Download - Heuristic reliability optimization by tabu search

Annals of Operations Research 63(1996)321-336 321

Heuristic reliability optimization by tabu search*

Pierre Hansen*

GERAD, Ecole des Hautes Etudes Commerciales, Montrdal, Canada

and

Keh-Wei Lih*

Bell Communications Research, Red Bank, N J, USA

A new heuristic algorithm, based on the tabu search methodology, is proposed for constrained redundancy optimization in series and in complex systems. It has the advantage of not being blocked as soon as a local optimum is found. Results given by the new method are compared with those of previous heuristics on a series of examples.

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

Reliability optimization of systems involving redundant components, subject to constraints on their size, weight, cost, etc., has attracted much attention. Many models, heuristics and exact algorithms have been proposed. A detailed survey classification and comparison of methods proposed up to 1979 is given in Tillman, Hwang and Kuo's book Optimization o f Systems Reliability. Several further heuristics [ 1 5 , 2 0 - 2 2 , 2 7 - 2 9 ] and exact algorithms [2-4] have been proposed, and some more comparisons of methods made [23]. Solution methods can be classified according to the type of problem solved (i.e., series or complex systems) and the techniques used (e.g., dynamic or integer programming, random or adaptive random search, direct ascent, etc.). While for series systems fairly large problems can be solved exactly, for the present it does not appear to be the case for complex systems. Heuristics are thus still needed. Moreover, they can also be useful in the case of series systems since knowledge of a good initial solution allows the simplification

*We are grateful to R. Bulfin for making the code for reliability optimization of series systems he wrote with C.-Y. Liu available to us.

'*Work of the first author was supported by NSERC Grant No. GP0105574, FCAR Grant No. 92EQ1048 and AFOSR Grant No. 90-0008 to Rutgers University.

*Work of the second author was partly supported by AFOSR Grant No. 90-0008 to Rutgers University while he was a graduate student.

© J.C. Baltzer AG, Science Publishers

322 P. Hansen, K.-W. Lih, Heuristic reliability optimization

of the problem by fixing some variables and curtailing the search if a branch-and- bound technique is used.

Many heuristics, such as [18, 26], proceed from an initial feasible solution to a local optimum by adding at each iteration one parallel redundant component to that subsystem for which this addition augments the most the system reliability (or, more generally, the value of some function guiding the search which takes into account reliability and tightness of the constraints). Such "direct ascent" or "forward" algorithms suffer from the defect of being blocked as soon as a local optimum is reached, i.e., when no further addition of a redundant component in parallel can improve reliability while still satisfying the constraints. So, if at some iteration the number of components in parallel in one subsystem exceeds the optimal number of that subsystem, the global optimum will not be reached.

The tabu search methodology, introduced in combinatorial optimization [9, 1 I, 12, 14], allows to avoid being blocked in local optima. The principle of tabu search algorithms is to follow a direction of steepest ascent until a local optimum is found (as do the forward heuristics) and then to take a step in the direction of mildest descent while forbidding the reverse move for a given number of iterations to avoid cycling, i.e., the endless repetition of the same sequence of iterations. The procedure is then iterated until no improved solution is found in a given number of steps. Further features [t0, 12] allow a more thorough exploration of the set of feasible solutions through intensification and diversification of the search.

In this paper, we show how this methodology can be used to solve heuristically but very efficiently, both in terms of value of the best solution obtained and of number of iterations, a variety of constrained redundancy optimization problems. The model considered and the notations used are described in the next section. The proposed tabu search algorithm is presented in section 3 and illustrated with two examples in section 4. Results of more extensive testing and comparison with other algorithms are given in section 5. Some improvements to the proposed algorithm are also discussed.

2. Redundancy optimization model and notations

The underlying system configuration in redundancy optimization problems can be classified into series systems and complex systems. Taking into consideration the possible use of device alternatives (i.e., different devices that provide the same functionality and can be used in the subsystems or stages) in each subsystem, a general model of redundancy allocation for reliability optimization problems can be written as follows:

Maximize R s = R(x)

n di subjectto ~ Z g i j k ( X i j ) <-- B k

i=1 j = l

k = 1,2 . . . . . m,

P. Hansen, K.-W. Lih, Heuristic reliability optimization 323

di

~_~ xij >1 j=l

Xij >-- 0

i = 1 , 2 . . . . . n,

j = 1,2 . . . . . di,

i - - 1 , 2 . . . . ,n ,

integers.

