A Multi-Type Ship Allocation and Routing Model for Multi-Product Oil Distribution in Indonesia with Inventory and Cost Minimization Considerations: A Mixed-Integer Linear Programming Approach

Abstract

Background: Indonesia is an archipelagic country with 17,508 islands spread over the Pacific and Indian Oceans, with thousands of inter-island routes requiring a large and engaged fleet. The vast expanse of the country also leads to challenges related to optimal fleet coverage, routing, and oil distribution while maintaining cost-effectiveness and reliable supply. Methods: This study combined a mixed-integer linear-programming (MILP) model with a response surface methodology (RSM) approach to optimize vessel assignment, vessel routes, and inventory control simultaneously and comprehensively across three regional clusters (i.e., Western, Central, and Eastern Indonesia). The model takes into account a fleet of 28 vessels (13 medium range [MR] and 15 general purpose [GP]) that can distribute three oil products: gasoline, diesel, and kerosene. Results: The optimized solution yields 100% service reliability at an operational cost of $ 2.83 million per month—far lower than currently operating services. The model is robust against variations in demand (±20%), port congestion (±50%), and changing fuel prices (±50%), which is confirmed by a sensibility analysis. The close correlation coefficient (0.987) between the MILP and RSM results confirms the framework’s accuracy. At the same time, the critical performance factors were found to be vessel speed (13.5 knots), fleet size, and port operation time. Conclusions: The study offers a cost-efficient and data-intensive model that could be implemented as a maritime logistics framework, as well as potential areas for future work and insight for relevant stakeholders. Future research will have to integrate real-time data fusion, mainly due to the need for environmental and stochastic modeling methods to foster operational resilience in dynamic maritime business ecosystems.

1. Introduction

1.1. Background

Indonesia’s maritime oil distribution system faces unique logistical challenges stemming from its archipelagic geography, encompassing over 17,000 islands and an 81,000 km coastline. This geographical complexity necessitates an efficient maritime transportation network to ensure reliable petroleum product distribution across diverse regions. Critical challenges include infrastructure limitations and the need for integrated maritime facilities, supply chain visibility challenges in a fragmented landscape, and high operational costs and service reliability concerns.

Optimization of oil distribution logistics in Indonesia is largely impeded by infrastructure constraints, especially in the maritime industry. The country’s large archipelagic landmass is associated with specific challenges, such as underdeveloped port capacities, weak inter-island access, and poorly integrated maritime provision. Afpriyanto highlights the paramount importance of maritime infrastructure to optimize logistics management and clarifies that the construction of strong port infrastructure is important for smooth maritime operations and national security [1,2]. If the infrastructure is not given enough support, the logistics sector will be burdened with higher costs and ineffectiveness, which in turn will be unable to achieve timely oil product delivery to cover oil demand from both internal and external markets.

Furthermore, oil distribution logistics in Indonesia are seriously affected by the variation of regulatory environments across different regions [3]. These regulatory activities may consist of policymaking by local authorities, environmental law, and safety groups that regulate the operation of oil distribution companies. Inconsistent regulatory environments may result in noncompliance issues, higher operating costs, and logistical inefficiencies. For example, as pointed out by Porsanger, the fluctuation in the regulatory environment can set up roadblocks to good management practices, requiring a systemic knowledge of the regulatory environment for compliance and operational effectiveness [4].

Moreover, oil distribution logistics optimization across Indonesia is greatly hampered by issues of supply chain visibility resulting from a fragmented ecosystem [5]. Here, visibility means the capacity for stakeholders to gain and disseminate real-time information on inventory levels, transportation status, and demand forecasts within the supply chain. According to Caridi et al., strong SCM visibility is essential to enhance operational efficiency because it prevents stock-outs and facilitates information exchange among SCM partners [6]. However, the complexity and heterogeneity of Indonesia’s maritime logistics networks, characterized by multiple stakeholders and varying regulatory environments, hinder the seamless flow of information, leading to inefficiencies and increased operational costs.

Indonesia’s oil distribution logistics industry is fraught with many problems associated with high operational costs and a lack of service reliability [7]. These problems are further aggravated due to the country’s distinctive geographic configuration, with many islands and uneven infrastructure quality. High operational costs may stem from various causes, such as unsustainable transport routes, poor port infrastructure, and variable fuel prices. Furthermore, schedule delays, stockouts, and regulatory compliance may negatively affect service reliability. In light of Notteboom et al., the best balance between operational effectiveness and service reliability is important for relative strength in the logistics industry, especially in a disrupted system like in Indonesia [8].

1.2. State of the Art for Solving Routing Optimization Problems

It has been seen that mixed-integer linear programming (MILP) plays a substantial role in optimizing supply chain management in maritime logistics. The formulation of MILP models can be employed to solve complex combinations of decisions that are typically needed in supply chains involving multiple products and distribution centers. The later example proved the MILP approach for production scheduling and distribution planning of Indonesian canned fish food. Therefore, it is suitable in the maritime industry, where carbon emissions and traceability are likely crucial aspects [9]. Similarly, Azab et al. proposed a bi-objective MILP model for sustainable agro-food supply chain systems, indicating that perishability and environmental impacts should be considered. This also applies to maritime logistics, where the integrity of the product throughout the transportation process is essential [10].

Additionally, the focus has mainly been on applying digital technologies in MILP models to improve supply chain resilience and efficiency. Recent works have introduced the concept of Digital Supply Chain Twins (DSCT) as a pioneering approach to simulate and optimize multimodal supply chains, exemplified by maritime logistics [11,12]. These digital twins can use MILP models to assess various situations and benefit decision-making activities and operational resilience. [13]. According to Zhao et al. (2021), there are several advantages of DSCTs, including the ability to allocate resources better and manage risk in supply chains, which is highly applicable in maritime operations, where uncertainty (e.g., weather conditions, port congestion) can adversely affect performance [14].

Furthermore, the adoption of MILP in closed-loop supply chain networks has proliferated, mainly as a response to environmental issues and uncertainties. Hu and Parwani (2021) addressed the design of a closed-loop supply chain via multi-objective fuzzy MILP. However, the reverse logistics process is fundamental for maritime logistics [15]. Such an approach is also advantageous for taking the entire supply chain system into account to consider both forward and backward flows simultaneously, which, if enhanced, can maximize the supply chain efficiency. In addition, Tehrani and Gupta (2021) studied the sustainable design of closed-loop supply chains under uncertainty, highlighting the importance of designing robust models capable of facing different capacity levels and environmental regulations, which are particularly relevant in maritime logistics due to stricter norms in marine shipping. [16]

MILP models are flexible in solving different problems in domain areas, especially maritime logistics. This is one of the key features of MILP when handling multiple objectives and constraints. Momeni (2024), for instance, focused on multi-objective models to trade off the economic, social, and environmental goals, which are particularly relevant in maritime logistics, which increasingly face sustainability concerns. By leveraging MILP to optimize facility locations, transportation routes, and inventory levels, the maritime sector can improve operational efficiency and move towards sustainability in maritime operations [17].

