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                                                         Reliability Engineering and System Safety 94 (2009) 830–837

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An efficient particle swarm approach for mixed-integer programming in
reliability–redundancy optimization applications
Leandro dos Santos Coelho Ã
                                                                                                       ´                         -a                                   ´
Industrial and Systems Engineering Graduate Program, LAS/PPGEPS, Pontifical Catholic University of Parana, PUCPR, Imaculada Conceic ˜o, 1155, 80215-901 Curitiba, Parana, Brazil

a r t i c l e in f o                                   a b s t r a c t

Article history:                                       The reliability–redundancy optimization problems can involve the selection of components with
Received 12 November 2007                              multiple choices and redundancy levels that produce maximum benefits, and are subject to the cost,
Received in revised form                               weight, and volume constraints. Many classical mathematical methods have failed in handling
29 August 2008
                                                       nonconvexities and nonsmoothness in reliability–redundancy optimization problems. As an alternative
Accepted 1 September 2008
Available online 16 September 2008
                                                       to the classical optimization approaches, the meta-heuristics have been given much attention by many
                                                       researchers due to their ability to find an almost global optimal solutions. One of these meta-heuristics
Keywords:                                              is the particle swarm optimization (PSO). PSO is a population-based heuristic optimization technique
Reliability–redundancy optimization                    inspired by social behavior of bird flocking and fish schooling. This paper presents an efficient PSO
Particle swarm optimization
                                                       algorithm based on Gaussian distribution and chaotic sequence (PSO-GC) to solve the reliability–
Evolutionary algorithm
                                                       redundancy optimization problems. In this context, two examples in reliability–redundancy design
                                                       problems are evaluated. Simulation results demonstrate that the proposed PSO-GC is a promising
                                                       optimization technique. PSO-GC performs well for the two examples of mixed-integer programming in
                                                       reliability–redundancy applications considered in this paper. The solutions obtained by the PSO-GC are
                                                       better than the previously best-known solutions available in the recent literature.
                                                                                                                      & 2008 Elsevier Ltd. All rights reserved.

1. Introduction                                                                           reliability–redundancy optimization problems. One of these
                                                                                          modern meta-heuristics is the particle swarm optimization (PSO).
   In 1952, the Advisory Group on the Reliability of Electronic                               PSO, first introduced by Kennedy and Eberhart [23,24], is a
Equipment defined the reliability in a broader sense: reliability                          stochastic global optimization technique inspired by social
indicates the probability implementing specific performance or                             behavior of bird flocking or fish schooling. It simulated the feature
function of products and achieving successfully the objectives                            of bird flocking and fish schooling to configure the heuristic
within a time schedule under a certain environment [1]. A design                          learning mechanism. PSO is initialized with a population of
engineer often tries to improve system reliability with a basic                           random solutions within the feasible range, called particles
design, to the largest extent possible subject to constraints on any                      (individual). The learning procedure of PSO is that the solution
component attributes (cost, weight, and volume) of system [2].                            of every individual particle is modified with the cause of its own
The problem is to select the optimal combination of components                            best experience and other individuals’ best experiences. In other
and redundancy levels to meet system level constraints while                              words, the particles fly through the search space influenced by
maximizing system reliability.                                                            two factors: one is the individual’s best position ever found
   Recently, many meta-heuristics [3,4], such as evolutionary                             (personal best); the other is the group’s best position (global best).
algorithms [5–13], tabu search [14,15], ant colony optimization                           In this case, each particle in PSO flies through the search space
[16–19], artificial immune system [20], fuzzy system [21],                                 with a velocity that is dynamically adjusted according to its own
and artificial neural networks [22] have been employed in                                  and its cognitive and social behaviors.
                                                                                              In canonical PSO, a uniform probability distribution to
                                                                                          generate random numbers is used. However, the use of other
                                                                                          probability distributions may improve the ability to fine-tuning or
  Abbrevations: PSO, particle swarm optimization; PSO-CA, canonical particle              even to escape from local optima. In the meantime, it has been
swarm optimization; PSO-CO, particle swarm optimization with constriction                 proposed the use of the Gaussian [25–27], Cauchy [28,29],
factor; PSO-GC, Gaussian probability distribution and also chaotic sequences in
                                                                                          exponential [30], Levy [31] probability distribution functions,
particle swarm optimization
  Ã Tel./fax: +55 41 327113 45.                                                           and chaotic sequences [32–35] to generate random numbers to
    E-mail address:                                               updating the velocity equation in PSO.

