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8th AIMMS/MOPTA
COMPETITION
Multiobjective network disruption
Team 4101
Advisor: Andrés Medaglia, Ph.D
Felipe Solano (Leader, M.Sc. student)
Diego Cely (Undergraduate student)
Santiago Cabrera (Undergraduate student)
Centro para la Optimización y Probabilidad Aplicada(COPA)
Departamento de Ingeniería Industrial
Universidad de los Andes (Colombia)
http://copa.uniandes.edu.co/
Agenda
• Problem description
• Part I solution strategy
• Part II solution strategy
• AIMMS user interface & results
2
http://copa.uniandes.edu.co/
Agenda
• Problem description
• Part I solution strategy
• Part II solution strategy
• AIMMS user interface & results
3
http://copa.uniandes.edu.co/
Competition
4
MOPTA 2016
competition
Part 2Part 1
MILP BendersModified
Benders
http://copa.uniandes.edu.co/
Part 1
5
MOPTA 2016
competition
Part 2Part 1
MILP BendersModified
Benders
http://copa.uniandes.edu.co/
Problem description
6
Part 1
• Undirected network
Set V
Transition nodes
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Problem description
7
Part 1
• Undirected network
• Capacitated edgesSet E
Transition nodes
http://copa.uniandes.edu.co/
Problem description
8
Part 1
• Undirected network
• Capacitated edges
Transition nodes
Source nodes
Subset S1
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Problem description
9
Part 1
∞
∞
∞
∞
• Undirected network
• Capacitated edges
• Infinite supply
Transition nodes
Source nodes
http://copa.uniandes.edu.co/
Problem description
10
Part 1
• Undirected network
• Capacitated edges
• Infinite supply
Transition nodes
Source nodes
Sink nodesSubset I1
http://copa.uniandes.edu.co/
Problem description
11
Part 1
∞
∞
∞
∞
𝒅𝟏
𝒅𝟐
𝒅𝟑
𝒅𝟒
𝒅𝟓
• Undirected network
• Capacitated edges
• Infinite supply
• Finite demandCan be
partially
satisfied
Transition nodes
Source nodes
Sink nodes
http://copa.uniandes.edu.co/
Problem description
12
Part 1
∞
∞
∞
∞
𝑑1
𝑑2
𝑑3
𝑑4
𝑑5
• Undirected network
• Capacitated edges
• Infinite supply
• Finite demand
Transition nodes
Source nodes
Sink nodes
Transporting
Agent
http://copa.uniandes.edu.co/
Problem description
13
Part 1
∞
∞
∞
∞
𝑑1
𝑑2
𝑑3
𝑑4
𝑑5
• Undirected network
• Capacitated edges
• Infinite supply
• Finite demand
Transition nodes
Source nodes
Sink nodes
Interdictor
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Problem description
14
Transition nodes
Source nodes
Sink nodes
• Undirected network
• Capacitated edges
• Infinite supply
• Finite demand
• Budget
∞
∞
∞
∞
𝑑1
𝑑2
𝑑3
𝑑4
𝑑5
Part 1
𝜷
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Part 2
15
MOPTA 2016
competition
Part 2Part 1
MILP BendersModified
Benders
http://copa.uniandes.edu.co/
Problem description
16
Part 2
∞
∞
∞
∞
𝑑1
𝑑2
𝑑3
𝑑4
𝑑5
Two agents
Share the
network
• Undirected network
• Capacitated arcs
• Infinite supply
• Finite demand
Transition nodes
Source nodes
Sink nodes
http://copa.uniandes.edu.co/
Problem description
17
Part 2
∞
∞
∞
∞
𝑑1
𝑑2
𝑑3
𝑑4
𝑑5
• Undirected network
• Capacitated arcs
• Infinite supply
• Finite demand
Transition nodes
Source nodes
Sink nodesSubset S1
http://copa.uniandes.edu.co/
Problem description
18
Part 2
∞
∞
∞
∞
𝑑1
𝑑2
𝑑3
𝑑4
𝑑5
• Undirected network
• Capacitated arcs
• Infinite supply
• Finite demand
Transition nodes
Source nodes
Sink nodesSubset I1
http://copa.uniandes.edu.co/
Problem description
19
Part 2
∞
∞
∞
∞
𝑑1
𝑑2
𝑑3
𝑑4
𝑑5
• Undirected network
• Capacitated arcs
• Infinite supply
• Finite demand
Transition nodes
Source nodes
Sink nodesSubset S2
http://copa.uniandes.edu.co/
Problem description
20
Part 2
∞
∞
∞
∞
𝑑1
𝑑2
𝑑3
𝑑4
𝑑5
• Undirected network
• Capacitated arcs
• Infinite supply
• Finite demand
Transition nodes
Source nodes
Sink nodesSubset I2
http://copa.uniandes.edu.co/
Problem description
21
𝐸 → 𝒜
Part 1
2 𝐸 = 𝒜
𝑖 𝑗
Undirected
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Problem description
22
𝐸 → 𝒜
Part 1
2 𝐸 = 𝒜
𝑖 𝑗
Directed
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Lemma
23
Part 1
[2] Lim & Smith (2007).
5 6
2 3
4
1
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Lemma
24
Part 1
[2] Lim & Smith (2007).
5 6
2 3
4
1
5 6
2 3
4
1
http://copa.uniandes.edu.co/ 25
Part 1
[2] Lim & Smith (2007).
5 6
2 3
4
1
5 6
2 3
4
1
Lemma
http://copa.uniandes.edu.co/ 26
Part 1
[2] Lim & Smith (2007).
