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					                                 European Journal of Operational Research 180 (2007) 1231–1244
                                                                                                           www.elsevier.com/locate/ejor

                                                       Decision Support

        Availability allocation and multi-objective optimization
                        for parallel–series systems
                               Cheng-Hsiung Chiang a, Liang-Hsuan Chen                               b,*

                          a
                           Department of Computer Science, Hsuan Chuang University, Hsinchu 300, Taiwan, ROC
           b
               Department of Industrial and Information Management, National Cheng Kung University, Tainan 701, Taiwan, ROC

                                            Received 17 October 2003; accepted 25 April 2006
                                                      Available online 7 July 2006




Abstract

   Availability allocation is required when the manufacturer is obliged to allocate proper availability to various compo-
nents in order to design an end product to meet specified requirements. This paper proposes a new multi-objective genetic
algorithm, namely simulated annealing based multi-objective genetic algorithm (saMOGA), to resolve the availability allo-
cation and optimization problems of a repairable system, specifically a parallel–series system. Compared with a general
multi-objective genetic algorithm, the major feature of the saMOGA is that it can accept a poor solution with a small prob-
ability in order to enlarge the searching space and avoid the local optimum. The saMOGA aims to determine the optimal
decision variables, i.e. failure rates, repair rates, and the number of components in each subsystem, according to multiple
objectives, such as system availability, system cost and system net profit. The proposed saMOGA is compared with three
other multi-objective genetic algorithms. Computational results showed that the proposed approach could provide higher
solution quality and greater computing efficiency.
Ó 2006 Elsevier B.V. All rights reserved.

Keywords: Availability allocation; Availability optimization; Multi-objective genetic algorithms; Simulated annealing; Parallel–series
system




1. Introduction                                                        meaningful measure than reliability to measure the
                                                                       effectiveness of maintained systems, because it
   Availability, which for years has been considered                   includes reliability as well as maintainability. On
important in the defense and aerospace industry,                       the subject of evaluating the availabilities of a sys-
has now become a very important design parameter                       tem and its components, there are commonly two
in many other fields, including commerce, commu-                        kinds of procedures. First, the aim of availability
nication, computer systems, and service industries.                    modeling is to develop an availability model to
For repairable systems, ‘‘availability’’ is a more                     appraise system availability. Second, availability
                                                                       allocation allocates the availability for each com-
 *
   Corresponding author. Tel.: +886 6 275 7575x53140; fax:
                                                                       ponent based on the system’s requirements or
+886 6 236 2162.                                                       objectives. In the existing literature, a number of
   E-mail address: lhchen@mail.ncku.edu.tw (L.-H. Chen).               researchers have investigated the theoretical

0377-2217/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.ejor.2006.04.037
1232              C.-H. Chiang, L.-H. Chen / European Journal of Operational Research 180 (2007) 1231–1244

problems of availability modeling (Srivasvata and                incorporates the advantages of the Pareto-optimal-
Fahim, 1988; Ibe and Wein, 1992; Wood, 1994;                     ity concept and simulated annealing approach, is
Zhao, 1994; Hassett et al., 1995; Varvarigou and                 presented to determine the optimal decision vari-
Ahuja, 1997; Lee, 2000; Sherwin, 2000; Wiley,                    ables based on multiple objectives, i.e. system avail-
2001; Zhang and Horigome, 2001; Cao et al.,                      ability, system cost, and system net profit. The
2002) and those for specific applications (Das                    initial population, generated according to the target
et al., 1993; Hariri and Mutlu, 1995; Robino and                 system availability, is made up of a number of fail-
Sericola, 1995; Grover, 1999; Ledoux, 1999; Seri-                ure and repair rates as well as the number of compo-
cola, 1999; Wikstrom et al., 2000; Ma et al., 2001;
                    }                                            nents in each subsystem.
Dewinter et al., 2002; Lai et al., 2002). This article               Two numerical examples of a parallel–series sys-
aims at both availability allocation and availability            tem are applied to demonstrate the proposed
optimization.                                                    approach. We compared the proposed model with
   Genetic algorithms (GAs) are usually used as an               other three methods, the Pareto-optimality based
optimization technique with good efficiency to                     multi-objective genetic algorithm (MOGA), the
search for the global optimum of a function in a                 NPGA (niched Pareto genetic algorithm) presented
specific problem (Kuo and Prasad, 2000). Recently,                by Horn et al. (1994), and a weighting technique
an increasing number of GA applications have been                based MOGA (Elegbede and Adjallah, 2003). The
presented to solve the reliability optimizations. Reli-          results showed that the proposed approach is more
ability optimization searches for the optimal reli-              efficient and effective than the others.
ability for each component in a system in order to                   This paper is organized as follows. Section 2 will
maximize or minimize the objective functions (e.g.               formulate the problem of a parallel–series system.
system safety or system cost), such as in Kumar                  Based on the problem, the architecture of saMOGA
et al. (1995), Painton and Cambell (1995), Coit                  is constructed and then the components are
and Smith (1996), Lisnianski et al. (1996), Brown                described in Sections 3 and 4, respectively. Using
et al. (1997), Taguchi et al. (1998), Cantoni et al.             the two numerical examples, the proposed approach
(2000), Marseguerra and Zio (2000) and Vidyarthi                 is shown to outperform the other three methods in
and Tripathi (2001). In contrast to reliability opti-            terms of efficiency and effectiveness in Section 5.
mization, fewer researchers have studied availability            In Section 6, the conclusions of this paper are
allocation and optimization to find out the optimal               provided.
failure and repair rates for each component in a sys-
tem for maximizing (or minimizing) the objectives,               2. Problem formulation of a parallel–series system
although notable papers are Srivasvata and Fahim
(1988), Varvarigou and Ahuja (1997), Busacca                        We use parallel–series systems to describe and
et al. (2001) and Elegbede and Adjallah (2003).                  demonstrate the proposed approaches, since such
   In most cases, the problem of availability alloca-            systems are well established (Elegbede and Adjallah,
tion and optimization can be defined as a multi-                  2003). The common structure of a parallel–series
objective optimization problem, which aims to                    system is illustrated in Fig. 1. The system is also
maximize system availability and minimize system                 employed to compare the efficiency and effectiveness
cost (Elegbede and Adjallah, 2003). In addition,                 of the proposed approach with those of three other
Busacca et al. (2001) presented another objective                methods. Without loss of generality, suppose that
to maximize the system net profit. The optimization               all components are identical (the components have
problem was proven as an NP-hard problem                         the same reliability and availability) in each subsys-
(Chern, 1992). Since a parallel–series system is well
established (Elegbede and Adjallah, 2003), this
paper proposes a new approach, i.e. the simulated                          1, 1           2, 1                     m, 1
annealing based multi-objective genetic algorithm
                                                                           …




