Two-dimensional Vector Packing Problem Research Articles

Hybrid branch-and-price-and-cut algorithm for the two-dimensional vector packing problem with time windows

This paper introduces the two-dimensional vector packing problem with time windows (2DVPPTW). It packs all items into identical bins to minimize the number of used bins. Items are characterized by their weights, volumes, and time windows. Items have different time requirements for delivery in many practical settings. For example, items have their time windows for packing and delivery. The 2DVPPTW is subjected to three constraints: weight, volume, and time windows. We present an integer programming (IP) model for the proposed 2DVPPTW. The IP model is reformulated into the master problem and the subproblem (SP). A hybrid branch-and-price-and-cut (H-BPC) algorithm is developed to solve the 2DVPPTW. On the basis of extensive computational experiments, the proposed H-BPC is significantly effective in comparison with the commercial branch-and-bound/cut solvers, such as CPLEX, and two exact methods. The primary reason for the computational efficiency of the H-BPC is the development of a heuristic algorithm, namely, adaptive large neighborhood search (ALNS), to solve the SP, which is usually computationally expensive. Developing the heuristic algorithm extends the works on the solution to the SPs of bin packing problems and two-dimensional vector packing problems. Two dynamic programming (DP) algorithms are also proposed to solve the SP optimally, namely, the label-setting algorithm and the maximal clique-based DP (MC_DP) algorithm. Moreover, two cooperative schemes are proposed and compared. One is composed of the ALNS and label-setting algorithm, which has a shorter computational time, and the other is composed of the ALNS and MC_DP, which can obtain more optimal solutions. Furthermore, subset-row inequalities, rounded capacity inequalities, multidimensional dual-feasible function, and incompatibility preprocessing policy help to decrease the computational burden dramatically.

An exact algorithm for two-dimensional vector packing problem with volumetric weight and general costs

The volumetric weight of a package has become an essential factor in calculating the delivery cost of shipments in the international logistics market. In this work, we extend the two-dimensional vector packing problem by considering a more realistic cost structure, which is a general function of volumetric weight. The problem is to pack a set of different items into some identical bins without violating weight limits and volume capacities so that the total delivery cost is minimized. We develop an exact approach based on a branch-and-price algorithm and subset-row inequalities for the problem. To efficiently solve the pricing problem in column generation, a label-setting algorithm with an effective label dominance rule and a bounding procedure is presented. A stronger label dominance rule is derived for the case where the cost function is convex. The computational results show that the exact method is effective in solving the various test instances of the problem. If the volumetric weight is not considered, the exact method can be adapted to solve the two-dimensional vector packing problem with piecewise linear cost function and outperformed the existing exact algorithm by computing 27 optimal solutions for previously open instances.

Bin packing with lexicographic objectives for loading weight- and volume-constrained trucks in a direct-shipping system

We consider a packing problem that arises in a direct-shipping system in the food and beverage industry: Trucks are the containers, and products to be distributed are the items. The packing is constrained by two independent quantities, weight (e.g., measured in kg) and volume (number of pallets). Additionally, the products are grouped into the three categories: standard, cooled, and frozen (the latter two require refrigerated trucks). Products of different categories can be transported in one truck using separated zones, but the cost of a truck depends on the transported product categories. Moreover, splitting orders of a product should be avoided so that (un-)loading is simplified. As a result, we seek for a feasible packing optimizing the following objective functions in a strictly lexicographic sense: minimize the (1) total number of trucks; (2) number of refrigerated trucks; (3) number of refrigerated trucks which contain frozen products; (4) number of refrigerated trucks which also transport standard products; (5) and minimize splitting. This is a real-world application of a bin-packing problem with cardinality constraints a.k.a. the two-dimensional vector packing problem with additional constraints. We provide a heuristic and an exact solution approach. The heuristic meta-scheme considers the multi-compartment and item fragmentation features of the problem and applies various problem-specific heuristics. The exact solution algorithm covering all five stages is based on branch-and-price using stabilization techniques exploiting dual-optimal inequalities. Computational results on real-world and difficult self-generated instances prove the applicability of our approach.

A branch-and-price algorithm for the two-dimensional vector packing problem

The two-dimensional vector packing problem is a well-known generalization of the classical bin packing problem. It considers two attributes for each item and bin. Two capacity constraints must be satisfied in a feasible packing solution for each bin. The objective is to minimize the number of bins used. To compute optimal solutions for the problem, we propose a new branch-and-price algorithm. A goal cut that sets a lower bound to the objective is used. It is effective in speeding up column generation by reducing the number of iterations. To efficiently solve the pricing problem, we develop a branch-and-bound method with dynamic programming, which first eliminates conflicts between two items through branching, and then solves the two-constraint knapsack problem at leaf nodes through dynamic programming. Extensive computational experiments were conducted based on 400 test instances from existing literature. Our algorithm significantly outperformed the existing branch-and-price algorithms. Most of the test instances were solved within just a few seconds.

