我正在为 154 名员工的工资对数建模。预测变量是一个二进制矩阵，其中行是员工，列是工作能力，交集为 1 或 0（即员工是否具有能力）。
我需要获取所有 63 个预测变量的整数系数。
对于线性规划问题，我得到了目标函数和约束：
目标函数：sum(n1+n2) --> min
限制：
我尝试编写 R 代码，但我不知道线性编程是如何工作的：
obj <- c(1, 1) # Minimize sum(n1 + n2)
A <- as.matrix(X)
E <- matrix(rep(1), nrow = 154, ncol = 63) # just randomly for the first time
b <- as.vector(offset_test) # just randomly for the first time
y <- as.vector(log(grades_and_salaries$salary))
lambda <- as.vector(lambda.min) # just randomly
mat <- cbind(A, E, -E, -y, rep(-lambda, nrow(A)))
dir <- rep("==", nrow(A))
x.dir <- rep(">=", length(obj)) # x >= 1
mat <- rbind(mat, diag(length(obj)))
dir <- c(dir, x.dir)
rhs <- c(b, rep(1, length(obj)))
lp.model <- lp("min", obj, mat, dir, rhs, all.int = TRUE)
我得到了一些奇怪的东西，但我期望得到 x 的整数系数向量。
如何在具有所有约束的情况下解决 R 中的任务并获得整数系数？
运行一长串假设和占位符参数，
import pandas as pd
import pulp
import numpy as np
rand = np.random.default_rng(seed=0)
m = 154
n = 63
A = rand.integers(low=0, high=2, size=(m, n))
x = pulp.LpVariable.matrix(
name='x', indices=np.arange(n), cat=pulp.LpInteger,
lowBound=1,
) # (n, 1)
E = np.identity(m) # (m, m)
n1 = pulp.LpVariable.matrix(
name='n1', indices=np.arange(m), cat=pulp.LpContinuous,
lowBound=0,
) # (m, 1)
n2 = pulp.LpVariable.matrix(
name='n2', indices=np.arange(m), cat=pulp.LpContinuous,
lowBound=0,
) # (m, 1)
salary = rand.uniform(low=100_000, high=300_000, size=m)
y = np.log(salary) # (m, 1)
lambd = 1
b = offset_test = 20
prob = pulp.LpProblem('log_wage_model')
prob.setObjective(pulp.lpSum(n1) + pulp.lpSum(n2))
for i, lhs in enumerate(A@x + E@n1 - E@n2 - y*lambd - b):
prob.addConstraint(
name=f'constraint_{i}',
constraint=lhs == 0,
)
print(prob)
prob.solve()
assert prob.status == pulp.LpStatusOptimal
df = pd.DataFrame({
'A@x': A@x,
'n1': n1,
'n2': n2,
'salary': salary,
'y': y,
}).map(pulp.value)
pd.options.display.max_rows = m
print('Solution rows:')
print(df)
print('x =')
print([xi.value() for xi in x])
log_wage_model:
MINIMIZE
1*n1_0 + 1*n1_1 + 1*n1_10 + 1*n1_100 + 1*n1_101 + 1*n1_102 + 1*n1_103 + 1*n1_104 + 1*n1_105 + 1*n1_106 + 1*n1_107 + 1*n1_108 + 1*n1_109 + 1*n1_11 + 1*n1_110 + 1*n1_111 + 1*n1_112 + 1*n1_113 + 1*n1_114 + 1*n1_115 + 1*n1_116 + 1*n1_117 + 1*n1_118 + 1*n1_119 + 1*n1_12 + 1*n1_120 + 1*n1_121 + 1*n1_122 + 1*n1_123 + 1*n1_124 + 1*n1_125 + 1*n1_126 + 1*n1_127 + 1*n1_128 + 1*n1_129 + 1*n1_13 + 1*n1_130 + 1*n1_131 + 1*n1_132 + 1*n1_133 + 1*n1_134 + 1*n1_135 + 1*n1_136 + 1*n1_137 + 1*n1_138 + 1*n1_139 + 1*n1_14 + 1*n1_140 + 1*n1_141 + 1*n1_142 + 1*n1_143 + 1*n1_144 + 1*n1_145 + 1*n1_146 + 1*n1_147 + 1*n1_148 + 1*n1_149 + 1*n1_15 + 1*n1_150 + 1*n1_151 + 1*n1_152 + 1*n1_153 + 1*n1_16 + 1*n1_17 + 1*n1_18 + 1*n1_19 + 1*n1_2 + 1*n1_20 + 1*n1_21 + 1*n1_22 + 1*n1_23 + 1*n1_24 + 1*n1_25 + 