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Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 396864, 16 pages

http://dx.doi.org/10.1155/2015/396864

## A New Biobjective Model to Optimize Integrated Redundancy Allocation and Reliability-Centered Maintenance Problems in a System Using Metaheuristics

^{1}Department of Industrial Engineering, Islamic Azad University, Science and Research Branch, Hesarak, Tehran 1477893855, Iran^{2}Center of Excellence in Advanced Manufacturing
and Optimization, School of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran

Received 5 February 2015; Revised 27 May 2015; Accepted 17 June 2015

Academic Editor: Babak Shotorban

Copyright © 2015 Shima MohammadZadeh Dogahe and Seyed Jafar Sadjadi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A novel integrated model is proposed to optimize the redundancy allocation problem (RAP) and the reliability-centered maintenance (RCM) simultaneously. A system of both repairable and nonrepairable components has been considered. In this system, electronic components are nonrepairable while mechanical components are mostly repairable. For nonrepairable components, a redundancy allocation problem is dealt with to determine optimal redundancy strategy and number of redundant components to be implemented in each subsystem. In addition, a maintenance scheduling problem is considered for repairable components in order to identify the best maintenance policy and optimize system reliability. Both active and cold standby redundancy strategies have been taken into account for electronic components. Also, net present value of the secondary cost including operational and maintenance costs has been calculated. The problem is formulated as a biobjective mathematical programming model aiming to reach a tradeoff between system reliability and cost. Three metaheuristic algorithms are employed to solve the proposed model: Nondominated Sorting Genetic Algorithm (NSGA-II), Multiobjective Particle Swarm Optimization (MOPSO), and Multiobjective Firefly Algorithm (MOFA). Several test problems are solved using the mentioned algorithms to test efficiency and effectiveness of the solution approaches and obtained results are analyzed.

#### 1. Introduction and Literature Review

In general, reliability is defined as ability of a system to meet required performance standards under specified conditions during a determined time horizon. It has a significant effect on manufacturing cost, company’s fame, production efficiency and environment, and so forth. There are two main approaches to enhance system reliability: implementing a proper maintenance policy and using effective redundancy strategies. Applying these approaches leads to increases in system reliability along with increasing costs of other resources. Thus, reaching a tradeoff between system reliability, cost, volume, weight, and so forth is significantly important [1].

There are two types of maintenance policies: corrective maintenance (CM) and preventive maintenance (PM). In corrective maintenance, the system is repaired or replaced after failure. However, prescheduled periodic maintenance actions are taken in preventive maintenance. It is obvious that preventive maintenance prevents major failures that impose high costs on the system. In recent years, many authors conducted variety of research on preventive maintenance scheduling problems. A mathematical model has been proposed by Goel and Gupta [2] for a multistate system with repair and replacement policy. Goel et al. [3] presented a formulation for design, production, and maintenance planning to incorporate the reliability allocation problem at the design stage. Then, a simultaneous optimization framework has been employed to solve the proposed model. Tsai et al. [4] studied the preventive maintenance scheduling problem for a multicomponent system by assuming three maintenance actions: mechanical service, repair, and replacement. Periodic preventive maintenance actions have been considered in order to maximize availability of the system. Mohanta et al. [5] have employed both GA and hybrid GA/SA techniques to optimize maintenance scheduling for a power plant and compared the obtained results by the algorithms.

Martorell et al. [6] proposed a multiobjective model to optimize the maintenance scheduling problem by integrating human and material resources. Reliability, availability, maintainability, and cost were considered as objectives of the model and the Genetic Algorithm was used to solve the problem. Another multiobjective maintenance model for a series-parallel system was investigated by Certa et al. [7]. They implemented periodic PM policy and considered maintenance cost and makespan as objectives of the problem. An effective Pareto optimal frontier approach was applied to solve the multiobjective problem. Moghaddass et al. [8] focused on finding an optimal tradeoff between design of a repairable multistate system with binary components and its maintenance strategy. Also, they considered both active and standby redundancy strategies. Doostparast et al. [9] developed a reliability-based approach to optimize preventive maintenance scheduling in coherent systems. A system is called coherent when its performance is related to all of the components. In other words, for coherent systems each component is relevant and system structure function is monotone and nondecreasing [10]. They studied periodic PM performance in three types of coherent systems and used a Simulated Annealing (SA) algorithm to solve the problem trying to minimize total costs along with meeting the minimum predetermined reliability level. Several other studies dealing with preventive maintenance scheduling problem have been done by [11–13] during the last two decades.

