programming, Robust optimization.

Since the 1970s, maritime transportation has increased sharply due to rising global trade. In 2017, approximately 17.1% of world seaborne trade was transported in containers.

In the global shipping industry, effective container positioning is paramount for minimizing operational costs and enhancing logistical efficiency [1].

Empty container positioning is a critical aspect of maritime logistics, directly influencing port efficiency and operational costs, which involves strategically distributing empty containers across ports based on fluctuating demand, shipping schedules, and economic factors. Ports experience variable requirements depending on seasonality, trade routes, and the influx of import/export activities.

When the number of import containers exceeds the number of export containers, some ports have a surplus of empty containers, while others have a deficit. When ports have a deficit of empty containers, shipping companies must lease or purchase them; when ports have a surplus, the empty containers are stored in depots. Strategically, positioning empty containers is therefore one of the most effective ways to address container imbalance [2].

One pivotal aspect of this optimization model is the decision about leasing versus purchasing containers. Leasing provides flexibility, allowing companies to adapt to rapidly changing market conditions without incurring the high costs of ownership (used to cover short-term empty container shortages). Purchasing, on the other hand, may offer long-term savings but ties up capital and assets, increasing financial risk in an unstable market (depending on the shipping company's strategy).

According to Kuzmicz and Pesch [3], containers spend about 56% of their 10-15-year lifespan stacked at depots or being repositioned empty. The key decision is when and how many empty containers are repositioned to a destination port [1].

Models used to simulate this industry's environment must incorporate constraints such as port storage capacity, maintenance costs, container turnaround times, etc. By systematically analyzing these factors, firms can develop a comprehensive strategy for empty container positioning that maximizes availability while minimizing costs.

The integration of robust optimization models into container positioning strategies offers an advanced approach to managing the complexities surrounding leasing versus purchasing. This method not only clarifies the decision-making process but also ensures that shipping companies remain agile and competitive in a dynamic maritime market, ultimately leading to enhanced resource allocation and improved service delivery across global ports.

The core of a robust optimization model is its ability to accommodate worst-case scenarios for uncertain parameters, such as fluctuations in demand for container leasing or purchasing, variable transportation costs, and changing maritime policies. The Bertsimas and Sim budget robust framework provides a structured approach to determine the portfolio of container acquisition strategies, whether leasing or purchasing, under limited budgets, while accounting for uncertainty.

In this paper, we consider the issue discussed by Moon et al. [2], namely, ocean-empty-container positioning planning. We introduce a robust model based on the Moon et al. [2] model and consider uncertainties in costs and rates associated with repositioning empty containers. In addition, to meet the demand, we also consider short-term leasing and purchasing factors. In long-term leasing, we can treat the leased container as owned, assuming the lease is for a long period. However, in short-term leasing, the lease duration is short (in several periods), and container handling is required when the container is returned. The objective of this study is to minimize the total relevant costs, including transportation, handling, inventory holding, leasing, and purchasing costs. The remaining content of the paper is organized as follows: in Section 2, we introduce the robust mathematical model based on Moon et al. [2]. In Section 3, we present the computational results for both the deterministic and robust models and, in the end, compare the two models. In Section 4, a sensitivity analysis of the effects of parameters related to leasing and returning leased containers, as well as to owned vessels' capacity, on total network costs has been presented. Finally, in Section 5, we present the conclusion and outline future research directions related to positioning empty containers.

Regarding the importance of positioning empty containers for ocean carriers and port authorities, numerous studies have been conducted over the past decade.

Wang et al. [4] examined the problem of locating shipping hub containers and positioning empty containers alongside them, accounting for collapsible containers. To this end, they present a mixed-integer programming model and subsequently address the pricing of empty container rental within it. Abdelshafie et al. [5] introduced a modeling framework based on agent-based modeling for Empty Container Repositioning (ECR). Song et al. [6] developed three optimization models based on three strategies in ECR among ports. The three Considered strategies are 1) full allocation, 2) no allocation, and 3) near allocation bound. The chain calculation method has been used in this paper. Bakir et al. [7] introduced a robust model that considers stochastic demand for positioning empty containers among ports, to minimize costs related to the repositioning process, and the methodology used in the paper is budget-robust optimization. Song et al. [6] considered ECR with stochastic demand and stochastic container supply across multiple ports. The two-stage model used in the paper has been calculated using the particle swarm optimization algorithm and the Markov chain. Feng et al. [8] introduced a multi-stage decision-making model for empty container positioning to minimize lorry costs and idle time. By treating trade imbalance as the primary driver of container repositioning, Gencer and Demir [9] developed deterministic and stochastic models to minimize the costs associated with container repositioning. The deterministic model has been solved using mixed-integer programming, and the stochastic model, considering uncertain demand, has been solved using scenario-based stochastic programming. Lu et al. [10] introduced a stochastic dynamic programming model with uncertain container demand to account for both repositioning empty containers and pricing factors, as well as their coordination, for a shipping company operating in a two-container depot system. Zhang and Facanha [11] conducted a case study on repositioning empty containers for one of the top shipping carriers in the world in the transpacific corridor, then, based on that, developed a model considering uncertain demand and supply at the strategic level. In the end, he offered three strategic propositions to enhance progress and reduce related costs. Zhang et al. [12] introduced a stochastic model to reduce costs associated with positioning empty containers, accounting for uncertain demand and lost sales. Di Francesco [13] studied positioning empty containers among ports considering uncertainty in port disruption and modeled the subject using multi-scenario stochastic programming. Dong et al. [14], after giving a literature review, intended to analyze two strategies: 1) origin-destination, and 2) dynamic rules based on conditions) for positioning empty containers and comparing them.