0951-8320/$ - see front matter & 2008 Elsevier Ltd. All rights reserved.
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  Nomenclature                                                                      tmax        the maximum number of allowable iterations
                                                                                    ud          a uniformly-distributed random number within the
  a                           ´
             a constant of Henon map                                                            range [0,1]
  b                           ´
             a constant of Henon map                                                vi          the volume of each component in subsystem i
  c1         the cognitive learning rate                                            vmax        the maximum velocity that each particle can make at
  c2         the social learning rate                                                           each iteration
  f( Á )     the objective function for the overall system relia-                   xi ¼ [xi1,xi2, y, xin]T the position of the ith particle of popula-
             bility                                                                             tion
  g          the set of constraint functions                                        wi          the weight of each component in subsystem i
  gbest      the global best particle                                               y1                           ´
                                                                                                the output of Henon map
  gi         the ith constraint function                                            y2                                ´
                                                                                                a signal of state in Henon map
  k                                      ´
             the iteration number in Henon map                                      C           the upper limit on the cost of the system
  l          the vector of resource limitation                                      F           the feasible region
  m          the number of subsystems in the system                                 Rs          the system reliability
  n ¼ (n1,n2,n3, y, nm) the vector of the redundancy allocation                     S           the search space
             for the system                                                         V           the upper limit on the sum of the subsystems’
  ni         the number of components in the ith subsystem                                      products of volume and weight
  pbest      the personal best particle                                             Ud          a uniformly-distributed random number within the
  pi ¼ [pi1,pi2, y, pin]T the best previous position of the ith                                 range [0,1]
             particle                                                               W           the upper limit on the weight of the system
  r ¼ (r1,r2,r3, y, rm) the vector of the component reliabilities for               li ¼ [li1,li2, y, lin]T the velocity of the ith particle
             the system                                                             j           a parameter of design
  ri         the reliability                                                        o           the inertia weight
  t          the iterations (generations)                                           w           the constriction coefficient

    This paper employs the Gaussian probability distribution and                    r ¼ (r1,r2,r3, y, rm) is the vector of the component reliabilities for
also chaotic sequences in PSO (PSO-GC) design to solve the                          the system, n ¼ (n1,n2,n3, y, nm) is the vector of the redundancy
reliability–redundancy optimization problems. In this context,                      allocation for the system; ri and ni are the reliability and the
two examples in reliability–redundancy design are evaluated. The                    number of components in the ith subsystem, respectively; f( Á ) is
results of proposed PSO-GC algorithm, canonical PSO (PSO-CA),                       the objective function for the overall system reliability; and l is the
and PSO with constriction factor (PSO-CO) are compared. The                         vector of resource limitation; m is the number of subsystems in
novel PSO-GC algorithm outperforms and provide solutions when                       the system. The goal is to determine the number of component
compared with PSO-CA, PSO-CO, and also other techniques                             and the components’ reliability in each system so as to maximize
presented in literature for the two reliability–redundancy opti-                    the overall system reliability. The problem belongs to the category
mization examples [36–38].                                                          of constrained nonlinear mixed-integer optimization problems.
    The remaining content of this paper is organized as follows. In
Section 2, the reliability–redundancy optimization problem is                       2.1. Example 1: complex (bridge) system
introduced, while the concepts of PSO approaches are explained in
Section 3. Section 4 presents the simulation results for two                           The first example problem used to demonstrate the efficiency
reliability–redundancy optimization problems. Finally, Section 5                    of PSO approaches were proposed in [38,40,41]. Fig. 1 represents
contains the concluding remarks and further research.                               the complex (bridge) system analyzed in this paper.
                                                                                       The complex (bridge) system optimization problem can be
                                                                                    stated as follows [38]:
2. Description of reliability–redundancy optimization problem
                                                                                    maximize         f ðr; nÞ ¼ R1 R2 þ R3 R4 þ R1 R4 R5 þ R2 R3 R5
   The goal of reliability engineering is to improve the reliability                                             À R1 R2 R3 R4 À R1 R2 R3 R5 À R1 R2 R4 R5
system. The reliability–redundancy optimizations are useful for                                                  À R1 R3 R4 R5 À R2 R3 R4 R5 þ 2R1 R2 R3 R4 R5   (3)
system designs that are largely assembled and manufactured
                                                                                    subject to
using off-the-shelf components, and also, have high reliability
requirements [39].                                                                                 X

   A reliability–redundancy optimization problem can be for-                        g 1 ðr; nÞ ¼         wi v2 n2 pV
                                                                                                             i i                                                 (4)
mulated with system reliability as the objective function or in the
constraint set. In this work, the reliability–redundancy allocation
problem of maximizing the system reliability subject to multiple
nonlinear constraints can be stated as a nonlinearly mixed-integer
programming model in general form as follows:
Maximize     Rs ¼ f ðr; nÞ,                                               (1)

subject to   gðr; nÞpl                                                    (2)