5 6
2 3
4
1
5 6
2 3
4
1
5 6
2 3
4
1
Lemma
http://copa.uniandes.edu.co/ 27
Part 1
[2] Lim & Smith (2007).
5 6
2 3
4
1
5 6
2 3
4
1
5 6
2 3
4
1
5 6
2 3
4
1
Lemma
http://copa.uniandes.edu.co/
Agenda
• Problem description
• Part I solution strategy
• Part II solution strategy
• AIMMS user interface & results
28
http://copa.uniandes.edu.co/
Part 1
29
MOPTA 2016
competition
Part 2Part 1
MILP BendersModified
Benders
http://copa.uniandes.edu.co/
Formulation
30
min𝑧
max𝑥
𝑖∈𝐼
𝑝𝑖𝑦𝑖 −
𝑖,𝑗 ∈𝒜
𝑏𝑖𝑗𝑥𝑖𝑗
Part 1
Enemy´s
profit
http://copa.uniandes.edu.co/
Formulation
31
min𝑧
max𝑥
𝑖∈𝐼
𝑝𝑖𝑦𝑖 −
𝑖,𝑗 ∈𝒜
𝑏𝑖𝑗𝑥𝑖𝑗
Part 1
Income
http://copa.uniandes.edu.co/
Formulation
32
min𝑧
max𝑥
𝑖∈𝐼
𝑝𝑖𝑦𝑖 −
𝑖,𝑗 ∈𝒜
𝑏𝑖𝑗𝑥𝑖𝑗
Part 1
Transport
cost
http://copa.uniandes.edu.co/
Formulation
33
min𝑧
max𝑥
𝑖∈𝐼
𝑝𝑖𝑦𝑖 −
𝑖,𝑗 ∈𝒜
𝑏𝑖𝑗𝑥𝑖𝑗
s.t,
𝑗: 𝑖,𝑗 ∈𝒜
𝑥𝑖𝑗 −
𝑗: 𝑗,𝑖 ∈𝒜
𝑥𝑗𝑖 = 0 ∀𝑖 ∈ 𝑉\ 𝑆 ∪ 𝐼−𝑦𝑖 ∀𝑖 ∈ 𝐼𝑦𝑖 ∀𝑖 ∈ 𝑆
(1)(2)(3)
𝑦𝑖 ≤ 𝑑𝑖 ∀𝑖 ∈ 𝐼 (4)
𝑥𝑖𝑗 + 𝑥𝑗𝑖 ≤ 𝑤𝑖𝑗(1 − 𝑧𝑖𝑗) ∀(𝑖, 𝑗) ∈ 𝒜|𝑖 < 𝑗 (5)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜; 𝑥𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝒜;
𝑦𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 ∪ 𝐼
(6)
Part 1
Balance constraints
http://copa.uniandes.edu.co/
Formulation
34
min𝑧
max𝑥
𝑖∈𝐼
𝑝𝑖𝑦𝑖 −
𝑖,𝑗 ∈𝒜
𝑏𝑖𝑗𝑥𝑖𝑗
s.t,
𝑗: 𝑖,𝑗 ∈𝒜
𝑥𝑖𝑗 −
𝑗: 𝑗,𝑖 ∈𝒜
𝑥𝑗𝑖 = 0 ∀𝑖 ∈ 𝑉\ 𝑆 ∪ 𝐼−𝑦𝑖 ∀𝑖 ∈ 𝐼𝑦𝑖 ∀𝑖 ∈ 𝑆
(1)(2)(3)
𝑦𝑖 ≤ 𝑑𝑖 ∀𝑖 ∈ 𝐼 (4)
𝑥𝑖𝑗 + 𝑥𝑗𝑖 ≤ 𝑤𝑖𝑗(1 − 𝑧𝑖𝑗) ∀(𝑖, 𝑗) ∈ 𝒜|𝑖 < 𝑗 (5)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜; 𝑥𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝒜;
𝑦𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 ∪ 𝐼
(6)
Part 1
Delivered units are
less than or equal to
demand
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Formulation
35
min𝑧
max𝑥
𝑖∈𝐼
𝑝𝑖𝑦𝑖 −
𝑖,𝑗 ∈𝒜
𝑏𝑖𝑗𝑥𝑖𝑗
s.t,
𝑗: 𝑖,𝑗 ∈𝒜
𝑥𝑖𝑗 −
𝑗: 𝑗,𝑖 ∈𝒜
𝑥𝑗𝑖 = 0 ∀𝑖 ∈ 𝑉\ 𝑆 ∪ 𝐼−𝑦𝑖 ∀𝑖 ∈ 𝐼𝑦𝑖 ∀𝑖 ∈ 𝑆
(1)(2)(3)
𝑦𝑖 ≤ 𝑑𝑖 ∀𝑖 ∈ 𝐼 (4)
𝑥𝑖𝑗 + 𝑥𝑗𝑖 ≤ 𝑤𝑖𝑗(1 − 𝑧𝑖𝑗) ∀(𝑖, 𝑗) ∈ 𝒜|𝑖 < 𝑗 (5)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜; 𝑥𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝒜;
𝑦𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 ∪ 𝐼
(6)
Part 1
The capacity of
each arc is not
exceeded
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Formulation
36
min𝑧
max𝑥
𝑖∈𝐼
𝑝𝑖𝑦𝑖 −
𝑖,𝑗 ∈𝒜
𝑏𝑖𝑗𝑥𝑖𝑗
s.t,
𝑗: 𝑖,𝑗 ∈𝒜
𝑥𝑖𝑗 −
𝑗: 𝑗,𝑖 ∈𝒜
𝑥𝑗𝑖 = 0 ∀𝑖 ∈ 𝑉\ 𝑆 ∪ 𝐼−𝑦𝑖 ∀𝑖 ∈ 𝐼𝑦𝑖 ∀𝑖 ∈ 𝑆
(1)(2)(3)
𝑦𝑖 ≤ 𝑑𝑖 ∀𝑖 ∈ 𝐼 (4)
𝑥𝑖𝑗 + 𝑥𝑗𝑖 ≤ 𝑤𝑖𝑗(1 − 𝑧𝑖𝑗) ∀(𝑖, 𝑗) ∈ 𝒜|𝑖 < 𝑗 (5)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜; 𝑥𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝒜;
𝑦𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 ∪ 𝐼
(6)
Part 1
Interdictor’s
plan
http://copa.uniandes.edu.co/
Formulation
37
min𝑧
max𝑥
𝑖∈𝐼
𝑝𝑖𝑦𝑖 −
𝑖,𝑗 ∈𝒜
𝑏𝑖𝑗𝑥𝑖𝑗
s.t,
𝑗: 𝑖,𝑗 ∈𝒜
𝑥𝑖𝑗 −
𝑗: 𝑗,𝑖 ∈𝒜
𝑥𝑗𝑖 = 0 ∀𝑖 ∈ 𝑉\ 𝑆 ∪ 𝐼−𝑦𝑖 ∀𝑖 ∈ 𝐼𝑦𝑖 ∀𝑖 ∈ 𝑆
(1)(2)(3)
𝑦𝑖 ≤ 𝑑𝑖 ∀𝑖 ∈ 𝐼 (4)
𝑥𝑖𝑗 + 𝑥𝑗𝑖 ≤ 𝑤𝑖𝑗(1 − 𝑧𝑖𝑗) ∀(𝑖, 𝑗) ∈ 𝒜|𝑖 < 𝑗 (5)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜; 𝑥𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝒜;
𝑦𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 ∪ 𝐼
(6)
Part 1
Variables’
domain
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Part 1
