Stacking outbound barge containers in an automated deep-sea terminal

Accepted Manuscript

Stacking Outbound Barge Containers in an Automated Deep-Sea Terminal Amir Gharehgozli , NimaZaerpour PII: DOI: Reference:

S0377-2217(17)31174-8 10.1016/j.ejor.2017.12.040 EOR 14899

To appear in:

European Journal of Operational Research

Received date: Revised date: Accepted date:

22 March 2017 14 September 2017 26 December 2017

Please cite this article as: Amir Gharehgozli , NimaZaerpour , Stacking Outbound Barge Containers in an Automated Deep-Sea Terminal, European Journal of Operational Research (2018), doi: 10.1016/j.ejor.2017.12.040

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The stacking problem of outbound containers in a deep-sea container terminal is studied. A shared policy is proposed to stack different container types in one pile with no reshuffling. Our heuristic uses vertical stacking to construct and Simulated Annealing to improve a solution. The results show that the shared heuristic outperforms the dedicated policy often used in practice. The shared heuristic is robust to realistic disturbances in the arrival and departure of barges.

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Stacking Outbound Barge Containers in an Automated Deep-Sea Terminal

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Amir Gharehgozli1, Nima Zaerpour2

1 Department of Maritime Administration, Texas A&M University at Galveston, Galveston, Texas, USA

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Corresponding author: Department of Maritime Administration, Texas A&M University at Galveston, PO Box 1675, Galveston, Texas 77553, USA. Tel: +1 (409) 740-4854, Fax: +1 (409) 741-4014, Email: [email protected].

2 College of Business Administration, California State University San Marcos, San Marcos, California, USA

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[email protected], [email protected]

Abstract

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In this paper, we study the stacking problem of outbound containers in a deep-sea container terminal. In such a terminal, outbound containers -to be transported by barge to the hinterland or other terminals within the port area- are stacked in a container stack.

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Reshuffling which is the process of removing interfering containers to access a desired container is one of the main challenges of such container terminals. In order to avoid

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reshuffling, container terminals commonly use a stacking policy in which each pile (column) accommodates the containers of the same barge with the same weight class, destination, etc. However, due to a limited number of containers per barge per weight class and destination, this policy might result in a low stack utilization. To address this issue, this study proposes an alternative stacking policy allowing different container types to share the same pile. We aim to minimize the total retrieval time of a set of containers. We show that the problem is strongly

-complete. Thus, we propose a heuristic to quickly solve the problem and we

compare the results with a lower bound. The results show that the proposed stacking 2

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heuristic can provide solutions with a gap of less than 10% with the lower bound for realsized instances with high utilization. In addition, the results show that the stacking heuristic can reduce the total retrieval time up to 30% compared to often-used in practice stacking policy. Furthermore, in order to investigate the performance of the proposed stacking policy under different settings, we perform a sensitivity analysis by varying block configuration, number of barges, barge size, and barge arrival time window.

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Keywords: OR in Maritime Industry; outbound barge containers; stacking operations; heuristic; value of information

1. Introduction

Oceanic transportation is the main driver of international freight transportation. According to UNCTAD

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(2015), seaborne trade reached 9.84 billion tons in 2014, divided across oil, major bulks, and dry cargo. The trends show that an increasing share of oceanic transportation is containerized freight. Due to globalization, the number of containers has increased significantly in recent years and the growth is expected in future. Further, container terminal throughputs has increased even faster due to an increase in the number of containers being transshipped. Thus, container terminals play a vital role in

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the organization of efficient global trade (Fransoo & Lee, 2013; Gorman et al., 2014; Gharehgozli et al., 2016). A large terminal handles millions of containers annually. For instance, container terminals in the

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Port of Rotterdam (Busiest European port) handled more than 12 million TEU in 2014 while those in Shanghai Port (Busiest port in the world) handled more than 35 million TEU in the same year. The top 20 world container ports in terms of throughput experienced an average growth of 487% from 2000 to

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2014 (American Association of Port Authorities, 2016; Bureau of Transportation Statistics, 2016). Handling the increased flow of containers puts burden not only on a terminal‟s seaside operations but

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also on the transport links with the hinterland. Barge, truck, and train are the three main modes of hinterland container transportation. Each transportation mode has key operational and commercial

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advantages. However, during the past decades, truck transport has been the dominant mode of transportation (INeS Danube, 2016). However, the current trend is prompting a modal shift from road to barge or train to reduce the pressure on the roads and to reduce greenhouse gas emissions. Barge has the lowest energy consumption and the lowest external costs compared to land transport modes (see Figure 1(a)). Furthermore, because of the physical properties of water conferring buoyancy and limited friction, barge has the ability to transport large quantities of cargo over long distances. In fact, with the same energy consumption, a barge can transport one ton of cargo almost four times farther than a truck (Viadonau, 2013 and 2015). Finally, inland navigation requires comparably low investment 3

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in maintaining and expanding its infrastructure. Figure 1(b) shows an example of infrastructure costs for different modes of transportation. The figure shows that infrastructure costs per ton-kilometer are roughly four times higher for road or rail than for waterways (Planco Consulting & Bundesanstalt für Gewässerkunde, 2007). Thus, port authorities have started with a model shift toward inland navigation. For example, one of the aims of the Port of Rotterdam authority is to change the current truck/barge/rail split of 45/40/15

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percent to 35/45/20 percent by 2035 (Port of Rotterdam Authority, 2012). Currently, in the Port of Rotterdam, 75–100 barges visit the port daily, visiting, on average, eight terminals out of the total 30 terminals in the port area (Douma et al., 2009). Another example is the increasing number of barges in moving containers between Bangkok and Thailand‟s gateway port of Laem Chabang, 90 km (60 miles)

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south of Bangkok (Mackey, 2015).

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(a) The sum of external costs for inland vessels is by (b) Comparison of infrastructure costs (example of far the lowest (average values for bulk goods) German inland transport modes) Figure 1. Costs associated with hinterland transportation

This model shift trend will result in an increasing number of barge calls at container terminals. On a daily basis, an enormous number of containers are unloaded from containerships arriving at a terminal

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and have to be transported to the hinterland by barge, after being temporarily stacked in the terminal for a period of time. The situation will exacerbate as the new 18,000 or 21,000 TEU megaships start to

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become prevalent. This trend imposes new operational challenges to terminal operators. One of the important operational problems is how to temporarily stack these barge containers in a terminal, while

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changing mode from a ship to a barge. In this paper, we study the stacking of outbound containers to be transported by barge to the hinterland or other terminals within the port area. We aim to minimize the total retrieval time of a set of containers from a stack (also known as block) in the container terminal. The total retrieval time can be used as a good proxy for other objectives such as waiting time, throughput, and stack utilization. Figure 2(a) shows a container block consisting of multiple container rows, tiers and bays. The containers are densely stacked in piles (A pile is a column in the container block). A pile can be empty or can consist of a group of containers stacked upon each other. A number of input/output (I/O) points 4

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are located at the seaside and landside of the block. An automated stacking cranes (ASC) stacks and retrieves containers from and to the I/O points. In recent years, in order to increase the container throughput at the seaside, (European) export and import terminals use twin ASCs to decouple landside and seaside operations. In such a configuration, the ASCs are non-passing, i.e. there are two identical ASCs which are unable to pass each other (see Figure 2(b)). In order to avoid ASC interferences, terminals have implemented other configurations with double or even triple ASCs, as shown in Figure passing the other ASC(s). Grand ASC

Seaside ASC ASC

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2(c) and (d), respectively. In double and triple ASCs, unlike twin ASCs, the grand ASC is capable of

Grand ASC

Seaside ASC

Landside ASC

Landside ASC Small (lanside and seaside) ASC

Tiers

e rsid Sea

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s Bay

Rows I/O point

(a) Single ASC

(c) Double ASCs

(b) Twin ASCs

(d) Triple ASCs

n cki Sta

ds Lan

rea ga

ide

Figure 2. A block of containers with different ASC configurations

In a container block, stacking containers on top of each other leads to reshuffling. A reshuffle is the removal of a container stacked on top of a desired container. Reshuffling is a time-consuming process

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and increases turnaround time of barges. Thus, reducing the number of reshuffles is a top priority for container terminals (Kim 1997; Zhao and Goodchild 2010). On the other hand, containers are usually

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categorized into different classes based on different criteria such as their weights and destinations. This imposes several restrictions on the stacking of containers in a barge. First, in order to ensure the barge‟s stability, heavier containers must be placed in lower tiers in a barge. Second, the containers of

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destinations that will be visited by the barge later should be loaded onto the barge earlier. Third, if containers of multiple barges are mixed in a pile, a container leaving earlier should be stacked at a

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higher tier in order to avoid reshuffling. This complicates the stacking process in a container block. Thus, terminal operators use a dedicated stacking policy to avoid reshuffling. In a dedicated stacking policy, every pile in the block accommodates only one type of containers with the same barge, weight

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class, and destination. In such a policy, always the container at the top tier of each pile needs to be accessed requiring no reshuffles in a block. However, this policy results in a low stack utilization as each pile only accommodates one type of containers. In addition, this policy does not use the planned arrival time information of the barges. To address these issues, this study proposes a mathematical model for a shared stacking policy that minimizes total retrieval time. This model is an extension of the one proposed by Zaerpour et al. (2015) to store pallets in a compact cross-dock center. The shared policy allows different container types to share the same pile. The successful implementation of the shared stacking policy depends on two 5

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factors: 1) how far in advance the arrival information of each barge is known; 2) how reliable the arrival information of each barge is. The historical data shows that although the planned arrival time window of a barge is known, the barge might still violate its planned time window. Even worse, terminal operators sometimes do not know which barge will pick up a container. This is specially the case for spot cargo for which a barge will be determined later in time. Furthermore, even if the barge operator is known, yet it might not have been determined when the barge will arrive to pick up a container. In

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such a case, that container cannot share a pile with the containers of other barges. This again might reduce the stack utilization. To study the tradeoff between the availability and reliability of information and stack utilization, we perform comprehensive numerical experiments for different levels of information availability and reliability.

