2010美赛数学建模
数学建模美赛翻译-关于2010年数学建模美国赛b题的翻译
2010年美国数学建模邀请赛试题2010-02-19 09:09PROBLEM A: The Sweet SpotExplain the “sweet spot” on a baseball bat.Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when hit. Why isn’t this spot at the end of the bat? A simple explanation based on torque might seem to identify the end of the bat as the sweet spot, but this is known to be empirically incorrect. Develop a model that helps explain this empirical finding.Some players believe th at “corking” a bat (hollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap) enhances the “sweet spot” effect. Augment your model to confirm or deny this effect. Does this explain why Major League Baseball prohibits “corking”?Does the material out of which the bat is constructed matter? That is, does this model predict different behavior for wood (usually ash) or metal (usually aluminum) bats? Is this why Major League Baseball prohibits metal bats?PROBLEM B: CriminologyIn 1981 Peter Sutcliffe was convicted of thirteen murders and subjecting a number of other people to vicious attacks. One of the methods used to narrow the search for Mr. Sutcliffe was to find a “center of mass” of the locations of the attacks. In the end, the suspect happened to live in the same town predicted by this technique. Since that time, a number of more sophisticated techniques have been developed to determine the “geographical profile” of a suspected serial criminal ba sed on the locations of the crimes.Your team has been asked by a local police agency to develop a method to aid in their investigations of serial criminals. The approach that you develop should make use of at least two different schemes to generate a geographical profile. You should develop a technique to combine the results of the different schemes and generate a useful prediction for lawenforcement officers. The prediction should provide some kind of estimate or guidance about possible locations of the next crime based on the time and locations of the past crime scenes. If you make use of any other evidence in your estimate, you must provide specific details about how you incorporate the extra information. Your method should also provide some kind of estimate about how reliable the estimate will be in a given situation, including appropriate warnings.In addition to the required one-page summary, your report should include an additional two-page executive summary. The executive summary should provide a broad overview of the potential issues. It should provide an overview of your approach and describe situations when it is an appropriate tool and situations in which it is not an appropriate tool. The executive summary will be read by a chief of police and should include technical details appropriate to the intended audience.。
2010美国数学建模B题论文
Team#8254page1of16 Play your best hit:The mystery in baseball batMCM Contest Question ATeam#8254February22,2010Contents1Introduction (2)1.1Problem Restatement (2)1.2What is the sweet spot on earth? (2)1.3How to cork a bat? (2)1.4The material of baseball bat matters? (2)2Assumption (3)3A Simplified Model (3)3.1Model Description (3)3.2Finding the”sweet spot” (5)4An Augmented Model (6)4.1Model Description (6)4.2Why no corked bats and metal bats? (8)4.2.1Qualitative Analysis (8)4.2.2Quantitative Analysis (9)5Sensitivity (12)6Conclusion (13)6.1Strength (13)6.2Weakness (13)6.3Recommendation (13)7References (13)8Appendix (14)Team#8254page2of16 1Introduction1.1Problem RestatementEvery experienced hitter knows that there is a spot on the baseball bat that when hitting with this spot,the hand of the hitter feels no pain and the ball can be hit farther.This is the”sweet spot”.According to the theory of torque,this spot should be at the end of the bat,but the experience of the hitters proves it wrong.The location of”sweet spot”on a given baseball bat is approximately 6-1/2”from the end of the bat[1].Our purpose is to establish a model to give a scientific explanation.In addition,some players believe that drilling a cylinder in the end of the bat andfilling it with cork or rubber,namely,to cork the bat [2],enhances the”sweet spot”effect.We would like to found our model and use it to interpret it.What’s more,the question that whether the material of bats matters is also a part discussed in this paper.1.2What is the sweet spot on earth?There are many definitions of”sweet spot”[3].Here in this paper,we define it as the location where maximum energy is transferred to the ball.1.3How to cork a bat?”Corking”is to drill a1inch diameter hole6inches longitudinally into the bat’s barrel end.The structure of the real bat and the corked bat can be showed in Fig1.In our augmented model,we will analyze the effect of corking.1.4The material of baseball bat matters?Compared to the wood bats,aluminum bats is not easily be broke,and as the aluminum bats are hollow,the thickness of the shell can be manipulated so that the center of mass may be more closer to the handle,and consequently reducing the perceived weight while swinging.In this way,it increases the mobility of(a)Normal baseball bat(b)Corked baseball batFig1:Normal baseball bat and corked batTeam#8254page3of16 the hitter.What’s more,using aluminum bat can largely improve the energy of the ejecting ball,which may be too hard to catch or even dangerous for players. All those features lead to the prohibition of the using of aluminum bat[4].We will explain it in details later in this paper.Fig2:Cross section of aluminum bat2AssumptionAs the hitting process is too complex,we make the following assumptions in order to simplify the problems and establish our models:1.The better hitting effect means a larger exit speed of the ball and a shorteracceleration time of the bat.2.When the hitter swings,hitter-bat system rotates around a vertical axis that”penetrates”the hitter.3.The collision of the ball and the bat is one-dimensional.4.The hitter always hits the ball at the”sweet spot”.3A Simplified Model3.1Model DescriptionIn our simplified model,we omit the situations that the player might modify the baseball bat and concentrate on explaining why the”sweet spot”is not at the end of the bat but a few distances from the end.When you hit a ball,the bat vibrates in response.These vibrations travel in waves up and down the length of the bat.At one point,called”the node”, the waves always cancel each other out.If you hit the ball on the bat’s node, the vibrations from the impact will cancel out,and you won’t feel any stinging or shaking in your hand.Since little of the bat’s energy is lost to vibrations when this spot is hit,more can go to the ball[5].We will found a model based onTeam#8254page4of16 the characteristics showed as italics.In our model,we ignore the swing of the hitter’s arms when hit.We denote the spot where the hitter holds the bat as pivot, assuming that the pivot isfixed,and the bat is mounted on the pivot so that it can swing around the pivot freely.The parameters we use are given in the tabular Notation,and the force and motion diagram is showed in Fig3.NotationSymbols MeaningF ball the applied force in the collisionF px,F py the vertical and horizontal component forces of the force in the pivotcm the center of mass of the batp the impact pointpivot the spot where the hitter holds the batI pivot the moment of inertia of the bat with respect to pivotL x−y the distances between x and yαthe rotation angular acceleration of the batm bat the mass of the batL sum the whole length of the batFig3:The force and motion diagram of the batNext,we will analyze the model above with the knowledge of kinetics in order tofind the”sweet spot”.Team #8254page 5of 163.2Finding the ”sweet spot”According to the theorem of moment of momentum we have:F ball ·L pivot =I pivot ·α(1)and based on the theorem of motion of center of mass:∑F =ma c we have:F ball +F px =m bat ·L pivot −cm ·α(2)Therefore,with equation (1)and (2)we have:F hand =F ball m bat ·L pivot −cm ·L pivot −p I pivot−1 Thus we know that the horizontal component force is 0whenL pivot −p =I pivotm bat L pivot −cmIn other words,when the distance between the pivot and the p is I pivot m bat ·L pivot −cm ,the p is the sweet spot.In order to find the specific location of the ”sweet spot”on a certain bat,we quote the data from [6]and choose a C243wooden bat,thus we get the parame-ters L pivot −cm =0.42m ,I pivot =0.208kg ·m 2,L knob −pivot =0.15m ,m bat =0.905kg ,L sum =0.86m .Then we calculate L pivot −p =0.55m ,so that L knob −p =0.70m .From the re-sult we can obviously see that the ”sweet spot”is about 0.16m far from the end of the fat part of the bat.The forces on different location of the bat can be showed in Fig4.Fig 4:The rotation system in a hitting processIn this way,we have successfully demonstrated that the experience of the hitters is right.Team#8254page6of16 4An Augmented Model4.1Model DescriptionThe previous model fails to take the situation that some hitter may modify the bat into account.What’s more,in real baseball game it is impossible that the arms of the hitter are stationary when hit.So we augmented ourfirst model. In this augmented model,we assume that the hitter and the bat form a rotation system in which the hitter stays vertical and the bat stays horizontal.The hitting process can thus be modeled as:the system gets a torque T generated by the hitter and rotates around the axis which vertically”penetrates”the hitter and then hits a ball.After hitting process,the ball ejects out with a velocity of v f. The distance between the impact spot and the axis is r.The process is described in Fig5.Fig5:The rotation system in a hitting processWe notice that the ball gets maximum velocity after hitting means the effect of hitting is optimal,so we should focus on the exit speed of the ball.Two processes are displayed as follows:Process1:The hitter swings the bat and accelerates the bat to an angular velocity ofω.Process2:After the imperfect inelastic collision of bat and ball,the ball ejects out with a velocity of v f.In process1,the moment of inertia of bat and the air resistance will hamper the rotation of the system of hitter-bat.It is easy to calculate the liner velocity of any spot on the bat by the kinematics formula v=ωr,and according to the formulas given by Keith Koenig in reference[7],we know that the angular ve-Team#8254page7of16 locity and the linear velocity of the impact spot on the bat before the collision in process1are:ω=TK Dtanhcosh−1expK DI hitter+I batθv bat=rω(3) where•r is the rotation radius,equals the length from the impact spot to the axis.•T is the torque applied to the hitter-bat system which is generated by the hitter.•K D is an aerodynamic parameter,which is given byK D=12ρC D DL44(for more please read reference[7])•tanh is hyperbolic tangent function.•cosh−1is anti-hyperbolic cosine function.•exp is exponential function.•θis the angle that the bat rotates.•I hitter is the hitter’s moment of inertia with respect to the axis.•I bat is the bat’s moment of inertia with respect to the axis.In our case,we treat T,K D,I hitter,θas constant parameters while r and I bat are the only variables.In process2,we assume that the bat and ball have an one-dimensional col-lision and they both have some extent of deformation.The deformation trans-forms a small part of the kinetic energy(about25%,reference[8])into potential energy stored in bat and ball,while most part(about75%)of it dismisses due to the friction and the oscillation of the bat.And referring to reference[7]we havev f=COR−r M1+r Mv ball+COR+11+r Mv bat(4)where•COR is the coefficient of restitution,it will be discussed in the next part.•v ball is the velocity of the ball right before the collision with the bat.Team #8254page 8of 16•v bat is the velocity of the bat right before the collision with the bat.by equation (3)and (4),we getv f =COR −r M 1+r M v ball +COR +11+r M r T K D tanh cosh −1 exp K D I hitter +I batθ (5)The parameters:COR ,I bat ,r M are explained in Appendix .4.2Why no corked bats and metal bats?In order to reveal the influence of each parameter on the velocity of the ball thus to figure out the effect of the corking behavior and predict different be-havior for wood and metal bats,we make some analysis about the application procedure of our augmented model.4.2.1Qualitative AnalysisWe assume that the velocity of the ball is constant before the collision with the bat,and the impact spots are on the same position of the bat,namely,r is a constant.The corking has two aspects of influences:one is the decrease of mass,the other is the deviation of the center of mass.1.Corked bat•The influence on v fDue to the lower density of material corked in the barrel,the whole mass of bat inclines and the center of mass moves to the rotation axis a little.On one hand,it will cause the decrease of I bat as I bat = r 2dm ,and according to the expression of v f :v f =COR −r M 1+r M v ball +COR +11+r M r T K D tanh cosh −1 exp K D I hitter +I batθ we know that the value of r T K D tanh cosh −1 exp K D I hitter +I batθ will increase.On the other hand,as we haver M =m ball (z −z p )2I cm +m bat L 2pivot −cmTeam#8254page9of16 we know that if m bat decreases and the center of mass deviates toknob,I cm and L pivot−cm will both decline,and that will make r M in-crease and lead to the decline of both COR−r M1+r M v ball and COR+11+r M.How-ever,it is impossible for us to ensure whether the value of v f willincrease or decrease without specific data.We will explain it later inthe Quantitative Analysis section.