An overview of the previous research
A substantial research has been dealing with the analysis and modelling of the liner container shipping networks and intensively published in the dedicated scientific and professional literature. In this rather narrow overview, this research has been broadly classified into three set categories dealing with: a) Analyzing of developments of the liner-shipping container networks including an overview of the research approach; b) Modelling and optimization of the operational and economic performances of the liner shipping networks and their components - nodes (ports) and links (lines/routes) connecting them; c) Design of MPC (Multi-Port-Calling) and H&S (Hub & Spoke) shipping networks; and d) The economics and geographical characteristics of the liner container shipping networks. In all considered cases, these networks have consisted of the large seaports connected by the intercontinental shipping lines/routes operated by the container ships of different size and payload capacity including the mega (ULC) ones.
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a)
The research on analyzing of developments of the liner-shipping container networks and the maritime transport overall from different perspectives including an overview of the research approach has been recently intensified based on the available evidence and data. For example, this has included analysis of development of different types of services provided by the liner shipping networks, proposing a global snapshot of the world-shipping network, and elaborating the interrelationship between developments of the liner shipping networks and the seaports (Ducruet and Notteboom 2012). In some way, this has been complemented by some kind of the multidisciplinary approach towards analyzing the maritime transport including its international and national characteristics, financial performances, skills of workforce involved, position of seaports in transition, logistics, and regulatory framework (Leggate et al. 2012). In addition, more the case-based and focused research has dealt with analyzing the spatial pattern of the China’s international shipping network along the so-called “the 21st Century Maritime Silk Road”. The attributes considered have been the network growth, hierarchy regarding types of shipping services, and phases/milestones of development. The navigational data for the period mid-1990s to 2016 has been used. The results have indicated that the spatial expansion of the China’s international shipping network has generally been taking place also driven by inclusion of more foreign seaports into the “Maritime Silk Road” network configuration. The role of the large seaports such as Hong Kong and Shanghai in such growing network has also changed including more diversification and consequent managerial implications (Wang et al. 2018). As well, the specific research has been dealing with the research of liner container shipping networks over the period 1967–2013. The source of information has been 294 papers published in the academic/scientific journals. The results have indicated the gradual diversification of the research from the prevailing economic consideration at the beginning to the more multi-disciplinary approach also integrating the seaport-related operations and multimodal networks serving the freight/cargo demand door-to-door towards the end of the considered period. Finally, the co-authorship in carrying out this research has been increasing over time although the approach has remained fragmented due to the strong disciplinary specialization (Lau et al. 2018; Wang et al. 2018).
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b)
Research on modelling, and optimization of the operational and economic performances of the liner shipping networks has generally dealt with explaining the factors influencing growth in the container ship size means by the liner service cash flow model. The results have shown that economies of scale have and will continue to be one of the main driving forces behind deployment of the larger container ships. In addition, the optimal ship size has been dependent of the characteristics of port terminals and trade route(s)/market(s) it has been deployed (Christa et al. 2008). The additional research has investigated viability of the mega-container ships operating in H&S compared to that of the conventional ships operating in MPC shipping networks by using a non-zero sum two-person game. The results have indicated that the viability of mega container ships operating in the H&S case-based network(s) has been strongly influenced by the relationships between ship size, the level and structure of costs of ship operator(s), and the freight rate influenced by the market competition (Imai et al. 2006). In addition, estimating the quality of liner shipping operations and particularly those at ports and their container terminals has also been highly on the research agenda. The corresponding research has resulted in developing the methodology based on the measures of the given port’s time and cost performances and their balancing indicating the need for eventual investments (in additional berths and/or terminals). The methodology has expected to be useful for the decision/policy makers in the given context (Bassan 2007). As well, the research has been carried out embracing: estimating and modelling possible ways of shortening if not optimizing the door-to-door delivery time of TEU flows - through shortening the ships’ sailing and berth time including use of the alternative routes; reducing the container transhipment time by applying different transhipment policies (existing forward and many-to-one and new backward and many-to-one); and through the innovative stacking and reshuffling strategies in the port container terminals. The developed empirical, analytical, and optimization models and their results have been expected to contribute to improvement of the route design, development of ship fleet, optimization of sailing speed, and disruption management (Grida et al. 2018). The research results have also indicated potential contribution to reducing the total costs of delivering TEU flows thanks to sailing along the alternative routes (Furuichi and Otsuka 2018), applying innovative transhipment policies (Du et al. 2017) and the innovative stacking and reshuffling strategies of containers (Gharehgozli et al. 2017). These all would substantively reduce the transhipment time of TEU flows and other cargo, in the former case at very low costs, and in the latter case by improving efficiency of handling mega ships at the port container terminals. Last but not least, the most recent research has dealt with different aspects of consideration including predominantly simulation and analytical approach to analyzing and modelling of the national and international maritime transport networks regarding their past and present development, topology and spatial distribution (particularly emergence and strengthening of the global container shipping networks), and influence on the regional development (Ducruet 2018).
