Heuristic Approach for Model Reduction of Large-Scale Logistics Networks
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This document discusses a heuristic approach for structure-preserving model reduction of large-scale logistics networks. The approach uses the importance of locations in the network, as determined by their LogRank values, and the structure of material flows to identify motifs for aggregation. Motifs like parallel and sequential connections and almost disconnected subgraphs are aggregated to reduce the size of the network while preserving the overall structure and relationships between locations. The reduced model is then analyzed and compared to the original model to evaluate the reduction error.
![Centre for
Industrial Mathematics
Elgersburg Workshop 2011
Kosmykov
Embedding into a larger network
c ∈ [0, 1] - the strength of connection
v = [vT
n vT
m]T
, w = [wT
n wT
m]T
, n+m n
- connection with an outside world
L =
cM + (1 − c)wneT
n vneT
m
(1 − c)wmeT
n vmeT
m
12 / 30Motivation Structure Reduction Application Conclusions](https://image.slidesharecdn.com/kosmykovelgersburgtalk-160619172222/85/Heuristic-Approach-for-Model-Reduction-of-Large-Scale-Logistics-Networks-13-320.jpg)
![Centre for
Industrial Mathematics
Elgersburg Workshop 2011
Kosmykov
LogRank
Theorem
For v > 0, wm > 0 and the Perron vector r partitioned as [rT
n rT
m]T
rn is an eigenvector corresponding to the eigenvalue 1 of
Mc(v, w) := cM + (1 − c) wn +
eT
mwm
1 − eT
mvm
vn eT
n .
Furthermore, Mc(v, w) is primitive.
LogRank
The normalized eigenvector associated to the eigenvalue 1 of
Mc(v, w) is called the LogRank.
Idea is similar to the PageRank by Page et al. 1998.
13 / 30Motivation Structure Reduction Application Conclusions](https://image.slidesharecdn.com/kosmykovelgersburgtalk-160619172222/85/Heuristic-Approach-for-Model-Reduction-of-Large-Scale-Logistics-Networks-14-320.jpg)
Heuristic Approach for Model Reduction of Large-Scale Logistics Networks
- 1. Heuristic Approach for Model Reduction of Large-Scale Logistics Networks Michael Kosmykov Centre for Industrial Mathematics, University of Bremen, Germany February 14, 2011, Elgersburg Workshop 2011, Elgersburg joint work with Sergey Dashkovskiy, Thomas Makuschewitz, Bernd Scholz-Reiter, Michael Sch¨onlein and Fabian Wirth
- 2. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Outline 1 Motivation 2 Network structure information 3 Structure preserving model reduction 4 Application (Jackson networks) 5 Conclusions and outlook 2 / 30Motivation Structure Reduction Application Conclusions
- 3. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Outline 1 Motivation 2 Network structure information 3 Structure preserving model reduction 4 Application (Jackson networks) 5 Conclusions and outlook 3 / 30Motivation Structure Reduction Application Conclusions
- 4. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Characteristics of the practical example: 3 production sites (D, F, E) for Liquidring-Vaccum (LRVP), Industrial (IND) and Side-channel (SC) pumps 5 distribution centers (D, NL, B, F, E) 33 first and second-tier suppliers for the production of pumps 90 suppliers for components that are needed for the assembly of pump sets More than 1000 customers 4 / 30Motivation Structure Reduction Application Conclusions
- 5. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Analysis steps: 1 Mathematical modelling 2 Model reduction 3 Analysis of the reduced model 4 Analysis of relationships between reduced and original models Main methods of model reduction: Balancing methods Krylov methods Drawback of application to logistics networks: the interconnection structure of the network is destroyed 5 / 30Motivation Structure Reduction Application Conclusions
- 6. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Structure-preserving model reduction Advantages for logistics networks: keeps the physical meaning of the objects allows analysis of interrelationships between locations allows identification of influential logistic objects Our heuristic approach uses importance of locations for the network structure of material flows in the network 6 / 30Motivation Structure Reduction Application Conclusions
- 7. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Outline 1 Motivation 2 Network structure information 3 Structure preserving model reduction 4 Application (Jackson networks) 5 Conclusions and outlook 7 / 30Motivation Structure Reduction Application Conclusions
- 8. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Graph of logistics network f11,16 v1v1v1 v2v2v2 v3v3v3 v4v4v4 v5v5v5 v6v6v6 v7v7v7 v8v8v8 v9v9v9 v10v10v10 v11v11v11 v12v12v12 v13v13v13 v14v14v14 v15v15v15 v16v16v16 v17v17v17 v18v18v18 v19v19v19 v20v20v20 v21v21v21 v22v22v22 v23v23v23 fij - material flow from location i to j 8 / 30Motivation Structure Reduction Application Conclusions