                                                                                                                     …
                                                                                           …




(saMOGA), to deal with the problem of availability                         1, j           2, j                      m, j
                                                                                                     ……
allocation and optimization of this system. Three
                                                                           …




                                                                                                                     …
                                                                                           …




important features of the system are considered as                        1, k1           2, k2                    m, km
decision variables, namely the number of compo-
                                                                       Subsystem 1     Subsystem 2             Subsystem m
nents, the failure rate and repair rate of each com-
ponent in each subsystem. The saMOGA, which                           Fig. 1. General structure of a parallel–series system.
                        C.-H. Chiang, L.-H. Chen / European Journal of Operational Research 180 (2007) 1231–1244               1233

tem. In general, the parallel–series system has two                    The assumptions of failure rate, repair rate and sys-
objectives, i.e. maximizing system availability (AS)                   tem availability are also the same as those in Case I.
and minimizing system cost (CS), which are pre-                        In contrast with Case I, Case II includes a new
sented by Elegbede and Adjallah (2003) and are                         objective, i.e. the system’s net profit G (Busacca
adopted in this paper as Case I. Moreover, we con-                     et al., 2001), to substitute for system cost CS in Case
sidered an additional objective, i.e. the system net                   I. The problem of Case II is formulated as follows:
profit, which was modified from Busacca et al.                            Max AS ðk; l; kÞ             and      Max Gðk; l; kÞ
(2001), and is introduced in Case II.
                                                                        Subject to AS P RS2
(1) Case I                                                                                                                     ð4Þ
                                                                        lki 6 ki 6 uki ; lli 6 li 6 uli ; lki 6 k i 6 uki
   The aim of the Case I problem is to maximize AS
and minimize CS, and the decision variables are the                        ði ¼ 1; 2; . . . ; mÞ:
number of components ki, the failure rate ki and the                   The definition of system availability AS of Eq. (4) is
repair rate li of each component for the ith subsys-                   the same as that in Eq. (2), and RS2 denotes the tar-
tem. We suppose that the failure rate and repair rate                  get system availability in Case II. The net profit
are constant, independent of time, so that the failure                 objective function G (Busacca et al., 2001) is defined
times (inter-arrival times) are exponentially distrib-                 as
uted. The formulation of this model presented by
                                                                       G ¼ P À ðC A þ C R þ C NS Þ:                            ð5Þ
Elegbede and Adjallah (2003) is shown below:
                                                                       The interest rate is not included in the net profit
Max AS ðk; l; kÞ          and       Min C S ðk; l; kÞ
                                                                       objective function. The variables in Eq. (5) are
Subject to AS P RS1                                                    briefly described as follows (Busacca et al., 2001)
           C S 6 C max                                       ð1Þ                     Z TM
lki 6 ki 6 uki ; lli 6 li 6 uli ; lki 6 k i 6 uki                      ðaÞ P ¼ P t Á      AS ðtÞ dt                     ð6Þ
                                                                                           0
   ði ¼ 1; 2; . . . ; mÞ:
                                                                       is the system profit and Pt (in $/year) indicates the
The symbol RS1 denotes the target system availabil-                    amount of money per unit time paid by the user.
ity in Case I, Cmax is the tolerable maximum cost,                     AS(t) is the system availability at time t, and TM is
and lki, uki, lli, uli, lki, and uki represent the lower               the time period being studied.
and upper bounds of ki, li and ki. k, l and k indicate
                                                                                    XX
                                                                                    m ki
the vectors associated with failure rates, repair rates                ðbÞ   CA ¼                    C iA;j                    ð7Þ
and numbers of components. For the parallel–series                                   i¼1       j¼1
system, the system’s asymptotic availability is used
                                                                       is the acquisition and installation cost of the m sub-
to evaluate the system availability (Elegbede and
                                                                       systems, and the.qffiffiffiffi is composed of ki compo-
                                                                                         ith one
Adjallah, 2003). The asymptotic availability and
cost are defined as
                                                                                 i
                                                                       nents. C A;j ¼ c i
                                                                                             kij is the cost incurred from
                                                                                                                    pffiffiffiffiffiffiffiffiffi
               Ym                    ki !                            component j in subsystem i and ci (in $/ yearÞ is
                                 ki
AS ðk; l; kÞ ¼       1À                     and      ð2Þ               a proportionality constant equal for all components
               i¼1
                             k i þ li                                  in the ith subsystem.
                  X
                  m
C S ðk; l; kÞ ¼
                                p           q
                        k i ðai ki i þ bi li i Þ;            ð3Þ                    XX
                                                                                    m ki
                                                                       ðcÞ C R ¼                     C iR;j                    ð8Þ
                  i¼1
                                                                                     i¼1       j¼1
where ai, bi and qi are positive real numbers, while pi
is negative, i = 1, 2, . . . , m. Eq. (2) assumes that re-             is the mean repair cost for all components of the
pair rates are independent of total repair activity le-                whole system. C iR;j is the mean value of repair cost
vel, i.e. the availability of the other components.                    for component j in the subsystem i, and it is defined by
The cost function, Eq. (3), from Elegbede and                                              !
                                                                         i             1
Adjallah (2003) is also used to this study.                                      i
                                                                       C R;j ¼ N R;j Á       Á C iR;j ;                    ð9Þ
                                                                                       lij
(2) Case II
    The system configuration of the Case II problem                     where C iR;j ¼ bi =lij is the yearly repair cost for com-
is the same as Case I, i.e. the parallel–series system,                ponent j in subsystem i. bi is (in $/year2) a propor-
and each component in the subsystem is identical.                      tionality constant equal for all components in the
1234                      C.-H. Chiang, L.-H. Chen / European Journal of Operational Research 180 (2007) 1231–1244
                                                              
