pythonoptimizationmathematical-optimizationjob-scheduling

Schedule Optimization Question - Variant of Job Shop


I am looking to build code on python for a variant of the job shop problem.

The differences are:

The goal is to combine as many tasks to the fewest number of machines without overlap while ensure the most efficient outcome

The machines have a capacity to run all day and will be measured in minutes 0-1440 minutes

there are multiple jobs, there is no order to the jobs, but each job has its own defined schedule

e.g. job 1: starts at 02:00 (140 minutes) and ends at 08:00 (480 minutes)

e.g. job 2: starts at 07:00 (420 minutes) and ends at 19:00 (1140 minutes)

e.g. job 3: starts at 08:00 (480 minutes) and ends at 20:00 (1200 minutes)

e.g. job 4: starts at 02:00 (140 minutes) and ends at 05:00 (300 minutes)

Can you help with the ideation / or code variation of the job shop problem to combine the jobs on the fewest number of machines?

Additionally as an extra request if possible (not too complex) would it be possible to incorporate jobs with different schedules during the week?

e.g.

Job 1: runs daily 02:00 - 08:00

job 2: runs Monday and Thursday only 07:00 - 19:00 etc.

In essence - assume I have a weekly Gannt chart / schedule with 50 machines - each with a single job, i want to compress the Gantt chart to reduce number of machines if they have space to run more than one job (simple illustration going from A to the more efficient B)

Have tried the job shop problem, and researching other problems but couldn't find a similar problem statement

Gantt chart


Solution

  • This is not a super difficult problem to solve.

    Instead of talking about jobs (which may repeat during the week), it may be easier to talk about individual sub-jobs. The first thing to do is to find out if two sub-jobs k and k' overlap. If this is the case, they cannot run on the same machine. The data structure to hold the overlap information can be calculated in advance (before the model runs).

    The best thing is to first write down on a piece of paper the mathematical model of the problem. I propose something like:

    enter image description here

    The objective is to minimize the number of machines we use (i.e. to which we assign sub-jobs to). The first constraint says: assign each sub-job to exactly one machine. The second constraint: if sub-jobs k and k' overlap, they can't run on the same machine. The third constraint: if a machine is not used, we can't assign any sub-jobs to it. The whole model is expressed in terms of individual sub-jobs instead of (job,day) combinations.

    This model can be implemented using any reasonable MIP solver. It is not very difficult to implement (it is rather simple: most of the work is to get the data in shape and calculate the overlap data) and not very difficult to solve (at least for the data set I used). The mathematical model above is sufficiently detailed and precise that you are advised to follow it as closely as possible.

    I tried it out with the following random data:

    ----     24 PARAMETER start  start time
    
    job1   206.000,    job2  1012.000,    job3   661.000,    job4   361.000,    job5   350.000,    job6   269.000,    job7   420.000
    job8  1028.000,    job9    80.000,    job10  600.000,    job11 1198.000,    job12  695.000,    job13 1190.000,    job14  915.000
    job15  156.000,    job16  768.000,    job17  191.000,    job18  300.000,    job19  803.000,    job20  522.000,    job21  432.000
    job22  422.000,    job23  157.000,    job24  180.000,    job25  707.000,    job26  997.000,    job27  277.000,    job28  799.000
    job29  931.000,    job30  364.000,    job31  132.000,    job32  603.000,    job33  192.000,    job34 1047.000,    job35  318.000
    job36  343.000,    job37  713.000,    job38  867.000,    job39  754.000,    job40  557.000,    job41  496.000,    job42  141.000
    job43  377.000,    job44   55.000,    job45  406.000,    job46  218.000,    job47  775.000,    job48  673.000,    job49  924.000
    job50  357.000
    