The rest of this paper is organized as follows. In Section 2, we review the literature on container stacking operations and barge transportation. In Section 3, we present a mathematical model for a

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shared stacking policy. Section 4 develops a solution method. In Section 5, we perform the numerical experiments (including sensitivity analysis) and give the managerial insights. Finally, in Section 6, we conclude the paper and present the potential future research topics. 2. Literature review

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For regions with easy access to waterways (e.g., the Netherlands, Germany, Southeast Asia, or Eastern United States), barge transport is an economical alternative to rail and road. However, in comparison with the extensive literature on the container handling operations at the seaside, landside and stacking

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area of a terminal (see e.g., the recent literature reviews by Gorman et al., 2014; Gharehgozli et al., 2016; Vis and De Koster, 2003; Carlo et al., 2013, 2014a, 2014b), the literature on loading and unloading barges

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or handling associated containers is narrow. Although barge and ship freight transportation are similar, specific practical constraints in barge transportation require new analytical methods (see e.g., Christiansen et al., 2004 and Christiansen et al., 2007). In the following, we first review the literature on

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barge operations and then focus on stacking operations. 2.1. Barge operations

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The academic literature on barge operations heavily revolves around the barge handling problem (BHP) which consists of routing and scheduling barges to make a tour to visit a set of terminals in a port. One of the key requirements for successful barge transportation is the effective coordination of terminal and barge operators. Currently, exchange of information is lacking and agreements and schedules are often not kept. A usual business practice is that the barge operator comes up with a rotation plan and later converse that to terminals. Nevertheless, in some instances, it might not be possible to perform the barge rotation as planned due to changes, disturbances, and miscommunication. 6

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The BHP has become a crucial problem in the last decade due to the emergence of megaships, deep-sea container terminal with high container turnovers and the immediate need for hinterland container transportation. Port of Rotterdam, being the largest ports in Europe, has been the pioneer in studying the BHP. One of the first studies, performed in 1998, resulted in agreements about the handling of barges at Europe Container Terminals (ECT) (RIL Foundation, 1998). Later, a next step was taken in a European project „Barge Planning Support‟ to investigate how publishing the quay schedules of

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terminals on the internet can help barge and terminal operators (RIL Foundation, 2000). As a result, an application (Barge Planning) was developed, to evaluate whether these operators hold their agreements. In the system, all information including the actual arrival times of barges, delays, and the causes of delays are registered. Despite being insightful, this project did not provide a general solution to the barge handling problem for barge operators visiting multiple terminal in the port area (Melis et al., 2003).

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Barge Planning Support was one of the first projects to consider a decentralized control structure for barge planning (Connekt, 2003; Melis et al., 2003; Schut et al., 2004). The aim was to create a planning platform, where one day in advance, barge operators planned and published their schedules. The plans could not be updated during execution. Obviously, in the dynamic business environment of the 21st century, an online centralized plan to coordinate the activities of barges and terminals might be more

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efficient. However, both terminals and barge operators prefer to be independent and are not willing to share information. Thus, online decentralized methods are more popular for barge planning. As a result,

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Douma et al. (2009, 2011a, 2011b) model the barge planning problem using agent-based planning and compare the results with the ones from a centralized method. The results show that sharing information on expected quay wait times can reduce the average tardiness per barge by 80% compared

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to a similar case with no information sharing. Furthermore, in spite of the partial information availability, their approach performs well compared to the centralized approach. The barge terminal 2012).

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multi-agent network (BATMAN) project aims to implement their results in the Port of Rotterdam (Mes,

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Portbase, with an investment of $100 million, is another attempt in the Netherlands for a comprehensive centralized approach for information gathering and sharing within the maritime industry (Zuidwijk, 2015). Portbase is a platform where all parties in the logistics chains of multiple Dutch ports can log in and access more than 40 services such as adding relevant information or tracking goods. This enhances communications, allows everyone to work efficiently, and reduces administrative costs. For example, the combined number of yearly phone calls has reduced by 30 million and the number of yearly emails by 100 million. Further, Port of Rotterdam has been able to reduce the total distance travelled by trucks by 30 million kilometers per year.

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Despite the intriguing findings and clear contributions, the present studies do not yet capture all aspects of the role of information in barge handling operations in practice, leaving room for further study. In this study we cover this gap by integrating barge arrival information with container stacking operations. A closely related field that can also benefit from the barge arrival information is the lockmaster‟s problem which concerns the optimal strategy for operating a lock (Petersen, 1988; Nauss, 2008; Smith et al., 2009, 2011; Verstichel et al., 2015, 2014a, 2014b; Passchyn et al., 2016a, 2016b). The

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same situation occurs when ships need to pass a narrow canal, and only a restricted number of wider areas is available where ships can pass each other (Griffiths, 1995). Based on our results, advance information, even with a certain level of inaccuracy, can provide benefits for finding better plans. Such a result is also supported by Srour et al. (2016) who study a pickup and delivery problem with the time at which requests will require service is only fully revealed during operations. Other studies in the same field studying the value of information include Ichoua et al. (2006), Thomas and White (2004),

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Thomas (2007), Cortés et al. (2009), Larsen et al. (2004), and Jaillet and Lu (2011). 2.2. Stacking operations

The information asymmetry inherent in barge transportation not only imposes many challenges to all the operations occurring outside the terminal borders but also inside the terminal borders. Within a terminal, depending on arrival schedule of barges, operators need to decide (1) in what sequence ASCs

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need to retrieve containers and (2) move them by AGVs to the quay so that (3) quay cranes can load them onto the barges. Unfortunately, in the literature not much research can be found on scheduling

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and routing container handling equipment at the seaside to load and unload barges. On the other hand, the focus on ships is ample in the literature. Gharehgozli et al. (2016) review the recent studies performed in this area. In most studies done on yard crane scheduling, the total time required to stack

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and retrieve containers (makespan) is the main objective function (Boysen and Stephan, 2016; Gharehgozli et al., 2014b; Kim and Kim, 1999; Narasimhan and Palekar, 2002; Chen and Langevin 2011).

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This also the case for problems where two or three yard cranes are assigned to a stack (Gharehgozli et al., 2015; Vis and Carlo, 2010). Bruns et al. (2016), Boysen and Stephan (2017), and Boysen et al. (2017)

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study the complexity of the crane scheduling problems. In configurations with multiple yard cranes, minimizing interferences is a major research topic (Speer and Fischer, 2017; Saini et al., 2017; Briskorn et al., 2016; Boysen, et al., 2015, 2017; Gharehgozli et al. 2015 and 2017c). However, compared to yard cranes, research on resolving interferences of quay cranes is much deeper (Bierwirth and Meisel, 2010 and 2015; Meisel and Bierwirth, 2013, Meisel, 2011; Kim and Park, 2004; Moccia et al., 2006; Choo et al., 2010, Lim et al, 2004, 2007; Lee et al., 2008; Chen et al., 2012, Liu et al., 2006, Agra and Oliveira, 2017; Zhang et al., 2017; Chen and Bierlaire, 2017; Türkogulları et al., 2016; Beens and Ursavas, 2016). New technologies such as the dual spreader cranes are also studied in academia (Lashkari et al., 2017). Next

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to analytical models, due to the sheer size of the facilities and complexity, simulation models have been also built to study the impact of terminal design and crane deployment on the overall performance of a terminal including the turnaround time of ships (Georigk et al., 2016; Dragović et al., 2016; Angeloudis and Bell, 2011; Kemme, 2012; Petering et al., 2009; Petering 2010, 2011; Petering and Murty, 2009; Gharehgozli et al, 2017c). In addition, in the literature, reshuffling has been considered as one of the main factors causing delay

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in retrieving containers and moving them to the quay. Papers dealing with container reshuffling separate it from all other operations and study it in three main categories (see also the recent literature reviews by Goerigk et al., 2016; Lehnfeld and Knust, 2014): (1) pre-marshalling, (2) relocating methods while retrieving containers, and (3) stacking methods. The objective function in these studies is to minimize the number of reshuffles by stacking containers in an appropriate order before the arrival of the ship. Gharehgozli et al. (2017a) name this prior-reshuffling and emphasize on the fact that ships delayed ships also need to be minimized.