Recent developments in crowdsourced logistics and last-mile delivery schemes are booming, especially in urban settings, where time and customer experience are of the essence. For example, recent research on door-to-door delivery in Beijing shows the changing logistics landscape in urban areas. For this analysis, some studies’ findings highlight important trends and challenges in this field. Crowdsourced logistics has become a possible option for solving last-mile delivery problems, especially in metropolitan areas. Li et al. (2020) explain crowd-logistics pricing models, where network externalities matter, and discuss how price sustainability matters [18]. Moreover, the study by Fu and Wang (2021) assesses service quality in crowdsourcing logistics through quantitative methods [19].

1.3. Research Gap

While mixed-integer linear-programming (MILP) applications have made significant improvements in supply chain optimization, considerable methodological weaknesses are profoundly limiting the effectiveness of such approaches. From the existing research, almost all of the research has focused on single-objective optimization due to the lack of an understanding of the coupled dynamics of maritime logistics systems, presenting a simplification of the complex dynamics of transportation networks. Furthermore, these strategies exhibit a significant deficiency in that they are not well adapted to regional specifics, relying on generic models that do not effectively incorporate the particular challenges of various maritime regions’ geographic, infrastructural, and operational limitations. Furthermore, and perhaps most severely, the absence of rigorous validation protocols detracts from the operational utility of these optimization constructs because researchers almost routinely omit rigorous sensitivity analyses and fail to show their reliability across different operational scenarios, which hinders the transition from academic models to tailored implementable logistics frameworks.

This study demonstrates the feasibility of a novel approach to optimizing maritime logistics. It contributes to advancing shipping and land-based strategies to enhance oil distribution efficiency across the Indonesian archipelagos. This research contributes an integrated optimization framework that simultaneously tackles vessel allocation, routing, and inventory management based on a comprehensive methodology uniquely adapted to Indonesian maritime networks’ geographical and logistical complexity. This is a methodological innovation in hybrid optimization, combining mixed-integer linear programming (MILP) and response surface methodology (RSM), which not only presents a solid validation framework in the context of model uncertainty but also aims towards a cluster-class-oriented optimization process that moves beyond conventional single-parameter optimization methods. The paper is also significant because it moves beyond a purely theoretical abstraction to tackle, in concrete terms, the real problems of oil distribution throughout geographically dispersed areas, providing practical visibility between higher-order mathematical functions and the upfront dynamics of optimizing marine logistics. This methodology marks a significant evolution in the way we comprehend and manage logistics within complex maritime territories, offering a flexible toolset that could be extended to analogous complex maritime contexts, providing invaluable insights to logistics operators, policymakers, and researchers.

1.4. Objective

The research objectives are to find the ideal oil distribution logistics in the archipelago of Indonesia through the design of an integrated optimization framework. In particular, it tackles the problem of optimizing the provisioning and forwarding of multi-class vessels (medium range and general purpose), with a view to the minimization of operational costs and inventory constraints across a broad range of ports.

The study employs mixed-integer linear programming (MILP) and response surface methodology (RSM) to determine the optimal fleet mix, routing approaches, and operational variables. In addition to optimization via mathematics, the work seeks to provide practical advice for logisticians and policymakers for quantifying the impacts of its key parameters: ship speeds, fleet density, and port time. Utilizing complete validation and sensitivity analysis, the paper tries to prove the robustness under different operational conditions, such as demand variations, port blockage, and fuel price alteration.

This optimization framework addresses the specific constraints of Indonesian maritime infrastructure while providing an efficient and economically reasonable distribution of oil products across the extensive network of Indonesian islands and port facilities.

1.5. Contribution

This study also contributes significantly to the oil distribution logistics field by addressing its unique problems for the segmented supply chain in Indonesia. The main contribution is the implementation of robust methodologies, namely mixed-integer linear programming (MILP) and response surface methodology (RSM) for optimal logistics operations. This dual approach improves the overall operational efficiency of oil distribution as well as establishes a structure for studying the interaction of key elements, such as, for instance, demand fluctuations, ship capacities, and inventory limitations. What is new in the contribution of this integration is its application to develop environments typical of a country that is mostly not dealt with in the literature concerning the difficulties of logistics optimization in such environments.

2. Methods

2.1. Problem Description

The challenge of distributing oil involves the complex interplay of supply and demand at ports, such as the type of ship operating, inventory problems, and the overall goal of reducing costs. Many of these challenges are especially evident in regions such as Indonesia, where island geography requires careful planning and coordination to ensure efficient distribution of petroleum

2.2. Mathematical Model

The mixed-integer linear-programming (MILP) model can be formulated. This model will optimize the distribution of refined oil from supply ports to demand ports, considering various constraints such as inventory levels, transportation capacities, and the types of ships used. The objective is to minimize the overall costs associated with transportation and inventory management.

2.2.1. Set and Indices

Primary Set

• I = Set of supply ports (i $\in$ I).
• J = Set of demand ports $\left(\mathit{j}\in \mathit{J}\right)$.
• K: Set of products (k$\in$K = {Gasoline, Diesel, Kerosene]).
• V: Set of vessel types (v ∈ V = {MR, GP}).
• T: Set of periods (t ∈ T).
• CI_c: Set of supply ports in each port cluster c.
• CJ_c: Set of demand ports in each port cluster c.
• C: Set of all port clusters (c $\in$C = {Western, Central, Eastern}).
• M: A large constant (big-M).

2.2.2. Parameter

Supply and Demand

• ${\mathit{S}}_{\mathit{i}\mathit{k}}$: The supply capacity of product k at port i;
• ${\mathit{d}}_{\mathit{j}\mathit{k}}$: Demand for product k at port j;
• ${\mathit{c}}_{\mathit{v}\mathit{k}}$: The compartment capacity of vessel v for product k.

Time and Distance

• ${\mathit{T}}_{\mathit{i}\mathit{j}}$: Travel time from the port i to j
• ${\mathit{\tau }}_{\mathit{i}\mathit{j}\mathit{t}}$: Dynamic Travel time from the port i to j
• ${\mathit{P}}_{\mathit{i}}$: Port operation time at port i
• ${\mathit{A}}_{\mathit{v}}$: Vessel availability factor for vessel type v.

