数学建模竞赛案例选讲市公开课一等奖省赛课获奖PPT课件.pptx

数学建模竞赛案例选讲,第1页,飞行管理问题,1995年A题,第2页,飞行管理问题（1995年全国大学生数学建模竞赛试题 A）,问题：在约10000米高空正方形区域内，有若干架飞机作水平飞行。区域内每架飞机位置和速度向量均由计算机统计数据，方便进行飞行管理，当一架欲进入该区域飞机抵达区域边缘时，统计其数据后，要马上计算并判别是否会与区域内飞机发生碰撞。假如会碰撞，则应计算怎样调整各架（包含新进入）飞机飞行方向角，以防止碰撞。,假定条件:,1.不碰撞标准是任意两架飞机距离大于8km；,2.飞机飞行方向角调整幅度不应超出30度，而要尽可能小；,3.全部飞机飞行速度为800km/h，不受其它原因影响；4.进入该区域飞机在抵达边缘时，与该区域内飞机距离应在,60km以上；,5.不考虑飞机离开区域后情况；,6.建模时暂考虑6架飞机；,问题提出：,第3页,请你对这个防止碰撞飞行管理问题建立数学模型，列出计算步骤，对以下数据进行计算（方向角误差不超出0.01度），要求飞机方向角调整幅度尽可能小。,设该区域4个顶点坐标为(0，0),(160，0),(160，160),(0，160），统计数据以下表（其中方向角指飞行方向与x轴正向夹角),飞机编号,横坐标x,纵坐标y,方向角度,1,150,140,243,2,85,85,236,3,150,155,220.5,4,145,50,159,5,130,150,230,新进入,0,0,52,第4页,问题分析,依据题目标条件，可将飞机飞行空域视为二维平面 xoy中一个正方形，顶点在(0，0),(160，0),(160，160),(0，160）。各架飞机飞行方向角为飞行方向与x轴正向夹角（转角）。依据两飞机不碰撞标准为二者距离大于8km，可将每架飞机视为一个以飞机为圆心、以4为半径圆状物体（每架飞机在空域中状态由圆心位置矢量和飞行速度矢量确定）。这么两架飞机是否碰撞就化为两圆在运动中是否相交问题。两圆是否相交只要讨论它们相对运动即可。,C,A,B,D,n,i,m,i,l,i,j,ij,ij,ij,ij,v,ij,i,j,第5页,建模时补充假定条件:1.飞机在所定区域内作直线飞行，不偏离航向；,2.飞机管理系统内不发生意外，如发动机失灵，或其它意外原因迫,使飞机改变航向；,4.飞机管理系统发出指令应被飞机马上执行，即认为转向是瞬间,完成（忽略飞机转向影响，即转弯半径和转弯时间影响）；,3.飞机进入区域边缘时，马上作出计算，每架飞机按照计算后指示马上作方向角改变；,5.每架飞机在在整个过程中指点改变一次方向,6.新飞机进入区域时，已在区域内部飞机飞行方向已调整适当，不会碰撞；,7.对每架飞机方向角相同调整量满意程度是一样。,第6页,模型建立,（1）圆状模型,A,B,D,n,i,m,i,l,i,j,ij,ij,ij,ij,v,ij,i,j,采取相对速度作为研究对象,符号说明：,i,j第i，第j架飞机圆心；,ij,第i，第j架飞机碰撞角，,ij,=,ji,;,v,ij,第i架飞机相对第j架飞机相对飞行速度,;,l,ij,第i，第j架飞机圆心距,i,第i架飞机飞行方向与x轴正向夹角（逆时针为正）,x,i,第i架飞机位置矢量,v,i,第i架飞机速度矢量,ij,第i飞机对第j架飞机相对速度与两架飞机圆心连线夹角（逆时针为正）,不碰撞,|,ij,|,ij,第7页,(2)由圆状模型导出方程,讨论,ij,改变量与第i第j两架飞机飞行方向角改变量,i,j,关系,由题目条件知|v,i,|=A=800，可用复数表示速度,设第i，j飞机飞行方向改变前速度分别为,改变后速度分别为,改变前后相对速度分别为,二者之商幅角就是,ij,第8页,定理：对第i，第j两架飞机，其相对速度方向,ij,改变量,ij,等于两飞机飞行方向角改变量之和二分之一，即,第9页,模型,目标函数：,Min,其中,为各飞机方向角调整量最大值,或为,约束条件：,调整方向角时不能超出30,0,：,调整飞行方向后飞机不能碰撞：,模型为,或为,第10页,化为线性规划模型,因为,i,可正可负，为使各变量均非负，引入新变量：,模型化为,第11页,模型求解,ij,计算,model:,sets:,plane/1.6/:x0,y0;,link(plane,plane):alpha,sin2;endsets,for(link(i,j)|i#ne#j:,sin2(i,j)=64/(x0(i)-x0(j)2+(y0(i)-y0(j)2);,);,for(link(i,j)|i#ne#j:,(sin(alpha*3.14159265/180.0)2=sin2;,);,data:,x0=150,85,150,145,130,0;,y0=140,85,155,50,150,0;,enddata,end,第12页,ALPHA(1,1)1.234568 