Tabu search for the redundancy allocation problem of homogenous series–parallel multi-state systems

This paper develops an efficient tabu search (TS) heuristic to solve the redundancy allocation problem for multi-state series–parallel systems. The system has a range of performance levels from perfect functioning to complete failure. Identical redundant elements are included in order to achieve a desirable level of availability. The elements of the system are characterized by their cost, performance and availability. These elements are chosen from a list of products available in the market. System availability is defined as the ability to satisfy consumer demand, which is represented as a piecewise cumulative load curve. A universal generating function technique is applied to evaluate system availability. The proposed TS heuristic determines the minimal cost system configuration under availability constraints. An originality of our approach is that it proceeds by dividing the search space into a set of disjoint subsets, and then by applying TS to each subset. The design problem, solved in this study, has been previously analyzed using genetic algorithms (GAs). Numerical results for the test problems from previous research are reported, and larger test problems are randomly generated. Comparisons show that the proposed TS out-performs GA solutions, in terms of both the solution quality and the execution time.     

Optimization of series–parallel multi-state systems under maintenance policies

In the redundancy optimization problem, the design goal is achieved by discrete choices made from components available in the market. In this paper, the problem is to find, under reliability constraints, the minimal cost configuration of a multi-state series–parallel system, which is subject to a specified maintenance policy. The number of maintenance teams is less than the number of repairable components, and a maintenance policy specifies the priorities between the system components. To take into account the dependencies resulting from the sharing of maintenance teams, the universal generating function approach is coupled with a Markov model. The resulting optimization approach has the advantage of being mainly analytical.     

Optimal multilevel protection in series–parallel systems

This paper considers vulnerable systems that can have different states corresponding to different combinations of available elements composing the system. Each state can be characterized by a performance rate, which is the quantitative measure of a system's ability to perform its task. Both the impact of external factors (attack) and internal causes (failures) affect system survivability, which is determined as the probability of meeting a given demand.In order to increase the system's survivability a multilevel protection is applied to its subsystems. This means that a subsystem and its inner level of protection are in their turn protected by the protection of an outer level. This double-protected subsystem has its outer protection and so forth. In such systems, the protected subsystems can be destroyed only if all of the levels of their protection are destroyed. Each level of protection can be destroyed only if all of the outer levels of protection are destroyed.In such systems, different protections play different roles in providing for the system's survivability. Subject to budget limitations a question arises which protections should be applied to obtain the desired survivability. An algorithm for solving the protection cost minimization problem subject to survivability constraint is presented in the paper. The algorithm is based on a universal generating function technique used for system survivability evaluation and on a genetic algorithm used as an optimization engine.Illustrative example is presented.     

Optimizing survivability of vulnerable series–parallel multi-state systems

In this paper we consider vulnerable systems which can have different states corresponding to different combinations of available elements composing the system. Each state can be characterized by a system performance rate, which is the quantitative measure of a system's ability to perform its task. Both the impact of external factors (attack) and internal causes (failures) affect system survivability which is determined as probability of meeting a given demand.We formulate the problem of finding structure of series–parallel multi-state system (including choice of system elements, their separation and protection) in order to achieve a desired level of system survivability by the minimal cost.An algorithm based on the universal generating function method is suggested for determination of the vulnerable series–parallel multi-state system survivability. A genetic algorithm is used as optimization tool in order to solve the structure optimization problem.     

Reliability optimization of series-parallel systems with a choice of redundancy strategies using a genetic algorithm

This paper proposes a genetic algorithm (GA) for a redundancy allocation problem for the series-parallel system when the redundancy strategy can be chosen for individual subsystems. Majority of the solution methods for the general redundancy allocation problems assume that the redundancy strategy for each subsystem is predetermined and fixed. In general, active redundancy has received more attention in the past. However, in practice both active and cold-standby redundancies may be used within a particular system design and the choice of the redundancy strategy becomes an additional decision variable. Thus, the problem is to select the best redundancy strategy, component, and redundancy level for each subsystem in order to maximize the system reliability under system-level constraints. This belongs to the NP-hard class of problems. Due to its complexity, it is so difficult to optimally solve such a problem by using traditional optimization tools. It is demonstrated in this paper that GA is an efficient method for solving this type of problems. Finally, computational results for a typical scenario are presented and the robustness of the proposed algorithm is discussed.     

