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Computers ind. Engng VoL 31, No. 1/'2, pp. 237 - 240, 1996 Copyright O 1996 E i t h e r Science Lid

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Optimal Design of System Reliability Using Interval Programming and Genetic Algorithms

Mitsuo Gen Runwei Cheng

Department of Industrial & Systems Engineering Ash~kaga Institute of Technology, Ashlkaga 326, Japan

A b s t r a c t : In this paper, we examine the op- timal design problem of system reliability with uncertain coefficients and formulate it as an in- terval programming model. Genetic algorithms are applied to the problem. The basic idea of the proposed method is first to transform the interval programming model into an equivalent bicriteria programming model and then to find the Pareto solutions of the bicriteria program- ming problem using genetic algorithms. Nu- merical examples are given to demonstrate the efficiency of the proposed approach.

K e y w o r d s : Reliability optimization, interval programming, bicriteria programming, and ge- netic algorithm.

1 I n t r o d u c t i o n

Traditional formulations on reliability opti- mization problems have assumed that the co- efficients of models are known as fixed quan- tities and reIiability optimization is treated as deterministic optimization problems. Because that the optimal design of system reliability is resolved in the same stage of overs]] system de- sign, model coefficients are usually uncertainty and imprecision during design phase. It is some- times very difficult for us to determine the pre- cise values for them.

In past three decades, three major ap- proaches have been proposed for decision mak- ing problems under uncertainties:

• stochastic programming , fuzzy programming • interval programming

In stochastic programming approach, the coef- ficients of mathematical model are viewed as random variable and their probability distribu- tions are assumed as to be known. In fuzzy pro- gramming approach, the coefficients are viewed

237

as fuzzy sets and their membership functions are assumed as to be known. However, in many application cases of real-world problems, it is not so easy for decision makers to specify either probability distributions or membership func- tions; on the contrary, such uncertainty can be easily represented as an interval of confidence. This is the motivation to developing interval arithmetic and interval programming [1].

In this paper, we have ,,Tamlned the optimal design problem of system reliability with uncer- tain coefficients. These coefficients are given as the intervals of confidence and the problem is formulated as an interval programming model. Genetic algorithms are applied to to the prob- lem. An adaptive hyperplane approach in crite- ria space is proposed to construct fitness func- tion so as to force genetic search to exploit the set of nondomlnated points. Adaptive penalty technique is used to guide genetic search to ap- proach the Pareto frontier from both side of fen- sibh and infeasible regions.

2 Interval Programming

An interval is defined as an ordered pair of real numbers as follows:

A -- [al~,a R]

= { z l a L < z < a R ; z E R }

where a L and a R are the left hound and right bound of interval A, respectively. The interval also can be defined as follows:

A = < a o, a w >

= { z l a c -a W<z<a c+aW;z~R} where a c and a W are the center and width of interval A, respectively. They are calculated as follows:

= ~(a R + a C a L ) 2

238 19th International Conference on Computers and Industrial Engineering

a w = ~(a/z- a/;)

The details on interval arithmetic can be found in [2].

The interval programming technique is de- veloped to deal with the mathematical pro- gramming problem with interval coefficients. Ishibuchi and Tanaka proposed a solution pro- cedure to interval programming problem, the essential of which is to transform the problem into an equivalent bicriteria programming prob- lem [3]. There are two key steps when per- forming such transformation: (1) using the def- iuition of the degree of inequality-holding-true for two intervals, transform interval constraints into equivalent crisp constraints, (2) using the definition of the order relation between inter- vals, transform interval objective into equiva- lent crisp two objectives.

Consider the following problem given by Fyife at al. [4]. Assume that system consists of N subsystem. Associated with each subsystem there exist several choices of design alternatives. The problem is to determine which design aiter- native to be selected and how many redundant units to be used in order to achieve the great- est reliability while keeping the total system cost and weight within the allowable amounts. It is formulated as integer nonlinear program- ming problem. For the uncertain circumstance, the parameters are replaced by interval num- bers and the problem can be formulated as an interval programming model:

N

m ~ R ( , , , , - ) = I I (1 - (1 - ~ ( ~ / ) ) " ) (1) i=1

N s.t. Gx ( - , , , , ) = ~ C ~ ( , ~ , ) , ~ _< c (2)

i=1 N

G2(,-.,.-.) = ~ W / ( a ~ ) ~ _< w (3) i=1

1 <_ a i ~_ ~i, integer, i = 1, ...,_N (4)

1 < mi <_ ul, integer, i = 1 , . . . , N (5)

where N is the total number of subsystems, mi the number of redundant units for subsystem i, ai the alternative design for subsystem i, ~i the upper bound of ai, ul the upper bound of mi, Ri(ai) = [rL(ai) ,r~(ai)] interval relia- bility, Ci(ai) = [cL(ai), c~(ai)] interval coeffi- cient, Wi(al) = [to~(al), toiR(ai)] interval coef- ficient, C = [c L, c/z] interval overail constraint on the system cost, and W = [to/;, to/z] interval overall constraint on the system weight.

