This section presents a detailed description of the problem, which is addressed in this study, including the following aspects: (1) liner shipping route description; (2) fleet deployment; (3) service of vessels at ports; (4) fuel consumption; (5) container demand sensitivity; (6) port service frequency; (7) container inventory; (8) modeling emissions; and (9) objective of the liner shipping company. Note that all the major tactical-level decisions, which have to be addressed by a given liner shipping company, are captured throughout this study.

### Liner shipping route description

Each liner shipping route is generally served either by an individual liner shipping company or a set of different liner shipping companies (or alliances). The routes consist of multiple ports of call. The number of ports to be called by vessels may vary from one liner shipping route to another. Let *R* = {1, …, *n*^{1}} denote the set of port rotations (i.e., the order of ports to be called), while the set of ports to be visited at port rotation *r* will be further referred to as \( {P}_r=\left\{1,\dots, {n}_r^2\right\},r\in R \). Two consecutive ports *p* and *p* + 1 are connected via voyage leg *p*. Each liner shipping company generally provides details about the port rotations that are currently served by the company, including the following information: (1) names of ports of call; (2) countries of the ports; (3) distance to the subsequent port to be visited; and others. Each port of call under every port rotation has fixed vessel schedules (e.g., arrival times, handling times, departure times).

A liner shipping network with four port rotations is illustrated in Fig. 1. The itinerary of each port rotation forms a loop – i.e., each port rotation starts and ends at the same port. For instance, in port rotation “1”, vessels start their voyage at Tokyo, then stop at Nagoya, Yokkaichi, Kobe, and return to Tokyo. The number of ports of call for port rotation “1” is 4 (i.e., \( {n}_1^2 \) = 4). Note that a given port can be included in multiple port rotations. Hong Kong, for example, is included in port rotations “2” and “3”. Furthermore, a given port can be visited more than once in a single port rotation. Based on the provided example, Penang is called twice in port rotation “4”. These calls can be distinguished by referring to the port calling sequence.

### Fleet deployment

At the fleet deployment stage, the liner shipping company assigns the available vessel types for service of the port rotations. Let *V* = {1, …, *n*^{3}} be the set of vessel types to be used by the liner shipping company to serve the port rotations. Let *d*_{vr} = 1 if vessel type *v* is assigned to serve port rotation *r*. Generally, liner shipping companies assign one vessel type for service of each port rotation (Wang and Meng, 2017). The latter condition can be enforced using the following relationship:

$$ \sum \limits_{v\in V}{\boldsymbol{d}}_{rv}=1\forall r\in R $$

(1)

A vessel string of the selected vessel type to be assigned for service of a given port rotation is assumed to be homogeneous (i.e., vessels of the same type are assumed to have the same technical properties, including the maximum sailing speed, fuel consumption rate, emissions produced by the main and auxiliary vessel engines, and others). Although such an assumption may not be applicable in some instances (e.g., technical properties for vessels of the same type may vary due to differences in age, vessel structure, capacity, past maintenance/repairs, and other reasons), it is commonly used at the fleet deployment and vessel scheduling stages in the liner shipping literature and in practice as well (Wang and Meng, 2017). If the required number of vessels of a specific type to be deployed for service of a given port rotation exceeds the total number of available vessels of that type (\( {q}_v^{own-m},v\in V \) – vessels), which are owned by the liner shipping company, a given liner shipping company may charter vessels from the other liner shipping companies. In such case, the liner shipping company incurs an additional cost (\( {c}_v^{char},v\in V \) – USD/day) for chartering vessels.