38
MOPTA 2016
competition
Part 2Part 1
MILP BendersModified
Benders
[1] Wood (1993)
http://copa.uniandes.edu.co/ 39
min𝑧
max𝑥
𝑖∈𝐼
𝑝𝑖𝑦𝑖 −
𝑖,𝑗 ∈𝒜
𝑏𝑖𝑗𝑥𝑖𝑗
s.t,
Part 1
Constraints 1 − (4)
𝑥𝑖𝑗 + 𝑥𝑗𝑖 ≤ 𝑤𝑖𝑗 1 − 𝑧𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜|𝑖 < 𝑗 (5.1)
Constraints (6)
Parameter
First approach – MILP
http://copa.uniandes.edu.co/ 40
min𝑧
max𝑥
𝑖∈𝐼
𝑝𝑖𝑦𝑖 −
𝑖,𝑗 ∈𝒜
𝑏𝑖𝑗𝑥𝑖𝑗
s.t,
Part 1
Constraints 1 − (4)
𝑥𝑖𝑗 + 𝑥𝑗𝑖 ≤ 𝑤𝑖𝑗 1 − 𝑧𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜|𝑖 < 𝑗 (5.1)
Constraints (6)
We will move
interdictor’s
decisions to the
objective function
First approach – MILP
http://copa.uniandes.edu.co/ 41
min𝑧
max𝑥
𝑖∈𝐼
𝑝𝑖𝑦𝑖 −
𝑖,𝑗 ∈𝒜
𝑏𝑖𝑗𝑥𝑖𝑗
s.t,
Part 1
Constraints 1 − (4)
𝑥𝑖𝑗 + 𝑥𝑗𝑖 ≤ 𝑤𝑖𝑗 1 − 𝑧𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜|𝑖 < 𝑗 (5.1)
Constraints (6)
We will move
interdictor’s
decisions to the
objective function
First approach – MILP
http://copa.uniandes.edu.co/ 42
min𝑧
max𝑥
𝑖∈𝐼
𝑝𝑖𝑦𝑖 −
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗 𝑧𝑖𝑗 𝑥𝑖𝑗 + 𝑥𝑗𝑖
s.t,
Part 1
Constraints 1 − (4)
𝑥𝑖𝑗 + 𝑥𝑗𝑖 ≤ 𝑤𝑖𝑗 1 − 𝑧𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜|𝑖 < 𝑗 (5.1)
Constraints (6)
Interdicted
arcs become
unprofitable
First approach – MILP
http://copa.uniandes.edu.co/ 43
min𝑧
max𝑥
𝑖∈𝐼
𝑝𝑖𝑦𝑖 −
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗 𝑧𝑖𝑗 𝑥𝑖𝑗 + 𝑥𝑗𝑖
s.t,
Part 1
Constraints 1 − (4)
𝑥𝑖𝑗 + 𝑥𝑗𝑖 ≤ 𝑤𝑖𝑗 1 − 𝑧𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜|𝑖 < 𝑗 (5.1)
Constraints (6)
[2] Lim &
Smith
First approach – MILPLim & Smith (2007)
http://copa.uniandes.edu.co/
First approach – MILP
44
min
𝑖∈𝑉
0𝑣𝑖 +
𝑖∈𝐼
𝑑𝑖𝑞𝑖 +
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑤𝑖𝑗𝑢𝑖𝑗
Part 1
Dual variable
for balance
constraints
http://copa.uniandes.edu.co/
First approach – MILP
45
min
𝑖∈𝑉
0𝑣𝑖 +
𝑖∈𝐼
𝑑𝑖𝑞𝑖 +
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑤𝑖𝑗𝑢𝑖𝑗
Part 1
Dual variable
of demand
constraints
http://copa.uniandes.edu.co/
First approach – MILP
46
min
𝑖∈𝑉
0𝑣𝑖 +
𝑖∈𝐼
𝑑𝑖𝑞𝑖 +
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑤𝑖𝑗𝑢𝑖𝑗
Part 1
Dual variable
of capacity
constraints
http://copa.uniandes.edu.co/ 47
min
𝑖∈𝑉
0𝑣𝑖 +
𝑖∈𝐼
𝑑𝑖𝑞𝑖 +
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑤𝑖𝑗𝑢𝑖𝑗
s.t,
Part 1
𝑣𝑖 − 𝑣𝑗 + 𝑢𝑖𝑗 + 𝑀𝑖𝑗 𝑧𝑖𝑗 ≥ −𝑏𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜| 𝑖 < 𝑗 (1)
𝑣𝑗 − 𝑣𝑖 + 𝑢𝑖𝑗 + 𝑀𝑖𝑗 𝑧𝑖𝑗 ≥ −𝑏𝑗𝑖 ∀(𝑖, 𝑗) ∈ 𝒜| 𝑖 < 𝑗 (2)
𝑣𝑖 + 𝑞𝑖 ≥ 𝑝𝑖 ∀𝑖 ∈ 𝐼 (3)
−𝑣𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 (4)
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑤𝑖𝑗 (5)
𝑞𝑖 ≥ 0 ∀𝑖 ∈ 𝐼; 𝑢𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝐸|𝑖 < 𝑗; 𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝐸 (6)
First approach – MILP
Dual
constraints of
flow variables
http://copa.uniandes.edu.co/ 48
min
𝑖∈𝑉
0𝑣𝑖 +
𝑖∈𝐼
𝑑𝑖𝑞𝑖 +
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑤𝑖𝑗𝑢𝑖𝑗
s.t,
Part 1
𝑣𝑖 − 𝑣𝑗 + 𝑢𝑖𝑗 + 𝑀𝑖𝑗 𝑧𝑖𝑗 ≥ −𝑏𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜| 𝑖 < 𝑗 (1)
𝑣𝑗 − 𝑣𝑖 + 𝑢𝑖𝑗 + 𝑀𝑖𝑗 𝑧𝑖𝑗 ≥ −𝑏𝑗𝑖 ∀(𝑖, 𝑗) ∈ 𝒜| 𝑖 < 𝑗 (2)
𝑣𝑖 + 𝑞𝑖 ≥ 𝑝𝑖 ∀𝑖 ∈ 𝐼 (3)