0pr i p1;    r i 2 <;   ni 2 Z þ ;   1pipm
where Rs is the reliability of system, g is the set of constraint
functions usually associated with system weight, volume and cost,                                  Fig. 1. Representation for the complex (bridge) system.
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Table 1                                                                                              Table 2
Data used in complex (bridge) system                                                                 Data used in overspeed protection system of gas turbine

Stage      105ai           bi               wivi2       wi         V           C          W          Stage    105aI     bi       vi       wi     V        C     W     T

1          2.330           1.5              1           7          110         175        200        1        1.0       1.5      1        6      250      400   500   1000 h
2          1.450           1.5              2           8                                            2        2.3       1.5      2        6
3          0.541           1.5              3           8                                            3        0.3       1.5      3        8
4          8.050           1.5              4           6                                            4        2.3       1.5      2        7
5          1.950           1.5              2           9

                              1000 bi                                                                ai ½ÀT= lnðri ފbi is the cost of each component with reliability ri at
g 2 ðr; nÞ ¼         ai À              ½ni þ e0:25ni ŠpC                                   (5)       subsystem i; T is the operating time during which the component
                              ln r i
                                                                                                     must not fail; and W is the upper limit on the weight of the
               m                                                                                     system. The input parameters defining the of the overspeed
g 3 ðr; nÞ ¼         wi ni e0:25ni pW                                                      (6)       protection system for a gas turbine are shown in Table 2. The data
               i¼1                                                                                   shown in Table 2 are also available in [36–38].
where V is the upper limit on the sum of the subsystems’ products
of volume and weight, C is the upper limit on the cost of the
system, and W is the upper limit on the weight of the system. Eqs.                                   3. Optimization using PSO algorithms
(4), (5) and (6) are constraints about the reliability, system cost
and weight, respectively. On the other words, constraint given by                                        PSO is a population-based heuristic global search algorithm
Eq. (4) is a combination of weight, redundancy allocation and                                        based on the social interaction and individual experience. In
volume. Eq. (5) is a cost constraint, while Eq. (6) is a weight                                      essence, PSO mimics the collective learning of individuals
constraint. In this context, niAZ, where Z is the space discrete of                                  when they are in groups, observed through natural behaviors
integers, and 0prip1, r i 2 <; where < is set of real numbers,                                       of bird flocks and fish schools. In these groups, there is a leader
0pipm. The parameters bi and ai are physical features of system                                      who guides the movement of the whole swarm. The movement
components. The input parameters defining the of the complex                                          of every individual is based on the leader and on his own
(bridge) system are shown in Table 1. The data shown in Table 1                                      knowledge. In general, it can be said that the model that PSO
are also available in [38,40,41].                                                                    is inspired assumes that the behavior of every particle is a
                                                                                                     compromise between its individual memory and a collective
2.2. Example 2: overspeed protection system for a gas turbine                                        memory.
                                                                                                         In PSO, a point (individual) in the problem space is referred to
   To evaluate the performance of the PSO approaches for the                                         as particles, is a candidate solution to the optimization problem at
mixed-integer nonlinear reliability design problem, the reliabili-                                   hand. Each particle in population (named swarm in PSO) is
ty–redundancy optimization problem of the overspeed protection                                       initialized with a random position and search velocity. Each
system for a gas turbine [36–38] is also considered. Overspeed                                       particle flies through the problem space and keeps track of its
detection is continuously provided by the electrical and mechan-                                     position, velocity and fitness. Moreover, its position (i.e. a
ical systems. When an overspeed occurs, it is necessary to cut off                                   solution) and velocity (i.e. change pattern of the solution) are
the fuel supply using control valves [36].                                                           adjusted according to its own experience and social cooperation
   This problem is formulated as the following mixed-integer                                         by its fitness to the environment.
nonlinear programming problem, i.e., the problem can be                                                  During this iterative process, the behavior of a particle is a
stated as                                                                                            compromise among three possible alternatives: (i) following its
                                                                                                     current pattern of exploration, (ii) going back towards its best
                                                                                                     previous position, and (iii) going towards the best historic value of
Maximize         f ðr; nÞ ¼             ½1 À ð1 À r i Þni Š                                (7)
                                                                                                     all the particles. A representation of procedure of a classical PSO is
                                                                                                     presented in Fig. 2.
subject to                                                                                               In this section, the PSO approaches validated in this work are
               m                                                                                     described. The first one is the canonical PSO, which is presented in
g 1 ðr; nÞ ¼         vi n2 pV
                         i                                                                 (8)       Section 3.1. In Section 3.2, the PSO based on constriction
                                                                                                     coefficient approach is detailed. The third one is the PSO
               m                                                                                     technique based on Gaussian distribution and chaotic sequence
g 2 ðr; nÞ ¼         Cðr i Þ½ni þ e0:25ni ŠpC                                              (9)       is described in Section 3.3.

