Gruobi多目标规划

某公司3个工厂生产的产品供应给4个客户,各工厂生产量、各用户需求量
及从各工厂到各用户的单位产品的运输费用由下表给出．

用户	1	2	3	4	生产量
工厂 1	5	2	6	7	300
工厂 2	3	5	4	6	200
工厂 3	4	5	2	3	400
需求量	200	100	450	250

(1)如果只考虑总运费最小,制定所有产品的最优运输方案.
(2)由于总生产量小于总需求量,公司经研究确定了调配方案的8项目标并规定了重要性的次序:第一目标:用户4为重要客户,需求量必须全部满足;第二目标:供应用户1的产品中,工厂3的产品不少于100个单位;第三目标:每个用户的满足率不低于80% ;第四目标:应尽量满足各用户的需求;第五目标:总运费不超过(1)中运输方案的10% ;第六目标:工厂2到用户4的路线应尽量避免运输任务;第七目标:用户1和用户3的需求满足率应尽量保持平衡;第八目标:力求减少总运费.
请列出相应的目标规划模型并进行求解.

第一问:

由于总生产量小于总需求量(相差100)个单位，因此虚设工厂4，具有100的生产力,到各用户之间的价格为0

设变量 X = [ x i j ] 3 ∗ 4 X = [x_{ij}]_{3*4} X=[xij]3∗4表示工厂 i i i向用户 j j j运送货物的量,那么针对第一问可建立模型如下
M i n z = ∑ i = 1 , j = 1 i = 3 , j = 4 x i j ∗ p r i c e i j S . T . { ∑ j = 1 j = 4 x i j < = m a k e [ i ] , i = 1 , 2 , 3 , 4 ∑ j = 1 j = 4 x i j > = n e e d [ j ] , i = 1 , 2 , 3 , 4 x i j > = 0 Min \quad z = \sum_{i=1,j=1}^{i=3,j=4}x_{ij}*price_{ij}\\ S.T.\left\{ \begin{aligned} \sum_{j=1}^{j=4}x_{ij}<=make[i],i=1,2,3,4\\ \sum_{j=1}^{j=4}x_{ij}>=need[j],i = 1,2,3,4\\ x_{ij}>=0\\ \end{aligned} \right. Minz=i=1,j=1∑i=3,j=4xij∗priceijS.T.⎩⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎧j=1∑j=4xij<=make[i],i=1,2,3,4j=1∑j=4xij>=need[j],i=1,2,3,4xij>=0

第二问:

由于总生产量小于总需求量(相差100)个单位，需求不可全部满足，因此新设置变量 b i − > = 0 b_{i}^{^-}>=0 bi−>=0与 b i + > = 0 b_{i}^{^+}>=0 bi+>=0满足 ∑ i = 1 i = 3 x i j + b j − − b j + > = n e e d [ j ] , j = 1 , 2 , 3 , 4 \sum_{i=1}^{i=3}x_{ij}+b_{j}^{^-}-b_{j}^{^+}>=need[j],\quad j=1,2,3,4 i=1∑i=3xij+bj−−bj+>=need[j],j=1,2,3,4
针对第四目标 b i − > = 0 b_{i}^{^-}>=0 bi−>=0与 b i + > = 0 b_{i}^{^+}>=0 bi+>=0分别表示不够的和超出的

针对第二目标设立松弛因子 s l a c k 2 − > = 0 , s l a c k 2 + > = 0 slack_2^{^-}>=0,slack_2^{^+}>=0 slack2−>=0,slack2+>=0

针对第三目标设立 c i − > = 0 c_{i}^{^-}>=0 ci−>=0与 c i + > = 0 c_{i}^{^+}>=0 ci+>=0

针对第五目标设立松弛因子 s l a c k 5 − > = 0 , s l a c k 5 + > = 0 slack_5^{^-}>=0,slack_5^{^+}>=0 slack5−>=0,slack5+>=0

针对第七目标设立松弛因子 s l a c k 7 − > = 0 , s l a c k 7 + > = 0 slack_7^{^-}>=0,slack_7^{^+}>=0 slack7−>=0,slack7+>=0

