R中的lp()线性求解器会找不到最优解吗?

问题描述 投票:0回答:1

给定一组 140 个选项,我的目标是选择一组选项,以最小化目标函数并实现权重必须 > 5584 个单位的约束。

标准 1 是要最小化的目标函数。

我的问题是,在最小化标准 2 的同时,我在尝试最小化标准 1 时偶然发现了一个比 lp() 返回的更好的解决方案。现在我对自己正在正确执行线性求解失去了信心。

请注意,我没有很强的数学背景,所以不太了解其背后的数学。

这是一个可重现的示例:


#First define the option set of 140 options, each with a weight and two criteria.  
DFToOptimizeFull <- structure(list(item_ID = c("Option1", "Option2", "Option3", "Option4", 
                           "Option5", "Option6", "Option7", "Option8", "Option9", "Option10", 
                           "Option11", "Option12", "Option13", "Option14", "Option15", "Option16", 
                           "Option17", "Option18", "Option19", "Option20", "Option21", "Option22", 
                           "Option23", "Option24", "Option25", "Option26", "Option27", "Option28", 
                           "Option29", "Option30", "Option31", "Option32", "Option33", "Option34", 
                           "Option35", "Option36", "Option37", "Option38", "Option39", "Option40", 
                           "Option41", "Option42", "Option43", "Option44", "Option45", "Option46", 
                           "Option47", "Option48", "Option49", "Option50", "Option51", "Option52", 
                           "Option53", "Option54", "Option55", "Option56", "Option57", "Option58", 
                           "Option59", "Option60", "Option61", "Option62", "Option63", "Option64", 
                           "Option65", "Option66", "Option67", "Option68", "Option69", "Option70", 
                           "Option71", "Option72", "Option73", "Option74", "Option75", "Option76", 
                           "Option77", "Option78", "Option79", "Option80", "Option81", "Option82", 
                           "Option83", "Option84", "Option85", "Option86", "Option87", "Option88", 
                           "Option89", "Option90", "Option91", "Option92", "Option93", "Option94", 
                           "Option95", "Option96", "Option97", "Option98", "Option99", "Option100", 
                           "Option101", "Option102", "Option103", "Option104", "Option105", 
                           "Option106", "Option107", "Option108", "Option109", "Option110", 
                           "Option111", "Option112", "Option113", "Option114", "Option115", 
                           "Option116", "Option117", "Option118", "Option119"), criteria_1 = c(0.19382357870646, 
                                                                                               0.406093545012686, 0.26444538198799, 0.26601032196915, 0.582727750839455, 
                                                                                               0.610218171520381, 0.278839154130233, 0.121920371729594, 0.394236226297326, 
                                                                                               0.377841864658525, 0.120477145449607, 0.26254096134174, 0.320858353224026, 
                                                                                               0.157895381699022, 0.367337498338429, 0.178897294673147, 0.206447474122835, 
                                                                                               0.125249409058597, 0.0964062671575285, 0.155883926971779, 0.745589005311248, 
                                                                                               0.271814782493108, 1, 0.709925503817279, 0.283967774188142, 0.465574672453751, 
                                                                                               0.747720902276497, 0.261597766848332, 0.223547497818285, 0.312169636303741, 
                                                                                               0.361688040733056, 0.33135717134122, 0.720315669627635, 0.597242543157505, 
                                                                                               0.793399950297349, 0.422337180472638, 0.319528402753296, 0.430606430135989, 
                                                                                               0.397399279889498, 0.343212756243173, 0.330215166243809, 0.266929589837542, 
                                                                                               0.341647931849574, 0.337592426703038, 0.42473989909206, 0.308525738460027, 
                                                                                               0.424706956637327, 0.357390032884661, 0.280438539204411, 0.409378774656271, 
                                                                                               0.346474174849303, 0.791793745557103, 0.450584584087061, 0.262056880638506, 
                                                                                               0.311923434799947, 0.448835397534517, 0.330430390281398, 0.141640071895463, 
                                                                                               0.1929124019673, 0.176881794381289, 0.379488987395177, 0.741791953949916, 
                                                                                               0.0370066289465928, 0.0768869958215097, 0.147257049396344, 0.570100734558947, 
                                                                                               0.36424969224812, 0.254741112761445, 0.339564812834843, 0.305964896057886, 
                                                                                               0.286239647689116, 0.326383438614336, 0.458683804448965, 0.567810482635859, 
                                                                                               0.235202885065509, 0.729063336203758, 0.341599616249299, 0.153703714406256, 
                                                                                               0.117457767195094, 0.134082379254345, 0.558969536898439, 0.286059793445029, 
                                                                                               0.322705673615406, 0.396612822128083, 0.100616428459969, 0.156259239780615, 
                                                                                               0.232564021060054, 0.104164041865814, 0.210723751509863, 0.192397806148102, 
                                                                                               0.166759676123656, 0.310175982060811, 0.200003698801935, 0.193767171976952, 
                                                                                               0.137411647758468, 0.369825867340157, 0.429786451982038, 0.202788665483821, 
                                                                                               0.786777129845286, 0.321634986042802, 0.496148738072809, 0.170282669379121, 
                                                                                               0.140613654358518, 0.155074004935589, 0.106219651041155, 0.121895751579215, 
                                                                                               0.144999508752868, 0.278420842748903, 0.61995781054043, 0.860520490784782, 
                                                                                               0.860520490784782, 0.894125146651717, 0.894125146651717, 0.894125146651717, 
                                                                                               0.443148836322235, 0.120796860641858, 0.0224245646683504, 0.223040877540759, 
                                                                                               0.195655525952297), criteria_2 = c(0.17641981214023, 0.240848545191324, 
                                                                                                                                  0.20835265524648, 0.248767529787931, 0.451793498520131, 0.381683037411269, 
                                                                                                                                  0.241976585373557, 0.116782623551492, 0.20673566370936, 0.286801382192965, 