The problem then can be transformed into an equivalent crisp bicriteria programming prob- lem [5]:

N m a x z ( m , . ) L - : E dL( m/ ' a i ) (6)

i=1 N

m a x , ( , , , . , , ) c = ~ ~ ( , , , , , . , / ) (z) i=1 N

s . t . g a ( m , ~ ) : ~ c i ( a i ) m , < c (8) i=1 N

g 2 ( m , ~ ) = ~ t o i ( ~ ) ~ < to (9) i=1

1 _< m/ _< ui, integer, i = 1, . . . ,N (10)

1 _< al < ~9i, integer, i = 1, . . . ,N (11)

where the coefficients in constraints are deter- mined as follows:

d~(mi ,a i ) : ln(1 - (1 - rL(al) 'm)

d/ /Z( rn i , a / ) = h i ( 1 - (1 - r~(al) 'n')

c/(~/) = h~c,n(~/) + (1 - ha)c~(,,/) t o , ( - / ) = h , t o ~ ( ~ / ) + (1 - h~)to?(~/) c = (1 - ha)c/z + hlc L

w = (1 - h2)to R + h2to L

ha and h2 are the degree of inequality-holding- true for constraints (2) and (3) given by decision makers. The problem is characterized as

• combinatorial nature • nonlinear in both objectives and constraints • multiple objectives

This makes the problem much hard to solve in general. In past few years, multiple objective optimization problem has became a new target for genetic algorithm community. GAs' abil- ity to find global optima while being able to cope with any kinds of objective functions and any kinds of constraints has motivated an in- creasing number of research works on applying genetic algorithms to solve multiple objective optimization problems.

3 G e n e t i c A l g o r i t h m

Now we discuss how to solve the problem (6)- (11) with GA. A chromosome is defined as fol- lows"

V k = I rk1 Vk2 . . . VkN ]

= [ ( ~ k ~ , ~ ) ( ~ , ~ , ) - " ("k. , '~hN)]

19th International Conference on Computers and Industrial Engineering 239

where subscript k is the index of chromosome. Initial population is generated randomly within the range [1,~] for all m~i and the range [l,~i] for all a~ . Crossover is implemented with uni- form crossover operator. Mutation is performed as random perturbation within the permissive range of integer variables. Deterministic selec- tion is used, that is, delete all duplicate among parents and offspring, then sort them in de- scending order and select the first pop_size chro- mosomes as the new population [6].

There are two main tasks involved in evalu- ation phase: (1) how to handle infeasible chro- mosomes and (2) how to determine fitness val- ues of chromosomes according to bicriteria. Let Vl, be the k-th chromosome in current genera- tion, the evaluation function is defined as fol- lows:

eval(Vk) = w(mk, ak)p(mk, ah) (12)

which contains two terms: weighted-sum objec- tive and penalty. The weighted-sum objective term tries to give a selection pressure to force genetic search to exploit the set of Pareto solu- tions and the penalty term tries to force ge- netic search to approach to Pareto solutions from both side of feasible and infeasible regions.

Comparing with conventional genetic algo- rithms, a distinctive feature of the implemen- tation for bicriteria programming is that it is necessary to maintain a set of nondomin~ted points along with the evolutionary process. Let E denote the set of nondominated solutions ex- amined so far. Two special points in E in- terest us. One point contains the mATi~um of zL(mk, a#,) among others in E and an- other contains the maximum of zO(m#,,a~,). Denote these two points as (z~in, z~mgx) and (ZOmjx, Z~in) , respectively, where

=~,, = -~ i " {zC(mk,ak) lmk,, ' ,~ ~ E} = l, k, k E E }

L

L max{zL(mt, ak) I ml,,a:, e E} z ~ u t =

Then we can make a new objective function based on these two special points as follows:

where wl = z°m~x - z ~ and w, = Z~x - z ~ n. Figure 1 gives an illustrative explanation of the objective. The line formed with points (zOmm, z~m~x) and (z°mw z~n) divides the criteria

space into two half spaces: one containing pos- itive ideal solution and another containing neg- ative ideal solution. The feasible solution space F is correspondin~y divided into two parts: F - and F +. It is easy to verify that all Pareto solu- tions lies in F +. At each generation, Pareto set E is updated and the two special points may be renewed. It means that along with the evolu- tionary process, the line formed with this two points will move gradually towards to Pareto frontier. In the other words, this fitness func- tion gives such selection pressure to force ge- netic search to exploit the nondominated points in the criteria space.