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c)
Research on explicitly design and implicitly optimizing particular performances of MPC and H&S shipping networks has been rather substantive as pointed out in some overview/review papers (Christiansen et al. 2013; Meng et al. 2014). This research has been roughly categorized into design of the line MPC and H&S shipping networks. As far as design of the former is concerned, the common objectives have been optimization of the network costs by ship scheduling and goods/freight/cargo routing separately (Agarwal and Ergun 2008) and jointly (Alvarez 2009), including providing an easy access to the sources of data needed for design (Brouer et al. 2014). A mixed-integer programming model and its modifications have been primarily used as the optimization tool. In addition, this has been research on design of the network(s) of liner shipping alliances respecting the costs due to sharing the ship capacities among alliance partners (Agarwal and Ergun 2010; Davison 2014). As well, a design problem of a single line shipping service route including repositioning of empty containers (Shintani et al. 2007; Song and Dong 2013), and compact formulation for the line shipping network design based on service flows (Plum et al. 2014) have been elaborated. In addition to the above-mentioned research by Imai et al. (2006), some research has also dealt with design of the line H&S shipping networks. This has included a competitive hub location problem for designing the network(s) (Gelareh et al. 2010), analysing the effects of ship fleet deployment on the network design respecting economies of scale in ship size (Gelareh and Pisinger 2011), a design problem with combined MPC and H&S network while considering empty container repositioning (Meng and Wang 2011), determining the optimal ship routing, size, and sailing frequency for the given H&S network (Hsu and Hsieh 2007), and design of the H&S networks by deploying the concept of the main port and few constraints such as the multi-type container shipment and transit time. A mixed-integer programming model with nonconvex multi-linear terms has been developed for the proposed problems and in some cases solved by an efficient genetic algorithm based on a multi-stage decomposition approach (Jianfeng et al. 2015).
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d)
Research on the economics and geographical characteristics of the liner container-shipping networks has been substantive too. For example, one of the most illustrative has provided a review of the characteristics of liner container-shipping networks regarding the types of services provided by shipping lines including dynamics of their design and configuration. In addition, a global overview of the world’s liner shipping networks based on the ships’ movement data and the position of seaports in the liner container-shipping networks regarding to the concept of centrality, hierarchy, and factors of selection have been elaborated (Ducruet and Notteboom 2012). The other has developed the model for predicting container flows round the world along shipping routes passing through 437 container ports and been served by 800 shipping companies. The profit maximization shippers have been assumed to make the route choice. The simple logit model containing the generalized user/shipper costs has been the basis for the route choice model. The generalized cost function has represented the cost of route including the port costs, the cost of transhipment and transport, and the value of transport time. The model has been estimated using the data from different sources/databases. Then, the scenario approach (8 scenarios) has been applied to test sensitivity of throughput of particular ports or transport services provided by shipping lines on the route choice. The results have mainly related to European ports (Tavasszy et al. 2011). As well, the specifically focused research has also investigating the impacts of fuel costs on the design of liner shipping networks. A set of simple analytical models has been developed to estimate: the container ships’ turnaround time depending on the length of route, the number and time of port calls, and sailing speed; the threshold route turnaround time depending on the number of deployed ships and transport service frequency; and the minimum required ships’ sailing speed needed to operate the liner services under above-mentioned conditions - service frequency, the number of port calls, turnaround time, and the size of deployed ship fleet. The models have been applied to the Europe-Far East container-shipping route. The results have indicated the permanent need for balancing between sailing speed, the turnaround time, the number of port calls, size of deployed ship fleet, and finally the overall design and costs of transport services offered to users/shippers shown to be highly dependent on the fuel costs (Notteboom and Vernimmen 2009). In addition, the research analysing the liner container-shipping alliances has been carried out. The three large alliances - Grand Alliance, New World Alliance, and CKYH Alliance have been analysed and compared mainly in terms of 17 indicators of operational performances and motives of shipping carriers to join the alliance depending on its market position in the international/intercontinental shipping markets (Panayides and Wiedmer 2011).
