On-line Packing Research Articles

A Study of the Capacitated Vehicle Routing Problem with Time-Window and Three-Dimensional Loading Constraints in Land–Sea Transport

This paper addresses the capacitated vehicle routing problem with time-window and three-dimensional loading constraints in land–sea transport (3L-CVRPTWLS, which is an extension of the 3L-CVRP) to minimize the total cost of land–sea transport. The 3L-CVRPTWLS considers the online packing environment and port-of-destination (POD) constraint, which are clearly of practical significance in freight distribution. Due to its high degree of combinatorial complexity, the literature on this problem is very limited. To solve the 3L-CVRPTWLS, we develop a general deepest-bottom-left-fill (DBLF) and layer heuristic for packing and a hybrid variable-neighborhood tabu search for the routing phase (HLVNTS) based on the “packing first, routing second” (P1R2) strategy. HLVNTS reduces the average total number of vehicles by 2.51% and the average total travel distance by 27.62% in a shorter amount of time. The experimental results show that the proposed algorithm performs well in the tested instances in terms of both computational efficiency and solution quality. Moreover, we evaluate the impact of the POD constraint on the total transportation cost. This study may provide some important support for the sustainable development of land–sea transport and help to protect the environment.

Comprehensive Review of Robotized Freight Packing

Background: This review addresses the emerging field of automated packing cells, which lies at the intersection of robotics and packing problems. Integrating these two fields is critical for optimizing logistics and e-commerce operations. The current literature focuses on packing problems or specific robotic applications without addressing their integration. Methods: To bridge this gap, we conducted a comprehensive review of 46 relevant studies, analyzing various dimensions, including the components of robotic packing cells, the types of packing problems, the solution approaches, and performance comparisons. Results: Our review reveals a significant trend towards addressing online packing problems, which reflects the dynamic nature of logistics operations where item information is often incomplete. We also identify several research gaps, such as the need for standardized terminologies, comprehensive methodologies, and the consideration of real-world constraints in robotic algorithms. Conclusions: This review uniquely integrates insights from robotics and packing problems, providing a structured framework for future research. It highlights the importance of considering practical robotic constraints. It proposes a research structure that enhances the reproducibility and comparability of results in real-world scenarios. By doing so, we aim to guide future research efforts and facilitate the development of more robust and practical automated packing systems.

Learning Physically Realizable Skills for Online Packing of General 3D Shapes

We study the problem of learning online packing skills for irregular 3D shapes , which is arguably the most challenging setting of bin packing problems. The goal is to consecutively move a sequence of 3D objects with arbitrary shapes into a designated container with only partial observations of the object sequence. We take physical realizability into account, involving physics dynamics and constraints of a placement. The packing policy should understand the 3D geometry of the object to be packed and make effective decisions to accommodate it in the container in a physically realizable way. We propose a Reinforcement Learning (RL) pipeline to learn the policy. The complex irregular geometry and imperfect object placement together lead to huge solution space. Direct training in such space is prohibitively data intensive. We instead propose a theoretically provable method for candidate action generation to reduce the action space of RL and the learning burden. A parameterized policy is then learned to select the best placement from the candidates. Equipped with an efficient method of asynchronous RL acceleration and a data preparation process of simulation-ready training sequences, a mature packing policy can be trained in a physics-based environment within 48 hours. Through extensive evaluation on a variety of real-life shape datasets and comparisons with state-of-the-art baselines, we demonstrate that our method outperforms the best-performing baseline on all datasets by at least 12.8% in terms of packing utility. We also release our datasets and source code to support further research in this direction. 1

Dynamic cloud manufacturing service composition with re-entrant services: an online policy perspective

