This document discusses optimizing the scheduling of teams in the third division of the Ecuadorian football league. It presents an integer programming formulation to partition teams into balanced zones while minimizing travel distances. Several solution methods are proposed, including heuristics that construct zones and local search, as well as a branch-and-bound approach with rounding. Computational results show the heuristic methods find good solutions faster than the exact branch-and-bound for various problem sizes.
APEX INSTITUTE was conceptualized in May 2008, keeping in view the dreams of young students by the vision & toil of Er. Shahid Iqbal. We had a very humble beginning as an institute for IIT-JEE / Medical, with a vision to provide an ideal launch pad for serious JEE students . We actually started to make a difference in the way students think and approach problems.
Pre-Calculus Midterm Exam
1
Score: ______ / ______
Name: ____________________________
Student Number: ___________________
Short Answer: Type your answer below each question. Show your work.
1 Verify the identity. Show your work.
cot θ ∙ sec θ = csc θ
2 A gas company has the following rate schedule for natural gas usage in single-family residences:
Monthly service charge $8.80
Per therm service charge
1st 25 therms $0.6686/therm
Over 25 therms $0.85870/therm
What is the charge for using 25 therms in one month? Show your work.
What is the charge for using 45 therms in one month? Show your work.
Construct a function that gives the monthly charge C for x therms of gas.
Pre-Calculus Midterm Exam
2
3 The wind chill factor represents the equivalent air temperature at a standard wind speed that would
produce the same heat loss as the given temperature and wind speed. One formula for computing
the equivalent temperature is
W(t) = {
𝑡
33 −
(10.45+10√𝑣−𝑣)(33−𝑡)
2204
33 − 1.5958(33 − 𝑡)
if 0 ≤ v < 1.79
if 1.79 ≤ v < 20
if v ≥ 20
where v represents the wind speed (in meters per second) and t represents the air temperature .
Compute the wind chill for an air temperature of 15°C and a wind speed of 12 meters per second.
(Round the answer to one decimal place.) Show your work.
4 Complete the following:
(a) Use the Leading Coefficient Test to determine the graph's end behavior.
(b) Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis
and turns around at each intercept. Show your work.
(c) Find the y-intercept. Show your work.
f(x) = x2(x + 2)
(a).
(b).
(c).
Pre-Calculus Midterm Exam
3
5 For the data set shown by the table,
a. Create a scatter plot for the data. (You do not need to submit the scatter plot)
b. Use the scatter plot to determine whether an exponential function or a logarithmic function is
the best choice for modeling the data.
Number of Homes Built in a Town by Year
6 Verify the identity. Show your work.
(1 + tan2u)(1 - sin2u) = 1
Pre-Calculus Midterm Exam
4
7 Verify the identity. Show your work.
cot2x + csc2x = 2csc2x - 1
8 Verify the identity. Show your work.
1 + sec2xsin2x = sec2x
Pre-Calculus Midterm Exam
5
9 Verify the identity. Show your work.
cos(α - β) - cos(α + β) = 2 sin α sin β
10 The following data represents the normal monthly precipitation for a certain city.
Draw a scatter diagram of the data for one period. (You
APEX INSTITUTE was conceptualized in May 2008, keeping in view the dreams of young students by the vision & toil of Er. Shahid Iqbal. We had a very humble beginning as an institute for IIT-JEE / Medical, with a vision to provide an ideal launch pad for serious JEE students . We actually started to make a difference in the way students think and approach problems.
Pre-Calculus Midterm Exam
1
Score: ______ / ______
Name: ____________________________
Student Number: ___________________
Short Answer: Type your answer below each question. Show your work.
1 Verify the identity. Show your work.
cot θ ∙ sec θ = csc θ
2 A gas company has the following rate schedule for natural gas usage in single-family residences:
Monthly service charge $8.80
Per therm service charge
1st 25 therms $0.6686/therm
Over 25 therms $0.85870/therm
What is the charge for using 25 therms in one month? Show your work.
