Multiobjective Network Disruption
8th AIMMS MOPTA Optimization Modeling
Competition
Varghese Kurian, Srinesh C., Venkata Reddy P., Sridharakumar
Narasimhan
Department of Chemical Engineering
Indian Institute of Technology Madras
Thursday 18th August, 2016
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Department of Chemical Engineering, IIT Madras
1
Problem Statement :1
Disruption of network to minimize the profit of the agent providing
supplies from nodes 15, 16 and 20
2
12
19
13
25
22
26
27
5
3
4
7
8
6
33
16
10
31
20
32
11
15
1
14
21
24
9
30
18
28
29
23
17
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Department of Chemical Engineering, IIT Madras
2
Problem for the Supplier Agent
(P 1)
max pT x − bT |y|
x,y
s.t
Flow balance constraint :
Maximum demand constraint :
Non negative demand constraint :
Edge capacity constraint :
Ay − x = 0
A : Incidence matrix
xi ≤ di
xi ≥ 0
i∈N
i ∈ N \{15, 16, 20}
|yj | ≤ wj
j∈E
xi : Units supplied to node i
pi : Profit per unit supply at node i
yj : Units transported through edge j
bj : Transportation cost per unit through edge j
Solution of this problem is integral! (Proof: By mapping onto
max flow problem after constructing super source and super sink)
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Department of Chemical Engineering, IIT Madras
3
Problem for the Disrupting Agent
The above problem is written as a min-max problem
With capacity of edge j is reduced to wj -cj
(P 2)
min
c
max pT x − bT |y|
s.t
x,y
Ay − x = 0
xi ≤ di
xi ≥ 0
|yj | ≤ wj − cj
i∈N
i ∈ N \{15, 16, 20}
j∈E
wj − cj ≥ 0
cj ≥ 0
Budget constraints :
X
cj ≤ budget
j
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Department of Chemical Engineering, IIT Madras
4
Key Results
Proposition 1: The marginal damages caused by disruption is a
non-increasing function of edge capacity wj , j ∈ E
Proposition 2: At most one edge is partially cut in the solution to P 2,
i.e., given a disrupting budget, in the optimal solution to P 2, amongst all
edges with cj > 0, at most one edge has cj < wj
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Department of Chemical Engineering, IIT Madras
5
Solution Approach - Problem 1
Solution to the supplier agent is mainly constrained by edge capacity constraints Lagrange multipliers can be therefore used for edge selection
Deterministic case:
1
2
3
4
Solve P1 and obtain the Lagrange multipliers of edge capacity
constraints
Arrange the edges in decreasing order of the Lagrange multipliers
Select the first edge for disruption
Repeat the previous steps until budget is reached
Randomized case:
1
2
3
4
5
Solve P1 and obtain the Lagrange multipliers of edge capacity
constraints
Arrange the edges in decreasing order of the Lagrange multipliers
Select an edge among the ordered edges with a descending
probability proportional to its value
Proceed until budget is reached
Repeat procedure (1)-(4) for a fixed number of user defined number
of iterations
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Department of Chemical Engineering, IIT Madras
6
Solution Approach - Problem 1
Solution to the supplier agent is mainly constrained by edge capacity constraints Lagrange multipliers can be therefore used for edge selection
Deterministic case:
1
2
3
4
Solve P1 and obtain the Lagrange multipliers of edge capacity
constraints
Arrange the edges in decreasing order of the Lagrange multipliers
Select the first edge for disruption
Repeat the previous steps until budget is reached
Randomized case:
1
2
3
4
5
Solve P1 and obtain the Lagrange multipliers of edge capacity
constraints
Arrange the edges in decreasing order of the Lagrange multipliers
Select an edge among the ordered edges with a descending
probability proportional to its value
Proceed until budget is reached
Repeat procedure (1)-(4) for a fixed number of user defined number
of iterations
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Department of Chemical Engineering, IIT Madras
6
Solution Approach - Problem 1
Solution to the supplier agent is mainly constrained by edge capacity constraints Lagrange multipliers can be therefore used for edge selection
Deterministic case:
1
2
3
4
Solve P1 and obtain the Lagrange multipliers of edge capacity
constraints
Arrange the edges in decreasing order of the Lagrange multipliers
