431

1 INTRODUCTION

The Liner Shipping Network Design (LSND) is the

strategic backbone of global supply chain connectivity,

facilitating over 80% of world trade [55]. Designing

these networks is inherently a multi-actor, multi-

constraint, and multi-objective optimization problem

influenced by conflicting demands from shipping

lines, regulators, and environmental policies [8].

Crucially, LSND requires balancing competing goals:

increasing service reliability often raises costs and

emissions, while fuel-saving strategies may prolong

transit time. This complexity demands optimization

approaches capable of simultaneously managing

economic efficiency, environmental sustainability, and

operational reliability

In recent years, research on LSND has advanced

rapidly, coinciding with the growing complexity of

global shipping and developments in optimization

technology. A wide range of mathematical and

computational approaches have been employed to

design efficient, adaptive, and sustainable shipping

networks. Classical methods, such as Mixed-Integer

Linear Programming (MILP), remain foundational due

to their precision in formulating complex decision

structures. However, as data scale and uncertainty

have increased, researchers have turned to heuristic

Challenges and Optimization in Liner Shipping Network

Design: A Systematic Mapping and Future Research

Agenda

S.D. Kumalasari

1,2

, K. Komarudin

& A. O. Moeis

Trisakti Institute of Transportation and Logistics, Jakarta, Indonesia

University of Indonesia, Depok, Indonesia

ABSTRACT: The Liner Shipping Network Design (LSND) is crucial for global maritime efficiency; however, the

field faces persistent trade-offs driven by escalating fuel costs, complex network demands, and geographical

challenges. This paper conducts a Systematic Literature Review (SLR) to explore how existing studies have

addressed these challenges through diverse optimization strategies. Using the PRISMA approach, 119 peer-

reviewed articles were reviewed and systematically mapped across several key dimensions: decision levels

(strategic, tactical, and operational), problem areas (shore-based and sea-based), service patterns (simple,

pendulum, butterfly, complex, and asymmetric), optimization approaches, and physical constraints. The analysis

shows that while mathematical programming, heuristics, metaheuristics, and hybrid models are widely applied,

their performance and applicability vary across different case studies. Strategic-level studies on network design

and fleet deployment dominate the existing research, whereas service patterns such as butterfly and asymmetric

networks remain underexplored despite their operational relevance for regions facing trade imbalances. This

review contributes to developing the links between optimization methods and specific LSND challenges. It

proposes a future research agenda that emphasizes uncertainty factors, green policy integration, and cooperative

network design, particularly relevant for developing adaptive and sustainable systems in complex archipelagic

or inter-regional contexts. By doing so, it provides a clearer understanding of the field landscape and outlines

potential pathways for advancing adaptive and sustainable shipping network optimization.

http://www.transnav.eu

the International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 20

Number 2

June 2026

DOI: 10.12716/1001.20.02.17

432

and metaheuristic approaches, including Genetic

Algorithm (GA), Variable Neighbourhood Search

(VNS), and Simulated Annealing (SA) to obtain near-

optimal solutions with improved computational

efficiency. Recently, hybrid models that integrate exact

and heuristic methods have emerged to strike a balance

between accuracy and computational speed. These

models are frequently applied to multi-stage problems

such as fleet deployment, schedule design, and port

assignment, which are interdependent within a

network system. Furthermore, recent studies have

expanded their focus toward stochastic and multi-

objective models, incorporating uncertainties in

demand, sailing time, and emission regulations. This

progression indicates that LSND research has evolved

from purely cost-based optimization toward more

holistic, sustainability-driven approaches.

Despite these developments, a comprehensive

global mapping of LSND challenges and their

corresponding optimization approaches remains

lacking, failing to connect the conceptual dimensions

systematically. Existing studies often examine LSND

issues, such as decision levels (strategic, tactical, and

operational), problem areas (shore versus sea), and

service patterns (simple, pendulum, butterfly,

complex, and asymmetric), in isolation, without

providing a longitudinal analysis to explain the

evolution of the research. As a result, LSND studies

tend to remain fragmented, with a focus on exploring

network design challenges and applying suitable

optimization methods.

Understanding the interconnections among

decision levels, problem areas, service patterns, and

optimization methods is crucial to designing liner

shipping networks that are both effective and

sustainable. Each decision level, from operational to

strategic, has distinct objectives and constraints that

influence one another. Meanwhile, problem areas on

the shore and seaside determine the decision context

and system complexity, while service patterns such as

simple, pendulum, butterfly, complex, and asymmetric

reflect network configurations requiring different

optimization strategies to balance uncertainty, cost,

and time. Integrating these conceptual dimensions is

therefore essential for developing LSND models that

are not only economically efficient but also adaptable

to operational dynamics and global policy frameworks.

This condition highlights the need for a systematic

mapping review capable of identifying global

challenges and aligning them with appropriate

optimization approaches to address the complex

problems of liner network design while shaping future

research directions. Accordingly, this study aims to

provide a comprehensive review of the evolution of

LSND research through a systematic mapping

approach. Specifically, the objectives of this study are

to:

1. Identify LSND challenges based on decision levels

(strategic, tactical, operational), problem areas

(shore-based and sea-based), and service pattern

characteristics within liner shipping networks.

2. Classify optimization approaches applied in LSND

studies, including the tendency toward linear,

nonlinear, and hybrid modeling.

3. Develop a future research agenda to map research

trajectories and identify emerging opportunities for

LSND advancement.

This paper is structured as follows: Section 1

presents the introduction, outlining the background

and objectives of the study. Section 2 describes the

research methodology and systematic review process.

Section 3 presents the results and analysis, followed by

Section 4, which provides an in-depth discussion and

interpretation of the findings. Section 5 presents the

future research directions, and finally, Section 6

concludes the paper.

2 RESEARCH METHOD

This study employs a Systematic Literature Review

(SLR) methodology, guided by the PRISMA (Preferred

Reporting Items for Systematic Reviews and Meta-

Analyses) framework, to ensure transparency, rigor,

and reproducibility throughout the review process.

Compared to narrative reviews, the SLR approach

provides a more structured mechanism for identifying,

evaluating, and synthesizing prior studies, thus

minimizing bias and enhancing methodological

robustness.

The review was structured around three key stages:

1. Planning the review – establishing research

objectives and developing the review protocol.

2. Conducting the review – systematically identifying,

selecting, and extracting data from relevant studies.

3. Synthesizing and reporting – organizing insights

into a coherent structure aligned with the research

question.

The SLR was designed to address the following

research questions:

1. What are the key challenges of Liner Shipping

Network Design (LSND) when analysed across

different decision levels (strategic, tactical, and

operational), problem areas (shore-based and sea-

based), and the service pattern characteristics

applied in liner shipping networks?

2. How have the optimization methods and modeling

approaches been applied to address LSND

challenges?

3. What are the future research directions and

opportunities for advancing LSND studies in the

coming years?

2.1 Data Collection Process

This study focuses on the fundamental issue of liner

shipping network design, namely the classification of

challenges. The final criterion is that the selected

articles must explicitly or implicitly examine the

correlation between challenges and optimization

within the scope of liner shipping network design.

This study exclusively utilizes the Scopus database,

which is justified by its comprehensive coverage of

high-impact journals in Operations Research, Logistics,

and Maritime Transportation, thereby ensuring a

robust representation of the LSND domain while

maintaining a methodological focus. Scopus was

utilized to collect relevant articles, ensuring that all

related publications were included and the

interdisciplinary perspectives of the topic under

review were accommodated. Search strings (a

combination of keywords) were selected for the

database to retrieve as many publications as possible

433

related to challenges and optimization (see Table 1 -

Search Strings Used in This Study). During the search

process, the publication period is not restricted to

capture all relevant articles. To ensure thorough

coverage of pertinent literature, a systematic search

strategy was employed using specific keywords

related to liner shipping network design. The primary

keywords included:

Table 1. Search Strings (SS) Used in This Study

Code

Used

Search String Used

SS - 1

"Shipping Route Design" OR "Maritime Route Planning" OR

"Shipping Route Planning" OR "Shipping Network Design"

SS - 2

"Liner Shipping" OR "Container Shipping"

The initial search query used for database searches

employed a Boolean combination of SS-1 OR SS-2 to

ensure broad coverage of relevant literature about

either shipping network design or scheduled liner

shipping services. This approach was adopted to

maximize the number of potentially relevant articles

during the identification stage of the systematic review

process. The keywords were adapted to the syntax

requirements of the Scopus database to ensure

consistency and accuracy. Subsequently, a structured

screening process was applied to assess titles, abstracts,

and full texts against predefined inclusion and

exclusion criteria. Only articles that explicitly or

implicitly addressed topics within the scope of liner

shipping network design were retained for further

analysis. The detailed assessment criteria are provided

in Table 2, "Screening Criteria Used to Select Papers."