Objec t ive funct ion R(x) and the lef t -hand side constraints gijk may be l inear or non-

linear. Th ey are assumed separable and nondecreas ing in the xij's, for j = 1, 2 . . . . . di, i = 1, 2 . . . . . n. These variables denote the number of al ternat ive devices j used in

subsys tem i. In the case of the series system, R(x) = [-I~.-_ 1 Ri(xi), where gi(xi) denotes rel iabi l i ty o f subsys tem (or stage) i. We may omit index j in those subsys tems

where only one device al ternat ive is present. Note that in rel iabi l i ty opt imiza t ion problems, the objec t ive funct ions are usually nonlinear. However , if the under ly ing

sys tem is series, the nonl inear objec t ive funct ion may be l inear ized by using = n = Y~i=l In Ri(xi). Notat ions used in logar i thmic , i.e., In Rs = In R(x) In ~Ii= ! Ri(xi) n

the above general model and throughout this paper are summar ized in table 1.

Table t

Notations used in this paper.

g s

R(x) gi (x i )

x

xi xij

n

m

di i

J k

system reliability reliability function of the system

reliability of subsystem i number of device vectors of the system

number of devices of subsystem i number of alternative devices j of subsystem i the kth resource consumption function for device alternative j of subsystem i

number of subsystems in system considered number of resource constraints

number of device alternatives in subsystem or stage i

index of stage or subsystem

index of device alternatives index of resource constraints

3. Tabu search

Tabu search methodology has been applied with success to many combinatorial

op t imiza t ion problems [ 6 - 8 ] . In this section, we descr ibe the concept o f tabu search and how to apply it to rel iabil i ty opt imizat ion problems.


It is clear that the optimization model defined in the previous section is a combinatorial optimization problem. A large class of algorithms, both heuristic and exact, for solving general combinatorial optimization problems generate a sequence of feasible or infeasible solutions until a stopping condition is satisfied. Thus, there are two problems to consider.

(1) How to obtain the next solution from the current one?

(2) How to judge if a good (or optimum) solution has been obtained and stop the whole process?

In the case of exact algorithms, we may follow a systematic way to generate solutions and stop the procedure when all the possible solutions have been considered explicitly or implicitly. In the case of a heuristic algorithm, a way of obtaining the next solution must be decided upon and a stopping rule must be established. Moreover, the possibility of cycling must be considered to ensure the finiteness of the algorithm.

Hansen and Jaumard [14] proposed the Steepest Ascent Mildest Descent procedure (SAMD) for combinatorial optimization to avoid being blocked in local optima when applying the steepest ascent method. This heuristic follows the direction of steepest ascent to a local optimum, then uses the mildest descent direction to escape this local optimum. To prevent returning to the local optimum that has just been explored, the reverse move along the descent direction just used is forbidden for some preselected number, say p, of iterations. The process stops if there is no improvement in the best known value during some preset number, say rep, of iterations. Although developed independently, this method should be viewed as a member of the tabu search family of algorithms. It is, in fact, quite close to the Simple Tabu Search algorithm described by Glover in [11].

We now describe how to apply tabu search to redundancy optimization. The objective function in such problems is nondecreasing. Increasing the number of devices used in any of the subsystems will always increase the system reliability. Following the steepest ascent direction means adding, at each iteration, one device to the subsystem for which this results in the largest increase in system reliability while retaining feasibility. In the case we have device alternatives in subsystems, we use the most reliable device compatible with feasibility. If no device can be added without violating feasibility, we withdraw one device from the subsystem for which this results in the least reduction of system reliability and forbid the addition of this same device during the following p steps. We stop the process if there is no improvement in system reliability R s in rep iterations. The Steepest Ascent Mildest Descent method [14], which corresponds to this case, only forbids reverse moves when a descent direction is followed. It may be of interest to also forbid, in a similar way, the reverse moves when an ascent direction is taken, for the following p ' steps. In fact, the Simple Tabu Search algorithm of Glover [11] does this with