### Service of vessels at ports

Certain arrival TWs are offered for the vessels by an MCT operator at each port of a given port rotation. Let \( {T}_{rp}=\left\{1,\dots, {n}_{rp}^4\right\},r\in R,p\in {P}_r \) be the set of available arrival TWs at port *p* of port rotation *r*. Two attributes are associated with each TW, offered at a given port, including the following: (1) \( {\tau}_{rp t}^{st},r\in R,p\in {P}_r,t\in {T}_{rp} \) – start of TW *t* at port *p* of port rotation *r* (hours); and (2) \( {\tau}_{rp t}^{end},r\in R,p\in {P}_r,t\in {T}_{rp} \) – end of TW *t* at port *p* of port rotation *r* (hours). The arrival TW is negotiated between the liner shipping company and each MCT operator. In certain cases, only one arrival TW can be offered to the liner shipping company at a given port (e.g., busy ports with frequent vessel arrivals). Vessels are expected to arrive within the agreed TW at each port. Similar to Fagerholt (2001), this study assumes that the arrival TWs at ports of call are “soft” (i.e., vessels are permitted to arrive before the TW starts and after the TW ends). In case of an early arrival, a vessel is expected to wait for service in the dedicated area of the port. In case of a late arrival, service of a vessel is expected to start upon its arrival; however, the MCT operator will impose a late arrival cost to the liner shipping company, which is proportional to the total late arrival hours (\( {\boldsymbol{\tau}}_{rp}^{late},r\in R,p\in {P}_r \) – hours) and the unit late arrival cost (\( {c}_{rp}^{late},r\in R,p\in {P}_r \) – USD/hour). The late arrival cost is imposed to the liner shipping company due to the fact that vessel arrival delays may significantly disrupt the port operations (e.g., cause congestion at a given port). The total late arrival cost (*LAC* – USD) can be estimated based on the following relationship:

$$ \boldsymbol{LAC}=\sum \limits_{r\in R}\sum \limits_{p\in {P}_r}{c}_{rp}^{late}{\boldsymbol{\tau}}_{rp}^{late} $$

(2)

Each MCT operator is assumed to offer different HRs to the liner shipping company for service of the arriving vessels. Let \( {H}_{rp t}=\left\{1,\dots, {n}_{rp t}^5\right\},r\in R,p\in {P}_r,t\in {T}_{rp} \) be the set of HRs at port *p* of port rotation *r* available during TW *t*. The handling productivity, associated with a given HR, determines the number of twenty-foot equivalent units (TEUs) to be handled at each vessel per hour and will be referred to as *ph*_{rpthv}, *r* ∈ *R*, *p* ∈ *P*_{r}, *t* ∈ *T*_{rp}, *h* ∈ *H*_{rpt}, *v* ∈ *V* (TEUs/hour). In certain cases, only one HR can be offered to the liner shipping company at a given port (e.g., the MCT operator may have to allocate the handling equipment for service of the other vessels during high-demand periods). If \( {\boldsymbol{QC}}_{rp}^{PORT},r\in R,p\in {P}_r \) (TEUs) is the quantity of containers to be handled at port *p* of port rotation *r*, the handling time for vessels of type *v* at port *p* of port rotation *r* during TW *t* under HR *h* will be \( {\boldsymbol{\tau}}_{rp t hv}^{hand}=\frac{{\boldsymbol{QC}}_{rp}^{PORT}}{ph_{rp t hv}}\forall r\in R,p\in {P}_r,t\in {T}_{rp},h\in {H}_{rp t},v\in V \) (hours). The handling productivity and the corresponding handling cost (\( {c}_{rp t hv}^{hand},r\in R,p\in {P}_r,t\in {T}_{rp},h\in {H}_{rp t},v\in V \) – USD/TEU) are negotiated between the liner shipping company and the MCT operator at each port of a given port rotation. Let *x*_{rpthv} = 1 if HR *h* is selected for service of type *v* vessels at port *p* of port rotation *r* during TW *t*; then, the total port handling cost (*PHC* – USD) can be estimated based on the following relationship:

$$ \boldsymbol{PHC}=\sum \limits_{r\in R}\sum \limits_{p\in {P}_r}\sum \limits_{t\in {T}_{rp}}\sum \limits_{h\in {H}_{rp t}}\sum \limits_{v\in V}{c}_{rp t hv}^{hand}{\boldsymbol{QC}}_{rp}^{PORT}{\boldsymbol{x}}_{rp t hv} $$

