Lecture 16: Network Models/
Scheduling Assignments
AGEC 352
Spring 2011 – March 28
R. Keeney
Olympic Swimming
Michael Phelps is the world’s greatest
swimmer
 If you need to win a race, you pick him
 If you need to win a relay race, you pick
him but where do you use him?
 Medley swimming

◦ Backstroke, Breaststroke, Butterfly, Freestyle
Medley Swimming (U.S. 2008)
Swimmer/
Stroke
Peirsol
Hansen
Phelps
Lezak
Back Breast B-fly Free
53.16
-53.11
--
-59.27
58.15
--
57.75
-54.25
-50.15 45.95
55.45 46.76
Sales Reps / Districts
Rep/Dist
A
B
C
1
100
90
85
2
80
75
75
3
40
20
10
If these people are paid on commission is
Seller A going to be happy about being the
highest rated rep?
Assignment Problems

Setup is identical to the transportation
problem we have considered
◦ Sources are people or things to be assigned
◦ Destinations are jobs or roles to be filled

Other applications
◦ Machines to tasks
 E.g. Airplanes to routes
◦ Sudoku (number to a cell)
Example

Umpiring in American League Baseball
◦ 14 teams
◦ 7 umpiring crews assigned to 7 games
◦ Minimum travel costs for crews going to
games, other constraints
◦ No afternoon games in city B if you worked a
night game in city A the previous day
◦ Day off required if leaving Pacific Time Zone
or Canada
◦ Crew must not work more than one week
straight on the same team’s games
Case

Mathematical allocation of ‘n’ objects or
agents to ‘n’ tasks
◦ Agents/objects are indivisible, one task only

Autopower Company audit of assembly
plants (destinations from transport)
◦ Leipzig, Nancy, Liege, Tilburg

VP’s to manage audit
◦ Finance, Marketing, Operations, Personnel
Considerations on Costs
Expertise relative to problem areas of
different plants
 Time demand of VP
 Language ability of VP

VP
Leipzig Nancy Liege Tilburg
Finance
24
10
21
11
Marketing
14
22
10
15
Operations
15
17
20
19
Personnel
11
19
14
13
Estimating the costs

Need something reliable for estimating
the opportunity cost of each VP in each
assignment
◦ E.g. A Dutch speaker in the French plant may
require a translator with him full-time
◦ E.g. The finance VP may need an human
resources specialist to assist her

Other measures:
◦ Swimming times, skill tests (ASVAB)
Solving
Simplex LP in Excel or by hand
 For small problems, enumeration
 An ‘n’ sized assignment problem has n!
possible solutions

◦ n! is called a factorial, multiply all the integers
up to n together to find the factorial
◦ E.g. 4! = 1*2*3*4 = 24
Setup

RHS values are always 1
◦ Sources (people)
 The total jobs must be <= 1
◦ Destinations (jobs)
 The number in the job must be >= 1

Balanced: Need someone for each job,
everyone needs a job
Algebraic Form
Let X i , j be the amount of i assigned to j
min
 C
i
i, j
X i, j
j
s.t.
Assignees :
X
i, j
1  i
j
Assignment s :  X i , j  1  j
i
Non - neg. : X i , j  0  i and j
Notes

Decision variables will be zero or one.
◦ Integers (but you don’t need integer
constraints)

Transportation problem with supply at
each source and demand at each
destination equal to one.