# Tabu Search-Part FRED GLOVER. (P) Minimize c(x): x X

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4 Copyright O 2001 All Rights Reserved Tabu Search. I1 7 prevent the cancellation of e(q) by e(q + I), if we can insure that at least one element in each successive C- Sequence will remain uncanceled, then no solution can ever repeat. (This is a sufficient but not a necessary condition to avoid repetition of solutions.) This means of avoiding a duplication of an earlier solution can be assured by specifying an attribute 2(i) to be tabu if and only if the associated element e(i) of ATL is the last remaining member of some C-Sequence, once all other members have been canceled. To enforce this condition successfully, a way must be afforded to remove each canceled element from every C-Sequence to which it belongs, and to identify when one or more of these sequences has been reduced to a single element. An efficient method for accomplishing this can be based on the observation that whenever one sequence lies within another, the larger sequence is dominated by the smaller and may be discarded, since retaining an element in the smaller sequence assures that one is also retained in the larger. (This observation is also central to establishing the sufficiency of nonempty C-Sequences to avoid duplicate solutions, as long as moves exist that permit this condition to be maintained.) Specifically, we introduce a data structure consisting of two arrays, startseq(e) and endseq(e), defined for each element e on the active tabu list ATL, where startseq(e) denotes the element f on ATL that starts the C-Sequence terminated by e (where f = void, a dummy element, if e does not terminate such a sequence), and endseq(e) denotes the element g on ATL that ends the C-Sequence initiated by e (where g = void if e does not initiate such a sequence). The dominance property implies that these two values are uniquely defined for each e on ATL (i.e., of two contending values for f or g, the element which is closer to e on ATL takes precedence). Moreover, the condition in which e is the only element of a C-Sequence is identified by startseq(e) = endseq(e) = e. The use of these arrays and the role of dominance is illustrated in Figures 1 and 2. Figure 1 shows the creation of a C-Sequence as a result of a cancellation step, together with the associated startseq and endseq assignments. Figure 2 provides an example of dominance, and of the creation of tabu status. (It may be noted in Figure 2 that the "old C-Sequence" will also dominate the new if the canceled element occurs as early in the ATL list as the starting element of the old C-Sequence, since the form of this sequence after the cancellation will still lie inside the new C-Sequence.) To complete the basis for identifying and updating the C-Sequences, we introduce the two arrays predecessor(e) and successor(e), which respectively identify the elements that precede and succeed e on ATL. Also, we introduce two elements, first-dummy and lastdummy, which respectively begin and end the ATL Figure I. Figure 2. ATL Before Cancellation ATL After Cancellation Creation of C-sequence by cancellation. New C-Sequence Dominated by Old (Hence New 1s Not Recorded) Old C-Sequence New C-Sequence Later ATL Before Creatina Tabu Element (Remalning C-Sequence Accentuated) Later ATL After Creating Tabu Element (Remalning C-Sequence Accentuated) startseq(4) = endseq(4) = 4 (Element T 1s tabu) Domination and creation of tabu status. New Element I New Element list, i.e., first-dummy precedes the first (oldest) element and last-dummy succeeds the last (most recent) element. Finally, we define iteration(e) = the iteration e was added to ATL, letting this value equal 0 if e is not on ATL. Then for two elements e and f on ATL, iteration(e) < iteration(f) indicates that e appears before f. The iteration array will be used to determine

7 10 Glover i Figure 3. Residual C-Sequence 3 5, 5 4, 5, 7 4, 5 (5 and 5 cancel) , 2 4 (3 and 3 cancel) 2 (i and 4 cancel) 1 Reverse elimination method. It is interesting to observe that the tabu classifications produced by the method for this example are the same as those produced by the more restrictive C- Sequence Method when the latter employs a minimum buffer size of tb = 1. However, with the addition of the element e(9) = '5 to TL, the Reverse Elimination Method continues to generate a set of restrictions which is exactly necessary and sufficient to prevent duplications, while the C-Sequence Method produces two additional (unnecessary) tabu restrictions. The precise details of the method that generates the sequences and tabu classifications shown in Figure 3 are as follows. REVERSE ELIMINATION METHOD AT EACH ITERATION comment: The following local initialization is implemented after selecting the move whose associated attribute is e(q). For each element e: designate e to be "not tabu", and set successor(e) := absent predecessor(1ast-dummy) := first-dummy successor(first-dummy) := last -dummy first-e := last-dummy n Tabu Status 3 Tabu i Tabu 5 Tabu n := 0 comment: Select the current size of TL by choosingp < q, and select the buffer size tb. Then trace the TL list in reverse order, as follows. for i := q until p do begin e := e(i) if successor(c) = absent then begin comment: Expansion Step: C is not in the current residual C-Sequence, hence e becomes the new first element of this sequence. successor(e) := first-e predecessor (first -e) := e first-e := e predecessor(e) = first-dummy n:=n+ 1 if n = 1 then make C tabu else begin comment: Elimination Step: P is in the current residual C-Sequence. Hence e is not added and C is eliminated. f := predecessor(2) g := successor(2) successor(f) := g predecessor(g) := f successor(^) := absent if P = first-e then first-e := g n:=n- 1 if n = 1 then make the complement of first-e tabu comment: if n = 0, the solutions x(i) and x(q + 1) are the same end; comment: create automatic tabu status for elements of the buffer if i > q - tb then make C tabu end; Reducing Effort and Related Strategic Considerations The foregoing procedure evidently requires more work than the C-Sequence procedure, since it traces the TL list at each iteration. However, some shortcuts are possible at the expense of more memory. For example, there is no need to continue the trace after adding an element on the Expansion Step which does not contain a complement among earlier elements of TL (a condition easily recorded). A process that may achieve greater savings is the following. On a given iteration, record a value least (n) for each value of n from 0 to a selected cut-off n*, which identifies the smallest value of i for which a Residual C-Sequence has a size of n(i) = n. This value may be determined automatically by starting with Co~vriaht O 2001 All Riahts Reserved -