    
    ----     24 PARAMETER end  finish time
    
    job1   841.663,    job2  1721.541,    job3  1270.409,    job4   702.414,    job5   537.411,    job6   468.961,    job7  1040.176
    job8  1573.341,    job9   224.589,    job10 1338.041,    job11 1374.758,    job12  952.016,    job13 1719.993,    job14 1620.162
    job15  414.936,    job16  914.630,    job17  767.402,    job18  904.559,    job19 1226.702,    job20  921.797,    job21  741.567
    job22  734.209,    job23  378.792,    job24 1028.091,    job25 1123.352,    job26 1728.052,    job27  631.027,    job28 1016.877
    job29 1635.122,    job30  538.001,    job31  409.572,    job32  726.951,    job33  522.298,    job34 1556.884,    job35  556.003
    job36  598.852,    job37 1090.897,    job38 1234.187,    job39 1125.228,    job40 1428.902,    job41 1391.010,    job42  549.524
    job43  787.853,    job44  777.143,    job45  835.414,    job46 1050.215,    job47  988.271,    job48 1366.674,    job49 1087.226
    job50  926.514
    
    
    ----     24 PARAMETER length  length of job (minutes)
    
    job1  635.663,    job2  709.541,    job3  609.409,    job4  341.414,    job5  187.411,    job6  199.961,    job7  620.176
    job8  545.341,    job9  144.589,    job10 738.041,    job11 176.758,    job12 257.016,    job13 529.993,    job14 705.162
    job15 258.936,    job16 146.630,    job17 576.402,    job18 604.559,    job19 423.702,    job20 399.797,    job21 309.567
    job22 312.209,    job23 221.792,    job24 848.091,    job25 416.352,    job26 731.052,    job27 354.027,    job28 217.877
    job29 704.122,    job30 174.001,    job31 277.572,    job32 123.951,    job33 330.298,    job34 509.884,    job35 238.003
    job36 255.852,    job37 377.897,    job38 367.187,    job39 371.228,    job40 871.902,    job41 895.010,    job42 408.524
    job43 410.853,    job44 722.143,    job45 429.414,    job46 832.215,    job47 213.271,    job48 693.674,    job49 163.226
    job50 569.514
    
    
    ----     24 SET jd  jobs need to run on certain days
    
                 day1        day2        day3        day4        day5        day6        day7
    
    job1          YES         YES                                             YES
    job2                      YES                                 YES
    job3                                  YES                     YES                     YES
    job4                                  YES         YES                                 YES
    job6                      YES                                 YES                     YES
    job7                                              YES                     YES         YES
    job8                                  YES         YES                     YES
    job9                      YES
    job10         YES         YES         YES                     YES                     YES
    job11                                                         YES
    job12                     YES         YES         YES         YES         YES         YES
    job13                                 YES         YES
    job14                                 YES
    job15                                                         YES
    job16                                             YES                                 YES
    job17                                                         YES
    job18                     YES
    job19         YES                                                                     YES
    job21                                 YES                     YES                     YES
    job22         YES                                                         YES
    job23                                             YES                     YES
    job24                                 YES                                 YES
    job25                                 YES
    job26                                 YES                     YES
    job27         YES                                             YES
    job28         YES                                             YES
    job30         YES                                             YES
    job31                                 YES
    job32                     YES                                 YES
    job33         YES                                                         YES         YES
    job34         YES                                                         YES         YES
    job35                     YES                                             YES
    job36                     YES                                                         YES
    job37         YES                                             YES
    job38         YES                     YES                     YES
    job39                                             YES         YES
    job40                     YES                                                         YES
    job42         YES                                 YES                     YES
    job44         YES
    job45                                                         YES
    job46                     YES
    job48                                             YES         YES
    job49                                                         YES                     YES
    