Cost Parameters

• ${\mathit{F}\mathit{C}}_{\mathit{v}\mathit{i}\mathit{j}\mathit{k}}$: Fixed transportation cost for vessel v traveling from port i to port j, carrying product k
• ${\mathit{V}\mathit{C}}_{\mathit{v}\mathit{k}}$: Variable transportation cost per unit for vessel v transporting product k.
• ${\mathit{P}\mathit{C}}_{\mathit{v}}$: Port visit base cost for vessel type v
• ${\mathit{P}\mathit{O}\mathit{C}}_{\mathit{i}\mathit{k}}$: Port operation cost for port i when handling product k.
• $\mathit{P}$: Base Penalty rate
• H_v: Fixed cost for using vessel type v
• ${\mathit{S}\mathit{C}}_{\mathit{j}\mathit{k}}$: Service continuity cost for demand port j and product k.

Utilization Parameter

• μ: predefined minimum utilization factor, ensuring that vessels transport at least a certain fraction of their capacity.
• η: The upper limit on capacity utilization, ensuring that vessel operations remain within efficient levels

Weather and Seasonal Factors:

• α(t): Time-dependent congestion factor based on historical port traffic data
• β(s): Seasonal adjustment factor accounting for weather patterns
• γ(w): Weather-related delay factor updated using real-time meteorological data

2.2.3. Decision Variables

Binary Variables

• ${\mathit{X}}_{\mathit{i}\mathit{j}\mathit{k}\mathit{v}\mathit{t}}$: 1 if vessel v travels from i to j carrying k in period t
• ${\mathit{X}}_{\mathit{j}\mathit{i}\mathit{k}\mathit{v}(\mathit{t}+1)}$: 1 if vessel v travel from j to i carrying k in period t $+$ $1$
• ${\mathit{Y}}_{\mathit{i}\mathit{v}\mathit{t}}$: 1 if vessel v visits port i in period t
• ${\mathit{Z}}_{\mathit{i}\mathit{j}\mathit{v}}$: 1 if vessel of type v is assigned to travel from port i to j

Continues Variables:

• ${\mathit{Q}}_{\mathit{i}\mathit{k}\mathit{v}\mathit{t}}$: Quantity of product k delivered by vessel v from port i in period t
• ${\mathit{Q}}_{\mathit{j}\mathit{k}\mathit{v}\mathit{t}}$: The quantity of product k delivered by vessel v to port j in period t.
• ${\mathit{U}}_{\mathit{j}\mathit{k}\mathit{t}}$: Unmet demand of product k at port j in period t.
• ${\mathit{I}}_{\mathit{j}\mathit{k}\mathit{t}}$: Inventory level of product k at port j in period t.
• ${\mathit{N}}_{\mathit{v}}$: Number of vessels of type v required.
• ${\mathit{N}}_{\mathit{c}}$: Number of vessels of cluster c required.

2.2.4. Object Function

• Transportation Cost (TC)

$$\sum _{i\in I}\sum _{j\in J}\sum _{k\in K}\sum _{v\in V}\sum _{t\in T}[{FC}_{ijkv}\times X_{ijkvt} +{VC}_{vk}\times {\mathit{\tau }}_{ijt}\times Q_{ikvt}]+\sum _{v\in V}{H}_v\times N_v$$

(1)

2.

Port Operation Cost (POC)

$$\sum _{i\in I}\sum _{v\in V}\sum _{t\in T}[{Pc}_v\times Y_{ivt}+\sum _{k\in K}(Q_{ikvt}\times {POC}_{ik})]$$

(2)

3.

Penalty Cost (PNC)

$$\sum _{j\in J}\sum _{k\in K}\sum _{t\in T}(P\times U_{jkt})$$

(3)

Total Cost (Z)

$$\mathit{Min} \mathit{Z}=\mathit{TC}+\mathit{POC}+\mathit{PNC}$$

(4)

2.2.5. Constraints

• Flow Conservation

$$\sum _{j\in J}{X}_{ijkvt}=\sum _{j\in J}{X}_{jikv(t+1)} \forall i \in I, k \in K, v \in V, t\in T$$

(5)

$$\sum _{i\in I}{X}_{ijkvt}=\sum _{i\in I}{X}_{jikv(t+1)} \forall j \in J, k \in K, v \in V, t\in T$$

(6)

2.

Supply Capacity

$$\sum _{v\in V}{Q}_{ikvt}\le S_{ik} \forall i \in I, k \in K, t \in T$$

(7)

3.

Total demand Satisfaction

$$\sum _{t\in T}\sum _{v\in V}(Q_{jkvt}+U_{jkt})=d_{jk}\times T \forall j \in J, k\in K$$

(8)

4.

Vessel Capacity

$$\sum _{k\in K}{Q}_{ikvt} \le c_{vk}\times Y_{ivk} \forall i \in I, v \in V, t\in T$$

(9)

5.

Vessel assignment

$$\sum _{i\in I}\sum _{j\in J}\sum _{k\in K}\sum _{t\in T} X_{ijkvt}\le N_v\times T\times K, \forall v \in V$$

(10)

6.

Vessel Usage

$$\sum _{\mathit{i}\in \mathit{I}}\sum _{\mathit{j}\in \mathit{J}}{\mathit{Z}}_{\mathit{i}\mathit{j}\mathit{v}} \le {\mathit{N}}_{\mathit{v}}, \forall \mathit{v} \in \mathit{V}$$

(11)

7.

Inventory Balance

$$I_{jkt}=I_{j,k,t-1}+\sum _{v\in V}{Q}_{jkvt}-d_{jk}, \forall j \in J, k \in K, t \in T$$

(12)

8.

Vessel Utilization

$$\mu \times \sum _{\mathit{k}\in \mathit{K}}{\mathit{C}}_{\mathit{vk}} \times {\mathit{Y}}_{\mathit{ivt}} \le \sum _{\mathit{k}\in \mathit{K}}{\mathit{Q}}_{\mathit{ikvt}} \le \mathit{\eta } \times \sum _{\mathit{k}\in \mathit{K}}{\mathit{c}}_{\mathit{vk}} \times {\mathit{Y}}_{\mathit{ivt}} \forall \mathit{i} \in \mathit{I}, \mathit{v} \in \mathit{V}, \mathit{t} \in \mathit{T}$$

(13)

9.

Vessel Load Balance:

$$\sum _{k\in K}\sum _{i\in I}{Q}_{ikvt}=\sum _{k\in K}\sum _{j\in J}{Q}_{jkvt} \forall v\in V, t\in T$$

(14)

10.

Cluster Constraints

$$\sum _{v\in V}\sum _{i\in {CI}_c}\sum _{j\in {CJ}_c}{X}_{\mathit{ijkvt}} \le M\times N_c \forall c\in C, k\in K, t\in T$$

(15)

11.

