# Uncertainties in Liner Shipping and Ship Schedule Recovery: A State-of-the-Art Review

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background

#### 1.2. Existing Research Gaps and Contributions of This Study

- ✓
- A comprehensive up-to-date review of the liner shipping literature is conducted with a specific emphasis on uncertainties in liner shipping operations and ship schedule recovery.
- ✓
- The collected studies are reviewed in a systematic way, capturing the main assumptions regarding sailing speed and port time modeling, objective(s) considered, key components of objective functions(s) considered, uncertain elements modeled, ship schedule recovery options modeled, solution approaches adopted, and certain specific considerations adopted.
- ✓
- A representative mathematical formulation is presented for the ship scheduling problem with uncertainties, which can be used by shipping lines to assess the impacts of uncertainties on liner shipping operations and design robust ship schedules. Moreover, the proposed mathematical formulation can serve as a foundation for future efforts that concentrate on uncertainties in liner shipping operations.
- ✓
- A set of representative mathematical formulations are presented for the ship schedule recovery problem with various recovery options (i.e., sailing speed adjustment, handling rate adjustment, port skipping, and port skipping with container diversion), which can be used by shipping lines to select the appropriate ship schedule recovery option(s). Furthermore, the proposed mathematical formulations can serve as a foundation for future efforts that concentrate on ship schedule recovery.
- ✓
- Research gaps in previous and contemporary studies on uncertainties in liner shipping operations and ship schedule recovery are clearly identified, and future research areas that should be considered in the following years are specifically underlined.

## 2. Literature Search

## 3. Uncertainties in Liner Shipping Operations

#### 3.1. Problem Description

#### 3.1.1. Liner Shipping Route and Ship Voyage

#### 3.1.2. Ship Service at Ports

#### 3.1.3. Fuel Consumption Estimation

#### 3.1.4. Port Service Frequency Determination

#### 3.1.5. Container Inventory Considerations

#### 3.2. Base Mathematical Model

**SSP-U**) can be formulated using the objective function (7) and constraints (8) through (26). Note that for a detailed description of all the notations used in the mathematical models presented in this manuscript, interested readers can refer to Appendix A that accompanies this manuscript. The bold notations are used for decision variables, auxiliary variables, and uncertain/stochastic parameters within the mathematical models, whereas the standard font is used for constant parameters. The proposed

**SSP-U**mathematical model assumes that the ship waiting and handling times are uncertain. The

**SSP-U**objective function (7) aims to maximize the total profit ($\mathit{\Pi}$—USD) that will be accumulated by the shipping line from the provided liner shipping service, which is estimated as a difference between the total revenue ($\mathit{R}$—USD) and the total cost associated with the service of the considered shipping route.

**SSP-U**model includes a total of four groups of constraint sets. The first constraint group, represented by constraints (8) through (11), estimates the sailing time of ships, taking into account the established speed bounds, and the consumption of fuel by the main engines of ships on voyage legs of the considered shipping route. In particular, constraints (8) and (9) assure that the sailing speed of ships remains within the established speed bounds on each voyage leg. Constraints (10) compute the sailing time of ships on each voyage leg based on the voyage leg length and the sailing speed of ships. On the other hand, the consumption of fuel by the main engines of ships on each voyage is computed by constraints (11) based on the sailing speed of ships, coefficients associated with the fuel consumption function, and ship payload.

**SSP-U**objective function (7).

#### 3.3. Review of the Relevant Studies

#### 3.4. Literature Summary and Research Gaps

#### 3.4.1. Summary of Findings

_{2}emissions. Song et al. [77] developed a multi-objective optimization model, where the first objective minimized the total cost of route service, whereas the second one aimed to maximize the schedule reliability.

#### 3.4.2. Limitations and Future Research Needs

- ➢
- More detailed and accurate historical data for liner shipping operations are required to model uncertainties associated with the main liner shipping processes and assess various mitigation strategies. The collected historical data can be further used in the development of statistical distributions for uncertain container demand, port time, and sailing time [68,81].
- ➢
- Future studies should concentrate on more detailed modeling of uncertain components in liner shipping operations [74]. For instance, the port time component can be disaggregated into various sub-components (e.g., port waiting time, handling time associated with offloading import containers, handling time associated with loading export containers). The effects of uncertainties can be further assessed for each sub-component.
- ➢
- A detailed evaluation of the existing studies on uncertainties in liner shipping operations indicates that many studies strictly concentrate on one source of uncertainties (i.e., uncertainty in demand or uncertainty in sailing time or uncertainty in port handling time or uncertainty in port waiting time). Holistic models that emulate multiples sources of uncertainties at the same time should be further explored by future studies.
- ➢
- Reliability of liner shipping services can be affected by a variety of factors [11,62], including geographical characteristics, the average age of deployed ships, previous maintenance activities of deployed ships, available handling resources of terminal operators and inland operations, and others. Future research should continue investigating the effects of these factors on liner shipping services and directly consider them for planning purposes.
- ➢
- Innovative policies should be explored to offset the effects of uncertainties in liner shipping operations. Dynamic decision-making policies can be promising [71]. As an example, delays due to uncertainties in sailing time could be mitigated by adjusting the ship sailing speed on consecutive voyage legs of a shipping route. However, the ship sailing speed adjustment decision should be made by taking into account other important operational factors (e.g., the number of remaining ports to be visited in a round voyage; increasing fuel cost due to speeding up the ships).
- ➢
- One of the common limitations in the existing liner shipping studies consists in the fact that the impacts of weather conditions are not directly accounted for when planning ship sailing speed decisions [76]. Future research efforts must focus on the development of models that directly capture the expected weather conditions on voyage legs when making ship sailing speed decisions.
- ➢
- Time-dependent port waiting and handling times should be taken into account in future studies. Waiting and handling times vary significantly at the ports depending on the day of the week and the time of day [80]. Ship arrivals should be planned for the time periods with a lower risk of congestion and port time delays.
- ➢
- Future studies should investigate various alternatives for mitigating the effects of significant delays during a ship voyage. In some instances, port skipping or partial loading/offloading of ships might be a promising decision in order to prevent the propagation of delays throughout the entire liner shipping network [81].
- ➢
- Strict port arrival TWs may not be feasible for shipping routes that often encounter uncertainties. Therefore, explicit modeling of soft arrival TWs, where ships are allowed to arrive outside the previously negotiated TW but penalized for TW violations, could be studied more as a part of future research [81].
- ➢
- It is known that fuel prices fluctuate regularly, and different ships need different types of fuel (e.g., ships sailing inside emission control areas must use low-sulfur fuel—[85,86,87,88,89]). Furthermore, ship sailing speed and fuel consumption are often subject to uncertainties [69]. Therefore, future studies should develop more comprehensive liner shipping operations planning models, which directly capture fuel price fluctuations, ship sailing speed uncertainty (and the associated fuel consumption uncertainty), fuel switching, and effective refueling policies.

## 4. Ship Schedule Recovery

#### 4.1. Problem Description

#### 4.2. Base Mathematical Models

#### 4.2.1. Sailing Speed Adjustment

**SSR-SSA**) can be formulated using the objective function (27) and constraints (28) through (48). The

**SSR-SSA**objective function (27) aims to minimize the total loss of profit that will be endured by the shipping line as a result of disruptions at the considered shipping route. The total profit loss is estimated as a difference between the total profit that was expected to be accumulated by the shipping line for the original ship schedule (${\mathit{\Pi}}^{0}$—USD) and the total profit that will be accumulated by the shipping line for the recovered schedule of ships ($\overline{\mathit{\Pi}}$—USD).

**SSR-SSA**model includes a total of three groups of constraint sets. The first constraint group, represented by constraints (28) through (32), estimates the recovered sailing time of ships, considering potential sailing speed adjustment to compensate for the effects of disruptions, and the recovered consumption of fuel by the main engines of ships on voyage legs of the considered shipping route. In particular, constraints (28) compute the recovered sailing speed of ships on each voyage leg, considering the expected change in sailing speed of ships due to a disruption and potential ship sailing speed adjustment. Note that constraints (28) assume that the ship sailing speed adjustment strategy could not be implemented on the voyage leg that experienced a disruption. Constraints (29) and (30) assure that the recovered sailing speed of ships remains within the established speed bounds on each voyage leg. Constraints (31) compute the recovered sailing time of ships on each voyage leg based on the voyage leg length and the recovered sailing speed of ships. Constraints (32) calculate the recovered consumption of fuel by the main engines of ships on each voyage based on the recovered sailing speed of ships, coefficients associated with the fuel consumption function, and ship payload.

**SSR-SSA**objective function (27), including the following: (1) total revenue that will be accumulated by the shipping line ($\overline{R}$—USD); (2) total cost associated with container handling at ports ($\overline{{{\rm K}}^{hand}}$—USD); (3) total cost associated with late ship arrivals ($\overline{{\mathit{{\rm K}}}^{late}}$—USD); (4) total cost associated with fuel consumption ($\overline{{\mathit{{\rm K}}}^{fuel}}$—USD); (5) total cost associated with basic ship operations ($\overline{{{\rm K}}^{oper}}$—USD); (6); total cost associated with chartering of ships ($\overline{{{\rm K}}^{char}}$—USD); (7) total cost associated with container inventory ($\overline{{\mathit{{\rm K}}}^{inv}}$—USD); and (8) total profit that will be accumulated by the shipping line ($\overline{\mathit{\Pi}}$—USD).

