measure ROI of a school.3.How to measure investment amount of a specific school.1.2Our ApproachWe offer a model of optimal investment which takes a great many factors in the College Scorecard Data Set into account. To begin with, we make a 1 to N optimized and prioritized candidate list of school we are recommending for investment by the AHP model. For the sake that we invest more students to better school, several factors are considered in the AHP model, such as SAT score, ACT score, etc. And then, we set investment amount of each university in the order of the list according to the standard of maximized ROI. The implement details of the model will be described in section 4.2.AssumptionsWe make the following basic assumptions in order to simplify the problem. And each of our assumptions is justified.1.Investment amount is mainly used for tuition and fees. Considering that theincome of an undergraduate is usually much higher than a high school students, we believe that it’s necessary to help more poor students have a chance to go to college.2.Bank rates will not change during the investment period. The variation ofthe bank rates have a little influence on the income we consider. So we make this assumption just to simplify the model.3.The employment rates and dropout rates will not change, and they aredifferent for different schools4.For return on investment, we only consider monetary income, regardlessof the intangible income.3.NotationsWe use a list of symbols for simplification of expression.4.The Optimal Investment ModelIn this section, we first prioritize schools by the AHP model (Section 4.1), and then calculate ROI value of the schools (Section 4.2). Ultimately, we identify investment amount of every candidate schools according to ROI (Section 4.3).4.1Analytic Hierarchy Process ModelIn order to prioritize schools, we must consider each necessary factor in the College Scorecard Data Set. For each factor, we calculate its weight value. And then, we can identify the investment necessity of each school. So, the model can be developed in 3 steps as follows:4.1.1Constructing the HierarchyWe consider 19 elements to measure priority of candidate schools, which can be seen in Fig 1. The hierarchy could be diagrammed as follows:Fig.1AHP for the investment decisionThe goal is red, the criteria are green and the alternatives are blue. All the alternatives are shown below the lowest level of each criterion. Later in the process, each alternatives will be rated with respect to the criterion directly above it.As they build their hierarchy, we should investigate the values or measurements of the different elements that make it up. If there are published fiscal policy, for example, or school policy, they should be gathered as part of the process. This information will be needed later, when the criteria and alternatives are evaluated.Note that the structure of the investment hierarchy might be different for other foundations. It would definitely be different for a foundation who doesn't care how much his score is, knows he will never dropout, and is intensely interested in math, history, and the numerous aspects of study[1].4.1.2Constructing the Judgement MatrixHierarchy reflects the relationship among elements to consider, but elements in the Criteria Layer don’t always weigh equal during aim measure. In deciders’ mind, each element accounts for a particular proportion.To incorporate their judgments about the various elements in the hierarchy, decision makers compare the elements “two by two”. The fundamental scale for pairwise comparison are shown in Fig 2.Fig 2Right now, let's see which items are compared. Our example will begin with the six criteria in the second row of the hierarchy in Fig 1, though we could begin elsewhere if we want. The criteria will be compared as to how important they are to the decisionmakers, with respect to the goal. Each pair of items in this row will be compared.Fig 3 Investment Judgement MatrixIn the next row, there is a group of 19 alternatives under the criterion. In the subgroup, each pair of alternatives will be compared regarding their importance with respect to the criterion. (As always, their importance is judged by the decision makers.) In the subgroup, there is only one pair of alternatives. They are compared as to how important they are with respect to the criterion.Things change a bit when we get to the alternatives row. Here, the factor in each group of alternatives are compared pair-by-pair with respect to the covering criterion of the group, which is the node directly above them in the hierarchy. What we are doing here is evaluating the models under consideration with respect to score, then with respect to Income, then expenditure, dropout rate, debt and graduation rate.The foundation can evaluate alternatives against their covering criteria in any order they choose. In this case, they choose the order of decreasing priority of the covering criteria.Fig 4 Score Judgement MatrixFig 5 Expenditure Judgement MatrixFig 6 Income Judgement MatrixFig 7 Dropout Judgement MatrixFig 8 Debt Judgement MatrixFig 9 Graduation Matrix4.1.3 Hierarchical RankingWhen the pairwise comparisons are as numerous as those in our example, specialized AHP software can help in making them quickly and efficiently. We will assume that the foundation has access to such software, and that it allows the opinions of various foundations to be combined into an overall opinion for the group.The AHP software uses mathematical calculations to convert these judgments to priorities for each of the six criteria. The details of the calculations are beyond the scope of this article, but are readily available elsewhere[2][3][4][5]. The software also calculates a consistency ratio that expresses the internal consistency of the judgments that have been entered. In this case the judgments showed acceptable consistency, and the software used the foundation’s inputs to assign these new priorities to the criteria:Fig 10.AHP hierarchy for the foundation investing decision.In the end, the AHP software arranges and totals the global priorities for each of the alternatives. Their grand total is 1.000, which is identical to the priority of the goal. Each alternative has a global priority corresponding to its "fit" to all the foundation's judgments about all those aspects of factor. Here is a summary of the global priorities of the alternatives:Fig 114.2 ROI Model4.2.1 Overview of the investment strategyConsider a foundation making investment on a set of N geographically dispersed colleges and university in the United States, D = {1, 2, 3……N }. Then we can select top N schools from the candidate list which has been sorted through analytic hierarchy process. The total investment amount is M per year which is donated by the Goodgrant Foundation. The investment amount is j m for each school j D ∈, satisfying the following balance constraint:j j D mM ∈=∑ (1)W e can’t invest too much or too little money to one school because we want to help more students go to college, and the student should have more choices. Then the investment amount for each school must have a lower limit lu and upper limit bu as follows:j lu m bu ≤≤ (2)The tuition and fees is j p , and the time duration is {1,2,3,4}j t ∈. To simplify ourmodel, we assume that our investment amount is only used for freshmen every year. Because a freshmen oriented investment can get more benefits compared with others. For each school j D ∈, the number of the undergraduate students who will be invested is j n , which can be calculated by the following formula ：,jj j j m n j D p t =∈⨯ (3)Figure12The foundation can use the ROI model to identify j m and j t so that it canmaximize the total net income. Figure1 has shown the overview of our investment model. We will then illustrate the principle and solution of this model by a kind of nonlinear programming method.4.2.2 Analysis of net income and investment costIn our return on investment model, we first focus on analysis of net income and investment cost. Obviously, the future earnings of undergraduate students are not only due to the investment itself. There are many meaning factors such as the effort, the money from their parents, the training from their companies. In order to simplify the model, we assume that the investment cost is the most important element and we don’t consider other possible influence factors. Then we can conclude that the total cost of the investment is j m for each school j D ∈.Figure 13For a single student, the meaning of the investment benefits is the expected earnings in the future. Assuming that the student is not going to college or university after graduating from high school and is directly going to work. Then his wage base is 0b as a high school graduate. If he works as a college graduate, then his wage base is 0a . Then we can give the future proceeds of life which is represented symbolically by T and we use r to represent the bank rates which will change over time. We assume that the bank rates will not change during the investment period. Here, we use bank rates in 2016 to represent the r . The future proceeds of life of a single undergraduate student will be different due to individual differences such as age, physical condition environment, etc. If we consider these differences, the calculation process will be complicated. For simplicity’s sake, we uniform the future proceeds of life T for 20 years. Then we will give two economics formulas to calculate the total expected income in the next T years for graduates and high school graduates:40(1)Tk k a u r +==+∑(4) 40(1)T kk b h r +==+∑(5) The total expected income of a graduate is u , and the total expected income of a highschool graduate is h .Then, we continue to analyze the net income. The net income can be calculated by the following formula:os NetIncome TotalIncome C t =- (6) For each school j D ∈, the net income is j P , the total income is j Q , and the cost is j m . Then we will get the following equation through formula (6):j j j P Q m =- (7)Therefore, the key of the problem is how to calculate j Q . In order to calculate j Q, weneed to estimate the number of future employment j ne . The total number of the invested is j n , which has been calculated above. Considering the dropout rates j α and the employment rates j β for each school j , we can calculate the number of future employment j ne through the following formula:(4)(1)jt j j j j n e n βα-=⨯⨯- (8)That way, we can calculate j Q by the following formula:()j j Q ne u h =⨯- (9)Finally, we take Eq. (2) (3) (4) (7) (8) into Eq. (6), and we will obtain Eq. (9) as follows:4(4)00400(1)()(1)(1)j TT t j j j j j k kk k j jm a b P m p t r r βα+-+===⨯⨯-⨯--⨯++∑∑ (10) We next reformulate the above equation of j P for concise presentation:(4)(1)j t j jj j j jc m P m t λα-⨯⨯=⨯-- (11)where jj j p βλ= and 400400(1)(1)TT k kk k a b c r r ++===-++∑∑ .4.2.3 Calculate Return On InvestmentROI is short of return on investment which can be determined by net income andinvestment cost [7]. It conveys the meaning of the financial assessment. For each schoolj D ∈ , the net income is j P , and the investment cost equals to j m . Then the j ROIcan be calculated by the following formula:100%j j jP ROI m =⨯ (12)We substitute Eq. (10) into Eq. (11), and we will get a new formula as follows:(4)((1)1)100%j t j j j jc ROI t λα-⨯=⨯--⨯ (13)4.2.4 Maximize the Total Net IncomeGiven the net income of each school, we formulate the portfolio problem that maximize the total net income, S=Max(4)((1))j t j jj j j j Dj Djc m P m t λα-∈∈⨯⨯=⨯--∑∑ (14)S. T.jj DmM ∈=∑,{1,2,3,4}t = ,j lu m bu ≤≤ ,Considering the constraint jj DmM ∈=∑, we can further simplify the model,S is equivalent to S’=Max(4)((1))j t j jj j j Dj Djc m P t λα-∈∈⨯⨯=⨯-∑∑ (15)S. T.jj DmM ∈=∑,{1,2,3,4t = ,j l u m b u ≤≤. By solving the nonlinear programming problem S’, we can get the sameanswer as problem S.5. Testing the Model 5.1 Error AnalysisSince the advent of analytic hierarchy process, people pay more attention to it due to the specific applicability, convenience, practicability and systematization of the method. Analytic hierarchy process has not reached the ideal situation whether in theory or application level because the results depend largely on the preference and subjective judgment. In this part, we will analyze the human error problem in analytic hierarchy process.Human error is mainly caused by human factors. The human error mainly reflects on the structure of the judgment matrix. The causes of the error are the following points:1. The number of times that human judge the factors’ importance is excessive.2. The calibration method is not perfect.Then we will give some methods to reduce errors:1. Reduce times of human judgment. One person repeatedly gave the samejudgment between two factors. Or many persons gave the same judgment between two factors one time. Finally, we take the average as result.2. Break the original calibration method. If we have defined the ranking vector111121(,...)n a a a a =between the factor 1A with others. Then we can get all theother ranking vector. For example : 12122111(,1...)na a a a a =.5.2 Stability AnalysisIt is necessary to analyze the stability of ranking result [6], because the strong subjectivefactors. If the ranking result changed a little while the judgment changed a lot, we can conclude that the method is effective and the result is acceptable, and vice versa. We assume that the weight of other factors will change if the weight of one factor changed from i ξ to i η:[8](1)(,1,2...,)(1)i j j i i j n i j ηξηξ-⨯==≠- (16)And it is simple to verify the equation:11nii η==∑ (17)And the new ranking vector ω will be:A ωη=⨯ (18)By this method, the Relative importance between other factors remain the same while one of the factor has changed.6. Results6.1 Results of Analytic Hierarchy ProcessWe can ranking colleges through the analytic hierarchy process, and we can get the top N = 20 schools as follows6.2 Results of Return On Investment ModelBased on the results above, we next use ROI model to distribute investment amountj m and time duration j t for each school j D ∈ by solving the following problem:Max (4)((1))j t j jj j j Dj Djc m P t λα-∈∈⨯⨯=⨯-∑∑S. T.jj DmM ∈=∑,{1,2,3,4t = , j l u m b u≤≤ . In order to solve the problem above, we collected the data from different sources. Inthe end, we solve the model with Lingo software. The program code is as follows:model: sets:roi/1..20/:a,b,p,m,t;endsets data:a = 0.9642 0.9250 0.9484 0.9422 0.9402 0.9498 0.90490.9263 0.9769 0.9553 0.9351 0.9123 0.9410 0.98610.9790 0.9640 0.8644 0.9598 0.9659 0.9720;b = 0.8024 0.7339 0.8737 0.8308 0.8681 0.7998 0.74920.6050 0.8342 0.8217 0.8940 0.8873 0.8495 0.87520.8333 0.8604 0.8176 0.8916 0.7527 0.8659;p = 3.3484 3.7971 3.3070 3.3386 3.3371 3.4956 3.22204.0306 2.8544 3.1503 3.2986 3.3087 3.3419 2.78452.9597 2.92713.3742 2.7801 2.5667 2.8058;c = 49.5528;enddatamax=@sum(roi(I):m(I)/t(I)/p(I)*((1-b(I))^4)*c*(1-a(I)+0.05)^(4-t(I)));@for(roi:@gin(t));@for(roi(I):@bnd(1,t(I),4));@for(roi(I):@bnd(0,m(I),100));@sum(roi(I):m(I))=1000;ENDFinally, we can get the investment amount and time duration distribution as follows:7.Strengths and Weaknesses7.1Strengths1.Fixing the bank rates during the investment period may run out, but it will haveonly marginal influences.2.For return on investment, we only consider monetary income, regardless of the3.intangible income. But the quantization of these intangible income is very importantand difficult. It needs to do too much complicated technical analysis and to quantify 4.too many variables. Considering that the investment persists for a short time, thiskind of random error is acceptable.5.Due to our investment which is freshmen oriented, other students may feel unfair.It is likely to produce adverse reaction to our investment strategy.6.The cost estimation is not impeccable. We only consider the investment amount andignore other non-monetary investment.5. AHP needs higher requirements for personnel quality.7.2Weaknesses1.Our investment strategy is distinct and clear, and it is convenient to implement.2.Our model not only identifies the investment amount for each school, but alsoidentifies the time duration that the organization’s money should be provide d.3.Data processing is convenient, because the most data we use is constant, average ormedian.4.Data sources are reliable. Our investment strategy is based on some meaningful anddefendable subset of two data sets.5.AHP is more simple, effective and universal.References[1] Saaty, Thomas L. (2008). Decision Making for Leaders: The Analytic Hierarchy Process for Decisions in a Complex World. Pittsburgh, Pennsylvania: RWS Publications. ISBN 0-9620317-8-X.[2] Bhushan, Navneet, Kanwal Rai (January 2004). Strategic Decision Making: Applying the Analytic Hierarchy Process. London: Springer-Verlag. ISBN 1-8523375-6-7.[3] Saaty, Thomas L. (2001). Fundamentals of Decision Making and Priority Theory. Pittsburgh, Pennsylvania: RWS Publications. ISBN 0-9620317-6-3.[4] Trick, Michael A. (1996-11-23). "Analytic Hierarchy Process". Class Notes. Carnegie Mellon University Tepper School of Business. Retrieved 2008-03-02.[5] Meixner, Oliver; Reiner Haas (2002). Computergestützte Entscheidungs-findung: Expert Choice und AHP – innovative Werkzeuge zur Lösung komplexer Probleme (in German). Frankfurt/Wien: Redline Wirtschaft bei Ueberreuter. ISBN 3-8323-0909-8.[6] Hazelkorn, E. The Impact of League Tables and Ranking System on Higher Education Decision Making [J]. Higher Education Management and Policy, 2007, 19(2), 87-110.[7] Leslie: Trainer Assessment: A Guide to Measuring the Performance of Trainers and Facilitors, Second Edition, Gower Publishing Limited, 2002.[8] Aguaron J, Moreno-Jimenea J M. Local stability intervals in the analytic hierarchy process. European Journal of Operational Research. 2000Letter to the Chief Financial Officer, Mr. Alpha Chiang. February 1th, 2016.I am writing this letter to introduce our optimal investment strategy. Before I describe our model, I want to discuss our proposed concept of a return-on-investment (ROI). And then I will describe the optimal investment model by construct two sub-model, namely AHP model and ROI model. Finally, the major results of the model simulation will be showed up to you.Considering that the Goodgrant Foundation aims to help improve educational performance of undergraduates attending colleges and universities in the US, we interpret return-on-investment as the increased income of undergraduates. Because the income of an undergraduate is generally much higher than a high school graduate, we suggest all the investment be used to pay for the tuition and fees. In that case, if we take both the income of undergraduates’ income and dropout rate into account, we can get the return-in-investment value.Our model begins with the production of an optimized and prioritized candidate list of schools you are recommending for investment. This sorted list of school is constructed through the use of specification that you would be fully qualified to provided, such as the score of school, the income of graduate student, the dropout rate, etc. With this information, a precise investment list of schools will be produced for donation select.Furthermore, we developed the second sub-model, ROI model, which identifies the investment amount of each school per year. If we invest more money in a school, more students will have a chance to go to college. However, there is an optimal investment amount of specific school because of the existence of dropout. So, we can identify every candidate school’s in vestment amount by solve a nonlinear programming problem. Ultimately, the result of the model simulation show that Washington University, New York University and Boston College are three schools that worth investing most. And detailed simulation can be seen in our MCM Contest article.We hope that this model is sufficient in meeting your needs in any further donation and future philanthropic educational investments within the United States.。