However, selecting a proper maintenance policy is not all we can do to maximize system reliability. Identifying and implementing the best redundancy strategy is another way to optimize system reliability. One of the famous problems in field of reliability optimization is redundancy allocation problem (RAP). Redundant components are incorporated into the system to back up different parts of the system and prevent system breakdown under different redundancy strategies. There are two main redundancy strategies: (1) active redundancy, in which all redundant components are implemented in a parallel structure together from time zero and only one component is required to work at any given time, and (2) standby redundancy, in which a sequential order is determined for using the redundant components at component failure time. Three variants of the standby redundancy strategy are called cold, warm, and hot. Each strategy can be implemented in a different part of a system. RAPs are proved to be NP-hard by Chern [14]. Therefore, metaheuristic algorithms have been widely used in the literature to solve such problems. Literature of the redundancy allocation problem (RAP) could be reviewed from several points of view. In this paper, literature of the problem has been accurately reviewed by assuming three main characteristics: objectives of the problem, applied solution algorithms, and considered redundancy strategies.

Coit [15] studied the redundancy optimization problem using integer programming approach and logarithm function to develop an equivalent formulation of the problem and obtained high quality solutions. He assumed nonconstant component hazard functions, Erlang distribution for component time-to-failure, imperfect switching, and multiple component choices for each subsystem. Cold standby redundancy strategy is considered for a nonrepairable series-parallel system. Zhao and Liu [16] proposed a stochastic model for the redundancy optimization problem aiming at maximizing system lifetime or system reliability by considering both active and standby redundancy strategies. They used stochastic simulation, Genetic Algorithm (GA), and Neural Network (NN) to develop a hybrid intelligent algorithm to solve the problem. Liang and Smith [17] proposed an Ant Colony Optimization (ACO) Algorithm to solve the redundancy allocation problem (RAP) for a series-parallel system. The objective is to maximize system reliability when active redundancy strategy has been implemented in the system. Restrictions are set on system cost and system weight, in addition. They found the ACO algorithm to be very effective and efficient for solving NP-hard reliability design problems because it brings GA flexibility, robustness, and ease of implementation along with improving its random behavior. Tavakkoli-Moghaddam et al. [18] studied RAP for a series-parallel system by considering both active and standby redundancy strategies. They formulated the problem as a nonlinear integer programming model and used a Genetic Algorithm to solve the NP-hard problem and maximize system reliability. Sadjadi and Soltani [19] proposed a heuristic and a hybrid GA for the RAP in a series-parallel system to maximize its reliability. Parameters of the proposed hybrid GA are calibrated using Taguchi’s robust design method to enhance efficiency and effectiveness of the algorithm. Solving numerical examples indicated that the proposed heuristic method is time-efficient and produces comparable solutions to the hybrid GA in terms of quality. Kumar et al. [20] studied a multiobjective multilevel RAP and proposed multiobjective hierarchical Genetic Algorithms to solve two numerical examples. They integrated the hierarchical genotype encoding scheme with two multiobjective Genetic Algorithms. Beji et al. [21] proposed a hybrid metaheuristic algorithm based on Particle Swarm Optimization (PSO) and local search for RAP in a series-parallel system and tried to maximize system reliability.

