Optimizing Spare Part Management for Vessels in Liner Shipping

Abstract

Seagoing vessels require regular maintenance. Preventive maintenance can be planned ahead of time, and can be executed either at sea or while visiting a port. The spare parts used when performing maintenance can come from warehouses that store the parts until needed at a port, but they can also come from on-vessel storages. Some spare parts must be available on a vessel at all times, in the case that corrective maintenance is required. This chapter considers liner shipping, where each vessel follows a pre-planned itinerary and a number of preventive maintenance tasks have been scheduled over time. A mathematical model is proposed that can be used to decide where to store spare parts, how many spare parts to keep in inventory, when to order spare parts from suppliers, and when and where to perform the scheduled maintenance tasks. Numerical experiments show that the model can be solved very quickly. The model can thus be used as a tool to support making decisions related to inventory management for spare parts.

1. Introduction

Maritime transportation is a vital mode for national and international trade, making it one of the most important means of transportation [1]. Its historical importance has persisted over time [2]. Moreover, the exploration of maritime transportation from various research angles underscores its relevance in today’s world.

Ensuring the reliability and safety of vessels is crucial in maritime transportation. Achieving this involves performing maintenance tasks that enable proper vessel functioning and prevent system failures. Effective maintenance requires a comprehensive approach with thorough long-term planning.

Maintenance planning has two key categories: Preventive and corrective maintenance [3]. Preventive maintenance occurs during system operation, aimed at preserving an item in a specific condition. On the other hand, corrective maintenance addresses system failures [4]. While preventive maintenance is important to maintenance planning, it cannot completely eliminate failures. Therefore, corrective maintenance has significant importance in maintenance planning as well [5].

When undertaking maintenance tasks, certain prerequisites come into play. Among the crucial aspects is the accessibility of spare parts. In the setting of maritime transportation, spare parts can be stored on board vessels or in warehouses, or they can be delivered directly from suppliers to a vessel visiting a port.

Addressing spare part inventory management becomes crucial when the option to store these components for future maintenance arises. Efficient spare part inventory management aligns with their availability, directly influencing vessel effectiveness and performance [6].

The demand for spare parts changes over time due to factors such as maintenance requirements [7]. To mitigate this variability, maintaining a stock of spare parts provides security [8]. Moreover, the number of items in inventory can be huge due to different demands [9].

While minimizing the number of spare parts is feasible, it should not compromise vessel availability [10]. Insufficient availability of spare parts raises the risk of vessel failures [11]. On the other hand, an excess of spare parts escalates holding costs [12].

In this research, spare part inventory management in maritime transportation is explored. The motivation is a liner shipping company, where vessels follow planned, cyclic itineraries, and where each vessel has a number of planned preventive maintenance tasks. A known number of spare parts are used when executing the maintenance tasks. A single type of spare part is considered, which is ordered from a supplier that can deliver the spare parts either to warehouses located near ports or directly to a vessel when visiting a port. The goal is to determine when to order from the supplier, how much inventory of spare parts to hold, and the exact timing of performing maintenance, so that the total costs are minimized. To address the problem, a new mathematical model is introduced.

This chapter continues by first giving a review of the related research literature. Then, the problem at hand is described. Subsequently, a mathematical model is formulated for the problem. This is followed by a discussion of computational experiments, where the model is tested to illustrate its capabilities to function as a decision support tool. Finally, concluding remarks are provided, including a discussion of directions for future research.

2. Literature review

Maintenance of vessels and spare part inventory management are the main focus of the following literature review. The literature underscores the critical role of maritime maintenance in strategic decision-making processes. Within these discussions, preventive and corrective maintenance are considered as two fundamental approaches highlighted in the literature. Additionally, spare part management is considered a critical issue in effective maintenance planning.

This research specifically addresses the connection between maintenance tasks within the maritime industry and the management of spare part inventories. The literature review is designed to establish this connection, in particular from the perspective of applying optimization models for decision support.

2.1 Importance of maritime maintenance

Maintenance can increase the efficiency of a vessel and the reliability of its services. Moreover, decisions related to maritime maintenance have a significant effect on other players in a vessel’s operation. Maintenance plays a crucial role in a ship’s total life cycle expenses, emphasizing the need for effective management. This cost encompasses various elements, spanning from spare parts to personnel expenses [13]. Moreover, the correlation between a vessel’s age and maintenance cost shows the significance of maintenance practices [14]. Turan et al. offered a model considering costs across a vessel’s life cycle, showing maintenance expenses accounting for approximately one-third of the total life cycle cost [15]. Their work underscores the importance of optimizing maintenance costs and their far-reaching impact on a vessel’s operations.

