Solving reliability redundancy allocation problems using an artificial bee colony algorithm
Introduction
System reliability optimization is very important in the real world and substantial effort has been made during the last two decades to develop reliability criteria to measure the quality of generation, transmission, and distribution in composite systems [1], [2].
Generally two major approaches have been used to achieve higher system reliability. The first way is increasing the reliability of system components, and the second way is using redundant components in various subsystems. In the first way, the system reliability can be improved to some degree, but the required reliability enhancement may be never attainable even though the most currently reliable elements are used. Using the second way requires one to choose the optimal element combination and redundancy-levels; the system reliability can be also enhanced, but the cost, weight, volume, etc. will be increased as well. Besides the above two routes, the conjunction of the two approaches and reassignment of interchangeable elements are alternative feasible ways to enhance the system reliability [2], [3]. Such problems of maximizing system reliability through redundancy and component reliability choices are called the “reliability-redundancy allocation problem” [2].
The redundancy allocation problem (RAP) may be the most common problem in the design-for-reliability approach [2]. It involves setting reliability objectives for components or subsystems in order to meet the resource consumption constraint, e.g. the total cost. Hence, RAP is becoming an increasingly important tool in the initial stages of or prior to the planning, designing and control of systems.
Based on the ways to enhance system reliability, the reliability design problems include the integer and mixed-integer reliability problems. In the integer reliability problems (redundancy allocation problems), the component redundancy allocation is to be decided while the component reliabilities are given. In optimizing the system reliability of mixed-integer reliability problems (reliability-redundancy allocation problems), the number of redundant components and the corresponding component reliabilities are to be decided simultaneously.
The major focus of recent work in the redundancy allocation problems has been on the development of heuristic/meta-heuristic algorithms for redundancy allocation [4], [5], [6], [10], [17]. A few works are directed toward exact solutions for such problems [2]. But a few approaches were proposed for the mixed-integer reliability problems of optimizing both the redundancy and component reliability [3], [10], [13], [20], [5] in literature. These system reliability problems are either subjected to the linear constraints [7], [15], [16] or to the nonlinear constraints [3], [8], [9], [11], [18], [20], [5]. Kuo and Prasad [2], Tillman et al. [19] and Yokota et al. [21] presented a good comprehensive survey of previous works.
In this paper, four mixed-integer reliability problems with multiple nonlinear constraints are considered and solved by using a novel meta-heuristic approach. The first three example problems include the series system, the series-parallel system [12], [14] and the complex (bridge) system [3], [11], [13], [20]. The last example problem is a gas turbine overspeed protection system [21], [22]. The four reliability-redundancy allocation problems of maximizing the system reliability subject to multiple nonlinear constraints can be stated as following nonlinearly mixed-integer programming model:where rj and nj are the reliability and the number of components in the jth subsystem, respectively; f(·) is the objective function for the overall system reliability; g(·) is the constraint function and l is the resource limitation; and m is the number of subsystems. The goal is to determine the number of components and the components’ reliability in each subsystem, so as to maximize the overall system reliability. This problem belongs to the category of constrained nonlinear mixed-integer optimization problems. For solving such kinds of mixed-integer reliability problems, most efforts have been devoted to nonlinearly-constrained reliability-redundancy allocation problems. It is known that the nonlinear mixed-integer programming problems are more difficult than pure redundancy allocation problems. Tillman et al. [19] and Gopal and Aggarwal [23] applied heuristic approaches to solve the reliability-redundancy allocation problem. Kuo et al. [24] demonstrated the Lagrange multipliers with the branch-and-bound method in a series system with four subsystems.
The iterative heuristic method and surrogate dual approach were used by Xu et al. [20] and Hikita et al. [3], respectively, to solve these mixed-integer reliability design problems. However, most of those require derivatives for all nonlinear constraint functions. That makes the exact optimal solutions to the reliability-redundancy allocation problem hard to derive because of the highly computational complexity. To overcome this difficulty, Yokota et al. [21] and Hsieh et al. [11] applied genetic algorithms to solve these mixed-integer reliability optimization problems.
Devised recently, the artificial bee colony algorithm is a new meta-heuristic approach, proposed by Basturk and Karaboga [25]. Because ABCs have the advantages of memory, multi-character, local search and the solution improvement mechanism, they are able to discover an excellent optimal solution. Although some applications using ABC are proposed in the literature, e.g., [29], to our knowledge it may be the first trial of application of ABCs to the reliability field in the literature. The meta-heuristic optimization approach employing penalty guided based ABCs to the mixed-integer reliability optimization problems is proposed in our work.
This paper is arranged as follows: Section 2 describes the acronyms and notations; Section 3 provides application of ABC to the mixed-integer reliability-redundancy optimization problems; Section 4 provides a description of the four test problems and the related data used for this study and the results of the experiments are also discussed. Finally, the conclusion of the paper is summarized and the directions for future research are described in Section 5.
Section snippets
Acronyms
- ABC
artificial bee colony
- RAP
redundancy allocation problem
Notations
the number of subsystems in the system
- ni
the number of components in subsystem i, 1≤i≤m
- n
≡(n1, n2, …, nm), the vector of the redundancy allocation for the system
- ri
the reliability of each component in subsystem i, 1≤i≤m
- r
≡(r1, r2, …, rm), the vector of the component reliabilities for the system
- qi
=1−ri, the failure probability of each component in subsystem i, 1≤i≤m
- Ri(ni)
, the reliability of subsystem i, 1≤i≤m
- R
the system reliability
- gi
the i
Penalty guided ABC approach
The artificial bee colony algorithm is a new population-based meta-heuristic approach proposed by Basturk and Karaboga [25] and further developed by Karaboga and the coauthors [26], [27], [28]. This approach is inspired by the intelligent foraging behavior of the honeybee swarm.
The foraging bees are classified into three categories—employed, onlookers and scouts. All bees that are busy in currently exploiting a food source are classified as ‘‘employed’’. The employed bees bring loads of nectar
Numerical examples
To evaluate the performance of the proposed approach in solving the mixed-integer nonlinear reliability design problems, four test problems are solved. The first three examples with nonlinear constraints used by Hikita et al. [3], Kuo et al. [13] Xu et al. [20] and Chen [5] were a series system, series-parallel system and complex (bridge) system, respectively. The fourth example is an overspeed protection system, which was investigated by Yokota et al. [21] Dhingra [22] and Chen [5].
All the
Conclusions
In this paper, we proposed the penalty guided artificial bee colony algorithm (ABC) to solve the nonlinearly mixed-integer reliability design problems, which include the series systems, the series-parallel system, the complex (bridge) system and the overspeed protection system. In these optimization problems, both the redundancy (number of redundant components) and the corresponding reliability of each component in each subsystem under multiple constraints are to be decided simultaneously.
Acknowledgements
The authors wish to thank the editor and anonymous referees for their constructive comments and recommendations, which have significantly improved the presentation of this paper. This study is supported by National Science Foundation under Grant (NSC 98-2221-E-007-051-MY3).
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