Unit 6

Unit 6

Structure

Assignment Problem

6.1. 6.2. 6.3. 6.4.

Introduction Mathematical formulation of the problem Hungarian method algorithm Routing problem 6.4.1. Unbalanced A.P 6.4.2 Infeasible Assignments 6.4.3 Maximization in A.P

6.5. 6.6.

Traveling salesmen problem Summary Terminal Questions Answers to SAQs and TQs

6.1 Introduction The assignment problem is a special case of the transportation problem, where the objective is to minimize the cost or time of completing a number of jobs by a number of persons and Maximize efficiently Revenue, sales etc In other words, when the problem involves the allocation of n different facilities to n different tasks, it is often termed as an assignment problem. This model is mostly used for planning. The assignment model is also useful in solving problems such as, assignment of machines to jobs, assignment of salesman to sales territories, traveling salesman problem etc. It may be noted that with n facilities and n jobs, there are n! possible assignments. One way of finding an optimal assignment is to write all the n! possible arrangement ,evaluate their total cost and select the assignment with minimum cost. But because of many computational procedures this method is not possible. In this unit we study an efficient method for solving assignment problems. There are n jobs for a factory and factory has n machines to process the jobs. A job i(=1,…,n) , when processed by machine j(=1,…,n) is assumed to incur a cost Cij. The assignment is to be made in such a way that each job can associate with one and only one machine Determine an assignment of jobs to machines so as to minimize the overall cost.

Sikkim Manipal University

98

Operations Research

Unit 6

Learning Objectives After studying this unit, you should be able to understand the following

1. At the end of this unit the students formulate a assignment problem Mathematically. 2. Solves a routing problem. 3. Analysis a traveling salesman problem. 4. Know the significance of the assignment problem. 5. Apply the Hungarian method to solve the problem. 6. Solve the practical problems like routing problem and traveling salesman problem. 6.2 Mathematical Formulation Of The Problem Let xij be a variable defined by ì0 if the i th job is not assigned to the j th machine ï x = í ij th th . ï1 if the i job is assigned to the j machine î Then, since only one job is to be assigned to each machine we have

n

n

xij = 1 and

å

n

z =

å

xij = 1

i =1

n

xij cij

j =1

Also the total assignment cost is given by

å å

j =1 i =1

Thus the assignment problem takes the following mathematical form Determine xij ≥ 0 (i , j =1,…, n) So as to minimize

n

z =

n j =1

xij cij

i =1

å å

Subject to the constraints

n

å

xij = 1 j =1, 2,…, n

i =1

Sikkim Manipal University

99

Operations Research

Unit 6

n

and

å

xij = 1 i = 1, 2,…, n

j =1

with xij = 0 or 1 Note: In an assignment problem if we add (or subtract) a real number to (from) every element of a row or column of the cost matrix, then an assignment which is optimum for the modified matrix is also optimum for the original one. Self Assessment Questions 1 State True or False

1. In A.P the constraints are of equality type. 2. The no. of facilities should be equal to no. of resources. 3. The decision variables can take any value. 6.3 Hungarian Method: Algorithm Step 1: Prepare Row ruled Matrix by selecting the minimum values for each row and subtract it from other elements of the row Step 2: Prepare column reduced Matrix by subtracting minimum value of the column from the other values of that column ...