A variant of an assignment problem

In one [1], the following problem is sketched:

We have different types of workers. In the data set below, I call them type-A and type-B workers.
We need to assign workers to jobs.
We have more workers than jobs, so we can execute all jobs.
This is a standard assignment problem: \[\begin{align} \min &\sum_{w,j} \color{darkblue}c_{w,j} \color{darkred}x_{w,j}\\ & \sum_j \color{darkred}x_{w,j} \le 1 &&\forall w \\ & \sum_w \color{darkred}x_{w,j} = 1 && \forall j \\ & \color{darkred}x_{w,j} \in [0,1] \end{align}\]
The complication is that we can form teams of two workers, one of type A and one of type B. Such a team can work faster and execute the job in less time.
How to handle this?

I don't know how to handle this within an assignment problem, but stating this as a MIP allows us to make some progress. Allowing two persons to work on a job is easy. Just replace the second constraint by \[1 \le \sum_w \color{darkred}x_{w,j} \le 2\] Unfortunately this ignores that we need one worker of each type in a team. Also, this makes it difficult to impose a cost for each specific team.

As an alternative one could explicitly use teams in the model. Here is my data set with random costs:

----     13 SET iworkers+teams

a1     ,    a2     ,    a3     ,    a4     ,    a5     ,    a6     ,    a7     ,    a8     ,    a9     ,    a10    
b1     ,    b2     ,    b3     ,    b4     ,    b5     ,    b6     ,    b7     ,    b8     ,    b9     ,    b10    
team1  ,    team2  ,    team3  ,    team4  ,    team5  ,    team6  ,    team7  ,    team8  ,    team9  ,    team10 
team11 ,    team12 ,    team13 ,    team14 ,    team15 ,    team16 ,    team17 ,    team18 ,    team19 ,    team20 
team21 ,    team22 ,    team23 ,    team24 ,    team25 ,    team26 ,    team27 ,    team28 ,    team29 ,    team30 
team31 ,    team32 ,    team33 ,    team34 ,    team35 ,    team36 ,    team37 ,    team38 ,    team39 ,    team40 
team41 ,    team42 ,    team43 ,    team44 ,    team45 ,    team46 ,    team47 ,    team48 ,    team49 ,    team50 
team51 ,    team52 ,    team53 ,    team54 ,    team55 ,    team56 ,    team57 ,    team58 ,    team59 ,    team60 
team61 ,    team62 ,    team63 ,    team64 ,    team65 ,    team66 ,    team67 ,    team68 ,    team69 ,    team70 
team71 ,    team72 ,    team73 ,    team74 ,    team75 ,    team76 ,    team77 ,    team78 ,    team79 ,    team80 
team81 ,    team82 ,    team83 ,    team84 ,    team85 ,    team86 ,    team87 ,    team88 ,    team89 ,    team90 
team91 ,    team92 ,    team93 ,    team94 ,    team95 ,    team96 ,    team97 ,    team98 ,    team99 ,    team100


----     13 SET wworkers

a1 ,    a2 ,    a3 ,    a4 ,    a5 ,    a6 ,    a7 ,    a8 ,    a9 ,    a10,    b1 ,    b2 ,    b3 ,    b4 ,    b5 
b6 ,    b7 ,    b8 ,    b9 ,    b10


----     13 SET atype-A workers

a1 ,    a2 ,    a3 ,    a4 ,    a5 ,    a6 ,    a7 ,    a8 ,    a9 ,    a10


----     13 SET btype-B workers

b1 ,    b2 ,    b3 ,    b4 ,    b5 ,    b6 ,    b7 ,    b8 ,    b9 ,    b10


----     13 SET tteams

team1  ,    team2  ,    team3  ,    team4  ,    team5  ,    team6  ,    team7  ,    team8  ,    team9  ,    team10 
team11 ,    team12 ,    team13 ,    team14 ,    team15 ,    team16 ,    team17 ,    team18 ,    team19 ,    team20 
team21 ,    team22 ,    team23 ,    team24 ,    team25 ,    team26 ,    team27 ,    team28 ,    team29 ,    team30 
team31 ,    team32 ,    team33 ,    team34 ,    team35 ,    team36 ,    team37 ,    team38 ,    team39 ,    team40 
team41 ,    team42 ,    team43 ,    team44 ,    team45 ,    team46 ,    team47 ,    team48 ,    team49 ,    team50 
team51 ,    team52 ,    team53 ,    team54 ,    team55 ,    team56 ,    team57 ,    team58 ,    team59 ,    team60 
team61 ,    team62 ,    team63 ,    team64 ,    team65 ,    team66 ,    team67 ,    team68 ,    team69 ,    team70 
team71 ,    team72 ,    team73 ,    team74 ,    team75 ,    team76 ,    team77 ,    team78 ,    team79 ,    team80 
team81 ,    team82 ,    team83 ,    team84 ,    team85 ,    team86 ,    team87 ,    team88 ,    team89 ,    team90 
team91 ,    team92 ,    team93 ,    team94 ,    team95 ,    team96 ,    team97 ,    team98 ,    team99 ,    team100