Table 2. Screening Criteria Used to Select Papers

Title and Abstract Assessment

Criteria

Full Text Assessment Criteria

Peer–reviewees only or final article

Provides empirical data, model-

based analysis, or case studies

relevant to network efficiency,

cost, or sustainability.

Limited to articles written in

English

Articles that focus on liner

network design, specifically

creating or designing new

routes using optimization

models

Contain keywords related to liner

shipping, network design,

container shipping, scheduling,

routing, fleet deployment,

container routing

The inclusion criteria were established to focus on

high-quality research that provides insights into liner

shipping network challenges and optimization

techniques. The search for relevant articles was limited

to peer-reviewed papers published in academic

journals and available in full-text English. Articles with

abstracts in English but without full text availability in

English were excluded from this study.

2.2 Data Analysis

As part of the systematic mapping review process, an

initial total of 2,313 articles were identified from the

Scopus database through both manual and automated

searches. After duplicate removal and language

filtering, 1,564 studies were retained for title and

keyword screening. During this stage, articles that did

not address liner shipping, network design, or

container routing were excluded, resulting in 450

relevant studies.

The second screening phase focused solely on

methodological relevance. Studies providing empirical

data, model-based analysis, or case studies related to

network efficiency, cost, or sustainability were

included. This step reduced the dataset to 244 studies.

Additionally, 12 more papers were identified through

backward snowballing, of which five met the inclusion

criteria.

Subsequently, during the eligibility assessment,

studies focusing on liner network analysis or liner

network operation without explicitly designing or

optimizing new routes were excluded. This process

resulted in 115 eligible articles, which, after

incorporating the five snowballed papers and

conducting a final quality assessment, yielded a total of

119 studies included in the final systematic mapping

analysis.

Figure 1. Article Screening (Based on the PRISMA Flow

Diagram)

From the selection process, a total of 119 papers

were identified as belonging to the category of liner

shipping network design employing optimization-

based approaches. These studies were subsequently

analysed based on their decision levels, problem areas,

and service patterns, and further classified according

to the optimization methods applied. Finally, the

analysis explored future research directions and

potential opportunities arising from the existing body

of literature.

2.3 Data Extraction Framework

Subsequently, data extraction was performed on the

119 selected papers by classifying each article

according to the analytical framework dimensions of

Liner Shipping Network Design (LSND). The

classification process was conducted based on the

following categories:

Table 3. Data Extraction Framework

Category

Details

Decision level

Strategic, tactical, operational

Problem area

1.Shore based (geopolitical & trade control,

port operation & congestion; infrastructure &

multimodal; market & competition)

2. Sea based (environmental & regulatory;

routing & navigational; operational at sea;

disruption & blockage; service network)

Service pattern

Simple, pendulum, complex, butterfly,

asymmetric

Regional

Asia, Europe, America, Africa, Australia

Input data type

Deterministic, stochastic

434

Optimization

model and

method

Nonlinear, linear, mixed (hybrid), exact,

heuristics, and metaheuristics

Software

CPLEX, XPRESS, MATLAB, GUROBI,

AIMMS

Future research

theme

Data & uncertainty,

Environmental & policy,

Network & operations,

Methodology,

Behavioral & strategic

The categorization served as a systematic guideline

to consistently organize and interpret information

from each of the reviewed papers. The grouping

process was conducted according to eight main

categories as follows:

1. Decision level: covering strategic, tactical, and

operational levels to identify the focus of planning

and decision-making in liner shipping network

design.

2. Problem area: classified into shore-based and sea-

based domains to capture the contextual focus of

issues influencing liner shipping network design.

3. Service pattern: including simple, pendulum,

butterfly, complex, and asymmetric configurations

to represent the network structures analysed in

each study.

4. Regional scope: categorizing the geographical

coverage of the routes analysed in the respective

studies.

5. Input data type: describing the characteristics of

input data used in optimization, such as

deterministic or stochastic parameters.

6. Optimization model and method: identifying the

modeling approaches adopted, whether linear,

nonlinear, or a combination (mixed or hybrid).

7. Software: listing the optimization tools or solvers

utilized (e.g., CPLEX, GUROBI, MATLAB) to

address modeling and computation processes.

8. Future research theme: mapping the research

trajectory and identifying future opportunities for

land development based on previous studies.

Through this data extraction framework, the study

provides a structured and systematic mapping of the

reviewed literature, enabling the identification of

linkages between key challenges, optimization

approaches, and future research directions in Liner

Shipping Network Design within a global context.

3 RESULTS AND ANALYSIS

The following section presents the results and analysis

derived from the categorization framework outlined in

Subsection 2.3, "Data Extraction Framework."

3.1 Decision Level

The mapping of decision levels in this study refers to

the analytical framework proposed by Brouer et al.

(2014) [1], which categorizes previous studies

according to their level of decision-making within the

context of Liner Shipping Network Design (LSND). In

general, research in this field can be classified into three

primary decision levels: strategic, tactical, and

operational. At the strategic level, studies focus on

long-term decisions such as determining markets to

serve, defining fleet size and mix, and designing the

overall shipping network configuration. The tactical

level involves medium-term planning related to

resource allocation and operational optimization,

including vessel scheduling, cargo routing, speed

optimization, service selection, and fleet deployment.

Meanwhile, the operational level addresses short-term,

technical decisions such as cargo stowage planning,

empty container repositioning, berth scheduling, and

disruption management. Thus, this framework

provides a systematic structure for mapping prior

research contributions, while simultaneously

illustrating the hierarchical complexity and temporal

horizon of decision-making processes within the

maritime transportation system.

Figure 2. Decision Level

Based on Figure 2, approximately 65% of the

reviewed studies were categorized at the strategic

level, with a particular focus on the network design

aspect, which emerged as the most frequently

discussed topic among the other decision levels. This

predominance can be explained by the fact that

network design constitutes a fundamental component

of the maritime transport system, as it determines the

routing structure, inter-port connectivity, and overall

configuration of cargo flows. Strategic decisions in this

context exert a long-term influence on operational

efficiency, logistics costs, and service performance.

A large portion of LSND studies also incorporates

tactical decision-making at the individual level. This

ensures that long-term decisions can be translated into

more detailed medium-term choices, especially

regarding the feasibility and efficiency of both levels of

decision-making. In contrast, not many LSND studies

consider operational-level decisions. This is because it

increases the size of the problem to a huge optimization

problem that is impractical to solve. Additionally, the

level of uncertainty at the operational level is high. As

a result, it is more suitable to be addressed when the

strategic and tactical decisions are fixed and only for a

short period ahead.

3.2 Problem Area (Shore vs Sea)

The problems encountered in liner shipping network

optimization can be broadly classified into two main

areas: the shore problem and the sea problem, as

depicted in Figure 3. The shore problem area

encompasses issues originating from land-based

systems, including policy, infrastructure, and market

dynamics that collectively influence the overall

performance of maritime transport networks. The first

subcategory, geopolitical and trade control, highlights

how political factors such as government regulations,

shipping alliances, the COVID-19 pandemic, and

geopolitical conflicts (e.g., the Russia–Ukraine war)

435

affect trade network stability and port selection

decisions [46]. The port operations and congestion

category addresses operational challenges at ports,

including congestion, queuing time, berthing delays,

and simultaneous container handling operations, all of

which impact the efficiency of logistics chains [41,55].

