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Túlio A. M. Toffolo http://www.toffolo.com.br
Otimização Linear e InteiraAula 10: Revisão
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2018/2 - PCC174/BCC464 Aula Prática - Laboratório COM30
/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão
Breve RevisãoThe football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
Modelagem
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
Método gráfico
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
O Algoritmo Simplex Forma padrão Soluções Básicas Método de Duas Fases
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
Dualidade
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
Análise de Sensibilidade
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Modelagem
/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão
Como modelar PLs?The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
O segredo é: praticar
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
Lembre-se: apenas consideramos modelos com variáveis contínuas por enquanto!
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
Dicas: Não se limite aos exemplos apresentados em aula. Os livros indicados na ementa da disciplina possuem exemplos e exercícios resolvidos.
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Método Gráfico
/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão
Método GráficoThe football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
Pode ser utilizado para resolver problemas com até três variáveis.
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
Muito útil para entender os principais conceitos envolvidos em Programação Linear (Inteira).
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/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão !7
x1
x2
�1 1 2 3 4 5 6 7 8 9 10 11 12�1
1
2
3
4
5
6
7
8
9
10
11
12
1
5x1+2x2 � 254x1�3x2 � �3x1 � 2x1, x2 � 0
5 / 48 Túlio Toffolo – Otimização Linear e Inteira – Aula 02: Algoritmo Simplex
/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão !8
O Gráfico
x1
x2
1 2 3 4 5 6 7 8
12345671720,4
Variáveis:x1: qtde de sojax2: qtde de milho
Restrições:Dinheiro (máx: 350)
soja: 70milho: 5070x1 + 50x2 350
Peso (máx: 400)soja: 50milho: 8050x1 + 80x2 400
Disponibilidadesoja: 4x1 4
Lucro (Objetivo):
Max. 300x1 + 280x2
26 / 30 Túlio Toffolo – Otimização Linear e Inteira – Aula 01: Introdução
Algoritmo Simplex
/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão
Formas canônica e padrão
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Ou ainda, considerando-se separadamente as forma padrão e canônica:
TABELA 4.2 FORMA PADRÃO E CANÔNICA
Primal Dual
Forma Canônica
Min z = cx
sujeito a:Ax ! bx ! 0
Max w = ub
sujeito a:uA " cu ! 0
Forma Padrão
Min z = cx
sujeito a:Ax = b
x ! 0
Max w = ub
sujeito a:uA " cu # R
4.2 – EXEMPLOS DE PARES PRIMAL X DUALExemplo 1: Determinar o dual do seguinte modelo de programação linear:
(P) Maximizar z = 6 x1 + 2 x2 + x3
sujeito a:
x1 – x2 + 7 x3 " 4
2 x1 + 3 x2 + x3 " 5
x1 ! 0 , x2 ! 0 , x3 ! 0
Aplicando as regras da dualidade temos que:
(D) Minimizar w = 4 u1 + 5 u2
sujeito a:
u1 + 2 u2 ! 6
– u1 + 3 u2 ! 2
7 u1 + u2 ! 1
u1 ! 0 , u2 ! 0
Exemplo 2: Determinar o dual do seguinte modelo de programação linear, representando graficamenteambos os modelos.
(P) Maximizar z = 3 x1 + 4 x2
sujeito a:
x1 – x2 " – 1
– x1 + x2 " 0
x1 ! 0 , x2 ! 0
D U A L I D A D E E S E N S I B I L I D A D E 1 3 1
Forma Canônica
Forma Padrão
Ou ainda, considerando-se separadamente as forma padrão e canônica:
TABELA 4.2 FORMA PADRÃO E CANÔNICA
Primal Dual
Forma Canônica
Min z = cx
sujeito a:Ax ! b
x ! 0
Max w = ub
sujeito a:uA " c
u ! 0
Forma Padrão
Min z = cx
sujeito a:Ax = b
x ! 0
Max w = ub
sujeito a:uA " c
u # R
4.2 – EXEMPLOS DE PARES PRIMAL X DUALExemplo 1: Determinar o dual do seguinte modelo de programação linear:
(P) Maximizar z = 6 x1 + 2 x2 + x3
sujeito a:
x1 – x2 + 7 x3 " 4
2 x1 + 3 x2 + x3 " 5
x1 ! 0 , x2 ! 0 , x3 ! 0
Aplicando as regras da dualidade temos que:
(D) Minimizar w = 4 u1 + 5 u2
sujeito a:
u1 + 2 u2 ! 6
– u1 + 3 u2 ! 2
7 u1 + u2 ! 1
u1 ! 0 , u2 ! 0
Exemplo 2: Determinar o dual do seguinte modelo de programação linear, representando graficamenteambos os modelos.