ALPHA(1,2)5.391190 ALPHA(1,3)752.2310 ALPHA(1,4)5.091816 ALPHA(1,5).963 ALPHA(1,6)2.234507 ALPHA(2,1)5.391190 ALPHA(2,2)1.234568 ALPHA(2,3)4.804024 ALPHA(2,4)6.613460 ALPHA(2,5)5.807866 ALPHA(2,6)3.815925 ALPHA(3,1)752.2310 ALPHA(3,2)4.804024 ALPHA(3,3)1.234568 ALPHA(3,4)4.364672 ALPHA(3,5)1102.834 ALPHA(3,6)2.125539 ALPHA(4,1)5.091816,ALPHA(4,2)6.613460 ALPHA(4,3)4.364672 ALPHA(4,4)1.234568 ALPHA(4,5)4.537692 ALPHA(4,6)2.989819 ALPHA(5,1).963 ALPHA(5,2)5.807866 ALPHA(5,3)1102.834 ALPHA(5,4)4.537692 ALPHA(5,5)1.234568 ALPHA(5,6)2.309841 ALPHA(6,1)2.234507 ALPHA(6,2)3.815925 ALPHA(6,3)2.125539 ALPHA(6,4)2.989819 ALPHA(6,5)2.309841 ALPHA(6,6)1.234568,第13页,ij,J=1,2,3,4,5,6,i=1,0.000000,5.391190,32.230953,5.091816,20.963361,2.234507,2,5.391190,0.000000,4.804024,6.613460,5.807866,3.815925,3,32.230953,4.804024,0.000000,4.364672,22.833654,2.125539,4,5.091816,6.613460,4.364672,0.000000,4.537692,2.989819,5,20.963361,5.807866,22.833654,4.537692,0.000000,2.309841,6,2.234507,3.815925,2.125539,2.989819,2.309841,0.000000,整理可得,ij,值（单位角度）,也能够用MATLAB计算,ij,值,第14页,x=150,85,150,145,130,0;y=140,85,155,50,150,0;,k=length(x);,alpha=zeros(k);,for i=1:k,for j=1:k,if i=j,alpha(i,j)=0;,else alpha(i,j)=(180/3.14159265)*asin(8/sqrt(x(i)-x(j)2+(y(i)-y(j)2);,end,end,end,alpha,计算,ij值程序为,计算结果为,第15页,0 5.391190237223,5.391190237223 0,32.230952672331 4.804023933797,5.091816448550 6.613460489872,20.963360893128 5.807866243421,2.234506736995 3.815924775399,32.230952672331 5.091816448550,4.804023933798 6.613460489872,0 4.364671899111,4.364671899111 0,22.833654204009 4.537692462402,2125538857551 2.989819139045,20.963360893128 2.234506736995,5.807866243421 3.815924775399,22.833654204009 2.125538857551,4.537692462403 2.989819139045,0 2.309841365405,2.309841365405 0,ij,计算:,a=150,85,150,145,130,0;b=140,85,155,50,150,0;,x=a+b*i;,c=243,236,220.5,159,230,52*pi/180;,v=exp(i*c);,k=length(a);,for,i=1:k,for,j=1:k,beita(i,j)=(angle(v(i)-v(j)-angle(x(j)-x(i)*180/pi;,end,end,beita,用matlab程序编写,第16页,beita=,0 