Objectives and assumptions
This above-mentioned rather narrowly focused literature overview has indicated that there has not been an explicit and straightforward academic and professional research dealing with the multidimensional examination of performances of the trunk lines/routes of liner container-shipping networks serving the intercontinental supply chains by the ships of different size and payload capacity. The present research aims at filling in this gap in some way. Therefore, its objectives are just to, according to the concept shown on Fig. 4, perform the multidimensional examination of performances of a trunk line/route of a given liner container-shipping network serving a has implies defining and developing the analytical models of indicators of these performances expected to enable their estimation and carrying out the sensitivity analysis respecting changes of the most influencing factors/variables, in this case the payload capacity and the sailing speed of container ships. Thus, the models of indicators of particular performances of the above-mentioned trunk line/route are based on the following assumptions (Daganzo 2005; Hall 1993; Janić et al. 1999; Janić 2005):
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The supplier seaport of a given trunk line/route is ultimately the “production location”, i.e. origin and the customer seaport is ultimately the “consumption location”, i.e., destination, of the given volumes of TEU flows;; these can be the main seaports or the hubs of corresponding networks of the liner shipping carriers or their alliances;
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The supplier seaport generates a series of successive “parcels” of the TEU flows to be transported to the customer seaport during the specified period of time exclusively by a given container ship fleet; this implies that, independently of the payload capacity of the ships in the fleet(s), there is always sufficient demand justifying always economically feasible direct and reasonably frequent transport services along the trunk line/route connecting its two end seaports; and vice versa, there is always the sufficient ship fleet capacity offered to satisfy the generated volumes of TEU flows;
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The size (volume, weight) of a single “parcel” of TEU flows is always less than or at most equal to the corresponding payload capacity of a deployed ship(s);
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The container ship fleet consisting of the ships of same size, payload capacity, and approximately the same average load factor perform direct liner shippingFootnote 2 between the supplier and the customer seaport without the multi-port calling;
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The indicators of infrastructural and technical/technological performances of the trunk line/route are considered as inputs for the models of its operational, economic, environmental, and social performances; and
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The exclusive use of a given ship fleet along the given trunk line/route implies “all-or-nothing principle” of transporting the given volumes of TEU flows under given conditions.
Structure of the models
Infrastructural performances and technical/technological performances
The infrastructural performances relate to the physical/spatial characteristics of the seaports and their container terminals at both ends of the given trunk line/route. These are the number and size of the berths to handle particular categories of ships and the size and capacity of the container terminals handling and storing the TEU flows. The indicators of these performances are not particularly modelled but assumed as given for modelling the indicators of the operational, economic, environmental, and social performances.
The technical/technological performances reflect the characteristics of the containers ships operating along the given trunk line/route. These are dimension (length, beam, height, and draft), payload capacity, and the number of engines and their power. In addition, these are the capacity of facilities and equipment for loading/unloading (cranes) and storing (dedicated space/depots at the container terminals) TEU flows at the seaports before and after their transportation along the given trunk line/route of the network. Similarly as at the infrastructural performances, their indicators are not particularly modelled but assumed as given for modelling the indicators of the operational, economic, environmental, and social performances.
Operational performances
The operational performances relate the volumes of TEU flows to be transported along the given trunk line/line during a given period of time; the frequency of transport services based on the size of parcels of TEU flows taken per service frequency; the required container ship fleet (i.e. the type and number of ships deployed); and the ship fleet’s transport work and technical productivity - all under given conditions. Therefore, the indicators of operational performances of the given trunk line/route are considered to be: i) Transport service frequency; ii) Size of ship fleet; iii) Transport work; and iv) Technical productivity.
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i)
Transport service frequency (dep/τ):
The transport service frequency can be set up either to serve the given volumes of TEU flows or enable the regular scheduled transport services during the given time.
