Profitability prospects for container roll-on/roll-off shipping on the Northern Sea Route (NSR)

Abstract

Due to their ability to transport both containerized and wheeled cargo, container roll-on/roll-off (ConRo) ships offer advantages in meeting and integrating the diverse transportation requirements of the Northern Sea Route (NSR). Against this backdrop, we provide an equilibrium diagram for the business prospects of RoRo shipping on the NSR. The paper builds a new mixed integer nonlinear programming (MINLP) model for the NSR single trade ship routing and scheduling problem (NSR-STSRSP), which considers integrated cargo transport and schedule planning. The model is solved within reasonable time by the Gurobi Optimizer through linearization of the MINLP. With the imminent introduction of the 1A super RoRo ship, featuring the world’s most advanced dual-fuel engine, we consider an international transit route along the NSR from China, passing by Japan, South Korea and Russia, before reaching Northwestern Europe. The probability of profit for ConRo shipping on the NSR is assessed through specially designed scenario experiments, where multiple factors are combined, including navigation season, ice class, fuel prices, cargo transport demand, ports and shippers. The results show that while developing a business strategy for ConRo shipping on the NSR, shipping companies must strike a balance between the following factors in the long run: investment costs in ice-class ships; operating costs of such ships; and revenues from contract- and spot cargoes during operations on the NSR. For a niche market such as that of the NSR, a business plan needs to be established on the premise of “on time delivery”, eyeing more long-term agreements for contract cargo shipping. We consider the introduction of ConRo shipping operations on the NSR to hold promising commercial prospects, offering a fresh transportation solution to the trade between Asia and Europe.

Appendix

The model linearization method is proposed so that the new model can be solved by the advanced linear programming optimizer Gurobi.

1.1 Discretization of fuel consumption functions

The speed-related formulas are replaced by Andersson et al.’s (2015) discretization method, that is, the voyage time and fuel consumption are described based on linear combinations of discretized speeds.

Let $\mu_{i}^{{{\text{ICE}}}} = \frac{1}{{v_{i}^{{{\text{ICE}}}} }}$, and $G_{i}^{{{\text{ICE}}}} \left( {\mu_{i}^{{{\text{ICE}}}} } \right) = \frac{1}{{\left( {u_{i}^{{{\text{ICE}}}} } \right)^{2} }}$. The range of $\mu_{i}^{{{\text{ICE}}}}$ can be divided into ${\rm K}$ identical segments, as shown in Fig. 

Fig. 12

Discretization of fuel consumption functions

12. The greater the value of ${\rm K}$, the more accurate the solution. The upper and lower limits of $\mu_{i}^{{{\text{ICE}}}}$ can be defined according to the speed range [$v^{{{\text{ICEMIN}}}}$, $v^{{{\text{ICEMAX}}}}$]. In this way, ${\rm K} + 1$ discrete points $\mu_{i}^{{\text{ICE - 0}}} ,\mu_{i}^{{\text{ICE - 1}}} , \ldots ,\mu_{i}^{{{\text{ICE - }}{\rm K}}}$ can be obtained, along with the corresponding ${\rm K} + 1$ fuel consumptions $G_{i}^{{{\text{ICE}}}} \left( {\mu_{i}^{{\text{ICE - 0}}} } \right),G_{i}^{{{\text{ICE}}}} \left( {\mu_{i}^{{\text{ICE - 1}}} } \right), \ldots ,G_{i}^{{{\text{ICE}}}} \left( {\mu_{i}^{{{\text{ICE - }}{\rm K}}} } \right)$. Then, a binary variable $\delta_{i}^{{{\text{ICE - }}\kappa }} ,\kappa = \left\{ {1,2, \ldots ,{\rm K}} \right\}$ is defined to suggest which speed will be used. If and only if the reciprocal $\mu_{i}^{{{\text{ICE}}}}$ of the speed is used in the leg i, the variable equals 1; otherwise, the variable equals 0.