摘要：第一段：写论文解决什么问题1．问题的重述a. 介绍重点词开头：例1：“Hand move” irrigation, a cheap but labor-intensive system used on small farms, consists of a movable pipe with sprinkler on top that can be attached to a stationary main.例2:……is a real-life common phenomenon with many complexi t ies.例3：An (effective plan) is crucial to………b. 直接指出问题：例1：We find the optimal number of tollbooths in a highway toll-plaza for a given number of highway lanes: the number of tollbooths that minimizes average delay experienced by cars.例2：A brand-new university needs to balance the cost of information technology security measures wi t h the potential cost of attacks on its systems.例3：We determine the number of sprinklers to use by analyzing the energy and motion of water in the pipe and examining the engineering parameters of sprinklers available in the market.例4: After mathematically analyzing the …… problem, our modeling group would like to present our conclusions, strategies, (and recommendations )to the …….例5：Our goal is... that (mini mizes the time )……….2．解决这个问题的伟大意义反面说明。

Tittle of paperSummary/Abstract Key words:I.Introduction（引言）Organ transplantation is a preferable treatment for the most serious forms of end-stage diseases. In recent years, advances in medical science and technology have made solid organ transplantation an increasingly successful and common medical procedure, a literal ''second chance at life". Not only does it offer the best hope for complete rehabilitation, but it has also proved to be the most cost-effective of all treatment options, including dialysis. Consequently, more and more people are benefiting from organ transplants and their survival rates are steadily improving. The surgical techniques involved have been mastered for half a century and are now considered as routine. The two main sources of kidneys for transplantation are deceased-donor kidneys and live-donations from family and friends. However, unfortunately, there is a considerable shortage of donor organs, compared to demands. As a matter of fact, efficient matching and allocation of organs donated has become an exigent problem.The United Network for Organ Sharing (UNOS), as the operator of the Organ Procurement and Transplantation Network (OPTN), is responsible for transplant organ distribution in the United States. UNOS oversees the allocation of many different types of transplants, including liver, kidney, pancreas, heart, lung, and cornea.Focusing on kidney transplantation, based on UNOS Kidney Allocation Model, we develop a mathematical model for US transplant networks. First, incomingorgans are matched with waiting candidates by medical institutions considering the factors as ABO blood compatibility, the degree of recipient major HLA mismatch in order to obtain a matching degree applied on the allocation part. After that, from the patients’perspective, on the basis of linear regression, priority weight is defined by pondering age, disease severity, time on waiting, PRA level, and region. Applying this mechanism of ranking, we realize MWBM (Maximum Weight Bipartite-graph Matching) and SMGS (Stable Matching based on Gale-Shapley algorithm). MWBM focuses on the optimal assignment of donors following the algorithm of bipartite-graph maximum weight matching; SMGS emphasizes the process of exchanges in order to obtain the stable exchanges between donors and candidates on the waiting list.II.The Description of Problem（问题重述)III.Basic Assumptions●The level of mismatch is only relative to the number of antigens.●The data and information are accurately registered according to the medicalmeasures●The data and information are refreshed in time according to the status of thepatients●No differences in the quality of the donor kidneys●The quality of the donor kidney is constantIV.Definitions and Notations●Kidney transplantation: A kidney transplant is a surgical procedure to implant ahealthy kidney into a patient with kidney failure.●Prioritization●MD: Matching Degree●PW: Prioritization weight●MWSM: Maximum Weight Bipartite Matching●SMGS: Stable Matching based on Gale-Shapley algorithm或V.ModelsThrough the investigation of US transplantation network, we draw a general picture of the mechanism. With reference to some resources available on the website of UNOS, a flow chart (Figure 1) is developed showing the procedure of the network.Currently, the initial waiting list is composed of patients who are waiting for a kidney or combined kidney-pancreas transplant. For the first time, the patients arerequested to show the correct and scientific information to the US kidney transplant network which is needed for donor-recipient matching, the ranking of patients on the waiting list, and determining the outcome of those transplanted. The patients’waiting lists are composed of initial patients, historical patients and unsuccessful recipient after transplantation. Historical patients refer to registered patients whose status have changed and have an influence on the procedure. A patient is taken off the waiting list when a graft is offered and accepted by that patient or the patient is dead while waiting for a transplant. Unsuccessful recipients refer to the patients who have a bad result of transplantation calling for transplantation again, as it is so-called relistFigure 1. A schematic depicting the steps occurring in the transplantation networks......Table 1.Survival rate involving HLA mismatchVI.Conclusions.Our model for the optimal allocation of the donor organs is established by three modules, procurement of MD and PW, optimal assignment by MWBM model and Stable Matching of Gale-Shapley algorithm. The model has offered a convincing procedure of the allocation with the ……VII.Strengths and weaknesses(模型优缺点)Strengths●……Weaknesses●VIII.References注意文献的积累，不要等到文章写完再去重新寻找文献。

