heuristic reliability optimization by tabu search
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Annals of Operations Research 63(1996)321-336 321
Heuristic reliability optimization by tabu search*
GERAD, Ecole des Hautes Etudes Commerciales, Montrdal, Canada
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. In t roduct ion
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 of Systems Reliability. Several further heuristics [15,20-22,27-29] and exact algorithms [2-4] have been proposed, and some more comparisons of methods made . 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 ~ Zgi jk (X i j )
P. Hansen, K.-W. Lih, Heuristic reliability optimization 323
~_~ xij >1 j=l
Xij >-- 0
i=1 ,2 . . . . . n,
j = 1,2 . . . . . di,
i - -1 ,2 .. . . ,n,
Object ive function R(x) and the left-hand side constraints gijk may be linear or non- linear. They are assumed separable and nondecreasing in the xij's, for j = 1, 2 ..... di, i = 1, 2 ..... n. These variables denote the number of alternative devices j used in subsystem i. In the case of the series system, R(x) = [-I~.-_ 1 Ri(xi), where gi(xi) denotes reliabil ity of subsystem (or stage) i. We may omit index j in those subsystems where only one device alternative is present. Note that in reliabil ity optimization problems, the objective functions are usually nonlinear. However, if the underlying system is series, the nonl inear objective function may be l inearized by using
= n = Y~i=l In Ri(xi). Notations used in logarithmic, i.e., In Rs = In R(x) In ~Ii= ! Ri(xi) n the above general model and throughout this paper are summarized in table 1.
Notations used in this paper.
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 optimization problems [6 -8] . In this section, we describe the concept of tabu search and how to apply it to reliability optimization problems.
324 P. Hansen, K.-W. Lih, Heuristic reliability optimization
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  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 .
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 , which corresponds to this case, only forbi