A bin packing game with cardinality constraints under the best cost rule

We consider the bin packing problem with cardinality constraints in a non-cooperative game setting. In the game, there are a set of items with sizes between 0 and 1, and a number of bins each of which has a capacity of 1. Each bin can pack at most [Formula: see text] items, for a given integer parameter [Formula: see text]. The social cost is the number of bins used in the packing. Each item tries to be packed into one of the bins so as to minimize its cost. The selfish behaviors of the items result in some kind of equilibrium, which greatly depends on the cost rule in the game. We say a cost rule encourages sharing if for an item, compared with sharing a bin with some other items, staying in a bin alone does not decrease its cost. In this paper, we first show that for any bin packing game with cardinality constraints under an encourage-sharing cost rule, the price of anarchy of it is at least [Formula: see text]. We then propose a cost rule and show that the price of anarchy of the bin packing game under the rule is [Formula: see text] when [Formula: see text].

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Improved Lower Bounds for the Online Bin Packing Problem with Cardinality Constraints

Lecture Notes in Computer Science - Computing and Combinatorics ◽

10.1007/978-3-642-38768-5_46 ◽

2013 ◽

pp. 518-530 ◽

Cited By ~ 2

Author(s):

Hiroshi Fujiwara ◽

Koji Kobayashi

Keyword(s):

Lower Bounds ◽

Bin Packing ◽

Packing Problem ◽

Cardinality Constraints ◽

Bin Packing Problem

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The integrated uncapacitated lot sizing and bin packing problem

RAIRO - Operations Research ◽

10.1051/ro/2021049 ◽

2021 ◽

Author(s):

Natã Goulart ◽

Thiago Ferreira de Noronha ◽

Martin Gomez Ravetti ◽

Mauricio Cardoso de Souza

Keyword(s):

Bin Packing ◽

Lot Sizing ◽

Packing Problem ◽

Planning Horizon ◽

Mixed Integer ◽

Holding Costs ◽

Bin Packing Problem ◽

Combinatorial Relaxation ◽

The Cost ◽

Inventory Holding

In the integrated uncapacitated lot sizing and bin packing problem, we have to couple lot sizing decisions of replenishment from single product suppliers with bin packing decisions in the delivery of client orders. A client order is composed of quantities of each product, and the quantities of such an order must be delivered all together no later than a given period. The quantities of an order must all be packed in the same bin, and may be delivered in advance if it is advantageous in terms of costs. We assume a large enough set of homogeneous bins available at each period. The costs involved are setup and inventory holding costs and the cost to use a bin as well. All costs are variable in the planning horizon, and the objective is to minimize the total cost incurred. We propose mixed integer linear programming formulations and a combinatorial relaxation where it is no longer necessary to keep track of the specific bin where each order is packed. An aggregate delivering capacity is computed instead. We also propose heuristics using different strategies to couple the lot sizing and the bin packing subproblems. Computational experiments on instances with different configurations showed that the proposed methods are efficient ways to obtain small optimality gaps in reduced computational times.

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fuzzy approach for the variable cost and size bin packing problem allowing incomplete packing

INTELIGENCIA ARTIFICIAL ◽

10.4114/intartif.vol24iss67pp71-89 ◽

2021 ◽

Vol 24 (67) ◽

pp. 71-89

Author(s):

Jorge Herrera-Franklin ◽

Alejandro Rosete ◽

Milton García-Borroto

Keyword(s):

Bin Packing ◽

Solution Method ◽

Optimization Problems ◽

Real Life ◽

Packing Problem ◽

Variable Cost ◽

Bin Packing Problem ◽

Trade Offs ◽

Np Hard Problem ◽

The Cost

The Variable Cost and Size Bin Packing Problem (VCSBPP) is a known NP-Hard problem that consists in minimizing the cost of all bins used to pack a set of items. There are many real-life applications of the VCSBPP where the focus is to improve the efficiency of the solution method. In spite of the existence of fuzzy approaches to adapt other optimization problems to real life conditions, VCSBPP has not been extensively studied in terms of relaxations of the crisp conditions. In this sense, the fuzzy approaches for the VCSBPP varies from relaxing the capacity of the bins to the items weights. In this paper we address a non-explored side consisting in relaxing the set of items to be packed. Therefore, our main contribution is a fuzzy version of VCSBPP that allows incomplete packing. The proposed fuzzy VCSBPP is solved by a parametric approach. Particularly, a fast heuristic algorithm is introduced that allows to obtain a set of solutions with interesting trade-offs between cost and relaxation of the original crisp conditions. An experimental study is presented to explore the proposed fuzzy VCSBPP and its solution.

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