MCM 2015 Summary Sheet for Team 35565For office use onlyT1________________ T2________________ T3________________ T4________________Team Control Number35565Problem ChosenBFor office use onlyF1________________F2________________F3________________F4________________ SummaryThe lost MH370 urges us to build a universal search plan to assist searchers to locate the lost plane effi-ciently and optimize the arrangement of search plans.For the location of the search area, we divided it into two stages, respectively, to locate the splash point and the wreckage‟s sunk point. In the first stage, we consider the types of crashed aircraft, its motion and different position out of contact. We also consider the Earth‟s rotation, and other factors. Taking all these into account, we establish a model to locate the splash point. Then we apply this model to MH370. we can get the splash point in the open water is 6.813°N 103.49°E and the falling time is 52.4s. In the second stage, considering resistances of the wreckage in different shapes and its distribution affected by ocean currents, we establish a wreckage sunk point model to calculate the horizontal displacement and the angle deviation affected by the ocean currents. The result is 1517m and 0.11°respectively. Next, we extract a satellite map of submarine topography and use MATLAB to depict seabed topography map, determining the settlement of the wreckage by using dichotomy algorithm under different terrains. Finally, we build a Bayesian model and calculate the weight of corresponding area, sending aircrafts to obtain new evidence and refresh suspected wreckage area.For the assignment of the search planes, we divide it into two stages, respectively, to determine the num-ber of the aircraft and the assignment scheme of the search aircraft. In the first stage, we consider the search ability of each plane and other factors. And then we establish global optimization model. Next we use Dinkelbach algorithm to select the best n search aircrafts from all search aircrafts. In the second stage, we divide the assignment into two cases whether there are search aircrafts in the target area. If there is no search aircraft, we take the search area as an arbitrary polygon and establish the subdivision model. Considering the searching ability of each plane, we divide n small polygons into 2n sub-polygons by using NonconvexDivide algorithm, which assigns specific anchor points to these 2n sub-polygons re-spectively. If there exist search aircrafts, we divide the search area into several polygons with the search aircrafts being at the boundary of the small polygons. To improve search efficiency, we introduce” ma x-imize the minimum angle strategy” to maximize right-angle subdivision so that we can reduce the turning times of search aircraft. When we changed the speed of the crashed plane about 36m/s, the latitude of the splash point changes about 1°.When a wreck landing at 5.888m out from the initial zone, it will divorce from suspected searching area, which means our models are fairly robust to the changes in parameters. Our model is able to efficiently deal with existing data and modify some parameters basing the practical situation. The model has better versatility and stability. The weakness of our model is neglect of human factors, the search time and other uncontrollable factors that could lead to deviation compared to practical data. Therefore, we make some in-depth discussions about the model, modifying assumptions establish-Searching For a Lost PlaneControl#35565February 10, 2014Team # 35565 Page 3 of 47 Contents1 Introduction (5)1.1 Restatement of the Problem (5)1.2 Literature Review (6)2 Assumptions and Justifications (7)3 Notations (7)4 Model Overview (10)5 Modeling For Locating the Lost Plane (10)5.1 Modeling For Locating the Splash Poin t (11)5.1.1 Types of Planes (11)5.1.2 Preparation of the Model—Earth Rotation (12)5.1.3 Modeling (13)5.1.4 Solution of The Model (14)5.2 Modeling For Locating Wreckage (15)5.2.1 Assumptions of the Model (16)5.2.2 Preparation of the Model (16)5.2.3 Modeling (21)5.2.4 Solution of the Model (25)5.3 Verification of the Model (26)5.3.1 Verification of the Splash Point (26)5.3.2 Verification of the binary search algorithm (27)6 Modeling For Optimization of Search Plan (29)6.1 The Global Optimization Model (29)6.1.1 Preparation of the Model (29)6.1.2 Modeling (31)6.1.3 Solution of the Model (31)6.2 The Area Partition Algorithm (33)6.2.1 Preparation of the Model (33)6.2.2 Modeling (34)6.2.3 Solution of the Model (35)6.2.4 Improvement of the Model (36)7 Sensitivity Analysis (38)8 Further Discussions (39)9 Strengths and Weaknesses (41)9.1 Strengths (41)9.2 Weaknesses (42)10 Non-technical Paper (42)1 IntroductionAn airplane (informally plane) is a powered, fixed-wing aircraft that is propelled for-ward by thrust from a jet engine or propeller. Its main feature is fast and safe. Typi-cally, air travel is approximately 10 times safer than travel by car, rail or bus. Howev-er, when using the deaths per journey statistic, air travel is significantly more danger-ous than car, rail, or bus travel. In an aircraft crash, almost no one could survive [1]. Furthermore, the wreckage of the lost plane is difficult to find due to the crash site may be in the open ocean or other rough terrain.Thus, it will be exhilarating if we can design a model that can find the lost plane quickly. In this paper, we establish several models to find the lost plane in seawater and develop an op-timal scheme to assign search planes to model to locate the wreckage of the lost plane.1.1 Restatement of the ProblemWe are required to build a mathematical model to find the lost plane crashed in open water. We decompose the problem into three sub-problems:●Work out the position and distributions of the plane‟s wreckage●Arrange a mathematical scheme to schedule searching planesIn the first step, we seek to build a model with the inputs of altitude and other factors to locate the splash point on the sea-level. Most importantly, the model should reflect the process of the given plane. Then we can change the inputs to do some simulations. Also we can change the mechanism to apply other plane crash to our model. Finally, we can obtain the outputs of our model.In the second step, we seek to extend our model to simulate distribution of the plane wreckage and position the final point of the lost plane in the sea. We will consider more realistic factors such as ocean currents, characteristics of plane.We will design some rules to dispatch search planes to confirm the wreckage and de-cide which rule is the best.Then we attempt to adjust our model and apply it to lost planes like MH370. We also consider some further discussion of our model.1.2 Literature ReviewA model for searching the lost plane is inevitable to study the crashed point of the plane and develop a best scheme to assign search planes.According to Newton's second law, the simple types of projectile motion model can work out the splash point on the seafloor. We will analyze the motion state ofthe plane when it arrives at the seafloor considering the effect of the earth's rotation,After the types of projectile motion model was established, several scientists were devoted to finding a method to simulate the movement of wreckage. The main diffi-culty was to combine natural factors with the movement. Juan Santos-Echeandía introduced a differential equation model to simplify the difficulty [2]. Moreover,A. Boultif and D. Louër introduced a dichotomy iteration algorithm to circular compu-ting which can be borrowed to combine the motion of wreckage with underwater ter-rain [3]. Several conditions have to be fulfilled before simulating the movement: (1) Seawater density keeps unchanged despite the seawater depth. (2) The velocity of the wreck stay the same compared with velocity of the plane before it crashes into pieces.(3) Marine life will not affect our simulation. (4) Acting forceof seawater is a function of the speed of ocean currents.However the conclusion above cannot describe the wreckage zone accurately. This inaccuracy results from simplified conditions and ignoring the probability distribution of wreckage. In 1989, Stone et.al introduced a Bayesian search approach for searching problems and found the efficient search plans that maximize the probability of finding the target given a fixed time limit by maintaining an accurate target location probabil-ity density function, and by explicitly modeling the target‟s process model [4].To come up with a concrete dispatch plan. Xing Shenwei first simulated the model with different kinds of algorithm. [5] In his model, different searching planes are as-sessed by several key factors. Then based on the model established before, he use the global optimization model and an area partition algorithm to propose the number of aircrafts. He also arranged quantitative searching recourses according to the maxi-mum speed and other factors. The result shows that search operations can be ensured and effective.Further studies are carried out based on the comparison between model andreality.Some article illustrate the random error caused by assumptions.2 Assumptions and JustificationsTo simplify the problem, we make the following basic assumptions, each ofwhich is properly justified.●Utilized data is accuracy. A common modeling assumption.●We ignore the change of the gravitational acceleration. The altitude of anaircraft is less than 30 km [6]. The average radius of the earth is 6731.004km, which is much more than the altitude of an aircraft. The gravitational accele-ration changes weakly.●We assume that aeroengine do not work when a plane is out of contact.Most air crash resulted from engine failure caused by aircraft fault, bad weather, etc.●In our model, the angle of attack do not change in an air crash and thefuselage don’t wag from side to side. We neglect the impact of natural and human factors●We treat plane as a material point the moment it hit the sea-level. Thecrashing plane moves fast with a short time-frame to get into the water. The shape and volume will be negligible.●We assume that coefficient of air friction is a constant. This impact is neg-ligible compared with that of the gravity.●Planes will crash into wreckage instantly when falling to sea surface.Typically planes travel at highly speed and may happen explosion accident with water. So we ignore the short time.3 NotationsAll the variables and constants used in this paper are listed in Table 1 and Table 2.Table 1 Symbol Table–ConstantsSymbol DefinitionωRotational angular velocity of the earthg Gravitational accelerationr The average radius of the earthC D Coefficient of resistance decided by the angle of attack ρAtmospheric densityφLatitude of the lost contact pointμCoefficient of viscosityS0Area of the initial wrecking zoneS Area of the wrecking zoneS T Area of the searching zoneK Correction factorTable 2 Symbol Table-VariablesSymbol DefinitionF r Air frictionF g Inertial centrifugal forceF k Coriolis forceW Angular velocity of the crash planev r Relative velocity of the crash planev x Initial velocity of the surface layer of ocean currentsk Coefficient of fluid frictionF f Buoyancy of the wreckagef i Churning resistance of the wreckage from ocean currents f Fluid resistance opposite to the direction of motionG Gravity of the wreckageV Volume of the wreckageh Decent height of the wreckageH Marine depthS x Displacement of the wreckageS y Horizontal distance of S xα Deviation angle of factually final position of the wreckage s Horizontal distance between final point and splash point p Probability of a wreck in a given pointN The number of the searching planeTS ' The area of sea to be searched a i V ˆ The maximum speed of each planeai D The initial distance from sea to search planeai A The search ability of each plane is),(h T L i The maximum battery life of each plane isi L The mobilized times of each plane in the whole search )1(N Q Q a a ≤≤ The maximum number of search plane in the searching zone T(h) The time the whole action takes4 Model OverviewMost research for searching the lost plane can be classified as academic and practical. As practical methods are difficult to apply to our problem, we approach theproblem with academic techniques. Our study into the searching of the lost plane takes several approaches.Our basic model allows us to obtain the splash point of the lost plane. We focus on the force analysis of the plane. Then we We turn to simple types of projectile motion model. This model gives us critical data about the movement and serves as a stepping stone to our later study.The extended model views the problem based on the conclusion above. We run diffe-rential equation method and Bayesian search model to simulate the movement of wreckage. The essence of the model is the way to combine the effect of natural factors with distribution of the wreckage. Moreover, using distributing conditions, we treat size of the lost plane as “initial wreckage zone” so as to approximately describe the distribution. Thus, after considering the natural factors, we name the distribution of wreckage a “wreck zone” to minimize searching zone. While we name all the space needed to search “searching zone”.Our conclusive model containing several kinds of algorithm attempts to tackle a more realistic and more challenging problem. We add the global optimization model and an area partition algorithm to improve the efficiency of search aircrafts according to the area of search zone. An assessment of search planes consisting of search capabili-ties and other factors are also added. The Dinkelbach and NonConvexDivide algo-rithm for the solutions of the results are also added.We use the extended and conclusive model as a standard model to analyze the problem and all results have this two model at their cores.5 Modeling For Locating the Lost PlaneWe will start with the idea of the basic model. Then we present the Bayesian search model to get the position of the sinking point.5.1 Modeling For Locating the Splash PointThe basic model is a academic approach. A typical types of projectile behavior con-sists of horizontal and vertical motion. We also add another dimension consider-ing the effect of the earth's rotation. Among these actions, the force analysis is the most crucial part during descent from the point out of contact to the sea-level. Types of plane might impact trajectory of the crashing plane.5.1.1 Types of PlanesWe classify the planes into six groups [7]:●Helicopters: A helicopter is one of the most timesaving ways to transfer be-tween the city and airport, alternatively an easy way to reach remote destina-tions.