----     13 SET jjobs

job1 ,    job2 ,    job3 ,    job4 ,    job5 ,    job6 ,    job7 ,    job8 ,    job9 ,    job10,    job11,    job12
job13,    job14,    job15


----     23 SET teamcomposition of teams

team1  .a1 ,    team1  .b1 ,    team2  .a1 ,    team2  .b2 ,    team3  .a1 ,    team3  .b3 ,    team4  .a1 
team4  .b4 ,    team5  .a1 ,    team5  .b5 ,    team6  .a1 ,    team6  .b6 ,    team7  .a1 ,    team7  .b7 
team8  .a1 ,    team8  .b8 ,    team9  .a1 ,    team9  .b9 ,    team10 .a1 ,    team10 .b10,    team11 .a2 
team11 .b1 ,    team12 .a2 ,    team12 .b2 ,    team13 .a2 ,    team13 .b3 ,    team14 .a2 ,    team14 .b4 
team15 .a2 ,    team15 .b5 ,    team16 .a2 ,    team16 .b6 ,    team17 .a2 ,    team17 .b7 ,    team18 .a2 
team18 .b8 ,    team19 .a2 ,    team19 .b9 ,    team20 .a2 ,    team20 .b10,    team21 .a3 ,    team21 .b1 
team22 .a3 ,    team22 .b2 ,    team23 .a3 ,    team23 .b3 ,    team24 .a3 ,    team24 .b4 ,    team25 .a3 
team25 .b5 ,    team26 .a3 ,    team26 .b6 ,    team27 .a3 ,    team27 .b7 ,    team28 .a3 ,    team28 .b8 
team29 .a3 ,    team29 .b9 ,    team30 .a3 ,    team30 .b10,    team31 .a4 ,    team31 .b1 ,    team32 .a4 
team32 .b2 ,    team33 .a4 ,    team33 .b3 ,    team34 .a4 ,    team34 .b4 ,    team35 .a4 ,    team35 .b5 
team36 .a4 ,    team36 .b6 ,    team37 .a4 ,    team37 .b7 ,    team38 .a4 ,    team38 .b8 ,    team39 .a4 
team39 .b9 ,    team40 .a4 ,    team40 .b10,    team41 .a5 ,    team41 .b1 ,    team42 .a5 ,    team42 .b2 
team43 .a5 ,    team43 .b3 ,    team44 .a5 ,    team44 .b4 ,    team45 .a5 ,    team45 .b5 ,    team46 .a5 
team46 .b6 ,    team47 .a5 ,    team47 .b7 ,    team48 .a5 ,    team48 .b8 ,    team49 .a5 ,    team49 .b9 
team50 .a5 ,    team50 .b10,    team51 .a6 ,    team51 .b1 ,    team52 .a6 ,    team52 .b2 ,    team53 .a6 
team53 .b3 ,    team54 .a6 ,    team54 .b4 ,    team55 .a6 ,    team55 .b5 ,    team56 .a6 ,    team56 .b6 
team57 .a6 ,    team57 .b7 ,    team58 .a6 ,    team58 .b8 ,    team59 .a6 ,    team59 .b9 ,    team60 .a6 
team60 .b10,    team61 .a7 ,    team61 .b1 ,    team62 .a7 ,    team62 .b2 ,    team63 .a7 ,    team63 .b3 
team64 .a7 ,    team64 .b4 ,    team65 .a7 ,    team65 .b5 ,    team66 .a7 ,    team66 .b6 ,    team67 .a7 
team67 .b7 ,    team68 .a7 ,    team68 .b8 ,    team69 .a7 ,    team69 .b9 ,    team70 .a7 ,    team70 .b10
team71 .a8 ,    team71 .b1 ,    team72 .a8 ,    team72 .b2 ,    team73 .a8 ,    team73 .b3 ,    team74 .a8 
team74 .b4 ,    team75 .a8 ,    team75 .b5 ,    team76 .a8 ,    team76 .b6 ,    team77 .a8 ,    team77 .b7 
team78 .a8 ,    team78 .b8 ,    team79 .a8 ,    team79 .b9 ,    team80 .a8 ,    team80 .b10,    team81 .a9 
team81 .b1 ,    team82 .a9 ,    team82 .b2 ,    team83 .a9 ,    team83 .b3 ,    team84 .a9 ,    team84 .b4 
team85 .a9 ,    team85 .b5 ,    team86 .a9 ,    team86 .b6 ,    team87 .a9 ,    team87 .b7 ,    team88 .a9 
team88 .b8 ,    team89 .a9 ,    team89 .b9 ,    team90 .a9 ,    team90 .b10,    team91 .a10,    team91 .b1 
team92 .a10,    team92 .b2 ,    team93 .a10,    team93 .b3 ,    team94 .a10,    team94 .b4 ,    team95 .a10
team95 .b5 ,    team96 .a10,    team96 .b6 ,    team97 .a10,    team97 .b7 ,    team98 .a10,    team98 .b8 
team99 .a10,    team99 .b9 ,    team100.a10,    team100.b10


