Optimal multilevel redundancy allocation in series and series–parallel systems

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Optimal multilevel redundancy allocation in series and series–parallel systems

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  Optimal multilevel redundancy allocation in series and series–parallel systems q Ranjan Kumar * , Kazuhiro Izui, Masataka Yoshimura, Shinji Nishiwaki Department of Aeronautics and Astronautics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan a r t i c l e i n f o  Article history: Received 21 November 2007Received in revised form 11 June 2008Accepted 4 November 2008Available online 18 November 2008 Keywords: Redundancy allocationMultilevel series systemMultilevel series–parallelModular redundancyComponent redundancyGenetic algorithms a b s t r a c t To achieve truly optimal system reliability, the design of a complex system must address multilevel reli-ability configuration concerns at the earliest possible design stage, to ensure that appropriate degrees of reliability are allocated to every unit at all levels. However, the current practice of allocating reliability ata single level leads to inferior optimal solutions, particularly in the class of multilevel redundancy allo-cation problems. Multilevel redundancy allocation optimization problems frequently occur in optimizingthe system reliability of multilevel systems. It has been found that a modular scheme of redundancy allo-cation in multilevel systems not only enhances system reliability but also provides fault tolerance to theoptimum design. Therefore, to increase the efficiency, reliability and maintainability of a multilevel reli-ability system, the design engineer has to shift away from the traditional focus on component redun-dancy, and deal more effectively with issues pertaining to modular redundancy. This paper proposes amethod for optimizing modular redundancy allocation in two types of multilevel reliability configura-tions, series and series–parallel. A modular design variable is defined to handle modular redundancyin these two types of multilevel redundancy allocation problem. A customized genetic algorithm, namely,a hierarchical genetic algorithm (HGA), is applied to solve the modular redundancy allocation optimiza-tion problems, in which the design variables are coded as hierarchical genotypes. These hierarchicalgenotypes are represented by two nodal genotypes, ordinal and terminal. Using these two genotypes isextremely effective, since this allows representation of all possible modular configurations. The numer-ical examples solved in this paper demonstrate the efficacy of a customized HGA in optimizing the mul-tilevel system reliability. Additionally, the results obtained in this paper indicate that achieving modularredundancy in series and series–parallel systems provides significant advantages when compared withcomponent redundancy. The demonstrated methodology also indicates that future research may yieldsignificantly better solutions to the technological challenges of designing more fault-tolerant systemsthat provide improved reliability and lower lifecycle cost.   2008 Elsevier Ltd. All rights reserved. 1. Introduction Modularity in product design is a crucial topic when developinghighly reliable product architectures. It is a key strategy for achiev-ing better serviceability and reliability, particularly when design-ing products whose lifetime operational costs exceed the initialacquisition cost, such as for airplanes, locomotives, power generat-ing plants, and major manufacturing equipment (Molina, Kusiak, &Sanchez, 1999). Most complex engineering systems of this kindcontain thousands of different components that function interde-pendently, while certain components are used only for a specificset of subtasks within the system. Such sets of components havingindependent functions can be accommodated within a simple sub-system, or sub-unit. Here, such a subsystem is called a module, asshown in Fig. 1. In system reliability theory, a module indicates agroup of components that has a single input from, and a single out-putto, the rest of the system(Kuo & Zuo, 2003). The contributionof all components in a module to the performance of the whole sys-tem can be represented by the state of the module. Once the stateof the module is known, one does not need to know the states of the components within the module to determine the states of the system.Systems that have modular subsystems usually have superiorfault tolerance, ease of maintenance, and allow modules to berecovered for possible further use when the system as a wholehas reached the end of its useful life (Rasmussen & Niles, 2005).Furthermore, a modular system is often simpler than a complexsystem built from single components. In essence, the modulararchitecture of reliability design reduces the number of parts inan optimal configuration by providing a modular redundancy. De-spite these subtle and