Similarly, we have $\mu_{i}^{{{\text{OUT}}}} = \frac{1}{{v_{i}^{{{\text{OUT}}}} }}$. Then, a binary variable $\delta_{i}^{{{\text{OUT - }}\lambda }} ,\lambda = \left\{ {1,2, \ldots ,\Lambda } \right\}$ is defined to determine the speed in ice-free water of leg i. The following constraints can be introduced:

$$\sum\limits_{\kappa = 0}^{\rm K} {\delta_{i}^{{{\text{ICE - }}\kappa }} } = 1,\quad \forall i \in \left\{ {1,2, \ldots ,P - 1} \right\}$$

(30)

$$\sum\limits_{\lambda = 0}^{\Lambda } {\delta_{i}^{{{\text{OUT - }}\lambda }} } = 1,\quad \forall i \in \left\{ {1,2, \ldots ,P - 1} \right\}$$

(31)

Thus, Eqs. (7) and (23) can be respectively converted to

$$\begin{gathered} C^{{{\text{FUEL}}}} = \sum\limits_{i = 1}^{P - 1} {\sum\limits_{f = 1}^{2} {F_{f} \frac{{sfc_{f} }}{{sfc_{{{\text{HFO}}}} }}} } \left\{ {\sum\limits_{\kappa = 0}^{\rm K} {\left[ {\alpha \rho \sigma_{i} L_{i}^{{{\text{NSR}}}} G_{i}^{{{\text{ICE}}}} \left( {\mu_{i}^{{{\text{ICE - }}\kappa }} } \right)\delta_{i}^{{{\text{ICE - }}\kappa }} } \right]} } \right. \\ \quad \left. { + \sum\limits_{\lambda = 0}^{\Lambda } {\left[ {\rho \left( {1 - \sigma_{i} } \right)L_{i}^{{{\text{NSR}}}} G_{i}^{{{\text{OUT}}}} \left( {\mu_{i}^{{{\text{OUT - }}\lambda }} } \right)\delta_{i}^{{{\text{OUT - }}\lambda }} } \right]} } \right\} \\ \end{gathered}$$

(32)

$$t_{i + 1}^{{{\text{ARI}}}} - t_{i}^{{{\text{DEP}}}} = \sum\limits_{\kappa }^{\rm K} {\sigma_{i} L_{i}^{{{\text{NSR}}}} } \mu_{i}^{{{\text{ICE - }}\kappa }} \delta_{i}^{{{\text{ICE - }}\kappa }} + \sum\limits_{\lambda }^{\Lambda } {\left( {1 - \sigma_{i} } \right)L_{i}^{{{\text{NSR}}}} \mu_{i}^{{{\text{OUT - }}\lambda }} \delta_{i}^{{{\text{OUT - }}\lambda }} } ,\quad \forall i \in \left\{ {1,2, \ldots ,P - 1} \right\}$$

(33)

1.2 Linearization of time window function

The penalty costs $C_{i}^{{{\text{ARITIME}}}}$ and $C_{i}^{{{\text{DEPTIME}}}}$ for violating the cargo delivery time window at port i are both piecewise functions. Continuous variables $w_{i1} ,w_{i2} ,w_{i3} ,w_{i4} \in \left[ {0,1} \right]$ are introduced to Eq. (11). Suppose that:

$$t_{i}^{{{\text{ARI}}}} = T_{i - 1}^{{{\text{LATE}}}} w_{i1} + T_{i}^{{{\text{EARLY}}}} w_{i2} + T_{i}^{{{\text{LATE}}}} w_{i3} + T_{i + 1}^{{{\text{EARLY}}}} w_{i4} ,\quad \forall i \in \left\{ {1,2, \ldots ,P} \right\}$$

(34)

Equation (11) can be rewritten as

$$C_{i}^{{{\text{ARITIME}}}} = S^{{{\text{EARLY}}}} \left( {T_{i}^{{{\text{EARLY}}}} - T_{i - 1}^{{{\text{LATE}}}} } \right)w_{i1} + S^{{{\text{LATE}}}} \left( {T_{i + 1}^{{{\text{EARLY}}}} - T_{i}^{{{\text{LATE}}}} } \right)w_{i4} ,\quad \forall i \in \left\{ {1,2, \ldots ,P} \right\}$$