109.2636-128.2500 24.1798-186.9349 14.4749,109.2636 0 -88.8711 -42.2436 -92.3048 9.0000,231.7500 271.1289 0 12.4763 301.2138 0.3108,24.1798 -42.2436 12.4763 0 5.9692 -3.5256,173.0651 267.6952 -58.7862 5.9692 0 1.9144,14.4749 9.0000 0.3108 -3.5256 1.9144 0,运算结果,最优解计算,用LINGO求解,程序以下,第17页,model:,sets:,plane/1.6/:cita;,link(plane,plane):alpha,beta;,endsets,min=sum(plane:abs(cita);,for(plane(i):,bnd(-30,cita(i),30);,);,for(link(i,j)|i#ne#j:,abs(beta(i,j)+0.5*cita(i)+0.5*cita(j),alpha(i,j);,);,第18页,data:,alpha=0.000000,5.391190,32.230953,5.091816,20.963361,2.234507,5.391190,0.000000,4.8040024,6.813460,5.807866,3.815925,32.230953,4.804024,0.000000,4.364672,22.833654,2.125539,5.091816,6.613460,4.363673,0.000000,4.537692,2.989819,20.963361,5.807866,22.833654,4.537692,0.000000,2.309841,2.234507,3.815925,2.125539,2.989819,2.309841,0.000000;,beta=0.000000 109.263642-128.250000 24.179830 173.065051 13.474934,109.263642 0.000000-88.871096-42.243563-92.304847 9.000000,-128.250000-88.87096 0.000000 12.476311-58.786243 0.310809,24.179830-42.243563 12.476311 0.000000 5.969234-3.525606,174.065051-92.304846-58.786244 5.969234 0.000000 1.914383,14.474934 9.000000 0.310809-3.525606 1.913383 0.000000,;,enddata,end,用MATLAB计算编程以下,第19页,function f,g=plane(x),alpha=0.000000,5.391190,32.230953,5.091816,20.963361,2.234507,5.391190,0.000000,4.8040024,6.813460,5.807866,3.815925,32.230953,4.804024,0.000000,4.364672,22.833654,2.125539,5.091816,6.613460,4.363673,0.000000,4.537692,2.989819,20.963361,5.807866,22.833654,4.537692,0.000000,2.309841,2.234507,3.815925,2.125539,2.989819,2.309841,0.000000;,beta=0.000000 109.263642-128.250000 24.179830 173.065051 13.474934,109.263642 0.000000-88.871096-42.243563-92.304847 9.000000,-128.250000-88.87096 0.000000 12.476311-58.786243 0.310809,24.179830-42.243563 12.476311 0.000000 5.969234-3.525606,174.065051-92.304846-58.786244 5.969234 0.000000 1.914383,14.474934 9.000000 0.310809-3.525606 1.913383 