16 Copyright O 2001 All Rights Reserved Tabu Search their identities, while preventing a dropped edge from being added back has an intensification role by limiting the search to the moves that incorporate the remaining non-tree edges. This complementarity of the intensification and diversification strategies suggests the value of not simply using a tabu list for a single type of attribute (when simpler lists are used), but of maintaining multiple lists whose sizes reflect the relative restrictiveness of their associated tabu conditions (as influenced by the relative number of choices available in different categories; e.g., non-tree versus tree edges). To date, applications involving multiple tabu lists have focused on creating different lists for different components or stages of search, or for preventing repetitions as well as reversals. By contrast, the use of multiple lists that incorporate different types of attributes has been largely unexplored, and the issue of exploiting tradeoffs between intensification and diversification by this means remains open to investigation. Such considerations, which invite a closer look at the standard tabu list structures, also invite examination of different types of dynamic strategies that maintain these structures. Accordingly, to conclude the treatment of dynamic tabu list strategies, we propose two alternative procedures that rely on the organization and processing techniques of standard tabu list implementations, but which introduce frequency criteria for determining the current tabu status of attributes on the list. In spirit, these procedures are related to those of probabilistic tabu search described in Section 9 of Part I, and the second procedure provides a natural accompaniment to a probabilistic tabu search approach. To describe - the first frequency-based method, consider the list ATL, which consists of the complements of the elements of ATL (hence consists of those elements that are potentially tabu in the C-Sequence Method). We may either consider the full set of such complements, or more - restrictively, by the use of multiple tabu lists, allow ATL to represent a subset that corresponds to attributes of a particular - classification. In the latter case, more than one such ATL may be handled simultaneously, selecting appropriate - parameters for each. The basic idea is to partition ATL into groups, bracketed according to the ages of their members, where tabu status for younger groups is maintained more stringently than for older groups. For example, the - first group may consist of the 3 youngest elements of ATL, which are required to maintain their tabu status without exception, the second - group may consist of the next 5 youngest elements of ATL, which are strongly but not invariably required to maintain their tabu status, and so on. (These divisions may be based on iterations rather than numbers of elements. For example, a group may consist of elements that were added between h and k iterations ago, and hence which may contain fewer than k h elements as cancellations occur.) A common approach for accomplishing such a progressive relaxation of tabu status is to apply successively smaller penalties to the elements in older groups, which often produces an effect similar to that of making all penalties uniform. By contrast, the approach adopted here allows elements of the groups to fully escape their tabu status, according to certain frequencies that increase with the age of the groups. Thus, rather than being subjected to diminishing penalties, elements are periodically allowed to be chosen freely, which requires the determination of an appropriate form of frequency measure and an associated means for implementing it. The Tabu Cycle Method The first of the two frequency-based procedures we propose is based on the use of iteration intervals called tabu cycles, which are made smaller for older groups than for younger groups. Specifically, if Group k has a tabu cycle of TC(k) iterations, then at each occurrence of this many iterations, on average, the elements of Group k escape their tabu status and are free to be chosen. To illustrate, suppose there are three groups (each older than the preceding), whose tabu cycles are 4, 3 and 2. Then, roughly speaking, an element could be selected from the first group once in every 4 iterations, from the second group once in every 3 iterations and the third group once in every 2 iterations. (A buffer group whose elements never escape tabu status has an implicit tabu cycle of infinity.) However, the process is not quite as simple as the illustration suggests. By selecting - elements in the manner indicated, an element of ATL could escape its tabu status on virtually every iteration: e.g., by picking an element of Group 3 on iteration 2, an element of Group 2 on iteration 3, then an element either of Group 3 or Group 1 on iteration 4, and so on. Moreover, there is no clear provision of how to handle the situation where no element is chosen from Group k for a duration of several tabu cycles, given that the goal is to allow an element to be selected, on average, once every cycle. To take care of the preceding considerations, we introduce a cycle count, CC(k), for Group k. Initially, CC(k) starts at 1 and is incremented by 1 at every iteration. Each group has three states, OFF, ON and FREE. We define Group k to be: OFF if CC(k) < TC(k) ON if CC(k) a TC(k) FREE if Group h is ON for all h a k.