第一问求得总运费为 2950.0 2950.0 2950.0

那么可以构建约束条件如下
S . T . { ∑ j = 1 j = 4 x i j < = m a k e [ i ] , i = 1 , 2 , 3 ∑ i = 1 i = 3 x i j + b j − − b j + > = n e e d [ j ] , j = 1 , 2 , 3 , 4 ∑ i = 1 i = 3 x i j + c j − − c j + > = n e e d [ j ] ∗ 0.8 , j = 1 , 2 , 3 , 4 x 31 + s l a c k 2 − − s l a c k 2 + > = 100 p r i c e a l l = ∑ i = 1 , j = 1 i = 3 , j = 4 x i j ∗ p r i c e i j p r i c e a l l + s l a c k 5 − − s l a c k 5 + = 2950 ∗ 1.1 b 1 − n e e d [ 1 ] + s l a c k 7 − − s l a c k 7 + = b 3 − n e e d [ 3 ] x i j > = 0 , b j − > = 0 , b j + > = 0 S.T.\left\{ \begin{aligned} \sum_{j=1}^{j=4}x_{ij}<=make[i],i=1,2,3\\ \sum_{i=1}^{i=3}x_{ij}+b_{j}^{^-}-b_{j}^{^+}>=need[j],j = 1,2,3,4\\ \sum_{i=1}^{i=3}x_{ij}+c_{j}^{^-}-c_{j}^{^+}>=need[j]*0.8,j = 1,2,3,4\\ x_{31}+slack_2^{^-}-slack_2^{^+}>=100\\ price_{all} = \sum_{i=1,j=1}^{i=3,j=4}x_{ij}*price_{ij}\\ price_{all}+slack_5^{^-}-slack_5^{^+} =2950*1.1\\ \frac{b_{1}^{^-}}{need[1]}+slack_7^{^-}-slack_7^{^+}=\frac{b_{3}^{^-}}{need[3]}\\ x_{ij}>=0,b_{j}^{^-}>=0,b_{j}^{^+}>=0\\ \end{aligned} \right. S.T.⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧j=1∑j=4xij<=make[i],i=1,2,3i=1∑i=3xij+bj−−bj+>=need[j],j=1,2,3,4i=1∑i=3xij+cj−−cj+>=need[j]∗0.8,j=1,2,3,4x31+slack2−−slack2+>=100priceall=i=1,j=1∑i=3,j=4xij∗priceijpriceall+slack5−−slack5+=2950∗1.1need[1]b1−+slack7−−slack7+=need[3]b3−xij>=0,bj−>=0,bj+>=0
那么这几个目标可以依次表达为
( 1 ) M i n b 4 − ( 2 ) M i n s l a c k 2 − ( 3 ) M i n ∑ j = 1 j = 4 c j − ( 4 ) M i n ∑ j = 1 j = 4 b j − ( 5 ) M i n s l a c k 5 − ( 6 ) M i n x 24 ( 7 ) M i n s l a c k 7 − + s l a c k 7 + ( 8 ) M i n p r i c e a l l (1)Min\quad b_{4}^{^-}\\ (2)Min\quad slack_2^{^-}\\ (3)Min\quad \sum_{j=1}^{j=4}c_{j}^{^-}\\ (4)Min\quad \sum_{j=1}^{j=4}b_{j}^{^-}\\ (5)Min\quad slack_5^{^-}\\ (6)Min\quad x_{24}\\ (7)Min\quad slack_7^{^-}+slack_7^{^+}\\ (8)Min\quad price_{all}\\ (1)Minb4−(2)Minslack2−(3)Minj=1∑j=4cj−(4)Minj=1∑j=4bj−(5)Minslack5−(6)Minx24(7)Minslack7−+slack7+(8)Minpriceall

import gurobipy as grb
import numpy as np
from gurobipy import *
model = grb.Model()
X = model.addVars(4,4,name = 'X')
price = np.array([[5,2,6,7],[3,5,4,6],[4,5,2,3],[0,0,0,0]])
make = [300,200,400,100]
need = [200,100,450,250]
model.addConstrs((X.sum(i,'*')<=make[i] for i in range(4)),name = "make")
model.addConstrs((X.sum('*',j)>=need[j] for j in range(4)),name = 'need')# 目标函数
model.setObjective(grb.quicksum(X[i,j]*price[i,j] for i in range(4) for j in range(4)), grb.GRB.MINIMIZE)
#grb.quicksum(X[i,j]*price[i,j] for i in range(3) for j in range(4))
model.write("test.lp")
# 求解
model.optimize()
for v in model.getVars():print(v.varName, '=', v.x)
print('目标函数值是：', model.objVal)
print("变量矩阵:\n",np.array(model.X).reshape(4,4))