profound benefits of a modular redundancythat enhances fault tolerance and reduces lifecycle costs, optimiz- 0360-8352/$ - see front matter    2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.cie.2008.11.008 q This manuscript was processed by Area Editor E.A. Elsayed. *  Corresponding author. Tel.: +81 75 753 5198; fax: +81 75 753 5857. E-mail addresses:  ranjan.k@ks3.ecs.kyoto-u.ac.jp (R. Kumar), izui@prec.kyoto-u.ac.jp (K. Izui), yoshimura@prec.kyoto-u.ac.jp (M. Yoshimura), shinji@prec.kyoto-u. ac.jp (S. Nishiwaki).Computers & Industrial Engineering 57 (2009) 169–180 Contents lists available at ScienceDirect Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie  ing modular-level allocation under resource constraints is a chal-lenging task for design engineers.Conventionally, redundancy is added either to a component le-vel or to a subsystem level, when optimizing system reliability. Theredundancy added at the component level is termed componentredundancy, and redundancy added at the modular level is termedmodular redundancy. Specifically, a redundant module is a similarmodule added in parallel to the existing module to increase its reli-abilitywithout altering its internal structure. Fig. 2 illustrates thesetwo redundancy schemes in a series system containing threecomponents.In other words, we preserve a module’s internal structure, suchas the arrangement of its sub-modules and components, while pro-viding modular redundancy. Thus, to know the status of the sys-tem, we need not know the status of its components. Modularredundancy therefore simplifies the complexity of the systemand makes it easier to isolate faults in case of failure.In the literature, we find that several models for several sys-tem configurations have been proposed, such as series, parallel,series–parallel, network, and  k-out-of-n  systems, and others. Tomaximize the system reliability of these models, a large numberof techniques have been proposed for optimal redundancy allo-cation problems. Most techniques for redundancy optimization,however, have been limited to single levels (Kuo & Prasad,2000). Boland and EL-Neweihi (1995) demonstrated that redun- dancy at the component level is not always more effective thanredundancy at the system level in cases of redundancy usingnon-identical parts. In addition, modular redundancy can helpa system become truly fault tolerant. For example, a modularsystem can shift operation from failed modules to healthy ones,allowing repairs to be carried out without downtime (Rasmussen& Niles, 2005). The design transition from component to modu-lar redundancy actually reduces costs and enhances efficiency,flexibility, and reliability. Despite the various benefits that mod-ularity offers, multilevel modular redundancy allocation optimi-zation has seldom been discussed in detail, or an appropriatemethodology provided. To leverage the merits of modular redun-dancy allocation, this paper presents a methodology for optimiz-ing the system reliability of a multilevel class of problems usinga modular redundancy allocation scheme.In a similar direction, Yun and Kim (2004) proposed a multi-level series redundancy allocation optimization model in whichthey considered that each unit of a three level series system issubjected to redundancy, and they optimized system reliabilityby using conventional genetic algorithms (GAs). Their methodcan solve certain problems based on the assumption where onlyone unit is allowed to have redundancy in a direct line. Thisassumption reduces the feasible design space and fails to yielda globally optimal solution, because conventional GAs requireone dimensional vector representation of the design variables.Later, Yun, Song, and Kim (2007) presented a formulation of multiple multilevel redundancy allocation problems in seriessystem and applied a GA with a sequential recording methodwithout reflecting the solution positions are used. However,the design variables in a multilevel system have hierarchicalrelationships, and the artificial transformation into vector codingleads to a reduced feasible design space and suboptimalsolutions.Therefore, this paper proposes a modular redundancy allocationoptimization methodology in which hierarchical design variablesare represented by hierarchical genotypes in the optimization. Thiscustomized methodology is based on a type of genetic algorithmproposed by Yoshimura and Izui (2002), in which the hierarchicalgenotype coding representation is used to exactly express theinternal structure and related hierarchical details, a techniqueusing so-called hierarchical genetic algorithms (HGAs). In orderto handle general multilevel redundancy allocation problems suchas series and series–parallel problems, this paper redefines a math-ematical expression of system reliability for series and series–par-allel and proposes a design-variable coding method usinghierarchical genotypes. This paper demonstrates that HGA canhandle both modular and component