Stowage planning in maritime container transportation

We consider a stowage-planning problem of arranging containers on a container ship in the maritime transportation system. Since containers are accessible only from the top of the stack, temporary unloading and reloading of containers, called shifting, is unavoidable if a container required to be unloaded at the current port is stacked under containers to be unloaded at later ports on the route of the ship. The objective of the stowage planning problem is to minimize the time required for shifting and crane movements on a tour of a container ship while maintaining the stability of the ship. For the problem, we develop a heuristic solution method in which the problem is divided into two subproblems, one for assigning container groups into the holds and one for determining a loading pattern of containers assigned to each hold. The former subproblem is solved by a greedy heuristic based on the transportation simplex method, while the latter is solved by a tree search method. These two subproblems are solved iteratively using information obtained from solutions of each other. To see the performance of the suggested algorithm, computational tests are performed on problem instances generated based on information obtained from an ocean container liner. Results show that the suggested algorithm works better than existing algorithms.     

A tree search method for the container loading problem with shipment priority

This paper addresses a special kind of container loading problem with shipment priority. We present a tree search method, which is based on a greedy heuristic. In the greedy heuristic, blocks made up of identical items with the same orientation are selected for packing into a container. Five evaluation functions are proposed for block selection, and the different blocks selected by each evaluation function constitute the branches of the search tree. A method of space splitting and merging is also embedded in the algorithm to facilitate efficient use of the container space. In addition, the proposed algorithm covers an important constraint called shipment priority to solve practical problems. The validity of the proposed algorithm is examined by comparing the present results with those of published algorithms using the same data.     

Heuristics with a new block strategy for the single and multiple containers loading problems

A heuristic algorithm using new block strategy for the heterogeneous single and multiple containers loading problem (CLP) is proposed in this paper. In order to solve the single CLP, this algorithm fills unused spaces with the homogeneous load-blocks of identically oriented boxes and splits residual space into three child-spaces starting with an empty container. An initial container pattern is first built applying this approach recursively until all boxes are stowed or no unused spaces are left. And then, alternative container patterns are generated after replacing the load-blocks of the pattern-determining spaces in the initial container pattern with the alternative-blocks previously stored. Finally, an improvement procedure compares these alternatives with the initial container pattern to identify an improved container pattern. An algorithm for the multiple CLP uses the single CLP algorithm to generate an initial solution and uses improvement procedures to improve the initial solution. Numerical experiments with 715 test cases for the single CLP and 47 test cases for the multiple the CLP revealed the excellent performance of this algorithm.     

Solving container loading problems by block arrangement

In order to solve heterogeneous single and multiple container loading problems, an algorithm is presented that builds homogeneous blocks of identically orientated items. First a greedy heuristic is presented that generates the desired block arrangements. Second the solutions provided by the greedy heuristic are improved by a tree search. Additional aspects such as load stability and weight distribution within the container are also taken into account. The test cases of Bischoff and Ratcliff are used for benchmarking purposes.     

A heuristic for the circle packing problem with a variety of containers

In this paper we present a heuristic algorithm based on the formulation space search method to solve the circle packing problem. The circle packing problem is the problem of finding the maximum radius of a specified number of identical circles that can be fitted, without overlaps, into a two-dimensional container of fixed size. In this paper we consider a variety of containers: the unit circle, unit square, rectangle, isosceles right-angled triangle and semicircle. The problem is formulated as a nonlinear optimization problem involving both Cartesian and polar coordinate systems.Formulation space search consists of switching between different formulations of the same problem, each formulation potentially having different properties in terms of nonlinear optimization. As a component of our heuristic we solve a nonlinear optimization problem using the solver SNOPT.Our heuristic improves on previous results based on formulation space search presented in the literature. For a number of the containers we improve on the best result previously known. Our heuristic is also a computationally effective approach (when balancing quality of result obtained against computation time required) when compared with other work presented in the literature.     