(P) Maximizar z = 3 x1 + 4 x2
sujeito a:
x1 – x2 " – 1
– x1 + x2 " 0
x1 ! 0 , x2 ! 0
D U A L I D A D E E S E N S I B I L I D A D E 1 3 1
/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão !11
Algoritmo Simplex - Implementações
GarantiasNo pior caso, o algoritmo tem complexidade exponencialNa prática, os problemas são resolvidos rapidamenteentre 2m e 3m iterações
“Programas Lineares com milhares ou milhões de variáveissão rotineiramente resolvidos utilizando-se o algoritmo Simplex
com computadores modernos.”
Overton, M.L. 1997
6 / 19 Túlio Toffolo – Otimização Linear e Inteira – Aula 01.2: Simplex e Modelagem
/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão !12
Algoritmo Simplex - Visão Geral
Lucro $1900
$200
$1400
O algoritmo inicia em algum vérticefactível; ex.: (0,0)
A cada iteração, são pesquisadosvértices vizinhos e o mesmo semove para o que oferecer melhora(método de subida - Hill Climbing);
Sem vizinhos de melhora? Soluçãoé ótima!
3 / 19 Túlio Toffolo – Otimização Linear e Inteira – Aula 01.2: Simplex e Modelagem
/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão !13
Algoritmo Simplex - Visão Geral
Maximize x1+6x2+13x3
Sujeito a x1 200x2 300
x1+ x2+ x3 400x2+ 3x3 600
x1, x2, x3 � 0
x1
x3
x2
Ótimo
4 / 19 Túlio Toffolo – Otimização Linear e Inteira – Aula 01.2: Simplex e Modelagem
/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão
O Algoritmo Simplex
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O Algoritmo Simplex
Passo 1 Converta o PL para a Forma Padrão.Passo 2 Obtenha uma Solução Básica Factível (se
possível) da Forma Padrão.Passo 3 Teste de Otimalidade: Determine se a Solução
Básica é Ótima. Se Ótima PARE.Passo 4 Caso não seja ótima - Mudança de Base:
determine:qual variável não básica irá entrar na base,com o intuito de melhorar a função objetivo;qual variável básica irá sair da base.
Passo 5 Utilize as operações elementares para computar aNova Solução Básica e volte para o Passo 3.
5 / 34 Túlio Toffolo – Otimização Linear e Inteira – Aula 03: Simplex e Modelagem
/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão
O Algoritmo Simplex
Dica: http://www.cos.ufrj.br/splint/
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Dualidade
/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão
Formas canônica e padrão
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Ou ainda, considerando-se separadamente as forma padrão e canônica:
TABELA 4.2 FORMA PADRÃO E CANÔNICA
Primal Dual
Forma Canônica
Min z = cx
sujeito a:Ax ! bx ! 0
Max w = ub
sujeito a:uA " cu ! 0
Forma Padrão
Min z = cx
sujeito a:Ax = b
x ! 0
Max w = ub
sujeito a:uA " cu # R
4.2 – EXEMPLOS DE PARES PRIMAL X DUALExemplo 1: Determinar o dual do seguinte modelo de programação linear:
(P) Maximizar z = 6 x1 + 2 x2 + x3
sujeito a:
x1 – x2 + 7 x3 " 4
2 x1 + 3 x2 + x3 " 5
x1 ! 0 , x2 ! 0 , x3 ! 0
Aplicando as regras da dualidade temos que:
(D) Minimizar w = 4 u1 + 5 u2
sujeito a:
u1 + 2 u2 ! 6
– u1 + 3 u2 ! 2
7 u1 + u2 ! 1
u1 ! 0 , u2 ! 0
Exemplo 2: Determinar o dual do seguinte modelo de programação linear, representando graficamenteambos os modelos.
(P) Maximizar z = 3 x1 + 4 x2
sujeito a:
x1 – x2 " – 1
– x1 + x2 " 0
x1 ! 0 , x2 ! 0
D U A L I D A D E E S E N S I B I L I D A D E 1 3 1
Forma Canônica
Forma Padrão
Ou ainda, considerando-se separadamente as forma padrão e canônica:
TABELA 4.2 FORMA PADRÃO E CANÔNICA
Primal Dual
Forma Canônica
Min z = cx
sujeito a:Ax ! b
x ! 0
Max w = ub
sujeito a:uA " c
u ! 0
Forma Padrão
Min z = cx
sujeito a:Ax = b
x ! 0
Max w = ub
sujeito a:uA " c
u # R
4.2 – EXEMPLOS DE PARES PRIMAL X DUALExemplo 1: Determinar o dual do seguinte modelo de programação linear:
(P) Maximizar z = 6 x1 + 2 x2 + x3
sujeito a:
x1 – x2 + 7 x3 " 4
2 x1 + 3 x2 + x3 " 5
x1 ! 0 , x2 ! 0 , x3 ! 0
Aplicando as regras da dualidade temos que:
(D) Minimizar w = 4 u1 + 5 u2
sujeito a:
u1 + 2 u2 ! 6
– u1 + 3 u2 ! 2
7 u1 + u2 ! 1
u1 ! 0 , u2 ! 0
Exemplo 2: Determinar o dual do seguinte modelo de programação linear, representando graficamenteambos os modelos.
