International Journal of Industrial Engineering Computations Reliability Optimization of Binary State Non-repairable Systems: a State of the Art Survey Keywords: Reliability Allocation Redundancy Allocation Reliability-redundancy Allocation Stochastic Uncertainty Fuzzy Uncertainty Interval Uncertain

The purpose of this paper is to discuss the state of the art on models and methods for reliability optimization problems (ROPs) including reliability allocation, redundancy allocation and reliability-redundancy allocation. There are literally few surveys for reviewing the literature of the ROPs. Tillman et al. (1980) classified the related papers by system structure, problem type, and solution methods, separately. In another work, Tzafestas (1980) reviewed system reliability optimization models and the optimization techniques. Yearout (1986) reviewed the literature related to standby redundancy. Kuo (2000) studied the system reliability optimization based on system structure and solution methods. Kuo and Prasad (2004) overviewed system reliability optimization methods. Later, Kuo (2007) reviewed recent advances in optimal reliability allocation problems. The present study adds to the previous literature surveys and focuses mainly on papers after year 2000 but with a quick review on the previous works so that the readers become familiar with the existing approaches. This research investigates the literature from system structure, system performance, uncertainty state and solution approach standpoints, simultaneously.


Introduction
To be more competitive in the market, the reliabilities of systems or products need to be optimized by system designers.This goal is accomplished by applying suitable reliability optimization techniques, which includes three main categories, i.e. the reliability allocation, the redundancy allocation and the reliability-redundancy allocation as presented in Fig. 1.Redundancy can be in system level, subsystem level (Module) and component level.

Fig. 1. Reliability optimization problems
In the reliability allocation problems, the reliability levels of the components is determined such that the consumption of a resource under a reliability constraint is minimized while the redundancy allocation problem generally involves the selection of components and redundancy levels to maximize the system reliability given various system-level's constraints.The reliability-redundancy allocation, which is the combination of two aforesaid problems, determines the reliability levels and the redundancy levels, simultaneously.

Fig. 2. RAP models
Fig. 2 presents taxonomy of redundancy allocation problems.By starting from the model and going through each path, a new model is resulted.The RAP is first classified and separated based on whether the problem is binary state or multi state.In the former, the components are in two states of functioning and non-functioning whilst in the latter, the states between them are also considered.Each of these categories can be studied with deterministic or non-deterministic parameters, which involve uncertainties such as stochastic uncertainty, interval uncertainty, fuzzy uncertainty, intuitionistic fuzzy or vague set, stochastic-fuzzy uncertainty, perturbation in a defined uncertainty set and chaos.Each problem can be modelled with a single objective or multiple objectives each of which with homogenous and heterogeneous components.The redundancy strategy can be either active or standby.The standby redundancy in turn is categorized into cold, hot and warm.The models have been formulated with different system structures and with different performance measures such as reliability, availability, mean time to failure (average life), percentile life, etc.For more information about the application of percentile life and reliability as performance measures readers are referred to the work by Kim and Kuo (2003).Note that fault-tolerance mechanisms, which are the ability of a system to continue performing its intended function in spite of faults, are not investigated in this research.Fig. 3 classifies the solution methods for RAPs.Mathematical programming methods, which include exact and approximation methods are the most suitable for small-sized problems.Chern (1992) proves that RAP in series system is an NP-Hard problem.As the system configuration becomes complex and other simplified assumptions are released the NP-Hardness of the problem increases.Therefore, heuristic and meta-heuristic approaches have been proposed to deal with RAPs.

Fig. 3. RAP solution methods
The rest of the paper is organized as follows.In section 2 the RAP is investigated which include deterministic and non-deterministic models.Section 3 is devoted to reliability allocation.Section 3 surveys the reliability-redundancy allocation problems.Finally, concluding remarks have been provided in section 4.

Redundancy allocation
As presented in Fig. 1, redundancy can be in system level (i.e.system redundancy), in subsystem level (i.e.modular redundancy) or in component level (i.e.component redundancy).It is a well-known principle to the design engineers that active redundancy at the component level is better than that at the system level in usual stochastic ordering (Nanda & Hazra, 2013;Barlow & Proschan, 1975).In the following subsections, modular and component redundancies are investigated regarding deterministic or non-deterministic parameters.Redundancy can be in active or standby forms.Yearout et al. (1986) and Li and Ding (2013) respectively reviewed standby and active redundancies in reliability.
Basic assumptions for RAP are as follows: 1) The availability of the components is unlimited; 2) Failures of individual components are independent; 3) All failed components will not damage the system and are not repaired.4) No preventive maintenance is considered.In the followings, other assumptions such as redundancy strategy, being deterministic or nondeterministic, etc. are mentioned.Furthermore, wherever a basic assumption is violated, it is mentioned in a column named other considerations.