Gurobi Optimizer version 9.1.2 build v9.1.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 8 rows, 16 columns and 32 nonzeros
Model fingerprint: 0x711bf819
Coefficient statistics:Matrix range     [1e+00, 1e+00]Objective range  [2e+00, 7e+00]Bounds range     [0e+00, 0e+00]RHS range        [1e+02, 5e+02]
Presolve time: 0.01s
Presolved: 8 rows, 16 columns, 32 nonzerosIteration    Objective       Primal Inf.    Dual Inf.      Time0    0.0000000e+00   1.000000e+03   0.000000e+00      0s9    2.9500000e+03   0.000000e+00   0.000000e+00      0sSolved in 9 iterations and 0.02 seconds
Optimal objective  2.950000000e+03
X[0,0] = 200.0
X[0,1] = 100.0
X[0,2] = 0.0
X[0,3] = 0.0
X[1,0] = 0.0
X[1,1] = 0.0
X[1,2] = 200.0
X[1,3] = 0.0
X[2,0] = 0.0
X[2,1] = 0.0
X[2,2] = 250.0
X[2,3] = 150.0
X[3,0] = 0.0
X[3,1] = 0.0
X[3,2] = 0.0
X[3,3] = 100.0
目标函数值是： 2950.0
变量矩阵:[[200. 100.   0.   0.][  0.   0. 200.   0.][  0.   0. 250. 150.][  0.   0.   0. 100.]]

import gurobipy as grb
import numpy as np
from gurobipy import *
model = grb.Model()
#添加变量
X = model.addVars(3,4,vtype=grb.GRB.INTEGER,name = 'X')
b1 = model.addVars(4,lb = 0,vtype=grb.GRB.INTEGER,name = 'b+')
b2 = model.addVars(4,lb = 0,vtype=grb.GRB.INTEGER,name = 'b-')
c1 = model.addVars(4,lb = 0,vtype=grb.GRB.INTEGER,name = 'c+')
c2 = model.addVars(4,lb = 0,vtype=grb.GRB.INTEGER,name = 'c-')
slack = model.addVars(3,2,lb=0,vtype=grb.GRB.INTEGER,name = 'slack')
price_all = model.addVar(lb=0,vtype=grb.GRB.INTEGER,name='Price_all')
#3行2列变量，第一列代表slack-，第二列代表slack+。第123行分别表示模型中的slack_2,slack_5,slack_7
price = np.array([[5,2,6,7],[3,5,4,6],[4,5,2,3]])
make = [300,200,400]
need = [200,100,450,250]
#添加约束
model.addConstrs((X.sum(i,'*')<=make[i] for i in range(3)),name = "make")
model.addConstrs((X.sum('*',j)-b1[j]+b2[j]>=need[j] for j in range(4)),name = 'need')
model.addConstrs((X.sum('*',j)-c1[j]+c2[j]>=need[j]*0.8 for j in range(4)),name = '80% need')
model.addConstr(X[2,0]+slack[0,1]-slack[0,0]>=100,name = '工厂3的产品不少于100个单位')
model.addConstr(grb.quicksum(X[i,j]*price[i,j] for i in range(3) for j in range(4))==price_all,name = 'price_all_con')
model.addConstr(price_all+slack[1,1]-slack[1,0]==2950*1.1,'Price_con')
model.addConstr(b2[0]/need[0]+slack[2,1]-slack[2,0]==b2[2]/need[2])
#设立多个目标，根据题意采用分层型
model.setObjectiveN(b2[3],index=0,priority=8,name="目标1")
model.setObjectiveN(slack[0,1],index=1,priority=7,name="目标2")
model.setObjectiveN(c2.sum(),index=2,priority=6,name="目标3")
model.setObjectiveN(b2.sum(),index=3,priority=5,name="目标4")
model.setObjectiveN(slack[1,1],index=4,priority=4,name="目标5")
model.setObjectiveN(X[1,3],index=5,priority=3,name="目标6")
model.setObjectiveN(slack.sum(2,'*'),index=6,priority=2,name="目标7")
model.setObjectiveN(price_all,index=7,priority=1,name="目标8")
#模型写入文件
model.write("test.lp")
# 求解
model.optimize()
for i in range(model.NumObj):model.setParam(grb.GRB.Param.ObjNumber, i)print('Obj%d = ' % (i + 1), model.ObjNVal)X_value = np.zeros((3,4))
for i in X.iterkeys():X_value[i] = (X.get(i).X)
print("运输矩阵为:\n",X_value,"最小价格为:",model.ObjNVal)

