数模竞赛-线性规划模型

线性规划

例题1：求解：

$$\begin{array}{c}maxz=70x_1+50x_2+60x_3,\\ \mathrm{s}.\mathrm{t}.\{\begin{array}{l}2x_1+4x_2+3x_3⩽150,\\ 3x_1+x_2+5x_3⩽160,\\ 7x_1+3x_2+5x_3⩽200,\\ x_i⩾0,i=1,2,3.\end{array}\end{array}$$

代码：

#pip install cvxpy cvxpy[GLPK_MI] cvxopt
import cvxpy,numpy
c=numpy.array([70,50,60])
a=numpy.array([[2,4,3],[3,1,5],[7,3,5]])
b=numpy.array([150,160,200])
x=cvxpy.Variable(3,pos=True) #决策变量 列向量
obj=cvxpy.Maximize(c@x)
cons=[a@x<=b]
prob=cvxpy.Problem(obj,cons)
prob.solve(solver='GLPK_MI')
print(x.value,prob.value) #最优解 和 最优值z

例题2：求解：

$$\begin{array}{c}maxz=1.15x_{41}+1.40x_{23}+1.25x_{32}+1.06x_{54},\\ \mathrm{s}.\mathrm{t}.\{\begin{array}{l}{x}_{11}+x_{14}=100000,\\ x_{21}+x_{23}+x_{24}=1.06x_{14},\\ x_{31}+x_{32}+x_{34}=1.15x_{21}+1.06x_{34},\\ x_{54}=1.15x_{31}+1.06x_{44},\\ x_{32}⩽40000,x_{23}⩽30000,\\ x_{ij}⩾0;i=1,2,3,4,5;j=1,2,3,4.\end{array}\end{array}$$

代码：

import cvxpy
x=cvxpy.Variable((5,4),pos=True)
obj=cvxpy.Maximize(1.15*x[3,0]+1.40*x[1,2]+1.25*x[2,1]+1.06*x[4,3])
cons=[x[0,0]+x[0,3]==100000,
      x[1,0]+x[1,2]+x[1,3]==1.06*x[0,3],
      x[2,0]+x[2,1]+x[2,3]==1.15*x[0,0]+1.06*x[1,3],
      x[3,0]+x[3,3]==1.15*x[1,0]+1.06*x[2,3],
      x[4,3]==1.15*x[2,0]+1.06*x[3,3],
      x[2,1]<=40000,x[1,2]<=30000]
prob=cvxpy.Problem(obj,cons)
prob.solve(solver='GLPK_MI')
print(prob.value,x.value)

例题3：求解：

$$\begin{array}{c}minz=2800(x_{11}+x_{21}+x_{31}+x_{41})+4500(x_{12}+x_{22}+x_{32})+6000(x_{13}+x_{23})+7300x_{14},\\ \mathrm{s}.\mathrm{t}.\{\begin{array}{l}{x}_{11}+x_{12}+x_{13}+x_{14}⩾15,\\ x_{12}+x_{13}+x_{14}+x_{21}+x_{22}+x_{23}⩾10,\\ x_{13}+x_{14}+x_{22}+x_{23}+x_{31}+x_{32}⩾20,\\ x_{14}+x_{23}+x_{32}+x_{41}⩾12,\\ x_{ij}⩾0;i=1,2,3,4;j=1,2,3,4.\end{array}\end{array}$$

代码：

import cvxpy
x=cvxpy.Variable((4,4),pos=True)
obj=cvxpy.Minimize(2800*sum(x[:,0])+4500*sum(x[:3,1])+6000*sum(x[:2,2])+7300*x[0,3])
cons=[sum(x[0,:])>=15,
      sum(x[0,1:])+sum(x[2,:3])>=10,
      sum(x[0,2:])+sum(x[1,1:3])+sum(x[2,:2])>=20,
      x[0,3]+x[1,2]+x[2,1]+x[3,0]>=12]
prob=cvxpy.Problem(obj,cons)
prob.solve(solver='GLPK_MI')
print(prob.value,x.value)

整数规划

例题：

$$\begin{array}{c}minz=\sum _{i=1}^6{x}_i,\\ \mathrm{s}.\mathrm{t}.\{\begin{array}{l}{x}_1+x_6⩾35,\\ x_1+x_2⩾40,\\ x_2+x_3⩾50,\\ x_3+x_4⩾45,\\ x_4+x_5⩾55,\\ x_5+x_6⩾30,\\ x_i⩾0,x_i\in \mathbb{Z},i=1,2,\cdots ,6.\end{array}\end{array}$$

代码：

import cvxpy,numpy
a=numpy.array([35,40,50,45,55,30])
x=cvxpy.Variable(6,integer=True)
obj=cvxpy.Minimize(sum(x))
cons=[x>=0]
for i in range(6):
	cons.append(x[(i-1)%6]+x[i]>=a[i])
prob=cvxpy.Problem(obj,cons)
prob.solve(solver='GLPK_MI')
print(prob.value,x.value)