# Explicit flow-risk allocation for cooperative maximum flow problems under interval uncertainty

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## Abstract

Many decision-making problems in transportation networks can be defined as maximum flow problems. During the last five decades, several efficient solution approaches have been proposed for the deterministic maximum flow problems. On the other hand, arc capacities of networks cannot be precisely defined in many real life settings. These networks are called uncertain. In this case, it becomes challenging to maintain a stable flow on the network. This paper presents a new approach based on the framework of interval analysis for the solution of maximum flow problems. We address a multiple-owners network problem by presenting a risk explicit interval linear programming model for the desired value of the system aspiration level. Afterwards, we employ a well-known collaborative game theoretic approach (the Shapley value) in a multiple-owners network under interval uncertainty in order to solve the maximum flow problem. A detailed numerical example is provided to present the suitability of the proposed approach in devising a stable network flow. The obtained numerical results and the trade-offs between decision risk and network flow information would be very valuable for supporting decision makers in resolving maximum flow problems when facing uncertainty.

## Keywords

Maximum flow problem Multiple-owners graph Cooperative games Risk explicit interval linear programming Risk allocation The Shapley value## Notes

### Acknowledgements

Authors are grateful to the anonymous reviewers and editor for their helpful suggestions that concretely have contributed to ameliorate the paper from its original version.

## References

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