Packing items to feed assembly lines

We treat a practical application of packing problems in feeding assembly lines. This study was motivated by a real situation encountered in the shop floor of a major automobile industry plant in Brazil. The assembly line feed problem (LFP) consists in how pack the items in the available containers to meet the line work centers’ requirements with a minimum total cost over the planning horizon. LFP is a variable-sized bin packing problem that has two special features: (i) a cardinality constraint on each bin’s size; and, (ii) a cost structure such that each bin’s cost varies according to the items that are packed in it. We propose an integer programming model and a GRASP heuristic for LFP. Numerical results on real-life test instances are reported.     

Port space allocation with a time dimension

In the Port of Singapore, as in many other ports, space has to be allocated in yards for inbound and transit cargo. Requests for container space occur at different times during the planning period, and are made for different quantities and sizes of containers. In this paper, we study space allocation under these conditions. We reduce the problem to a two-dimensional packing problem with a time dimension. Since the problem is NP-hard, we develop heuristic algorithms, using tabu search, simulated annealing, a genetic algorithm and ‘squeaky wheel’ optimization, as solution approaches. Extensive computational experiments compare the algorithms, which are shown to be effective for the problem.     

Three-stage heuristic algorithm for three-dimensional irregular packing problem

This paper studies a new practical problem which can be decomposed into three three-dimensional packing problems: three-dimensional irregular packing with variable-size cartons problem, three-dimensional variable-size bin packing problem, and the single container loading problem. Since the three sub-problems are NP-hard, searching a good solution becomes more difficult. In this paper, mathematical models of each sub-problem are developed and three-stage heuristic algorithms are proposed to solve this new problem. Experiments are conducted with random instances generated by real-life case. Computational results indicate that the proposed algorithm is efficient and can yield satisfactory results.     

Storage space allocation models for inbound containers in an automatic container terminal

This paper studies the problem of improving the operations efficiency for retrieving inbound containers in a modern automatic container terminal. In the terminal, when an external truck arrives to collect a container stored in a specific container block, it waits at one end of the block where an automatic stack crane will retrieve the container and deliver it to the truck. With the aim of reducing the expected external truck waiting time which is determined by how the containers are stored in a block, we propose two correlated approaches for the operations efficiency improvement, (1) by designing an optimized block space allocation to store the inbound containers after they are discharged from vessels, and (2) by conducting overnight re-marshaling processes to re-organize the block space allocation after some containers are retrieved. For the block space allocation problem, we consider three optimization models under different strategies of storing containers, namely, a non-segregation model, a single-period segregation model, and a multiple-period segregation model. Optimal solution methods are proposed for all three models. For the re-marshaling problem with a given time limit, we find that the problem is NP-hard and develop a heuristic algorithm to solve the problem. We then use simulation to validate our models and solution approaches. Simulation results reveal important managerial insights such as the advantage of the multiple-period segregation over the myopic single-period segregation, the possibility of overflow of the segregation model, and the benefit of re-marshaling.     

A hybrid genetic algorithm for the container loading problem

This paper presents a hybrid genetic algorithm (GA) for the container loading problem with boxes of different sizes and a single container for loading. Generated stowage plans include several vertical layers each containing several boxes. Within the procedure, stowage plans are represented by complex data structures closely related to the problem. To generate offspring, specific genetic operators are used that are based on an integrated greedy heuristic. The process takes several practical constraints into account. Extensive test calculations including procedures from other authors vouch for the good performance of the GA, above all for problems with strongly heterogeneous boxes.     

A Lagrangian relaxation-based heuristic for the vehicle routing with full container load

We address a problem of vehicle routing that arises in picking up and delivering full container load from/to an intermodal terminal. The substantial cost and time savings are expected by efficient linkage between pickup and delivery tasks, if the time of tasks and the suitability of containers for cargo allow. As this problem is NP-hard, we develop a subgradient heuristic based on a Lagrangian relaxation which enables us to identify a near optimal solution. The heuristic consists of two sub-problems: the classical assignment problem and the generalized assignment problem. As generalized assignment problem is also NP-hard, we employ an efficient solution procedure for a bin packing based problem, which replaces the generalized assignment problem. The heuristic procedure is tested on a wide variety of problem examples. The test results demonstrate that the procedure developed here can efficiently solve large instances of the problem.     