               m                                                                                     3.1. Standard or canonical PSO approach (CA-PSO)
g 3 ðr; nÞ ¼         wi ni e0:25ni pW                                                     (10)
                                                                                                         The procedure for implementing the global version of
                              þ                                                                      canonical PSO is given by the following steps [33,34]:
1pni p10;            ni 2 Z
                                                                                                         Step 1. Initialization random of positions and velocities: Initialize
where Z is the space discrete of positive integers,                                                  a population of particles with random positions and velocities in
                                                                                                     the n-dimensional problem space using a uniform probability
0:5pr i p1 À 10À6 ;               ri 2 <
                                                                                                     distribution function at iteration t ¼ 1.
where vi is the volume of each component in subsystem i; V is the                                        Step 2. Evaluation of particle’s fitness: Evaluate each particle’s
upper limit on the sum of the subsystems’ products of volume and                                     objective function value (fitness). In this paper, the deal of PSO-CA
weight; C is the upper limit on the cost of the system; Cðr i Þ ¼                                    is the maximization of objective function.
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                                                                                               Fig. 3. Representation of modification in a search by PSO algorithm.

                                                                                        pbest and gbest, respectively. The index of the best particle among
                                                                                        all the particles in the population is represented by the symbol g.
                                                                                        Factors ud and Ud are uniformly-distributed random numbers
                                                                                        within the range [0,1].
                                                                                            The first relation in Eq. (11) is used to calculate ith particle’s
                                                                                        new velocity by taking into consideration three terms: the
                                                                                        particle’s previous velocity, the distance between the particle’s
                                                                                        best previous and current position, and, lastly, the distance
                                                                                        between swarm’s best experience (the position of the best particle
                                                                                        in the swarm) and ith particle’s current position. The velocity in
                                                                                        Eq. (11) is also limited by a maximum, vmax, meaning the
                                                                                        maximum jump that each particle can make at each iteration
                                                                                        (generation). The selected value for vmax should not be too high to
                                                                                        avoid oscillations, or too low to explore sufficiently and the
                                                                                        particle becomes trapped in a local optimal solution.
                                                                                            In Eq. (11), the value given to the inertia weight affects the type
                                                                                        of search in the following way: a large w will direct the PSO for a
                                                                                        global search while a small w will direct the PSO for a local search.
                  Fig. 2. Procedure of a classical PSO approach.
                                                                                        This parameter can vary linearly from a larger value to a smaller
                                                                                        value in order to make the search global in the early run and local
                                                                                        in the end of the run.
    Step 3. Comparison to pbest (personal best): Compare each                               Considering these concerns, Shi and Eberhart [42] have found
particle’s fitness with the particle’s pbest. If the current value is                    improvements in the performance of the PSO with a linearly
better than pbest, then set the pbest value equal to the current                        varying inertia weight over the generations. The mathematical
value and the pbest location equal to the current location in                           representation of this concept is given by
n-dimensional space.
                                                                                                           ðt max À tÞ
    Step 4. Comparison to gbest (global best): Compare the fitness                       o ¼ ðo1 À o2 Þ                 þ o2                                          (13)
                                                                                                               t max
with the population’s overall previous best. If the current value is
better than gbest, then reset gbest to the current particle’s array                     where o1 and o2 are the initial and final values of the inertia
index and value.                                                                        weight, respectively. Through empirical studies, Shi and Eberhart
    Step 5. Updating of each particle’s velocity and position: Each                     [42] have observed that the good solution can be improved by
particle tries to modify its position using the current velocity and                    varying the value of o from 0.9 at the beginning of the search to
its distance from its own best position and the global best particle.                   0.4 at the end of the search for most problems.
The modification of the velocity, vi, and position of the particle, xi,                      Then by Eq. (12) the ith particle flies towards a new position
can be represented by Eqs. (11) and (12)                                                according to its previous position and its velocity, considering
                                                                                        Dt ¼ 1. Fig. 3 shows a representation of a searching point by PSO
li ðt þ 1Þ ¼ oli ðtÞ þ c1 udðpi ðtÞ À xi ðtÞÞ þ c2 Udðpg ðtÞ À xi ðtÞÞ       (11)
                                                                                            Step 6. Repeating the evolutionary cycle: If a predefined stopping
xi ðt þ 1Þ ¼ xi ðtÞ þ Dt li ðt þ 1Þ                                          (12)
                                                                                        criterion is met, usually a sufficiently good fitness or a maximum
where o is the inertia weight; i ¼ 1,2, y, N indicates the index of                     number of iterations (generations), then output gbest and its
particles; t ¼ 1,2, y, tmax indicates the iterations (generations);                     objective value; otherwise set t ¼ t+1 and go back to Step 2.
tmax is the maximum number of allowable iterations; li ¼ [li1,li2,
y, lin]T stands for the velocity of the ith particle, xi ¼ [xi1,xi2, y,
xin]T stands for the position of the ith particle of population, and                    3.2. PSO using constriction coefficient approach (PSO-CO)
pi ¼ [pi1,pi2, y, pin]T represents the best previous position of the
ith particle. Positive constants c1 and c2 are the cognitive learning                      Eq. (11) of velocity updating for the canonical PSO algorithm
and social learning rates, respectively, which are the acceleration                     can be considered as a kind of difference equation. A particle’s
constants responsible for varying the particle velocity towards                         velocity is an important parameter because it determines the
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resolution about the solution regions. The choice of a too small                                                                  ´
                                                                                         where k is the iteration number. The Henon map is used in this
value for vmax can cause very small updating of velocities and                                                                                   ´
                                                                                         work for a ¼ 1.4 and b ¼ 0.3 (the values for which the Henon map
positions of particles at each iteration. Hence, the algorithm may                       has a strange attractor).
take a long time to converge and faces the problem of getting                               The proposed approach called PSO-GC is given by the following
stuck to local minima. To overcome these situations, researchers                         pseudocode:
[43,44] have recently proposed improved velocity update rules
employing a constriction factor w. In doing so, the velocity
equation is updated according to