Warning: constraint name "80% need[0]" has a space
Gurobi Optimizer version 9.1.2 build v9.1.2rc0 (win64)
Thread count: 4 physical cores, 8 logical processors, using up to 8 threads
Optimize a model with 15 rows, 35 columns and 75 nonzeros
Model fingerprint: 0x8b575f13
Variable types: 0 continuous, 35 integer (0 binary)
Coefficient statistics:Matrix range     [2e-03, 7e+00]Objective range  [1e+00, 1e+00]Bounds range     [0e+00, 0e+00]RHS range        [8e+01, 3e+03]---------------------------------------------------------------------------
Multi-objectives: starting optimization with 8 objectives ...
---------------------------------------------------------------------------Multi-objectives: applying initial presolve ...
---------------------------------------------------------------------------Presolve removed 1 rows and 1 columns
Presolve time: 0.00s
Presolved: 14 rows and 34 columns
---------------------------------------------------------------------------Multi-objectives: optimize objective 1 (目标1) ...
---------------------------------------------------------------------------Found heuristic solution: objective 250.0000000
Presolve removed 7 rows and 21 columns
Presolve time: 0.00s
Presolved: 7 rows, 13 columns, 24 nonzeros
Variable types: 0 continuous, 13 integer (0 binary)Root relaxation: objective 0.000000e+00, 3 iterations, 0.00 secondsNodes    |    Current Node    |     Objective Bounds      |     WorkExpl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time*    0     0               0       0.0000000    0.00000  0.00%     -    0sExplored 0 nodes (3 simplex iterations) in 0.08 seconds
Thread count was 8 (of 8 available processors)Solution count 2: 0 250 Optimal solution found (tolerance 1.00e-04)
Best objective 0.000000000000e+00, best bound 0.000000000000e+00, gap 0.0000%
---------------------------------------------------------------------------Multi-objectives: optimize objective 2 (目标2) ...
---------------------------------------------------------------------------Loaded user MIP start with objective 100Presolve removed 8 rows and 22 columns
Presolve time: 0.00s
Presolved: 7 rows, 12 columns, 24 nonzeros
Variable types: 0 continuous, 12 integer (0 binary)Root relaxation: objective 0.000000e+00, 2 iterations, 0.00 secondsNodes    |    Current Node    |     Objective Bounds      |     WorkExpl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time*    0     0               0       0.0000000    0.00000  0.00%     -    0sExplored 0 nodes (2 simplex iterations) in 0.12 seconds
Thread count was 8 (of 8 available processors)Solution count 2: 0 100 Optimal solution found (tolerance 1.00e-04)
Best objective 0.000000000000e+00, best bound 0.000000000000e+00, gap 0.0000%
---------------------------------------------------------------------------Multi-objectives: optimize objective 3 (目标3) ...
---------------------------------------------------------------------------Loaded user MIP start with objective 440Presolve removed 6 rows and 16 columns
Presolve time: 0.00s
Presolved: 10 rows, 18 columns, 38 nonzeros
Variable types: 0 continuous, 18 integer (0 binary)Root relaxation: objective 0.000000e+00, 8 iterations, 0.00 secondsNodes    |    Current Node    |     Objective Bounds      |     WorkExpl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time*    0     0               0       0.0000000    0.00000  0.00%     -    0sExplored 0 nodes (8 simplex iterations) in 0.17 seconds
Thread count was 8 (of 8 available processors)Solution count 2: 0 440 Optimal solution found (tolerance 1.00e-04)
Best objective 0.000000000000e+00, best bound 0.000000000000e+00, gap 0.0000%
---------------------------------------------------------------------------Multi-objectives: optimize objective 4 (目标4) ...
---------------------------------------------------------------------------Loaded user MIP start with objective 150Presolve removed 6 rows and 18 columns
Presolve time: 0.00s
Presolved: 11 rows, 16 columns, 39 nonzeros