2.2 Preventive maintenance

In maritime operations, preventive maintenance is essential, planned to avoid expected breakdowns. Challenges include securing skilled staff, managing spare parts, and overseeing inventories in multiple locations. Pillay et al. analyzed operational delays, a key factor in assessing preventive maintenance [16]. Optimizing these tasks and schedules assists decision-makers in considering cost-effectiveness [17].

Sustainability influences maintenance strategies. Franciosi et al. explored sustainability in maintenance models, emphasizing emission reduction [18]. Liu et al. focused on environmental optimization in maintenance, particularly on engine performance’s impact on greenhouse gas emissions [19].

2.3 Spare part management

The maritime industry relies heavily on spare parts for efficient vessel maintenance, emphasizing the importance of their reliability and availability. These components can be strategically stored in warehouses or on board vessels, with the capability of direct restocking by suppliers as needed. Proper spare part management is essential for enhancing vessel reliability, ensuring safety, and reducing maintenance costs. Rinaldi et al. conducted research demonstrating a correlation between the total cost and the maximum number of spare parts storages, underscoring the significance of spare part management in the broader context of asset management [20].

The delivery of spare parts has a critical role in effective spare part management, influencing vessel reliability and delivery costs. Wagner et al. addressed logistical concerns, presenting a strategic framework applicable to diverse businesses and scenarios [7]. This research highlighted the potential of logistical-based planning in spare part management. Vukić et al. proposed an optimal spare parts delivery method for vessels, considering various transportation scenarios and establishing a routing solution [21]. Their study emphasized the significance of spare part availability for shipping companies and underscored the interconnectedness of different supply chain elements in ensuring accessibility.

The demand for spare parts, varying in quantity and type, requires forecasting based on available information. Wang and Syntetos introduced an approach to forecast spare parts demand, emphasizing the critical role of demand information in optimizing spare part inventory management [22]. Van der Auweraer and Boute considered maintenance plans and system failure behavior in forecasting spare parts demand, highlighting the important role of information in accurate predictions [23].

Managing a vast number of spare parts in the maritime industry introduces complexity to spare part management. Sheikh-Zadeh et al. proposed a grouping model for spare parts management, underscoring the importance of categorization in enhancing inventory management efficiency [24]. Additionally, Cakmak and Guney focused on spare part classification as a means to reduce inventory management time, addressing the challenge posed by the industry’s extensive spare parts inventory [25].

Effectively managing spare parts inventory in maritime operations involves balancing the costs of excess storage against the risks of stock-outs. Turrini and Meissner emphasized the high penalties associated with stock-out situations, highlighting the need for a strategic approach [26]. Zheng et al. proposed a solution by integrating ordering and maintenance optimization to strike a balance between storage and stock-out costs [27]. Anglou et al. contributed by presenting an approach for maritime companies to improve order management, considering factors like supplier selection, which significantly influences cost estimates and overall effectiveness [28]. This research underscored the critical role of strategic supplier decisions in reducing costs, ensuring quality, and optimizing spare parts inventory in maritime settings.

Nenni and Schiraldi emphasized the importance of optimized inventory management to address the costly storage of spare parts in maritime operations [29]. On a technological front, Kostidi et al. presented the potential of additive manufacturing to improve spare part availability and reliability while minimizing storage space and costs [30].

Sleptchenko et al. emphasized the importance of strategic planning in repairing spare parts to reduce time and costs [31]. Sustainability concerns were tackled by Driessen et al. [32] and Pater and Mitici [33], showcasing the efficiency gains and cost reductions achievable through effective spare part management. Huiskonen’s work highlighted the managerial challenge of selecting spare parts, emphasizing the role of logistical considerations in this process [34].

2.4 Maintenance optimization

Optimization of maritime maintenance tasks, integral to a vessel’s lifecycle, demands a comprehensive examination of its interplay with spare parts management. Notably, the scheduling of maintenance tasks emerges as a critical facet of this optimization, with Kian et al. proposing a mathematical model that strategically considers location, timing, and predictive accuracy, emphasizing its role in cost-effective and timely spare parts management [35].

The relationship between maintenance and spare part management is an important point. Wang introduced a mathematical optimization model that considered the connection between these two components [36]. Also, research by Abderrahmane et al. focused on optimal maintenance frequencies within a finite time horizon, recognizing the complex relationship between maintenance and spare part management [37]. Similarly, Eruguz et al. focused on the impact of spare part management on maintenance tasks, emphasizing the comprehensive nature of their integration and its consequential effect on operational efficiency [38].