•The influence onflexibilityWe regard the procedure that the hitter-bat system is accelerated fromstationary situation to rotating with an angular velocity ofωas auniformly accelerated motion procedure.By kinematic formulas weknow that the acceleration time is t=2θω,and from the expression ofωmentioned in page6,section4.1,we know that as the I bat decreasesdue to corking,the value ofωwill increase,so that the accelerationtime will be shorten and in this way theflexibility develops.2.Metal bats vs.wood batsUsing our augmented model we are able to analyze the behavior of bats that have different composition material.•The influence on v fMetal bats(usually aluminum bats)have trampoline effect due togood elasticity,so that it has larger COR than wood bats(usually ashbats).In addition,as the aluminum bats have less mass,the influenceof material is just like that in discussing corked bats.It is obvious thatin order tofind which kind of bat’s v f is larger we need the supportof data.•The influence onflexibilityThe aluminum bats are lighter than wood bats so that the influenceonflexibility is the same as corked bats.4.2.2Quantitative Analysis1.Data usingAs the qualitative analysis cannot show the exact differences between wood bats,corked wood bats(corked bats)and aluminum bats,we quote the data from reference[7][12][13][14]to verify our augmented model.Table1:Value of ConstantsTeam#8254page10of16Constants Valuem ball/(g)134.2v ball/(m/s)25(assumption)T316N·mK D/(N·m·s2)0.00544θ 2.36radr/(m)0.762R hitter−pivot/(m)0.305I hitter/(kg·m2)0.444Table2:Value of VariablesValues Wood bat Corked bat Aluminum batCOR0.500.500.58mass/(g)876834827I p/(kg·m2)0.2110.1950.17L cm−pivot/(cm)424039.5I bat/(kg·m2)0.5160.4760.446z−z p/(m)0.470.460.45r M0.14030.14540.1596 Using the data above we can calculate the three relative values of v f that are47.1m/s,47.6m/s and51.3m/s.The I bat is calculated based on parallel axis theorem.For a better compare of the hitting effects of the three bats,we use com-puter to make a hitting experiment simulation about theflying trajectories of the ball after hitting:Assuming that the ball dose a slanting parabolic motion and the launch angle of the ball after hitting with the bat isθ=30◦.Fig6shows the trajec-tories of balls that are hit by the three kinds of bats.2.The influence of three kinds of bats onflexibilityCombine the formula t=2θωwith the expression ofω,we have the expres-sion of the acceleration time t:t=2θTK Dtanhcosh−1expK DI hitter+I batθAfter using the values given in the table1and table2,we get:Kinds of bats t/(s)Wood Bats0.121Corked Bats0.118Aluminum Bats0.116Team#8254page11of16Fig6:The trajectories of the balls hit by the three kinds of bats It is easy to see that the acceleration time of corked bat is0.003s shorter than that of wood bat.During this time,the incoming ball moves0.075m farther,that is to say,theflying distance of the incoming ball from the ser-vice point to the collision point when using a corked bat is0.075m farther than that of wood bat,so that the hitter would feel more easy to deal with the incoming ball as he or she has more time to react and accelerate the bat when using a corked bat.In the same way we know that when usinga aluminum bat rather than a wood bat,the hitter has0.005s longer and0.125m farther to handle the bat.3.Summary of our analysisAt this point,we are able to answer the second and third questions about the corking behavior and the matters of different materials.•Why does Major League Baseball prohibit”corking”?From the analysis above we know that for a baseball bat used in ourmodel,if it is corked,the swinging velocity of bat right before hittingthe ball will increase0.5m/s and theflexibility of swinging can alsobe developed,as the mass of the bat decreases and the center of massof it deviates.•Why does Major League Baseball prohibit metal bats?According to our model,the hitting effect is closely linked with thematerials.We know from our qualitative verification that the hittingvelocity of using a aluminum bat is4.2m/s more than that of woodbat,and the deviation of the center of mass further gains0.005s reac-tion time for the hitter.As the ball hit by aluminum bat has a muchbigger velocity,it makes the catching of the ball difficult and evendangerous.Team#8254page12of16 5SensitivityAfter analyzing the application of our augmented model,wefind it neces-sary to analyze the sensitivity of our model,aiming at implementing it more effectively.We choose to research the exit speed of normal wood bat with differ-ent parameters.1.The influence of massThe change of mass will influence both r M and I bat,but as the value of them are too hard to obtain by direct calculating,we make some simplificationsbelow:r M=m ball (z−z p)2I p=m ball(z−z p)2I cm+m bat L2pivot−cmIn this expression,as I cm is very small(about0.04kg·m2),we treat it as a constant one.It is the same to z−z p(about0.47m)and L pivot−cm(about 0.42m).Hence,the value of r M only correlates to m bat,and we haver M=0.1340.4720.04+m bat×0.422then in I bat=I cm+m bat·L2hitter−cm,we also treat L hitter−cm(about0.725m) as a constant,so we haveI bat=0.04+m bat×0.7252Relating to equation(5),we denote COR=0.5and get:m bat(kg)0.830.850.870.89v f(m/s)46.846.7646.7146.672.The influence of materialThe major influence of different materials is they have different COR s.When the mass is constant,we have I bat=0.516kg·m2,and r M=0.1403, andfinally we get:COR0.40.450.50.550.6v f/(m/s)42.344.747.149.551.9From the results above,we can see that COR seems a more prominent influence.Team#8254page13of16 6Conclusion6.1StrengthFirst of all,this paper solves the problem of”sweet spot”,and we give an easy formula to calculate the position of”sweet spot”.Secondly,we analyze the hitting process and divide it into two stages,and discuss the factors that affect the exit speed in details while giving a formula that can describe this stage.Then we answer the questions through qualitative analysis and quantitative calculation.Finally,we make a analysis about the sensitivity and prove the rationality by comparing the results.6.2WeaknessFirst of all,as the real hitting process is too complex to analyze,we make several simplifications in order to facilitate the founding of model.In model 1,we just regard the bat as a pendulum rod with one endfixed,which is a little different from the real situation.And in model2,we simplify the complex process of hitting into two stages.Secondly,the data wefind are not precise,especially for the value of COR which we regard as constant.Additionally,the calculations in this paper are also simplified,thus the accu-racy of our results declines.6.3RecommendationThe biggest disadvantage of our model is lacking experiments,and if we have time and facilities to do some experiments,the result must be more reliable.For example,the equation(6)in Appendix can be used to measure COR,and in order to measure the value of I pivot,we could refer to[15]and use the method to obtain data.With this data we can verify our model in a better way.7References[1]/wiki/Sweet spot[2]/drussell/bats-new/corkedbat.html[3]/drussell/bats-new/sweetspot.html[4]/wiki/Aluminum Bats vs.Wood Bats[5]/baseball/sweetspot.htmlTeam#8254page14of16[6]/sysengr/slides/baseballBat.ppt[7]Keith Koenig,Nan Davis Mitchell,Thomas E.Hannigan,J.Keith Clutter.The influence of moment of inertia on baseball/softball bat swing speed.SportsEngineedng(2004)7,105-117.[8]Alan M.Nathana.Characterizing the performance of baseball bats.Am.J.Phys.,Vol.71,No.2,February2003134-143[9]/wiki/Coefficient of restitution[10]Lv ZhongjieHuang Fenglei.Coefficient of Restitution of a Circular PlateDuring Inelastic Collision.Transactions of Beijing Institute of Technology.Vol.28No.4.[11]P.J.Drane and J.A.Sherwood.Characterization of the effect of temperatureon baseball COR performance.[12]/sysengr/slides/baseballBat.ppt[13]/docs/621958/How-Does-a-Baseball-Bat-Work[14]/wiki/moment of inertia[15]/drussell/bats-new/bat-moi.html8AppendixParameters ExplanationIn expression(5),page8,section4.1,there are several parameters that will influence thefinal velocity of the ball:1.COR•What is COR?The coefficient of restitution(COR),or bounciness of an object is afractional value representing the ratio of velocities after and before animpact[9].Fig7:The one-dimensional collision processTeam #8254page 15of 16The coefficient of restitution is given byCOR =v 1f −v 2f v 1−v 2(6)where–v 1is the velocity of object 1before the collision.–v 2is the velocity of object 2before the collision.–v 1f is the velocity of object 1after the collision.–v 2f is the velocity of object 2after the collision.All the parameters above are scalars.In the ideal situations,we may have a so-called plastic collision when COR =0,namely the deformation of the material cannot re-store.And when COR =1,called perfectly elastic collision,is a situa-tion that the deformation can restore entirely.In general,the value of COR varies from (0,1).•What factors affect COR ?MaterialCOR represents the deformation recovery ability of the material.Gen-erally speaking,the more elastic the material is,the higher the value of COR will be.Impact velocityCOR decreases when the impact velocity increases.[10]The Temperature and Relative Humility of The EnvironmentCompared to the factors above,another two factors,the tempera-ture and relative humility have a relatively smaller influence.COR decreases when the temperature decreases and it decreases when the relative humility increases.[11]2.I bat•MOI (moment of inertia)[14]Moment of inertia is a measure of an object’s resistance to changes in its rotation rate.It is the rotational analog of mass,the inertia of a rigid rotating body with respect to its rotation.The moment of iner-tia plays much the same role in rotational dynamics as mass does in linear dynamics,determining the relationship between angular mo-mentum and angular velocity,torque and angular acceleration,and several other quantities.It is denoted asI = r 2dmwhere m is mass and r is the perpendicular distance to the axis of rotation.Team#8254page16of16•Parallel Axis TheoremI z=I cm+mL2where I cm is the moment of inertia of the rotor with respect to thecenter of mass,and the L is the distance from the center of mass toaxis z.•The factors affect MOIFigure:It influences the location of the center of mass,thereby affectsthe distance from the center of mass to rotation axis.Mass:Its increase is proportional to the increase of MOI.3.r MIn reference[8],Alan M.Nathan develops a formula relating v f to the initial speed of the ball v ball and the initial speed of the bat at the impact pointv bat as:r M=m ball (z−z p)2I pwhere•m ball and m bat are the ball and the bat’s mass respectively.•Z is the location of the impact point.•Z p is the location of the pivot point.•I p is the moment of inertia of the bat with respect to the pivot point. From the expression above we can see that reducing the mass of the bat m bat while keeping the other parameters constant will lead to a augment of r m.。
2010年北美数学建模竞赛中英文A、B题目
2010年北美数学建模比赛中英文题目(MCM)2010 MCM题目A题：棒球棒上的最佳击球点Explain the “sweet spot” on a baseball bat.Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when hit. Why isn’t this spot at the end of the bat? A simple explanation based on torque might seem to identify the end of the bat as the sweet spot, but this is known to be empirically incorrect. Develop a model that helps explain this empirical finding.Some players believe that “corking” a bat (hollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap) enhances the “sweet spot” effect. Augment your model to confirm or deny this effect. Does this explain why Major League Baseball prohibit s “corking”?Does the material out of which the bat is constructed matter? That is, does this model predict different behavior for wood (usually ash) or metal (usually aluminum) bats? Is this why Major League Baseball prohibits metal bats?中文翻译：解释棒球棒上的“最佳击球点”。
