On the Use of Equivalence Classes for Optimal and Suboptimal Bin Packing and Bin Covering

Bin packing and bin covering are important optimization problems in many industrial fields, such as packaging, recycling, and food processing. The problem concerns a set of items, each with its own value, that are to be sorted into bins in such a way that the total value of each bin, as measured by the sum of its item values, is not above (for packing) or below (for covering) a given target value. The optimization problem concerns minimizing, for bin packing, or maximizing, for bin covering, the number of bins. This is a combinatorial NP-hard problem, for which true optimal solutions can only be calculated in specific cases, such as when restricted to a small number of items. To get around this problem, many suboptimal approaches exist. This article describes the formulations of the bin packing and covering problems that allow finding the true optimum, for instance, counting hundreds of items using general-purpose MILP-solvers. Also presented are suboptimal solutions that come within less than 10% of the optimum while taking significantly less time to calculate, even ten to 100 times faster, depending on the required accuracy. Note to Practitioners - A typical case for bin covering is in food processing where food items are automatically sorted into trays of similar weight so that the overweight is minimized. Another application is in recycling, where items such as batteries should be put in crates of similar weight, so that the crates do not exceed a target weight due to later manual handling, but, at the same time, we want as few crates as possible. This is a bin packing problem. On an industrial scale, these tasks are fully automated. Though modern software tool\u27s efficiency to solve bin sorting problems has increased significantly in later years, the problems are inherently tough in the sense that the solution time grows exponentially with the number of items. This limits the problem sizes that can be solved to optimality within a reasonable time. Therefore, much research has focused on heuristic rules that give reasonable solving times while not giving the true optimal number of bins. However, in many cases, the true optimal solution is preferable, and sometimes even necessary, so this is an industrially interesting problem. This article describes an approach to solve the bin packing and covering problems to the true optimum that increases the limit of the number of items that can typically be handled. This is done by observing that items of the same value need not be distinguished. Instead, we can formulate packing/covering problems over item values rather than individual items and sort integer numbers of these values into bins, which allows us to solve to optimum for more than 500 items in a reasonable time. In addition, by redefining what we mean by the same value, we can consider more items to have the same value and achieve even better calculation efficiency

Constraint handling strategies in Genetic Algorithms application to optimal batch plant design

Optimal batch plant design is a recurrent issue in Process Engineering, which can be formulated as a Mixed Integer Non-Linear Programming(MINLP) optimisation problem involving specific constraints, which can be, typically, the respect of a time horizon for the synthesis of various products. Genetic Algorithms constitute a common option for the solution of these problems, but their basic operating mode is not always wellsuited to any kind of constraint treatment: if those cannot be integrated in variable encoding or accounted for through adapted genetic operators, their handling turns to be a thorny issue. The point of this study is thus to test a few constraint handling techniques on a mid-size example in order to determine which one is the best fitted, in the framework of one particular problem formulation. The investigated methods are the elimination of infeasible individuals, the use of a penalty term added in the minimized criterion, the relaxation of the discrete variables upper bounds, dominancebased tournaments and, finally, a multiobjective strategy. The numerical computations, analysed in terms of result quality and of computational time, show the superiority of elimination technique for the former criterion only when the latter one does not become a bottleneck. Besides, when the problem complexity makes the random location of feasible space too difficult, a single tournament technique proves to be the most efficient one