Multiobjective Network Disruption 8th AIMMS MOPTA Optimization Modeling Competition

Transcription

1 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 18 th August, th AIMMS MOPTA Optimization Modeling Competition 1

2 Problem Statement :1 Disruption of network to minimize the profit of the agent providing supplies from nodes 15, 16 and th AIMMS MOPTA Optimization Modeling Competition 2

3 Problem for the Supplier Agent (P 1) max x,y s.t p T x b T y Flow balance constraint : Ay x = 0 A : Incidence matrix Maximum demand constraint : x i d i i N Non negative demand constraint : x i 0 i N\{15, 16, 20} Edge capacity constraint : y j w j j E x i : Units supplied to node i p i : Profit per unit supply at node i y j : Units transported through edge j b j : 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) 8th AIMMS MOPTA Optimization Modeling Competition 3

4 Problem for the Disrupting Agent The above problem is written as a min-max problem With capacity of edge j is reduced to w j -c j (P 2) min c Budget constraints : max x,y pt x b T y s.t Ay x = 0 x i d i i N x i 0 i N\{15, 16, 20} y j w j c j j E w j c j 0 c j 0 c j budget j 8th AIMMS MOPTA Optimization Modeling Competition 4

5 Key Results Proposition 1: The marginal damages caused by disruption is a non-increasing function of edge capacity w j, 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 c j > 0, at most one edge has c j < w j 8th AIMMS MOPTA Optimization Modeling Competition 5

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 Solve P1 and obtain the Lagrange multipliers of edge capacity constraints 2 Arrange the edges in decreasing order of the Lagrange multipliers 3 Select the first edge for disruption 4 Repeat the previous steps until budget is reached Randomized case: 1 Solve P1 and obtain the Lagrange multipliers of edge capacity constraints 2 Arrange the edges in decreasing order of the Lagrange multipliers 3 Select an edge among the ordered edges with a descending probability proportional to its value 4 Proceed until budget is reached 5 Repeat procedure (1)-(4) for a fixed number of user defined number of iterations 8th AIMMS MOPTA Optimization Modeling Competition 6

7 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 Solve P1 and obtain the Lagrange multipliers of edge capacity constraints 2 Arrange the edges in decreasing order of the Lagrange multipliers 3 Select the first edge for disruption 4 Repeat the previous steps until budget is reached Randomized case: 1 Solve P1 and obtain the Lagrange multipliers of edge capacity constraints 2 Arrange the edges in decreasing order of the Lagrange multipliers 3 Select an edge among the ordered edges with a descending probability proportional to its value 4 Proceed until budget is reached 5 Repeat procedure (1)-(4) for a fixed number of user defined number of iterations 8th AIMMS MOPTA Optimization Modeling Competition 6

8 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 Solve P1 and obtain the Lagrange multipliers of edge capacity constraints 2 Arrange the edges in decreasing order of the Lagrange multipliers 3 Select the first edge for disruption 4 Repeat the previous steps until budget is reached Randomized case: 1 Solve P1 and obtain the Lagrange multipliers of edge capacity constraints 2 Arrange the edges in decreasing order of the Lagrange multipliers 3 Select an edge among the ordered edges with a descending probability proportional to its value 4 Proceed until budget is reached 5 Repeat procedure (1)-(4) for a fixed number of user defined number of iterations 8th AIMMS MOPTA Optimization Modeling Competition 6

9 Results - for 15% Budget Figure: Solution by deterministic algorithm at 15 % budget Figure: Solution by randomized algorithm at 15 % budget Budget for disruption Profit - Deterministic Profit- Randomized 0 % of the total capacity % of the total capacity th AIMMS MOPTA Optimization Modeling Competition 7

10 Results - for 20% Budget Figure: Solution by deterministic algorithm at 20 % budget Figure: Solution by randomized algorithm at 20 % budget Budget for disruption Profit -Deterministic Profit- Randomized 0 % of the total capacity % of the total capacity th AIMMS MOPTA Optimization Modeling Competition 8

11 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 8th AIMMS MOPTA Optimization Modeling Competition 9

12 Features of GUI 8th AIMMS MOPTA Optimization Modeling Competition 10

13 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 th AIMMS MOPTA Optimization Modeling Competition 11

14 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 8th AIMMS MOPTA Optimization Modeling Competition 12

15 Problem 2 - Formulation (M 1 ) max x 1,y 1 p T x 1 b T e y 1 s.t Ay 1 x 1 = 0 x 1i d 1i x 1i 0 min c s.t (M) αm 2 (1 α)m 1 c j budget j y 1j + y 2j w j c j w j c j 0 i N i N\{20} y 1j w j c j j E c j 0 (M 2 ) max x 2,y 2 p T x 2 b T e y 2 s.t Ay 2 x 2 = 0 x 2i d 2i i N x 2i 0 i N\{15, 16} y 2j w j c j j E m 1 and m 2 are optimal solutions to the optimization problems M 1 and M 2 respectively 8th AIMMS MOPTA Optimization Modeling Competition 13

16 Results Figure: Best cut at 15 % budget Figure: Best cut at 20 % budget Budget for disruption profit made by profit made by.5 m 2 J + (m 1 ) J (m 2 ).5 m 1 No disruption % of the total capacity % of the total capacity th AIMMS MOPTA Optimization Modeling Competition 14

17 Features of GUI In addition to features in part 1, new features are included specific to part 2 8th AIMMS MOPTA Optimization Modeling Competition 15

18 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 8th AIMMS MOPTA Optimization Modeling Competition 16