The combination of maintenance and spare part management enhances the reliability of a system [39]. This provides a practical solution but also poses challenges for managers dealing with the complexity of the issue. Furthermore, the incorporation of uncertainty into maritime maintenance optimization, as explored by Manea et al. [40], underscored the adaptability of strategies when faced with uncertainty in costs and labor resource limitations.

In this situation, the comprehensive strategy of optimizing maintenance, along with managing spare parts inventory, emphasizes the crucial importance of maintenance planning. Basten and Ryan explore the consequences of delayed maintenance on spare parts inventory management, considering costs and highlighting the interconnected relationship of these aspects in the operational environment of the maritime industry [41].

2.5 Spare part inventory optimization

The main goal of spare part inventory optimization is to find the best stock level for spare parts, closely linked to maintenance needs [42]. Spare part inventories within the maritime industry, spanning vessels to warehouses, enhance overall availability, as demonstrated by Zhu and Zhou, who highlighted the use of multiple inventories with varying spare part levels across industries [43].

Louit et al. contributed three key optimization criteria for inventory management: Minimizing costs, maximizing availability, and achieving predefined reliability, accounting for assumed demand rates [44]. Zhang et al. structure their inventory optimization model around minimizing spare part inventory levels and total costs, encompassing transportation, inventory holding, and time-related expenses [45]. Transportation costs are a significant consideration, as demonstrated by Levner et al., who emphasized the impact of transportation costs on spare part inventory optimization [46].

Considering optimizing spare part inventory management with maintenance, Eruguz et al. proposed an integrated approach that minimizes delivery, replacement, and inventory holding costs [11]. Jiang et al. [47] and Zhang et al. [48] extended this integration by considering preventive intervals, inspection, and maximum inventory levels as decision variables.

The large quantity of spare parts on ships shows the importance of categorization to simplify the optimization process. Ben Hmida et al. [49] classified critical spare parts to decrease downtime costs in failure scenarios, while Muniz et al. [50] focused on minimizing inventory levels while maximizing criticality.

Demand dynamics, influenced by maintenance tasks and system failures, are critical considerations in spare part inventory management [51]. Zhu et al. focused on demand forecasting driven by planned maintenance, emphasizing the significance of integrating maintenance planning into spare part demand analysis [52].

3. Problem description

Liner shipping companies define routes and schedules for their vessels traveling between different ports. During each port visit, ships have the capability to pick up or deliver goods or passengers and receive various services. The routes are predetermined with fixed schedules. Figure 1 provides an illustration of a liner service with 13 ports, a fixed 42-day route duration, and the distances and durations between ports are known. The fixed nature of the routes and port visits allows for advanced planning and scheduling.

Figure 1.

An example of a liner shipping service [53].

For each vessel, preventive maintenance tasks are planned ahead of time. Adequate spare parts are crucial for maintenance, emphasizing the importance of knowing the location and quantity of each stored spare part for accessibility. Both the itineraries of the vessels and their maintenance plans are considered as inputs. There is some flexibility in determining the exact timing of performing the preventive maintenance. However, if a task is performed later than initially scheduled, it increases the risk of needing more expensive corrective maintenance, and if a task is performed earlier than initially scheduled, it means that the next preventive maintenance may be needed earlier than it would otherwise be.

The planner must determine the optimal number of spare parts on vessels and at warehouses, focusing on planned (preventive) maintenance and associated spare parts inventory management. It is crucial to strike a balance between having enough spare parts to execute the planned maintenance and minimize the number of spare parts on a vessel, and in warehouses, together with the associated storage costs. There are also space limitations for storage, and the vessel stops at different ports with varying access to warehouses. In the problem at hand, only one type of spare parts is considered.

The problem scenario involves the requirement for spare parts across different maintenance tasks for various vessels, each with distinct spare part needs and maintenance task due dates. Spare parts may be stored on vessels, in warehouses, or ordered and delivered directly from suppliers, each with associated storage costs and capacity constraints.

Maintenance tasks can occur at sea or in ports, with spare parts sourced directly from vessel or warehouse storage, suppliers, or a combination of these. The initial inventory levels on vessels and in warehouses are predefined, influencing the required number of stored spare parts. Known values for inventory costs, restocking costs, and maintenance task costs, varying across vessels and ports, contribute to the decision-making process.

The timing of maintenance task execution is critical, impacting the cost-effectiveness of operations. The consideration of planned maintenance tasks on their due dates aims to prevent potential future corrective maintenance needs. Decisions revolve around determining the number of stored spare parts and restocking strategies, with Table 1 providing a comprehensive overview of the problem’s key information and decision points.