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delay and posterior-reshuffles which are the extra reshufflings required to remove containers of the

In pre-marshalling, minimizing the number of reshuffles is achieved by pre-marshalling containers based on the retrieval plan (Cordeaua, et al., 2015; Lee and Hsu, 2007; Lee and Chao, 2009; Caserta and Voß, 2009b; Expósito-Izquierdo et al., 2012; Bortfeldt and Forster, 2012; Huang and Lin, 2012). We also

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consider the papers dealing with estimating the number of reshuffles given a stack configuration in this category (Kang et al., 2006; Kim, 1997; Lee and Kim, 2010; De Castillo and Daganzo, 1993). Branch

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and bound is one of the techniques used to study the pre-marshalling problem (Tanaka and Tierney, 2017; Van Brin and Vaner Zwaan, 2014; Expósito-Izquierdo et al., 2012; Tierney et al., 2016; Zhang et al, 2015). However, in order to deal with the complexity of the problem, heuristic algorithms have been

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developed for instance by Hottung and Tierney, 2016; Wang et al., 2015; Wang, et al., 2017; Bortfeldt and Forster, 2012; Expósito-Izquierdo et al., 2012; Jovanovic et al., 2017.

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The problem of minimizing the number of reshuffles of a container stack while containers are retrieved is called the block (container) relocation problem (BRP), which is proven to be

-hard by Caserta et

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al. (2012). Recent studies can include Ji et al. (2015), Jin et al. (2015), Ku and Arthanari (2016), Caserta et al. (2009, 2011), Caserta and Voß (2009a, 2009b, 2009c), Forster and Bortfeldt (2012), Lee and Lee (2010), Zehendner and Feillet (2014) and Zehendner et al. (2015). Similar to pre-marshalling problem, heuristic algorithms or branch and bound improved with heuristics are among the techniques used to solve the BRP (Tanaka and Mizuno, 2015; Tanaka and Takii, 2016; Zhang et al, 2010a; Zhu et al., 2012; Kim and Hong, 2006; Rei & Pedroso, 2012; Ünlüyurt & Aydin, 2012; Ting and Wu, 2016). The dynamic BRP, where arrival and retrieval times of containers are given, has been studied by Akyüz and Lee (2014) and Wan et al. (2009). Ku and Arthanari (2016) study the time-based BRP where each retrieval and stacking 9

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operation is completed within a given service time window. Zehendner et al. (2017) introduce the online BRP. They give worst-case analysis of the problem and derive its competitive ratio. Premarshalling and relocating can minimize the number of reshuffles while containers are retrieved. However, while stacking containers, a good stacking policy is required to minimize the handling effort in later stages. The following studies focus on developing methods to properly allocating containers (Dekker et al., 2007; Kim and Park, 2003; Kim et al., 2000; Zhang et al., 2010b, 2014a, 2014b; Yu and Qi,

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2013; Casey and Kozan, 2012; Bruns et al., 2016; Boysen and Emde, 2016; Jang et al., 2013). A quick look on the literature of container stacking, reviewed in this section, reveals that almost all papers focus on minimizing the number of reshuffles (see the papers on pre-marshalling, re-locating, and stacking methods). The time element has been mainly used to schedule yard cranes stacking and retrieving containers on top of a container block and has not been directly used to model stacking containers. This leaves a gap in the literature in terms of the fact that minimizing the number of

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reshuffles is the only one factor to minimize the total time required to retrieve containers from a container block. The other important factor is the travel time required by the yard crane to move between retrieval containers. We cover this gap in this paper.

We contribute to the literature by introducing the objective function of minimizing retrieval time while proposing a shared stacking policy and avoiding reshuffling. Among all, the previous work

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directly related to our research is by Gharehgozli et al. (2014a) and Zaerpour et al. (2015). Gharehgozli et al. (2014a) is among the first to introduce the notion of shared stacking policy for container stacking.

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They show that the shared stacking policy significantly outperforms the dedicated stacking policy. The same as the previous studies, they only consider minimizing the number of reshuffles as the objective function. Zaerpour et al. (2015) study the shared stacking policy in the field of warehousing. They

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model the temporary storage of pallets in a cross-dock center. We extend their model by adding constraints specific to container terminal stacking operations and propose a new

-completeness

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proof. Furthermore, we propose a new heuristic algorithm to solve the model which is inspired by real heuristics used in container terminals.

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Last but not least, an overview of the literature shows that due to the sheer complexity of a container terminal system, researchers have mostly analyzed and optimized subsystems in isolation. There exist few studies in the literature where the container terminals have been studied in an integrated manner. Cao et al. (2010) propose an integrated model for scheduling yard trucks and yard cranes in a container terminal. Chen et al. (2013) formulate a similar problem as a constraint programming model and solve it by developing a three-stage solution method. First, yard cranes are scheduled. Second, yard trucks are routed, and finally, the complete solution is obtained. Jiang and Jin (2017) study how the container allocation problem impacts yard crane deployment problem in a transshipment container terminal. 10

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They argue that the number of containers allocated to each block affects the traffic congestion and determines the number of yard cranes required in each block. Zeng and Yang (2009) consider the holistic problem of determining the loading or unloading sequence, scheduling, and dispatching QCs, yard cranes, and yard trucks simultaneously. Zuidwijk and Veenstra (2015) argue that the timely arrival of containers at their final destinations is challenged by uncertainty regarding the transit time from deep-sea to hinterland terminals. This results in considering more slack time to plan deliveries to the

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final destinations. By provisioning information, uncertainty can be reduced and consequently performance can be improved (Lee et al., 2000). This paper is among a handful of studies which attempts to integrate container terminal operations by proposing a model to stack containers based on a shared stacking policy and considering information inaccuracy. 3. Problem description

containers in a single container stack (also known as block) by minimizing the

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We study how to stack

total retrieval time of the ASC while reshuffling is avoided. The stack consists of tiers; the total number of piles is landside, rows by require that

. Bays are indexed by

from left to right, and tiers by

, i.e., the stack cannot accommodate more than

Bays,

Rows, and

from seaside to from bottom to top. We containers. The ASC can

move along both bays and rows simultaneously in Chebyshev distance. There is also time needed for

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the hoist to roll up or down to pick up or drop off a container. As a result, the total time required to retrieve container and

*

are the horizontal travel times along the bays and rows, respectively and

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, where

from bay , row and tier can be calculated as

+ is the

vertical travel time which accounts for the necessary hoisting actions. barges. All these containers are pre-marshalled and stacked in the

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Containers belong to a set of

sequence that will be picked up by barges the next day. The sequence is determined based on the

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planned arrival times of the associated barges. In other words, each container has a time window which specifies when it will be picked up. The time window is constructed based on the associated barges. In practice, due to limited berthing space, only a few barges can be loaded and unloaded at the same time.

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So if a barge delays, the next barge available at the terminal will be assigned to the berth to avoid the berth and equipment to be idle which is very costly. Thus, the terminal may not have an available slot for a delayed barge in the next few hours. This means that a small delay may result in a huge waiting time for the barge operator. As a result, barges try to keep to their schedules as much as possible (In Section 5.2, we discuss the situations where the barge operators cannot meet the planned arrival time windows and the impacts on the performance of shared stacking policy). In addition, in reality there exist instances where the arrival information of the assigned barge to a departing container is not known sufficiently in advance. In such a situation, the containers of this barge cannot share a pile with 11

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other containers in order to avoid reshuffling (The effect of information availability on the performance of shared stacking policy has been discussed in Section 5.3). The time windows can be used to avoid reshuffling while retrieving containers. For now, we assume that the barges will certainly arrive within their planned arrival time windows (this assumption will be relaxed in Section 5). Let ,

- and

,

- be the arrival time windows of barge

,

. If the time windows do not overlap (i.e.

or

and

) then one barge

on top of containers of the later barge. For example, if stacked on top of barge

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definitely arrives earlier than the other barge. Therefore, containers of the earlier barge can be stacked , then all containers of barge

can be

without creating any reshuffles. It should be noted that other constraints

such as weight constraint is also considered in constructing the time windows and stacking of the containers. On the hand, if the time windows overlap (i.e.,

), there is a chance of arrival of any of two barges earlier than the other

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, or

,

one. Thus, stacking the containers of the two barges on top of each other might result in reshuffling. In our study, we avoid such a stacking. If the arrival time information of the barge for a container is not known, we virtually consider a very wide time window for the container. Thus, this container cannot share a pile with the containers of other barges. Based on the time windows, for container we construct set

which includes all containers that cannot be stacked underneath container since

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it results in reshuffling.

We use Figure 3 to explain the parameters of the model. It shows a stack of containers with tiers. The total number of piles is

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rows, and

example, we focus on bay 1 where

bays,

. For the sake of this

containers of

barges are stacked. Location of each

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container can be found using the bay, row, and tier numbers. For example, container 1 is stacked in bay 1, row 6, and tier 3, i.e., (

). Containers

(

stacked on barge 3. The time window of barge 1 is ,

have to be stacked on barge 1,

have to be stacked on barge 2, and finally containers

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container

)

-, and bage 3 is

,

-

,

,

-

have to be

-, barge 2 is

,

,

-

-. Container of barge 1 can be stacked on top of

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containers of barge 2, since their time windows do not overlap. Note that barge 1 arrives earlier (

), so its containers have to be on top. On the other hand, time windows of barge 1 and

barge 3 overlap (i.e., other.

