Parallel genetic algorithm with a knowledge base for a redundancy allocation problem considering the sequence of heterogeneous components
Introduction
Designing engineering and service systems, the reliability has become a more critical indicator because the system complexity and market competition is dramatically increased. The failure of products or the incompleteness of service could lead to not only direct or indirect forms of financial loss of a company but also fatal accidents in systems such as medical, nuclear, aerospace, and an unmanned vehicle. In the recent, as corporations are under pressure to furnish a logistics service with high reliability and consistency in delivery performance, the optimal reliability design has also been introduced to the field of supply chain management (e.g. see Christopher, 2016, Gaonkar, Xie, Ng, Habibullah, 2011, Pasandideh, Niaki, Asadi, 2015, Sadeghi, Sadeghi, Niaki, 2014, Sadic, de Sousa, Crispim, 2017, Taghizadeh, Hafezi, 2012).
For the optimal reliability design of a system, a traditional redundancy allocation problem (RAP) involves the selections of components and levels-of-redundancy for each subsystem to maximize or minimize some defined objective function, and Coit (2003) proposed a RAP with further decision variables for the choice of redundancy strategies either active or standby. Since then, Kim and Kim (2017) suggested the Markov models for evaluating the reliability of systems with heterogeneous components. Though Coit, 2001, Coit, 2003 proposed the approximated model for a standby redundant system, they were devised to estimate accurate reliability. Moreover, based on the Markov models, Kim and Kim (2017) suggested an advanced RAP with mixed component (RAPMC) which has the choice of redundancy strategies, and it is formulated as a binary integer programming with a nonlinear objective function. However, as the RAP develops, its is complexity does also exponentially increase, and thus it becomes more difficult to solve.
Due to the non-convexity, non-smoothness and high dimension of the problems for the optimal reliability design, many classical mathematical methods fail to attain acceptable solutions (Wang & Li, 2012), and recently, meta-heuristic algorithms have mainly attempted as the solution methodology for the problems. They include a genetic algorithm (Ardakan, Hamadani, 2014, Kim, Jeon, 2012, Soltani, Sadjadi, Tavakkoli-Moghaddam, 2014), simulated annealing (Chambari, Najafi, Rahmati, Karimi, 2013, Zaretalab, Hajipour, Sharifi, Shahriari, 2015), particle swarm optimization (Kong, Gao, Ouyang, Li, 2015, Yeh, 2014), memetic algorithm (Ramezani & Pourdarvish, 2016), electromagnetism-like mechanism (Teimouri, Zaretalab, Niaki, & Sharifi, 2016), simulation-based optimization (Guilani, Azimi, Niaki, & Niaki, 2016), harmony search (Valaei & Behnamian, 2017), etc.
The central purpose of this study is to suggest a parallel genetic algorithm with a knowledge base (PGAKB) to efficiently solve a RAPMC with a choice of redundancy strategies, and the sequence of components is also considered in a standby redundant subsystem. That is, for each subsystem, the RAPMC will determine the redundancy strategy, the types and numbers of components, and their sequence. Furthermore, in order to improve the algorithm performance such as solution quality, robustness, and computational cost, the PGAKB is designed based on two strategies, the ‘parallel operations’ of some genetic algorithms (GAs) and the utilization of a ‘knowledge base.’ The parallelization strategy is implemented as a procedure in which several subgroups that divide the whole population perform evolutionary operations independently during appointed generations and periodically exchange superior solutions. The knowledge base stores useful information generated by each member in the evolution process, and it is employed by colleagues and descendants to avoid duplicative tasks. The knowledge accumulated over all generations is also exploited in the local search to improve the quality of the final solution.
Furthermore, for solving a complex problem, the PGAKB operates in the form of an expert system emulating the decision-making ability of experts. In general expert systems as shown in Fig. 1, users and experts are different groups of humans. However, in the PGAKB, individuals stand in for the humans. The knowledge is represented by their genotype and the value of a fitness function, and the GA operators and fitness function play the role of knowledge acquisition system. Each creature creates and exploits the knowledge, and the PGAKB uses a deterministic inference engine by forward chaining. In conclusion, the PGAKB takes the form of an expert system in which all members are experts and users, and it also describes a society that develops for itself by creating knowledge.
This article is organized as follows. Section 2 reviews the system reliability models, which are formulated by a structured continuous-time Markov chain (CTMC), and influence of the system reliability by the component sequence. Section 3 introduces the mathematical model for a RAPMC proposed in this study, and it is represented by a binary integer nonlinear programming. In Section 4, a PGAKB designed for solving the RAPMC is suggested, and Section 5 discusses the results analyzing the performance of the PGAKB by numerical experiments on well-known benchmark problems. Finally, Section 6 offers the conclusion and future studies.
Section snippets
Reliability and component sequencing
In this paper, the system reliability functions suggested by Kim and Kim (2017) are applied. They have considered that the time-to-failure (TTF) of a component is distributed according to a phase-type distribution, which is denoted by PH(π, D).
Redundancy allocation problem with mixed components
In this section, the RAPMC for a series-parallel system as shown in Fig. 4 is introduced. For each subsystem, the problem has decision variables involving the redundancy strategy, the types and numbers of components, and their sequence. Herein, the redundancy strategy is selected either active or standby.
The vector xi of components used for subsystem i consists of binary integer variables xijk as Eq. (11), and thus it concurrently includes information about types, number, and sequence of the
Parallel genetic algorithm with a knowledge base
This study suggests a PGAKB for solving an RAPMC, and the main aim is to reduce computational cost and to improve the solution quality. The PGA consists of several GAs with each sub-population, and it iterates the cooperation process, a migration operator, that individual GAs independently search for solution during designated generations and then exchange superior solutions of them. Furthermore, in the evolutionary process of the GAs, the empirical knowledge of individuals accumulates in the
Benchmark problem
For evaluating the utility of the PGAKB, numerical experiments were performed with a favorite benchmark problem (Coit, 2001, Coit, 2003). A system in the example has 14 subsystems as shown in Fig. 4, and there are three or four components which can equivalently function for each subsystem. For individual component, the price, weight and the parameters (m, λ) of TTF distribution (i.e., m-Erlang) are presented in Table 3.
The objective of the benchmark is to maximize system reliability at h.
Conclusion and future studies
This study has proposed the newest version of a RAPMC, and to maximize system reliability, it simultaneously determines the redundancy strategy, the types, number, and sequence of components for each subsystem. Namely, it includes more variables than traditional RAPs for optimal reliability design, and herein, the redundancy strategy is selected either active or standby redundancy. Through these considerations, the RAPMC provided a system configuration with exceedingly higher reliability than
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