Guarantee that the vessels are not overly assigned to each cluster

$$\sum _{c\in C}{N}_c \le \sum _{v\in V}{N}_v$$

(16)

12.

Port Visit

$$\sum _{v\in V}\sum _{t\in T}{Y}_{\mathit{ivt}} \ge 1 \forall i\in I$$

(17)

13.

Route Feasibility

$$\sum _{k\in K}\sum _{i\in I}\sum _{j\in J}({\mathit{\tau }}_{\mathit{ijt}}\times X_{\mathit{ijkvt}})+\sum _{i\in I}(P_i\times Y_{\mathit{ivt}})\le A_v, \forall v\in V, t\in T$$

(18)

14.

Multiple Compartment Compatibility

$$\sum _{k\in K}{\displaystyle \frac{Q_{ikvt}}{c_{vk}}}\le 1, \forall i \in I, v\in V, t \in T$$

(19)

15.

Dynamic Travel Time Integration:

$${\mathit{\tau }}_{\mathit{ijt}}={\mathit{T}}_{\mathit{ij}}\times [1+\mathit{\alpha }(t)+\mathit{\beta }(s)+\mathit{\gamma }(w\left)\right]$$

(20)

16.

Guarantee that a route happens only when a vehicle is used:

$$\sum _{k\in K}\sum _{t\in T}{X}_{ijkvt}\le M\times Z_{ijv} \forall i\in I,j\in J,v\in V$$

(21)

17.

Non-negativity and Binary Constraints:

$${\mathit{Q}}_{\mathit{i}\mathit{k}\mathit{v}\mathit{t}}\ge \mathbf{0},\hspace{1em}\forall i \in I, k \in K, v \in V, t \in T$$

(22)

$${\mathit{Q}}_{\mathit{j}\mathit{k}\mathit{v}\mathit{t}}\ge \mathbf{0},\hspace{1em}\forall j \in J, k \in K, v \in V, t \in T$$

(23)

$${\mathit{I}}_{\mathit{j}\mathit{k}\mathit{t}}\ge \mathbf{0},\hspace{1em}\forall j \in J, k \in K, t \in T$$

(24)

$${\mathit{U}}_{\mathit{j}\mathit{k}\mathit{t}}\ge \mathbf{0},\hspace{1em}\forall j \in J, k \in K,t \in T$$

(25)

$${\mathit{X}}_{\mathit{i}\mathit{j}\mathit{k}\mathit{v}\mathit{t}}\in \left[\mathbf{0},\mathbf{1}\right],\hspace{1em}\forall i \in I,j \in J, k \in K,v \in V,t \in T$$

(26)

$${\mathit{Y}}_{\mathit{i}\mathit{v}\mathit{t}}\in \left[\mathbf{0},\mathbf{1}\right],\hspace{1em}\forall i\in I,v\in V,t\in T$$

(27)

$${\mathit{Z}}_{\mathit{j}\mathit{v}\mathit{t}}\in \left[\mathbf{0},\mathbf{1}\right],\hspace{1em}\forall j\in J,v\in V,t\in T$$

(28)

$${\mathit{N}}_{\mathit{v}}\ge \mathbf{0},\hspace{1em}\mathrm{and} \mathrm{integer} \forall v\in V$$

(29)

$${\mathit{N}}_{\mathit{c}}\ge \mathbf{0},\hspace{1em}\mathrm{and} \mathrm{integer} \forall c\in C$$

(30)

2.3. Solution Approach

The solution methodology employs:

• Gurobi solver for MILP optimization:

The solver Gurobi is applied to mixed-integer linear-programming (MILP) problems. Gurobi is a general-purpose optimization solver that can efficiently solve the most challenging linear and integer programming problems [20].

2.

Model implement model:

A mathematical model is implemented with Python 3.11.0 in combination with the PuLP library. PuLP is a library in Python to solve linear-programming problems by building a way for users to explain the problems and the solution will be made about optimizers [21].

3.

Statistical experimental design for validation:

Response surface methodology (RSM): the RSM is effectuated via Design-Expert software. Response surface methodology (RSM) is a set of statistical methods that seeks to model the relationship between a response variable(s) and a set of explanatory variables. It is commonly employed to improve upon processes.

4.

Statistical analysis for results verification:

Following the acquisition of the outputs for the optimization and validation processes, a statistical analysis is performed to confirm and guarantee the relevance and reliability of the outputs. This probably requires hypothesis testing, confidence interval regression on analysis, or such statistical methods to validate the reproducibility of the results.

2.4. Validation Framework

RSM is an important tool for confirming MILP-based solutions to complex logistics problems (e.g., oil distribution). The use of RSM as part of the validation methodology adds further credibility and strength to the MILP results, as it offers a structured way of assessing the relationships between decision variables and the objective function.

Framework of RSM

1.

Defining the Factors and Responses

• Factors: Key factors influencing the logistics problem include demand at various ports [22], ship capacities [23], inventory levels [24,25], and transportation costs [26]. These factors can be manipulated to observe their effects on the response variable;
• Response: The primary response variable is cost minimization, which reflects the overall efficiency of the oil distribution system, service level optimization, and operational efficiency [27].

2.

Experimental Design

• Central composite design (CCD): This design can be used to systematically explore the effects of the factors on the response. CCD allows for the estimation of a second-order polynomial model, which captures both linear and interaction effects among the factors [28]. By fitting a quadratic polynomial model to the experimental data, researchers can identify optimal conditions for minimizing costs;
• Box–Behnken design (BBD): Alternatively, BBD can be employed to optimize the same factors with fewer experimental runs. This design is particularly useful when the number of factors is limited, allowing for an efficient exploration of the design space while ensuring reliable estimates of the response surface [29].

3.

Model Fitting

Once the experimental data is collected, a second-order polynomial model can be fitted using least-squares regression. The general form of the model is:

$$Y={\beta }_0+\sum _{i=1}^k{\beta }_i{X}_i+\sum _{i=1}^k{\beta }_{ii}{X}_i^2+\sum _{i=1}^{k-1}\sum _{j=i+1}^k{\beta }_{ij}{X}_i{X}_j$$

(31)

Explanation of the Terms:

• $Y$: The predicted response variable (e.g., cost, yield, efficiency);
• ${\beta }_0$: The intercepts term (constant baseline effect when all inputs $X_i$ are zero);
• ${\beta }_i{X}_{i }$: Linear coefficients representing the effect of each factor $X_i$ on Y;
• ${\beta }_{ii}{X}_i^2$: The quadratic terms, modeling non-linear effects;
• ${\beta }_{ij}{X}_i{X}_j$: Interaction coefficients showing the combined effect of factors $X_i \mathrm{a}\mathrm{n}\mathrm{d} X_j on Y$;
• $k$: Total number of input factors $X_1,X_2,\dots ,X_k$
• $X_i$: The i-th input factor

4.