Table 2

Procedure TABU.

procedure T A B U logical change; Select an integer feasible solution (z, R,), e.g., one unit of the first

admissible device in each subsystem; Let fli and f[j denote the number of iterations to forbid in using the ascent

aald the decent directions, respectively, where i j E I J denotes the index set of the search directions;

Set flj *" 0 and ][j ~- O, i j E I J; change ~- t rue; (z ' , R:) ~- (z, R.); while change do

change *-- false; r e p e a t rep t imes

Let 6;i ~- max{6ij : 6,j = R(k) - R(z), i j e I J, f,j = 0} where ~ is obtained from z by adding one unit in direction i j while maintaining feasibility;

if B6( j then Update (z, Ro) using index/'j and 6/3; f/) ~- p~ + 1; /¢j , - / ' ~ - 1, ( i . / • Z J : f ' j > 0}; if R . > R ; then

(~-, R;) .-- (~, R.); change *-- t rue;

end_if; else

Let 6:j *-- m ~ { 6 , j : 6,~ = R ( e ) - R ( ~ ) , i j e H , / ' j = 0} where ~: is obtained from z by subtracting one unit in direction ij;

Update (z, R,) using index/'j and 6;i; f~ ~-- p + 1;

endAf; f~ j ~ - I , j - 1, {ij e sJ : f~j > 0};

end - r epea t ; end_while; Output (z ' , R:) as the maximizer and system reliability.

p = p ' . We consider these parameters as separate. Moreover, in the treatment of p" we decrease the tabu search status only when further ascent moves are taken. It therefore remains longer than the tabu on descent moves and diversifies the search direction. The steps of the procedure are summarized in table 2.


4. Examples

We consider two examples: a series system with several alternative devices at some stages and a complex system (bridge system). All constraints are assumed to be linear for simplicity, but nonlinear constraints can also be easily taken into account.

EXAMPLE I

This problem is due to Chern and Jan [4]. Dominated alternative devices in stage 1 are omitted. The system consists of three stages in series. There is one device in stage 1, three devices in stage 2, and two devices in stage 3. At least one working device is needed in each stage in order for the sys tem to work. The resource consumpt ion functions are the sum of the cost and the sum of the weight of devices used. The goal is to maximize system reliability under constraints on resources used and min imum requirements in each stage. The opt imizat ion model for this example is given below. Table 3 summarizes the numerical data.

Maximize

subject to

Rs(x ) = [1 - (1 - 0.99) x'~ ]

x [1 - (1 - 0.98) x~ (1 - 0.80) x22 (1 - 0.90) x23 ]

x [1 - (1 - 0.98) x3' (1 - 0.92) x32 ]

4Xll + 8x21 + 3x22 + 3x23 + l lx31 + 5x32 _< 30,

2Xll + 3x21 + 3x22 + 9x23 + 4x31 + 6x32 _< 17,

Xll _> 1,

x21 + x22 + X23 --> 1,

X31 4- x32 --> l,

xij>O integer Vi, j .

Table 3

Data for example 1.

Stage Device Reliability Cost Weight

1 1 0.99 4 2

2 1 0.98 8 3 2 0.80 3 3 3 0.90 3 9

3 1 0.98 11 4 2 0.92 5 6

Resource: 30 17


Table 4

Search steps for example 1.