#### 4.2.2. Handling Rate Adjustment

**SSR-HRA**) can be formulated using the objective function (49) and constraints (29)–(34), (36)–(41), (43)–(47), and (50)–(54). Similar to the

**SSR-SSA**mathematical model, the

**SSR-HRA**objective function (49) aims to minimize the total loss of profit that will be endured by the shipping line as a result of disruptions at the considered shipping route. The total profit loss is estimated as a difference between the total profit that was expected to be accumulated by the shipping line for the original ship schedule (${\Pi}^{0}$—USD) and the total profit that will be accumulated by the shipping line for the recovered schedule of ships ($\overline{\mathit{\Pi}}$—USD).

#### 4.2.3. Port Skipping

**SSR-PS**) can be formulated using the objective function (55) and constraints (29)–(34), (36)–(40), (43)–(47), and (56)–(62). Similar to the

**SSR-SSA**mathematical model, the

**SSR-PS**objective function (55) aims to minimize the total loss of profit that will be endured by the shipping line as a result of disruptions at the considered shipping route. The total profit loss is estimated as a difference between the total profit that was expected to be accumulated by the shipping line for the original ship schedule (${\Pi}^{0}$—USD) and the total profit that will be accumulated by the shipping line for the recovered schedule of ships ($\overline{\mathit{\Pi}}$—USD).

#### 4.2.4. Port Skipping and Container Diversion

**SSR-PSCD**) can be formulated using the objective function (63) and constraints (29)–(34), (36)–(40), (43)–(47), (56)–(58), and (64)–(74). Similar to the

**SSR-SSA**mathematical model, the

**SSR-PSCD**objective function (63) aims to minimize the total loss of profit that will be endured by the shipping line as a result of disruptions on the considered shipping route. The total profit loss is estimated as a difference between the total profit that was expected to be accumulated by the shipping line for the original ship schedule (${\Pi}^{0}$—USD) and the total profit that will be accumulated by the shipping line for the recovered schedule of ships ($\overline{\mathit{\Pi}}$—USD).

#### 4.3. Review of the Relevant Studies

#### 4.4. Literature Summary and Research Gaps

#### 4.4.1. Summary of Findings

#### 4.4.2. Limitations and Future Research Needs

- ➢
- Ship schedule recovery is associated with conflicting decisions. In particular, sailing speed adjustment may allow for partially compensating delays during the voyage and maintaining adequate service levels for customers. However, such a recovery option will increase the amount of required fuel and fuel costs. There is a lack of multi-objective mathematical formulations for ship schedule recovery that are able to assist with the analysis of conflicting objectives [94,106,111]. Future research should focus more on multi-objective ship schedule recovery.
- ➢
- Future studies should concentrate on the development of innovative forecasting methods that could predict the occurrence of disruptive events and their duration [110]. The outcomes from these methods could be further used by shipping lines in the selection of the appropriate recovery strategies and offset the effects of disruptive events.
- ➢
- The shipping industry has been facing many challenges in recent years (e.g., COVID-19), and the cost of ship schedule recovery would add additional pressure on shipping lines. Risk-sharing mechanisms between carriers and shippers should be investigated further in future studies to alleviate the pressure on shipping lines and enable them to maintain a high level of customer service [110].
- ➢
- Sailing speed adjustment can serve as an effective recovery option but incurs additional fuel costs. The fuel consumption of ships depends on some other attributes as well, including previous maintenance activities, ship payload, ship age, and ship geometric characteristics [15,107]. Future research on ship schedule recovery should account for the aforementioned attributes and accurately quantify the amount of required fuel for the recovered ship schedules.
- ➢
- Decentralized decision-making with several shipping lines should be studied more in depth. A freight forwarder, for example, may arrange transshipment between the ships of two different shipping lines. These two shipping lines would coordinate their ship recovery schedules for transshipment in an ideal world. Nevertheless, since each shipping line must minimize its cost function, centralized and optimized scheduling would be difficult to execute in practice. Game-theoretic models for ship schedule recovery in decentralized settings would be suitable in such scenarios [103].
- ➢
- Sailing speed adjustment was identified as the most popular ship schedule recovery strategy. However, sailing speed adjustment alone may not be able to fully offset the effects of a disruptive event. Therefore, future studies should focus on the development of more advanced mathematical models and solution methods that consider a simultaneous implementation of various recovery strategies (e.g., sailing speed adjustment + port skipping or port swapping—[106]).
- ➢
- Certain extreme ship schedule recovery options (e.g., “cut-and-go” when a ship can leave a given port without completing its service) should be better explored by future research efforts to determine the scenarios when these options might be viable and reduce potential monetary losses due to disruptive events.
- ➢
- The effects of disruptions at ports and sea may influence not only shipping lines but other major supply chain players as well, including marine terminal operators, logistics companies, and inland operators [47]. Future mathematical models should evaluate various recovery strategies, considering the entire intermodal network effects—not just ship schedules.
- ➢
- Drones have been widely used for monitoring various assets, including the assessment of infrastructure damages as a result of disruptive events [115,116,117,118,119]. The deployment of drones for the assessment of disruptive events in liner shipping operations should be investigated as a part of future research. Drones can be used to accurately determine the effects of damages to the port infrastructure and the expected duration of port closures.

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Notations Adopted in the Proposed Mathematical Formulations

Set | Description of Sets | Remarks |
---|---|---|

$P=\left\{1,\dots ,{n}^{1}\right\}$ | set of ports for the considered shipping route (ports) | All models |

${H}_{p}=\left\{1,\dots ,{n}_{p}^{2}\right\},p\in P$ | set of handling rates that can be requested by the shipping line at port p (handling rates) | SSR-HRA |

**SSR-HRA**—ship schedule recovery with handling rate adjustment.

Decision Variable | Description of Decision Variables | Remarks |
---|---|---|

${\mathit{s}}_{p}\in {\mathbb{R}}^{+}\forall p\in P$ | sailing speed of ships on voyage leg $p$ (knots) | SSP-U |

${\mathit{q}}^{tot}\in \mathbb{N}$ | number of ships to be deployed (ships) | SSP-U |

${\mathit{\sigma}}_{p}^{sea}\in \mathbb{R}\forall p\in P$ | adjustment of ship sailing speed on voyage leg $p$ (knots) | SSR-SSA |

${\mathit{x}}_{ph}\in \mathbb{B}\forall p\in P,h\in {H}_{p}$ | =1 if handling rate h will be used for ship service at port p (otherwise = 0) | SSR-HRA |

${\mathit{x}}_{p}^{skip}\in \mathbb{B}\forall p\in P$ | =1 if port $p$ will be skipped by the shipping line (otherwise = 0) | SSR-PS and SSR-PSCD |

${\mathit{x}}_{p{p}^{*}}^{div}\in \mathbb{B}\forall p,{p}^{*}\in P,p\ne {p}^{*}$ | =1 if containers will be diverted from port $p$ that experienced a disruption to alternative port ${p}^{*}$(otherwise = 0) | SSR-PSCD |

**SSP-U**—ship scheduling problem with uncertainties;

**SSR-SSA**—ship schedule recovery with sailing speed adjustment;

**SSR-HRA**—ship schedule recovery with handling rate adjustment;

**SSR-PS**—ship schedule recovery with port skipping;

**SSR-PSCD**—ship schedule recovery with port skipping and container diversion.

Auxiliary Variable | Description of Auxiliary Variables | Remarks |
---|---|---|

${\mathit{q}}^{own}\in \mathbb{N}$ | number of own ships to be deployed (ships) | SSP-U |

${\mathit{q}}^{char}\in \mathbb{N}$ | number of chartered ships to be deployed (ships) | SSP-U |

${\mathit{\tau}}_{p}^{arr}\in {\mathbb{R}}^{+}\forall p\in P$ | arrival time of ships at port $p$ (hours) | SSP-U |

$\overline{{\mathit{\tau}}_{p}^{arr}}\in {\mathbb{R}}^{+}\forall p\in P$ | recovered arrival time of ships at port $p$ (hours) | SSR |

$\overline{{\mathit{\tau}}_{p}^{wait}}\in {\mathbb{R}}^{+}\forall p\in P$ | recovered waiting time of ships at port $p$ (hours) | SSR |

$\overline{{\mathit{\tau}}_{p}^{hand}}\in {\mathbb{R}}^{+}\forall p\in P$ | recovered handling time of ships at port $p$ (hours) | SSR |

${\mathit{\tau}}_{p}^{dep}\in {\mathbb{R}}^{+}\forall p\in P$ | departure time of ships from port $p$ (hours) | SSP-U |

$\overline{{\mathit{\tau}}_{p}^{dep}}\in {\mathbb{R}}^{+}\forall p\in P$ | recovered departure time of ships from port $p$ (hours) | SSR |

$\overline{{\mathit{s}}_{p}}\in {\mathbb{R}}^{+}\forall p\in P$ | recovered sailing speed of ships on voyage leg $p$ (knots) | SSR |

${\mathit{\tau}}_{p}^{sail}\in {\mathbb{R}}^{+}\forall p\in P$ | sailing time of ships on voyage leg $p$ (hours) | SSP-U |

$\overline{{\mathit{\tau}}_{p}^{sail}}\in {\mathbb{R}}^{+}\forall p\in P$ | recovered sailing time of ships on voyage leg $p$ (hours) | SSR |