Available information	Decisions
Planning horizon	Inventory levels on vessels
Lists of ports, vessels, and maintenance plans	Inventory levels at warehouses
Current locations of vessels	Restocking decisions for vessels
Number of spare parts per maintenance	Restocking decisions for warehouses
Initial inventory levels	When to perform maintenance
Inventory holding costs	Where to perform maintenance
Restocking costs	Source of spare parts for maintenance
Maintenance costs
Upper and lower inventory limits

Table 1.

Overview of available information (left) and the different decisions required (right).

Figure 2 illustrates the flow of spare parts. The supplier can deliver spare parts directly to a vessel, when the vessel is in a port, at a high cost. Alternatively, the spare parts can be shipped to a warehouse, and then from the warehouse to the vessel when visiting a nearby port.

Figure 2.

Spare parts flow.

4. Methodology

The spare part inventory management problem described above is modeled using mathematical programming. This section introduces the notation used in the mathematical model, followed by a detailed explanation of the mathematical model.

4.1 Notation

This section introduces the sets, the parameters, and then the variables used in the mathematical model. The planning horizon is divided into equal periods, so that the whole planning horizon consists of the periods in $T=\left\{1,2,\dots ,H\right\}$. The set of ports which a vessel can visit is denoted by $P$. The set of vessels is $V$, and the set of maintenance tasks to be done on vessel $v\in V$ is dented by $M_v$. The model has the following parameters:

$H$: The number of time periods in the planning horizon.

$I_{0v}^V$: The initial inventory level on board vessel $v\in V$.

$I_{0p}^W$: The initial inventory level on board vessel $v\in V$.

$L_{\mathit{tpv}}$: Indicator for whether or not a given vessel $v\in V$ is located at a given port $p\in P$ at a given time period $t\in T$.

$N_{\mathit{vm}}$: Number of spare parts needed for maintenance $m\in M_v$ for vessel $v\in V$.

$C_v^{\mathit{IV}}$: Inventory cost for a spare part on board vessel $v\in V$.

$C_p^{\mathit{IW}}:$ Inventory cost for a spare part at a warehouse at port $p\in P$.

$C_p^{\mathit{RV}}:$ Restocking cost of a spare part from a supplier directly to a vessel when the vessel visits port $p\in P$.

$C_p^{\mathit{RW}}:$ Restocking cost of a spare part from a supplier to a warehouse at port $p\in P$.

$C_p^{\mathit{RWV}}:$ Restocking cost of a spare part from a warehouse at port $p\in P$ to a vessel visiting the port.

$C_v^{\mathit{MV}}$: Maintenance cost for tasks done by the crew of vessel $v\in V$ at sea.

$C_p^{\mathit{MP}}$: Maintenance cost for tasks done at port $p\in P$.

$C_{\mathit{vmt}}^{\mathit{MT}}$: Penalty cost for performing maintenance $m\in M_v$ for vessel $v\in V$ in period $t\in T$.

$I_v^{\mathit{\text{maxV}}}$: Maximum number of spare parts stored on vessel $v\in V$.

$I_p^{\mathit{\text{maxW}}}$: Maximum number of spare parts stored in the warehouse at port $p\in P$.

$I_v^{\mathit{\text{minV}}}$: Minimum number of spare parts which should be available on board vessel $v\in V$.

$I_p^{\mathit{\text{minW}}}$: The minimum number of spare parts which should be stored in the warehouse at port $p\in P$.

The decision variables considered in the mathematical model are as follows:

$I_{\mathit{tv}}^V$: The inventory level of vessel $v\in V$ in period $t\in T$.

$I_{\mathit{tp}}^w$: The inventory level of a warehouse in port $p\in P$ in period, $t\in T$.

$X_{\mathit{tpv}}^{\mathit{RV}}$: Number of spare parts to be restocked at period $t\in T$ from a supplier directly to vessel $v\in V$ when the vessel visits port $p\in P$.

$Y_{\mathit{tpv}}^{\mathit{RV}}$: Binary variable equal to 1 if and only if a spare part is restocked at period $t\in T$ from a supplier directly to vessel $v\in V$ when the vessel visits port $p\in P$.

$X_{\mathit{tp}}^{\mathit{RW}}$: Number of spare parts to be restocked at time period $t\in T$ from a supplier to a warehouse at port $p\in P$.

$Y_{\mathit{tp}}^{\mathit{RW}}$: Binary variable equal to 1 if and only if a spare part is restocked at period $t\in T$ from a supplier to a warehouse at port $p\in P$.