2010MCM(数学建模美赛)
2010 Mathematical Contest in Modeling (MCM) Summary Sheet(Attach a copy of this page to each copy of 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.SummaryTo reduce the hunting time for investigation and ensure safety of the general public, we optimized the Circle Principle by Golden-section search, so that an accurate geographical profile was able to be drawn in a short time, through error testing by the title given case of Peter Sutcliffe, our models are really useful, time-saving and accurate.Task I: we use the basic Circle Principle confirming a primary geographical profile, combining and the method of probability. Then, a smaller profile appeared, and after computer programming, the result is very reasonable.Task II: in this question, we mainly take Gray Forecast and GM[1,1] predicting the next location. But the highlight is the conjunction with our balance theory, factors influencing the prediction were all set as weight, How to keep balance is the special way to forecast the next spot.Task III: same as Task II, the only different is the number of weight was raised.Task IV: the error is the distance between the realistic location and predicting location, in addition. it’s quite low.Key word: Golden-section Search Gray Forecast Balance TheoryThe Center of Geographical Profile Primary Geographical ProfileCircle PrinciplePROBLEM B: An Easy Profiling ModelContent1. Introduction & Assumption (2)1.1 Introduction (2)1.2 Assumption (3)2. Solution of Task I (3)2.1 Analysis (3)2.2 Profiling model (3)2.2.1 Primary scheme (3)2.2.2 Target scheme (4)2.3 Model Testing (4)3. Solution of Task II (6)3.1 Predicting model (6)4. Solution of Task III (9)4.1 Intensive model (9)5. Solution of Task IV (9)5. Sensitivity Analysis (9)6. Executive summary (13)7. Reference (14)8. Appendix (15)1. Introduction & Assumption1.1 IntroductionCriminology helps the police dealing with cases more effectually, especially for arresting suspect of serial criminal. Reducing hunt time is very imperative for the public safety, consequently a new word was born among a long period of criminal investigation, which is “geographical profile” [1]. And it is an investigative methodology that makes use of the locations of a connected series ofcrimes to determine the most probable area of offender residence. Challenge to predict the residence of a offender accurately and quickly is the most important problemIn our paper, we mainly try to put forward a method to ascertain the geographical profile easily and credibly. In addition, combining the results of profiling model, factors and extra information, we can get a better result by verification.1.2 Assumption·Suppose a continues offend will not change his criminal type .2. Solution of Task I2.1 AnalysisAccording to request of this problem, at least two different schemes were proposed to generate the geographical profile. In order to facilitate Task I, we put forward two schemes, the former one is mainly about to confirm a wide area which has the maximum possibility to find out the suspect, in other words, we could get a primary geographical profile; the latter one could dwindle the range of the profile. Through the schemes above, the final geographical profile has been generated. The schemes above will involve the following a principle of geographical profile.Theoretical Principles of geographical profile1. Circle hypothesis theoryMark all the crime on the map. Suppose that these crimes were committed by a single person who has not yet been found. And to find the distance between two furthest location of crime, Make a circle including all the crime sites, which diameter is the distance between the two locations. Assumption is that the offenders live in the circle. May be somewhere near the center of the circle. People will find that 80% of rapists truly live within the circumference. More than 60% offenders live in this center area of the circle.2.2 Profiling model2.2.1 Primary schemeThis part is primarily about drawing a wide profile of the potential suspect’s residence, and the method that we hold is based on the theory of Circle hypothesis, the process presents as followStep 1. Confirming each past criminal geographical position and figuring all the positions aspoints , then we need to find out the longest distance among each pair of two points in the whole figure so that we could get the criminal center of circle based on circle hypothesis[2].Step 2. Expressing the method in Step 1 into the form of mathematics, we could transform the geographical coordinate into a planar plane. Let x denotes the longitude, y denotes the latitude. To simplify the type of value, if '''o x a b c =,'''o y d e f =,then we convert degree to united form by formula ,603600603600o o b c e f X a Y d =++=++. so that we get all the planar coordinates of criminal locations, due to circle hypothesis, the longest distance of every two points among the figure is able to be found. Getting the coordinates of the two points, we eventually reach the criminal center of circle. For instance, the coordinates of the two points are respectively 11(,)x y and 22(,)x y So the center of circle is1212(,)22x x y y C ++=. Step 3. Drawing a circle of radius with half of the maximum distance above, the primary geographical profile has been made.2.2.2 Target schemeStep 1. As we know that suspect may appear everywhere in this circle, but it ’s boring and time-consuming to check out every place. To find out the residence quickly and accurately, we want to confirm a smaller circle in the primary circle based on the Golden-section search [3]. Then, drawing the smaller circle of 0.618 times the radius, we finally get a smaller geographical profile.Step 2. Determining the final profile, we decide to use a special method that the inspiration is from a TV Series which called Numb3rs. First, supposing (,)E x y is any point on the smaller circle and there are n points on the figure. Second, we require all the distances between the n points and E . At last, through matlab programming, the minimum standard deviation of distances is able to calculate so that we could get the best point.Step 3. Taking the best point as a new center of circle, draw another circle of 0.618 times the radius, In the end, the final circle is our accurate geographical profile.2.3 Model TestingIn order to check out whether the accuracy of our profiling model, we decided to use the case ofPeter Sutcliffe testing our method model. Through the above steps, we have gotten the primary geographical profile which is the big circle in the follow drawing. Then，by means of Target scheme, the circle has been curtailed gradually. And finally, we get the prediction location which is the center of last geographical profile.Flowchart of geographic profile generate algorithm23456789102345678910Geographical ProfilexyFig 1 geographical profile by two schemes3. Solution of Task II3.1 Predicting modelFirst of all, we need to confirm that our purpose is predicting the next distance between criminal location and the center of circle by the method of circle hypothesis and locations of serial past criminal cases. Ronald M. and others have assessed more than 800 murders, additionally, an amusing finding about walking distance of the criminal perpetrators was found. With the increase of the criminal number, his guts and experience have been improved a lot so that his walking distance and the scope of victims are all on the climb. [4]Consequently, as time goes by, the distances between criminal locations and center of circle is an increasing sequence, gray time series forecasting using the system the quantity of the time series forecast. That is the case with the known locations to the center of a circle from the time series to construct the gray prediction model to predict the next location of the crime to the distance from the center of the circle.To predict the next case centered on the distance to pitch again after the next place is to determine the exact location.According to the ecological balance of the biosphere principles of energy and matter in the biosphere constant flow and circulation. Solar energy is the energy source of all life activities, throughout the biosphere at the center.For the next location prediction of crime in order to ensure that the entire system is in equilibrium principle, be determined through experiments. The initial outline of the offender can be seen as activities of the criminals circle, if the offender to a continuous crime, and criminal activities on the need to ensure that insiders are in a state of equilibrium flows. The center circle of the sun is similar to the biosphere.To center of a circle as a fulcrum, with locations in every known case under the hanging weight balance of the whole circle. Weights left on behalf of a criminal information, the more information weights have increased, suggesting that this crime for the entire system, the greater the proportion and significance.Known place of the incident location and the weight each time the number of cases, but also determined the next case of the distance from the center circle, the very next case to determine location. The algorithm in detail is as follow(0)(0)(0)(0),...,(1),(2)(1).d d d d n +(n)，computingInitial Sequence (0)(0)(0)(0){(1),(2),...,()}D d d d n =Normalized (0)(0)(0)(0)(0)(0)(0)(2)(3)(){1,,,...,}(1)(1)(1)d d d n UDd dd = Cumulative (1)(0)(1)(0)(0)(0)(1)(0)(0)(0)(0)(0)(1)(0)(0)(0)(1)1,(2)(2)1,(1)(2)(3)(3)1,(1)(1)......(2)(3)()()1....(1)(1)(1)d d d d d d d d d d d d n d n d d d ==+=++=++++ )1(z means value for the (1)d series,(1)(1)(1)1()()(1),2,3,4,...,2z k d k d k k n ⎡⎤=+-=⎣⎦ then(1)(0)(1)(1)(1)((2),(3),(4),...,()).z z z z z n =Define Gray differential equation model GM(1,1) as(0)(1)()()d k az k b +=let 2,3,4,...,k n = substitute into GM(1,1) model(0)(1)(0)(1)(0)(1)(0)(1)(2)(2),(3)(3),(4)(4),......()(),d az b d az b d az b d n az n b +=+=+=+=let (0)(0)(0)(0)((2),(3),(4),...,())T N Y d d d d n =，T b a u ),(=，(1)(1)(1)(2)1(3)1......()1z z B z n ⎡⎤-⎢⎥-⎢⎥=⎢⎥⎢⎥-⎣⎦then express GM(1,1) model as a matrix equation N Y B u =⋅the Least-squares method is used to conform the value of u .^1^ˆ()T T N u B B B Y a b -⎡⎤⎢⎥==⎢⎥⎢⎥⎣⎦，^^,.a b a b == Discrete solution of differential equations gray concrete expression is(1)(0)(1)((1))ak b bd k de a a -+=-⋅+k n =，st. (1)(1)d n +，calculate (0)(1)(1)(1)(1)().d n d n d n +=+- working out (0)(1)d n +。
2010 美赛 MCM 优秀论文
3 Center of Minimum Distance Model.................................. 5
数学建模竟赛的情况简介
第1教案数学建模及竞赛知识介绍目的要求:1. 了解数学建模的基础知识、相关的基本概念；2. 了解数学模型的特点和学习方法；3. 掌握数学建模的具体过程和步骤，教学重点及难点:重点：了解数学建模的一般步骤和方法，体会如何用数学的语言和方法表述和解决实际问题。
难点：体会如何用数学的语言和方法表述和解决实际问题。
教学方法手段:讲授法，案例教学法，多媒体创新点:应用和创新是数学建模的特点，也是素质教育的灵魂；不论用数学方法解决哪类实际问题，还是与其他学科想结合形成交叉学科，首先的和关键的一步是用数学的语言表述所研究的对象，即建立数学模型。
在高科技，特别是计算机技术迅速发展的今天，计算和建模正成为数学科学技术转化的主要途径。
教学过程:1．1 从现实对象到数学模型本节先讨论原型和模型，特别是数学模型的关系，再介绍数学模型的意义。
原型和模型原型（prototype）和模型（model）是一对对偶体。
原型指人们在现实世界里关心、研究或者从事生产、管理的实际对象。
在科技领域通常使用系统（system）、过程（process）等词汇，如机械系统、电力系统、生态系统、生命系统、社会经济系统，又如钢铁冶炼过程、导弹飞行过程、化学反应过程、污染扩散过程、生产销售过程、计划决策过程等。
本书所述的现实对象、研究对象、实际问题等均指原型。
模型则是指为某个特定目的将原型的某一部分信息减缩、提炼而构成的原型替代物。
特别强调构造模型的目的性。
模型不是原形原封不动的复制品，原型有各个方面和各种层次的特征，而模型只要求反映与某种目的有关的那些方面和层次。
一个原型，为了不同的目的可以有很多不同的模型，模型的基本特征是由构造模型的目的决定的。
例如：展厅里的飞机模型：外形上逼真，但是不一定会飞；航模竞赛的模型飞机：具有良好的飞行性能，在外观上不必苛求；飞机设计、试制过程中用大的数学模型和计算机模拟：要求在数量规律上真实反映飞机的飞行动态特征，毫不涉及飞机的实体。
2010数学建模竞赛B题分析
选取近3次综合世博会作为上海世博会纵向影响力的参考，它们分 别是1992年西班牙塞维利亚世博会，2000德国汉诺威世博会以及2005 年日本爱知世博会，将从上述城市竞争力的各个方面进行讨论，表现上 海世博会对城市竞争力影响的大小。其数据如下表：
年份
主办方
世博投资 （亿元）
世博收益 （亿元）
举办市第三产 业产值增长率
C
I f In If
100%
If
In
正常举行世博会上海净投资额 假设不举办世博会上海净投资额
通过查找相关资料我们得出本次上海市由亍世博会而进行的基础配 套设施投资（丌包括园区投资）总额预测值为3200亿元左右[1]。而丌丼 办世博会的投资额预测值为2346.8亿元（见表2）。利用上述公式计算得 到世博会的相关净投资率约为0.363。 带动投资在上海丌丼办世博会情况下和丼办世博会情况下的比较值 为 0.637:1。
灰色系统(Grey System)理论是我国著名学者邓聚龙教授2O世纪8O年代初 创立的一种兼备软硬科学特性的新理论．该理论将信息完全明确的系统定义为 白色系统，将信息完全丌明确的系统定义为黑色系统，将信息部分明确、部分 丌明确的系统定义为灰色系统．由亍客观世界中，诸如工程技术、社会、经济 、农业、环境、军事等许多领域，大量存在着信息丌完全的情况．要么系统因 素戒参数丌完全明确，因素关系丌完全清楚；要么系统结构丌完全知道，系统 的作用原理丌完全明了等，从而使得客观实际问题需要用灰色系统理论来解决 。
现在来比较世博会的海外影响力，首先注意到在世博会开幕乊前， 关键字Shanghai expo的点击指数较为稳定，保持在7左右。为了比 较世博会前后点击率的差别，我们对世博会期间点击率的积分结果取 平均数，得到世博会当年，关键字Shanghai expo的平均点击指数 24.5657。 上述结果表明：上海世博对上海的知名度影响力贡献比达 1:3=0.3333:1。
2010A美赛特等奖
Every hitter will have such feeling of vibration in hand at the bat gripping part, when the hitting spot is Spot A at the end of bat or at an upper Spot C. And sometimes it is a strong vibration but the ball can not go very far. And when the ball and the bat hit directly at the Spot B at the proper position of the bat (as shown in the following Fig. 1), the ball can go faster and farther without giving the hitter the feeling of strong vibration.