Cloud manufacturing (CMfg) emerges as a promising manufacturing paradigm, where service composition (SC) is a critical process concentrating on matching tasks and services. Existing studies usually ignore the dynamic nature of the CMfg environment, where task information is not always known before. Moreover, CMfg services are re-entrant, i.e. after being occupied for a period of service time, these services re-enter the CMfg platform (i.e. be available again). Re-entrant services significantly complicate CMfg platform revenue management. In this regard, we study the dynamic SC problem of CMfg (CMfg-DSC) incorporating re-entrant services within an online setting for the first time. CMfg-DSC is reformulated as an online packing problem. If a task is accepted, each requested service will be occupied until service time terminates. We propose online policies with performance guarantees, namely, static online packing policy (Static), opportunity-cost-based policy (Oppo), and dual-based policy with/without known distribution (Dual-k & Dual-u). Experiment results show that (1) Static is applicable for most cases; (2) Oppo has the potential for decent performance but at the cost of time; (3) Dual-u is reliable when only past observations are available; (4) Dual-k performs well given abundant service provision, but its performance would deteriorate if we lower the reward-cost threshold.

A deep reinforcement learning hyper-heuristic with feature fusion for online packing problems

In recent years, deep reinforcement learning has shown great potential in solving computer games with sequential decision-making scenarios. Hyper-heuristic is a generic search framework, capable of intelligently selecting or generating algorithms to solve a class of optimisation problems with stochastic or dynamic settings. This paper proposes a new general framework for solving online packing problems using deep reinforcement learning hyper-heuristics. Although analytical approaches can address most offline packing problems successfully, their online versions have proved much more challenging and the performance of the existing methods is often not satisfactory. In this paper, we extend a recent deep reinforcement learning hyper-heuristic framework by fusing the visual information of real-time packing with distributional information of random parameters of the problem. Computational experiments show that our method outperforms the state of the art online methods with reductions in optimality gap between 2%–19% for knapsack problem and 0.7% for the online strip packing problem. In addition, a new visual analysis presentation is also devised to better interpret the learned packing strategies, which can reveal more information than the widely used landscape analysis. As online packing problems are widely available in production environments, the proposed approach can serve as an important reference to solve other similar combinatorial optimisation problems for which visual layout inputs would aid learning.

Online Allocation and Pricing: Constant Regret via Bellman Inequalities

We develop a framework for designing simple and efficient policies for a family of online allocation and pricing problems that includes online packing, budget-constrained probing, dynamic pricing, and online contextual bandits with knapsacks. In each case, we evaluate the performance of our policies in terms of their regret (i.e., additive gap) relative to an offline controller that is endowed with more information than the online controller. Our framework is based on Bellman inequalities, which decompose the loss of an algorithm into two distinct sources of error: (1) arising from computational tractability issues, and (2) arising from estimation/prediction of random trajectories. Balancing these errors guides the choice of benchmarks, and leads to policies that are both tractable and have strong performance guarantees. In particular, in all our examples, we demonstrate constant-regret policies that only require resolving a linear program in each period, followed by a simple greedy action-selection rule; thus, our policies are practical as well as provably near optimal.

The Bayesian Prophet: A Low-Regret Framework for Online Decision Making

We develop a new framework for designing online policies given access to an oracle providing statistical information about an off-line benchmark. Having access to such prediction oracles enables simple and natural Bayesian selection policies and raises the question as to how these policies perform in different settings. Our work makes two important contributions toward this question: First, we develop a general technique we call compensated coupling, which can be used to derive bounds on the expected regret (i.e., additive loss with respect to a benchmark) for any online policy and off-line benchmark. Second, using this technique, we show that a natural greedy policy, which we call the Bayes selector, has constant expected regret (i.e., independent of the number of arrivals and resource levels) for a large class of problems we refer to as “online allocation with finite types,” which includes widely studied online packing and online matching problems. Our results generalize and simplify several existing results for online packing and online matching and suggest a promising pathway for obtaining oracle-driven policies for other online decision-making settings. This paper was accepted by George Shanthikumar, big data analytics.

