IIND4622 Optimización 2: Solución Tarea 1 - AIMMS · Agenda •Problem description •Part I solution strategy •Part II solution strategy •AIMMS user interface & results 3

http://copa.uniandes.edu.co/http://copa.uniandes.edu.co/

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


Agenda





3


Competition

4

MOPTA 2016

competition

Part 2Part 1

MILP BendersModified

Benders


Part 1

5

MOPTA 2016

competition

Part 2Part 1


Benders


Problem description

6

Part 1

• Undirected network

Set V

Transition nodes


Problem description

7

Part 1


• Capacitated edgesSet E

Transition nodes


Problem description

8

Part 1


• Capacitated edges

Transition nodes

Source nodes

Subset S1


Problem description

9

Part 1

∞

∞

∞

∞



• Infinite supply

Transition nodes

Source nodes


Problem description

10

Part 1



• Infinite supply

Transition nodes

Source nodes

Sink nodesSubset I1


Problem description

11

Part 1

∞

∞

∞

∞

𝒅𝟏

𝒅𝟐

𝒅𝟑

𝒅𝟒

𝒅𝟓



• Infinite supply

• Finite demandCan be

partially

satisfied

Transition nodes

Source nodes

Sink nodes


Problem description

12

Part 1

∞

∞

∞

∞

𝑑1

𝑑2

𝑑3

𝑑4

𝑑5



• Infinite supply

• Finite demand

Transition nodes

Source nodes

Sink nodes

Transporting

Agent

https://www.google.com.co/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwj684WFgbnLAhXFbB4KHUE4DHwQjRwIBw&url=https://en.wikipedia.org/wiki/Mario&bvm=bv.116573086,d.dmo&psig=AFQjCNGzsKTeVMYesZjZLPU0Oc8ObsFZgA&ust=1457798561955383



Problem description

13

Part 1

∞

∞

∞

∞

𝑑1

𝑑2

𝑑3

𝑑4

𝑑5



• Infinite supply

• Finite demand

Transition nodes

Source nodes

Sink nodes

Interdictor



https://www.google.com.co/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0ahUKEwiXn-iwg7nLAhXBKx4KHQh4AgkQjRwIBw&url=https://en.wikipedia.org/wiki/Bowser_(character)&bvm=bv.116573086,d.dmo&psig=AFQjCNFM7j7G7yi4oQMhrU4kMyYOQl0tuA&ust=1457799201914704



Problem description

14

Transition nodes

Source nodes

Sink nodes



• Infinite supply

• Finite demand

• Budget

∞

∞

∞

∞

𝑑1

𝑑2

𝑑3

𝑑4

𝑑5

Part 1

𝜷






Part 2

15

MOPTA 2016

competition

Part 2Part 1


Benders


Problem description

16

Part 2

∞

∞

∞

∞

𝑑1

𝑑2

𝑑3

𝑑4

𝑑5

Two agents

Share the

network


• Capacitated arcs

• Infinite supply

• Finite demand

Transition nodes

Source nodes

Sink nodes






Problem description

17

Part 2

∞

∞

∞

∞

𝑑1

𝑑2

𝑑3

𝑑4

𝑑5



• Infinite supply

• Finite demand

Transition nodes

Source nodes

Sink nodesSubset S1






Problem description

18

Part 2

∞

∞

∞

∞

𝑑1

𝑑2

𝑑3

𝑑4

𝑑5



• Infinite supply

• Finite demand

Transition nodes

Source nodes

Sink nodesSubset I1






Problem description

19

Part 2

∞

∞

∞

∞

𝑑1

𝑑2

𝑑3

𝑑4

𝑑5



• Infinite supply

• Finite demand

Transition nodes

Source nodes

Sink nodesSubset S2






Problem description

20

Part 2

∞

∞

∞

∞

𝑑1

𝑑2

𝑑3

𝑑4

𝑑5



• Infinite supply

• Finite demand

Transition nodes

Source nodes

Sink nodesSubset I2






Problem description

21

𝐸 → 𝒜

Part 1

2 𝐸 = 𝒜

𝑖 𝑗

Undirected


Problem description

22

𝐸 → 𝒜

Part 1

2 𝐸 = 𝒜

𝑖 𝑗

Directed


Lemma

23

Part 1

[2] Lim & Smith (2007).