----     28 PARAMETER ccost coefficients

               job1        job2        job3        job4        job5        job6        job7        job8        job9

a1           17.17584.32755.03830.11429.22122.40534.98385.6276.711
a2           63.97215.95225.00866.89343.53635.97035.14413.14915.010
a3           11.04950.23816.01787.24626.51128.58159.39672.27262.825
a4           18.21064.57356.07576.99629.78166.11175.58262.74528.386
a5            7.27717.56652.56375.02117.8123.41458.51362.12338.936
a6           78.34030.00312.54874.8876.92320.2020.50726.96149.985
a7           99.36036.99037.28977.19839.66891.31011.95873.5485.542
a8           22.57539.61227.60115.23793.63242.26613.46638.60637.463
a9           10.16938.38932.40919.21311.23759.65651.1454.50778.310
a10          50.65915.92565.68952.38812.44098.67222.81267.56577.678
b1           73.4978.54415.03543.41918.69469.26976.29715.48138.938
b2            8.71254.04012.68673.40011.32348.83579.56049.20553.356
b3            2.46317.7826.1321.66483.56560.1662.70219.60995.071
b4           39.33480.54654.09939.07255.78293.27634.8770.82994.884
b5           58.03216.64264.33634.43191.23390.0061.62636.86366.438
b6           49.6624.49377.37053.29774.67772.00563.16011.49297.116
b7           79.07061.0225.43148.5185.25569.85819.47822.60381.364
b8           81.95386.04121.26945.6793.83632.30043.98831.53313.477
b9            6.44141.46034.16146.82964.26764.35833.76110.08290.583
b10          40.41911.16775.11380.3402.36648.08827.85990.1611.759
team1        50.42183.12660.2148.22557.77659.31868.37715.87733.178
team2        57.62471.98368.3731.98583.98071.00515.55161.07166.155
team3         1.2521.01795.20397.66896.63285.62814.1614.97355.303
team4        34.96811.73458.59844.55341.23291.45121.37822.41754.233
team5        31.0144.02082.11723.09641.00330.25844.49271.60059.315
team6        68.63967.46333.21375.99417.67868.24867.29983.12151.517
team7         8.46957.2162.20674.20490.51056.08247.28371.75651.301
team8        96.55295.78989.92332.75545.71059.61887.86217.06763.360
team9        33.62658.86457.43954.34257.81697.72232.14776.29796.251
team10       27.01675.84861.74329.10074.0700.77686.6511.51442.828
team11       93.34746.92821.36151.07836.57293.5406.80150.38739.241
team12       54.7485.82737.77297.40737.98215.62947.22639.70020.552
team13        7.21276.06529.01524.49543.59836.88155.3427.46690.944
team14       86.56897.53257.23831.39745.50337.10241.9858.53081.454
team15       44.43169.54767.66957.15417.35260.43558.59972.77824.623
team16       99.60246.44953.04919.03819.90664.49079.90858.63697.157
team17        0.30961.8020.66640.55865.62961.55026.3417.0454.597
team18       35.53533.78748.64925.93789.11984.93363.45298.64775.792
team19       60.35897.70555.40793.50441.32080.6450.07845.52641.917
team20       18.98742.80616.65188.62667.65486.3349.27699.64861.681
team21       44.19965.90648.44531.81991.40418.42387.21145.61045.892
team22       82.29448.05291.37692.92810.81135.06085.99834.05021.309
team23       29.67611.55039.97891.17625.15510.29814.63338.6304.641
team24       97.21524.34336.19763.03424.28510.10640.59347.90614.495
team25       72.04287.19529.95275.39084.37646.82365.49337.80535.887
team26       80.64973.26192.00133.16420.9784.71588.3049.28183.691
team27       57.99696.21151.41959.03056.7462.99096.89059.3755.962
team28       99.2955.53361.90880.44429.65995.89269.47563.14161.598
team29       35.58276.10813.63671.66396.37044.19226.44844.41839.670
team30        6.91093.85784.39795.04457.9383.52240.7435.74353.197
team31       18.61450.06893.88268.15718.52387.28816.50143.08377.173
team32       74.91464.76074.83852.31768.61169.10786.56579.57621.755
team33       32.48239.01229.93221.89682.17215.26695.0553.19273.446
team34       97.1089.63347.89272.21743.32015.81810.06680.55139.869
team35       90.91872.30916.63532.75558.13557.75462.7582.67312.942
team36       33.86122.42490.00382.93831.62295.22225.66962.61297.126
team37       59.51860.63863.36595.8248.22612.53760.52274.14884.752
team38       34.62140.95993.98660.29389.95428.47322.22457.48450.950
team39       75.58247.4917.6199.75032.96720.0619.07544.87746.281