${\mathit{\tau}}_{p}^{late}\in {\mathbb{R}}^{+}\forall p\in P$ | late arrival hours of ships at port $p$ (hours) | SSP-U |

$\overline{{\mathit{\tau}}_{p}^{late}}\in {\mathbb{R}}^{+}\forall p\in P$ | recovered late arrival hours of ships at port $p$ (hours) | SSR |

$\overline{{\mathit{\tau}}^{stt}}\in {\mathbb{R}}^{+}$ | turnaround time of ships for the recovered ship schedule (hours) | SSR |

${\mathit{\phi}}_{p}\in {\mathbb{R}}^{+}\forall p\in P$ | consumption of fuel by the main engines of ships on voyage leg $p$ (tons/nmi) | SSP-U |

$\overline{{\mathit{\phi}}_{p}}\in {\mathbb{R}}^{+}\forall p\in P$ | recovered consumption of fuel by the main engines of ships on voyage leg $p$ (tons/nmi) | SSR |

${\mathit{\delta}}_{p}^{port}\in {\mathbb{R}}^{+}\forall p\in P$ | number of containers to be handled at port $p$ (TEUs) | SSP-U |

${\mathit{\delta}}_{p}^{sea}\in {\mathbb{R}}^{+}\forall p\in P$ | number of containers to be carried on voyage leg $p$ (TEUs) | All models |

${\mathit{x}}_{p}^{dd}\in \mathbb{B}\forall p\in P$ | =1 if containers diverted from a port that experienced a disruption will be handled at alternative port $p$ (otherwise = 0) | SSR-PSCD |

${\mathit{\delta}}_{p{p}^{*}}^{div}\in \mathbb{N}\forall p,{p}^{*}\in P,p\ne {p}^{*}$ | number of containers diverted from port $p$ that experienced a disruption to alternative port ${p}^{*}$ (TEUs) | SSR-PSCD |

${\mathit{{\rm K}}}^{hand}\in {\mathbb{R}}^{+}$ | total cost associated with container handling at ports (USD) | SSP-U |

$\overline{{\mathit{{\rm K}}}^{hand}}\in {\mathbb{R}}^{+}$ | total cost associated with container handling at ports for the recovered schedule of ships (USD) | SSR |

${\mathit{{\rm K}}}^{late}\in {\mathbb{R}}^{+}$ | total cost associated with late ship arrivals (USD) | SSP-U |

$\overline{{\mathit{{\rm K}}}^{late}}\in {\mathbb{R}}^{+}$ | total cost associated with late ship arrivals for the recovered schedule of ships (USD) | SSR |

${\mathit{{\rm K}}}^{fuel}\in {\mathbb{R}}^{+}$ | total cost associated with fuel consumption (USD) | SSP-U |

$\overline{{\mathit{{\rm K}}}^{fuel}}\in {\mathbb{R}}^{+}$ | total cost associated with fuel consumption for the recovered schedule of ships (USD) | SSR |

${\mathit{{\rm K}}}^{oper}\in {\mathbb{R}}^{+}$ | total cost associated with basic ship operations (USD) | SSP-U |

${\mathit{{\rm K}}}^{char}\in {\mathbb{R}}^{+}$ | total cost associated with chartering of ships (USD) | SSP-U |

${\mathit{{\rm K}}}^{inv}\in {\mathbb{R}}^{+}$ | total cost associated with container inventory (USD) | SSP-U |

$\overline{{\mathit{{\rm K}}}^{inv}}\in {\mathbb{R}}^{+}$ | total cost associated with container inventory for the recovered schedule of ships (USD) | SSR |

$\overline{{\mathit{{\rm K}}}^{div}}\in {\mathbb{R}}^{+}$ | total cost associated with container diversion for the recovered schedule of ships (USD) | SSR-PSCD |

$\mathit{R}\in {\mathbb{R}}^{+}$ | total revenue that will be accumulated by the shipping line (USD) | SSP-U |

$\overline{\mathit{R}}\in {\mathbb{R}}^{+}$ | total revenue that will be accumulated by the shipping line for the recovered schedule of ships (USD) | SSR |

$\mathit{\Pi}\in {\mathbb{R}}^{+}$ | total profit that will be accumulated by the shipping line (USD) | SSP-U |

$\overline{\mathit{\Pi}}\in {\mathbb{R}}^{+}$ | total profit that will be accumulated by the shipping line for the recovered schedule of ships (USD) | SSR |

**SSP-U**—ship scheduling problem with uncertainties;

**SSR**—ship schedule recovery models (i.e.,

**SSR-SSA**and

**SSR-PS**and

**SSR-PSCD**and

**SSR-HRA**);

**SSR-SSA**—ship schedule recovery with sailing speed adjustment;

**SSR-HRA**—ship schedule recovery with handling rate adjustment;

**SSR-PS**—ship schedule recovery with port skipping;

**SSR-PSCD**—ship schedule recovery with port skipping and container diversion.

Parameter | Description of Parameters | Remarks |
---|---|---|

${\tau}_{p}^{st}\in {\mathbb{R}}^{+}\forall p\in P$ | start of the arrival TW at port $p$ (hours) | SSR |

${\tau}_{p}^{end}\in {\mathbb{R}}^{+}\forall p\in P$ | end of the arrival TW at port $p$ (hours) | SSP-U |

${\tau}_{p}^{arr}\in {\mathbb{R}}^{+}\forall p\in P$ | arrival time of ships at port $p$ for the original ship schedule (hours) | SSR |

${\chi}_{p}\in {\mathbb{R}}^{+}\forall p\in P$ | handling productivity for ship service at port $p$ (TEU/hour) | SSR |

${\chi}_{ph}\in {\mathbb{R}}^{+}\forall p\in P,h\in {H}_{p}$ | handling productivity for ship service at port $p$ when handling rate $h$ is requested (TEU/hour) | SSR-HRA |

$\varphi \in \mathbb{N}$ | frequency of port service for the considered shipping route (days) | All models |

${q}^{own-max}\in \mathbb{N}$ | maximum number of own ships that could be deployed for the considered shipping route (ships) | SSP-U |

${q}^{char-max}\in \mathbb{N}$ | maximum number of chartered ships that could be deployed for the considered shipping route (ships) | SSP-U |

$\overline{{q}^{own}}\in \mathbb{N}$ | number of own ships to be deployed for the original ship schedule (ships) | SSR |

$\overline{{q}^{char}}\in \mathbb{N}$ | number of chartered ships to be deployed for the original ship schedule (ships) | SSR |

$\overline{{\delta}_{p}^{port}}\in {\mathbb{R}}^{+}\forall p\in P$ | number of containers to be handled at port $p$ for the original ship schedule (TEUs) | SSR |

${l}_{p}\in {\mathbb{R}}^{+}\forall p\in P$ | length of voyage leg $p$ for the considered shipping route (nmi) | All models |

$\alpha ,\gamma \in {\mathbb{R}}^{+}$ | coefficients associated with the fuel consumption function | All models |

$\omega \in {\mathbb{R}}^{+}$ | average cargo weight within a standard TEU (tons) | All models |

${\delta}^{empty}\in {\mathbb{R}}^{+}$ | weight of a ship without containers (tons) | All models |

${\delta}^{cap}\in {\mathbb{R}}^{+}$ | maximum weight of containers that could be loaded on a ship (tons) | All models |

${s}^{max}\in {\mathbb{R}}^{+}$ | maximum sailing speed that could be set for ships (knots) | SSR |

${s}^{min}\in {\mathbb{R}}^{+}$ | minimum sailing speed that could be set for ships (knots) | SSR |

${s}_{p}\in {\mathbb{R}}^{+}\forall p\in P$ | sailing speed of ships on voyage leg $p$ for the original ship schedule (knots) | SSR |

$\tilde{{\mathit{\tau}}_{p}^{wait}}\in {\mathbb{R}}^{+}\forall p\in P$ | expected waiting time of ships at port $p$ (hours) | SSP-U |

$\tilde{{\mathit{\tau}}_{p}^{hand}}\in {\mathbb{R}}^{+}\forall p\in P$ | expected handling time of ships at port $p$ (hours) | SSP-U |

$\tilde{{\mathit{\tau}}_{p}^{d-port}}\in {\mathbb{R}}^{+}\forall p\in P$ | expected duration for a disruption at port $p$ (hours) | SSR |

$\tilde{{\mathit{\sigma}}_{p}^{d-sea}}\in \mathbb{R}\forall p\in P$ | expected change in sailing speed of ships due to a disruption on voyage leg $p$ (knots) | SSR |

$\tilde{{\mathit{z}}_{p}^{port}}\in \mathbb{B}\forall p\in P$ | =1 if a disruption happened at port $p$ (otherwise = 0) | SSR |

$\tilde{{\mathit{z}}_{p}^{sea}}\in \mathbb{B}\forall p\in P$ | =1 if a disruption happened on voyage leg $p$ (otherwise = 0) | SSR |

${z}_{p}^{skip}\in \mathbb{B}\forall p\in P$ | =1 if the port skipping would be a feasible option for port $p$ as a result of disruption occurrence (otherwise = 0) | SSR-PS and SSR-PSCD |

