An augmented beam search-based algorithm for the circular open dimension problem

In this paper, we discuss the circular open dimension problem (CODP); that is a problem of the cutting/packing family. In CODP, we are given an initial strip of fixed width W and unlimited length, as well as a finite set N of n circular pieces C_i of known radius r_i,i ∈ N. The objective is to search for a global optimum corresponding to the minimum length of the initial strip containing the n pieces. We propose an augmented algorithm for solving the CODP which combines a beam search, a binary search and the well-known multi-start strategy. In addition, in order to increase the efficiency of the algorithm, we incorporate a strategy based on the separate beams instead of the pooled ones. The performance of the proposed algorithm is evaluated on a set of benchmark instances composed of a group taken from the literature and another group of randomly generated instances. The results show that the proposed algorithm is able to improve several best known solutions of the literature and it remains competitive for the new generated ones.     

Solving the circular open dimension problem by using separate beams and look-ahead strategies

In this paper, a constructive method is investigated for solving the circular open dimension problem (CODP), a problem of the Cutting and Packing family. CODP is a combinatorial optimization problem which is characterized by a set of circular pieces of known radii and a strip of fixed width W and unlimited length. The objective is to determine the smallest rectangle of dimensions (L, W), where L is the length of the rectangle, that will contain all the pieces such that there is no overlapping between the placed pieces and all the demand constraints are satisfied. The method combines the separate-beams search, look-ahead, and greedy procedures. A study concerning both restarting and look-ahead strategies is undertaken to determine the best tuning for the method. The performance of the method is computationally analyzed on a set of instances taken from the literature and for which optimal solutions are not known. Best-known solutions are obtained.     

Adaptive beam search lookahead algorithms for the circular packing problem

This paper addresses the circular packing problem (CPP), which consists in packing n circles C_i, each of known radius r_i, i∈N={1, …, n}, into the smallest containing circle C. The objective is to determine the radius r of C as well as the coordinates (x_i, y_i) of the center of C_i, i∈N. CPP is solved using two adaptive algorithms that adopt a binary search to determine r, and a beam search to check the feasibility of packing n circles into C when the radius is fixed at r. A node of level ?, ?=1, …, n, of the beam search tree corresponds to a partial packing of ? circles of N into C. The potential of each node of the tree is assessed using a lookahead strategy that, starting with the partial packing of the current node, assigns each unpacked circle to its maximum hole degree position. The beam search stops either when the lookahead strategy identifies a feasible packing or when it has fathomed all nodes. The computational tests on a set of benchmark instances show the effectiveness of the proposed adaptive algorithms.     

Solving the irregular strip packing problem via guided local search for overlap minimization

The irregular strip-packing problem (ISP) requires a given set of non-convex polygons to be placed without overlap within a rectangular container having a fixed width and a variable length, which is to be minimized. As a core sub-problem to solve ISP, we consider an overlap minimization problem (OMP) whose objective is to place all polygons into a container with given width and length so that the total amount of overlap between polygons is made as small as possible. We propose to use directional penetration depths to measure the amount of overlap between a pair of polygons, and present an efficient algorithm to find a position with the minimum overlap for each polygon when it is translated in a specified direction. Based on this, we develop a local search algorithm for OMP that translates a polygon in horizontal and vertical directions alternately. Then we incorporate it in our algorithm for OMP, which is a variant of the guided local search algorithm. Computational results show that our algorithm improves the best-known values of some well-known benchmark instances.     

The multiperiod two‐dimensional non‐guillotine cutting stock problem with usable leftovers

A mixed integer linear programing model for the two‐dimensional non‐guillotine cutting problem with usable leftovers was recently introduced by Andrade et al. The problem consists in cutting a set of ordered items using a set of objects of minimum cost and, within the set of solutions of minimum cost, maximizing the value of the usable leftovers. Since the concept of usable leftovers assumes they can potentially be used to attend new arriving orders, the problem is extended to the multiperiod framework in this work. In this way, the decision at each instant does not minimize in a myopic way the cost of the objects required to attend the orders of the current instant; but it aims to minimize the overall cost of the objects up to the considered time horizon. Some variants of the proposed model are analyzed and numerical results are presented.     

