A parallel simulated annealing algorithm as a tool for fitness landscapes exploration

Solving a discrete optimization problem consists in finding a solution which maximizes (or minimizes) an objective function. The function is often called the fitness and the corresponding landscape the fitness landscape. We are concerned with statistical measures of a fitness landscape in the context of the vehicle routing problem with time windows (VRPTW). The measures are determined by using a parallel simulated annealing algorithm as a tool for exploring a solution space. This chapter summarizes our experience in designing parallel simulated annealing algorithms and investigating fitness landscapes of a sample NP-hard bicriterion optimization problem. Since 2002 we have developed several versions of the parallel simulated annealing (SA) algorithm (11)-(19). Each of these versions comprises a number of parallel SA processes which co-operate periodically by passing and exploiting the best solutions found during the search. For this purpose a specific scheme of co-operation of processes has been devised. The methods of parallelization of simulated annealing are discussed in Aarts and van Laarhoven (2), Aarts and Korst (1), Greening (20), Abramson (3), Boissin and Lutton (8), and Verhoeven and Aarts (35). Parallel simulated annealing to solve the VRPTW is applied by Arbelaitz et al. (4). Onbaşoğlu and Özdamar (26) present the applications of parallel simulated annealing algorithms in various global optimization problems. The comprehensive study of parallelization of simulated annealing is given by Azencott et al. (5) The parallel SA algorithm allowed us to discover the landscape properties of the VRPTW benchmarking tests (33). This knowledge not only increased our understanding of processes which happen during optimization, but also helped to improve the performance of the parallel algorithm. The usage of the landscape notion is traced back to the paper by Wright (37). The more formal treatments of the landscape properties are given by Stadler (32), Hordijk and Stadler (22), Reidys and Stadler (31). Statistical measures of a landscape are proposed by Weinberger (36). The reviews of the landscape issues are given by Reeves (30) and Reeves and Rowe (29). Section 2 of this chapter formulates the optimization problem which is solved. Section 3 describes a sequential SA algorithm. In section 4 two versions of the parallel SA algorithm, called independent and co-operating searches, are presented. Section 5 is devoted to the statistical measures of the fitness landscapes in the context of the VRPTW. In subsections 5.1-5.2 some basic notions are introduced, and in subsection 5.3 the results of the experimental study are discussed. Section 6 concludes the chapter. 13