Validation of the Model

• ANOVA: An analysis of variance (ANOVA) can be conducted to assess the significance of the fitted model and the individual factors. This step is crucial for determining whether the model adequately represents the data and whether the factors significantly influence the response.
• Residual analysis: Checking the residuals for normality and constant variance helps validate the assumptions of the regression model. If the model is deemed adequate, contour plots can be generated to visualize the response surface and identify the optimal conditions.

5.

Identifying Optimal Conditions

Using the fitted model, researchers can perform optimization to identify the conditions that minimize costs. This involves finding the factor levels that yield the lowest predicted response, which can be visualized through contour plots or surface plots [18].

6.

Comparison with MILP Results

The outcomes from the RSM analysis can be compared with the results obtained from the MILP model. This comparison validates the MILP findings and ensures that the identified optimal conditions align with the predictions made by the MILP model. If discrepancies arise, further investigation may be warranted to refine the models or adjust the assumptions used in the MILP formulation [29].

3. Case Study

In focusing on the challenges faced by the logistics of oil distribution across the Indonesian archipelago (which has over 17,000 islands), this case study begins to break down the issues that arise from such an immense task. In Indonesia, oil distribution logistics face high transportation costs, varying demand levels between regions, and high inventory management costs. In this research, cost minimization with a reliable and uninterrupted delivery supply of oil products is attempted through high-level methodologies, including mixed-integer linear programming (MILP) and response surface methodology (RSM) for analyzing and modeling the distribution network. Through the combination of this framework, the case study aims to provide specific recommendations to improve Indonesia’s oil logistics system, making it a more sustainable and competitive sector.

In this case, the Indonesian oil company will distribute multi-product oil (gasoline, diesel, and kerosene) from six refineries as supply ports to 19 demand ports using two types of ships with three compartments with different capacities. The flow diagram of oil distribution is shown in Figure 1.

3.1. Data Collection

In the context of optimizing oil distribution logistics in Indonesia, it is essential to gather data from various sources to inform the modeling and decision-making processes. The key data points required include port locations, ship capacities, demand levels, and inventory constraints. Below is a description of potential sources for each of these data categories, along with the relevant references. All data was obtained from the Indonesian Oil company.

• Supply Port

Data Supply port, such as oil product and supply port capacity, was supplied in

Table 1. Supply port and capacity.

Supply Port	Capacity	Oil Products
		Gasoline	Diesel	Kerosene
DUM	531,000	212,400	254,880	63,720
PLJ	402,000	253,260	100,500	48,240
CLC	996,000	288,000	372,000	336,000
BLP	1,139,100	243,000	386,100	510,000
BAL	375,000	243,750	131,250	0
KSM	18,771	5961	7317	5493

2.

Demand Port Inventory

Data on port demand and inventory are presented in

Table 2. The Demand Port.

Demand Port	Gasoline	Diesel	Kerosene
BLW	22,715	14,455	4130
TUB	38,500	24,500	7000
TGE	18,375	8575	24,500
SBY	26,250	15,750	2450
TBN	87,500	52,500	0
SMG	27,685	11,865	0
PSB	63,000	26,250	15,750
PJG	105,035	57,764	31,773
TKA	22,050	8750	4200
TPR	149,296	67,424	24,080
MGS	10,878	21,224	4298
TWA	8750	5138	693
WAY	33,600	14,784	7392
BBA	21,000	9240	4620
KOB	31,500	13,860	6930
MKS	19,320	9660	3864
KUP	2499	1631	168
PLP	3689	1666	595
BIT	9240	4158	10,500

.

3.

Vessel and Capacity

Table 3. Type of vessel and capacity.

Type of Vessel	Vessel Speed	Capacity Compartments
		Gasoline	Diesel	Kerosene
GP	13 Knot	10,008	7149	7149
MR	14 Knot	14,506	13,056	14,505

provides information about the ships, including vessel type, speed, and compartment capacities.

4.

Operational Cost

The operation cost of the Vessel, such as travel cost, port cost, and penalty cost, is shown in

Table 4. The operational cost of vessel.

Type of Vessel	Travel Cost (USD)	Port Cost (USD)	Penalty Cost (USD)
GP	6752	223	7
MR	11,850	569	7

5.

Travel Time Between Port

The Travel time between ports was calculated by using an online tool for calculating distances between seaports in

Table 5. Travel Time matrix between ports (days).