(3)

### Fuel consumption

As previously discussed, the liner shipping company deploys a homogenous vessel fleet at each port rotation. The fuel consumption rate of the main vessel engines, which turn vessel propellers, is expected to be the same for homogenous vessels (Wang and Meng, 2017). Note that the fuel consumption rate is influenced by a number of different factors, including weather conditions, payload, sailing speed, and vessel geometric characteristics (Wang and Meng, 2012a; Kontovas, 2014). However, the vessel sailing speed was found to be the major factor influencing the fuel consumption (Wang and Meng, 2012a). The design fuel consumption per nautical mile (nmi) for vessels of type *v* at voyage leg *p* of port rotation *r* (\( {\boldsymbol{f}}_{rpv}^{design},r\in R,p\in {P}_r,v\in V \) – tons/nmi) can be estimated based on the following relationship (Wang and Meng, 2012a):

$$ {\boldsymbol{f}}_{rp v}^{design}=\frac{\gamma_v{\left({\boldsymbol{\vartheta}}_{rp}\right)}^{\left({\alpha}_v-1\right)}}{24}\forall r\in R,p\in {P}_r,v\in V $$

(4)

where:

*α*_{v}, *γ*_{v}, *v* ∈ *V* – are the coefficients of the fuel consumption function for vessel type *v*;

*ϑ*_{rp}, *r* ∈ *R*, *p* ∈ *P*_{r} – is the vessel sailing speed at voyage leg *p* of port rotation *r* (knots).

Some of the liner shipping studies have highlighted that payload of vessels is one of the major predictors of fuel consumption by vessel engines (Psaraftis and Kontovas, 2013; Kontovas, 2014). Assuming that the design fuel consumption is based on the maximum payload capacity, the fuel consumption of vessels (*f*_{rpv}, *r* ∈ *R*, *p* ∈ *P*_{r}, *v* ∈ *V* – tons/nmi) can be estimated from the following relationship (MAN Diesel & Turbo, 2012; Kontovas, 2014; Adland and Jia, 2016):

$$ {\displaystyle \begin{array}{c}{\boldsymbol{f}}_{rp v}={\boldsymbol{f}}_{rp v}^{design}\bullet {\left(\frac{{\boldsymbol{QC}}_{rp}^{SEA}\bullet AWC+{LWT}_v}{TWC_v+{LWT}_v}\right)}^{\frac{2}{3}}=\frac{\gamma_v{\left({\boldsymbol{\vartheta}}_{rp}\right)}^{\left({\alpha}_v-1\right)}}{24}\bullet {\left(\frac{{\boldsymbol{QC}}_{rp}^{SEA}\bullet AWC+{LWT}_v}{TWC_v+{LWT}_v}\right)}^{\frac{2}{3}}\\ {}\forall r\in R,p\in {P}_r,v\in V\end{array}} $$

(5)

where:

\( {\boldsymbol{QC}}_{rp}^{SEA},r\in R,p\in {P}_r \) – is the quantity of containers to be transported at voyage leg *p* of port rotation *r* (TEUs);

*AWC* – is the average weight of cargo inside a standard 20-ft container (tons);

*LWT*_{v}, *v* ∈ *V* – is the empty weight of a vessel of type *v* (tons);

*TWC*_{v}, *v* ∈ *V* – is the total weight of containers that can be loaded on a vessel of type *v* (tons).

