Dynamic determination of vessel speed and selection of bunkering ports for liner shipping under stochastic environment

Abstract

In this work, we study a liner shipping operational problem which considers how to dynamically determine the vessel speed and refueling decisions, for a single vessel in one service route. Our model is a multi-stage dynamic model, where the stochastic nature of the bunker prices is represented by a scenario tree structure. Also, we explicitly incorporate the uncertainty of bunker consumption rates into our model. As the model is a large-scale mixed integer programming model, we adopt a modified rolling horizon method to tackle the problem. Numerical results show that our framework provides a lower overall cost and more reliable schedule compared with the stationary model of a related work.

Appendices

Appendix 1: Modified version of the stationary model

1.1 Notations

Following notations are used to express model 1:

Parameters

$n$ :

number of port of calls;

$d_{i,j}$ :

distance between port $i$ and port $j$ (nautical miles);

Parameters

$t$ :

total cycle time (h);

$t_i$ :

port time (time one ship spends on entering, unloading and loading cargo, idling and exiting) at port $i$ (h);

$e_i$ :

earliest arrival time at port $i$;

$l_i$ :

latest arrival time at port $i$;

$C_i$ :

bunker fuel consumption when the ship is at port i;

$w$ :

bunker fuel capacity for a single ship;

$v_\mathrm{min}$ :

minimum ship sailing speed (nautical miles/h);

$v_\mathrm{max}$ :

maximum ship sailing speed (nautical miles/h);

$k_1,k_2$ :

bunker fuel consumption coefficients;

$P_i$ :

bunker price for port $i$;

$f$ :

fixed bunkering cost;

$h$ :

inventory holding cost pmt for bunker;

$\gamma$ :

coefficient to control the service level;

Decision variables

$V_{i,j}$ :

ship speed between port $i$ and $j$;

$S_i$ :

bunker fuel-up-to level for the ship at port $i$;

$B_i$ :

bunkering decision variable. $=1$ if bunkering at port $i;\ =0$, otherwise;

Dependent variables

$I_i$ :

bunker fuel inventory when the ship reaches port $i$;

Dependent variables

$F_{i,j}$ :

daily bunker consumption rate for a ship travels from port $i$ to $j$;

$A_i$ :

ship arrival time at port $i$;

 

1.2 Model

 

$$\begin{aligned}&\min \sum _{i=1}^n[(S_i-I_i)P_i+f\cdot B_i+(S_i-C_i)\cdot h]-P_1\cdot I_{n+1} \nonumber \\&I_{1}=0; \end{aligned}$$

(18)

$$\begin{aligned}&I_{i}=R_{i-1}-a_{i-1}-F_{i-1,i}\cdot d_{i-1,i}/24 \cdot V_{i-1,i} \quad i \in 2,3,\ldots ,n+1 \end{aligned}$$

(19)

$$\begin{aligned}&R_{i}-I_{i} \le B_i \cdot w \quad i \in 1,2,\ldots ,n \end{aligned}$$

(20)

$$\begin{aligned}&R_{i} \le w \quad i \in 1,2,\ldots ,n \end{aligned}$$

(21)

$$\begin{aligned}&I_{i} \ge \gamma \cdot w \quad i \in 1,2,\ldots ,n \end{aligned}$$

(22)

$$\begin{aligned}&F_{i,i+1}=k_1(V_{i,i+1})^3+k_2 \quad i \in 1,2,\ldots ,n \end{aligned}$$

(23)

$$\begin{aligned}&v_\mathrm{min}\le V_{i,i+1}\le v_\mathrm{max}\quad i \in 1,2,\ldots ,n \end{aligned}$$

(24)

$$\begin{aligned}&A_i+t_i +d_{i,i+1}/V_{i,i+1}=A_{i+1}\quad i \in 1,2,\ldots ,n \end{aligned}$$

(25)

$$\begin{aligned}&e_i \le A_i \le l_i \quad i \in 1,2,\ldots ,n \end{aligned}$$

(26)

$$\begin{aligned}&A_{n+1}=t \end{aligned}$$

(27)

$$\begin{aligned}&B_{i}=0 \text{ or} 1 \quad i \in 1,2,\ldots ,n \end{aligned}$$

(28)

$$\begin{aligned}&F_{n,n+1}=F_{n,1},\quad d_{n,n+1}=d_{n,1},\quad V_{n,n+1}=V_{n,1} \end{aligned}$$

(29)

The objective function is to minimize the expected total cost, which includes the fixed and variable bunkering cost and inventory holding cost. Bunker left at the end of the service loop is refunded. Constraint (18) sets the initial inventory to be 0. Constraint (19) is a flow conservation constraint. Constraints (20) and (21) ensure that the maximum bunkering amount and bunker-up-to level are less than the bunker fuel capacity. Constraint (22) controls the minimum bunker inventory to be a fixed percentage of the total bunker capacity. Constraint (23) expresses the daily consumption rate at a certain speed between port $i$ and $i+1$. Constraint (24) is simply to limit the ship speed within a reasonable range, while constraint (25) to (27) are about time window constraints. Constraint (28) is a binary constraint.

Appendix 2: Comparison between the scenario reduction approach with the modified rolling horizon approach

We applied the fast forward selection algorithm in [1] to reduce the bunker price scenario tree size in our first case study, MAX service route, and compared the results with that of our modified rolling horizon approach. As mentioned, the size of the MAX service route allows us to solve the whole dynamic model by CPLEX directly, so we can easily derive the optimality gap of the scenario reduction method and our modified rolling horizon approach, respectively.

Table 18 shows the optimality gap of the scenario reduction method under different parameter settings. There are altogether 256 price scenarios initially and we still look at 3 different cases of bunker price fluctuation.“Number of scenarios” means the total number of scenarios retained after reduction and these percentage numbers in the table denote the optimality gap between the scenario reduction method and the direct solving of the dynamic model.

Table 18 Optimality gap of the scenario reduction method

Table 19 shows the optimality gap of our modified rolling horizon approach under three different cases of bunker price percentage change. Table 20 is a comparison of the solving time between these two methods under case 1.

Table 19 Optimality gap of the modified rolling horizon approach

Table 20 Solving time comparison of these two methods

Comparing the results in Tables 18, 19 and 20, we can see that the modified rolling horizon approach is a good approach to be used for our problem. Under all three cases, the modified rolling horizon approach is better than the scenario reduction method when the scenario reduction method retains less than 100 scenarios. Moreover, the solving time for the modified rolling horizon approach remains unchanged while the solving time for the scenario reduction method increases considerably with the number of scenarios. And under all these 3 cases, the optimality gap of our modified rolling horizon approach is under 5%, which is encouraging.

When it comes to our second case study where there are a total of 4$^{16}$ price scenarios, the implementation of the scenario reduction method becomes even harder. As CPLEX can only solve the problem with less than 500 price scenarios, this means only 5 out of 4$^{14}$ scenarios is retained. Not only will the scenario reduction algorithm take a long time to reduce the scenarios, but the optimality gap of the reduced tree might be big based on our study of the small size problem.

In summary, we feel that the scenario reduction technique might not work well in our problem. However, having said so, we still feel that there is a potential in this method to be applied to this type of the problem, but this will need an in-depth research work. As for our proposed method, it can be viewed as a special type of scenario reduction technique in the sense that all the branches in the near future are enumerated, but the branches far away from the decision point are not enumerated fully and so we have a reduction in scenarios. Moreover, by forcing it to solve at every decision point, we are able to obtain decisions using the most updated information.