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Minimum concave cost multicommodity network design

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Abstract

The minimum concave cost multicommodity network design problem (MCMNDP) arises in many application areas, such as transportation planning, energy distribution systems and especially in the design of both packet and circuit switching backbone networks. Exact concave cost optimization algorithms have been developed but they are applicable only if the network size is small. Therefore, MCMNDP is usually solved using non-exact iterative methods. In this paper, such heuristic techniques proposed within the context of circuit switching and packet switching network design are evaluated in detail. Following a comprehensive literature survey, Yaged’s linearization, Minoux’s greedy and Minoux’s accelerated greedy methods have been selected for the circuit switching network design case for further investigation. Minoux’s greedy methods are found to create routes that include cycles causing degradation in the quality of the solution; therefore, we propose a simple but effective modification scheme as a cycle elimination strategy. Similarly, but within the context of packet switching network design, Gerla and Kleinrock’s concave branch elimination, Gersht’s greedy, and Stacey’s concave link elimination methods have been selected for further investigation. All of these methods consider aggregate flows on each link, simultaneously re-routing more than one commodity in one step. In this paper, we propose an alternative disaggregate approach, where only one commodity is handled at a time. Our final proposal is the adaptation of the algorithms proposed for circuit switching network design to the packet switching case. Then an extensive comparative computational study is performed for a number of networks and cost structures to help establish the best method with respect to time and solution quality. Our computational results have shown that the performances of the MCMNDP algorithms heavily depend on the network type and the cost structure. The results have also revealed that our proposed modification to Minoux greedy to eliminate cycles leads to considerable improvements and our proposed disaggregate approach gives the best result in some networks with certain cost structures.

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Abbreviations

G :

Undirected graph

G D :

Directed graph

X :

Set of n nodes

U :

Set of m links

U D :

Set of m arcs

b k :

Demand vector

r k :

Demand value

K :

Set of commodities

Φ u :

Concave-cost function for link (arc) u

φ u k :

k-th individual flow component on link (arc) u

ψ u :

Aggregate flow on link (arc) u

A :

Node-link (node-arc) incidence matrix of G(G D )

F u :

Fixed installation cost

α u :

Link (arc) concavity value

c u :

Capacity of link (arc) u

γ :

Total network traffic

T max  :

Allowed maximum average network delay

T :

Average network delay

L :

Chain of links

ε :

Flow pattern

S k :

Set of all possible flow solutions φ k corresponding to elementary chains only

L k :

Chain along which the k-th commodity flows

N(ε):

Neighborhood of ε

C i :

Total network cost obtained by using method i

C D :

Total network cost obtained by disaggregate local search

MCMNDP:

Minimum concave cost multicommodity network design problem

MCMNDPC :

Circuit switching MCMNDP

MCMNDPP :

Packet switching MCMNDP

M.G.:

Minoux greedy

A.M.G.:

Accelerated Minoux greedy

M.M.G.:

Modified Minoux greedy

M.A.M.G.:

Modified accelerated Minoux greedy

CBE:

Concave branch elimination

CLE:

Concave link elimination

CLE Inc.:

Concave link elimination incremental

DLS:

Disaggregate local search

ADLS:

Adapted disaggregate local search

References

  1. 1.

    Amiri, A., & Pirkul, H. (1997). New formulation and relaxation to solve a concave-cost network flow problem. Journal of Operational Research Society, 48/3, 278–287.

  2. 2.

    An, L. T. H., & Tao, P. D. (2002). D. C. programming approach for multicommodity network optimization problems with step increasing cost functions. Journal of Global Optimization, 22(1–4), 205–232.

  3. 3.

    Balakrishnan, A., Magnanti, T. L., & Wong, R. T. (1989). A dual-ascent procedure for large scale uncapacitated network design. Operational Research, 37, 716–740.

  4. 4.

    Balakrishnan, A., & Graves, C. S. (1989). A composite algorithm for a concave-cost network flow problem. Networks, 19, 175–202.

  5. 5.

    Balakrishnan, A., Magnanti, T. L., Shulman, A., & Wong, R. T. (1991). Models for planning capacity expension in local access telecommunication networks. Annals of Operations Research, 33, 239–284.

  6. 6.

    Balakrishnan, A., Magnanti, T. L., & Wong, R. T. (1995). A decomposition algorithm for local access telecommunications network expanding planning. Operational Research, 43, 58–76.

  7. 7.

    Baybars, I., & Edahl, R. H. (1988). A heuristic method for facility planning in telecommunications networks with multiple alternate routes. Naval Research Logistics, 35, 503–528.

  8. 8.

    Bazlamaçcı, C. F., & Hindi, K. S. (1996). Enhanced adjacent extreme-point search and tabu search for the minimum concave-cost uncapacitated transshipment problem. Journal of Operational Research Society, 47, 1150–1165.

  9. 9.

    Bazlamaçcı, C. F. (1996). Optimised network design: minimum spanning trees and minimum concave-cost problems. Ph.D. Thesis, University of Manchester Institute of Science and Technology (UMIST), Manchester.

  10. 10.

    Bienstock, D. (1993). Computational experience with an effective heuristic for some capacity expansion problems in local access networks. Telecommunication Systems, 1, 379–400.

  11. 11.