Hybrid approach for the two‐dimensional bin packing problem with two‐staged patterns

This paper presents a two‐phase heuristic approach for the two‐dimensional bin packing problem with two‐staged patterns and nonoriented items. A solution is generated in each phase and the better one is selected. Residual problems are solved by column generation in the first phase, where a partial admitting procedure is used to admit some of the patterns into the phase‐1 solution. The second solution is obtained from solving an integer linear programming problem over the set of all patterns generated in the first phase, where a time limit is used and subsequently the solution may not be optimal over the pattern set. The computational results indicate that the approach yields the best solution quality among the heuristics that use two‐staged or more complex patterns.     

Models for the two‐dimensional rectangular single large placement problem with guillotine cuts and constrained pattern

In this paper, we address the constrained two‐dimensional rectangular guillotine single large placement problem (2D_R_CG_SLOPP). This problem involves cutting a rectangular object to produce smaller rectangular items from orthogonal guillotine cuts. In addition, there is an upper limit on the number of copies that can be produced of each item type. To model this problem, we propose a new pseudopolynomial integer nonlinear programming (INLP) formulation and obtain an equivalent integer linear programming (ILP) formulation from it. Additionally, we developed a procedure to reduce the numbers of variables and constraints of the integer linear programming (ILP) formulation, without loss of optimality. From the ILP formulation, we derive two new pseudopolynomial models for particular cases of the 2D_R_CG_SLOPP, which consider only two‐staged or one‐group patterns. Finally, as a specific solution method for the 2D_R_CG_SLOPP, we apply Benders decomposition to the proposed ILP formulation and develop a branch‐and‐Benders‐cut algorithm. All proposed approaches are evaluated through computational experiments using benchmark instances and compared with other formulations available in the literature. The results show that the new formulations are appropriate in scenarios characterized by few item types that are large with respect to the object's dimensions.     

Constrained two‐dimensional guillotine cutting problem: upper‐bound review and categorization

In the two‐dimensional (2D) cutting (2DC) problem, a large rectangular sheet has to be dissected into smaller rectangular pieces with the aim of maximizing the total profit associated with the extracted pieces. When the number of copies of each piece to be extracted is bounded, it is referred to as constrained 2DC (C2DC) problem. The C2DC has been widely studied by the operations research community for its applications and theoretical issues. In this work, we recall the best exact and heuristic solving approaches for the C2DC and we provide a review and a categorization of the available upper bounds. We also discuss improvements and combinations of these upper bounds and give directions for their effective exploitation. Finally, we demonstrate the loss of accuracy of several exact methods present in literature because of the effect of the used antiredundancy strategies on the implemented bounding criteria. This work, based on more than 90 contributions, has a twofold target. For researchers working in C2DC, it provides a useful insight on the topic. For expert practitioners, it represents a systematic collection of the main findings and achievements, posing also the basis for future research.     

A biased‐randomized algorithm for the two‐dimensional vehicle routing problem with and without item rotations

This paper proposes an efficient algorithm, with a reduced number of parameters, for solving the two‐dimensional loading‐capacitated vehicle routing problem (2L‐CVRP). This problem combines two of the most important issues in logistics, that is, vehicle routing and packing problems. Our approach contemplates unrestricted loading including the possibility of applying 90° rotations to each rectangular‐shaped item while loading it into the vehicle, which is a realistic assumption seldom considered in the existing literature. The algorithm uses a multistart approach that is designed to avoid local minima and also to make the algorithm an easily parallelizable one. At each restart, a biased randomization of a savings‐based routing algorithm is combined with an enhanced version of a classical packing heuristic to produce feasible good solutions for the 2L‐CVRP. The proposed algorithm has been compared with the classical benchmarks for two different 2L‐CVRP variants, that is, with and without item rotations. Experimental results show that our approach outperforms several best‐known solutions from previous work, both in terms of quality and the computational time needed to obtain them.     

