Assignment

1. Assignment Chapter 6 Sponsored by the National Science Foundation, Directorate for Education and Human Resources, project DRL-0733137

2. Chapter 6 Organization • 6.1 Coach Bass’s Problem • 6.2 The Medley Relay Team • Hungarian Algorithm • Excel Solver

3. 6.1 Coach Bass’s Problem • Coach Bass is a swim coach who must assign four swimmers to compete in the 400 yard medley • 100 yard Butterfly • 100 yard Backstroke • 100 yard Breaststroke • 100 yard Freestyle

4. 6.2 Medley Relay Team • Who should swim what leg of the relay?

5. 6.2 Medley Relay Team – Hungarian Algorithm • The Hungarian Algorithm will help us easily determine which swimmer to swim which leg of the relay • The algorithm is designed specifically to solve assignment problems

6. 6.2 Medley Relay Team – Hungarian Algorithm • Using the flow chart on p. 3 (Figure 5.2.1), follow the algorithm to determine the swimmers assignments • n denotes the size of the n x n square matrix • The algorithm only works for square matrices

7. 6.2 Medley Relay Team – Final Matrix • But what does this mean?

8. 6.2 Medley Relay Team – Solution

9. 6.2 Medley Relay Team – Complication • Swimmer Wu has joined the team • but the algorithm only works for square matrices

10. 6.2 Medley Relay Team – Complication • Add a dummy task (or people if necessary) and continue with the Hungarian Algorithm

11. 6.2 Medley Relay Team – Even More Complication

12. 6.2 Medley Relay Team – Even More Complication • The new matrix would require performing the Hungarian Algorithm on a 8x8 matrix • Is there a better way? • Excel Solver!

13. 6.2 Medley Relay Team – LP Formulation • Write out • Decision Variables • Objective Function • Constraints • Create Excel Spreadsheet • Input decision objective function, constraints, RHS constraint values • Use Solver to find the solution • Output final values, objective function value, LHS constraint values

14. 6.2 Medley Relay Team – Decision Variables and Objective Function • What values are changing in this assignment problem? • Who swims which leg • How are we going to determine this assignment? • Using binary variables • How many decision variables are there? • 32 • What is Coach Bass trying to accomplish when assigning swimmers a leg of the relay? • Coach Bass is trying to minimize the total time of the 400 yard Medley Relay

15. 6.2 Medley Relay Team – Constraints • Know the decision variables are binary • takes care of non-negativity • How many swimmers can swim each stroke in the race? • Will all the swimmers swim?

16. 6.2 Medley Relay Team – Yucky Excel Spreadsheet

17. 6.2 Medley Relay Team – A Better Excel Spreadsheet Every swimmer swims at most one leg. Every leg has exactly one swimmer.

18. 6.2 Medley Relay Team – Excel Spreadsheet = SUMPRODUCT(C7:F14,C19:F26) Every swimmer swims at most one leg. = SUM(C19:F19) = SUMC26:F26) Every leg has exactly one swimmer. = SUM(C19:C26) … = SUM(F19:F26)

19. 6.2 Medley Relay Team – Solver

20. In Summary • The Hungarian Algorithm is a useful tool for assigning tasks in the optimal scenario • However, the Hungarian Algorithm can blow up quickly – but Excel Solver is there for the rescue • Can solve assignment problems by setting up a binary linear programming formulation

21. Practice – 6.2.3 • Solve for the minimum total relay time using the Hungarian Algorithm

22. 6.2.3 Solution

23. Practice – 6.HW.1 • 6.HW.1 is on p. 15 • Use the Excel template from 6.2 to solve

24. 5.HW.1 Solution