The infrastructure & multimodal category emphasizes

the importance of integrating sea and land transport

through multimodal infrastructure development (road

and rail), LNG bunkering station placement, and port

infrastructure investment [4]. Meanwhile, the market

& competition subcategory explains the dynamics of

fluctuating demand and fuel prices, competitive

behaviour among carriers, and shipper preferences

that influence service pattern design in liner shipping

networks.

In contrast, the sea problem area encompasses

challenges that arise in the marine environment,

directly linked to vessel operations, navigational

conditions, and environmental factors. The ecological

& regulatory category addresses sustainability issues,

such as sulphur emission restrictions under IMO

regulations, the environmental impacts of alternative

routes (e.g., the Suez Canal vs. the Cape of Good Hope)

[47], and the implementation of environmentally

friendly scheduling (green schedules). The routing &

navigational subcategory is the most extensively

studied, covering the design of port call sequences,

cargo allocation, and voyage planning to optimize

operational and navigational efficiency. The

operational at sea category focuses on technical aspects

such as sailing speed, fuel consumption, voyage time

uncertainty due to weather conditions, and

heterogeneous fleet management. The disruption &

blockage category highlights the risks of extreme

disruptions, including blockages in critical passages

like the Suez Canal or severe weather events that cause

delays and logistical instability [47]. Lastly, the service

network subcategory discusses how multiple shipping

routes are integrated to form cohesive service networks

that can adapt to variations in cargo demand and

service frequency, such as coastal liner services and

feeder shipping systems. Overall, this framework

provides a comprehensive depiction of the

multifaceted challenges in global liner shipping

systems, demonstrating the interconnectedness

between shore-based and sea-based dimensions in

achieving network efficiency, sustainability, and

resilience across the maritime transport sector.

Figure 3. Problem Area Liner Shipping Network Design

Based on the literature mapping presented in Figure

4. The problem area of liner shipping network design

reveals that issues within the Sea Problem area,

particularly those related to Routing and Navigational

aspects, dominate the research landscape in liner

shipping network design. This dominance arises from

the strategic significance associated with routing

decisions, which directly determine operational

efficiency, fuel consumption, voyage time, and service

schedule reliability. The processes of route planning

and port sequence determination involve numerous

interrelated decision variables, including vessel

compatibility, weather conditions, ocean currents,

navigational constraints, vessel capacity, and

interactions between mainline and feeder routes. These

interdependencies make routing and navigation one of

the most challenging and influential research areas in

liner network design. The substantial proportion of

studies focusing on Routing and Navigational

optimization reflects a clear tendency among

researchers and practitioners to prioritize sea-based

operational efficiency as a critical determinant of

maritime logistics performance. Decisions made at this

level have a direct and measurable impact on cost

efficiency, emission reduction, and service reliability

across the global supply chain.

3.3 Problem Characteristics (Regional and Service

Patterns)

The classification of studies based on service patterns

and route coverage identified in the literature reveals

four main types of liner service configurations [8]:

simple service, pendulum service, complex service,

butterfly service, and asymmetric service as a new era

of service pattern [88]. These patterns illustrate how

vessels connect ports within scheduled loops to

achieve efficient and reliable cargo movements. The

simple service represents the most basic configuration,

linking a series of ports sequentially and returning to

the origin without any branches, ensuring operational

stability and predictability. The pendulum service

resembles a swinging route that connects two major

regions through a central hub port, allowing vessels to

proceed in one direction without retracing their path.

The butterfly service combines two interconnected

loops extending from a single hub, forming a structure

similar to butterfly wings, often designed to serve two

sub-regions efficiently. A more intricate configuration,

the complex service, integrates multiple loops and

branches within a vast, interconnected network,

commonly adopted in large-scale global operations to

optimize fleet utilization and network coverage.

Finally, the asymmetric service features unequal routes

or voyage durations between outbound and return

legs, reflecting trade imbalances or differing cargo

demands across regions. Together, these

configurations demonstrate the strategic diversity and

operational sophistication that underpin modern liner

network design.

The mapping indicates that most studies focus on

the Asia-Europe and America–Asia–Europe trade

routes, which represent the world's major shipping

corridors characterized by the highest cargo volumes

and the most significant operational complexity.

Among these, the pendulum and complex service

patterns are the most frequently examined

configurations, as they closely reflect the real

operational structures of global container shipping,

where vessels operate in multi-port networks with

436

balanced directional flows between continents,

mirroring the back-and-forth motion of a pendulum.

In contrast, studies analyzing simple service

patterns are generally confined to regional contexts

such as Asia, Europe, or America, where liner

networks are more straightforward and exhibit higher

service frequency, typically connecting a limited

number of major ports with relatively balanced

demand. Meanwhile, butterfly and asymmetric service

patterns have received comparatively less attention.

Yet, they remain relevant in asymmetric network

structures, particularly on routes with imbalanced

export-import flows or unidirectional market

dominance. These findings suggest that network

complexity, route coverage, and cargo distribution are

the primary determinants shaping the selection and

design of service patterns in liner shipping network

operations.

Table 4. Service Patterns and Regional Liner Shipping

Network Design

Simple

Service

Pendulum

Service

Complex

Service

Butterfly

Service

Asymmetric

Service

Asia

(Regional)

[2], [3],

[4], [5],

[6], [7],

[8], [9],

[10], [11]

[12], [13], [14],

[15], [16], [17],

[18]

[19], [20],

[21], [22],

[23],[24],

[25], [26],

[27], [28],

[29]

[30]

Europe

(Regional)

[31]

[32], [33]

[31]

America

(Regional)

[34]

[35]

Africa

(Regional)

[36], [37]

Asia -

Africa

[38]

[39]

[40]

Asia -

America

[41]

[42], [43], [44]

[45]

Asia –

Europe

[46],

[47],

[48],

[49],

[50],

[51],

[52], [53]

[54], [55], [56],

[57], [58], [59],

[60], [61], [62],

[63], [50], [64],

[65], [66], [67],

[68], [69], [70],

[71], [72], [73],

[74], [75]

[76], [77],

[78],[79],

[80], [81],

[82], [83],

[84], [89]

[51], [85],

[86], [87]

[88]

Asia –

Australia

[64], [8]

[90]

[91], [92]

Asia –

Europe -

Africa

[93]

Asia –

Australia -

Europe

[94], [95],

[96]

America –

Asia -

Europe

[97], [98]

[99]

Europe -

Africa

[100],

[101]

[102], [103]

[101]

Europe –

America

[104], [105]

[106]

Unknown

[107],

[108],

[109]

[110],

[111],

[112],

[113],

[114],

[115],

[116], [117]

[107]

The butterfly and asymmetric service patterns

represent more complex network configurations

compared to other service types. The butterfly pattern

is characterized by two route branches originating

from a single central hub and extending toward two

distinct destination regions. This configuration is

particularly suitable for areas with dispersed cargo

demand and moderate shipment volumes, such as

inter-regional or feeder networks. In contrast, the

asymmetric service pattern reflects an imbalance

between outbound and inbound directions, whether in

terms of cargo volume, number of port calls, or voyage

distance. This pattern is often suitable when combined

with routes that exhibit uneven cargo flows, which can

be stabilized through integration with major trade

lanes, such as the Asia–Europe or Asia–America

routes. Both service patterns have received limited

attention in the literature due to their high modeling

complexity, lack of empirical data, and the dynamic

nature of global trade imbalances, which are difficult to

capture using conventional optimization models.

Nevertheless, studies focusing on these two service

configurations hold significant potential for future

research, particularly in the development of adaptive,

efficient, and sustainable liner shipping networks.

3.4 Input Data Optimization

Research on liner shipping network optimization can

generally be divided into two main modeling

approaches: the deterministic and the stochastic. In the

deterministic approach, all model parameters and

conditions are assumed to be known and fixed in

advance, including cargo demand, sailing time, vessel

capacity, and operational cost. This type of Model is

commonly used to address problems such as network

design, fleet deployment, and service scheduling,

under the assumption of stable market and operational

conditions. The main advantage of the deterministic

approach lies in its mathematical simplicity and its

ability to produce optimal solutions with high

computational efficiency. However, deterministic

models often fail to adequately represent fundamental

world uncertainties in liner shipping systems, such as

demand fluctuations, extreme weather events, or

operational disruptions at ports.

