Topology optimisation of repairable flow networks for a maximum average availability

摘要

We state and prove a theorem regarding the average production availability of a repairable flow network, composed of independently working edges, whose failures follow a homogeneous Poisson process. The average production availability is equal to the average of the maximum output flow rates on demand from the network, calculated after removing the separate edges with probabilities equal to the edges unavailabilities. This result creates the basis of extremely fast solvers for the production availability of complex repairable networks, the running time of which is independent of the length of the operational interval, the failure frequencies, or the lengths of the downtimes for repair. The computational speed of the production availability solver has been extended further by a new algorithm for maximising the output flow in a network after the removal of several edges, which does not require determining the feasible edge flows in the network. The algorithm for maximising the network flow is based on a new theorem, referred to as 'the maximum flow after edge failures theorem', stated and proved for the first time. Finally, unlike heuristic optimisation algorithms, the proposed algorithm for a topology optimisation of the network always determines the optimal solution. The high computational speed of the developed production availability solver created the possibility for embedding it in simulation loops, performing a topology optimisation of large and complex repairable networks, aimed at attaining a maximum average availability within a specified budget for building the network. An exact optimisation method has been proposed, based on pruning the full-complexity network by using the branch and bound method as a way of exploring possible network topologies. This makes the proposed algorithm much more efficient, compared to an algorithm implementing a full exhaustive search. In addition, the proposed method produces an optimal solution compared to heuristic optimisation methods. The application of the bound and branch method is possible because of the monotonic dependence of the production availability on the number of the edges pruned from the full-complexity network.