    
    ----     29 PARAMETER counts  
    
    jobs    43.000,    subjobs 93.000
    
    
    ----     49 SET k  subjobs
    
    subjob1 ,    subjob2 ,    subjob3 ,    subjob4 ,    subjob5 ,    subjob6 ,    subjob7 ,    subjob8 ,    subjob9 ,    subjob10
    subjob11,    subjob12,    subjob13,    subjob14,    subjob15,    subjob16,    subjob17,    subjob18,    subjob19,    subjob20
    subjob21,    subjob22,    subjob23,    subjob24,    subjob25,    subjob26,    subjob27,    subjob28,    subjob29,    subjob30
    subjob31,    subjob32,    subjob33,    subjob34,    subjob35,    subjob36,    subjob37,    subjob38,    subjob39,    subjob40
    subjob41,    subjob42,    subjob43,    subjob44,    subjob45,    subjob46,    subjob47,    subjob48,    subjob49,    subjob50
    subjob51,    subjob52,    subjob53,    subjob54,    subjob55,    subjob56,    subjob57,    subjob58,    subjob59,    subjob60
    subjob61,    subjob62,    subjob63,    subjob64,    subjob65,    subjob66,    subjob67,    subjob68,    subjob69,    subjob70
    subjob71,    subjob72,    subjob73,    subjob74,    subjob75,    subjob76,    subjob77,    subjob78,    subjob79,    subjob80
    subjob81,    subjob82,    subjob83,    subjob84,    subjob85,    subjob86,    subjob87,    subjob88,    subjob89,    subjob90
    subjob91,    subjob92,    subjob93
    
    
    ----     49 SET map  mapping between jobs/days and subjobs
    
    job1 .day1.subjob1 ,    job1 .day2.subjob2 ,    job1 .day6.subjob3 ,    job2 .day2.subjob4 ,    job2 .day5.subjob5 
    job3 .day3.subjob6 ,    job3 .day5.subjob7 ,    job3 .day7.subjob8 ,    job4 .day3.subjob9 ,    job4 .day4.subjob10
    job4 .day7.subjob11,    job6 .day2.subjob12,    job6 .day5.subjob13,    job6 .day7.subjob14,    job7 .day4.subjob15
    job7 .day6.subjob16,    job7 .day7.subjob17,    job8 .day3.subjob18,    job8 .day4.subjob19,    job8 .day6.subjob20
    job9 .day2.subjob21,    job10.day1.subjob22,    job10.day2.subjob23,    job10.day3.subjob24,    job10.day5.subjob25
    job10.day7.subjob26,    job11.day5.subjob27,    job12.day2.subjob28,    job12.day3.subjob29,    job12.day4.subjob30
    job12.day5.subjob31,    job12.day6.subjob32,    job12.day7.subjob33,    job13.day3.subjob34,    job13.day4.subjob35
    job14.day3.subjob36,    job15.day5.subjob37,    job16.day4.subjob38,    job16.day7.subjob39,    job17.day5.subjob40
    job18.day2.subjob41,    job19.day1.subjob42,    job19.day7.subjob43,    job21.day3.subjob44,    job21.day5.subjob45
    job21.day7.subjob46,    job22.day1.subjob47,    job22.day6.subjob48,    job23.day4.subjob49,    job23.day6.subjob50
    job24.day3.subjob51,    job24.day6.subjob52,    job25.day3.subjob53,    job26.day3.subjob54,    job26.day5.subjob55
    job27.day1.subjob56,    job27.day5.subjob57,    job28.day1.subjob58,    job28.day5.subjob59,    job30.day1.subjob60
    job30.day5.subjob61,    job31.day3.subjob62,    job32.day2.subjob63,    job32.day5.subjob64,    job33.day1.subjob65
    job33.day6.subjob66,    job33.day7.subjob67,    job34.day1.subjob68,    job34.day6.subjob69,    job34.day7.subjob70
    job35.day2.subjob71,    job35.day6.subjob72,    job36.day2.subjob73,    job36.day7.subjob74,    job37.day1.subjob75
    job37.day5.subjob76,    job38.day1.subjob77,    job38.day3.subjob78,    job38.day5.subjob79,    job39.day4.subjob80
    job39.day5.subjob81,    job40.day2.subjob82,    job40.day7.subjob83,    job42.day1.subjob84,    job42.day4.subjob85
    job42.day6.subjob86,    job44.day1.subjob87,    job45.day5.subjob88,    job46.day2.subjob89,    job48.day4.subjob90
    job48.day5.subjob91,    job49.day5.subjob92,    job49.day7.subjob93
    