5 6

2 3

4

1


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




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




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




Agenda





28


Part 1

29

MOPTA 2016

competition

Part 2Part 1


Benders


Formulation

30

min𝑧

max𝑥

𝑖∈𝐼

𝑝𝑖𝑦𝑖 −

𝑖,𝑗 ∈𝒜

𝑏𝑖𝑗𝑥𝑖𝑗

Part 1

Enemy´s

profit






Formulation

31

min𝑧

max𝑥

𝑖∈𝐼


𝑖,𝑗 ∈𝒜


Part 1

Income






Formulation

32

min𝑧

max𝑥

𝑖∈𝐼


𝑖,𝑗 ∈𝒜


Part 1

Transport

cost






Formulation

33

min𝑧

max𝑥

𝑖∈𝐼


𝑖,𝑗 ∈𝒜


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






Formulation

34

min𝑧

max𝑥

𝑖∈𝐼


𝑖,𝑗 ∈𝒜


s.t,


𝑥𝑖𝑗 −



(1)(2)(3)

𝑦𝑖 ≤ 𝑑𝑖 ∀𝑖 ∈ 𝐼 (4)


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






Formulation

35

min𝑧

max𝑥

𝑖∈𝐼


𝑖,𝑗 ∈𝒜


s.t,


𝑥𝑖𝑗 −



(1)(2)(3)

𝑦𝑖 ≤ 𝑑𝑖 ∀𝑖 ∈ 𝐼 (4)


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






Formulation

36

min𝑧

max𝑥

𝑖∈𝐼


𝑖,𝑗 ∈𝒜


s.t,


𝑥𝑖𝑗 −



(1)(2)(3)

𝑦𝑖 ≤ 𝑑𝑖 ∀𝑖 ∈ 𝐼 (4)


𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜; 𝑥𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝒜;

𝑦𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 ∪ 𝐼

(6)

Part 1

Interdictor’s

plan






Formulation

37

min𝑧

max𝑥

𝑖∈𝐼


𝑖,𝑗 ∈𝒜


s.t,


𝑥𝑖𝑗 −



(1)(2)(3)

𝑦𝑖 ≤ 𝑑𝑖 ∀𝑖 ∈ 𝐼 (4)


𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜; 𝑥𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝒜;

𝑦𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 ∪ 𝐼

(6)

Part 1

Variables’

domain






Part 1

38

MOPTA 2016

competition

Part 2Part 1


Benders

[1] Wood (1993)


min𝑧

max𝑥

𝑖∈𝐼


𝑖,𝑗 ∈𝒜


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




min𝑧

max𝑥

𝑖∈𝐼


𝑖,𝑗 ∈𝒜


s.t,

Part 1



Constraints (6)

We will move

interdictor’s

decisions to the

objective function





min𝑧

max𝑥

𝑖∈𝐼


𝑖,𝑗 ∈𝒜


s.t,

Part 1



Constraints (6)

We will move

interdictor’s

decisions to the

objective function





min𝑧

max𝑥

𝑖∈𝐼


𝑖,𝑗 ∈𝒜|𝑖<𝑗

𝑏𝑖𝑗 + 𝑀𝑖𝑗 𝑧𝑖𝑗 𝑥𝑖𝑗 + 𝑥𝑗𝑖

s.t,

Part 1



Constraints (6)

Interdicted

arcs become

unprofitable





min𝑧

max𝑥

𝑖∈𝐼




s.t,

Part 1



Constraints (6)

[2] Lim &

Smith

First approach – MILPLim & Smith (2007)