Variable types: 0 continuous, 16 integer (0 binary)Root relaxation: objective 1.000000e+02, 10 iterations, 0.00 secondsNodes    |    Current Node    |     Objective Bounds      |     WorkExpl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time0     0  100.00000    0    2  150.00000  100.00000  33.3%     -    0s
H    0     0                     137.0000000  100.00000  27.0%     -    0s
H    0     0                     100.0000000  100.00000  0.00%     -    0s0     0  100.00000    0    2  100.00000  100.00000  0.00%     -    0sExplored 1 nodes (10 simplex iterations) in 0.23 seconds
Thread count was 8 (of 8 available processors)Solution count 3: 100 137 150 Optimal solution found (tolerance 1.00e-04)
Best objective 1.000000000000e+02, best bound 1.000000000000e+02, gap 0.0000%
---------------------------------------------------------------------------Multi-objectives: optimize objective 5 (目标5) ...
---------------------------------------------------------------------------Loaded user MIP start with objective 0Explored 0 nodes (0 simplex iterations) in 0.25 seconds
Thread count was 1 (of 8 available processors)Solution count 1: 0 Optimal solution found (tolerance 1.00e-04)
Best objective 0.000000000000e+00, best bound 0.000000000000e+00, gap 0.0000%
---------------------------------------------------------------------------Multi-objectives: optimize objective 6 (目标6) ...
---------------------------------------------------------------------------Loaded user MIP start with objective 200Presolve removed 10 rows and 21 columns
Presolve time: 0.01s
Presolved: 9 rows, 13 columns, 39 nonzeros
Variable types: 0 continuous, 13 integer (0 binary)Root relaxation: objective 0.000000e+00, 7 iterations, 0.00 secondsNodes    |    Current Node    |     Objective Bounds      |     WorkExpl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time*    0     0               0       0.0000000    0.00000  0.00%     -    0sExplored 0 nodes (7 simplex iterations) in 0.30 seconds
Thread count was 8 (of 8 available processors)Solution count 2: 0 200 Optimal solution found (tolerance 1.00e-04)
Best objective 0.000000000000e+00, best bound 0.000000000000e+00, gap 0.0000%
---------------------------------------------------------------------------Multi-objectives: optimize objective 7 (目标7) ...
---------------------------------------------------------------------------Loaded user MIP start with objective 0Explored 0 nodes (0 simplex iterations) in 0.32 seconds
Thread count was 1 (of 8 available processors)Solution count 1: 0 Optimal solution found (tolerance 1.00e-04)
Best objective 0.000000000000e+00, best bound 0.000000000000e+00, gap 0.0000%
---------------------------------------------------------------------------Multi-objectives: optimize objective 8 (目标8) ...
---------------------------------------------------------------------------Loaded user MIP start with objective 3414Presolve removed 12 rows and 21 columns
Presolve time: 0.00s
Presolved: 9 rows, 13 columns, 40 nonzeros
Variable types: 0 continuous, 13 integer (0 binary)Root relaxation: infeasible, 14 iterations, 0.00 secondsNodes    |    Current Node    |     Objective Bounds      |     WorkExpl Unexpl |  Obj  Depth IntInf | Incumbent    BestBd   Gap | It/Node Time0     0 infeasible    0      3414.00000 3414.00000  0.00%     -    0sExplored 0 nodes (14 simplex iterations) in 0.37 seconds
Thread count was 8 (of 8 available processors)Solution count 1: 3414 Optimal solution found (tolerance 1.00e-04)
Best objective 3.414000000000e+03, best bound 3.414000000000e+03, gap 0.0000%---------------------------------------------------------------------------
Multi-objectives: solved in 0.38 seconds, solution count 7Obj1 =  0.0
Obj2 =  0.0
Obj3 =  0.0
Obj4 =  100.0
Obj5 =  0.0
Obj6 =  0.0
Obj7 =  0.0
Obj8 =  3414.0000000000005
运输矩阵为:[[ -0.  91.  -0. 209.][ 72.  -0. 128.  -0.][100.  -0. 259.  41.]] 最小价格为: 3414.0000000000005

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