1. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org A SSIGNMENT Sponsored by the National Science Foundation, Directorate for Education and Human Resources, project DRL-0733137 CHAPTER 6

2. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org Chapter 6 Organization 6.1 Coach Bass’s Problem 6.2 The Medley Relay Team ▪Hungarian Algorithm ▪Excel Solver

3. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 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. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 6.2 Medley Relay Team Who should swim what leg of the relay? Swimmer 100yd Fly (sec) 100yd Back (sec) 100yd Breast (sec) 100yd Free (sec) Schmidt 59.5959.8372.8352.61 Reid 60.4559.5674.1453.31 Sanchez 61.8464.6373.6953.70 Lamartina 62.3759.1374.3654.77

5. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 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. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 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. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 6.2 Medley Relay Team – Final Matrix But what does this mean? Swimmer 100yd Fly (sec) 100yd Back (sec) 100yd Breast (sec) 100yd Free (sec) Schmidt 02.620.390.16 Reid 01.490.840 Sanchez 16.1700 Lamartina 0.86000.4

8. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 6.2 Medley Relay Team – Solution Swimmer 100yd Fly (sec) 100yd Back (sec) 100yd Breast (sec) 100yd Free (sec) Schmidt  Reid  Sanchez  Lamartina 

9. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 6.2 Medley Relay Team – Complication Swimmer Wu has joined the team ▪but the algorithm only works for square matrices Swimmer 100yd Fly (sec) 100yd Back (sec) 100yd Breast (sec) 100yd Free (sec) Schmidt 59.5959.8372.8352.61 Reid 60.4559.5674.1453.31 Sanchez 61.8464.6373.6953.70 Lamartina 62.3759.1374.3654.77 Wu 60.3364.3072.7454.05

10. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 6.2 Medley Relay Team – Complication Add a dummy task (or people if necessary) and continue with the Hungarian Algorithm Swimmer 100yd Fly (sec) 100yd Back (sec) 100yd Breast (sec) 100yd Free (sec) Dummy Schmidt 59.5959.8372.8352.610 Reid 60.4559.5674.1453.310 Sanchez 61.8464.6373.6953.700 Lamartina 62.3759.1374.3654.770 Wu 60.3364.3072.7454.050

11. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 6.2 Medley Relay Team – Even More Complication Swimmer 100yd Fly (sec) 100yd Back (sec) 100yd Breast (sec) 100yd Free (sec) Schmidt 59.5959.8372.8352.61 Reid 60.4559.5674.1453.31 Sanchez 61.8464.6373.6953.70 Lamartina 62.3759.1374.3654.77 Wu 60.3364.3072.7454.05 Greene 62.4164.3466.1956.61 Kleinfeld 62.4367.6374.0555.55 Lepinski 59.4465.0672.7452.49

12. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 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. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 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. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 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. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 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. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 6.2 Medley Relay Team – Yucky Excel Spreadsheet

17. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 6.2 Medley Relay Team – A Better Excel Spreadsheet Every swimmer swims at most one leg. Every leg has exactly one swimmer.

18. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 6.2 Medley Relay Team – Excel Spreadsheet = SUMPRODUCT(C7:F14,C19:F26) = SUM(C19:F19) = SUMC26:F26) = SUM(C19:C26) … = SUM(F19:F26) Every swimmer swims at most one leg. Every leg has exactly one swimmer.

19. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 6.2 Medley Relay Team – Solver

20. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 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. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org Practice – 6.2.3 Solve for the minimum total relay time using the Hungarian Algorithm Swimmer 100yd Fly (sec) 100yd Back (sec) 100yd Breast (sec) 100yd Free (sec) Schmidt 59.5959.8372.8352.61 Reid 60.4559.5674.1453.31 Sanchez 61.8464.6373.6953.70 Lamartina 62.3759.1374.3654.77 Wu 60.3364.3072.7454.05

22. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 6.2.3 Solution Swimmer 100yd Fly (sec) 100yd Back (sec) 100yd Breast (sec) 100yd Free (sec) Dummy Schmidt  Reid  Sanchez  Lamartina  Wu 

23. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org Practice – 6.HW.1 6.HW.1 is on p. 15 Use the Excel template from 6.2 to solve

24. MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) www.MINDSETproject.org 5.HW.1 Solution