ith subsystem, and N iR;j ¼ T M                     1
                                                    ki
                                                           1
                                                         þ li         is the      emulates the theory of biological evolution, and this
                                                     j      j
                                                                                  algorithm has been applied to many fields success-
mean number of failures during the time period TM.                                fully. In order to deal with several objective functions
                 Z TM
                                                                                  simultaneously, a number of multi-objective optimi-
ðdÞ C NS ¼ C U Á      ½1 À AS ðtފ dt          ð10Þ                               zation techniques have been developed. The MOGA
                            0
is the penalty cost during downtime. CU (in $/year)                               approaches can be divided into two categories: (1)
is the economic penalty per unit time, which is                                   non-Pareto approaches, e.g. VEGA (vector evalu-
caused by the undelivered service due to the system                               ated genetic algorithm) method (Schaffer, 1985); (2)
being unavailable.                                                                Pareto-based approaches, e.g. NPGA (niched Pareto
    The parameters of the above two problems will                                 genetic algorithm) method (Horn et al., 1994).
be given in Section 5.                                                                In this paper, we present a new MOGA, namely
                                                                                  saMOGA (simulated annealing based multi-objec-
3. The architecture of saMOGA                                                     tive genetic algorithm), by employing the Pareto-
                                                                                  optimality concepts. Pareto-optimality, proposed by
    To resolve the availability allocation and optimi-                            Vilfred Pareto (Petrie et al., 1995), is an economic
zation problems as introduced in Section 2, the                                   term for describing the solution properties for multi-
saMOGA (simulated annealing based multi-objec-                                    ple objectives. The Pareto solutions are non-domi-
tive genetic algorithm) approach is presented. As                                 nated; that is, no parts of a Pareto optimal solution
shown in Fig. 2, the optimal decision variables to                                can be improved without making some other parts
be determined are the optimal failure rates, repair                               worse. The proposed saMOGA is applied to resolve
rates and number of components in subsystems,                                     the availability allocation and optimization prob-
i.e. kà ; là and k à (i = 1, 2, . . . , m). The objective func-                   lems. The structure of the proposed saMOGA com-
      i    i       i
tions, AS and CS, are used in Case I, and AS and G                                bines the annealing concepts of the simulated
are applied in Case II. We proposed the saMOGA,                                   annealing (Kirkpatrick et al., 1983) and the neighbor-
which has the following features, to tackle the prob-                             hood design introduced in Section 4.4. The algorithm
lem as described in Section 2. (1) It is the Pareto-                              is developed with five features: (1) Pareto-optimality
optimality based approach for the multi-objective                                 based approach; (2) simulated annealing based
optimization problems; (2) it combines the simu-                                  searching strategy; (3) multiple objective functions;
lated annealing aspect to the searching strategy;                                 (4) neighborhood design for evolutionary mecha-
(3) the neighborhood design is used for the evolu-                                nism; and (5) multiple genetic operators.
tionary mechanism. The detailed design of saM-
OGA will be presented next in Section 4.                                          4.1. Representation mechanism and generation of
                                                                                  initial population
4. Design of saMOGA
                                                                                     The decision variables of an optimization prob-
   The genetic algorithm, first proposed by Holland                                lem can be represented as an artificial chromosome
(1975), provides a powerful, general-purpose optimi-                              consisting of numerous artificial genes. Several genes
zation paradigm in which the computational process                                are used to express a decision variable. Genes can be
                                                                                  encoded in several ways for a specific problem, such
                                                                                  as binary-encoded genes (Goldberg, 1989), discrete-
                                  Objectives:                                     encoded genes (Chen and Chiang, 2003), real-
                                  Max. AS (system availability)
                                  Min. CS (system cost)                           encoded genes (Chen et al., 2001), gray-encoded
Generate initial population of    Max. G (system’s net profit)                    genes (Andre et al., 2001), symbol-encoded genes
    saMOGA randomly                                                               (Juidette and Youlal, 2000), hybrid-encoded genes
                                                                λ1 , μ 1* , k1*
                                                                 *
                                                                                  combining numbers and symbols (Lazzerini and
                                     Simulated Annealing
                                                                λ* , μ 2 , k 2
                                                                 2
                                                                       *     *
                                                                                  Marcelloni, 2000), etc. This paper uses the binary-
                                    based Multi-Objective
                                                                                  encoded genes, since the encoding is easy and there
                                      Genetic Algorithm           ...
                                                                                  is high precision in representing a parameter. We
                                          (saMOGA)              λ* , μ m , k m
                                                                 m
                                                                       *     *