The two-dimensional vector packing problem with general costs

The two-dimensional vector packing problem with general costs (2DVPP-GC) arises in logistics where shipping items of different weight and volume are packed into cartons before being transported by a courier company. In practice, the delivery cost of a carton of items is usually retrieved from a cost table. The costs may not preserve any known mathematical function since it could specify arbitrary price at any possible weight. Such a general pricing scheme meets a majority of real-world bin packing applications, where the price of delivery service is determined by many complicated and correlated factors. Compared to the classical bin packing problem and its variants, the 2DVPP-GC is more complex and challenging. To solve the 2DVPP-GC with minimizing the total cost, we propose a memetic algorithm to compute solutions of high quality. Fitness functions and improved operators are proposed to achieve effectiveness. Computational experiments on a variety of test instances show that the algorithm is competent to solve the 2DVPP-GC. In particular, optimal solutions are found in a second for all the test instances that have a known optimal solution.

A branch-and-price algorithm for the two-dimensional vector packing problem with piecewise linear cost function

The two-dimensional vector packing problem with piecewise linear cost function (2DVPP-PLC) models a practical packing problem faced by many companies that use courier services. We propose a branch-and-price algorithm to solve the 2DVPP-PLC exactly. The column generation procedure is a key component that affects the performance of a branch-and-price algorithm. The pricing problem exhibits an interesting structure that allows us to decompose it into subproblems that form a lattice. We explore dominance relations on the lattice and design an efficient algorithm for the pricing problem. Experimental results show that our branch-and-price algorithm is capable of solving 2DVPP-PLC test instances effectively.

A single machine scheduling problem with two-dimensional vector packing constraints

Abstract We consider a scheduling problem where jobs consume a perishable resource stored in vials. It leads to a new scheduling problem, with two-dimensional jobs, one dimension for the duration and one dimension for the consumption. Jobs have to be finished before a given due date, and the objective is to schedule the jobs on a single machine so that the maximum lateness does not exceed a given threshold and the number of vials required for processing all the jobs is minimized. We propose a two-step approach embedding a Recovering Beam Search algorithm to get a good-quality initial solution reachable in short time and a more time consuming matheuristic algorithm. Computational experiments are performed on the benchmark instances available for the two-dimensional vector packing problem integrated with additional due dates to take into account the maximum lateness constraints. The computational results show very good performances of the proposed approach that remains effective also on the original two-dimensional vector packing instances without due dates where 7 new bounds are obtained.

The two-dimensional vector packing problem with piecewise linear cost function

The two-dimensional vector packing problem with piecewise linear cost function (2DVPP-PLC) is a practical problem faced by a manufacturer of children׳s apparel that ships products using courier service. The manufacturer must ship a number of items using standard-sized cartons, where the cost of a carton quoted by the courier is determined by a piecewise linear function of its weight. The cost function is not necessarily convex or concave. The objective is to pack all given items into a set of cartons such that the total delivery cost is minimized while observing both the weight limit and volume capacity constraints. This problem is commonly faced by many manufacturers that ship products using courier service. We formulate the problem as an integer programming model. Since the 2DVPP-PLC generalizes the classical bin packing problem, it is more complex and challenging. Solving it directly using CPLEX is successful only for small instances. We propose a simple heuristic that is extremely fast and produces high-quality solutions for instances of practical size. We develop an iterative local search algorithm to improve the solution quality further. We generate two categories of test data that can serve as benchmark for future research.

큰 사이즈 아이템들에 대한 2차원 벡터 패킹문제의 어려움

We consider a two-dimensional vector packing problem in which each item has size in x- and y-coordinates. The purpose of this paper is to provide a ground work on how hard two-dimensional vector packing problems are for large items. We prove that the problem with each item greater than 1/2-e either in x- or y-coordinates for 0 < e ≤ 1/6 has no APTAS unless P = NP.

A two-dimensional vector packing model for the efficient use of coil cassettes

We consider the problem of efficiently packing steel products, known as coils, into special containers, called cassettes for shipping. The objective is to minimize the number of cassettes used for packing all the given coils where each cassette has capacity limits on both total payload weight and size. We model this problem as a two-dimensional vector packing problem and propose a heuristic. We also analyze the worst-case performance of the proposed algorithm under a special condition which, in fact, holds for the particular real-world case that we handled. Our computational experiment with real production data shows that the proposed algorithm performs quite satisfactorily in practice.

An approximation algorithm with absolute worst-case performance ratio 2 for two-dimensional vector packing

The two-dimensional vector packing problem is the generalization of the classical one-dimensional bin packing problem to two dimensions. While an asymptotic polynomial time approximation scheme has been designed for one-dimensional bin packing, the existence of an asymptotic polynomial time approximation scheme for two dimensions would imply P= NP. The existence of an approximation algorithm for the two-dimensional vector packing problem with an asymptotic performance guarantee 2 was an open problem so far. In this paper we present an O(n log n) time algorithm for two-dimensional vector packing with absolute performance guarantee 2.

A branch-and-bound algorithm for the two-dimensional vector packing problem

The two-dimensional vector packing (2DVP) problem can be stated as follows. Given are N objects, each of which has two requirements. The problem is to find the minimum number of bins needed to pack all objects, where the capacity of each bin equals 1 in both requirements. A heuristic adapted from the first fit decreasing rule is proposed, and lower bounds for optimal solutions to the 2DVP problem are investigated. Computing one of these lower bounds is shown to be equivalent to computing the largest number of vertices of a clique of a 2-threshold graph (which can be done in polynomial time). These lower bounds are incorporated into a branch-and-bound algorithm, for which some limited computational experiments are reported.