Fuzzy global optimization of complex system reliability ... - IEEE Xplore

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 8, NO. 3, JUNE 2000

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Fuzzy Global Optimization of Complex System Reliability V. Ravi, P. J. Reddy, and Hans-Jürgen Zimmermann

Abstract—In this paper, the problem of optimizing the reliability of complex systems has been modeled as a fuzzy multi-objective optimization problem where apart from the system reliability, system cost, weight, and volume are all considered as fuzzy goals/objectives. Three different kinds of optimization problems: 1) reliability optimization of a complex system with constraints on cost and weight; 2) optimal redundancy allocation in a multistage mixed system with constraints on cost and weight; and 3) optimal reliability allocation in a multistage mixed system with constraints on cost, weight, and volume have been solved. Four numerical examples have been solved to demonstrate the effectiveness of the present methodology. The influence of various kinds of aggregators such as: 1) product operator; 2) min operator; 3) the arithmetic mean operator; 4) fuzzy and 5) a convex combination of the min and the max operators; and 6) compensatory and ( -operator) on the quality of the solutions is also studied. The inefficiency of the noncompensatory min operator has been demonstrated. One of the well-known global optimization meta-heuristics—threshold accepting—has been invoked to take care of the optimization part of the model because it is a variant of the simulated annealing algorithm and, hence, can tackle the nonconvex optimization problems very well, unlike the modified steepest-ascent method [6], [8]. Linear membership functions have been assumed for the all the goals/objectives. A software has been developed to implement the above model. The results are encouraging because in the case of some problems investigated here they coincided with those yielded in the crisp single-objective environment. Also, fuzzy optimization techniques can be used as viable and useful alternatives to the goal programming approaches for this kind of problems posed in an ill-structured environment. Index Terms—Complex systems, multi-objective optimization, optimal redundancy allocation, Pareto optimal solution, reliability, fuzzy optimization, threshold accepting algorithm.

I. INTRODUCTION

F

UZZY set theory has been applied to reliability theory/engineering with great success in the past two decades. The incorporation of the fuzzy set theoretic concepts into the multidisciplinary area of reliability theory has been done by modifying the basic assumptions underlying the definition of reliability of a component or system. Conventional reliability theory is based on, among others, the following two fundamental assumptions [1]. 1) Binary state assumption: the system can only be in either of the two crisp states viz. fully functioning or fully failed. Manuscript received July 8, 1998; revised February 28, 2000. V. Ravi and H.-J. Zimmermann are with the Lehrstuhl fuer Unternehmensforschung, RWTH, D-52056, Aachen, Germany. P. J. Reddy is with the Computer Center, Indian Institute of Chemical Technology, Hyderabad 500 007 (AP), India. Publisher Item Identifier S 1063-6706(00)05111-0.

2) Probability assumption: the system failure behavior is fully characterized by the probability measures. Although, these two assumptions are often valid, they are not reasonable in a large variety of cases. This called for the incorporation of the concepts of fuzzy set theory into these assumptions. Thus, 1) and 2) are modified as follows [1]. 1 ) Fuzzy state assumption: the system success and failure are characterized by fuzzy states. At any given time the system can be viewed as being in one of the two states to some extent. Thus, system failure is not defined in a binary way, but in a fuzzy way. 2 ) Possibility assumption: the system failure behavior is fully characterized by the possibility measures. For the sake of simplicity, the conventional reliability theory is called “PROBIST reliability theory,” since it is based on assumptions 1) and 2). When 1) is replaced by 1 ) and/or 2) is replaced by 2 ), the resultant is called fuzzy reliability theory. Thus fuzzy reliability theory manifests itself in three different forms viz. PROFUST reliability theory, POSBIST reliability theory, and POSFUST reliability theory [1]. Fuzzy reliability theory in its various forms found applications, especially in fault tree analysis [2], [3], reliability optimization [4] and risk analysis [5]. However, fuzzy mathematical techniques can be successfully applied to conventional reliability theory, without taking recourse to any form of the fuzzy reliability theories. This was first demonstrated by Park [6], who applied fuzzy optimization techniques to the problem of reliability apportionment for a simple two-component series system. Later, Dhingra [7] and Rao and Dhingra [8] worked on reliability and redundancy apportionment for a four-stage and a five-stage overspeed protection systems, respectively, using crisp and fuzzy multi-objective optimization approaches. To summarize, all the three works [6]–[8] use the noncompensatory min operator as the aggregator. Park [6] used a linear membership function for the goals, whereas Dhingra [7] and Rao and Dhingra [8] used linear membership functions for the constraints and nonlinear membership function for the reliability. All the three works do not make use of any of the random search based global optimization methods. Thus, the whole family of meta-heuristics that can efficiently solve highly nonlinear nonconvex mixed-integer optimization problems, is overlooked. The primary goal of the present study is to stress the deficiencies of the earlier works [6]–[8] and suggest ways to extend them: 1) by employing various aggregators instead of the noncompensatory min operator and 2) by focusing on the need for the application of a global optimization technique to solve problems involving complex systems. Also, the methodology