−𝑣𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 (4)
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑤𝑖𝑗 (5)
𝑞𝑖 ≥ 0 ∀𝑖 ∈ 𝐼; 𝑢𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝐸|𝑖 < 𝑗; 𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝐸 (6)
First approach – MILP
Dual
constraints of
𝑦𝑖
http://copa.uniandes.edu.co/ 49
min
𝑖∈𝑉
0𝑣𝑖 +
𝑖∈𝐼
𝑑𝑖𝑞𝑖 +
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑤𝑖𝑗𝑢𝑖𝑗
s.t,
Part 1
𝑣𝑖 − 𝑣𝑗 + 𝑢𝑖𝑗 + 𝑀𝑖𝑗 𝑧𝑖𝑗 ≥ −𝑏𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜| 𝑖 < 𝑗 (1)
𝑣𝑗 − 𝑣𝑖 + 𝑢𝑖𝑗 + 𝑀𝑖𝑗 𝑧𝑖𝑗 ≥ −𝑏𝑗𝑖 ∀(𝑖, 𝑗) ∈ 𝒜| 𝑖 < 𝑗 (2)
𝑣𝑖 + 𝑞𝑖 ≥ 𝑝𝑖 ∀𝑖 ∈ 𝐼 (3)
−𝑣𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 (4)
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗 𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗 (5)
𝑞𝑖 ≥ 0 ∀𝑖 ∈ 𝐼; 𝑢𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝒜|𝑖 < 𝑗; 𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜 (6)
First approach – MILP
Variables’
domain
http://copa.uniandes.edu.co/ 50
min
𝑖∈𝑉
0𝑣𝑖 +
𝑖∈𝐼
𝑑𝑖𝑞𝑖 +
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑤𝑖𝑗𝑢𝑖𝑗
s.t,
Part 1
𝑣𝑖 − 𝑣𝑗 + 𝑢𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 ≥ −𝑏𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜| 𝑖 < 𝑗 (1)
𝑣𝑗 − 𝑣𝑖 + 𝑢𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 ≥ −𝑏𝑗𝑖 ∀(𝑖, 𝑗) ∈ 𝒜| 𝑖 < 𝑗 (2)
𝑣𝑖 + 𝑞𝑖 ≥ 𝑝𝑖 ∀𝑖 ∈ 𝐼 (3)
−𝑣𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 (4)
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗 (5)
𝑞𝑖 ≥ 0 ∀𝑖 ∈ 𝐼; 𝑢𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝒜|𝑖 < 𝑗; 𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜 (6)
First approach – MILP
𝑧𝑖𝑗 → 𝑧𝑖𝑗
http://copa.uniandes.edu.co/ 51
min
𝑖∈𝑉
0𝑣𝑖 +
𝑖∈𝐼
𝑑𝑖𝑞𝑖 +
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑤𝑖𝑗𝑢𝑖𝑗
s.t,
Part 1
𝑣𝑖 − 𝑣𝑗 + 𝑢𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 ≥ −𝑏𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜| 𝑖 < 𝑗 (1)
𝑣𝑗 − 𝑣𝑖 + 𝑢𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 ≥ −𝑏𝑗𝑖 ∀(𝑖, 𝑗) ∈ 𝒜| 𝑖 < 𝑗 (2)
𝑣𝑖 + 𝑞𝑖 ≥ 𝑝𝑖 ∀𝑖 ∈ 𝐼 (3)
−𝑣𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 (4)
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗 (5)
𝑞𝑖 ≥ 0 ∀𝑖 ∈ 𝐼; 𝑢𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝒜|𝑖 < 𝑗; 𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜 (6)
First approach – MILP
Interdictor’s
budget
constraint
http://copa.uniandes.edu.co/ 52
min
𝑖∈𝑉
0𝑣𝑖 +
𝑖∈𝐼
𝑑𝑖𝑞𝑖 +
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑤𝑖𝑗𝑢𝑖𝑗
s.t,
Part 1
𝑣𝑖 − 𝑣𝑗 + 𝑢𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 ≥ −𝑏𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜| 𝑖 < 𝑗 (1)
𝑣𝑗 − 𝑣𝑖 + 𝑢𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 ≥ −𝑏𝑗𝑖 ∀(𝑖, 𝑗) ∈ 𝒜| 𝑖 < 𝑗 (2)
𝑣𝑖 + 𝑞𝑖 ≥ 𝑝𝑖 ∀𝑖 ∈ 𝐼 (3)
−𝑣𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 (4)
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗 (5)
𝑞𝑖 ≥ 0 ∀𝑖 ∈ 𝐼; 𝑢𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝒜|𝑖 < 𝑗; 𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜 (6)
First approach – MILP
http://copa.uniandes.edu.co/