A large number of studies on redundancy allocation problem have been conducted after 2010. Among those who studied multiobjective RAP (MORAP), Zio and Bazzo [22] used a level diagram analysis of Pareto solutions to assist the decision maker in selecting his/her preferred system design in terms of reliability and availability. Soylu and Ulusoy [23] applied UTADIS sorting procedure to categorize the Pareto solutions obtained by augmented epsilon constraint method into preference ordered classes. They considered maximization of the minimum system reliability along with minimization of the overall system cost and weight as objectives. Safari [24] studied a MORAP by considering system reliability and overall system cost as objectives and both active and standby strategies as candidate redundancy strategies. He used a Nondominated Sorting Genetic Algorithm (NSGA-II) to solve the multiobjective RAP. Khalili-Damghani and Amiri [25] applied three solution methods, epsilon constraint, multistart partial bound enumeration algorithm, and Data Envelopment Analysis (DEA) to optimize system reliability, cost, and weight in a RAP for a series-parallel system. Chambari et al. [26] studied a biobjective RAP trying to maximize system reliability and minimize overall cost along with making decision about using active and/or standby redundancy strategies for a system with nonrepairable components. They proposed two metaheuristics, NSGA-II and MOPSO, to solve the problem. However, Zoulfaghari et al. [27] considered system reliability and availability as objectives of a RAP with both repairable and nonrepairable components and proposed a mixed integer nonlinearprogramming (MINLP) model for the problem. Cao et al. [28] used a decomposition-based exact approach to solve a multiobjective RAP of a mixed system trying to optimize system reliability, cost, and weight. Garg and Sharma [29] studied a multiobjective RAP with nonstochastic uncertain parameters by considering system reliability and cost as objectives. They formulated the problem as a fuzzy multiobjective optimization problem.

Firefly Algorithm as a new metaheuristic optimization method was introduced in 2008 by Yang [30]. dos Santos Coelho et al. introduced a modified FA approach combined with chaotic sequences to optimize reliability-redundancy problem [31]. Many other authors proposed metaheuristic algorithms for RAP. Sadjadi and Soltani [32] developed a heuristic method and a honey bee mating algorithm to solve the large-scale RAP. Hsieh and Yeh [33] applied a penalty guided bee colony algorithm for RAP in a series-parallel system. Several other metaheuristic algorithms have been proposed by other researchers in [34–38].

In this paper, a novel mathematical model of a system of repairable and nonrepairable components is formulated. The model contains two objectives: firstly, it aims to select a proper redundancy strategy for nonrepairable part of the system and secondly, it offers a maintenance policy for repairable part of the system. Minimizing net present value of total cost and maximizing system reliability are objectives of the problem. In addition, different types of redundancy strategies, repair, and replacement actions are considered in order to model the problem as realistic as possible. Other practical constraints such as available budget for purchasing redundant components, volume, weight, and maximum allowed failure rate in each inspection period are taken into account. Due to NP-hardness of the problem, the authors tried to employ metaheuristic methods to solve proposed model. Three common solution approaches called NSGA-II, MOPSO, and MOFA were selected based on the Vanoye and Parra classification. Ruiz-Vanoye and Díaz-Parra [39] classified metaheuristics into three groups: metaheuristics based on gene transfer (like Genetic Algorithms), metaheuristics based on interactions among individual insects (e.g., Ant Colony, Honey Bees, and Firefly Algorithms) and metaheuristics based on biological aspects of alive beings (such as Simulated Annealing, Tabu Search, and Particle Swarm Optimization Algorithms).

Remainder of the paper is organized as follows. In Section 2, mathematical formulation of the problem is proposed followed by detailed explanation of objective functions and constraints. Three metaheuristic algorithms are presented in Section 3 to solve the proposed model. A set of numerical examples have been solved using the metaheuristics in Section 4. Then, obtained results are indicated and computational analysis is carried out. Finally, a summary of the paper and conclusions have been presented in Section 5.

#### 2. Problem Formulation

In this section, a new integrated mathematical model is proposed for redundancy allocation and reliability-centered maintenance problems. Objective of the reliability problems could be one or a set of the following objectives: maximizing system reliability and minimizing cost, weight, and volume of the system. In this paper, system reliability and costs including maintenance and operational costs are considered as objectives.

In most articles, the system under study includes either repairable or nonrepairable components. However, systems usually consist of repairable and nonrepairable components simultaneously in real world [27]. Generally, components of electronic devices are not repairable and should be replaced by new ones after failure. However, components of mechanical systems are usually repairable and repairing or replacing the broken component after failure brings the system back to the normal condition.

In this paper, a system of electronic and mechanical components has been considered. Figure 1 represents configuration of the system. Two approaches are applied to achieve the highest possible system reliability: (1) maximizing reliability of each component by using a diverse set of high quality and reliable redundant parallelized components (heterogeneous redundancy); (2) choosing optimal maintenance policies. It is obvious that the first approach can be used for nonrepairable electronic components and the second one is applied on repairable mechanical components.