Multi-objective availability and cost optimization by PSO and COA for series-parallel systems with subsystems failure dependencies
Introduction
The design of any system in the context of dependability involves many challenges and the objectives may be conflicting. System dependability includes system reliability, availability, maintainability, safety, and cost (RAMS+C) [1], [2], [3], [4]. Several works have been devoted to system dependability and most of the solution approaches are based on bio-inspired optimization algorithms. The problems have been formulated as single- or multi-objective optimization problems. For single-objective optimization, in Refs. [5,6], the system reliability has been investigated as a single-objective optimization problem of the reliability-redundancy allocation. The problem has been solved by using the genetic algorithm and penalty-guided stochastic fractal search, respectively. Various bio-inspired optimization techniques have been used in Refs. [7,8] to solve this kind of problem. In Refs. [9], [10], [11], [12], the optimization of system availability and cost of a parallel-series system has been investigated with three sets of decision variables to be allocated: redundancy, failure rate, and repair rate. The cost has been fixed as an objective in Refs. [9,10], whereas the target was the availability in Ref. [11]. For multi-objective optimization, in Refs. [13,14] the Pareto fronts for two objectives have been generated: system reliability vs revenue and system reliability vs cost, while three objectives in Ref. [15]. A choice of redundancy strategies has been introduced in Ref. [14]. In Ref. [16], various objectives have been considered by using the weighed-sum methods, which allows handling any priority prior preference on the objective by the decision-maker when solving the problem.
Most of the previous works have investigated systems with independent failures between their components and subsystems. The problem of minimum system cost under system availability with failure dependencies has been solved in Refs. [17,18]. In Ref. [19], the authors addressed the multi-objective failure dependency problem using the flower pollination algorithm (FPA) and plant propagation algorithm (PPA) under linear dependency.
Addressing the system cost as a single target or as a multi-objective problem with a linear dependency has an impact on the system availability. Furthermore, many systems involve subsystems with strong failure dependency. Therefore, the present work aims to investigate the system availability and cost optimization problem presented in Refs. [17,18] as a multi-objective problem by considering strong dependency. The problem is formulated by resorting to the weighted-sum method and solved using the particle swarm optimization (PSO) algorithm and the cuckoo optimization algorithm (COA). The results of a series-parallel system consisting of six subsystems are compared.
The remainder of the paper is organized as follows. Section 2 formulates the problem. Section 3 presents the numerical case study. Brief descriptions of the implemented PSO and COA are given in Section 4. The results are given in Section V with a discussion. Finally, the last section concludes the paper.
Section snippets
Problem description
The system cost and availability of a system consisting of subsystems under failure dependencies can be investigated at the design stage by addressing the optimal redundant components to add in parallel and repair teams to allocate at each subsystem.
According to Refs. [17,18], the design of a series-parallel system (see Figure 1) under weak and strong failure dependencies can be described as follows:
System cost
System availability with strong failure dependency
Numerical case study
In this section, an illustrative numerical application consisting of a series-parallel system with six subsystems is considered. Data of this system are reported in Table 1 [17], [18], [19]. The considered allowable values are given as follows: Cs*=1100, As*=0.90, and Nmax=15. The costs are given in arbitrary units. On the other hand, the weights for each objective are considered equal, i.e. w1= w2=0.5.
Solution approaches
To solve the above-described problem, two bio-inspired optimization techniques are used: particle swarm optimization (PSO) and cuckoo optimization algorithm (COA). The main characteristics of these techniques are summarized in Table 2 [11], whereas Table 3 illustrates some of their applications in engineering. The redundancy and repair variables are rounded to the nearest integer values. The constraints of system cost and availability are handled using the penalty function method [6,11].
Results
The implemented PSO and COA have been programmed using MATLAB 2017 and run on a PC with Intel Core I5-7300U vPro 7th Gen (2.7GHz, 8GB of RAM). Due to the high computation time, the algorithms are run five times to undertake some minimum statistical comparison. Table 6 reports the minimum values of z (see Eq. 4) obtained by each algorithm after five independent runs. The best values for each run are highlighted in bold.