1*n1_26 + 1*n1_27 + 1*n1_28 + 1*n1_29 + 1*n1_3 + 1*n1_30 + 1*n1_31 + 1*n1_32 + 1*n1_33 + 1*n1_34 + 1*n1_35 + 1*n1_36 + 1*n1_37 + 1*n1_38 + 1*n1_39 + 1*n1_4 + 1*n1_40 + 1*n1_41 + 1*n1_42 + 1*n1_43 + 1*n1_44 + 1*n1_45 + 1*n1_46 + 1*n1_47 + 1*n1_48 + 1*n1_49 + 1*n1_5 + 1*n1_50 + 1*n1_51 + 1*n1_52 + 1*n1_53 + 1*n1_54 + 1*n1_55 + 1*n1_56 + 1*n1_57 + 1*n1_58 + 1*n1_59 + 1*n1_6 + 1*n1_60 + 1*n1_61 + 1*n1_62 + 1*n1_63 + 1*n1_64 + 1*n1_65 + 1*n1_66 + 1*n1_67 + 1*n1_68 + 1*n1_69 + 1*n1_7 + 1*n1_70 + 1*n1_71 + 1*n1_72 + 1*n1_73 + 1*n1_74 + 1*n1_75 + 1*n1_76 + 1*n1_77 + 1*n1_78 + 1*n1_79 + 1*n1_8 + 1*n1_80 + 1*n1_81 + 1*n1_82 + 1*n1_83 + 1*n1_84 + 1*n1_85 + 1*n1_86 + 1*n1_87 + 1*n1_88 + 1*n1_89 + 1*n1_9 + 1*n1_90 + 1*n1_91 + 1*n1_92 + 1*n1_93 + 1*n1_94 + 1*n1_95 + 1*n1_96 + 1*n1_97 + 1*n1_98 + 1*n1_99 + 1*n2_0 + 1*n2_1 + 1*n2_10 + 1*n2_100 + 1*n2_101 + 1*n2_102 + 1*n2_103 + 1*n2_104 + 1*n2_105 + 1*n2_106 + 1*n2_107 + 1*n2_108 + 1*n2_109 + 1*n2_11 + 1*n2_110 + 1*n2_111 + 1*n2_112 + 1*n2_113 + 1*n2_114 + 1*n2_115 + 1*n2_116 + 1*n2_117 + 1*n2_118 + 1*n2_119 + 1*n2_12 + 1*n2_120 + 1*n2_121 + 1*n2_122 + 1*n2_123 + 1*n2_124 + 1*n2_125 + 1*n2_126 + 1*n2_127 + 1*n2_128 + 1*n2_129 + 1*n2_13 + 1*n2_130 + 1*n2_131 + 1*n2_132 + 1*n2_133 + 1*n2_134 + 1*n2_135 + 1*n2_136 + 1*n2_137 + 1*n2_138 + 1*n2_139 + 1*n2_14 + 1*n2_140 + 1*n2_141 + 1*n2_142 + 1*n2_143 + 1*n2_144 + 1*n2_145 + 1*n2_146 + 1*n2_147 + 1*n2_148 + 1*n2_149 + 1*n2_15 + 1*n2_150 + 1*n2_151 + 1*n2_152 + 1*n2_153 + 1*n2_16 + 1*n2_17 + 1*n2_18 + 1*n2_19 + 1*n2_2 + 1*n2_20 + 1*n2_21 + 1*n2_22 + 1*n2_23 + 1*n2_24 + 1*n2_25 + 1*n2_26 + 1*n2_27 + 1*n2_28 + 1*n2_29 + 1*n2_3 + 1*n2_30 + 1*n2_31 + 1*n2_32 + 1*n2_33 + 1*n2_34 + 1*n2_35 + 1*n2_36 + 1*n2_37 + 1*n2_38 + 1*n2_39 + 1*n2_4 + 1*n2_40 + 1*n2_41 + 1*n2_42 + 1*n2_43 + 1*n2_44 + 1*n2_45 + 1*n2_46 + 1*n2_47 + 1*n2_48 + 1*n2_49 + 1*n2_5 + 1*n2_50 + 1*n2_51 + 1*n2_52 + 1*n2_53 + 1*n2_54 + 1*n2_55 + 1*n2_56 + 1*n2_57 + 1*n2_58 + 1*n2_59 + 1*n2_6 + 1*n2_60 + 1*n2_61 + 1*n2_62 + 1*n2_63 + 1*n2_64 + 1*n2_65 + 1*n2_66 + 1*n2_67 + 1*n2_68 + 1*n2_69 + 1*n2_7 + 1*n2_70 + 1*n2_71 + 1*n2_72 + 1*n2_73 + 1*n2_74 + 1*n2_75 + 1*n2_76 + 1*n2_77 + 1*n2_78 + 1*n2_79 + 1*n2_8 + 1*n2_80 + 1*n2_81 + 1*n2_82 + 1*n2_83 + 1*n2_84 + 1*n2_85 + 1*n2_86 + 1*n2_87 + 1*n2_88 + 1*n2_89 + 1*n2_9 + 1*n2_90 + 1*n2_91 + 1*n2_92 + 1*n2_93 + 1*n2_94 + 1*n2_95 + 1*n2_96 + 1*n2_97 + 1*n2_98 + 1*n2_99 + 0
SUBJECT TO
constraint_0: n1_0 - n2_0 + x_0 + x_1 + x_10 + x_11 + x_12 + x_13 + x_14
+ x_15 + x_16 + x_17 + x_18 + x_19 + x_2 + x_21 + x_22 + x_25 + x_26 + x_28
+ x_29 + x_30 + x_33 + x_35 + x_45 + x_46 + x_47 + x_49 + x_50 + x_53 + x_54
+ x_55 + x_57 + x_58 + x_59 + x_60 + x_61 + x_62 + x_9 = 32.3426223434
...
Result - Optimal solution found
