6 Assignment Problem

Operations Research Unit 6

Sikkim Manipal University 98

Unit 6 Assignment Problem

Structure

6.1. Introduction

6.2. Mathematical formulation of the problem

6.3. Hungarian method algorithm

6.4. Routing problem

6.4.1. Unbalanced A.P

6.4.2 Infeasible Assignments

6.4.3 Maximization in A.P

6.5. Traveling salesmen problem

6.6. 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.



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

= . machine j the to assigned is job i the if 1

machine j the to assigned not is job i the if 0 x th th

th th ij

Then, since only one job is to be assigned to each machine we have

∑=

n

1 i xij = 1 and ∑

=

n

1 j xij = 1

Also the total assignment cost is given by

z = ∑ ∑ = =

n

1 i

n

1 j xij cij

Thus the assignment problem takes the following mathematical form

Determine xij ≥ 0 (i , j =1,…, n)

So as to minimize

z = ∑ ∑ = =

n

1 j

n

1 i xij cij

Subject to the constraints

∑=

n

1 i xij = 1 j =1, 2,…, n



and ∑=

n

1 j xij = 1 i = 1, 2,…, n

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

Step 3: First rowwise assign a zero by if there is only one zero in the row and cross (X) other zeros in that column.

Step 4: Now assign column wise if there is only one zero in that column and cross other zeros in that row.

Repeat Step 3 and 4 till all zeros are either assigned or crossed. If the number of assignments

made is equal to number of rows present, then it is the optimal solution otherwise proceed as

follows. Step 5: Mark (P) the row which is not assigned. Look for crossed zero in that row. Mark the column containing the crossed zero. Look for assigned zero in that column. Mark the row

containing assigned zero. Repeat this process till all makings are over. Step 6: Draw straight line through unmarked rows and marked column. The number of straight line drawn will be equal to number of assignments made. Step 7: Examine the uncovered elements. Select the minimum.



a. Subtract it from uncovered elements.

b. Add it at the point of intersection of lines.

c. Leave the rest as it is.

Prepare a New Table. Step 8: Repeat Steps 3 to 7 till number of allocations = Number of rows.

Example 1: Find the optimum assignment so as to minimize the cost.

Persons Jobs 1 2 3 4 5

A

B

C

D

E

8 4 2 6 1

0 9 5 5 4

3 8 9 2 6

4 3 1 0 3

9 5 8 9 5

Example: Consider the problem of assigning five jobs to five persons. The assignment costs are given as

follows.

Persons Job

1 2 3 4 5 A 8 4 2 6 1 B 0 9 5 5 4 C 3 8 9 2 6 D 4 3 1 0 3 E 9 5 8 9 5

Determine the optimum assignment schedule.

Solution:

Persons Jobs 1 2 3 4 5

A B C

7 3 1 5 0 0 9 5 5 4 1 6 7 0 4



D E

4 3 1 0 3 4 0 3 4 0

Solution: Applying Hungarian Method Row reduced Matrix

Persons Jobs

1 2 3 4 5

A

B

C

D

E

7 3 1 5 0

0 9 5 5 4

1 6 7 0 4

4 3 1 0 3

4 0 3 4 0

Column reduced Matrix: Will be same as each column’s minimum value is zero. We start

assigning the jobs

Jobs P

Persons 1 2 3 4 5

A

B

PC

PD

E

7 3 0 5

9 4 5 4

1 6 6 4

4 3 x 3

4 2 4 x

Since the number of assignments = 5. It is the optimum solution.

Optimum solution

A to 5 1

B to 1 0

C to 4 2

D to 3 1

E to 2 5

9 Total cost Self Assessment Questions 2

0 0

0

0 0



State Yes / No

1. In Hungarian method first we prepare rowreduced matrix 2. The number of assignments should be equal to number of rows for optimum solution. 3. There can be more than one allocation in a row.

6.4 Routing Problem Network scheduling is a technique for planning and scheduling of large projects. It has

successfully been applied in transportation and communication problems .A typical network

problem consists of finding route from one node(origin) to another (destination) between which

alternative paths are available at various stages of the journey. The problem is to select the route

that yields minimum cost. A number of different constraints may be placed on acceptable routes

e.g.: not returning to the node already passed through or passing through every node once and

only once. Problems of such type are called as routing problems.