                                                                                                                                  0.070652688912388, 0.364572800178567, 0.269242816166067, 0.1189751136006, 
                                                                                                                                  0.323905107166701, 0.193872198256378, 0.147740819150117, 0.0860132098261335, 
                                                                                                                                  0.0580418616854518, 0.109446297259946, 0.500647366537576, 0.180393459509617, 
                                                                                                                                  0.653491503844082, 0.472660220201084, 0.163162249237616, 0.283984633127125, 
                                                                                                                                  0.415992733266123, 0.164994074648886, 0.167340730610985, 0.268800486663044, 
                                                                                                                                  0.32576183372859, 0.187746554917954, 0.513868883897277, 0.407993976147579, 
                                                                                                                                  0.520823834574429, 0.346647590611749, 0.249144414555891, 0.36143709685635, 
                                                                                                                                  0.275195240878429, 0.210744193768878, 0.276940989909035, 0.213850998663282, 
                                                                                                                                  0.189043772333223, 0.256031575641972, 0.267888778202805, 0.299825443704003, 
                                                                                                                                  0.232862594751094, 0.293319667536792, 0.165555116687722, 0.249950801232123, 
                                                                                                                                  0.185427153206608, 0.450646068867545, 0.283324056730209, 0.215590084378473, 
                                                                                                                                  0.168950797268736, 0.305193207638845, 0.190241720123152, 0.168636009700241, 
                                                                                                                                  0.140235480483511, 0.116295393379208, 0.254990237458096, 0.534546957175079, 
                                                                                                                                  0.0342844548026959, 0.147617877226165, 0.0856553598188811, 0.360464565082389, 
                                                                                                                                  0.26523128883826, 0.157202592030299, 0.189919356942208, 0.16741285463652, 
                                                                                                                                  0.184588573334462, 0.194498543981115, 0.290262033461934, 0.376277472517838, 
                                                                                                                                  0.167260290450748, 0.494939091832352, 0.312304468360028, 0.125687206435128, 
                                                                                                                                  0.124559809278627, 0.10118904396013, 0.436260473197599, 0.268812179666543, 
                                                                                                                                  0.28390152035242, 0.319129746519373, 0.0887131242082749, 0.0936087834963992, 
                                                                                                                                  0.146028350082448, 0.0852325359256434, 0.141593369478712, 0.142175267274303, 
                                                                                                                                  0.119637448691408, 0.213864087532372, 0.143729711242536, 0.240836706142566, 
                                                                                                                                  0.0857609795196696, 0.284704814576988, 0.321668930739399, 0.178845649197193, 
                                                                                                                                  0.893388564922643, 0.22274883820528, 0.43912198342029, 0.174430954851812, 
                                                                                                                                  0.118927528091718, 0.100615369508259, 0.0691456056815229, 0.0878417595367306, 
                                                                                                                                  0.133033845100036, 0.262412843717465, 0.320020039396849, 0.440301379519025, 
                                                                                                                                  0.440301379519025, 0.705078014946413, 0.705078014946413, 0.705078014946413, 
                                                                                                                                  0.478037536193209, 0.0604065918539912, 0.0112204438672376, 0.117238408833876, 
                                                                                                                                  0.131639362146937), itemWeight = c(151, 482, 309, 323, 648, 766, 
                                                                                                                                                                     309, 29, 385, 232, 114, 104, 326, 172, 69, 133, 130, 107, 222, 
                                                                                                                                                                     118, 657, 152, 1150, 723, 339, 516, 1126, 300, 247, 372, 286, 
                                                                                                                                                                     380, 935, 849, 1239, 571, 344, 639, 640, 476, 456, 336, 563, 
                                                                                                                                                                     536, 682, 463, 477, 518, 350, 695, 384, 953, 457, 284, 345, 549, 
                                                                                                                                                                     418, 196, 276, 217, 569, 657, 37, 78, 151, 613, 392, 280, 389, 
                                                                                                                                                                     320, 332, 366, 584, 619, 258, 903, 349, 160, 124, 141, 671, 282, 
                                                                                                                                                                     333, 428, 97, 57, 251, 62, 214, 194, 167, 316, 198, 231, 135, 
                                                                                                                                                                     345, 514, 188, 787, 359, 522, 168, 128, 165, 114, 129, 131, 92, 
                                                                                                                                                                     374, 307, 307, 307, 307, 307, 86, 73, 7, 109, 208)), row.names = c(NA, 
                                                                                                                                                                                                                                        -119L), class = c("tbl_df", "tbl", "data.frame"))