team40       70.31587.49255.51325.56325.91835.51213.68880.70732.598
team41       40.71216.16286.17537.77188.86126.99877.73942.27942.985
team42       96.85127.00231.81588.33858.62238.20997.29867.08395.115
team43       11.83496.53884.35572.13396.44436.45776.83133.24854.208
team44       69.1565.47424.02088.27510.05867.83610.1384.28175.345
team45       18.59579.01686.45274.79338.94979.50462.09876.13758.026
team46       12.29161.78653.39514.40839.2281.47687.07724.55280.318
team47        3.95291.29354.25022.09992.55779.27873.93867.77450.119
team48       84.20265.73811.65127.4258.31650.53123.26879.85798.805
team49       34.05025.64199.30724.77912.14894.37125.32036.33539.410
team50       91.41819.24114.41476.03733.0580.01954.34353.50686.860
team51       34.19286.53746.08542.10672.58366.9599.29052.65989.543
team52       51.5182.49944.81850.12968.11979.90487.29528.29891.049
team53        1.81446.04162.39771.63633.5161.28835.62136.81791.687
team54       91.7033.81230.2387.00346.64956.07046.50954.73317.797
team55       70.7108.98340.81013.68373.46359.27735.7467.36863.081
team56       64.88012.22276.00463.31182.90190.88285.85696.6243.575
team57       28.70468.66227.52739.0289.47021.75684.35384.22688.356
team58       57.29533.57439.98452.03821.88070.11135.38930.44395.367
team59       13.34876.0509.41644.53779.38346.26132.45035.38917.424
team60       60.26629.78650.11548.84278.45262.11160.55880.20665.321
team61       48.29325.07648.48080.98851.94497.47310.24546.9951.720
team62       55.26288.31557.95178.82611.30965.8446.64189.80688.661
team63       83.95292.29823.66144.99427.2025.52996.6591.7751.772
team64       40.23190.44744.81734.87950.56139.5295.69471.67117.966
team65       18.09063.98414.54566.33213.19370.74257.0573.34128.519
team66       89.69286.51854.62132.15996.33591.0538.30163.8717.836
team67       70.98839.98521.74379.52572.17878.87542.39988.49665.532
team68       28.22039.85065.03028.16073.64966.88948.94035.90967.851
team69       15.37212.77586.77276.50457.89678.85689.68043.89658.892
team70        9.64825.85813.71329.32359.71768.81454.85516.83281.477
team71       19.76171.65731.43862.19147.63279.54052.10147.06161.375
team72       70.55457.88827.6083.98026.70623.41769.72896.83867.951
team73       25.74647.55787.61040.2298.16770.93285.33329.27527.718
team74       12.42219.26068.59379.72665.10446.90052.86951.10040.703
team75       21.64473.24557.38017.48022.51298.8578.09871.53571.020
team76       63.6990.91541.98451.11877.63769.81658.57056.03662.796
team77       25.20880.88381.77617.89587.6114.85236.31168.44971.587
team78        7.30677.86762.60817.97970.58444.2126.01752.78741.516
team79       90.69037.16686.20141.74528.67418.11461.10474.74984.652
team80        5.56619.2716.2174.9312.81259.61452.06645.39293.225
team81       10.01112.27156.49166.26986.98472.97070.64351.23866.539
team82       51.27936.20294.21785.48182.9295.23692.3376.47848.276
team83       30.50356.25138.43816.20268.00728.0494.74421.57775.036
team84       87.86427.18214.75618.2791.90664.28173.22256.71757.004
team85       73.60779.4754.99740.3724.53937.5618.75326.09045.468
team86        2.42432.72354.4122.8841.61642.87169.67929.67270.529
team87       41.06531.77450.93423.79926.39469.3869.29013.32010.917
team88       65.12528.14111.85929.77448.32125.05828.43681.41480.148
team89       72.42084.14467.92474.00952.94938.86229.61093.23681.799
team90       40.9645.39856.44913.81242.35854.71837.84556.76772.899
team91       29.23121.86251.14369.91156.02733.61651.87624.92241.096
team92       66.7647.57941.14151.8400.69018.66518.36154.36555.272
team93       36.44360.22483.94768.47988.23961.7479.94338.9102.983
team94        6.39612.78728.33291.48584.99321.96626.91746.70251.649
team95       29.60062.57993.92979.01234.29258.55760.67525.5883.830
team96       96.94557.3038.34134.51989.81254.27540.90132.75598.966
team97        5.72076.32787.03478.06117.19869.22963.38761.9130.038
team98       21.2778.25228.34197.28447.14622.19656.53789.96615.708
team99       77.38512.01578.86179.37583.14611.3793.41372.92588.689
team100      11.05020.27621.44827.92815.07376.67191.57494.4987.094

