文章目录
- 简单回顾
- 线性规划LP
- 整数规划IP
- 0-1规划
简单回顾
线性规划是数学规划中的一类最简单规划问题，常见的线性规划是一个有约束的，变量范围为有理数的线性规划。如：
使用matlab的linprog函数即可求解简单的线性规划问题，可以参考这篇博客：
MATLAB求解线性规划（含整数规划和0-1规划）问题
matlab求解线性规划LP问题需要化为最小化问题，所有约束条件必须为≤类型，限制较多。本文介绍使用python+gurobi进行求解。
python+gurobi介绍参考这篇博客：
gurobi最新下载安装教程 2023.11
线性规划LP
import gurobipy
from gurobipy import GRB
# 创建模型
c = [7, 12]
a = [[9, 4],
[4, 5],
[3, 10]]
b = [300, 200, 300]
MODEL = gurobipy.Model("Example")
# 创建变量
x = MODEL.addVars(2, lb=0, ub=gurobipy.GRB.INFINITY, name='x')
# 更新变量环境
MODEL.update()
# 创建目标函数
MODEL.setObjective(x.prod(c), gurobipy.GRB.MAXIMIZE)
# 创建约束条件
MODEL.addConstrs(x.prod(a[i]) <= b[i] for i in range(3))
# 执行线性规划模型
MODEL.optimize()
print("Obj:", MODEL.objVal)
for v in MODEL.getVars():
print(f"{v.VarName}：{round(v.X,3)}")
Gurobi Optimizer version 10.0.3 build v10.0.3rc0 (win64)
CPU model: Intel(R) Core(TM) i7-8565U CPU @ 1.80GHz, instruction set [SSE2|AVX|AVX2]
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 3 rows, 2 columns and 6 nonzeros
Model fingerprint: 0x6b25b35d
Coefficient statistics:
Matrix range [3e+00, 1e+01]
Objective range [7e+00, 1e+01]
Bounds range [0e+00, 0e+00]
RHS range [2e+02, 3e+02]
Presolve time: 0.01s
Presolved: 3 rows, 2 columns, 6 nonzeros
Iteration Objective Primal Inf. Dual Inf. Time
0 3.2500000e+30 2.812500e+30 3.250000e+00 0s
2 4.2800000e+02 0.000000e+00 0.000000e+00 0s
Solved in 2 iterations and 0.01 seconds (0.00 work units)
Optimal objective 4.280000000e+02
Obj: 428.0
x[0]：20.0
x[1]：24.0
最终可得最优解为x = 20, y = 24, 最优值为428。
gurobi对最大化问题、最小化问题，大于等于和小于等于约束都支持。
整数规划IP
import gurobipy
from gurobipy import GRB
import numpy as np
# 创建模型
c = [3, 2]
a = [[2, 3],
[4, 2]]
b = [14, 18]
MODEL = gurobipy.Model("Example")
# 创建变量
#x = MODEL.addVars(2, lb=0, ub=gurobipy.GRB.INFINITY, name='x')
x1 = MODEL.addVar(vtype=GRB.INTEGER,lb=0,ub=GRB.INFINITY, name='x1')
x2 = MODEL.addVar(vtype=GRB.INTEGER,lb=0,ub=GRB.INFINITY, name='x2')
# 更新变量环境
MODEL.update()
# 创建目标函数
MODEL.setObjective(c[0]*x1+c[1]*x2, gurobipy.GRB.MAXIMIZE)
# 创建约束条件
for i in range(2):
MODEL.addConstr(a[i][0]*x1 + a[i][1]*x2 <= b[i])
# 执行线性规划模型
MODEL.optimize()
print("Obj:", MODEL.objVal)
for v in MODEL.getVars():
print(f"{v.VarName}：{round(v.X,3)}")
Gurobi Optimizer version 10.0.3 build v10.0.3rc0 (win64)
CPU model: Intel(R) Core(TM) i7-8565U CPU @ 1.80GHz, instruction set [SSE2|AVX|AVX2]
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 2 rows, 2 columns and 4 nonzeros
Model fingerprint: 0x15a6e8bd
Variable types: 0 continuous, 2 integer (0 binary)
Coefficient statistics:
Matrix range [2e+00, 4e+00]
Objective range [2e+00, 3e+00]
Bounds range [0e+00, 0e+00]
RHS range [1e+01, 2e+01]
Found heuristic solution: objective 14.0000000
Presolve time: 0.00s
Presolved: 2 rows, 2 columns, 4 nonzeros
Variable types: 0 continuous, 2 integer (0 binary)
Explored 0 nodes (0 simplex iterations) in 0.00 seconds (0.00 work units)
Thread count was 8 (of 8 available processors)
Solution count 1: 14
Optimal solution found (tolerance 1.00e-04)