    
    ----     64 PARAMETER start2  start time of subjob
    
    subjob1   206.000,    subjob2  1646.000,    subjob3  7406.000,    subjob4  2452.000,    subjob5  6772.000,    subjob6  3541.000
    subjob7  6421.000,    subjob8  9301.000,    subjob9  3241.000,    subjob10 4681.000,    subjob11 9001.000,    subjob12 1709.000
    subjob13 6029.000,    subjob14 8909.000,    subjob15 4740.000,    subjob16 7620.000,    subjob17 9060.000,    subjob18 3908.000
    subjob19 5348.000,    subjob20 8228.000,    subjob21 1520.000,    subjob22  600.000,    subjob23 2040.000,    subjob24 3480.000
    subjob25 6360.000,    subjob26 9240.000,    subjob27 6958.000,    subjob28 2135.000,    subjob29 3575.000,    subjob30 5015.000
    subjob31 6455.000,    subjob32 7895.000,    subjob33 9335.000,    subjob34 4070.000,    subjob35 5510.000,    subjob36 3795.000
    subjob37 5916.000,    subjob38 5088.000,    subjob39 9408.000,    subjob40 5951.000,    subjob41 1740.000,    subjob42  803.000
    subjob43 9443.000,    subjob44 3312.000,    subjob45 6192.000,    subjob46 9072.000,    subjob47  422.000,    subjob48 7622.000
    subjob49 4477.000,    subjob50 7357.000,    subjob51 3060.000,    subjob52 7380.000,    subjob53 3587.000,    subjob54 3877.000
    subjob55 6757.000,    subjob56  277.000,    subjob57 6037.000,    subjob58  799.000,    subjob59 6559.000,    subjob60  364.000
    subjob61 6124.000,    subjob62 3012.000,    subjob63 2043.000,    subjob64 6363.000,    subjob65  192.000,    subjob66 7392.000
    subjob67 8832.000,    subjob68 1047.000,    subjob69 8247.000,    subjob70 9687.000,    subjob71 1758.000,    subjob72 7518.000
    subjob73 1783.000,    subjob74 8983.000,    subjob75  713.000,    subjob76 6473.000,    subjob77  867.000,    subjob78 3747.000
    subjob79 6627.000,    subjob80 5074.000,    subjob81 6514.000,    subjob82 1997.000,    subjob83 9197.000,    subjob84  141.000
    subjob85 4461.000,    subjob86 7341.000,    subjob87   55.000,    subjob88 6166.000,    subjob89 1658.000,    subjob90 4993.000
    subjob91 6433.000,    subjob92 6684.000,    subjob93 9564.000
    
    
    ----     64 PARAMETER end2  end time of subjob
    
    subjob1    841.663,    subjob2   2281.663,    subjob3   8041.663,    subjob4   3161.541,    subjob5   7481.541
    subjob6   4150.409,    subjob7   7030.409,    subjob8   9910.409,    subjob9   3582.414,    subjob10  5022.414
    subjob11  9342.414,    subjob12  1908.961,    subjob13  6228.961,    subjob14  9108.961,    subjob15  5360.176
    subjob16  8240.176,    subjob17  9680.176,    subjob18  4453.341,    subjob19  5893.341,    subjob20  8773.341
    subjob21  1664.589,    subjob22  1338.041,    subjob23  2778.041,    subjob24  4218.041,    subjob25  7098.041
    subjob26  9978.041,    subjob27  7134.758,    subjob28  2392.016,    subjob29  3832.016,    subjob30  5272.016
    subjob31  6712.016,    subjob32  8152.016,    subjob33  9592.016,    subjob34  4599.993,    subjob35  6039.993
    subjob36  4500.162,    subjob37  6174.936,    subjob38  5234.630,    subjob39  9554.630,    subjob40  6527.402
    subjob41  2344.559,    subjob42  1226.702,    subjob43  9866.702,    subjob44  3621.567,    subjob45  6501.567
    subjob46  9381.567,    subjob47   734.209,    subjob48  7934.209,    subjob49  4698.792,    subjob50  7578.792
    subjob51  3908.091,    subjob52  8228.091,    subjob53  4003.352,    subjob54  4608.052,    subjob55  7488.052
    subjob56   631.027,    subjob57  6391.027,    subjob58  1016.877,    subjob59  6776.877,    subjob60   538.001
    subjob61  6298.001,    subjob62  3289.572,    subjob63  2166.951,    subjob64  6486.951,    subjob65   522.298
    subjob66  7722.298,    subjob67  9162.298,    subjob68  1556.884,    subjob69  8756.884,    subjob70 10196.884
    subjob71  1996.003,    subjob72  7756.003,    subjob73  2038.852,    subjob74  9238.852,    subjob75  1090.897
    subjob76  6850.897,    subjob77  1234.187,    subjob78  4114.187,    subjob79  6994.187,    subjob80  5445.228
    subjob81  6885.228,    subjob82  2868.902,    subjob83 10068.902,    subjob84   549.524,    subjob85  4869.524
    subjob86  7749.524,    subjob87   777.143,    subjob88  6595.414,    subjob89  2490.215,    subjob90  5686.674
    subjob91  7126.674,    subjob92  6847.226,    subjob93  9727.226 
    

