M T TODINOV: TOPOLOGY OPTIMISATION OF REPAIRABLE FLOW NETWORKS AND . . . . IJSSST, Vol. 11, No. 3 ISSN: 1473-804x online, 1473-8031 print 75 Topology Optimization of Repairable Flow Networks and Reliability Networks M.T.Todinov Department of Mechanical Engineering and Mathematical Sciences, Oxford Brookes University Oxford OX33 1HX, UK e-mail: mtodinov@brookes.ac.uk Abstract—A framework for topology optimization of repairable flow networks and reliability networks is presented. The optimization consists of determining the optimal network topology with a maximum transmitted flow achieved within a specified budget for building the network. A method for a topology optimization of reliability networks of safety-critical systems has also been developed. The proposed method solves the important problem related to how to build within a fixed budget the system characterised by the smallest risk of failure. Both methods are based on pruning the full complexity network by using the bound and branch method as a way of exploring possible network topologies. The methods have a superior running time compared to methods based on a full exhaustive search. A method has also been developed for assessing the threshold flow rate reliability of repairable flow networks with complex topology. The method is based on estimating the probability that the output flow will be equal to or greater than a specified threshold value. The method is based on a Monte Carlo simulation and is superior to alternative methods based on minimal paths and cut sets. It can be applied for assessing the quality of service of telecommunication networks. Keywords- repairable flow networks, flow network optimisation, maximum flow, flow path, specific flow resistance, minimal paths I. INTRODUCTION Repairable flow networks are a very important class of networks (e.g. production networks, communication networks, transportation networks, energy distribution networks and supply networks). An essential feature of the repairable flow networks, is that a renewal of failed components is taking place after a certain delay for repair. This feature distinguishes repairable flow networks from stochastic flow networks [1-4] and static flow networks [5- 11]. Analysis and optimisation of repairable flow networks is a new area of research initiated in [12]. A repairable flow network can be presented as a directed graph G=(V,E) where V denotes the set of nodes and E denotes the set of edges (components). The nodes are notional and cannot fail while the edges represent the separate components and are characterized by a flow capacity, cost, hazard (failure rate) and time for repair. The problem related to the probability that on demand, the flow from the stochastic network will be equal or greater than a specified level was discussed in [1-4]. The proposed approaches however have been based on minimal paths and cut sets. Although for small size networks this approach is acceptable, with increasing the size of the network, the number of minimal paths and cut sets increases exponentially and this approach is no longer feasible. This point can be illustrated with the example in Fig.1. The flow network in the figure has N N N  minimal cut sets and 1  N N minimal paths. Even for the moderate 10  N , the storage and manipulation of the minimal paths and cut sets is no longer possible. ... ... ... ... ... ... ... 1 2 2 1 N N ... ... ... 1 2 2 1 N N S T Figure 1. An example of a flow network where the number of minimal paths and minimal cut sets increases exponentially with increasing the size of the system. As we show later, this limitation can be avoided if a simulation method is used for revealing the reliability of the output flow rate. Every repairable flow network is also associated with a specific capital cost for building it. A significant part of this cost is the sum of the costs of its components. A very important objective here is to achieve a desired amount of output flow, at a minimal cost for building the network. Currently, no algorithms are known to be capable of achieving this objective. Existing software tools, handling flows in networks operate on a fixed network topology. They do not perform a repeated modification of the network topology and calculation of the flow in the presence of failures, in order to determine the right topology combining a minimum cost for building the network, improved resistance of the flow to component failures and maximizing the transmitted flow. Another important aspect of the topology optimization of networks is the topology optimization of reliability