Heuristic reliability optimization by tabu search

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<ul><li><p>Annals of Operations Research 63(1996)321-336 321 </p><p>Heuristic reliability optimization by tabu search* </p><p>Pierre Hansen* </p><p>GERAD, Ecole des Hautes Etudes Commerciales, Montrdal, Canada </p><p>and </p><p>Keh-Wei Lih* </p><p>Bell Communications Research, Red Bank, N J, USA </p><p>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. </p><p>1. In t roduct ion </p><p>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 [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 </p><p>*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. </p><p>'*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. </p><p>*Work of the second author was partly supported by AFOSR Grant No. 90-0008 to Rutgers University while he was a graduate student. </p><p> J.C. Baltzer AG, Science Publishers </p></li><li><p>322 P. Hansen, K.-W. Lih, Heuristic reliability optimization </p><p>of the problem by fixing some variables and curtailing the search if a branch-and- bound technique is used. </p><p>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. </p><p>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. </p><p>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. </p><p>2. Redundancy optimization model and notations </p><p>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: </p><p>Maximize R s = R(x) </p><p>n di subjectto ~ Zgi jk (X i j ) </p></li><li><p>P. Hansen, K.-W. Lih, Heuristic reliability optimization 323 </p><p>di </p><p>~_~ xij &gt;1 j=l </p><p>Xij &gt;-- 0 </p><p>i=1 ,2 . . . . . n, </p><p>j = 1,2 . . . . . di, </p><p>i - -1 ,2 .. . . ,n, </p><p>integers. </p><p>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 </p><p>= 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. </p><p>Table t </p><p>Notations used in this paper. </p><p>g s </p><p>R(x) gi(xi) </p><p>x </p><p>xi xij </p><p>n </p><p>m </p><p>di i J k </p><p>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 </p><p>3. Tabu search </p><p>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. </p></li><li><p>324 P. Hansen, K.-W. Lih, Heuristic reliability optimization </p><p>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. </p><p>(1) How to obtain the next solution from the current one? </p><p>(2) How to judge if a good (or optimum) solution has been obtained and stop the whole process? </p><p>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. </p><p>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]. </p><p>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 </p></li><li><p>P. Hansen, K.-W. Lih, Heuristic reliability optimization 325 </p><p>Table 2 </p><p>Procedure TABU. </p><p>procedure TABU logical change; Select an integer feasible solution (z, R,), e.g., one unit of the first </p><p>admissible device in each subsystem; Let fli and f[j denote the number of iterations to forbid in using the ascent </p><p>aald the decent directions, respectively, where i j E I J denotes the index set of the search directions; </p><p>Set flj *" 0 and ][j ~- O, i j E I J; change ~- true; (z' , R:) ~- (z, R.); while change do </p><p>change *-- false; repeat rep t imes </p><p>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; </p><p>if B6( j then Update (z, Ro) using index/'j and 6/3; f/) ~- p~ + 1; /j , - / '~ - 1, ( i . / Z J : f' j &gt; 0}; if R .&gt;R; then </p><p>(~-, R;) .-- (~, R.); change *-- true; </p><p>end_if; else </p><p>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; </p><p>Update (z, R,) using index/'j and 6;i; f~ ~-- p+ 1; </p><p>endAf; f~j ~- I , j - 1, {ij e sJ : f~j &gt; 0}; </p><p>end-repeat; end_while; Output (z', R:) as the maximizer and system reliability. </p><p>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. </p></li><li><p>326 P. Hansen, K.-W. Lih, Heuristic reliability optimization </p><p>4. Examples </p><p>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. </p><p>EXAMPLE I </p><p>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 system to work. The resource consumption 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 minimum requirements in each stage. The optimization model for this example is given below. Table 3 summarizes the numerical data. </p><p>Maximize </p><p>subject to </p><p>Rs(x ) = [1 - (1 - 0.99) x'~ ] x [1 - (1 - 0.98) x~ (1 - 0.80) x22 (1 - 0.90) x23 ] </p><p>x [1 - (1 - 0.98) x3' (1 - 0.92) x32 ] </p><p>4Xll + 8x21 + 3x22 + 3x23 + llx31 + 5x32 _&lt; 30, </p><p>2Xll + 3x21 + 3x22 + 9x23 + 4x31 + 6x32 _&lt; 17, </p><p>Xll _&gt; 1, </p><p>x21 + x22 + X23 --&gt; 1, </p><p>X31 4- x32 --&gt; l, </p><p>xij&gt;O integer Vi, j. </p><p>Table 3 </p><p>Data for example 1. </p><p>Stage Device Reliability Cost Weight </p><p>1 1 0.99 4 2 </p><p>2 1 0.98 8 3 2 0.80 3 3 3 0.90 3 9 </p><p>3 1 0.98 11 4 2 0.92 5 6 </p><p>Resource: 30 17 </p></li><li><p>P. Hansen, K.-W. Lih, Heuristic reliability optimization 327 </p><p>Table 4 </p><p>Search steps for example 1. </p><p>Step xN x21 xzz x~3 x31 x32 Reliability Improved/Reduced </p><p>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 </p><p>4 2 1 1 0 1 0 0.9759824 (Heuristic solution) </p><p>We now apply procedure TABU to this example and summar ize the search steps in table 4. The parameters are p ' = 0, p = 2, and rep = 4. The best so lut ion is found in four steps. S ince no better solut ion is obta ined after rep steps, the search stops with the current best solut ion as the heurist ic one. </p><p>EXAMPLE 2 </p><p>Th is example is due to Aggarwa l [1 ] and also studied in [16, 27]. It is a br idge sys tem with one dev ice in each subsystem and one l inear cost constraint . F igure 1 shows the under ly ing sys tem conf igurat ion. Table 5 summar izes the data. </p><p>The redundancy opt imizat ion mode l is as fo l lows: </p><p>where </p><p>Max imize </p><p>subject to </p><p>Rs(x ) = R1R 2 + R3R4 + RIR4R 5 + R2R3R 5 </p><p>- R1R2R3R 4 - RIR2R3R 5 - R1R2R4R 5 </p><p>- R 1 R3 R4 R5 - R 2 R3 R4 R5 </p><p>+ 2 R1 R2 R3 R4 R5 </p><p>5 Gc(x ) -- Z cixi 1, integers i = 1, 2, 3, 4, 5, </p><p>R i =[1 - (1 -p i ) xi ] i=1 ,2 ,3 ,4 ,5 . </p></li><li><p>328 P. Hansen, K.-W. Lih, Heuristic reliability optimization </p><p>in out O o </p><p>Figure 1. A complex system - bridge system. </p><p>Table 5 </p><p>Data for example 2. </p><p>Subsystem Reliability Cost </p><p>1 0.70 2 </p><p>2 0.85 3 </p><p>3 0.75 2 </p><p>4 0.80 3 </p><p>5 0.90 1 </p><p>Resource: 20 </p><p>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. </p><p>5. Modifications and results </p><p>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 al...</p></li></ul>


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