${z}_{p{p}^{*}}^{div}\in \mathbb{B}\forall p,{p}^{*}\in P,p\ne {p}^{*}$ | =1 if containers can be potentially diverted from port $p$ that experienced a disruption to alternative port ${p}^{*}$ (otherwise = 0) | SSR-PSCD |

${\delta}_{p}^{term}\in {\mathbb{R}}^{+}\forall p\in P$ | available container terminal capacity for accommodating the containers diverted at port $p$ (TEUs) | SSR-PSCD |

${\delta}_{p}^{land}\in {\mathbb{R}}^{+}\forall p\in P$ | available inland transport capacity for accommodating the containers diverted at port $p$ (TEUs) | SSR-PSCD |

${\kappa}_{p}^{hand}\in {\mathbb{R}}^{+}\forall p\in P$ | unit cost associated with container handling at port $p$ (USD/TEU) | All models |

${\kappa}_{ph}^{hand}\in {\mathbb{R}}^{+}\forall p\in P,h\in {H}_{p}$ | unit cost associated with container handling at port $p$ when handling rate $h$ is requested (USD/TEU) | SSR-HRA |

${\kappa}_{p}^{late}\in {\mathbb{R}}^{+}\forall p\in P$ | unit cost associated with late ship arrivals at port $p$ (USD/hour) | All models |

${\kappa}^{fuel}\in {\mathbb{R}}^{+}$ | unit cost associated with fuel consumption (USD/ton) | All models |

${\kappa}^{oper}\in {\mathbb{R}}^{+}$ | unit cost associated with basic ship operations (USD/day) | All models |

${\kappa}^{char}\in {\mathbb{R}}^{+}$ | unit cost associated with chartering of ships (USD/day) | All models |

${\kappa}^{inv}\in {\mathbb{R}}^{+}$ | unit cost associated with container inventory (USD/TEU/hour) | All models |

${\kappa}_{p}^{rev}\in {\mathbb{R}}^{+}\forall p\in P$ | unit cost associated with transporting the cargo for the considered shipping route, i.e., freight rate (USD/TEU) | All models |

${\kappa}_{p}^{skip}\in {\mathbb{R}}^{+}\forall p\in P$ | cost associated with skipping port $p$ for the considered shipping route (USD) | SSR-PS and SSR-PSCD |

${\kappa}_{p}^{mis}\in {\mathbb{R}}^{+}\forall p\in P$ | unit cost associated with misconnected cargo at port $p$ for the considered shipping route (USD/TEU) | SSR-PSCD |

${\kappa}_{p{p}^{*}}^{d-term}\in {\mathbb{R}}^{+}\forall p,{p}^{*}\in P,$ $p\ne {p}^{*}$ | unit cost associated with handling the containers diverted from port $p$ that experienced a disruption at alternative port ${p}^{*}$ (USD/TEU) | SSR-PSCD |

${\kappa}_{p{p}^{*}}^{d-land}\in {\mathbb{R}}^{+}\forall p,{p}^{*}\in P,$ $p\ne {p}^{*}$ | unit cost associated with inland transport cost of the containers diverted from port $p$ that experienced a disruption at alternative port ${p}^{*}$ (USD/TEU) | SSR-PSCD |

${\Pi}^{0}\in {\mathbb{R}}^{+}$ | total profit that was expected to be accumulated by the shipping line for the original ship schedule (USD) | SSR |

$\overline{{{\rm K}}^{oper}}\in {\mathbb{R}}^{+}$ | total cost associated with basic ship operations for the recovered schedule of ships (USD) | SSR |

$\overline{{{\rm K}}^{char}}\in {\mathbb{R}}^{+}$ | total cost associated with chartering of ships for the recovered schedule of ships (USD) | SSR |