●Twins Pistons: An economical aircraft range suitable for short distance flights.Aircraft seating capacity ranging from 3 to 8 passengers.●Turboprops: A wide range of aircraft suitable for short and medium distanceflights with a duration of up to 2-4 hours. Aircraft seating capacity ranging from 4 to 70 passengers.●Executive Jets:An Executive Jet is suitable for medium or long distanceflights. Aircraft seating capacity ranging from 4 to 16 passengers●Airliners:Large jet aircraft suitable for all kinds of flights. Aircraft seatingcapacity ranging from 50 to 400 passengers.●Cargo Aircrafts:Any type of cargo. Ranging from short notice flights carry-ing vital spare parts up to large cargo aircraft that can transport any volumin-ous goods.The lost plane may be one of these group. Then we extract the characteristics of planes into three essential factors: mass, maximum flying speed, volume. We use these three factors to abstract a variety of planes:●Mass: Planes of different product models have their own mass.●Maximum flying speed: Different planes are provided with kinds of me-chanical configuration, which will decide their properties such as flying speed.●Volume: Planes of distinct product models have different sizes and configura-tion, so the volume is definitive .5.1.2 Preparation of the Model —Earth RotationWhen considering the earth rotation, we should know that earth is a non-inertial run-ning system. Thus, mobile on the earth suffers two other non-inertial forces except air friction F r . They are inertial centrifugal force F g and Coriolis force F k . According to Newton ‟s second law of motion, the law of object relative motion to the earth is:Rotational angular velocity of the earth is very small, about .For a big mobile v r , it suffers far less inertial centrifugal force than Coriolis force, so we can ignore it. Thus, the equation can be approximated as follows:Now we establish a coordinate system: x axis z axis pointing to the east and south re-spectively, y axis vertical upward, then v r , ω and F r in the projection coordinate system are as follows:⎪⎪⎩⎪⎪⎨⎧++=⋅⋅-⋅⋅=++=kdt dz j dt dy i dt dx m v k j w kF j F i F F r rz ry rx r φωφωcos sinφis the latitude of the lost contact point of the lost plane. Put equation 1-3 and equa-tion 1-2 together, then the component of projectile movement in differential equation is:ma FF F k g r=++srad ⋅⨯=-5103.7ωmamv F r r =+ω2⎪⎪⎪⎩⎪⎪⎪⎨⎧+⋅=+⋅=+⎪⎭⎫ ⎝⎛+⋅-=m F dt dx w dt z d m F dt dx w dt y d m F dt dz dt dy w dtx d rz ry rx φφφφsin 2cos 2sin cos 22222225.1.3 ModelingConsidering the effect caused by earth rotation and air draught to plane when crashing to sea level, we analyze the force on the X axis by using Newton ‟s second law, the differential equation on x y and axis, we can conclude:In conclusion, we establish the earth rotation and types of projectile second order dif-ferential model:()⎪⎩⎪⎨⎧+-⋅'⋅⋅=''-⋅'+⋅'⋅⋅-=''-⋅'⋅⋅=''m gf y w m z m f z x w m y m f y w m x m obj 321cos 2cos sin 2sin 2.φφφφAccording to Coriolis theorem, we analyze the force of the plane on different direc-tions. By using the Newton ‟s laws of motion, we can work out the resultant accelera-tion on all directions:⎪⎪⎪⎪⎪⎪⎪⎪⎩⎪⎪⎪⎪⎪⎪⎪⎪⎨⎧'+'+'⋅'⋅⋅⨯+='+'+'⋅'⋅⋅⨯+='+'+'⋅'⋅⋅⨯+=⋅⋅-⋅⋅=⋅⨯=⋅'''⋅===-2222222225)()()(21)()()(21)()()(21cos sin 103.704.022z y x z c F f z y x y c F f z y x x c F f k j w s rad S y x F c D rz D ryD rx D ρρρφωφωωμφC D is the angle of attack of a plane flew in the best state, w is the angular speed of a moving object, vector j and k are the unit vector on y and z direction respectively,μisrx F y w m x m -⋅'⋅⋅⨯=''φsin 2()ry F z x w m y m -'+⋅'⋅⨯-=''φφcos sin 2mg F y w m z m rz +-⋅'⋅⋅⋅=''φcos 2the coefficient of viscosity of the object.5.1.4 Solution of the ModelWhen air flows through an object, only the air close to layer on the surface of the ob-ject in the laminar airflow is larger, whose air viscosity performance is more noticea-ble while the outer region has negligible viscous force [8]. Typically, to simplify cal-culation, we ignore the viscous force produced by plane surface caused by air resis-tance.Step 1: the examination of dimension in modelTo verify the validity of the model based on Newton ‟s second theorem, first, we standardize them respectively, turn them into the standardization of dimensionless data to diminish the influence of dimensional data. The standard equation is:Step 2: the confirmation of initial conditionsIn a space coordinate origin based on plane, we assume the earth's rotation direc-tion for the x axis, the plane's flight heading as y axis, the vertical downward di-rection for z axis. Space coordinate system are as follows:Figure 1 Space coordinate systemStep 3: the simplification and solutionAfter twice integrations of the model, ignoring some of the dimensionless in thesxx y i -=integral process, we can simplify the model and get the following:⎪⎪⎪⎩⎪⎪⎪⎨⎧+'⋅⋅⋅-⋅'⋅⨯='''-⋅⋅⋅-⋅'⋅⨯-=''⋅'⋅⨯=''g z m s c y w z y v m s c z w y y w x D D 220)(2cos 2)(2cos 2sin 2ρφρφφWe can calculate the corresponding xyz by putting in specific data to get the in-formation about the point of losing contact.Step 4: the solution of the coordinateThe distance of every latitude on the same longitude is 111km and the distance ofevery longitude on the same latitude is 111*cos (the latitude of this point) (km). Moreover, the latitude distance of two points on the same longitude is r ×cos(a ×pi/180) and the longitude distance of two points on the same latitude is: r ×sin(a ×pi/180)[9].We assume a as the clockwise angle starting with the due north direction and r as the distance between two points; X 、Y are the latitude and longitude coordinates of the known point P respectively; Lon , Lat are the latitude and longitude coordi-nates of the unknown point B respectively.Therefore, the longitude and latitude coordinates of the unknown point Q is:⎪⎪⎩⎪⎪⎨⎧⨯⨯+=⨯⨯⨯⨯+=111)180/cos()180/cos(111)180/sin(pi a r Y Lat pi Y pi a r X LonThus, we can get coordinates of the point of splash by putting in specific data.5.2 Modeling For Locating WreckageIn order to understand how the wreckage distributes in the sea, we have to understand the whole process beginning from the plane crashing into water to reaching the seaf-loor. One intuition for modeling the problem is to think of the ocean currents as astochastic process decided by water velocity. Therefore, we use a differential equation method to simulate the impact on wreckage from ocean currents.A Bayesian Searching model is a continuous model that computing a probability dis-tribution on the location of the wreckage (search object) in the presence of uncertain-ties and conflicting information that require the use of subjective probabilities. The model requires an initial searching zone and a set of the posterior distribution given failure of the search to plan the next increment of search. As the search proceeds, the subjective estimates of the detection will be more reliable.5.2.1 Assumptions of the ModelThe following general assumptions are made based on common sense and weuse them throughout our model.●Seawater density keeps unchanged despite the seawater depth.Seawater density is determined by water temperature, pressure, salinity etc.These factors are decided by or affected by the seawater density. Considering the falling height, the density changes slightly. To simplify the calculation, we consider it as a constant.●The velocity of the wreck stay the same compared with velocity of theplane before it crashes into pieces. The whole process will end quickly witha little loss of energy. Thus, we simplify the calculation.●Marine life will not affect our simulation.Most open coast habitats arefound in the deep ocean beyond the edge of the continental shelf, while the falling height of the plane cannot hit.●Acting force of seawater is a function of the speed and direction of oceancurrents. Ocean currents is a complicated element affected by temperature, wide direction, weather pattern etc. we focus on a short term of open sea.Acting force of seawater will not take this factors into consideration.5.2.2 Preparation of the Model●The resistance of objects of different shapes is different. Due to the continuityof the movement of the water, when faced with the surface of different shapes, the water will be diverted, resulting in the loss of partial energy. Thus the pressure of the surface of objects is changed. Based on this, we first consider the general object, and then revise the corresponding coefficients.●Ocean currents and influencing factorsOcean currents, also called sea currents, are large-scale seawater movements which have relatively stable speed and direction. Only in the land along the coast, due to tides, terrain, the injection of river water, and other factors, the speed and direction of ocean currents changes.Figure 2Distribution of world ocean currentsIt can be known from Figure 2 that warm and cold currents exist in the area where aircraft incidences happened. Considering the fact that the speed of ocean currents slows down as the increase of the depth of ocean, the velocity with depth sea surface currents gradually slowed down, v x is set as the initial speed of ocean currents in subsequent calculations.●Turbulent layerTurbulent flow is one kind of state of the fluid. When the flow rate is very low, the fluid is separated into different layers, called laminar flow, which do not mix with each other. As the flow speed increases, the flow line of the fluid begins to appear wavy swing. And the swing frequency and amplitude in-creases as the flow rate increases. This kind of stream regimen is called tran-sition flow. When the flow rate becomes great, the flow line is no longer clear and many small whirlpools, called turbulence, appeared in the flow field.Under the influence of ocean currents, the flow speed of the fluid changes as the water depth changes gradually, the speed and direction of the fluid is un-certain, and the density of the fluid density changes, resulting in uneven flow distribution. This indirectly causes the change of drag coefficient, and the re-sistance of the fluid is calculated as follows:2fkvGLCM texture of submarine topographyIn order to describe the impact of submarine topography, we choose a rectan-gular region from 33°33…W, 5°01…N to 31°42‟W , 3°37‟N. As texture is formed by repetitive distribution of gray in the spatial position, there is a cer-tain gray relation between two pixels which are separated by a certain dis-tance, which is space correlation character of gray in images. GLCM is a common way to describe the texture by studying the space correlation cha-racter of gray. We use correlation function of GLCM texture in MATLAB:I=imread ('map.jpg'); imshow(I);We arbitrarily select a seabed images and import seabed images to get the coordinate of highlights as follows：Table 1Coordinate of highlightsNO. x/km y/km NO. x/km y/km NO. x/km y/km1 154.59 1.365 13 91.2 22.71 25 331.42 16.632 151.25 8.19 14 40.04 18.12 26 235.77 13.93 174.6 14.02 15 117.89 14.89 27 240.22 17.754 172.38 19.23 16 74.51 12.29 28 331.42 24.455 165.71 24.82 17 45.6 8.56 29 102.32 19.486 215.75 26.31 18 103.43 5.58 30 229.1 18.247 262.46 22.96 19 48.934 3.51 31 176.83 9.188 331.42 22.34 20 212.42 2.85 32 123.45 3.239 320.29 27.55 21 272.47 2.48 33 32.252 11.7910 272.47 27.55 22 325.85 6.45 34 31.14 27.811 107.88 28.79 23 230.21 7.32 35 226.88 16.0112 25.579 27.05 24 280.26 9.93 36 291.38 5.46Then we use HDVM algorithm to get the 3D image of submarine topography, which can be simulated by MATLAB.Figure 3 3D image of submarine topographyObjects force analysis under the condition of currentsf is the resistance, f i is the disturbance resistance, F f is the buoyancy, G isgravity of object.Figure 4Force analysis of object under the conditions of currentsConsidering the impact of currents on the sinking process of objects, wheninterfered with currents, objects will sheer because of uneven force. There-。