      +       job10       job11       job12       job13       job14       job15

a1           50.02199.81257.87399.11376.22513.069
a2           58.91183.08923.08266.57377.58630.366
a3           46.38041.33111.77031.4214.65533.855
a4            8.64210.25164.12554.5313.15279.236
a5           35.87124.30324.64213.05093.34537.994
a6           15.12917.41733.06431.69132.20996.398
a7           57.6305.1410.60140.12351.98862.888
a8           26.84894.83718.89429.7517.45540.135
a9           94.57559.64660.73436.25159.40767.985
a10          93.24520.12429.71419.72324.63564.648
b1           69.54384.58161.27297.5972.68918.745
b2            1.06254.38745.11397.53318.38516.353
b3           33.55459.42625.91964.06315.52546.002
b4           57.19233.36398.37576.64611.00999.480
b5           59.3383.45784.18293.20850.79629.960
b6           70.67498.62785.48262.14470.13170.089
b7           99.17375.06771.8340.05926.38682.382
b8           81.09641.67914.17846.55328.29989.568
b9           21.73591.88745.1758.99337.42041.499
b10          68.10495.09190.01889.88087.44639.100
team1        31.58651.99336.37916.77668.30950.539
team2        19.43736.35262.39073.13941.39715.749
team3        18.40399.41780.90930.6218.74043.050
team4        63.10632.74314.87892.91525.1036.259
team5        13.11916.12531.56357.20626.8723.639
team6        28.30355.54241.3997.34180.60133.272
team7        88.70877.15214.01226.45268.25644.980
team8        77.15956.9452.76881.09927.89343.335
team9        94.89925.58932.49521.47917.39673.125
team10       35.86570.48741.58754.98034.50469.958
team11       20.48552.95558.90934.58025.28854.766
team12       62.7340.35550.3950.22352.13783.612
team13        4.87082.03679.29565.95439.17741.315
team14       50.92073.44282.43841.45392.44339.415
team15       14.20889.11944.27611.43290.36233.385
team16       49.10937.25482.71782.0873.13492.520
team17       16.02745.24816.88757.16085.8713.587
team18       50.78876.77783.21358.39457.50655.671
team19        1.5668.19859.88155.51861.80088.509
team20        0.07760.89417.37131.11972.8888.475
team21       12.41010.38023.48934.01687.30784.479
team22       92.63636.07387.59914.81445.59826.825
team23        2.63815.4007.27182.86439.97741.717
team24       50.96988.5285.54850.74076.41197.900
team25       25.44825.54845.06294.48533.5534.849
team26       46.3810.01715.65870.50328.03563.558
team27       54.41112.73642.04988.14571.65931.358
team28       52.30214.10383.67389.20969.64731.179
team29       36.69262.12685.97375.50190.08143.805
team30       11.24189.38524.16964.22429.07881.356
team31       97.80594.44624.88488.64688.44396.495
team32       58.96292.26260.2903.57091.11256.501
team33       11.04948.77536.10421.63992.37144.996
team34       11.70987.43514.48617.77454.52046.860
team35        6.41331.11057.85180.98067.92073.567
team36       96.20842.52810.5397.70864.41331.222
team37       35.24664.14189.57338.81727.34097.040
team38       55.74834.41739.82777.6232.82336.238
team39       81.19645.00095.42612.26440.66088.637
team40       42.8840.89622.42966.06728.74013.112
team41       24.90738.1777.09871.56370.2907.016
team42       71.82644.36087.97346.98146.38537.143
