Lecture 23:
The Timetable Constrained
Distance Minimization Problem
© J. Christopher Beck 2008
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Outline
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The timetable constrained distance
minimization problem (TCDMP)
The problem
IP Model
CP Model
CP/IP Hybrid
Results
© J. Christopher Beck 2008
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Readings
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Rasmussen, R. & Trick, M.,
The timetable constrained distance
minimization problem, Annals of
Operations Research, in press.
© J. Christopher Beck 2008
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TCDMP
Pattern set
(HAPS!)
Timetable
© J. Christopher Beck 2008
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TCDMP
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Given:
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a timetable for a double round robin
distance matrix between each city
upper bound on the # of consecutive
home/away games for a team
Find:
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a feasible pattern set which minimizes the
total distance travelled by all teams
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assign all home/away games
© J. Christopher Beck 2008
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Notation
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T: set of teams, |T| = n
S: set of slots, S0 = S U {0}
D: distance matrix
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entry Di1i2 is the distance between home of
team i1 and home of team i2
TT: timetable
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entry TTis is the opponent of team i in slot
s
© J. Christopher Beck 2008
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IP Formulation
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his = 1 if team i plays at home in slot s
= 0 if it plays away
dis = distance team i travels between
slot s and s + 1
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slot 0 and 2n-1 are “home”
© J. Christopher Beck 2008
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IP Formulation
away/away
home/away
away/home
starts at home (in slot 0)
end at home (in last slot)
right # of home games
© J. Christopher Beck 2008
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one team must be home
one home one away
against a given other team
max consecutive home games
max consecutive away games
© J. Christopher Beck 2008
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CP Formulation
logical
constraints to
determine
travel
distance
© J. Christopher Beck 2008
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Sequence Constraint
sequence(min,max,width,base,value,card)
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GCC
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min,max,width: integers
base: vector of variables
value: vector of values
card: vector of cardinalities
value[i] is taken by card[i] elements of
base, and
for any sub-sequence of base of size width
each value[i] is present [min .. max] times
© J. Christopher Beck 2008
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CP Formulation
logical
constraints to
determine
travel
distance
team i must play exactly n-1 home games
(1 appears n-1 times)
in UB+1 consecutive games, must have at
© J. one
Christopher
2008
least
and atBeck
most
UB home games
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CP Formulation
one home, one away
against a given other team
team and opponent not both home or both away
starts at home (in slot 0)
end at home (in last slot)
© J. Christopher Beck 2008
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Results
Time limit per run = 1800 seconds
IP
n
6
8
10
12
14
16
CP
# Solved
Mean CPU (s)
# Solved
Mean CPU (s)
10
10
1
0
0
0
22.5
397.4
1273.7
-
10
0
0
0
0
0
17.0
-
© J. Christopher Beck 2008
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CP/IP Hybrid
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Phase 1: generate all feasible patterns
for each team
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remember we already have the timetable
Phase 2: find optimal pattern set by
assigning each team to a pattern from
Phase 1
© J. Christopher Beck 2008
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Phase 1: CP Formulation
same sequence constraint as in full CP model
team i must play exactly n-1 home games (1 appears n-1 times)
in UB+1 consecutive games, must have at least one
and at most UB home games
one home, one away
against a given other team
one problem per team (same formulation solved n times)
© J. Christopher Beck 2008
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Phase 1: CP Formulation
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Find all satisfying solutions to model
Pi: set of satisfying patterns for team i
hjs: hs entry in pattern j є Pi
dij: distance team i must travel if it uses
pattern j є Pi
© J. Christopher Beck 2008
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Phase 2: IP Formulation
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xij = 1 iff team i takes pattern j є Pi
each team takes one pattern
when teams meet, one is home, one away
© J. Christopher Beck 2008
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Results
Time limit per run = 1800 seconds
IP
n
6
8
10
12
14
16
CP
# Solved
Mean CPU (s)
# Solved
Mean CPU (s)
10
10
1
0
0
0
22.5
397.4
1273.7
-
10
0
0
0
0
0
17.0
-
© J. Christopher Beck 2008
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Results
Time limit per run = 1800 seconds
IP
n
6
8
10
12
14
16
CP
CP/IP
# Solved
Mean CPU (s)
# Solved
Mean CPU (s)
# Solved
Mean CPU (s)
10
10
1
0
0
0
22.5
397.4
1273.7
-
10
0
0
0
0
0
17.0
-
10
10
10
10
10
10
0.07
0.44
2.02
12.2
38.1
197.2
© J. Christopher Beck 2008
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Discussion
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Clever decomposition!
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Generate-and-choose is interesting
CP not strong at optimization but good
at satisfaction
Similar to 3 phase breakdown we’ve
been looking at, but finds optimal
solutions
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why?
© J. Christopher Beck 2008
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