Unbalanced Assignment Model

Lecture 24

By

Dr. Arshad Zaheer

RECAP

Assignment Model (Maximization) Hungarian Method Steps InvolvedIllustrations Optimal Assignment

Unbalanced Assignment Problem

Case 1This is the case when the total number of machines exceeds total number of jobs. In this case, introduce required number of fictitious or dummy jobs at ‘0’ cost or at the cost stated in the problem to get the balanced assignment problem. Then use the assignment technique to obtain the optimal assignment. The fictitious or dummy jobs assigned to the machines mean that the corresponding machine will not be assigned any job.

Unbalanced Assignment Problem

Case 2This is the case when the total number of jobs exceeds total number of machines. In this case, introduce required number of fictitious or dummy machines at ‘0’ cost or at the cost stated in the problem to get the balanced assignment problem. Then use the assignment technique to obtain the optimal assignment. The jobs which are assigned fictitious or dummy machines will be left over.

IllustrationMachines exceeds Jobs

ProblemJ1 J2 J3 J4

M1 2 4 3 5

M2 3 8 9 12

M3 4 12 11 10

M4 3 18 15 12

M5 2 22 20 18

M6 3 25 15 20

Requirement: Which job is to assign which machine to get the minimum cost

In this given problem, there are only four jobs while six machines. In an ideal condition there should be equal no of jobs so we need to make them equal. For this purpose we will introduce two fictitious jobs at zero cost.

J1 J2 J3 J4 J5 J6

M1 2 4 3 5 0 0

M2 3 8 9 12 0 0

M3 4 12 11 10 0 0

M4 3 18 15 12 0 0

M5 2 22 20 18 0 0

M6 3 25 15 20 0 0

Balanced Problem

J1 J2 J3 J4 J5 J6 Min. Row

M1 2 4 3 5 0 0 0

M2 3 8 9 12 0 0 0

M3 4 12 11 10 0 0 0

M4 3 18 15 12 0 0 0

M5 2 22 20 18 0 0 0

M6 3 25 15 20 0 0 0

Step 1. Identify minimum of each row

Subtract identified no from each and every entry of corresponding row

J1 J2 J3 J4 J5 J6 Min. Row

M1 2 4 3 5 0 0 0

M2 3 8 9 12 0 0 0

M3 4 12 11 10 0 0 0

M4 3 18 15 12 0 0 0

M5 2 22 20 18 0 0 0

M6 3 25 15 20 0 0 0

Step 2. identify minimum of columnJ1 J2 J3 J4 J5 J6

M1 2 4 3 5 0 0

M2 3 8 9 12 0 0

M3 4 12 11 10 0 0

M4 3 18 15 12 0 0

M5 2 22 20 18 0 0

M6 3 25 15 20 0 0

Min. column

2 4 3 5 0 0

Subtract identified no from each and every entry of corresponding column

J1 J2 J3 J4 J5 J6

M1 0 0 0 0 0 0

M2 1 4 6 7 0 0

M3 2 8 8 5 0 0

M4 1 14 12 7 0 0

M5 0 18 17 13 0 0

M6 1 21 12 15 0 0

Min. column

2 4 3 5 0 0

J1 J2 J3 J4 J5 J6

M1 0 0 0 0 0 0

M2 1 4 6 7 0 0

M3 2 8 8 5 0 0

M4 1 14 12 7 0 0

M5 0 18 17 13 0 0

M6 1 21 12 15 0 0

J1 J2 J3 J4 J5 J6

M1

M2

M3

M4

M5

M6

J1 J2 J3 J4 J5 J6

M1

M2

M3

M4

M5

M6

J1 J2 J3 J4 J5 J6

M1

M2

M3

M4

M5

M6

Original TableauJ1 J2 J3 J4 J5 J6

M1 2 4 3* 5 0 0

M2 3 8* 9 12 0 0

M3 4 12 11 10* 0 0

M4 3 18 15 12 0* 0

M5 2* 22 20 18 0 0

M6 3 25 15 20 0 0*

Optimal Distribution

• M1 ------- J3=3• M2 ------- J2=8• M3 ------- J4=10• M4 ------- J5=0• M5 ------- J1=2• M6 ------- J6=0TOTAL =23

Illustration Jobs exceeds Machines

ProblemJ1 J2 J3 J4 J5 J6

M1 2 4 3 5 2 3

M2 3 8 9 12 15 12

M3 4 12 11 10 18 16

M4 3 18 15 12 20 15

Requirement:Which job is to assign which machine to get the minimum cost

In this given problem, there are only four machines but six jobs. In an ideal condition there should be equal no of jobs so we need to make them equal. For this purpose we will introduce two fictitious Machines at zero cost.

Balanced Problem

J1 J2 J3 J4 J5 J6

M1 2 4 3 5 2 3

M2 3 8 9 12 15 12

M3 4 12 11 10 18 16

M4 3 18 15 12 20 15

M5 0 0 0 0 0 0

M6 0 0 0 0 0 0

Step 1. Identify minimum of each row

J1 J2 J3 J4 J5 J6 Min.Row

M1 2 4 3 5 2 3 2

M2 3 8 9 12 15 12 3

M3 4 12 11 10 18 16 4

M4 3 18 15 12 20 15 3

M5 0 0 0 0 0 0 0

M6 0 0 0 0 0 0 0

Subtract identified no from each and every entry of corresponding row

J1 J2 J3 J4 J5 J6 Min.Row

M1 0 2 1 3 0 1 2

M2 0 5 6 9 12 9 3

M3 0 8 7 6 14 12 4

M4 0 15 12 9 17 12 3

M5 0 0 0 0 0 0 0

M6 0 0 0 0 0 0 0

Step 2. identify minimum of columnJ1 J2 J3 J4 J5 J6

M1 0 2 1 3 0 1

M2 0 5 6 9 12 9

M3 0 8 7 6 14 12

M4 0 15 12 9 17 12

M5 0 0 0 0 0 0

M6 0 0 0 0 0 0

Min.Column

0 0 0 0 0 0

J1 J2 J3 J4 J5 J6

M1 0 2 1 3 0 1

M2 0 5 6 9 12 9

M3 0 8 7 6 14 12

M4 0 15 12 9 17 12

M5 0 0 0 0 0 0

M6 0 0 0 0 0 0

J1 J2 J3 J4 J5 J6

M1

M2

M3

M4

M5

M6

J1 J2 J3 J4 J5 J6

M1

M2

M3

M4

M5

M6

Original TableauJ1 J2 J3 J4 J5 J6

M1 2 4 3 5 2* 3

M2 3 8* 9 12 15 12

M3 4 12 11 10* 18 16

M4 3* 18 15 12 20 15

M5 0 0 0* 0 0 0

M6 0 0 0 0 0 0*

Optimal Distribution

• M1 ------- J5=2• M2 ------- J2=8• M3 ------- J4=10• M4 ------- J1=3• M5 ------- J3=0• M6 ------- J6=0TOTAL =23

Thank You