Step xN x21 xzz x~3 x31 x32 Reliability Improved/Reduced

0 1 1 0 0 1 0 0.9507960 (Initial solution)

1 1 1 0 0 1 1 0.9686477 Improved

2 I 1 0 0 1 0 0.9507960 Reduced

3 1 1 1 0 1 0 0.9663192 Improved

4 2 I 1 0 1 0 0.9759824 Improved

5 1 1 1 0 1 0 0.9663192 Reduced

6 1 1 2 0 1 0 0.9694238 Improved

7 1 1 1 0 I 0 0.9663192 Reduced

8 2 I 1 0 1 0 0.9759824 Improved

4 2 1 1 0 1 0 0.9759824 (Heuristic solution)

W e n o w a p p l y p r o c e d u r e T A B U to this e x a m p l e and s u m m a r i z e the s ea r ch

s t eps in t ab le 4. T h e p a r a m e t e r s are p ' = 0, p = 2, and rep = 4. T h e bes t so lu t i on is

f o u n d in f o u r s teps . S i n c e no be t t e r so lu t ion is o b t a i n e d a f t e r rep s teps , the s ea r ch

s tops wi th the cu r r en t be s t so lu t ion as the heur i s t i c one.

EXAMPLE 2

T h i s e x a m p l e is due to A g g a r w a l [1 ] and a l so s tud ied in [16, 27]. It is a b r i d g e

s y s t e m wi th o n e d e v i c e in each s u b s y s t e m and one l i nea r cos t cons t r a in t . F i g u r e 1

s h o w s the u n d e r l y i n g s y s t e m c o n f i g u r a t i o n . Tab l e 5 s u m m a r i z e s the data .

T h e r e d u n d a n c y o p t i m i z a t i o n m o d e l is as f o l l ows :

w h e r e

M a x i m i z e

subjec t to

R s ( x ) = R1R 2 + R3R4 + RIR4R 5 + R2R3R 5

- R1R2R3R 4 - RIR2R3R 5 - R1R2R4R 5

- R 1 R3 R4 R5 - R 2 R3 R4 R5

+ 2 R1 R2 R3 R4 R5

5 Gc(x ) -- Z cixi <_ C,

i=1

xi > 1, integers i = 1, 2, 3, 4, 5,

R i = [ 1 - ( 1 - p i ) xi ] i = 1 , 2 , 3 , 4 , 5 .


in out O o

Figure 1. A complex system - bridge system.

Table 5

Data for example 2.

Subsystem Reliability Cost

1 0.70 2

2 0.85 3

3 0.75 2

4 0.80 3

5 0.90 1

Resource: 20

Applying procedure TABU to this system with parameters p" = 1, p = 2, and rep = 6 leads to the search steps summarized in table 6. Again, the optimal solution is obtained at iteration 12.

5. Modifications and results

The TABU procedure of section 3 was applied to a series of test problems from the literature corresponding to both series and complex systems. Results are presented in tables 7 and 8. Optimal values were determined for all the test problems


Table 6

Search steps for example 2.

Step xl x2 x3 x4 x5 Reliability Improved/Reduced

0 1 1 1 1

1 1 1 2 1 2 1 1 2 2

3 1 I 3 2

4 2 1 3 2

5 2 1 2 2

6 3 1 2 2

7 3 1 t 2

8 4 1 1 2

9 4 1 1 1

10 4 2 1 1

11 3 2 1 1

12 3 2 2 1

13 2 2 2 1

14 2 2 3 1

15 1 2 3 1

16 1 2 3 1

17 1 2 3 1

18 1 2 2 1

12 3 2 2 1

0 .891325 (Initial solution) 0 .946506 Improved 0 .973776 Improved 0 .988179 Improved 0 .992096 Improved 0.987303 Reduced 0.991361 Improved 0 .983719 Reduced 0 .988396 Improved 0 .964957 Reduced 0 .992919 Improved 0 .987947 Reduced 0 .993216 I m p r o v e d

0 .988148 Reduced 0 .992342 Improved 0.985041 Reduced

2 0 .990254 Improved 3 0 .990779 Improved 3 0 .976779 Reduced

1 0 .993216 (Heuristic solution)

using an enumerative algorithm. In all cases, the tabu search procedure reached the global optimum. In three cases for complex systems, the solution obtained is better than the best one previously known.

To test the limits of the method, 240 larger test problems corresponding to series systems with 3, 5, 8 and 10 constraints and with 5 to 30 stages were randomly generated in a similar way as in Ghare and Taylor [5] and Bulfin and Liu [2]. Global optimal solutions were determined by a branch-and-bound algorithm. Results are presented in tables 9 through 12, which give average relative error (are), maximum relative error (mre), optimality rate (or) and average execution time (aet), in seconds, for a series of 10 problems each time. All computational experiments were done on a SUN SPARC 2 workstation with 32MB RAM.

