scispace - formally typeset
Search or ask a question
Journal ArticleDOI

A Branch-and-Price Algorithm to Solve the Integrated Berth Allocation and Yard Assignment Problem in Bulk Ports

TL;DR: In this article, two crucial optimization problems of berth allocation and yard assignment in the context of bulk ports are studied, and an exact solution algorithm based on a branch and price framework is proposed to solve the integrated problem.
About: This article is published in European Journal of Operational Research.The article was published on 2014-06-01 and is currently open access. It has received 80 citations till now. The article focuses on the topics: Branch and price & Assignment problem.

Summary (4 min read)

1 Introduction

  • In the last decade, the shipping tonnage for dry bulk and liquid bulk cargo has risen by 52% and 48% respectively.
  • This paper focuses on the tactical level decision planning for the integrated berth and yard management in the context of bulk ports.
  • In contrast in bulk ports, depending on the vessel requirements and cargo properties, a wide variety of equipments are used for discharging or loading operations.
  • The tactical yard assignment problem refers to decisions that concern the storage location and the routing of materials.
  • This affects the travel distance between the assigned berth to the vessel and storage location of the cargo type of the vessel on the yard, and furthermore determines the storage efficiency of the yard.

2 Literature Review

  • From the past OR literature on container terminal operations, it is well established that integrated planning of operations can allow port terminals to reduce congestion, lower delay costs and enhance efficiency.
  • Works on integrated problems related to yard management in container terminals include Bish et al. (2001) and Kozan and Preston (2006) who propose the integration of yard allocation and container transfers, whereas Chen et al. (2007) and Lau and Zhao (2007) study the integrated scheduling of handling equipment in a container terminal.
  • An evolutionary heuristic is developed to solve larger instances obtained from the real port data.
  • To the best of their knowldge, operations research problems have received almost no attention thus far in the context of bulk port terminals.

3 Problem Statement

  • In this section the authors elaborate on the background for the integrated berth allocation and yard assignment problem in the context of bulk ports.
  • In the present work, the authors extend the berth allocation problem to account for the assignment of different yard locations to specific cargo types and vessels berthing at the port.
  • A major difference between bulk port and container terminal operations is the need to explicitly account for the cargo type on the vessel in bulk ports.
  • In the integrated model, the authors assume that each vessel has a single cargo type but that can be discharged and transferred to (from) multiple yard locations.

4.2 Mathematical Model

  • The objective function (1) minimizes the total service time of all vessels, which is the sum of total delays and total handling time of vessels berthing at the port.
  • Constraints (3)-(5) are the non-overlapping restrictions for any two vessels berthing at the port.
  • Constraints (21) state that the yard locations have dedicated cargo types, and given yard location can be assigned to at most one cargo type.

5 Solution Approach

  • The mixed integer programming formulation of the integrated model is extremely complex and unwieldy, and could not be used to solve even small sized instances, as was validated by numerical experiments.
  • Thus in the following section, the authors decompose the mixed integer model and formulate it as a set partitioning problem.

5.1 Set Partitioning Model

  • Let Ω be the set of feasible berthing assignments of all vessels berthing at the port in the given planning horizon.
  • Constraints (36) ensure that there is exactly one feasible berthing assignment for each vessel in the optimal solution.
  • Constraints (38) and (39) state that a given cargo location on the yard can store at most one cargo type.
  • The above formulation was tested for small instances inspired from real port data.
  • Hence in order to avoid the "explosion" of the solution time, the authors propose to solve the linear programming relaxation of the above problem using column generation, as described in the next section.

5.2 Column Generation

  • The main idea behind the column generation method is to keep the set partitioning model as the so called "restricted master problem" (RMP).
  • Moreover, the binary decision variables λa and µpw are relaxed in order to speed up the convergence of the process.
  • In the first iteration of column generation, the RMP is solved using the setΩ1 consisting of vessel assignments in the initial feasible solution provided by the greedy heuristic (Section 5.2.1).
  • Since there is no direct effect on the objective function when the µ variable value(s) are adjusted in constraints 39 and 40, it implies that the dual variable values for these constraints would be always zero.
  • When there are no columns with negative reduced cost for any subproblem to enterΩ1, the column generation terminates.

5.2.2 Sub–Problem

  • In each subproblem, the objective is to identify the feasible assignment for that particular vessel with the most negative reduced cost to be added to the current pool of active columns Ω1 in the restricted master problem.
  • Constraints (46) are used to determine the total handling time of the vessel.
  • Constraints (51) ensure that a vessel is assigned to a yard location only if that yard location stores the cargo type on the vessel.
  • Since the sub-problem is solved separately for each vessel, there are no non-overlapping constraints in the above formulation and the complexity of the problem is significantly reduced.
  • Thus in the proposed algorithm, the sub-problems are solved as mixed integer programs directly using CPLEX solver.