In contrast, the stochastic approach explicitly

incorporates uncertainty into model parameters,

covering factors such as cargo demand, vessel arrival

times, weather conditions, and fuel costs. Stochastic

models typically employ probability distributions to

capture such variability and aim to generate robust

solutions that remain effective under changing real-

world conditions. Within the context of liner shipping

network design, this approach has been widely applied

to studies on speed optimization, disruption

management, and schedule reliability. Although

stochastic models offer more realistic representations

of operational environments, they require more

complex probabilistic data and significantly greater

computational effort. Overall, the dominance of

deterministic models in the literature indicates that

most studies remain focused on developing efficient

foundational models, while stochastic approaches are

emerging as a promising research direction to enhance

the resilience and adaptability of liner shipping

networks under operational uncertainty.

437

Table 5. Input Data Optimization

Input Data

Deterministic

Stochastic

Demand

[46], [54], [30], [47], [97], [100], [2], [38],

[19], [101], [56], [20], [42], [76], [49],

[59], [35], [12], [37], [5], [62], [23], [31],

[112], [39], [64], [77], [34], [10], [98],

[89], [43], [65], [93], [113], [102], [103],

[26], [11], [81], [115], [15], [85], [68],

[92], [94], [16], [82], [45], [83], [8], [99],

[86], [79], [27], [53], [33], [69], [84], [70],

[17], [75], [88], [44], [40], [71], [28], [87],

[85], [106], [9], [95], [73], [18], [109],

[96], [116], [74], [91]

[48], [45], [57],

[32], [21], [58],

[63], [50], [23],

[6], [25], [25],

[90], [91], [107],

[14], [7], [118],

[53], [29], [66],

[9]

Distance

[56], [104], [57], [32], [42], [76], [4], [21],

[58], [22], [36], [49], [59], [35], [60], [12],

[37], [110], [5], [61], [63], [111], [23],

[31], [6], [64], [34], [10], [98], [89], [43],

[65], [93], [25], [90], [91], [107], [117],

[102], [105], [78], [14], [7], [13], [114],

[26], [11], [81], [52], [15], [92], [16], [53],

[33], [69], [70], [17], [75], [88], [44],

[119], [66], [71], [106], [109], [120], [96],

[74]

Operational

Cost (Shore)

[47], [55], [38], [19], [45], [20], [57], [42],

[4], [21], [58], [22], [36], [49], [35], [60],

[12], [110], [63], [50], [111], [23], [31],

[112], [39], [6], [64], [77], [98], [93], [25],

[90], [107], [102], [105], [103], [7], [121],

[26], [81], [51], [52], [115], [15], [108],

[85], [92], [118], [16], [82], [45], [83], [8],

[99], [86], [79], [53], [33], [69], [70], [17],

[75], [44], [29], [40], [66], [28], [95],

[120]

[24], [66]

Voyage Cost

(Sea)

[46], [54], [30], [47], [2], [55], [38], [3],

[19], [45], [20], [57], [32], [76], [4], [21],

[22], [36], [49], [59], [35], [60], [37],

[110], [5], [61], [50], [111], [31], [39], [6],

[64], [77], [34], [98], [89], [43], [65], [24],

[25], [90], [107], [113], [102], [105], [78],

[103], [14], [7], [13], [121], [26], [11],

[81], [52]¸ [15], [108], [85], [68], [92],

[118], [94], [16], [82], [45], [8], [99], [86],

[79], [27], [53], [69], [70], [17], [75], [44],

[29], [40], [66], [71], [87], [85], [18],

[109], [120], [96], [74], [91]

[97], [23], [90],

[91]

Hub

Location

(Port)

[20], [76], [4], [21], [62], [111], [31], [10],

[24], [117], [113], [8], [84], [84]

Ship's

Parameter

[46], [54], [30], [100], [2], [55], [3], [19],

[45], [104], [57], [32], [42], [21], [58],

[36], [49], [59], [35], [60], [12], [37],

[110], [61], [63], [50], [23], [31], [112],

[39], [6], [64], [77], [34], [10], [98], [89],

[43], [65], [93], [93], [90], [91], [107],

[117], [113], [102], [105], [78], [103],

[14], [7], [13], [121], [114], [26], [11],

[81], [51], [52], [15], [92], [118], [94],

[16], [82], [45], [83], [8], [99], [86], [79],

[27], [53], [33], [69], [84], [70], [17], [75],

[88], [44], [119], [40], [66], [106], [95],

[73], [74]

[90], [91]

Port's

Parameter

[46], [30], [100], [2], [38], [3], [4], [22],

[59], [37], [61], [62], [77], [25], [25], [83],

[27], [17], [119], [109]

[55]

Time

Parameter

[30], [100], [2], [3], [19], [45], [58], [22],

[49], [12], [63], [31], [112], [39], [64],

[89], [43], [93], [90], [121], [11], [81],

[51], [115], [15], [108], [68], [94], [82],

[45], [99], [84], [70], [88], [29], [40], [85],

[120], [91]

[55], [41], [48],

[6]

Fixed Cost

[101], [56], [4], [36], [49], [35], [5], [61],

[63], [10], [25], [7], [13], [114], [11], [81],

[92], [82], [99], [84], [73]

Route

[101], [57], [58], [22], [49], [35], [60],

[12], [37], [110], [5], [61], [62], [31], [14],

[118], [84], [119], [29], [18]

[50], [24], [13]

Weather

[104], [13], [29]

Port Time

[4], [61], [50], [23], [112], [39], [6], [34],

[10], [91], [113], [102], [78]

[104], [63]

Carbon

Emission

[32], [42], [4], [59], [60], [5], [63], [91],

[105], [108], [91]

[32], [91]

Recovery

Time

[32]

Choice

Inertia

(Shipper

Preference)

[59]

[63]

Penalty

[118]

[23]

The dominance of deterministic models in liner

shipping network optimization research, accounting

for approximately three-quarters of all studies, is

primarily attributed to data availability, formulation

simplicity, and computational efficiency. In maritime

research, most operational parameters, such as sailing

schedules, vessel capacities, and fuel costs, are

typically assumed to be constant or can be estimated

using average values. These deterministic assumptions

enable researchers to construct simpler and more stable

mathematical models that can be effectively solved

using classical optimization techniques, such as mixed-

integer linear programming (MILP). Moreover, the

limited availability of stochastic or probabilistic data,

such as daily cargo demand variations, weather

disruptions, or fuel price fluctuations, makes it difficult

to accurately implement stochastic approaches without

sufficient historical data support.

From a computational standpoint, deterministic

models are also considerably more efficient, producing

optimal results that can be validated through

sensitivity analyses without the need for repeated

simulation runs. This makes the deterministic

approach particularly appealing for preliminary

studies focusing on basic network structures, service

design, or strategic policy evaluation. Conversely,

stochastic models require greater computational time

and algorithmic complexity, as they must account for

multiple possible realizations of uncertain parameters.

Therefore, the predominance of deterministic

modeling reflects researchers' tendency to balance

system realism with analytical tractability, before

advancing toward more realistic yet technically

demanding stochastic approaches.

3.5 Optimization Model, Method and Software

Most studies in the field of liner shipping network

design optimization employ linear models (do not

encompass nonlinearity). These linearity assumption is

used particularly to formulate problems characterized

by proportional relationships between decision

variables, costs, and operational constraints. Linear

models assume that any change in a decision variable

such as fleet size, cargo volume, or route configuration

results in a proportional change in the objective

function, for instance in terms of total cost, voyage

time, or service level. The most commonly used model

types are mixed-integer linear programming (MILP)

and linear programming (LP), which are widely

applied to problems involving network design, fleet

deployment, and service scheduling. The inclusion of

integer and binary variables helps in modelling

complex relationships among variables and represents

the complex constraints.

Linear optimization offers several advantages,

including computational efficiency and ease of

interpretation, making it particularly suitable for large-

scale problems with complex decision structures.

Several studies also combine linear optimization with

decomposition techniques, such as Benders

decomposition and column generation, to further

enhance computational performance. From a software

438

perspective, linear models are typically implemented

using solvers such as CPLEX, GUROBI, and AIMMS,

which support both integer and mixed-integer

formulations.

