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...programming, heuristics and meta-heuristics, all of which have given either optimal or very good solutions. Tillman et al. (1997), Kuo and Prasad (2000) and Kuo et al. (2001) provide good overview of these techniques. Recent work on redundancy optimization has been directed towards maximizing the percentile life at a specified failure probability since reliability maximization problems are no longer applicable when there is no specific mission time (Coit and Smith, 1995; Kim and Kuo, 2003).
Traditional binary reliability modeling assumes that a system/component is either working or has failed, however, in some cases this is not an accurate representation, since systems/components are able to perform their task but at a degraged level. The failure of some components leads only to the degradation of the system performance and as such, a range of states governs the reliability behavior of the system. Multi-state reliability assumes that both the state of the system and/or its components can have a range of performance levels varying from perfectly functioning to complete failure.
The multi-state concept was introduced in the mid-1970s. Barlow and Wu (1978), El-Neveihi et al. (1978), and Ross (1979) introduced the basic concepts of multi-state system reliability. Aven (1985, 1993) presented two efficient algorithms for reliability evaluation and gave the relevant performance measures for monotone multi-state systems. Ushakov (1986, 1987, 1988) introduced the concept of a Universal Generating Function (UGF). Later Lisnianski et al. (1996), Levitin et al. (1997) and Lisnianski and Levitin (2003), handled the redundancy optimization problem for series, parallel, series-parallel, and bridge structures using the UGF method and Genetic Algorithms (GAs). GA can be a useful and effective optimization tool for many applications; however, it does have the disadvantages that it can be cumbersome and involve long running times.
In this paper a heuristic is developed to solve a redundancy allocation problem for a multi-state series-parallel system whose component capacity is the performance measure. The objective is to minimize the cost subject to system-level-performance constraints (to provide a desired level of reliability). System reliability is defined as the ability to satisfy consumer demands which is presented as a piecewise cumulative load curve. The components are binary and chosen from a list of products available on the market. They are characterized in terms of their feeding capacity, reliability and cost. The solution approach makes use of a homogeneous collection of components to provide redundancy in a subsystem. This problem was initially introduced by Ushakov (1987) and solved by Lisnianski et al. (1996) and Levitin et al. (1997) using the UGF and GA methods, and also by Ramirez-Marquez and Coit (2004) who proposed a heuristic called RAMC.
Our proposed heuristic is motivated by RAMC. A sensitivity factor is defined, which helps in making the selection of component type. First an initial solution is constructed and then at every subsystem, called a node, branching is performed. In the branching phase, the average component reliability required is computed, and each node is re-explored for a better solution. The selection of the component type is made based on the defined, sensitivity factor, giving a new solution at every branch. If during an execution an excursion into an infeasible region is made, then the solution is improved by moving towards the feasible region. Since the optimal solution lies near the boundary of the feasible and infeasible regions, excursions are helpful by alleviating the risk of becoming trapped in a local optimum (Kim and Yum, 1993). After a series of possible improvements in the solution, the selection of the optimal/nearly optimal solution is made from the set consisting of the initial solution and solutions obtained at every branch. The approach is simple and easy to use.
The algorithm has been applied to the power systems studied by Lisnianski et al. (1996) and Levitin et al. (1997) for various levels of reliability requirements. The proposed heuristic to be referred as MARG, is appealing because of its simplicity, computational efficiency and observed solution quality. The algorithm is simple to understand and hence easy to code and apply. The results obtained are equivalent to the results obtained using GAs (Lisnianski et al., 1996; Levitin et al., 1997) but better than the heuristic solutions (Ramirez-Marquez and Coit, 2004). Also, the running time is significantly less than those for the GA and RAMC approaches. As such, the heuristic developed is attractive, and can be easily and efficiently applied to numerous real-life systems.
2. Assumptions
1. The states of the components are binary (i.e., either working or failed) with known cost, reliability and feeding capacity; the capacity of components is the performance measure.
2. The component reliabilities are known and deterministic.
3. The multi-state series-parallel system has a finite number of system states that are s-independent.
4. Failed components do not damage the system and are non-repairable.
5. Homogeneous redundancy is used.
3. Notation
n = number of subsystems connected in series;
[m.sub.i] = number of versions available for each subsystem i, i = 1, 2,..., n;
[x.sub.ij] = number of type j components in parallel used in subsystem i;
x = system design vector ([x.sub.11], [x.sub.12],..., [x.sub.1m.sub.1]; [x.sub.21], [x.sub.22],..., [x.sub.2m.sub.2];...; [x.sub.n1], [x.sub.n2],..., [x.sub.nm.sub.n]);
[x.sup.I] = initial solution vector;
[x.sup.Uu] = solution vector at branch u, u = 1, 2,..., n;
[c.sub.ij] = cost of type j component used in subsystem i;
[r.sub.ij] = reliability of type j component used in subsystem i;
[C.sub.ij] = cost of subsystem i having [x.sub.ij] components in parallel;
[R.sub.ij] = reliability of subsystem i having [x.sub.ij] components in parallel;
C(x) = system cost at vector x;
[g.sub.ij] = capacity of type j component used in subsystem i;
[G.sub.i] = capacity of subsystem i;
[G.sub.s] = capacity of the series system;
[D.sub.ij] = sensitivity factor defined as: ([R.sub.ij]/[C.sub.ij]) of subsystem i having [x.sub.ij] components in parallel;
E(x) = multi-state series-parallel system reliability index;
[E.sub.o] = minimum system reliability index;
[E.sub.o.sup.av] = "Average" multi-state series-parallel system reliability index;
[T.sub.l] = time duration in the interval l, l = 1, 2,..., L;
[W.sub.l] = demand level in the interval l, l = 1, 2,..., L.
4. Problem formulation and system reliability analysis
The problem as defined in Lisnianski et al. (1996) and Ramirez-Marquez and Coit (2004), considers a system consisting of n subsystems connected in series. Different components are available for each subsystem i.e., each subsystem has [m.sub.i] versions of the component types available on the market, and they...
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