5.3 Branch and Bound

  • Since the authors solve the linear relaxation of the restricted master problem, the final solution obtained after the convergence of the column generation process is typically not an integer solution.
  • In the "worst" case, the solution of the relaxed version of the restricted master problem provides a lower bound to the solution of the original problem and an integer solution can be obtained by applying the branch and bound algorithm (for more details on branch and bound, please refer to Hillier and Lieberman (2001)) to the obtained solution.
  • Therefore to find the optimal solution, column generation has to be executed again at every node of the branch and bound tree (Barnhart et al. (1998)).
  • When the list of µ variables has been exhausted, i.e. there are no fractional µ variables, then the authors start branching on the λ variables.
  • Note that while branching, both the RMP and the subproblem(s) have to be modified by adding extra constraints.

6 Improvement Methods

  • The branch and price method has been designed to tackle computationally heavy problems and to achieve a tight bound (lower bound in case of minimization).
  • The recommended method is to decompose the model in a way such that the sub-problem has a block angular structure (see Alvelos (2005) and Desaulniers et al. (2005)), so that each subproblem can be solved independently of each other.
  • Each sub-problem is trying to find the berthing and yard assignment of a given vessel that has the most negative reduced cost for a given set of dual variables reported from the solution output of the restricted master problem in the last iteration.
  • There are many other techniques to improve the convergence of the column generation process such as stabilization of the dual variables (Pigatti et al. (2005)) and dynamic constraint aggregation (Elhallaoui et al. (2005) and Elhallaoui et al. (2008)).

7.1 Generation of Instances

  • The algorithm was implemented in Java Programming language.
  • The test instances are based on a sample of real port data obtained from the SAQR port, RasAl-Khaimah, UAE, the biggest bulk port in the middle east.
  • In the tested instances, the authors consider five cargo types - liquid bulk that needs the pipeline facility for discharging, rock aggregates that need the conveyor facility for loading and three other general type dry cargoes (clay, coal and cement).
  • In all the tested instances, the planning horizon is 5 days or 120 hours.

7.2 Computational Results and Analysis

  • In Tables 6, 8 and 9, the results obtained from the column generation method for instances containing |N|=10, 25 and 40 vessels respectively are shown.
  • As discussed earlier, the solution value obtained from the column generation method provides a lower bound to the optimal solution of the original problem.
  • In the tables shown, the optimality gap between the lower bound and the optimal solution is represented by gap1, whereas gap2 represents the optimality gap between the optimal solution and the upper bound.
  • As shown in Table 6, for |N| = 10 vessels, for instances 7 and 8, the optimal solution is obtained directly from the column generation method, and hence there is no need for running the branch and bound algorithm for these instances .
  • On the other hand, the total optimality gap represented by gap in the tables, is smaller for the instances with higher level of congestion.

Did you find this useful? Give us your feedback

Citations
More filters
Journal ArticleDOI
TL;DR: This paper classify this new literature according to the features of models considered for berth allocation, quay crane scheduling and integrated approaches by using the classification schemes proposed in the preceding survey.

408 citations

Journal ArticleDOI
TL;DR: In this paper, the authors conduct a systematic literature review to analyze operational strategies (e.g., peak shaving, operations optimization), technology usage, alternative fuels and energy management systems for improving the energy efficiency and environmental performance of ports and terminals.
Abstract: Many ports and terminals endeavor to enhance energy efficiency as energy prices have increased through years and climate change mitigation is a key target for the port industry. Stricter environmental regulations are adopted by authorities to limit pollutants and GHG emissions arising from energy consumption. Increasingly, port operational strategies and energy usage patterns are under scrutiny. To ingrain sustainability and environmental protection of ports, the use of innovative technology appears as a critical conduit in achieving a transition from a carbon-intensive port industry (dependent on fossil fuels) to a low-carbon port model by harnessing renewable energy, alternative fuels (e.g. LNG, hydrogen, biofuel), smarter power distribution systems, energy consumption measurement systems. In this context, this paper conducts a systematic literature review to analyze operational strategies (e.g. peak shaving, operations optimization), technology usage (e.g. electrification of equipment, cold-ironing, energy storage systems), renewable energy, alternative fuels and energy management systems (e.g. smart grid with renewable energy) for improving the energy efficiency and environmental performance of ports and terminals. Research gaps and future research directions are identified. Analysis shows that there is a great potential for ports to achieve further energy efficiency and researchers have many impactful research opportunities.