                                                                                  express the decision variables in a chromosome as
                                                              Optimal
                                                         Decision Variables       Chromosome structure :
 Fig. 2. Architecture of saMOGA for a parallel–series system.                       ½k1 l1 k2 l2 . . . . . . km lm k 1 k 2 . . . k m Š:
                  C.-H. Chiang, L.-H. Chen / European Journal of Operational Research 180 (2007) 1231–1244         1235

Each variable in a chromosome is encoded as sev-                 genetic operations. However, with regard to the
eral binary bits (one bit is ‘0’ or ‘1’).                        evolutionary mechanism of neighborhood design,
   In general, the initial population of MOGA is                 only the chromosomes in the boundary around the
generated randomly based on the encoding of chro-                specific chromosome have to execute the genetic
mosomes. We also generate the initial population                 operations, so that the executing efficiency increases
randomly, since the proposed saMOGA can drive                    significantly (Goldberg, 1989; Horn et al., 1994).
the searching to an enlarged space when finding                   Neighborhood design is often used to search for
the global optimum.                                              the Pareto optimal solutions (Fonseca and Fleming,
                                                                 1998; Tan et al., 2001). In this paper, we use such
4.2. Ranking the individuals                                     concepts for evolutionary strategy.
                                                                    The ‘‘neighborhood’’ means the boundary around
   For finding out the Pareto optimal solutions in                a chromosome, called the central chromosome Yc in
the population, we present two ranking algorithms                a population, as shown in Fig. 5. According to the
for Case I and Case II, as described in Section 2,               use of neighborhood design in the literature (Gold-
respectively. The ranking algorithms can help us                 berg, 1989; Horn et al., 1994), the Yc is selected
make a comparison between two individuals to                     from a population randomly. In Fig. 5, the neigh-
determine the non-dominated individuals. In order                borhood size of central chromosome Yc is 6, i.e.
to find more Pareto optimal solutions, the variety                Nc = 6. If the Nc is specified larger, the solutions
of solution space needs to be enhanced, so that                  produced may be higher quality, but more computa-
the current infeasible solutions remain in the rank-             tional time is needed. This study treats the neighbor-
ing algorithms. This gives the infeasible solutions              hood as a sub-population in the whole population.
chances to improve into better, more feasible solu-              In order to enhance the computational efficiency,
tions. The basic principles of the ranking algorithm             the basic operations of GA are performed in the
are divided into three parts.                                    sub-population.
   Part A, separate all individuals of a population
into two groups: the feasible (which satisfy the con-            4.4. Genetic operators
straints, i.e. Eqs. (1) and (4)) and infeasible groups.
   Part B, rank the feasible/infeasible individuals                 In general, GA performs three basic genetic
using the bubble sort approach based on the priority             operators, namely selection, crossover, and muta-
of objective functions.                                          tion (Goldberg, 1989). Nowadays, several new
   Part C, merge the ranked feasible and infeasible              genetic operators have been developed. Nearchou
individuals.                                                     (1999) proposed three different operators, swap,
   The two ranking algorithms, namely RANKING                    insertion, and deletion. Chen and Chiang (2003)
I and RANKING II, are illustrated in Figs. 3 and 4,              also presented the shift operator, which has shown
which have similar structures in Part A. If an indi-             good performance. This study employs four oper-
vidual at least achieves tolerable levels of all objec-          ators for the proposed GA, namely crossover,
tive functions, such as RS1 , RS2 and Cmax, this                 mutation, swap, and shift, as described in the
individual is added to the feasible fitness value set             following.
fitpass. Otherwise, the individual is put into the infea-
sible fitness value set fitunpass. After that, Part B sorts          (1) Crossover: The two-point crossover is acti-
the individuals using the bubble sort approach                         vated according to a probability Pc. Inter-
based on the priority of objective functions. The                      change of the genes between the two strings
                                                 _
prior orders of objective functions are C S 1 AS for                   is based on the selected pairs of cross points.
the Case I problem and G 1    _ AS for the Case II prob-           (2) Mutation: Flips the value of each gene of a
                               _
lem, where the symbol ‘‘1’’ means ‘‘is preferred                       chromosome randomly according to a small
than’’. Part C merges the sorted feasible and infea-                   probability Pm.
sible individuals.                                                 (3) Swap: Exchanges the two-side (left and right)
                                                                       genes of a swap point in a chromosome based
4.3. Neighborhood design                                               on a probability Ps.
                                                                   (4) Shift: A sequence of genes is shifted by one or
  With the general evolutionary strategy, all chro-                    more positions to the left or right according to
mosomes in a population have to carry out the                          the probability Psh.
1236             C.-H. Chiang, L.-H. Chen / European Journal of Operational Research 180 (2007) 1231–1244




