Bin Capacity Research Articles

A hybrid VNS matheuristic for a bin packing problem with a color constraint

We study a new variant of the bin packing problem with a color constraint. Given a finite set of items, each item has a set of colors. Each bin has a color capacity, the total number of colors for a bin is the unification of colors for its items and cannot exceed the bin capacity. We need to pack all items into the minimal number of bins. For this NP-hard problem, we present approximability results and design a hybrid matheuristic based on the column generation technique. A hybrid VNS heuristic is applied to the pricing problem. The column generation method provides a lower bound and a core subset of the most promising bin patterns. Fast heuristics and exact solution for this core produce upper bounds. Computational experiments for test instances with number of items up to 500 illustrate the efficiency of the approach.

Bin Packing Problem with Conflicts and Item Fragmentation

In this paper, we study the Bin Packing Problem with Conflicts and Item Fragmentation (BPPC-IF) which has applications in the delivery and storage of items that cannot be packed together. Given a set of items each with a certain size, the goal in BPPC-IF is to pack these items into a minimum number of fixed-capacity bins while not packing fragments of conflicting items into the same bin. We assume a size-preserving fragmentation, i.e., the total size of fragments of an item packed into the bins has to be equal to the item’s original size. We first prove that BPPC-IF is still NP-hard even though items can be fragmented. Unlike the Bin Packing Problem with Item Fragmentation (BPPIF), we show that BPPC-IF does not necessarily admit optimal solutions with a special structure. Moreover, we show that preprocessing an instance with oversized items (items with size greater than bin capacity) by packing a fragment of such items with size equal to bin capacity to a single bin does not necessarily yield an optimal solution. Using this observation, we develop a lower bounding procedure. Finally, we propose a heuristic algorithm which sequentially packs items into the bins using the observation about the oversized items. Through an extensive computational study, we demonstrate the superior performance of the proposed solution approach over the existing algorithms in the literature.

An Arable Field for Benchmarking of Metaheuristic Algorithms for Capacitated Coverage Path Planning Problems

This study specifies an agricultural field (Latitude = 56°30′0.8″ N, Longitude = 9°35′27.88″ E) and provides the absolute optimal route for covering that field. The calculated absolute optimal solution for this field can be used as the basis for benchmarking of metaheuristic algorithms used for finding the most efficient route in the field. The problem of finding the most efficient route that covers a field can be formulated as a Traveling Salesman Problem (TSP), which is an NP-hard problem. This means that the optimal solution is infeasible to calculate, except for very small fields. Therefore, a range of metaheuristic methods has been developed that provide a near-optimal solution to a TSP in a “reasonable” time. The main challenge with metaheuristic methods is that the quality of the solutions can normally not be compared to the absolute optimal solution since this “ground truth” value is unknown. Even though the selected benchmarking field requires only eight tracks, the solution space consists of more than 1.3 billion solutions. In this study, the absolute optimal solution for the capacitated coverage path planning problem was determined by calculating the non-working distance of the entire solution space and determining the solution with the shortest non-working distance. This was done for four scenarios consisting of low/high bin capacity and short/long distance between field and storage depot. For each scenario, the absolute optimal solution and its associated cost value (minimum non-working distance) were compared to the solutions of two metaheuristic algorithms; Simulated Annealing Algorithm (SAA) and Ant Colony Optimization (ACO). The benchmarking showed that neither algorithm could find the optimal solution for all scenarios, but they found near-optimal solutions, with only up to 6 pct increasing non-working distance. SAA performed better than ACO, concerning quality, stability, and execution time.

Selfish colorful bin packing games

We consider selfish colorful bin packing games in which a set of items, each one controlled by a selfish player, are to be packed into a minimum number of unit capacity bins. Each item has one of $$m\ge 2$$ colors and no items of the same color may be adjacent in a bin. All bins have the same unitary cost which is shared among the items it contains, so that players are interested in selecting a bin of minimum shared cost. We adopt two standard cost sharing functions, i.e., the egalitarian and the proportional ones. Although, under both cost functions, these games do not converge in general to a (pure) Nash equilibrium, we show that Nash equilibria are guaranteed to exist. We also provide a complete characterization of the efficiency of Nash equilibria under both cost functions for general games, by showing that the prices of anarchy and stability are unbounded when $$m\ge 3$$ , while they are equal to 3 when $$m=2$$ . We finally focus on the subcase of games with uniform sizes (i.e., all items have the same size). We show a tight characterization of the efficiency of Nash equilibria and design an algorithm which returns Nash equilibria with best achievable performance. All of our bounds on the price of anarchy and stability hold with respect to both their absolute and asymptotic version.