17 20 Glover An element is allowed to be chosen from Group k only if it is FREE, hence only if its cycle count equals or exceeds its tabu cycle value (qualifying the group as ON), and only if this same condition holds for all older groups. The ON and FREE states are equivalent for the oldest group. As implied by our earlier discussion, we assume TC(k) < TC(k - 1) for all k > 1. The definition of the FREE state derives from the fact that each CC(k) value should appropriately be interpreted as applying to the union of Group k with all groups younger than itself. Accordingly, once an element is selected from Group k, the cycle count CC(k) is re-set by the operation CC(h) := CC(h) - TC(h) for all h 3 k, whereupon the cycle counts are incremented again by 1 at each succeeding iteration. By this rule, if Group k becomes FREE as soon as it is ON, and if an element is chosen from Group k at that point, then CC(k) is re-set to 0 (which gives it a value 1 the iteration after it receives the value TC(k)). On the other hand, if no element is selected from Group k (or any younger group) until CC(k) is somewhere between TC(k) and 2TC(k) iterations, the rule for re-setting CC(k) assures that Group k will again become ON when the original (unadjusted) cycle count reaches 2TC(k) iterations. Thus, on the average, this allows the possibility of choosing an element from Group k once during every TC(k) iteration. This process is illustrated in Figure 8. An additional buffer group ("Group 0") may be assumed to be included, although not shown, whose elements are never allowed to escape tabu status, hence which is always OFF. For convenience, a sequence of iterations is shown starting from a point where all cycle counts have been re-set to 1 (iteration 6 1 in the illustration). Since a group is OFF until its cycle count reaches its tabu cycle value, each group begins in the OFF state. A group that is both ON and FREE is shown as FREE, and a FREE group from which an element is selected is indicated by an asterisk. Figure 8 discloses how the choice of an element from a FREE group affects the cycle counts, and hence the states, of each group. Thus, for example, on iteration 64 the choice of an element from Group 2 reduces the cycle counts of both Group 2 and Group 3 by the rule Hence on iteration 65, where these counts are again incremented by 1, their values are shown as CC(2) = 2 and CC(3) = 1. There are a few additional features of this process to emphasize. First the tabu cycles do not have to be integers. A value such as TC(k) = 3.5 can be selected Figurel. Tabu cycle method illustrated. (*New element is selected from the indicated group.) to allow the method, on average, to choose an element from Group k once every 3.5 iterations (hence twice every 7 iterations). The rules remain exactly as specified. A slight elaboration of the rules is required, however, to handle a situation that may occur if no element is selected from Group k or any younger group for a relatively large number of iterations. In this case, CC(k) may attain a value which is several times that of TC(k), causing Group k to remain continuously ON, and hence potentially FREE, until a sufficient number of its elements (or of younger groups) are chosen to bring its value back below TC(k). This leads to the possibility that a series of iterations will occur where elements are repeatedly selected from Group k. (The frequency of selection will be limited however, by the cycle count values of older groups. Hence the greatest risk of inappropriate behavior occurs - when elements are selected entirely outside of ATL for a fairly high number of iterations.) Such a "statistically exceptional" outcome can be guarded against by bounding the value of CC(k), preventing CC(k) from being incremented once it reaches a specified multiple of TC(k)-for example, upon reaching some fraction f of the number of elements assigned to Group k. Similarly, CC(k) should initially be bounded by (fy + l)tc(k) until Group k has acquired y elements. (The multiple fy + I allows Group k to be ON, and hence permits elements to be selected from younger groups, during the initial period where y = 0.) It should be noted for the case of multiple tabu lists that the definition of "iteration" may need to be varied for different lists, if not all types of attributes are present in each move. The Conditional Probability Method The second frequency-based tabu list approach has an orientation similar to the first, but chooses elements from different groups by establishing a probability P(k) that Group k will be FREE on a given iteration. The probability assigned to Group k may be viewed concep- C m t 0 7n111 All W t 4 R P ~ P ~, P ~

