An approach to model reduction of logistic networks based on ranking

1. An approach to model reduction of logistic
networks based on ranking
Bernd Scholz-Reiter1
Fabian Wirth2
Sergey Dashkovskiy3
Michael Kosmykov3
Thomas Makuschewitz1
Michael Sch¨onlein2
1
BIBA - Bremer Institut f¨ur Produktion und Logistik GmbH
at the University of Bremen, Bremen, Germany
2
Institute of Mathematics, University of W¨urzburg, W¨urzburg, Germany
3
Centre of Industrial Mathematics, University of Bremen, Bremen, Germany
17-21 August 2009, LDIC’09, Bremen, Germany

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Outline
1 Introduction
2 Ranking in logistic networks
3 Model Reduction
4 Conclusions and further research
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Characteristics of the practical example
3 production sites for pumps
5 distribution centres
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
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Approach for the analysis of large logistic networks
1 Starting from a real world logistic
network a model is obtained
2 Identification of important and
less important locations with
ranking
3 Dependent on the rank of
locations a model of lower size is
set up
4 Analysing of the model of lower
size
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Modelling
We model a logistic network as a
weighted graph where only mate-
rial flows are taken into account.
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Modelling
We model a logistic network as a
weighted graph where only mate-
rial flows are taken into account.
Weighted adjacency matrix A of the graph:
A =








0 0 0 0 0 6
7 0 0 8 0 0
4 5 0 0 7 0
0 0 2 0 0 0
0 11 0 0 0 0
6 0 0 0 0 0








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Rank of location
The importance of logistic net-
work location depends on:
importance of receiving
locations
material flows to these
locations
NRi = α
n
j=1
aij NRj + (1 − α)1
n , 0 ≤ α ≤ 1
Formulation as an eigenvector problem:
NR = G · NR, NR = (0.1040; 0.2455; 0.2016; 0.2056; 0.1778; 0.0655)T
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Matrix properties
A matrix A is called column-normalized if for all i = 1, . . . , n
n
j=1
aij =
1, if there exists j such that aij = 0
0, otherwise
A matrix A is column-stochastic if
n
j=1
aij = 1 for all i = 1, . . . , n.
It is called primitive if there exists a positive integer k such that the
matrix Ak
has only positive elements.
We call an adjacency matrix A irreducible if the corresponding graph
is strongly connected, i.e., for every nodes i and j of the graph there
exists a sequence of directed edges connecting i to j. Note that any
primitive matrix is irreducible.
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Transformation steps
Column-normalization:
H = Hij , Hij =



aij
k∈Id
j
akj
= aij , if
k∈Id
j
akj = 0
0, otherwise
Make the matrix stochastic:
S = H + ebT
n , e = (1, . . . , 1)T
∈ Rn
, bi = 1, if
n
j=1
Hji = 0 and bi = 0
otherwise.
Making the matrix primitive:
G = αS + (1 − α)E, E = eeT
n
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Eigenvector problem
The problem of finding a right-eigenvector:
NR = G · NR
is equivalent to the system of linear equations
NRi = α
n
j=1
aij NRj + (1 − α)1
n .
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Eigenvector problem
The problem of finding a right-eigenvector:
NR = G · NR
is equivalent to the system of linear equations
NRi = α
n
j=1
aij NRj + (1 − α)1
n .
As matrix G is stochastic and primitive from Perron-Frobenius theorem it
follows that the solution exists and is unique.
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Rules for model reduction
1 Exclusion of low-ranked locations connected to only one
location
Locations with low ranks that are connected to only one location
might by excluded in the reduced model
2 Aggregation of low-ranked locations
Low-ranked locations are aggregated with each other by summing up
the weights of their links accordingly if they are connected by one of
the following two types to the network:
paralell
sequential
3 Exclusion of sub-networks of low-ranked locations connected
to only one important location
Sub-networks of locations that are connected to the rest of the
network only by one location can be modeled as a single location.
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Exclusion of low-ranked locations connected to only
one location
Figure: Original network.
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Exclusion of low-ranked locations connected to only
one location
Figure: Original network. Figure: Reduced network.
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Aggregation of low-ranked locations connected
parallely
Figure: Original network.
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Aggregation of low-ranked locations connected
parallely
Figure: Original network. Figure: Reduced network.
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Aggregation of low-ranked locations connected
sequentially
Figure: Original network.
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Aggregation of low-ranked locations connected
sequentially
Figure: Original network. Figure: Reduced network.
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Exclusion of sub-networks of low-ranked locations
connected to only one important location
Figure: Original network.
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Exclusion of sub-networks of low-ranked locations
connected to only one important location
Figure: Original network.
Figure: Reduced network.
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Complex example
Figure: Original network.
Figure: Reduced network.
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Conclusions and further research
Large-logistic network has been
modeled as a weighted graph
An approach of node ranking
was proposed
Introduced 4 rules allows reduce
the size of the model
A theoretical analysis of
reduction rules has to be
provided
The reduced model can be
studied
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Thank you for your attention
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