MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) A SSIGNMENT...

MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)

www.MINDSETproject.org

ASSIGNMENT

Sponsored by the National Science Foundation, Directorate for Education and Human Resources, project DRL-0733137

CHAPTER 6



Chapter 6 Organization

• 6.1 Coach Bass’s Problem• 6.2 The Medley Relay Team

▪ Hungarian Algorithm▪ Excel Solver



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



6.2 Medley Relay Team

• Who should swim what leg of the relay?

Swimmer100yd Fly

(sec)100yd Back

(sec)100yd Breast

(sec)100yd Free

(sec)

Schmidt 59.59 59.83 72.83 52.61

Reid 60.45 59.56 74.14 53.31

Sanchez 61.84 64.63 73.69 53.70

Lamartina 62.37 59.13 74.36 54.77



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



6.2 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



6.2 Medley Relay Team – Final Matrix

• But what does this mean?

Swimmer100yd Fly

(sec)100yd Back

(sec)100yd Breast

(sec)100yd Free

(sec)

Schmidt 0 2.62 0.39 0.16

Reid 0 1.49 0.84 0

Sanchez 1 6.17 0 0

Lamartina 0.86 0 0 0.4



6.2 Medley Relay Team – Solution

Swimmer100yd Fly

(sec)100yd Back

(sec)100yd Breast

(sec)100yd Free

(sec)

Schmidt Reid Sanchez Lamartina



6.2 Medley Relay Team – Complication

• Swimmer Wu has joined the team ▪ but the algorithm only works for square

matrices

Swimmer100yd Fly

(sec)100yd Back

(sec)100yd Breast

(sec)100yd Free

(sec)

Schmidt 59.59 59.83 72.83 52.61

Reid 60.45 59.56 74.14 53.31

Sanchez 61.84 64.63 73.69 53.70

Lamartina 62.37 59.13 74.36 54.77

Wu 60.33 64.30 72.74 54.05



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 59.59 59.83 72.83 52.61 0

Reid 60.45 59.56 74.14 53.31 0

Sanchez 61.84 64.63 73.69 53.70 0

Lamartina 62.37 59.13 74.36 54.77 0

Wu 60.33 64.30 72.74 54.05 0



6.2 Medley Relay Team – Even More Complication


100yd Back(sec)

100yd Breast(sec)

100yd Free(sec)

Schmidt 59.59 59.83 72.83 52.61

Reid 60.45 59.56 74.14 53.31

Sanchez 61.84 64.63 73.69 53.70

Lamartina 62.37 59.13 74.36 54.77

Wu 60.33 64.30 72.74 54.05

Greene 62.41 64.34 66.19 56.61

Kleinfeld 62.43 67.63 74.05 55.55

Lepinski 59.44 65.06 72.74 52.49



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!



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



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



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?



6.2 Medley Relay Team – Yucky Excel Spreadsheet



6.2 Medley Relay Team – A Better Excel Spreadsheet

Every swimmer swims at most one leg.

Every leg has exactly one swimmer.



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.



6.2 Medley Relay Team – Solver



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



Practice – 6.2.3

• Solve for the minimum total relay time using the Hungarian Algorithm

Swimmer100yd Fly

(sec)100yd Back

(sec)100yd Breast

(sec)100yd Free

(sec)

Schmidt 59.59 59.83 72.83 52.61

Reid 60.45 59.56 74.14 53.31

Sanchez 61.84 64.63 73.69 53.70

Lamartina 62.37 59.13 74.36 54.77

Wu 60.33 64.30 72.74 54.05



6.2.3 Solution


100yd Back(sec)

100yd Breast(sec)

100yd Free(sec) Dummy

Schmidt Reid Sanchez Lamartina Wu



Practice – 6.HW.1

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



5.HW.1 Solution

MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) A SSIGNMENT...

Documents

Transcript of MINDSET ( M athematics IN struction using D ecision S cience and E ngineering T ools ) A SSIGNMENT...