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a)
Serving the given volumes of TEU flows in the single direction:
$$ {f}_{ij}\left(\tau \right)=\frac{Q_{ij}\left(\tau \right)}{\lambda_{ij}\bullet {PL}_{ij}} $$
(1)
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b)
Enabling the regular scheduled transport services in the single direction during the given time:
$$ {f}_{ij}^{\ast}\left(\tau \right)=\frac{\tau }{h_{ij}\left(\tau \right)} $$
(2)
From Eq. 2, if the ship’s payload per frequency is considered as a TEU parcel, the total volumes of TEU flows, which can be transported during the given time, is estimated as:
$$ {Q}_{ij}^{\ast}\left(\tau \right)={\beta}_{ij}\left(\tau \right)\bullet \left[\mathit{\min}\left({f}_{ij}\left(\tau \right);{f}_{ij}^{\ast}\left(\tau \right)\right)\right]\bullet \left({\lambda}_{ij}\bullet {PL}_{ij}\right) $$
(3)
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ii)
Size of ship fleet (the number of deployed ships)
The size of ship fleet, i.e., the number of ships deployed along the given trunk line during the given time is estimated as follows:
$$ {N}_{ij}\left(\tau \right)={\beta}_{ij}\left(\tau \right)\bullet \left[\mathit{\min}\left({f}_{ij}\left(\tau \right);{f}_{ij}^{\ast}\left(\tau \right)\right)\right]\bullet \left[{t}_{i\mathrm{j}}\left({d}_{ij}\right)\right] $$
(4)
If each ship operates along the line/route in both directions at the approximately same load factor, the average ship’s turnaround time [tij(dij)] in Eq. 4 is estimated as follows:
$$ {\displaystyle \begin{array}{l}{t}_{ij}\left({d}_{ij}\right)={\tau}_{ij}+{\tau}_{ji}={\Delta}_{ij/1}+\frac{\lambda_{ij}\cdot {PL}_{ij}}{u_{i1}\cdot {\mu}_{i1}}+{D}_{di}+\frac{d_{ij}}{\left[{s}_{ij}\cdot {v}_{ij}\left({d}_{ij}\right)\right]}+{D}_{ij}+{D}_{aj}+{\Delta}_{ji/1}+\frac{\lambda_{ij}\cdot {PL}_{ij}}{u_{j1}\cdot {\mu}_{j1}}+\\ {}\kern0.84em +{D}_{dj}+{\Delta}_{ji/2}+\frac{\lambda_{ji}\cdot {PL}_{ij}}{u_{j2}\cdot {\mu}_{j2}}+\frac{d_{ji}}{\left[{s}_{ji}\cdot {v}_{ji}\left({d}_{ji}\right)\right]}+{D}_{ji}+{D}_{ai}+{\Delta}_{ij/2}+\frac{\lambda_{ji}\cdot {PL}_{ij}}{u_{i2}\cdot {\mu}_{i2}}\end{array}} $$
(5)
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iii)
Transport work (TEU-km/τ)
The transport work is the product between the total number of transported TEU parcels during the given time (τ), i.e., transport service frequency, and the corresponding distance. It is estimated as follows:
$$ {TW}_{ij}\left(\tau \right)={\beta}_{ij}\left(\tau \right)\bullet \left[\mathit{\min}\left({f}_{ij}\left(\tau \right);{f}_{ij}^{\ast}\left(\tau \right)\right)\right]\bullet \left({\lambda}_{ij}\bullet {PL}_{ij}\right)\bullet {d}_{ij} $$
(6)
iv) Technical productivity (TEU /τ
2
).
The technical productivity is the product between the total number of transported TEU parcels during the given time, transport service frequency, and the speed of their delivery throughout the chain. It is estimated as follows:
$$ {TPC}_{ij}\left(\tau \right)={\beta}_{ij}\left(\tau \right)\bullet \left[\mathit{\min}\left({f}_{ij}\left(\tau \right);{f}_{ij}^{\ast}\left(\tau \right)\right)\right]\bullet \left({\lambda}_{ij}\bullet {PL}_{ij}\right)\bullet \left({d}_{ij}/{\tau}_{ij}\right) $$
(7)
Where.