li ðt þ 1Þ ¼ w½li ðtÞ þ c1 udðpi ðtÞ À xi ðtÞÞ þ c2 Udðpg ðtÞ À xi ðtÞފ      (14)              If t41
By using a constriction coefficient w expressed as                                                   If f1 À ððminðf Þ À f i ðr; nÞÞ=ðminðf Þ À maxðf ÞÞÞgox
                                                                                                       % chaotic sequence based on Henon map in velocity
                 2                                                                                  updating
w¼               pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi                                         (15)
      j2 À j À     j2 À 4jj
                                                                                                    li ðt þ 1Þ ¼ w½li ðtÞ þ c1 hi ðpi ðtÞ À xi ðtÞÞ þ c2 Hi ðpg ðtÞ
with the parameter j ¼ c1+c2, j44 and w is a function of c1 and
c2. Typically, c1 and c2 are both set to be 2.05. Thus, j is set to 4.1                                           À xi ðtÞފ                                          (18)
and the constriction coefficient w is 0.729. Clerc and Kennedy [43]                                  Else If
found that the system behavior could be controlled so that the                                        % Gaussian distribution in velocity updating
system behavior has the following features: (i) the system does
not diverge in a real value region and finally can converge; and (ii)
                                                                                                    li ðt þ 1Þ ¼ w½li ðtÞ þ c1 Gi ðpi ðtÞ À xi ðtÞÞ þ c2 Udðpg ðtÞ
the system can search different regions efficiently by avoiding                                                    À xi ðtÞފ                                          (19)
premature convergence. The PSO-CO uses Eqs. (14), (15), and                                        End If
also Eq. (12) to update the position of particles in swarm. This                                Else If
modification are simplistic in a sense that they only (gradually)                                  % use Eq. (19) in iteration t ¼ 1
reduce the parameter value(s) without affecting the structure
of PSO.                                                                                         li ðt þ 1Þ ¼ w½li ðtÞ þ c1 Gi ðpi ðtÞ À xi ðtÞÞ þ c2 Udðpg ðtÞ À xi ðtÞފ
                                                                                                End If
3.3. Framework of novel PSO method using Gaussian distribution
and chaotic sequences (PSO-GC)
                                                                                         where min(f) and max(f) are the worst and best objective function
    In PSO-CA and PSO-CO methods, a uniform probability                                  values (maximization problem) of the swarm at iteration t,
distribution to generate random numbers ud and Ud is                                     respectively. The fi(r, n) is the objective function value of ith
adopted in velocity equation. The use of Gaussian probability                            particle; x is a constant positive value in range [0,1]; Gi denotes a
distribution [25–27] can improve the ability to fine-tuning or                            Gaussian random number scaled in the range [0;1] and is
even to escape from local optima in PSO design. The mechanisms                           generated a new for each ith particle; hi and Hi are generated by
of Gaussian mutation operations have been studied by Yao et al.                            ´
                                                                                         Henon map given by Eqs. (18) and (19) normalized in the
[45] and Chellapilla [46]. They pointed out that Gaussian                                range [0,1]. Occasionally, the swarm of particles can converge
mutation in evolutionary algorithms is promising at fine-grained                          onto one point, which means min(f) ¼ max(f). In this case,
search.                                                                                  the value {1À[min(f)Àfi(r, n)]/[min(f)Àmax(f)]} of arbitrary
    In other way, the PSO-CA and PSO-CO methods cannot ensure                            particle is set to 1. The best particle presents the smaller value
the optimization’s ergodicity entirely in phase space, because it is                     than the other particles in swarm for the expression described
random in this method. An alternative is the implementation of a                         by {1À[min(f)Àfi(r, n)]/[min(f)Àmax(f)]}.
PSO approach based on chaotic sequences. Chaos is a kind of                                  Based on the pseudocode of PSO-GC, it can be noted that the
characteristic that often exists in nonlinear dynamic systems,                           good particles in swarm tend to perform exploitation to refine
which has been studied and applied in many fields, such as                                results by local search using Gaussian distribution, while bad
engineering and mathematics [47–49]. Chaotic motion is a kind of                         particles tend to perform large modification to explore space with
highly unstable motion of deterministic systems in finite phase                           large step using a chaotic sequence since the essence of keeping
space and it is especially used as a component of effective                              diversity is to prevent or perturb the swarm. In other words,
optimization algorithms [32–35,50–54] relying on its university,                         PSO-GC can provide a way to maintain population diversity and to
randomcity and sensitivity dependence on the initial conditions of                       sustain good convergence capacity.
chaotic mapping.
    Inspired by the Gaussian probability distribution and chaotic                        3.4. Constraints handling with evaluated PSO approaches
motion ideas, this paper provides a novel combined optimization
method, which introduces chaotic mapping into PSO based on                                   A key factor in the application of PSO algorithms to solve the
Henon map [55] so as to improve the global convergence and also                          reliability–redundancy optimization problems is how the algo-
a Gaussian probability distribution to improve the local conver-                         rithm handles the constraints relating to the problem. Over the
gence. It is a promising to achieve trade-off between exploration                        last few decades, several methods have been proposed to handle
and exploitation and, moreover, effective way of dealing with the                        constraints in evolutionary algorithms [56–59]. These methods
updating of velocity in PSO-CO.                                                          can be grouped into four categories: methods that preserve the
      ´                                                ´
    Henon’s map is a simplified version of the Poincare map of the                        feasibility of solutions, penalty-based methods, methods that
Lorenz system [47]. The Henon map [55] is given by                                       clearly distinguish between feasible and unfeasible solutions and
                                                                                         hybrid methods.
y1 ðkÞ ¼ 1 À aðy1 ðk À 1ÞÞ2 þ y2 ðk À 1Þ                                      (16)           When PSO algorithms are used for constrained optimization
                                                                                         problems, it is common to handle constraints using concepts of
y2 ðkÞ ¼ by1 ðk À 1Þ                                                          (17)       penalty functions (which penalize unfeasible solutions), i.e., one
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attempt to solve an unconstrained problem in the search space S                          approaches uses the variables vectors n and r, where the
using a modified objective function f (we are maximizing function                         boundaries are given in Section 2. In this paper, during the
for the overall system reliability in this paper) such as                                evolution process, the integer variables ni are treated as real
              (                                                                          variables, and in evaluating the objective function, the real values
                 f ðxi Þ;                if xi 2 F
max f ðxi Þ ¼                                               (20)                         are transformed to the nearest integer values.
                 f ðxi Þ À penaltyðxi Þ; otherwise:                                         Each optimization method was implemented in Matlab
                                                                                         (MathWorks). All the programs were run on a 3.2 GHz Pentium
where xi are solutions obtained by PSO approaches, i.e. the
                                                                                         IV processor with 2 GB of Random Access Memory (RAM). In order
position of the ith particle of population; penalty(xi) is zero and no
                                                                                         to eliminate stochastic discrepancy, in each case study, 50
constraint is violated; otherwise it is positive. The penalty
                                                                                         independent runs were made for each of the optimization
function is usually based on a distance measured to the
                                                                                         methods involving 50 different initial trial solutions for each
nearest solution in the feasible region F or to the effort to repair
                                                                                         optimization method.
the solution.
                                                                                            For each testing problem, the parameters of the PSO-CA are set
    In this work, the penalty-based method proposed in [38]
                                                                                         as follows: c1 ¼ c2 ¼ 2.05, o linearly decreases from o1 ¼ 0.9 to
(details in Section 2.3 of [38]) was used in all PSO designs for
infeasible solutions (constraint violation). The adopted approach
                                                                                         o2 ¼ 0.4. Moreover, the maximum velocity vmax at each particle
                                                                                         are set as 20% of the search space of particles. The parameters of
is used to convert a constrained problem to an unconstrained one
                                                                                         the PSO-CO are set as follows: c1 ¼ c2 ¼ 2.05. In, PSO-GC, it
by modifying the search space. A penalty value is defined to take
                                                                                         adopted c1 ¼ c2 ¼ 2.05 and x ¼ 0.2.
the constrained violation into account. In method proposed in
[38], the terms l are subtracted (maximization problem) of
objective function f(r, n) if g(r, n)4l.                                                 4.2. Results and discussion for the complex (bridge) system