A reference length approach for the 3D strip packing problem

In the three-dimensional strip packing problem (3DSP), we are given a container with an open dimension and a set of rectangular cuboids (boxes) and the task is to orthogonally pack all the boxes into the container such that the magnitude of the open dimension is minimized. We propose a block building heuristic based on extreme points for this problem that uses a reference length to guide its solution. Our 3DSP approach employs this heuristic in a one-step lookahead tree search algorithm using an iterative construction strategy. We tested our approach on standard 3DSP benchmark test data; the results show that our approach produces better solutions on average than all other approaches in literature for the majority of these data sets using comparable computation time.     

A goal-driven prototype column generation strategy for the multiple container loading cost minimization problem

In the multiple container loading cost minimization problem (MCLCMP), rectangular boxes of various dimensions are loaded into rectangular containers of various sizes so as to minimize the total shipping cost. The MCLCMP can be naturally modeled as a set cover problem. We generalize the set cover formulation by introducing a new parameter to model the gross volume utilization of containers in a solution. The state-of-the-art algorithm tackles the MCLCMP using the prototype column generation (PCG) technique. PCG is an effective technique for speeding up the column generation technique for extremely hard optimization problems where their corresponding pricing subproblems are NP-hard. We propose a new approach to the MCLCMP that combines the PCG technique with a goal-driven search. Our goal-driven prototype column generation (GD-PCG) algorithm improves the original PCG approach in three respects. Computational experiments suggest that all three enhancements are effective. Our GD-PCG algorithm produces significantly better solutions for the 350 existing benchmark instances than all other approaches in the literature using less computation time. We also generate two new set instances based on industrial data and the classical single container loading instances.     

Transportation problem with nonlinear side constraints a branch and bound approach

In a container terminal management, we are often confronted with the following problem: how to assign a reasonable depositing position for an arriving container, so that the efficiency of searching for and loading of a container later can be increased. In this paper, the problem is modeled as a transportation problem with nonlinear side constraints (TPNSC). The reason of nonlinear side constraints arising is that some kinds of containers cannot be stacked in the same row (the space of storage yard is properly divided into several rows). A branch and bound algorithm is designed to solve this problem. The algorithm is based on the idea of using disjunctive arcs (branches) for resolving conflicts that are created whenever some conflicting kinds of containers are deposited in the same row. During the branch and bound, the candidate problems are transformed into classical transportation problems, so that the efficient transportation algorithm can be applied, at the same time the reoptimization technique is employed during the branch and bound. Further, we design a heuristic to obtain a feasible initial solution for TPNSC in order to prune some candidates as early and/or as much as possible. We report computational results on randomly generated problems.     

Planning Activities in a Network of Logistic Platforms with Shared Resources

This paper has been motivated by the study of a real application, the transshipment container terminal of Gioia Tauro in Italy. The activities in a container terminal concern with the movement of containers from/to mother vessels and feeders and with the handling and storage of containers in the yard. For such type of applications both operational (e.g., scheduling) and tactical (e.g., planning) models, currently available in the literature, are not useful in terms of operations management and resources optimization. Indeed, the former models are too detailed for the complexity of the systems, while the latter are not able to capture the operational constraints in representing those activities which limit the nominal capacity. Herein, the container terminal, or more in general a service or production system, is represented as a network of complex substructures or platforms. The idea is to formalize the concept of platform capacity, which is used to represent the operational aspects of the container terminal in a mathematical model for the tactical planning. The problem, which consists in finding an allocation of resources in each platform in order to minimize the total delay on the overall network and on the time horizon, is modelled by a mathematical programming formulation for which we carry out a computational analysis using CPLEX-MIP solver. Moreover, we present a dynamic programming based heuristic to solve larger instances in short computational time. On all but one of the smaller instances, the heuristic solutions are also optimal. On the larger instances, the maximum gap, i.e. the percentage deviation, between the heuristic solutions and the best solutions computed by CPLEX-MIP within the time limit of 3600 s, has been 6.3%.     