Select the first edge for disruption
Repeat the previous steps until budget is reached
Randomized case:
1
2
3
4
5
Solve P1 and obtain the Lagrange multipliers of edge capacity
constraints
Arrange the edges in decreasing order of the Lagrange multipliers
Select an edge among the ordered edges with a descending
probability proportional to its value
Proceed until budget is reached
Repeat procedure (1)-(4) for a fixed number of user defined number
of iterations
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Department of Chemical Engineering, IIT Madras
6
Results - for 15% Budget
Figure: Solution by
deterministic algorithm at 15 %
budget
Budget for disruption
0 % of the total capacity
15 % of the total capacity
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Figure: Solution by randomized
algorithm at 15 % budget
Profit - Deterministic
76790
36978
Profit- Randomized
76790
29152
Department of Chemical Engineering, IIT Madras
7
Results - for 20% Budget
Figure: Solution by
deterministic algorithm at 20 %
budget
Budget for disruption
0 % of the total capacity
20 % of the total capacity
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Figure: Solution by randomized
algorithm at 20 % budget
Profit -Deterministic
76790
35050
Profit- Randomized
76790
19060
Department of Chemical Engineering, IIT Madras
8
Results
Series of solutions obtained by the randomized algorithm:
Figure: Profits made by the
supplier agent with iterations at
15 % budget
Figure: Profits made by the
supplier agent with iterations at
20 % budget
The randomized algorithm performs better with larger disruption
budget
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Department of Chemical Engineering, IIT Madras
9
Features of GUI
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Department of Chemical Engineering, IIT Madras
10
Problem Statement :2
Disruption of network to minimize the profit of the agent providing
supplies from nodes 15, 16 while maintaining the profit of the agent
supplying from node 20
2
12
19
13
25
22
26
27
5
3
4
7
8
6
33
16
10
31
20
32
11
15
1
14
21
24
9
30
18
28
29
23
17
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Department of Chemical Engineering, IIT Madras
11
Solution Approach - Problem 2
1
Solve the problems of supplier agents independently
2
Reorganize the flows by retaining profit
3
Check if any edge is over the capacity? If yes- Solve the congestion
pricing problem
4
Edges are disrupted following the randomized algorithm
5
Proceed until the budget is met
6
Repeat the steps (1)-(5) for a fixed user defined number of iterations
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Department of Chemical Engineering, IIT Madras
12
Problem 2 - Formulation
min
c
s.t
(M )
αm2 − (1 − α)m1
X
cj ≤ budget
j
|y1j | + |y2j | ≤ wj − cj
wj − cj ≥ 0
cj ≥ 0
(M1 )
(M2 )
max pT x1 − bT
e |y1 |
max pT x2 − bT
e |y2 |
x1 ,y1
x2 ,y2
s.t Ay1 − x1 = 0
s.t Ay2 − x2 = 0
x1i ≤ d1i
x1i ≥ 0
i∈N
i ∈ N \{20}
|y1j | ≤ wj − cj
j∈E
x2i ≤ d2i
x2i ≥ 0
i∈N
i ∈ N \{15, 16}
|y2j | ≤ wj − cj
j∈E
m1 and m2 are optimal solutions to the optimization problems M1 and M2
respectively
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Department of Chemical Engineering, IIT Madras
13
Results
Figure: Best cut at 15 %
budget
Budget for disruption
No disruption
15 % of the total capacity
20 % of the total capacity
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Figure: Best cut at 20 %
budget
profit made by
J + (m1 )
27385
21850
24265
profit made by
J − (m2 )
31625
1340
1340
.5 × m2
−.5 × m1
2120
-10255
-11465
Department of Chemical Engineering, IIT Madras
14
Features of GUI
In addition to features in part 1, new features are included specific
to part 2
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Department of Chemical Engineering, IIT Madras
15
Conclusions
A randomized algorithm for network disruption involving a single
agent is proposed
The profit of the supplier agent is reduced to almost one-fourth by
disruption of 20% of the network
The algorithm is extended to the case where there are multiple
supplier agents present
An interface is developed on AIMMS platform
Team IIT Madras
8th AIMMS MOPTA Optimization Modeling Competition
Department of Chemical Engineering, IIT Madras
16