An Optimization Algorithm for the Ordered Open-End Bin-Packing Problem

Abstract

The ordered open-end bin packing problem is a variant of the bin packing problem in which the items to be packed are sorted in a given order and the capacity of each bin can be exceeded by the last item packed into the bin. We present a branch-and-price algorithm for its exact optimization. The pricing subproblem is a special variant of the binary knapsack problem, in which the items are ordered and the last one does not consume capacity. We present a specialized optimization algorithm for this subproblem. The speed of the column generation algorithm is improved by subgradient optimization steps, allowing for multiple pricing and variable fixing. Computational results are presented on instances of different size and items with different correlations between their size and their position in the given order.

Modeling and Solving the Open-End Bin Packing Problem

“…Thereafter, Yang and Leng [4] studied the Ordered Open Bin Packing Problem which extends the requirement corresponding to the order of passenger's itinerary in subway station problem. The same problem, which is the Ordered Open-End Bin Packing Problem, have been also studied by Cesili and Righini [1] where authors present branch and price algorithm for its exact optimization.…”

“…Ceselli's lower bound: Ceselli and Righini [1] proposed a combinatorial lower bound algorithm called in what follows CBA. This lower bound computes a set of overflow items iteratively in an optimal fractional packing by the following way; the authors define R to be the overall residual capacity of all the bins already initialized, whenever an item j is found, whose size is greater than R, a new bin is initialized and the overflow item of this bin is selected as the largest item among those already packed but not yet used as overflow items, then insert item j into the set O of overflow items in order to yield the maximum residual capacity for the next iteration.…”

“…In this section, we propose a modification of the integer linear program initially proposed by Cesili and Righini [1]. The authors studied a variant of the open bin packing problem called the Ordered Open-End Bin Packing Problem (OOEBPP) in which items to be packed are sorted in a given order.…”

“…Whilst, a few work carried out on this problem, we present a modification of the 0-1 linear programming formulation proposed by Ceselli and Righini [1] in which the authors studied a variant of the open-end bin packing called Ordered Open-End Bin Packing Problem (OOEBBP). Then we present our contributions based on the proposition of two greedy heuristics for solving the problem in an offline mode while ensuring high quality solutions in very short computational time compared to solving the problem in optimal way.…”

“…The objective is to minimize the number of used bins. In this paper, we present our Integer Linear Program model based on a modification of Cesili and Righini model [1]. Also, we propose two greedy heuristics to solve a problem.…”