team43       13.84431.3677.98893.41549.42567.891
team44       98.7041.48414.68699.09813.2198.543
team45       66.3568.21456.68644.32332.82533.060
team46        8.40398.72255.82768.19285.95858.673
team47       43.11257.00811.10498.70151.94070.649
team48       32.50585.53778.99832.10921.28752.089
team49       83.63444.40092.63047.92190.38850.953
team50       45.45435.83311.38966.95493.8011.507
team51       99.01350.68664.45076.03090.60816.629
team52       45.08687.40121.6898.01370.52990.050
team53       99.01320.76935.03948.67664.3865.633
team54       38.19651.29295.65560.22945.48269.008
team55       77.60142.04880.74360.46565.90379.763
team56       96.2490.30711.84975.86025.55091.410
team57       15.02575.07452.29427.73862.18596.519
team58       20.16792.25846.87280.25233.4778.141
team59       98.77512.93614.74965.75916.31681.181
team60       45.92859.69697.95218.54498.21895.787
team61       30.61630.09328.04543.32650.26320.108
team62       73.26362.78736.5213.72172.86661.417
team63       14.84947.07372.49216.5418.85826.414
team64       62.30940.15211.72872.43071.5737.407
team65       40.58383.0849.44115.55290.33045.777
team66       33.25710.30325.40974.7209.08559.351
team67       76.15220.12335.18556.50276.92871.735
team68       34.54156.67380.03222.24742.67168.199
team69       83.58642.0606.9772.52085.09834.960
team70       55.37144.28216.56952.01582.6874.228
team71       81.40482.44350.15878.51273.53941.490
team72       59.11450.26865.87633.90325.40417.194
team73       17.48087.13627.0015.61557.78888.698
team74       53.11042.12356.28518.66375.41971.128
team75       56.39135.59448.2401.75090.26797.382
team76       99.21455.1623.34581.59387.44266.650
team77       46.95383.7491.1467.4894.97780.452
team78       45.7244.45047.4999.82329.86982.283
team79       39.59475.00288.49222.72382.52597.911
team80       81.16942.27975.64789.44011.41639.335
team81       64.25039.29949.81857.86496.40295.548
team82       94.84124.04011.9585.31859.96910.779
team83       92.9331.02681.71097.46478.84157.344
team84       36.4508.39569.40086.81236.59716.455
team85       71.75024.88430.63674.94679.26377.997
team86       99.36291.16846.24026.16859.7111.738
team87       83.33434.91688.32085.14625.04320.107
team88        2.64331.24924.75767.15283.60431.640
team89       99.15645.08144.85827.23088.12983.337
team90       65.14124.61058.16771.45918.80186.023
team91       51.62876.88666.3951.91690.86445.338
team92       40.15587.42862.17741.00727.3071.300
team93       58.29420.85740.27328.3081.95312.584
team94        2.41839.41967.58710.24237.15657.060
team95       67.27849.56638.35077.75589.9270.652
team96        4.01689.29730.99362.08517.33618.329
team97       21.72161.05846.02420.75350.34433.152
team98       29.13314.52785.9092.35584.51427.198
team99       26.20279.05463.52211.88352.11617.222
team100      81.17348.66046.72724.45584.13238.186

As I have 10 type-A workers and 10 type-B workers, we end up with 100 possible teams. As we can still use individual workers, our set $i$ has $10+10+10\times 10= 120$ elements. As you can see the model will quickly become rather large due to the enumeration of all possible teams.