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presented here will be illustrated in the case of multistage mixed systems comprising many stages and other types of complex systems such as the bridge network system, for instance. In what follows, Section II presents the problem description. In Section III describes the formulation of the problem into a fuzzy optimization model, definitions of the membership functions for the fuzzy goals and the crisp equivalent formulation of the fuzzy optimization problem. Sections IV and V present results, discussion and conclusions respectively. The numerical examples in their original form are described in detail in the Appendix.

Fig. 1.

An example of a complex system.

A. Notation [reliability, cost] of the system; [reliability , cost] of the th component; Reliability of the th stage; number of constraints; number of decision variables (number of components in a complex system or the number of stages in a multi stage mixed system); lower bound on the reliability of the th component; lower bound on the system reliability; constants associated with cost function of the th component; number of the redundancies of the th component; variables defining the volume, weight and product of volume and weight, respectively, of the th component [13]; constants associated with the volume, weight and product of volume and weight respectively of the th component [13]; th constraint; [volume, weight, Product of volume and weight] of the system [13]; membership function of the fuzzy set “decision”; membership functions of the system (reliability, cost, weight, volume and the product of weight and volume); membership functions of the fuzzy decision, th fuzzy constraint, and th fuzzy goal. II. PROBLEM DESCRIPTION In the parlance of reliability engineering, a complex system consists of several components connected to one another neither purely serially nor purely parallely. Two such complex systems

Fig. 2. A mixed system having at each stage.

Fig. 3.

N -stages in series and components in parallel

Complex bridge network system.

and two multistage mixed systems studied here are depicted in Figs. 1–3. In this paper, basically three types of problems are studied. Model Type 1: (Cost Optimization in Complex systems): Minimize subject to

and

where, is the system cost, and are, respectively, the lower bounds on the reliabilities of the th component and system. Model Type 2: (Optimal Redundancy Allocation in Multistage Mixed Systems): Find the optimal number of components , which maximize system reliability given by

subject to

RAVI et al.: FUZZY GLOBAL OPTIMZIATION OF COMPLEX SYSTEM RELIABILITY

Model Type 3: (Multi-Objective Optimization in Multistage Mixed Systems): Maximize Minimize subject to

Minimize Minimize and Minimize and

where, is the system cost, is the system reliability, is the system weight, is the system volume and is the and product of the system weight and system volume, are respectively the lower bounds on the reliabilities of the th component and system. III. FUZZY OPTIMIZATION MODEL In this paper, the multiple conflicting objectives inherently present in the three types of models stated in the previous section have been given equal weights and, thus, all of them have been modeled as fuzzy multi-objective optimization models. In order to accomplish this, all the conflicting goals/objectives have been fuzzyfied and the well-known Bellman and Zadeh [9] model has been invoked, which is briefly described as follows. Bellman and Zadeh [9], while formulating their famous model assumed that the objectives as well as constraints in an ill-structured situation could be represented by fuzzy sets. A decision is then defined as the intersection of all the fuzzy sets represented by objectives and the constraints and is represented by its membership function as follows [10]:

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Problem 3: “Maximize system reliability as close as possible to 1 with approximate system cost of 5.019 93 (cost units).” Problem 4: “Maximize system reliability as close as possible to 1 with approximate system cost of 450 (cost units), approximate system weight of 250 (weight units), approximate system volume of 65 (volume units) and approximate value of the product of system weight and system volume of 12000 (units)." Though earlier workers [6]–[8] followed a similar approach as presented here, some important issues were left out there, which are addressed in this paper. First, empirical research in fuzzy decision-making (optimization) showed that the min operator is noncompensatory in nature and, thus, other aggregators were suggested, which are compensatory [12]. The compensatory and and fuzzy and are examples in this class. Second, use of these compensatory aggregators for some membership functions makes the fuzzy set of “decision,” in the Bellman and Zadeh [9] sense nonconvex. Hence, to solve the crisp equivalent of the original fuzzy optimization problem, a robust global optimization technique is needed. The authors in this paper prefer using the threshold accepting algorithm, owing to the encouraging results presented in [11] and their own numerical experience over a range of test problems taken from the literature. The modified steepest-ascent method [6], [8] cannot handle the optimization problems arising from the use of the compensatory aggregators. Linear membership functions defined for all the fuzzy goals in all the problems are described as follows. Problem 1: Case (i) if

where, represents the fuzzy decision, represents the th and represents the th fuzzy fuzzy constraint , is the membership function goal is the membership function of th of the decision and is the membership function of the th constraint and goal and is an appropriate “aggregator” or connective. Within the framework of Bellman and Zadeh [9] model, and following Zimmermann [10] the optimal solution is obtained by maxi, subject to all the crisp constraints in the model. mizing To solve it, one of the well-known global optimization metaheuristics viz. threshold accepting algorithm [11] is invoked. This algorithm was empirically proved to be superior to the traditional simulated annealing in the case of large-scale combinatorial optimization problems such the travelling salesman problem [11]. A. Formulation of the Numerical Examples as Fuzzy Optimization Problems Fuzzy version of all the examples are given as follows. Problem 1:: Case (i) “Maximize system reliability as close as possible to 1 with approximate system cost of 641.8 (cost units).” Case (ii) “Maximize system reliability as close as possible to 1 with approximate system cost of 390.57 (cost units).” Problem 2: “Maximize system reliability above 0.95 with approximate system cost of 400 (cost units) and approximate system weight of 414 (weight units).”

if if if if if Case (ii) if if if if if if Problem 2: if if if if if if if if if

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TABLE I(a) RESULTS OF PROBLEM 1-(i)

IV. RESULTS AND DISCUSSION

Problem 3: if if if if if if Problem 4: if if if if if if if if if if if if if if The crisp equivalent of the fuzzy multi-objective optimization Problem 4, which implies different utilities for the compromise solution, is as follows. The crisp equivalents for other problems, which can be formulated the same way, are omitted here for the sake of brevity Maximize subject to all the crisp constraints given in the example 4 in the Appendix. In this formulation, it is obvious that is the aggregator connecting the membership functions of all the goals.

Six types of aggregators viz. (i) product operator; (ii) min operator; (iii) the arithmetic mean operator; (iv) fuzzy and (v) a convex combination of the min and the max operators; and (vi) compensatory and ( -operator) have been employed and their influence on the results is investigated. The results for the Problems 1–4 are summarized in Tables I–IV, respectively. Each table contains six cases, where case (i) corresponds to the product operator; case; (ii) corresponds to the min operator case; (iii) corresponds to the arithmetic mean operator; case; (iv) corresponds to the fuzzy and; case(v) corresponds to a convex combination of the min and the max operators; and, finally, case (vi) corresponds to the compensatory and ( operator). The parameter governing the threshold accepting algorithm namely eps, which specifies by how much factor one should reduce the threshold during the course of the algorithm, has been chosen to be 0.01 throughout the study for all the problems and in cases. A computer program by name FUZ_OPT_REL has been developed in C language (using Visual C++ 5.0 compiler) to implement the model. It has been executed on a Pentium machine with 133-MHz clock speed in the Windows NT 4.0 environment. Results of Problem 1-(i) (see Table I(a)), indicate that the cases (i) and (iii) use less computational time and provide one Pareto optimum each. Case (ii) also produced one Pareto optimum, but took more CPU time. Case (iv) irrespective of the value of , gives rise to identical Pareto optimal solution. Case (v)-(a) and (v)-(b) include all the Pareto optimal solutions, but consume more CPU time. Case (vi) leads to different Pareto optimal solutions depending on the value of . The value of the obtained is not relevant because it is determined by the aggregating operator that was used in the modeling. It is only important that it reaches the maximum in a give case. As regards the cost objective, the solution provided by case (v)-(a) turned out to be much better than the one reported by Ravi et al. [15], who solved the problem in a “crisp single-objective optimization” environment. Also, the fuzzy and operator [case (iv)] yielded efficient solution compared to the min operator [case (ii)].


