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MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
CHAPTER 6
ASSIGNMENT
Sponsored by the National Science Foundation, Directorate for Education and Human Resources, project DRL-0733137
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MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
Chapter 6 Organization
• 6.1 Coach Bass’s Problem
• 6.2 The Medley Relay Team
▪ Hungarian Algorithm
▪ Excel Solver
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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
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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
Reid
Sanchez
Lamartina
59.59
60.45
61.84
62.37
59.83
59.56
64.63
59.13
72.83
74.14
73.69
74.36
52.61
53.31
53.70
54.77
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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
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MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
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
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6.2 Medley Relay Team – Final Matrix
Swimmer
Schmidt
Reid
Sanchez
Lamartina
100yd Fly
(sec)
100yd Back
(sec)
100yd Breast
(sec)
100yd Free
(sec)
0
0
2.62
1.49
0.39
0.84
0.16
0
1
0.86
6.17
0
0
0
0
0.4
• But what does this mean?
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6.2 Medley Relay Team – Solution
Swimmer
Schmidt
Reid
Sanchez
Lamartina
100yd Fly
(sec)
100yd Back
(sec)
100yd Breast
(sec)
100yd Free
(sec)




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6.2 Medley Relay Team – Complication
• Swimmer Wu has joined the team
▪ but the algorithm only works for square
matrices
100yd Fly
(sec)
100yd Back
(sec)
100yd Breast
(sec)
100yd Free
(sec)
Schmidt
Reid
Sanchez
59.59
60.45
61.84
59.83
59.56
64.63
72.83
74.14
73.69
52.61
53.31
53.70
Lamartina
Wu
62.37
60.33
59.13
64.30
74.36
72.74
54.77
54.05
Swimmer
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6.2 Medley Relay Team – Complication
• Add a dummy task (or people if necessary) and
continue with the Hungarian Algorithm
Swimmer
100yd Fly 100yd Back 100yd Breast
(sec)
(sec)
(sec)
100yd Free
(sec)
Dummy
Schmidt
Reid
Sanchez
59.59
60.45
61.84
59.83
59.56
64.63
72.83
74.14
73.69
52.61
53.31
53.70
0
0
0
Lamartina
Wu
62.37
60.33
59.13
64.30
74.36
72.74
54.77
54.05
0
0
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6.2 Medley Relay Team – Even More
Complication
100yd Fly
(sec)
100yd Back
(sec)
100yd Breast
(sec)
100yd Free
(sec)
Schmidt
Reid
59.59
60.45
59.83
59.56
72.83
74.14
52.61
53.31
Sanchez
Lamartina
Wu
Greene
Kleinfeld
Lepinski
61.84
62.37
60.33
62.41
62.43
59.44
64.63
59.13
64.30
64.34
67.63
65.06
73.69
74.36
72.74
66.19
74.05
72.74
53.70
54.77
54.05
56.61
55.55
52.49
Swimmer
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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!
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MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
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
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MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
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
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MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
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?
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6.2 Medley Relay Team – Yucky Excel
Spreadsheet
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6.2 Medley Relay Team – A Better Excel
Spreadsheet
Every swimmer
swims at most
one leg.
Every leg has exactly one swimmer.
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MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
6.2 Medley Relay Team – Excel
Spreadsheet
= SUMPRODUCT(C7:F14,C19:F26)
Every swimmer
swims at most
one leg.
= SUM(C19:F19)
Every leg has exactly one swimmer.
= SUM(C19:C26) … = SUM(F19:F26)
= SUMC26:F26)
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6.2 Medley Relay Team – Solver
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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
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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
Reid
Sanchez
Lamartina
59.59
60.45
61.84
62.37
59.83
59.56
64.63
59.13
72.83
74.14
73.69
74.36
52.61
53.31
53.70
54.77
Wu
60.33
64.30
72.74
54.05
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6.2.3 Solution
Swimmer
Schmidt
Reid
Sanchez
Lamartina
Wu
100yd Fly 100yd Back 100yd Breast
(sec)
(sec)
(sec)
100yd Free
(sec)
Dummy





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MINDSET (Mathematics INstruction using Decision Science and Engineering Tools)
Practice – 6.HW.1
• 6.HW.1 is on p. 15
• Use the Excel template from 6.2 to solve
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5.HW.1 Solution