My first thought was to add a side constraint to the assignment problem:

Assignment with side-constraint
\[\begin{align}\min&\sum_{i,j} \color{darkblue}c_{i,j}\color{darkred}x_{i,j}\\ & \sum_j \color{darkred}x_{i,j} \le 1 &&\forall i\\ & \sum_i \color{darkred}x_{i,j} = 1 &&\forall j \\ & \sum_j \left[\color{darkred}x_{w,j} + \sum_{t\|\color{darkblue}{\mathit{team}}(t,w)} \color{darkred}x_{t,j} \right] \le 1 && \forall w \\ & \color{darkred}x_{i,j}\in \{0,1\}\end{align}\]

The last constraint says: a worker $w$ can be used at most once, either individually or as part of a team. This problem is now a full-blown MIP as we can no longer assume that the decision variables $\color{darkred}x_{i,j}$ are automatically integer-valued.

If we look at the model more carefully, we see that the last constraint implies the first constraint: any worker or any team can be used only once. That means we can drop the first constraint, and we end up with:

Final model
\[\begin{align}\min&\sum_{i,j} \color{darkblue}c_{i,j}\color{darkred}x_{i,j} \\ & \sum_j \left[\color{darkred}x_{w,j} + \sum_{t\|\color{darkblue}{\mathit{team}}(t,w)} \color{darkred}x_{t,j} \right] \le 1 && \forall w \\ & \sum_i \color{darkred}x_{i,j} = 1 &&\forall j \\ & \color{darkred}x_{i,j}\in \{0,1\}\end{align}\]

Results

To obtain a reference point, I first solved the pure assignment problem using only individuals (no teams). This gave:

----    103 VARIABLE x.L  assignment

           job1        job2        job3        job4        job5        job6        job7        job8        job9

a5                                                                        1
a6                                                                                    1
b1                                    1
b3                                                1
b4                                                                                                1
b6                        1
b8                                                            1
b9            1
b10                                                                                                           1

  +       job10       job11       job12       job13       job14       job15

a1                                                                        1
a4                                                            1
a7                                    1
b2            1
b5                        1
b7                                                1


----    103 VARIABLE z.L                   =       59.379total cost

When we add teams to the model, we see:

----    142 VARIABLE x.L  assignment

               job1        job2        job3        job4        job5        job6        job7        job8        job9

a1                                                                                                                1
a5                                                                            1
a6                                                                                        1
b3                                                    1
b4                                                                                                    1
b9                1
b10                                                               1
team17                                    1
team28                        1

      +       job10       job11       job12       job13       job14       job15

a4                                                                1
a7                                        1
b2                1
b5                            1
team86                                                                        1
team91                                                1


----    142 VARIABLE z.L                   =       40.057total cost

The objective goes down. That makes sense: we can select some really cheap teams. To verify that we do not select a worker twice, I also generated a different solution report:

----    153 SET sol alternative solution report

                  job1        job2        job3        job4        job5        job6        job7        job8        job9

team17.a2                                  YES
team17.b7                                  YES
team28.a3                      YES
team28.b8                      YES
-     .a1                                                                                                          YES
-     .a5                                                                      YES
-     .a6                                                                                  YES
-     .b3                                              YES
-     .b4                                                                                              YES
-     .b9          YES
-     .b10                                                         YES

         +       job10       job11       job12       job13       job14       job15

team86.a9                                                                      YES
team86.b6                                                                      YES
team91.a10                                             YES
team91.b1                                              YES
-     .a4                                                          YES
-     .a7                                  YES
-     .b2          YES
-     .b5                      YES

This confirms that workers are not duplicated in the solution.

Finally, some performance numbers. All these models are easy to solve, even the larger ones.

----    144 PARAMETER resultscompare statistics

                assign     sidecon       final

type              RMIP         MIP         MIP
vars           301.0001801.0001801.000
  discr        300.0001800.0001800.000
equs            36.000156.00036.000
status         Optimal     Optimal     Optimal
obj             59.37940.05740.057
time             0.1720.4220.515
nodes               NA
iterations      12.00085.00085.000

For the MIP models, we see that the solver did not need to use any branching (zero nodes).

Conclusion

I am not sure this model can be used as-is for the purpose described in [1]. It is a good tool to think a bit about the problem, however. Furthermore, it is a good way to assess the quality of solutions when developing heuristics. MIP solvers have the special property that they can provide proven optimal solutions or a gap that gives some indication about the quality of the solution.

