Mathematical Modeling: 11 Mathematical Programming Models

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 线性规划：
[x,fval] = linprog(c, A, b, Aeq, beq, lb, ub, x0);

% c是目标函数各个变量的系数组成的列向量
% lb是各个变量的下界组成的列向量
% 默认求的是目标函数的min、不等式约束是小于等于

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 整数规划：
[x,fval] = intlinprog(c, intcon, A, b, Aeq, beq, lb, ub);

% intcon 行向量或数字，哪个变量要求是整数 就把它的下标加进来

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 01规划：
% 在整数规划基础上 把lb ub限制为0 1

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 非线性规划：
[x,fval] = fmincon(@fun, x0, A, b, Aeq, beq, lb, ub, @nonlfun, option);

% x0表示给定的初始值（用行向量或者列向量表示），必须得写
% option 表示求解非线性规划使用的方法
option = optimoptions('fmincon','Algorithm','interior-point')           % 内点法
option = optimoptions('fmincon','Algorithm','sqp')                      % SQP（序列二次规划法）
option = optimoptions('fmincon','Algorithm','active-set')               % active set（有效集法）
option = optimoptions('fmincon','Algorithm','trust-region-reflective')  % trust region reflective (信赖域反射算法)

%%%%%%%%%%%%%% 使用蒙特卡罗的方法来找初始值(推荐）
clc,clear;
n=10000000;
x1=unifrnd(-100,100,n,1);  % 生成在[-100,100]之间均匀分布的随机数组成的n行1列的向量构成x1
x2=unifrnd(-100,100,n,1); 
fmin=+inf; % 初始化函数f的最小值为正无穷（后续只要找到一个比它小的我们就对其更新）
for i=1:n
    x = [x1(i), x2(i)];  %构造x向量, 这里千万别写成了：x =[x1, x2]
    if 约束条件1 & 约束条件1
        result = 目标函数;  % 如果满足条件就计算目标函数值
        if  result  < fmin 
            fmin = result; 
            x0 = x;
        end
    end
end

%%%%%%%%%%%%%%%% fun.m
function f = fun(x)
% 目标函数：max  f(x) = x1^2 +x2^2 -x1*x2 -2x1 -5x2
    f = -x(1)^2-x(2)^2 +x(1)*x(2)+2*x(1)+5*x(2) ; 
end

%%%%%%%%%%%%%%%% nonlfun.m
function [c,ceq] = nonlfun(x)
    % 非线性不等式约束，多个要用列向量
    c = [-x(1)^2+x(2)-x(3)^2;  
         x(1)+x(2)^2+x(3)^2-20];
    % 非线性等式约束
    ceq = [-x(1)-x(2)^2+2;
           x(2)+2*x(3)^2-3]; 
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 最大最小化模型：
[x,fval] = fminimax(@Fun, x0, A, b, Aeq, beq, lb, ub, @nonlfun, option);

% Fun是函数向量 Fun.m
function f = Fun(x)
    a=[1 4 3 5 9 12 6 20 17 8];
    b=[2 10 8 18 1 4 5 10 8 9];
    %  函数向量
    f=zeros(10,1);
    for i = 1:10
        f(i) = abs(x(1)-a(i))+abs(x(2)-b(i));  
    end
end 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 多目标规划：
[x,fval] = fgoalattain(func,goal,weight,x0,A,b,Aeq,beq,lb,ub)
% 具体用法看后面的例子

 

Table of contents

Overview

Classification

linear programming

example

nonlinear programming

example

Site selection problem

National Competition Question A: Flight Management

integer programming

example

backpack problem

assignment problem

Steel pipe cutting problem

max-min model

example

multi-objective programming model

References

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Overview

The goal of mathematical programming: the problem of finding the extreme value of the objective function under certain constraints

Composition: decision variables, objective function, constraints

Classification

• Linear programming: The objective function and constraints are both linear expressions of decision variables (the simplex method can definitely solve linear programming and has unique/multiple solutions)
• Nonlinear programming: One of the objective functions and constraints is a nonlinear expression of the decision variable (the initial value of the decision variable needs to be given, and only approximate solutions can be found)
• Integer programming: require variables to be integers
  • linear integer programming
  • Nonlinear integer programming: There is currently no accurate solution, Monte Carlo can be used
• 01 planning: a special case of integer planning, which can only take 0/1

linear programming

Reference: matlab linear programming--a simple introduction_Lao Cai's blog-CSDN blog_matlab linear programming

 

The initial value x0 only speeds up the result, and the result has nothing to do with x0

example

Matlab linear programming examples_zz_Outlier's blog-CSDN blog_matlab linear programming examples

clear,clc
format long g   %可以将Matlab的计算结果显示为一般的长数字格式（默认会保留四位小数，或使用科学计数法）
% (1) 系数向量
a=[1.25  8.75  0.5  5.75  3  7.25];  % 工地的横坐标
b=[1.25  0.75  4.75	5  6.5  7.25];   % 工地的纵坐标
x = [5  2];  % 料场的横坐标
y = [1  7];  % 料场的纵坐标
c = [];  % 初始化用来保存工地和料场距离的向量 (这个向量就是我们的系数向量）
for  j =1:2
    for i = 1:6
        c = [c;  sqrt( (a(i)-x(j))^2 + (b(i)-y(j))^2)];  % 每循环一次就在c的末尾插入新的元素
    end
end
% (2) 不等式约束
A =zeros(2,12);
A(1,1:6) = 1;
A(2,7:12) = 1;
b = [20,20]';
% (3) 等式约束
Aeq = zeros(6,12);  
for i = 1:6
    Aeq(i,i) = 1;  Aeq(i,i+6) = 1;
end
% Aeq = [eye(6),eye(6)]  % 两个单位矩阵横着拼起来
beq = [3 5 4 7 6 11]';  % 每个工地的日需求量
%（4）上下界
lb = zeros(12,1);

