Principles of Tabu Search

ion and decomposition (through explicit and attributive memory) Timing Recency of events Frequency of events Differentiation between short term and long term Quality and impact Relative attractiveness of alternative choices Magnitude of changes in structure or constraining Relationships Context Regional interdependence Structural interdependence Sequential interdependence Responsive Exploration Strategically imposed restraints and inducements (tabu conditions and aspiration levels) Concentrated focus on good regions and good solution features (intensification processes) Characterizing and exploring promising new regions (diversification processes) Nonmonotonic search patterns (strategic oscillation) Integrating and extending solutions (path relinking) In this chapter we will describe some key aspects of this methodology, as the use of memory structures and search strategies, and illustrate them in an implementation to solve the linear ordering problem (LOP). 23.2 Memory Structures Tabu search begins in the same way as ordinary local or neighborhood search, proceeding iteratively from one point (solution) to another until a chosen termination criterion is satisfied. Each solution x has an associated neighborhood N (x) ⊂ X , and each solution x ′ ∈ N (x) is reached from x by an operation called a move. We may contrast TS with a simple descent method where the goal is to minimize f (x). Such a method only permits moves to neighbor solutions that improve the current objective function value and ends when no improving solutions can be found. The final x obtained by a descent method is called a local optimum, since it is at least as good as or better than all solutions in its neighborhood. The evident shortcoming of a descent method is that such a local optimum in most cases will not be a global optimum, that is, it usually will not minimize f (x) over all x ∈ X. Tabu search permits moves that deteriorate the current objective function value but the moves are chosen from a modified neighborhood N∗(x). Shortand long-term memory structures are responsible for the specific composition of N∗(x). In other words, the modified neighborhood is the result of maintaining a selective history of the states encountered during the search. In the TS strategies based on short-term considerations, N∗(x) characteristically is a subset of N(x), and the tabu classification serves to identify elements of N(x) excluded from N∗(x). In TS strategies that include longer term considerations, N∗(x) may also be expanded to include solutions not ordinarily found in N(x), such as © 2007 by Taylor & Francis Group, LLC C5505 C5505 ̇C023 March 20, 2007 13:28 23-4 Handbook of Approximation Algorithms and Metaheuristics TABLE 23.3 Examples of Recency-Based Memory Context Attributes To Record the Last Time . . . Binary problems Variable index (i) Variable i changed its value from 0 to 1 or 1 to 0 (depending on its current value) Job sequencing Job index ( j ) Job j changed positions Job index ( j ) and position (p) Job j occupied position p Pair of job indexes (i , j ) Job i exchange positions with job j Graphs Arc index (i) Arc i was added to the current solution Arc i was dropped from the current solution solutions found and evaluated in past search, or identified as high-quality neighbors of these past solutions. Characterized in this way, TS may be viewed as a dynamic neighborhood method. This means that the neighborhood of x is not a static set, but rather a set that can change according to the history of the search. The structure of a neighborhood in TS differs from that used in local search in an additional manner, by embracing the types of moves used in constructive and destructive processes (where the foundations for such moves are accordingly called constructive neighborhoods and destructive neighborhoods). Such expanded uses of the neighborhood concept reinforce a fundamental perspective of TS, which is to define neighborhoods in dynamic ways that can include serial or simultaneous consideration of multiple types of moves. Tabu search uses attributive memory for guiding purposes (i.e., to compute N∗(x)). Instead of recording full solutions, attributive memory structures are based on recording attributes. This type of memory records information about solution properties (attributes) that change in moving from one solution to another. The most common attributive memory approaches are recencyand frequency-based memory. Recency, as its name suggests, keeps track of solutions attributes that have changed during the recent past. Frequency typically consists of ratios about the number of iterations a certain attribute has changed or not (depending whether it is a transition or a residence frequency). Some examples of recencyand frequencybased memory are shown in Table 23.3 and Table 23.4 respectively. Characteristically, a TS process based strictly on short-term strategies may allow a solution x to be visited more than once, but it is likely that the corresponding reduced neighborhood N∗(x) will be different each time. With the inclusion of longer term considerations, the likelihood of duplicating a previous neighborhood upon revisiting a solution, and more generally of making choices that repeatedly visit only a limited subset of X , is all but nonexistent. Recency-based memory is the most common memory structure used in TS implementations. As its name suggests, this memory structure keeps track of solutions attributes that have changed during the recent past. To exploit this memory, selected attributes that occur in solutions recently visited are labeled TABLE 23.4 Examples of Frequency-Based Memory Context Residence Measure Transition Measure Binary problems Number of times variable i has been Number of times variable i has changed assigned the value of 1 values Job sequencing Number of times job j has occupied Number of times job i has exchanged position p positions with job j Average objective function value Number of times job j has been moved to when job j occupies position p an earlier position in the sequence Graphs Number of times arc i has been part Number of times arc i has been deleted of the current solution from the current solution when arc j has been added Average objective function value Number of times arc i has been added when arc i is part of the solution during improving moves © 2007 by Taylor & Francis Group, LLC C5505 C5505 ̇C023 March 20, 2007 13:28 Principles of Tabu Search 23-5 tabu-active, and solutions that contain tabu-active elements, or particular combinations of these attributes, are those that become tabu. This prevents certain solutions from the recent past from belonging to N∗(x) and hence from being revisited. Other solutions that share such tabu-active attributes are also similarly prevented from being visited. Note that while the