References

Assignment problem with 2 workers per job, https://stackoverflow.com/questions/68482524/assignment-problem-with-2-workers-per-job

Appendix: GAMS model

$ontext

A variant of an assignment problem.

References:
     https://stackoverflow.com/questions/68482524/assignment-problem-with-2-workers-per-job
     https://yetanothermathprogrammingconsultant.blogspot.com/2021/07/a-variant-of-assignment-problem.html

$offtext

*------------------------------------------------------------------
* basic data
*------------------------------------------------------------------

set
    i 'workers+teams'/a1*a10, b1*b10, team1*team100/
    w(i) 'workers'/a1*a10,b1*b10/
    a(w) 'type-A workers'/a1*a10/
    b(w) 'type-B workers'/b1*b10/
    t(i) 'teams'/team1*team100/
    team(t,w) 'composition of teams'
    j 'jobs'/job1*job15/
;

display i,w,a,b,t,j;

*------------------------------------------------------------------
* enumerate possible teams
*------------------------------------------------------------------

set t1(t) /team1/;
loop((a,b),
   team(t1,a) = yes;
   team(t1,b) = yes;
* next element:
   t1(t) = t1(t-1);
);
option team:0:0:8;
display team;

*------------------------------------------------------------------
* random cost coefficients
*------------------------------------------------------------------

parameter c(i,j) 'cost coefficients';
c(i,j) = uniform(0,100);
display c;

*------------------------------------------------------------------
* reporting macros
*------------------------------------------------------------------

acronym TimeLimit, Optimal, Error;
acronym MIP,RMIP;

parameter results(*,*) 'compare statistics';
$macro report(m,label,modeltype) \
    results('type',label) = modeltype; \
    results('vars',label) = m.numVar; \
    results(' discr',label) = m.numDVar; \
    results('equs',label) = m.numEqu; \
    results('status',label) = Error; \
    results('status',label)$(m.solvestat=1) = Optimal; \
    results('status',label)$(m.solvestat=3) = TimeLimit; \
    results('obj',label) = z.l; \
    results('time',label) = m.resusd; \
    results('nodes',label) = m.nodusd; \
    results('iterations',label) = m.iterusd; \
    results('gap%',label)$(m.solvestat=3) = 100*abs(m.objest - m.objval)/abs(m.objest);   \
    display results;

*------------------------------------------------------------------
* assignment problem (don't use teams)
* model uses w instead of i
*------------------------------------------------------------------

binaryvariable x(i,j) 'assignment';
variable z 'total cost';

equations
   obj1          'assignment objective'
   assign1a(w)   'assignment constraint'
   assign1b(j)   'assignment constraint'
;

obj1..        z =e= sum((w,j),c(w,j)*x(w,j));
assign1a(w).. sum(j, x(w,j)) =l= 1;
assign1b(j).. sum(w ,x(w,j)) =e= 1;

model assign1 /all/;

* solve as RMIP: automatically integer
solve assign1 minimizing z using rmip;

option x:0;
display x.l,z.l;

report(assign1,'assign',RMIP)

*------------------------------------------------------------------
* assignment problem with side constraint
* assignment constraints now use i
*------------------------------------------------------------------

equations
   obj2         'assignment objective'
   assign2a(i)   'assignment constraint'
   assign2b(j)   'assignment constraint'
   side(w)      'side-constraint'
;

obj2..         z =e= sum((i,j),c(i,j)*x(i,j));
assign2a(i).. sum(j, x(i,j)) =l= 1;
assign2b(j).. sum(i ,x(i,j)) =e= 1;
side(w)..      sum(j, x(w,j) + sum(team(t,w),x(t,j))) =l= 1;

option optcr=0,threads=8;

model assign2 /all-assign1/;
solve assign2 minimizing z using mip;

display x.l,z.l;

report(assign2,'sidecon',MIP)

*------------------------------------------------------------------
* drop first assignment constraint
*------------------------------------------------------------------

model assign3 /assign2-assign2a/;
solve assign3 minimizing z using mip;

display x.l,z.l;

report(assign3,'final',MIP)

*------------------------------------------------------------------
* solution report with team composition
*------------------------------------------------------------------

set sol(*,w,j) 'alternative solution report';
sol(team(t,w),j)$(x.l(t,j)>0.5) = yes;
sol('-',w,j)$(x.l(w,j)>0.5) = yes;
display sol;

A variant of an assignment problem

Results

Conclusion

References

Appendix: GAMS model

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