Although a wide variety of problems other than routing may be developed in connection with the construction and utilization of networks. Here we consider the special type of routing problem that occurs frequently in O .R. the travelling salesman problem.

6.4.1. Unbalanced A.P

If the number of rows ≠ number of column then it is an unbalanced A.P. We introduce either a dummy row or dummy column and proceed as usual.

Example 2:

Operations Machines M1 M2 M3 M4

01

02

03

10 15 12 11

9 10 9 12

15 16 16 17

Solution: Introducing a dummy row and applying Hungarian method, we have


01 10 15 12 11



02

03

04

9 10 9 12

15 16 16 16

0 0 0 0

Row reduced Matrix and assigning


01

02

03

04

5 2 1

x 1 3

x 1 1 1

x x x

Iteration 1


01

02

03

04

4 1 x

1 1 3

x x x

1 x x

Hungarian Method leads to Multiple solution. Selecting (03, M2) arbitrarily

The optimum solution is

01 to M1 10

02 to M3 9

03 to M2 16

04 to M4 0

35 Total Wastages

0

0

0

0

0

0

0



6.4.2. Infeasible Assignments Certain jobs cannot be assigned to certain Machines or operators. We insert a high value α at

these cells and Apply Hungarian Method.

Example 3:

Solve the following A.P

Territory P1 P2 P3 P4

T1

T2

T3

T4

20 32 27

15 20 17 18

16 18 20

20 18 24

Solution: Introducing “α” at dashes places and applying Hungarian Method. We have

P1 P2 P3 P4

T1

T2

T3

T4

20 α 32 27

15 20 17 18

16 18 α 20

α 20 18 24

Rowreduced Matrix

P1 P2 P3 P4

T1

T2

T3

T4

0 α 12 7

0 5 2 3

0 2 α 4

α 2 0 6 Column reduced Matrix

P1 P2 P3 P4

T1

T2

T3

T4

α 12 4

x 3 2

x α 1

α x 3

∴Optimum assignment is

0

0

0

0



T1 to P1 20

T2 to P4 18

T3 to P2 18

T4 to P3 18

74 Total Cost

6.4.3. Maximization In A.P

Hungarian Method cannot be applied to maximization problems. Therefore we reduce it to

maximization problem by subtracting every value form the highest value of the Matrix and then

proceed as usual.

Example 4: Solve the following A.P Efficiency Matrix

Classes Professors P1 P2 P3 P4

C1

C2

C3

C4

70 75 80 65

80 70 65 75

75 76 85 80

60 65 70 90

Solution: Since it is a Maximization problem we subtract every value from the maximum value of

90. We have.

Professors P1 P2 P3 P4

C1

C2

C3

C4

20 15 10 25

10 20 25 15

15 20 5 10

30 25 20 0

Row reduced Matrix

P1 P2 P3 P4

C1

C2

C3

10 5 0 15

0 10 15 5

10 15 0 5



C4 30 25 20 0

Column Reduced Matrix

P1 P2 P3 P4

C1

C2

C3

C4

10 12 4

5 15 5

10 10 5

30 20 20

∴ Optimum Solution

C1 – P2 75

C2 – P1 80

C3 – P3 85

C4 – P4 90

330 Max Efficiency

Self Assessment Questions 3 Fill in the blanks

1. In unbalanced T.P number of rows__________ to number of column. 2. Hungarian method cannot be applied directly to _________ problem. 3. If some jobs cannot be assigned to some M/cs then it is called _________ assignment

problem.

6.5 Traveling Salesman Problem Suppose the salesman has to visit n cities .He wishes to start from a particular city, visit each city once, and then return to his starting point. The objective is to select the sequence in which the cities are visited in such a way that his total traveling time is minimized. .Clearly starting from a given city, the salesman will have total of (n1)! different sequences .Further, since the salesman has to visit all the n cities; the optimal solution remains independent of selection of the starting point.

The problem can be represented as a network where the nodes and arcs represent the cities and the distance between them respectively. Assume that in a five city problem, a round trip of the

0

0

0

0



salesman is given by the following arcs.