    Some jobs, due to my random data, don't appear in any of the days. So we don't have 50 jobs here, but just 43. The number of subjobs is 93.

    This model solves very fast: 0.06 seconds. Here is the solution:

    ----    106 VARIABLE x.L  assign job to machine
    
                machine1    machine2    machine3    machine4    machine5    machine6    machine7    machine8    machine9   machine10
    
    subjob1        1.000
    subjob2        1.000
    subjob3        1.000
    subjob4        1.000
    subjob5                    1.000
    subjob6        1.000
    subjob7        1.000
    subjob8        1.000
    subjob9                    1.000
    subjob10       1.000
    subjob11                   1.000
    subjob12                   1.000
    subjob13       1.000
    subjob14       1.000
    subjob15                   1.000
    subjob16                   1.000
    subjob17                               1.000
    subjob18                   1.000
    subjob19       1.000
    subjob20       1.000
    subjob21                   1.000
    subjob22                   1.000
    subjob23                   1.000
    subjob24                               1.000
    subjob25                               1.000
    subjob26                                           1.000
    subjob27                                           1.000
    subjob28                               1.000
    subjob29                                           1.000
    subjob30                               1.000
    subjob31                   1.000
    subjob32                               1.000
    subjob33                                                       1.000
    subjob34                                           1.000
    subjob35                   1.000
    subjob36                                                       1.000
    subjob37                               1.000
    subjob38       1.000
    subjob39                   1.000
    subjob40                                           1.000
    subjob41                                           1.000
    subjob42                               1.000
    subjob43                                                                   1.000
    subjob44                                                       1.000
    subjob45                                                       1.000
    subjob46                                                                   1.000
    subjob47                               1.000
    subjob48                                           1.000
    subjob49                   1.000
    subjob50                               1.000
    subjob51                                                                   1.000
    subjob52                                                       1.000
    subjob53                                                                               1.000
    subjob54                                                                                           1.000
    subjob55                                                                   1.000
    subjob56                                           1.000
    subjob57                                                                   1.000
    subjob58                                           1.000
    subjob59                                           1.000
    subjob60                   1.000
    subjob61                   1.000
    subjob62                               1.000
    subjob63                                                       1.000
    subjob64                                                                               1.000
    subjob65                                                       1.000
    subjob66                                                                               1.000
    subjob67                                           1.000
    subjob68       1.000
    subjob69                   1.000
    subjob70                   1.000
    subjob71                               1.000
    subjob72                                                                   1.000
    subjob73                                                       1.000
    subjob74                                                       1.000
    subjob75                                                       1.000
    subjob76                                                                                           1.000
    subjob77                                                                   1.000
    subjob78                                                                                                       1.000
    subjob79                                                       1.000
    subjob80                                           1.000
    subjob81                                                                               1.000
    subjob82                                                                   1.000
    subjob83                                                                               1.000
    subjob84                                                                   1.000
    subjob85                               1.000
    subjob86                                                                                           1.000
    subjob87                                                                               1.000
    subjob88                                                                                                       1.000
    subjob89                                                                               1.000
    subjob90                                                       1.000
    subjob91                                                                                                                   1.000
    subjob92                                                                                                       1.000
    subjob93                                                                                           1.000
    