The tabu search algorithm was used in two ways. A first variant, called auto_TABU, applied the procedure with all possible values of p and p' such that 0 < p, p ' < 0.4n, where n denotes the number of stages of the system. This always led to the globally optimal solution. A second variant, called gen_TABU, aimed at removing the problem of choosing adequate values for p and p' without enumerating a large number of them. To this effect, p and p' were set at 0, their minimum value,


Table 7

Summary of solutions for series examples; all the solutions were found by tabu search with p ' = 0.

Example Tabu solution Best previous Reference Stages Constraints and its reliability solution known

15 2 linear (3,4,6,4,3,2,4,5,4,2,3,4,5,4,5) same as [17, 15] R s = 0.945613 tabu solution

4 2 linear (6,6,5,4) same as [ 17, 15] R~. = 0.997726 tabu solution

5 2 linear (3,4,5,4,3) same as [26] Rs = 0.984952 tabu solution

Stage Device Times

1 1 2 3 2 linear 2 1 1 same as [4]

2 2 1 tabu solution 3 1 1

Rs = 0.975982

Stage What to do

1 use device 3 same as [24, 16] 3 3 nonlinear 2 3 times tabu solution

3 5 times

Rs = 0.970240

5 3 nonlinear (3,2,2,3,3) same as [26] Rs = 0.904467 tabu solution

when the search started. The i r values were increased by 1 every 3 or 4 i terat ions, for p ' and p, respect ively. They were a l lowed to increase to 0.4n and were then

re turned to their min imum values. Thus, the values of p ' and p would osci l la te

be tween 0 and 0.4n th roughout the search.

The gen_TABU variant is faster than the au to_TABU one and yields as good results for small p roblems (see tables 9 through 11) and less good ones for large

problems. For compar i son purposes , all test p roblems were also so lved with the

heur is t ics o f Bulf in and Liu [2] (denoted by B L H in tables 9 through 12), based

on using a surrogate constraint , and o f Sharma and Venkateswaran [26] (denoted

by SV in tables 9 through 12), a s imple ascent procedure . Whi le the fo rme r heuris t ic

is as good as gen_TABU (but not as au to_TABU), the latter one gives much worse results excep t for the smal ler problems. This c lear ly i l lustrates the main advantage

o f T A B U search methods , i.e., being able to t ranscend local optimali ty.

P. H a n s e n , K . -W. Lih , H e u r i s t i c re l iab i l i t y o p t i m i z a t i o n 331

Table 8

S u m m a r y of so lu t ions for com p lex examples (*: p ' = 1).

E x a m p l e Tabu so lu t ion Bes t p rev ious Re fe r ence S u b s y s t e m s Cons t r a in t s and its re l iabi l i ty so lu t ion k n o w n

10 1 l inear (1 ,1 ,2 ,2 ,1 ,1 ,1 ,2 ,3 ,2) s ame as [29] R~ = 0 .983708 tabu so lu t ion

4 2 l inear (3 ,1 ,1 ,1) (2 ,2 ,1 ,3) [27] R~ = 0 . 9 9 7 3 7 0 R, = 0 .9970

b r idge 1 l inear (3,2,2,1,1)* same as [ 1,16, 27] R s = 0 .993216 tabu so lu t ion

b r idge 1 l inear (2 ,1 ,1 ,1 ,1) same as [19] R s = 0 .951174 tabu so lu t ion C = 50

b r idge t l inear (2 ,1 ,2 ,1 ,1) s ame as [ 19] R s = 0 .976084 tabu so lu t ion C = 60

b r idge 1 l inear (1,1,3,2,2)" (2 ,2 ,2 ,1 ,1) [ 19] R~ = 0 .990188 R~. = 0 .987 C = 70

b r idge 1 l inear (4 ,2 ,2 ,1 ,1) same as [ 19] R s = 0 .997658 tabu so lu t ion C = 80

