Application of Linear Programming on Example of Relationship
between Two Types of Activity and Optimized Dietary Supplement
Intake
Ana Špirelja Gruić
1
and Igor Gruić
2
1
V. Gymnasium, Klaićeva 1, 10000 Zagreb, Croatia
2
Faculty of Kinesiology, University of Zagreb, Horvaćanski zavoj 15, 10000 Zagreb, Croatia
Keywords: Mathematics, Linear Programming, Optimization, Nutrition, Health, Training, Kinanthropology.
Abstract: Kinesiology, as a science on movement, can use reductive and constructive logic and tools to inspect, analyse
and produce phenomena related to human, activity. Deterministic and stochastic nature of kinanthropological
phenomena are often analysed by complex statistical methods. Application of linear programming for
optimization in producing simple decision and recommendation regarding intake of exact proportion of
recovery dietary supplements complexes in two different activities (aerobic and anaerobic) revealed elegance
of the method, and revealed prospective practical implication in sport practice, rehabilitation process, and in
everyday life.
1 INTRODUCTION
Decision making is mostly rational act, based on good
estimates and facts, but in real life it often depends on
feelings, intuition, and happiness. It becomes harder
to decide when more different conditions have to be
fulfilled in order to reach the best possible (optimal)
decision. Mathematical modelling and programming
can be helpful, to different varying degree, depending
on the nature of the observed phenomenon and the
complexity of the problem (Špirelja, 2007).
Within general knowledge of multicriteria linear
optimizations (Ehrgott, 2005, Neralić, 2003, Steuer,
1986, Špirelja, 2007), analysis of relations and
differences between dietary/nutrition regimes (Aird
et al., 2018, Denham, 2017, Ferguson et al., 2004,
Henson, 1991, Rawson et al., 2018 ), and influences
of different training/exercise regimes (Pasiakos et al.,
2015, Patel et al., 2017), specific optimizations by
implementation of mathematical tools were feasible
and applicable (Asano et al., 2018, Briend et al., 2018,
Persson et al., 2018,)
The aim of this paper was, through a cross-section
of mathematical methods and two concrete example
of linear programming, to provide a practical tool for
optimizing simple and everyday needs for sport, but
also for everyday life, out of the relationship between
two types of activity and optimized input of dietary
supplement (DS).
2 METHODS
2.1 Linear Programming with Two
Variables
For those problems which have linear nature and
require only two variables (e.g. dietary supplements),
it is sufficient to know the graphical solution of linear
inequalities and the mathematical fact that the
optimum solution lies in one of the vertices of a
feasible set, defined by the constraints (cross section
of linear inequalities).
2.2 Linear Programming with More
Variables – A Simplex Method
Finding the vertex of the feasible set by the graphical
method in a more-than-two-dimensional space is
often demanding and time-consuming. A simpler
approach is to apply the Simplex Method, which is an
iterative method, i.e. step-by-step method of
improvements of the basic feasible solution, until the
final step results with optimal feasible solution (if it
Grui
´
c, A. and Grui
´
c, I.
Application of Linear Programming on Example of Relationship between Two Types of Activity and Optimized Dietary Supplement Intake.
DOI: 10.5220/0007234601970202
In Proceedings of the 6th International Congress on Sport Sciences Research and Technology Support (icSPORTS 2018), pages 197-202
ISBN: 978-989-758-325-4
Copyright © 2018 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
197
exists). It mostly implies application of Gauss-Jordan
transformations of matrices which can be solved
manually and by different software tools.
2.3 Experimental Design
Dietary supplements (DS) for recovery after intensive
activity have the function of supplying body with
calories, proteins, amino acids (BCAA, Glutamine),
electrolytes (e.g. magnesium ions), vitamins (B1, B2,
B6) etc. The goal of the ‘optimization’ was to
calculate minimum intake of DS as possible within
the default features of the training and the limit on the
input.
Table 1: An example of a composition of three related DS
(mass in one portion of DS1:91g, DS2:75g, DS3:65g).
portion DS 1 DS 2 DS 3
calories 320 270 235
carbohydrates 60 53 28
proteins 20 13 27
B1 0.008 0.005 0.0033
B2 0.0085 0.004 0.0066
B6 0.008 0 0.0033
Mg 0.250 0.160 0.205
BCAA 4.5 0.98 5.98
Glutamine 6 1.56 0
In addition to regular activity and controlled diets,
for the purpose of this example, constraints for intake
of a part of the vitamin B complex are:
B1 less than 7mg
B2 less than 7mg
B6 less than 7mg
2.3.1 Anaerobic Training
Consumption in the chosen anaerobic training in the
example assumes intake of:
more than 280kcal
more than 15g of protein
more than 200mg of magnesium
more than 2g of BCAA amino acids
more than 3g amino acids Glutamine
2.3.2 Aerobic Training
Consumption in the chosen aerobic training in this
example assumes intake of:
more than 320kcal
more than 55g of carbohydrates
more than 15g of protein
more than 200mg of magnesium
more than 2g of BCAA amino acids
more than 3g amino acids Glutamine
3 OPTIMIZATION OF INTAKE
OF TWO DIETARY
SUPPLEMENTS
3.1 Relationship between Anaerobic
Activity and Optimized Intake of
Two Dietary Supplements
For variables X1 - the mass of portion intake of the
DS1, X2 - the mass of portion intake of the DS2, with
regard to default constraints and parameters, and Z -
objective (goal) function linear optimization problem
for anaerobic training was set:
MINIMIZE: Z = 91 X1 + 75 X2 (1)
320 X1 + 270 X2 280
20 X1 + 13 X2 15
0.25 X1 + 0.16 X2 0.2
4.5 X1 + 0.98 X2 2
6 X1 + 1.56 X2 3
0.008 X1 + 0.005 X2 0.007
0.0085 X1 + 0.004 X2 0.007
0.008 X1 0.007
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
X1, X2 0 (10)
The feasible set is pentagon (Figure 1), and the
candidates for the optimal solution are in vertices of
that pentagon. The optimal solution is in the vertex in
which the objective function reaches the minimum
value.