Moon et al. [15] calculated and compared the total costs of positioning containers among ports by considering two scenarios: 1) using foldable containers, and 2) using standard containers. The model heuristic algorithm has been used. Meng and wang [16] in their work involve designing a network model considering the flow passage through hub ports, multiple regular ports, and the positioning of empty containers. In the first step, pairs of ports served by a shipping company are considered, and a mixed-integer linear programming model is then defined. Real operational data from Pacific-Asia-Europe shipping lines are applied to this model, and results are obtained based on real-world data.

Additionally, to demonstrate the net capital preservation of the model across the entire network, the model was finally assessed without considering the allocation of empty containers. Moon et al. [2] developed an optimization model to account for costs associated with positioning empty containers, including fixed and variable container shipping costs, Terminal Handling Costs (THC) at ports, and other related expenses, as well as the costs of renting and purchasing empty containers. In the end, it utilized heuristic, genetic, and hybrid algorithms to solve the problem.

As our contribution is to develop a robust optimization model based on Moon et al. [2], we adopt the same problem definition as demonstrated in Moon et al. [2]. We consider a scenario for describing a shipping company's activities. The shipping company receives an order at a port from a customer and, after that, attempts to send an empty container to the customer. The customer's cargo will then be loaded into the received empty containers. After staffing, the containers are completed, and the full containers are sent to the nearest port container terminal, loaded onto a vessel, and carried to the destination port. At the destination port, after unloading the full delivered containers, they are then transported to the receiver. By the time the receiver completes stripping, the containers are delivered back empty to the depot. After inspection and maintenance, those empty containers are ready for another trip. When full containers are received from another port, they are imported; if they are transported to other ports, they are exported. If the number of imported containers is either fewer or more than the number of exported containers, there will be an imbalance. If the number of exported containers exceeds the number imported at a port, there will be a shortage of empty containers.

On the other hand, for ports with more imports than exports, there will be a surplus of empty containers. At surplus ports, surplus empty containers are stored in empty container depots, and a certain cost is incurred for holding them, in addition to the fact that these valuable assets will remain idle for some time. At the deficit ports, there is no choice but to lease or purchase empty containers to meet demand. The shipping company must find a solution to reduce those costs. One solution could be "empty container positioning," which is transporting empty containers from surplus to deficit ports. By positioning empty containers at ports, costs related to leasing, purchasing, and inventory are reduced, but there will be transportation costs. Moreover, due to demand fluctuations, a port might have a surplus at some times and a deficit at others. Therefore, the question for the shipping company is: How many empty containers of different types, and when and where are they supposed to be positioned? We therefore aim to propose a mathematical model to answer this question. The rest of the problem definition can be seen in [2].

After introducing the robust model in Section 2, we now present the results of both models and, in the end, compare them to understand how uncertainty can affect deterministic model results.

We now attempt to analyze the effects of some parameters in the deterministic model on the objective function and total costs in this section. We consider three parameters.

In this article, we used the deterministic model of Moon et al. [2] as the basis for our developed robust optimization model. A robust method has been used to minimize the effects of uncertainty in the worst-case scenario. Seven parameters, including variable handling costs at ports for both owned and rented vessels, carriable transportation costs for both owned and rented vessels, variable leasing costs for containers, container purchasing costs, and demand, were considered uncertain in our developed robust model. Based on results from both deterministic and robust models, and by understanding the importance of considering uncertainty in mathematical models, our developed robust model produced more realistic results than the deterministic model. Capacity of leasing and returning leased containers plays a vital role in repositioning costs, however, it cannot completely replace purchasing containers if we are eager to achieve the optimal solution (In the real world environment, based on the strategy of each shipping company, a shipping company might choose to rent all of its assets including vessels, containers, depots and, etc. while another company chooses only to purchase assets permanently and some others might choose to both purchase and lease at the same time). Fleet capacity also plays an important role. Shipping companies with larger fleets tend to leverage economies of scale to gain an advantage over the competition and reduce costs effectively.

For future research, considering foldable containers in the model would elevate the value of the introduced model. Using other methods of robust optimization could also be another possible extension. In the end, developing a model with two objective functions to minimize costs and maximize profit simultaneously would definitely help shipping companies achieve optimal solutions.

The authors appreciate the valuable insights and contributions of the experts who participated in this study.

This research received no external funding.

The data used in this study are available from the corresponding author upon reasonable request.