b r idge 1 l inear ( I ,1 ,5 ,3 ,1 ) ° (4 ,3 ,2 ,1 ,1) [ 19] R s = 0 .998995 R, = 0 .9988 C = 90

b r i d g e 1 l inear (5 ,3 ,2 , l , 1 ) same as [19] R s = 0.999561 tabu so lu t ion C = 100

b r idge 1 l inear (6 ,3 ,2 ,1 ,2) same as [19] R s = 0.999811 tabu so lu t ion C = 110

b r idge I l inear (2,1,7,4,1 ) s ame as [ 19] R s = 0 .999959 tabu so lu t ion C = 120

b r idge 1 l inear (4,3,1,1,1)" (2 ,2 ,3 ,1 ,1) [ 13] R s = 0 .999607 R s = 0 .99234

332 P Hansen, K.-W. Lih, Heuristic reliability optimization

Table 9

Results for randomly generated problems with three resource constraints.

Problem size m x n

3 x 5 3 x 8 3 x 1 0 3 x 1 5 3 x 2 0 3 x 3 0

auto_TABU are 0.000000 0.000000 0,000000 0.000000 0.000000 0.000000 rare 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 or 10/10 10/10 10/10 10/10 10/10 10/10

aet 0.322000 1.320000 3.069000 14.91400 43.81900 208,2710

gen_TABU are 0.000000 0.000000 0.000322 0.000000 0.000405 0.000743 mre 0.000000 0.000000 0,002494 0.000000 0.001542 0.004597 or 10/10 10/10 8/10 10/10 3/10 3/10 aet 0.168000 0.412000 0.603000 1.385000 2.692000 6.245000

BLH are 0.000000 0,000038 0.000229 0.000214 0.000014 0.000050 mre 0.000000 0.000382 0.001768 0.002120 0,000122 0.000439 or 10/I0 9 / I 0 8/10 7/10 8/10 7 / I 0 aet 0.039000 0.060000 0.077000 0.125000 0.166000 0.251000

SV are 0.000334 0.000113 0.002334 0.000391 0.002047 0.002884 mre 0.003128 0.000500 0.016035 0.001810 0.011588 0.009802 or 8/10 7/10 6/10 3/10 2/10 1/10 aet 0.001000 0.006000 0,010000 0.020000 0.033000 0.071000

Table l0

Results for randomly generated problems with five resource constraints.

Problem size m x n

5 x 5 5 x 8 5 x 1 0 5 x 1 5 5 x 2 0 5 x 3 0

auto_TABU are 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 mre 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 or 10/10 10/10 10/10 10/10 10/I0 10/10 aet 0.404000 1.554000 3.903000 18.88000 56.56800 259.6230

genTABU are 0.000000 0.000000 0.000007 0.000009 0.000086 0.000177 mre 0.000000 0.000000 0,000071 0.000091 0.000557 0,000697 or 10/10 10/10 9/10 9/10 7/10 4/10 aet 0.208000 0.525000 0.766000 1.676000 3.322000 7.947000

BLH a~ 0.000000 0.000000 0.000003 0.000009 0.000000 0.000064 mre 0,000000 0.000000 0.000030 0.000091 0.000003 0.000635 or t0/10 10/10 9/10 8/10 9/10 8/10 aet 0,053000 0.097000 0.122000 0.165000 0,233000 0,352000

SV are 0.000363 0.000147 0.000144 0.001418 0.000558 0.002807 mre 0.001907 0.001280 0.000809 0.010069 0.002466 0.012721 or 6/10 7/10 5/10 4/10 4/10 0/10 aet 0.003000 0.009000 0,009000 0.020000 0.038000 0.082000

P. Hansen, K.-W. Lih, Heuris t ic reliability optimization 333

Table 11

Results for randomly generated problems with eight resource constraints.

