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January  2016, 12(1): 117-140. doi: 10.3934/jimo.2016.12.117

## A compaction scheme and generator for distribution networks

 1 Department of Industrial and Information Management, National Cheng Kung University, Tainan, 701, Taiwan

Received  March 2014 Revised  November 2014 Published  April 2015

In a distribution network, materials or products that go through a decomposition process can be considered as flows entering a specialized node, called D-node, which distributes each decomposed flow along an outgoing arc. Flows on each arc emanating from a D-node have to obey a pre-specified proportional relationship, in addition to the capacity constraints. The solution procedures for calculating optimal flows over distribution networks in literature often assumes D-nodes to be disjoint, whereas in reality D-nodes may often connect to each other and complicate the problem. In this paper, we propose a polynomial-time network compaction scheme that compresses a distribution network into an equivalent one of smaller size, which can then be directly solved by conventional solution methods in related literature. In order to provide test cases of distribution networks containing D-nodes for computational tests in related research, we implement a random network generator that produces a connected and acyclic distribution network in a compact form. Mathematical properties together with their proofs are also discussed to provide more insights in the design of our generator.
Citation: I-Lin Wang, Ju-Chun Lin. A compaction scheme and generator for distribution networks. Journal of Industrial and Management Optimization, 2016, 12 (1) : 117-140. doi: 10.3934/jimo.2016.12.117
##### References:
 [1] R. K. Ahuja, T. Magnanti and J. Orlin, Network Flows: Theory, Algorithms and Applications, Prentice Hall, Englewood Cliffs, New Jersey, 1993. [2] R. J. Anderson and J. C. Setubal, Goldberg's algorithm for maximum flow in perspective: A computatioinal study, in Network flows and matching: First DIMACS implementation challenge (eds. D. S. Johnson and C. McGeoch), 12, American Mathematical Society, (1993), 1-17. [3] U. Bahceci and O. Feyzioglu, A network simplex based algorithm for the minimum cost proportional flow problem with disconnected subnetworks, Optimization Letters, 6 (2012), 1173-1184. doi: 10.1007/s11590-011-0356-5. [4] M. D. Chang, C. H. J. Chen and M. Engquist, An improved primal simplex variant for pure processing networks, ACM Transactions on Mathematical Software, 15 (1989), 64-78. doi: 10.1145/62038.62041. [5] C. H. J. Chen and M. Engquist, A primal simplex approach to pure processing networks, Management Science, 32 (1986), 1582-1598. doi: 10.1287/mnsc.32.12.1582. [6] B. V. Cherkassky and A. V. Goldberg, On implementing push-relabel method for the maximum flow problem, Algorithmica, 19 (1997), 390-410. doi: 10.1007/PL00009180. [7] B. T. Denton, J. Forrest and R. J. Milne, Ibm solves a mixed-integer program to optimize its semiconductor supplychain, Interfaces, 36 (2006), 386-399. [8] S. C. Fang and L. Qi, Manufacturing network flows: A generalized network flow model for manufacturingprocess modeling, Optimization Methods and Software, 18 (2003), 143-165. doi: 10.1080/1055678031000152079. [9] D. Goldfarb and M. D. Grigoriadis, A computational comparison of the dinic and network simplex methods formaximum flow, Annals of Operations Research, 13 (1988), 83-123. doi: 10.1007/BF02288321. [10] D. Klingman, A. Napier and J. Stutz, Netgen: A program for generating large scale capacitated assignment, transportation and minimum cost flow networks, Management Science, 20 (1974), 814-820. [11] J. Koene, Minimal Cost Flow in Processing Networks, a Primal Approach, PhD thesis, Eindhoven University of Technology, Eindhoven, The Netherlands, 1983. [12] L.-C. Kung and C.-C. Chern, Heuristic factory planning algorithm for advanced planning and scheduling, Computers and Operations Research, 36 (2009), 2513-2530. doi: 10.1016/j.cor.2008.09.013. [13] Y.-K. Lin, C.-T. Yeh and C.-F. Huang, Reliability evaluation of a stochastic-flow distribution network with delivery spoilage, Computers and Industrial Engineering, 66 (2013), 352-359. doi: 10.1016/j.cie.2013.06.019. [14] H. Lu, E. Yao and L. Qi, Some further results on minimum distribution cost flow problems, Journal of Combinatorial Optimization, 11 (2006), 351-371. [15] P. Lyon, R. J. Milne, R. Orzell and R. Rice, Matching assets with demand in supply-chain management at ibm microelectronics, Interfaces, 31 (2001), 108-124. doi: 10.1287/inte.31.1.108.9693. [16] R. L. Sheu, M. J. Ting and I. L. Wang, Maximum flow problem in the distribution network, Journal of Industrial and Management Optimization, 2 (2006), 237-254. doi: 10.3934/jimo.2006.2.237. [17] J. Shu, M. Chou, Q. Liu, C.-P. Teo and I.-L. Wang, Models for effective deployment and redistribution of bicycles within public bicycle-sharing systems, Operations, 61 (2013), 1346-1359. doi: 10.1287/opre.2013.1215. [18] I. L. Wang and S. J. Lin, A network simplex algorithm for solving the minimum distribution cost problem, Journal of Industrial and Management Optimization, 5 (2009), 929-950. doi: 10.3934/jimo.2009.5.929. [19] I. L. Wang and Y. H. Yang, On solving the uncapacitated minimum cost flow problems in a distribution network, International Journal of Reliability and Quality Performance, 1 (2009), 53-63.