In

other *

to construct the

). Therefore, their containers cannot be stacked on top of each

words,

*

+,

*

+

*

+,

and

+. For simplicity, in this example, only the time windows of barges are considered ‟s. other constraints such as weight constraint can be also considered.

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Figure 3. A block of containers with

Containers of

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barges stacked in bay 1

The problem of stacking container in the stack by minimizing the total retrieval time of the ASC and model. Let

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avoiding reshuffling can now be formulated mathematically as the following integer programming be the binary decision variable which equals 1 if and only if container

stacked at bay

row

, and tier

equals 0.

(i.e., location (

is )), otherwise, it

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The objective function below minimizes the total retrieval time of the ASC.

1

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∑∑∑∑

Constraints 2 and 3 ensure that each location is occupied by at most one container and each container

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is located at exactly one location.

2

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∑

3

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∑∑∑

Constraints 4 ensure that containers are stacked on top of each other without adding any reshufflings to the system. The containers which might need earlier retrieval than the container c are not allowed to be stacked underneath container c. ∑

4

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Constraints 5 guarantee that if a container is not stacked on the ground, it is then stacked on top of another container. In other words, no gaps are allowed within each pile and no container can be stacked in a position above an empty slot. Such constraints are not required in the model of Zaerpour et al. (2015) as the pallets are stored in warehouse racks. These constraints increase the complexity of our model. ∑

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5

In Section 4.3., we develop a Lagrangean Relaxation approach to find a lower bound for the problem and evaluate the performance of our heuristic algorithm. In this approach, we relax Constraints 4 and 5 by introducing non-negative Lagrange multipliers. Constraints 4 and 5 are constructed in the same way

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so that consistent multipliers can be used in the Lagrangean Relaxation approach. Finally, Constraints 6 are the integrality constraints. *

+

6

Consequently, the proposed mathematical model is given by the formulation P. ∑∑∑∑

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|

( )

and

containers, the model generates

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Given a stack of

( ) binary variables. For example,

to formulate stacking of 160 containers in a stack with 4 bays, 10 rows, and 4 tiers,

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binary variables are generated.

Theorem 1. The problem of stacking outbound containers formulated as P is strongly

-complete.

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The proof can be found in Appendix A. Due to the complexity of the problem, solving practical instances in a reasonable time is not possible.

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Therefore, in the following section, we propose a heuristic algorithm to solve the problem. 4. A Shared Stacking heuristic In this section, we propose a shared stacking heuristic to solve our problem. Our proposed shared stacking heuristic consists of two parts: 1) initial solution construction part done by vertical stacking heuristic and 2) improvement part done by an adapted simulated annealing (SA) heuristic. In Section 4.1, we discuss the adapted SA heuristic algorithm adjusted to our problem. In Section 4.2, we present the vertical stacking heuristic which provides the initial solution for the SA heuristic. Finally in Section 14

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4.3, we develop a Lagrangean Relaxation approach which is used to construct a lower bound for the results of our proposed heuristic. 4.1. Simulated Annealing heuristic In practice, stacking decisions have to be made quickly. However, the complexity of the problem means that only relatively small instances can be solved to optimality within a reasonable time. Therefore, looking at the container stacking literature shows that many focus on a small problem with a single bay

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(see, for example, Lee and Hsu, 2007; Lee and Chao, 2009; Kim et al, 2001). Furthermore, heuristic algorithms have been used to deal with the complexity of the problem (see the literature review section). To solve real-size problems with a given set of containers and block dimensions, we develop an SA heuristic. SA is a powerful heuristic framework to efficiently obtain high quality solutions for combinatorial problems, as shown in the numerical experiments of this paper and also in the marmite

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literature and other fields of study. SA, initially proposed by Kirkpatrick et al. (1983), is an improvement algorithm inspired by the annealing process in metallurgy. In the context of container terminal operations, it has been shown that SA can provide near optimal solution for different terminal problems such as yard crane scheduling (Hu et al., 2016; Vis and Carlo, 2010; Lee et al., 2007; Cao et al., 2008; Jung and Kim, 2006), berth scheduling (Kim and moon, 2003; Moorthy and Teo, 2006; Emde and Boysen, 2016; Legato et al., 2010 and 2014; Zhen et al, 2011), and gate management (Chen et al, 2016).

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Braekers et al. (2013) and (2014) show that the deterministic annealing heuristic, a variant of SA, is efficient in finding high quality solution for a full truckload vehicle routing problem in drayage

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operations around intermodal container terminals. Furthermore, SA has been applied by several authors in a variety of contexts including facility layout by Meller and Bozer (1996) and Tompkins et al. (2003), and crossdocking by Bozer and Carlo (2008). Finally, other heuristics, such as the adaptive large

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neighborhood search (ALNS) heuristic, also use an SA scheme to obtain a near optimal solution. The ALNS, initially developed by Ropke and Pisinger (2006a) and (2006b), has been successfully applied to

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different operational problems in container terminals (see, for example, Gharehgozli et al, 2015, Cordeau et al, 2011, Hansen et al, 2008)

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A simple form of local search (a descent algorithm) starts with an initial solution. A neighbor of this solution is then generated by a suitable mechanism and the change in the objective function is calculated. If a reduction in the objective function is found, the current solution is replaced by the generated neighbor, otherwise the current solution is retained. The process is repeated until no further improvement can be found in the neighborhood of the current solution. Although such an algorithm is simple and quick to execute, the disadvantage of the method is that the local minimum found may be far from any global minimum.

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The basic outline of the SA heuristic is presented in Algorithm 1 (Eglese, 1990). SA is a type of local search algorithm. In SA, the algorithm attempts to avoid becoming trapped in a local optimum by occasionally accepting a neighborhood move which increases the value of the objective function. This is called Metropolis Criterion (Metropolis et al., 1953), in which solutions that do not lower the total retrieval time are accepted. This results in exploring more of the potential solution space. The acceptance or rejection of an uphill move is determined by a sequence of random numbers, but with a

Algorithm 1. SA heuristic scheme Require: A feasible solution ; Ensure: A near optimal feasible solution

;

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controlled probability.

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1: procedure SA( ) 2: ; 3: while termination criterion is not met do 4: generate solution , a neighbor of ; ( ) then 5: if ( ) 6: , ; 7: else if satisfies the acceptance criterion then 8: ; 9: end if 10: end while 11: return ; 12: end procedure

In the followings, we explain in details how we apply SA to our problem:

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Step 1. Set ; i.e., the initial feasible solution. Although SA is designed to converge irrespective of its initial solution, we opted to use the feasible solution of the vertical stacking heuristic (explained in

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section 4.2) as an initialization for SA to improve its convergence speed. Also, we set the temperature and iteration counter

.

Step 2a. Select a random container. Swap the location of this container with the location of another

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randomly selected container. If swapping results in reshuffling, the solution becomes infeasible. Therefore, discard the swap and continue with selecting another container. Set the new solution as Step 2b. Compute the decrease in the retrieval time; i.e., set

( )

( ). If

go to step 3;

otherwise go to step 2c. Step 2c. Sample a random number

,

-. If

(

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.

), go to step 3; otherwise go to step 4.

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Step 3. Accept solution

; i.e., set

“current best” solution; i.e., set

( )

and and (

Step 4. Update the iteration counter,

)

( ). If

( )

(

), then update the

( ).

, and the temperature,

a cooling rate. To specify a cooling rate parameter, we calculate

, where

is

and

by using the initial

solution obtained by the algorithm. We assume that the start temperature is

of the initial solution

while the ending temperature is

of the initial solution. Then, the cooling factor becomes a

iterations. To further diversify the search, temperature

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parameter which guarantees the convergence from the starting to ending temperature after is reheated back to start temperature (

)

every iterations. This means that depending on the maximum number of iterations ( ), the algorithm is reheated ⁄ times. best

is the container allocation with no reshuffling.

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As a result of this algorithm,

4.2. Vertical stacking heuristic

Vertical stacking heuristic is one of the algorithms used in practice to stack containers arriving at the terminal in order to minimize the number of reshuffles while retrieving them. Caserta et al. (2012) have used this heuristic to generate solutions for the blocks relocation problem (BRP), which can be defined as follows: given a set of stacked containers, which relocations are necessary to retrieve the containers

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from the stack in a predefined order while minimizing the number of relocations?

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Simply stated, the vertical stacking policy assigns each container to a pile if reshuffling is avoided. If such a pile does not exist in the stack, the policy assigns the container to an empty pile. Finally, if any empty pile cannot be found, a non-full pile is randomly selected for stacking the container. As the main

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objective is to minimize the number of reshuffles, the containers can be assigned to any pile without taking the ASC retrieval time into consideration. We now discuss a modified version of the vertical

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stacking heuristic which accounts for the retrieval time of the ASC and the information of departure time of containers. Zaerpour et al. (2015) also develop a heuristic algorithm which is based on the same “construction and improvement” idea. Our proposed heuristic differs from the one proposed by

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Zaerpour et al. (2015) from various perspectives. From the system layout perspective, the heuristic by Zaerpour et al. (2015) deals with a storage system with only one I/O point on one side of the aisle (onesided system). Our proposed heuristic deals with a block of container with I/O points on two sides of the system (landside and seaside). In addition, in order to avoid reshuffling while assigning the loads to the locations, the heuristic by Zaerpour et al. (2015) only considers whether two loads belong to a truck or not. Our proposed heuristic additionally considers other criteria such as the gap between containers, weight class and destination of each container while stacking it in a pile. Moreover, the heuristic by Zaerpour et al. (2015) uses several preparatory steps (e.g. ideal boundary construction, graph 17

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representation) before constructing a feasible solution. This increases the computational time of the heuristic especially for large instance problems. Our heuristic starts with feasible solution construction reducing the computational time for large instance problems. Step 1. Sorting containers. Sort the containers based on the arrival times of their associated barges in the ascending sequence of arrival, i.e., picking up the

, -

, -,

, -

where

, -

is the arrival time of the barge

container in the sequence.