    Chang, S. G., & Gavish, B. (1993). Telecommunications network topological design and capacity expansion: formulations and algorithms. Telecommunication Systems, 1, 99–131.

  12. 12.

    Chang, S. G., & Gavish, B. (1995). Lower bounding procedures for multiperiod telecommunications network expansion problems. Operations Research, 43, 43–57.

  13. 13.

    Chattopadhyay, G. N., Morgan, T. W., & Raghuram, A. (1989). An innovative technique for backbone network design. IEEE Transactions on Systems, 19(5), 1122–1132.

  14. 14.

    Cheng, S. T. (1998). Topological optimization of a reliable communication network. IEEE Transactions on Reliability, 47/3, 225–233.

  15. 15.

    Dutta, A., & Lim, J. I. (1992). A multiperiod capacity planning model for backbone computer communication networks. Operations Research, 40, 689–705.

  16. 16.

    Gallo, G., & Sodini, C. (1979). Concave cost minimization on networks. European Journal of Operations Research, 3, 239–249.

  17. 17.

    Gallo, G., & Sodini, C. (1979). Adjacent extreme flows and application to minimum concave cost flow problems. Networks, 9, 95–121.

  18. 18.

    Gavish, B., & Neuman, I. (1989). A system for routing and capacity assignment in computer communication networks. IEEE Transactions on Communications, 5 COM-37, 360–366.

  19. 19.

    Gavish, B., & Altinkemer, K. (1990). Backbone network design tools with economic tradeoffs. ORSA Journal on Computing, 2, 236–245.

  20. 20.

    Gerla, M., & Kleinrock, L. (1987). On the topological design of distributed computer networks. IEEE Transactions on Communications, COM-25, 48–60.

  21. 21.

    Gersht, A., & Weihmayer, R. (1990). Joint optimization of data network design and facility selection. IEEE Journal on Selected Areas in Communications, 8(9), 1667–1681.

  22. 22.

    Goel, A., & Estrin, D. (2005). Simultaneous optimization for concave costs: single sink aggregation or single source buy-at-bulk. Algorithmica, 43(1–2), 5–15.

  23. 23.

    Gondran, M., & Minoux, M. (1984). Graphs and algorithms. New York: Wiley.

  24. 24.

    Kleinrock, L. (1976). Queuing Systems, vol. 2. Computer Applications. New York: Wiley.

  25. 25.

    Luss, H. (1982). Operations research and the capacity expansion programs: a survey. Operations Research, 30, 907–974.

  26. 26.

    Magnanti, T. L., & Wong, R. T. (1984). Network design and transportation planning: models and algorithms. Transportation Science, 18(1), 1–55.

  27. 27.

    Magnanti, T. L., Mirchandani, P., & Vachani, R. (1995). Modeling and solving the two-facility capacitated network loading problem. Operations Research, 43, 142–157.

  28. 28.

    Minoux, M. (1976). Multinots de cout minimal avec functions de cout concaves. Annales des Telecommunications, 31, 77–92.

  29. 29.

    Minoux, M. (1987). Network synthesis and dynamic optimization. Annals of Discrete Mathematics, 31, 283–324.

  30. 30.

    Minoux, M. (1989). Network synthesis and optimum network design problems: Models, solution methods and applications. Networks, 9, 313–360.

  31. 31.

    Muriel, A., & Munsh, F. (2004). Capacitated multicommodity network flow problems with piecewise linear concave costs. IIE Transactions, 36(7), 683–696.

  32. 32.

    Parrish, S. H., Cox, T., Kuehner, W., & Qui, Y. (1987). Planning for optimal expansion of leased line communications network. IEEE Transactions on Communications, COM-35, 202–209.

  33. 33.

    Say, F. (2005). Minimum concave cost multicommodity network design. M.Sc. Thesis, Middle East Technical University, Ankara.

  34. 34.

    Sharma, R. L. (1990). Network topology optimization: The art and science of network design. New York: Reinhol.

  35. 35.

    Stacey, C. H. E., Eyers, T., & Anido, G. J. (2000). A concave link elimination (CLE) procedure and lower bound for concave topology, capacity and flow assignment network design problems. Telecommunication Systems, 13, 351–372.

  36. 36.

    Ward, J. (1999). Minimum-aggregate-concave-cost multicommodity flows in strong-series-parallel networks. Mathematics of Operations Research, 24(1), 106–129.

  37. 37.

    Yaged, B. A. Jr. (1971). Minimum cost routing for static network models. Networks, 1, 139–172.

  38. 38.

    Yaged, B. A. Jr. (1973). Minimum cost routing for dynamic network models. Networks, 3, 193–224.

  39. 39.

    Zadeh, N. (1973). On building minimum cost communication networks. Networks, 3, 315–331.

  40. 40.

    Zangwill, W. I. (1968). Minimum concave cost flows in certain networks. Management Science, 14, 429–450.

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Correspondence to Cüneyt F. Bazlamaçcı.

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Bazlamaçcı, C.F., Say, F. Minimum concave cost multicommodity network design. Telecommun Syst 36, 181–203 (2007). https://doi.org/10.1007/s11235-008-9068-2

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Keywords

  • Concave cost network design
  • Circuit switching and packet switching network design
  • Multicommodity flow problem