Exact algorithms for class-constrained packing problems

We consider a generalization of the classic Bin Packing Problem, called the Class-Constrained Bin Packing Problem (CCBPP), in which given a set of items, each one with a size and a class, one must pack the items in the least amount of bins respecting the bin capacity and the number of different classes that it can hold. We also consider the Class-Constrained Knapsack Problem (CCKP), in which items also have a value and one must select a subset of items with a maximum value which fits in a single bin. We consider a Branch-and-Price framework for CCBPP, exploring several options in the design of a Branch-and-Price algorithm. For the Pricing Problem, which is equivalent to the CCKP, we present two dynamic programming algorithms and a Branch-and-Bound algorithm. We also present other exact algorithms for generalizations of CCKP. These generalized problems also appear as Pricing Problems when solving the CCBPP. Our best algorithm for CCBPP was able to solve 95% of the instances considered in less than 15 min.

On Fault-Tolerant Bin Packing for Online Resource Allocation

We study an online fault-tolerant bin packing problem that models reliable resource allocation. In this problem, each item is replicated and has f + 1 replicas including one primary and f standbys. The packing of items is required to tolerate up to f faulty bins, i.e., to guarantee that at least one correct replica of each item is available regardless of which f bins turn to be faulty. Any feasible packing algorithm must satisfy an exclusion constraint and a space constraint. The exclusion constraint is generalized from the fault tolerance requirement and the space constraint comes from the capacity planning. The target of bin packing is to minimize the number of bins used. We first derive a lower bound on the number of bins needed by any feasible packing algorithm. We then study three heuristic algorithms named mirroring, shifting and mixing under a particular setting where all items have the same size. The mirroring algorithm has a low utilization of the bin capacity. Compared with the mirroring algorithm, the shifting algorithm requires fewer bins. However, in online packing, the process of opening bins by the shifting algorithm is not smooth. It turns out that even for packing a few items, the shifting algorithm needs to quickly open a large number of bins. The mixing algorithm adopts a dual average strategy to gradually open new bins for incoming items. We prove that the mixing algorithm is feasible and show that it balances the number of bins used and the process of opening bins. Finally, to pack items with different sizes, we extend the mirroring algorithm by adopting the First-Fit strategy and extend both the shifting and mixing algorithms by involving the harmonic strategy. The asymptotic competitive ratios of the three extended algorithms are analyzed respectively.

Online bin packing with cardinality constraints resolved

Bin packing with cardinality constraints is a basic bin packing problem. In the online version with the parameter k≥2, items having sizes in (0,1] associated with them are presented one by one to be packed into unit capacity bins, such that the capacities of bins are not exceeded, and no bin receives more than k items. We resolve the online problem and prove a lower bound of 2 on the overall asymptotic competitive ratio. Additionally, we significantly improve the known lower bounds on the asymptotic competitive ratio for every specific value of k. The novelty of our constructions is based on full adaptivity that creates large gaps between item sizes. Last, we show a lower bound strictly larger than 2 on the asymptotic competitive ratio of the online 2-dimensional vector packing problem, where no such lower bound was known even for fixed high dimensions.

Metaheuristic Approaches for One-Dimensional Bin Packing Problem: A Comparative Performance Study

Nature-inspired metaheuristic algorithms have steadily gained popularity over the last two decades. They have been applied to a plethora of optimization problems both in continuous and combinatorial domains. In this paper, the one-dimensional bin packing problem is solved through the implementation of two underlying heuristics, namely, best fit and better fit, and four representative state-of-the-art global metaheuristic algorithms, namely, firefly algorithm, genetic algorithm, adaptive cuckoo search algorithm, and artificial bee colony algorithm. The underlying best fit and better fit heuristics are employed by the four aforementioned global metaheuristic algorithms as reordering heuristics and local search improvement mechanisms are used for generating good packing schema. Furthermore, these two local heuristics possess special characteristics which, when incorporated into the global metaheuristics, allow them to escape local optimums and avoid getting stuck, and more so, enables the algorithms to generate good quality solutions. The main focus of this paper is the presentation of a systematic performance evaluation study for the representative algorithms, with some initial computational results that show the effectiveness of the respective algorithms and their ability to achieve promising solutions. The experiments conducted here were carried out using three standard bin packing problem datasets categories with over 1,210 instances in total, each with differing capacities, number of items and distributions between item weights. The numerical results of the representative algorithms were compared with the solutions achieved with the underlying heuristic techniques, in this case, the best fit and better fit heuristics, respectively. Similarly, the analysis of the initial computational results obtained revealed the superior performance of the individual algorithm implementation. Moreover, performance was established by taking into account both the algorithms computational time and the solution quality. Overall, several observations regarding the solution and the underlying heuristics were made. It is worth noting that by utilizing best fit heuristic, the algorithms attained optimal solutions for instances of the easy dataset requiring smaller capacity bins. However, as the complexity of the instances increased, the ability to produce high-quality results decreased. Nevertheless, this heuristic can produce near-optimal solutions and a good packing schema for most instances. On the other hand, utilizing better fit as the underlying heuristic results in optimal solutions almost all the time, regardless of capacity and number of items.