(35)

where,

$$w_{i1} + w_{i2} + w_{i3} + w_{i4} = 1,\quad \forall i \in \left\{ {1,2, \ldots ,P} \right\}$$

(36)

In addition, binary variables $z_{i1} ,z_{i2} ,z_{i3} ,z_{i4} \in \left\{ {0,1} \right\}$ are introduced to satisfy:

$$w_{i1} \le z_{i1} ,w_{i2} \le z_{i1} + z_{i2} ,w_{i3} \le z_{i2} + z_{i3} ,w_{i4} \le z_{i4} ,\quad \forall i \in \left\{ {1,2, \ldots ,P} \right\}$$

(37)

$$z_{i1} + z_{i2} + z_{i3} + z_{i4} = 1,\quad \forall i \in \left\{ {1,2, \ldots ,P} \right\}$$

(38)

Equation (12) is treated in a similar manner. By introducing continuous variables $u_{i1} ,u_{i2} ,u_{i3} ,u_{i4} \in \left[ {0,1} \right]$ and binary variables $s_{i1} ,s_{i2} ,s_{i3} ,s_{i4} \in \left\{ {0,1} \right\}$, we have:

$$t_{i}^{{{\text{DEP}}}} = T_{i - 1}^{{{\text{LATE}}}} u_{i1} + T_{i}^{{{\text{EARLY}}}} u_{i2} + T_{i}^{{{\text{LATE}}}} u_{i3} + T_{i + 1}^{{{\text{EARLY}}}} u_{i4} ,\forall i \in \left\{ {1,2, \cdots ,P} \right\}$$

(39)

$$C_{i}^{{{\text{DEPTIME}}}} = S^{{{\text{EARLY}}}} \left( {T_{i}^{{{\text{EARLY}}}} - T_{i - 1}^{{{\text{LATE}}}} } \right)u_{i1} + S^{{{\text{LATE}}}} \left( {T_{i + 1}^{{{\text{EARLY}}}} - T_{i}^{{{\text{LATE}}}} } \right)u_{i4} ,\quad \forall i \in \left\{ {1,2, \ldots ,P} \right\}$$

(40)

$$u_{{i1}} + u_{{i2}} + u_{{i3}} + u_{{i4}} = 1,\quad \forall i \in \left\{ {1,2, \ldots ,P} \right\}$$

(41)

$$u_{i1} \le s_{i1} ,u_{i2} \le s_{i1} + s_{i2} ,u_{i3} \le s_{i2} + s_{i3} ,u_{i4} \le s_{i4} ,\quad \forall i \in \left\{ {1,2, \ldots ,P} \right\}$$

(42)

$$s_{i1} + s_{i2} + s_{i3} + s_{i4} = 1,\quad \forall i \in \left\{ {1,2, \ldots ,P} \right\}$$

(43)

1.3 Model transformation

Through the above steps, model [M1] can be transformed into model [M2]:

$$\begin{gathered} \max Z = \sum\limits_{{c = 1}}^{C} {\sum\limits_{{d = 1}}^{D} {\sum\limits_{{i = 1}}^{{P - 1}} {\sum\limits_{{j = 2}}^{P} {\left[ {R_{{ijc}}^{{{\text{MC}}}} M_{{ijc}}^{{\text{C}}} + R_{{ijc}}^{{{\text{OC}}}} \left( {x_{{ijcd}} - M_{{ijc}}^{{\text{C}}} } \right)} \right]} } } } \hfill \\ + \sum\limits_{{b = 1}}^{B} {\sum\limits_{{i = 1}}^{{P - 1}} {\sum\limits_{{j = 2}}^{P} {\left[ {R_{{ijb}}^{{{\text{MB}}}} M_{{ijb}}^{{\text{B}}} + R_{{ijb}}^{{{\text{OB}}}} \left( {y_{{ijb}} - M_{{ijb}}^{{\text{B}}} } \right)} \right]} } } \hfill \\ - \sum\limits_{{i = 1}}^{{P - 1}} {\sum\limits_{{f = 1}}^{2} {F_{f} \frac{{sfc_{f} }}{{sfc_{{{\text{HFO}}}} }}} } \left\{ {\sum\limits_{{\kappa = 0}}^{{\rm K}} {\left[ {\alpha \rho \sigma _{i} L_{i}^{{{\text{NSR}}}} G_{i}^{{{\text{ICE}}}} \left( {\mu _{i}^{{{\text{ICE - }}\kappa }} } \right)\delta _{i}^{{{\text{ICE - }}\kappa }} } \right]} } \right. \hfill \\ \left. { + \sum\limits_{{\lambda = 0}}^{\Lambda } {\left[ {\rho \left( {1 - \sigma _{i} } \right)L_{i}^{{{\text{NSR}}}} G_{i}^{{{\text{OUT}}}} \left( {\mu _{i}^{{{\text{OUT - }}\lambda }} } \right)\delta _{i}^{{{\text{OUT - }}\lambda }} } \right]} } \right\} - \sum\limits_{{i = 1}}^{{P - 1}} {\sum\limits_{{j = 2}}^{P} {\sum\limits_{{c = 1}}^{C} {\sum\limits_{{d = 1}}^{D} {\left( {H_{i}^{{{\text{CLOAD}}}} M_{c} x_{{ijcd}} + H_{j}^{{{\text{CUNLOAD}}}} M_{c} x_{{ijcd}} } \right)} } } } \hfill \\ - \sum\limits_{{i = 1}}^{{P - 1}} {\sum\limits_{{j = 2}}^{P} {\sum\limits_{{b = 1}}^{B} {\left( {H_{i}^{{{\text{BLOAD}}}} y_{{ijb}} + H_{j}^{{{\text{BUNLOAD}}}} y_{{ijb}} } \right)} } } - 0.0158C_{{\varpi \tau }} - \sum\limits_{{i = 1}}^{P} {\left[ {S^{{{\text{EARLY}}}} \left( {T_{i}^{{{\text{EARLY}}}} - T_{{i - 1}}^{{{\text{LATE}}}} } \right)} \right.} w_{{i1}} \hfill \\ \left. { + S^{{{\text{LATE}}}} \left( {T_{{i + 1}}^{{{\text{EARLY}}}} - T_{i}^{{{\text{LATE}}}} } \right)w_{{i4}} + S^{{{\text{EARLY}}}} \left( {T_{i}^{{{\text{EARLY}}}} - T_{{i - 1}}^{{{\text{LATE}}}} } \right)u_{{i1}} + S^{{{\text{LATE}}}} \left( {T_{{i + 1}}^{{{\text{EARLY}}}} - T_{i}^{{{\text{LATE}}}} } \right)u_{{i4}} } \right] \hfill \\ - \frac{{1.3\left( {C^{{{\text{CAPITAL}}}} - C^{{{\text{RESIDUAL}}}} } \right)}}{{365 \times 30}}\frac{{t_{P}^{{{\text{DEP}}}} }}{{24}} \hfill \\ {\text{s}}{\text{.t}}{\text{. }}\left( {14} \right)\sim \left( {22} \right),\left( {24} \right)\sim \left( {25} \right),\left( {28} \right)\sim \left( {31} \right),\left( {33} \right), \hfill \\ {\text{ }}\left( {36} \right)\sim \left( {38} \right),\left( {41} \right)\sim \left( {43} \right), \hfill \\ {\text{ }}w_{{i1}} ,w_{{i2}} ,w_{{i3}} ,w_{{i4}} ,u_{{i1}} ,u_{{i2}} ,u_{{i3}} ,u_{{i4}} \in \left[ {0,1} \right], \hfill \\ {\text{ }}z_{{i1}} ,z_{{i2}} ,z_{{i3}} ,z_{{i4}} ,s_{{i1}} ,s_{{i2}} ,s_{{i3}} ,s_{{i4}} \in \left\{ {0,1} \right\}. \hfill \\ \end{gathered}$$

(M2)