To simplify the problem, the ball is still before hitting, and then:
v=
(1 + e)ω 0 x p 1+ mx 2 p Jz
The momentum
P of the ball after being bit is: (1 + e ) m ω 0 x p P = mv = mx 2 p 1+ Jz
ω0
are the revolving angular velocity of
the bat on Point O before and after hitting respectively, v0 and v are the tangential velocity of the ball along the bat before and after hitting respectively, u0 and u are the linear velocity of the ball before and after hitting respectively. M and m are the mass of the bat and the ball respectively. Taking axle Z as the revolving axle, x p = H as the distance from the center point O to the contact point of the bat and the ball, xc = R as the distance from the center point O to the mass center of the bat, the following equation can be obtained according to the theorem of moment of momentum of collision. That is Use
2010年数学建模B题
2010年上海世博会影响力的定量评估摘要世博会是一项享誉全球的大型活动，素有“经济奥林匹克盛会”之称，其规模之大、参赛人数之多、影响力之大对东道国和举办城市的旅游业的影响是一般单项活动所不能匹敌的，这些通过历史数据和资料可以得到印证。
世博会所具有的国际影响力，为上海成为现代化国际旅游城市提供了很好的契机，其蕴含的意义和影响是极其深远的。
针对该题我们选择从上海旅游业的发展来评估上海世博会的影响力。
首先为评价上海至申办世博成功前后，世博效应对上海旅游产业的拉动作用，建立评价指标体系，取2000年到2009各年数据为样本，建立评价模型（模型一），采用投影寻踪方法，运用DPS 8.01数据处理软件。
结论如下:变量投影方向分别为x1= 0.1793，x2=0.1482，x3=0.1581，x4=0.2557，x5=0.403，x6=0.4347，x7=0.3138，x8=0.0996，x9=0.3166，x10=0.2909，x11=0.4053，x12=0.216;样本投影值为（-3.8312，-3.2739，-2.5318，-2.5318，-0.7344，0.5714，1.6351，2.9655， 3.8656，3.8656）。
从中可以看出：从2002年上海市申请世博会成功后，随着大量资金的投入，其对上海市旅游业的拉动作用越来越显著。
然后通过预测数据，对历届世博会对举办城市旅游业的影响，世博园的游客量，上海举办世博与否对上海旅游业的影响，世博会的负面影响分析等方面进行研究。
可以将上述过程分为三个阶段。
第一阶段：从已知的2010年5月到8月进世博园参观人数（图形1）分析，建立GM（1，1）模型，预测出上海世博园的游客总量约为7208.196万人次。
又查得相关数据，分析历届世博会对举办城市旅游业的影响（表1），运用文献分析法研究世博会对举办城市旅游业产生的影响。
第二阶段：结合已知的4月、5月、6月、7月上海旅游人数的数据资料，建立GM（1，1）模型，预测出2010年上海市8月、9月、10月的游客总量分别为775773人、794463人、813603人，又查出2006年到2009年各月来沪旅游总人数，建立表2：2006-2010年上海市旅游人数，使其与2010年同期作比较做出折线图（图形2），并对图形分析得：随着年份的增长，上海市的游客数量也在不停增长，且世博会期间的游客量增长较大。
2010年数模美赛参赛要求(中英文)
2010国际大学生数学建模竞赛参赛帮助中英文对照MCM: The Mathematical Contest in ModelingICM: The Interdisciplinary Contest in ModelingMCM：数学建模竞赛ICM：交叉学科建模竞赛Contest Registration and Instructions竞赛注册和指导(All instructions and rules apply to ICM as well as to MCM, except where otherwise noted.)（所有MCM的说明和规则除特别说明以外都适用于ICM）To participate in a contest, each team must be sponsored by a faculty advisor from its institution.每个MCM的参赛队需有一名所在单位的指导教师负责。
Team Advisors: Please read these instructions carefully. It is your responsibility to make sure that teams are correctly registered and that all of the following steps required for participation in the contest are completed:Please print a copy of these contest instructions for reference before, during, and after the contest. Click here for the printer friendly version.指导老师：请认真阅读这些说明，确保完成了所有相关的步骤。
2010年数学建模试题(全部)
2010高教社杯全国大学生数学建模竞赛题目（请先阅读“全国大学生数学建模竞赛论文格式规范”）A 题 储油罐的变位识别与罐容表标定通常加油站都有若干个储存燃油的地下储油罐，并且一般都有与之配套的“油位计量管理系统”，采用流量计和油位计来测量进/出油量与罐内油位高度等数据，通过预先标定的罐容表（即罐内油位高度与储油量的对应关系）进行实时计算，以得到罐内油位高度和储油量的变化情况。
许多储油罐在使用一段时间后，由于地基变形等原因，使罐体的位置会发生纵向倾斜和横向偏转等变化（以下称为变位），从而导致罐容表发生改变。
按照有关规定，需要定期对罐容表进行重新标定。
图1是一种典型的储油罐尺寸及形状示意图，其主体为圆柱体，两端为球冠体。
图2是其罐体纵向倾斜变位的示意图，图3是罐体横向偏转变位的截面示意图。
请你们用数学建模方法研究解决储油罐的变位识别与罐容表标定的问题。
（1）为了掌握罐体变位后对罐容表的影响，利用如图4的小椭圆型储油罐（两端平头的椭圆柱体），分别对罐体无变位和倾斜角为α=4.10的纵向变位两种情况做了实验，实验数据如附件1所示。
请建立数学模型研究罐体变位后对罐容表的影响，并给出罐体变位后油位高度间隔为1cm 的罐容表标定值。
（2）对于图1所示的实际储油罐，试建立罐体变位后标定罐容表的数学模型，即罐内储油量与油位高度及变位参数（纵向倾斜角度α和横向偏转角度β ）之间的一般关系。
请利用罐体变位后在进/出油过程中的实际检测数据（附件2），根据你们所建立的数学的罐地平线 图1 储油罐正面示意图 油位探针2010高教社杯全国大学生数学建模竞赛题目 （请先阅读“全国大学生数学建模竞赛论文格式规范”）B 题 2010年上海世博会影响力的定量评估 20101851年伦互联网数据，定量评估2010年上海世博会的影响力。
2010高教社杯全国大学生数学建模竞赛题目（请先阅读“全国大学生数学建模竞赛论文格式规范”）C 题 输油管的布置某油田计划在铁路线一侧建造两家炼油厂，同时在铁路线上增建一个车站，用来运送成品油。
2010年数学建模C、D题解析
4 − k2 l a+b k 1 P =( (b − a) + , − l ) ，此时 f min = [k (a + b) + l 4 − k 2 ] ； 2k 2 2 2 2 4 − k2
*
ab 4 − k2 （3）当 l ≥ (b + a ) 时， P* = ( , 0) ，此时 f min = (a + b) 2 + l 2 . a+b k 对共用管道与非共用管道相同的情况，只需令 k = 1 即可. 此问的优化模型实际上有两个约束条件 x ≥ 0, y ≥ 0 . 上面得到（1）（2）（3） 、 、 ，
关于学生论文中的几种情形
1．讨论不全面，很多队没有给出什么条件下不用公 用管道；有些虽有讨论，但不清晰，一定要用已知参数的 关系来讨论. 2．有些同学用镜面反射，Ferma定理，这些只适用于 各种管道费用相同的情形，即只适用第1问中的特殊情形 ，后面仍然要建立优化模型，对整个论文并没有增添什么 色彩. 之所以采用这两种方法，主要是参考了《两城镇取 水管线的最短铺设》和《泵站选址与水管铺设》两篇论文 . 不少学生根本就没有弄清Ferma定理，叙述Ferma定理就 不正确. 另外，用了文章就应引用为参考文献. 3．不少论文叙述混乱，符号混淆，图形不合理.
