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Algorithms for an Integer Multicommodity Network Flow Problem with Node Reliability Considerations

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Abstract

In this paper, we consider the problem of sending a set of multiple commodities from their origin to destination nodes via intermediate hubs. Each hub node is associated with a reliability function, which depends on the total flow that crosses that hub. The probability that each commodity is successfully relayed from its origin to its destination is given by the product of hub reliabilities on the commodity’s path. The problem we consider seeks to find minimum-cost commodity paths such that each commodity reaches its destination with a sufficiently large probability. We first formulate the problem as a nonlinear multicommodity network-flow problem and prove that it is strongly NP-hard. We then present two linearization techniques for this formulation, and propose a pair of lower- and upper-bounding formulations, which can then be used within an exact cutting-plane algorithm to solve the problem. Finally, we analyze the computational effectiveness of our proposed strategies on a set of randomly generated instances.

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Acknowledgements

The authors are grateful for the comments of two anonymous referees, whose remarks led to an improved version of this paper. Furthermore, we sincerely appreciate the cooperation of the GAMS Development Corporation in providing an evaluation copy of GAMS 24.1.1 and GloMIQO 2 for the purposes of our computational investigation. The authors also gratefully acknowledge the support of the National Science Foundation under grant CMMI-1100765, the Defense Threat Reduction Agency under grant HDTRA-10-01-0050, the Air Force Office of Scientific Research under grant FA9550-12-1-0353, and the Office of Naval Research under grant N000141310036.

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  1. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA

    Bita Tadayon & J. Cole Smith

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  1. Bita Tadayon
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  2. J. Cole Smith
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Correspondence to J. Cole Smith.

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Communicated by Christodoulos Floudas.

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Tadayon, B., Smith, J.C. Algorithms for an Integer Multicommodity Network Flow Problem with Node Reliability Considerations. J Optim Theory Appl 161, 506–532 (2014). https://doi.org/10.1007/s10957-013-0378-5

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