$X_{\mathit{tpv}}^{\mathit{RWV}}$: Number of spare parts to be restocked at period $t\in T$ from a warehouse at port $p\in P$ to vessel $v\in V$ visiting the port.

$Y_{\mathit{tpv}}^{\mathit{RWV}}$: Binary variable equal to 1 if and only if a spare part is restocked in period $t\in T$ from a warehouse at port $p\in P$ to vessel $v\in V$ visiting the port.

$X_{\mathit{vmt}}^{\mathit{MV}}$: Number of spare parts coming from storage on board vessel $v\in V$ used for maintenance task $m\in M_v$ at period $t\in T.$

$Y_{\mathit{vmt}}^{\mathit{MV}}$: Binary variable equal to 1 if and only if maintenance task $m\in M_v$on vessel $v\in V$ is performed in period $t\in T$ at sea using the vessel’s own crew.

$X_{\mathit{\text{vpmt}}}^{\mathit{MW}}$: Number of a spare parts coming from a warehouse at port $p\in P$ to be used for maintenance task $m\in M_v$, $v\in V$, at period $t\in T$.

$Y_{\mathit{\text{vpmt}}}^{\mathit{MP}}$: Binary variable equal to 1 if and only if maintenance task $m\in M_v$ on vessel $v\in V$ is performed by the personnel of port $p\in P$ when the vessel visits the port at time period $t\in T$.

4.2 Mathematical model

Using the mathematical notation defined above, the following mathematical programming model is proposed.

$$\begin{array}{ll}min\sum _{t\in T}\sum _{v\in V}{I}_{\mathit{tv}}^V{C}_v^{\mathit{IV}}& +\sum _{t\in T}\sum _{p\in P}{I}_{\mathit{tp}}^w{C}_p^{\mathit{Iw}}+\sum _{t\in T}\sum _{p\in P}\sum _{v\in V}{X}_{\mathit{tpv}}^{\mathit{RV}}{C}_p^{\mathit{RV}}+\sum _{t\in T}\sum _{p\in P}{X}_{\mathit{tp}}^{\mathit{RW}}{C}_p^{\mathit{RW}}\\ & +\sum _{t\in T}\sum _{p\in P}\sum _{v\in V}{X}_{\mathit{tpv}}^{\mathit{RWV}}{C}_p^{\mathit{RWV}}+\sum _{v\in V}\sum _{m\in M_v}\sum _{t\in T}{Y}_{\mathit{vmt}}^{\mathit{MV}}{C}_v^{\mathit{MV}}\\ & +\sum _{v\in V}\sum _{p\in P}\sum _{m\in M_v}\sum _{t\in T}{Y}_{\mathit{\text{vpmt}}}^{\mathit{MP}}{C}_p^{\mathit{MP}}+\sum _{v\in V}\sum _{m\in M_v}\sum _{t\in T}{Y}_{\mathit{vmt}}^{\mathit{MV}}{C}_{\mathit{vmt}}^{\mathit{MT}}\\ & +\sum _{v\in V}\sum _{p\in P}\sum _{m\in M_v}\sum _{t\in T}{Y}_{\mathit{\text{vpmt}}}^{\mathit{MP}}{C}_{\mathit{vmt}}^{\mathit{MT}}\end{array}$$E1

The objective function in Eq. (1) consists of inventory costs for vessels and warehouses; restocking costs between suppliers, vessels, and warehouses; and maintenance costs that depend on whether the maintenance is performed at sea or at port, as well as the time period in which the maintenance is performed. The constraints of the model follow:

$$I_{\mathit{tv}}^V\le I_v^{\mathit{\text{maxV}}}t\in T,v\in V$$E2

Constraints (2) make sure that the number of spare parts on board of a vessel never exceeds the considered upper limit for this spare part.

$$I_{\mathit{tp}}^w\le I_p^{\mathit{\text{maxW}}}t\in T,p\in P$$E3

The number of spare parts at each warehouse at a port never exceeds the corresponding capacity at the warehouse. This is modeled using constraints (3), which can also be used to indicate that a warehouse is not available, by setting the right hand side to zero.

$$I_{\mathit{tv}}^V\ge I_v^{\mathit{\text{minV}}}t\in T,v\in V$$E4

$$I_{\mathit{tp}}^w\ge I_p^{\mathit{\text{minW}}}t\in T,p\in P$$E5

There are also lower limits for the number of spare parts on board of vessels and in warehouses, as indicated in constraints (4) and (5).