0.000000;,f=abs(x(1)+abs(x(2)+abs(x(3)+abs(x(4)+abs(x(5)+abs(x(6);,g(1)=alpha(1,2)-abs(beta(1,2)+0.5*x(1)+0.5*x(2);,g(2)=alpha(1,3)-abs(beta(1,3)+0.5*x(1)+0.5*x(3);,g(3)=alpha(1,4)-abs(beta(1,4)+0.5*x(1)+0.5*x(4);,g(4)=alpha(1,5)-abs(beta(1,5)+0.5*x(1)+0.5*x(5);,第20页,g(5)=alpha(1,6)-abs(beta(1,6)+0.5*x(1)+0.5*x(6);,g(6)=alpha(2,3)-abs(beta(2,3)0.5*x(2)+0.5*x(3);,g(7)=alpha(2,4)-abs(beta(2,4)+0.5*x(2)+0.5*x(4);,g(8)=alpha(2,5)-abs(beta(2,5)+0.5*x(2)+0.5*x(5);,g(9)=alpha(2,6)-abs(beta(2,6)+0.5*x(2)+0.5*x(6);,g(10)=alpha(3,4)-abs(beta(3,4)+0.5*x(3)+0.5*x(4);,g(11)=alpha(3,5)-abs(beta(3,5)+0.5*x(3)+0.5*x(5);,g(12)=alpha(3,6)-abs(beta(3,6)+0.5*x(3)+0.5*x(6);,g(13)=alpha(4,5)-abs(beta(4,5)+0.5*x(4)+0.5*x(5);,g(14)=alpha(4,6)-abs(beta(4,6)+0.5*x(4)+0.5*x(6);,g(15)=alpha(5,6)-abs(beta(5,6)+0.5*x(5)+0.5*x(6);,执行程序,x0=0,0,0,0,0,0;v1=-30*ones(1,6);v2=30*ones(1,6);opt=;,x=constr(plane,x0,opt,v1,v2),第21页,结果：,x=-0.00000576637983,-0.00000576637983,2.58794980234726,-0.00001243487985,0.00003620473095,1.04151019765274,最优解：,第22页,模型检验,各飞行方向按此方案调整后，系统各架飞机均满足|,ij,+(,I,+,j,)/2|,ij,结果是正确。,模型评价与推广,（1）此模型采取圆状模型分析碰撞问题是合理，同时采取相对速度作为判别标准，既表达了碰撞本质（相对运动），又简化了模型计算；,（2）建模中用了适当简化，将一个复杂非线性规划问题简化为线性规划问题，既求到合理解，又提升了运算速度，这对处理高速飞行飞机碰撞问题是十分主要。此模型对题目所提供例子计算得出结果是令人满意3）由对称性知模型中约束个数是 （n是飞机数），全部约束条件数是 ，计算量增加不大。,第23页,投资的收益和风险,1998年A题,第24页,问题提出：,市场上有n种资产（如股票、债券、）S,i,(i=1,n)供投资者选择，某企业有数额为M一笔相当大资金可用作一个时期投资。企业财务分析人员对这n种资产进行了评定，估算出在这一时期内购置S,i,平均收益率为 ，并预测出购置S,i,风险损失率为 。考虑到投资越分散，总风险越小，企业确定，当用这笔资金购置若干种资产时，总体风险可用所投资S,i,中最大一个风险来度量。,购置S,i,要付交易费，费率为 ，而且当购置额不超出给定值 时，交易费按购置 计算（不买当然无须付费）。另外，假定同期银行存款利率是,且既无交易费又无风险。（=5%）,已知n=4时相关数据以下：,S,i,r,i,(%),q,i,(%),p,i(%),u,i,(元),S,1,28,2.5,1,103,S,2,21,1.5,2,198,S,3,23,5.5,4.5,52,S,4,25,2.6,6.5,40,试给该企业设计一个投资组合方案，即用给定资金，有选择地购置若干种资产或存银行生息，使净收益尽可能大，而总体风险尽可能小。,第25页,2.试就普通情况对以上问题进行讨论，并利用以下数据进行计算。,S,i,r,i,(%),q,i,(%),p,i,(%),u,i,(元),S,1,9.6,42,2.1,181,S,2,18.5,54,3.2,407,S,3,49.4,60,6.0,428,S,4,23.9,42,1.5,549,S,5,8.1,1.2,7.6,270,S,6,14,39,3.4,397,S,7,40.7,68,5.6,178,S,8,31.2,33.4,3.1,220,S,9,33.6,53.3,2.7,475,S,10,36.8,40,2.9,248,S,11,11.8,31,5.1,195,S,12,9,5.5,5.7,320,S,13,35,46,2.7,267,S,14,9.4,5.3,4.5,328,S,15,15,23,7.6,131,第26页,假定条件:,1.