#Minimize criteria 1 on the full set
minimizedC_1FullSet <- lp(direction = "min",
   objective.in = 
     DFToOptimizeFull$criteria_1,
   const.mat = matrix(
     DFToOptimizeFull$itemWeight, nrow = 1), 
   
   const.dir = ">=", #firm energy must be greater than or equal to
   const.rhs = 5584, #the energy target
   all.bin = TRUE)

#Now check the summed criteria 1. 
minimizedC_1FullSet

#--
#Now here is the problem, I can minimize criteria 1 on a subset from the fullset, and find a lower summed criteria 1. 

DFToOptimizeSelect <- DFToOptimizeFull %>% filter(item_ID %in% c( "Option19", "Option27", "Option35", 
                                            "Option39", "Option43", "Option45", 
                                            "Option50", "Option57"))


minimizedC_1Select <- lp(direction = "min",
                             objective.in = 
                           DFToOptimizeSelect$criteria_1,
                             const.mat = matrix(
                               DFToOptimizeSelect$itemWeight, nrow = 1), 
                             
                             const.dir = ">=", #firm energy must be greater than or equal to
                             const.rhs = 5584, #the energy target
                             all.bin = TRUE)

minimizedC_1Select

请帮助我理解为什么这是可能的???这个优化是我博士后的一个关键分析,我需要确信优化工作正常。

我预计最小化标准 1 会最小化标准 1。但我找到了更好的解决方案。请帮我理解为什么。

r optimization linear-programming knapsack-problem lpsolve
1个回答
0
投票

给定

import pandas as pd
import pulp

item_ID = pd.RangeIndex(name='item_ID', start=1, stop=120)
df = pd.DataFrame(
    index=item_ID,
    data={
        'criteria_1': (
            0.19382357870646,
            0.406093545012686, 0.26444538198799, 0.26601032196915, 0.582727750839455,
            0.610218171520381, 0.278839154130233, 0.121920371729594, 0.394236226297326,
            0.377841864658525, 0.120477145449607, 0.26254096134174, 0.320858353224026,
            0.157895381699022, 0.367337498338429, 0.178897294673147, 0.206447474122835,
            0.125249409058597, 0.0964062671575285, 0.155883926971779, 0.745589005311248,
            0.271814782493108, 1, 0.709925503817279, 0.283967774188142, 0.465574672453751,
            0.747720902276497, 0.261597766848332, 0.223547497818285, 0.312169636303741,
            0.361688040733056, 0.33135717134122, 0.720315669627635, 0.597242543157505,
            0.793399950297349, 0.422337180472638, 0.319528402753296, 0.430606430135989,
            0.397399279889498, 0.343212756243173, 0.330215166243809, 0.266929589837542,
            0.341647931849574, 0.337592426703038, 0.42473989909206, 0.308525738460027,
            0.424706956637327, 0.357390032884661, 0.280438539204411, 0.409378774656271,
            0.346474174849303, 0.791793745557103, 0.450584584087061, 0.262056880638506,
            0.311923434799947, 0.448835397534517, 0.330430390281398, 0.141640071895463,
            0.1929124019673, 0.176881794381289, 0.379488987395177, 0.741791953949916,
            0.0370066289465928, 0.0768869958215097, 0.147257049396344, 0.570100734558947,
            0.36424969224812, 0.254741112761445, 0.339564812834843, 0.305964896057886,
            0.286239647689116, 0.326383438614336, 0.458683804448965, 0.567810482635859,
            0.235202885065509, 0.729063336203758, 0.341599616249299, 0.153703714406256,
            0.117457767195094, 0.134082379254345, 0.558969536898439, 0.286059793445029,
            0.322705673615406, 0.396612822128083, 0.100616428459969, 0.156259239780615,
            0.232564021060054, 0.104164041865814, 0.210723751509863, 0.192397806148102,
            0.166759676123656, 0.310175982060811, 0.200003698801935, 0.193767171976952,
            0.137411647758468, 0.369825867340157, 0.429786451982038, 0.202788665483821,
            0.786777129845286, 0.321634986042802, 0.496148738072809, 0.170282669379121,