美赛文献综述开头
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本文将结合最新的研究进展，就美国数学建模大赛（MCM/ICM）的相关文献进行综述，希望能为广大参赛者提供一些有益的启发。

首先，美赛作为一个知名的数学建模竞赛，每年吸引了来自全球各地的数千名参赛者。

参赛者需要在规定的时间内，针对所提出的问题进行深入的研究和分析，最终给出合理的建模方案和解决方案。

因此，对于参赛者而言，对前人的研究成果进行充分的了解是至关重要的。

其次，通过对美赛历年获奖作品的文献进行分析，可以发现一些共性和规律。

比如，优秀的团队往往具有良好的团队协作能力和高效的分工合作方式；在问题建模和解决过程中，他们通常能够充分挖掘数据的潜在规律，结合数学工具和计算机算法，提出创新性的解决方案。

这些都是参赛者可以借鉴和学习的经验和教训。

最后，希望参赛者在阅读文献综述的过程中，不仅能够了解前人的研究成果和方法，还要结合自身的实际情况和研究对象，找到适合自己的解题思路和技巧。

同时，要注重团队协作能力的培养和提升，不断学习和改进自己的数学建模能力，为未来的参赛之路奠定坚实的基础。

综上所述，文献综述在美赛准备过程中具有重要的指导意义，通过对前人的研究成果进行深入的分析和总结，可以帮助参赛者更好地把握竞赛的要点和难点，为取得优异的成绩提供有力的支持。

希望广大参赛者能够认真对待文献综述，积极学习和借鉴前人的经验，不断提升自身的数学建模能力，为美赛的成功参赛打下坚实的基础。

美赛写作及参赛经验分享 Revised by Jack on December 14,2020Tittle of paperSummary/Abstract Key words:I.Introduction（引言）Organ transplantation is a preferable treatment for the most serious forms of end-stage diseases. In recent years, advances in medical science and technology have made solid organ transplantation an increasingly successful and common medical procedure, a literal ''second chance at life". Not only does it offer the best hope for complete rehabilitation, but it has also proved to be the most cost-effective of all treatment options, including dialysis. Consequently, more and more people are benefiting from organ transplants and their survival rates are steadily improving. The surgical techniques involved have been mastered for half a century and are now considered as routine. The two main sources of kidneys for transplantation are deceased-donor kidneys and live-donations from family and friends. However, unfortunately, there is a considerable shortage of donor organs, compared to demands. As a matter of fact, efficient matching and allocation of organs donated has become an exigent problem.The United Network for Organ Sharing (UNOS), as the operator of the Organ Procurement and Transplantation Network (OPTN), is responsible for transplant organ distribution in the United States. UNOS oversees the allocation of many different types of transplants, including liver, kidney, pancreas, heart, lung, and cornea.Focusing on kidney transplantation, based on UNOS Kidney Allocation Model, we develop a mathematical model for US transplant networks. First, incoming organs are matched with waiting candidates by medical institutions considering the factors as ABO blood compatibility, the degree of recipient major HLA mismatch in order to obtain a matching degree applied on the allocation part. After that, from the patients’perspective, on the basis of linear regression, priority weight is defined by pondering age, disease severity, time on waiting, PRA level, and region. Applying this mechanism of ranking, we realize MWBM (Maximum Weight Bipartite-graph Matching) and SMGS (Stable Matching based on Gale-Shapley algorithm). MWBMfocuses on the optimal assignment of donors following the algorithm of bipartite-graph maximum weight matching; SMGS emphasizes the process of exchanges in order to obtain the stable exchanges between donors and candidates on the waiting list.II.The Description of Problem（问题重述)III.Basic Assumptions●The level of mismatch is only relative to the number of antigens.●The data and information are accurately registered according to the medicalmeasures●The data and information are refreshed in time according to the status of thepatients●No differences in the quality of the donor kidneys●The quality of the donor kidney is constantIV.Definitions and Notations●Kidney transplantation: A kidney transplant is a surgical procedure to implant a healthykidney into a patient with kidney failure.●Prioritization●MD: Matching Degree●PW: Prioritization weight●MWSM: Maximum Weight Bipartite Matching●SMGS: Stable Matching based on Gale-Shapley algorithm或V.ModelsThrough the investigation of US transplantation network, we draw a general picture of the mechanism. With reference to some resources available on the website of UNOS, a flow chart (Figure 1) is developed showing the procedure of the network.Currently, the initial waiting list is composed of patients who are waiting for a kidney or combined kidney-pancreas transplant. For the first time, the patients are requested to show the correct and scientific information to the US kidney transplant network which is needed for donor-recipient matching, the ranking of patients on the waiting list, and determining the outcome of those transplanted. The patients’waiting lists are composed of initial patients, historical patients and unsuccessful recipient after transplantation. Historical patients refer to registered patients whose status have changed and have an influence on the procedure. A patient is taken off the waiting list when a graft is offered and accepted by that patient or the patient is dead while waiting for a transplant. Unsuccessful recipients refer to the patients who have a bad result of transplantation calling for transplantation again, as it is so-called relistFigure 1. A schematic depicting the steps occurring in the transplantation networks......Table rate involving HLA mismatchVI.Conclusions.Our model for the optimal allocation of the donor organs is established by three modules, procurement of MD and PW, optimal assignment by MWBM model and Stable Matching of Gale-Shapley algorithm. The model has offered a convincing procedure of the allocation with the ……VII.Strengths and weaknesses(模型优缺点)Strengths●……Weaknesses●VIII.References注意文献的积累，不要等到文章写完再去重新寻找文献。

TitileSummaryDuring cell division, mitotic spindles are assembled by microtubule-based motor proteins1, 2. The bipolar organization of spindles is essential for proper segregation of chromosomes, and requires plus-end-directed homotetrameric motor proteins of the widely conserved kinesin-5 (BimC) family3. Hypotheses for bipolar spindle formation include the 'push−pull mitotic muscle' model, in which kinesin-5 and opposing motor proteins act between overlapping microtubules2, 4, 5. However, the precise roles of kinesin-5 during this process are unknown. Here we show that the vertebrate kinesin-5 Eg5 drives the sliding of microtubules depending on their relative orientation. We found in controlled in vitro assays that Eg5 has the remarkable capability of simultaneously moving at 20 nm s-1 towards the plus-ends of each of the two microtubules it crosslinks. For anti-parallel microtubules, this results in relative sliding at 40 nm s-1, comparable to spindle pole separation rates in vivo6. Furthermore, we found that Eg5 can tether microtubule plus-ends, suggesting an additional microtubule-binding mode for Eg5. Our results demonstrate how members of the kinesin-5 family are likely to function in mitosis, pushing apart interpolar microtubules as well as recruiting microtubules into bundles that are subsequently polarized by relative sliding. We anticipate our assay to be a starting point for more sophisticated in vitro models of mitotic spindles. For example, the individual and combined action of multiple mitotic motors could be tested, including minus-end-directed motors opposing Eg5 motility. Furthermore, Eg5 inhibition is a major target of anti-cancer drug development, and a well-defined and quantitative assay for motor function will be relevant for such developmentsContentTitile (1)Summary (1)1Introduction (1)1.1Restatement of the Problem (1)1.2Background (1)1.1.1Common Solving Technique (1)1.1.2Previous Works (1)1.3Example (1)2Analysis of the Problem (1)2.1Outline of the Approach (1)2.2Basic Assumptions (2)2.3Definitions and Key Terms (2)3Calculating and Simplifying the Model (2)4The Model Results (3)5Validating the Model (3)6Strengths and Weaknesses (3)6.1Strengths (3)6.2Weaknesses (3)7Food for Thought (3)8Conclusion (3)References (4)Appendices (4)Appendix A Source Code (4)Appendix B (4)1Introduction1.1Restatement of the Problem …1.2Background…1.1.1Common Solving Technique…1.1.2Previous Works…1.3Example…2Analysis of the Problem …2.1Outline of the Approach…2.2Basic Assumptions●●●●●2.3Definitions and Key Terms●●●●Table 1.…Symbol Meaning Unit3Calculating and Simplifying the Model …4The Model Results……5Validating the Model…6Strengths and Weaknesses6.1S trengths●●●●6.2W eaknesses●●●●7Food for Thought…8Conclusion….References…AppendicesAppendix A Source CodeHere are the simulation programmes we used in our model as follow. Input matlab source:……….Appendix B…….Input C++ source：…………..…………..。

2020年美赛C题论文引言在2020年的美赛C题中，我们将研究某城市的停车问题。

停车问题在现代城市中非常普遍，而且经常引起交通拥堵和资源浪费。

因此，寻找一种合理的停车方案对于城市的可持续发展至关重要。

本文将介绍我们对该停车问题的建模过程、假设和模型结果。

问题描述该城市位于一个山区，拥有许多旅游景点，吸引了大量游客。

然而，停车场的数量有限，传统的交通管理方式导致了拥堵和停车困难。

因此，我们需要提出一种新的停车方案，以改善交通状况和资源利用。

我们需要回答以下问题：1.如何确定合理的停车位价格以确保公平性和减少拥堵？2.如何确定合理的停车位数量以满足游客的需求？3.如何指导游客选择合适的停车场？数据处理和建模为了解决上述问题，我们从该城市收集了大量的交通数据和停车场信息。