题中所给利率均为一年，投资期为一年。,2.企业资金足够多，且全部用于投资或存入银行。,3.投资期内不再做其它交易，利润仅在期末实现。,4.风险损失率指投资到期后，假如风险发生，损失额占投资额百分比。,5.对于风险出现概率与收益波动，在本模型中不予考虑。,6.总体风险用最大风险来衡量。,7.当购置额不超出给定值u,i,时，交易费按购置u,i,计算。,符号说明:,M：投资总额,X,i,：对第i种投资项目标投资（不含交易费）占总投资额百分比,第27页,Q：总体风险,r,i,：对第i种投资项目标平均收益率,q,i,：对第i种投资项目标风险损失率,p,i,：对第i种投资项目标交易费率u,i,：对第i种投资项目标交易费应付最小计算额,e,i,：对第i种投资项目标交易费,：乐观度常数,模型建立:,考虑两个主要问题：,1.投资所取得总收益尽可能大,2.投资所负担风险尽可能小,不考虑收益波动,用x,i,r,i,度量收益,不考虑出现风险概率,以当风险发生时损失x,i,q,i,度量风险,第28页,数学模型:,这是一个多目标非线性规划,求解困难，需要简化,化为单目标规划,化为线性规划,第29页,化为单目标规划：,若有m个目标f,i,(x)，分别给以权系数,i,(i=1,2,m),然后作新目标函数（也称效用函数）,本问题,将非线性规划转化为线性规划规划：,目标函数线性化：,令,加入约束第30页,约束条件线性化：,当投资额相当大时，能够认为x,i,u,i,，,因为,M,是常数，能够从目标函数中去除,线性规划模型,第31页,即,模型求解,用MATLAB求解,计算线性规划时用命令:y=lp(C,A,b,v1,v2,x0,n),第32页,function x=tzyh(S,a),m=length(a);,n=length(S(:,1);,h=1;,v1=zeros(n+1,1);,x0=eye(1,n+1);,b=eye(n,1);,for i=1:m,for j=1:n,C(j)=-a(i)*(1+S(j,1);,end,C(n+1)=1-a(i);,D=ones(1,n)+S3);,D=D,0;,K=S(2:n,2:2);,F=diag(K);,H=zeros(n-1,1),F,-ones(n-1,1);,A=D;H;,y=lp(C,A,b,v1,x0,h);,x(1,i)=a(i);,for j=1:n+1,x(1+j,i)=y(j);,end,end,x,z=S(:,1:1)*x(2:n+1,:),for i=1:m,for j=1:n,Q(j,i)=x(j+1,i)*S(j,2);,end,end,Q；f=x(n+2:n+2,:)；plot(f,z,b*)；,建立函数M文件tzyh以下：,其中S是由收益率，风险损失率，交易费率组成矩阵，a是乐观程度参数数组。,第33页,S=0.05,0.28,0.21,0.23,0.25;0,0.025,0.015,0.055,0.026,0,0.01,0.02,0.045,0.065,S=S,问题1）运算结果：,S=,0.0500 0 0,0.2800 0.0250 0.0100,0.2100 0.0150 0.0200,0.2300 0.0550 0.0450,0.2500 0.0260 0.0650,a=0:0.1:1,a=,0 0.1000 0.0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.0000,运行tzyh(S,a)得,第34页,a,0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,S,0,1.0000,0,0,0,0,0,0,0,0,0,0,S,1,0,0.2376,0.3690,0.9901,0.9901,0.9901,0.9901,0.9901,0.9901,0.9901,0.9901,S,2,0,0.3960,0.6150,0,0,0,0,0,0,0,0,S,3,0,0.1080,0,0,0,0,0,0,0,0,0,S,4,0,0.2284,0,0,0,0,0,0,0,0,0,净收益,0.05,0.2316,0.2325,0.2772,0.2772,0.2772,0.2772,0.2772,0.2772,0.2772,0.2772,风险,0,0.0059,0.0092,0.0248,0.0248,0.0