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Step 2. Sorting piles. Sort all empty piles based on the shortest time required for the ASC to retrieve a container from a location in that pile to the seaside I/O point in an ascending sequence, i.e., , -

,

- , where , -

is the ASC retrieval time for

, -

pile in the sequence.

Note that using the retrieval time equation, piles will be sorted from the seaside to the landside. The reason is that the ASC retrieves containers to the seaside where barges are waiting to be loaded. The contains the term

*

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I/O point is located in the center of the seaside-end of the stack. Since the retrieval time equation + for a location (

), for bays near the seaside with

rows result in shorter retrieval time whereas for bays further away from the seaside with

, central , all

rows have the same priority in terms of retrieval time . For instance, Figure 4(a) shows the heat map for the priority of piles, for a stack with 40 bays, 10 rows and 4 tiers. Piles with the shortest to longest

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retrieval time are indicated by different colors from dark blue to dark red. Figure 4(b) clearly shows that for a stack with 10 rows and 40 bays, the travel time along the rows impacts the total travel time

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before the 10th row; after this row the travel time along the bays is the dominant travel time.

(a) Travel time of the ASC for different piles of a container stack

(b) Impact of the travel time along the bays and rows on the total travel time

Figure 4. A thermal plot indicating stack piles with different ASC retrieval times by means of different colors Note. Color scale is , where dark blue shows the shortest retrieval time and dark red shows the longest retrieval time. To generate the thermal plot, it has been assumed that containers have the same size in all dimensions and the ASC has the same speed along the bays, rows. Location of the seaside I/O point is in the middle of the stack in front of row 1.

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Step 3. Locating containers. Containers with the latest arrival times are assigned to piles with the shortest retrieval times. Note that the locations with the shortest retrieval times in a pile are on higher tiers. If containers are assigned to these locations, Constraints 5 will be violated and we obtain a solution in which there is a gap (empty location) between two containers stacked in a pile or containers are not stacked on the ground. Therefore, containers are stacked in the lowest tiers first. Step 4. Avoiding reshuffling. Two containers can be assigned to the same pile, if no reshuffles will be

, -

, -, can be assigned to the same pile if

, where barges

, -, respectively. On the hand, if

,

then these containers cannot be assigned to the same pile.

and

with the sequence

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necessary while retrieving the containers. In other words, two containers

and

move containers , - and , or

,

Step 5. Finalizing the solution. While locating a container, if reshuffling cannot be avoided by stacking a

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container in a pile with the shortest travel time location, the container will be located in another pile with the next shortest travel time. A solution with no reshuffling can be always found since the information of all containers are known in advance. As explained in section 3, terminal operators do a housekeeping using the night during which containers are stacked in the sequence that will be picked up by barges the next day assuming the complete information is available. In the worst case scenario, our heuristic can result in a dedicated stacking policy which is a feasible solution for our model. In

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Section 5, we will perform numerical experiments where partial information is available.

containers belonging to

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Figure 5 shows an illustrative example of how the algorithm works. Consider an instance with barges each of which with a capacity of 4, 4, 3, and 3 containers. The

time windows of these barges are

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have to be stacked in one bay,

,

-,

, with

,

-

rows, and

,

-, and tiers (in total

,

-. All containers piles).

Following the steps of the algorithm, containers are first sorted. The sequence is shown in the

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parenthesis in the figure. The other number is the barge number on which the container has to be loaded. Second, piles are sorted from 1 to 4. Based on Figure 4, piles 1 and 2 can be the middle piles of a bay and piles 3 and 4 are the outer piles. Finally, the containers are allocated to the piles. Using this

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heuristic, after allocating containers of barges 1 and 3 to piles 1 and 2, containers of barge 2 and 4 have to be allocated (they both have the same time windows). The heuristic randomly starts with the containers of barge 2. The first container is assigned to the top tier of pile 2 and the remaining 3 containers to pile 3. Finally, containers of barge 4 have to be allocated. However, since the time windows of barge 4 overlaps with barge 2, none of its containers can be assigned to the top tier of pile 3. Therefore, all containers are assigned to pile 4.

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tier

Pile

1

4 2 (4) 2 (5) 2 (6)

4 (1) 4 (2) 4 (3)

3

4

2

3 2 Empty spot 1

Stacked container

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1 2 (11) (7) 1 3 (12) (8) 1 3 (13) (9) 1 3 (14) (10)

Figure 5. An example of the solution of the adjusted vertical stacking heuristic with barges stacked in

piles

Containers of

The adjusted vertical stacking heuristic discussed here obtains a feasible solution that will be improved

4.3. Lagrangean Relaxation Approach

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by the SA algorithm discussed in Section 4.1.

To be able to evaluate the performance of our proposed heuristic, we use a Lagrangean relaxation approach to obtain a lower bound for our original optimization model. This approach has been used in different context such as storing pallets in a cross-dock center (Zaerpour et al., 2015), production and distribution planning (Fumero and Vercellis, 1999), and inventory routing problems (Yu et al., 2008).

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The constraints that complicate the resolution of our model P are constraints (4) and (5). They can be relaxed by introducing non-negative Lagrange multipliers. By introducing Lagrange multipliers

|

∑∑∑∑

|

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ED

, we define the Lagrangean relaxation model as:

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and

∑∑∑∑ ∑

∑∑∑∑ ∑

∑

)

∑

( (2), (3), and (6)

)

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|

(

The Lagrangean relaxation model provides a lower bound for the original optimization model. Our Lagrangean relaxation model has an assignment problem structure that can be easily solved, but Lagrange multipliers make the problem complex. It is clear that the best choice for ‟s and ‟s would be an optimal solution to the dual problem

(

). The Lagrangean dual problem is usually

solved by using the subgradient method (see for example, Bertsekas, 1999, and 2015; Zhao et al., 1999). Although the subgradient method is not an ascending algorithm, the subgradient direction is in acute angle with the direction toward the optimal multipliers, and the distance between the current 20

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multipliers and the optimal ones is decreased step by step. We explain the steps of the subgradient approach as follows. Step 0. Initialize (

,

,

, and

, and

at

where is the iteration step and

is a parameter for step sizing

).

Step 1. Solve the Lagrangean Model to obtain

that is the current dual value with a given

and

.

The model is an assignment problem that can be solved in polynomial time (Dell‟Amico et al., 2001;

Step 2. Set the step size

(

in iteration by

)

, where

of the original optimization model is estimated by ( obtained prior to iteration ; parameters -; the value of

in iteration

)

⁄

, -

;

, -

∑∑∑∑ ∑ (

∑

, -

- and

, otherwise

) ∑∑∑∑ ∑ (

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{

,

) if

(

;

is the best dual value

are random numbers where

is given by

; and

∑

is a parameter with

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,

and

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Martello and Toth, 1987; Kuhn, 1955).

) }

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The values of these parameters are set in line with the literature (see, for example, Yu et al., 2008, 2012, Abdul Rahim 2014, and Zaerpour et al., 2015)

/

} and

{

(

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∑

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Step 3. Update the Lagrange multipliers in iteration

,

{

∑

)

Step 4. Check the stop criterion, which is given by either (i) The dual value

.

}. has not improved for a

given number of iterations, or (ii) A given maximum number of iterations has been reached. If the criterion is met, stop and output the results, otherwise set

and go to Step 1.

5. Numerical experiments In this section, we perform several numerical experiments. In section 5.1., we evaluate the performance of the SA heuristic for different scenarios. For each scenario, the results of the heuristic are compared 21

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with those obtained by Lagrangean relaxation. In sections 5.2 and 5.3, a detailed comparison of the shared and dedicated stacking policies is performed. Furthermore, we investigate the value of information accuracy and advance information availability on the performance of these policies. Table 1 summarizes the set of parameters that are used in our numerical experiments. In our numerical analysis, we start with a base example which is inspired by a real container stack at the seaside. Then, to evaluate the performance of our heuristic for different sources of variation, a sensitivity analysis is parameters are fixed.

Parameter Number of barges ( ) Barge size (# containers) Size of the stack ( ) Time window length (hours) Speed of the ASC (m/min)1

Base example 10 5 40 4

Range of scenarios [1, 20] [1, 11] [20,40] [2,4]

1

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Table 1. Parameters used in the numerical experiments

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performed. In Table 1, the range of scenarios shows how each parameter is varied while the other

Fixed parameters

[0.5, 8]

Gantry ( ) = 240, Trolley ( ) = 60, Hosting ( ) = 72 Length=5.89, Width=2.33, Height=2.38

Container size (m) 2, 3 1 No acceleration or deceleration is considered. 2 All containers are 20 feet long. 3 No separating space between containers is considered.