                             Fig. 3. Pseudo-code of ranking algorithm for the Case I Problem.



                                                                regular crystalline structure. The important feature
4.5. Algorithm of the saMOGA                                    is that accepting poorer solutions is allowed with a
                                                                small probability based on the reducing tempera-
   Fig. 6 shows the scheme of the proposed saM-                 ture. In the past two decades, simulated annealing
OGA, which combines the design of Pareto-opti-                  has made great contributions to various fields.
mality based MOGA, as described in Goldberg                         The proposed saMOGA algorithm in Fig. 6 is
(1989), and the concept of simulated annealing                  applied to solve both Case I and Case II problems.
approach (Kirkpatrick et al., 1983). The basic prin-            In the applications, only the conditions of the
ciple of simulated annealing simulates the processes            While-Loop, the fitness function, and the ranking
of cooling metals in aligning the atoms to form a               algorithms are different, due to different constraints
                     C.-H. Chiang, L.-H. Chen / European Journal of Operational Research 180 (2007) 1231–1244            1237




                                    Fig. 4. Pseudo-code of ranking algorithm for the Case II Problem.



                                                                        and objective functions in Eqs. (1) and (4). Before
                                                                        executing the algorithm, some parameters, Nc, Pc,
                     1111010011                                                                                        ^
                        .   .   .
                                                                        Pm, Ps, Psh, T0 (the annealing temperature), g (the
                                                                        reducing rate of temperature), Npop and s (the toler-
                     1010110010
                                                                        ance of error function) have to be specified before-
Central chromosome   1111001101
         Υc          0010000100              Υc ' s neighborhood
                                                                        hand. The sensitivity analysis of the parameters
                     0101001101              with size = 6 treated as   for saMOGA is provided in Section 5.1.
                                             a sub-population
                     1010111000                                            As shown in Fig. 6, in the beginning of the saM-
                     0101010111
                                                                        OGA algorithm the initial population Wp is gener-
                     0111110001
                                                                        ated randomly. Afterwards, we assess each
                        .   .   .
                                                                        individual’s objective functions, i.e. AS, CS and G,
                     1011011011
                                                                        which are regarded as the fitness values. Using the
Fig. 5. An example of neighborhood for central chromosome Yc.           fitness values of each individual, we can apply the
1238             C.-H. Chiang, L.-H. Chen / European Journal of Operational Research 180 (2007) 1231–1244




                                            Fig. 6. Pseudo-code of saMOGA.



ranking algorithm to sort the individuals in a popu-            solution is feasible, we then stop the algorithm
lation. Thus, the best and worst individuals can                and return the best solution. Otherwise, we do the
be easily found as the BEST_OF_(Wp) and WOR-                    inner procedure of the While-Loop. Within the
ST_OF_(Wp), respectively. The information on the                While-Loop, we first select the central individual
best and worst individuals is used to calculate the             Yc and build its neighborhood WN. For each indi-
error function Er, which is used as the STOP condi-             vidual X(d), d = 1, 2, . . . , Nc, in WN, determine
tion of the While-Loop. If the difference between the            whether the Yc dominates X(d). If Yc dominates
best and worst solutions is tolerable and the best              X(d), i.e. Yc 1 X(d), do the following four opera-
                 C.-H. Chiang, L.-H. Chen / European Journal of Operational Research 180 (2007) 1231–1244          1239

