美国大学生数学建模竞赛培训材料共299页.pdf
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1、2009cI?)?m?2008 c 12?8 F i c?2009cI?)?m3=?mc?o?n?d?F?mk?d?o?:1?I?0?S?;1?9?(?d?dI?n?http:/www.carroll.edu/kcline/mcm.pdf;1n?D?k?0308Cc?A?k?5g?L?n#?1o?A0?matlab!lingo?Nk?SKud?|?ka?P?AO?a?fi.?U?,?a?I?9?IS?p?J?6?m?D?1XJ?k?fl?X?mcm-dlu-?08c1?14F ii 888I?III?10.1I?)?m?m?o?.30.2mc?S?.4II?999?(?71?mmm?91.1S
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3、A Convenient Truth?A model for sea level rise forecast 1789The Most Expensive is not the Best21510?mmmKKK26210.1 2003 A-The Stunt Person.26210.2 2005 B-Tollbooths.26210.3 2005 C-Nonrenewable Resources.26310.4 2006 A-Positioning and Moving Sprinkler Systems forIrrigation.26410.5 2007 A-Gerrymandering
4、.26510.6 2008 A-Take a Bath.26510.7 2008 C-Finding the Good in Health Care Systems.266IV?AAA26911 Matlab?AAA27111.1 Cellular Automata in Matlab.27111.1.1 Matlab code considerations.27111.1.2 Examples.27211.2 Queuing Model in Matlab.27411.3 matlab?SK.279 v 12 lingo?AAA28312.1?55y.28312.2 015y.28412.3
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11、?yfi?5?nflK?fi?U/?flK?k?N?(J?J?L?“XJ?.&?.?y?z?k?I?ATL?/“5?w?L?I?3L?e?)”5?o?u“?16 12?(?)?N(J?JJJ?Nm=?.J?)?Y$1?u?(J?(?7L?1g/?7L?(?)?Y?A?Cz?CzN?,U?(?(J4?w?)?Y(?“/?3?A?e?U?w?2.7(?.?dddUUU?YYYkJ?(?=fi3?J?LX?”l?N?w?A?1?u?B 34?u?C67?”I?i5V)k?fl?z?A?J?i5?1?XJ3(J?pfiJ?”?A?N?wu?B?B?kgC?:”3(?c?”a?”?I?3?Lfi?N!?
12、(?.?d?)”?/?I?U?/?w“?L?V)5?i?L?)”:(J?*:?3pJ9?J?:9?5?b?y?nflK?J?U?Y?7I?k?du?vk5?93O?y”mk?m?/?wflK?N?r”2.8ooo?fi!L?qN?”?N?5?)IK!w“?L!L?!/?i?y?!?m?C?H?U?U?AT?i?!?AT?5K5/?mLIK?LiL9?U?C?k?2.8 o?17 IKIK?XJ?K?k?IK?5?j?wIK?L?)?6AT?J?Ng?ATS?IKIKfIK?IKNr?z?k(?8I?y?vkIK?=?k|u?K8?rKXeSN?”?.?I?)4?*:”1?*:1?*:1n?*:
13、1o?*:?a.?L?+?kn?1.?y?a2.rN?g?3.?N?5?XJ?U?$1?O?S?L?$1Azgzg?,?LXJU?L?/“|?u?=?U3e?+=yxplazaWWyxii where 1()denotes an indicator function and plaza denotes the matrix of cells.The Booth Tolls for Thee.75.Simulation and Results To determine the optimal number of tollbooths for a given number of highway l
14、anes,the cellular automata simulation is run for relevant combinations of the two.Recall one of our general assumptions is that the number of tollbooths in any plaza is at least equal to the number of highway lanes that are feeding it.An optimal tollbooth number is selected for a given number of hig
15、hway lanes when the system cost optimization method discussed earlier is applied.Recall the cost optimization method defines total cost as follows:()QBLBWNCtotal+=,Using the cellular automata model,we compute waiting time as a function of both the number of lanes and the number of tollbooths.For a f
16、ixed L,we compare all values of Ctotal and choose the lowest one.The results of this method are presented in Table 6.Table 6:Optimization for Cellular Automata Model Highway LanesTypical DayRush Hour122244356477589610117121381415162729Optimal#Booths As indicated in Table 6,there is fairly good agree
17、ment between the recommended number of booths for a typical day and for peak hours.However,we note that the optimal booth number for a typical day never exceeds that for rush hour.Rush hour seems to require slightly more booths than a typical day in order for the plaza to operate most efficiently.Ea
18、ch value in Table 6 is representative of approximately 20 trials.Through these trials,we noted a remarkable stability in our model.Despite the stochastic nature of our algorithm,each number of lanes was almost always optimized to the same number of tollbooths.There were a handful of exceptions;they
19、occurred exclusively for small numbers of highway lanes(3 lanes).Integer values are presented in Table 6 only because fractional tollbooths have no physical meaning.The Booth Tolls for Thee.76.Example As an example of one optimization using the cellular automata model,let us consider the instance of
20、 six highway lanes.For comparison,the analogous optimization is carried out previously in models 1 and 2.Figure 11:Minimization of mean waiting time for six lane roadway.Use of 10 tollbooths minimizes the mean wait for customers.Figure 11 is created by running the simulation repeatedly for six lanes
21、 and varying the number of tollbooths.A choice of ten tollbooths provides the lowest mean wait time for vehicles.However,ten is not necessarily the optimal number of tollbooths for the system.To determine the optimal solution,we must refer to the cost optimization function developed previously.The B
22、ooth Tolls for Thee.77.Figure 12:Minimization of total daily system cost for six lane roadway.Use of 10 tollbooths minimizes both the mean wait for customers and the system cost.As seen in Figure 12,ten tollbooths minimizes system cost as well as mean waiting time(Figure 11).Thus,ten tollbooths is t
23、he optimal number for a toll plaza with six incoming lanes(given our selection of parameters).Although the curves in Figures 11 and 12 look very similar,they are indeed more than scalar multiples.One notes that the differences between the two curves is most pronounced at the two ends.Discussion of C
24、ellular Automata Traffic Model Evaluation of Assumptions Let us now consider the assumptions made in the development of the cellular automata traffic model.In what way have these assumptions been either confirmed or discredited?Has there been an assumption which has proven to be particularly limitin
25、g?We first assumed that the plaza contains only three types of cells occupied,vacant,and forbidden.In fact,there are two other kinds of cells(flagged cells and incrementing booth cells).These arose as artifacts of the nature of the computer program and did not affect the dynamics of the system.Altho
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