Research Article Optimizing Ship Speed to Minimize Total Fuel Consumption with Multiple Time Windows Jae-Gon Kim, 1 Hwa-Joong Kim, 2 Hong Bae Jun, 3 and Chong-Man Kim 4 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 Correspondence should be addressed to Chong-Man Kim; chongman@mju.ac.kr Received 8 April 2016; Accepted 22 September 2016 Academic Editor: Miguel A. Salido Copyright © 2016 Jae-Gon Kim et al. is 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. 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. ere are two kinds of time windows: hard time window and soſt one [3, 4]. A hard time window should be kept at all costs while the soſt one can be violated with appropriate penalties. ere 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 draſt of ships that may safely enter [6, 7]. Many ports have time-dependent draſt restrictions due to the tide that leads to multiple time windows at each port. erefore, 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. ere 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 soſt 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 Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2016, Article ID 3130291, 7 pages http://dx.doi.org/10.1155/2016/3130291