Approximate Solution Scheme for Inverse Bin-Packing Problem Subject to Decision Maker’s Preferences

The problem of packing a maximal number of items in the given set of equal capacity bins subject to Decision Maker’s (DM’s) preferences over the items is under consideration. The solution of this problem must satisfy the following conditions: (1) the total weight of items in a bin is not to be greater than the bin capacity; and (2) for each unpacked item there are no packed items less preferable for DM, instead of which it may be packed without violating the capacity limit. The approximate solution scheme for this problem based on modified First Fit Decreasing algorithm is proposed.

A New Branch-and-Price-and-Cut Algorithm for One-Dimensional Bin-Packing Problems

In this paper, a new branch-and-price-and-cut algorithm is proposed to solve the one-dimensional bin-packing problem (1D-BPP). The 1D-BPP is one of the most fundamental problems in combinatorial optimization and has been extensively studied for decades. Recently, a set of new 500 test instances were proposed for the 1D-BPP, and the best exact algorithm proposed in the literature can optimally solve 167 of these new instances, with a time limit of 1 hour imposed on each execution of the algorithm. The exact algorithm proposed in this paper is based on the classical set-partitioning model for the 1DBPPs and the subset row inequalities. We describe an ad hoc label-setting algorithm to solve the pricing problem, dominance, and fathoming rules to speed up its computation and a new primal heuristic. The exact algorithm can easily handle some practical constraints, such as the incompatibility between the items, and therefore, we also apply it to solve the one-dimensional bin-packing problem with conflicts (1D-BPPC). The proposed method is tested on a large family of 1D-BPP and 1D-BPPC classes of instances. For the 1D-BPP, the proposed method can optimally solve 237 instances of the new set of difficult instances; the largest instance involves 1,003 items and bins of capacity 80,000. For the 1D-BPPC, the experiments show that the method is highly competitive with state-of-the-art methods and that it successfully closed several open 1D-BPPC instances.

A branch-and-price algorithm for the temporal bin packing problem

We study an extension of the classical Bin Packing Problem, where each item consumes the bin capacity during a given time window that depends on the item itself. The problem asks for finding the minimum number of bins to pack all the items while respecting the bin capacity at any time instant. A polynomial-size formulation, an exponential-size formulation, and a number of lower and upper bounds are studied. A branch-and-price algorithm for solving the exponential-size formulation is introduced. An overall algorithm combining the different methods is then proposed and tested through extensive computational experiments.

The identification of the domestic waste collection system associated with the least operative musculoskeletal disorders using human Resource absence data

With increasing pressures around public sector costs, UK Local Authorities (LAs) and waste collection companies, are under pressure to reduce absence rates due to ill health. The identification of the 'safest' method of waste collection in the UK has been largely unresolved with many different types of waste and recycling receptacles used and deemed acceptable. The purpose of the study was to investigate the relationships between domestic waste collection methods and absence due to Musculoskeletal Disorders (MSDs) through the comparison of absence rates for different activity. Studies based upon ergonomic theory had suggested the use of wheeled bins is better than the use of boxes, but this has not been tested empirically. Absence data was obtained from 15 LAs who allocated a more detailed activity role to their records, allowing for activity absence rates to be calculated. The outputs were collated and analysed using SPSS to identify statistically significant relationships between types of waste collection services. The results confirm that wheeled bins are associated with less proxy measures of MSD than boxes, baskets and sacks with even lower absence rates associated with 1100 L capacity bins, when handled by two workers. Findings also indicates that there is a level where MSD absence interventions are unlikely to be sustainable.In conclusion these findings should help LAs better understand some critical factors regarding waste collection strategies and MSD absence and inform HSE enforcement strategies. Employers should interrogate their own ill health data and seek to move to systems that create less MSDs.

