Introduction

Abstract

AbstractThis chapter introduces the basic definitions, main elements, applications, and instances of two optimization problems, the linear ordering problem (LOP), and the maximum diversity problem (MDP). We will use them in the next chapters to describe heuristics, meta-heuristics and exact approaches, and to report our experiments. The LOP is one of the classical combinatorial optimization problems which was already classified as NP-hard in 1979 by Garey and Johnson [64]. It has received considerable attention in various application areas ranging from archeology and scheduling to economics. Solution methods for the LOP have been proposed since 1958, when Chenery and Watanabe outlined some ideas on how to obtain solutions for this problem. The interest in this problem has continued over the years, resulting in the book [143] and many recent papers in scientific journals. This chapter surveys the main LOP applications and benchmark library of instances LOLIB. The challenge of maximizing the diversity of a collection of points arises in a variety of settings, from location to genetics. The growing interest of dealing with diversity translated into mathematical models and computer algorithms in the late eighties, when Michael Kuby studied dispersion maximization in general graphs [54]. The MDP is the first model proposed and the most studied to deal with diversity, and was classified as NP-hard. Many optimization methods have been proposed to obtain efficient solutions to this problem, which makes it especially convenient as an illustrative example in this book. This chapter surveys the different models proposed to maximize diversity, their applications, and the benchmark library of instances MDPLIB.