Assignment model

Assignment model

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  TOPIC: ASSIGNMENT MODEL OPERATIONRESEARCH
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  ASSIGNMENT MODEL
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  ASSIGNMENT MODEL • Assigningof jobs to factors (men or machine) to get most optimum output or get least cost. • Hungarian method is the mostly used method of solving assignment problems. • Types of Assignment problems: (i) Balanced (ii) Unbalanced
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  OBJECTIVES (i) No workersis more than one job. (ii) No job is allotted to more than one worker. (iii)Total time taken to complete a job is minimum. (iv)The work done is cost effective.
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  BALANCED ASSIGNMENT • Anassignment is called balanced assignment problem if the number of persons (factors) is same as the number of jobs. J1 J2 J3 _ _ _ _ _ _ _ _ _ _ _ _ Jn P1 T11 T12 T13 _ _ _ _ _ _ _ _ _ _ _ _ T1n P2 T21 T22 T23 _ _ _ _ _ _ _ _ _ _ _ _ T2n P3 T31 T32 T33 _ _ _ _ _ _ _ _ _ _ _ _ T3n I I I I _ _ _ _ _ _ _ _ _ _ _ _ I I I I I _ _ _ _ _ _ _ _ _ _ _ _ I Pn Tn1 Tn2 Tn3 _ _ _ _ _ _ _ _ _ _ _ _ Tnn
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  EXAMPLE OF BALANCEDPROBLEM 1 2 3 4 A 1 8 4 1 B 5 7 6 5 C 3 5 4 12 D 3 1 6 3 PROFITS (IN 1000’s) PLANTS Solve the following assignment problem to get maximum profit:
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  METHOD STEP 1 • Subtractall the element of each row from the largest elements of respective rows. STEP 2 • Subtract the least element of each column from other elements of respective column and allot jobs. 1 2 3 4 A 7 0 4 7 B 2 0 1 2 C 9 7 8 0 D 3 5 0 3 PROFIT S PLANTS PROFITS PLANTS 1 2 3 4 A 5 0 4 7 B 0 0 1 2 C 7 7 8 0 D 1 5 0 3 0
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  METHOD • Hence theoptimum solution is: (i) A2 (ii) B1 (iii) C4 (iv) D3 Hence the total profit will be 8000+ 5000+ 12000 +6000 = 31000 8000 8000 12000 6000
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  UNBALANCED ASSIGNMENT • Anassignment is called unbalanced assignment problem if the number of persons (factors) is not same as the number of jobs. J1 J2 J3 _ _ _ _ _ _ _ _ _ _ _ _ Jn-1 P1 T11 T12 T13 _ _ _ _ _ _ _ _ _ _ _ _ T1(n-1) P2 T21 T22 T23 _ _ _ _ _ _ _ _ _ _ _ _ T2(n-1) P3 T31 T32 T33 _ _ _ _ _ _ _ _ _ _ _ _ T3(n-1) I I I I _ _ _ _ _ _ _ _ _ _ _ _ I I I I I _ _ _ _ _ _ _ _ _ _ _ _ I Pn Tn1 Tn2 Tn3 _ _ _ _ _ _ _ _ _ _ _ _ Tn(n-1)
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  EXAMPLE OF UNBALANCEDPROBLEM Solve the following assignment problem to get it completed in least time: A B C D E 1 62 78 50 101 82 2 71 84 61 73 59 3 87 92 111 71 81 4 48 64 87 77 80 JOBS MACHINES 5 0 0 0 0 0
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  METHOD Subtract all theelement of each row from the largest elements of respective rows. •STEP 1 A B C D E 1 39 23 51 0 19 2 13 0 23 11 25 3 24 19 0 40 30 4 39 23 0 10 7 5 0 0 0 0 0 JOBS MACHINES
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  METHOD Subtract the leastelement of each column from other elements of respective column and allot jobs. •STEP 2 V A B C D E 1 39 23 51 0 19 2 13 0 23 11 25 3 24 19 0 40 30 4 39 23 0 10 7 5 0 0 0 0 0 JOBS MACHINES 0 0 0 0 0
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  METHOD Cancel all zerosby drawing lines less than the number of rows or columns. •STEP 3 A B C D E 1 39 23 51 0 19 2 13 0 23 11 25 3 24 19 0 40 30 4 39 23 0 10 7 5 0 0 0 0 0 JOBS MACHINES
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  METHOD Choose the leastuncovered element and add it to all intersection points and subtract it from all uncovered elements. •STEP 4 A B C D E 1 32 23 51 0 12 2 6 0 23 11 18 3 17 19 0 40 23 4 32 23 0 10 0 5 0 7 7 7 0 JOBS MACHINES 0 0
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  METHOD • Hence theoptimum solution is: (i) A5 0 (ii) B2 84 (iii) C3 111 (iv) D1 101 (v) E4 80 Hence the total profit will be 0+ 84+ 111 +101+ 80 = 376
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