Local search heuristics are an important class of algorithms for obtaining goodsolutions for hard combinatorial optimization problems. Important issues forthese heuristics are solution quality and the time needed to obtain a good solution.
Roughly speaking, a local search heuristic starts with an initial solution, e.g.obtained with the help of a constructive heuristic. The local search heuristic then iteratively explores a neighborhood of the current solution and chooses a new solution out of this neighborhood to become the next current solution. Often, only new solutions are considered, which improve on the current one. The local search heuristic halts if some stopping criterion is met, e.g. there exists no better solution in the neighborhood.
The neighborhood is a critical issue in a local search heuristic. It directly influences the local behavior of a local search heuristic, since it restricts the choice of a new solution in a single iteration. This determines the local efficiency of a local search heuristic. On the other side, this local behavior also influences the global behavior, i.e. the sequence of feasible solutions obtained by the local search heuristic. This sequence of solutions represents the navigational behavior of the local search heuristic and determines the global efficiency of a local search heuristic.
A key element is the size of a neighborhood. The size determines how many choices there are to find a better solution. This also designates the time the heuristic needs in every iteration. Since with smaller neighborhoods we may not have (directly or indirectly) considered many solutions at the end of a local search heuristic, the size may also have an influence on the solution quality from a global point of view.
In the first part of the thesis, we consider neighborhoods of large size that can be explored in a fast manner, in order to reach local optima of good quality in less time. We consider the scheduling problems of minimizing the sum of job completion times on a single machine with the presence of release dates, and of minimizing the sum of weighted job completion times on parallel and identical machines. By concentrating on the local efficiency, we develop very large-scale neighborhoods for these two scheduling problems. For all the introduced neighborhoods, we explain what solutions they contain and how the neighborhood is efficiently explored to obtain a better neighbor. We also compare their performance to basic neighborhoods regarding solution quality and running time by conducting computational tests.
For the single machine problem, we first introduce a neighborhood that bases on combining independent operators. These operators form a smaller neighborhood, and by combining several operators we obtain a very large-scale neighborhood. A best neighbor in this very large-scale neighborhood can be obtained by determining a shortest path in an improvement graph. In the computational test, this neighborhood does not perform well. We present a second very large-scale neighborhood that bases on a dominance rule for the considered problem.Here, a best neighbor can be obtained by sorting. In the computational test,this neighborhood is fast but regarding solution quality, it is only successful in combination with another neighborhood.
For the parallel machine problem, we also present a very large-scale neighborhood that bases on combining independent operators of a smaller neighborhood. But here, the best neighbor is obtained by determining a maximum weight matching in an improvement graph. In the computational test, this neighborhood is slightly faster than the underlying smaller neighborhood. A second very large-scale neighborhood is introduced that bases on a restricted version of the considered problem, which can be solved in polynomial time. The best neighbor of this neighborhood is obtained by determining a minimum weight perfect matching. In the computational test, this neighborhood performs comparable to the smaller neighborhood used for obtaining the first very-large scale neighborhood.
In the second part of the thesis, we examine the global efficiency in terms of solution quality obtained at the end of a local search heuristic. We consider the problems of scheduling jobs on parallel identical machines in order to minimize the sum of weighted job completion times and of minimizing the makespan (completion time of last planned job). For these problems, we consider certain neighborhoods and analyze the quality of their local optima. Although the found performance guarantees are worse compared to constructive heuristics (however, for our cases the differences are quite small), the advantage of giving performance guarantees for local optima is that these guarantees hold for all solutions which are locally optimal and not only for a single solution as for most constructive heuristics. It is interesting to see that already simple structured solutions like the local optima allow a reasonable performance guarantee. Since local search heuristics like simulated annealing or tabu search tend to visit a lot of local optima during the search process, we may expect that the best found solution has a much better quality than that given by the performance guarantee.