The solution is in vertex F (0.56, 0.37), which
would mean that the optimal combination of two
preparations is: DS1 51.36 g and DS2 28.58 g.
K-BioS 2018 - Special Session on Kinesiology in Sport and Medicine: from Biomechanics to Sociodynamics
198
Point X coordinate
(
X1
)
Y coordinate
(
X2
)
Value of the ob
j
ective function
(
Z
)
E 0.75862068965517 0.13793103448276 79.379310344828
F 0.56441717791411 0.3680981595092 78.969325153374
N 0.66666666666667 0.33333333333333 85.666666666667
Q
0.23287671232877 1.027397260274 98.246575342466
Y 0.29473684210526 0.78947368421053 86.031578947368
Figure 1: Anaerobic training parameters.
3.2 Relationship between Aerobic
Activity and Optimized Intake of
Two Dietary Supplements
For variables X1 - the mass of portion intake of the
DS1, X2 - the mass of portion intake of the DS2, with
regard to default constraints and parameters, and Z -
objective (goal) function linear optimization problem
for aerobic training was set:
MINIMIZE: Z = 91 X1 + 75 X2
(1)
320 X1 + 270 X2 320
(2)
20 X1 + 13 X2 15
(3)
0.25 X1 + 0.16 X2 0.2
(4)
4.5 X1 + 0.98 X2 2
(5)
6 X1 + 1.56 X2 3
(6)
60 X1 + 53 X2 55
(7)
0.008 X1 + 0.005 X2 0.007
(8)
0.0085 X1 + 0.004 X2 0.007
(9)
0.008 X1 0.007
(10)
X1, X2 0
(11)
The feasible set is triangle (Figure 2), and the
candidates for the optimal solution are vertices of that
triangle. The optimal solution is the vertex in which
the function of the target reaches the minimum value.
The solution is at point H (0.28, 0.86) which would
mean that the optimal combination of two
preparations: DS1 25.27 g, and DS2 64.24 g.
4 OPTIMIZATION OF INTAKE
OF MORE DIETARY
SUPPLEMENTS
In next example three different dietary supplements
were included. Graphical method, presented in
previous examples, would here require plotting linear
inequalities in 3-D and defining feasible region.
Simplex method will then be more appropriate.
Application of Linear Programming on Example of Relationship between Two Types of Activity and Optimized Dietary Supplement Intake
199
Point X coordinate (X1) Y coordinate (X2) Value of the objective function (Z)
C 0.51785714285714 0.57142857142857 89.982142857143
H 0.27730192719486 0.85653104925054 89.474304068522
S 0.23287671232877 1.027397260274 98.246575342466
Figure 2: Aerobic training parameters.
4.1 Relationship between Anaerobic
Activity and Optimized Intake of
Three Dietary Supplements
For variables X1 - the mass of portion intake of the
DS1, X2 - the mass of portion intake of the DS2, X3
- the mass of portion intake of the DS3, with regard
to default constraints and parameters, and Z -
objective (goal) function linear optimization problem
for anaerobic training was set:
MINIMIZE: Z = 91 X1 + 75 X2 + 65 X3
(1)
320 X1 + 270 X2 + 235 X3 280
(2)
20 X1 + 13 X2 + 27 X3 15
(3)
0,25 X1 + 0,16 X2 + 0,205 X3 0,2
(4)
4,5 X1 + 0,98 X2 + 5,98 X3 2
(5)
6 X1 + 1,56 X2 + 0 X3 3
(6)
0,008 X1 + 0,005 X2 + 0,0033 X3 0,007
(7)
0,0085 X1 + 0,004 X2 + 0,0066 X3 0,007
(8)
0,008 X1 + 0 X2 + 0,0033 X3 0,007
(9)
X1, X2, X3 0
(10)
The optimal solution (in table 2) is (0.3864, 0.4368,
0.1634) which would mean that the optimal
combination of three preparations is: DS1 35.16 g,
DS2 32.76 g and DS3 10.62 g.
Table 2: Final solution shown in transformed matrix after six iterations (by using simplex method).