The best value of z found by each algorithm, CPU time, required number of
Discussion
From Table 6, it can be observed that the best values of z are 532.0468 (PSO) and 529.5485 (COA). The CPU times are 5580.03 s (PSO) and 4350.08 s (COA), as reported in Table 7. The value obtained by the COA is better than that of PSO. Furthermore, the required number of function evaluations is 8660 and 3740, respectively (see Table 7). The five runs reported in Table 6 correspond to the following standard deviations: 2.9150 for the PSO and 1.0001 for the COA (see Table 7). These results
Conclusions
The goal of this paper was to investigate the multi-objective availability and cost optimization for a series-parallel system with subsystems failure dependencies by considering strong dependency. The weighted-sum method has been used and two bio-inspired optimization techniques have been applied, the particle swarm optimization (PSO) and the cuckoo optimization algorithm (COA). The addressed design strategy considers both objectives. The numerical results for a system consisting of six
Declaration of Competing Interest
We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.
Mohamed Arezki Mellal is an Associate Professor at the Department of Mechanical Engineering, Faculty of Technology, M'Hamed Bougara University, Algeria and was a Visiting Scholar at the Center for Advanced Life Cycle Engineering, Department of Mechanical Engineering, University of Maryland, College Park, MD, USA. Likewise, he was a Visiting Scholar at various universities. He has published in several journals and conference proceedings. He has edited six books and authored nine book chapters.
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Cited by (0)
Mohamed Arezki Mellal is an Associate Professor at the Department of Mechanical Engineering, Faculty of Technology, M'Hamed Bougara University, Algeria and was a Visiting Scholar at the Center for Advanced Life Cycle Engineering, Department of Mechanical Engineering, University of Maryland, College Park, MD, USA. Likewise, he was a Visiting Scholar at various universities. He has published in several journals and conference proceedings. He has edited six books and authored nine book chapters. He is a member of the Algerian National Laboratory for Maintenance Education in conjunction with the European Union (Erasmus+). He has also been a committee member for over seventy international conferences. He serves as a regular reviewer for thirty-six SCIE-indexed journals and an editorial board member in seven peer-reviewed international journals.
Enrico Zio received the M.Sc. degree in nuclear engineering from the Politecnico di Milano, Milan, Italy, in 1991, the M.Sc. degree in mechanical engineering from the University of California at Los Angeles, Los Angeles, CA, USA, in 1995, the Ph.D. degree in nuclear engineering from the Politecnico di Milano, in 1996, and the Ph.D. degree in probabilistic risk assessment from the Massachusetts Institute of Technology (MIT), Cambridge, MA, USA, in 1998. He is currently a Full Professor with the Centre for Research on Risk and Crises, Ecole de Mines, ParisTech, PSL University, Sophia Antipolis, France, a Full Professor and the President of the Alumni Association at Politecnico di Milano, an Eminent Scholar with Kyung Hee University, Seoul, South Korea, a Distinguished Guest Professor with Tsinghua University, Beijing, China, an Adjunct Professor with the City University of Hong Kong, Hong Kong, Beihang University, Beijing, and Wuhan University, Wuhan, China, and the Co-Director of the Center for Reliability and Safety of Critical Infrastructures and the SinoFrench Laboratory of Risk Science and Engineering, Beihang University. He has authored or coauthored over seven books and 300 papers on international journals. His current research interests include modeling of the failure-repair-maintenance behavior of components and complex systems, analysis of their reliability, maintainability, prognostics, safety, vulnerability, resilience, and security characteristics, and development and use of the Monte Carlo simulation methods, artificial techniques, and optimization heuristics. Dr Zio is the Chairman and Co-Chairman of several international conferences, an Associate Editor of several international journals, and a referee of more than 20.