Table 6. Linear Optimization

Type of Model

Optimization Approach

Software

Reference

Linear

Two Stage

Exact

Branch and Bound

Column Generation

[107],

Branch and Price

GUROBI

[112], [121]

Mixed Integer Linear

Programming

CPLEX

[34]

Benders Algorithm

[56], [113]

Branch and Bound

[43]

Karush Kuhn Tucker

[65]

Branch and Cut

[102], [40]

Cutting Plane

[111]

Integer Programming

GUROBI, CPLEX

[7], [26], [85], [92], [99], [28]

CPLEX

[43], [114], [51], [52], [15], [17],

[119], [95], [109]

Multi-

objective

Metaheuristic

Clarke & Wright

Greedy Randomized

Variable

Neighbourhood

Descent

CPLEX

[117]

The Grey Wolf

Optimizer Algorithm

Non-Dominated

Sorting

Genetic

Algorithm II

MATLAB

[41], [91]

Genetic Algorithm

LINGO 11

[11]

Simulated Annealing

[44]

Bi Level

Mixed Integer Linear

Programming

(MILP)

Integrated

Repositioning Sub

Model

MATLAB

[57]

Genetic Algorithm

CPLEX, XPRESS

VERSION,

MATLAB R15,

GUROBI

[54], [30], [47], [59], [62], [23], [39],

[91], [13], [68]

Mix Vehicle Routing

Problem (FSMVRP)

[38]

Adaptive

Neighbourhood Search

(ANS)

[79]

Multinomial Logit

(MNL)

[58]

Heuristic

[76], [49], [35], [110], [61], [50], [31],

[10], [65], [103], [14], [118], [94],

[82], [45], [8], [86], [27], [53], [70],

[88], [66], [71], [73], [116]

Variable

Neighbourhood

Decomposition

(VNDS)

[111], [98], [90]

Column Generation

[34], [33]

Expanding Horizon

Heuristic

GUROBI

[100]

Multi-Objective Robust

Model

[32]

Lagrangian

Decomposition

CPLEX

[84], [72]

Cooperative Game

Theory

[87]

Benders

Decomposition

CPLEX

[106]

Greedy Algorithm

LINDO

[12]

Branch and Bound

MATLAB

[77]

Robust Optimization

Minimax Regret

Approach

AIMMS

[97]

Benders Algorithm

Primal Dual &

Multiple Cut

GUROBI

[6]

Column Generation

Tabu Search

CPLEX

[96]

Branch and Price

[36]

Genetic Algorithm

[24]

Branch and Bound

MATLAB

[74]

Overall, the dominance of linear optimization and

mixed-integer approaches in the literature reflects a

prevailing tendency among researchers to prioritize

model transparency and computational efficiency,

particularly during the early stages of network design

formulation. Linear models are capable of providing

optimal solutions under deterministic assumptions

while maintaining a stable and mathematically

tractable structure. However, this approach has

inherent limitations in representing nonlinear

operational phenomena, such as the relationship

between vessel speed and fuel consumption, or voyage

time variability under environmental conditions.

In line with this, studies employing nonlinear

optimization approaches within the context of liner

network design demonstrate considerable variation in

both model types and solution techniques. Nonlinear

439

models are adopted when the relationship between

decision variables and objective functions cannot be

adequately expressed through linear formulations, for

example, the exponential or cubic relationship between

vessel speed and fuel consumption. An increase in

vessel speed by 10%, for instance, can lead to a 30–35%

rise in fuel consumption. Similar nonlinearities also

emerge in the relationship between speed and carbon

emission costs, as well as in inter-route interactions

that influence energy efficiency and total operational

cost. Therefore, the application of nonlinear

optimization becomes essential to capture the realistic

dynamics of modern liner shipping systems, which are

increasingly complex and sustainability-oriented.

Table 7. Nonlinear Optimization

Type of Model

Optimization Approach

Softwar

Referen

Nonline

Hybrid

Algorith

Mixed

Integer

Nonlinear

Programmi

MATLA

[3], [42]

CPLEX

[45],

[55], [4],

[22]

Multi-

objective

Stochastic

Programmi

ng Models

Conditional

Value at

Risk (Cvar)

[48]

Bi Level

User

Equilibrium

with

Variational

Inequality

Network

Design

Investment

Algorithm

CONOP

[37]

Mathematic

al Program

with

Equilibriu

Constraint

(MPEC)

[5]

Heurist

Genetic

Algorithm

Frank

Wolfe

Algorithm

MATLA

[46],

[63]

Adaptive

Heuristic

Genetic

Algorithm

CPLEX

[101]

Mixed

Integer

Nonlinear

Programmi

MATLA

[75]

Column

Generation

[69]

Knapsack

Problem

[18]

Greedy

Algorithm

[72]

Brute Force

Enumeratio

[105]

Fuzzy AHP

Ant Colony

[108],

[81],

Continuous

Optimizatio

[120]

Exact

Augmented

Lagrangian

MATLA

[115]

Branch and

Cut

CPLEX

[93]

The optimization approaches adopted in nonlinear

problems vary widely, ranging from mathematical

programming techniques, such as nonlinear

programming (NLP) and mixed-integer nonlinear

programming (MINLP), to metaheuristic methods,

including the genetic algorithm (GA). The use of

metaheuristic approaches is particularly prevalent in

studies addressing large-scale problems with

numerous interdependent decision variables and

constraints. In terms of computational tools, nonlinear

models are commonly solved using software such as

CPLEX, MATLAB, or CONOPT, which are capable of

handling nonlinear objective functions and multi-

objective optimization tasks.

In general, these findings indicate that the

application of nonlinear models in liner shipping

network research remains relatively limited compared

with linear models, primarily due to their higher

mathematical complexity and the need for more

detailed datasets. Nevertheless, nonlinear approaches

tend to produce more realistic and representative

results of actual operational conditions, particularly in

optimizing vessel speed, fuel consumption, and carbon

emissions under dynamic environmental settings.

Recent research trends also reveal a growing interest in

integrating nonlinear models with robust or stochastic

optimization techniques to develop adaptive solutions

that can respond to operational uncertainties and

evolving global environmental policies.

In line with this development, an emerging trend

involves the formulation of hybrid linear–nonlinear

and multi-objective optimization models, which

combine the computational efficiency of linear

approaches with the representational accuracy of

nonlinear formulations. This hybrid approach is

expected to bridge the gap between formulation

simplicity and system realism, thereby enabling the

generation of more accurate, flexible, and applicable

solutions for planning and operating increasingly

complex liner shipping networks.

Table 8. Mix Optimization

Type of Model

Optimization Approach

Softwar

Referen

Line

ar +

Non

Line

Two-Step

Optimizati

Metaheuris

tic

Mixed

Integer

Linear

Programmi

Nonlinear

Programmi

CPLEX,

GURO

BI,

MATL

[2], [20],

[21]

Benders

Decomposit

ion

[89],

[85]

Inverse

Optimizatio

[60],

[83]

Bi Level

Heuristic

Fuzzy Bi

Objective

Fuzzy

Reformulati

CPLEX

[104]

Two Stage

Mixed

Integer

Programmi

Convex

Optimizatio

GURO

[64],

[29]

Genetic

Algorithm

[16]

Mixed

Integer

Nonlinear

Programmi

Knapsack

Problem

[9]

Gravity

Model

Carrier

Profit +

Welfare

Maximizati

MATL

[25]

Research employing hybrid optimization models in

liner shipping network studies has emerged as a

response to the inherent limitations of single-method

approaches, whether purely linear or nonlinear in

representing the complexity of modern maritime

systems. In general, hybrid models combine the

strengths of two or more techniques, such as

integrating linear mathematical structures to ensure

computational efficiency with nonlinear elements that

enhance the realism of results. Several studies also

merge deterministic and stochastic optimization

approaches to better capture demand uncertainty and

operational disruptions, or integrate Mixed-Integer

440

Linear Programming (MILP) with Nonlinear

Programming (NLP) to accelerate solution

convergence in large-scale problems.