$$I_{\mathit{tp}}^W=I_{\mathit{tp}}^W+X_{\mathit{tp}}^{\mathit{RW}}-\sum _{v\in V}{X}_{\mathit{tpv}}^{\mathit{RWV}}-\sum _{v\in V}\sum _{m\in M_v}{X}_{\mathit{\text{vpmt}}}^{\mathit{MW}}t\in T,p\in P$$E6

The inventory level of a spare part at each warehouse at a port during each period is related to the inventory level of the previous time period. Also, each warehouse can receive spare parts from a supplier in each period. Moreover, spare parts can be sent from a warehouse to a vessel for a maintenance task, and to restock the inventory on board of the vessel. All these considerations are represented in constraints (6).

$$X_{\mathit{tp}}^{\mathit{RW}}\le I_p^{\mathit{\text{maxW}}}{Y}_{\mathit{tp}}^{\mathit{RW}}t\in T,p\in P$$E7

Constraints (7) make sure that the binary variable $Y_{\mathit{tp}}^{\mathit{RW}}$ takes a value of 1 if any units are sent from a supplier to the warehouse at port $p$ in time period $t$. It also limits the quantity that can be sent in a single period, which is set equal to the capacity of the warehouse.

$$X_{\mathit{\text{vpmt}}}^{\mathit{MW}}\le I_p^{\mathit{\text{maxW}}}{Y}_{\mathit{\text{vpmt}}}^{\mathit{MW}}t\in T,v\in V,p\in P,m\in M_v$$E8

As the previous constraints, constraints (8) are used to set binary variables to 1 to indicate the presence of a flow of spare parts. In this case, the flow is from a warehouse to a vessel where the parts are used immediately in a maintenance operation.

$$Y_{\mathit{\text{vpmt}}}^{\mathit{MW}}\le L_{\mathit{tpv}}t\in T,v\in V,p\in P,m\in M_v$$E9

A spare part can only be picked from a warehouse for a maintenance task on a vessel when the vessel visits a port that has access to the considered warehouse, as enforced by constraints (9).

$$I_{\mathit{tv}}^V=I_{\mathit{tv}}^V+\sum _{p\in P}{X}_{\mathit{tpv}}^{\mathit{RV}}+\sum _{p\in P}{X}_{\mathit{tpv}}^{\mathit{RWV}}-\sum _{m\in \mathit{Mv}}{X}_{\mathit{vmt}}^{\mathit{MV}}t\in T,v\in V$$E10

Constraints (10) define the level of inventory of each vessel at the end of each period, which has a direct relationship with the previous inventory level. Each vessel can receive spare parts directly from a supplier or a warehouse when it visits a port that has access to them. Moreover, spare parts on board of a vessel can be directly used for a maintenance task.

$$Y_{\mathit{tpv}}^{\mathit{RV}}\le L_{\mathit{tpv}}t\in T,p\in P,v\in V$$E11

A vessel can only receive spare parts from a supplier when it visits a port that has access to the supplier, as enforced by constraints (11). Furthermore, there is a possibility for a vessel of receiving spare parts from a warehouse when it visits a port that has access to the warehouse, as given by constraints (12).

$$Y_{\mathit{tpv}}^{\mathit{RWV}}\le L_{\mathit{tpv}}t\in T,p\in P,v\in V$$E12

$$X_{\mathit{tpv}}^{\mathit{RV}}\le I_v^{\mathit{\text{maxV}}}{Y}_{\mathit{tpv}}^{\mathit{RV}}t\in T,p\in P,v\in V$$E13

When spare parts are sent to a vessel from a supplier, a corresponding binary variable $Y_{\mathit{tpv}}^{\mathit{RV}}$ must be forced to 1, so that the related costs can be calculated in the objective function. This is ensured in constraints (13). A similar connection is made in constraints (14), regarding spare parts that are sent from a warehouse to a vessel.

$$X_{\mathit{tpv}}^{\mathit{RWV}}\le I_v^{\mathit{\text{maxV}}}{Y}_{\mathit{tpv}}^{\mathit{RWV}}t\in T,p\in P,v\in V$$E14

$$X_{\mathit{vmt}}^{\mathit{MV}}\le I_v^{\mathit{\text{maxV}}}{Y}_{\mathit{vmt}}^{\mathit{MV}}t\in T,v\in V,m\in M_v$$E15

It is possible to get the spare parts from storage areas on board of a vessel to be used directly for a maintenance task. Then, the corresponding binary variable $Y_{\mathit{vmt}}^{\mathit{MV}}$ should be set to 1, as ensured by constraints (15).

