Modelling optimum fraction assignment in the 4x100 m
relay race by integer linear programming
Francesco Masedu1, Massimo Angelozzi2
1
2
Department of Information Technology, University of L'Aquila, Italy
PhD in Physical Activity Sciences, University of L'Aquila, Italy
[email protected]
ABSTRACT
Masedu F, Angelozzi M
Modelling optimum fraction assignment in the 4x100 m relay race by integer linear programming
Ital J Sport Sci 2006: 13: 74-77
We present and discuss a novel application of established methods of operative research to a strategy directed at obtaining the best
possible time performance in the 4x100 m track and field relay. The assignation of fractions to athletes that will maximize the final
performance and the formalisation of the relevant constraints are addressed. After describing the mathematical background, a problem is
proposed and solved. This strategy is compared with the more intuitive approach based on the selection of the athlete who obtained the
best performance in the relevant fraction, evidencing the advantage of applying the operative research method.
STRUMENTI E METODI
KEYWORDS: integer linear programming, operative research, optimum assignment, relay race,
training
INTRODUCTION
Relay is a fascinating discipline in which both
individual and team performances are involved. The
most important relay races are undoubtedly the
4x100 m and 4x400 m track and field and the 4x100
m and 4x200 m free style and 4x100 medley in
swimming. In relay races four different athletes run
or swim one of four equal fractions of the distance.
In track and field races a baton is exchanged, while
in swimming relays the touch of the edge of the lane
by the previous athlete launches the next.
Mathematical models have been devised to study the
4x100 m track and field relay (Ward-Smith and
Radford, 2002; Redford and Ward-Smith, 2003) and
to optimise relay start in swimming (McLean et al.,
2000), but to date no study has used operative
research methods.
Training of sprinters for relay involves optimum
assignment of fractions. Such decisions are usually
pragmatic, i.e. the athlete who made the best time in
a fraction is chosen to run it. Though intuitive, this
method is not the best. Of course, all sprinters need
specific training to make a good start, or to run curve
fractions faster, but all these features eventually need
to be included in a final racing strategy. Operative
74
research, in the present case integer linear
programming, may be a suitable tool to make
optimal assignment decisions.
A further application of this assignment strategy may
be in the selection of the athletes who participate in
team championships. Since at these events each
athlete receives a score for each performance and can
take part in only two races and a relay, the
assignment strategy is obviously crucial.
METHOD
We assume a situation wherein 4 sprinters, one for
each fraction of a 4x100 m track and field relay, are
to be selected from a group of 6 eligible athletes to
obtain the fastest possible team. As stated above,
each will perform differently in each fraction of the
track. Acting on purely combinatorial considerations,
the trainer would be required to test 360 possible
assignments. This option is energy - and time consuming when 6 athletes are involved, but since
these quantities rise as factorials a larger group (say
15) from which to choose (32,760 assignments) will
make the problem intractable, or at least
impracticable to solve with this method. A useful
ITALIAN JOURNAL of SPORT SCIENCES
means to circumvent it is provided by integer linear
programming techniques with the setting of a
suitable function to be minimised and the
identification of the constraints to which the problem
is subject.
This method requires the trainer to note in a table the
performance of each sprinter in each fraction, so that
each cell will report the time taken by the subject i to
run the fraction j. A possible outcome of this
operation is reported in table 1:
By making explicit the equations mentioned above,
our situation results to be a problem that can be
solved using integer linear programming:
CHRONOMETRIC OUTCOMES
4 fractions x100 m
Sprinter 1
Sprinter 2
…
Sprinter n
I
II
III
IV
τ11
τ21
…
τn1
τ12
τ22
…
τn2
τ13
τ23
…
τn3
τ14
τ24
…
τn4
Table 1 - Formal definition of the distribution values
The 0-1 variable xij is defined as:
if the sprinter i runs the fraction j
Then an objective is set, i.e. the total race time that
needs be reduced, as follows:
Function (2) will be subject to two sets of constraints
(3) (4) that respectively characterise the following
requirements:
each athlete can be assigned to only one fraction;
some athlete must be excluded, i.e. each fraction can
be run by only one sprinter.
The constraint xij
produces definition (1).
Making decisions: a computer-aided example
To make a concrete example of the strategy
described above, let us consider the hypothetical race
performances of 6 sprinters (table 1) and analyse
them using software LINDO, release 6.1, which
launches a branch and bound algorithm (Schrage,
2004).
By substituting the values of table 2 into the
equations and disequations listed above, the
computation gives the output reported in fig. 1,
which shows the data relevant to the solution of our
problem.
FRACTION
Athlete
Sprinter 1
Sprinter 2
Sprinter 3
Sprinter 4
Sprinter 5
Sprinter 6
Fraction I Fraction II Fraction III
12.27 s
11.34 s
11.29 s
12.54 s
12.20 s
11.54 s
11.57 s
11.45 s
11.50 s
12.34 s
11.22 s
11.48 s
11.54 s
12.45 s
11.45 s
12.32 s
12.07 s
11.56 s
Fraction IV
12.07 s
12.34 s
11.52 s
11.57 s
12.03 s
12.30 s
Table 2 - Times in seconds (s) made by each athlete in each fraction
This computation gives the optimum solution, i.e. the
optimum assignment of each sprinter that will allow
to reduce total race time.
