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> Operational Research >> Integer Programming >>> Branch and Bound method

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> Operational Research >> Integer Programming >>> Branch and Bound method. John.woodward@nottingham.edu.cn. Person-Task Problem. 4 people {A,B,C,D} and 4 tasks {1,2,3,4} The table below shows the number of minutes for each person to complete each task. Each person does one task.

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  1. > Operational Research>> Integer Programming >>> Branch and Bound method John.woodward@nottingham.edu.cn

  2. Person-Task Problem • 4 people {A,B,C,D} and 4 tasks {1,2,3,4} • The table below shows the number of minutes for each person to complete each task. • Each person does one task. • Each task needs an assigned a person. • Minimize total time taken. • How do we assign people to tasks? • E.g. ACDB =9+1+2+6 = 18 minutes in total. John Woodward Branch and Bound

  3. Example of Bounding Function • Example, calculate best value given partial assignment A??? • (A does task 1, other tasks are not yet assigned)? • The cost of assigning person A to task 1 is 9 minutes • Best unassigned person for • 2 is C (1), 3 is D (2), 4 is C (2) in (minutes) • Note that C is assigned twice! • Total time = (9+1+2+2) = 14 minutes. • We want to minimize so the bounding function is an underestimate. John Woodward Branch and Bound

  4. Branch and Bound Stage 1 Task 0 Task 1 Task 2 Task 3 Incumbent (best complete) solution None yet. PRUNED FEASIBLE PRUNED & FESIBLE John Woodward Branch and Bound

  5. Branch and Bound Stage 2 Task 0 Task 1 Task 2 Task 3 ACDC=14 Incumbent (best complete) solution CBDA = 13 Pruned as > 13 DCCC=8 BCDC=9 CBDA=13 promising Feasible so is1st incumbent. There is no point growing this node any more. PRUNED promising PRUNED & FESIBLE FEASIBLE John Woodward Branch and Bound

  6. Branch and Bound Stage 3 Task 0 Task 1 Task 2 Task 3 ACDC=14 Incumbent (best complete) solution CBDA = 13 DCCC=8 CBDA=13 BCDC=9 DACC=12 No new feasible solutions, therefore no new incumbent PRUNED DBCC=10 FEASIBLE PRUNED & FESIBLE DCAA=12 John Woodward Branch and Bound

  7. Branch and Bound Stage 4 Task 0 Task 1 Task 2 Task 3 ACDC=14 BADC=13 Incumbent (best complete) solution BCDA = 12 DCCC=8 DBCC=10 BCDA=12 BCDC=9 Feasible but Pruned as > 12 BDCC=13 CBDA=13 DACC=12 Pruned as > 12 PRUNED promising FEASIBLE PRUNED & FESIBLE DCAA=12 Pruned as > 12 John Woodward Branch and Bound

  8. Branch and Bound Stage 5 Task 0 Task 1 Task 2 Task 3 ACDC=14 BADC=13 Incumbent (best complete) solution DBAC = 11 CBDA=13 DBCC=10 DCCC=8 BCDC=9 BCDA=12 Pruned BDCC=13 DACC=12 DBAC=11 PRUNED FEASIBLE PRUNED & FESIBLE DBCA=13 DCAA=12 John Woodward Branch and Bound

  9. The End • Thank you for you attention • Any questions. John Woodward Branch and Bound