Part 1
53
MOPTA 2016
competition
Part 2Part 1
MILP BendersModified
Benders
[3] Wood (2011)
http://copa.uniandes.edu.co/
Second approach - Benders
54
Part 1
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Second approach - Benders
55
Part 1
Transportation plan
𝑥𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜
http://copa.uniandes.edu.co/
Second approach - Benders
56
Part 1
Transportation plan
𝑥𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜Interdiction plan
𝑧𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜
http://copa.uniandes.edu.co/
Second approach - Benders
57
Part 1
Profit of transporting agentmax
subject to
• Flow constraints
• Arc capacity constraints
decision
• Transportation plan
http://copa.uniandes.edu.co/
Second approach - Benders
58
Part 1
Best solution of the enemymin
subject to
• Best solution is greater than or
equal to any solution
considering interdiction plan.
• Interdiction budget
decision
• Interdiction plan
http://copa.uniandes.edu.co/
Second approach - Benders
59
Part 1
Best solution of the enemymin
subject to
• Best solution is greater than or
equal to any solution
considering interdiction plan.
• Interdiction budget
decision
• Interdiction plan
Profit of the enemymax
subject to
• Flow constraints
• Arc capacity constraints
decision
• Transportation plan
http://copa.uniandes.edu.co/
Second approach - Benders
60
Part 1
Best solution of the enemymin
subject to
• Best solution is greater than or
equal to any solution
considering interdiction plan.
• Interdiction budget
decision
• Interdiction plan
Profit of the enemymax
subject to
• Flow constraints
• Arc capacity constraints
decision
• Transportation plan
http://copa.uniandes.edu.co/
Second approach - Benders
61
max𝑥
𝑖∈𝐼
𝑝𝑖𝑦𝑖 −
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗 𝑧𝑖𝑗 𝑥𝑖𝑗 + 𝑥𝑗𝑖
s.t,
𝑗: 𝑖,𝑗 ∈𝒜
𝑥𝑖𝑗 −
𝑗: 𝑗,𝑖 ∈𝒜
𝑥𝑗𝑖 = 0 ∀𝑖 ∈ 𝑉\ 𝑆 ∪ 𝐼−𝑦𝑖 ∀𝑖 ∈ 𝐼𝑦𝑖 ∀𝑖 ∈ 𝑆
(1)(2)(3)
𝑦𝑖 ≤ 𝑑𝑖 ∀𝑖 ∈ 𝐼 (4)
𝑥𝑖𝑗 + 𝑥𝑗𝑖 ≤ 𝑤𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜|𝑖 < 𝑗 (5)
𝑥𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝒜; 𝑦𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 ∪ 𝐼 (6)
Part 1
Profit of transporting agent
• Flow constraints
• Transportation plan
• Arc capacity constraints
http://copa.uniandes.edu.co/
Second approach - Benders
62
max𝑥
𝑖∈𝐼
𝑝𝑖𝑦𝑖 −
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗 𝑧𝑖𝑗 𝑥𝑖𝑗 + 𝑥𝑗𝑖
s.t,
𝑗: 𝑖,𝑗 ∈𝒜
𝑥𝑖𝑗 −
𝑗: 𝑗,𝑖 ∈𝒜
𝑥𝑗𝑖 = 0 ∀𝑖 ∈ 𝑉\ 𝑆 ∪ 𝐼−𝑦𝑖 ∀𝑖 ∈ 𝐼𝑦𝑖 ∀𝑖 ∈ 𝑆
(1)(2)(3)
𝑦𝑖 ≤ 𝑑𝑖 ∀𝑖 ∈ 𝐼 (4)
𝑥𝑖𝑗 + 𝑥𝑗𝑖 ≤ 𝑤𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜|𝑖 < 𝑗 (5)
𝑥𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝒜; 𝑦𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 ∪ 𝐼 (6)
Part 1
http://copa.uniandes.edu.co/
Second approach - Benders
63
min𝑧,𝜑−
𝜑−
s.t,
𝜑− ≥
𝑖∈𝐼
𝑝𝑖 𝑦𝑖 −
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛 + 𝑥𝑗𝑖𝑛 ∀𝑛 ∈ 𝑌
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜
Part 1
Best
solution of
the enemy
• Best solution is greater than or
equal to any solution
considering interdiction plan.