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The study is performed on a Notebook with 2.30 GHz Intel® Core™ i7 processor, with 16 GB of RAM and the programming language is MATLAB® 2015a. For each scenario (including base example), 100

(

.

)

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realizations are generated through a Monte Carlo simulation. The number of realizations satisfies / with a 90% confidence level (

and

is the

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variance and mean of the objective values,

) where

and

are respectively the

percentile of the normal distribution,

is the relative error (Law and Kelton, 2007).

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5.1. Performance evaluation of the solution method In this section, we evaluate the performance of the SA heuristic. In order to run the SA heuristic, the

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parameters are first tuned using a set of 10 randomly generated tuning instances. The tuning instances are different from the instances used later in this section. As Ropke and Pisinger (2006a) and Gharehgozli et al. (2015) suggest, we apply an ad hoc trial and error strategy to set the parameters. In other words, we fix all parameters and change only one parameter at a time. The SA heuristic was applied 10 times and the parameter value resulting in the best objective was selected. The parameters used for all scenarios are as follows: (

)

(

). For example, Figure

6a shows the progress of the algorithm over 10000 iterations starting from a random initial solution for the given parameters and a randomly generated tuning instance of the base example with 22

,

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and

, each with 5 containers. Among all, the parameter

has the largest impact in the SA

search. As shown in Figure 6b, by setting a higher value for , the search will be more diversified. Thus,

1300

1250

1250

1200

1200

1150 1100 1050 1000

1150 1100 1050 1000

950

950

900

900

850

850 2000

4000

6000

8000

Number of iterations

10000

0

2000

4000

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0

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1300

Best objective function (sec)

Current objective function (sec)

more iterations may be required which can slow down the search.

6000

8000

10000

Number of iterations

)

(

)

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(a) the performance of the SA heuristic when (

(b) the performance of the SA heuristic when (

)

(

)

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Figure 6. The performance of the SA heuristic for different parameters

Table 2 compares the results of the SA heuristic and Lagrangean relaxation approach for the base ) shows the results of SA heuristic (shared stacking

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example. A one-tailed paired t-test (Ha:

policy) and dedicated stacking policy significantly differ at a 5% level for all instances tested. The results show that SA can quickly find solution which have a gap of 13.68% compared to results obtained by the Lagrangean relaxation. This gap can be explained due to the poor performance of the Lagrangean relaxation, when the stack utilization is low (

). Later, in Table 3,

we can see when the utilization increases, the gap becomes only 3.5%. Since the Lagrangean relaxation approach provides a close lower bound for the problem, therefore we can claim that SA can obtain a near-optimal solution. 23

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Table 2. Comparison of solutions for the base example Barge TW p-value size (sec) (sec) (%) (sec) (sec) (%) (sec) 10 5 1 11.38 <0.05 974.65 863.72 0.54 745.58 -15.85 6.56 (1) Let be the total average retrieval time for method x (in seconds), where x = D (dedicated storage), H (SA heuristic), L (Lagrangean relaxation). (2) Let be the gap between the SA heuristic and method x (%), where x = D (dedicated storage), L (Lagrangean relaxation). Gaps are calculated as

and

.

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(3) Let be the average computational time per instance for method x (in seconds), where x = H (SA heuristic), L (Lagrangean relaxation). Table 3. Sensitivity analysis for a varying number of barges, barge sizes, and stack sizes

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Barge p-value size (sec) (sec) (%) (sec) (sec) (%) (sec) 1 5 -43.88 0.16 71.97 71.97 0.00 0.88 -64.03 5 5 311.48 1.82 436.92 382.44 12.47 <0.05 0.45 -22.78 15 5 1249.17 14.62 1605.54 1386.80 13.62 <0.05 0.80 -11.02 20 5 1812.60 25.58 2294.40 1975.86 13.88 <0.05 1.27 -9.01 10 1 224.43 149.19 33.52 <0.05 0.57 102.60 -45.41 0.68 10 3 554.28 485.70 12.37 <0.05 0.44 392.86 -23.63 2.61 <0.05 10 7 1350.13 1270.53 5.90 0.73 1142.00 -11.25 12.71 <0.05 10 9 1882.79 1729.33 8.15 1.03 1581.93 -9.32 21.02 <0.05 10 11 2294.06 2205.98 3.84 1.44 2061.50 -7.01 31.30 10 5 1026.24 871.73 15.06 <0.05 1.28 794.27 -9.75 3.81 10 5 1018.79 871.79 14.43 <0.05 0.85 779.89 -11.78 4.18 10 5 974.65 865.26 11.22 <0.05 0.71 748.55 -15.59 4.98 10 5 974.65 864.02 11.35 <0.05 0.62 748.56 -15.42 5.82 10 5 <0.05 835.96 782.60 6.38 0.79 756.37 -3.47 0.94 10 5 <0.05 861.54 799.18 7.24 0.58 745.55 -7.19 3.10 (1) Bold numbers represent the parameter being varied. (2) In the first set of scenario, a maximum of 20 barges each of which with 5 containers are considered. The reason is that in the dedicated stacking, each barge needs 2 piles. In other words, 40 piles are required to stack all containers which is the dimension of the base scenario.

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Table 3 shows the results for a sensitivity analysis for a varying number of barges, barge sizes, and stack sizes. Based on the results, we make the following observations.

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Observation 1. The shared stacking heuristic (SA heuristic) results in high quality solutions. Almost in all instances, the gap between the solution of the heuristic and the solution of the Lagrangean relaxation approach is very small. Note that the results show that the gap increases when the utilization of the stack decreases. The reason is that in these instances, the performance of the Lagrangean relaxation approach drops. Observation 2. The shared stacking heuristic outperforms the dedicated policy. More specifically, by increasing the number of barges, the gap between the dedicated policy and shared policy increases. These barges move containers of different weight groups and destinations. Therefore, the 24

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outperformance becomes even larger as the shared stacking policy can increase stack utilization compared to the dedicated stacking policy. Such scenarios can be simulated by considering more barges. Thus we have considered additional scenarios with 15 barges (i.e., 5 barges with containers of 3 weight groups) and 20 bares (i.e., 5 barges with containers 3 weight groups and some with 2 destinations). The results show that the performance of the shared stacking improves further in more practical instances (up to 14%, see Table 3). The share stacking heuristic also outperforms the dedicated

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policy when containers of smaller barges share piles. In such a scenario, containers of different categories can share a few piles near the I/O points, whereas the dedicated policy spread them over multiple piles.

Observation 3. The height of the stack significantly impacts the gap between the shared stacking policy and the dedicated policy. By increasing the number of tiers, the gap between the shared and dedicated policies increases. The reason is that in a pile with more tiers, more types of containers can

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share the pile. The number of piles does not seem to impact the performance of the shared or dedicated policy. The slight improvement in the performance of the shared stacking policy by decreasing the number of piles is due to the fact that we reduce the number bays from 8 to 5 while the number of rows remain constant. Note that the ASC gantry speed along the bays is higher than the ASC trolley speed along the rows.

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Last but not least, we evaluate the performance of the heuristic algorithm in this section by comparing the heuristic and exact results for 5 randomly selected instances. The exact results are obtained by

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programming the model in the AIMMS® environment where CPLEX® 12.6.3 is used as the solver. The results presented in Table 4 show that the heuristic algorithm can find an optimal solution for small

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size problems in less than a second. The experiments are carried out on small instances, since CPLEX has long computation time. For instances with more than 5 barges, CPLEX takes more than a day to find

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an optimal solution, only if the computer does not run out of memory. Table 4. Comparison of the heuristic and exact solutions

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Barge # # size variables constraints (sec) (sec) (sec) (sec) 1 5 71.97 800 1366 1.63 71.97 0.91 2 5 146.84 1600 3771 1.89 146.84 0.70 3 5 221.43 2400 6176 19.94 221.43 0.66 4 5 294.51 3200 9781 19.56 294.51 0.48 5 5 382.44 4000 10986 31.42 382.44 0.70 (1) Columns “# variables” and “# constraints” show the number of variables and constraints in the mixed integer programming model, respectively. Note that the number of constraints depends on . Therefore, different instances generated for the same parameters might have different number of constraints. (2) Let be the optimal solution obtained by CPLEX.

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(3) Let

be the CPLEX computation time.

5.2. Impact of inaccuracy in barge arrival information In this section, we discuss the impact of inaccuracy in barge arrival information on the performance of shared stacking heuristic. We take two different approaches: 1) inspired by empirical rule (Zaerpour et

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al., 2015), and 2) inspired by credit risk analysis techniques in financial risk domain. Inspired by empirical rule: Barge operators try their best to stick to their time windows. However, in reality barges might arrive outside their time windows due to reasons such as congestion, weather conditions, accidents, etc. We investigate the impact of the earliness and lateness of barges on the performance measures. In order to do this experiment, we use the empirical rule concepts. We assume the actual arrival time of each barge is normally distributed with known mean

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deviation . The assigned time window to each barge can capture

and standard

and

percentile

of the actual arrival time instants depending on the level of uncertainty. Barges are loaded in a first come, first served manner. Barges arriving early need to wait for their turn. If a barge arrives outside its time window, reshuffling may occur. Reshuffled containers are stacked in a temporary location in the middle of the stack (bay 15 as we consider a stack with 30 bays). Based on empirical rule,

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mean that 69%, 95%, and 99% of barges arrive within their assigned time windows. Table 5 summarize the results for different scenarios when there is uncertainty in barge arrival times.