2012-4-28 费浦生 feipusheng@
关于C题《输油管的布置》的第3问
3．设输送 A 厂的管线为 k1 = 5.6 万元／km，输送 B 厂的管线为 k2 = 6.0 万 元／km，共用管线费用为 k3 = 7.2 万元／km，拆迁等附加费用同上. 请给出管 线最佳布置方案及相关费用. 总费用表达式为
2010 -2014MCM Problems建模竞赛美赛题目重点
2010 MCM ProblemsPROBLEM A: The Sweet SpotExplain the “sweet spot” on a baseball bat.Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when hit. Why isn’t this spot at the end of t he bat? A simple explanation based on torque might seem to identify the end of the bat as the sweet spot, but this is known to be empirically incorrect. Develop a model that helps explain this empirical finding.Some players believe that “corking” a bat (h ollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap enhances the “sweet spot” effect. Augment your model to confirm or deny this effect. Does this explain why Major League Baseball prohibits “corking”?Does the material out of which the bat is constructed matter? That is, does this model predict different behavior for wood (usually ash or metal (usually aluminum bats? Is this why Major League Baseball prohibits metal bats?MCM 2010 A题：解释棒球棒上的“最佳击球点”每一个棒球手都知道在棒球棒比较粗的部分有一个击球点，这里可以把打击球的力量最大程度地转移到球上。
连环犯罪预测-2010美国大学生数学建模B题(MCM)
For office use onlyT1________________ T2________________ T3________________ T4________________ Team Control Number7370Problem ChosenBFor office use onlyF1________________F2________________F3________________F4________________2010 Mathematical Contest in Modeling (MCM) Summary Sheet(Attach a copy of this page to each copy of 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.SummaryIn our paper, we define two types of serial murder, the single-hunting area type and the multi-hunting area type, and we mainly do research on serial murder of these two types.We work out a method to determine if a given serial murder belongs to the single-hunting area type or the multi-hunting area type, or neither of them.To do research on the cases of the single-hunting area type, we developed two sub-models, the Concentration Driven Model (CDM) and the Centro-based Model (CM). We use the “center of mass” and the search-radius calculated by an algorithm to figure out the scope of hunting area.We discretize the hunting area by divided it into small action spaces and all of our models depend on the discretized one.The CDM is based a truth that the distribution of preys will affect the criminal’s moving to a certain extent. We develop a discrete algorithm to calculate variation of the prey’s distribution, and an algorithm to search for the next possible crime locations.The CM is based on the fact that the criminal tends to move to some places far away from those latest crime locations to avoid be arrested, and CM also includes the random factors. We develop a partially-random algorithm to determine the next possible directions the criminal may choose based on the time and locations of the past crimes. We determine the next possible distance the criminal will go basing on a reference.We then develop an algorithm to solve cases of multi-hunting area based on the sub-models in our model development.For the sub-Models have take into consideration some direct driven factors, in our Combined Model, we consider some indirect factors that may affect the moving of criminals. We develop a method to figure out the main common properties among the locations of past crimes, and then use an matching-algorithm to figure out the superiority for each result that figured out by the sub-models, basing on the assumption that if a place have much common property with the past crime locations, there’s more chances that it will be the next crime location. And the police can select some places with high superiority to investigate.In our paper, we study three cases of single-hunting area type and one case of multi-hunting area type to determine the best value of the key coefficients; we also use them to do the Sensitivity and Reliability Analysis, and we find that after the sample size approximately increases to 7, our model will give out relatively accurate results.We write a report to the police whether our model can be used to solve a given case, and if yes, when the model will give accurate results in terms of the sample size. We also give some suggestions that should be taken care of to the executive officer.Key words: serial-crime; prey-concentration; single-hunting area; multi-hunting areaTo Seize the Devils!Content1. Introduction (2)2. Terminology (2)3. Assumptions and Symbol (3)4. Sub-Models (3)4.1 Preparation (3)4.1.1 Determine the Search Radius and the Center of Mass (4)4.1.2 Hunting Area Division (5)4.1.3 Normalized Distance (ND) (6)4.2 Concentration Driven Model (CDM) (6)4.2.1 Overview (6)4.2.2 Frame of CDM (6)4.2.3 Model Definition (7)4.2.3.1 Concentration Calculator (7)4.2.3.2 Points Generator (8)4.3 Centro-based Model (CM) (10)4.3.1 Overview (10)4.3.2 Model Definition (10)4.3.2.1 Random Direction Algorithm (10)4.3.2.2 Random Distance Algorithm (12)4.4 Residence of the criminal (13)4.5 Parameter Determination (13)5. Model Comparison (13)6. Sub-Model Development (14)6.1 Problem Addressing (14)6.2 Identify the type of a case (14)6.3 Use the Basic Model (16)7. Model Combining (16)7.1 Overview (16)7.2 The Property Vector (16)7.3 Model Definition (17)8. Strengths and Weaknesses of the Model (18)8.1 Strengths (18)8.2 Weaknesses (18)9. Case Study (19)10. Sensitivity Analysis (20)11. Model Reliability (21)12. Executive Summary (22)13. References (24)Appendix (25)Appendix I (25)Appendix II (25)Appendix III (26)Appendix IV (26)1.I ntroductionSerial crime is generally defined as crimes in a serial or repetitive nature. [1]Serial murder, serial rape and serial arson are the crimes regarded as serial crimes. Serial crime is a frightening and perplexing phenomenon that has proven to be a difficult puzzle for both criminal investigators and criminological researchers. Despite being a rare event, this crime has a broad-based impact on the larger community (Jenkins, 1992a; Silverman & Kennedy, 1993). [2] In our paper, we try to develop a method to aid in a local police agency’s investigations of serial criminals.The objectives in our study are:z Make use of at least two different schemes to generate a geographical profile of serial criminals; z Develop a technique to combine the results of the different schemes and generate a useful prediction;z Provide some kind of estimate about how reliable the estimate will be in a given situation, including appropriate warnings;z Provide an additional two-page executive summary, which provides a broad overview of the potential issues and an overview of your approach and describe situations when it is an appropriate tool and situations in which it is not an appropriate tool;Firstly, we determine what the meaning of the geographical profiling is. Geographic profiling is a criminal investigative methodology that analyzes the locations of a connected series of crimes to determine the most probable area of the murderer’s residence or his next crime location. [3] To generate a geographical profile of serial criminal, we should provide the murderer’s sphere of activities; predict the murderer’s next crime location.In our paper, firstly we work out a Concentration Driven Model (CDM) to make a prediction providing some kind of guidance about possible locations of the next crime from the microscopic point of view. Secondly, we add the concept of the center of hunting area and build a Centro-based Model (CM) to macroscopically predict the possible locations of the next crime. Thirdly, we combine both models and come up with a combined model aiming at generate a more reliable prediction of the possible locations of the next crime. Finally, we analyze the sensitivity and the reliability of our models. At the end of this paper, we provide an executive summary for a chief of police on the strategy for investigations of serial criminals.2.T erminologyz Crime location: a place where has just experienced a crime actionz Hunting area: a concentrated area that has experience a series of crimes. An area can be assumed to be hunting area only when the distance between two successive crime locations doesn’t vary much as the number of the crimes increases. In our model, the hunting area is a square.z Action space: we divided the hunting area into several small parts, and each part is an action space. The action space represents a district, a city, a block, or someplace else. It depends on the scale of a certain serial crime.z Single-hunting area and multi-hunting area: in our paper, we just investigate two types of serial crime. One is single-hunting area and the other is multi-hunting area.¾The type of single-hunting area contains only one hunting area.¾The type of multi-hunting area contains multiple hunting areas, and the distance of each pair of hunting areas is much larger than the distance between two successive crime locations within one hunting area.z The search-radius and the core-radius:¾The search-radius limits the search scale of the criminal, and the criminal cannot reachbeyond it. In our paper, the search-radius is half the side-length of the hunting area.