首先，我们对数据进行处理，包括数据清洗、整理和校验。

然后，我们使用Python编程语言对数据进行分析和建模。

下面是我们的建模过程：1.确定停车需求模型：我们将游客的停车需求建模为一个随机变量，可以以概率密度函数的形式表示。

为了准确地估计需求模型，我们使用了大量的历史停车数据和游客统计数据。

2.确定停车位定价模型：我们考虑了停车位价格对停车需求的影响，并建立了一个停车位定价模型。

该模型将考虑停车位的成本、游客的支付意愿和其他相关因素。

3.确定停车场选择模型：我们使用了多属性决策分析方法来确定游客选择停车场的因素和权重。

通过评估每个停车场的特点和游客的偏好，我们可以为游客提供选择停车场的指导。

假设为了简化问题和建立数学模型，我们做出了以下假设：1.停车需求是服从某种概率分布的随机变量。

2.停车位定价的主要目标是确保公平性和减少拥堵。

3.游客的停车选择主要受停车位价格和距离的影响。

4.停车场之间没有容量限制。

这些假设可以帮助我们建立合理的模型和解决方案，但也需要在实际应用中考虑其他可能的因素。

模型结果基于我们的建模过程和假设，我们得到了以下模型结果：1.停车需求模型：通过对历史停车数据和游客统计数据的分析，我们得到了停车需求的概率密度函数模型。

Tittle of paperSummary/Abstract Key words:I.Introduction（引言）Organ transplantation is a preferable treatment for the most serious forms of end-stage diseases. In recent years, advances in medical science and technology have made solid organ transplantation an increasingly successful and common medical procedure, a literal ''second chance at life". Not only does it offer the best hope for complete rehabilitation, but it has also proved to be the most cost-effective of all treatment options, including dialysis. Consequently, more and more people are benefiting from organ transplants and their survival rates are steadily improving. The surgical techniques involved have been mastered for half a century and are now considered as routine. The two main sources of kidneys for transplantation are deceased-donor kidneys and live-donations from family and friends. However, unfortunately, there is a considerable shortage of donor organs, compared to demands. As a matter of fact, efficient matching and allocation of organs donated has become an exigent problem.The United Network for Organ Sharing (UNOS), as the operator of the Organ Procurement and Transplantation Network (OPTN), is responsible for transplant organ distribution in the United States. UNOS oversees the allocation of many different types of transplants,including liver, kidney, pancreas, heart, lung, and cornea.Focusing on kidney transplantation, based on UNOS Kidney Allocation Model, we develop a mathematical model for US transplant networks. First, incoming organs are matched with waiting candidates by medical institutions considering the factors as ABO blood compatibility, the degree of recipient major HLA mismatch in order to obtain a matching degree applied on the allocation part. After that, from the patients’perspective, on the basis of linear regression, priority weight is defined by pondering age, disease severity, time on waiting, PRA level, and region. Applying this mechanism of ranking, we realize MWBM (Maximum Weight Bipartite-graph Matching) and SMGS (Stable Matching based on Gale-Shapley algorithm). MWBM focuses on the optimal assignment of donors following the algorithm of bipartite-graph maximum weight matching; SMGS emphasizes the process of exchanges in order to obtain the stable exchanges between donors and candidates on the waiting list.II.T he Description of Problem（问题重述)III.Basic Assumptions●The level of mismatch is only relative to the number of antigens.●The data and information are accurately registered according tothe medical measures●The data and information are refreshed in time according to thestatus of the patients●No differences in the quality of the donor kidneys●The quality of the donor kidney is constantIV.D efinitions and Notations●Kidney transplantation: A kidney transplant is a surgical procedure to implant a healthykidney into a patient with kidney failure.●Prioritization●MD: Matching Degree●PW: Prioritization weight●MWSM: Maximum Weight Bipartite Matching●SMGS: Stable Matching based on Gale-Shapley algorithm或V.ModelsThrough the investigation of US transplantation network, we draw a general picture of the mechanism. With reference to some resources available on the website of UNOS, a flow chart (Figure 1) is developed showing the procedure of the network.Currently, the initial waiting list is composed of patients whoare waiting for a kidney or combined kidney-pancreas transplant. For the first time, the patients are requested to show the correct and scientific information to the US kidney transplant network which is needed for donor-recipient matching, the ranking of patients on the waiting list, and determining the outcome of those transplanted. The patients’waiting lists are composed of initial patients, historical patients and unsuccessful recipient after transplantation. Historical patients refer to registered patients whose status have changed and have an influence on the procedure.A patient is taken off the waiting list when a graft is offered and accepted by that patient or the patient is dead while waiting for a transplant. Unsuccessful recipients refer to the patients who have a bad result of transplantation calling for transplantation again, as it is so-called relistFigure 1. A schematic depicting the steps occurring in thetransplantation networks......Table 1.Survival rate involving HLA mismatchVI.C onclusions.Our model for the optimal allocation of the donor organs is established by three modules, procurement of MD and PW, optimal assignment by MWBM model and Stable Matching of Gale-Shapley algorithm. The model has offered a convincing procedure of the allocation with the ……VII.Strengths and weaknesses(模型优缺点)Strengths……WeaknessesVIII.References注意文献的积累，不要等到文章写完再去重新寻找文献。

Title Abstract:Key words:Contents1. Introduction (3)1.1 Why does toll way collects toll? (3)1.2 Toll modes (3)1.3 Toll collection methods (3)1.4 Annoyance in toll plazas (3)1.5 The origin of the toll way problem (3)1.6 Queuing theory (4)2. The Description of Problem (5)2.1 How do we approximate the whole course of paying toll? (5)2.2 How do we define the optimal configuration? (5)2.2.1 From the perspective of motorist (5)2.2.2 From the perspective of the toll plaza (6)2.2.3 Compromise (6)2.3 Overall optimization and local optimization (6)2.4 The differences in weights and sizes of vehicles (7)2.5 What if there is no data available? (7)3. Models (7)3.1 Basic Model (7)3.1.1 Symbols and Definitions (7)3.1.2 Assumptions (8)3.1.3 The Foundation of Model (9)3.1.4 Solution and Result (11)3.1.5 Analysis of the Result (11)3.1.6 Strength and Weakness (13)3.2 Improved Model (14)3.2.1 Extra Symbols (14)3.2.2 Additional Assumptions (14)3.2.3 The Foundation of Model (14)3.2.4 Solution and Result (15)3.2.5 Analysis of the Result (18)3.2.6 Strength and Weakness (19)4. Conclusions (19)4.1 Conclusions of the problem (19)4.2 Methods used in our models (19)4.3 Application of our models (19)5. Future Work (19)5.1 Another model (19)5.2 Another layout of toll plaza (23)5.3 The newly- adopted charging methods (23)6.References (23)7.Appendix (23)Programs and codes (24)I. IntroductionIn order to indicate the origin of the toll way problems, the following background is worth mentioning.1.11.21.31.41.51.6II. The Description of the Problem2.1 How do we approximate the whole course of paying toll?●●●●2.2 How do we define the optimal configuration?1) From the perspective of motorist:2) From the perspective of the toll plaza:3) Compromise:2.3 The local optimization and the overall optimization●●●Virtually:2.4 The differences in weights and sizes of vehicles2.5 What if there is no data available?III. Models3.1 Basic Model3.1.1 Terms, Definitions and SymbolsThe signs and definitions are mostly generated from queuing theory.●●●●●3.1.2 Assumptions●●●●●3.1.3 The Foundation of Model1) The utility function●The cost of toll plaza:●The loss of motorist:●The weight of each aspect:●Compromise:2) The integer programmingAccording to queuing theory, we can calculate the statistical properties as follows.3)The overall optimization and the local optimization●The overall optimization:●The local optimization:●The optimal number of tollbooths:3.1.4 Solution and Result1) The solution of the integer programming:2) Results:3.1.5 Analysis of the Result●Local optimization and overall optimization:●Sensitivity: The result is quite sensitive to the change of the threeparameters●Trend:●Comparison:3.1.6 Strength and Weakness●Strength: In despite of this, the model has proved that . Moreover, wehave drawn some useful conclusions about . T he model is fit for, such as●Weakness: This model just applies to . As we have stated, .That’sjust what we should do in the improved model.3.2 Improved Model3.2.1 Extra SymbolsSigns and definitions indicated above are still valid. Here are some extra signs and definitions.●●●●3.2.2 Additional Assumptions●●●Assumptions concerning the anterior process are the same as the Basic Model.3.2.3 The Foundation of Model1) How do we determine the optimal number?As we have concluded from the Basic Model,3.2.4 Solution and Result1) Simulation algorithmBased on the analysis above, we design our simulation arithmetic as follows.●Step1:●Step2:●Step3:●Step4:●Step5:●Step6:●Step7:●Step8:●Step9:2) Flow chartThe figure below is the flow chart of the simulation.3) Solution3.2.5 Analysis of the Result3.2.6 Strength and Weakness●Strength: The Improved Model aims to make up for the neglect of .The result seems to declare that this model is more reasonable than the Basic Model and much more effective than the existing design.●Weakness: . Thus the model is still an approximate on a large scale. Thishas doomed to limit the applications of it.IV. Conclusions4.1 Conclusions of the problem●●●4.2 Methods used in our models●●●4.3 Applications of our models●●●V. Future Work5.1 Another model5.1.1The limitations of queuing theory 5.1.25.1.35.1.41)●●●●2)●●●3)●●●4)5.2 Another layout of toll plaza5.3 The newly- adopted charging methodsVI. References[1][2][3][4]VII. Appendix。

全国流行的艾滋病分析摘要本文全面的考虑了那些患艾滋病最严重的国家。

利用短期的阻滞微分方程模型，建立了可控的生命期望值，在数值上定义了各个国家的严重程度。

最后得到结论：艾滋病情况最严重的国家是：博茨瓦纳、泰国、汤加、乌克兰、巴哈马群岛、圭亚那(拉丁美洲)。

本文运用了通用的计算机模拟方式，通过建立微分方程模型分析了那些艾滋病情况最严重的国家来直接处理不同人群的艾滋病情况。

治疗分析包括2055年国际援助估算总量，对ARV治疗效果的预测，研发出预防艾滋病疫苗的可能性。

同时，本文还考虑了药物所带来的副作用。

本文最后提出了一系列如何最优分配资源的建议，即在短期艾滋病的项目研究以及疫苗的发展上投入较高的资金，同时要较好的确定维持药物有效条件下ARV的全球覆盖率。

严重程度的定义方法严重性是由什么导致的？最明显的答案就是一个国家庞大的艾滋病感染人数，或者说是艾滋病感染人群在所有人口中所占的较高比例，但是，这并不是一个完整的分析。

存在一种严重的情况便意味着要寻找解决的方案。

基于这点，对艾滋病感染患者，我们能做的就只有是治疗。

而有着高治疗率的国家能为被感染的人群做很多，所以这些国家并不能被认为是情况最严重的。

严重性还包括行动的迫切性，因为艾滋病病毒在短期内得不到抑制也是十分有害的，我们相信对艾滋病患者最有效的方法是测定预测出每年由于感染而艾滋病而失去生命的增长人数。

必要的假设●接受ARV治疗的患者都是100%的坚持治疗——要么接受治疗，要么就不接受，没有所谓的中间状态。

●在未来5年内，各方面因素没有受到干涉。

●接受ARV治疗的比例是一个定值。

●在本文所做的短期预测的这段时间内，没有其他的能引起人口重大死亡的事件发生，如自然灾害，战争，全国流行性疾病等一些能够对人口产生重大影响的事件。

●People-year ：一个人一年的时间，所有人口一年的时间等于所有个人一年时间的总和。

●为了预测在在未来5年内没有其他因素影响下艾滋病对人口的直接影响程度，本文定义了艾滋病的严重性程度：绝对严重：在未来5年内，因为感染艾滋病病毒而失去的总的寿命值相对严重：在未来5年内，每个人因为感染艾滋病病毒而失去的平均寿命值。