What is the charge for using 45 therms in one month? Show your work.
Construct a function that gives the monthly charge C for x therms of gas.
Pre-Calculus Midterm Exam
2
3 The wind chill factor represents the equivalent air temperature at a standard wind speed that would
produce the same heat loss as the given temperature and wind speed. One formula for computing
the equivalent temperature is
W(t) = {
𝑡
33 −
(10.45+10√𝑣−𝑣)(33−𝑡)
2204
33 − 1.5958(33 − 𝑡)
if 0 ≤ v < 1.79
if 1.79 ≤ v < 20
if v ≥ 20
where v represents the wind speed (in meters per second) and t represents the air temperature .
Compute the wind chill for an air temperature of 15°C and a wind speed of 12 meters per second.
(Round the answer to one decimal place.) Show your work.
4 Complete the following:
(a) Use the Leading Coefficient Test to determine the graph's end behavior.
(b) Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis
and turns around at each intercept. Show your work.
(c) Find the y-intercept. Show your work.
f(x) = x2(x + 2)
(a).
(b).
(c).
Pre-Calculus Midterm Exam
3
5 For the data set shown by the table,
a. Create a scatter plot for the data. (You do not need to submit the scatter plot)
b. Use the scatter plot to determine whether an exponential function or a logarithmic function is
the best choice for modeling the data.
Number of Homes Built in a Town by Year
6 Verify the identity. Show your work.
(1 + tan2u)(1 - sin2u) = 1
Pre-Calculus Midterm Exam
4
7 Verify the identity. Show your work.
cot2x + csc2x = 2csc2x - 1
8 Verify the identity. Show your work.
1 + sec2xsin2x = sec2x
Pre-Calculus Midterm Exam
5
9 Verify the identity. Show your work.
cos(α - β) - cos(α + β) = 2 sin α sin β
10 The following data represents the normal monthly precipitation for a certain city.
Draw a scatter diagram of the data for one period. (You
In Job Sequencing Deadline Problem, the
the objective is to find the sequence of jobs,
which is completed within their deadline
and gives maximum profit.
5/2/2020 MyOpenMath
https://www.myopenmath.com/assess2/?cid=69772&aid=4975705#/print 1/9
Chapter 7 Review Courtney Garrity
Question 1 0/10 pts 5 99
Question 2 0/10 pts 5 99
Question 3 0/10 pts 5 99
Question 4 0/10 pts 5 99
Rewrite in terms of and
Next Question
cos(x + )
π
3
sin(x) cos(x)
Question Help: Video
Solve for the smallest positive solution.
x =
Give your answer accurate to two decimal places.
Next Question
sin(5x)cos(9x) − cos(5x)sin(9x) = − 0.45
Question Help: Video
Rewrite as
A =
=
Note: should be in the interval
Next Question
−5 sin(x) + 1 cos(x) A sin(x + ϕ)
ϕ
ϕ −π < ϕ < π
Write the product as a sum:
18 cos(41w)cos(15w) =
5/2/2020 MyOpenMath
https://www.myopenmath.com/assess2/?cid=69772&aid=4975705#/print 2/9
Question 5 0/10 pts 5 99
Question 6 0/10 pts 5 99
Question 7 0/10 pts 5 99
Question 8 0/10 pts 5 99
Next Question
Question Help: Video
Write the sum as a product:
Next Question
cos(23.6s) − cos(8.6s) =
Question Help: Video
Find all solutions to on
=
Give your answers as a list separated by commas
Next Question
cos(7x) − cos(x) = sin(4x) 0 ≤ x <
2π
3
x
Question Help: Video
Simplify to an expression involving a single trigonometric function.
