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Minimum concave cost multicommodity network design

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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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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


Neighborhood of ε

C i :

Total network cost obtained by using method i

C D :

Total network cost obtained by disaggregate local search


Minimum concave cost multicommodity network design problem


Circuit switching MCMNDP


Packet switching MCMNDP


Minoux greedy


Accelerated Minoux greedy


Modified Minoux greedy


Modified accelerated Minoux greedy


Concave branch elimination


Concave link elimination

CLE Inc.:

Concave link elimination incremental


Disaggregate local search


Adapted disaggregate local search


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