¾The core-radius is an abstract valuable; it varies as the number of crimes increases. We will define it later in our paper.Here is a graph to illustrate the terms above:1 Graph 1z Latest crime location: the location of the latest crimez Prey: certain kind of people that the murderer tends to kill3.A ssumptions and SymbolAssumptions:z We treat the action space as a unit. A murderer’s basic step is to stay in a action space, or to move to the very neighboring action space.z The prey concentration of each action space is the natural distribution through the hunting area before the very first crime of the serial crime happened. Also, we consume that the natural distribution remains constant.Symbol:z K: the number of parts we divided the length of the hunting area intoz , … , … :the action space located in i row and the j column in the hunting areaz , :the concentration or the number of prey in Sq ,z … … :the array of concentration through the hunting areaz : the longitude of the location of i crimez : the latitude of the location of i crimez : the longitude of the “mass of center”z : the latitude of the “mass of center”4.S ub-Models4.1PreparationWe develop two sub-models in this part, both of which are based on a single-hunting area. In this part, we introduce several important concepts, such as the search radius, the center of mass, the ND, etc. Now, we are going to definite the scope of the hunting area. And the search radius and thecenter of mass are both used to describe the scope of the hunting area. One thing that we must bear in mind is that the center of mass is used to determine the scope of hunting area, rather than a place that the murderer will come back to after a crime action.4.1.1 Determine the Search Radius and the Center of MassThe search radiusThe search radius of the murderer’s hunting area is measured as of the maximum distance from “center of mass” to crime location. The following regression equation reflects the maximum distance from “center of mass” to crime site against maximum distance between crime sites y a x 0.61(Equation 4.1.1)Where:y is the maximum distance in miles from the “center of mass” to crime site; and x is the maximum distance in miles between crime sites.The gradient of a in Equation 4.1.1 indicates an eccentric placement of the “center of mass” vis-à-vis the crime sites (a perfect centric placement would yield a gradient of 0.5). Regressions for U.S. and British serial murder crime location data yield values of 0.81 and 0.79, respectively, and an Australian study found gradients of 0.77 for rape, 0.60 for arson, and 0.65 for burglary. [2] The center of massCalculate the “center of the mass” by using Centroid-Formula: Longitude∑ ,Latitude ∑ VerificationWe review the serial murder of Peter Sutcliffe. Graph 2 shows Yorkshire Ripper locations within West Yorkshire (Victims 6 & 9 are off this map to the south west). In Graph 2 the labels of 1-13 were the victims’ locations where the bodies were found. The labels of 14-17 were the survivors and the label of 18 was the place where he was arrested.2 Graph 2Figure 1shows the quantified original image of the crimes, and Figure 2 shows the image without singularities.3 Figure 14 Figure 2Take the order of occurrence of the murder into consideration; we do the following works as the serial number n increases:1) Calculate the “center of the mass” by using Centroid-Formula: Longitude∑ ,Latitude ∑ ; 2) Calculate the maximum distance from the “center of mass” to crime site by using Equation4.1.1.Figure 3 shows all the positions of the “center of mass”, from which we can find that the “center of mass” tend to be lumped as we take more crimes into consideration.Figure 4 shows the variation of y as the serial number n increases. As is shown in 4.2, y is the maximum distance from the “center of mass” to crime location. Y tends to be stable as n increases. It means that the murder’s hunting area is limited to an area that centered with the “center of mass” and that has a radius of y.5 Figure 36 Figure 4All the above proves that it is reasonable to describe the scope of a hunting area by the center of mass and the search radius.4.1.2 Hunting Area DivisionIn order to apply our model to different scales of serial crime, we should normalize the hunting area, and treat it as a square 1 on a side. The procedure is as follows: We define the normalizationparameter as 2 times the length of search radius, and then get the scope of the hunting area normalized by dividing the distance by the normalization parameter. We divide the hunting area into k k action spaces. Each of the action spaces is a square on a side.4.1.3 Normalized Distance (ND)It is difficult to deal with problems of different order of magnitude. As the scope of the murderer’s hunting area differs at a wide range, we introduce the concept of Normalized Distance (ND) to make our model more tractable.We get the ND of a crime location by dividing the actual distance by 2 times the length of search radius.4.2 Concentration Driven Model (CDM)4.2.1 OverviewThe Concentration Driven Model (CDM) is a discrete model. In this model, we suppose that the murderer’s moving from one action space to another depends on preys’ distribution through the hunting area. Meanwhile, as the crimes being committed sequentially, the distribution will change.We develop an algorithm (we call it concentration calculator) to calculate the current preys’ distribution using the happening time and the location of the past crimes scenes. Then, we develop an algorithm (we call it points-generator) to search the next possible crime locations. The points-generator is a ‘discrete ‘steepest descent method.4.2.2 Frame of CDM7 Frame of CDM4.2.3Model Definition4.2.3.1 Concentration Calculatorz Change of the Prey’s DistributionAccording to our assumption above, each action space own a property of prey concentration. So we investigate the transfer of preys among the many action spaces to estimate the change of the prey distribution.z Causes for Transfer¾ A crime has just been committedAccording to our assumptions, when a crime has just been committed in an action space, all the preys in the action space will soon run away to other action spaces. This assumption is reasonable. In fact, people who will emerge in a certain place always for the reason that he will pass through the place on the way to work or some places else in his daily life, and he will go round if an crime has just happened in the place. This is always true when people choose the way to work or people choose a place for purchase, etc.Suppose the current crime location is represented by C , . We let the preys transfer to the eight neighboring action spaces according to a certain principle. We limit the transfer just among the crime location .We divided the eight spaces into two groups of A and B, Figure 5 can illustrate it:8 Figure 5We assume that each action space of group A will receives a% of the preys in the crime location, and each action space of group B receives b%, where a ,a b 25.This is for the reason that action space of group A connects with the crime location basing on edge, while group B basing on angle, and we think that the larger the contact area is, the more preys will transfer.Therefore, after a crime is committed, the instant transfer follows the principle below:1)When the coordinates of an action space i′,j′ is one of i 1,j , i 1,j , i,j 1 ,i,j 1 , C ′, ′ a% C , ;2)When the coordinates of an action space i′,j′ is one of （i 1,j 1）, i 1,j 1 ,i 1,j 1 , i 1,j 1 , C ′, ′ b% C , ;3)C ′, ′ 0.¾As time goes by, preys that run away to other action spaces will come back After a crime being committed, the number of the preys in the crime location will reduce, but as time goes by, the influence will gradually reduce, preys will gradually come back because the way contain the crime location or the supermarket they choose in their daily life before the crime’s happening is their usual choice and it must be the best one. Thus the distribution will recover. We put forward assumptions below:In every time unit, there are a preys come back from each action space of group A, whileb preys from each action space of group B. We assume that the preys don’t transfer on largescale any more until the distribution recover to the normal level.Apart from the ending situation above, there is another ending situation, that is, when the concentration or number of preys in an action space of A or B group being affected by another crime later, that means it also exchange preys with other action spaces later, it may not return all the preys who come from the crime location .Under this circumstances, the return or the transfer between the certain action space and the crime location will terminate.¾Exception on the borderFor two kinds of points on the border shown in Figure 6 and Figure 7, we also divided the action spaces around the crime location into group of A and B, and then figure deal with it in the similar way:9Figure 6 10Figure 7z Current Distribution of the PreysAccording to our assumptions, the distribution of the preys through the hunting area is already known, and represented by a two-dimensional array C . The distribution after the ith crime action represented by C (i=1…n), and we will figure out C during the simulation, the process as below:1)After the i th crime action, we can calculate the C based on the C2)According to the algorithm above, and repeat n times, we could figure out C .4.2.3.2 Points GeneratorWe develop an algorithm to search for the next possible action spaces that would welcome a crime action; we called the algorithm points-generator. The algorithm derives from Gradient descent, but we improve it so that it could be used in a discrete problem.z Gradient DescentGradient descent is a first-order optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or of the approximate gradient) of the function at the current point. If instead one takes steps proportional to the gradient, one approaches a local maximum of that function; the procedure is then known as gradient ascent.z Gradient descent in discrete useWe replace the Gradient in the Gradient descent with the difference of concentration.Though we use a method developed from an optimization algorithm, we don’t mean to search for the local optimization result, we just use the method to simulate the moving of the criminal.Why not to search for the optimization result will be explained below. And we will obtain a series of action space that will welcome a crime action.Why not search for the optimization result?There two reasons to explain it:1)The first reason is that the total iteration number to find a local maximum result would bevery large, and