Online maximum matching with recourse

We study the online maximum matching problem in a model in which the edges are associated with a known recourse parameter $k$. An online algorithm for this problem has to maintain a valid matching while edges of the underlying graph are presented one after the other. At any moment the algorithm can decide to include an edge into the matching or to exclude it, under the restriction that at most $k$ such actions per edge take place, where $k$ is typically a small constant. This problem was introduced and studied in the context of general online packing problems with recourse by Avitabile et al. [Information Processing Letters, 2013], whereas the special case $k=2$ was studied by Boyar et al. [WADS 2017]. In the first part of this paper we consider the edge arrival model, in which an arriving edge never disappears from the graph. Here, we first show an improved analysis on the performance of the algorithm AMP of Avitabile et al., by exploiting the structure of the matching problem. In addition, we show that the greedy algorithm has competitive ratio $3/2$ for every even $k$ and ratio $2$ for every odd $k$. Moreover, we present and analyze an improvement of the greedy algorithm which we call $L$-Greedy, and we show that for small values of $k$ it outperforms the algorithm AMP. In terms of lower bounds, we show that no deterministic algorithm better than $1+1/(k-1)$ exists, improving upon the known lower bound of $1+1/k$. The second part of the paper is devoted to the edge arrival/departure model, which is the fully dynamic variant of online matching with recourse. The analysis of $L$-Greedy and AMP carry through in this model; moreover we show a lower bound of $(k^2-3k+6) / (k^2-4k+7)$ for all even $k \ge 4$. For $k\in\{2,3\}$, the competitive ratio is $3/2$.

Symmetry Exploitation for Online Machine Covering with Bounded Migration

Online models that allow recourse can be highly effective in situations where classical online models are too pessimistic. One such problem is the online machine covering problem on identical machines. In this setting, jobs arrive one by one and must be assigned to machines with the objective of maximizing the minimum machine load. When a job arrives, we are allowed to reassign some jobs as long as their total size is (at most) proportional to the processing time of the arriving job. The proportionality constant is called the migration factor of the algorithm. Using a rounding procedure with useful structural properties for online packing and covering problems, we design first a simple (1.7 + ε)-competitive algorithm using a migration factor of O(1/ε), which maintains at every arrival a locally optimal solution with respect to the Jump neighborhood. After that, we present as our main contribution a more involved (4/3+ε)-competitive algorithm using a migration factor of Ō (1/ε 3 ). At every arrival, we run an adaptation of the Largest Processing Time first (LPT) algorithm. Since the new job can cause a complete change of the assignment of smaller jobs in both cases, a low migration factor is achieved by carefully exploiting the highly symmetric structure obtained by the rounding procedure.

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.

Combining Mobile Robotics and Packing for Optimal deliveries

The problem addressed is taken from warehousing and distribution. It concerns the order preparation in e-commerce. An order is a set of lines. Each line describes the ordered product (box) with its corresponding quantity. The problem consists of packing boxes subject to orientation, fragility, stackability, weight and volume constraints of various sizes into available containers (trays, cartons) in a way which optimizes the total number of containers to integrate in a number of delivery trips in e-commerce. The article addresses the problem where the packing is performed by a mobile robot. The aim of the work carried out in the context of the project INCASE[0] which is a plat-form of the INTERREG V [1] is to conduct, among other experiments[2] a set of experiments where the packing activity is performed by a mobile robot in order to pick up items from shelves and then pack items in boxes. To keep the problem simple, we assume that positions of items to pick up and to place them in the box are known; they are computed by efficient bin packing optimization algorithms (2D, 3D) of Optim Suite of KLS OPTIM in production since many years. There are many ways of using bin packing algorithms. The first possible use is the online packing of items based on a fixed order defined by the host system. This approach has several drawbacks, among them the inefficient handling of stability constraint and sub-use of available box volumes. The second approach is to allow packing algorithms to compute optimal or sub-optimal solutions whilst respecting all constraints; this approach is used in production. One drawback of this approach is such solutions are not always feasible by a mobile robot. The aim of our work is that the mobile robot must be able to take into account optimization constraints in order to place objects on the pallets. The objective is to find a trade off between optimality and feasibility. The work is to conduct a set of real-life experiments and simulations by mobile robots to identify failure cases like falling, unsteady, damaged items due to robot arm forces, instability or space. The aim is to learn from these simulations, introduction of machine learning, and to derive a set of rules to integrate in the bin packing optimization algorithms as strategies to build robust packing solution for autonomous mobile robots.