220 citations

Journal ArticleDOI
Lu Zhen1
TL;DR: This study finds that the robust method can derive a near optimal solution to the stochastic model in a fast way, and also has the benefit of limiting the worst-case outcome of the tactical BAP decisions.

147 citations

Journal ArticleDOI
TL;DR: In this article, the authors extended the current state-of-the-art by proposing novel set partitioning models and analyzed the effects of different discretization schemes and the impact of using a time-variant/invariant quay crane allocation policy.
Abstract: Most of the operational problems in container terminals are strongly interconnected. In this paper, we study the integrated Berth Allocation and Quay Crane Assignment Problem in seaport container terminals. We will extend the current state-of-the-art by proposing novel set partitioning models. To improve the performance of the set partitioning formulations, a number of variable reduction techniques are proposed. Furthermore, we analyze the effects of different discretization schemes and the impact of using a time-variant/invariant quay crane allocation policy. Computational experiments show that the proposed models significantly improve the benchmark solutions of the current state-of-art optimal approaches.

134 citations

Journal ArticleDOI
Lu Zhen1
TL;DR: In this paper, the authors investigated the concept of yard congestion quantitatively in the context of yard truck interruptions, and developed a combination of probabilistic and physics-based models for truck interruption.
Abstract: As a tactical-level plan, a yard template determines the assignment of spaces in a container port yard for arriving vessels. This paper investigates the concept of yard congestion quantitatively in the context of yard truck interruptions, and develops a combination of probabilistic and physics-based models for truck interruptions. The above work enables us to exactly evaluate the expected link travel time, which then acts as the basis for proposing a mixed-integer programming model that minimizes the total expected travel time of moving containers around the yard. A Squeaky Wheel Optimization based meta-heuristic is developed to solve the model. Experiments are also conducted to validate the effectiveness of the model and the solution method.

112 citations

References
More filters
Journal ArticleDOI
TL;DR: In this paper, column generation methods for integer programs with a huge number of variables are discussed, including implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branch-and-bound tree.
Abstract: We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branch-and-bound tree. We present classes of models for which this approach decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. We then discuss computational issues and implementation of column generation, branch-and-bound algorithms, including special branching rules and efficient ways to solve the LP relaxation. We also discuss the relationship with Lagrangian duality.

2,248 citations

Book
01 Jan 1968

1,360 citations

Journal ArticleDOI
TL;DR: The main logistics processes and operations in container terminals are described and classified and a survey of methods for their optimization is presented.
Abstract: In the last four decades the container as an essential part of a unit-load-concept has achieved undoubted importance in international sea freight transportation. With ever increasing containerization the number of seaport container terminals and competition among them have become quite remarkable. Operations are nowadays unthinkable without effective and efficient use of information technology as well as appropriate optimization (operations research) methods. In this paper we describe and classify the main logistics processes and operations in container terminals and present a survey of methods for their optimization.

1,158 citations

Journal ArticleDOI
TL;DR: An earlier survey which proved to be of utmost importance for the community is updated and extended to provide the current state of the art in container terminal operations and operations research.
Abstract: The current decade sees a considerable growth in worldwide container transportation and with it an indispensable need for optimization. Also the interest in and availability of academic literatures as well as case reports are almost exploding. With this paper an earlier survey which proved to be of utmost importance for the community is updated and extended to provide the current state of the art in container terminal operations and operations research.

1,016 citations

Journal ArticleDOI
TL;DR: New classification schemes for berth allocation problems and quay crane scheduling problems are developed and particular focus is put on integrated solution approaches which receive increasing importance for the terminal management.

722 citations

Frequently Asked Questions (2)
Q1. What contributions have the authors mentioned in the paper "A branch-and-price algorithm to solve the integrated berth allocation and yard assignment problem in bulk ports" ?

In this research, two crucial optimization problems of berth allocation and yard assignment in the context of bulk ports are studied. The authors discuss how these problems are interrelated and can be combined and solved as a single large scale optimization problem. The authors propose an exact solution algorithm based on branch and price framework to solve the integrated problem. 

In the past, few scholars have attempted to study the berth allocation problem in integration with the yard assignment problem in the context of container terminals, while in the context of bulk ports the problem has not been studied at all. Thus t may be worth investigating the reduction in the solution time of the sub-problems by using heuristic methods such as dynamic programming, instead of using exact methods and optimization solvers.