From an implementation perspective, software

packages such as GUROBI, CPLEX, and MATLAB are

commonly used due to their flexibility in solving multi-

objective and multi-method optimization problems.

The mixed optimization approach effectively bridges

the gap between solution accuracy and computational

efficiency, producing results that are more adaptive to

dynamic and uncertain operational environments.

Consequently, recent research trends reveal a growing

interest in hybrid models as an evolutionary step

beyond purely linear or nonlinear methods,

particularly in addressing the challenges of achieving

efficiency, sustainability, and resilience within global

liner shipping networks.

3.6 Optimization Model

The optimization model in liner shipping network

design (LSND) is typically constructed with three key

components: objective function, decision variables, and

constraints.

The objective function serves as the core component

that defines the direction and purpose of the entire

Model. It specifies the desired outcomes of the

optimization system, such as minimizing total

operating costs, reducing emissions, or maximizing

service quality. The majority of existing research

continues to focus on cost minimization, particularly in

relation to fuel costs, vessel operating expenses, and

port charges. This focus reflects the characteristics of

deterministic base models, which typically assume

stable market and environmental conditions.

Over time, the focus on optimization has evolved to

incorporate multi-dimensional objectives, including

additional variables such as service level, delivery

time, and energy efficiency. This stage marks a

conceptual shift from a cost-centered orientation to a

balanced trade-off between cost, time, and service

quality. Recent studies have further expanded this

scope by emphasizing sustainability and operational

resilience, integrating objective functions that account

for carbon emissions, disruption risks, and long-term

profitability.

Table 9. Objective Functions

Objective

Functions

Step 1

Step 2

Step

Minimize

Total Fixed

Cost

[30], [97], [100], [55], [38], [41],

[48], [3], [104], [112], [6], [64],

[34], [10], [43], [24], [90], [102],

[78], [103], [114], [11], [81], [92],

[118], [94], [45], [83], [8], [79],

[33], [70], [88], [119], [40], [28],

[109], [120], [74], [91]

[20], [39], [65],

[117], [108], [118],

[94], [8], [79], [33],

[44], [40], [106],

[109]

[9]

Minimize

Total

Variable

Cost

[46], [30], [97], [100], [55], [41],

[48], [3], [19], [101], [20], [104],

[57], [32], [76], [21], [58], [35],

[12], [37], [110], [5], [50], [111],

[23], [31], [112], [6], [64], [77],

[34], [10], [89], [43], [24], [90],

[107], [102], [105], [78], [103],

[121], [114], [26], [11], [81], [15],

[92], [118], [94], [16], [82], [45],

[83], [8], [99], [79], [33], [84], [70],

[17], [75], [88], [119], [29], [40],

[66], [71], [28], [95], [73], [109],

[120], [96], [74], [91]

[38], [20], [57],

[76], [4], [21], [35],

[37], [50], [39],

[10], [65], [90],

[107], [117], [105],

[26], [108], [92],

[118], [94], [16],

[8], [99], [79], [27],

[33], [84], [44],

[29], [40], [106],

[73], [109], [91]

[4],

[9]

Minimize

Total

Voyage

Distance

[117]

Minimize

Total

Carbon

Emissions

[57], [4]

[41], [32], [5], [74]

Minimize

Risk

Function

[48]

Maximize

Total Profit

[47], [2], [42], [49], [60], [61], [62],

[63], [98], [65], [93], [91], [14], [7],

[51], [52], [115], [68], [86], [27],

[53], [69], [44], [87], [106], [9],

[18], [116]

[54], [97], [2], [36],

[60], [62], [6], [64],

[98], [25], [113],

[78], [13], [121],

[51], [52], [85],

[83], [9]

[25]

Maximize

Total

Revenue

[54], [45], [36], [59], [39], [13], [85]

[46], [59], [68],

[18], [96], [116]

Maximize

Reduced

Cost

[69]

Maximize

Total Load

(cargo)

[72]

[93], [72]

Maximize

Service

Frequency

[56], [22], [113]

[100], [56], [66]

[56]

Minimize

Lateness

[104]

Maximum

Shipper

Utility

[58], [87]

Minimize

Hub Nodes

[111]

Consistent with the formulation of the objective

function that defines the direction of optimization, the

next critical component in liner network design models

is the decision variable, representing the controllable

parameters used to achieve the desired optimization

goals. As illustrated in the table above, the

development of decision variables in the literature has

also evolved through three primary stages.

Initially, decision variables were primarily focused

on network design and basic resource allocation,

including route selection, fleet size and deployment,

cargo flow, and service frequency. These variables

directly contribute to minimizing operational costs,

aligning with the early-stage objective of cost-based

optimization. Over time, the research focus has shifted

toward operational adjustment and performance

efficiency, encompassing variables such as sailing

speed, service scheduling, and container or cargo

allocation. This stage supports the transition toward

multi-dimensional optimization, emphasizing a

balance between cost, time, and service level.

Table 10. Decision Variables

Decision

Variables

Step 1

Step 2

Step

Cargo Flow

[30], [47], [100], [2], [55], [48],

[3], [19], [45], [101], [56], [20],

[32], [76], [22], [49], [60], [12],

[37], [110], [5], [62], [63], [50],

[112], [77], [34], [10], [98], [89],

[43], [65], [93], [25], [91], [103],

[14], [7], [13], [114], [26], [81],

[51], [15], [85], [92], [94], [82],

[45], [83], [8], [79], [27], [53],

[84], [70], [75], [44], [119], [29],

[40], [71], [28], [87], [72], [106],

[95], [120], [116], [91]

[46], [54], [38],

[56], [56], [20],

[57], [21], [36],

[59], [50], [39], [6],

[64], [107], [113],

[105], [121], [51],

[85], [68], [92],

[83], [99], [33],

[84], [40], [66],

[87], [72], [9], [73],

[18], [96], [116]

[76]

Vessel

Routing

[46], [54], [30], [47], [45], [101],

[104], [57], [32], [42], [58], [59],

[37], [63], [50], [111], [31], [112],

[39], [64], [34], [10], [98], [43],

[60], [111], [6],

[93], [108], [85],

[8], [79], [69], [40],

[18], [96], [74]

441

[65], [93], [24], [91], [113], [102],

[14], [121], [114], [11], [81],

[115], [68], [94], [16], [82], [83],

[84], [70], [17], [75], [88], [119],

[40], [66], [71], [87], [9], [95],

[73], [18], [109], [91]

Hub

Location

[30], [56], [76], [21], [35], [28],

[72]

[56], [56], [76],

[35], [72]

Port

Selection

[47], [20], [4], [36], [110], [61],

[62], [6], [98], [105], [115], [118],

[82], [86]

[100], [2], [48],

[56], [20], [50],

[52], [16], [79]

[25]

Fleet Size

[47], [38], [101], [20], [21], [50],

[31], [15], [68], [83], [79], [53],

[75], [120]

[97], [50], [112],

[25], [26], [92],

[118], [94], [83],

[79], [9], [18]

Vessel

Assignment

[30], [2], [55], [48], [3], [45], [57],

[42], [36], [49], [59], [35], [63],

[23], [31], [34], [98], [90], [117],

[92], [118], [33], [69], [75], [44],

[40], [96]

[100], [38], [35],

[62], [98], [107],

[117], [40], [9]

Vessel Speed

[47], [2], [48], [3], [45], [104],

[57], [42], [59], [60], [63], [34],

[10], [91]

[41], [32], [64],

[93], [25], [90],

[85], [18], [74]

Number of

Vessels

[41], [20], [42], [21], [58], [110],

[77], [98], [90], [91], [107], [78],

[13], [121], [52], [15], [68], [92],

[16], [82], [45], [83], [99], [86],

[79], [27], [69], [84], [70], [75],

[88], [106], [73]

[20], [4], [62],

[107], [78], [13],

[84], [87], [74]

Sailing

Frequency

[47], [97], [45], [12], [61], [31],

[43], [121], [15], [99], [75], [40],

[66], [73], [120]

[38], [48], [21],

[62], [112], [90],

[11], [108], [118],

[16], [8], [29]

[76]

Delay Time

[69]

[104], [10], [91]

[9]

Choice

Inertia

[39]

[58]

Emission

[32]

Fixed Cost

(Investment)

[37]

The last component of the liner shipping network

design model involves the constraints, which represent

a set of conditions ensuring that the Model's solutions

remain realistic, feasible, and operationally applicable

within maritime transport systems. Most constraints

used in the literature are deterministic, encompassing

vessel capacity limits, port handling capacities, cargo

flow balance, and fleet or route availability. These

constraints ensure that optimization outcomes do not

exceed the existing system's capabilities, thereby

maintaining the representativeness and practicality of

the Model under real-world operational conditions.