Next Question
sin(6w) − sin(4w)
cos(6w) + cos(4w)
Question Help: Video
5/2/2020 MyOpenMath
https://www.myopenmath.com/assess2/?cid=69772&aid=4975705#/print 3/9
Question 9 0/10 pts 5 99
Question 10 0/10 pts 5 99
Question 11 0/10 pts 5 99
Solve for the four smallest positive solutions
=
Give your answers accurate to at least two decimal places, as a list separated by commas
Next Question
sec(4x) − 6 = 0
x
Solve for all solutions
=
Give your answers accurate to 2 decimal places, as a list separated by commas
Next Question
4 sin2(x) − 10 sin(x) + 4 = 0 0 ≤ x < 2π
x
Question Help: Video
Solve for all solutions
=
Give your answers accurate to 2 decimal places, as a list separated by commas
Next Question
8 sin2(t) − 2 cos(t) − 5 = 0 0 ≤ t < 2π
t
Question Help: Video
If , x in quadrant I, then �nd (without �nding x)
sin x =
2
5
sin(2x) =
cos(2x) =
tan(2x) =
5/2/2020 MyOpenMath
https://www.myopenmath.com/assess2/?cid=69772&aid=4975705#/print 4/9
Question 12 0/10 pts 5 99
Question 13 0/10 pts 5 99
Question 14 0/10 pts 5 99
Question 15 0/10 pts 5 99
Next Question
Solve for all solutions
=
Give your answers accurate to at least 2 decimal places, as a list separated by commas
Next Question
4 sin(2ϕ) + 2 cos(ϕ) = 0 0 ≤ ϕ < 2π
ϕ
Question Help: Video
Solve for all solutions
=
Give your answers accurate to at least 2 decimal places, as a list separated by commas
Next Question
6 cos(2β) = 6 cos2(β) − 1 0 ≤ β < 2π
β
Question Help: Video Video
If for then
Next Question
csc(x) = 4, 90 ∘ < x < 180 ∘ ,
sin( ) =
x
2
cos( ) =
x
2
tan( ) .
Evaluation: The performance of supervised learning models is typically evaluated using various metrics such as accuracy, precision, recall, or mean squared error, depending on the nature of the problem (classification or regression
Practical and Worst-Case Efficient ApportionmentRaphael Reitzig
Proportional apportionment is the problem of assigning seats to parties according to their relative share of votes. Divisor methods are the de-facto standard solution, used in many countries.
In recent literature, there are two algorithms that implement divisor methods: one by Cheng and Eppstein (ISAAC, 2014) has worst-case optimal running time but is complex, while the other (Pukelsheim, 2014) is relatively simple and fast in practice but does not offer worst-case guarantees.
This talk presents the ideas behind a novel algorithm that avoids the shortcomings of both. We investigate the three contenders in order to determine which is most useful in practice.
Read more over here: http://reitzig.github.io/publications/RW2015b
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
"Protectable subject matters, Protection in biotechnology, Protection of othe...
particionamiento
1. Optimizing the Third Division of the Ecuadorian Football
League.
Diego Recalde, Ramiro Torres, Polo Vaca
Escuela Politécnica Nacional
Quito, Ecuador
XIII Encuentro de Matemáticas. Quito-2012– p. 1
2. Outline
• Motivation.
• The Problem.
• Integer Formulation.
• Algorithms and solution.
• Computacional Results
• Conclusions
XIII Encuentro de Matemáticas. Quito-2012– p. 2
3. Outline
• Motivation.
• The Problem.
• Integer Formulation.
• Algorithms and solution.
• Computacional Results
• Conclusions
XIII Encuentro de Matemáticas. Quito-2012– p. 2
4. Motivation
• Professional football leagues
1. Real Madrid ($ 479,5)
2. FC Barcelona ($ 450,7)
3. Manchester United ($ 367)
4. Bayern Munich ($ 321,4 )
5. Arsenal ($ 251,1 )
XIII Encuentro de Matemáticas. Quito-2012– p. 3
5. Motivation
• Professional football leagues
1. Real Madrid ($ 479,5)
2. FC Barcelona ($ 450,7)
3. Manchester United ($ 367)
4. Bayern Munich ($ 321,4 )
5. Arsenal ($ 251,1 )
• No Professional football leagues
!Only in Quito
◦ Asociacion de Ligas de Pichincha(27 leagues)
◦ Union de Ligas Independientes (64 leagues)
◦ Federacion Ligas de Quito(84 leagues)
!each one with at least 20 teams.