schemes of redundancy allo-cation easily by using two newly defined genotypes, nodal andterminal.This paper is organized as follows. Section 2 describes the de-tailed mathematical formulation for the multilevel redundancyallocation optimization problems. In Section 3, HGA conceptsare explained, and a HGA coding method for modular redun-dancy allocation optimization problems is proposed. In Section4, we solve two multilevel redundancy optimization problems,one series and one series–parallel, each having four hierarchicallevels. In this section, the input data used and the results aresummarized. The results obtained in Section 4 are explainedand discussed in Section 5. Finally, Section 6 concludes the paper. Nomenclature U  s  system level unit U  i  i th unit; a common name for system, subsystem, andcomponent R i  reliability of   U  i  x i  number of components used in  U  i  x   set of design variables  x i  f  (  x  ) reliability function C  (  x  ) cost function n i  number of sub-units in  U  i U  i ; n i  n i th sub-unit of   U  i U   ji ; m  j th redundant unit of   m th sub-unit of   U  i R  ji ; m  reliability of   U   ji ; m  x  ji ; m  number of sub-units of unit  U   ji ; m ; a nonnegative integer C   ji ; m  cost of unit  U   ji ; m C  i  cost of   i th unit C  s  system cost C  0  threshold cost k i  additional costs of   i th unit when adding a redundantunit to a unit N  i  ordinal genotype node of   i th unit N  t   terminal genotype node of   i th unit Product Module 1Module 2Module3 System Modules Components Fig. 1.  Multilevel configuration of system reliability.170  R. Kumar et al./Computers & Industrial Engineering 57 (2009) 169–180  2. Problem description  2.1. Multilevel redundancy allocation problems A multilevel redundancy allocation optimization problem isstructurally hierarchical, with the system level topmost andthe component level at the very bottom. The subsystems inbetween the top and the lowest levels are the so-called mod-ules. Each of these modules and their components are termeda unit. Fig. 3 is a schematic diagram of a general multilevelredundancy allocation configuration. In this figure,  U  1  is a sys-tem unit containing  U  11  to  U  1 ; n 1  units as modules at its nextlower hierarchical level. Similarly, the  U  11 , which is actuallythe second level of the system hierarchy module contains  n 11 sub-units as modules or components at its next lower level,represented as  U  111  to  U  11 n 11 . This structure is replicated untilthe lowest level of system hierarchy is reached. The connectinglines in the diagram imply the logical relationships among theunits at different levels, relationships that may be in series, inparallel, or combinations of these two. Redundancy at all levelsis assumed to be active and failures are statisticallyindependent.In general, a given unit  U  i  in the multilevel system has  n i  sub-units,  U  i 1 ; U  i 2 ;  . . .  ; U  in i , which must be connected either in seriesor in parallel. When  x i  is the number of   U  i  redundant units, thereare  n i  x i  sub-units in the level below  U  i . A unit in the  j th redundantunit of the  m th sub-unit of   U  i  is denoted  U   ji ; m . Thus, the reliability  R i of unit  U  i  for multilevel series and parallel configurations can becalculated using the following equations: R i  ¼ Y n i m 1  Y  x i  j ð 1   R  ji ; m Þ " #  ð 1 Þ R i  ¼  1  Y n i m Y  x i  j ð 1   R  ji ; m Þ " #  ð 2 Þ where  R  ji ; m  are reliability values of the sub-unit  U   ji ; m . Each  R  ji ; m  valueis calculated using the above equation at the level immediately be-low the unit, and these calculations are recursively iterated to thelevel just above the very lowest hierarchical level. At the very low-est level, where there are no sub-units belonging to unit  U  i , the reli-ability can be obtained as follows: R i  ¼  1  Y  x i  j ð 1   R  ji Þ ð 3 Þ Fig. 4 shows an example of redundancy allocation in a unit  U  1 .Fig. 4a and b illustrates the redundancy allocation in series andparallel system, respectively.The cost constraint of a multilevel redundancy allocation modelalso reveals hierarchical relationships among the multilevel units.The system cost is essentially the sum of the component and mod-ule costs. For example, the cost of   U  1  is the sum of the costs of  U  11 ; U  12 ;  . . .  ; U  1 n 1  and the assembly costs. In general, the redun-dancy cost of a unit  U  i  is calculated as  ð  x i Þð the cost of   U  i 1 þ the cost of   U  i 2  þþ the cost of   U  in i  þ  assembly costs Þ . The ass-embly costs represent the sum of the costs of adding, duplicatingor repairing the module or component. Note that there are definiteadvantages to using modular redundancy, because the cost of add-ing, duplicating, or repairing a module is lower than carrying out asimilar action upon a component. This is because the lower the le-vel in a system, the more costly the repair job. The expressed costfunction will differ depending upon the arrangement of differentstructures.  