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TABLE I(b) RESULTS OF PROBLEM 1-(ii)

TABLE II RESULTS OF PROBLEM 2

As regards Problem 1-(ii) (see Table I(b)), the cases(i) and (ii) use less computational time but provide one Pareto optimum each. Cases (ii) and (iv)-(b) produced the same Pareto optimal solution. Again, cases (iii) and (iv)-(a) produced the same solution which is better than the crisp solution of Ravi et al. [15], but took more CPU time. However, case (v)-(a) outperformed both

of them. Cases (v)-(a) and (v)-(b) include all the Pareto optimal solutions, but consume more CPU time. Case (vi) leads to different Pareto optimal solutions depending on the value of . The results of Problem 2 (Table II) show that the cases (i) and (vi)-(a) provide the same Pareto optimal solution. Similarly, cases(ii) and (iv)-(b) produced the same Pareto solution.

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TABLE III RESULTS OF PROBLEM 3

TABLE IV RESULTS OF PROBLEM 4

Again, cases (iii) and (iv)-(a) produced the same Pareto optimal solution. Further, cases(v)-(a) and (vi)-(b) resulted in the same Pareto optimal solutions. Cases (v)-(b) and (vi)-(b) gave rise to two different Pareto optimal solutions. Regarding Problem 3 (see Table III) Cases (i) and (ii) provided different Pareto optimal solutions, although, case (ii) came very close to the crisp solution reported by Ravi et al. [15]. Cases (iii), (iv)-(a), (v)-(b) and (vi)-(b) all provided same Pareto optimal solution, whereas cases (iv)-(b) and (v)-(a) gave rise to very close Pareto optimal solutions. All cases consumed considerable amount of CPU time. Results of Problem 4 (see Table IV) show that all cases produced different Pareto optimal solutions. However, cases (ii) and (v)-(b) produced same Pareto optimal solutions. All cases produced efficient solutions compared to the one reported by

Sakawa [13]. The convex combination of min and max operators [case (v)-(a) and (b)) outperformed the min operator (case (ii)] by producing an efficient solution. V. CONCLUSION The problem of optimizing the reliability of complex systems has been modeled as a fuzzy multi-objective optimization problem, where the reliability, cost, weight, and volume of the system are considered as fuzzy objectives. Four optimization problems involving different kinds of complex systems and multistage mixed systems have been successfully solved using the model. Threshold accepting, a global optimization meta-heuristics has been used in the optimization part of the model owing to its ability in solving the nonconvex problems. The influence


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TABLE V THE CONSTANTS FOR PROBLEM 2

.

of various kinds of aggregators such as: (i) product operator; (ii) min operator; (iii) the arithmetic mean operator; (iv) fuzzy and (v) a convex combination of the min and the max operators; and (vi) compensatory and ( -operator) on the solutions is also studied mainly to see their advantage over the noncompensatory min operator. It was found that in some problems illustrated here, other aggregators such as fuzzy and and convex combination of min and max operators yielded efficient solutions compared to the min operator. Linear membership functions have been assumed for the all the goals/objectives. The results are encouraging and they indicate that fuzzy goal programming techniques can be employed as viable alternatives to the traditional goal programming approaches to the kind of problems solved in this paper.