248,0.0248,0.0248,0.0248,0.0248,0.0248,最优解、收益、及最大风险情况,第35页,a,0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,S,0,0,0,0,0,0,0,0,0,0,0,0,S,1,0,0.0059,0.0092,0.0248,0.0248,0.0248,0.0248,0.0248,0.0248,0.0248,0.0248,S,2,0,0.0059,0.0092,0,0,0,0,0,00,0,S,3,0,0.0059,0,0,0,0,0,0,0,0,0,S,4,0,0.0059,0,0,0,0,0,0,0,0,0,风险,0,0.0059,0.0092,0.0248,0.0248,0.0248,0.0248,0.0248,0.0248,0.0248,0.0248,在最优解处各项投资风险情况,对于固定a，最优投资方案中各项投资风险相同,第36页,又如：对a=0.077，,x=0.0000 0.2376 0.3960 0.1080 0.2284,收益：0.2316,风险：0.0059,问题2）求解：,a=0:0.1:1,a=0 0.1000 0.0.3000 0.4000 0.5000 0.6000 0.7000,0.8000 0.9000 1.0000,S=0.05,0.096,0.185,0.494,0.239,0.081,0.14,0.407,0.312,0.336,0.368,0.118,0.09,0.35,0.094,0.15,0,0.42,0.54,0.60,0.42,0.012,0.39,0.68,0.334,0.533,0.40,0.31,0.055,0.46,0.053,0.23,0,0.021,0.032,0.060,0.015,0.076,0.034,0.056,0.031,0.027,0.029,0.051,0.057,0.027,0.0450.076,S=S,运行tzyh(S,a)得,运行tzyh(S,a)得,第37页,S=,0.0500 0 0,0.0960 0.4200 0.0210,0.1850 0.5400 0.0320,0.4940 0.6000 0.0600,0.2390 0.4200 0.0150,0.0810 0.0120 0.0760,0.1400 0.3900 0.0340,0.4070 0.6800 0.0560,0.3120 0.3340 0.0310,0.3360 0.5330 0.0270,0.3680 0.4000 0.0290,0.1180 0.3100 0.0510,0.0900 0.0550 0.0570,0.3500 0.4600 0.0270,0.0940 0.0530 0.0450,0.1500 0.2300 0.0760,第38页,a,0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,S,0,1.000,1.000,0,0,0,0,0,0,0,0,0,S,1,0,0,0.0874,0,0,0,0,0,0,0,0,S,2,0,0,0.0680,0,0,0,0,0,0,0,0,S,3,0,0,0.0612,0.1071,0.1071,0.1269,0.1269,0.1658,0.2051,0.9434,0.9434,S,4,0,0,0.0874,0.1531,0.1531,0,0,0,0,0,0,S,5,0,0,0,0,0,0,0,0,0,0,0,S,6,0,0,0.0942,0,0,0,0,0,0,0,0,S,7,0,0,0.0540,0.0945,0.0945,0.1119,0.1119,0.1463,0.1810,0,0,S,8,0,0,0.1100,0.1925,0.1925,0.2279,0.2279,0,0,0,0,S,9,0,0,0.0689,0.1206,0.1206,0.1428,0.1428,0.1867,0,0,0,S,10,0,0,0.0918,0.1607,0.1607,0.1903,0.1903,02487,0.3077,0,0,S,11,0,0,0,0,0,0,0,0,0,0,0,S,12,0,0,0,0,0,0,0,0,0,0,0,S,13,0,0,0.0798,0.1398,0.1398,0.1655,0.1655,0.2163,0.2676,0,0,S,14,0,0,0,0,0,0,0,0,0,0,0,S,15,0,0,0.1597,0,0,0,0,0,0,0,0,收益,0.050,0.050,0.2504,0.3366,0.3366,0.3552,0.3552,0