Engineering an efficient two‐phase local search algorithm for the co‐rotating twin‐screw extruder configuration problem

The twin‐screw configuration problem arises during polymer extrusion and compounding. It consists in defining the location of a set of pre‐defined screw elements along the screw axis in order to optimize different, typically conflicting objectives. In this paper, we present a simple yet effective stochastic local search (SLS) algorithm for this problem. Our algorithm is based on efficient single‐objective iterative improvement algorithms, which have been developed by studying different neighborhood structures, neighborhood search strategies, and neighborhood restrictions. These algorithms are embedded into a variation of the two‐phase local search framework to tackle various bi‐objective versions of this problem. An experimental comparison with a previously proposed multi‐objective evolutionary algorithm shows that a main advantage of our SLS algorithm is that it converges faster to a high‐quality approximation to the Pareto front.     

A parallel hybrid local search algorithm for the container loading problem

In this contribution, a parallel hybrid local search algorithm for the three‐dimensional container loading problem (CLP) is proposed. First a simulated annealing method for the CLP is developed, which is then combined with an existing tabu search algorithm to form a hybrid metaheuristic. Finally, parallel versions are introduced for these algorithms. The emphasis is on CLP instances with a weakly heterogeneous load. Numerical tests based on the well‐known 700 test instances from Bischoff and Ratcliff are performed, and the outcome is compared with methods from other authors. The results show a high solution quality obtained with reasonable computing time.     

Variable neighborhood search algorithms for the vehicle routing problem with two‐dimensional loading constraints and mixed linehauls and backhauls

In this paper, we explore a vehicle routing problem with two‐dimensional loading constraints and mixed linehauls and backhauls. The addressed problem belongs to a subclass of general pickup and delivery problems. Two‐dimensional loading constraints are also considered. These constraints arise in many real‐world situations and can improve efficiency since backhaul customers do not need to be delayed in a route when it is possible to load their items earlier and without rearrangements of the items. To tackle this problem, we report extensive computational experiments to assess the performance of different variants of the variable neighborhood search approaches. The initial solution relies on an insertion heuristic. Both shaking and local search phases resort to 10 neighborhood structures. Some of those structures were specially developed for this problem. The feasibility of routes is heuristically obtained with a classical bottom‐left based method to tackle the explicit consideration of loading constraints. All the strategies were implemented and exhaustively tested on instances adapted from the literature. The results of this computational study are discussed at the end of this paper.     

An evolutionary algorithm for the one‐dimensional cutting stock problem

This paper deals with the one‐dimensional integer cutting stock problem, which consists of cutting a set of available objects in stock in order to produce ordered smaller items in such a way as to minimize the waste of material. The case in which there are various types of objects available in stock in limited quantities is studied. A new heuristic method based on the evolutionary algorithm concept is proposed to solve the problem. This heuristic is empirically analyzed by solving randomly generated instances and the results are compared with other methods from the literature.     

Extended guided tabu search and a new packing algorithm for the two-dimensional loading vehicle routing problem

In this paper, we develop an extended guided tabu search (EGTS) and a new heuristic packing algorithm for the two-dimensional loading vehicle routing problem (2L-CVRP). The 2L-CVRP is a combination of two well-known NP-hard problems, the capacitated vehicle routing problem, and the two-dimensional bin packing problem. It is very difficult to get a good performance solution in practice for these problems. We propose a meta-heuristic methodology EGTS which incorporates theories of tabu search and extended guided local search (EGLS). It has been proved that tabu search is a very good approach for the CVRP, and the guiding mechanism of the EGLS can help tabu search to escape effectively from local optimum. Furthermore, we have modified a collection of packing heuristics by adding a new packing heuristic to solve the loading constraints in 2L-CVRP, in order to improve the cost function significantly. The effectiveness of the proposed algorithm is tested, and proven by extensive computational experiments on benchmark instances.     