% 进行求解
[x fval] = linprog(c, A, b, Aeq, beq, lb)
x = reshape(x,6,2)  % 将x变为6行2列便于观察（reshape函数是按照列的顺序进行转换的，也就是第一列读完，读第二列，即x1对应x_1,1，x2对应x_2,1）

% fval =
%           135.281541790676

nonlinear programming

First use Monte Carlo to find a solution as the initial value x0 for nonlinear programming, matlab: fmincon

example

Site selection problem

National Competition Question A: Flight Management

See pdf for details

integer programming

 

example

backpack problem

assignment problem

%% 指派问题（选择队员去进行游泳接力比赛）
clear;clc
c = [66.8 75.6 87 58.6 57.2 66 66.4 53 78 67.8 84.6 59.4 70 74.2 69.6 57.2 67.4 71 83.8 62.4]';  % 目标函数的系数矩阵（先列后行的写法）
intcon = [1:20];  % 整数变量的位置(一共20个决策变量，均为0-1整数变量)
% 线性不等式约束的系数矩阵和常数项向量（每个人只能入选四种泳姿之一，一共五个约束）
A = [1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0;
       0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0;
       0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0;
       0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1];
% A = zeros(5,20);
% for i = 1:5
%     A(i, (4*i-3): 4*i) = 1;
% end
b = [1;1;1;1;1];
% 线性等式约束的系数矩阵和常数项向量 （每种泳姿有且仅有一人参加，一共四个约束）
Aeq = [   1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0;
          0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0;
          0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0;
          0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1];
% Aeq = [eye(4),eye(4),eye(4),eye(4),eye(4)];  % 或者写成 repmat(eye(4),1,5)  
beq = [1;1;1;1];
lb = zeros(20,1);  % 约束变量的范围下限
ub = ones(20,1);  % 约束变量的范围上限
%最后调用intlinprog()函数
[x,fval] = intlinprog(c,intcon,A,b,Aeq,beq,lb,ub)
% reshape(x,4,5)'
%      0     0     0     1    甲自由泳
%      1     0     0     0    乙蝶泳
%      0     1     0     0    丙仰泳
%      0     0     1     0    丁蛙泳
%      0     0     0     0    戊不参加

Steel pipe cutting problem

The enumeration method is used here: a 6.9m piece can only cut out 2 2.9m pieces, so the enumeration range of 2.9m is: [0, 2]; similarly, 2.1m: [0, 3]; 0.9m: [0 , 6]

for i = 0: 2  % 2.9m长的圆钢的数量 最多只能切出2根
    for j = 0: 3  % 2.1m长的圆钢的数量
        for k = 0:6   % 1m长的圆钢的数量
            if 2.9*i+2.1*j+1*k >= 6 && 2.9*i+2.1*j+1*k <= 6.9
                disp([i, j, k])
            end
        end
    end
end

c = ones(7,1);  % 目标函数的系数矩阵
intcon=[1:7];  %  整数变量的位置(一共7个决策变量，均为整数变量)
A = -[   1 2 0 0 0 0 1;  
         0 0 3 2 1 0 1;
         4 1 0 2 4 6 1];  % 线性不等式约束的系数矩阵
b = -[100 100 100]'; %  线性不等式约束的常数项向量
lb = zeros(7,1); % 约束变量的范围下限
[x,fval]=intlinprog(c,intcon,A,b,[],[],lb)

 

max-min model

example

 

multi-objective programming model

References

[Mathematical Modeling] Multi-objective Planning_SuperSources' Blog-CSDN Blog_Multi-Objective Planning

Numerical Analog 3—Matlab linear programming, nonlinear programming, multi-objective programming (super complete solution collection)_day by day get on's blog-CSDN blog_Nonlinear programming matlab programming to find the maximum value

Source of the following examples: https://shenlong.blog.csdn.net/article/details/110501496

Multi-objective genetic algorithm:  https://zstar.blog.csdn.net/article/details/119302954

function f = mutiplesubjiect(x)
  f(1)=3*x(1)-2*x(2);
  f(2)=-4*x(1)-3*x(2);

% 单独对每个目标线性规划：
f1=[3;-2];
a=[2  3;2 1];
b=[18;10];
lb=[0;0];
ub=[];
[x,favl]=linprog(f1,a,b,[],[],lb,ub);
% 输出结果为：x=0.0000 6.0000，favl=-12.0000，所以最大值为12.0000

f2=[-4;-3];
[x,favl]=linprog(f2,a,b,[],[],lb,ub);
% 输出结果为：x=3.0000 4.0000，favl=-24.0000，所以最大值为24.0000

% 所以得到目标goal和权重
goal=[12,24];
weight=[12,24];
x0=[0;0];
[x,fval] = fgoalattain('mutiplesubjiect',goal,weight,x0,a,b,[],[],lb,[])