ANNO 13 - NUMERO 1 2006
75
12
= 45.5799980
LP OPTIMUM FOUND AT STEP
OBJECTIVE VALUE
45.5800018
0 PIVOT
12
RE-INSTALLING BEST SOLUTION...
NEW INTEGER SOLUTION OF
OBJECTIVE FUNCTION VALUE
AT BRANCH
Value
0.000000
0.000000
0.000000
0.000000
1.000000
0.000000
0.000000
0.000000
0.000000
0.000000
1.000000
0.000000
0.000000
0.000000
0.000000
1.000000
0.000000
1.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
Reduced Cost
12.270000
11.570000
11.540000
12.070000
11.340000
11.450000
12.450000
12.340000
11.290000
11.500000
11.450000
11.520000
12.540000
12.340000
12.320000
11.570000
12.200000
11.220000
12.070000
12.030000
11.540000
11.480000
11.560000
12.300000
ROW
SLACK OR SURPLUS
DUAL PRICES
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
0.000000
0.000000
0.000000
0.000000
1.000000
0.000000
0.000000
0.000000
0.000000
1.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
Figure 1 - LINDO software output
Fraction III
100 m
Athlete 2
Athlete 5
Athlete 3
Fraction IV Total time
100 m
400 m
Athlete 4
[Time 11.34 s] [Time 11.22 s] [Time 11.45 s] [Time 11.57 s] 45.58 s
Table 3 - Optimum fraction assignment
The intuitional strategy that chooses the athlete based
on the lowest time measured in a given fraction
would result in the fractions being assigned as shown
in table 4.
76
Fraction III
100 m
Athlete 3
Athlete 5
Athlete 1
Fraction IV Total time
100 m
400 m
Athlete 4
Table 4 - Assignments based on the lowest times in each fraction
Variable
X11
X12
X13
X14
X21
X22
X23
X24
X31
X32
X33
X34
X41
X42
X43
X44
X51
X52
X53
X54
X61
X62
X63
X64
Fraction II
100 m
Fraction II
100 m
[Time 11.29 s] [Time 11.22 s] [Time 11.54 s] [Time 11.57 s] 45.62 s
1) 45.58000
Fraction I
100 m
Fraction I
100 m
DISCUSSION
The comparison between assignment strategies
reported in tables 3 and 4 illustrates the advantages
of the operative research-based method.
The most important factors in 4x100 m track and
field relay are the starts of the last three runners (Salo
and Bezodis, 2004), the stick exchange technique
(Match, 1991), the distance between carrier and
receiver (Boyadjian and Bootsma, 1999) and, last but
not least, fraction assignment.
In the 4x100 m race, fraction assignment is
complicated by the fact that a 100 m race is different
from a 100 m relay fraction, highlighting the
inadvisability of assigning fractions only based on
performance in individual races. Indeed, the choice
of the fraction runners should not rely on trial
fraction time either, because too few relays are run in
a year for athletes to perform at top level in trials,
and because time recording may be affected by the
three baton exchanges, which strongly affect the final
race time.
In longer relay races, where the baton exchange is
less crucial and fraction performances mirror the
athlete's best track times, the assignment strategy is
less important. In these cases, there is a stronger
correlation between performance in individual
races and in relay, so greater care needs to be
exercised to select the best performing runner for
each fraction.
Different issues are involved in other relay
disciplines, such as Nordic ski races, particularly the
4x10 km men and 4x5 km women. In such
disciplines, although the selection of the fraction
runners is crucial to optimise final race time, each
racing field and course is unique. Indeed, in selecting
athletes the trainer should consider not only their best
performances but also their individual attitudes in
coping with the course.
A further context for the application of the method
described above is team composition for team
championships. In track and field sports, which are
eminently individual, important championships are
organised ever more frequently and typically include
team championships. In Italy, teams are required to
enter competitors in all the disciplines envisaged,
with athletes allowed to participate in no more than
ITALIAN JOURNAL of SPORT SCIENCES
two races and one relay. Each athlete receives a score
for placement or performance and the final score of
the team is the sum of the individual scores. In this
context, the assignment of athletes to each discipline
is crucial.
In conclusion, operative research may be a useful
technical instrument to improve individual and team
performances in sports.
REFERENCES
Boyadjian, A., Bootsma, R.J. (1999). Timing in relay
running. Percept Mot Skills, 88, 1223-1230
Match, G. (1991). The 4x100 metres relay with the push-
ANNO 13 - NUMERO 1 2006
forward pass. New Studies in Athletics, 1, 67-73
McLean, S.P., Holthe, M.J., Vint, P.F., Beckett, K.D.,
Hinrichs, R.N. (2000). Addition of an approach to a
swimming relay start. J Applied Biomechanics, 16, 342355
Radford, P.F., Ward-Smith, A.J. (2003). The baton
exchange during the 4 x 100 m relay: a mathematical
analysis, J Sports Sci, 21, 493-501
Salo, A., Bezodis, I. (2004). Which starting style is faster
in sprint running-standing or crouch start? Sports
Biomech, 3, 43-53
Schrage, L. (2004). Optimization Modeling with LINDO,
Fifth Edition, LINDO Systems
Ward-Smith, A.J., Radford, P.F. (2002). A mathematical
analysis of the 4 x 100 m relay. J Sports Sci, 20, 369-381
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