# Chapter 2 The carrier optimisation problem

## 2.1 Introduction

We introduce our mixed-integer nonlinear optimization problem , which represents the starting point of this analysis, as it allow us to compute the business as usual emissions levels. We start by presenting the game in the absence of coalition formation. This will be useful as a baseline.

## 2.2 Model variables and parameter

For ease of reference, relevant notation is summarized in the following table:

Model Parameters definition
$$N^{firms}$$ Number of Firms in the Simulation Framework
$$T= {2018, .., 2042}$$ Planing Horizon
$$i \in [1, N^{firms} ]$$ Firm Index
$$t \in [1,T]$$ Period Index
$$d$$ Cycle Distance in Nautical Miles
$$Y_t^{AB}$$ Annual Market Demand in $$TEU$$ in Period $$t$$
$$s_{i}$$ Firm $$i$$ ’s Market Share
$$X^{AB}_{i,t}$$ Firm $$i$$ ’s Annual Demand in $$TEU$$ in Period $$t$$
$$\phi_{i}$$ Firm $$i$$ ’s Fleet’s Main Engine Energy Efficiency
$$\eta_t^{fuel}$$ The Chosen Fuel’s Price in Period $$t$$ in ($/ton) $$T_{port}$$ Average Port Time During the Cycle (including Pilotage, Mooring, Time buffers, etc.) in $$(hours)$$ $$CO_{i}$$ Annual Non-Fuel Fixed Operating Cost Per Vessel in ($/vessel)
$$\tau$$ Average Annual Working Time Per Vessel in $$(hours)$$
$$k_i$$ Vessel Capacity in $$TEU$$ Per Firm
$$r$$ Discount Rate
$$N_{i,t}$$ Number of ships deployed in the service to satisfy the firm’s demand at the chosen speed level
$$V_{i,t}$$ Average sailing speed in nautical miles $$nm$$ per hour $$[knots]$$

## 2.3 Containership Simulated Market Transport Demand for the European-Asian Route

$$$Y_t^{Europe-Asia} = (\frac{GDP_t}{GDP_0})^v \times (\frac{\rho_t}{\rho_0})^η \times Y_0^{AB}$$$

Containerized Trade for the Europe–Far East service (Million TEU) are adapted from UNCTAD (2021) from 2018 till 2021. Assuming $$\frac{\rho_t}{\rho_0} = 1$$ :

$$$Y_t^{Europe-Asia} = \biggl(\frac{GDP_t}{GDP_{2021}}\biggr)^{0.8} ~\ Y_{2021}^{Europe-Asia} ~\ ~\ ~\ \forall t \geq 2021$$$

## 2.4 The carrier’s optimisation problem

\begin{aligned} \max_{{ N_{i,t} ,V_{i,t} }} \pi_{i} = {} \sum_{t=1}^{T} {(1+r)^{-t}} \bigg[ ~\ \rho_t ~\ X^{AB}_{i,t} - \bigg( \eta_t^{fuel} ~\ \frac{X^{AB}_{i,t}} {k_i} ~\ d ~\ \phi_i ~\ (V_{i,t})^2 \bigg) - \bigg( \eta_t^{MGO} ~\ \frac{X^{AB}_{i,t}} {k_i} ~\ d ~\ F_A ~\ \frac{1}{V_{i,t}} \bigg) - \bigg( CO_{i,t} ~\ N_{i,t} \bigg) \bigg] \forall i =1,.., N^{ firms} \end{aligned}

subject to: $$$\mbox{Firm Level Demand}: X_{i,t}^{AB}= s_{i} Y_t^{AB} = s_{i} \times (\frac{GDP_t}{GDP_0})^v \times (\frac{\rho_t}{\rho_0})^η \times Y_0^{AB}$$$
$$$\mbox{Firm Level Supply and Operational Constraints}: \begin{cases} V_i^{min} \leq V_{i,t} \leq V_i^{max} \\ \\ N_{i,t} \in Z^+ \\ N_{i,t} \geq N_{{min}_{i,t}}^{vessel} = \frac{X^{AB}_{t,i} \times \bigg( \frac{d }{V_{i,t}} +t_{port} \bigg)} {k_i \times \tau } \\ \end{cases}$$$

$$$\mbox{Energy Efficiency and Fuel Consumption} \begin{cases} \phi_i = SFOC_0^M \times EL^M \times PS^M \times 10^{-6} \times (\frac{1}{V_i^{s}})^3 \\ F_A = SFOC^A \times EL^A \times PS^A \times 10^{-6}\\ \end{cases}$$$