τ is the given time (day, week, year);
TU is the time unit (h, day);
Qij(τ) is the volume of TEU flows to be transported from the supplier seaport (i) to the customer seaport (j) of the trunk line/route during time (τ) (TEU/τ);
λij, PLij is the average load factor and the payload capacity, respectively, of a ship serving the line/route (ij) (the product of both is considered as a TEU parcel (λij ≤ 1.0) (−; TEU/ship);
hij(τ) is the average time between scheduling the regular transport services between the supplier seaport (i) and the customer seaport (j) during time (τ), (TU/dep);
βij(τ) is the proportion of planned/scheduled transport services realized between the supplier seaport (i) and the customer seaport (j) during time (τ) (βij(τ) ≤ 1.0) (−);
τij, τji is the average sailing time of a ship between the supplier seaport (i) and the customer seaport (j), and vice versa, respectively (TU/ship);
Δij/1, Δji/1 is the time before starting loading a TEU parcel at the supplier seaport (i) for the customer seaport (j) and before its unloading at the customer seaport (j) after arrived from the supplier seaport (i), respectively (TU);
Δji/2, Δij/2 is the time before starting loading a TEU parcel at the customer seaport (j) for the supplier seaport (i) and before its unloading at the supplier seaport (i) after arrived from the customer seaport (j), respectively (TU);
dij, dji is the length of trunk line/route, i.e., the sailing distance between the supplier seaport (i) and the customer seaport (j), and vice versa, respectively (nm) (nm - nautical mile; 1 nm = 1.852 km);
vij(dij), vji(dji) is the average (planned) ship’s steaming speed on the line/route (dij) and (dji), respectively (kts) (kt - knot; h - hour; 1 kt = 1 nm/h);
Ddi, DajDdj, Dai is the average departure delay of a given transport service in the supplier seaport (i) and the average arriving delay at the customer seaport (j), respectively, and vice versa, (TU/ship);
Dij, Dji is the average delay per transport service along the trunk line/route between the supplier seaport (i) and the customer seaport (j), and back, respectively (TU/ship);
μi1, μj1 is the capacity of loading and unloading ships at the supplier seaport (i) and the customer seaport (j), respectively (TEU/TU);
ui1, uj1 is the proportion of used loading and unloading capacity at the supplier seaport (i) and at the customer seaport (j), respectively (≤ 1.0);
μj2, μi2 is the loading and unloading capacity of ships at the customer seaport (j) and at the supplier seaport (i), respectively (TEU/TU);
uj2, ui2 is the proportion of used loading and unloading capacity at the customer seaport (j) and the supplier seaport (i), respectively (≤ 1.0); and.
sij, sjiis the portion of maintained average ship’s planned operating speed under given operating conditions along the trunk line/route (dij) and (dji), respectively (sij, sji ≤ 1.0) (−).
In Eq. 1 (a-d), the loading and unloading capacity of a ship at the supplier and/or the customer seaport (i) and (j) is estimated as follows:
$$ {\mu}_{i/j/1/2}={n}_{c/i/j}\bullet {m}_{c/i/j}\bullet {k}_{i/j}\bullet {\rho}_{c/i/j} $$
(8)
Where.
nc/i/j is the number of cranes per berth in the seaport (i) and/or in the seaport (j) (units/berth);
mc/i/j is the loading/unloading rate, i.e., the number of moves per crane per unit of time in the seaport (i) and/or in the seaport (j) (moves/h);
ki/j is the TEU-factorFootnote 3 in the seaport (i) and/or in the seaport (j) (−);
ρc/i/jis the average crane utilization rate in the seaport (i) and/or in the seaport (j) (≤ 1.0);
Economic performances
The economic performances generally relate to the total and the average inventory, handling, transport, and the external costs (externalities) along the given trunk line/route under given conditions. Therefore, the considered indicators of economic performances are: i) Inventory; ii) Handling; iii) Transport; iv) External; and v) Total costs, of a given TEU parcel transported along the given trunk line/route, estimated as follows:
$$ {\displaystyle \begin{array}{l}{C}_{ij/ INV}\left({\lambda}_{ij},{PL}_{ij}\right)={IT}_i\left({\lambda}_{ij},{PL}_{ij}\right)\cdot {\alpha}_i+\left({\lambda}_{ij}\cdot {PL}_{ij}\right)\cdot \left(\frac{d_{ij}}{s_{ij}\cdot {v}_{ij}\left({d}_{ij}\right)}+{D}_{ij}\right)\cdot {\alpha}_{ij}+\\ {}\kern13em +{IT}_j\left({\lambda}_{ij},{PL}_{ij}\right)\cdot {\alpha}_j\end{array}} $$
(9)