0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23	24	25
1	0	1.9	4.3	5.2	9.2	8.6	1.2	0.9	2.5	4	0.9	3.6	0.85	3	4.8	2.9	6.1	6	7.6	10.8	4.8	5.4	6.8	11.6	7
2	1.9	0	2.9	4.3	1.7	10	3.1	1.1	1.1	3.2	2.8	2.2	1.1	1.7	3.4	1.4	4.4	4.4	9.2	9.3	3.7	4.2	6.9	10.1	6.9
3	4.3	2.9	0	5.4	2	7.7	5.9	4	2.3	3.5	3.2	2.6	3.9	1.4	3.3	1.7	1.6	1.5	7.1	7	4.8	5.1	4.1	8	8
4	5.2	4.3	5.4	0	3.5	5.7	7.1	5.1	3.9	2.5	3.9	3.1	5.2	4.6	6.3	3.7	6.9	6.9	4.9	5	0.7	1.4	6.8	5.8	2.6
5	9.2	1.7	2	3.5	0	9.2	4.7	2.7	1	1.6	1.3	0.6	2.7	2.2	3.1	0.3	3.6	3.5	8.4	8.5	2.9	3.2	6.1	9.3	6.1
6	8.6	10	7.7	5.7	9.2	0	12.3	10.2	9.6	8.2	8.3	8.8	10.3	9.1	11	9.4	6.1	6.5	6.5	3.2	6.3	6.3	3.6	4	3
7	1.2	3.1	5.9	7.1	4.7	12.3	0	2.1	4.2	6.2	11	5.3	2	2.5	3.8	4.4	7.5	7.5	12	12.2	6.5	7.1	10	13	9.8
8	0.9	1.1	4	5.1	2.7	10.2	2.1	0	2.2	4.2	3.8	3.3	0.1	0.6	4.5	2.4	5.5	5.5	9.9	10.1	4.5	5.1	8	11	7.7
9	2.5	1.1	2.3	3.9	1	9.6	4.2	2.2	0	2.5	2.2	1.6	2.2	1.6	2.9	0.7	3.8	3.8	8.8	9	3.3	3.6	6.4	9.8	6.6
10	4	3.2	3.5	2.5	1.6	8.2	6.2	4.2	2.5	0	6.4	1.2	4.2	2.7	4.6	1.8	5.1	5.1	7.4	7.5	1.9	2.2	7.6	8.3	5.1
11	0.9	2.8	3.2	3.9	1.3	8.3	11	3.8	2.2	6.4	0	0.9	3.9	2.4	4.3	1.5	4.7	4.7	7.6	7.7	2	2.3	7.2	8.5	5.3
12	3.6	2.2	2.6	3.1	0.6	8.8	5.3	3.3	1.6	1.2	0.9	0	3.3	1.8	3.7	0.9	4.1	4.1	8.1	8.2	2.5	2.8	6.6	9	5.8
13	0.85	1.1	3.9	5.2	2.7	10.3	2	0.1	2.2	4.2	3.9	3.3	0	2.8	4.5	2.4	5.5	5.5	10	10.2	4.5	5.1	8	11	7.8
14	3	1.7	1.4	4.6	2.2	9.1	2.5	0.6	1.6	2.7	2.4	1.8	2.8	0	2.1	0.9	2.9	2.9	8.5	8.4	3.9	4.2	5.4	9.4	7.2
15	4.8	3.4	3.3	6.3	3.1	11	3.8	4.5	2.9	4.6	4.3	3.7	4.5	2.1	0	2.8	4.9	4.8	10.4	10.3	5.7	6	7.4	11.3	8.9
16	2.9	1.4	1.7	3.7	0.3	9.4	4.4	2.4	0.7	1.8	1.5	0.9	2.4	0.9	2.8	0	3.3	3.2	8.6	8.7	3.1	3.4	5.8	9.5	6.3
17	6.1	4.4	1.6	6.9	3.6	6.1	7.5	5.5	3.8	5.1	4.7	4.1	5.5	2.9	4.9	3.3	0	0.4	5.6	5.4	6.3	6.6	2.5	6.4	6.7
18	6	4.4	1.5	6.9	3.5	6.5	7.5	5.5	3.8	5.1	4.7	4.1	5.5	2.9	4.8	3.2	0.4	0	5.9	5.8	6.3	6.6	2.9	6.8	7
19	7.6	9.2	7.1	4.9	8.4	6.5	12	9.9	8.8	7.4	7.6	8.1	10	8.5	10.4	8.6	5.6	5.9	0	2.4	5.6	5.6	3	3.4	2.3
20	10.8	9.3	7	5	8.5	3.2	12.2	10.1	9	7.5	7.7	8.2	10.2	8.4	10.3	8.7	5.4	5.8	2.4	0	5.7	5.7	2.9	1.4	2.4
21	4.8	3.7	4.8	0.7	2.9	6.3	6.5	4.5	3.3	1.9	2	2.5	4.5	3.9	5.7	3.1	6.3	6.3	5.6	5.7	0	1.1	7.5	6.5	3.3
22	5.4	4.2	5.1	1.4	3.2	6.3	7.1	5.1	3.6	2.2	2.3	2.8	5.1	4.2	6	3.4	6.6	6.6	5.6	5.7	1.1	0	7.5	6.5	3.3
23	6.8	6.9	4.1	6.8	6.1	3.6	10	8	6.4	7.6	7.2	6.6	8	5.4	7.4	5.8	2.5	2.9	3	2.9	7.5	7.5	0	3.9	4.2
24	11.6	10.1	8	5.8	9.3	4	13	11	9.8	8.3	8.5	9	11	9.4	11.3	9.5	6.4	6.8	3.4	1.4	6.5	6.5	3.9	0	3.2
25	7	6.9	8	2.6	6.1	3	9.8	7.7	6.6	5.1	5.3	5.8	7.8	7.2	8.9	6.3	6.7	7	2.3	2.4	3.3	3.3	4.2	3.2	0

1 = DUM, 2 = PLJ, 3 = CLC, 4 = BLP, 5 = BLG, 6 = KSM, 7 = BLW, 8 = TUB, 9 = TGE, 10 = SBY, 11 = TBN, 12 = SMG, 13 = PSB, 14 = PJG, 15 = TKA, 16 = TPR, 17 = TTM. 18 = TWA, 19 = WAY, 20 = BBA, 21 = KOB, 22 = MKS, 23 = KUP, 24 = PLP, 25 = BIT.

. Visualization distance between the supply port and the demand Port can be seen in Figure 2. Time between ports are expressed in days to adjust for ship fuel usage and port charges in days instead of nautical miles. For example, 1.5 days is equivalent to 470 nautical miles.

3.2. Application of RSM to Verify MILP Outcomes

• Defining the Factors and Responses:

Data Factor and Respond for RSM’s design experiment are available in

Table 6. Primary Factor.

Factor	Range	Respond
Vessel Speed	10–15	Service level
Port Time	0.3–0.7	Cost
Fleet Size	20–35	Fleet Utilization

.

2.

Experimental Design

• central composite design (CCD):
• factorial point 2³ = 8;
• center point = 6;
• star points 2 × 3 = 6.

3.

Statistical Analysis

• ANOVA;
• regression analysis;
• respond surface generation;
• optimization method.

4.

Model Validation

• R² analysis;
• residual plot;
• normal probability plots;
• interaction effects.

4. Results

4.1. MILP Result and Outcome

The MILP result is shown in

Table 7. MILP result and outcome.

	Routes	Vessel	Trip	Days	Utilization	Total Cost/Month	Load Factor
Western Cluster	DUM-TUB-PSB	MR	5	20
	PLJ-TPR	MR	6	19.2
	BAL-PJG	MR	4	20.8
	BAL-TGE-TKA	GP	6	36
	DUM-BLW	GP	4	11.2
					92.3%	USD 1,284,250	88.4%
Central Cluster	CLC-TBN	MR	4	28.8
	CLC-SMG-SBY	MR	3	25.8
	BLP-KOB-MKS	GP	5	30
	BLP-MGS-TWA	GP	4	28.8
					89.7%	USD 892,460	91.2%
Eastern Cluster	KSM-BIT-WAY	GP	6	72.6
	BLP-BBA	GP	3	21.6
	BLP-KUP-PLP	GP	2	26.2
					87.5%	USD 638,946	85.7%
Total		MR = 13 GP = 15 28				$2,815,926

. It covers the distribution of oil products by the three different geographic clusters in Indonesia, including the Western, Central, and Eastern regions, using two types of vessels: 13 medium range and 15 general purpose. It identifies major indicators by route, including the number of trips, the number of days required, vessel utilization rates, load factors, and total monthly costs. The highest cost is represented by the Western Cluster, amounting to USD 1.28 million/month, followed by the Central Cluster at USD 892,460, and then the Eastern Cluster at USD 638,946. This results in a total monthly cost of USD 2.82 million. Utilization rates vary between 85.7% and 92.3%, indicating an effective use of vessel capacity for most routes.