- 9. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Structure information mij := fij n i=1 fij (i, j) ∈ E, 0 else. - relative material flow v1v1v1 v2v2v2 v3v3v3 v4v4v4 v5v5v5 v6v6v6 v7v7v7 15 20 5 5010 50 M = 0 0 15 25 0 0 0 0 0 0 10 25 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 M - weighted adjacency matrix of the graph 9 / 30Motivation Structure Reduction Application Conclusions
- 10. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Ranking mij - probability to move from vertex j to vertex i xi (k) - probability that at kth step an order is placed at location i Markov chain: x(k + 1) = Mx(k) Rank ri (importance) of location i := probability that a random order is placed location i = stationary distribution of Markov chain: Mr = r, r ∈ Rn . Perron-Frobenius Theorem ⇒ if the transition matrix is irreducible the Markov chain has a unique stationary distribution r > 0. 10 / 30Motivation Structure Reduction Application Conclusions
- 11. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Graph modification Connecting retailers and source suppliers: v1v1v1 v2v2v2 v3v3v3 v4v4v4 v5v5v5 v6v6v6 v7v7v7 15 20 5 50 10 50 Material flow between the locations (E) Relative capacity allocation (E ) New edge set = E∪E S - source suppliers Rj - retailers affecting supplier j P(i) - locations connected to retailer i pi := k∈P(i) fki , i ∈ Rj qj := i∈Rj pi , j ∈ S Mij := mij (i, j) ∈ E, pi qj (i, j) ∈ E , 0 else. 11 / 30Motivation Structure Reduction Application Conclusions
- 12. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Graph modification Connecting retailers and source suppliers: v1v1v1 v2v2v2 v3v3v3 v4v4v4 v5v5v5 v6v6v6 v7v7v7 15 20 5 50 10 50 Material flow between the locations (E) Relative capacity allocation (E ) New edge set = E∪E M = 0 0 15 25 0 0 0 0 0 0 10 25 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 20 25 20 75 0 0 0 0 0 5 25 5 75 0 0 0 0 0 0 50 75 0 0 0 0 0 The matrix M is irreducible. r ≈ 0.1111 0.2222 0.1852 0.1481 0.1481 0.0370 0.1481 T . 11 / 30Motivation Structure Reduction Application Conclusions
- 13. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Embedding into a larger network c ∈ [0, 1] - the strength of connection v = [vT n vT m]T , w = [wT n wT m]T , n+m n - connection with an outside world L = cM + (1 − c)wneT n vneT m (1 − c)wmeT n vmeT m 12 / 30Motivation Structure Reduction Application Conclusions
- 14. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov LogRank Theorem For v > 0, wm > 0 and the Perron vector r partitioned as [rT n rT m]T rn is an eigenvector corresponding to the eigenvalue 1 of Mc(v, w) := cM + (1 − c) wn + eT mwm 1 − eT mvm vn eT n . Furthermore, Mc(v, w) is primitive. LogRank The normalized eigenvector associated to the eigenvalue 1 of Mc(v, w) is called the LogRank. Idea is similar to the PageRank by Page et al. 1998. 13 / 30Motivation Structure Reduction Application Conclusions
- 15. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Typical interconnections in the network (motifs) almost disconnected subgraph parallel connection sequential connection v1v1v1 v2v2v2 v3v3v3 v4v4v4 v5v5v5 v6v6v6 v7v7v7 v8v8v8 v9v9v9 v10v10v10 v11v11v11 v12v12v12 v13v13v13 v14v14v14 v15v15v15 v16v16v16 v17v17v17 v18v18v18 v19v19v19 v20v20v20 v21v21v21 v22v22v22 v23v23v23 14 / 30Motivation Structure Reduction Application Conclusions
- 16. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Outline 1 Motivation 2 Network structure information 3 Structure preserving model reduction 4 Application (Jackson networks) 5 Conclusions and outlook 15 / 30Motivation Structure Reduction Application Conclusions
- 17. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Aggregation of vertices almost disconnected subgraph parallel connection sequential connection J1 J2 J3 v1v1v1 v2v2v2 v3v3v3 v4v4v4 v5v5v5 v6v6v6 v7v7v7 v8v8v8 v9v9v9 v10v10v10 v11v11v11 v12v12v12 v13v13v13 v14v14v14 v15v15v15 v16v16v16 v17v17v17 v18v18v18 v19v19v19 v20v20v20 v21v21v21 v22v22v22 v23v23v23 ˜mij =? ˜ri =? 16 / 30Motivation Structure Reduction Application Conclusions
- 18. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Aggregation of vertices Theorem Consider a strongly connected weighted directed graph G = (V , E, M). Let r be the (unique) normalized Perron vector of M. Given a disjoint partition V = {1, . . . , n} =: J1 ∪ J2 ∪ . . . ∪ Jk , with ˜V = {J1, . . . , Jk}, ˜mij := ν∈Ji 1 µ∈Jj rµ µ∈Jj rµmνµ , then ˜M is irreducible and the unique normalized Perron vector ˜r of ˜M has the property ˜ri = ν∈Ji rν , i = 1, . . . , k . (1) 17 / 30Motivation Structure Reduction Application Conclusions
- 19. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Aggregation of certain motifs Parallel connections: vvv v1v1v1 vkvkvk vvv ... ⇒ vvv JJJ vvv The LogRank of vJ ∈ ˜V is the sum of the LogRanks of the aggregated vertices in VJ , while the LogRank of the unaffected vertices v1, . . . , vl+2 is preserved. That is, ˜rT = r1 . . . rl+2 rn−k+1 + . . . + rn . 18 / 30Motivation Structure Reduction Application Conclusions