For office use onlyT1________________ T2________________ T3________________ T4________________ Team Control Number27328Problem ChosenAFor office use onlyF1________________F2________________F3________________F4________________ 2014Mathematical Contest in Modeling (MCM/ICM) Summary Sheet(Attach a copy of this page to your solution paper.)Type a summary of your results on this page. Do not includethe name of your school, advisor, or team members on this page.2014 A: The Keep-Right-Except-To-Pass RuleIt is very important to obey the traffic rules when you drive, especially in which direction do you drive. The rules can avoid the chaos and traffic accidents.To judge whether the rule is good, we can consider traffic flow and safety. So, how to assess the performance of the role? How to improve and perfect it?We build this model mainly from four aspects.For the first question we mainly take relevant factors into consideration. At the beginning, we divide the traffic condition into four categories. That is: Whether obeying the rule in heavy or light traffic. Then for the four categories, we analyze the problem in six aspects. At last,we get some traffic performance evaluation scores. Therefore we can evaluate the rule in a mathematical point of view. For the second question we think about the relationship between whether obeying the rule, the distance between two vehicles and the amount of vehicles passed by in an hour. We calculate the traffic fl ow based on the previous research’s formula. With the help of the statistics we measured before, we can calculate the score under the four circumstances. For the third question, we think about the impact of driving direction on vehicle’s speed. If the spee d changes, then the score will change, too. Our model also need to modify correspondingly. For the fourth question, We analyze the difference between artificial system and intelligent system. We adopt the intelligent system to modify our model because the traffic flow and safety index change. To sum up, when the traffic is heavy the rule performs well and is effective to promote traffic flow. However, when the traffic is light the rule performs weaker and has no significant effect to promote traffic flow. But the conclusion is exactly opposite in the left-driving countries. Intelligent system doesn't have a big impact on our conclusion. Our advise is when the traffic is light, we can leave one lane just for passing vehicles.COVERCOVER 0Summary (2)Key word (3)Problem background (3)Problem analysis (4)Assumptions (5)Symbols description (5)Model design and solving (6)Model 1: Introduction of vehicle length, weather, frictions and actual speed (6)Model 2: Calculating the distance between two vehicles (8)Model 3: Scores considering S(the traffic flow within an hour) and I(the number of traffic accident every 100000 kilometers) .. 9 Model 4: Improving traffic flow (13)Model 5: Can our new model be applied to driving-in-the-left countries ? (14)Model 6: Intelligent system model (14)Weaknesses and strengths of the model (15)Reference (15)Appendix (16)2014 A: The Keep-Right-Except-To-Pass RuleSummaryIt is very important to obey the traffic rules when you drive, especially in which direction do you drive. The rules can avoid the chaos and traffic accidents.To judge whether the rule is good, we can consider traffic flow and safety. So, how to assess the performance of the role? How to improve and perfect it?We build this model mainly from four aspects.For the first question we mainly take relevant factors into consideration. At the beginning, we divide the traffic condition into four categories. That is: Whether obeying the rule in heavy or light traffic. Then for the four categories, we analyze the problem in six aspects. At last,we get some traffic performance evaluation scores. Therefore we can evaluate the rule in a mathematical point of view. For the second question we think about the relationship between whether obeying the rule, the distance between two vehicles and the amount of vehicles passed by in an hour. We calculate the traffic flow based on the previous research’s formula. With the help of the statistics we measured before, we can calculate the score under the four circumstances. For the third question, we think about the impact of driving direction on vehicle’s speed. If the speed changes, then the score will change, too. Our model also need to modify correspondingly. For the fourth question, We analyze the difference between artificial system and intelligent system. We adopt the intelligent system to modify our model because the traffic flow and safety index change. To sum up, when the traffic is heavy the rule performs well and is effective to promote traffic flow. However, when the traffic is light the rule performs weaker and has no significant effect to promote traffic flow. But the conclusion is exactly opposite in the left-driving countries. Intelligent system doesn't have a big impact on our conclusion. Our advise is when the traffic is light, we can leave one lane just for passing vehicles.Key wordfactor analysis, optimization methods, traffic flow, the number of traffic accidents, scoreProblem backgroundIt is very important to obey the traffic rules when you drive, especially in which direction do you drive. The rules can avoid the chaos and traffic accidents. We can drive on the left or on the right.34% countries drive on the left and 66% countries drive on the right. The biggest advantage of driving on the left is human’s instinct to avoid its evils. In the case of fast movement, when you find it’s dangerous in the front, you will instinctively tilt to the left to protect your heart. The advantage of driving on the right is that drivers can take charge of the steering wheel by the left hand and can change the shift flexibly. Most of the countries adopt the rule that driving on the right. Drivers accustomed to the right don’t need to spare time on learning the left rules. At the same time, the seats drivers sit are usually on the left all over the world. If the car is the same, it is cheaper to buy the car that the seat is on the left than on the right. In our country, as the result of politics, economy and culture, we obey to the rule that drive on the right unless passing anther car.To judge whether the rule is good, we can consider traffic flow and safety. Generally speaking, the traffic rule will have good to promoting traffic flow, but when the vehicles are keeping increasing, the traffic must be heavy ,so the number of traffic accidents must increase, too. So, if we want to find a better way ,we must make the rules considering traffic flow and safety.So, regardless of the different driving directions, when we make the traffic rules, we must consider these factors. To ensure safety, some roads are under the control of intelligent systems. This can avoid driving after drinking, driving with tiredness. Theintelligent system can avoid some human judgment and forecast the traffic condition. The road will not be too crowded under the control of it. So this system can be good to the traffic.Based on this background, we build a mathematical model to check if the rule is rather good. At the same time ,we will answer the questions below:When we take safety, weather, road condition, traffic flow into consideration, how is the performance of this rule in light and heavy traffic?Is this rule effective in promoting better traffic flow? If not, how to check our model? Is our model applying to the driving on the left countries?If vehicle transportation on the same roadway was fully under the control of an intelligent system –either part of the road network or imbedded in the design of all vehicles using the roadway–to what extent would this change the results of your earlier analysis?Problem analysisAfter reading the question, we will split the problem into some small problems to solve.●Consider the road friction, weather, traffic condition, weather obey the rule’simpact on the vehicle’s speed.●Consider the relationship between the distance of two vehicles and speed.●Consider the relationship between traffic flow and the number of traffic accidents.We have two conditions: heavy traffic and light traffic. We calculate the score in different conditions when obeying the rule or not obeying the rule.●If our model can be applied to other countries?●Consider the intelligent system’s effect. Under the intelligent system, do we needto check our model?In question 1,2,3,4,we build a score to discuss the rule’s impact on traffic condition in heavy or light traffic. In the score, the traffic flow index takes up 50%,the safety index takes up 50%.In the same way, we can calculate the score when not obeying the rule.After preliminary analysis, when the traffic is light, if we don’t obey the rule, the score is higher. So, we take three road lanes for instance, when the traffic is heavy, we obey the rule. when the traffic is light, we don’t obey the rule. In question 5,We take human body’s inflexibility into consideration because it will affect the speed. In question 6,with the help of intelligent system, we can avoid heavy traffic and traffic accidents, we will have a new score and then check the rules.AssumptionsIn the optimization methods, we just think the road is straight to build a model.We set medium size vehicle as a standard. The stable speed of the vehicle in the free state(medium-car standard model)is X.Speed conversion coefficient under the standard of medium size car’s speed is A.We consider the vehicles all drive on the right, then we have four conditions, obey the rules or not, heavy traffic or light.The traffic flow is within an our. We take one normal day for instance. From 7a.m. to 9a.m and4:30p.m to 6p.m it is heavy traffic. From 23p.m to 4a.m it is light traffic. On this basis, we can build our model.Symbols descriptionModel design and solvingModel 1: Introduction of vehicle length, weather, frictions and actual speedIt covers 6 variable: vehicle length(c),the weather, road conditions and friction impact on the speed(K),the traffic impact on the speed(α),whether obeying rule impact on the speed(β),The stable speed of the vehicle in the free state(medium-car standard model)(X) and Speed conversion coefficient under the standard of medium size car’s speed(A).According to the research, we can know K and A.To K, we have five conditions:Level one: The road condition is good and vehicles can pass away normally.Level two: The road condition is slightly bad and vehicles can pass slowly.Level three: The road condition is bad and cars can pass slowly with a reasonable distance.Level four: The road condition is worse and vehicles can’t pass safely.Level five: The road condition is worst and no vehicles can pass.（Sheet 1）To A, we divide vehicles into 8 kinds. They are cars, van, motor coach, small trucks, freight train, large trucks, tractor and trailer. The stable speed of the vehicle in the free state(medium-car standard model) is X and Speed conversion coefficient under the standard of medium size car’s speed is A.Sheet 2 shows the vehicle speed Conversion Factor A：（Sheet 2）Considering these factors, the speed X is affected by α，β，A and K. According to the optimization methods, we just think these factors are proportional to X. Above all, we can define the actual speed of the vehicle.Model 2: Calculating the distance between two vehiclesWe think the least distance between two vehicles can ensure safety when braking.The vehicle brake deceleration process can be seen as linear motion.The acceleration is K ·g(g is acceleration due to gravity ). Then:22220122V kgt Vt kgt L V KXKX L gβαβα-=⎧⎪⎨-=⎪⎩=∴=Model 3: Scores considering S(the traffic fl ow within an hour) and I(thenumber of traffic accident every 100000 kilometers)Our model will give us a score considering S(the traffic flow within an hour) and I(thenumber of traffic accident every 100000 kilometers).If the score is big enough, thenthe rule is effective in promoting traffic condition.Firstly, we deduce the formula of the traffic flow. The most widely used formula in ourcountry is: time*V/(C+L).Considering heavy and light traffic, we just need to chooseone our in heavy time and light time.Secondly, we deduce the formula of the number of traffic accident every 100000kilometers(I).Obviously, the actual speed affects I. Here are some reasons below:(1)affecting eyesight. When speed is increasing, the drivers eyesight will be poorerthan the still eyesight.(2)affecting visual range. When speed is increasing, the visual range will be small andnarrow.(3)affecting recognizable sight. When driving, the driver needs to identify varioustraffic signs or traffic environment. The identification distance will be different under the different speed. When the speed is too high, the identification distance will be small. It will be hard to be aware of the road condition.(4)affecting judgment. It is according to the position change that drivers recognizethe object.When the vehicle is in a traveling state , the objects outside the vehicle change the position slowly and small.If the speed is high, then the change is slow and small. It is really hard to identify. Therefore, when the driver is driving, his ability to distinguish an object will fall.(5)impact on the braking distance and security region. As picture 3 shows,a vehiclewith the speed of 30km / h can stop his car in 13 meters. The passers-by will be safe outside the 13 meters. The damage is zero and this area is safe to passers-by.However, a vehicle with the speed of 50km / h can stop his car in 26 meters. If the passers-by still walk in 13metres,it will be dangerous. The safety index is zero.(picture 3)Sheet 4 is an ordinary road braking distances on different speed：Sheet 5 :stopping sight distance in a highway：In 2004, after analyzing the traffic accidents, Chinese scholars think accident rate increases when the standard deviation of speed increases. The bigger the standard deviation is, the higher the accident rate is. At the same time, research indicates that speed standard deviation and the accident rate are exponential relationship.With the improvement in the level of speed’s discrete, accident rate will be growing exponentially .Below are the picture 6:(picture 6)In 1993,Monash University Accident Research Centre summed a function model to tell the relationship between the speed level and traffic accidents rate.The function is :23=+∆+∆5000.80.014I V VIn the formula,I means the number of traffic accident every 100000 kilometers.-,Our model is to use this formula to calculate the number of V∆means V Xaccidents .At last, how do we calculate the score P ? That is the traffic flow index takes up 50% and the safety index takes up 50%.111V L + 1S )221V L +2S )331V L +S )10.8(V )X -(1I ) By linear regression, we can calculate αand β.Under the circumstance of heavy traffic, we take the scores when obeying the rule or not obeying the rule for comparison. We can have a conclusion that in this case the score is higher when obeying the rule. Similarly, under the circumstance of light traffic, we take the scoreswhen obeying the rule or not obeying the rule for comparison. We can have a conclusion that in this case the score is higher when not obeying the rule. Therefore, we can know the rule performs well in heavy traffic but performs bad in light traffic.Model 4: Improving traffic flowThe model here is similar the model above when calculating traffic flow./(C )S t V L =+.Similarly, we have to conditions. That is heavy traffic and light traffic, then we calculate the traffic flow when obeying the rule or not obeying rule to see if the rule is effective to promote the traffic flow.111V L +1S ) 221V L +2S ) 331V L +3S )According to the α、β ,we can calculate the traffic flow S easily.We find it obviously when the traffic is heavy, we will have a bigger traffic flow when obeying the rule. When the traffic is light, the traffic flow is bigger if not obeying the rule.Therefore, We promote When the traffic is heavy, we need to obey the previous rule--- drivers need to drive in the right-most lane unless they are passing another vehicle. When the traffic is light, we take two lanes in the right side for driving and the left one side for passing vehicles.(3 lanes for instance)At this time we control the vehicle’s speed in a reasonable range and then insureβ1βin a flexible range. Then the traffic flow is bigger when not obeying the and2previous rule.Model 5: Can our new model be applied to driving-in-the-left countries ? Whether our new model can be applied to driving-in-the-left country depends on the traffic flow and safety index. At this moment ,our model becomes that keep left when driving unless passing vehicles in heavy traffic, while in light traffic, we retained two lanes on the left as travel lanes and one left lane as a passing lane. Similarly, taking score into consideration, which is composed of traffic and security Index. In fact, it is easy to find that β1 and β2 exchanged themselves, so w e have enacted new rules about reciprocity. It is that we retained two lanes on the left as travel lanes and one remaining lane as a passing lane when the traffic is heavy. In addition, maintaining the previous rule that keep left except passing in light traffic.Model 6: Intelligent system modelIf the intelligent system replace the artificial system, the safety index will increase in a big extent. At the same time, the traffic flow will increase, too. However,βwill not change because this factor depends on human’s performance. So we need to modify our model partly. When the traffic is heavy, we keep the previous rule. When the traffic is light, we make one lane for passing vehicles. With the help of intelligent system, we can make two lanes in the most side for passing and the lanes left for driving.Weaknesses and strengths of the modelAt the very beginning, we analyze some factors that will affect the result. We just think the problem is not very complex, so we simplify our work. We give all the index a reasonable flexible range. We also consult some authority research.However, we also have some disadvantages. We don’t take the traffic lights and some more complex road condition into consideration.Time is urgent, we don’t find more statistics to test our model accurately. Reference[1] [J] The model of the relationship between vehicle speed and traffic flow, Wei Wang, Nan Jing ,Jiangsu province[2] [D] The research of intelligent system and traffic flow, Jianhui Dai, Tian Jin[3] Addison, Low Paul S, David J. Order and chaos in the dynamics of vehicle platoons[ J] . Traffic Engineering & Control,1996, 37( 7 8) : 456 -459.[4] Daganzo C F, Cassidy M J, Bertini R L. Possible explanations of phase transitions in highway traffic[ J] . Transportation Research , Part A, 1999, 33( 5) : 365- 379.[ 5] Jiang Y. Traffic capacity speed and queue- discharge rate of Indiana. s four- lane freeway work zones[ A] . In: Transportation Research Record 1657, TRB , National Research Council [ C] . Washington D C, 1999. 39 -44.[6] Schonfeld P, Chien S. Optimal work zone lengths for two lane highways [ J] . Journal of Transportation Engineering , Urban Transportation Division, ASCE , 1999, 125( 1) : 21-29.[ 7] Nam D D, Drew D R. Analyzing freeway traffic under congestion: traffic dynamics approach [ J] . Journal of Transportation Engineering, Urban Transportation Division, ASCE,1998, 124( 3) : 208 -212.Appendix[1]alpha1=0.56;alpha2=1.76;beta1=1.16;beta2=0.98;X=20;K=0.65;g=10;C=5;V1=beta1*alpha1*K*X;V2=beta1*alpha2*K*X;V3=beta2*alpha1*K*X;V4=beta2*alpha2*K*X;L1=((beta1)^2*(alpha1)^2*K*X^2)/(2*g);L2=((beta1)^2*(alpha2)^2*K*X^2)/(2*g);L3=((beta2)^2*(alpha1)^2*K*X^2)/(2*g);L4=((beta2)^2*(alpha2)^2*K*X^2)/(2*g);S1=((1*V1)/(C+L1))*3600S2=((1*V2)/(C+L2))*3600S3=((1*V3)/(C+L3))*3600S4=((1*V4)/(C+L4))*3600W1=S1/(S1+S3);W2=S2/(S2+S4);W3=S3/(S1+S3);W4=S4/(S2+S4);I1=500+0.8*(abs(V1-X))^2*3.6^2+0.014*(abs(V1-X))^3*3.6^3;I2=500+0.8*(abs(V2-X))^2*3.6^2+0.014*(abs(V2-X))^3*3.6^3;I3=500+0.8*(abs(V3-X))^2*3.6^2+0.014*(abs(V3-X))^3*3.6^3;I4=500+0.8*(abs(V4-X))^2*3.6^2+0.014*(abs(V4-X))^3*3.6^3;Q1=1-(I1/(I1+I3));Q2=1-(I2/(I2+I4));Q3=1-(I3/(I1+I3));Q4=1-(I4/(I2+I4));P1=(W1+Q1)/2P2=(W2+Q2)/2P3=(W3+Q3)/2P4=(W4+Q4)/2S1 =2.8993e+003S2 =1.6144e+003S3 =2.8809e+003S4 =1.8482e+003P1 =0.5283P2 =0.4011P3 =0.4717P4 = 0.5989[2]alpha1=0.56;alpha2=1.76;beta1=1.16;beta2=0.98;X=8;K=0.65;g=10;C=5; V1=beta1*alpha1*K*X;V2=beta1*alpha2*K*X;V3=beta2*alpha1*K*X;V4=beta2*alpha2*K*X;L1=((beta1)^2*(alpha1)^2*K*X^2)/(2*g);L2=((beta1)^2*(alpha2)^2*K*X^2)/(2*g);L3=((beta2)^2*(alpha1)^2*K*X^2)/(2*g);L4=((beta2)^2*(alpha2)^2*K*X^2)/(2*g);S1=((1*V1)/(C+L1))*3600S2=((1*V2)/(C+L2))*3600S3=((1*V3)/(C+L3))*3600S4=((1*V4)/(C+L4))*3600W1=S1/(S1+S3);W2=S2/(S2+S4);W3=S3/(S1+S3);W4=S4/(S2+S4);I1=500+0.8*(abs(V1-X))^2*3.6^2+0.014*(abs(V1-X))^3*3.6^3;I2=500+0.8*(abs(V2-X))^2*3.6^2+0.014*(abs(V2-X))^3*3.6^3;I3=500+0.8*(abs(V3-X))^2*3.6^2+0.014*(abs(V3-X))^3*3.6^3;I4=500+0.8*(abs(V4-X))^2*3.6^2+0.014*(abs(V4-X))^3*3.6^3;Q1=1-(I1/(I1+I3));Q2=1-(I2/(I2+I4));Q3=1-(I3/(I1+I3));Q4=1-(I4/(I2+I4));P1=(W1+Q1)/2P2=(W2+Q2)/2P3=(W3+Q3)/2P4=(W4+Q4)/2S1 =2.0689e+003S2 = 2.7959e+003S3 =1.8259e+003S4 =2.8860e+003P1 = 0.5274P2 = 0.4795P3 = 0.4726P4 = 0.5205[3]alpha1=0.56;alpha2=1.76;beta1=1.16;beta2=0.98;X=30;K=0.65;g=10;C=5;V1=beta1*alpha1*K*XV2=beta1*alpha2*K*XV3=beta2*alpha1*K*XV4=beta2*alpha2*K*XL1=((beta1)^2*(alpha1)^2*K*X^2)/(2*g)L2=((beta1)^2*(alpha2)^2*K*X^2)/(2*g)L3=((beta2)^2*(alpha1)^2*K*X^2)/(2*g)L4=((beta2)^2*(alpha2)^2*K*X^2)/(2*g)S1=((1*V1)/(C+L1))*3600S2=((1*V2)/(C+L2))*3600S3=((1*V3)/(C+L3))*3600S4=((1*V4)/(C+L4))*3600W1=S1/(S1+S3)W2=S2/(S2+S4)W3=S3/(S1+S3)W4=S4/(S2+S4)I1=500+0.8*(abs(V1-X))^2*3.6^2+0.014*(abs(V1-X))^3*3.6^3 I2=500+0.8*(abs(V2-X))^2*3.6^2+0.014*(abs(V2-X))^3*3.6^3 I3=500+0.8*(abs(V3-X))^2*3.6^2+0.014*(abs(V3-X))^3*3.6^3 I4=500+0.8*(abs(V4-X))^2*3.6^2+0.014*(abs(V4-X))^3*3.6^3 Q1=1-(I1/(I1+I3))Q2=1-(I2/(I2+I4))Q3=1-(I3/(I1+I3))Q4=1-(I4/(I2+I4))P1=(W1+Q1)/2P2=(W2+Q2)/2P3=(W3+Q3)/2P4=(W4+Q4)/2S1 =2.6294e+003S2 =1.1292e+003S3 =2.7898e+003S4 =1.3159e+003P1 =0.5243P2 =0.3510P3 =0.4757P4 =0.6490。