The Bayesian Prophet

Motivated by the success of using black-box predictive algorithms as subroutines for online decision-making, we develop a new framework for designing online policies given access to an oracle providing statistical information about an offline benchmark. Having access to such prediction oracles enables simple and natural Bayesian selection policies, and raises the question as to how these policies perform in different settings. Our work makes two important contributions towards tackling this question: First, we develop a general technique we call compensated coupling which can be used to derive bounds on the expected regret (i.e., additive loss with respect to a benchmark) for any online policy and offline benchmark; Second, using this technique, we show that the Bayes Selector has constant expected regret (i.e., independent of the number of arrivals and resource levels) in any online packing and matching problem with a finite type-space. Our results generalize and simplify many existing results for online packing and matching problems, and suggest a promising pathway for obtaining oracle-driven policies for other online decision-making settings.

Near-Optimum Online Ad Allocation for Targeted Advertising

Motivated by Internet targeted advertising, we address several ad allocation problems. Prior work has established that these problems admit no randomized online algorithm better than (1 - 1/ e )-competitive (see Karp et al. (1990) and Mehta et al. (2007)), yet simple heuristics have been observed to perform much better in practice. We explain this phenomenon by studying a generalization of the bounded-degree inputs considered by Buchbinder et al. (2007), graphs which we call ( k , d )- bounded . In such graphs the maximal degree on the online side is at most d and the minimal degree on the offline side is at least k . We prove that, for such graphs, these problems’ natural greedy algorithms attain a competitive ratio of 1 - d -1/ k + d -1, tending to 1 as d / k tends to zero. We prove this bound is tight for these algorithms. Next, we develop deterministic primal-dual algorithms for the above problems, achieving a competitive ratio of 1−(1 - 1/ d ) k > 1 - &1/ e k / d , or exponentially better loss as a function of k / /d , and strictly better than 1 - 1/e whenever k ≥ d . We complement our lower bounds with matching upper bounds for the vertex-weighted problem. Finally, we use our deterministic algorithms to prove by dual-fitting that simple randomized algorithms achieve the same bounds in expectation. Our algorithms and analysis differ from previous ad allocation algorithms, which largely scale bids based on the spent fraction of their bidder’s budget, whereas we scale bids according to the number of times the bidder could have spent as much as her current bid. Our algorithms differ from previous online primal-dual algorithms in that they do not maintain dual feasibility but only a primal-to-dual ratio and only attain dual feasibility upon termination. We believe our techniques could find applications to other well-behaved online packing problems.

Efficient online packing of 4-dimensional cubes into the unit cube

Any sequence of 4-dimensional cubes of total volume not greater than 1/8 can be online packed into the unit cube.

The Bayesian Prophet: A Low-Regret Framework for Online Decision Making

We develop a new framework for designing online policies given access to an oracle providing statistical information about an offline benchmark. Having access to such prediction oracles enables simple and natural Bayesian selection policies, and raises the question as to how these policies perform in different settings. Our work makes two important contributions towards this question: First, we develop a general technique we call compensated coupling which can be used to derive bounds on the expected regret (i.e., additive loss with respect to a benchmark) for any online policy and offline benchmark. Second, using this technique, we show that a natural greedy policy, which we call the Bayes Selector, has constant expected regret (i.e., independent of the number of arrivals and resource levels) for a large class of problems we refer to as Online Allocation with finite types, which includes widely-studied Online Packing and Online Matching problems. Our results generalize and simplify several existing results for Online Packing and Online Matching, and suggest a promising pathway for obtaining oracle-driven policies for other online decision-making settings.

On-line Packing Cubes into $n$ Unit Cubes

If $n\geq 3$ and $d\in \{3,4\}$ or if $n\geq 1$ and $d\geq 5$, then any sequence of $d$-dimensional cubes of edge lengths not greater than $1$ whose total volume does not exceed $(n+1)\cdot 2^{-d}$ can be on-line packed into $n$ unit $d$-dimensional cubes