Table 11. Constraints

Constraints

Step 1

Step 2

Step

Demand

[100], [38], [56], [57], [32], [76],

[21], [60], [63], [23], [10], [65],

[91], [78], [103], [14], [13], [11],

[81], [108], [94], [82], [45], [83],

[8], [27], [53], [33], [17], [29], [71],

[87], [95], [109], [96]

[46], [54], [97],

[56], [76], [21],

[58], [62],

[113], [16], [69]

[76]

Cargo Flow

[54], [30], [47], [100], [2], [19],

[101], [20], [42], [76], [21], [58],

[36], [49], [35], [60], [12], [37],

[37], [110], [5], [62], [63], [50],

[31], [112], [64], [77], [34], [10],

[98], [95][89], [43], [93], [24], [25],

[90], [91], [107], [113], [103], [14],

[7], [121], [114], [81], [51], [115],

[15], [85], [92], [94], [82], [45],

[83], [79], [27], [53], [33], [69],

[84], [70], [17], [75], [88], ,[87],

[44], [119], [40], [71], [28], [72],

[106], [9], [120], [96], [116], [91]

[46], [54], [2],

[38], [57], [21],

[58], [36], [59],

[62], [50],

[112], [6], [64],

[93], [25],

[107], [105],

[51], [85], [68],

[94], [16], [99],

[119], [66],

[72], [9], [73],

[18], [116], [74]

[76],

[9]

Voyage Cost

(Sea)

[46], [59], [39] , [68], [120]

[20], [59], [60],

[5], [25], [8]

[56]

Operational

Cost (Shore)

[46], [4], [59], [39], [39], [120]

[20], [60], [5],

[8]

[56]

Ship

Assignment

[30], [48], [20], [21], [49], [60],

[62], [50], [31], [112], [10], [43],

[90], [91], [26], [11], [92], [83],

[99], [86], [66], [106], [109]

[100], [20],

[112], [117],

[92], [87], [109]

Port

Assignment

[30], [48], [56], [20], [76], [45], [4],

[22], [36], [35], [37], [111], [23],

[64], [77], [93], [24], [91], [117],

[102], [105], [78], [114], [16], [27],

[33], [28], [72], [106], [73]

[56], [35],

[111], [107],

[117], [72]

[76],

[25]

Service

Frequency

[46], [30], [41], [3], [45], [104],

[32], [58], [36], [49], [61], [98],

[89], [43], [91], [52], [15], [16],

[82], [83], [99], [86], [27], [69],

[75], [119], [95], [73]

[56], [104], [4],

[21], [112],

[90], [107],

[108], [79],

[44], [119],

[74], [91]

[56]

Fuel

Consumption

[55]

[2], [41], [4]

Vessel Speed

[55], [41], [48], [42], [60], [63], [91]

[64], [25], [90],

[78]

Number of

Vessels

[47], [55], [45], [20], [58], [110],

[61], [39], [77], [98], [93], [102],

[78], [103], [17], [66], [87], [96]

[38], [20], [36],

[107], [121]

Vessel's

Capacity

[97], [2], [38], [48], [3], [19], [45],

[101], [57], [32], [42], [21], [58],

[12], [63], [50], [31], [77], [98],

[89], [43], [65], [91], [117], [102],

[103], [7], [13], [121], [51], [15],

[85], [118], [94], [82], [45], [83],

[8], [79], [27], [53], [69], [84], [70],

[17], [75], [88], [44], [29], [40],

[71], [72], [106], [95], [18], [96],

[116], [91]

[100], [57],

[21], [62], [50],

[39], [6], [64],

[25], [117],

[113], [78],

[13], [26], [85],

[92], [118],

[16], [99], [33],

[84], [66], [72],

[9], [73], [18]

[56]

Handling

Time

[46], [30], [47], [55], [3], [45],

[104], [31], [77]

[2], [48], [118]

Voyage

Duration

[46], [30], [47], [55], [45], [104],

[49], [61], [31], [39], [10], [11],

[79], [33], [66]

[48], [6], [64],

[108], [118],

[29], [73], [96]

Transit Time

[30], [47], [113], [78], [81], [92],

[82]

[41], [20], [78],

[51], [8], [99],

[40], [91]

Vessel

Compatibility

[16]

[2]

Disruption

Risk

[3]

[32]

Emission

[3], [104], [74]

[32]

Route

[56], [57], [22], [5], [63], [111],

[23], [6], [64], [34], [10], [65], [90],

[113], [105], [7], [94],[13], [114],

[52], [115], [15], [92], [82], [69],

[84], [75], [88], [44], [95], [120]

[35], [50],

[111], [39], [6],

[64], [93],

[113], [68],

[69], [29], [9],

[18], [74]

[76]

Choice Inertia

[59], [68]

Fixed Cost

(Investment)

[5]

[37]

Time

Windows

[89]

[10]

Subsequent developments in liner network design

research have expanded the formulation of constraints

to incorporate temporal and operational factors, such

as sailing time limits, service frequency schedules, and

fuel consumption restrictions. More advanced and

adaptive constraints have also been introduced,

accounting for choice inertia and environmental

considerations, including carbon emission constraints.

These additions reflect the ongoing evolution of liner

network optimization models toward more dynamic,

sustainability-oriented, and behaviourally realistic

frameworks that better capture the complexities of

modern maritime operations.

4 DISCUSSION

The evolution of Liner Shipping Network Design

(LSND) research has demonstrated a significant

methodological and thematic transformation, driven

by evolving global supply chain demands and

442

technological advancements in optimization. This

discussion synthesizes the shift across three distinct

periods, connecting specific challenges and

optimization strategies based on the analytical

framework. During the 2000–2010 period, studies

primarily focused on developing deterministic models

with relatively simple mathematical structures,

predominantly employing Integer Programming (IP)

and Mixed-Integer Programming (MIP) approaches.

The main objective at this stage was to identify optimal

network configurations, including hub port selection

and the design of hub-and-spoke routes aimed at

minimizing operational costs. Several studies also

began to introduce dynamic models that accounted for

fluctuations in demand and freight rates, while

integrating scheduling and network design within a

unified optimization framework. Furthermore, game

theory and inverse optimization approaches were

applied to balance capacity utilization and resource

allocation among competing liner operators. Although

the research scope during this period remained micro-

level and deterministic, it laid the foundational

structure for the development of modern LSND

models.

Entering the 2011–2020 period, LSND research

began to shift toward system complexity and multi-

dimensional integration. The research focus expanded

beyond cost efficiency to encompass the integration of

routing, scheduling, and sailing speed within dynamic

operational planning. The adoption of multi-objective

models (MILP) and game-theoretic approaches grew

substantially, reflecting efforts to balance economic,

technical, and cooperative goals among shipping

alliances. Moreover, sustainability considerations

emerged through the concept of Green and Sustainable

Network Design, incorporating fuel consumption

factors, Emission Control Area (ECA) policies, and

carbon emission reduction into optimization models

[42]. Other significant trends included the integration

of intermodal transport, empty container

repositioning, and the development of hybrid and

metaheuristic models that combine exact and

metaheuristic methods to address large-scale and

complex optimization problems. During this period,

research also expanded regionally and cross-

continentally (e.g., Norway, China–Europe,

Indonesia), indicating broader empirical coverage and

enhanced policy relevance.