For office use onlyT1________________ T2________________ T3________________ T4________________Team Control Number 26282Problem ChosenAFor office use onlyF1________________F2________________F3________________F4________________2014 Mathematical Contest in Modeling (MCM) Summary Sheet (Attach a copy of this page to your solution paper.)1.Introduction近年来，世界上的交通拥堵问题越来越严重，严重的交通拥堵问题引发了人们的对现行交通规则的思考。

在汽车驾驶规则是右侧的国家多车道高速公路经常遵循除非超车否则靠右行驶的交通规则，那么这个交通规则是否能够对交通拥堵起着什么作用呢？在汽车驾驶规则是右侧的国家多车道高速公路经常遵循以下原则：司机必须在最右侧驾驶，除非他们正在超车，超车时必须先移到左侧车道在超车后再返回。

根据这个规则，在美国单向的3车道高速公路上，最左侧的车道是超车道，这条车道的目的就是超车。

现在我们提出了4个问题：1、什么是低负荷和高负荷，如何界定他们？2、这条规则在提升车流量的方面是否有效？3、这条规则在安全问题上所起的作用？4、这条规则对速度的限制？1.1 Survey of Previous Research1.2 Restatement of the problem本题需要我们建立一个数学模型对这个规则进行评价。

我们需要解决的问题如下：●什么是低负荷和高负荷，如何界定他们？●这条规则在提升车流量的方面是否有效？●这条规则在安全问题上所起的作用？●这条规则对速度的限制？●对于靠左行的规则，该模型能否可以使用？？（待定）●如果交通运输完全在智能系统的控制下，会怎样影响建立的模型？针对以上问题，我们的解题思路和方法如下所示：◆我们根据交通密度对低负荷和高负荷进行界定，交通密度是指：在某时刻，每单位道路长度内一条道路的车辆数。