a large iteration number may not be realistic because the criminal’s ability, physical strength, or his financial condition is limited and always cannot afford to go so far.2)The second reason is that even though a criminal is deeply affected by the preyconcentration, his destination may not be the optimal solution but a place that is above his standard. This is because the criminal is somewhat shortsighted and he may satisfy with the places above his standard.z Figure out the next action spaceWe replace the gradient in the Gradient descent with the difference of prey numbers between the adjacent action spaces, as follows:We still use the local area as is shown in Figure 8 (the action space on the border of the hunting area can be proposed in the similar way)11 Figure 8For that in every iterative round, the algorithm will generate a series of points, for each point/current iterative point, we do things as follows:For each action space marked from 1 to 8 around current iterative point in the figure above, we calculate the difference of prey concentration between the marked action space and the current iterative point, choose those action spaces with a corresponding difference that is larger than a certain value of h to be a part of the iterative points of the next round.In fact, the threshold h can describe the criminal’s sensibility degree to prey concentration, and if threshold h is large, indicating that the criminal is sensitive to the difference of prey concentration, otherwise the opposite.z When to Terminate the IterationAccording to the analysis above, we terminate the iteration when the number of iteration steps has reached a constant value of d, the value of d is constant as the scale of the crime varies, and in a given serial murder, it is also constant.Why it is constant as scale varies?In fact, no matter it is crime in a large scale with automobiles as its transportation or a crime in small scale with transportation just base on foot, the d is constant among all kinds of cases because we have normalized the scope of the case, thus for a large scale case one iterative step (move from current action space to another action space) represents a longer distance than in a small scale case.Evidence for a constant d in a given serial murderWe also have found that in one case, the distance between two crime-sites that happened in succession don’t vary largely, that means that to assume that the value of d is constant in a given serial murder is reasonable.We studied the case of Peter Sutcliff, and his moving track during the case is in Figure 9.12Figure 9In this case, there are 15 distances, we calculate the mean m and the standard deviation σ of them, and =0.2903, the value is minor and that means the distance don’t vary very much. On the other hand, the conclusion above can be explained as that the criminal’s physicalstrength, or his financial condition or something else would limit him.4.3 Centro-based Model (CM)4.3.1 OverviewThis is a macro-dynamic mode. We divide the hunting area into k action spaces, in which we find the possible locations of the next crime based on the time and locations of the past crime scenes. We determine the next possible crime locations by confirming the murderer’s choice of the next crime location’s direction and distance.In CM, the murderer’s choice of the direction of the next possible crime location is based on the time and locations of the past crime scenes. The direction is a stochastic variable, following a certain distribution. The distribution is influenced by the time and locations of the past crime scenes. We design the probability density function of the murderer’s next direction choice on the basis of his past crime scenes’ time and locations. And then we generate a crime location’s direction through a simulation on the basis of the distribution.We got a buffered distance-decay function [2], and construct a probability density function of the ND of the murderer’s next crime location in accordance with the curve’s variation law. Then we run a simulation to get the murderer’s next crime location’s distance.4.3.2 Model Definition4.3.2.1 Random Direction Algorithm1. Calculate the weights f t of the murderer’s past crime scene at the time of t . We opt for thefunction defined as follows:f t a lg b t 1 ,a 0, 0(4.3.2.1-1) Wheret is the time of the murderer’s past crime scenes. The crime happened earlier as the valueof t getting larger. a and b are determined by concrete serial crime case.13Figure 10Figure 10 shows the function curve of this function.We can see that f 0 0, which means the murdererwon’t choose the same direction as that of his latest crimelocation. This may be on account of the large possibility ofbeing arrested at the very direction. We can also find thatf t tends to get larger as t is increasing. This is becausethat as the crime happened earlier the impact of itsdirection on the murderer’s next direction choice is gettingsmaller and smaller. 2. Using the weights f t direction choice as follows:g θ w f t 1 cos θ θ,θ π,π(4.3.2.1-2) Whereθ is the argument that can uniquely determine an action space;θ is the argument of the murderer’s past crime location at the time of t ; and w ∑.Figure 11 shows the definition of the argument.14 Figure 113. Calculate the function of possibility distribution as follows:G θ g θ d θθ π,θ π,π(4.3.2.1-3) 4. Divide the range of θ into h equal non-overlapping intervals. Accordingly, we divide the rangeof G θ into h non-overlapping intervals that are not always of equal width.5. We run a simulation to get a random sampling on an interval 0,1 . If this sampling fall into thei th interval of G θ , we calculate the average of the two endpoints of the i th interval of θ. And it is the direction of the murderer’s next crime location.4.3.2.2 Random Distance AlgorithmBuffered distance-decay function:It is a dynamic process of a murderer’s target selection. Crimes occur in those areas where suitable targets overlap the offender’s awareness space. Offenders then search outward from these areas, the search behavior following some form of distance-decay function. There is usually a “buffer zone,” however, centered the criminal’s residence. Within this zone, targets are viewed as less desirable because of the perceived level of risk associated with operating too close to home. For the offender, this area represents an optimized balance between the maximization of opportunity and the minimization of risk. Figure 12 shows an example, derived from a serial rape case, of a typical buffered distance-decay function .Algorithm:1. Confirm the type of the serial crime, and determine the gradient of a in Equation 4.1.1;2. Calculate the maximum distance from residence to crime site y on the basis of the maximumdistance between the past crime sites x, using the Equation 4.1.1;3. Calculate the ND of the actual distance d by using the equation ND; 4. Construct a probability density function d ND of the ND of the murderer’s next crime locationin accordance with the buffered distance-decay function curve’s variation law. We study Figure 12, and design the probability density function in form of the formula below: d ND q 1 q 2 10 ND q 3 2,0 ND 0.2k x b,0.2 1 (4.3.2.2-1) Figure 13 shows the probability density function curve. We can find that it preserves the approximate characters of Figure 12, and we can use this probability density function to simulate the ND of the murderer’s next crime location.15 Figure 1216Figure 135. Calculate the function of possibility distribution as follows:D ND D ND d ND D,ND 0,1 (4.3.2.2-2)6. Divide the range of ND into h equal non-overlapping intervals. Accordingly, we divide the rangeof D ND into h non-overlapping intervals that are not always of equal width;7. We run a simulation to get a random sampling on an interval 0,1 . If this sampling fall into thei th interval of D ND , we calculate the average of the two endpoints of the i th interval of ND ;8. Calculate the result of ND y , and it is the distance of the murderer’s next crime location.。
2010数学建模优秀论文
上海世博会影响力的定量评估摘要本文是一个对上海世博会影响力的定量评估问题，首先我们收集了与世博会有关的数据，如国内来沪旅游人数，国外来沪旅游人数等。
并用灰色预测对相应的数据进行了预处理，然后我们从横向（本届世博对上海的影响）和纵向（本届世博和历届世博的影响比较）两个角度对世博影响力进行了研究，最后还应用了多目标优化模型求出在不同投资增长系数下上海世博对当地旅游经济最大影响力系数。
第一步，我们横向考虑世博会对本地旅游业的影响力，并将该影响分为对旅游经济的影响和对旅游文化的影响两方面。
首先应用本底趋势线模型得出相应数据的本底值，再分别建立对旅游经济和旅游文化的影响力系数模型，然后利用本底值和统计值得出相底值增加了579.39亿元的旅游收入。
而世博对旅游文化的影响力系数为1.29。
第二步，我们纵向考虑上海世博会与历届世博会相比的影响力。
根据收集的历届世博会相关的规模数据，将世博会影响力等级从低到高分为1-5等，从而建立了世博会综合影响力的模糊评价模型。
对历届世博会的影响力做出综合评价并得出了相应的综合影第三步，我们从环保，旅游收入以及后世博效应三个角度对上海世博的影响重新进行了思考。
综合权衡这三个方面因素，我们建立了一个多目标优化的模型。
得出了在不同投资增长系数下的一个合理的旅游经济影响力系数和世博年最优的旅游者的人数。
当投资增长系数为0.4时，其对旅游经济的影响力系数为1.297，则该年最大的旅客人数为13415.54万人。
而我们根据预测值得出2010年总旅客人数为12695万人，说明预测的旅客人数未超过最大人数限制。
最后，我们根据所求得的影响力系数，对上海世博会写了一篇影响力评估报告。
关键词：本底趋势线模型模糊评价模型多目标优化旅游文化影响力系数1.问题重述1.1问题背景中国2010年上海世界博览会（Expo 2010），是第41届世界博览会。
于2010年5月1日至10月31日期间，在中国上海市举行。
2010年数学建模C题参考答案
