1 概述
规划问题的数学模型一般由三个因素构成
决策变量 目标函数 约束条件
数学规划是运筹学的一个重要分支，线性规划是数学规划的一个重要分支
线性规划即以线性函数为目标函数，线性条件为约束条件
建立线性规划模型的基本步骤
（1）分析问题，找出决策变量
（2）根据问题，找出决策变量必须满足的一组线性等式或者不等式约束，即为约束条件
（3）根据问题的目标，构造关于决策变量的一个线性函数，即为目标函数
2 算例及Matlab代码实现
2.1 算例
2.2 Python代码实现
from gurobipy import * # 在Python中调用gurobi求解包 Max_Obj=1 # 用于判定目标函数求最大值还是求最小值 M_LP=Model("LP") #线性规划问题 # 变量声明 OP =M_LP.addVar(lb=-GRB.INFINITY,ub=GRB.INFINITY, name="OP") x1 =M_LP.addVar(lb=0,ub=5, name="x1") x2 =M_LP.addVar(lb=0,ub=3, name="x2") x3 =M_LP.addVar(lb=0,ub=2, name="x3") # 对变量边界进行限定的第二种写法：以x1为例 # M_LP.addConstr(x1>=0,"Bound_Con11") # M_LP.addConstr(x1<=5,"Bound_Con12") # 设置目标函数 if Max_Obj==0: M_LP.setObjective(x1+2*x2-3*x3,GRB.MINIMIZE) else: M_LP.setObjective(x1+2*x2-3*x3,GRB.MAXIMIZE) # 添加约束 M_LP.addConstr(x1+2*x2<=3,"Con1") M_LP.addConstr(x2+x3<=2,"Con2") M_LP.addConstr(x1+x2+x3==4,"Con3") # Optimize model M_LP.optimize() M_LP.write("LP.lp") print('**************') print(' The optimal solution ') print('**************') print('OP is :',M_LP.ObjVal) # 输出目标值 print('x1 is :',x1.x) # .x 用于输出 X1 的值 print('x2 is :',x2.x) print('x3 is :',x3.x)
from gurobipy import * # 在Python中调用gurobi求解包 Max_Obj=1 # 用于判定目标函数求最大值还是求最小值 M_LP=Model("LP") #线性规划问题 # 变量声明 OP =M_LP.addVar(lb=-GRB.INFINITY,ub=GRB.INFINITY, name="OP") x1 =M_LP.addVar(lb=0,ub=5, name="x1") x2 =M_LP.addVar(lb=0,ub=3, name="x2") x3 =M_LP.addVar(lb=0,ub=2, name="x3") # 对变量边界进行限定的第二种写法：以x1为例 # M_LP.addConstr(x1>=0,"Bound_Con11") # M_LP.addConstr(x1<=5,"Bound_Con12") # 设置目标函数 if Max_Obj==0: M_LP.setObjective(x1+2*x2-3*x3,GRB.MINIMIZE) else: M_LP.setObjective(x1+2*x2-3*x3,GRB.MAXIMIZE) # 添加约束 M_LP.addConstr(x1+2*x2<=3,"Con1") M_LP.addConstr(x2+x3<=2,"Con2") M_LP.addConstr(x1+x2+x3==4,"Con3") # Optimize model M_LP.optimize() M_LP.write("LP.lp") print('**************') print(' The optimal solution ') print('**************') print('OP is :',M_LP.ObjVal) # 输出目标值 print('x1 is :',x1.x) # .x 用于输出 X1 的值 print('x2 is :',x2.x) print('x3 is :',x3.x)
2.3 求解结果
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64) Thread count: 8 physical cores, 16 logical processors, using up to 16 threads Optimize a model with 3 rows, 4 columns and 7 nonzeros Model fingerprint: 0xdcaef29a Coefficient statistics: Matrix range [1e+00, 2e+00] Objective range [1e+00, 3e+00] Bounds range [2e+00, 5e+00] RHS range [2e+00, 4e+00] Presolve removed 1 rows and 2 columns Presolve time: 0.00s Presolved: 2 rows, 2 columns, 4 nonzeros Iteration Objective Primal Inf. Dual Inf. Time 0 1.0100000e+00 1.004000e+00 0.000000e+00 0s 1 -0.0000000e+00 0.000000e+00 0.000000e+00 0s Solved in 1 iterations and 0.00 seconds (0.00 work units) Optimal objective -0.000000000e+00 ************** The optimal solution ************** OP is : -0.0 x1 is : 3.0 x2 is : 0.0 x3 is : 1.0 Process finished with exit code 0
Gurobi Optimizer version 9.5.2 build v9.5.2rc0 (win64) Thread count: 8 physical cores, 16 logical processors, using up to 16 threads Optimize a model with 3 rows, 4 columns and 7 nonzeros Model fingerprint: 0xdcaef29a Coefficient statistics: Matrix range [1e+00, 2e+00] Objective range [1e+00, 3e+00] Bounds range [2e+00, 5e+00] RHS range [2e+00, 4e+00] Presolve removed 1 rows and 2 columns Presolve time: 0.00s Presolved: 2 rows, 2 columns, 4 nonzeros Iteration Objective Primal Inf. Dual Inf. Time 0 1.0100000e+00 1.004000e+00 0.000000e+00 0s 1 -0.0000000e+00 0.000000e+00 0.000000e+00 0s Solved in 1 iterations and 0.00 seconds (0.00 work units) Optimal objective -0.000000000e+00 ************** The optimal solution ************** OP is : -0.0 x1 is : 3.0 x2 is : 0.0 x3 is : 1.0 Process finished with exit code 0