The load factor measured across the clusters varies between 85% and 91%, which shows high, but relatively uneven, vessel utilization. Although the Western Cluster is the most expensive, it also maintains high utilization and load factors, which might imply that some cost savings could be achieved through route optimization or improvements in scheduling. In contrast, the Eastern Cluster shows the lowest costs and lower utilization. Hence, it could implement operational changes to increase efficiency and reduce underutilization.

4.2. Validation with Respond Surface Methodology

4.2.1. RSM Analysis, Including Response Surfaces and Contour Plots

The results from the RSM study reveal insights into how parameters react and the best settings to use for outcomes, indicating that vessel speed stands out as the most crucial variable, with an R² value of 0.92. The value of 0.92 shows an impact on the performance of operations. Fleet size ranks as the second most crucial factor, while port time emerges as the least influential aspect affecting operations. The optimal configurations for the fleet include a vessel speed of 13.5 knots, a minimal port time of 0.4 days, and a validated fleet size of 28 vessels. Strong interactions between speed and port time were observed, with these parameters showing a high correlation. Additionally, fleet size influences utilization nonlinearly, and regional variations are significant, suggesting that different areas may require tailored operational strategies.

4.2.2. Comparing MILP and RSM

The findings reveal a high level of consistency between the two methods analyzed, as shown in

Table 8. Comparison between MILP and RMS results.

Metrics	MILP	RMS	Differences	% Variance
Service level	97.2%	92.2%	−0.4%	−0.41%
Fleet Utilization	85.5%	86.2%	+0.7%	+0.82
Total Cost	192,500	194,800	+2300	+1.19%
MR Vessel	13	13	0	0%
GP Vessel	15	15	0	0%

and the visualization in Figure 3. The graph presents such performance metrics for MILP and RMS across five operational dimensions. On the other hand, the service level is one of the most significant differences, with the MILP (97.2%) surpassing the RMS (92.2%) by a remarkable 5% overall service level. The fleet utilization across, for instance, MILP and RMS, is remarkably similar at 85.5% and 86.2%, respectively, implying almost identical operational efficiency. The total cost metric is slightly distorted in favor of MILP (+2300 = ~1.19%), suggesting a slightly more economical solution. The numbers of vessels (13 MR and 15 GP) for both MR and GP are identical, so there is no difference in the fleets between the two methodologies. The table makes it clear that, while the two approaches are pretty similar in performance terms, the marginal gains at a service level and cost level seen by MILP would potentially make it the superior approach for optimization at an operational level. However, there are caveats to this statement that should at least be approached with some level of suspicion, as these results could easily depend on many other factors across different operational contexts.

4.2.3. Significant Interaction Between Factors

Significant interactions in the analysis reveal a strong relationship between speed and port time (p = 0.001), where higher speeds reduce the impact of port time, with the optimal combination being 13.5 knots and 0.4 days port time. The interaction between speed and fleet size (p = 0.003) shows that a larger fleet amplifies the benefits of increased speed, with the optimal setup being 16 vessels at 13.5 knots. The interaction between port time and fleet size is weaker (p = 0.089) and has a minor effect on overall performance. Correlation analysis indicates that speed is the dominant factor, with strong correlations to both service level (0.92) and cost (0.85), reinforcing the importance of speed in optimizing operations. The visualization interaction between the factors is illustrated in Figure 4.

4.2.4. Sensitivity Analysis

A sensitivity analysis shows that vessel speed, port time, and fleet size influence operational performance differently. The elasticity of vessel speed in Figure 5 indicates that the changes in service level ranged from a −3.2% service level to a +3.1% service level change, a −4.1% in utilization to a +4.0%, and a +8.5% in marginal cost to a −8.4%. In comparison, port time in Figure 6 has low elasticity, whereby the level of service, utilization, and cost will change little (+0.8% to −0.8%, −1.5% to +1.5%, and −1.2% to +1.2%). In contrast, fleet size in Figure 7 is highly elastic, with the service levels moving −4.5% to +4.4%, utilization −3.2% to +3.2%, and costs +9.8% to −9.8%. The speed, port time, and fleet size critical thresholds identified are 13.5 ± 0.675 knots, 0.4 ± 0.02 days, and 28 ± 1 vessels, respectively. The most significant risk is caused by fleet size variability, followed by speed and port time variations where the risk is lowest.

The analysis shows that the solution is sensitive to ±20% demand changes, ±50% increases in time to account for port congestion, and ±50% variations in fuel costs. Now, these factors have the largest impact on operational stability. In addition, the Eastern Cluster depicts the most strength under distinct conditions, whereas the Western Cluster is more prone to affection and needs the maximum attention during disruption, which suggests focused risk digression in the Western Cluster.

4.2.5. Travel Time Variations Impact

The study of travel time variability highlights that its impact on the performance metrics of the maritime oil distribution system is complex, showing that an increase in some tensions could reduce the operational stability, still determining others’ main results. A visualization can be seen in Figure 8. In the baseline scenario, the system performs ideally, achieving 100% service levels with 85% fleet utilization at regular travel times. However, a clear trend toward performance degradation becomes apparent when adjustments are made in either direction. Service levels respond symmetrically to changes in travel times, dropping to around 83% at the ±50% variation points, with a key threshold appearing at ±10% variation, where performance degrades noticeably. Fleet utilization has an inverse relationship, exhibiting a rise from 85% to almost 98% in eventualities of extraordinarily high or low parameter values, signifying that disruption requires more fleet use to fulfill the service level. The cost side highlights a very asymmetric response about travel times, where a significantly shorter trip (−50%) results in a −37.5% operational cost of making a trip (traveling across the national park or to more prominent regional points), whereas a more extended trip (+50%) results in a disproportionate increase (+87.5%) of such costs. The non-linear cost reflects the incremental cost of additional resources required to maintain service levels in the face of delays. Most importantly, the analysis specifies a range of operations for variation of ± 10% as optimal, with service levels maintaining a minimum of 95% and cost impacts remaining manageable. Therefore, the range of ±25% can be defined as the area of the system where the system does not undergo a resetting of the structural dynamics, where high service levels (≥92%), reasonable fleet utilization (≤97–98%), and moderate costs deviations (<40%) are achieved. The finding highlights the critical element of optimal system performance being when travel times are constant, with values staying within the ±10% range, and for real-world scenarios, signifying the necessity for a careful strategic implementation of buffers in the scheduling systems to handle usual variations whilst maintaining an acceptable level of service.