44

min

𝑖∈𝑉

0𝑣𝑖 +

𝑖∈𝐼

𝑑𝑖𝑞𝑖 +


𝑤𝑖𝑗𝑢𝑖𝑗

Part 1

Dual variable

for balance

constraints





45

min

𝑖∈𝑉

0𝑣𝑖 +

𝑖∈𝐼

𝑑𝑖𝑞𝑖 +



Part 1

Dual variable

of demand

constraints





46

min

𝑖∈𝑉

0𝑣𝑖 +

𝑖∈𝐼

𝑑𝑖𝑞𝑖 +



Part 1

Dual variable

of capacity

constraints




min

𝑖∈𝑉

0𝑣𝑖 +

𝑖∈𝐼

𝑑𝑖𝑞𝑖 +



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


𝑤𝑖𝑗 (5)

𝑞𝑖 ≥ 0 ∀𝑖 ∈ 𝐼; 𝑢𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝐸|𝑖 < 𝑗; 𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝐸 (6)


Dual

constraints of

flow variables




min

𝑖∈𝑉

0𝑣𝑖 +

𝑖∈𝐼

𝑑𝑖𝑞𝑖 +



s.t,

Part 1



𝑣𝑖 + 𝑞𝑖 ≥ 𝑝𝑖 ∀𝑖 ∈ 𝐼 (3)

−𝑣𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 (4)




𝑤𝑖𝑗 (5)

𝑞𝑖 ≥ 0 ∀𝑖 ∈ 𝐼; 𝑢𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝐸|𝑖 < 𝑗; 𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝐸 (6)


Dual

constraints of

𝑦𝑖




min

𝑖∈𝑉

0𝑣𝑖 +

𝑖∈𝐼

𝑑𝑖𝑞𝑖 +



s.t,

Part 1



𝑣𝑖 + 𝑞𝑖 ≥ 𝑝𝑖 ∀𝑖 ∈ 𝐼 (3)

−𝑣𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 (4)

(𝑖,𝑗)∈𝒜| 𝑖<𝑗

𝑤𝑖𝑗 𝑧𝑖𝑗 ≤ 𝛽


𝑤𝑖𝑗 (5)

𝑞𝑖 ≥ 0 ∀𝑖 ∈ 𝐼; 𝑢𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝒜|𝑖 < 𝑗; 𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜 (6)


Variables’

domain




min

𝑖∈𝑉

0𝑣𝑖 +

𝑖∈𝐼

𝑑𝑖𝑞𝑖 +



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)


𝑤𝑖𝑗𝑧𝑖𝑗 ≤ 𝛽


𝑤𝑖𝑗 (5)



𝑧𝑖𝑗 → 𝑧𝑖𝑗






min

𝑖∈𝑉

0𝑣𝑖 +

𝑖∈𝐼

𝑑𝑖𝑞𝑖 +



s.t,

Part 1



𝑣𝑖 + 𝑞𝑖 ≥ 𝑝𝑖 ∀𝑖 ∈ 𝐼 (3)

−𝑣𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 (4)




𝑤𝑖𝑗 (5)



Interdictor’s

budget

constraint






min

𝑖∈𝑉

0𝑣𝑖 +

𝑖∈𝐼

𝑑𝑖𝑞𝑖 +



s.t,

Part 1



𝑣𝑖 + 𝑞𝑖 ≥ 𝑝𝑖 ∀𝑖 ∈ 𝐼 (3)

−𝑣𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 (4)




𝑤𝑖𝑗 (5)








Part 1

53

MOPTA 2016

competition

Part 2Part 1


Benders

[3] Wood (2011)