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The results show that the performance of the shared stacking policy measured in terms of the utilization and total retrieval time depends on the time windows agreed with barge operators. Containers of reliable barge operators can be stacked using a shared policy without being concerned

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about reshuffling. Furthermore, the wider the time windows are, the smaller the gap is between the shared and dedicated policies. Obviously, when the time windows are wide, the shared stacking policy

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cannot stack containers together and therefore improve the solution compared with the dedicated policy. Note that in our experiments, we assume that containers are reshuffled to the middle of the stack. Using a bay near the bays that contain containers can improve the results of the shared stacking

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policy in terms of the time required for reshuffling. Table 5. Impact of arrival time uncertainty on results

1 5 15 20

Barge size 5 5 5 5

TW 1 1 1 1

71.97 436.92 1605.54 2294.40

71.97 382.44 1386.80 1975.86

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# Res 0.00 0.00 0.00 0.00

T_Res (sec) 0.00 0.00 0.00 0.00

# Res 0.00 0.00 3.40 1.40

T_Res (sec) 0.00 0.00 152.02 64.22

# Res 0.00 0.80 10.10 3.90

T_Res (sec) 0.00 37.06 451.35 178.87

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1 10 1 224.43 149.19 0.00 0.00 0.00 0.00 1.70 83.58 1 10 3 554.28 485.70 0.00 0.00 0.00 0.00 0.90 40.00 1 10 7 1350.13 1270.53 0.00 0.00 1.00 43.16 2.80 120.22 1 10 9 1882.79 1729.33 0.00 0.00 0.00 0.00 2.00 90.95 1 10 11 2294.06 2205.98 0.00 0.00 0.00 0.00 2.60 110.98 10 5 1 1026.24 871.73 0.00 0.00 0.00 0.00 2.60 120.74 10 5 1 1018.79 871.79 0.00 0.00 2.10 96.87 3.50 162.26 10 5 1 974.65 865.26 0.00 0.00 0.00 0.00 1.50 68.33 10 5 1 974.65 864.02 0.00 0.00 0.00 0.00 4.30 201.57 10 5 1 835.96 782.60 0.00 0.00 0.50 21.13 1.20 50.41 10 5 1 861.54 799.18 0.00 0.00 0.00 0.00 2.50 105.15 5 .5 10 974.65 864.02 0.00 0.00 0.00 0.00 1.10 50.74 5 2 10 974.65 869.61 0.00 0.00 0.00 0.00 1.20 57.73 5 4 10 974.65 903.51 0.00 0.00 0.00 0.00 0.80 37.37 5 6 10 974.65 962.55 0.00 0.00 0.00 0.00 0.00 0.00 5 8 10 974.65 974.65 0.00 0.00 0.00 0.00 0.00 0.00 (1) #Res and T_Res represent the number of reshuffles and reshuffling time, respectively. (2) Bold numbers, in the first four columns, represent the parameter being varied. Bold numbers, in the other columns, represent the instances where the dedicated stacking policy outperforms the shared stacking policy.

Inspired by the Credit Risk Analysis techniques: One of the main conclusions from Table 5 is that the number of reshuffles is related to the likelihood of barge delay. In general, terminal operators perform reshuffles (i.e., during the pre-marshalling operations) prior to the arrival of barges, known as “prior reshuffling.” In that case, containers will be stacked in the desired sequence, and no reshuffling occurs if

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containers arrive on time. However, reshuffles may occur when there is a delay in the arrival of a barge. This introduces a whole new wave of reshuffles to terminals, known as “posterior reshuffles” (See Figure

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7). This latter reshuffling, though not as common as former one, is another source of time delays. Based on this idea, we try to estimate the expected number of reshuffles (ER) in case of barge delay, using the

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model initially proposed by Gharehgozli et al. (2017a) based on the credit scoring and rating of

Posterior reshuffles

Delayed arrival time

Prior reshuffles

Expected arrival time

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Observation moment

creditworthiness used in the banking industry.

Time

Figure 7. Two types of reshuffles: prior and posterior reshuffles.

In the financial risk literature, expected loss is the value of a possible loss times the probability of that loss occurring (Basel Committee on Banking Supervision, 2001). Inspired by this idea, the ER model is defined based on three variables, probability of delay (PD), which is the probability of delay of a barge; 27

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reshuffles given delay (RGD), which is the likely number of reshuffles; and call size at delay (CSD), which is the number of containers to be loaded on the barge, to estimate ER. The combination of the three variable gives the expected number of reshuffles: 7

( ) where, ( ) is the expected number of reshuffles, delay, and

is the probably of delay;

is the reshuffles given

is the call size at delay.

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These variables can be estimated based on real data collected in container terminals. Computers record all barge arrival times, container movements, container handling equipment movements, gate arrivals, and all the other departures and arrivals. Such databases can be used in order to forecast and mitigate disruption. However, container terminal operators are not willing to reveal such confidential data as the release may impact their competitive advantage. Furthermore, generalizing the results driven from

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the data obtained from a specific terminal in a country to all terminals around the globe is not feasible. Therefore, a simulation of various data points is used to perform a sensitivity analysis to show the impact of varying parameters on the expected number of reshuffles.

Figure 8(a) shows the expected number of reshuffles in case of barge delays for different values of variables. In the heat map, the values of ER are shown by means of different colors, where dark blue

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shows the smallest expected number of reshuffles and dark red shows the largest expected number of reshuffles. In order to perform the sensitivity analysis, we consider 1 delayed barge with a call size,

,

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which can vary in a range from 1 to 20 TEU. Furthermore, the probability of delay, , and reshuffles given delay, , are in a range from 0% to 100%. The conclusions are in line with the ones made using the

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empirical rule method.

Impact of probability of delay (PD): If a container terminal works with a set of reliable barge operators who are always on time (i.e.,

), no extra reshuffles will be occurred in case of no delay (ER=0). On the

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other hand, as the customers become less reliable (i.e.,

), then ER gradually increases. If based

on historical data, container terminals can conclude that the contracted carriers are totally unreliable

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and usually delay (i.e.,

), then it is recommended that containers of different carriers,

weights, sizes, and destinations have to be stacked in separate piles. As it will be discussed later, by doing so, reshuffles given delay (RGD) becomes small (i.e.,

) which minimizes ER.

Impact of reshuffles given delay (RGD): If container terminal operators use a dedicated stacking policy (i.e.,

=0%), then ER equals 0 in case of delay. On the other hand, when different container groups start sharing piles, RGD increases. In this case, if the barge delays ER will increase. In other words, the value

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of ER for a specific barge with a specific call size depends on both PD and RGD. Figure 8(b) shows the impact of PD and RDG on ER for a barge with

TEU.

Impact of call size at delay (CSD): The other point that container terminal operators have to pay attention is CSD. If the number of containers to be loaded on a delayed barge is small, then ER will not be as big as a large barge. Therefore, from the viewpoint of the expected number of reshuffles, containers of smaller barges can be mixed whereas mixing containers of larger barges has larger consequences.

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Looking at practice, one can observe that this in fact happens in container terminals. Therefore, in order to minimize cost and shorten the turnaround of larger barges, terminal operators stack the

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associated containers separately.

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(a) A thermal plot indicating the expected number of (b) The impact of varying PD ( ) and RGD ( ) on ER for reshuffles by means of different colors. a barge with CSD ( ) = 10 TEU. Figure 8. The expected number of reshuffles

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Note. Color scale is , where dark blue shows the smallest expected number of reshuffles and dark red shows the largest expected number of reshuffles.

5.3. Impact of information availability

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In container terminals, sometimes at the last moments it becomes known when or which barge will pick up a container. This impacts the stacking policy used and consequently the number of reshuffles and stack utilization in the container terminal. Figure 9 shows the impact of chosen stacking policy on

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the utilization and the number of reshuffles. 

Figure 9(a) shows the utilization for stacks with sizes 40, 80, 120, and 160 piles (with 4 tiers) and

under dedicated stacking policy. Dedicated stacking policy is used to stack varying types of containers (from 1 to a theoretical value of 1,000 different types of containers). It can be observed that in the dedicated policy, the utilization decreases rapidly by increasing the types of containers in one stack. When one type of containers is stacked, the stack can be fully utilized, i.e. 100% utilization. In the other extreme case, all containers are of different types. The utilization can then

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theoretically drop to

. In other words, each of 40 piles accommodates one container of

a specific type. Other containers cannot be stacked on top of the container of each pile. So out of 

positions are occupied.

Figure 9(a) also shows the number of extra piles required to stack all containers of the examples with 40, 80, 120, 160 piles. In all instances dedicated stacking policy is used to stack containers. When only one type of containers is stacked, no extra piles are required. However, by stacking

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more types of containers, extra piles are necessary. If 40 piles of containers (160 containers) need to be stacked together (each randomly selected from 1000 types of containers), 120 extra piles are required (in total 160 piles to stack all 160 containers). 