• Interdiction plan
• Interdiction budget
http://copa.uniandes.edu.co/
Second approach - Benders
64
min𝑧,𝜑−
𝜑−
s.t,
𝜑− ≥
𝑖∈𝐼
𝑝𝑖 𝑦𝑖 −
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛 + 𝑥𝑗𝑖𝑛 ∀𝑛 ∈ 𝑌(1)
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜
Part 1
Add a cut such that 𝜑−
is greater than or equal
to any solution n
• Interdiction plan
• Interdiction budget
http://copa.uniandes.edu.co/
𝜑− ≥
𝑖∈𝐼
𝑝𝑖 𝑦𝑖 −
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛 + 𝑥𝑗𝑖𝑛 ∀𝑛 ∈ 𝑌(1)
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗(2)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜
𝜑− ≥ 0
(3)
Second approach - Benders
65
min𝑧,𝜑−
𝜑−
s.t,
Part 1
By interdicting arcs, the
right side of the
constraint is minimized
• Interdiction plan
• Interdiction budget
http://copa.uniandes.edu.co/
𝜑− ≥
𝑖∈𝐼
𝑝𝑖 𝑦𝑖 −
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛 + 𝑥𝑗𝑖𝑛 ∀𝑛 ∈ 𝑌(1)
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗(2)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜
𝜑− ≥ 0
(3)
Second approach - Benders
66
min𝑧,𝜑−
𝜑−
s.t,
Part 1
Interdictor’s
budget
constraint
• Interdiction plan
http://copa.uniandes.edu.co/
Second approach - Benders
67
min𝑧,𝜑−
𝜑−
s.t,
𝜑− ≥
𝑖∈𝐼
𝑝𝑖 𝑦𝑖 −
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛 + 𝑥𝑗𝑖𝑛 ∀𝑛 ∈ 𝑌(1)
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗(2)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜𝜑− ∈ ℝ1
(3)
Part 1
Variables’
domain
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Part 2
68
MOPTA 2016
competition
Part 2Part 1
MILP BendersModified
Benders
[3] Wood (2011)
&
http://copa.uniandes.edu.co/
Idea
69
Part 2
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Idea
70
Part 2
𝑓𝑙𝑜𝑤 + 𝑓𝑙𝑜𝑤≤ 𝑤?
http://copa.uniandes.edu.co/
Idea
71
Part 2
𝑓𝑙𝑜𝑤 + 𝑓𝑙𝑜𝑤≤ 𝑤?
No No ↑ $↑ $
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Idea
72
Part 2
𝑓𝑙𝑜𝑤 + 𝑓𝑙𝑜𝑤≤ 𝑤?
No No ↑ $↑ $
http://copa.uniandes.edu.co/
Idea
73
Part 2
𝑓𝑙𝑜𝑤 + 𝑓𝑙𝑜𝑤≤ 𝑤?
Yes
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Idea
74
Part 2
𝑓𝑙𝑜𝑤 + 𝑓𝑙𝑜𝑤≤ 𝑤?
Yes
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Idea
75
Part 2
𝑓𝑙𝑜𝑤 + 𝑓𝑙𝑜𝑤≤ 𝑤?
No No
Yes
↑ $↑ $
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Modified Benders
76
Part 2
Profit of the enemymax
subject to
• Flow constraints
• Arc capacity constraints
decision
• Transportation plan of enemy.
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Modified Benders
77
Part 2
subject to
decision
Profit of the allymax
• Flow constraints
• Arc capacity constraints
• Transportation plan of ally.
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Modified Benders
78
Part 2
Profit of the enemymax
subject to
• Flow constraints
• Arc capacity constraints
decision
• Transportation plan of enemy.
Profit of the allymax
• Flow constraints
• Arc capacity constraints
• Transportation plan of ally.
and
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Enemy and Ally problems
79
max𝑥
𝑖∈𝐼
𝑝𝑖𝑦𝑖 −
𝑖,𝑗 ∈𝐸|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗 𝑧𝑖𝑗 𝑥𝑖𝑗 + 𝑥𝑗𝑖
s.t,
Part 2
𝑗: 𝑖,𝑗 ∈𝒜
𝑥𝑖𝑗 −
𝑗: 𝑗,𝑖 ∈𝒜
𝑥𝑗𝑖 = 0 ∀𝑖 ∈ 𝑉\ 𝑆 ∪ 𝐼−𝑦𝑖 ∀𝑖 ∈ 𝐼𝑦𝑖 ∀𝑖 ∈ 𝑆
(1)(2)(3)
𝑦𝑖 ≤ 𝑑𝑖 ∀𝑖 ∈ 𝐼 (4)
𝑥𝑖𝑗 + 𝑥𝑗𝑖 ≤ 𝑤𝑖𝑗(1 − 𝑧𝑖𝑗) ∀(𝑖, 𝑗) ∈ 𝒜|𝑖 < 𝑗 (5)
𝑥𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝒜; 𝑦𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 ∪ 𝐼 (6)
http://copa.uniandes.edu.co/
Modified Benders
80
Part 2
Solution of enemy
– Solution of ally
min
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Modified Benders
81
Part 2
Solution of enemy
– Solution of ally
min
subject to
• Solution of enemy is greater
than or equal to any solution
considering interdiction plan.