Second approach - Benders

54

Part 1







55

Part 1

Transportation plan

𝑥𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜







56

Part 1

Transportation plan

𝑥𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜Interdiction plan

𝑧𝑖𝑗 ∀(𝑖, 𝑗) ∈ 𝒜







57

Part 1

Profit of transporting agentmax

subject to

• Flow constraints

• Arc capacity constraints

decision

• Transportation plan





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





59

Part 1


subject to





decision


Profit of the enemymax

subject to



decision








60

Part 1


subject to





decision



subject to



decision








61

max𝑥

𝑖∈𝐼




s.t,


𝑥𝑖𝑗 −



(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








62

max𝑥

𝑖∈𝐼




s.t,


𝑥𝑖𝑗 −



(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





63

min𝑧,𝜑−

𝜑−

s.t,

𝜑− ≥

𝑖∈𝐼

𝑝𝑖 𝑦𝑖 −


𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛 + 𝑥𝑗𝑖𝑛 ∀𝑛 ∈ 𝑌




𝑤𝑖𝑗

𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜

Part 1

Best

solution of

the enemy










64

min𝑧,𝜑−

𝜑−

s.t,

𝜑− ≥

𝑖∈𝐼



𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛 + 𝑥𝑗𝑖𝑛 ∀𝑛 ∈ 𝑌(1)




𝑤𝑖𝑗

𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜

Part 1

Add a cut such that 𝜑−

is greater than or equal

to any solution n






𝜑− ≥

𝑖∈𝐼







𝑤𝑖𝑗(2)

𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜

𝜑− ≥ 0

(3)


65

min𝑧,𝜑−

𝜑−

s.t,

Part 1

By interdicting arcs, the

right side of the

constraint is minimized






𝜑− ≥

𝑖∈𝐼







𝑤𝑖𝑗(2)

𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜

𝜑− ≥ 0

(3)


66

min𝑧,𝜑−

𝜑−

s.t,

Part 1

Interdictor’s

budget

constraint






67

min𝑧,𝜑−

𝜑−

s.t,

𝜑− ≥

𝑖∈𝐼







𝑤𝑖𝑗(2)

𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜𝜑− ∈ ℝ1

(3)

Part 1

Variables’

domain




Part 2

68

MOPTA 2016

competition

Part 2Part 1


Benders

[3] Wood (2011)

&


Idea

69

Part 2




Idea

70

Part 2

𝑓𝑙𝑜𝑤 + 𝑓𝑙𝑜𝑤≤ 𝑤?




Idea

71

Part 2


No No ↑ $↑ $




Idea

72

Part 2


No No ↑ $↑ $




Idea

73

Part 2


Yes






Idea

74

Part 2


Yes






Idea

75

Part 2


No No

Yes

↑ $↑ $






Modified Benders

76

Part 2


subject to



decision

• Transportation plan of enemy.




Modified Benders

77

Part 2

subject to

decision

Profit of the allymax



• Transportation plan of ally.


Modified Benders

78

Part 2


subject to



decision






and




Enemy and Ally problems

79

max𝑥

𝑖∈𝐼


𝑖,𝑗 ∈𝐸|𝑖<𝑗


s.t,

Part 2


𝑥𝑖𝑗 −



(1)(2)(3)

𝑦𝑖 ≤ 𝑑𝑖 ∀𝑖 ∈ 𝐼 (4)


𝑥𝑖𝑗 ≥ 0 ∀ 𝑖, 𝑗 ∈ 𝒜; 𝑦𝑖 ≥ 0 ∀𝑖 ∈ 𝑆 ∪ 𝐼 (6)


Modified Benders

80

Part 2

Solution of enemy

– Solution of ally

min




Modified Benders

81

Part 2

Solution of enemy


min

subject to

• Solution of enemy is greater

than or equal to any solution





Modified Benders

82

Part 2

Solution of enemy


min

subject to




• Solution of ally is smaller than

or equal to the associated

solution of the interdiction plan.




