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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 ðtÞ 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. kÃ ; lÃ and k Ã (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 : 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. 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