XIII Encuentro de Matemáticas. Quito-2012– p. 3
6. Motivation
1. Successful implementation of Mathematical Programming
for scheduling the first division of the Ecuadorian Football
League.
2. Managers of Ecuadorian Football Federation (FEF) were
pleased with the results of the Mathematical Approach.
• Inclusion of the new characteristics of equity and
attractiveness in the schedules.
• The ease of having a computational tool to generate
schedules, as opposed to the tedious manual task.
• Unintentional errors of the empirical method are avoided.
XIII Encuentro de Matemáticas. Quito-2012– p. 4
7. Outline
• Motivation.
• The Problem.
• Integer Formulation.
• Algorithms and solution.
• Computacional Results
• Conclusions
XIII Encuentro de Matemáticas. Quito-2012– p. 5
8. The Third Division Ecuadorian Championship
• Every Provincial Association has a championship.
• Who participates in the Third Division Championship? the
two best positioned teams of every provincial championship.
• By regulation, 4 zones must be created.
Current partition in zones empirically created by FEF:
zona 1: Pichincha, Imbabura, Cotopaxi, Tungurahua, Chimborazo,
Bolivar
zona 2: Orellana, Sucumbios, Napo, Pastaza, Morona Santiago
zona 3: Loja, Azuay, Cañar, El Oro, Guayas
zona 4: Esmeraldas, Santo Domingo, Los Ríos, Manabí,
Santa Elena
XIII Encuentro de Matemáticas. Quito-2012– p. 6
9. Summary
Province teams Cities Zone Province teams Cities Zone
Azuay 6 3 3 Loja 12 5 3
Bolivar 6 4 1 Manabí 27 12 4
Cañar 12 5 3 Morona Santiago 5 2 2
Chimborazo 8 2 1 Sucumbios 7 2 2
Cotopaxi 7 3 1 Tungurahua 9 3 1
Esmeraldas 18 4 4 Orellana 6 1 2
El Oro 16 9 3 Pastaza 10 3 2
Guayas 14 3 3 Pichincha 11 5 1
Imbabura 18 5 1 Santa Elena 8 3 4
Los Rios 9 3 4 Santo Domingo 12 1 4
Province teams Cities Zone
Total 20 221 78 4
XIII Encuentro de Matemáticas. Quito-2012– p. 7
10. Our Problem
To find a partition of provinces in zones such that:
• Displacement of every team must be minimized in every
zone.
• Zones (partition) must be balanced in some way (ranking of
teams)
• Fixed number of teams on each zone.
XIII Encuentro de Matemáticas. Quito-2012– p. 8
11. Outline
• Motivation.
• The Problem.
• Integer Formulation.
• Algorithms and solution.
• Computacional Results
• Conclusions
XIII Encuentro de Matemáticas. Quito-2012– p. 9
17. Our Problem
Given:
Undirected Graph, distance-matrix(epresenting the length of the
arc), node activity measures, and an integer number p ∈ N
Task:
Find p sets of nodes(cliques) in such a way that the total cost of
the formed sets is minimized.
Moreover, the sets must be balanced according to different node
activity measures:
! football level ∈ [L0,L1]
! number of teams ∈ [,
19. Notation
Let:
• G = (V,E) : a graph, with the set of nodes V (teams) and
set of edges E.
• dij : represents the distance between nodes i and j.
• fi : denotes the football level of team i ∈ V .
• p : number of cliques to be partitioned.
XIII Encuentro de Matemáticas. Quito-2012– p. 12
20. Notation
Let:
• G = (V,E) : a graph, with the set of nodes V (teams) and
set of edges E.
• dij : represents the distance between nodes i and j.
• fi : denotes the football level of team i ∈ V .
• p : number of cliques to be partitioned.