Deterministic models
Deterministic models assume that all parameters including component reliability, cost, weight, volume, etc. are known, precisely.The RAP is restricted by available system cost, weight, volume, etc.In the field of system-reliability, it has been reported that various exact/mathematical programming and heuristic/meta-heuristic techniques have been used to cope with RAPs.For review on the work before year 2000 readers are referred to the works by Tillman et al. (1980), Kuo (2000), Kuo and Prasad (2004) and Kuo (2007).In the area of single objective models with active strategy, Fyffe et al. (1968) were the first who proposed a model for RAP where system's reliability is maximized subject to constraints on cost and weight.They used a Lagrangian multiplier in the objective function to handle multiple constraints.Nakagawa and Miyazaki (1981) showed that, sometimes, the use of a Lagrangian multiplier becomes inefficient.Therefore, they proposed the surrogate constraint approach.Other exact and mathematical programming techniques include DP (e.g.Bellman & Dreyfus, 1958;Fyffe et al., 1968;Jianping, 1996;Misra, 1971), Integer programming (e.g.Ghare, 1969;Misra & Sharma, 1991), Partial bound enumeration (e.g.Jianping & Xishen , 1997), Branch and bound (e.g.Bulfin & Liu,1985;Sup & Kwon,1999), GP (e.g.Federowicz & Mazumdar,1968;Govil, 1983;Misra & Sharma, 1973), Lagrange Multiplier (Govil & Agarwala,1983), etc.
Table 1 lists the related works in the literature after year 2000.Note that the references are not arranged in chronological order but they are arranged based on the solution method.As it is observed, most works have been carried out in active redundancy area except the work by Amari (2010) and Tannous et al. (2011) which consider both active and/or warm standby redundancy strategies and seems like more researches are needed with respect to redundancy strategy.

Non-deterministic models
Non-deterministic models assume that at least one of the design parameters is not known, precisely.
Regarding the level of access to the historical data the uncertainty can be considered in the following forms.

Stochastic Uncertainty
In the stochastic uncertainty either the distribution of the data is specified or mean and standard deviation of the data are known.One way to formulate these problems is to maximize the expected value of system reliability (e.g.Rubinstein et al., 1997).However, this method is not suitable for designers who are risk-averse.Therefore, both expected value and variance of the system reliability (e.g.Marseguerra et al., 2005) or a function of both such as coefficient of variation (e.g.Tekiner & Coit, 2011) is taken into consideration.In case where both expected value and variance of the system reliability are considered the problem is treated as a multi-objective problem.

Interval Uncertainty
In the interval uncertainty it is assumed that just the information about the lower and upper bounds of the interval is known.Yokota et al. (1995) are the first who formulated the optimal design problem of system reliability as a nonlinear integer programming problem with interval coefficients.They transformed the problem into a bi-criteria 0-1 nonlinear programming problem without interval coefficients, and solve it directly through GA.In general, there are some ways to formulate interval reliability problems, which are maximization of the left bound and/or center of the interval (Yokota et al., 1996;Taguchi et al., 1998), nonlinear goal programming formulation (Taguchi et al., 1997), etc.There are some methods to deal with such problems such as GA (Taguchi & Gen, 1997;Yokota et al., 1996), hybrid GA and SA (Taguchi et al., 1998), Improved GA (Taguchi & Yokota, 1999), etc. Table 4 lists the related works after year 2000.

