问题描述：
工厂需要把 N 份工作分配给 N 个工人，每份工作只能由一个工人做，且每个工人也只能做一份工作。假设工人 i i i 处理工作 j j j 需要的时间是 T i j T_{ij} Tij ，创造的利润是 C i j C_{ij} Cij，那么需要如何安排才能使总耗时最小且总利润最大？
代码：
import gurobipy as gp
from gurobipy import GRB
import numpy as np
N = 10 # Set the number of workers and jobs
np.random.seed(1234) # To generate the same data each time
# Generate the time and cost data using the dict
# An index i corresponds to a worker
# An index j corresponds to a job
T = {(i, j): np.random.randint(0, 100) for i in range(N) for j in range(N)}
C = {(i, j): np.random.randint(0, 100) for i in range(N) for j in range(N)}
# Establish the model
m = gp.Model('Assignment_problem_with_multiple_objectives')
# Add decision variables
x = m.addVars(T.keys(), vtype=GRB.BINARY, name='x')
# Add constraints
m.addConstrs((x.sum(i, '*') == 1 for i in range(N)), 'Constraint_of_worker')
m.addConstrs((x.sum('*', j) == 1 for j in range(N)), 'Constraint_of_job')
m.setObjectiveN(x.prod(T), index=0, priority=1, abstol=0, reltol=0, name='minimize_time')
m.setObjectiveN(-x.prod(C), index=1, priority=2, abstol=100, reltol=0, name='maximize_profit')
# Solve the model
m.optimize()
# Display the results
print('*' * 60)
for i in T.keys():
if x[i].x > 0.9:
print("The job %d is allocated to the worker %d" % (i[1]+1, i[0]+1))
print("\nThe objective function values are:")
print('Obj1: the total time is: {}'.format(m.ObjNVal))
m.setParam(GRB.Param.ObjNumber, 1)
print('Obj2: the total profit is: {}'.format(-m.ObjNVal))
控制台输出结果：
Using license file C:\Users\86158\gurobi.lic
Gurobi Optimizer version 9.1.1 build v9.1.1rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 20 rows, 100 columns and 200 nonzeros
Model fingerprint: 0x65ad441f
Variable types: 0 continuous, 100 integer (100 binary)
Coefficient statistics:
Matrix range [1e+00, 1e+00]
Objective range [1e+00, 1e+02]
Bounds range [1e+00, 1e+00]
RHS range [1e+00, 1e+00]
---------------------------------------------------------------------------
Multi-objectives: starting optimization with 2 objectives ...
---------------------------------------------------------------------------
Multi-objectives: applying initial presolve ...
---------------------------------------------------------------------------
Presolve time: 0.02s
Presolved: 20 rows and 100 columns
---------------------------------------------------------------------------
Multi-objectives: optimize objective 1 (maximize_profit) ...
---------------------------------------------------------------------------
Found heuristic solution: objective -424.0000000
Presolve time: 0.00s
Presolved: 20 rows, 100 columns, 200 nonzeros
Variable types: 0 continuous, 100 integer (100 binary)
Root relaxation: objective -8.630000e+02, 18 iterations, 0.00 seconds
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
* 0 0 0 -863.0000000 -863.00000 0.00% - 0s
Explored 0 nodes (18 simplex iterations) in 0.02 seconds
Thread count was 8 (of 8 available processors)
Solution count 2: -863 -424
No other solutions better than -863
Optimal solution found (tolerance 1.00e-04)
Best objective -8.630000000000e+02, best bound -8.630000000000e+02, gap 0.0000%
---------------------------------------------------------------------------
Multi-objectives: optimize objective 2 (minimize_time) ...
---------------------------------------------------------------------------
Loaded user MIP start with objective 687
Presolve time: 0.00s
Presolved: 21 rows, 100 columns, 288 nonzeros
Variable types: 0 continuous, 100 integer (100 binary)
Root relaxation: objective 3.596170e+02, 30 iterations, 0.00 seconds
Nodes | Current Node | Objective Bounds | Work
Expl Unexpl | Obj Depth IntInf | Incumbent BestBd Gap | It/Node Time
0 0 359.61702 0 12 687.00000 359.61702 47.7% - 0s
H 0 0 396.0000000 359.61702 9.19% - 0s
H 0 0 373.0000000 359.61702 3.59% - 0s
Cutting planes:
Gomory: 1
Explored 1 nodes (30 simplex iterations) in 0.02 seconds
Thread count was 8 (of 8 available processors)
Solution count 3: 373 396 687
Optimal solution found (tolerance 1.00e-04)
Best objective 3.730000000000e+02, best bound 3.730000000000e+02, gap 0.0000%
---------------------------------------------------------------------------
Multi-objectives: solved in 0.03 seconds, solution count 4
************************************************************
The job 8 is allocated to the worker 1
The job 10 is allocated to the worker 2
The job 9 is allocated to the worker 3
The job 3 is allocated to the worker 4
The job 2 is allocated to the worker 5
The job 4 is allocated to the worker 6
The job 5 is allocated to the worker 7
The job 7 is allocated to the worker 8
The job 1 is allocated to the worker 9
The job 6 is allocated to the worker 10
The objective function values are:
Obj1: the total time is: 373.0
Obj2: the total profit is: 768.0
Process finished with exit code 0