Variables:
xci
j =
(
1 , if nodes i, j ∈ V are assigned to clique c
0 , otherwise
xci
i =
(
1 , if node i ∈ V belongs to clique c
0 , otherwise
XIII Encuentro de Matemáticas. Quito-2012– p. 12
21. Integer Programming Formulation
m´ın
p
X
c=1
X
(i,j)2E
dijxc
ij
subject to
+ xc
jk − xc
ij + xc
ik ≤ 1, ∀1 ≤ i j k ≤ |V |, c = {1, 2, . . . , p}
+ xc
jk + xc
ij − xc
ik ≤ 1, ∀1 ≤ i j k ≤ |V |, c = {1, 2, . . . , p}
− xc
jk + xc
ij + xc
ik ≤ 1, ∀1 ≤ i j k ≤ |V |, c = {1, 2, . . . , p}
p
X
c=1
xc
ii = 1, ∀i ∈ V
ii ≤Xi
j
xc
xc
ij +Xi
j
xc
ji ≤
22. xc
ii, ∀i ∈ V, c = {1, 2, . . . , p}
L0 ≤X
i2V
fixc
ii ≤ L1, ∀c = {1, 2, . . . , p}
xc
ij ∈ {0, 1}, ∀1 ≤ i ≤ j ≤ |V |, c = {1, 2, . . . , p}
XIII Encuentro de Matemáticas. Quito-2012– p. 13
23. Integer Programming Formulation
m´ın
p
X
c=1
X
(i,j)2E
dijxc
ij
The model is NP-hard!clique partitioning problem
subject to
+ xc
jk − xc
ij + xc
ik ≤ 1, ∀1 ≤ i j k ≤ |V |, c = {1, 2, . . . , p}
+ xc
jk + xc
ij − xc
ik ≤ 1, ∀1 ≤ i j k ≤ |V |, c = {1, 2, . . . , p}
− xc
jk + xc
ij + xc
ik ≤ 1, ∀1 ≤ i j k ≤ |V |, c = {1, 2, . . . , p}
p
X
c=1
xc
ii = 1, ∀i ∈ V
ii ≤Xi
j
xc
xc
ij +Xi
j
xc
ji ≤
24. xc
ii, ∀i ∈ V, c = {1, 2, . . . , p}
L0 ≤X
i2V
fixc
ii ≤ L1, ∀c = {1, 2, . . . , p}
xc
ij ∈ {0, 1}, ∀1 ≤ i ≤ j ≤ |V |, c = {1, 2, . . . , p}
XIII Encuentro de Matemáticas. Quito-2012– p. 13
25. Outline
• Motivation.
• The Problem.
• Integer Formulation.
• Algorithms and solution.
• Computacional Results
• Conclusions
XIII Encuentro de Matemáticas. Quito-2012– p. 14
26. Relaxed Problem
v1
v2
v3
v4
v6 v5
v7
XIII Encuentro de Matemáticas. Quito-2012– p. 15
27. Relaxed Problem
v1
v2
v3
v4
v6 v5
v7
XIII Encuentro de Matemáticas. Quito-2012– p. 15
28. Relaxed Problem
v1
v2
v3
v4
v6
v5
v7
XIII Encuentro de Matemáticas. Quito-2012– p. 15
29. Relaxed Problem
v1
v2
v3
v4
v6
v5
v7
Given: A set of node sources C, where |C| = p
xic =
(
1 , if nodes i ∈ V are assigned to source c ∈ C
0 , otherwise
XIII Encuentro de Matemáticas. Quito-2012– p. 15
30. Relaxed Model(RM)
m´ın
X
c∈C
X
i∈V rC
dicxic
subject to
X
c∈C
xic = 1, ∀i ∈ V r C
≤
X
i∈V rC
xic ≤
31. , ∀c ∈ C
L0 ≤
X
i∈V rC
fixic ≤ L1, ∀c ∈ C
xic ∈ {0, 1}, ∀i ∈ V r C, c ∈ C
XIII Encuentro de Matemáticas. Quito-2012– p. 16
32. Heuristics
The proposed algorithms consists of two phases(Kalcsics,
Nickel and Schöder):
1. Construct the set C of source nodes.
2. Assign v ∈ V r C to the corresponding source c ∈ C.
H1) Choose p random nodes to be included in C.