The total vessel fuel consumption cost (*FCC* – USD) can be estimated based on the total fuel consumption, the unit fuel cost (*c*^{fuel} – USD/ton), and the voyage leg length (*l*_{rp}, *r* ∈ *R*, *p* ∈ *P*_{r} – nmi) using the following relationship:

$$ \boldsymbol{FCC}={c}^{fuel}\sum \limits_{r\in R}\sum \limits_{p\in {P}_r}\sum \limits_{v\in V}{l}_{rp}{\boldsymbol{f}}_{rp v}{\boldsymbol{d}}_{rv} $$

(6)

Note that the fuel consumption of the auxiliary vessel engines (that are commonly used to provide power on board the vessels) does not significantly fluctuate throughout the vessel voyage at a given port rotation and is accounted for in the vessel operational cost. Once a sailing speed is selected for a given voyage leg, it is assumed to remain the same while the vessel is sailing at that voyage leg. This study does not account for the factors that may result in fluctuations of the vessel sailing speed along a given voyage leg (e.g., wind speed, adverse weather conditions, height of waves, main engine malfunctioning, errors of the crew).

Several factors influence the vessel sailing speed choice. The highest possible sailing speed is selected according to the capacity of the main vessel engines (Psaraftis and Kontovas, 2013). The lowest sailing speed, on the other hand, is typically selected to decrease potential wear of the main vessel engines (Wang et al., 2013). Decreasing the vessel sailing speed will lead to reduction in the fuel consumption, and, thereby, the fuel consumption cost as well as the amount of emissions that are produced in sea. However, reducing the sailing speed will further cause an increase in the total transit time of containers along with the associated inventory costs. The latter will cause an increase in the total turnaround time of vessels, which will further necessitate the liner shipping company to deploy more vessels in order to guarantee a certain port service frequency. If more vessels are deployed, the total vessel operational/chartering cost will become higher. The total vessel turnaround time can be decreased by requesting HRs with higher handling productivities at ports of call. However, the latter will lead to increasing port handling cost at each port rotation and increasing amount of emissions from the handling equipment. Therefore, vessel sailing speeds at each port rotation should be selected by considering all the aforementioned factors and associated tradeoffs. The sailing time at voyage leg *p* of port rotation *r* (\( {\boldsymbol{\tau}}_{rp}^{sail},r\in R,p\in {P}_r \) – hours) can be estimated based on the voyage leg length and the vessel sailing speed using the following relationship:

$$ {\boldsymbol{\tau}}_{rp}^{sail}=\frac{l_{rp}}{{\boldsymbol{\vartheta}}_{rp}}\forall r\in R,p\in {P}_r $$

(7)

### Container demand sensitivity

Inspired by Cheaitou and Cariou (2019), this study considers the container demand at ports (\( {\boldsymbol{QC}}_{rp}^{PORT},r\in R,p\in {P}_r \) – TEUs) to be elastic and sensitive to the vessel sailing speed. Generally, customers transport a larger amount of cargo if the vessel sailing speed at a voyage leg is higher. Thus, the relationship between the container demand at ports and the vessel sailing speed can be expressed by the following equation (Cheaitou and Cariou, 2019):

$$ {\boldsymbol{QC}}_{rp}^{PORT}={\alpha}_{rp}^{dem}-\frac{\beta_{rp}^{dem}}{{\boldsymbol{\vartheta}}_{rp}}\forall r\in R,p\in {P}_r $$

(8)

where:

\( {\alpha}_{rp}^{dem},{\beta}_{rp}^{dem},r\in R,p\in {P}_r \) – are the coefficients of the container demand sensitivity to the vessel sailing speed at port *p* of port rotation *r*.

Upon arrival at a port of call, the import containers of the respective port (from the perspective of the MCT operator) will be unloaded from the vessels, and the export containers of the respective port will be loaded on board the vessels. Therefore, the quantity of containers to be transported at the voyage legs (\( {\boldsymbol{QC}}_{rp}^{SEA},r\in R,p\in {P}_r \) – TEUs) can be determined from the following relationships:

$$ {\displaystyle \begin{array}{c}{\boldsymbol{QC}}_{r\left(p+1\right)}^{SEA}={\boldsymbol{QC}}_{rp}^{SEA}-{\boldsymbol{QC}}_{r\left(p+1\right)}^{PORT}\bullet {Import}_{r\left(p+1\right)}+{\boldsymbol{QC}}_{r\left(p+1\right)}^{PORT}\bullet \left(1-{Import}_{r\left(p+1\right)}\right)\forall r\in R,\\ {}p\in {P}_r,p<{n}_r^2\end{array}} $$