During the 2021–2025 phase, LSND research has

entered a strategic transformation stage, characterized

by a strong focus on resilience, sustainability, and

geopolitical adaptation. Emerging issues such as Asia–

Europe route realignments driven by geopolitical

conflicts (Suez vs. Cape vs. Arctic routes) [47], as well

as ASEAN's RCEP (Regional Comprehensive

Economic Partnership) policies and carbon taxation

[47,32,22], have stimulated the development of more

complex and adaptive models. Robust and stochastic

optimization approaches have become increasingly

prominent, often combined with hybrid metaheuristics

to handle demand uncertainty, port congestion, and

operational disruptions [3]. The focus on

decarbonization and green shipping has intensified

through the inclusion of emission caps, LNG

bunkering station planning, and fuel efficiency

strategies [4]. Recent studies also highlight multi-actor

interactions within liner shipping alliances by

integrating profit-sharing mechanisms, choice inertia,

and service reliability into decision frameworks

[58,59,68]. The integration of sea, land, and inland

waterway networks has been strengthened through

multimodal optimization approaches, supporting

global supply chain connectivity and resilience [13,15].

Overall, this period reflects a paradigm shift in LSND

research from static deterministic models toward

adaptive, sustainable, and resilient systems that not

only optimize cost efficiency but also address

environmental challenges and dynamic market

conditions in global maritime logistics.

5 FUTURE RESEARCH AGENDA

In addressing the global challenges outlined in Section

4, discussion, including geopolitical uncertainty,

decarbonization regulations, and digital

transformation within the maritime sector, Liner

Shipping Network Design (LSND) research stands at a

critical juncture, poised for a paradigm shift. Future

studies must move beyond the traditional cost

efficiency orientation and advance toward a more

holistic optimization framework that integrates

sustainability, system resilience, and multimodal

connectivity. Accordingly, this section presents a

future research agenda designed to strengthen both the

scientific relevance and the strategic contribution of

LSND to the sustainability and connectivity of the

global maritime supply chain.

As LSND research increasingly evolves toward

sustainability, resilience, and global complexity, future

directions must be formulated in a more

comprehensive and interdisciplinary manner. The

growing intricacy of modern shipping systems

demands multi-dimensional approaches that extend

beyond technical and economic considerations to

encompass data uncertainty, actor behaviour,

environmental policy, and methodological innovation.

In light of these dynamics, the proposed future

research framework for LSND can be categorized into

five major domains, as illustrated in Figure 4.

Figure 4. Domain Future Research Agenda

443

Future research directions in Liner Shipping

Network Design (LSND) can be categorized into five

major domains: data and uncertainty, environmental

and policy, network and operations, methodology, and

behavioural and strategic.

The evolution of LSND research demonstrates that

while various conceptual and methodological

dimensions have been extensively examined,

substantial opportunities remain for further

exploration. Within the data and uncertainty domain,

most existing studies still rely on deterministic

approaches, assuming fixed parameters such as cargo

demand and sailing time. Only a limited number of

studies have integrated uncertainty into their models,

for instance, through stochastic or basic sensitivity

analyses. Therefore, future research should expand the

application of stochastic programming, probabilistic

demand modeling, and robust optimization to capture

better dynamic variations in demand, fuel prices,

weather disruptions, and port congestion. This

transition will enhance the resilience and adaptability

of liner shipping networks in the face of operational

uncertainty.

In the environmental and policy domain, previous

studies have begun incorporating emission and energy

efficiency components into network optimization

models, particularly following the implementation of

Emission Control Areas (ECA) and carbon tax policies.

However, most analyses remain limited to emission

cost calculations and have yet to fully integrate more

complex decarbonization frameworks such as

Emission Trading Schemes (ETS), carbon quotas, or the

adoption of alternative fuels like LNG and biofuels.

Hence, future research should emphasize the dynamic

interplay between network design, carbon incentive

policies, and investment analysis in green maritime

technologies within a sustainable policy framework.

For the network and operations dimension, existing

studies have explored mainline and feeder network

optimization, as well as the integration of scheduling

and fleet deployment. Nevertheless, most models

remain static and focus on single-route configurations

without addressing operational adaptability or

network complexity. In this regard, butterfly and

asymmetric service patterns remain underexplored

despite their high potential for addressing cargo and

demand imbalance across regions. The butterfly

pattern offers flexibility to serve dual destinations from

a central hub, while the asymmetric pattern reflects the

global trade imbalance between outbound and

inbound routes. Future studies should therefore

advance toward dynamic and cooperative network

optimization, integrating these complex service

structures with adaptive container repositioning,

multi-operator collaboration, and post-disruption

recovery strategies. Furthermore, the integration of

inland and river transport within multimodal

transport systems represents an up-and-coming area

for strengthening maritime connectivity and

enhancing supply chain sustainability.

From a methodological perspective, most LSND

studies continue to rely on Mixed-Integer Linear

Programming (MILP) for its computational efficiency.

Although the use of metaheuristics, such as Genetic

Algorithms and the Non-Dominated Sorting Genetic

Algorithm (NSGA-II), is increasing, hybrid

optimization models that combine exact and heuristic

approaches remain limited. Future research should

integrate machine learning and real-time dynamic

scheduling to improve predictive capability and

computational efficiency for large-scale systems under

high uncertainty.

Finally, the behavioural and strategic domain

remains relatively new and underdeveloped in LSND

research. Existing studies tend to focus on technical

aspects, while behavioural factors, such as shipper

choice inertia, inter-operator interactions, and alliance

collaboration dynamics, are rarely incorporated into

optimization frameworks. Future research should

expand toward behaviour-based and market-oriented

modeling, integrating customer preference analysis,

dynamic pricing strategies, and trade-offs between

customer satisfaction, profitability, and carbon

reduction.

Overall, these findings suggest that although LSND

research has reached methodological maturity in

deterministic and technical aspects, future

opportunities lie in integrating uncertainty,

sustainability, multi-actor collaboration, and human

behaviour into an adaptive framework capable of

addressing the challenges of decarbonization and

global disruption in the maritime sector.

6 CONCLUSION

First, this study successfully identified and classified

the main challenges in Liner Shipping Network Design

(LSND) across three decision levels: strategic, tactical,

and operational, and linked them with problem areas

(shore-based and sea-based) and service patterns

(simple, pendulum, butterfly, complex, and

asymmetric). The analysis reveals that strategic-level

issues, particularly network design and fleet

deployment, dominate the literature due to their direct

impact on cost structures and global logistics

efficiency. Meanwhile, sea-based problems, especially

those related to routing and navigational constraints,

emerge as the central operational challenges. Among

various service configurations, the pendulum service

pattern remains the most widely implemented,

particularly on Asia–Europe routes, owing to its

balanced and stable cargo flows.

Second, this study classified the optimization

approaches employed in LSND research and found

that linear optimization models still dominate due to

their computational efficiency. However, recent trends

indicate a methodological shift toward nonlinear,

stochastic, and hybrid optimization models that more

accurately reflect real-world operational conditions

such as demand uncertainty, voyage time variability,

and carbon emission regulations. This methodological

evolution marks a transition of LSND research from

deterministic modeling to more adaptive,

collaborative, and sustainability-oriented systems.

Third, this study developed a future research

agenda outlining the prospective directions of LSND

research across four major domains: data and

uncertainty, environmental and policy, network and

operations, and methodological and behavioural

approaches. This agenda emphasizes the importance of

developing models that incorporate dynamic

444

uncertainty, integrate decarbonization policies, and

promote collaborative strategies among operators to

create resilient and green shipping networks.

Contextually, the findings of this study hold

strategic relevance for the development of maritime

logistics policy and further research in regions

characterized by dispersed islands and asymmetric

cargo flows. The asymmetric service model identified

in this research provides an ideal framework for

designing efficient yet inclusive shipping systems

tailored to archipelagic geographies, particularly when

implemented through stochastic modeling to better

approximate real-world conditions. By integrating the

classification of LSND challenges, optimization

approaches, and the proposed future research agenda,

this study provides both conceptual and

methodological foundations to support the

formulation of adaptive and sustainable maritime

policy blueprints for the global supply chain.

ACKNOWLEDGMENTS

This work is supported by the Seed Funding Grant 2024,

funded by the Faculty of Engineering Universitas Indonesia,

No. NKB-3429/UN2.F4.D/PPM.00.00/2024.

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