ASSIGNMENT MODEL

The Assignment is a special type of transportation model where the resources (rows) are collected to the destinations (columns) on a one-to-one correspondence so as to minimize cost.  This may refer, in particular, to teams assigned to projects; jobs assigned to workers; machines assigned to laborers; etc.  It should be observed that the number of rows is equal to the number of columns in the Assignment Table to ensure that no row cannot be assigned to a column, or vice-versa on a one-to-one correspondence.

1.  The Atlantic Coast Conference has four basketball games on a particular night.  The conference office wants to assign four teams of officials to the four games in a way that will minimize the total distance traveled by the officials.  The distances in miles for each team of officials to each game location are shown below:

Officials                                        Game Sites
                                     Raleigh               Atlanta                  Durham                 Clemson
A                                       210                     90                          180                      160
B                                       100                     70                          130                      200
C                                       175                     105                        140                      170
D                                         80                       65                        105                     120

2.  A metal parts shop  has three operators and three machines: a drill press, a lathe, and a grinder.  Each operator is qualified to operate each machine.  The following table shows the time ( in minutes) required by each operator to produce a part on each machine.

Operator                                          Machine
                                       Press                      Lathe                  Grinder
1                                       22                           18                           35
2                                       41                           30                           28
3                                       25                           36                           18

Determine the optimal assignment that will minimize the total machine time.

3.  A plant has four operators to be assigned to four machines.  The time (minutes) required by each worker to produce a product on each machine is shown below.

Operator                                        Machine
                                        A                     B                    C                      D
1                                     10                     12                   9                       11
2                                      5                      10                   7                       8
3                                     12                     14                   13                      11
4                                      8                      15                    11                     9

4.  A shop has four machinists to be assigned to four machines.  The hourly cost of having each machine operated by each machinist is as follows.

Machinist                                        Machine
                                        A                    B                        C               D
1                                     12                   11                       8               14
2                                     10                     9                       10             8
3                                     14                    8                         7               11
4                                       6                    8                        10              9

5.  The Omega pharmaceutical firm has five salespersons, whom the firm wants to assign to five sales regions.  Given their various previous contacts, the salespersons are able to cover the regions in different amounts of time.  The amount of time (days) required by each salesperson to cover each city is shown below.

Salesperson                                      Region
                                   A                B                    C                    D               E
1                                17               10                   15                   16             20
2                                12                9                    16                    9              14
3                                11                16                   14                   15            12
4                                14                10                   10                    18            17
5                                13                12                     9                   15             11

6.  The Bunker Manufacturing firm has five employees and six machines, and wants to assign the employees to the machines so as to minimize cost.  A cost table showing the cost incurred by each employee on each machine is given below.

Employee                                      Machine
                                A              B             C             D              E           F
1                              12             7            20            14              8          10
2                              10             14          13            20              9          11
3                               5               3             6             9               7           10
4                               9               11           7             16             9           10
5                              10              6            14             8              10          12

7.  Given the following cost table for an assignment problem, determine the optimal assignment and compute total minimum cost.  Identify all alternative solutions if there are multiple optimal solutions,

Operator                                        Machine
                               A                B               C            D
1                            10                2                 8            6
2                             9                 5                 11          9
3                             12              7                 14          14
4                             3                 1                  4            2

8.  An electronics firm produces electronic components, which it supplies to various electrical manufacturers.  Quality control records indicate that different employees produce different numbers of defective items.  The average number of defects produced by each employee for each of six components is given in the following table.

Employee                                 Component
                             A            B        C        D         E             F
1                           30           24       16      26        30            22
2                           22           28        14     30        20             13
3                           18           16        25     14        12             22
4                           14            22       18      23        21            30
5                           25            18        14      16        16           28
6                           32            14        10      14       18           20

9.  A dispatcher for the Citywide Taxi Company presently has six taxicabs at different locations and five customers who have called for service.  The mileage from each taxi's present location to each customer is shown below.

Cab                                        Customer
                               1               2                3                     4               5
A                          7                2                4                    10              7
B                          5                1                5                      6              6
C                          8                7                6                      5              5
D                          2                5                2                      4              5
E                           3               3                 5                      8              4
F                           6               2                 4                     3               4

10.  The Southwest Athletic Conference has six basketball officials who must be assigned to three conference games, two to each game.  The conference office wants to assign the officials so that the total distance they travel will be minimized.  The distance each official would have to travel to each game is given in the following table:

Official             Game
                           Austin                Houston                Lubbock
1                              20                      45                           10
2                              40                      90                           70
3                               60                     70                            30
4                               30                      60                           40
5                               70                      15                           50
6                               80                      25                           35

11.  A university department head has five instructors to be assigned to four different courses.  All of the instructors have taught the courses in the past and have been evaluated by the students.  The rating for each instructor for each course is given in the following table

Instructor               Course
                                  A           B            C             D
1                               80           75           90           85
2                               95           90           90           97
3                              85            95           88            92
4                               93           91            80           84
5                               91           92            93           88