Fuzzy Uncertainty
The concept of fuzzy reliability has been introduced and formulated in the book by Cai (1996).One form of fuzzy uncertainty is having flexible goals and constraints (e.g.Park, 1987).Another form is having imprecise data for components reliabilities/lifetimes, cost, weight, etc. Cheng and Mon (1993) used the α-cut of a triangular fuzzy number to get the interval and find the fuzzy reliability of the series system and the parallel system.Chen (1994) used fuzzy numbers to find the fuzzy reliability of both the series system and the parallel system.Mon and Cheng (1994) used the α -cut of fuzzy number to derive a non-linear program of the fuzzy system in both the series and parallel cases.Singer (1990) considered the fuzzy reliability of both series and parallel systems using an approximation of a fuzzy binary operation  with two L-R type fuzzy numbers.Chen (1996) presented a method for fuzzy system reliability analysis based on fuzzy time series and the  -cuts operations of fuzzy numbers.Hong et al. (1997) analyzed fuzzy system reliability by the use of t-norm based convolution of fuzzy arithmetic operation, where the reliability of each component is represented by L-R type fuzzy numbers.Table 5 lists other works after year 2000.

Intuitionistic fuzzy and vague sets
Kumar et al. ( 2011) extended the concept of fuzzy set by idea of triangular intuitionistic fuzzy sets (IFS) and proposed a general procedure to construct the membership function and non-membership function of the reliability function using intuitionistic fuzzy failure rate.The major advantage of using intuitionistic fuzzy sets over fuzzy sets is that intuitionistic fuzzy sets separate the positive and the negative evidence for the membership of an element in a set (Mahapatra & Roy, 2009).Table 6 lists the works related to intuitionistic fuzzy and vague sets.

Fuzzy-random Uncertainty
Due to subjective judgment, imprecise human knowledge and perception in capturing statistical data, the real data of lifetimes in many systems are both random and fuzzy in nature.Table 7 lists the related work to fuzzy-random uncertainty.

Robust optimization
Robust approaches are based on the idea that the resulted robust solution would be immune against different realizations of uncertain parameters.There are two main robustness measures: one based on regret (e.g.Feizollahi & Modarres, 2012;Soltani et al., 2013) and the other one based on the realized performance (Feizollahi et al., 2014;Soltani & Sadjadi, 2014).Table 8 lists the existing works in the literature.Amongst the works, just one work, i.e. the work by Soltani et al. (2013) considers cold standby strategy and more researches are needed with respect to redundancy strategy.The author has currently an under review work considering the redundancy strategy choices.Developing other robust optimization techniques is of interest and the author is currently working on this subject matter.

Chaos uncertainty
Chaos means the oscillations, which seem random but in fact they are generated by the deterministic nonlinear model.The idea of chaos theory application in the reliability modeling appears in Zou and Li (2001).The integration of fuzzy reliability model with the phase portraits of variables creates preconditions for the construction of phase portrait reflecting the system reliability dynamics (Rotshtein et al., 2012).To the best of our knowledge, in the area of RAP just one work for chaos uncertainty has been reported as presented in Table 9 and more researches in this area would be interesting.

Multi-level redundancy (Modular redundancy)
As mentioned earlier, redundancy at the component level is more effective than redundancy at the system level.This is true under some specific assumptions but Boland and EL-Neweihi (1995) showed that this was not true in the case of redundancy with non-identical spare parts.Modular redundancy can be more effective than component redundancy.In other words, providing redundancy at high levels such as modules or subsystems can be more economical than providing redundancy at low level of components.Multi-level redundancy allocation problems (MRAP) have been addressed, considering the redundancy at the subsystem level (modular redundancy).Another form of this problem is multiple multi-level redundancy allocation problem (MMRAP) in which all available items for redundancy (system, module and component) can be simultaneously chosen.During the optimization process, the values of design variables expressing redundancy are interdependent, since they are hierarchical, i.e., the reliability of an upper level unit depends on the reliability of its lower level units.In the hierarchical structure of a reliability system, the system level is the top most level and the component level is the lowest.Subsystem or module levels are located between the top level and the bottom level.Table 10 lists the related works to multi-level or modular redundancy.To the best of our knowledge, just one work, i.e. the work by Pourdarvish and Ramezani (2013), considers cold standby redundancy.Therefore, more researches on redundancy strategy are needed.More researches are also required in the area of multi-objective multi-level redundancy.Non-deterministic multi-level redundancy is an interesting area too.

Reliability Allocation
The reliability allocation problems assume fix or known redundancy levels and they aim at determining the reliability levels.There are some methods for determining reliability levels such as Hooke and Jeeves (H-J) pattern search (Tillman et al., 1977), GAG2 (Gopal et al., 1980), to name a few.For more study on these methods, the readers can view the work by Tillman (1985).Table 11 lists the reliability allocation models existed in the literature.