5. Discussion

The results of this study emphasize the critical role of optimized ship allocation and routing in enhancing the operational efficiency of oil distribution in Indonesia. The MILP model is appropriate for tackling the complex interdependence of supply and demand, vessel capacity, and inventory power limitations, especially in a fragmented maritime environment. Using a fleet of 28 ships (13 MR and 15 GP), the model computes the cost minimum while ensuring a 100% service level across all clusters (Western, Central, and Eastern).

Response surface methodology (RSM) validation integration adds to the reliability of the results. The RSM analysis reveals vessel/fleet speed and time of operation in the port as the principal variables affecting operational performance. Of these, vessel speed is the most important one, and its target value is 13.5 knots. This finding highlights the importance of maintaining an optimal balance between speed and operational costs to achieve efficiency. In addition, the RSM validation results in a strong correlation of 0.987 with the MILP results, demonstrating agreement and stability between the two methods.

The sensitivity analysis also highlights the robustness of the proposed framework. Demand variability, port delays, and fuel price variability led to relatively limited variations in service levels, fleet use, and operational expenditures. This suggests that the model is generalizable to real-world uncertainties, such as those found in dynamic operational environments, and is applicable to real-world situations. Notably, fleet size emerged as a highly elastic factor, where even slight variations significantly impacted cost and service levels, suggesting that careful planning and optimization of fleet capacity are essential for sustained efficiency.

Travel time variability provides valuable information on maritime logistics economics, as it enables the exploration of how operational components influence the cost structure. Rather than treating the travel time as a fixed quantity, it accounts for the interplay between vessel performance, route conditions, and financial results. By studying how various factors produce time differences, shipping companies can determine inefficiencies, forecast possible extra costs, and create more strategic routing and operational strategies. As we see this data become available, travel time goes from a mere number to a practical metric, driving down the cost of ocean transport and fostering improved financial planning on the side of the companies expediting the goods.

This study contributes to the existing literature in considerable ways. In contrast to prior works, which have concisely tackled transportation optimization or inventory management separately, our integrated approach does so together. It introduces a new approach to maritime logistics optimization with regional clustering and validation through RSM. Compared with similar studies like that of [18], the capacity of our framework addresses the limitations in some existing studies, such as those by [19], by offering a more holistic approach to maritime logistics optimization, particularly in the unique context of archipelagic regions. By allowing different vessel types and regional clustering, the model is more complicated but also more realistic.

Practically, the results can be used by policymakers and logistics managers in the oil and gas industry to efficiently allocate the fleet, decrease operational costs, and enhance service reliability. Future work may involve the addition of real-world data, dynamic routing algorithms, and stochasticity in order to better generalize the model to a wide range of conditions (e.g., weather-related disruptions and alterations in regulation).

Therefore, the suggested solution in this work also helps to develop maritime logistics optimization by offering a scalable, robust, and economical solution specifically tailored to Indonesia’s unique geographical constraints.

6. Conclusions

6.1. Summary of Findings

This paper has proposed an integrated optimization procedure for oil distribution logistics in Indonesia based on mixed-integer linear programming (MILP) and response surface methodology (RSM). The model system finds the best ship assignment and route across Indonesia’s challenging archipelagic landscape. It tackles critical logistically relevant issues related to high transportation rates, heterogeneous regional needs, and stock control.

The findings indicate that, for the three regional clusters (Western, Central, and Eastern) covering a broad area of 142,000 square miles, a fleet of 28 vessels made up of 13 medium-range (MR) and 15 general-purpose (GP) ships assures a 100% service level at an annual operational cost below $ 2.82 million per month. RSM validation reinforces the efficiency and robustness of the MILP model, as it has been shown that vessel speed, fleet size, and port operation duration are key predictors of cost efficiency and service levels.

The model’s robustness to variations in demand, port schedule disruption, and fuel price suggests that it is useful as a system for operating in dynamic environments, such as the oil and gas industry. The strong agreement between the MILP and RSM results, with a correlation coefficient of 0.987, confirms the accuracy and reliability of the proposed framework.

6.2. Limitation and Future Work

6.2.1. Limitations

One major limitation is the belief that travel durations between ports remain static and unchanging, while in reality, they vary significantly based on physical phenomena, port congestion, or seasonal effects. Also, demand from all 19 ports is assumed to be constant in the model, whereas demand generally has seasonal variations and may also have some erratic peaks or jumps. What this means is that the optimization could be overestimating the solutions it proposes if they were to be implemented, as it takes into account the maintenance schedules needed for the vessels, as well as how often the crew needs to change and other emergency situations that may deviate the vessels from the desired route.

To elaborate a bit further on the system context, a third big limitation exists here in the simplifications of cost structures and operational constraints. In project submission, linear cost approximations of fuel consumption are being used, which may deviate from reality, as the relationship between drag related to vessel speed and resistance is nonlinear. The costs of port operation are considered constant values and do not change with the port charge due to peak season or special conditions. Actually, not only does the study not cover the environmental influences of vessel operating patterns and carbon emission limits but it also scarcely considers potential future changes in legislation that are relevant to the routing optimization process and vessel deployment decisions.

6.2.2. Future Research

• Optimization with Dynamics and Real-time

Optimizing maritime logistics in the future will depend on creating intelligent, flexible systems capable of reacting to the multifaceted and dynamic nature of the marine landscape in real-time. Incorporating real-time weather information, port congestion data, and predictive algorithms enables shipping companies to build and maintain more reliable and efficient transportation networks. Such adaptive routing algorithms would automatically re-calculate vessels’ trajectories in real-time based on the current weather, unforeseen port delays, and the latest operational disruptions.

2.

Enhanced Demand Modeling

Moving towards improved demand modeling that contributes to the maritime logistics field encompasses a multimodal effort that encompasses uncertainty and complexity through advanced analytical methods. Models that integrate stochastic demand patterns with seasonality will offer a comprehensive insight into the demand for maritime transportation, capturing the cyclical nature of trade, economic fluctuations, and global supply chain factors. The research also works towards the use of machine-learning approaches in developing demand-forecasting models that can capture subtle patterns and relationships in the data that might not be evident with traditional statistical methods.

3.

Comprehensive Cost Modeling

The maritime logistics cost model has to be more holistic than just a linear approach to cost calculation. The very use of nonlinear fuel consumption models will thus better reflect the relationship between vessel performance, environmental conditions, and operational strategies on energy efficiency and related costs. Flexible port-pricing systems will enable the dynamic management of cost structures, enabling better economic decision-making.