-91,0000 -75,0000 -65,0000 0 0 0 0 0 0 0 0 0
Base Cb P0 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12
P2 -75,0000 0,4368 0 1 0 -0,0128 0 14,7160 0 0,0715 0 0 0 0
P7 0,0000 1,1444 0 0 0 0,0380 0 -72,7481 1 0,2537 0 0 0 0
P3 -65,0000 0,1634 0 0 1 0,0059 0 -11,6977 0 0,1701 0 0 0 0
P1 -91,0000 0,3864 1 0 0 0,0033 0 -3,8262 0 -0,1853 0 0 0 0
P5 0,0000 2,8198 0 0 0 0,0605 1 -201,0520 0 1,8176 0 0 0 0
P9 0,0000 0,0012 0 0 0 178533408,4789 0 -0,0044 0 0,0006 1 0 0 0
P10 0,0000 0,0009 0 0 0 -162846765,6084 0 0,0509 0 0,0002 0 1 0 0
P11 0,0000 0,0034 0 0 0 -46333521,1973 0 0,0692 0 0,0009 0 0 1 0
P12 0,0000 0,0000 0 0 0 0,0000 0 0,0000 0 0,0000 0 0 0 1
Z
-78,5482 0 0 0 0,2724 0 4,8281 0 0,4383 0 0 0 0
K-BioS 2018 - Special Session on Kinesiology in Sport and Medicine: from Biomechanics to Sociodynamics
200
4.2 Relationship between Aerobic
Activity and Optimized Intake of
Three Dietary Supplement
For variables X1 - the mass of portion intake of the
DS1, X2 - the mass of portion intake of the DS2, X3
- the mass of portion intake of the DS3, with regard
to default constraints and parameters, and Z -
objective (goal) function linear optimization
problem for aerobic training was set:
MINIMIZE: Z = 91 X1 + 75 X2 + 65 X3
(1)
320 X1 + 270 X2 + 235 X3 320
(2)
20 X1 + 13 X2 + 27 X3 15
(3)
0,25 X1 + 0,16 X2 + 0,205 X3 0,2
(4)
4,5 X1 + 0,98 X2 + 5,98 X3 2
(5)
6 X1 + 1,56 X2 + 0 X3 3
(6)
0,008 X1 + 0,005 X2 + 0,0033 X3 0,007
(7)
0,0085 X1 + 0,004 X2 + 0,0066 X3 0,007
(8)
0,008 X1 + 0 X2 + 0,0033 X3 0,007
(9)
60 X1 + 53 X2 + 28 X3 55
(10)
X1, X2, X3 0
(11)
The optimal solution (in table 3) is (0.2773, 0.8565,
0) which would mean that the optimal combination
of three preparations is: DS1 25.23 g, DS2 64.24 g
and DS3 0 g.
Table 3: Final solution shown in transformed matrix after eleven iterations (by using simplex method).
-91,0000 -75,0000 -65,0000 0 0 0 0 0 0 0 0 0
Base Cb P0 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12
P6 0,0000 0,0064 0 0 -0,0855 -0,0005 0 1 0 -0,0145 0 0 0 0
P2 -75,0000 0,8565 0 1 1,2580 -0,0054 0 0 0 0,2855 0 0 0 0
P7 0,0000 0,0873 0 0 -6,2190 0,0010 0 0 1 -0,8042 0 0 0 0
P10 0,0000 0,0012 0 0 0,0043 95824411,1349 0 0 0 0,0009 0 1 0 0
P12 0,0000 7,0343 0 0 19,0503 -0,2002 0 0 0 0,6781 0 0 0 1
P9 0,0000 0,0005 0 0 -0,0004 156316916,4882 0 0 0 0,0005 1 0 0 0
P1 -91,0000 0,2773 1 0 -0,3271 0,0014 0 0 0 -0,2409 0 0 0 0
P11 0,0000 0,0048 0 0 0,0059 -111349036,4026 0 0 0 0,0019 0 0 1 0
P5 0,0000 1,6809 0 0 -17,1874 -0,0418 1 0 0 -1,1064 0 0 0 0
Z
-89,4743 0 0 0,4127 0,2748 0 0 0 0,5086 0 0 0 0
5 CONCLUSIONS
The problem of choosing a suitable method in sports
research, in this case the introduction of recovery
preparations and dietary supplements, is a key issue
because of the often stochastic nature of the observed
variables.
In the latest trends in research in sports and
kinesiology, there is the concept of 'vicarianza'
(Sibilio, 2017), through which different variables of
input are set into the relationship with the rules of the
observed activity (e.g. rules of handball, tactics, or
verified protocol of therapeutic procedure after
operative procedure, etc.), then through decision-
making mechanisms, all the way to last and finite,
mostly measurable effects of the activity described by
input variables. In this context, linear multicriteria
optimizations tool was useful for introducing DS3 as
appropriate for recovery after anaerobic training, but
not necessary for recovery after aerobic training.
ACKNOWLEDGEMENTS
This paper is announcement of future cooperation
between laboratories of Institute of Kinesiology,
Faculty of Kinesiology University of Zagreb, and
high school of natural and mathematical sciences V.
Gymnasium Zagreb within project “School for life”
supported by Ministry of Science and Education,
Republic of Croatia.
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