The first and the third term in Eq. 9 represent the inventory costs of a TEU parcel at the supplier seaport (i) and at the customer seaport (j), respectively. The second term represents the inventory, i.e., the cost of time of a TEU parcel while being in transportation between the seaports (i) and (j). From Fig. 3, the inventory time of a TEU parcel (λij·PLij) at the seaport (i) before being sent to the seaport (j) is estimated as follows:
$$ {IT}_i\left({\lambda}_{ij},{PL}_{ij}\right)=\frac{1}{2}\bullet {\left({\lambda}_{ij}\bullet {PL}_{ij}\right)}^2\bullet \left(\frac{1}{\theta_{ij}}+\frac{1}{u_{i1}\bullet {\mu}_{i1}}\right)+\left({\lambda}_{ij}\bullet {PL}_{ij}\right)\bullet \left({\overline{\tau}}_{d/ ij}+{\Delta }_{ij/1}+{D}_{di}\right) $$
(10)
Similarly, the inventory time of a TEU parcel (λij·PLij) at the seaport (j) after arrived from the seaport (i) is estimated as follows:
$$ {IT}_j\left({\lambda}_{ij},{PL}_{ij}\right)=\frac{1}{2}\bullet {\left({\lambda}_{ij}\bullet {PL}_{ij}\right)}^2\bullet \left(\frac{1}{u_{j1}\bullet {\mu}_{j1}}+\frac{1}{\theta_{ji}}\right)+\left({\lambda}_{ij}\bullet {PL}_{ij}\right)\bullet \left({D}_{aj}+{\Delta }_{ji/1}+{\overline{\tau}}_{d/ ji}\right) $$
(11)
The handling and transport costs of a TEU parcel include those of handling it at the supplier seaport (i) and at the customer seaport (j), and transporting between these seaports as follows:
$$ {C}_{ij/ TRA}\left({\lambda}_{ij},{PL}_{ij}\right)=\left({\lambda}_{ij}\bullet {PL}_{ij}\right)\cdotp \left[{c}_i+{c}_{ij}\left({\lambda}_{ij},{PL}_{ij}\right)\bullet \left(\frac{d_{ij}}{s_{ij}\bullet {v}_{ij}\left({d}_{ij}\right)}\right)+{c}_j\right] $$
(12)
The externalities are expressed in terms of the cost of emitted GHG from the fuel consumption as follows:
$$ {C}_{ij/e}\left({\lambda}_{ij},{PL}_{ij}\right)=\left\{ FC\left[{PL}_{ij};{v}_{ij}\left({d}_{ij}\right)\right]\bullet \sum \limits_{k=1}^K{e}_k\bullet {C}_{CO_2}\right\}\bullet \left({\lambda}_{ij}\bullet {PL}_{ij}\right) $$
(13)
The total costs are estimated as the sum of inventory, handling, transport, and external costs as follows:
$$ {C}_{ij}\left({\lambda}_{ij},{PL}_{ij}\right)={C}_{ij/ INV}\left({\lambda}_{ij},{PL}_{ij}\right)+{C}_{ij/ TRA}\left({\lambda}_{\mathrm{i}j},{PL}_{ij}\right)+{C}_{ij/e}\left({\lambda}_{ij},{PL}_{ij}\right) $$
(14)
Where.
θij, θji is the intensity of collecting and distributing the TEU parcels at the supplier seaport (i) and the customer seaport (j), respectively (TEU/TU) (TU ≡ day);
ci, cj is the average handling (loading/unloading/transhipment/storage) cost of a TEU parcel at the supplier seaport (i) and the customer seaport (j), respectively ($US/TEU);
cij(λij, PLij) is the average transport cost of a TEU parcel between the seaports (i) and (j) ($US/(TEU-day);
αi, αij, αj is the cost of a TEU parcel inventory time while at the supplier seaport (i), in transportation, and at the customer seaport (j), respectively ($US/TEU-TU).
\( {\overline{\tau}}_{d/ ij} \), \( {\overline{\tau}}_{d/ ji} \) is the average dwell time of a TEU parcel at the container terminal of the supplier seaport (i) before being sent to the customer seaport (j) and at the customer port (j) before being distributed to the ultimate customers, respectively (TU/TEU).
FC[PLij, vij(dij)] is the average fuel consumption by a container ship of the payload capacity (PLijj) operating along the trunk line/route (dij) at the steaming speed [vij(dij)] (tons/day);
ek is the emission rate of the (k)-th GHG from the fuel consumed by the container ship operating along the trunk line/route (dij) (ton of GHG/ton of fuel); and.
K is the number of various GHG emitted from the fuel consumed by the container ships operating along the trunk line/route (dij); and.
\( {C}_{CO_2} \) is the average social cost of emissions of GHG (CO2) ($US/ton).
The other symbols are analogous to those in Eq. 1 (a, b, c, d).
The average transport costs [cij(λij, PLij)] in Eq. 12 consist of the main components approximately including the ship’s: i) Fixed depreciation and insurance costs; ii) Fuel costs; and iii) Operational costs (crew, insurance, stores and lubes repairs and maintenance)Footnote 4 (Christa et al. 2008; Drewry 2017; UNCTAD 2017). By replacing the size of a TEU parcel (λij·PLij) by the volumes of TEU flows generated during time (τ) [Qij(τ) + Qji(τ)], the corresponding indicators of economic performances can be estimated from Eq. 2 (a, b, c, d, e, f).