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Modified Benders
82
Part 2
Solution of enemy
– Solution of ally
min
subject to
• Solution of enemy is greater
than or equal to any solution
considering interdiction plan.
• Solution of ally is smaller than
or equal to the associated
solution of the interdiction plan.
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Modified Benders
83
Part 2
Solution of enemy
– Solution of ally
min
subject to
• Solution of enemy is greater
than or equal to any solution
considering interdiction plan.
• Solution of ally is smaller than
or equal to the associated
solution of the interdiction plan.
• Interdiction budget
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• Solution of enemy is greater
than or equal to any solution
considering interdiction plan.
• Solution of ally is smaller than
or equal to the associated
solution of the interdiction plan.
• Interdiction budget
Modified Benders
84
Part 2
Solution of enemy
– Solution of ally
min
subject to
decision
• Interdiction plan
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Modified Benders
85
Part 2
Profit of the enemymax
subject to
• Flow constraints
• Arc capacity constraints
decision
• Transportation plan of enemy.
Profit of the allymax
• Flow constraints
• Arc capacity constraints
• Transportation plan of ally.
and
Solution of enemy
– Solution of ally
min
subject to
• Solution of enemy is greater
than or equal to any solution
considering interdiction plan.
• Solution of ally is smaller than
or equal to the associated
solution of the interdiction plan.
• Interdiction budget
decision
• Interdiction plan
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Interdiction on Enemy
86
Part 2
min𝑧,𝜑−
𝜑−
s.t,
𝜑− ≥
𝑖∈𝐼
𝑝𝑖 𝑦𝑖 −
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛 + 𝑥𝑗𝑖𝑛 ∀𝑛 ∈ 𝑌(1.1)
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗(2)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜
𝜑− ∈ ℝ1(4)
Solution of the enemy
• Interdiction plan
• Solution of J- is greater than or
equal to any solution
considering interdiction plan.
• Interdiction budget
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Part 2
min𝑧,𝜑−
𝜑−
s.t,
𝜑− ≥
𝑖∈𝐼
𝑝𝑖 𝑦𝑖 −
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛 + 𝑥𝑗𝑖𝑛 ∀𝑛 ∈ 𝑌(1.1)
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗(2)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜
𝜑− ∈ ℝ1(4)
Interdiction on Enemy
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max𝑧,𝜑+
𝜑+
s.t,
𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +
𝑖∈𝐼
𝑝𝑖 𝑦𝑖+ −
𝑖,𝑗 ∈𝐸|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛+ + 𝑥𝑗𝑖𝑛
+ ∀𝑛 ∈ 𝑌 (1.2)
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑤𝑖𝑗(2)
𝑛
𝛿𝑛 ≥ 1 (3)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝐸
𝛿𝑛 ∈ 0,1 ∀𝑛 ∈ 𝑌𝜑+ ∈ ℝ1
(4)
Part 2
Interdiction on Ally
Solution of ally
Interdiction plan
• Solution of ally is smaller than
or equal to the chosen solution
of the interdiction plan.
decision
• Interdiction budget
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𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +
𝑖∈𝐼
𝑝𝑖 𝑦𝑖+ −
𝑖,𝑗 ∈𝐸|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛+ + 𝑥𝑗𝑖𝑛
+ ∀𝑛 ∈ 𝑌 (1.2)
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑤𝑖𝑗(2)
𝑛
𝛿𝑛 ≥ 1 (3)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝐸
𝛿𝑛 ∈ 0,1 ∀𝑛 ∈ 𝑌𝜑+ ∈ ℝ1
(4)
Part 2
• Solution of J+ is smaller than or
equal to the associated solution
of the interdiction plan.
• Interdiction budget
min𝑧,𝜑+
−𝜑+
s.t,
Interdiction on Ally
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𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +
𝑖∈𝐼
𝑝𝑖 𝑦𝑖+ −
𝑖,𝑗 ∈𝐸|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛+ + 𝑥𝑗𝑖𝑛
+ ∀𝑛 ∈ 𝑌 (1.2)
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑤𝑖𝑗(2)
𝑛
𝛿𝑛 ≥ 1 (3)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝐸
𝛿𝑛 ∈ 0,1 ∀𝑛 ∈ 𝑌𝜑+ ∈ ℝ1
(4)
Part 2
Binary variable that
relates solutions of
the ally and the
enemy
Interdiction on Ally
• Solution of J+ is smaller than or
equal to the associated solution
of the interdiction plan.