(P) Maximizar z = 3 x1 + 4 x2
sujeito a:
x1 – x2 " – 1
– x1 + x2 " 0
x1 ! 0 , x2 ! 0
D U A L I D A D E E S E N S I B I L I D A D E 1 3 1
/ 12/ 47 Túlio Toffolo — Otimização Linear e Inteira — Aula 05: Dualidade
Dualidade
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
O dual de um problema de maximização é um problema de minimização.
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
As m restrições primais estão em correspondência com as m variáveis duais.
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
As n restrições duais estão em correspondência com as n variáveis primais.
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
O coeficiente de cada variável na FO, primal ou dual, aparece no outro problema como lado direito da restrição correspondente.
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
A matriz de coeficientes do primal é transposta no dual.
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/ 12/ 47 Túlio Toffolo — Otimização Linear e Inteira — Aula 05: Dualidade
DualidadeThe football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
Em um problema de minimização….
x* = u*
f(x)
fD(x)
(valor do primal)
(valor do dual)
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/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão
DualidadeThe football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
Relações entre o primal e o dual:
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Ótimo Ilimitado Inviável
Ótimo Possível Nunca Nunca
Ilimitado Nunca Nunca Possível
Inviável Nunca Possível Possível
/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão
Transformação Primal x Dual
MIN
Restrição≤ ≤
Variável
MAX
= qq.≥ ≥
Variável≤ ≥
Restriçãoqq. =≥ ≤
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Análise de Sensibilidade
/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão
Custo Reduzido
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
O custo reduzido de uma variável é o custo (coeficiente) dela na linha da função objetivo no tableau do Simplex
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
!23
Após Pivoteamento: Nova SBF
0: 5x2 + 10x5 + 10x6 = 280 z = 280
1: � 2x2 + x4 + 2x5 � 8x6 = 24 x4 = 24
2: � 2x2 + x3 + 2x5 � 4x6 = 8 x3 = 8
3: x1 + 1, 25x2 + � 0, 5x5 + 1, 5x6 = 4 x1 = 4
4: x2 + x7 = 5 x7 = 5
Função objetivo: max z = �5x2 � 10x5 � 10x6
Não há mais variáveis atrativas
SOLUÇÃO ÓTIMA (x1, . . . , x3): (2 , 0 , 8)BASE ÓTIMA (x1, . . . , x7): (2 , 0 , 8 , 24 , 0 , 0 , 5)
41 / 48 Túlio Toffolo – Otimização Linear e Inteira – Aula 02: Algoritmo Simplex
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
O custo reduzido é também o shadow price das restrições de não-negatividade
/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão
Variáveis de FolgaThe football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
As variáveis de folga indicam restrições ativas e inativas
!24
Após Pivoteamento: Nova SBF
0: 5x2 + 10x5 + 10x6 = 280 z = 280
1: � 2x2 + x4 + 2x5 � 8x6 = 24 x4 = 24
2: � 2x2 + x3 + 2x5 � 4x6 = 8 x3 = 8
3: x1 + 1, 25x2 + � 0, 5x5 + 1, 5x6 = 4 x1 = 4
4: x2 + x7 = 5 x7 = 5
Função objetivo: max z = �5x2 � 10x5 � 10x6
Não há mais variáveis atrativas
SOLUÇÃO ÓTIMA (x1, . . . , x3): (2 , 0 , 8)BASE ÓTIMA (x1, . . . , x7): (2 , 0 , 8 , 24 , 0 , 0 , 5)
41 / 48 Túlio Toffolo – Otimização Linear e Inteira – Aula 02: Algoritmo Simplex
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
As restrições ativas (com variáveis de folga iguais a zero) são as únicas limitando o lucro.
/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão
Shadow PriceThe football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
Também conhecido como custo dual
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
O shadow price de uma restrição é o valor que a variável dual referente à restrição assume.
Indica o ganho na função objetivo por unidade aumentada no RHS da restrição. O ganho é válido dentro dos limites estabelecidos pelo RHS da restrição.
!25
/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão
Allowable increase/decreaseThe football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
Indica o aumento/redução permitidos em coeficientes da função objetivo ou no RHS de uma restrição sem que a base deixe de ser ótima!
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
O cálculo é feito tomando como base o custo reduzido (de variáveis primais e duais).
Regra geral:
The football leagues grouping problem
Problem constraints:
Leagues must comprise between m� and m+ teams.At most 2 teams from the same club can be in a league.There is a limit on the level difference between teams inthe same league.There is a limit on the travel time/distance between teamsin the same league.
It’s a generalization of the clique partitioning problem withminimum clique size requirement.
7 / 25 Toffolo et al. – IP heuristics for nesting problems
Pequenas modificações nos dados geralmente não alteram o conjunto de variáveis básicas.
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/ 12/ 27 Túlio Toffolo — Otimização Linear e Inteira — Aula 09: Revisão
Perguntas?
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