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**Unbalanced Assignment Model**

Lecture 24 By Dr. Arshad Zaheer

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**RECAP Assignment Model (Maximization) Hungarian Method Steps Involved**

Illustrations Optimal Assignment

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**Unbalanced Assignment Problem**

Case 1 This 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.

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**Unbalanced Assignment Problem**

Case 2 This 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.

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**Illustration Machines exceeds Jobs**

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Problem J1 J2 J3 J4 M1 2 4 3 5 M2 8 9 12 M3 11 10 M4 18 15 M5 22 20 M6 25 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. Requirement: Which job is to assign which machine to get the minimum cost

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**Balanced Problem J1 J2 J3 J4 J5 J6 M1 2 4 3 5 M2 8 9 12 M3 11 10 M4 18**

M2 8 9 12 M3 11 10 M4 18 15 M5 22 20 M6 25

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**Step 1. Identify minimum of each row**

J1 J2 J3 J4 J5 J6 Min. Row M1 2 4 3 5 M2 8 9 12 M3 11 10 M4 18 15 M5 22 20 M6 25

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**Subtract identified no from each and every entry of corresponding row**

J1 J2 J3 J4 J5 J6 Min. Row M1 2 4 3 5 M2 8 9 12 M3 11 10 M4 18 15 M5 22 20 M6 25

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**Step 2. identify minimum of column**

J1 J2 J3 J4 J5 J6 M1 2 4 3 5 M2 8 9 12 M3 11 10 M4 18 15 M5 22 20 M6 25 Min. column

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**Subtract identified no from each and every entry of corresponding column**

J1 J2 J3 J4 J5 J6 M1 M2 1 4 6 7 M3 2 8 5 M4 14 12 M5 18 17 13 M6 21 15 Min. column 3

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J1 J2 J3 J4 J5 J6 M1 M2 1 4 6 7 M3 2 8 5 M4 14 12 M5 18 17 13 M6 21 15

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J1 J2 J3 J4 J5 J6 M1 M2 M3 M4 M5 M6

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J1 J2 J3 J4 J5 J6 M1 M2 M3 M4 M5 M6

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J1 J2 J3 J4 J5 J6 M1 M2 M3 M4 M5 M6

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**Original Tableau J1 J2 J3 J4 J5 J6 M1 2 4 3* 5 M2 3 8* 9 12 M3 11 10***

M2 3 8* 9 12 M3 11 10* M4 18 15 0* M5 2* 22 20 M6 25

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**Optimal Distribution M1 ------- J3=3 M2 ------- J2=8**

TOTAL =23

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**Illustration Jobs exceeds Machines**

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Problem J1 J2 J3 J4 J5 J6 M1 2 4 3 5 M2 8 9 12 15 M3 11 10 18 16 M4 20 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. Requirement: Which job is to assign which machine to get the minimum cost

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**Balanced Problem J1 J2 J3 J4 J5 J6 M1 2 4 3 5 M2 8 9 12 15 M3 11 10 18**

16 M4 20 M5 M6

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**Step 1. Identify minimum of each row**

J1 J2 J3 J4 J5 J6 Min. Row M1 2 4 3 5 M2 8 9 12 15 M3 11 10 18 16 M4 20 M5 M6

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**Subtract identified no from each and every entry of corresponding row**

J1 J2 J3 J4 J5 J6 Min. Row M1 2 1 3 M2 5 6 9 12 M3 8 7 14 4 M4 15 17 M5 M6

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**Step 2. identify minimum of column**

J1 J2 J3 J4 J5 J6 M1 2 1 3 M2 5 6 9 12 M3 8 7 14 M4 15 17 M5 M6 Min. Column

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J1 J2 J3 J4 J5 J6 M1 2 1 3 M2 5 6 9 12 M3 8 7 14 M4 15 17 M5 M6

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J1 J2 J3 J4 J5 J6 M1 M2 M3 M4 M5 M6

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J1 J2 J3 J4 J5 J6 M1 M2 M3 M4 M5 M6

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**Original Tableau J1 J2 J3 J4 J5 J6 M1 2 4 3 5 2* M2 8* 9 12 15 M3 11**

10* 18 16 M4 3* 20 M5 0* M6

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**Optimal Distribution M1 ------- J5=2 M2 ------- J2=8**

TOTAL =23

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Linear Programming: An Algebraic Approach 4 The Simplex Method with Standard Maximization.

Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Linear Programming: An Algebraic Approach 4 The Simplex Method with Standard Maximization.

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