    
    ----    106 VARIABLE used.L  
    
    machine1  1.000,    machine2  1.000,    machine3  1.000,    machine4  1.000,    machine5  1.000,    machine6  1.000
    machine7  1.000,    machine8  1.000,    machine9  1.000,    machine10 1.000
    
    
    ----    106 VARIABLE num.L                 =       10.000  number of machines needed
    
    ----    110 PARAMETER assignments  
    
                  machine1    machine2    machine3    machine4    machine5    machine6    machine7    machine8    machine9   machine10
    
    job1 .day1       1.000
    job1 .day2       1.000
    job1 .day6       1.000
    job2 .day2       1.000
    job2 .day5                   1.000
    job3 .day3       1.000
    job3 .day5       1.000
    job3 .day7       1.000
    job4 .day3                   1.000
    job4 .day4       1.000
    job4 .day7                   1.000
    job6 .day2                   1.000
    job6 .day5       1.000
    job6 .day7       1.000
    job7 .day4                   1.000
    job7 .day6                   1.000
    job7 .day7                               1.000
    job8 .day3                   1.000
    job8 .day4       1.000
    job8 .day6       1.000
    job9 .day2                   1.000
    job10.day1                   1.000
    job10.day2                   1.000
    job10.day3                               1.000
    job10.day5                               1.000
    job10.day7                                           1.000
    job11.day5                                           1.000
    job12.day2                               1.000
    job12.day3                                           1.000
    job12.day4                               1.000
    job12.day5                   1.000
    job12.day6                               1.000
    job12.day7                                                       1.000
    job13.day3                                           1.000
    job13.day4                   1.000
    job14.day3                                                       1.000
    job15.day5                               1.000
    job16.day4       1.000
    job16.day7                   1.000
    job17.day5                                           1.000
    job18.day2                                           1.000
    job19.day1                               1.000
    job19.day7                                                                   1.000
    job21.day3                                                       1.000
    job21.day5                                                       1.000
    job21.day7                                                                   1.000
    job22.day1                               1.000
    job22.day6                                           1.000
    job23.day4                   1.000
    job23.day6                               1.000
    job24.day3                                                                   1.000
    job24.day6                                                       1.000
    job25.day3                                                                               1.000
    job26.day3                                                                                           1.000
    job26.day5                                                                   1.000
    job27.day1                                           1.000
    job27.day5                                                                   1.000
    job28.day1                                           1.000
    job28.day5                                           1.000
    job30.day1                   1.000
    job30.day5                   1.000
    job31.day3                               1.000
    job32.day2                                                       1.000
    job32.day5                                                                               1.000
    job33.day1                                                       1.000
    job33.day6                                                                               1.000
    job33.day7                                           1.000
    job34.day1       1.000
    job34.day6                   1.000
    job34.day7                   1.000
    job35.day2                               1.000
    job35.day6                                                                   1.000
    job36.day2                                                       1.000
    job36.day7                                                       1.000
    job37.day1                                                       1.000
    job37.day5                                                                                           1.000
    job38.day1                                                                   1.000
    job38.day3                                                                                                       1.000
    job38.day5                                                       1.000
    job39.day4                                           1.000
    job39.day5                                                                               1.000
    job40.day2                                                                   1.000
    job40.day7                                                                               1.000
    job42.day1                                                                   1.000
    job42.day4                               1.000
    job42.day6                                                                                           1.000
    job44.day1                                                                               1.000
    job45.day5                                                                                                       1.000
    job46.day2                                                                               1.000
    job48.day4                                                       1.000
    job48.day5                                                                                                                   1.000
    job49.day5                                                                                                       1.000
    job49.day7                                                                                           1.000
    

    We need 10 machines for the assignments to be non-overlapping. The number of sub-jobs for some of the machines is very small. Machine 10 has only one sub-job. The results look fine at first sight:

    enter image description here

    The numbers are the job numbers. E.g. job 1 appears in day 1, 2 and 6 all on machine 1.