            0.140613654358518, 0.155074004935589, 0.106219651041155, 0.121895751579215,
            0.144999508752868, 0.278420842748903, 0.61995781054043, 0.860520490784782,
            0.860520490784782, 0.894125146651717, 0.894125146651717, 0.894125146651717,
            0.443148836322235, 0.120796860641858, 0.0224245646683504, 0.223040877540759,
            0.195655525952297,
        ),
        'criteria_2': (
            0.17641981214023, 0.240848545191324,
            0.20835265524648, 0.248767529787931, 0.451793498520131, 0.381683037411269,
            0.241976585373557, 0.116782623551492, 0.20673566370936, 0.286801382192965,
            0.070652688912388, 0.364572800178567, 0.269242816166067, 0.1189751136006,
            0.323905107166701, 0.193872198256378, 0.147740819150117, 0.0860132098261335,
            0.0580418616854518, 0.109446297259946, 0.500647366537576, 0.180393459509617,
            0.653491503844082, 0.472660220201084, 0.163162249237616, 0.283984633127125,
            0.415992733266123, 0.164994074648886, 0.167340730610985, 0.268800486663044,
            0.32576183372859, 0.187746554917954, 0.513868883897277, 0.407993976147579,
            0.520823834574429, 0.346647590611749, 0.249144414555891, 0.36143709685635,
            0.275195240878429, 0.210744193768878, 0.276940989909035, 0.213850998663282,
            0.189043772333223, 0.256031575641972, 0.267888778202805, 0.299825443704003,
            0.232862594751094, 0.293319667536792, 0.165555116687722, 0.249950801232123,
            0.185427153206608, 0.450646068867545, 0.283324056730209, 0.215590084378473,
            0.168950797268736, 0.305193207638845, 0.190241720123152, 0.168636009700241,
            0.140235480483511, 0.116295393379208, 0.254990237458096, 0.534546957175079,
            0.0342844548026959, 0.147617877226165, 0.0856553598188811, 0.360464565082389,
            0.26523128883826, 0.157202592030299, 0.189919356942208, 0.16741285463652,
            0.184588573334462, 0.194498543981115, 0.290262033461934, 0.376277472517838,
            0.167260290450748, 0.494939091832352, 0.312304468360028, 0.125687206435128,
            0.124559809278627, 0.10118904396013, 0.436260473197599, 0.268812179666543,
            0.28390152035242, 0.319129746519373, 0.0887131242082749, 0.0936087834963992,
            0.146028350082448, 0.0852325359256434, 0.141593369478712, 0.142175267274303,
            0.119637448691408, 0.213864087532372, 0.143729711242536, 0.240836706142566,
            0.0857609795196696, 0.284704814576988, 0.321668930739399, 0.178845649197193,
            0.893388564922643, 0.22274883820528, 0.43912198342029, 0.174430954851812,
            0.118927528091718, 0.100615369508259, 0.0691456056815229, 0.0878417595367306,
            0.133033845100036, 0.262412843717465, 0.320020039396849, 0.440301379519025,
            0.440301379519025, 0.705078014946413, 0.705078014946413, 0.705078014946413,
            0.478037536193209, 0.0604065918539912, 0.0112204438672376, 0.117238408833876,
            0.131639362146937,
        ),
        'itemWeight': (
            151, 482, 309, 323, 648, 766,
            309, 29, 385, 232, 114, 104, 326, 172, 69, 133, 130, 107, 222,
            118, 657, 152, 1150, 723, 339, 516, 1126, 300, 247, 372, 286,
            380, 935, 849, 1239, 571, 344, 639, 640, 476, 456, 336, 563,
            536, 682, 463, 477, 518, 350, 695, 384, 953, 457, 284, 345, 549,
            418, 196, 276, 217, 569, 657, 37, 78, 151, 613, 392, 280, 389,
            320, 332, 366, 584, 619, 258, 903, 349, 160, 124, 141, 671, 282,
            333, 428, 97, 57, 251, 62, 214, 194, 167, 316, 198, 231, 135,
            345, 514, 188, 787, 359, 522, 168, 128, 165, 114, 129, 131, 92,
            374, 307, 307, 307, 307, 307, 86, 73, 7, 109, 208,
        ),
        'select': pulp.LpVariable.matrix(
            name='select', indices=item_ID, cat=pulp.LpBinary,
        ),
    },
)