A modified squirrel search algorithm based on improved best fit heuristic and operator strategy for bin packing problem

Bin Packing Problem (BPP) is one of the main optimization problems in which a set of items of known sizes should be packed into a minimum number of bins. In which the total size of items in each bin is not exceeding the bin capacity. Several approximation algorithms have been suggested that provide an approximate solution to the BPP. Nevertheless, they are not applicable to solve large-scale instances within a reasonable time or obtain an optimal number of bins. In this paper, we propose a modified version of the most recent optimization algorithm squirrel search algorithm (SSA) for solving the one-dimensional bin packing problem (MSBPP). The proposed algorithm is based on an improved best fit heuristic for generating a feasible initial packing of items into bins. During improving the solution, the operator’s strategy can take place to obtain an optimized solution for the BPP. To the best of our knowledge, this paper is the first work of applying squirrel search algorithm in a bin packing problem as an example of the optimization problem. The modified algorithm has been tested on various benchmark datasets ranged from easy to hard class dataset. As well as, we compare it against other popular algorithms such as Particle swarm optimization (PSO), African buffalo optimization (ABO) and Crow search algorithm (CSA) to demonstrate its efficiency and accuracy. According to the results of an easy class, the proposed algorithm has achieved an improvement ranged from 16% to 32% for N2C1W1_A instance when compared with other algorithms to reach an optimal result with the least number of iterations. In addition, the average execution time of it beats other compared algorithms with improvement reached up to 91% faster than PSO, 98% faster than ABO and 94% faster than CSA. For medium class, the bin efficiency of the proposed algorithm has been improved by 26% during iterations for N1W1B2R9 instance with minimum number of iterations and least execution time while for hard class, it has obtained an average efficiency 0.175 which is lower than other algorithms and average execution time 99.8 as a result of HARD3 instance. The experimental results validate the superiority of the proposed MSBPP in terms of not only the solution accuracy but also both the execution time and the number of iterations.

Enhanced Pseudo-polynomial Formulations for Bin Packing and Cutting Stock Problems

We study pseudo-polynomial formulations for the classical bin packing and cutting stock problems. We first propose an overview of dominance and equivalence relations among the main pattern-based and pseudo-polynomial formulations from the literature. We then introduce reflect, a new formulation that uses just half of the bin capacity to model an instance and needs significantly fewer constraints and variables than the classical models. We propose upper- and lower-bounding techniques that make use of column generation and dual information to compensate reflect weaknesses when bin capacity is too high. We also present nontrivial adaptations of our techniques that solve two interesting problem variants, namely the variable-sized bin packing problem and the bin packing problem with item fragmentation. Extensive computational tests on benchmark instances show that our algorithms achieve state of the art results on all problems, improving on previous algorithms and finding several new proven optimal solutions.

A variable neighborhood search algorithm for the bin packing problem with compatible categories

In this paper, we address the Bin Packing Problem with Compatible Categories (BPCC), a challenging optimization problem that arises in the context of last mile distribution to nanostores in large cities, particularly in developing countries. By introducing the concept of incompatible categories of items to be delivered (i.e., types of items that cannot be transported together, such as food and cleaning products), as opposed to the item-by-item incompatibility found in previous literature, we aim to determine the best assignment of deliveries of distinct products to a homogeneous fleet of capacitated vehicles in order to minimize the number of required vehicles (bins). To solve large instances of the BPCC that are commonly found in practice, we propose an efficient variable neighborhood search (VNS) metaheuristic that relies on a simple greedy heuristic to generate initial solutions and on simple and efficient neighborhoods as well as problem-tailored shaking procedures. We perform extensive computational experiments on a very large set of 8000 instances derived from benchmark datasets. The tested instances differ in terms of number of items (ranging from 201 to 1002), bin capacities, compatibility matrices and proportion of items that belong to the different categories. We also present two new mathematical formulations for the BPCC and compare them to the item-by-item bin packing formulation (Gendreau et al., 2004) using a high performance computer (HPC). The computational experiments indicate that our VNS algorithm can effectively solve the BPCC in very short CPU times.