## 2.5 Emissions Model

we set sulfur and carbon BAU emission levels $$\overline{e^{CO_2}_{i,t}}$$ ,$$\overline{e^{SO_x}_{i,t}}$$ as follows: \begin{aligned} \overline{e^{CO_2}_{i,t}} [ tonnes _{(CO_2)} ] & = \epsilon^{CO_2 } \bigg[ \frac{tonnes _{(CO_2)}}{tonnes_{(fuel)}} \bigg] \times F_{i,t} [tonnes_{(fuel)}] \\ & = \epsilon^{CO_2 }_{fuel} \times \bigg( \frac{X^{AB}_{i,t}}{k_i} \times d \times \phi_i \times (V_{i,t}^{BAU})^2 \bigg)+ \epsilon^{CO_2 }_{MGO} \times \bigg( \frac{X^{AB}_{i,t}}{k_i} \times F_A \times \frac{d}{V_{i,t}^{BAU}} \bigg) \\ & = \frac{X^{AB}_{i,t}} {K_i} \times d \bigg( \epsilon^{CO_2 }_{fuel} \times \phi_i \times (V_{i,t}^{BAU})^2 + \epsilon^{CO_2 }_{MGO} \times \frac{F_A}{V_{i,t}^{BAU}} \bigg) \end{aligned}\begin{aligned} \overline{e^{SO_x}_{i,t}} [ tonnes _{(SO_x)} ] & = \epsilon^{SO_x } \bigg[ \frac{tonnes _{(SO_x )}}{tonnes_{(fuel)}} \bigg] \times F_{i,t} [tonnes_{(fuel)}] \\ & = \epsilon^{SO_x }_{fuel} \times \bigg( \frac{X^{AB}_{i,t}}{k_i} \times d \times \phi_i \times (V_{i,t}^{BAU})^2 \bigg)+ \epsilon^{SO_x }_{MGO} \times \bigg( \frac{X^{AB}_{i,t}}{k_i} \times F_A \times \frac{d}{V_{i,t}^{BAU}} \bigg) \\ & = \frac{X^{AB}_{i,t}} {K_i} \times d \bigg( \epsilon^{SO_x }_{fuel} \times \phi_i \times (V_{i,t}^{BAU})^2 + \epsilon^{SO_x }_{MGO} \times \frac{F_A}{V_{i,t}^{BAU}} \bigg) \end{aligned}

## 2.6 Simulation Parameters

### 2.6.1 Firm level characteristics and Service/trade characteristics

Parameters Notation Firm 1 Firm 2 Firm 3
Cycle Distance between Europe and East Asia $$(nm)$$ $$^a$$ $$d$$ $$23,000$$ $$23,000$$ $$23,000$$
Average time Port $$(days)$$ $$^a$$ $$T_{port}$$ 10 10 10
Vessel capacity $$(TEU)$$ $$^b$$ $$k_i$$ $$14,000$$ $$8000$$ $$6000$$
Main engine power $$(kW)$$ $$^b$$ $$PS_i^M$$ $$89700$$ $$68,500$$ $$57,100$$
Auxiliary engine power $$(kW)$$ $$^b$$ $$PS_i^A$$ $$14,000$$ $$12,000$$ $$12,900$$
Design speed $$(knots)$$ $$^b$$ $$V^s_i$$ $$25.0$$ $$25.0$$ $$25.0$$
Specific fuel oil consumption (main engine) $$(g/kWh)$$ $$^b$$ $$SFOC_{0,i}^M$$ $$175$$ $$133$$ $$114$$
Specific fuel oil consumption (auxiliary engine)$$(g/kWh$$) $$^b$$ $$SFOC_{0,i}^A$$ $$32$$ $$24$$ $$26$$
Average Engine load factor (main engine) $(%)$ $$^b$$ $$EL^M$$ $$0.8$$ $$0.8$$ $$0.8$$
Average Engine load factor (auxiliary engine) $(%)$ $$^b$$ $$EL^A$$ $$0.5$$ $$0.5$$ $$0.5$$
Maximum Vessel speed $$(knots)$$ $$^b$$ $$V_i^{max}$$ $$28.0$$ $$28.0$$ $$28.0$$
Minimum Vessel speed$$(knots)$$ $$^b$$ $$V_i^{min}$$ $$12.0$$ $$12.0$$ $$12.0$$
Total Annual Cost $$(million ~\ USD/year)$$ $$^b$$ $$CO_{i}$$ $$18.25$$ $$13.731$$ $$12.22$$

### 2.6.2 Energy Efficiency and Fuel Consumption per firm

Parameters Notation Firm 1 Firm 2 Firm 3
Main Engine in $$ton/hour.knot^3$$ $$\phi_i$$ 0.000803712 0.000466458 0.000333281
Auxiliary Engine in $$ton/hour$$ $$F_{A_{i}}$$ 0.224 0.144 0.1677

### 2.6.3 Fuel and Emission Parameters

fuel type HFO (3.55%) ULSFO (0.5%) MGO (0.1%)
Global 20 Ports Average fuel Prices $422.5 / tonne 525.5$/tonne 597 \$/tonne
Carbon Emission Factor 3.114 (g/g of Fuel) 3.206 (g/g of Fuel) 3.206 (g/g of Fuel)
Sulfur Emission Factor 0.07 (g/g of Fuel) 0.01 (g/g of Fuel) 0.002 (g/g of Fuel)