scholarly journals The integrated uncapacitated lot sizing and bin packing problem

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.

scholarly journals Optimizing Inventory Replenishment for Seasonal Demand with Discrete Delivery Times

2021 ◽  
Vol 11 (23) ◽  
pp. 11210
Author(s):  
Mohammed Alnahhal ◽  
Diane Ahrens ◽  
Bashir Salah
Keyword(s):  
Lot Sizing ◽  
Planning Horizon ◽  
Service Level ◽  
Mixed Integer ◽  
Lot Size ◽  
Dynamic Planning ◽  
Holding Costs ◽  
Delivery Times ◽  
Mip Model ◽  
Total Inventory

This study investigates replenishment planning in the case of discrete delivery time, where demand is seasonal. The study is motivated by a case study of a soft drinks company in Germany, where data concerning demand are obtained for a whole year. The investigation focused on one type of apple juice that experiences a peak in demand during the summer. The lot-sizing problem reduces the ordering and the total inventory holding costs using a mixed-integer programming (MIP) model. Both the lot size and cycle time are variable over the planning horizon. To obtain results faster, a dynamic programming (DP) model was developed, and run using R software. The model was run every week to update the plan according to the current inventory size. The DP model was run on a personal computer 35 times to represent dynamic planning. The CPU time was only a few seconds. Results showed that initial planning is difficult to follow, especially after week 30, and the service level was only 92%. Dynamic planning reached a higher service level of 100%. This study is the first to investigate discrete delivery times, opening the door for further investigations in the future in other industries.

Models and Algorithms for the Bin-Packing Problem with Minimum Color Fragmentation

2021 ◽  
Author(s):  
Saharnaz Mehrani ◽  
Carlos Cardonha ◽  
David Bergman
Keyword(s):  
Integer Programming ◽  
Bin Packing ◽  
Packing Problem ◽  
Mixed Integer ◽  
Decision Diagrams ◽  
Branch And Price ◽  
Recursive Decomposition ◽  
Computational Performance ◽  
Bin Packing Problem ◽  
Decomposition Strategies

In the bin-packing problem with minimum color fragmentation (BPPMCF), we are given a fixed number of bins and a collection of items, each associated with a size and a color, and the goal is to avoid color fragmentation by packing items with the same color within as few bins as possible. This problem emerges in areas as diverse as surgical scheduling and group event seating. We present several optimization models for the BPPMCF, including baseline integer programming formulations, alternative integer programming formulations based on two recursive decomposition strategies that utilize decision diagrams, and a branch-and-price algorithm. Using the results from an extensive computational evaluation on synthetic instances, we train a decision tree model that predicts which algorithm should be chosen to solve a given instance of the problem based on a collection of derived features. Our insights are validated through experiments on the aforementioned applications on real-world data. Summary of Contribution: In this paper, we investigate a colored variant of the bin-packing problem. We present and evaluate several exact mixed-integer programming formulations to solve the problem, including models that explore recursive decomposition strategies based on decision diagrams and a set partitioning model that we solve using branch and price. Our results show that the computational performance of the algorithms depends on features of the input data, such as the average number of items per bin. Our algorithms and featured applications suggest that the problem is of practical relevance and that instances of reasonable size can be solved efficiently.

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

2019 ◽  
Vol 11 (02) ◽  
pp. 1950022
Author(s):  
Qingqin Nong ◽  
Jiapeng Wang ◽  
Suning Gong ◽  
Saijun Guo
Keyword(s):  
Cooperative Game ◽  
Bin Packing ◽  
Social Cost ◽  
Price Of Anarchy ◽  
Packing Problem ◽  
Cardinality Constraints ◽  
Bin Packing Problem ◽  
The Social ◽  
The Cost

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].

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

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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