                                                                                             The swarm size is set to 90 and the stopping criterion tmax was
4. Computation results
                                                                                         150 iterations in all PSO algorithms for the complex (bridge)
                                                                                         system. In other words, all PSO algorithms adopt 13,500 cost
   In this section, it turns to the description and analysis of the
                                                                                         function evaluations in each run.
results obtained by the optimization tests.
                                                                                             Simulation results of all PSO schemes for the complex
                                                                                         (bridge) system are listed in Table 3. In terms of mean and
4.1. Parameter settings                                                                  best (gbest of all runs) f(r, n) results, the PSO-GC approach
                                                                                         outperforms PSO-CA and PSO-CO. The best results obtained for
   Two examples described in Sections 2.1 and 2.2 are employed.                          the complex (bridge) system using PSO-GC was 0.99988957, as
Each particle of swarm in PSO-CA, PSO-CO, and PSO-GC                                     shown in Table 4.
                                                                                             Table 5 compared the results obtained in this paper for
                                                                                         the complex (bridge) system with those of other studies reported
Table 3                                                                                  in the literature. Note that in complex (bridge) system case,
Convergence results of f(r, n) (50 runs) for the complex (bridge) system using PSO       the best result reported here using PSO-GC was comparatively
algorithms                                                                               lower than recent studies presented in the recent literature.
                                                                                         By using Maximum Possible Improvement (MPI) index given
Optimization       f(r, n)
method                                                                                   by MPI (%) ¼ [Rs(PSO-GC)ÀRs (other)]/[1ÀRs (other)] and pre-
                   Maximum         Minimum           Mean            Standard            sented in Table 5, it shows the PSO-GC made improvements in
                   (best)          (worst)                           deviation           terms of reliability found by other optimization approaches in
PSO-CA             0.99988891      0.99944273        0.99981782      0.00002290
PSO-CO             0.99988946      0.99980434        0.99988464      0.00000377
PSO-GC             0.99988957      0.99987750        0.99988594      0.00000069