**SSP-U**—ship scheduling problem with uncertainties;

**SSR**—ship schedule recovery models (i.e.,

**SSR-SSA**and

**SSR-PS**and

**SSR-PSCD**and

**SSR-HRA**);

**SSR-SSA**—ship schedule recovery with sailing speed adjustment;

**SSR-HRA**—ship schedule recovery with handling rate adjustment;

**SSR-PS**—ship schedule recovery with port skipping;

**SSR-PSCD**—ship schedule recovery with port skipping and container diversion.

## References

- McLean, C. Government action and the new blue economy. In Preparing a Workforce for the New Blue Economy; Elsevier: Amsterdam, The Netherlands, 2021; pp. 513–525. [Google Scholar]
- Christiansen, M.; Fagerholt, K.; Nygreen, B.; Ronen, D. Maritime transportation. Handb. Oper. Res. Manag. Sci.
**2007**, 14, 189–284. [Google Scholar] - Zhang, Y.; Sun, Z. The Coevolutionary Process of Maritime Management of Shipping Industry in the Context of the COVID-19 Pandemic. J. Mar. Sci. Eng.
**2021**, 9, 1293. [Google Scholar] [CrossRef] - Zhu, J.; Wang, H.; Xu, J. Fuzzy DEMATEL-QFD for Designing Supply Chain of Shipbuilding Materials Based on Flexible Strategies. J. Mar. Sci. Eng.
**2021**, 9, 1106. [Google Scholar] [CrossRef] - Lezhnina, E.A.; Balykina, Y.E. Cooperation between sea ports and carriers in the logistics chain. J. Mar. Sci. Eng.
**2021**, 9, 774. [Google Scholar] [CrossRef] - Svanberg, M.; Holm, H.; Cullinane, K. Assessing the Impact of Disruptive Events on Port Performance and Choice: The Case of Gothenburg. J. Mar. Sci. Eng.
**2021**, 9, 145. [Google Scholar] [CrossRef] - Wang, S.; Meng, Q. Robust schedule design for liner shipping services. Transp. Res. Part E Logist. Transp. Rev.
**2012**, 48, 1093–1106. [Google Scholar] [CrossRef] - How Bad Weather Impacts Shipping (and How to Deal With It). Available online: https://www.icecargo.com.au/weather-impacts-shipping/ (accessed on 1 January 2022).
- Li, C.; Qi, X.; Song, D. Real-time schedule recovery in liner shipping service with regular uncertainties and disruption events. Transp. Res. Part B Methodol.
**2016**, 93, 762–788. [Google Scholar] [CrossRef] - Slagen, D. The Operational Guide to Weather Excellence: Intermodal. Available online: https://www.tomorrow.io/blog/the-operational-guide-to-weather-excellence-intermodal/ (accessed on 1 January 2022).
- Notteboom, T.E. The time factor in liner shipping services. Marit. Econ. Logist.
**2006**, 8, 19–39. [Google Scholar] [CrossRef] - Dadashi, A.; Dulebenets, M.A.; Golias, M.M.; Sheikholeslami, A. A novel continuous berth scheduling model at multiple marine container terminals with tidal considerations. Marit. Bus. Rev.
**2017**, 2, 142–157. [Google Scholar] [CrossRef] [Green Version] - Liu, B.; Li, Z.-C.; Wang, Y.; Sheng, D. Short-term berth planning and ship scheduling for a busy seaport with channel restrictions. Transp. Res. Part E Logist. Transp. Rev.
**2021**, 154, 102467. [Google Scholar] [CrossRef] - Review of Maritime Transport. 2020. Available online: https://unctad.org/en/PublicationsLibrary/rmt2020_en.pdf (accessed on 1 January 2022).
- Pasha, J.; Dulebenets, M.A.; Fathollahi-Fard, A.M.; Tian, G.; Lau, Y.-Y.; Singh, P.; Liang, B. An integrated optimization method for tactical-level planning in liner shipping with heterogeneous ship fleet and environmental considerations. Adv. Eng. Inform.
**2021**, 48, 101299. [Google Scholar] [CrossRef] - Chen, Q.; Ge, Y.-E.; Lau, Y.-Y.; Dulebenets, M.A.; Sun, X.; Kawasaki, T.; Mellalou, A.; Tao, X. Effects of COVID-19 on passenger shipping activities and emissions: Empirical analysis of passenger ships in Danish waters. Marit. Policy Manag.
**2022**, 1–21. [Google Scholar] [CrossRef] - Dulebenets, M.A. Multi-objective collaborative agreements amongst shipping lines and marine terminal operators for sustainable and environmental-friendly ship schedule design. J. Clean. Prod.
**2022**, 342, 130897. [Google Scholar] [CrossRef] - Millefiori, L.M.; Braca, P.; Zissis, D.; Spiliopoulos, G.; Marano, S.; Willett, P.K.; Carniel, S. COVID-19 impact on global maritime mobility. Sci. Rep.
**2021**, 11, 1–16. [Google Scholar] - Wetzel, D.; Tierney, K. Integrating fleet deployment into liner shipping vessel repositioning. Transp. Res. Part E Logist. Transp. Rev.
**2020**, 143, 102101. [Google Scholar] [CrossRef] - Zhang, E.; Chu, F.; Wang, S.; Liu, M.; Sui, Y. Approximation approach for robust vessel fleet deployment problem with ambiguous demands. J. Comb. Optim.
**2020**, 1–15. [Google Scholar] [CrossRef] - Zhen, L.; Wu, Y.; Wang, S.; Laporte, G. Green technology adoption for fleet deployment in a shipping network. Transp. Res. Part B Methodol.
**2020**, 139, 388–410. [Google Scholar] [CrossRef] - Chen, J.; Zhuang, C.; Yang, C.; Wan, Z.; Zeng, X.; Yao, J. Fleet co-deployment for liner shipping alliance: Vessel pool operation with uncertain demand. Ocean Coast. Manag.
**2021**, 214, 105923. [Google Scholar] [CrossRef] - Rodriguez, M.H.; Agrell, P.J.; Manrique-de-Lara-Peñate, C.; Trujillo, L. A multi-criteria fleet deployment model for cost, time and environmental impact. Int. J. Prod. Econ.
**2022**, 243, 108325. [Google Scholar] [CrossRef] - Lin, D.-Y.; Tsai, Y.-Y. The ship routing and freight assignment problem for daily frequency operation of maritime liner shipping. Transp. Res. Part E Logist. Transp. Rev.
**2014**, 67, 52–70. [Google Scholar] [CrossRef] - Zhang, A.; Lee Lam, J.S. Impacts of schedule reliability and sailing frequency on the liner shipping and port industry: A study of Daily Maersk. Transp. J.
**2014**, 53, 235–253. [Google Scholar] [CrossRef] - Giovannini, M.; Psaraftis, H.N. The profit maximizing liner shipping problem with flexible frequencies: Logistical and environmental considerations. Flex. Serv. Manuf. J.
**2019**, 31, 567–597. [Google Scholar] [CrossRef] [Green Version] - Pasha, J.; Dulebenets, M.A.; Kavoosi, M.; Abioye, O.F.; Theophilus, O.; Wang, H.; Kampmann, R.; Guo, W. Holistic tactical-level planning in liner shipping: An exact optimization approach. J. Shipp. Trade
**2020**, 5, 1–35. [Google Scholar] [CrossRef] - Lee, H.; Aydin, N.; Choi, Y.; Lekhavat, S.; Irani, Z. A decision support system for vessel speed decision in maritime logistics using weather archive big data. Comput. Oper. Res.
**2018**, 98, 330–342. [Google Scholar] [CrossRef] - Mallidis, I.; Iakovou, E.; Dekker, R.; Vlachos, D. The impact of slow steaming on the carriers’ and shippers’ costs: The case of a global logistics network. Transp. Res. Part E Logist. Transp. Rev.
**2018**, 111, 18–39. [Google Scholar] [CrossRef] [Green Version] - Li, X.; Sun, B.; Guo, C.; Du, W.; Li, Y. Speed optimization of a container ship on a given route considering voluntary speed loss and emissions. Appl. Ocean Res.
**2020**, 94, 101995. [Google Scholar] [CrossRef] - Wu, W.-M. The optimal speed in container shipping: Theory and empirical evidence. Transp. Res. Part E Logist. Transp. Rev.
**2020**, 136, 101903. [Google Scholar] [CrossRef] - Wang, S.; Alharbi, A.; Davy, P. Liner ship route schedule design with port time windows. Transp. Res. Part C Emerg. Technol.
**2014**, 41, 1–17. [Google Scholar] [CrossRef] [Green Version] - Alharbi, A.; Wang, S.; Davy, P. Schedule design for sustainable container supply chain networks with port time windows. Adv. Eng. Inform.
**2015**, 29, 322–331. [Google Scholar] [CrossRef] - Dulebenets, M.A. A comprehensive multi-objective optimization model for the vessel scheduling problem in liner shipping. Int. J. Prod. Econ.
**2018**, 196, 293–318. [Google Scholar] [CrossRef] - Ozcan, S.; Eliiyi, D.T.; Reinhardt, L.B. Cargo allocation and vessel scheduling on liner shipping with synchronization of transshipments. Appl. Math. Model.
**2020**, 77, 235–252. [Google Scholar] [CrossRef] - Zhang, B.; Zheng, Z.; Wang, D. A model and algorithm for vessel scheduling through a two-way tidal channel. Marit. Policy Manag.
**2020**, 47, 188–202. [Google Scholar] [CrossRef] - Zheng, J.; Hou, X.; Qi, J.; Yang, L. Liner ship scheduling with time-dependent port charges. Marit. Policy Manag.
**2020**, 49, 18–38. [Google Scholar] [CrossRef] - Zhuge, D.; Wang, S.; Zhen, L.; Laporte, G. Schedule design for liner services under vessel speed reduction incentive programs. Nav. Res. Logist. (NRL)
**2020**, 67, 45–62. [Google Scholar] [CrossRef] [Green Version] - Dulebenets, M.A. The vessel scheduling problem in a liner shipping route with heterogeneous fleet. Int. J. Civ. Eng.
**2018**, 16, 19–32. [Google Scholar] [CrossRef] - Wang, Y.; Wang, S. Deploying, scheduling, and sequencing heterogeneous vessels in a liner container shipping route. Transp. Res. Part E Logist. Transp. Rev.
**2021**, 151, 102365. [Google Scholar] [CrossRef] - Zheng, J.; Ma, Y.; Ji, X.; Chen, J. Is the weekly service frequency constraint tight when optimizing ship speeds and fleet size for a liner shipping service? Ocean Coast. Manag.
**2021**, 212, 105815. [Google Scholar] [CrossRef] - Pantuso, G.; Fagerholt, K.; Hvattum, L.M. A survey on maritime fleet size and mix problems. Eur. J. Oper. Res.
**2014**, 235, 341–349. [Google Scholar] [CrossRef] [Green Version] - Meng, Q.; Wang, S.; Andersson, H.; Thun, K. Containership routing and scheduling in liner shipping: Overview and future research directions. Transp. Sci.
**2014**, 48, 265–280. [Google Scholar] [CrossRef] [Green Version] - Wang, S.; Meng, Q. Container liner fleet deployment: A systematic overview. Transp. Res. Part C Emerg. Technol.
**2017**, 77, 389–404. [Google Scholar] [CrossRef] - Meng, Q.; Zhao, H.; Wang, Y. Revenue management for container liner shipping services: Critical review and future research directions. Transp. Res. Part E Logist. Transp. Rev.
**2019**, 128, 280–292. [Google Scholar] [CrossRef] - Christiansen, M.; Hellsten, E.; Pisinger, D.; Sacramento, D.; Vilhelmsen, C. Liner shipping network design. Eur. J. Oper. Res.
**2020**, 286, 1–20. [Google Scholar] [CrossRef] - Dulebenets, M.A.; Pasha, J.; Abioye, O.F.; Kavoosi, M. Vessel scheduling in liner shipping: A critical literature review and future research needs. Flex. Serv. Manuf. J.
**2021**, 33, 43–106. [Google Scholar] [CrossRef] - Song, D. A literature review, container shipping supply chain: Planning problems and research opportunities. Logistics
**2021**, 5, 41. [Google Scholar] [CrossRef] - Krippendorff, K. Content Analysis: An Introduction to Its Methodology; Sage Publications: Thousand Oaks, CA, USA, 2018. [Google Scholar]
- Dulebenets, M.A. The green vessel scheduling problem with transit time requirements in a liner shipping route with Emission Control Areas. Alex. Eng. J.
**2018**, 57, 331–342. [Google Scholar] [CrossRef] - Dulebenets, M.A. Green vessel scheduling in liner shipping: Modeling carbon dioxide emission costs in sea and at ports of call. Int. J. Transp. Sci. Technol.
**2018**, 7, 26–44. [Google Scholar] [CrossRef] - Christiansen, M.; Fagerholt, K. Robust ship scheduling with multiple time windows. Nav. Res. Logist. (NRL)
**2002**, 49, 611–625. [Google Scholar] [CrossRef] - Fagerholt, K. Ship scheduling with soft time windows: An optimisation based approach. Eur. J. Oper. Res.
**2001**, 131, 559–571. [Google Scholar] [CrossRef] - Ronen, D. The effect of oil price on containership speed and fleet size. J. Oper. Res. Soc.
**2011**, 62, 211–216. [Google Scholar] [CrossRef] - Psaraftis, H.N.; Kontovas, C.A. Speed models for energy-efficient maritime transportation: A taxonomy. Transp. Res. Part C Emerg. Technol.
**2013**, 26, 331–351. [Google Scholar] [CrossRef] - Kontovas, C.A. The green ship routing and scheduling problem (GSRSP): A conceptual approach. Transp. Res. Part D Transp. Environ.
**2014**, 31, 61–69. [Google Scholar] [CrossRef] - Ferrari, C.; Parola, F.; Tei, A. Determinants of slow steaming and implications on service patterns. Marit. Policy Manag.
**2015**, 42, 636–652. [Google Scholar] [CrossRef] - De, A.; Mamanduru, V.K.R.; Gunasekaran, A.; Subramanian, N.; Tiwari, M.K. Composite particle algorithm for sustainable integrated dynamic ship routing and scheduling optimization. Comput. Ind. Eng.
**2016**, 96, 201–215. [Google Scholar] [CrossRef] - Wen, M.; Pacino, D.; Kontovas, C.; Psaraftis, H. A multiple ship routing and speed optimization problem under time, cost and environmental objectives. Transp. Res. Part D Transp. Environ.
**2017**, 52, 303–321. [Google Scholar] [CrossRef] - Reinhardt, L.B.; Pisinger, D.; Sigurd, M.M.; Ahmt, J. Speed optimizations for liner networks with business constraints. Eur. J. Oper. Res.