• Interdiction budget
min𝑧,𝜑+
−𝜑+
s.t,
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𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +
𝑖∈𝐼
𝑝𝑖 𝑦𝑖+ −
𝑖,𝑗 ∈𝐸|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛+ + 𝑥𝑗𝑖𝑛
+ ∀𝑛 ∈ 𝑌 (1.2)
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑤𝑖𝑗(2)
𝑛
𝛿𝑛 ≥ 1 (3)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝐸
𝛿𝑛 ∈ 0,1 ∀𝑛 ∈ 𝑌𝜑+ ∈ ℝ1
(4)
Part 2
Relax right side
except for at least
one solution
Interdiction on Ally
• Interdiction budget
min𝑧,𝜑+
−𝜑+
s.t,
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𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +
𝑖∈𝐼
𝑝𝑖 𝑦𝑖+ −
𝑖,𝑗 ∈𝐸|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛+ + 𝑥𝑗𝑖𝑛
+ ∀𝑛 ∈ 𝑌 (1.2)
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑤𝑖𝑗(2)
𝑛
𝛿𝑛 ≥ 1 (3)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝐸
𝛿𝑛 ∈ 0,1 ∀𝑛 ∈ 𝑌𝜑+ ∈ ℝ1
(4)
Part 2
Relax right side
except for at least
one solution
At least one
is selected
Interdiction on Ally
min𝑧,𝜑+
−𝜑+
s.t,
http://copa.uniandes.edu.co/ 93
𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +
𝑖∈𝐼
𝑝𝑖 𝑦𝑖+ −
𝑖,𝑗 ∈𝐸|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛+ + 𝑥𝑗𝑖𝑛
+ ∀𝑛 ∈ 𝑌 (1.2)
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑤𝑖𝑗(2)
𝑛
𝛿𝑛 ≥ 1 (3)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝐸
𝛿𝑛 ∈ 0,1 ∀𝑛 ∈ 𝑌𝜑+ ∈ ℝ1
(4)
Part 2
Profit of the
ally
Interdiction on Ally
min𝑧,𝜑+
−𝜑+
s.t,
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Interdiction on Ally
94
𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +
𝑖∈𝐼
𝑝𝑖 𝑦𝑖+ −
𝑖,𝑗 ∈𝐸|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛+ + 𝑥𝑗𝑖𝑛
+ ∀𝑛 ∈ 𝑌 (1.2)
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑤𝑖𝑗𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝐸| 𝑖<𝑗
𝑤𝑖𝑗(2)
𝑛
𝛿𝑛 ≥ 1 (3)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝐸
𝛿𝑛 ∈ 0,1 ∀𝑛 ∈ 𝑌𝜑+ ∈ ℝ1
(4)
Part 2
Budget
min𝑧,𝜑+
−𝜑+
s.t,
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Interdictor’s problem
95
Part 2
min𝑧,𝜑−
𝜑− − 𝜑+
s.t,
𝜑− ≥ 1 − 𝛿𝑛 +
𝑖∈𝐼
𝑝𝑖 𝑦𝑖 −
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛 + 𝑥𝑗𝑖𝑛 ∀𝑛 ∈ 𝑌(1.1)
𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +
𝑖∈𝐼
𝑝𝑖 𝑦𝑖+ −
𝑖,𝑗 ∈𝐸|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛+ + 𝑥𝑗𝑖𝑛
+ ∀𝑛 ∈ 𝑌 (1.2)
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗 (2)
𝑛
𝛿𝑛 ≥ 1 (3)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜
𝛿𝑛 ∈ 0,1 ∀𝑛 ∈ 𝑌𝜑−,𝜑+ ∈ ℝ1
(4)
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Part 2
min𝑧,𝜑−
𝜑− − 𝜑+
s.t,
𝜑− ≥ 1 − 𝛿𝑛 +
𝑖∈𝐼
𝑝𝑖 𝑦𝑖 −
𝑖,𝑗 ∈𝒜|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛 + 𝑥𝑗𝑖𝑛 ∀𝑛 ∈ 𝑌(1.1)
𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +
𝑖∈𝐼
𝑝𝑖 𝑦𝑖+ −
𝑖,𝑗 ∈𝐸|𝑖<𝑗
𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛+ + 𝑥𝑗𝑖𝑛
+ ∀𝑛 ∈ 𝑌 (1.2)
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗𝑧𝑖𝑗 ≤ 𝛽
(𝑖,𝑗)∈𝒜| 𝑖<𝑗
𝑤𝑖𝑗 (2)
𝑛
𝛿𝑛 ≥ 1 (3)
𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜
𝛿𝑛 ∈ 0,1 ∀𝑛 ∈ 𝑌𝜑−,𝜑+ ∈ ℝ1
(4)
Interdictor’s problem
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Agenda
• Problem description
• Part I solution strategy
• Part II solution strategy
• AIMMS user interface & results
97
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Part 2
AIMMS user interface
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Part 2
AIMMS user interface
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Part 2
AIMMS user interface
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Part 2
AIMMS user interface
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Part 2
AIMMS user interface
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Results
103
Part 1
62% 70%
Profit of the enemy
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Results
104
Part 1
62% 70%
Profit of the enemy
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Results
105
Part 1
62% 76%
Profit of the enemy
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Part 2
AIMMS user interface
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Part 2
AIMMS user interface
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Part 2
AIMMS user interface
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Results
109
Part 2
Profit of the transporting agents
18%
-94% -100%
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Results
110
Part 2
Profit of the transporting agents
18%
-94% -100%
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Results
111
Part 2
Profit of the transporting agents
18%
-94% -100%
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Results
112
Part 2
Computational Time
𝜷 𝟎.𝟏𝟓 𝟎.𝟐𝟎
Part 1 MILP 0.22 s 0.44 s
Benders 583 s 1514 s
Part 2 Benders* 189 s 92 s
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AIMMS user interface
113
Thank you!
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Literature Review
114
[1] Wood, R. (1993). Deterministic network interdiction. Mathematical and
Computer Modelling, 17(2), 1-18. doi:10.1016/0895-7177(93)90236-r
[2] Lim, C., & Smith, J. C. (2007). Algorithms for discrete and continuous
multicommodity flow network interdiction problems. IIE
Transactions, 39(1), 15-26. doi:10.1080/07408170600729192
[3] Wood, R. K. (2011). Bilevel Network Interdiction Models: Formulations
and Solutions. Wiley Encyclopedia of Operations Research and
Management Science. doi:10.1002/9780470400531.eorms0932
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