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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 J1J1 J2J2 J3J3 J4J4 M1M M2M M3M M4M M5M M6M 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.

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J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 M1M M2M M3M M4M M5M M6M Balanced Problem

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J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 Min. Row M1M M2M M3M M4M M5M M6M Step 1. Identify minimum of each row

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Subtract identified no from each and every entry of corresponding row J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 Min. Row M1M M2M M3M M4M M5M M6M

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Step 2. identify minimum of column J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 M1M M2M M3M M4M M5M M6M Min. column

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Subtract identified no from each and every entry of corresponding column J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 M1M M2M M3M M4M M5M M6M Min. column

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J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 M1M M2M M3M M4M M5M M6M

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J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 M1M1 M2M2 M3M3 M4M4 M5M5 M6M6

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J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 M1M1 M2M2 M3M3 M4M4 M5M5 M6M6

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J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 M1M1 M2M2 M3M3 M4M4 M5M5 M6M6

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Original Tableau J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 M1M1 243*3* 5 00 M2M2 38*8* M3M * 00 M4M *0*0 M5M5 2*2* M6M *0*

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Optimal Distribution M J3=3 M J2=8 M J4=10 M J5=0 M J1=2 M J6=0 TOTAL =23

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Illustration Jobs exceeds Machines

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Problem J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 M1M M2M M3M M4M 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.

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Balanced Problem J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 M1M M2M M3M M4M M5M M6M

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Step 1. Identify minimum of each row J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 Min. Row M1M M2M M3M M4M M5M M6M

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Subtract identified no from each and every entry of corresponding row J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 Min. Row M1M M2M M3M M4M M5M M6M

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Step 2. identify minimum of column J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 M1M M2M M3M M4M M5M M6M Min. Column

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J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 M1M M2M M3M M4M M5M M6M

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J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 M1M1 M2M2 M3M3 M4M4 M5M5 M6M6

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J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 M1M1 M2M2 M3M3 M4M4 M5M5 M6M6

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Original Tableau J1J1 J2J2 J3J3 J4J4 J5J5 J6J6 M1M *2*3 M2M2 38*8* M3M * 1816 M4M4 3*3* M5M5 000*0*000 M6M *0*

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Optimal Distribution M J5=2 M J2=8 M J4=10 M J1=3 M J3=0 M J6=0 TOTAL =23

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