(9)

$$ {\boldsymbol{QC}}_{r(1)}^{SEA}={QC}_r^{SEA-0}-{\boldsymbol{QC}}_{r(1)}^{PORT}\bullet {Import}_{r(1)}+{\boldsymbol{QC}}_{r(1)}^{PORT}\bullet \left(1-{Import}_{r(1)}\right)\forall r\in R $$

(10)

where:

*Import*_{rp}, *r* ∈ *R*, *p* ∈ *P*_{r} – is the portion of import containers at port *p* of port rotation *r*;

\( {QC}_r^{SEA-0},r\in R \) – is the total quantity of containers on board the vessel before it is moored at the first port of call (i.e., port “1”) under port rotation *r* (TEUs).

Note that the quantity of containers on board a given vessel must not exceed the total cargo carrying capacity of that vessel type. Hence, the following relationship must be maintained:

$$ {\boldsymbol{QC}}_{rp}^{sea}\bullet AWC\le {TWC}_v+M\left(1-{\boldsymbol{d}}_{rv}\right)\forall r\in R,p\in {P}_r,v\in V $$

(11)

where:

*M* – is a large positive number.

### Port service frequency

Generally, the port service frequency is established based on the existing demand at each port of a given port rotation. In order to guarantee the agreed port service frequency at each port rotation, the liner shipping company has to assure that the following relationship will be maintained (Wang et al., 2014; Alharbi et al., 2015; Dulebenets and Ozguven, 2017):

$$ 24{\boldsymbol{\phi}}_r{\boldsymbol{q}}_r=\sum \limits_{p\in {P}_r}{\boldsymbol{\tau}}_{rp}^{sail}+\sum \limits_{p\in {P}_r}\sum \limits_{t\in {T}_{rp}}\sum \limits_{h\in {H}_{rp t}}\sum \limits_{v\in V}{\boldsymbol{\tau}}_{rp t hv}^{hand}{\boldsymbol{x}}_{rp t hv}+\sum \limits_{p\in {P}_r}{\boldsymbol{\tau}}_{rp}^{wait}\forall r\in R $$

(12)

where:

*q*_{r}, *r* ∈ *R* – is the total number of vessels that will be deployed for service of port rotation *r* (vessels);

*ϕ*_{r}, *r* ∈ *R* – is the port service frequency for port rotation *r* (days);

24 – is the total number of hours in a day (hours/day)

\( {\boldsymbol{\tau}}_{rp}^{wait},r\in R,p\in {P}_r \) – is the vessel waiting time at port *p* of port rotation *r* (hours).

The right-hand side of equality (12) represents the total vessel turnaround time at port rotation *r* (i.e., the total time required by the vessels, which were allocated for that port rotation, to visit all the ports of call and return to the first port of call). The vessel turnaround time at each port rotation is composed of the following components: (i) total vessel sailing time over all the voyage legs of a given port rotation; (ii) total port handling time over all the ports of a given port rotation; and (iii) total port waiting time over all the ports of a given port rotation. The required number of vessels to be deployed for service of a given port rotation can be further determined by dividing the total turnaround time by the port service frequency and the integer “24” (which denotes the total number of hours in a day). As previously discussed, the vessels, used for service of a given port rotation, can be owned by the liner shipping company and/or chartered from other liner shipping companies. It is assumed that the vessels, deployed for a given port rotation, are homogeneous and have the same unit vessel operational cost (\( {c}_v^{oper},v\in V \) – USD/day). The total vessel operational cost (*VOC* – USD) and the total number of vessels that will deployed for service of port rotation *r* (*q*_{r}, *r* ∈ *R* – vessels) can be estimated based on the following relationships:

$$ \boldsymbol{VOC}=\sum \limits_{r\in R}\sum \limits_{v\in V}{c}_v^{oper}{\boldsymbol{q}}_{rv}^{own}{\boldsymbol{\phi}}_r $$