An adaptive guidance meta-heuristic for the vehicle routing problem with splits and clustered backhauls

This paper presents the case study of an Italian carrier, Grendi Trasporti Marittimi, which provides freight transportation services by trucks and containers. Its trucks deliver container loads from a port to import customers and collect container loads from export customers to the same port. In this case study, all import customers in a route must be serviced before all export customers, each customer can be visited more than once and containers are never unloaded or reloaded from the truck chassis along any route. We model the problem using an Integer Linear Programming formulation and propose an Adaptive Guidance metaheuristic. Our extensive computational experiments show that the adaptive guidance algorithm is capable of determining good-quality solutions in many instances of practical or potential interest for the carrier within 10?min of computing time, whereas the mathematical formulation often fails to provide the first feasible solution within 3?h of computing time.     

Packing non-identical circles within a rectangle with open length

Packing non-identical circles inside a rectangle witnesses a wide range of industrial applications. However, the non-convex constraints in this problem make it intractable using exact analytical approaches. Even via heuristic methods, the solution time for industrial-scale instances sometimes is too long to be acceptable. This article aims to challenge the existing methods for the benchmark instances. The most significant contributions of this work are: firstly, we proposed three types of packing positions for selection and used human intelligence to convert an arbitrary circle sequence into a feasible compact layout; secondly, diverse position selection criteria have been tested, and it is found that the criterion commonly used in the literature is not the best; thirdly, the traditional genetic algorithm is adapted with lower crossover rate but higher mutation rate particularly, and a minor-adjustment operator with the purpose of exploring the neighborhood of the current best solutions is introduced.     

Sub-exponential graph coloring algorithm for stencil-based Jacobian computations

Partial differential equations can be discretized using a regular Cartesian grid and a stencil-based method to approximate the partial derivatives. The computational effort for determining the associated Jacobian matrix can be reduced. This reduction can be modeled as a (grid) coloring problem. Currently, this problem is solved by using a heuristic approach for general graphs or by developing a formula for every single stencil. We introduce a sub-exponential algorithm using the Lipton–Tarjan separator in a divide-and-conquer approach to compute an optimal coloring. The practical relevance of the algorithm is evaluated when compared with an exponential algorithm and a greedy heuristic.     

A multi-heuristic approach for solving the pre-marshalling problem

Minimizing the number of reshuffling operations at maritime container terminals incorporates the pre-marshalling problem (PMP) as an important problem. Based on an analysis of existing solution approaches we develop new heuristics utilizing specific properties of problem instances of the PMP. We show that the heuristic performance is highly dependent on these properties. We introduce a new method that exploits a greedy heuristic of four stages, where for each of these stages several different heuristics may be applied. Instead of using randomization to improve the performance of the heuristic, we repetitively generate a number of solutions by using a combination of different heuristics for each stage. In doing so, only a small number of solutions is generated for which we intend that they do not have undesirable properties, contrary to the case when simple randomization is used. Our experiments show that such a deterministic algorithm significantly outperforms the original nondeterministic method. The improvement is twofold, both in the quality of found solutions, and in the computational effort.     

The rollon–rolloff waste collection vehicle routing problem with time windows

This study introduces a rollon–rolloff waste collection vehicle routing problem involving large containers that accumulate huge amounts of garbage at construction sites and shopping districts. In this problem, tractors move one container at a time between customer locations, a depot, disposal facilities, and container storage yards. The complicated constraints discussed in this study arise from having multiple disposal facilities, multiple container storage yards, seven service types of customer demands, different time windows for customer demands and facilities, various types and sizes of containers, and the lunch break of tractor drivers. In addition, real-world issues, such as changing service types, multiple demands at a customer’s location, and tractors with different work schedules, are dealt with. This study proposes a large neighborhood search based iterative heuristic approach consisting of several algorithms for the problem. The effectiveness of the proposed methods is demonstrated by computational experiments using benchmark data, some instances of which are derived from real-world problems.