Exact and heuristic solutions for combinatorial optimization problems

Abstract

This is a summary of the Ph.D. thesis defended by the author on July 2012 at the University of Modena and Reggio Emilia. The thesis was supervised by Mauro Dell’Amico and co-supervised by Manuel Iori. The manuscript is written in English and is available from the author upon request at jose.diaz@unimore.it. The thesis addresses relevant optimization problems: the Bin Packing Problem, the Quadratic Assignment Problem, the large-scale Energy Management Problem, and the Node, Edge and Arc Routing Problem. To solve these problems exact, heuristic and local search techniques are proposed. The first problem studied is the Bin Packing Problem with Precedence Constraints (BPP-P): given a set of identical capacitated bins, a set of weighted items and a set of precedences among those items, we are interested in determining the minimum number of bins that can accommodate all items and can be ordered in such a way that all precedences are satisfied. The problem has a very intriguing combinatorial structure and models many assembly and scheduling issues. According to our knowledge the BPP-P has received little attention in the literature, and in the thesis we address it for the first time with exact solution methods. In particular, we develop reduction criteria, a large set of lower bounds, a variable neighborhood search upper bounding technique and a branch-and-bound algorithm (see Dell’Amico M., Diaz Diaz J.C. and Iori M. (2012): The Bin Packing Problem with Precedence Constraints, Operations Research, doi:10.1287/opre.1120.1109). We show the effectiveness of the proposed algorithms by means of extensive computational tests on benchmark instances and comparison with standard integer linear programming techniques. The second problem studied is to find the Friendly Bin Packing Instances without Integer Round-up Property. It is well known that the gap between the optimal values of bin packing and fractional bin packing, if the latter is rounded up to the closest integer, is almost always null. Known counterexamples to this for integer input values involve fairly large numbers. Specifically, the first one was derived in 1986 and involved a bin capacity of the order of a billion. Later in 1998 a counterexample with a bin capacity of the order of a million was found. In the thesis we show a large number of counterexamples with bin capacity of the order of a hundred, showing that the gap may be positive even for numbers which arise in customary applications. The associated instances are constructed starting from the Petersen graph and taking advantage of the fact that it is fractionally, but not integrally, 3-edge colorable. The third problem studied is the Single-finger Keyboard Layout Problem. The problem of designing new keyboards layouts able to improve the typing speed of an