Mathematical Problems in Engineering

Volume 2016, Article ID 3130291, 7 pages

http://dx.doi.org/10.1155/2016/3130291

## Optimizing Ship Speed to Minimize Total Fuel Consumption with Multiple Time Windows

^{1}Department of Industrial and Management Engineering, Incheon National University, Incheon, Republic of Korea^{2}Graduate School of Logistics, Inha University, Incheon, Republic of Korea^{3}Department of Industrial Engineering, Hongik University, Seoul, Republic of Korea^{4}Department of Industrial and Management Engineering, Myongji University, Yongin, Republic of Korea

Received 8 April 2016; Accepted 22 September 2016

Academic Editor: Miguel A. Salido

Copyright © 2016 Jae-Gon Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study the ship speed optimization problem with the objective of minimizing the total fuel consumption. We consider multiple time windows for each port call as constraints and formulate the problem as a nonlinear mixed integer program. We derive intrinsic properties of the problem and develop an exact algorithm based on the properties. Computational experiments show that the suggested algorithm is very efficient in finding an optimal solution.

#### 1. Introduction

According to the report of World Shipping Council in 2008, fuel cost represents as much as 50–60% of total ship operating cost. Since fuel consumption is known to be the third power function of ship speed [1], many global shipping companies are trying to reduce fuel consumption by slowing down ship speed (called slow steaming). In this study, we consider the ship speed optimization problem with the objective of minimizing total fuel consumption of a (tramp or liner) ship operated on a given route. For a tramp ship, the ship speed optimization problem is a tactical problem which should be solved for every sailing, while it is a strategic problem to be solved just one time when designing a shipping route for a liner ship [2]. We determine ship speed on each leg under time window restrictions related with port calls. Time window sizes are narrow for congested ports, while they are wide for noncongested ones.

There are two kinds of time windows: hard time window and soft one [3, 4]. A hard time window should be kept at all costs while the soft one can be violated with appropriate penalties. There usually exist multiple time windows for each port call depending on the available service time of the port. Most ports have restricted operating hours since they are closed for service at night and during weekends. In this case, the wide time windows can be regarded as multiple time windows [5]. Also, ports have restrictions on the draft of ships that may safely enter [6, 7]. Many ports have time-dependent draft restrictions due to the tide that leads to multiple time windows at each port. Therefore, our aim in this study is to develop a mathematical model and an exact algorithm for the ship speed optimization problem with multiple hard time windows.

There are some previous studies on the ship speed optimization problem which are related to our problem although they consider different objective functions, decision variables, and constraints. Ronen [1] performs pioneering research on determining optimal ship speed by considering the tradeoff between fuel savings by slow steaming and the loss of revenues due to the resulting voyage extension on the other hand. Ting and Tzeng [8] propose a dynamic programming model for the ship scheduling problem with both soft and hard time window constraints with the objective of meeting the time window constraints as closely as possible. Brown et al. [9] suggest a linear programming model for optimizing operation modes of a naval ship to minimize the fuel consumption when the ship transit time is given. Corbett et al. [10] evaluate the effect of the speed reduction on mitigation of CO_{2} emissions quantitatively. Ronen [11] determines the optimal speed and fleet size (the number of deployed ships) for containerships on a single route with a weekly service cycle. Wang and Meng [12] propose a model and an algorithm to determine an optimal sailing speed of container ships on each leg of each ship route in a liner shipping network while considering transshipment and container routing. Previous literature on the speed optimization problem can also be found in the airline industry such as Lovegren and Hansman [13], Jensen et al. [14], and Aktürk et al. [15].

Our study stems from the ship speed optimization problem to minimize total fuel consumption with hard time windows studied by Fagerholt et al. [16], Norstad et al. [17], Hvattum et al. [18], Kim et al. [2], and Zhang et al. [19]. Fagerholt et al. [16] formulate the problem as a nonlinear programming model and propose a heuristic algorithm by discretizing the time window and reformulating the problem as the shortest path model. Norstad et al. [17] develop a recursive smoothing algorithm (RSA) for speed optimization and present a multistart local search heuristic for the tramp ship routing and scheduling problem with speed optimization. Hvattum et al. [18] prove that the RSA of Norstad et al. [17] guarantees optimality. Kim et al. [2] also propose an exact algorithm for the ship speed optimization problem. Zhang et al. [19] establish the optimality properties for the problem. Although the suggested algorithms and properties are very efficient for the problem with a single time window constraint, they cannot handle multiple time windows which are common in maritime transportation.

In this study, we extend the studies of Hvattum et al. [18] and Kim et al. [2] to solve a more general ship speed optimization problem with multiple time window constraints. The remainder of this paper is organized as follows: the next section presents a mathematical formulation of the problem with a small example problem. In Section 3, we derive optimality properties of the problem. We present an exact algorithm based on the optimality properties in Section 4. Section 5 reports the computational test results on randomly generated test instances followed by concluding remarks in the last section.

#### 2. Problem Description

The notations used throughout the paper are shown in the Notations.

The daily fuel consumption function can be approximated by a well-known cubic function of speed as , where is a factor of converting speed to the fuel consumption [1]. Then, the total amount of fuel consumption of a vessel sailing from port to port can be expressed as

Using (1), the considered problem can be formulated as the following nonlinear mixed integer programming model. We set since port 1 is the start port.

Objective function (2) expresses the total amount of fuel consumption of the ship. Constraint (3) ensures that cargo service at each port starts after the ship arrives at the port. Constraints (4) and (5) ensure that the start time of cargo services lies in exactly one of the time windows at each port. Constraint (6) represents that the ship speed on a leg is bounded by its allowable minimum and maximum speeds. Constraints (7) and (8) specify feasible ranges of decision variables.

A small example with four ports is used throughout the paper to help readers understand the problem and suggested properties and algorithm. In the example, , , , and cargo service time is assumed to be zero at all ports. Figure 1 depicts time windows at ports, distances between ports, and a feasible solution of the example. In the feasible solution, , , and .