A Tabu Search Framework for Dynamic Combinatorial Optimization Problems

Combinatorial Optimization problems are often computationally expensive. Due to the NP-complete nature of such problems, finding an optimal solution is impractical. Many generic techniques have been developed to approximate such problems and find reasonably good or partial solutions. A few solutions examined apply these techniques to dynamic optimization problems where the domain is subject to frequent change. Dynamic problems are much closer to the real world and span many disciplines. However, most solutions are highly specific and work only within a precise domain with a predetermined set of changes. We provide a generic object-oriented approach for solving dynamic combinatorial optimization problems using a reactive Tabu Search. We claim that changes in a problem set, in the real world, do not always affect an entire solution set and should therefore not mandate restarting the algorithm. We show an example of this approach used in a real world dynamic vehicle scheduling and routing problem and define a class of problems, spanning many disciplines including operations research and logistics, that will lend themselves well to this approach. DEFINITION OF COMBINATORIAL OPTIMIZATION RELATED TO MANY DISCIPLINES A common definition of combinatorial optimization problems are those problems that are concerned with the efficient allocation of limited resources to meet desired objectives when the values of some or all of the variables are restricted to be integral (Hoffman, 2005). For the purpose of this work we take a similar but more simplified view. Combinatorial optimization is the process of finding the best combination of a set of atomic entities. This is summed up in the following: Given a set S = {a1, a2, ..., an} where ai is an atomic entity, find an optimal ordering or subset of S. This process can easily be seen as exponential since the number of possible sub-sets of any set is equal to 2 where n is the number of atomic entities in the set. Thus looking at a fairly simple example of twenty possible entities results in over a million possible combinations. Real-world problems typically have far more than twenty atomic entities being optimized. We consider a vehicle routing problem that can easily have several thousand entities defined as trips. This quickly results in an astronomical number of combinations. Considering all possible solutions is impossible within a reasonable amount of time. This is not a new problem and is a large area of study in computer science and other fields (Crescenzi, et al., 1997). It turns out that satisficing solutions are possible that can produce 'good enough' solutions. The Web Dictionary for Cybernetics and Systems (2006) provides the following definition of satisficing. Satisficing is an alternative to optimization for cases where there are multiple and competitive objectives in which one gives up the idea of obtaining a "best" solution. In this approach one sets lower bounds for the various objectives that, if attained, will be "good enough" and then seeks a solution that will exceed these bounds. The satisficer's philosophy is that in real-world problems there are too many uncertainties and conflicts in values for there to be any hope of obtaining a true optimization and that it is far more sensible to set out to do "well enough" (but better than has been done previously). Heuristics-based approaches are often used that provide several satisfactory solutions based on multiple objectives (Eracar, 2005). Glover and Laguna (1993, 1997) have generalized much of the functionality of these methods into a general use search algorithm entitled Tabu Search (Glover, 1986). Most of the solutions we have examined assume a static world. They compute on a set of input data that is not expected to change. Some work has been done on extending these solutions to allow for dynamic behavior (Bartel, 2004; Randall, 2002; Qili, 2000). There has been some work on