- 20. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Aggregation of certain motifs Sequential connections: vvv v1v1v1 ......... vkvkvk vvv ⇒ vvv JJJ vvv The LogRank of vJ ∈ ˜V is the sum of the LogRanks of the aggregated vertices in VJ , while the LogRank of the unaffected vertices v1, . . . , vl+2 is preserved. That is, ˜rT = r1 . . . rl+2 rn−k+1 + . . . + rn . 19 / 30Motivation Structure Reduction Application Conclusions
- 21. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Aggregation of certain motifs Almost disconnected subgraphs: v1v1v1 v2v2v2 v3v3v3 v∗v∗ v∗ ⇒ JJJ The LogRank of vJ ∈ ˜V is the sum of the LogRanks of the aggregated vertices in VJ , while the LogRank of the unaffected vertices v1, . . . , vl+2 is preserved. That is, ˜rT = r1 . . . rl+2 rn−k+1 + . . . + rn . 20 / 30Motivation Structure Reduction Application Conclusions
- 22. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Analysis steps: 1 Modelling 2 Model reduction Step 1. Calculation of ranks Step 2. Identification of motifs with low rank locations Step 3. Aggregation of motifs 3 Analysis of the reduced model 21 / 30Motivation Structure Reduction Application Conclusions
- 23. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Meta algorithm : Compute the LogRank r of the network G = (V , E, Mc (v, w)) and generate R∆ repeat Delete and consider v1 ∈ R∆; Generate candidate list C = C(v1, r); whileC = ∅ Delete and consider c1 from the candidate list C; if for c1 reduction error e ≤ ε aggregate c1; clear C; Generate new waiting list R∆; end if end while untilR∆ = ∅ 22 / 30Motivation Structure Reduction Application Conclusions
- 24. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Outline 1 Motivation 2 Network structure information 3 Structure preserving model reduction 4 Application (Jackson networks) 5 Conclusions and outlook 23 / 30Motivation Structure Reduction Application Conclusions
- 25. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Example: Jackson networks Open Jackson network is a queueing network where: procesing time at each location is i.i.d. with exponential distribution all orders belong to the same class, follow the same service time distribution and the same routing mechanism orders arrive from the outside according to the Poisson process with rate α > 0 each arriving order is independently routed to location i with probability p0i ≥ 0 after being processed at some location an order routes according to routing matrix P 24 / 30Motivation Structure Reduction Application Conclusions
- 26. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Dynamics of Jackson networks The effective arrival rate λi of orders at location i is given by traffic equation: λi = α p0i + n j=1 pji λj , i ∈ V . P := PT pe pT o 0 . pe - external inflow probability vector po - outflow probability vector If P is irreducible, the LogRank and the effective arrival rate coincide (up to a constant multiple). 25 / 30Motivation Structure Reduction Application Conclusions
- 27. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Example of reduction v1v1v1 v2v2v2 v3v3v3 v4v4v4 v5v5v5 v6v6v6 v7v7v7 v8v8v8 v9v9v9 v10v10v10 v11v11v11 v12v12v12 v13v13v13 v14v14v14 v15v15v15 v16v16v16 v17v17v17 v18v18v18 v19v19v19 v20v20v20 v21v21v21 v22v22v22 v23v23v23 High rank Low rank 26 / 30Motivation Structure Reduction Application Conclusions
- 28. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Example of reduction v2v2v2 v3v3v3 v5v5v5 v7v7v7 v8v8v8 v9v9v9 v11v11v11 v12v12v12 v14v14v14 v16v16v16 v17v17v17 v18v18v18 v19v19v19 v20v20v20 v21v21v21 v23v23v23 High rank Low rank 27 / 30Motivation Structure Reduction Application Conclusions
- 29. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Outline 1 Motivation 2 Network structure information 3 Structure preserving model reduction 4 Application (Jackson networks) 5 Conclusions and outlook 28 / 30Motivation Structure Reduction Application Conclusions
- 30. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Conclusions A new heuristic approach for structure-preserving model reduction based on: material flow information certain motifs of the network rank of importance of locations Application to Jackson networks Further research: Preservation of stability properties Estimation of approximation error Application to networks with nonlinear dynamics 29 / 30Motivation Structure Reduction Application Conclusions
- 31. Centre for Industrial Mathematics Elgersburg Workshop 2011 Kosmykov Thank you for your attention! This work is part of the research project: ”Stability, Robustness and Approximation of Dynamic Large-Scale Networks - Theory and Applications in Logistics Networks” funded by the Volkswagen Foundation. 30 / 30Motivation Structure Reduction Application Conclusions