**2020**, 285, 1127–1140. [Google Scholar] [CrossRef] [Green Version] - Wu, M.; Li, K.X.; Xiao, Y.; Yuen, K.F. Carbon Emission Trading Scheme in the shipping sector: Drivers, challenges, and impacts. Mar. Policy
**2022**, 138, 104989. [Google Scholar] [CrossRef] - Vernimmen, B.; Dullaert, W.; Engelen, S. Schedule unreliability in liner shipping: Origins and consequences for the hinterland supply chain. Marit. Econ. Logist.
**2007**, 9, 193–213. [Google Scholar] [CrossRef] - Chuang, T.-N.; Lin, C.-T.; Kung, J.-Y.; Lin, M.-D. Planning the route of container ships: A fuzzy genetic approach. Expert Syst. Appl.
**2010**, 37, 2948–2956. [Google Scholar] [CrossRef] - Meng, Q.; Wang, T. A chance constrained programming model for short-term liner ship fleet planning problems. Marit. Pol. Mgmt.
**2010**, 37, 329–346. [Google Scholar] [CrossRef] - Qi, X.; Song, D.-P. Minimizing fuel emissions by optimizing vessel schedules in liner shipping with uncertain port times. Transp. Res. Part E Logist. Transp. Rev.
**2012**, 48, 863–880. [Google Scholar] [CrossRef] - Wang, S.; Meng, Q. Liner ship route schedule design with sea contingency time and port time uncertainty. Transp. Res. Part B Methodol.
**2012**, 46, 615–633. [Google Scholar] [CrossRef] - Di Francesco, M.; Lai, M.; Zuddas, P. Maritime repositioning of empty containers under uncertain port disruptions. Comput. Ind. Eng.
**2013**, 64, 827–837. [Google Scholar] [CrossRef] - Halvorsen-Weare, E.E.; Fagerholt, K.; Rönnqvist, M. Vessel routing and scheduling under uncertainty in the liquefied natural gas business. Comput. Ind. Eng.
**2013**, 64, 290–301. [Google Scholar] [CrossRef] [Green Version] - Du, Y.; Meng, Q.; Wang, Y. Budgeting fuel consumption of container ship over round-trip voyage through robust optimization. Transp. Res. Rec.
**2015**, 2477, 68–75. [Google Scholar] [CrossRef] - Kepaptsoglou, K.; Fountas, G.; Karlaftis, M.G. Weather impact on containership routing in closed seas: A chance-constraint optimization approach. Transp. Res. Part C Emerg. Technol.
**2015**, 55, 139–155. [Google Scholar] [CrossRef] - Lee, C.-Y.; Lee, H.L.; Zhang, J. The impact of slow ocean steaming on delivery reliability and fuel consumption. Transp. Res. Part E Logist. Transp. Rev.
**2015**, 76, 176–190. [Google Scholar] [CrossRef] - Ng, M. Container vessel fleet deployment for liner shipping with stochastic dependencies in shipping demand. Transp. Res. Part B Methodol.
**2015**, 74, 79–87. [Google Scholar] [CrossRef] - Norlund, E.K.; Gribkovskaia, I.; Laporte, G. Supply vessel planning under cost, environment and robustness considerations. Omega
**2015**, 57, 271–281. [Google Scholar] [CrossRef] - Song, D.-P.; Li, D.; Drake, P. Multi-objective optimization for planning liner shipping service with uncertain port times. Transp. Res. Part E Logist. Transp. Rev.
**2015**, 84, 1–22. [Google Scholar] [CrossRef] - Wang, S. Optimal sequence of container ships in a string. Eur. J. Oper. Res.
**2015**, 246, 850–857. [Google Scholar] [CrossRef] - Aydin, N.; Lee, H.; Mansouri, S.A. Speed optimization and bunkering in liner shipping in the presence of uncertain service times and time windows at ports. Eur. J. Oper. Res.
**2017**, 259, 143–154. [Google Scholar] [CrossRef] [Green Version] - Song, D.-P.; Li, D.; Drake, P. Multi-objective optimization for a liner shipping service from different perspectives. Transp. Res. Procedia
**2017**, 25, 251–260. [Google Scholar] [CrossRef] - Ng, M.; Lin, D.-Y. Fleet deployment in liner shipping with incomplete demand information. Transp. Res. Part E Logist. Transp. Rev.
**2018**, 116, 184–189. [Google Scholar] [CrossRef] - Tan, Z.; Wang, Y.; Meng, Q.; Liu, Z. Joint ship schedule design and sailing speed optimization for a single inland shipping service with uncertain dam transit time. Transp. Sci.
**2018**, 52, 1570–1588. [Google Scholar] [CrossRef] - Gürel, S.; Shadmand, A. A heterogeneous fleet liner ship scheduling problem with port time uncertainty. Cent. Eur. J. Oper. Res.
**2019**, 27, 1153–1175. [Google Scholar] [CrossRef] - Tierney, K.; Ehmke, J.F.; Campbell, A.M.; Müller, D. Liner shipping single service design problem with arrival time service levels. Flex. Serv. Manuf. J.
**2019**, 31, 620–652. [Google Scholar] [CrossRef] - Liu, M.; Liu, X.; Chu, F.; Zhu, M.; Zheng, F. Liner ship bunkering and sailing speed planning with uncertain demand. Comput. Appl. Math.
**2020**, 39, 1–23. [Google Scholar] [CrossRef] - Ding, J.; Xie, C. Stochastic Programming for Liner Ship Routing and Scheduling under Uncertain Sea Ice Conditions. Transp. Res. Rec.
**2021**. [Google Scholar] [CrossRef] - Liu, D.; Shi, G.; Hirayama, K. Vessel Scheduling Optimization Model Based on Variable Speed in a Seaport with One-Way Navigation Channel. Sensors
**2021**, 21, 5478. [Google Scholar] [CrossRef] - Dulebenets, M.A. Advantages and disadvantages from enforcing emission restrictions within emission control areas. Marit. Bus. Rev.
**2016**, 2, 302–330. [Google Scholar] [CrossRef] [Green Version] - Ma, D.; Ma, W.; Jin, S.; Ma, X. Method for simultaneously optimizing ship route and speed with emission control areas. Ocean Eng.
**2020**, 202, 107170. [Google Scholar] [CrossRef] - Wang, K.; Li, J.; Huang, L.; Ma, R.; Jiang, X.; Yuan, Y.; Mwero, N.A.; Negenborn, R.R.; Sun, P.; Yan, X. A novel method for joint optimization of the sailing route and speed considering multiple environmental factors for more energy efficient shipping. Ocean Eng.
**2020**, 216, 107591. [Google Scholar] [CrossRef] - Zhen, L.; Hu, Z.; Yan, R.; Zhuge, D.; Wang, S. Route and speed optimization for liner ships under emission control policies. Transp. Res. Part C Emerg. Technol.
**2020**, 110, 330–345. [Google Scholar] [CrossRef] - Ma, W.; Hao, S.; Ma, D.; Wang, D.; Jin, S.; Qu, F. Scheduling decision model of liner shipping considering emission control areas regulations. Appl. Ocean Res.
**2021**, 106, 102416. [Google Scholar] [CrossRef] - Teodorović, D.; Stojković, G. Model for operational daily airline scheduling. Transp. Plan. Technol.
**1990**, 14, 273–285. [Google Scholar] [CrossRef] - Rosenberger, J.M.; Johnson, E.L.; Nemhauser, G.L. Rerouting aircraft for airline recovery. Transp. Sci.
**2003**, 37, 408–421. [Google Scholar] [CrossRef] - Barnhart, C. Irregular operations: Schedule recovery and robustness. Glob. Airl. Ind.
**2009**, 253–274. [Google Scholar] [CrossRef] - Clausen, J.; Larsen, A.; Larsen, J.; Rezanova, N.J. Disruption management in the airline industry—Concepts, models and methods. Comput. Oper. Res.
**2010**, 37, 809–821. [Google Scholar] [CrossRef] [Green Version] - Cheraghchi, F.; Abualhaol, I.; Falcon, R.; Abielmona, R.; Raahemi, B.; Petriu, E. Big-data-enabled modelling and optimization of granular speed-based vessel schedule recovery problem. In Proceedings of the 2017 IEEE International Conference on Big Data (Big Data), Boston, MA, USA, 11–14 December 2017; pp. 1786–1794. [Google Scholar]
- Dienst, D. Airline Disruption Management-The Aircraft Recovery Problem; Technical University of Denmark, DTU, DK-2800 Kgs.: Lyngby, Denmark, 2010. [Google Scholar]
- Thengvall, B.G.; Yu, G.; Bard, J.F. Multiple fleet aircraft schedule recovery following hub closures. Transp. Res. Part A Policy Pract.
**2001**, 35, 289–308. [Google Scholar] [CrossRef] - Thengvall, B.G.; Bard, J.F.; Yu, G. A bundle algorithm approach for the aircraft schedule recovery problem during hub closures. Transp. Sci.
**2003**, 37, 392–407. [Google Scholar] [CrossRef] [Green Version] - Marla, L.; Vaaben, B.; Barnhart, C. Integrated disruption management and flight planning to trade off delays and fuel burn. Transp. Sci.
**2017**, 51, 88–111. [Google Scholar] [CrossRef] [Green Version] - Brouer, B.D.; Dirksen, J.; Pisinger, D.; Plum, C.E.; Vaaben, B. The Vessel Schedule Recovery Problem (VSRP)–A MIP model for handling disruptions in liner shipping. Eur. J. Oper. Res.
**2013**, 224, 362–374. [Google Scholar] [CrossRef] - Paul, J.A.; Maloni, M.J. Modeling the effects of port disasters. Marit. Econ. Logist.
**2010**, 12, 127–146. [Google Scholar] [CrossRef] - Jones, D.A.; Farkas, J.L.; Bernstein, O.; Davis, C.E.; Turk, A.; Turnquist, M.A.; Nozick, L.K.; Levine, B.; Rawls, C.G.; Ostrowski, S.D. US import/export container flow modeling and disruption analysis. Res. Transp. Econ.
**2011**, 32, 3–14. [Google Scholar] [CrossRef] - Li, C.; Qi, X.; Lee, C.-Y. Disruption recovery for a vessel in liner shipping. Transp. Sci.
**2015**, 49, 900–921. [Google Scholar] [CrossRef] - Qi, X. Disruption management for liner shipping. In Handbook of Ocean Container Transport Logistics; Springer: Berlin/Heidelberg, Germany, 2015; pp. 231–249. [Google Scholar]
- Fischer, A.; Nokhart, H.; Olsen, H.; Fagerholt, K.; Rakke, J.G.; Stålhane, M. Robust planning and disruption management in roll-on roll-off liner shipping. Transp. Res. Part E Logist. Transp. Rev.
**2016**, 91, 51–67. [Google Scholar] [CrossRef] [Green Version] - Hasheminia, H.; Jiang, C. Strategic trade-off between vessel delay and schedule recovery: An empirical analysis of container liner shipping. Marit. Policy Manag.
**2017**, 44, 458–473. [Google Scholar] [CrossRef] - Cheraghchi, F.; Abualhaol, I.; Falcon, R.; Abielmona, R.; Raahemi, B.; Petriu, E. Modeling the speed-based vessel schedule recovery problem using evolutionary multiobjective optimization. Inf. Sci.
**2018**, 448, 53–74. [Google Scholar] [CrossRef] - Abioye, O.F.; Dulebenets, M.A.; Pasha, J.; Kavoosi, M. A vessel schedule recovery problem at the liner shipping route with emission control areas. Energies
**2019**, 12, 2380. [Google Scholar] [CrossRef] [Green Version] - Mulder, J.; Dekker, R. Designing robust liner shipping schedules: Optimizing recovery actions and buffer times. Eur. J. Oper. Res.
**2019**, 272, 132–146. [Google Scholar] [CrossRef] [Green Version] - Mulder, J.; van Jaarsveld, W.; Dekker, R. Simultaneous optimization of speed and buffer times with an application to liner shipping. Transp. Sci.
**2019**, 53, 365–382. [Google Scholar] [CrossRef] - Xing, J.; Wang, Y. Disruption Management in Liner Shipping: A Service-Cost Trade-off Model for Vessel Schedule Recovery Problem. In Proceedings of the 2019 5th International Conference on Transportation Information and Safety (ICTIS), Liverpool, UK, 14–17 July 2019; pp. 794–799. [Google Scholar]
- Cheraghchi, F.; Abualhaol, I.; Falcon, R.; Abielmona, R.; Raahemi, B.; Petriu, E. Distributed Multi-Objective Cooperative Coevolution Algorithm for Big-Data-Enabled Vessel Schedule Recovery Problem. In Proceedings of the 2020 IEEE Conference on Cognitive and Computational Aspects of Situation Management (CogSIMA), Victoria, BC, Canada, 24–29 August 2020; pp. 90–97. [Google Scholar]
- Abioye, O.F.; Dulebenets, M.A.; Kavoosi, M.; Pasha, J.; Theophilus, O. Vessel schedule recovery in liner shipping: Modeling alternative recovery options. IEEE Trans. Intell. Transp. Syst.
**2021**, 22, 6420–6434. [Google Scholar] [CrossRef] - De, A.; Wang, J.; Tiwari, M.K. Fuel bunker management strategies within sustainable container shipping operation considering disruption and recovery policies. IEEE Trans. Eng. Manag.
**2019**, 68, 1089–1111. [Google Scholar] [CrossRef] [Green Version] - Du, J.; Zhao, X.; Guo, L.; Wang, J. Machine Learning-Based Approach to Liner Shipping Schedule Design. J. Shanghai Jiaotong Univ. (Sci.)
**2021**, 1–13. [Google Scholar] [CrossRef] - Li, G.; Zhou, X.; Yin, J.; Xiao, Q. An UAV scheduling and planning method for post-disaster survey. Int. Arch. Photogramm. Remote Sens. Spat. Inf. Sci.
**2014**, 40, 169. [Google Scholar] [CrossRef] [Green Version] - Ejaz, W.; Ahmed, A.; Mushtaq, A.; Ibnkahla, M. Energy-efficient task scheduling and physiological assessment in disaster management using UAV-assisted networks. Comput. Commun.
**2020**, 155, 150–157. [Google Scholar] [CrossRef] - Pasha, J.; Elmi, Z.; Purkayastha, S.; Fathollahi-Fard, A.M.; Ge, Y.E.; Lau, Y.Y.; Dulebenets, M.A. The Drone Scheduling Problem: A Systematic State-of-the-Art Review. IEEE Trans. Intell. Transp. Syst.
**2022**. [Google Scholar] [CrossRef] - Macrina, G.; Pugliese, L.D.P.; Guerriero, F.; Laporte, G. Drone-aided routing: A literature review. Transp. Res. Part C Emerg. Technol.
**2020**, 120, 102762. [Google Scholar] [CrossRef] - Chung, S.H.; Sah, B.; Lee, J. Optimization for drone and drone-truck combined operations: A review of the state of the art and future directions. Comput. Oper. Res.
**2020**, 123, 105004. [Google Scholar] [CrossRef]