输油管的布置摘要本文建立了关于布置输油管管线费用最省的优化模型，针对问题，我们做 出了合理的简化假设，利用lingo 软件，最终对问题进行了求解.对于问题一，我们从非共用管道和共用管道（费用相同与不同）考虑一炼油厂1A 、另一炼油厂2A 和车站k 看成平面上三点，构建动态三角形k A A 21.求出费马点P 的具体位置.使其在费用相同情况下得出总费用最小值S ：12311323213212322212/)()()()(3[S X X X X X X X X X X X X X X X S ⨯++⨯-+⨯-+⨯+++++= 费用在不同情况下，假设费用为1S 和2S ，与S 关系式为：271274328)2(S X S X X X X S ⨯+⨯⨯-++= 对于问题二，在城区铺设管道的建设附加费用以经验法得出为21.4（万元/千米）.我们还是通过对非共用管道和共用管道进行分析建立模型，铺设费用均相同，计算得出非共用管道费用最小=S 337.5362，共用管道费用最小8.281=S ，比较可得出当两炼油厂共用管道时，共用管道费用最小.通过检验可确定为最优解，得到最佳管线布置方案.对于问题三，我们可以应用前面模型解答，改变铺设费用的系数，代入前面模型可得费用取得最小值为210.84，即可得到最佳设计方案.该模型用图表与文字结合来说明求解，直观、通俗易懂.关键词 费马点 经验法 共用管道 lingo一、问题的重述某油田计划在铁路线一侧建造两家炼油厂,同时在铁路线上增建一个车站，用来运送成品油.由于这种模式具有一定的普遍性,油田设计院希望建立管线建设费用最省的一般数学模型与方法.1. 针对两炼油厂到铁路线距离和两炼油厂间距离的各种不同情形,提出你的设计方案.在方案设计时,若有共用管线,应考虑共用管线费用与非共用管线费用相同或不同的情形.2. 设计院目前需对一更为复杂的情形进行具体的设计.两炼油厂的具体位置由附图所示,其中A厂位于郊区（图中的I区域）,B厂位于城区（图中的II区域）,两个区域的分界线用图中的虚线表示.图中各字母表示的距离（单位：千米）分别为a= 5,b = 8,c = 15,l = 20.若所有管线的铺设费用均为每千米7.2万元. 铺设在城区的管线还需增加拆迁和工程补偿等附加费用,为对此项附加费用进行估计,聘请三家工程咨询公司（其中公司一具有甲级资质,公司二和公司三具有乙级资质）进行了估算.估算结果如下表所示：工程咨询公司公司一公司二公司三附加费用（万元/千米）21 24 203. 在该实际问题中.为进一步节省费用,可以根据炼油厂的生产能力,选用相适应的油管.这时的管线铺设费用将分别降为输送A厂成品油的每千米5.6万元,输送B厂成品油的每千米6.0万元,共用管线费用为每千米7.2万元,拆迁等附加费用同上.请给出管线最佳布置方案及相应的费用.二、问题的分析本文是一个关于输油管的布置以及建设费用最省的优化问题,建设总费用与输油管的长度和输油管的铺设费用有关，对于问题一，我们可以从非共用管道和共用管道（费用相同与不同）考虑，费用相同的情况：一炼油厂1A 、另一炼油厂2A 和车站k 可看成平面上三点，可构建动态三角形k A A 21，在不同三角形中费马点的位置不同.可把三角形形状分为三类，分别求出费马点的具体位置. 距离最短即建设总费用最小，费用不同情况：假设费用分别为1S ，2S .可得到总费用S 与1S ，2S 之间的关系式.对于问题二，已知两炼油厂的具体位置，我们还可从非共用管道和共用管道进行分析建立模型. 城区建设附加费可通过经验法求得. 最短路程不一定是最少费用非共用管道和共用管分别可以按X 的范围分别分为两类（c X ≤≤0和l X c ≤≤）.最终可求最小费用S ，得到最佳布置方案.对于问题三，我们可以应用前面模型，分别以不同的铺设费用，代入前面模型可得费用取得最小值.得到最佳设计方案.三、模型的假设1.假设所选的区域地势平坦， 没有障碍物；2.假设铺设工程顺利进行，不再因其他因素而增加铺设管道的费用；3.假设不考虑市场因素对输油管价格的影响；4.假设铁路和两炼油厂两两之间至少保持安全距离.四、符号的定义1. 铺设每公里非共用输油管线的费用为1S ；2. 铺设每公里共用输油管线的总费用为2S ；3. 总费用为S ；4. 在城区所增加的附加费用为1W ；5. 铺设输油管线的总长为X ；五、模型的建立与求解建设总费用与输油管的长度和输油管的铺设费用有关.模型一的建立与求解针对两炼油厂到铁路线距离和两炼油厂间距离的各种不同情形去建立模型,根据铺设每公里共用管线与非共用管线的费用相同与否.对其进行分类.当铺设每公里共用管线与非公用管线的费用相同时,根据“费马点”（证明见附录）在车站和两炼油厂三点所构成的三角形中的不同位置,我们可以将其分成三类.设甲厂1A 到车站k 的距离为1X ,乙厂2A 到车站k 的距离为2X ,甲厂到乙厂之间的距离为3X ,铺设每公里非共用管线的费用为1S ,铺设每公里共用管线的费用为2S ,总费用为S 总路程为X .（1）当甲乙两厂和车站的相对位置如下图所示,角1A 大于或等于120时.根据费马点的证明（见附录）我们可以算出甲,乙两厂到车站的最短距离31X +X =X最少费用2113S S S ⨯X +⨯X =（2）当两个厂按角k 大于或等于120度分布时,如下图所示;甲厂1A 车站k 乙厂2A1X2X3X同样，根据费马点的原理可知,其最短路程21X +X =X .其费用()121S S ⨯X +X =.(3) 当两炼油厂和车站三点所构成的三角形中的三个角均小于120度分布时,如下图所示;根据“费马点的证明（见附录）”可以求出最短路程2/)()()()(3[23113232132123222121X X X X X X X X X X X X X X X Pk PA PA ++⨯-+⨯-+⨯+++++=++其费用12311323213212322212/)()()()(3[S X X X X X X X X X X X X X X X S ⨯++⨯-+⨯-+⨯+++++= 当铺设每公里的共用输油管与非共用管线的费用不同时,假设两炼油厂的位置任意如图)1(分布,车站设在铁路线上的任意一点k 上.设共用管线的距离为7X .A 和1A 点关于铁路线对称,11k A 交12k A 相交于点1B .1A 到铁路线的垂直距离为3X ,2A 到铁路线的垂直距离为4X ,两炼油厂投影在铁路线上的水平距离为8X ,设每公里非共用管线的费用为1S ,铺设每公里共用管线的费用为2S ,总费用为S .非共用管线的长度11k A +12k A =2AA =274328)2(X X X X ⨯-++ (370X X ≤≤)共用管线的长度：7X车站k甲厂1A乙厂2AP 甲厂1A车站k 乙厂2Ak1X 2X因此,当铺设每公里的共用输油管与非共用管线的费用不同时.其最省费用为.271274328)2(S X S X X X X S ⨯+⨯⨯-++=模型二的建立与求解由题已知两炼油厂的具体位置为点A 和B 点，在非共用管线情况下：如下图所示 车站F 在如图特殊位置时，得到结果为：4.212.0202.0246.0211=⨯+⨯+⨯=W假设以点C 为原点，建立坐标，车站F 的坐标为（X ，0），经过交界线的点为H 坐标为（C ，H ）)80(≤≤H当150≤≤X ，如图（一）利用几何中勾股定理，可得：22x a AF +=，22)(x c h FH -+=，22)()(h b c l HB -+-=，1)(W EB C HB HF AF S ⨯+⨯++=，A1Ak1k 3X 4X5X 6X 7X2A8X图（1）（一） （二） （三）利用lingo 软件，解得：=S 337.5362 25.7=X 94.6=H当2015≤≤X ，如图(二)利用几何中勾股定理，可得： 22)(h a c AH -+=22)(x c h FH -+=22)(b x l FB +-=()()1W FB HF C FB HF AH S ⨯++⨯=+=利用lingo 软件，解得：=S 87942.4 99480.19=X 4739589.0=H模型二的结果分析和检验在非共线条件下，当25.7=X 94.6=H 时.模型能得到最小值=S 337.5362，此方案即为所求设计方案.我们为了检验模型结果的可靠性，故选择三个特殊点代入，得：当车站B 在C 点时，即（0=X ），=S 372.44当车站B 在分界线时，即（15=X ），=S 348.729当车站B 在D 点，即（20=X ），=S 502.94由上述三点可知模型结果为最小点.此设计方案可行.C AD C F B c lC A E DB模型三的建立于求解存在共用管道的情况下； 设共用管道EF 的距离为Y ，共用管道与非共用管道的交点F ，坐标为（Y X ,）（80≤≤Y ），经过交界线的点为H ，坐标为（C ，H ）)80(≤≤H .（1）当150≤≤X ，利用几何中勾股定理，可得：22)(X Y a AE +-=22)()(X c Y H FH -+-=，22)()(c l H b HB -+-=，1)(W HB C HB FH Y AF S ⨯+⨯+++=，利用lingo 软件，解得：8.281=S 30114.14=X 74.6=Y 14.7=H当2015≤<X ；利用几何中勾股定理，可得：22)(C H a AH +-=，22)()(C X Y H HF -+-=，22)()(X b Y b FB -+-=，1)()(W FB FH C FB HF AH S ⨯++⨯++=利用lingo 软件，解得：3088.302=S 15=X 309.1==Y HC A E BD C AE B D C A DE B (1)模型三的结果分析和检验存在共用管道的情况下，当30114.14=X 74.6=Y 14.7=H 时，模型能得到最小值8.281=S .为了检验模型结果的可靠性我们将特殊点带入检验：如图（1）（1）当共用管道为AC 时，S =342（2）当共用管道为HF 时，312=S（3）当共用管道为FD 时，290=S通过比较可知模型结果准确无误.在模型二和模型三的比较中，模型三中费用最小值8.281=S 为最佳设计方案.模型四的建立于求解模型四可在模型三的基础上求解.在其它情况不变的条件下，只改变铺设费用的大少.得:FE HB FH AF S ⨯++⨯+⨯=2.7)(0.66.5 （150≤≤X ） （1）Y X l Y b c X Y H H a c S ⨯+-+-⨯+-+-+-+⨯=2.7)()(0.6))()()((6.5222222（1520≤≤X ） （2）解（1）得：0617.221=S 867288.6=X 042522.0=Y 0678.7=H(2) 得：84.210=S 15=X 0=Y 23.0=H由此我们可以得知，在15=X , 0=Y , 23.0=H 时，费用取得最小值为210.84，即可得到最佳设计方案.(6) FC AE D C AF DB HH B E(5)六、模型的进一步讨论在模型一中，我们采用的是优化模型，也就是说我们假设了厂址的可选区域是地势平坦的，对正常的管线铺设施工的基本是没影响的。
2010年美国数学建模A题论文连环罪犯居住地及作案时间地点的预测
连环罪犯居住地及作案时间地点的预测摘要本文主要通过“圆周假设理论”的改进行地理轮廓预测，根据Rossmo公式预测出了罪犯居住地的可能范围。
对时间和地点运用灰度预测方法预测了下次案发时间地点。
对于发展一种辅助警察调查方法，并运用这种方法生成地理轮廓，讨论引入了“圆周假设理论”。
在“圆周假设理论”的基础上，对该理论进行不同角度的改进，最后总结出三个确定地理轮廓的方案：改进圆周假设理论，中心图解法，最匹配圆改进方法，对Peter Sutcliffe的案例进行检验得到三个可能居住地坐标为：(0.9062,0.4051),(0.8872,0.3390),(0.8930,0.3460)都接近实际居住坐标(0.88,0.45)。
然后运用Rossmo公式求的概率分布矩阵并生成二维伪彩色图和灰度图，以此预测出最可能的居住范围，预测范围准确并且很小，可以很有效的缩小警察的排查范围。
通过对已有案例的时间和地点分析预测下一次案例的发生时间和地点。
通过GM(1,1)模型对案发的时间间隔以及案发地与居住点的距离进行预测，以Peter Sutcliffe 的案例进行检验，最后一次作案实际时间间隔为46，预测的时间间隔为63，误差17天，准确性为63%。
预测最后5次案发地与居住点的距离，与实际情况比较后，发现准确度为60%左右。
已经可以很有效的缩小警察的搜索预警范围。
关键词犯罪地理分析 Rossmo模型 GM（1,1）一、问题重述在Peter Sutcliffe13起谋杀案中，一种用来缩小搜索罪犯所在范围的方法是找到这些罪犯的点的“重心”。
从那时开始更多更复杂的的技术被发展起来通过系列犯罪的地点用来确认罪犯的“地理轮廓”。
为一个地方警署发展一种辅助他们调查连环犯罪的方法。
这种方法至少用两种不同的方案生成“地理轮廓”运用一种方法结合其他方法的结果生成一个对警察有用的预测。
根据以前的作案时间和地点对下一次可能的作案时间地点进行预测。
2010mcmA 美国大学生数学建模竞赛 baseball sweetspot
m1v1 sin m2u2 m1v2 m2u2 m1v1r sin J 1 m1v2 r J 2
(1)
-4-
整个系统的恢复系数等于击球前后球和棒的速度差的比值：
e u2 r2 v2 v1 sin u1 r1
.,
(2)
因为不同材料的球棒在振动时产生的能量损耗不同，所以我们在速度表达式中 引入了参数 来表示材料对能量损耗的影响，由（1）和（2）可得到 v2 的最终表达 式为：
三、问题假设
1. 2. 3. 4. 5. 6. 棒球和球棒的碰撞在同一个平面。 碰撞过程，球棒不会折断也不会发生永久性的形变。 不考虑球的旋转对于碰撞的影响。 碰撞点在棒球的球心和球棒的中心轴上。 碰撞瞬间忽略击球者作用力的影响。 忽略球与棒分离时，棒球的势能损失。
四、符号说明
符号 说明
e
J
系统的恢复系数 转动惯量 质心 固有频率
-6-
b. 对于球棒质心的影响 因为软木的密度小于原本木质球棒的材料密度，所以在棒球棒尾部添加软木塞 会使得球棒的质心向握柄处移动。 c. 对于转动惯量的影响 转动惯量的计算公式：
图 4 结点集中区示意图
-3-
对于长度为 30 ~ 34.6inches 76 ~ 88cm 的球棒，实验表明，其结点集中区为距球 棒顶端 4 ~ 7inches 10.16 ~ 17.78cm 的区域，即距球棒顶端 15.7 ~ 20.2% 的部分。 能量转移模型的建立 根据比赛规则，击球员击球后开始跑垒，跑垒的时间主要由棒球的飞行时间来决定 的，而决定飞行的时间的直接因素就是出球速度，所以能给棒球最大出球速度的击 球点是比较合理的最佳击球位置。我们针对速度的研究模拟了一张击球过程图（图 5） ，主要针对球棒的转动角速度和棒球的往返速度进行分析。
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连环作案嫌疑人“地理轮廓”估计的统计模型摘要：在不同地点发生的系列谋杀案对社会的危害极大，如何确定其犯罪嫌疑人的住所是破案的关键。
对犯罪嫌疑人住所的估计，给出了3种方法，即圆周假设、质心法和最小距离法，通过对比验证表明，最小距离法较优；一般犯罪分子都会选择”不近不远”的作案地点，基本上服从二维正态分布。
同时考虑犯罪分子的心理特征，作案方式等进一步对模型进行优化，得出更合理的模型，提高对连环作案的案件的效率。
关键字：连环作案、圆周假设、质心法、最小距离、概率分布一．问题叙述：1.1问题重述1981年Peter Sutcliffe(萨克利夫)被判刑因为他参与了十三起谋杀和对其他人的恶毒攻击。
缩小搜索Sutcliffe的方法之一是发现一个攻击位置的“质心”.最终犯罪嫌疑人恰好生活在该方法预测的同一个小镇。
从那时起，已经发展出一系列更加复杂的技术用来预测基于犯罪地点的具有地理效应（地理轮廓）的系列犯罪行为。
你的团队被一个当地警察局要求发展出一种方法用来帮助他们的系列犯罪调查。
（1）你们的方法应该至少需要利用两种不同的情景以生成地理效应（地理轮廓），进而根据不同情况下的分析结果对执法人员提供有效的预测。
（2）基于以往犯罪的时间和位置，预测信息应该提供一些估计或指导下次可能的犯罪地点。
如果在预测中用到了其它的信息，必须提供特别的细节说明告诉我们这些信息是如何被整合的。
（3）你们的方法中也应该包括在给定条件下（包括适当警告信息）下预测的可靠性估计。
1.2“地理画像”地理轮廓是是一项刑事调查方法,分析确定最有可能的罪犯居住面积确定的罪行的连接一连串的位置。
通过采用定性和定量的方法，它有助于理解空间的违法行为和针对较小的区域的社会调查。
通常用于在连环谋杀或强奸（但还纵火，轰炸，抢劫及其他犯罪）的情况下，技术可帮助警方侦探优先考虑大规模的重大犯罪调查，往往需要成百上千个嫌疑人和提示中的信息。
基本原则是犯罪相关的地点提供关于受害人的资料和罪犯的与地理环境的相互作用。
它甚至可以显示罪犯对周边地理情况的熟悉程度和罪犯对安全距离的界定，可以反映了他的非刑事空间生活方式方面即其居住地的规划。
二．模型假设：由于犯罪活动的巨大变动以及几乎所有连环杀人案的凶手通常患有心理疾病，所以采用相对简单的计算机模型去预测连环杀人案一般都面临几个障碍。
以下是应用我们模型的对犯罪行为所采取的假设：•犯罪是单独作案的。
我们假设模型中的案件都是由单独个体作案的，我们的模型不对有组织的犯罪，团伙犯罪和暴动进行分析。
•案件的性质为连环杀人案件;•作案地点在被罪犯被发现地点的附近。
在凶杀案中这意味着尸体的发现地点就是作案地点。
这不是不合理的假设，因为多数连环案件的凶手在作案后会把受害人尸体丢到作案地点附近。
在连环案件中，比如连环强奸案，爆窃或纵火案，作案地点与警方发现地点是无区别的。
•假设罪犯作案动机相对稳定。
•犯罪发生在一个狭小的区域，比如一个市或县。
这里忽略案件发生在州或国家之间的情况。
在那些案件中，我们的模型必须被应用于每一个犯罪群体。
•假设案件的作案地点服从正态分布。
四.犯罪分子的居住地范围的预测4.1圆周假设“犯罪圆周假设”基本原理：假设一个犯罪嫌疑人连续作案，其作案地点分布广泛，若找出两个最远的犯罪位置，将二者连接起来，并以此连线为直径，画出一个包括所有犯罪地点的圆周，多数情况下，犯罪嫌疑人就住在圆周里，而且可能就住在靠近圆周中心的地区，这就是西方犯罪情报分析理论著名的“圆周假设”。
将犯罪嫌疑人两次最远处作案地点之间连线，并以连线为直径,画一个圆周，就可以把所选的作案地点包括在内。
犯罪嫌疑人住所大致处于圆周的中的地带，这与“圆周假设”理论基本吻合。
若以犯罪分子的居住地为起点，根据大致的方位，将发案地点连接起来，就会发现，在早期，犯罪分子主要围绕自己的居住地作案。
到了后来，有了较大的改变，基本上是从居住地出发，放射状向四周散开，由此可以推测出犯罪分子是沿着公路或铁路远行，选择到离家较远的地方作案，然后再按原路返回居住地。
首先，第一次作案在Leeds（利兹），这是值得关注的重点。
其次，以第一次发案点为圆点，向其他发案地区连线就会发现，案件大多发生在两条红线构成的扇区内，其中多数案件都沿公路展开。
再次，曼彻斯特最远处作案点。
最后，将两个最远的作案地点连线，并以最长线的中点为圆心画圆，则可将所有的作案地点全部包括在内，符合犯罪的“圆周假说”理论，犯罪分子有可能居住在圆周内。
结论：两条红线的交叉点，又是第一次发案地点，据此可以预测，以Bradford(布拉德福德)、Leeds （利兹）为中心的地区，应是犯罪嫌疑人的原始居住地，这符合犯罪人员首次作案的心理，即犯罪人员早期作案大多选择其居住地附近。
这符合事实，查资料知道彼得萨克利夫（Peter Sutcliffe ）是居住在Bradford(布拉德福德)的。
且圆心的纬度和经度为（53.77255，1.6869）与实际居住地距离相差6724.8米其中计算不同经纬度之间的公式对上面的公式解释如下：t1 Lung1 表示A 点经纬度，Lat2 Lung2 表示B 点经纬度；2.a=Lat1 – Lat2 为两点纬度之差 b=Lung1 -Lung2 为两点经度之差；3.6378.137为地球半径，单位为千米； 计算出来的结果单位为千米。
4.2.质心求法在一个陌生的地方作案比在一个自己熟悉的地方作案危险; 在自己藏匿的地方周围作案比在相对较远的地方作案更容易暴露自己的行踪以及在在一个地方连续作案比较危险等因素，可以初步利用质心法求得罪犯居住地大致的犯罪在哪个区域内。
(W N ,)=()∑=ni i i W N n 1,1纬度N 经度W 犯罪顺序53.8178 1.5428 1 53.8083 1.5311 2 53.8334 1.5002 3 53.8109 1.7633 4 53.8179 1.5325 5 53.4324 2.2506 6 53.8001 1.7721 753.6544 1.78 8 53.4599 2.2225 9 53.7117 1.8736 10 53.79 1.764 11 53.8085 1.6715 12 53.82281.5781 13纬度的均值，经度的均值,知道PeterSutcliffe 的居住地在布拉德福德伊顿区 6 花园巷且查资料得出其纬经度为（53.81307，1.7893）计算出质点与其居住地的距离相差4902米。
4.3.最近距离法由犯罪心理学，可以对犯罪分子在选择连环作案地点时的心理特征进行分析，他们通常考虑考虑三点：一是对作案地点要熟悉；二是要避开熟人；三是不能离居住地太远。
所以犯罪分子选择的作案地点一般会是离居住地不近不远。
因此可以利用最小距离法()()∑==ni i z d z d 1。
又因犯罪分子以往作案大致在leeds ，manchester ，halifax ， bradford ， huddersfield 这五个市区内，可分别计算这五个市区到13个犯罪地点的最小距离，从而确定犯罪分子的最小距离，即可初步确定犯罪分子在哪个市区。
地区 leeds manchester halifax bradford huddersfield纬度 经度53.80128 1.548567 53.47925 2.247926 53.72702 1.85754 53.79598 1.759398 53.64579 1.785635与第1，2，3，4，5，6，7，8，9，10，11，12，13犯罪地点的距离1874.441 1386.546 4775.786 14131.25 2126.655 61818.64 14670.67 22313.34 58404.56 23559.96 14195.64 8106.886 3077.934 59768.5 59721.34 63034.16 48757.02 60304.06 5209.192 47474.31 36501.85 2729.908 35729.76 47000.25 52729.28 58340.35 23000 23265.3 26272.69 11188.81 23614.48 41764.43 9871.416 9547.353 38215.59 2003.334 9313.444 15208.05 21213.62 14418.12 15044.67 17505.95 1677.162 15086.57 51776.22 950.8893 15791.32 48229.25 12000.73 730.3533 5933.683 12263.35 24902.97 24619.63 28045.43 18405.8325350.6838795.2117170.121026.01935474.49335.956 16087.74 19575.69 23937.12距离总和 230442.3 577300 254478.5 211408.3 282726.8 由以上数据可以看出，犯罪距离之和最小值是211408.3米，初步认定在bradford市区，知道Peter Sutcliffe的居住地在布拉德福德伊顿区 6 花园巷且查资料得出其纬经度为（53.81307，1.7893）计算出质点与其居住地的距离相差2730.7米。
4.4模型的改进由于以上三种方法都有各自的优缺点，且忽略了一些因素如时间间隔因素，距离波动因素等，所以我们可以对其模型进行优化。
4.4.1时间间隔因素得到z2•基于理论：两次作案时间间隔越小，两次作案点越有价值。
•所谓时间间隔：每个作案点时间间隔为它与前一次的间隔•处理权重时间时间间隔30 October 1975020 January 1976825 February 197738123 April 19777826 June 1977641 October 19779221 January 197811231 January 19781016 May 19781044 April 19793222 September 197915020 August 198035217 November 198088由时间间隔可知案件7，8时间间隔最小，两次作案点最有价值。
取两者的中心为z2。
4.4.2距离波动因素得到z3计算初始z 初始a•每个点的距离波动定义为|d-a|•权重处理办法与时间的相同•加权平均得到新的z根据最小距离法得到z3的纬度和经度分别是53.79598和1.759398 4.4.3最大概率圆的算法•认为初始点包含某些重要信息，并且可能在加权之后有信息丢失所以不舍弃z1•用z1，z2，z3确定最小覆盖圆•缺点：•改进：对犯罪地点的纬度经度进行分析，找出质心点，时间间隔因素，距离波动因素的点再次寻找最小覆盖园，使得预测犯罪分子居住地的范围更加精确。