Survivability of series–parallel systems with multilevel protection

In this paper, we consider vulnerable systems which can have different states corresponding to different combinations of available elements composing the system. Each state can be characterized by a performance rate, which is the quantitative measure of a system's ability to perform its task. Both the impact of external factors (attack) and internal causes (failures) affect system survivability, which is determined as the probability of meeting a given demand.In order to increase the system's survivability a multilevel protection can be applied to its subsystems. In such systems, the protected subsystems are destroyed by external impacts only if all of the levels of their protection are destroyed.The paper describes an algorithm for evaluating the survivability of series–parallel systems with arbitrary configuration of multilevel protection. The algorithm is based on a composition of Boolean and the Universal Generating Function techniques. The adaptation of the algorithm for numerical implementation is suggested.Illustrative examples are presented.     

New method to minimize the preventive maintenance cost of series–parallel systems

General preventive maintenance model for input components of a system, which improves the reliability to ‘as good as new,’ was used to optimize the maintenance cost. The cost function of a maintenance policy was minimized under given availability constraint. An algorithm for first inspection vector of times was described and used on selected system example. A special ratio-criterion, based on the time dependent Birnbaum importance factor, was used to generate the ordered sequence of first inspection times. Basic system availability calculations of the paper were done by using simulation approach with parallel simulation algorithm for availability analysis. These calculations, based on direct Monte Carlo technique, were applied within the programming tool Matlab. A genetic algorithm optimization technique was used and briefly described to create the Matlab's algorithm to solve the problem of finding the best maintenance policy with a given restriction. Adjacent problem, which we called ‘reliability assurance,’ was also theoretically solved, concerning the increase of the cost when asymptotic availability value conforms to a given availability constraint.     

New methods to minimize the preventive maintenance cost of series–parallel systems using ant colony optimization

This article is based on a previous study made by Bris, Châtelet and Yalaoui [Bris R, Chatelet E, Yalaoui F. New method to minimise the preventive maintenance cost of series–parallel systems. Reliab Eng Syst Saf 2003;82:247–55]. They use genetic algorithm to minimize preventive maintenance cost problem for the series–parallel systems. We propose to improve their results developing a new method based on another technique, the Ant Colony Optimization (ACO). The resolution consists in determining the solution vector of system component inspection periods, T_P. Those calculations were applied within the programming tool Matlab. Thus, highly interesting results and improvements of previous studies were obtained.     

Coupling ant colony and the degraded ceiling algorithm for the redundancy allocation problem of series–parallel systems

The redundancy allocation problem (RAP) is a well known NP-hard problem which involves the selection of elements and redundancy levels to maximize system reliability given various system-level constraints. As telecommunications and internet protocol networks, manufacturing and power systems are becoming more and more complex, while requiring short developments schedules and very high reliability, it is becoming increasingly important to develop efficient solutions to the RAP. This paper presents an efficient algorithm to solve this reliability optimization problem. The idea of a heuristic approach design is inspired from the ant colony meta-heuristic optimization method and the degraded ceiling local search technique. Our hybridization of the ant colony meta-heuristic with the degraded ceiling performs well and is competitive with the best-known heuristics for redundancy allocation. Numerical results for the 33 test problems from previous research are reported and compared. The solutions found by our approach are all better than or are in par with the well-known best solutions.     

A combined approach to solve the redundancy optimization problem for multi-state systems under repair policies

This paper combines universal moment generating function technique with stochastic Petri nets to solve the redundancy optimization problem for multi-state systems under repair policies. Redundant elements are included in order to achieve a desirable level of production availability. The elements of the system are characterized by their cost, performance and availability. These elements are chosen from a list of products available on the market. The number of repair teams is less than the number of reparable elements, and a repair policy specifies the maintenance priorities between the system elements. A heuristic is proposed to determine the minimal cost system configuration under availability constraints. This heuristic, first applies universal moment generating function technique to evaluate the system availability, assuming unlimited maintenance resources. Once a preliminary solution is found by the optimization algorithm, stochastic Petri nets are used to model different repair policies, and to find the best system configuration (architecture and number of repairmen) in terms of global performance (availability and cost). This combined procedure is applied to a reference example.     