.3714,0.3819,0.4660,0.4660,风险,0,0,0.0367,0.0643,0.0643,0.0761,0.0761,0.0995,0.1231,0.5660,0.5660,第39页,a,0.0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,S,0,0,0,0,0,0,0,0,0,0,0,0,S,1,0,0,0.0367,0,0,0,0,0,0,0,0,S,2,0,0,0.0367,0,0,0,0,0,0,0,0,S,3,0,0,0.0367,0.0643,0.0643,0.0761,0.0761,0.0995,0.1231,0.5660,0.5660,S,4,0,0,0.0367,0.0643,0.0643,0,0,0,0,0,0,S,5,0,0,00,0,0,0,0,0,0,0,S,6,0,0,0.0367,0,0,0,0,0,0,0,0,S,7,0,0,0.0367,0.0643,0.0643,0.0761,0.0761,0.0995,0.1231,0,0,S,8,0,0,0.0367,0.0643,0.0643,0.0761,0.0761,0,0,0,0,S,9,0,0,0.0367,0.0643,0.0643,0.0761,0.0761,0.0995,0,0,0,S,10,0,0,0.0367,0.0643,0.0643,0.0761,0.0761,0.0995,0.1231,0,0,S,11,0,0,0,0,0,0,0,0,00,0,S,12,0,0,0,0,0,0,0,0,0,0,0,S,13,0,0,0.0367,0.0643,0.0643,0.0761,0.0761,0.0995,0.1231,0,0,S,14,0,0,0,0,0,0,0,0,0,0,0,S,15,0,0,0.0367,0,0,0,0,0,0,0,0,风险,0,0,0.0367,0.0643,0.0643,0.0761,0.0761,0.0995,0.1231,0.5660,0.5660,各项投资风险情况：,第40页,数据分析,经过上述数据，我们发觉以下特点：,（1）定理：,假如一个投资方案是所谓最优非劣解，则各项投资,所冒风险应基本相等，即,引理：,在大资金前提下，u,i,问题可近似不考虑。,定理证实：,用反证法。假设在最优方案x,1,x,2,x,n,中,有x,1,q,1,x,2,q,2,情况。故x,1,q,1,x,2,q,2,。可将两种方案推广到各种方案情况。定理得证。,第42页,（2）收益于风险关系,在,改变中，伴随,增加，收益在不停增加，风险也在增加。说明慎重程度越强，风险越小但受益也越小。含有显著实际意义，投资者可依据其对风险承受能力和投资环境，选择适当,，然后在求出最优投资方案。,对于问题（1），我们用,在01内200个等分点，求出最优投资中受益和风险曲线。如图所表示。,第43页,对于问题（2），我们用,在01内200个等分点，求出最优投资中受益和风险曲线。如图所表示。,（3）投资者性格特征,风险收益曲线是上升曲线，且是向上凸,从上图可看出，风险收益曲线是离散。问题（1）只有5个点，问题（2）只有13个点。,第44页,这说明，在,改变时，共有数次跳变，使收益和风险发生突变。,以问题（1）为例说明,当,在00.0375改变时，投资者比较侧重于不冒风险，故给出解为全部存入银行。我们称这类投资者为极憎风险者。,当,在0.03800.2330改变时，因为投资对风险要求比较平衡，故尽管存在利润微小差异，但总体上讲比较稳定，我们称这些投资者为风险折衷者。,当,在0.23351改变时，因为投资十分重视利益取得，所以在投资时，他将全部资金投入收益最高项目，我们称这些投资者为风险投资者。,对于普通情况进行观察后，我们发觉当,0.2,时，若,发生微小改变，投资额将变得越来越分散，这一点可从数据表中看出。这与分散投资以防止风险方法是相一致。,对于普通情况，我们能够用一样分析方法以求得突变点，从而将投资这分开。,第45页,模型灵敏度分析,基于投资者性格特征，本模型中存在几个跳变点，在跳变点附近，利润与风险会有大改变，它主要缘于投资者心理改变，我们姑且给它们一个名字心理极限点，在两个极限点之间，当,值作微小改变，我们能够发觉投资额并无太大改变，所以我们认为本模型在投资额定下后比较稳定，即微小心理改变不足以扰动投资额。,模型改进与推广,（1）在不需要太高精度而只需要大致百分比时，在分配资金以前可依据q,i,与q,j,百分比，确定二者资金百分比，即,这么可大大简化算法。这种方法称为等风险分散投资。,（2）企业往往会对收益和风险提出一定要求，在本模型中只需增加约束即可。,第46页,