Transportation & Assignment Solution Methodsls

Z07_TAYL4367_10_SE_ModB.QXD

B-22

Module B

1/9/09

8:19 AM

Page B-22

Transportation and Assignment Solution Methods

Solution of the Assignment Model
An assignment model is a special
form of the transportation model
in which all supply and
demand values equal one.

Table B-34
The Travel Distances to Each
Game for Each Team of
Officials

An opportunity cost table is
developed by first substracting
the minimum value in each
row from all other row values and
then repeating this process for
each column.

Table B-35
The Assignment Tableau with
Row Reductions

The assignment model is a special form of a linear programming model that is similar to the transportation model. There are differences, however. In the assignment model, the supply at each source and the demand at each destination are limited to one unit each. The following example from the text will be used to demonstrate the assignment model and its special solution method. 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 in Table B-34.

Game Sites
Officials

RALEIGH

ATLANTA

DURHAM

CLEMSON

A
B
C
D

210
100
175
80

90
70
105
65

180
130
140
105

160
200
170
120

The supply is always one team of officials, and the demand is for only one team of officials at each game. Table B-34 is already in the proper form for the assignment. The first step in the assignment method of solution is to develop an opportunity cost table. We accomplish this by first subtracting the minimum value in each row from every value in the row. These computations are referred to as row reductions. We applied a similar principle in the VAM method when we determined penalty costs. In other words, the best course of action is determined for each row, and the penalty or “lost opportunity” is developed for all other row values. The row reductions for this example are shown in Table B-35. Game Sites

Officials

RALEIGH

ATLANTA

DURHAM

CLEMSON

A
B
C
D

120
30
70
15

0
0
0
0

90
60
35
40

70
130
65
55

Next, the minimum value in each column is subtracted from all column values. These computations are called column reductions and are shown in Table B-36, which represents the completed opportunity cost table for our example. Assignments can be made in this table wherever a zero is present. For example, team A can be assigned to Atlanta. An optimal solution results when each of the four teams can be uniquely assigned to a different game. Table B-36

The Tableau with
Column Reductions

Game Sites
Officials

RALEIGH

ATLANTA

DURHAM

CLEMSON

A
B
C
D

105
15
55
0

0
0
0
0

55
25
0
5

15
75
10
0

Z07_TAYL4367_10_SE_ModB.QXD

1/9/09

8:19 AM

Page B-23

Solution of the Assignment Model
Assignments are made to locations
with zeros in the opportunity cost
table.
An optimal solution occurs when
the number of independent unique
assignments equals the number of
rows or columns.

Table B-37
The Opportunity Cost Table
with the Line Test

Notice in Table B-36 that the assignment of team A to Atlanta means that no other team can be assigned to that game. Once this assignment is made, the zero in row B is infeasible, which indicates that there is not a unique optimal assignment for team B. Therefore, Table B-36 does not contain an optimal solution.

A test to determine whether four unique assignments exist in Table B-36 is to draw the minimum number of horizontal or vertical lines necessary to cross out all zeros through the rows and columns of the table. For example, Table B-37 shows that three lines are required to cross out all zeros.

Game Sites
Officials

RALEIGH

ATLANTA...