Developing a New Approach to Solve Solid Assignment Problems Under Intuitionistic Fuzzy Environment

1. Introduction

Assignment problem (AP) is a special type of a transportation problem (TP) in which the aim of the decision maker (DM) is to assign n number of machines to n number of jobs at a minimum time / minimum cost / maximum profit / maximum production. An assignment problem can be viewed as a transportation problem in which all supplies and demands equal to one. Assignment problem is used worldwide in solving real world problems. An assignment problem plays a vital role in assigning of accountants to accounts of the clients, trucks to drivers, contracts to bidders by systematic bid-evaluation, jobs to persons, routes to trucks, research teams to problems, sales/marketing people to sales territories, machines to operators, or police vehicles to patrolling areas, and so on.

An assignment problem has been used in a variety of application contexts such as resource allocation, manpower planning, personnel scheduling and so on. The standard assignment problem consists of assigning a number of tasks to an equal number of agents. Each agent is assigned to exactly one task and each task has exactly one agent assigned to perform it in such a way to minimize the overall cost (or time) of assigning agents to tasks. When the standard assignment problem is considered with the minimization of assignment cost as the objective function, it is called the cost minimizing assignment problem. The well- studied Hungarian method developed by Kuhn (1955) is recognized to be the first practical method for solving the standard two-dimensional assignment problem.

To solve the classical assignment problem using Hungarian method, one should know time of completion or cost of making all the possible assignments. Each assignment problem has a two-dimensional table, jobs (or tasks) one wishes to assign are represented in the rows and persons (or machines) to be assigned are expressed in the columns. Cost / time for each particular assignment is in the numbers in the two-dimensional table. Particularly, this table is referred as cost / time matrix of the two-dimensional assignment problem or classical assignment problem. Hungarian method is based on the principle that if a constant is added (or subtracted) to the elements of cost / time matrix, the optimum solution (or optimum assignment) of the assignment problem is the same as the original problem. Original cost / time matrix is reduced to another cost / time matrix by adding (or subtracting) a constant value to the elements of rows and columns of cost / time matrix where the total cost or total completion time of an assignment is zero. Further, this assignment is referred as the optimum assignment (or optimal assignment) since the optimum assignment remains unchanged after the reduction.

Let us consider n jobs J₁, J₂, …, J_n and n machines M₁, M₂, …, M_n (one machine to one job). Let c_ij be the unit cost of assigning i^th machine to the j^th job and let x_ij denotes that j^th job is to be assigned to i^th machine. Our objective is to determine the assignment of jobs to machines so that the total cost of completing all the jobs is minimum.

Now, the mathematical model of the above assignment problem is given by:Minimize (i)subject to: (Row restriction)(1) (Column restriction)(2)

if the job j is assigned to the machine i, and , otherwise.

Now, the above problem can be put in the following assignment table (see Table 1).