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A multi-objective mathematical model to select fleets and maritime routes in short sea shipping: a case study in Chile


This paper proposes a mathematical model for intermodal chains with seaborne transport, in which the optimization of a multi-objective model enables conflicting objectives to be handled simultaneously. Through the assessment of ‘door-to-door’ transport in terms of costs, time, and environmental impact, the most suitable maritime route and the optimized fleet are jointly proposed to maximize the opportunities for success of intermodal chains versus trucking. The NSGA-II algorithm is applied to resolve the model. The Pareto fronts obtained not only permit decision-making in the short-term but also enable long-term strategies to be defined according to the behaviour of these frontiers when sensitivity analysis is undertaken. A real-life case in Chile is studied to test the usefulness of the model. Aside from identifying the most suitable Motorway of the Sea with its optimized fleet for Chile, the application case has provided several significant findings to promote the intermodal option regardless of its location.

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Correspondence to Alba Martínez-López.

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Appendix A


A = {1,...,a}: Different legs for the intermodal chains: capillary hauls (road haulage in both costs) and the trunk haul (maritime route).

BB = {1,...,b}: Installation of bow thruster: yes or no.

C = {1,...,c}: Cost inputs to reach the minimum required freight: depreciation costs, financing costs, insurance costs, maintenance costs, crew costs, fuel costs, and port tariff costs (ship dues, cargo dues, pilot tariff, towing tariff, mooring dues, and loading/unloading costs).

DD = {1,...,d}: Final destinations on land (nodes). For the transport network of the application case, Iquique and Antofagasta are used in the northern region and Concepción and Temuco are used in the southern region.

EE = {1,...,e}: Kind of main engines: diesel engines and turbines.

GG = {1,...,g}: Cargo-handling systems: vessel cranes or port cranes.

H = {1,...,h}: Possible propellers: screws or waterjets.

I = {1,...,i}: Number of main engines.

J = {1,...,j}: Direction for the transport (north–south and south–north).

K = {1,...,k}: Possible spoke ports. For the application case, they are Arica, Iquique, Mejillones, and Antofagasta in the northern region and San Vicente and Coronel in the southern region.

M = {1,...,m}: Possible hub ports. For the application case, they are Valparaíso and San Antonio in the V region.

N = {1,...,n}: Number of shaft lines in the machine room.

PP = {1,...,p}: Types of cargo units for container vessels: TEUs and FEUs.

SS = {1,...,s}: Stages during maritime transport: free sailing, maneuvering (pilotage time, towing time, and mooring time), and berthing (loading and unloading operations).

U = {1,...,u}: Group of evaluated pollutants: SO2, NOx, PM2.5, and CO2.

V = {1,...,v}: Classification of the zones according to the harmful impact of the emissions: metropolitan zone and urban zone.

WW = {1,...,w}: Port services for the maneuvering stage: pilotage, towing, and mooring services.

Z = {1,...,z}: The origins on land (nodes). For the transport network of the application case, in the central region, Santiago, Valparaíso hub port (Valparaiso or San Antonio), La Serena, and Rancagua are used.


ST = {c}: The MoS analyzed: MoS North and MoS South.

DIS = {d}: Compulsory driving with two drivers (yes and no).


CF1u: Unitary costs for the pollutants during free sailing (€/); ∀u ∈ U.

CFsuv: Unitary costs for the pollutants considering the maritime stages and the affected zones (€/); ∀s ∈ SS∧ ∀u ∈ U∧ ∀v ∈ V.

CTc: Cost of the items for the maritime required freight of the trunk haul (€); ∀c ∈ C.

CKdp: Unitary cost per kilometre by road (unimodal; the value is dependent on the required number of drivers and the cargo unit transported (€/km)); ∀p ∈ PP∧ ∀d ∈ DIS.

CMU: Overall transport costs for the intermodal chain (€/(t × trip)).

CU: Overall transport costs for the unimodal alternative (€/(t × trip)).

DMmk: Maritime distance for the trunk haul (km); ∀m ∈ M ∧∀k ∈ K.

DRazd: Road distance for the unimodal alternative (km); ∀z ∈ Z ∧ ∀d ∈ DD.

DRbzm: Road distance for the capillary hauls in the intermodal chains from/to hub ports (km); ∀z ∈ Z ∧ ∀m ∈ M.

DRbkd: Road distance for the capillary hauls in the intermodal chains from/to spoke ports (km); ∀k ∈ K ∧ ∀d ∈ DD.

EGsu: Emission coefficients for container vessels during the different maritime stages (kg/nm and in kg/h); ∀s ∈ SS ∧ ∀u ∈ U.

EGUu: Emission coefficients for trucking (gr of pollutant/kg of fuel consumed); ∀u ∈ U.

FCp: Fuel consumption for trucks by considering the cargo unit transported (gr fuel/km); ∀p ∈ PP.

CNk: Number of cranes per vessel ∀k ∈ K.

CNm: Number of cranes per vessel ∀m ∈ M.

MUE: Environmental costs for the intermodal chains (€/(t × trip)).

RE: Environmental costs (€/(t × trip)) for the road transport.

REa: Environmental costs (€/(t × trip)) for the stretches of the intermodal chain ∀a ∈ A.

Xd: Relative probability of delivering/receiving a load for each node of the northern and southern regions (%) regarding other alternative nodes ∀d ∈ DDMoS North: Iquique (Xd = X1 = 43%) and Antofagasta (Xd = X2 = 57%).

MoS South: Concepción (Xd = X3 = 75.95%) and Temuco (Xd = X4 = 24.04%).

Xcjz: Relative probability of delivering/receiving a load for each node of the central region (%) regarding other alternative nodes for each MoS (MoS North and MoS South) and direction (north–south and south–north) ∀z ∈ Z∧∀c ∈ ST∧∀j ∈ J.

TSw: Time for every port operation during the maneuvering stage (h); ∀w ∈ WW.

TVM: Time invested in intermodal transport (h).

TVU: Time invested in the unimodal alternative (road haulage) (h).

Vk: Average speed of the cranes for the spoke ports ∀k ∈ K (cycles/h).

Vm: Average speed of the cranes for the hub ports ∀m ∈ M (cycles/h).

VT: Speed of the truck (km/h).

Appendix B

See Table 12.

Table 12 Unitary port costs for the preliminary and current scenarios

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Martínez-López, A. A multi-objective mathematical model to select fleets and maritime routes in short sea shipping: a case study in Chile. J Mar Sci Technol 26, 673–692 (2021).

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  • Short sea shipping
  • Motorways of the sea
  • Intermodal chains
  • Multi-objective optimization
  • Analysis of sensitivity
  • Decision support tool