An approach to model reduction of logistic networks based on ranking
0 likes51 views
This document proposes an approach to model reduction of logistic networks based on node ranking. It introduces a method to rank the importance of locations in a logistic network based on material flows. Three rules are described for reducing the model based on excluding, aggregating, or replacing subnetworks of low-ranked nodes. Examples are provided to illustrate how the rules can be applied to simplify large logistic network models. The approach allows analysis of reduced models to gain insights while decreasing computational requirements for large, real-world networks.
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
- 2. Centre for Industrial Mathematics Outline 1 Introduction 2 Ranking in logistic networks 3 Model Reduction 4 Conclusions and further research 2 / 17Introduction Ranking Reduction rules Conclusions and further research
- 3. Centre for Industrial Mathematics 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 3 / 17Introduction Ranking Reduction rules Conclusions and further research
- 4. Centre for Industrial Mathematics 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 4 / 17Introduction Ranking Reduction rules Conclusions and further research
- 5. Centre for Industrial Mathematics Modelling We model a logistic network as a weighted graph where only mate- rial flows are taken into account. 5 / 17Introduction Ranking Reduction rules Conclusions and further research
- 6. Centre for Industrial Mathematics 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 5 / 17Introduction Ranking Reduction rules Conclusions and further research
- 7. Centre for Industrial Mathematics 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 6 / 17Introduction Ranking Reduction rules Conclusions and further research
- 8. Centre for Industrial Mathematics 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. 7 / 17Introduction Ranking Reduction rules Conclusions and further research
- 9. Centre for Industrial Mathematics 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 8 / 17Introduction Ranking Reduction rules Conclusions and further research
- 10. Centre for Industrial Mathematics 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 . 9 / 17Introduction Ranking Reduction rules Conclusions and further research
- 11. Centre for Industrial Mathematics 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. 9 / 17Introduction Ranking Reduction rules Conclusions and further research
- 12. Centre for Industrial Mathematics 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. 10 / 17Introduction Ranking Reduction rules Conclusions and further research
- 13. Centre for Industrial Mathematics Exclusion of low-ranked locations connected to only one location Figure: Original network. 11 / 17Introduction Ranking Reduction rules Conclusions and further research
- 14. Centre for Industrial Mathematics Exclusion of low-ranked locations connected to only one location Figure: Original network. Figure: Reduced network. 11 / 17Introduction Ranking Reduction rules Conclusions and further research
- 15. Centre for Industrial Mathematics Aggregation of low-ranked locations connected parallely Figure: Original network. 12 / 17Introduction Ranking Reduction rules Conclusions and further research
- 16. Centre for Industrial Mathematics Aggregation of low-ranked locations connected parallely Figure: Original network. Figure: Reduced network. 12 / 17Introduction Ranking Reduction rules Conclusions and further research
- 17. Centre for Industrial Mathematics Aggregation of low-ranked locations connected sequentially Figure: Original network. 13 / 17Introduction Ranking Reduction rules Conclusions and further research
- 18. Centre for Industrial Mathematics Aggregation of low-ranked locations connected sequentially Figure: Original network. Figure: Reduced network. 13 / 17Introduction Ranking Reduction rules Conclusions and further research
- 19. Centre for Industrial Mathematics Exclusion of sub-networks of low-ranked locations connected to only one important location Figure: Original network. 14 / 17Introduction Ranking Reduction rules Conclusions and further research
- 20. Centre for Industrial Mathematics Exclusion of sub-networks of low-ranked locations connected to only one important location Figure: Original network. Figure: Reduced network. 14 / 17Introduction Ranking Reduction rules Conclusions and further research
- 21. Centre for Industrial Mathematics Complex example Figure: Original network. Figure: Reduced network. 15 / 17Introduction Ranking Reduction rules Conclusions and further research
- 22. Centre for Industrial Mathematics 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 16 / 17Introduction Ranking Reduction rules Conclusions and further research
- 23. Centre for Industrial Mathematics Thank you for your attention 17 / 17Introduction Ranking Reduction rules Conclusions and further research