Environmental performances
The environmental performances relate to the energy (fuel) consumption and the consequent direct and indirect emissions of GHG (Green House Gases) by the ships performing transport services and the area of land used by the trunk line/route infrastructure, in this case parts of the seaports and their container terminals enabling handling both ships and their payload (TEU parcels). Therefore, the indicators of environmental performances are considered to be: i) The ships’ fuel consumption and related emissions of GHG (Green House Gases); and ii) The land used. The ships’ and seaports’ waste and used water also having an impact on the environment are not considered.
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i)
Fuel consumption and emissions of GHG (Green House Gases) (tons)
$$ {FC}_{ij}\left({\lambda}_{ij},{PL}_{ij}\right)= FC\left[{PL}_{ij};{v}_{ij}\left({d}_{ij}\right)\right]/\left({\lambda}_{ij}\bullet {PL}_{ij}\right) $$
(15)
And
$$ {EM}_{ij}\left({\lambda}_{ij},{PL}_{ij}\right)={FC}_{ij}\left({\lambda}_{ij}\bullet {PL}_{ij}\right)\bullet \sum \limits_{k=1}^K{e}_k $$
(16)
Where all symbols are as in the previous Equations.
By combining Eqs. 5, 15, and 16, the corresponding total amounts can be estimated for the trunk line/route under given conditions.
ii) Land use
Container ships are handled at the berths (port’s seaside area) facing the seaport container terminals, which indirectly or directly occupy the land on the shore (landside). This area of land used by these terminals generally includes: i) The apron area; ii) The container storage and transfer area; and iii) The area for different buildings (container freight station, offices, workshops, etc.) (Gharehgozli et al. 2016; Ligteringen and Veslsink 2012). The areas of land used for the apron and the container storage are only estimated.
The land used for the berth area in the supplier seaport (i) and/or in the consumer seaport (j) is estimated as follows:
$$ {LU}_{ba/i/j}={L}_{q/i/j}\bullet {S}_{i/j} $$
(17)
Where.
Lq/i/j is the length of quay along the berths in the seaport (i) and/or in the seaport (j) (m); and.
Si/j is the width of a berth as the right-angle distance between the waterfront and the line of the storage yard in the seaport (i) and/or in the seaport (j) (m).
The length of quay (Lq/i/j) in Eq. 17 is estimated as follows (Ligteringen and Veslsink 2012):
$$ {\displaystyle \begin{array}{l}{L}_{q/i/j}=\left\{{L}_{s/\max }+2\cdot 15, for\kern0.5em {n}_{b/i/j}=1\right\}\\ {}\kern5em \left\{1.1\cdot {N}_{b/i/j}\cdot \left(\overline{L_s}+15\right)+15, for\kern0.5em {n}_{b/i/j}>1\right\}\end{array}} $$
(18)
Where.
Ls/max,\( {\overline{L}}_s \) is length of the largest and the average ship, respectively, using a single and few berths, respectively, in any seaport (m); and.
Nb/i/j is the number of berths in the seaport (i) and/or the seaport (j) intended to the container ships operating along the given trunk line/route (ij) (−).
The number of berths (Nb/i/j) in Eq. 18 is estimated based on Eq. 8 as follows:
$$ {N}_{b/i/j}=\frac{Q_{ij}\left(\tau \right)+{Q}_{ji}\left(\tau \right)}{\mu_{i/j/1/2}\bullet {n}_{h/i/j}\bullet {n}_{dyr/i/j}\bullet {\rho}_{b/i/j}} $$
(19)
Where.
nh/i/j is the berth’s operational time during the day in the seaport (i) and/or in the seaport (j) (h/TU) (h - hour; TU ≡ day);
ndyr/i/j is the berth’s operation time during the year in the seaport (i) and/or in the seaport (j) (TU/τ) (TU ≡ day; τ ≡ year); and.
ρb/i/j is the average utilization rate of berths in the seaport (i) and/or in the seaport (j) (≤ 1.0).
The other symbols are analogous to those in the previous Equations.
The land used for storing containers in the container terminals at the seaport (i) and/or the seaport (j) can be estimated as follows (Ligteringen and Veslsink 2012):
$$ {LU}_{cs/i/j}=\frac{\left[{Q}_{ij}\left(\tau \right)+{Q}_{ji}\left(\tau \right)\right]\bullet {\overline{t}}_{d/i/j}\bullet {A}_{TEU/i/j}}{n_{ct s/i/j}\bullet {n}_{ct/ dyr/i/j}\bullet {\rho}_{ct/i/j}} $$
(20)
ATEU/i/j is the area occupied by a container in the container terminal of the seaport (i) and/or the seaport (j) (m2/TEU);
ncts/i/j is the stacking height of containers in the container terminal of the seaport (i) and/or the seaport (j) (−); and.