tions (crossover, mutation, swap and shift opera-               analyses in applying saMOGA algorithms. In order
tions) and find a new individual Nnew. If the Nnew               to formulate the parallel–series problems, some
dominates X(d), replace X(d) by Nnew; otherwise,                parameter values used in Elegbede and Adjallah’s
a chance is given to accept this poorer solution                study (2003) are also adopted in Case I, listed as
based on the small probability. When a gener-                   (1) and (2) below. Except for the ranges of failure
ated random number h is smaller than
        pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi                         rates, repair rates and the number of components
expfÀð kNnew À X ðdÞkÞ2 =T 0 g, which is frequently             in each subsystem, the other parameters of Case II
used in simulated annealing (Kirkpatrick et al.,                are defined according to Busacca et al.’s study
1983), the Nnew is accepted to replace the X(d). After          (2001), as shown in (3) in the following:
examining all individuals in the neighborhood WN,
the best and worst individuals are updated, the error             (1) The ranges of the decision variables in Eq. (1):
function is recalculated, and the temperature is                      i = 1, 2, . . . , 5
reduced.                                                              lki = 10À3 · (0.4, 0.4, 0.5, 0.3, 0.5)/
   According to the neighborhood design of the                        hour, uki = 10À3 · (2, 2, 2, 2, 2)/hour,
For-Loop, the fitness of the individuals will be grad-                 lli = 0.5 · (0.4, 0.7, 0.9, 0.8, 0.7)/hour,
ually improved. Obviously, the computational effort                    uli = 0.85 · (0.4, 0.7, 0.9, 0.8, 0.7)/hour,
for the sub-population is less than that of the whole                 lki = (1, 1, 1, 1, 1), uki = (9, 9, 9, 9, 9),
population. In addition, the strategy of condition-                   RS1 ¼ 0:999 and Cmax = 600 units.
ally accepting unsatisfactory solutions can avoid                 (2) The system cost in Eq. (3): i = 1, 2, . . . , 5
finding only the locally optimal solutions.                            ai = 0.01 · (4, 2, 5, 8, 12),
                                                                      bi = 0.1 · (0.4, 0.2, 1.0, 0.8, 1.2),
5. Computational results                                              pi = À0.8 · (0.4, 0.2, 1.0, 0.8, 1.2) and
                                                                      qi = 0.85 · (0.4, 0.2, 1.0, 0.8, 1.2).
   This section demonstrates our proposed                         (3) The parameters used in Eqs. (4)–(10): i =
approach to the two problems: Case I (Eq. (1))                        1, 2, . . . , 5
and Case II (Eq. (4)). For Case I, we compare our                     lki = 5.5 · 10À3 · (1, 1, 1, 1, 1)/
approach with three other methods: namely, (1)                        year, uki = 10À1 · (1, 1, 1, 1, 1)/year,
the weighting technique based MOGA (wMOGA),                           lli = 0.7416 · (1, 1, 1, 1, 1)/year,
proposed by Elegbede and Adjallah (2003), (2) the                     uli = 3.1623 · (1, 1, 1, 1, 1)/year,
Pareto-optimality based MOGA (cMOGA) (Gold-                           lki = (1, 1, 1, 1, 1), uki = (9, 9, 9, 9, 9),
                                                                                                  pffiffiffiffiffiffiffiffiffi
berg, 1989), and (3) niched Pareto genetic algorithm                  ci = (33, 34, 33, 33, 34)$/ year,
(NPGA), proposed by Horn et al. (1994). The                           bi = (207, 204, 205, 205, 201)$/year2,
cMOGA approach is based on the Pareto-optimal-                        TM = 30 (year), Pt = 100 ($/year), CU = 200
ity concept with the neighborhood design. In fact,                    ($/year) and RS2 ¼ 0:95.
the cMOGA algorithm is similar to the proposed
saMOGA approach. The differences are (1)                             Based on the above settings, the parametric sen-
cMOGA only uses two basic operators, the cross-                 sitivity analyses for the saMOGA algorithms are
over and mutation operators, in GA, while saM-                  described as follows. The tolerance of error function
OGA applies four operators; (2) saMOGA further                  s is determined based on the decision maker’s pref-
employs the simulated annealing based mechanism.                erence, and thus we do not need to investigate it
In regard to Case II, the cMOGA and NPGA are                    here. With regard to Npop, obviously the larger pop-
compared. For comparisons, the above approaches,                ulation size Npop requires more CPU time. However,
excluding wMOGA, are programmed by MATLAB                       the larger Npop does not imply the saMOGA can
software in the Windows XP operating system.                    produce a larger number of Pareto solutions as well
Before comparisons, we first investigate the proper              as better objective function values. In the Case I
parameters used for saMOGA.                                     problem, according to our experiments of popula-
                                                                tion size from 10 to 100, we select Npop = 80
5.1. Parameter settings and sensitivity analyses                (Npop = 70 for Case II) since it can result in the low-
                                                                est system cost (higher system net profit for Case II)
   In this section, we first define the parameters con-           and the larger number of Pareto solutions. Regard-
tained in Case I and Case II problems, and then per-            ing the neighborhood size Nc, the executed CPU
form the corresponding parametric sensitivity                   time will increase and the number of Pareto solu-
1240                  C.-H. Chiang, L.-H. Chen / European Journal of Operational Research 180 (2007) 1231–1244

