(A) formulation and heuristic algorithms of bin packing problems based on the fuzzy set theory

Abstract

Many interesting problems in computer systems, operations research, and production systems are transformed into rectangle packing problems known as bin packing problems. Provided that a list of rectangles and a rectangular space called a bin are given, the objective of the bin packing problem is to minimize the total height of the packing. As an optimization problem over geometric objects, the bin packing problem assumes strict satisfaction of constraints. In real world situations, however, a certain level of inaccuracy or violation among constraints is usually permitted because estimation is uncertain and some constraints do not need to be satisfied strictly. This thesis proposes a formulation of the bin packing problem based on the fuzzy set theory, named as the fuzzy bin packing problem. The problem is defined as a fuzzy optimization problem, which maximizes the user``s satisfaction level, which is defined in terms of fuzzy sets. The fuzzy bin packing problem uses fuzzy sets for representing the widths and heights of the rectangles. The boundaries of the bin are also represented by fuzzy sets. The objective of the fuzzy bin packing problem is to maximize the satisfaction level, which compromises the packing height with the violations among constraints. Heuristics for finding feasible solutions are also proposed. These heuristics search for the lowest and leftmost positions which do not decrease the total satisfaction when the given piece is placed. An application to an industrial scheduling problem, called nozzle allocation scheduling problem, is presented as an illustration of the usefulness of the proposed problem and related algorithms.