Modified Benders

83

Part 2

Solution of enemy


min

subject to


















Modified Benders

84

Part 2

Solution of enemy


min

subject to

decision





Modified Benders

85

Part 2


subject to



decision






and

Solution of enemy


min

subject to








decision







Interdiction on Enemy

86

Part 2

min𝑧,𝜑−

𝜑−

s.t,

𝜑− ≥

𝑖∈𝐼



𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛 + 𝑥𝑗𝑖𝑛 ∀𝑛 ∈ 𝑌(1.1)




𝑤𝑖𝑗(2)

𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜

𝜑− ∈ ℝ1(4)

Solution of the enemy


• Solution of J- is greater than or





Part 2

min𝑧,𝜑−

𝜑−

s.t,

𝜑− ≥

𝑖∈𝐼







𝑤𝑖𝑗(2)

𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜

𝜑− ∈ ℝ1(4)

Interdiction on Enemy


max𝑧,𝜑+

𝜑+

s.t,

𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +

𝑖∈𝐼

𝑝𝑖 𝑦𝑖+ −


𝑏𝑖𝑗 + 𝑀𝑖𝑗𝑧𝑖𝑗 𝑥𝑖𝑗𝑛+ + 𝑥𝑗𝑖𝑛

+ ∀𝑛 ∈ 𝑌 (1.2)




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


or equal to the chosen solution

of the interdiction plan.

decision



𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +

𝑖∈𝐼




+ ∀𝑛 ∈ 𝑌 (1.2)




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



min𝑧,𝜑+

−𝜑+

s.t,



𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +

𝑖∈𝐼




+ ∀𝑛 ∈ 𝑌 (1.2)




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


• Solution of J+ is smaller than or

equal to the associated solution



min𝑧,𝜑+

−𝜑+

s.t,


𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +

𝑖∈𝐼




+ ∀𝑛 ∈ 𝑌 (1.2)




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



min𝑧,𝜑+

−𝜑+

s.t,


𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +

𝑖∈𝐼




+ ∀𝑛 ∈ 𝑌 (1.2)




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


min𝑧,𝜑+

−𝜑+

s.t,


𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +

𝑖∈𝐼




+ ∀𝑛 ∈ 𝑌 (1.2)




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


min𝑧,𝜑+

−𝜑+

s.t,



94

𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +

𝑖∈𝐼




+ ∀𝑛 ∈ 𝑌 (1.2)




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,


Interdictor’s problem

95

Part 2

min𝑧,𝜑−

𝜑− − 𝜑+

s.t,

𝜑− ≥ 1 − 𝛿𝑛 +

𝑖∈𝐼




𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +

𝑖∈𝐼




+ ∀𝑛 ∈ 𝑌 (1.2)




𝑤𝑖𝑗 (2)

𝑛

𝛿𝑛 ≥ 1 (3)

𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜

𝛿𝑛 ∈ 0,1 ∀𝑛 ∈ 𝑌𝜑−,𝜑+ ∈ ℝ1

(4)




Part 2

min𝑧,𝜑−

𝜑− − 𝜑+

s.t,

𝜑− ≥ 1 − 𝛿𝑛 +

𝑖∈𝐼




𝜑+ ≤ 𝑀 1 − 𝛿𝑛 +

𝑖∈𝐼




+ ∀𝑛 ∈ 𝑌 (1.2)




𝑤𝑖𝑗 (2)

𝑛

𝛿𝑛 ≥ 1 (3)

𝑧𝑖𝑗 ∈ 0,1 ∀ 𝑖, 𝑗 ∈ 𝒜

𝛿𝑛 ∈ 0,1 ∀𝑛 ∈ 𝑌𝜑−,𝜑+ ∈ ℝ1

(4)

Interdictor’s problem




Agenda





97


Part 2

AIMMS user interface


Part 2



Part 2



Part 2



Part 2



Results

103

Part 1

62% 70%

Profit of the enemy


Results

104

Part 1

62% 70%

Profit of the enemy


Results

105

Part 1

62% 76%

Profit of the enemy


Part 2



Part 2



Part 2



Results

109

Part 2

Profit of the transporting agents

18%

-94% -100%


Results

110

Part 2


18%

-94% -100%


Results

111

Part 2


18%

-94% -100%


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



113

Thank you!


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

Documents

IIND4622 Optimización 2: Solución Tarea 1 - AIMMS · Agenda •Problem description •Part I solution strategy •Part II solution strategy •AIMMS user interface & results 3