(13)

$$ {\boldsymbol{q}}_r=\sum \limits_{v\in V}\left({\boldsymbol{q}}_{rv}^{own}+{\boldsymbol{q}}_{rv}^{char}\right)\forall r\in R $$

(14)

where:

\( {\boldsymbol{q}}_{rv}^{own},r\in R,v\in V \) – is the total number of vessels of type *v*, owned by the liner shipping company, that will be deployed for service of port rotation *r* (vessels);

\( {\boldsymbol{q}}_{rv}^{char},r\in R,v\in V \) – is the total number of vessels of type *v*, chartered by the liner shipping company, that will be deployed for service of port rotation *r* (vessels).

The total vessel chartering cost (*VCC* – USD) can be estimated based on the unit chartering cost (\( {c}_v^{char},v\in V \) – USD/day), the number of chartered vessels, and the established port service frequency using the following relationship:

$$ \boldsymbol{VCC}=\sum \limits_{r\in R}\sum \limits_{v\in V}{c}_v^{char}{\boldsymbol{q}}_{rv}^{char}{\boldsymbol{\phi}}_r $$

(15)

### Container inventory

Two types of inventory costs are considered in this study: (i) inventory cost in sea; and (ii) inventory cost at ports of call. The inventory cost in sea is proportional to the total transit time of containers, transported between consecutive ports of call. Similarly, the inventory cost at ports is proportional to the total waiting time and the total handling time of containers at ports. Generally, sailing time of vessels in sea is larger as compared to waiting and handling times of vessels at ports; therefore, the inventory cost in sea is expected to be larger than the inventory cost at ports of call. The total container inventory costs in sea and at ports of call (denoted as *CIC*^{SEA} – USD and *CIC*^{PORT} – USD, respectively) can be estimated based on the following relationships (Wang et al., 2014; Dulebenets, 2018a):

$$ {\boldsymbol{CIC}}^{SEA}=\sum \limits_{r\in R}\sum \limits_{p\in {P}_r}{c}^{inv}{\boldsymbol{QC}}_{rp}^{SEA}{\boldsymbol{\tau}}_{rp}^{sail} $$

(16)

$$ {\boldsymbol{CIC}}^{PORT}=\sum \limits_{r\in R}\sum \limits_{p\in {P}_r}{c}^{inv}{\boldsymbol{QC}}_{rp}^{SEA}{\boldsymbol{\tau}}_{rp}^{wait}+\sum \limits_{r\in R}\sum \limits_{p\in {P}_r}\sum \limits_{t\in {T}_{rp}}\sum \limits_{h\in {H}_{rp t}}\sum \limits_{v\in V}{c}^{inv}\left({\boldsymbol{QC}}_{rp}^{SEA}-{\boldsymbol{QC}}_{rp}^{PORT}\right){\boldsymbol{\tau}}_{rp t hv}^{hand}{\boldsymbol{x}}_{rp t hv} $$

(17)

where:

*c*^{inv} – is the unit inventory cost (USD/TEU/hour).

### Modeling emissions

In order to improve the environmental sustainability of container shipping, the liner shipping company has to account for the emissions (e.g., *CO*_{2}, *SO*_{x}, *NO*_{x}, etc.), produced in sea and at ports of call throughout container handling. The amount of emissions produced at voyage leg *p* of port rotation *r* (\( {\boldsymbol{EP}}_{rp}^{SEA},r\in R,p\in {P}_r \) – tons) can be determined based on the fuel consumption and the emission factor in sea (*EF*^{SEA} – tons of emissions/ton of fuel) using the following relationship (Psaraftis and Kontovas, 2013; Kontovas, 2014; Dulebenets, 2018a):

$$ {\boldsymbol{EP}}_{rp}^{SEA}=\sum \limits_{v\in V}{EF}^{SEA}{l}_{rp}{\boldsymbol{f}}_{rp v}{\boldsymbol{d}}_{rv}\forall r\in R,p\in {P}_r $$