prob = pulp.LpProblem(name='backpack', sense=pulp.LpMinimize)

firm_energy = pulp.lpDot(df['itemWeight'], df['select'])
prob.addConstraint(
    name='min_firm_energy', constraint=firm_energy >= 5584,
)

第一个解决方案是

# Minimize criteria 1 on the full set
prob.setObjective(pulp.lpDot(df['criteria_1'], df['select']))

prob.solve()
assert prob.status == pulp.LpStatusOptimal
firm_energy = firm_energy.value()
df['select'] = df['select'].apply(pulp.value)
df = df.loc[df['select'] > 0.5, ['criteria_1', 'criteria_2', 'itemWeight']]
print(df)
Objective value:                3.48857926
Enumerated nodes:               0
Total iterations:               42
Time (CPU seconds):             0.03
Time (Wallclock seconds):       0.03

Option for printingOptions changed from normal to all
Total time (CPU seconds):       0.03   (Wallclock seconds):       0.03

         criteria_1  criteria_2  itemWeight
item_ID                                    
19         0.096406    0.058042         222
35         0.793400    0.520824        1239
39         0.397399    0.275195         640
43         0.341648    0.189044         563
44         0.337592    0.256032         536
45         0.424740    0.267889         682
46         0.308526    0.299825         463
50         0.409379    0.249951         695
61         0.379489    0.254990         569

第二个解决方案是

Objective value:                2.16717874
Enumerated nodes:               0
Total iterations:               0
Time (CPU seconds):             0.01
Time (Wallclock seconds):       0.01

Option for printingOptions changed from normal to all
Total time (CPU seconds):       0.02   (Wallclock seconds):       0.02

         criteria_1  criteria_2  itemWeight
item_ID                                    
19         0.096406    0.058042         222
27         0.747721    0.415993        1126
35         0.793400    0.520824        1239
39         0.397399    0.275195         640
43         0.341648    0.189044         563
45         0.424740    0.267889         682
50         0.409379    0.249951         695
57         0.330430    0.190242         418

鉴于不同的目标,这两种解决方案会有所不同,这是完全可以预料的。

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