tions will have the increasing trend, if the Nc                         I to meet the requirement of a large sample in statis-
increases. However, Nc does not have an exact rela-                     tics. In order to compare the computational effi-
tionship with the two objective function values. In                     ciency, the STOP criterions of saMOGA, NPGA
Case I, we choose Nc = 24 (Nc = 30 for Case II)                         and cMOGA were all set as 200 iterations. To begin
since it can generate the minimal system cost (higher                   with the saMOGA, the initial population was gener-
system net profit) and higher system availability.                       ated randomly, consisting of three kinds of decision
   There are no significant trends for the required                      variables, i.e. failure, repair rates and the numbers
CPU time and number of Pareto solutions, as the                         of components for all subsystems.
annealing temperature T0 and temperature reduced                            Table 2 lists the results of the three approaches,
     ^                              ^
rate g are increased. T0 = 12 and g ¼ 0:6 for Case                      saMOGA, NPGA and cMOGA, from the 30 simu-
               ^
I (T0 = 4 and g ¼ 0:5 for Case II) are chosen, since                    lations, including the averages and standard devia-
they can produce better performance. Similarly, the                     tion of CPU time and the two objective function
probability parameters Pc, Pm, Ps and Psh do not                        values based on feasible solutions. Obviously, the
have a close relation with the required CPU time                        average system availabilities from the three
and number of Pareto solutions. We select the                           approaches are almost equivalent, while saMOGA
parameters that can achieve the better values of                        is the best in terms of the mean C S and standard
objective function. The above parameter settings                        deviation rCS of the system cost. As to the computa-
are listed in Table 1. Although the investigations                      tion time, NPGA takes the most time among the
of parameter settings are performed for saMOGA,                         three approaches; moreover, saMOGA and
the associated settings can also be applied to the                      cMOGA are approximately equal on average, while
other GA-based approaches to be compared in the                         saMOGA has the smaller standard deviation. As to
following sections, since they use the same                             other comparisons, the best solutions of objective
parameters.                                                             functions (AS, CS) for saMOGA, NPGA and
                                                                        cMOGA from 30 simulations are provided by
5.2. Comparison for the problem of Case I                               adopting the system cost as the first priority (refer
                                                                        to RANKING I), namely (0.99911, 122.07487),
  For comparing with wMOGA, NPGA and                                    (0.99995, 142.4814), and (0.99998, 149.54307),
cMOGA, 30 simulations were performed for Case                           respectively. Note that the system availabilities of
                                                                        the above best solutions are a little less than the cor-
                                                                        responding averages. Among the three approaches,
Table 1
Parameter settings of saMOGA for Case I and II problems                 saMOGA has the least cost with the system avail-
                                                                        ability greater than the minimum requirement. The
Parameter                      Parameter setting
                                                                        optimal solutions found by saMOGA are k =
                               Case I                     Case II
                                                                        (2, 6, 2, 1, 2), k = 10À3 · (0.47109, 1.85439, 1.82845,
Npop                           80                         70            0.59038, 1.95806)/hour and l = (0.28964, 0.48867,
Nc                             24                         30
                                                                        0.71727, 0.67973, 0.47382)/hour, where the mea-
Pc                              0.4                        0.6
Pm                              0.4                        0.2          surement units of ki and li are frequencies per hour.
Ps                              0.8                        0.9          Physically, three subsystems have two components
Psh                             0.4                        1.0          individually, while the second and fourth ones have
T0                             12                          4            six and one component, respectively. Each subsys-
^
g                               0.6                        0.5
                                                                        tem is designed to have different (or approximately


Table 2
The simulation results of the three approaches in the Case I problem
Approach         CPU time                                    Objective function value                     No. of avg. feasible sol.b
                                          a
                 Average        Std. dev.       AS           rAS                CS          rC S
saMOGA            58.96445      1.12912         0.99999      3.8242 · 10À5      302.23872    90.69897     52.53
NPGA             125.39965      9.09658         0.99999      2.2386 · 10À5      354.33664   104.044       50.27
cMOGA             58.33488      2.53784         0.99999      5.481 · 10À5       410.61301    97.67376     48.73
 a
     Std. dev.: standard deviation.
 b
     Indicates the average number of feasible solutions found for each simulation.
                      C.-H. Chiang, L.-H. Chen / European Journal of Operational Research 180 (2007) 1231–1244                                       1241

equivalent) failure and repair rates based on the                                   solutions can be found. Fig. 7 illustrates the Pareto
parameter settings in the previous section to achieve                               optimal solution sets for saMOGA.
the conflicting objectives. Regarding the optimal
solutions obtained by wMOGA (Elegbede and                                           5.3. Comparison for the problem of Case II
Adjallah, 2003), the two objective values are
AS = 0.9993 and CS = 546.43 units, significantly                                         In this case, saMOGA is also compared with
worse than the other approaches.                                                    NPGA and cMOGA approaches. The parameter
   Besides the comparisons of CPU time and objec-                                   settings are the same as those in the previous case,
tive function values, saMOGA can produce more                                       except for those in the objective function of the sys-
feasible solutions, as listed in Table 2, and more                                  tem’s net profit. Thirty simulations are also per-
Pareto optimal solutions (non-dominated solutions)                                  formed by setting STOP criterion as Case I. Table
from 30 simulations. The proposed saMOGA has                                        3 lists the simulation outcomes in terms of the aver-
83 Pareto solutions, while NPGA and cMOGA                                           age and standard deviation of computation time
have 48 and 36, respectively. This is because saM-                                  and objective function values. From the table, the
OGA includes the simulated annealing method to                                      system availabilities are almost equivalent among
enlarge the search space so that more Pareto                                        the three approaches. However, the proposed
optimal solutions can be searched and then better                                   saMOGA outperforms the other approaches in the


                                                     CS (Units)
                                                         600
                                                         500
                                                         400
                                                         300
                                                         200
                                                         100
                                                           0
                                                          0.999    0.9992   0.9995       0.9997
                                                                                                  AS 1
                                                                            (a)

                       CS (Units)                                                 CS (Units)
                        600                                                        600
                        500                                                        500
                        400                                                        400
                        300                                                        300
                        200                                                        200
                        100                                                        100
                             0                                                        0
                           0.9997        0.9998           0.9999     AS 1           0.99997       0.99998      0.99999          1
                                                   (b)                                                   (c)
                                                                                                                           AS

                        Fig. 7. Pareto optimal set of the Case I problem: (a) saMOGA, (b) NPGA, (c) cMOGA.