(3,1) ,(1,2) ,(2,4) ,(4,5) ,(5,3)

These arcs taken in order are the first, second, third, fourth and fifth directed arcs for the trip. Generally the k th directed arc represents the k th leg of the trip. i.e. on leg k

The salesman travels from city i to city j ( i , j = 1, 2,……, n; j i ≠ )

To formulate the problem whose solution will yield the minimum traveling time ,let the variables xijk be defined as

= otherwise 0,

j city to i city from is are directed k if 1, x th

ijk

Where i, j and k are integers that vary between 1 and n.

Following are the constraints of the problem. (a) Only one directed arc may be assigned to a specific k ,

Thus

∑ ∑ j k

xijk = 1 k =1, 2, 3…n

j i ≠

(b) Only one other city may be reached from a specific city i, thus

∑ ∑ j k

xijk =1, i = 1, 2…, n

(c) Only one other city can initiate a direct arc to a specified city j,thus

∑ ∑ i k

xijk =1, j =1, 2, .., n

(d) Given the k th directed arc ends at some specific city j,the (k+1) th directed arc must start at

the same city j;thus

∑ i

xijk = ∑ r

xjr ( k +1) for all j and k.

j i ≠ j r ≠

These constraints ensure that the round trip will consist of connected arcs. The objective function

is to minimize

z = ∑i

∑ ∑ j k

dij xijk j i ≠



Where dij is the distance from city i to city j.

Self Assessment Questions 4 Fill in the blanks

1. In traveling salesmen problem the objective is to visit each cities ________ __________. 2. Salesman has ________ different sequences if n is the number of cities to be visited.

6.6 Summary In this unit, we focused on a special type of transportation problem. where the objective was to

allocate n different facilities to n different tasks .Although an assignment problem can be

formulated as a linear programming problem, it is solved by special method know as Hungarian

method. If the number of persons is the same as the number of jobs, the assignment problem is

said to be balanced. Lastly a brief introduction was given about the traveling salesman problem.

Terminal Questions 1. Four jobs are to be done on four different machines. The cost in(rupees) of producing i th on

the j th machine is given below:

Machines M1 M2 M3 M4

J1 15 11 13 15

J2 17 12 12 13

J3 14 15 10 14

J4 16 13 11 17

Assign the jobs to different machines so as to minimize the total cost.

2. A marketing manager has 5 salesmen and 5 sales districts .considering the capabilities of the

salesman and the nature of districts, the marketing manager estimates that sales per month

(in hundred rupees) for each salesman in each district would be as follows.

Jobs



Sales Districts A B C D E

1 32 38 40 28 40

2 40 24 28 21 36

3 41 27 33 30 37

4 22 38 41 36 36

5 29 33 40 35 39

Find the assignment of salesman to districts that will result in maximum sales.

3. In a plant layout there are five vacant places. The plant orders four Machines to be installed

in these places. The cost of installing is as follows find the optimum assignment.

M/G A B C D E M1 9 11 15 10 11 M2 12 9 10 9 M3 11 14 11 7 M4 14 8 12 7 8

4. Find the assignment that maximizes the total sale.

Zone

Sales men 1 2 3 4

M1 42 35 28 21

M2 30 25 20 15

M3 30 25 20 15

M4 24 20 16 12

Answers For Self Assessment Questions Self Assessment Questions 1

1. True 2. True 3. False

Salesman




1. True 2. True 3. False


1. ≠

2. Maximisation problem

3. Infeasible


1. Only once 2. (n 1)!

Answer to Terminal Questions (1) The optimum assignment policy is

Job1 to machine 2, Job 2 to machine 4

Job 3 to machine1, Job 4 to machine3.

and the minimum assignment Cost = Rs. (11+13+14+11) = Rs. 49.

(2) Optimal assignment policy is salesman 1 to district B, 2 to A, 3 to E, 4 to C and 5 to D.

Hence the maximum sales = Rs. (38+40+37+41+35)×100 =

Rs. 19,100.

(3) M1 – A2; M2 – B; M3 – E; M4 – D Total 38

(4) M1 – 1; M2 – 2; M3 – 3; M4 – 4 Total 99

6 Assignment Problem

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