2.2. Redundancy allocation optimization formulation The redundancy allocation optimization problem in a reliabilitysystem consisting of a set of design variables is expressed as Maximize  R s  ¼  F  ð  x  Þ ð 4 Þ Subject to  C  ð  x  Þ 6 C  0 ;  ð 5 Þ (a) Basic reliability block diagram (b) Component level allocation (c) Modular level allocation 1 2 3 11 12 21 22 31 32 11 21 31 12 22 32 Fig. 2.  Redundancy allocation in a series system containing three components. 1 U  1 1 n U  11 U  111 U  112 U  11 11 n U  12 U  121 U  122 U  12 12  n U  11 1 n U  111 1 n nn U  Fig. 3.  A general multilevel redundancy allocation configuration. R. Kumar et al./Computers & Industrial Engineering 57 (2009) 169–180  171  where  R s ,  F  (  x  ),  C  (  x  ), and  x   are the system reliability, reliability func-tion, cost function, and a set of design variables, respectively.  C  0  is agiven fixed positive value for the cost constraint. For example, the problem of optimizing a 2-level series redun-dancy allocation, as shown in Fig. 5, can be stated mathematicallyas follows: R s  ¼ ½ 1  ð 1  f 1  ð 1   R 11 Þ  x 11 g  f 1  ð 1   R 12 Þ  x 12 Þg  x 1  ð 6 Þ where  x 1 ,  x 11 , and  x 12  are the numberof redundancy of units U  1 ,  U  11 ,and  U  12 , respectively. The values of the design variables,  x 11 , and  x 12 depend on the value of   x 1 , the design variable at the parent unit. If the number of redundancies represented by  x 1  is two, then theredundancies of   x 11 , and  x 12  should be at least two, however, thevalues of design variables  x 11 , and  x 12  are independent of eachother. In this paper, the following cost functions have been applied tocalculate the costs for modules and components: C  i  ¼ X n i m ¼ 1 X  x i  j ¼ 1 C   ji ; m  ð 7 Þ C  i  ¼  c  i  x i  þ  k  x i i  ð 8 Þ where  C   ji ; m  are modular cost of the sub-unit  U   ji ; m . The symbols  x i ,  c  i ,and  k i , respectively represent the number of redundancy, the unitcost, and the assembly cost for  i th unit. Each  C   ji ; m  value is calculatedusing Eq. (7) at the level immediately below the unit, and these cal- 111 U  1 U  11 11  x  U  112 U  12 12  x  U  11 U  12 U  1 U  111 U  1 U  11 11  x  U  112 U  12 12  x  U  11 U  12 U  1 U  (a) Series configuration (b) Parallel configuration Using Eq. (1), system reliability is ])1(1[])1(1[ 1211 12111  x  x   R R R  −−×−−= Using Eq. (2), system reliability is ])1()1(1[ 1211 12111  x  x   R R R  −−−= 11 U  12 U  11 U  12 U  Fig. 4.  Series and parallel redundancy allocation in a unit  U  1 . 111 U  11 U  1111  x  U  112 U  1212  x  U  111 U  11  x  U  1111  x  U  112 U  1212  x  U  System 11 U  11 U  11 U  11 U  Fig. 5.  An example of series redundancy allocation in a unit  U  1 .172  R. Kumar et al./Computers & Industrial Engineering 57 (2009) 169–180  culations are recursively iterated to the level just above the verylowest hierarchical level. At the very lowest level, where there areno sub-units belonging to unit  U  i , the cost is calculated by Eq. (8).Eq. (8) has been taken from the paper by Yun and Kim (2004). Thus, the total cost for a multilevel structure is calculated by using Eqs.(7) and (8). 3. Hierarchical genetic algorithms Hierarchical genetic algorithms (Yoshimura & Izui, 2002) arecustomized and applied to solve the multilevel redundancy allo- Original Structure Mutated offspring 13124312Parent 1Parent 2Offspring 1Offspring 22131211431432312(b) Mutation operation (a)Crossover operation Fig. 6.  Crossover and mutation operators for hierarchical genotype.  Table 1 Hierarchical genotype representation. Ordinal genotype node  N  i  Terminal genotypenode  N  t  i Designvariable  x  ji ; m : the number of subordinate modules forthe  m th moduleParameter  T  : unit type  k : the redundancyfor component  U  i k : the redundancy for unit  U  i  r  i : unit reliability n : the number of sub-modules  c  i : unit cost (a) An example of a multilevel reliability system U  1 (b) Design variables and parameters at each ordinal and terminal node U 11 U 12 U 1 U 1211 U 1212 U 1212 U 121 U 122 U 1222 U 1221 U 122 U 111 U 112 U 112 U 11 U 111 U 111 U 112 U 1221 U 1222 U 1222 1 11, 11 = U   x  2 12, 11 = U   x  U  11 2 21, 11 = U   x  1 22, 11 = U   x  1 11, 12 = U   x  2 12, 12 = U   x  U  12 2 11, 1 = U   x  1 12, 1 = U   x  U  1 System 1 11,  = sys  x   n = 1 k  = 1 n = 2 k  = 1 n = 2 k  = 2 n = 2 k  = 1 k   = 1 k   = 2 U  112  U  111 k   = 2 k  = 1 U  112 U  1211 k  = 1  k  = 2 U  1212 T  = S T  =S  T  =P 1 11, 121 = U   x  2 12, 121 = U   x  U  121 n = 2 k  = 1 T  = S 1 11, 122 = U   x  1 12, 122 = U   x  U  122 1 21, 122 = U   x  2 22, 122 = U   x   n = 2 k  = 2 T  = S U  1221 k  = 1  k  = 1 U  1222  U  1221 k   = 1 k  = 2 U  1222 Fig. 7.  Hierarchical genotype representation in system  U  1 . R. Kumar et al./Computers & Industrial Engineering 57 (2009) 169–180  173
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