INTRODUCTION
A special type of problem called the assignment problem is also an allocation problem. Here we have n
jobs to perform with n persons and the problem is how to distribute the jobs to the different persons involved.
Depending on the intrinsic capacity or merit or potential of the individual, he will be able to accomplish the task in
different times. Then the objective function in assigning the different jobs to different persons is to find the optimal
assignment that will minimize the total time taken to finish all the jobs by the individuals. For example, we have
four different building activities say, construction of a hotel, a theatre, a hospital and a multistoried building and
there are four contractors competing for these jobs. Each contractor has to be assigned only one job. The allocation
should aim to minimize the total time taken to complete the construction of all four activities after assigning only
one job to one individual. In fact there are (4!) permutations possible for allocating 4 jobs to 4 contractors. We have
24 possible ways and it is tiresome to list all the possible ways and find the best one. If we have more jobs to be
allocated, it is even difficult to list out the different permutations of allocations, then what to speak of choosing the
best combinations!
The problem may be stated formally as follows. Given an nxn array of real numbers representing the
individual return associated with assigning one item to one person. We have to find the best assignment so that the
total return is optimal.
Consider the following example, given below in the table 1
Table 1
Jobs
Men
A B C D
1
5
6
8
7
2
4
7
6
6
3
5
4
6
5
4
6
7
4
6
In the above example, the elements of the matrix represent the times taken by A, B, C and D in accomplishing the
jobs 1, 2, 3 and we have to find which job is to be assigned to whom so that the total time taken will be minimum.
This is the objective function. Thus, this is also an allocation problem. A solution can be found to the above problem
by the algorithem used to solve the transportation problem of degenerate transportation problem. In this way only 4
cells will be allocated. This leads to problem of degenerate transportation problem. There should be (4 + 4 + 1) = 7
allocations in the initial basic feasible solution, but we have only 4 allocations. Hence it is the degeneracy.
MATHEMATICAL FORMULATION OF THE PROBLEM
Considering the above example, we have four jobs and four persons. We want to allot one job to one person
so that the total time taken will be minimum. We shall formulate a mathematical model for the problem.
Let the decision variable xij be the assignment of ith job to jth person, and cij be the time taken for ith job by
the jth person.
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