## 2 Problem Statement  
In the proposed scenario, the geographical location of a petrochemical plant in the UAE requires transporting products from a manufacturing plant near one port to two other ports for international shipping. The fundamental system structure is based on the system studied in [26]. Figure 1 depicts the schematic structure of the considered system. For domestic shipping, different types of vessels (i.e., small and large) transport products from the port at the manufacturing plant to other international ports and return to the origin port for another shipmen. Such a back-and-forth voyage continues, forming a closed-loop system. This study incorporates uncertainties from bad weather and round-trip time into the system. It also considers new decision factors, such as the number of each type of vessel and forward and backward travel speeds. We have employed a few assumptions to construct the problem. First, we assume that the products are packed in a container and are ready to be loaded onto a ship. Second, the vessels berthed at the origin port are assumed to wait until they are fully loaded. Third, according to the field experts' comments, vessels on the voyage are assumed to continue to travel to the destination port, regardless of weather conditions. Lastly, we assume that only one aggregated type of product is transported. A vessel's trip begins from the waiting area in near-coastal water from the origin port, Port A. Because Port A has three docks, up to three vessels of any size can be berthed simultaneously and load products packaged in a container. The vessels in the waiting area stay there until any dock becomes available. A specific sea area adjacent to Port A is dedicated for use by arriving and departing tugging. For safety in the coastal waters of the origin port, both arriving and departing tugging is performed one by one in each 3 h time slot like a single runway at an airport. Once a vessel departs, it travels to Port B or Port C. Because the destination ports are managed by other organizations and used by many other companies, the manufacturingcompany has no control over the operations at the other ports. In contrast, it can control and manage all the operations at Port A and related infrastructure. Thus, unloading at Ports B and C is often delayed due to the uncertainties of terminal availability. Accordingly, a round-trip time, which includes times for the forward and backward voyages and unloading at the destination port, is somewhat uncertain. After unloading the products and loading empty containers, the vessel returns to Port A. Then, one cycle of a vessel trip is completed. Because the manufactured products always need to be transported to the international ports (i.e., Ports B and C), the travel cycle is repeated as often as needed. The controllable operations at the origin port (e.g., tugging, loading and unloading, and waiting) follow the operation schedules determined by a manager and the inter-returning pattern of vessels. Therefore, the uncertain returning pattern may impose difficulties on scheduling vessel operations at the origin port. In particular, the voyage could be interrupted by uncertain weather conditions. In the considered system, vessels cannot depart from ports when a wind speed exceeds 35 km/h or a visibility distance is less than 1 km. This study also considers such an uncertain voyage interruption along with the uncertain delay at the destination ports. Currently, the manufacturing company rents several vessels for a specific period (three years) and operates vessels at the origin port based on the first-available-first-use (FAFU) policy. The vessels usually travel at 22.2 km/h (12 knots). Thus, it would be worth examining the impact of differentiating the numbers of each type of vessel and forward and backward travel speeds on the system performance compared with the current operating conditions. Eventually, the optimal operating conditions should be prescribed to minimize the total costs considering the environmental effect.