Algorithms for a two‐machine flowshop problem with jobs of two classes

We consider a problem of scheduling jobs of two classes of urgencies in a two‐machine flowshop with the objective of minimizing total tardiness of one class for urgent jobs and the maximum completion time of the other class for non‐urgent jobs. Urgent jobs are an important consideration in the real manufacturing systems, but it has not been studied due to the difficulty of the problem. In this research, a branch‐and‐bound (B&B) algorithm is proposed through developing lower bounds, dominance properties, and heuristic algorithms for obtaining an initial feasible solution. To evaluate the performance of the proposed algorithms, computational experiments on randomly generated instances are performed. Results of the experiments show that the suggested B&B algorithm can solve problems with up to 20 jobs in a reasonable amount of CPU time.     

A new heuristic algorithm for the circular packing problem with equilibrium constraints

The circular packing problem with equilibrium constraints is an optimization problem about simplified satellite module layout design.A heuristic algorithm based on tabu search is put forward for solving this problem.The algorithm begins from a random initial configuration and applies the gradient method with an adaptive step length to search for the minimum energy configuration.To jump out of the local minima and avoid the search doing repeated work,the algorithm adopts the strategy of tabu search.In the pr...     

Hybrid greedy heuristics based on linear programming for the three‐dimensional single bin‐size bin packing problem

In this paper, we propose to solve the three‐dimensional single bin‐size bin packing problem (3D‐SBSBPP) using a simple strategy based on integer linear programming (ILP) heuristics, without using any improvement based on metaheuristics. We first propose an ILP that is converted into a series of three‐dimensional single knapsack problems (3D‐SKP). Then, the first tailored heuristic can be viewed as a hybrid approach in which both “selection” and “positioning” phases are combined. The first phase serves to select a subset of items where each of these items is susceptible to belonging to an active container. The positioning phase serves to pack a subset of items already preselected by the selection phase. Then, both phases cooperate till packing all items into their corresponding containers. The second heuristic can be viewed as an extended version of the first one. Indeed, before deciding whether the current container is closed or a new container is activated, “a local reoptimization phase” is considered. Finally, both proposed heuristics are evaluated on a set of random instances obtained by using the standard generator scheme of the literature. The provided results show that both proposed heuristics remain competitive when compared to the results obtained by one of the best methods of the literature.     

基于禁忌搜索的启发式算法求解带平衡约束的圆形装填问题

带平衡约束的圆形装填(Packing)问题是一类简化的卫星舱布局优化问题.现提出一个基于禁忌搜索的启发式(TSH)算法对该问题进行求解.算法从任一初始格局出发,应用基于自适应步长的梯度法进行能量极小化.为了使计算能有效地逃离局部极小点的陷阱且避免迂回搜索,算法采用了禁忌搜索的策略.在禁忌搜索的过程中,我们对传统的邻域解、禁忌对象以及当前解接受原则进行了有效的改进.对两组共11个有代表性的算例进行了实算.计算结果表明,TSH算法刷新了其中7个算例的当今国际上的最好纪录,对于其余4个算例,该算法均达到问题的最优解.     

Approximate algorithms for the container loading problem

In this paper we develop several algorithms for solving three–dimensional cutting/packing problems. We begin by proposing an adaptation of the approach proposed in Hifi and Ouafi (1997) for solving two–staged unconstrained two–dimensional cutting problems. We show how the algorithm can be polynomially solved for producing a constant approximation ratio. We then extend this algorithm for developing better approximate algorithms. By using hill–climbing strategies, we construct some heuristics which produce a good trade–off between the computational time and the solution quality. The performance of the proposed algorithms is evaluated on different problem instances of the literature, with different sizes and densities (a total of 144 problem instances).