Best objective 1.400000000000e+01, best bound 1.400000000000e+01, gap 0.0000%
Obj: 14.0
x1：4.0
x2：1.0
可得该整数规划问题的最优解为x1 = 4, x2 = 1,最优值为14。
如果解其对应的松弛问题：
import gurobipy
from gurobipy import GRB
import numpy as np
# 创建模型
c = [3, 2]
a = [[2, 3],
[4, 2]]
b = [14, 18]
MODEL = gurobipy.Model("Example")
# 创建变量
x = MODEL.addVars(2, lb=0, ub=gurobipy.GRB.INFINITY, name='x')
# 更新变量环境
MODEL.update()
# 创建目标函数
MODEL.setObjective(x.prod(c), gurobipy.GRB.MAXIMIZE)
# 创建约束条件
MODEL.addConstrs(x.prod(a[i]) <= b[i] for i in range(2))
# 执行线性规划模型
MODEL.optimize()
print("Obj:", MODEL.objVal)
for v in MODEL.getVars():
print(f"{v.VarName}：{round(v.X,3)}")
Gurobi Optimizer version 10.0.3 build v10.0.3rc0 (win64)
CPU model: Intel(R) Core(TM) i7-8565U CPU @ 1.80GHz, instruction set [SSE2|AVX|AVX2]
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 2 rows, 2 columns and 4 nonzeros
Model fingerprint: 0x15a42e7d
Coefficient statistics:
Matrix range [2e+00, 4e+00]
Objective range [2e+00, 3e+00]
Bounds range [0e+00, 0e+00]
RHS range [1e+01, 2e+01]
Presolve time: 0.01s
Presolved: 2 rows, 2 columns, 4 nonzeros
Iteration Objective Primal Inf. Dual Inf. Time
0 5.0000000e+30 2.750000e+30 5.000000e+00 0s
2 1.4750000e+01 0.000000e+00 0.000000e+00 0s
Solved in 2 iterations and 0.01 seconds (0.00 work units)
Optimal objective 1.475000000e+01
Obj: 14.75
x[0]：3.25
x[1]：2.5
可以发现对应的解是x1 = 3.25, x2 = 2.5, 最优值为14.75。松弛问题的最优解总是优于整数规划问题的。
0-1规划
无论是matlab的linprog函数还是gurobi，0-1规划实际上只需要在整数规划的基础上，让决策变量的定义域在0-1之间即可。
仍然是上面的同一个问题：
## 0-1规划
import gurobipy
from gurobipy import GRB
# 创建模型
c = [3, 2]
a = [[2, 3],
[4, 2]]
b = [14, 18]
MODEL = gurobipy.Model("Example")
# 创建变量
#x = MODEL.addVars(2, lb=0, ub=gurobipy.GRB.INFINITY, name='x')
x1 = MODEL.addVar(vtype=GRB.INTEGER,lb=0,ub=1, name='x1')
x2 = MODEL.addVar(vtype=GRB.INTEGER,lb=0,ub=1, name='x2')
# 更新变量环境
MODEL.update()
# 创建目标函数
MODEL.setObjective(c[0]*x1+c[1]*x2, gurobipy.GRB.MAXIMIZE)
# 创建约束条件
for i in range(2):
MODEL.addConstr(a[i][0]*x1 + a[i][1]*x2 <= b[i])
# 执行线性规划模型
MODEL.optimize()
print("Obj:", MODEL.objVal)
for v in MODEL.getVars():
print(f"{v.VarName}：{round(v.X,3)}")
Gurobi Optimizer version 10.0.3 build v10.0.3rc0 (win64)
CPU model: Intel(R) Core(TM) i7-8565U CPU @ 1.80GHz, instruction set [SSE2|AVX|AVX2]
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 3 rows, 2 columns and 6 nonzeros
Model fingerprint: 0x6b25b35d
Coefficient statistics:
Matrix range [3e+00, 1e+01]
Objective range [7e+00, 1e+01]
Bounds range [0e+00, 0e+00]
RHS range [2e+02, 3e+02]
Presolve time: 0.01s
Presolved: 3 rows, 2 columns, 6 nonzeros
Iteration Objective Primal Inf. Dual Inf. Time
0 3.2500000e+30 2.812500e+30 3.250000e+00 0s
2 4.2800000e+02 0.000000e+00 0.000000e+00 0s
Solved in 2 iterations and 0.01 seconds (0.00 work units)
Optimal objective 4.280000000e+02
Obj: 428.0
x[0]：20.0
x[1]：24.0
Gurobi Optimizer version 10.0.3 build v10.0.3rc0 (win64)
CPU model: Intel(R) Core(TM) i7-8565U CPU @ 1.80GHz, instruction set [SSE2|AVX|AVX2]
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 3 rows, 6 columns and 18 nonzeros
