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
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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

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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
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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).
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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
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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
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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