18 Tabu Search. I1 21 tually as the inverse of the tabu cycle value TC(k), i.e., P(k) = l/tc(k). As with the treatment of cycle counts in the Tabu Cycle Method, the appropriate treatment of probabilities in the Conditional Probability Method is based on interpreting each P(k) as applicable to the union of Group k with all groups younger than itself. Also, Group k can only be FREE by implicitly requiring all older groups likewise to be FREE. Under the assumption P(k) > P(k - 1) for all k > 1, we generate a conditional probability, CP(k), as a means of determining whether Group k can be designated as FREE. (We may again suppose the existence of a buffer, Group 0, which admits no choice of its elements, hence for which P(0) = 0.) The rule for generating CP(k) is as follows: P(k) - P(k - 1) CP(k) = for k > P(k- 1) Then at each iteration, the process for determining the state of each group starts at k = 1, and proceeds to larger values in succession, designating Group k to be FREE with probability CP(k). If Group k is designated FREE, then all groups with larger k values are also designated FREE and the process stops. Otherwise, the next larger k is examined until all groups have been considered. The derivation of CP(k) is based on the following argument. By interpretation, P(k) represents the probability that Group k or some younger group is the first FREE group, while CP(k) designates the probability that Group k is the first FREE group but no younger will generally fall below P(k), without any compensating increase in freedom to choose elements from these groups (which would potentially allow the average per iteration to come closer to P(k)). Moreover, the same result will occur if for some number of iterations no element is selected from a group as young as Group k. To handle this, the original P(k) values may be replaced by "substitute probabilities" P*(k) in the determination of CP(k). These substitute probabilities make use of the same cycle count values CC(k) used in the Tabu Cycle Method, invoking the relationship TC(k) = l/p(k). To begin, P*(k) = P(k) until CC(k) exceeds TC(k) (bounding CC(k) in early iterations as specified previously). Then P*(k) is allowed to exceed P(k) as an increasing function of the quantity CC(k)/TC(k). In contrast to the Tabu Cycle Method, a negative value can result for CC(k) in this approach, as a result of the update CC(k) := CC(k) - TC(k) (which occurs whenever an element is selected from a Group h, for h =S k). A variation with an interesting interpretation resets the cycle count CC(k) to 0 instead of decrementing it by TC(k). Then CC(k) counts consecutive iterations where no element is chosen from any Group h, h 6 k, an event which may be (loosely) construed as occuning with probability (1 - P(k))', for r = CC(k). Then the substitute probability P*(k) on the following iteration may be established by setting P*(k) = 1 - (1 - P(k))'+' (Note this gives P*(k) = P(k) when r = 0.) Finally, to provide a valid basis for computing CP(k), we require P*(k) 2 P*(k - 1) for k > 1. Hence, group is FREE- The CP(l) = P(l) is appro- beginning with the largest k and working backward, we priate since it is not possible for any group younger bc L than Group 1 to be FREE. In general, for larger values of k, the event that Group k or a younger group is P*(k - 1) := Minimum(P*(k - I), P*(k)). FREE derives from the two exclusive events (assumed independent) where either (1) Group k - 1 or some younger group is the first FREE group, which occurs with probability P(k - l), or (2) this is not the case and Group k is the first FREE group. This gives rise to the formula and solving for CP(k) gives the value specified. Consideration of the rationale underlying the Tabu Cycle Method described earlier, however, shows that an appropriate characterization of the CP(k) probabilities is not yet complete. Designating Group k to be FREE does not imply an element will be selected from this group. - For some number of iterations, elements outside of ATL may be accepted regardless - of the FREE state of groups of elements within ATL. When this occurs, the expected number of elements per iteration chosen from groups no older than any given Group k The substitute probability approach has the advantage of increasing the probability of choosing elements of neglected groups as long as they remain neglected. The variation that re-sets CC(k) = 0 on the update step also automatically avoids the creation of inappropriately high probabilities for the type of situation handled in the Tabu Cycle Method by capping CC(k). By assigning numerical values to tabu attributes, with an associated limit on the sum (or other function) of these values to determine the tabu status of a move, both the Tabu Cycle Method and the Conditional Probability Method can be applied to multiple attribute moves as readily as to single attribute moves. 2. Structured Move Sets and Staged Tabu Search The dynamic tabu list strategies described in the preceding sections offer a variety of opportunities for tailoring methods to particular problem settings by Copyright O 2001 All Rights Reserved

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