Table 3
The simulation results of the three approaches in the Case II problem
Approach         CPU time (in seconds)                               Objective function value                                   No. of avg. feasible sol.b
                                               a
                 Average            Std. dev.            AS          rAS                    G                  rG
saMOGA            61.01841           1.74557             0.96538     4.823 · 10À3           1412.17891          77.35633        59.73
NPGA             183.61770          10.59654             0.97285     1.3148 · 10À2           813.89557         162.23907        20.6
cMOGA             66.06266           1.21468             0.96298     1.0386 · 10À2           920.53123         151.57847         0.4
 a
     Std. dev.: standard deviation.
 b
     Indicates the average number of feasible solutions found for each simulation.
1242             C.-H. Chiang, L.-H. Chen / European Journal of Operational Research 180 (2007) 1231–1244


                                            G ($)
                                           1500

                                           1250

                                           1000

                                            750

                                            500
                                              0.95   0.96    0.97          0.98     0.99     1
                                                                     (a)                    AS

                    G ($)                                              G ($)
                   1500                                                1500

                   1250                                                1250

                   1000                                                1000

                    750                                                    750

                    500                                                    500
                      0.95   0.96   0.97     0.98    0.99        1           0.95    0.96    0.97      0.98   0.99        1
                                           (b)              AS                                   (c)                 AS

                  Fig. 8. Pareto optimal set of the Case II problem: (a) saMOGA, (b) NPGA, (c) cMOGA.



computation time and the system net profit. NPGA                               From the solution quality and the efficiency mea-
performs the worst for computational efficiency and                          sures for the above two cases, the performance of
the system net profit.                                                      the proposed saMOGA is the most promising of
   Adopting the system net profit as the first priority                      the three approaches. Thus, these results have con-
(refer to RANKING II), the best solutions (AS, G)                          vincingly demonstrated the effectiveness and effi-
of saMOGA, NPGA and cMOGA from the simula-                                 ciency of the proposed saMOGA.
tions are obtained as (0.95918, 1510.72039),
(0.96960, 1346.23344), and (0.95959, 1205.26413),                          6. Conclusions
respectively. Obviously, saMOGA also has the best
system net profit with system availability almost                              This paper presents a novel approach, i.e. the
equivalent to the others. The best solutions found                         saMOGA (simulated annealing based multi-objec-
by saMOGA are: k = (1, 1, 1, 1, 1), k = (0.02462,                          tive genetic algorithm), for availability allocation
0.02360, 0.02601, 0.02434, 0.03127)/year and                               and optimization problems. The saMOGA is devel-
l = (3.11734, 3.08658, 3.08658, 3.15283, 3.07948)/                         oped based on the Pareto-optimality concept and
year. Physically, the structure of best solution is a                      the neighborhood design for evolutionary mecha-
series system since each subsystem has only one                            nism. Furthermore, the concept from simulated
component (ki = 1 for all i). The results for failure                      annealing that can accept the poorer solutions into
and repair rates are almost equivalent for all subsys-                     the next generation is included in saMOGA in order
tems, intuitively reflecting the fact that the parame-                      to avoid reaching the local optimum.
ter settings are approximately equivalent for each                            The parallel–series system with two kinds of
subsystem in Section 5.1. In addition, the proposed                        problem, Case I and Case II, is used to demonstrate
saMOGA not only has the maximal number of fea-                             the proposed approach. For comparisons, we con-
sible solutions (59.73 in average), but the maximal                        ducted 30 simulations in the two cases. In the Case
number, 37, of Pareto solutions due to the applica-                        I problem, we compared the saMOGA with three
tion of the simulated annealing method, as demon-                          other methods, wMOGA, NPGA and cMOGA
strated in Case I. In this case, NPGA and cMOGA                            methods. For the Case II problem, we compared
only have 9 and 5 Pareto solutions, respectively.                          the proposed saMOGA with the NPGA and
These Pareto solutions are shown in Fig. 8.                                cMOGA methods. From the simulations, the solu-
                     C.-H. Chiang, L.-H. Chen / European Journal of Operational Research 180 (2007) 1231–1244                        1243

tion quality of saMOGA is the best among all meth-                   Grover, W.D., 1999. High availability path design in ring-based
ods, and the required CPU time for saMOGA is                            optical networks. IEEE/ACM Transactions on Networking 7
                                                                        (4), 558–574.
also satisfactory. Furthermore, the saMOGA can                       Hariri, S., Mutlu, H., 1995. Hierarchical modeling of availability
reach the most number of Pareto optimal solutions.                      in distributed systems. IEEE Transactions on Software
All of these simulation results demonstrated that the                   Engineering 21 (1), 50–56.
proposed saMOGA is effective and efficient for the                      Hassett, T.F., Dietrich, D.L., Szidarovszky, F., 1995. Time-
problems.                                                               varying failure rates in the availability and reliability analysis
                                                                        of repair systems. IEEE Transaction on Reliability 44 (1),
                                                                        155–160.
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