$$\sum _{t\in T}{Y}_{\mathit{vmt}}^{\mathit{MV}}+\sum _{t\in T}\sum _{p\in P}{Y}_{\mathit{\text{vpmt}}}^{\mathit{MP}}=1v\in V,m\in M_v$$E16

A maintenance task can either be done by a vessel’s crew and their facilities, or by a port’s personnel and their facilities when the vessel visits the port. Constraints (16) make sure that one of these options is selected for every maintenance task.

$$\sum _{p\in P}{X}_{\mathit{\text{vpmt}}}^{\mathit{MW}}+X_{\mathit{vmt}}^{\mathit{MV}}\ge N_{\mathit{vm}}v\in V,m\in M_v,t\in T$$E17

For each planned maintenance task, the number of spare parts used must be sufficient. Constraints (17) make sure that this is the case after considering that the spare parts can be taken either directly from the vessel, or from a nearby warehouse.

5. Computational experiments

To evaluate the potential to use the mathematical model presented in the previous section for decision support in spare part management, a series of computational experiments are conducted. These involve solving the model for different instances, which are divided into two groups. The difference between the groups lies in the planning horizon, which is either 1 year, with 365 time periods corresponding to days, or 3 years, with 156 time periods corresponding to weeks.

Within each group of instances, a base case is defined using artificial data. This instance considers 10 vessels, 20 ports, and 2 warehouses. Each vessel has three planned maintenances in the one-year instances and six maintenances in the three-year instances. Each maintenance requires one spare part, and the capacity of each vessel is five items, whereas the capacity of each warehouse is ten items. The cost of sending spare parts directly from the supplier to a vessel is twice the cost of sending the parts to a warehouse, which again is ten times the cost of restocking a vessel using the nearby warehouse. The initial inventories are assumed to be empty, corresponding to the minimum allowed inventory levels.

Starting from the base instances, further instances are considered by varying one aspect of the underlying base instance. In particular, we investigate the effect of the following parameters:

• Number of ports with access to warehouses (increased to three)

• Number of moving vessels (reduced to nine)

• Number of maintenance tasks for each vessel (reduced by one-third)

• Number of required spare parts for each maintenance task (increased to two)

• Initial inventory level on vessels (increased by one)

• Initial inventory level at warehouses (increased by one)

• Ratio of inventory to ordering cost (halved)

• Ratio of maintenance cost by vessel’s crew and facilities to maintenance cost by port’s personnel and facilities (increased from two to three)

• Minimum inventory level on vessels (increased by one)

The mathematical model was implemented in AMPL and solved using the CPLEX 20.1.0 solver on a computer running macOS Ventura equipped with 16 GB RAM and an Apple M2 processor. The CPLEX provides a mixed integer programming solver based on the branch-and-bound method, applying the simplex method to solve linear programming relaxations, and using advanced heuristics and cut-generation.

The generated instances can be solved to optimality quickly, with running times ranging from 1.7 to 15.6 seconds. Table 2 shows the individual running times as well as the objective function values for each instance solved.

		One-year horizon		Three-year horizon
Instance	Parameter varied	Obj. fun	Seconds	Obj. fun	Seconds
Base	None	8954.0	3.1	15585.0	1.9
1	#Ports	8834.0	3.2	13830.0	1.8
2	#Vessels	8194.0	2.9	14055.0	1.7
3	#Tasks	5987.0	2.1	10391.0	1.3
4	#Parts per task	12538.0	12.2	22087.8	8.4
5	Vessel inventory	9459.0	15.6	16802.0	6.4
6	Warehouse inventory	8765.8	3.3	15803.4	1.8
7	Inventory/ordering cost	11970.0	3.2	21685.0	2.3
8	Maintenance cost	9854.0	3.0	16185.0	1.8
9	Minimum inventory	12604.0	3.3	26508.0	1.8

Table 2.

Numerical results from solving twenty test instances.

Examining the instances with a one-year planning horizon, the frequency of direct spare part orders from suppliers to vessels is greatly influenced by factors such as the quantity of needed spare parts and the related ordering costs. The solutions show a sensitivity to ordering costs, avoiding direct orders when costs are increased. In contrast, with a three-year planning horizon, the ordering patterns are seen to vary based on factors such as the number of needed spare parts and the availability of warehouses. Additionally, the analysis indicates that there are more deliveries to warehouses when there are more warehouses available. An increase in the initial inventory levels on vessels and warehouses reduces the necessity for placing orders with the supplier.