Objective value: 474.91974913
Enumerated nodes: 0
Total iterations: 11
Time (CPU seconds): 0.01
Time (Wallclock seconds): 0.01
Option for printingOptions changed from normal to all
Total time (CPU seconds): 0.01 (Wallclock seconds): 0.01
Solution rows:
A@x n1 n2 salary y
0 38.0 0.000000 5.657378 229262.369129 12.342622
1 33.0 0.000000 1.050450 154747.529521 11.949550
2 35.0 0.000000 2.437057 285770.012422 12.562943
3 32.0 0.555584 0.000000 283674.850854 12.555584
4 34.0 0.000000 1.883355 182890.970369 12.116645
5 36.0 0.000000 3.563959 251712.122649 12.436041
6 40.0 0.000000 7.532326 259801.652211 12.467674
7 32.0 0.284543 0.000000 216326.286444 12.284543
8 32.0 0.211185 0.000000 201025.020215 12.211185
9 34.0 0.000000 1.895780 180632.479017 12.104220
10 34.0 0.000000 1.743287 210388.906366 12.256713
11 29.0 2.991654 0.000000 161402.104187 11.991654
12 35.0 0.000000 3.125621 143541.359510 11.874378
13 39.0 0.000000 6.708636 217806.899571 12.291364
14 32.0 0.000000 0.403295 108738.869969 11.596705
15 35.0 0.000000 2.785452 201702.194806 12.214548
16 33.0 0.000000 0.503001 267533.168507 12.496999
17 33.0 0.000000 1.430002 105873.287671 11.569998
18 27.0 5.495064 0.000000 267016.144404 12.495064
19 34.0 0.000000 1.414815 292197.448173 12.585185
20 30.0 1.688665 0.000000 119212.703460 11.688665
21 26.0 5.648508 0.000000 114520.336704 11.648508
22 35.0 0.000000 3.428236 106060.364293 11.571764
23 37.0 0.000000 4.715235 216374.222582 12.284765
24 29.0 3.106623 0.000000 181067.076894 12.106623
25 28.0 4.366097 0.000000 234707.990334 12.366097
26 26.0 5.825136 0.000000 136644.271579 11.825136
27 38.0 0.000000 5.612143 239871.171418 12.387857
28 39.0 0.000000 6.872818 184828.264931 12.127182
29 36.0 0.000000 3.422853 289858.060771 12.577147
30 37.0 0.000000 4.620211 237943.686629 12.379789
31 33.0 0.000000 0.712445 216978.804476 12.287555
32 28.0 4.237749 0.000000 206436.608219 12.237749
33 31.0 1.116685 0.000000 182898.220970 12.116685
34 29.0 2.711578 0.000000 121975.752339 11.711578
35 33.0 0.000000 0.529253 260601.218073 12.470747
36 37.0 0.000000 4.656043 229568.457005 12.343957
37 38.0 0.000000 6.143451 141004.825949 11.856549
38 35.0 0.000000 3.412180 107777.081859 11.587820
39 29.0 2.704954 0.000000 121170.495922 11.704954
40 40.0 0.000000 8.458001 102949.979789 11.541999
41 28.0 4.588149 0.000000 293064.793331 12.588149
42 28.0 4.290188 0.000000 217550.833317 12.290188
43 31.0 1.117903 0.000000 183121.116609 12.117903
44 41.0 0.000000 8.930394 174487.026246 12.069606
45 34.0 0.000000 1.515953 264090.505973 12.484047
46 32.0 0.170466 0.000000 193003.982220 12.170466
47 33.0 0.000000 0.435772 286137.584139 12.564228
48 32.0 0.348152 0.000000 230533.600009 12.348152