                                                                                         Table 5
                                                                                         Comparison of best result for the complex (bridge) system with other results
                                                                                         presented in literature
Table 4
Best result (50 runs) for the complex (bridge) system                                    Parameter          Hikita et al.     Hsieh et al.   Chen [19]     This paper
                                                                                                            [40]              [41]                         (using PSO-
Parameter               PSO-CA              PSO-CO                PSO-GC                                                                                   GC)

f(r, n)                  0.99988891         0.99988946            0.99988957             f(r, n)             0.9997894         0.99987916     0.99988921   0.99988957
n1                       3                  3                     3                      n1                  3                 3              3            3
n2                       3                  3                     3                      n2                  3                 3              3            3
n3                       3                  2                     2                      n3                  2                 3              3            2
n4                       3                  4                     4                      n4                  3                 3              3            4
n5                       1                  1                     1                      n5                  2                 1              1            1
r1                       0.815457           0.830911              0.826678               r1                  0.814483          0.814090       0.812485     0.826678
r2                       0.872681           0.857333              0.857172               r2                  0.821383          0.864614       0.867661     0.857172
r3                       0.856412           0.912827              0.914629               r3                  0.896151          0.890291       0.861221     0.914629
r4                       0.703959           0.647359              0.648918               r4                  0.713091          0.701190       0.713852     0.648918
r5                       0.768451           0.697767              0.715291               r5                  0.814091          0.734731       0.756699     0.715290
MPI (%)a                 0.594              0.099                 –                      MPI (%)a           47.564             8.615          0.325        –
Slack (g1)b             18                  5                     5                      Slack (g1)b        18                18             19            5
Slack (g2)b              0.000778           0.030411              0.000339               Slack (g2)b         1.854075          0.376347       0.001494     0.000339
Slack (g3)b              4.264769           1.560466              1.560466               Slack (g3)b         4.264770          4.264770       4.264770     1.560466