ρct/i/j is the average utilization of the available space in the container terminal(s) of the seaport (i) and/or the seaport (j) (≤ 1.0).
The other symbols are analogous to those in the previous Equations.
The “total” land used for berths and storage areas of the container terminals at the seaport (i) or the seaport (j) can be estimated by Eqs. 17 and 19 as follows:
$$ {LU}_{TOT/i/j}={LU}_{ba/i/j}+{LU}_{cs/i/j} $$
(21)
Where all symbols are as in the previous Eqs.
This area of land used does not include that for transferring containers between the storage yard and the inland transport modes (road and rail), and that for different buildings (container freight station, offices, workshops, etc.).
Social performances
The social performances relate exclusively to the impacts of the given trunk line/route on the society, i.e., population, such as generally noise, congestion, and the safety of operations. The excessive noise generated by delivering the TEU flows between the ultimate suppliers and customers to/from the corresponding seaports by the inland transport modes and their systems can burden the neighbouring population. At the same time, congestion and delays can also occur at these modes and systems. In addition, some congestion and delays can also occur in handling ships at the line’s seaports. The safety of operations implies an inherent risk of losing ships, their payload, and lives of staff due to the traffic incidents/accidents happening at the seaports and along the trunk line/route connecting them. Consequently, the indicators of social performances considered to relate exclusively to its impacts are: i) Noise; ii) Congestion; and iii) Traffic incidents/accidents (i.e., Safety) (Janić, 2014). At the given trunk line/route, the exclusively considered relevant impacts are congestion and traffic incidents/accidents (i.e., safety). The effects such as its contribution to the GDP (Gross Domestic Product), employment, profitability of the particular main actors/stakeholders involved (shipping companies, ports, inland freight transport operators, etc.) are not considered.
i) Congestion and delays.
The congestion and delays of the liner container-shipping services can happen around the supplier and the consumer seaports due to many reasons. At the supplier seaport, particular departing ships can be delayed generally due to unplanned insufficiency of their loading capacity. At the consumer seaport, the container ships performing scheduled regular services have been given in advance the time slots for entry and using the infrastructure (berths and container terminals). They can be handled without substantive delays before entry if arrive at the seaport almost close or exactly within the assigned time slots and if the already served ships leave the seaport according to the planned schedule implying freeing its capacity for handling the new comers. However, if there is stochasticity/irregularity in both processes the arriving ships can be imposed delays before entering or leaving the given seaport. For example, from G/G/N queuing system model, the average waiting/delay and the total time of a ship arriving at the customer seaport of a given line/route is estimated as follows (Ligteringen and Veslsink 2014):
$$ {\overline{D}}_q\approx \frac{\lambda \cdot \left({\sigma}_X^2+{\sigma}_S^2/{N}_b\right)}{2\ \left(1-\lambda /{N}_b\cdot \mu \right)}\mathrm{and}\kern0.5em \overline{D}\approx \frac{1}{N_b\mu }+\frac{\lambda \cdot \left({\sigma}_X^2+{\sigma}_S^2/{N}_b\right)}{2\ \left(1-\lambda /{N}_b\cdot \mu \right)} $$
(22)
Where.
λ is the intensity of arriving ships at the customer seaport (ships/day or week) (λ = 1/ha) (ha - is the average inter-arrival time (TU/ship; TU ≡ days or weeks);
μ is the capacity of a single berth (ships/day or a week) (μ = 1/τd) (τd - is the average service time of a ship at the berth (TU/ship));
Nb is the number of free berths for handling the arriving ships; and.
σX, σS is the standard deviation of the probability distribution of the arrival times and the service times of arriving ships, respectively (TU/ship).
As can be seen, the ship’s average waiting time increases more than proportionally with increasing of the berths’ utilization rate [ρ = λ/(Nb ∙ μ)], thus implying keeping this utilization rate rather low in order to reduce the ships’ delays before entering or leaving the port, and vice versa.
ii) Safety.
The safety can be expressed by the number of expected or potential losses of container ships while operating along the given trunk line/route during time (τ) as as follows:
$$ {n}_{SL/ ij}\left(\tau \right)=p\bullet \left[{Q}_{ij}\left(\tau \right)+{Q}_{ji}\left(\tau \right)\right]\bullet {d}_{ij} $$
(23)
Where.
p is the average rate of accidents/losses of container ships during time (τ) (number/TEU-mile).
The other symbols are analogous to those in the previous Equations.