Reliability-Redundancy Allocation
The reliability-redundancy problem was initially introduced by Misra and Ljubojevic (1973).They treated the number of redundancies as real variables and solved the problem as a reliability allocation problem by using Lagrange multipliers.A trial and error approach was used afterward to obtain the integer solution for the number of redundancies.The problem has been studied on four typical system configurations such as series, series-parallel, complex and over-speed protection system for a gas turbine.The aim is to determine reliability levels and redundancy levels to maximize system reliability subject to constraints on cost, weight and volume usually in nonlinear forms.The techniques for solving RRAP mainly combine the methods for reliability allocation and the methods for redundancy allocation.Tillman et al. (1977) were among the first to solve RRAP using THK method, which is based on a combination of Hooke-Jeeves (H-J) pattern search (Hooke & Jeeves, 1961) for reliability allocation and the Aggrawal-Gupta-Misra (A-G-M) greedy heuristic (Aggrawal et al., 1975) for redundancy allocation in a series-parallel system.Other combinations include H-J&GAG (Gopal et al., 1978), H-J&S-V (Sharama & Venkateswaran, 1971), H-J&N-N (Nakagawa & Nakashima, 1977), H-J&K-I (Kohda & Inoue, 1982), GAG2 (Gopal et al., 1980) and GAG, GAG2&S-V, GAG2&NN, GAG2&K-I, where the first heuristic finds reliability levels and the second heuristic finds redundancy levels.Besides, Xu et al. (1990) proposed XKL method, which uses a heuristic to find redundancy levels and an analytical approach to find reliability levels.Jacobson and Arora (1996) proposed a Simplex search for solving RRAP.Mathematical programming methods include LMBB method (Kuo et al., 1987) which combines the branch-and-bound and Lagrange multipliers methods, and surrogate constraint (Hikita et al., 1992).Meta-heuristic approaches include GA (Yokota et al., 1996;Hsieh et al., 1998) and the evolutionary algorithm developed by Prasad and Kuo (1997).After year 2000 with the advent of the new heuristic and meta-heuristics, methods such as Column Generation, IA, SA, PSO, HS, ICA, cuckoo search separately or in hybrid forms have been implemented to this problem.
Regarding the method for multi-objective reliability-redundancy problems before year 2000 we can refer to goal programming and goal attainment formulations (e.g.Dhingra, 1992;Rao & Dhingra, 1992).

Concluding remarks
In the present research, three types of reliability optimization methods including redundancy allocation, reliability allocation and reliability-redundancy allocation have been investigated from different perspectives, which have been collected in a table format.Since the area of the reliability optimization problems is vast and so many works have been carried out in this area, the focus of this study has been on binary state non-repairable systems and the problems have been categorized with respect to different criteria.Scholars and researchers who wish to study on each of the mentioned area, need to flashback to the initial works and see the changing trend of the subjects from the scratch so that they can perceive the need for developing new models and methods in reliability.For this purpose, for each area the author has tried to have a quick review on early works and for each classification some of the prominent works in the literature have been introduced so that the interested readers can refer them and find other related papers to study in depth.This study has classified the literature in a novel way and at the end of each classification research guides for further researches have been proposed to help researchers and seems like the same is needed for multistate and/or repairable systems.

Table 1 (a)
Exact and Mathematical programming approaches for RAP (Single Objective Problems)

Table 2 (b)
Heuristic and Meta-heuristic approaches for RAP (Multi Objective Problems)

Table 3
Stochastic uncertainty

Table 4 (b)
Redundancy allocation under interval uncertainty (Multi-objective Problems)

Table 5
Fuzzy uncertainty

Table 6
Intuitionistic fuzzy and vague sets

Table 7
Fuzzy-random uncertainty

Table 8
Robust RAP

Table 9
Chaos uncertainty for RAP

Table 11
As observed, the problem of reliability allocation has gained less attention amongst researchers in comparison with RAP or RRAP.The uncertainty type considered in the literature for this kind of problem is of fuzzy uncertainty.Considering other forms of uncertainty can be an interesting research area.
(Ardakan & Hamadani, 2014;Liu, 2006)after year 2000.Note that the table is sorted based on the solution method.To the best of our knowledge, amongst the paper investigated, just two works have studied cold standby strategy(Ardakan & Hamadani, 2014;Liu, 2006)and a few works are in non-deterministic area.Considering other redundancy strategies and other forms of uncertainties can be a fruitful area of research.