APPENDIX

Problem 2: 15-Stage Mixed System [16]: Find the optimal number of components ; which maximize system reliability subject to

and The constants for the 15-stage problem are shown in Table V at the top of the page. The crisp global maximum of the system reliability (0.945) was reported by Ravi et al. [15] and Luus [16]. The schematic for this problem is in Fig. 2. Problem 3: Here a bridge network system as shown in Fig. 3 has been considered, each having a component reliability, , . The system reliability is given by [15]

Problem 1: Life Support System in a Space Capsule: The schematic is in Fig. 1. The system reliability and the problem formulation are as follows [14]

Minimize

subject to , and where, is the system cost, and are, respectively, the lower bounds on the reliabilities of the th component and system. Two different forms of system cost functions are considered in the following cases [14]. Case (i) pp. 187–191 , , , , and for . The crisp global minimum of the system cost of 641.83 was obtained by Ravi et al. [15]. Case (ii) where

pp. 146–152 , , , , , and for . Ravi et al. [13] report 390.57 as the crisp global minimum of system cost.

where

The Problem is to find the decision variables which minimize

,

subject to , and , where and , for . The crisp global minimum of the system cost of 5.019 93 was reported by Ravi et al. [13]. Problem 4: This problem is taken from Sakawa [13]. The block diagram for this problem is in Fig. 2. This involves multistage mixed system, where the problem is to determine (alloof four compocate) the optimal reliabilities , nents whose redundancies are specified in order to achieve the following five goals: Maximize minimize Subject to

minimize minimize and minimize

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where

and

and

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[5] K. B. Misra and G. G. Weber, “Use of fuzzy set theory for level-1 studies in probabilistic risk assessment,” Fuzzy Sets Syst., vol. 37, pp. 139–160, 1990. [6] K. S. Park, “Fuzzy apportionment of systems reliability,” IEEE Trans. Rel., vol. 36, pp. 129–132, 1987. [7] A. K. Dhingra, “Optimal apportionment of reliability and redundancy in series systems under multiple objectives,” IEEE Trans. Rel., vol. 41, pp. 576–582, 1992. [8] S. S. Rao and A. K. Dhingra, “Reliability and redundancy apportionment using crisp and fuzzy multi-objective optimization approaches,” Reliability Eng. Syst. Safety, vol. 37, pp. 253–261, 1992. [9] R. E. Bellman and L. Z. Zadeh, “Decision making in a fuzzy environment,” Management Sci., vol. 17, pp. B141–164, 1970. [10] H.-J. Zimmermann, Fuzzy Set Theory and Applications, 2nd ed. Boston, MA: Kluwer, 1991. [11] G. Dueck and T. Scheuer, “Threshold Accepting: A general-purpose optimization algorithm appearing superior to Simulated Annealing,” J. Computat. Phys., vol. 90, pp. 161–175, 1990. [12] H.-J. Zimmermann and P. Zysno, “Latent connectives in human decision making,” Fuzzy Sets Syst., vol. 4, pp. 37–51, 1980. [13] M. Sakawa, “Interactive multi-objective optimization by the sequential proxy optimization technique,” IEEE Trans. Rel., vol. 31, pp. 461–464, 1982. [14] F. A. Tillman, C.-L. Hwang, and W. Kuo, Optimization of System Reliability. New York: Marcel Dekker, 1980. [15] V. Ravi, B. S. N. Murty, and P. J. Reddy, “Nonequilibrium simulated annealing-algorithm applied to reliability optimization of complex systems,” IEEE Trans. Rel., vol. 46, pp. 233–239, 1997. [16] R. Luus, “Optimization of system reliability by a new nonlinear integer programming procedure,” IEEE Trans. Rel., vol. 24, pp. 14–16, 1975.

V. Ravi, biography and photograph not available at the time of publication.

P. J. Reddy, biography and photograph not available at the time of publication.

Hans-Jürgen Zimmermann, biography and photograph not available at the time of publication.