Extended great deluge algorithm for the imperfect preventive maintenance optimization of multi-state systems

This paper deals with preventive maintenance optimization problem for multi-state systems (MSS). This problem was initially addressed and solved by Levitin and Lisnianski [Optimization of imperfect preventive maintenance for multi-state systems. Reliab Eng Syst Saf 2000;67:193–203]. It consists on finding an optimal sequence of maintenance actions which minimizes maintenance cost while providing the desired system reliability level. This paper proposes an approach which improves the results obtained by genetic algorithm (GENITOR) in Levitin and Lisnianski [Optimization of imperfect preventive maintenance for multi-state systems. Reliab Eng Syst Saf 2000;67:193–203]. The considered MSS have a range of performance levels and their reliability is defined to be the ability to meet a given demand. This reliability is evaluated by using the universal generating function technique. An optimization method based on the extended great deluge algorithm is proposed. This method has the advantage over other methods to be simple and requires less effort for its implementation. The developed algorithm is compared to than in Levitin and Lisnianski [Optimization of imperfect preventive maintenance for multi-state systems. Reliab Eng Syst Saf 2000;67:193–203] by using a reference example and two newly generated examples. This comparison shows that the extended great deluge gives the best solutions (i.e. those with minimal costs) for 8 instances among 10.     

An efficient heuristic versus a robust hybrid meta-heuristic for general framework of serial–parallel redundancy problem

We present a heuristic approach to solve a general framework of serial–parallel redundancy problem where the reliability of the system is maximized subject to some general linear constraints. The complexity of the redundancy problem is generally considered to be NP-Hard and the optimal solution is not normally available. Therefore, to evaluate the performance of the proposed method, a hybrid genetic algorithm is also implemented whose parameters are calibrated via Taguchi's robust design method. Then, various test problems are solved and the computational results indicate that the proposed heuristic approach could provide us some promising reliabilities, which are fairly close to optimal solutions in a reasonable amount of time.     

Minmax defense strategy for complex multi-state systems

This paper presents a general optimization methodology that merges game theory and multi-state system survivability theory. The defender has multiple alternatives of defense strategy that presumes separation and protection of system elements. The attacker also has multiple alternatives of its attack strategy based on a combination of different possible attack actions against different groups of system elements. The defender minimizes, and the attacker maximizes, the expected damage caused by the attack (taking into account the unreliability of system elements and the multi-state nature of complex series-parallel systems). The problem is defined as a two-period minmax non-cooperative game between the defender who moves first and the attacker who moves second. An exhaustive minmax optimization algorithm is presented based on a double-loop genetic algorithm for determining the solution. A universal generating function technique is applied for evaluating the losses caused by system performance reduction. Illustrative examples with solutions are presented.     

Comparing effectiveness and efficiency in technical specifications and maintenance optimization

Optimization of technical specification requirements and maintenance (TS&M) has been found interesting from the very beginning at Nuclear Power Plants (NPPs). However, the resolution of such a kind of optimization problem has been limited often to focus only on individual TS&M-related parameters (STI, AOT, PM frequency, etc.) and/or adopting an individual optimization criterion (availability, costs, plant risks, etc.). Nevertheless, a number of reasons exist (e.g. interaction, similar scope, etc.) that justify the interest to focus on the coordinated optimization of all of the relevant TS&M-related parameters based on multiple criteria.The purpose of this paper is on signifying benefits and improvement areas in performing the coordinated optimization of TS&M through reviewing the effectiveness and efficiency of common strategies for optimizing TS&M at system level. A case of application is provided for a stand-by safety-related system to demonstrate the basic procedure and to extract a number of conclusions and recommendations from the results achieved. Thus, it is concluded that the optimized values depend on the particular TS&M-related parameters being involved and the solutions with the largest benefit (minimum risk or minimum cost) are achieved when considering the simultaneous optimization of all of them, although increased computational resources are also required. Consequently, it is necessary to analyze not only the value reached but also the performance of the optimization procedure through effectiveness and efficiency measures which lead to recommendations on potential improvement areas.     

A framework to practical predictive maintenance modeling for multi-state systems

A simple practical framework for predictive maintenance (PdM)-based scheduling of multi-state systems (MSS) is developed. The maintenance schedules are derived from a system-perspective using the failure times of the overall system as estimated from its performance degradation trends.

The system analyzed in this work is a flow transmission water pipe system. The various factors influencing PdM-based scheduling are identified and their impact on the system reliability and performance are quantitatively studied. The estimated times to replacement of the MSS may also be derived from the developed model.

The results of the model simulation demonstrate the significant impact of maintenance quality and the criteria for the call for maintenance (user demand) on the system reliability and mean performance characteristics. A slight improvement in maintenance quality is found to postpone the system replacement time by manifold. The consistency in the quality of maintenance work with minimal variance is also identified as a very important factor that enhances the system's future operational and downtime event predictability.

The studies also reveal that in order to reduce the frequency of maintenance actions, it is necessary to lower the minimum user demand from the system if possible, ensuring at the same time that the system still performs its intended function effectively.

The model proposed can be utilized to implement a PdM program in the industry with a few modifications to suit the individual industrial systems’ needs.     