**Figure 6.**Distribution of the reviewed studies on uncertainties in liner shipping operations by various attributes: (

**a**) model objective; (

**b**) objective components; (

**c**) uncertain elements; and (

**d**) solution approach.

**Figure 7.**Ship schedule recovery options considered: (

**a**) sailing speed adjustment; (

**b**) handling rate adjustment; (

**c**) port skipping; and (

**d**) port skipping and container diversion.

**Figure 8.**Distribution of the reviewed studies on ship schedule recovery by various attributes: (

**a**) model objective; (

**b**) objective components; (

**c**) recovery strategies; and (

**d**) solution approach.

a/a | Authors | Sailing Speed | Port Time | Objective | Objective Components | Uncertain Elements | Solution Approach | Notes/Important Considerations |
---|---|---|---|---|---|---|---|---|

1 | Notteboom (2006) [11] | N/A | N/A | Assess the causes that might influence the reliability of shipping services between Northern Europe and East Asia | N/A | N/A | Case Study | Reliability of ship schedules |

2 | Vernimmen et al. (2007) [62] | N/A | N/A | Assess the causes of liner schedule unreliability and how these causes could impact supply chain players | N/A | N/A | Case Study | Reliability of ship schedules |

3 | Chuang et al. (2010) [63] | U | U | Total profit maximization | REV; TFC; TOC; TPC | Handling time; Sailing time; Container demand | Metaheuristic | Proposing a fuzzy Genetic Algorithm for liner shipping planning |

4 | Meng and Wang (2010) [64] | F | U | Total route service cost minimization | MSC; REV; TOC | Container demand | CPLEX | Ship fleet planning with uncertainty in container demand |

5 | Qi and Song (2012) [65] | V | U | Total route service cost minimization | TFC; TVC | Handling time | Iterative Optimization Algorithm; Sample Average Approximation | Minimizing the total expected fuel consumption (and emissions) |

6 | Wang and Meng (2012) [7] | V | U | Total route service cost minimization | TFC; TOC; TVC | Waiting time; Handling time | Iterative Optimization Algorithm | Consideration of waiting time and handling time uncertainties due to port congestion |

7 | Wang and Meng (2012) [66] | U | U | Total route service cost minimization | TFC; TOC | Waiting time; Handling time; Sailing time | Iterative Optimization Algorithm | Sea contingency time and uncertainty in port time |

8 | Di Francesco et al. (2013) [67] | V | U | Total route service cost minimization | MSC; TIC; TFC; TOC; TPC; TVC | Waiting time; Handling time; Container demand | CPLEX | Consideration of uncertain port service times and empty container repositioning |

9 | Halvorsen-Weare et al. (2013) [68] | U | U | Total route service cost minimization | MSC; TIC; TFC; TOC; TPC; TVC | Sailing time; Demand | Xpress-IVE | Considered changing weather conditions |

10 | Du et al. (2015) [69] | U | F | Total fuel consumption minimization | MSC | Fuel consumption | Iterative Optimization Algorithm | Considering the impacts of adverse weather on the total fuel consumption |

11 | Kepaptsoglou et al. (2015) [70] | U | F | Total route service cost minimization | TFC; TOC; TPC | Sailing time | Metaheuristic | Consideration of potential impacts of severe weather on ship sailing times |

12 | Lee et al. (2015) [71] | V | U | Analyze the relationship amongst shipping time, bunker cost, and cargo delivery reliability | TFC | Waiting time; Handling time | Analytical Method | Proposed a methodology for ship scheduling with guaranteed reliability of cargo delivery |

13 | Ng (2015) [72] | F | U | Total route service cost minimization | MSC; REV; TOC | Container demand | CPLEX | Consideration of uncertain shipping demand in ship fleet deployment |