Shrinking Maxima, Decreasing Costs: New Online Packing and Covering Problems

We consider two new variants of online integer programs that are duals. In the packing problem we are given a set of items and a collection of knapsack constraints over these items that are revealed over time in an online fashion. Upon arrival of a constraint we may need to remove several items (irrevocably) so as to maintain feasibility of the solution. Hence, the set of packed items becomes smaller over time. The goal is to maximize the number, or value, of packed items. The problem originates from a buffer-overflow model in communication networks, where items represent information units broken into multiple packets. The other problem considered is online covering: there is a universe to be covered. Sets arrive online, and we must decide for each set whether we add it to the cover or give it up. The cost of a solution is the total cost of sets taken, plus a penalty for each uncovered element. The number of sets in the solution grows over time, but its cost goes down. This problem is motivated by team formation, where the universe consists of skills, and sets represent candidates we may hire. The packing problem was introduced in Emek et al. (SIAM J Comput 41(4):728---746, 2012) for the special case where the matrix is binary; in this paper we extend the solution to general matrices with non-negative integer entries. The covering problem is introduced in this paper; we present matching upper and lower bounds on its competitive ratio.

Online algorithms for 1-space bounded 2-dimensional bin packing and square packing

In this paper, we study 1-space bounded 2-dimensional bin packing and square packing. A sequence of rectangular items (square items) arrive one by one, each item must be packed into a square bin of unit size on its arrival without any information about future items. When packing items, 90°-rotation is allowed. 1-space bounded means there is only one “active” bin. If the “active” bin cannot accommodate the coming item, it will be closed and a new bin will be opened. The objective is to minimize the total number of bins used for packing all items in the sequence.Our contributions are as follows: For 1-space bounded 2-dimensional bin packing, we propose an online packing algorithm with a tight competitive ratio of 5.06. A lower bound of 3.17 on the competitive ratio is proven. Moreover, we study 1-space bounded square packing, where each item is a square with side length no more than 1. A 4.3-competitive algorithm is achieved, and a lower bound of 2.94 on the competitive ratio is given. All these bounds surpass the previously best known results.

The Geometry of Online Packing Linear Programs

We consider packing linear programs with m rows where all constraint coefficients are normalized to be in the unit interval. The n columns arrive in random order and the goal is to set the corresponding decision variables irrevocably when they arrive to obtain a feasible solution maximizing the expected reward. Previous (1 − ϵ)-competitive algorithms require the right-hand side of the linear program to be Ω((m/ϵ2)log(n/ϵ)), a bound that worsens with the number of columns and rows. However, the dependence on the number of columns is not required in the single-row case, and known lower bounds for the general case are also independent of n. Our goal is to understand whether the dependence on n is required in the multirow case, making it fundamentally harder than the single-row version. We refute this by exhibiting an algorithm that is (1 − ϵ)-competitive as long as the right-hand sides are Ω((m2/ϵ2)log(m/ϵ)). Our techniques refine previous probably approximately correct learning based approaches that interpret the online decisions as linear classifications of the columns based on sampled dual prices. The key ingredient of our improvement comes from a nonstandard covering argument together with the realization that only when the columns of the linear program belong to few one-dimensional subspaces we can obtain such small covers; bounding the size of the cover constructed also relies on the geometry of linear classifiers. General packing linear programs are handled by perturbing the input columns, which can be seen as making the learning problem more robust.

A Bin Packing Heuristic for On-Line Service Placement and Performance Control

The ever-increasing size and complexity of cloud computing, data centers, virtualization, web services, and other forms of distributed computing make automated and effective service management increasingly important. This article treats the service placement problem as a novel generalization of the on-line vector packing problem. This generalization of the service placement problem does not require a priori knowledge of the service resource profiles, allows for resource profiles to change over time, and allows services to be moved once placed on a server. An on-line self-organizing model profiles resource supplies and demands arranging services in a placement based on their resulting quality rating. A policy-driven asymmetric matrix norm quantifies the quality of the placement allowing for administrative preferences regarding service performance versus service inclusion. Service resource usage profiles' variations cause changes in their assigned placement quality; forcing new, better server placements to be found. Because some placements perform better, a proportional integral derivative controller for performance feedback adjusts the services' actual profile according to service's individual response times. This large scale system autonomically organizes placement of services in response to changes in demand and network disruptions. This article presents theorems which demonstrate the theoretical basis for the model. The article includes empirical results from the implementation of this model in a self-organizing testbed of web servers and services.