tabu classification strictly refers to solutions that are forbidden to be visited, by virtue of containing tabu-active attributes (or more generally by violating certain restriction based on these attributes), moves that lead to such solutions are also often referred to as being tabu. Frequency-based memory provides a type of information that complements the information provided by recency-based memory, broadening the foundation for selecting preferred moves. Like recency, frequency often is weighted or decomposed into subclasses. Also, frequency can be integrated with recency to provide a composite structure for creating penalties and inducements that modify move evaluations. Frequencies typically consist of ratios, whose numerators represent counts expressed in two different measures: a transition measure—the number of iterations where an attribute changes (enters or leaves) the solutions visited on a particular trajectory, and a residence measure—the number of iterations where an attribute belongs to solutions visited on a particular trajectory, or the number of instances where an attribute belongs to solutions from a particular subset. The denominators generally represent one of three types of quantities: (1) the total number of occurrences of all events represented by the numerators (such as the total number of associated iterations); (2) the sum (or average) of the numerators; and (3) the maximum numerator value. In cases where the numerators represent weighted counts, some of which may be negative, denominator (3) is expressed as an absolute value and denominator (2) is expressed as a sum of absolute values (possibly shifted by a small constant to avoid a zero denominator). The ratios produce transition frequencies that keep track of how often attributes change, and residence frequencies that keep track of how often attributes are members of solutions generated. In addition to referring to such frequencies, thresholds based on the numerators alone can be useful for indicating when phases of greater diversification are appropriate. 23.3 Search Strategies The use of recency and frequency memory in TS generally fulfills the function of preventing searching processes from cycling, that is, from endlessly executing the same sequence of moves (or more generally, from endlessly and exclusively revisiting the same set of solutions). More broadly, however, the various manifestations of these types of memory are designed to impart additional robustness or vigor to the search. A key element of the adaptive memory framework of TS is to create a balance between search intensification and diversification. Intensification strategies are based on modifying choice rules to encourage move combinations and solution features historically found good. They may also initiate a return to attractive regions to search them more thoroughly. Diversification strategies, however, seek to incorporate new attributes and attribute combinations that were not included within solutions generated previously. In one form, these strategies undertake to drive the search into regions dissimilar to those already examined. It is important to keep in mind that intensification and diversification are not mutually opposing, but are rather mutually reinforcing. Most types of intensification strategies require a means for identifying a set of elite solutions as basis for incorporating good attributes into newly created solutions. Membership in the elite set is often determined by setting a threshold that is connected to the objective function value of the best solution found during the search. A simple instance of the intensification strategy is shown in Figure 23.1. Two simple variants for elite solution selection have proved quite successful. One introduces a diversification measure to assure the solutions recorded differ from each other by a desired degree, and then erases all short-term memory before resuming from the best of the recorded solutions. The other keeps a bounded length sequential list that adds a new solution at the end only if it is better than any previously seen, and the short-term memory that accompanied this solution is also saved. © 2007 by Taylor & Francis Group, LLC C5505 C5505 ̇C023 March 20, 2007 13:28 23-6 Handbook of Approximation Algorithms and Metaheuristics Apply TS short-term memory Apply an elite selection strategy. do { Choose one of the elite solutions. Resume short-term memory TS from chosen solution. Add new solutions to elite list when applicable. } while (iterations < limit and list not empty) FIGURE 23.1 Simple TS intensification approach. Diversification is automatically created in TS (to some extent) by short-term memory functions, but is particularly reinforced by certain forms of longer term memory. TS diversification strategies are often based on modifying choice rules to bring attributes into the solution that are infrequently used. Alternatively, they may introduce such attributes by periodically applying methods that assemble subsets of these attributes into candidate solutions for continuing the search, or by partially or fully restarting the solution process. Diversification strategies are particularly helpful when better solutions can be reached only by crossing barriers or “humps” in the solution space topology. The incorporation of modified choice rules can be moderated by using the following penalty function: MoveValue′ = MoveValue + d ∗ Penalty This type of penalty approach is commonly used in TS, where the Penalty value is often a function of frequency measures such as those indicated in Table 23.2, and d is an adjustable diversification parameter. Larger d values correspond to a desire for more diversification. 23.4 Advanced Designs: Strategic Oscillation and Path Relinking There are many forms in which a simple TS implementation can be improved by adding long-term elements. In this paper we restrict our attention to two of the most used methods, namely strategic oscillation and path relinking (PR), which constitute the core of many adaptive memory programming algorithms. Strategic oscillation operates by orienting moves in relation to a critical level, as identified by a stage of construction or a chosen interval of functional values. Such a critical level or oscillation boundary often represents a point where the method would normally stop. Instead of stopping when this boundary is reached, however, the rules for selecting moves are modified, to permit the region defined by the critical level to be crossed. The approach then proceeds for a specified depth beyond the oscillation boundary, and turns around. The oscillation boundary again is approached and crossed, this time from the opposite direction, and the method proceeds to a new turning point (see Figure 23.2). Iterations 0 Oscillation boundary Depth L ev el o r fu nc tio na l v al ue