49 34.0 0.000000 2.130800 142799.942126 11.869200
50 29.0 3.530806 0.000000 276732.277534 12.530806
51 28.0 4.420649 0.000000 247867.221786 12.420648
52 29.0 2.860175 0.000000 141516.958482 11.860175
53 31.0 1.312630 0.000000 222488.350025 12.312630
54 36.0 0.000000 3.768053 205242.470788 12.231947
55 36.0 0.000000 4.192889 134203.310824 11.807111
56 34.0 0.000000 1.784131 201968.932779 12.215869
57 25.0 7.535059 0.000000 277911.821637 12.535059
58 33.0 0.000000 0.517574 263662.781035 12.482426
59 20.0 12.529270 0.000000 276307.529033 12.529270
60 24.0 8.596200 0.000000 295433.861909 12.596200
61 32.0 0.609968 0.000000 299529.357840 12.609968
62 31.0 1.435923 0.000000 251682.269557 12.435923
63 26.0 6.599394 0.000000 296378.778383 12.599394
64 27.0 4.759453 0.000000 127957.402323 11.759453
65 25.0 7.154548 0.000000 189956.041211 12.154548
66 31.0 0.739899 0.000000 125479.677889 11.739899
67 34.0 0.000000 1.851949 188725.897408 12.148051
68 29.0 2.607723 0.000000 109943.649877 11.607723
69 28.0 3.551812 0.000000 103965.276043 11.551812
70 26.0 6.209559 0.000000 200698.541801 12.209559
71 33.0 0.000000 0.883416 182879.742710 12.116584
72 32.0 0.452657 0.000000 255929.462230 12.452657
73 33.0 0.000000 1.152469 139738.959296 11.847531
74 31.0 0.745966 0.000000 126243.297582 11.745966
75 31.0 0.739157 0.000000 125386.644699 11.739157
76 31.0 1.352831 0.000000 231614.891180 12.352831
77 34.0 0.000000 2.181739 135708.016320 11.818261
78 34.0 0.000000 2.043448 155834.916203 11.956552
79 35.0 0.000000 2.625445 236701.396495 12.374555
80 29.0 3.193201 0.000000 197442.217332 12.193201
81 29.0 2.945328 0.000000 154095.475075 11.945328
82 37.0 0.000000 4.779799 202845.746939 12.220201
83 29.0 2.774968 0.000000 129958.126942 11.774968
84 33.0 0.000000 0.993478 163819.764330 12.006522
85 25.0 7.478763 0.000000 262698.566072 12.478763
86 35.0 0.000000 2.469327 276695.510238 12.530673
87 33.0 0.000000 0.422917 289839.695523 12.577083
88 26.0 6.068594 0.000000 174310.572694 12.068594
89 32.0 0.000000 0.058682 153478.841506 11.941318
90 36.0 0.000000 4.417523 107202.721563 11.582477
91 30.0 2.321097 0.000000 224380.183706 12.321097
92 38.0 0.000000 5.978384 166311.150704 12.021616
93 30.0 1.596947 0.000000 108765.255675 11.596947
94 36.0 0.000000 3.757704 207377.495516 12.242296
95 29.0 2.807464 0.000000 134250.649582 11.807464
96 29.0 2.788956 0.000000 131788.753134 11.788956
97 27.0 4.606063 0.000000 109761.222428 11.606063
98 29.0 3.549336 0.000000 281908.003156 12.549336
99 32.0 0.250926 0.000000 209174.990413 12.250926
100 34.0 0.000000 2.261386 125318.482481 11.738614
101 28.0 3.771038 0.000000 129448.421070 11.771038
102 33.0 0.000000 0.430775 287570.942573 12.569225