A new multi-objective optimization model for preventive maintenance and replacement scheduling of multi-component systems

In this article, a new multi-objective optimization model is developed to determine the optimal preventive maintenance and replacement schedules in a repairable and maintainable multi-component system. In this model, the planning horizon is divided into discrete and equally-sized periods in which three possible actions must be planned for each component, namely maintenance, replacement, or do nothing. The objective is to determine a plan of actions for each component in the system while minimizing the total cost and maximizing overall system reliability simultaneously over the planning horizon. Because of the complexity, combinatorial and highly nonlinear structure of the mathematical model, two metaheuristic solution methods, generational genetic algorithm, and a simulated annealing are applied to tackle the problem. The Pareto optimal solutions that provide good tradeoffs between the total cost and the overall reliability of the system can be obtained by the solution approach. Such a modeling approach should be useful for maintenance planners and engineers tasked with the problem of developing recommended maintenance plans for complex systems of components.     

Maintenance free operating period – an alternative measure to MTBF and failure rate for specifying reliability?

The paper analyses the concept of maintenance free operating period (MFOP), the reliability requirement driven by the Ministry of Defence (UK) for the next generation of future aircraft to be included in the fleet. Since the traditional reliability requirement MTBF (mean operating time between failure) has several drawbacks, the immediate reaction would be to analyse the credibility of the new measure MFOP against MTBF. The paper discusses various issues associated with MFOP. Two mathematical models are developed to predict the maintenance free operating period survivability (MFOPS), one using mission reliability approach and the other using alternating renewal theory. The paper also analyses cost implications of MFOP to the customer and to the producer.     

Modelling and optimization of proof testing policies for safety instrumented systems

This paper introduces a new development for modelling the time-dependent probability of failure on demand of parallel architectures, and illustrates its application to multi-objective optimization of proof testing policies for safety instrumented systems. The model is based on the mean test cycle, which includes the different evaluation intervals that a module goes periodically through its time in service: test, repair and time between tests. The model is aimed at evaluating explicitly the effects of different test frequencies and strategies (i.e. simultaneous, sequential and staggered). It includes quantification of both detected and undetected failures, and puts special emphasis on the quantification of the contribution of the common cause failure to the system probability of failure on demand as an additional component. Subsequently, the paper presents the multi-objective optimization of proof testing policies with genetic algorithms, using this model for quantification of average probability of failure on demand as one of the objectives. The other two objectives are the system spurious trip rate and lifecycle cost. This permits balancing of the most important aspects of safety system implementation. The approach addresses the requirements of the standard IEC 61508. The overall methodology is illustrated through a practical application case of a protective system against high temperature and pressure of a chemical reactor.     

An integrated probabilistic risk analysis decision support methodology for systems with multiple state variables

Probabilistic risk analysis (PRA) methods have been proven to be valuable in risk and reliability analysis. However, a weak link seems to exist between methods for analysing risks and those for making rational decisions. The integrated decision support system (IDSS) methodology presented in this paper attempts to address this issue in a practical manner. In consists of three phases: a PRA phase, a risk sensitivity analysis (SA) phase and an optimisation phase, which are implemented through an integrated computer software system. In the risk analysis phase the problem is analysed by the Boolean representation method (BRM), a PRA method that can deal with systems with multiple state variables and feedback loops. In the second phase the results obtained from the BRM are utilised directly to perform importance and risk SA. In the third phase, the problem is formulated as a multiple objective decision making problem in the form of multiple objective reliability optimisation. An industrial example is included. The resultant solutions of a five objective reliability optimisation are presented, on the basis of which rational decision making can be explored.     

Availability and cost functions for periodically inspected preventively maintained units

Unavailability and cost rate functions are developed for components whose failures can occur randomly but they are detected only by periodic testing or inspections. If a failure occurs between consecutive inspections, the unit remains failed until the next inspection. Components are renewed by preventive maintenance periodically, or by repair or replacement after a failure, whichever occurs first (age-replacement). The model takes into account finite repair and maintenance durations as well as costs due to testing, repair, maintenance and lost production or accidents. For normally operating units the time-related penalty is loss of production. For standby safety equipment it is the expected cost of an accident that can happen when the component is down due to a dormant failure, repair or maintenance. The objective of maintenance optimization is to minimize the total cost rate by proper selection of two intervals, one for inspections and one for replacements. General conditions and techniques are developed for solving optimal test and maintenance intervals, with and without constraints on the production loss or accident rate. Insights are gained into how the optimal intervals depend on various cost parameters and reliability characteristics.