(18)

The amount of emissions produced at ports is dependent on the type of equipment used by the MCT operators. Handling productivity is also a major factor influencing the amount of emissions produced. If an HR with a high handling productivity is requested by the liner shipping company at a given port of call, more equipment will be required for container handling operations, which will increase the amount of emissions produced. The amount of emissions produced at port *p* of port rotation *r* (\( {\boldsymbol{EP}}_{rp}^{PORT},r\in R,p\in {P}_r \) – tons) can be determined based on the quantity of containers to be handled at the port, the emission factor at the port (\( {EF}_{rphv}^{PORT},r\in R,p\in {P}_r,h\in {H}_{rpt},v\in V \) – tons of emissions/TEU), and the selected HR using the following relationship (Tran et al., 2017; Dulebenets, 2018c):

$$ {\boldsymbol{EP}}_{rp}^{PORT}={\boldsymbol{QC}}_{rp}^{PORT}\sum \limits_{t\in {T}_{rp}}\sum \limits_{h\in {H}_{rp t}}\sum \limits_{v\in V}\left({EF}_{rp hv}^{PORT}{\boldsymbol{x}}_{rp t hv}\right)\forall r\in R,p\in {P}_r $$

(19)

The total emission cost in sea (*EC*^{SEA} – USD) and the total emission cost at ports of call (*EC*^{PORT} – USD) can be calculated based on the total amount of emissions, produced in sea as well as at ports of call, and the unit emission cost (*c*^{emis} – USD/ton) using the following relationships:

$$ {\boldsymbol{EC}}^{SEA}={c}^{emis}\sum \limits_{r\in R}\sum \limits_{p\in {P}_r}{\boldsymbol{EP}}_{rp}^{SEA} $$

(20)

$$ {\boldsymbol{EC}}^{PORT}={c}^{emis}\sum \limits_{r\in R}\sum \limits_{p\in {P}_r}{\boldsymbol{EP}}_{rp}^{PORT} $$

(21)

Note that eqs. (18)–(21) can be applied for modeling the major pollutants, produced by vessels in sea and by the container handling equipment at ports.

### Objective of the liner shipping company

This study assumes that the objective of the liner shipping company is to maximize the total profit. Such an assumption is in line with the previously conducted liner shipping studies (Giovannini and Psaraftis, 2019; Dulebenets et al., 2019). The total profit can be calculated as a difference between the total revenue and the total route service cost. The total revenue (*REV* – USD) can be estimated based on the quantity of containers to be delivered to ports of call at the considered port rotations and the unit freight rate (\( {c}_{rp}^{rev},r\in R,p\in {P}_r \) – USD/TEU) using the following relationship (Giovannini and Psaraftis, 2019):

$$ \boldsymbol{REV}=\sum \limits_{r\in R}\sum \limits_{p\in {P}_r}{c}_{rp}^{rev}{\boldsymbol{QC}}_{rp}^{PORT} $$

(22)

As for the total route service cost, the key route service cost components, found in the liner shipping literature, are considered in this study, which include: (1) vessel operational cost; (2) vessel chartering cost; (3) port handling cost; (4) port late arrival cost; (5) fuel consumption cost; (6) container inventory costs in sea and at ports of call; and (7) emission costs in sea and at ports of call. In order to maximize the total profit, the following major decisions have to be taken into consideration by the liner shipping company: (1) the number of vessels of each type, owned by the liner shipping company, that will be deployed for each port rotation; (2) the number of vessels of each type, chartered by the liner shipping company, that will be deployed for each port rotation; (3) port service frequency at each port rotation; (4) the sailing speed at each voyage leg of each port rotation; (5) the arrival TW at each port of call for each port rotation; and (6) the HR at each port of call for each port rotation.