103 28.0 3.653770 0.000000 115124.563443 11.653770
104 32.0 0.317166 0.000000 223499.806337 12.317166
105 29.0 2.571004 0.000000 105979.850267 11.571004
106 37.0 0.000000 4.919856 176335.560069 12.080144
107 30.0 2.572370 0.000000 288476.950924 12.572370
108 27.0 5.063236 0.000000 173379.131488 12.063236
109 28.0 4.149042 0.000000 188913.068814 12.149042
110 39.0 0.000000 7.354497 114176.760050 11.645503
111 30.0 1.646951 0.000000 114342.260909 11.646952
112 28.0 4.277224 0.000000 214748.661155 12.277224
113 36.0 0.000000 3.419639 290791.265371 12.580361
114 31.0 1.518807 0.000000 273431.609899 12.518807
115 32.0 0.346233 0.000000 230091.695188 12.346233
116 28.0 4.597144 0.000000 295712.813714 12.597144
117 35.0 0.000000 2.864618 186349.952387 12.135382
118 32.0 0.582983 0.000000 291554.622472 12.582983
119 30.0 2.229906 0.000000 204823.881443 12.229906
120 32.0 0.000000 0.367599 112690.586456 11.632401
121 34.0 0.000000 1.830026 192908.986342 12.169974
122 34.0 0.000000 1.410835 293362.630228 12.589165
123 27.0 5.119152 0.000000 183349.874215 12.119151
124 37.0 0.000000 4.398223 297086.063654 12.601777
125 40.0 0.000000 7.911671 177784.750619 12.088329
126 37.0 0.000000 5.206957 132328.502235 11.793043
127 33.0 0.000000 1.075586 150906.229495 11.924414
128 29.0 2.904558 0.000000 147939.422941 11.904558
129 33.0 0.000000 0.714688 216492.630817 12.285312
130 34.0 0.000000 2.210345 131880.932157 11.789655
131 35.0 0.000000 2.946624 171677.993782 12.053376
132 26.0 6.591676 0.000000 294100.319926 12.591676
133 32.0 0.000000 0.279193 123106.697198 11.720807
134 32.0 0.348017 0.000000 230502.475417 12.348017
135 37.0 0.000000 5.113488 145293.624324 11.886512
136 33.0 0.000000 0.893438 181056.077762 12.106562
137 31.0 0.657601 0.000000 115566.445335 11.657601
138 33.0 0.000000 0.567016 250943.712535 12.432984
139 31.0 1.505715 0.000000 269875.220048 12.505715
140 31.0 1.435954 0.000000 251690.151225 12.435954
141 35.0 0.000000 2.981896 165728.138731 12.018104
142 29.0 3.246344 0.000000 208218.626320 12.246344
143 36.0 0.000000 4.000142 162731.610654 11.999858
144 33.0 0.000000 0.402298 295877.908802 12.597702
145 34.0 0.000000 1.869567 185430.084963 12.130433
146 35.0 0.000000 2.517524 263675.966099 12.482476
147 34.0 0.000000 2.321838 117967.243305 11.678162
148 31.0 1.575606 0.000000 289411.867920 12.575606
149 34.0 0.000000 1.468080 277040.844396 12.531920
150 36.0 0.000000 3.619432 238129.090308 12.380568
151 38.0 0.000000 5.812084 196401.410072 12.187916
152 29.0 3.371170 0.000000 235901.626528 12.371170
153 41.0 0.000000 8.504428 267151.750313 12.495572
x =
[1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 2.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0]