Model fingerprint: 0x157f6a4a
Coefficient statistics:
Matrix range [9e+00, 6e+01]
Objective range [6e+00, 1e+01]
Bounds range [0e+00, 0e+00]
RHS range [6e+01, 2e+02]
Presolve time: 0.01s
Presolved: 3 rows, 6 columns, 18 nonzeros
Iteration Objective Primal Inf. Dual Inf. Time
0 0.0000000e+00 4.187500e+01 0.000000e+00 0s
3 2.9842520e+01 0.000000e+00 0.000000e+00 0s
Solved in 3 iterations and 0.01 seconds (0.00 work units)
Optimal objective 2.984251969e+01
Obj: 29.84251968503937
x[0]：0.0
x[1]：0.433
x[2]：0.0
x[3]：4.252
x[4]：0.0
x[5]：0.0
Gurobi Optimizer version 10.0.3 build v10.0.3rc0 (win64)
CPU model: Intel(R) Core(TM) i7-8565U CPU @ 1.80GHz, instruction set [SSE2|AVX|AVX2]
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 2 rows, 2 columns and 4 nonzeros
Model fingerprint: 0x15a6e8bd
Variable types: 0 continuous, 2 integer (0 binary)
Coefficient statistics:
Matrix range [2e+00, 4e+00]
Objective range [2e+00, 3e+00]
Bounds range [0e+00, 0e+00]
RHS range [1e+01, 2e+01]
Found heuristic solution: objective 14.0000000
Presolve time: 0.00s
Presolved: 2 rows, 2 columns, 4 nonzeros
Variable types: 0 continuous, 2 integer (0 binary)
Explored 0 nodes (0 simplex iterations) in 0.00 seconds (0.00 work units)
Thread count was 8 (of 8 available processors)
Solution count 1: 14
Optimal solution found (tolerance 1.00e-04)
Best objective 1.400000000000e+01, best bound 1.400000000000e+01, gap 0.0000%
Obj: 14.0
x1：4.0
x2：1.0
Gurobi Optimizer version 10.0.3 build v10.0.3rc0 (win64)
CPU model: Intel(R) Core(TM) i7-8565U CPU @ 1.80GHz, instruction set [SSE2|AVX|AVX2]
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 2 rows, 2 columns and 4 nonzeros
Model fingerprint: 0x15a42e7d
Coefficient statistics:
Matrix range [2e+00, 4e+00]
Objective range [2e+00, 3e+00]
Bounds range [0e+00, 0e+00]
RHS range [1e+01, 2e+01]
Presolve time: 0.01s
Presolved: 2 rows, 2 columns, 4 nonzeros
Iteration Objective Primal Inf. Dual Inf. Time
0 5.0000000e+30 2.750000e+30 5.000000e+00 0s
2 1.4750000e+01 0.000000e+00 0.000000e+00 0s
Solved in 2 iterations and 0.01 seconds (0.00 work units)
Optimal objective 1.475000000e+01
Obj: 14.75
x[0]：3.25
x[1]：2.5
Gurobi Optimizer version 10.0.3 build v10.0.3rc0 (win64)
CPU model: Intel(R) Core(TM) i7-8565U CPU @ 1.80GHz, instruction set [SSE2|AVX|AVX2]
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 2 rows, 2 columns and 4 nonzeros
Model fingerprint: 0xdff3d373
Variable types: 0 continuous, 2 integer (0 binary)
Coefficient statistics:
Matrix range [2e+00, 4e+00]
Objective range [2e+00, 3e+00]
Bounds range [1e+00, 1e+00]
RHS range [1e+01, 2e+01]
Found heuristic solution: objective 5.0000000
Explored 0 nodes (0 simplex iterations) in 0.00 seconds (0.00 work units)
Thread count was 1 (of 8 available processors)
Solution count 1: 5
Optimal solution found (tolerance 1.00e-04)
Best objective 5.000000000000e+00, best bound 5.000000000000e+00, gap 0.0000%
Obj: 5.0
x1：1.0
x2：1.0
可得0-1规划的最优解是x1 = x2 = 1, 最优值 = 5。
当然0-1规划的典型应用场景是指派问题、运输问题、排班问题等。