Element-capacity difference algorithm for optimal bin packing

In this paper, the NP-complete problem of 1D bin packing is considered. In the 'bin packing problem', a set of numbers or elements that need to be loaded in the bins is considered. The total load of the elements in any bin must not exceed the bin size or bin capacity. These elements must be packed or loaded in the bins in such a way that there is minimum amount of space left in each bin and the number of bins to be used is minimised. A new algorithm, which helps in tackling the bin packing problem, by giving optimal solutions in minimum time, is presented. The algorithms currently in use, requires the elements to be sorted before the loading. The algorithm discussed in this paper avoids the sorting operation before the loading while giving the same best fit results.

Quality of strong equilibria for selfish bin packing with uniform cost sharing

The bin packing problem deals with packing items of sizes no larger than 1 into unit capacity bins. Here, we analyze a class of bin packing games where the cost of an item is 1 over the total number of items packed into its bin, which is a bin packing congestion game. We study strong equilibria and find the tight values of the SPoA and SPoS, that is, asymptotic approximation ratios of the worst and best strong equilibria. We show that these values are approximately 1.69103 and 1.611824, respectively. In particular, we observe that the two values are not equal, showing a difference from other known kinds of cost sharing approaches.

New formulations for Variable Cost and Size Bin Packing Problems with Item Fragmentation

In the Bin Packing Problem with Item Fragmentation a set of items of known weight and a set of bins of limited capacity are given; the task is to find the minimum cost assignment of items to bins without exceeding their capacity. However, contrary to the classical Bin Packing Problem, items can be split and fractionally assigned to different bins at a cost. In this paper we generalize models and properties from the literature by considering a set of heterogeneous bins, possibly having different cost and capacity. We prove that such a natural extension changes substantial features of the problem. We propose both compact and extended formulations and a branch-and-price algorithm that combines column generation techniques and implicit enumeration strategies to achieve guarantees on the optimality of the solutions. We present the results of an extensive experimental campaign proving that our algorithm outperforms general purpose commercial solvers by orders of magnitude.

The Subset Assignment Problem for Data Placement in Caches

We introduce the subset assignment problem in which items of varying sizes are placed in a set of bins with limited capacity. Items can be replicated and placed in any subset of the bins. Each (item, subset) pair has an associated cost. Not assigning an item to any of the bins is not free in general and can potentially be the most expensive option. The goal is to minimize the total cost of assigning items to subsets without exceeding the bin capacities. The subset assignment problem models the problem of managing a cache composed of banks of memory with varying cost/performance specifications. The ability to replicate a data item in more than one memory bank can benefit the overall performance of the system with a faster recovery time in the event of a memory failure. For this setting, the number n of data objects (items) is very large and the number d of memory banks (bins) is a small constant (on the order of 3 or 4). Therefore, the goal is to determine an optimal assignment in time that minimizes dependence on n. The integral version of this problem is NP-hard since it is a generalization of the knapsack problem. We focus on an efficient solution to the LP relaxation as the number of fractionally assigned items will be at most d. If the data objects are small with respect to the size of the memory banks, the effect of excluding the fractionally assigned data items from the cache will be small. We give an algorithm that solves the LP relaxation and runs in time $$O(\left( {\begin{array}{c}3^d d+1\end{array}}\right) {\text {poly}}(d) n \log (n) \log (nC) \log (Z))$$ , where Z is the maximum item size and C the maximum storage cost.

Disposal of Household Waste: Practices, Challenges & Management

Household waste comprises of garbage and rubbish from individual home and flats/ apartments. This waste consists of fruits and vegetables, food scrapes, papers, cans and bottles, clothes, chocolate wrappers, shampoo bottles and sachets etc. These are non- hazardous wastes generated by household and can be recycled. Hazardous wastes are those from pest control, home cleaners, batteries, garage oils which have adverse impact on the environment and threatens human life and other living organisms. The improper disposal of household hazardous waste by pouring waste oils and cleaners on the ground or disposing the containers along with other wastages affects the people living in that area and also affects the health of sewage cleaners. Dumping of solid wastes in the outskirts or open area has created huge landfills which pose serious threats on environment causing air pollution, contamination of ground water. Another serious problem in improper disposal of domestic waste is in regard to the process of collecting household waste. Lack of equipment and infrastructure for collection, inadequate number of bins and bin capacity, lack of manpower and technical skills and high rate of generation of waste from household sector. The present paper is intended to investigate the practices in the disposal of household waste, problems and challenges of improper disposal and measures for effective household waste management. The paper also highlights suggestions for proper disposal of household waste.