Problem size m x n

8 x 5 8 x 8 8 x 1 0 8×15 8 × 2 0 8 x 3 0

auto_TABU are 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

mre 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 or 10/10 10/10 10/10 10/10 10/10 10/10 aet 0,519000 2.192000 5.056000 24.58300 74.59100 346.3470

gen_TABU are 0.000000 0.000000 0.000000 0.000000 0.000000 0.000195 mre 0.000000 0.000000 0.00000 0.000000 0.000000 0.000536 or 10/10 10/10 10/10 1{3/10 9/10 2/10 act 0,273000 0.652000 1.016000 2.117000 4.935000 11,43000

BLH are 0.000000 0,000001 0.000001 0.000091 0.000000 0.000167 mre 0.000000 0,000009 0.000010 0,000472 0,000000 0.001222 or 10/10 9 / I 0 9/10 6/10 9/10 6/10 aet 0.078000 0.120000 0.164000 0.240000 0.331000 0.517000

SV are 0.000334 0.001290 0.001659 0.001170 0.001372 0.001533 mre 0.000000 0.005571 0.014801 0.003362 0.008469 0.004163 or 10/10 6/10 5/10 3/10 3/10 0/10 aet 0.003000 0.004000 0.012000 0.025000 0.047000 0.104000

Table 12

Results for randomly generated problems with ten resource constraints.

Problem size m x n

10x5 10x8 1 0 x l 0 10x15 10x20 10x30

auto_TABU are 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 mre 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 or 10/10 10/10 10/10 10/10 10/10 10/10 aet 0,584000 2.451000 5.947000 28.47300 86,59800 386.6050

genTABU are 0.000000 0.000174 0.000000 0.000000 0.000017 0.000373 mre 0.000000 0,001737 0.000000 0.000000 0.000112 0.002679 or 10/10 9/10 t 0 / I 0 10/10 8/10 5/10 aet 0.298000 0.794000 1.170000 2.580000 4.569000 11.09400

BLH are 0.000000 0.000000 0.000000 0.000007 0.000044 0.000042 mre 0.000000 0.000000 0.000000 0.000073 0.000443 0.000387 or 10/10 10/10 10/10 9/10 9/10 8/10 aet 0.095000 0.157000 0.188000 0.285000 0.392000 0.591000

SV are 0.002021 0.000188 0.000234 0.000213 0.001658 0.003988 mre 0,020206 0.001737 0.002090 0.000780 0.013052 0.014792 or 9/10 8/10 6/10 5/10 3/10 2/10 aet 0.005000 0.009000 0.012000 0.028000 0.049000 0.109000


i l l 0

Figure 2. A 20-subsystem complex system.

out 0

Table 13

Data for 20-subsystem complex system.

Subsystem Reliability Cost Subsystem Reliablity Cost

1 0.95 2 11 0.70 1

2 0.75 1 12 0.75 3

3 0.80 2 13 0.95 4

4 0.85 4 14 0.70 1

4 0.80 1 15 0.80 1

6 0.90 3 16 0.75 3

7 0.75 3 17 0.85 2

8 0.80 1 18 0.90 4

9 0.70 2 19 0.95 5

10 0.85 2 20 0.75 3


Table 14

Summary of solutions for 20-subsystem complex system.

Example Tabu solution RHS Time Subsystems Constraints and its reliability value (seconds)

20 1 (1,2,1,1,1,1,1,1,1,1,1,1,1,5,2,1,1,1,1,3) 60 14.04 R s = 0.99456711525989

20 1 (4,1,5,1,5,4,1,1,1,1,1 ,I,1,3,1,1,1,1,1,2) 80 19.21 R.~ = 0.99996319937110

20 1 (3,1,5,1,5,4,1,1,1,1,1,1,1,8,2,1,3,1,1,6) 100 28.69 Rs = 0.99999977780974

20 1 (7,1,11,1,11,8,1,1,1,1,1,1,1,4,1,1,1,1,1,3) 120 29.08 Rs = 0.99999999900134

We cons t ruc ted a com p l ex sys tem with 20 subsys tems and one l inear cost

constra int . There is one device in each subsys tem. Figure 2 shows the sy s t em

conf igura t ion . Table 13 summar i ze s the numer ica l data. Table 14 summar i ze s the

resul ts us ing the T A B U procedure . The a lgor i thm provides solut ions which appea r

to be of good qual i ty within a reasonable solving t ime.

References

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