!probability is proportional to football level.
Solve RM.
H2) Choose nodes (i, j) where dij is the largest distance.
Include (i, j) in set C.
for l = 3, . . . p do
Find the farthest node k to all nodes in C.
C = C ∪ {k}.
end for
Solve RM.
Exclude dij
XIII Encuentro de Matemáticas. Quito-2012– p. 17
33. Local Search
Let Q = {Q1, . . . ,Qp} be the subsets obtained to solving RM.
Compute clique(Q).
for all Qi ∈ Q do
Exchange the source ci with j ∈ Qi
if c(Q′
i) c(Qi) then
ci = j
Solve RM !Q′.
Compute clique(Q′).
if clique(Q′) clique(Q) then
Q = Q′
end if
end if
end for
XIII Encuentro de Matemáticas. Quito-2012– p. 18
34. Branch Bound + Rounding
ii = 0 and xc
xc
ii = 1 xc
ik = 0, ∀k ∈ V
XIII Encuentro de Matemáticas. Quito-2012– p. 19
35. Branch Bound + Rounding
ii = 0 and xc
xc
ii = 1 xc
ik = 0, ∀k ∈ V
xc
ij = 1
✻
xc
jj = 1
✏✮✏
xc
jj = 0
xc
jk = 0, ∀k ∈ V
XIII Encuentro de Matemáticas. Quito-2012– p. 19
36. Branch Bound + Rounding
ii = 0 and xc
xc
ii = 1 xc
ik = 0, ∀k ∈ V
xc
ij = 1
✻
xc
jj = 1
✏✮✏
xc
jj = 0
xc
jk = 0, ∀k ∈ V
Moreover:
(
if xci
i ≥
=⇒ round up variable xci
i = 1
if xci
i ≤ 1 −
=⇒ round down variable xci
i = 0
XIII Encuentro de Matemáticas. Quito-2012– p. 19
37. Outline
• Motivation.
• The Problem.
• Integer Formulation.
• Algorithms and solution.
• Computacional Results
• Conclusions
XIII Encuentro de Matemáticas. Quito-2012– p. 20
38. Computacional Results
• Heuristic method implemented in C++.
• Core i5 with 4Gb RAM
• Gurobi and SCIP
XIII Encuentro de Matemáticas. Quito-2012– p. 21
39. Computacional Results
• Heuristic method implemented in C++.
• Core i5 with 4Gb RAM
• Gurobi and SCIP
IP Heur 1 Heur 2 B B
# nodes p obj t(seg) obj t(seg) obj t(seg) obj t(seg)
11 3 3 2094 44.22 2134 1.66 2181 0.02 2094 1.18
15 3 5 4460 17.44 4535 1.96 4579 0.21 4460 1.42
18 3 6 6230 88.22 6526 2.17 6526 0.34 6238 13.52
26 4 6 – M 8094 3.03 8998 1.05 7354 210.17
20 4 5 8852.4 185 8852.4 2.71 9089.9 0.45 8852.4 0.62
42 8 5 – M 12667.8 9.08 13295.5 1.29 12288 616.05
XIII Encuentro de Matemáticas. Quito-2012– p. 21
42. Outline
• Motivation.
• The Problem.
• Integer Formulation.
• Algorithms and solution.
• Computacional Results
• Conclusions
XIII Encuentro de Matemáticas. Quito-2012– p. 23
43. Conclusions
! Equitable and competitive championships.
! An external entity gives transparency to this process.
! Optimization methods vs empirical methods
Future Work
! Improve the current algorithms.
! Develop new algorithms.
! Analize the problem in special graphs.
XIII Encuentro de Matemáticas. Quito-2012– p. 24
44. For your attention
Thank you !!!
XIII Encuentro de Matemáticas. Quito-2012– p. 25