Based on the results presented in Figure 9(a), the dedicated policy results in low utilization. Inspired by this idea, we perform a set of new analyses to study how the shared policy can be used

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in a terminal by dedicating some parts of the stack to containers with no information. For containers with full information, shared stacking policy is used. Containers with no information are stacked using a random stacking policy. Obviously, later when it becomes clear when or which barge will pick up these containers, reshuffles may become necessary. The results in Figure 9(b) show that the number of reshuffles increases when the percentage of containers with no

information, 80 containers (

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information grows. For example, for a case with 40 piles, when 50% of containers have no ) need to be stacked in 20 piles (i.e.,

piles

of 4 tiers) using the random policy. Figure 9(c) shows how the random stacking policy performs for

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different stack sizes (in number of piles) and types of containers. With increasing the types of containers, the number of reshuffles increases and this increase is more significant for larger 

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stacks.

The final analysis is done to compare a combined shared and random stacking policy with a combined shared and dedicated policy. In the first policy, containers with information are stacked

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using a shared policy and containers with no information are stacked using a random policy. In the latter policy, instead of the random stacking policy, the dedicated policy is used to stack containers

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with no information. We assume that we know which barges will pick them up but we do not know when. If both pieces of information are missing, then it is not possible to stack them other than using a random policy. The results in Figure 9(d) show that for the base example with 40 piles when only information of a few containers is missing the combined shared and dedicated policy performs better. However, as the percentage of missing information increases, the combined shared and random stacking policy takes over as the other policy spreads container over many piles and as a result the retrieval time increases. Note that this result depends on several factors such as the location of the bay for reshuffling and the number of types of containers that are stacked together. 30

(b) The number of reshuffles required for different stack sizes (in number of piles), under the combined shared and random policy where the random policy is used to stack containers with no information

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(a) The average utilization and extra piles required for different stack sizes (in number of piles), under the dedicated stacking policy

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(c) The average number of reshuffles in a stack with different (d) comparing the extra retrieval time for the combined number of piles, under the combined shared and random shared and dedicated policy with the combined shared and policy where the random policy is used to stack containers random policy (average on 1000 types of containers) with no information (only 100 types since the trends reach a plateau) Figure 9. Utilization and number of reshuffles in the dedicated and shared stacking policies

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5.4. Managerial insights In this section, we discuss the insights developed in this paper and how they can be used by terminal operators to run terminals more efficiently. Shared and dedicated stacking. Dedicated policy can be considered as a special case of shared policy where each pile accommodates only one container type. The shared stacking policy reduces the total retrieval time of the ASC up to 30% compared to the dedicated policy. In fact, the relative improvement increases with an increasing number of barges. In case of a very shallow stack with a few containers of a few barges to be stacked, the difference between shared storage and dedicated storage is negligible. In 31

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such a scenario, both stacking policies can often fully use the piles closer to the I/O point and reduce the ASC total retrieval time. Stack utilization. The results show that the dedicated stacking policy results in a low stack utilization. For stacking 90 types of containers (i.e., 10 barges with containers of 3 weight groups and 3 destinations), the utilization drops to less than 30%. The results show that the shared stacking policy can alternatively be used to increase the utilization and save space.

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Information accuracy. The results show that the shared stacking policy is reasonably robust to inaccuracy in barge arrival information. The policy performs well, even when the time windows constructed for arrival times of barges capture only 70% (i.e.,

) of actual arrival times. If barge operators are less

reliable in terms of arrival times (e.g., when the time windows capture only 30% of the actual arrival times), a dedicated policy is recommended.

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Advance information. Terminal operators may not have full information on all barges that will pick up the containers. In such a situation, using the dedicated policy results in occupying too much land. In order to increase the land utilization, we can use a combined stacking policy which stacks containers with information based on the shared policy and containers without information using a random or dedicated stacking policy. The random policy may result in reshuffling whereas the dedicated policy

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may result in low utilization. Thus, the choice of random or dedicated in the combined policy depends on several factors including the types of containers, the percentage of containers with no information,

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and the location of the reshuffling bay. 6. Conclusion

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Handling the increasing number of containers has put an extra burden on container terminals. Despite the flexibility that trucks offer for transporting containers to the hinterland, the current trend suggests

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a modal shift from truck to train and barge in order to reduce the pressure on the current road infrastructure, to lower the costs, and to reduce greenhouse gas emissions. This modal shift trend will result in an increasing number of barge calls at container terminals. Thus, one of the important

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operational problems is how to temporarily stack barge containers in a terminal. In this paper, we use mathematical modeling to answer this question. Nowadays, terminals aim to stack containers in highly dense and multi-high stacks. Such a configuration not only results in a lower footprint but also in a lower total retrieval time if reshuffling can be avoided. In practice, the dedicated stacking policy is used to avoid reshuffling which subsequently results in partially filled piles. As an alternative policy, we propose a shared stacking policy to increase the stack utilization and reduce the total retrieval time. For each barge, a time window is defined such that it captures the actual arrival time of the barge. We use the time windows 32

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to mathematically formulate the operational problem of stacking outbound containers to be transported to the hinterland by barges. Since an exact pile-sharing model is strongly

-complete,

an adapted simulated annealing heuristic is proposed to solve the problem. The performance of the heuristic algorithm is compared to a lower bound obtained by a Lagrangean relaxation approach. The gap between the heuristic and Lagrangean relaxation of the exact model appears to be small for real size instances with high utilization, indicating that the heuristic algorithm

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can obtain near-optimal solutions. In addition to testing the performance of the heuristic algorithm, the numerical experiments contains several managerial insights. The results show that the shared stacking policy outperforms the dedicated policy in terms of two performance measures including the total retrieval time and utilization. The reason is that the shared stacking policy fully occupies the piles closer to the I/O point with containers of different barges. As a result, shorter response times for barges

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can be achieved, a prime concern for managers, in particular for busy deep-sea terminals.

In addition, due to higher utilization, more space will be available for serving new barge. Using space better is also a prime objective for terminal managers, as it reduces the investment cost. For example, lack of space has made Port of Rotterdam to invest around €3 billion in the Maasvlakte 2 project to extend the port by approximately 1000 hectares (Gharehgozli et al., 2017b). Finally, the performance of the shared stacking policy depends on the information accuracy measured by the percentage of barges

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arrive outside agreed time windows, and advance information measure by the percentage of container with no assigned barge. Shared stacking policy is robust to small and medium disturbances but when

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the uncertainty in arrival information of barges increases the performance of the shared stacking policy drops.

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Our study can be considered as the building block for other stacking problems such as the block relocation problem or the pre-marshalling problem. In such problems, containers are already stacked

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and have to be relocated to new locations with the minimum number of reshuffles. On the other hand, in the problem studied in this paper, we are looking for suitable locations to stack containers. So, our problem can be used as an input for the block relocation problem. It would be interesting to see an

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integrated approach where the number of reshuffles and the total retrieval time is minimized while rehandling containers. Another potential direction for future research is to study a problem in which containers arrive dynamically and their information is revealed in real time. New models and solution methods are required for such a problem (see for example, Akyüz and Lee, 2014; Wan et al., 2009; Ku and Arthanari, 2016; Zehendner et al., 2017; Gharehgozli et al, 2014a). Finally, our model and solution method are general and can be used to stack containers of other modes of transport including trucks with a capacity of one or two containers and vessels with a capacity of hundreds or thousands of containers. The larger number of individual containers and the larger groups of containers can make 33

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the problem more complex. Therefore, it is suggested to study whether our model or solution method needs to be modified in order to efficiently deal with individual containers or with groups of hundreds of containers to be stacked in one block. Acknowledgment The authors would like to thank Prof. Nils Boysen, for his insights in the complexity proof. The authors suggestions helped us to improve the quality of this paper. References

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also would like to thank the editor and anonymous reviewers whose constructive comments and

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Appendix A. Proof of Theorem 1 We prove that our problem is

-complete based on a reduction from the numerical 3-dimensional

matching which is well-known to be strongly

-complete (Garey and Johnson, 1979). This problem is

defined as follows: Given three multisets of integers , bound . Does there exist a subset once and that for every triple (

of

and , each containing

such that every integer in ,

) in the subset

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and

elements, and a occurs exactly

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To generate an instance of our problem from an instance of numerical 3-dimensional matching, we introduce a block which has exactly

piles with

one barge each to be supplied with exactly

tiers per pile. For each integer value

we introduce

containers. All these containers receive an identical time

window, which depends on the multiset the integer value originates from. All containers referring to an integer from multiset

,

and

receive time window ,

-,

- and ,

- respectively. All

containers referring to the same integer value have to be stacked in the same (single) pile. If the further destination to be served

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containers are distributed among two or more piles, we still have

simultaneously but not enough remaining piles to store them without blocking. Therefore, each pile has to receive all containers referring to three integer values one of each multiset ,

and , because

only integers of different subsets can be stored together in the same pile. We, thus, have three integers per pile loading all b tiers to capacity, which obviously leads to a one-to-one mapping between both

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problems.

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