    a                                                                                        a
        MPI (%) ¼ [Rs(PSO-GC)ÀRs(other)]/[1ÀRs(other)].                                          MPI (%) ¼ [Rs(PSO-GC)ÀRs(other)]/[1ÀRs(other)].
    b                                                                                        b
        Slack is the unused resources.                                                           Slack is the unused resources.
                                                                         ARTICLE IN PRESS

836                                                     L.S. Coelho / Reliability Engineering and System Safety 94 (2009) 830–837

Table 6
Convergence results of f(r, n) (50 runs) for the overspeed protection system for a gas turbine using PSO algorithms

Optimization method                     f(r, n)

                                        Maximum (best)                        Minimum (worst)                     Mean                                 Standard deviation

PSO-CA                                  0.999943                              0.999733                            0.999901                             0.000008
PSO-CO                                  0.999944                              0.999746                            0.999878                             0.000010
PSO-GC                                  0.999953                              0.999638                            0.999907                             0.000011

Table 7                                                                                        of the results by PSO approaches in 50 independent runs is also
Best result (50 runs) for the overspeed protection system for a gas turbine                    very small.
                                                                                                  The best results obtained for the overspeed protection system
Parameter                  PSO-CA                      PSO-CO                PSO-GC
                                                                                               using PSO-GC was 0.999953, as shown in Table 7. From Table 8, it
f(r, n)                     0.999943                    0.999944              0.999953         can conclude that a best solution found by PSO-GC for the
n1                          5                           5                     5                overspeed protection system has a slight advantage over the other
n2                          6                           5                     6
                                                                                               solvers reported in the literature.
n3                          4                           4                     4
n4                          5                           6                     5
r1                          0.897525                    0.885764              0.902231
                                                                                               5. Conclusion and further research
r2                          0.857242                    0.878177              0.856325
r3                          0.935091                    0.955408              0.948145
r4                          0.885802                    0.865562              0.883156            PSO is one of the most recent stochastic methods developed for
MPI (%)a                   17.0                        16.071                 –                solving optimization problems. The current study investigates the
Slack (g1)b                55                          55                    55
                                                                                               combination of Gaussian distribution and chaotic sequence in PSO
Slack (g2)b                15.467870                    0.173677              0.975465
Slack (g3)b                24.801883                   15.363463             24.801882         design, and the performance of the proposed PSO-GC is compared
                                                                                               with PSO-CA, PSO-CO, and other results presented in literature for
          MPI (%) ¼ [Rs(PSO-GC)ÀRs(other)]/[1ÀRs(other)].                                      two cases studies including discrete and continuous decision
          Slack is the unused resources.                                                       variables in reliability engineering field.
                                                                                                  Simulation results presented in Tables 3–8 reveal that PSO-GC
Table 8
                                                                                               scheme presented improvements upon the effectiveness and
Comparison of best result for the overspeed protection system for a gas turbine                efficiency and also obtained a good trade-off between exploitation
with other results presented in literature                                                     in Gaussian distribution and exploration by chaotic sequences.
                                                                                                  The PSO-GC was demonstrated to be a promising and viable
Parameter            Dhingra [36]      Yokota et al.      Chen [38]        This paper
                                                                                               tool to solve reliability–redundancy optimization problems.
                                       [37]                                (using PSO-
                                                                           GC)                 Furthermore, a number of improvements and extensions are
                                                                                               currently being investigated by the author related to benchmark
f(r, n)               0.99961            0.999468          0.999942         0.999953           problems in reliability engineering field.
n1                    6                  3                 5                5
n2                    6                  6                 5                6
n3                    3                  3                 5                4
n4                    5                  5                 5                5
r1                    0.81604            0.965593          0.903800         0.902231
r2                    0.80309            0.760592          0.874992         0.856325              This work was supported by the National Council of Scientific
r3                    0.98364            0.972646          0.919898         0.948145           and Technologic Development of Brazil—CNPq—under Grant
r4                    0.80373            0.804660          0.890609         0.883156
MPI (%)a             87.948             91.165            18.965            –
Slack (g1)b          65                 92                50               55
Slack (g2)b           0.064            À70.733576          0.002152         0.975465
Slack (g3)b           4.348            127.583189         28.803701        24.801882

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