(,c., ~..o

+~. ~ ~( r , , •

(~=,,. ~ )

Figure 1: mustrative explanation of new objec- tive

The penalty function is constructed as fol- low S:

where Abj(m~,ak) = msx{O, gj(mh, a h ) - = =

1,...,pop_size}. Abj(mk,at) is the value of violation for constraint 3" for the k-th chromo- some, Aj/~ ~x is the maximum of violation for constraint j among current population, and e is a small positive number used to avoid zero- division.

The penalty term can be viewed as s mea- sure of the degree of infeasihiUty for a chro- mosome. Chromosomes generated from either initial phase or reproduction phase may violate system constraints. The special measure is used to evaluate how far an infeasible chromosome separates from the feasible region. Usually op- timum occurs at the boundary between feasible and infeasible area. When we simply give a big and constant penalty to each infeasible one~ it will be rejected from evolutionary process and genetic search will approach to optimum only from feasible side. The adaptive penalty ap- proach can adjust the ratio of penalty adap- tively at each generation in order to make s

240 19th International Conference on Computers and Industrial Engineering

balance between the preservation of informa- tion and the pressure for infeasibility and avoid over-penalty [7].

4 N u m e r i c a l E x a m p l e

The numerical example is an extension of the problem in reference [4]. The system consists of 14 subsystems. Each subsystem has 3 or 4 alternative designs and the possible maYimum of redundant units for each subsystem is 6. The interval coefficients of the problem were given in [5].

The parameters for genetic algorithm was set as follows: population size 40, crossover and mutation ratio 0.4, maximum generation 2000, and hi = h2 = 0.5.

The Pareto solutions found by the proposed algorithm are depicted in Figure 2 and the corresponding intervals (interval objectives) are given in Figure 3. Prom the figure we can know that these intervals can not be compared with each other under the order relation ~_LC.

and a solution procedure based on interval pro- gramming and genetic algorithm is developed to solve the problem.

The proposed GA can be directly applied to other bicriteria programming problems and also can be generalized into general multiple objec- tive optlmi~ation problems. This is our interest of further studies.

A c k n o w l e d g m e n t

This work was supported in part by a research grant from the Ministry of Education, Science and Culture, Japanese Government: Grant-in- Aid for Scientific Research, the International Scientific Research Program (No. 07045032).

References

[1] Ishibuchi, H. and H. Tanaka, Multiobjec- tive programming in optimization of the in- terval objective function, European Journal of Operational Research, vol.48, pp.219-225, 1990.

I

e , v r ~ 0.9~5 t r e e t g q s ~ t ~ t ~ 8 #

[2]

[3]

Hansen, E., Global Optimization Using In. terval Analysis, Marcel dekker Inc., New York, 1992.

Ishibuchi, H. and H. Tanaka, Formulation and analysis of linear prograraming problem with interval coefficients, Journal of Japan Industrial Management Association, vo1.40, no.5, pp.320-329, 1989, in Japanese.

Figure 2: Pareto solutions in criteria space

e . ~ ~ o.m ~ e . ~ o : m o.,,ns

um.,, , ,n , , , , b ) , . ~ , , . o re

Figure 3: Interval objective values

[4] Fyfl'e, D., W. Hines, and N. Lee, System re- liability allocation and a computational al- gorithm, IEEE Transactions on Reliability, vol.R-l?, pp.64-69, 1968.

[5] Gen, M. and R. Cheng, Optimal design of system reliability under uncertainty us- ing interval programming and genetic algo- rithm, Technical report, ISE94-6, Ashikaga Institute of Technology, Japan, 1995.

[6] Gen, M. and R. Cheng, Genetic Algorithms and Engineering Design, John Wiley & Sons, New York, 1996.

5 Conclus ion

In this paper, we have investigated the opti- mal design problem of system reliability with uncertain coefficients. The problem is formu- lated as nonlinear interval programming model

[7] Gen, M. and R. Cheng, Interval program- wing using genetic algorithms, In Pro- ceeding of Sizth International Symposiem on Robotics & Manufacturing, Montpellier, France, 1996.