14 | Norlund et al. (2015) [73] | U | U | Total route service cost minimization | TFC; TOC | Waiting time; Handling time; Sailing time | Simulation-Optimization | Acceptable levels of emissions and costs could be achieved with the adequate robustness level |

15 | Song et al. (2015) [74] | V | U | Total route service cost minimization; Average schedule unreliability minimization; Annual total CO _{2} emissionminimization | MSC; TEC; TFC; TOC; TPC; TVC | Waiting time; Handling time | Metaheuristic | Developing a method to optimize the multiple objectives simultaneously |

16 | Wang (2015) [75] | F | U | Total delay minimization | N/A | Container demand | Heuristic | Consideration of heterogeneous fleet and container demand uncertainty |

17 | Aydin et al. (2017) [76] | V | U | Total route service cost minimization | TFC; TVC | Waiting time; Handling time | Dynamic Programming | Speed bunkering and optimization under port time uncertainty |

18 | Song et al. (2017) [77] | U | U | Total route service cost minimization; Reliability maximization | MSC; TFC; TIC; TOC; TVC | Waiting time; Handling time; Sailing time | Metaheuristic | A multi-objective model consisted of three service-reliability KPIs and two cost-related KPIs |

19 | Ng and Lin (2018) [78] | F | U | Total route service cost minimization | MSC; REV; TOC | Container demand | CPLEX | Conditional information was assumed to be available for container demand |

20 | Tan et al. (2018) [79] | U | F | Total fuel consumption cost minimization; Total ship turnaround time minimization | TFC; TOC | Sailing time | Analytical Method | Joint service schedule design and ship sailing speed optimization problem for inland shipping |

21 | Gurel and Shadmand (2019) [80] | V | U | Total route service cost minimization | TFC | Waiting time; Handling time | CPLEX | Heterogeneous ship fleet considerations |

22 | Tierney et al. (2019) [81] | U | V | Total route service cost minimization | TFC; TOC | Sailing time | GUROBI | Ship journey times were examined for a real-life liner shipping network |

23 | Liu et al. (2020) [82] | V | U | Total route service cost minimization | TFC; TIC; TOC | Container demand | Iterative Optimization Algorithms | Solving liner ship bunkering and speed optimization problem |

24 | Ding and Xie (2021) [83] | U | V | Total profit maximization | REV; TFC; TOC; TVC | Sailing time | Iterative Optimization Algorithm | Balancing chances of unexpected delays and tight timelines |

25 | Liu et al. (2021) [84] | U | V | Average waiting time minimization | N/A | Sailing time | Metaheuristic | Introduced the notion of a minimum safety time interval (MSTI) in order to decrease the number of constraints |

**Sailing Speed and Port Time**[V—Variable; F—Fixed; U—Uncertain];

**Objective Components**[MSC—Miscellaneous Costs; REV—Total Revenue; TEC—Total Ship Emission Cost; TFC—Total Fuel Consumption Cost; TIC—Total Container Inventory Cost; TOC—Total Ship Operational Cost; TPC—Total Port Handling Cost; TVC—Total Cost Associated with Violation of Port Time Windows].

a/a | Authors | Sailing Speed | Port Time | Objective | Objective Components | Recovery Strategies | Solution Approach | Notes/Important Considerations |
---|---|---|---|---|---|---|---|---|

1 | Paul and Maloni (2010) [100] | F | V | Total route service cost minimization | MSC; TFC; TOC; TPC | Ship re-routing | Heuristic | Disruptive event modeling at ports |

2 | Jones et al. (2011) [101] | V | V | Total cost minimization | MSC | Ship re-routing; Port skipping; Port skipping with container diversion | Heuristic | Proposed a decision support tool to emulate disruptions affecting the USA freight intermodal network |

3 | Brouer et al. (2013) [99] | V | V | Total route service cost minimization | MSC; TFC; TOC; TPC; TVC | Speed adjustment; Port skipping; Port swapping | CPLEX | Proving the VSRP to be NP-complete |

4 | Li et al. (2015) [102] | V | V | Total route service cost minimization | TFC; TVC | Speed adjustment; Port skipping; Port swapping | Dynamic Programming | Finding a suitable delay penalty function |

5 | Qi (2015) [103] | V | V | Total route service cost minimization | TFC; TVC | Speed adjustment; Port skipping; Port swapping | Dynamic Programming | Introduction of two major models, one for a single ship and another one for multiple ships in one network |

6 | Fischer et al. (2016) [104] | V | F | Total route service cost minimization | MSC; TFC; TOC; TVC | Speed adjustment; Rewards for early arrivals; Penalization of risky voyage start times | Heuristic | Addressing disruptions in the fleet deployment for roll-on roll-off shipping |

7 | Li et al. (2016) [9] | V | V | Total route service cost minimization | MSC; TFC; TPC; TVC | Speed adjustment; Handling rate adjustment; Port skipping | Dynamic Programming | Considering both regular and unexpected uncertain events |

8 | Cheraghchi et al. (2017) [94] | V | F | Total monetary loss minimization; Total delay minimization; Average speed compliance maximization | MSC; TFC; TVC | Speed adjustment | Metaheuristics | Ship schedule delays were analyzed using the historical AIS data |

9 | Hasheminia and Jiang (2017) [105] | V | U | Total delay minimization | N/A | N/A | Analytical Method | Ships with a larger number of containers had a lower risk of delays at the terminal |

10 | Cheraghchi et al. (2018) [106] | V | F | Total monetary loss minimization; Total delay minimization | TFC; TVC | Speed adjustment | Metaheuristics | Problem evaluation in three scenarios (i.e., scalability analysis, ship steaming policies, and voyage distance analysis) |

11 | Abioye et al. (2019) [107] | V | V | Total monetary loss minimization | REV; TFC; TIC; TOC; TPC; TVC | Speed adjustment; Port skipping | CPLEX | Capturing enforced regulations within emission control areas |

12 | Mudler and Dekker (2019) [108] | V | V | Total route service cost minimization | TFC; TVC | Speed adjustment; Port skipping; Extreme recovery actions | Heuristic | Allocation of buffer times |

13 | Mulder et al. (2019) [109] | V | V | Total route service cost minimization | MSC; TFC; TVC | Speed adjustment; Extreme recovery actions | Iterative Optimization Algorithm | Proposing a method for the integrated development of ship timetables; allocation of buffer times |

14 | Xing and Wang (2019) [110] | V | V | Total route service cost minimization | MSC; TFC; TPC; TVC | Speed adjustment; Handling rate adjustment; Port swapping; Port skipping without container diversion; Port skipping with container diversion | LINGO | The proposed ship schedule recovery options were categorized into three tiers |

15 | Cheraghchi et al. (2020) [111] | V | F | Total monetary loss minimization; Total delay minimization; Average speed compliance maximization | MSC; TFC; TVC | Speed adjustment | Metaheuristics | A multi-objective optimization model was developed using the Automatic Identification System (AIS) data |

16 | Abioye et al. (2021) [112] | V | V | Total monetary loss minimization | MSC; REV; TFC; TOC; TPC; TVC | Speed adjustment; Handling rate adjustment; Port skipping without container diversion; Port skipping with container diversion | BARON | Capturing various realistic scenarios of disruptions |

17 | De et al. (2021) [113] | V | V | Total profit maximization | REV; TEC; TFC; TPC | Ship re-routing; Port swapping | Heuristic | Deciding on ship scheduling actions and container operations; deciding on the location and amount of marine diesel oil and heavy fuel oil to be bunkered |

18 | Du et al. (2021) [114] | V | U | Total route service cost minimization | TFC; TOC; TVC | Speed adjustment | Heuristic | Development of a machine learning-based model |

**Sailing Speed and Port Time**[V—Variable; F—Fixed; U—Uncertain];

**Objective Components**[MSC—Miscellaneous Costs; REV—Total Revenue; TEC—Total Ship Emission Cost; TFC—Total Fuel Consumption Cost; TIC—Total Container Inventory Cost; TOC—Total Ship Operational Cost; TPC—Total Port Handling Cost; TVC—Total Cost Associated with Violation of Port Time Windows].

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Elmi, Z.; Singh, P.; Meriga, V.K.; Goniewicz, K.; Borowska-Stefańska, M.; Wiśniewski, S.; Dulebenets, M.A.
Uncertainties in Liner Shipping and Ship Schedule Recovery: A State-of-the-Art Review. *J. Mar. Sci. Eng.* **2022**, *10*, 563.
https://doi.org/10.3390/jmse10050563

**AMA Style**

Elmi Z, Singh P, Meriga VK, Goniewicz K, Borowska-Stefańska M, Wiśniewski S, Dulebenets MA.
Uncertainties in Liner Shipping and Ship Schedule Recovery: A State-of-the-Art Review. *Journal of Marine Science and Engineering*. 2022; 10(5):563.
https://doi.org/10.3390/jmse10050563

**Chicago/Turabian Style**

Elmi, Zeinab, Prashant Singh, Vamshi Krishna Meriga, Krzysztof Goniewicz, Marta Borowska-Stefańska, Szymon Wiśniewski, and Maxim A. Dulebenets.
2022. "Uncertainties in Liner Shipping and Ship Schedule Recovery: A State-of-the-Art Review" *Journal of Marine Science and Engineering* 10, no. 5: 563.
https://doi.org/10.3390/jmse10050563