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##### References:
 [1] R. K. Ahuja, T. Magnanti and J. Orlin, Network Flows: Theory, Algorithms and Applications, Prentice Hall, Englewood Cliffs, New Jersey, 1993. [2] R. J. Anderson and J. C. Setubal, Goldberg's algorithm for maximum flow in perspective: A computatioinal study, in Network flows and matching: First DIMACS implementation challenge (eds. D. S. Johnson and C. McGeoch), 12, American Mathematical Society, (1993), 1-17. [3] U. Bahceci and O. Feyzioglu, A network simplex based algorithm for the minimum cost proportional flow problem with disconnected subnetworks, Optimization Letters, 6 (2012), 1173-1184. doi: 10.1007/s11590-011-0356-5. [4] M. D. Chang, C. H. J. Chen and M. Engquist, An improved primal simplex variant for pure processing networks, ACM Transactions on Mathematical Software, 15 (1989), 64-78. doi: 10.1145/62038.62041. [5] C. H. J. Chen and M. Engquist, A primal simplex approach to pure processing networks, Management Science, 32 (1986), 1582-1598. doi: 10.1287/mnsc.32.12.1582. [6] B. V. Cherkassky and A. V. Goldberg, On implementing push-relabel method for the maximum flow problem, Algorithmica, 19 (1997), 390-410. doi: 10.1007/PL00009180. [7] B. T. Denton, J. Forrest and R. J. Milne, Ibm solves a mixed-integer program to optimize its semiconductor supplychain, Interfaces, 36 (2006), 386-399. [8] S. C. Fang and L. Qi, Manufacturing network flows: A generalized network flow model for manufacturingprocess modeling, Optimization Methods and Software, 18 (2003), 143-165. doi: 10.1080/1055678031000152079. [9] D. Goldfarb and M. D. Grigoriadis, A computational comparison of the dinic and network simplex methods formaximum flow, Annals of Operations Research, 13 (1988), 83-123. doi: 10.1007/BF02288321. [10] D. Klingman, A. Napier and J. Stutz, Netgen: A program for generating large scale capacitated assignment, transportation and minimum cost flow networks, Management Science, 20 (1974), 814-820. [11] J. Koene, Minimal Cost Flow in Processing Networks, a Primal Approach, PhD thesis, Eindhoven University of Technology, Eindhoven, The Netherlands, 1983. [12] L.-C. Kung and C.-C. Chern, Heuristic factory planning algorithm for advanced planning and scheduling, Computers and Operations Research, 36 (2009), 2513-2530. doi: 10.1016/j.cor.2008.09.013. [13] Y.-K. Lin, C.-T. Yeh and C.-F. Huang, Reliability evaluation of a stochastic-flow distribution network with delivery spoilage, Computers and Industrial Engineering, 66 (2013), 352-359. doi: 10.1016/j.cie.2013.06.019. [14] H. Lu, E. Yao and L. Qi, Some further results on minimum distribution cost flow problems, Journal of Combinatorial Optimization, 11 (2006), 351-371. [15] P. Lyon, R. J. Milne, R. Orzell and R. Rice, Matching assets with demand in supply-chain management at ibm microelectronics, Interfaces, 31 (2001), 108-124. doi: 10.1287/inte.31.1.108.9693. [16] R. L. Sheu, M. J. Ting and I. L. Wang, Maximum flow problem in the distribution network, Journal of Industrial and Management Optimization, 2 (2006), 237-254. doi: 10.3934/jimo.2006.2.237. [17] J. Shu, M. Chou, Q. Liu, C.-P. Teo and I.-L. Wang, Models for effective deployment and redistribution of bicycles within public bicycle-sharing systems, Operations, 61 (2013), 1346-1359. doi: 10.1287/opre.2013.1215. [18] I. L. Wang and S. J. Lin, A network simplex algorithm for solving the minimum distribution cost problem, Journal of Industrial and Management Optimization, 5 (2009), 929-950. doi: 10.3934/jimo.2009.5.929. [19] I. L. Wang and Y. H. Yang, On solving the uncapacitated minimum cost flow problems in a distribution network, International Journal of Reliability and Quality Performance, 1 (2009), 53-63.
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