Based on the analysis, the highest overall number of maintenance tasks carried out by the crews of the vessels at sea is observed in the one-year planning horizon when there is an initial inventory of spare parts on board of vessels. On the other hand, in the instance where the need for maintenance tasks is lower, the number of maintenance tasks performed at sea is the minimum among all instances.

In the three-year planning horizon, the pattern of the greatest number of maintenance tasks performed at sea aligns with the observations for the one-year planning horizon. Conversely, the instance with the highest number of warehouses has the lowest number of maintenance tasks performed at sea.

According to the findings, the instances with a non-zero initial inventory level for vessels exhibit the highest total number of time periods where vessels maintain an inventory of spare parts exceeding the minimum level, observed in both the one-year and three-year planning horizons. This shows that it is unnecessary to keep more than the minimum inventory level on board of vessels, as in the case studied the vessels are visiting ports with warehouses relatively frequently.

In summary, the mathematical model provided can be solved very efficiently, even for long planning horizons. This allows decision makers to test different scenarios, to see how different situations may affect the need for keeping different amounts of spare parts in inventory.

6. Conclusion

Transportation is an essential part of our lives. Our daily plans are affected directly and indirectly by transportation which exists in different forms. Among transportation modes, maritime transportation plays an important role which expands different areas of research around it. The vessels used in maritime transportation require regular maintenance, which involves the use of spare parts, some of which are critical to the operation of the vessels.

Inventory management of spare parts for vessels has not been studied in depth in the research literature [54]. The planning involved can be complex, as spare parts can be stored both on board vessels and in warehouses located near ports. To minimize costs, the planners must balance the need for having spare parts available at the times of performing planned preventive maintenance tasks, while balancing the costs of ordering from suppliers and the cost of holding inventory.

This chapter discusses one particular optimization problem arising under certain assumptions. The setting is taken from liner shipping, where each vessel has a predetermined itinerary that it follows in a cyclic manner. This means that decisions regarding routing and scheduling have already been fixed. Each vessel also has a set of planned maintenance tasks, but with some flexibility in deciding the exact timing of performing each of them.

The contribution of this chapter is to propose a mathematical programming model for the problem considered and to show that the model is solvable using commercially available software. Several artificially generated instances are solved, which gives some insights into how the structure of the solution may change depending on the cost structure and other aspects of the particular instance.

This research defined artificial instances to test the model to be close to a real-world setting. The model and the presented results show the potential of using optimization for spare part inventory management in maritime maintenance.

However, there are many directions for future developments. First, the model only considers preventive maintenance tasks. In reality, the need for corrective maintenance tasks arises dynamically and must be handled. While the current model can be adopted by enforcing a safety stock (setting the lower limit of inventory at vessels and warehouses to equal the safety stock), it may be better to explicitly model the uncertainty, and to formulate this as a dynamic optimization problem.

Second, in this context, liner shipping is an easy mode, as the routes and schedules of each vessel are planned well in advance. For tramp shipping and industrial shipping, the routes are much more dynamic. In those areas, it may be more pertinent to plan the timing of preventive maintenance simultaneously while planning the routes and schedules of vessels. This increases the difficulty of the planning problem significantly.

Third, the problem considered in this chapter is a tactical problem, and there are strategic decisions that play an important role. This concerns for example the location of warehouses, the capacity of warehouses, and the selection of suppliers.

Fourth, the considered problem takes into account only a single type of spare parts. In reality, each maintenance task requires a large number of different spare parts, of different complexity. Thus, the timing of the maintenance tasks and the ordering from suppliers may need to be coordinated. This is particularly true if a single supplier can provide several different types of spare parts, and the ordering costs are non-linear in the number of spare parts ordered.

In practice, future research is required to see how maritime transportation companies can apply planning tools as presented here. In the case of incorporating corrective maintenance, these companies require good data collection abilities to ensure good managerial decisions.

As environmental concerns increase every day, there is a need to extend the knowledge in the maritime industry in a way of respecting this issue. This can be a good way of thinking for future research by including fuel consumption and the environmental impact of spare part inventory management in the presented model.

As technology arises by exploring different areas of knowledge, it is necessary to adopt new technologies and use them in the best way of respecting the environment and society. A fast-growing technology is additive manufacturing. This can be also a good way of extending the research in the future to make a connection between additive manufacturing and spare part inventory management. The circularity of a business can be also a good way to expand the presented model for future research which can be done by working on the reverse logistics of unused or damaged spare parts.

Acknowledgments

The authors thank the editor for valuable comments that helped to improve this chapter.

Conflict of interest

The authors declare no conflict of interest.