The assignment model

Consider the situation of assigning m jobs (or workers) to n machines. A job i(= 1,2,3 ...m) when assigned to machine j(= 1,2,3 ...n)  acquires a cost Cij. The objective is to assign the jobs to the machines (on job per machine) at the least total cost. The situation is known as the assignment problem.

The formulation of this problem may be regarded as a special case of the transportation model. These jobs represent "sources" and machines represent "destinations". The supply obtainable at each source is 1; which is  ai = 1 for all i. Likewise, the demand needed at each destination is 1, which is bj = 1 for all j. The cost of "transporting" (assigning) job i to machine j is Cij.  If a job cannot be assigned to a certain machine, the corresponding Cij is taken equal to M, a very high cost. We represent in the table the general ideal of the model.

Before the model can be solved by the transportation technique, it is necessary to balance the problem by adding fictitious jobs or machines, depending on whether m < n or m > n.  It will be assumed that
m = n without loss of generality.

The assignment model can be stated mathematically as shown below:

Let Xij  = 0 if the jth job is not assigned to the ith machine.
          = 1 if the jth job is allocated to the ith machine.

The model is thus specified by:

Minimize: 

Subject to :

Xij = 0 or 1