MCM北美数学建模论文模版.doc

Team # 4488 Page 9 of 9TitleAbstract: Key words: Contents1. Introduction.3 1.1 Why does toll way collects toll? .3 1.2 Toll modes3 1.3 Toll collection methods.3 1.4 Annoyance in toll plazas.31.5 The origin of the toll way problem.31.6 Queuing theory.42. The Description of Problem.5 2.1 How do we approximate the whole course of paying toll? .52.2 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 plaza6 2.2.3 Compromise.62.3 Overall optimization and local optimization.62.4 The differences in weights and sizes of vehicles.72.5 What if there is no data available? .73. Models.73.1 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.133.2 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.194. Conclusions.19 4.1 Conclusions of the problem.194.2 Methods used in our models.19 4.3 Application of our models.195. Future Work.19 5.1 Another model19 5.2 Another layout of toll plaza.23 5.3 The newly- adopted charging methods.236. References.237. Appendix.23 Programs and codes.24I. IntroductionIn order to indicate the origin of the toll way problems, the following background is worth mentioning.1.1 1.2 1.3 1.4 1.5 1.6 II. The Description of the Problem2.1 How do we approximate the whole course of paying toll?llll2.2 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 optimizationlll Virtually: 2.4 The differences in weights and sizes of vehicles 2.5 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. l l l l l 3.1.2 Assumptionsl l l l l3.1.3 The Foundation of Model1) The utility functionl The cost of toll plaza: l The loss of motorist: l The weight of each aspect: l Compromise: 2) The integer programmingAccording to queuing theory, we can calculate the statistical properties as follows.3) The overall optimization and the local optimizationl The overall optimization: l The local optimization: l 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 Resultl Local optimization and overall optimization: l Sensitivity: The result is quite sensitive to the change of the three parameters l Trend: l Comparison: 3.1.6 Strength and Weaknessl Strength: In despite of this, the model has proved that . Moreover, we have drawn some useful conclusions about . The model is fit for, such as l Weakness: This model just applies to . As we have stated, . Thats just 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.l l l l3.2.2 Additional Assumptionslll 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.l Step1: l Step2: l Step3: l Step4: l Step5: l Step6: l Step7: l Step8: l 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 Weaknessl 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.l Weakness: . Thus the model is still an approximate on a large scale. This has doomed to limit the applications of it. IV. Conclusions4.1 Conclusions of the problemll l 4.2 Methods used in our modelsll l 4.3 Applications of our modelslllV