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Abstract

Military technological advancements are crucial for meeting evolving operational needs. Submarines are essential naval assets due to their difficulty in detection. Selecting appropriate equipment for mission efficiency is essential, and automating these tasks with a Decision Support System (DSS) is vital. Air Independent propulsion (AIP) is critical for covert submarine operations, allowing propulsion underwater without surfacing for oxygen. Choosing the right AIP system is challenging because it requires balancing operational efficiency, economic cost-effectiveness, and environmental consequences, all of which entail complex factors that are difficult to quantify or compare. The developed DSS is based on a two-stage Fuzzy Mamdani Model (FMM), applied to perform inference over a submarine AIP. Three reference values were obtained in the first stage to aid operational, economic, and environmental decision-making. In the second stage, these three values create a recommendation value using a percentage system selection. To prove its effectiveness, the DSS was used to evaluate four types of AIPs to establish the level of recommendation and the values of the subcategories in the initial phase. Experts from the Portuguese Navy (PoN) assisted with the system development by defining decision-making rules. The FMM offers advantages over traditional decision-making models by effectively managing vague and qualitative data, enabling reasoning that reflects expert judgment instead of relying on deterministic models. This DSS not only enhances the accuracy and transparency of AIP selection but also provides a scalable and adaptable framework applicable to various complex decision-making scenarios in other engineering applications and fields.

Keywords

decision support system Fuzzy Mamdani Model equipment selection air independent propulsion submarines

1. Introduction

The Armed Forces (AF) play a key role in technological innovation and scientific development (Chin, 2019). Technological superiority is vital for battlefield advantage, requiring careful equipment selection due to economic and technological constraints. War impacts territories, populations, and the economy, as significant resources divert from essential areas of a country (Liadze et al., 2023).

The underwater realm has perpetually captivated humanity, with the development of improved exploration systems being a longstanding objective (Marx, 1990). Two primary concepts for underwater navigation have emerged (Moorhouse, 2015): submersibles and submarines. Submersibles require a principal vessel for launch and recovery due to their limited power capabilities, whereas submarines can operate independently. The first successful deployment of submersibles in warfare occurred during World War II, conferring a substantial advantage to their users (Jones, 2022). The concept of submarines was first conceived by William Bourne in 1578, although practical testing was not conducted then (Avallone, 2000). The inaugural functional submarine, the Turtle, was constructed in 1776, paving the way for numerous technological advancements to refine this crucial instrument in military operations (Lautenschläger, 1987).

Submarines are efficient platforms with superior stealth, giving them a strategic advantage over other naval vessels (Haffa & Patton, 1991). Conventional submarines need to maximize underwater time for stealth. For instance, diesel-electric submarines must frequently surface to snorkel for fresh air to operate their Diesel Engines (DE) and recharge batteries, compromising their discretion (Brighton et al., 1994). This challenge led to the development of nuclear-powered submarines (Chopra, 2023), despite the high costs and nuclear waste disposal issues (Friedman-Jimenez et al., 2022).

Recent submarine advancements aim to reduce energy use, improve battery storage, and integrate Air Independent Propulsion (AIP) systems (Başhan, 2022). AIP extends a submarine’s submerged operation time and is a cost-effective alternative to diesel generators, which need air to charge batteries. It allows for a longer submerged duration while maintaining stealth (Menon et al., 2020). Standard AIP systems include Fuel Cells (FC), Stirling Engines (SE), closed-cycle Steam Turbines (ST), and closed-cycle DE (Kordesch & Simader, 1996; Menon et al., 2020; Wang et al., 2023).

The increasing complexity of real-world systems and the many variables involved in analyzing them have led to developing methods and frameworks known as Decision Support Systems (DSSs). These DSSs are designed to explore a wide range of variables and, through logical reasoning, offer insights that can assist in a particular decision-making process.

Aristotle founded classical logic, which derives binary true or false conclusions (Gabbay & Woods, 2004). Over time, probability theory and fuzzy sets introduced by (Zadeh, 1965) offered a classification method aligned with human thinking, addressing uncertainties. Fuzzy logic effectively manages imprecise qualitative data and improves decision-making where traditional logic is inadequate (Zadeh, 1988). Initially used in control systems (Stelzer et al., 2007), fuzzy logic applications have expanded across other domains.

The Mamdani model is widely used for deriving conclusions from logical rules (Dadios, 2012). Ebrahim Mamdani initially developed it to control a steam engine and boiler using linguistic control rules from experienced human operators (Mamdani, 1976). The model has since been applied in various areas, such as medical diagnosis systems (Gupta et al., 2022), management (Pourjavad & Shahin, 2018), among many others.

This work develops a DSS to select an AIP system for submarines, enhancing submerged autonomy. A two-stage Fuzzy Mamdani Model (FMM) assigns a recommendation value to each AIP system based on operational, economic, and ambient factors. It aims to choose the most suitable AIP system, reducing human subjectivity and improving decision-making. Experts from the Portuguese Navy (PoN) assisted in defining the decision-making rules to enhance accuracy. The implemented system can also be easily adapted to analyze other systems and cases.

Selecting an appropriate AIP system involves navigating complex trade-offs between qualitative and quantitative factors, often in environments of uncertainty and limited data, which makes conventional DSS approaches less effective. Fuzzy logic is particularly well-suited for this context because it facilitates decision-making based on imprecise, linguistic, and expert-derived information, commonly encountered in defense applications. This is especially true when evaluating submarine propulsion systems, where data may be scarce, uncertain, or subjective. DSS methodologies like Neural Networks (NN), Analytic Hierarchy Processes (AHP), and Bayesian models can also aid in selecting AIP systems. NNs excel in recognizing complex patterns and learning, but usually require extensive, high-quality datasets, which are lacking in this defense decision context (Shafi et al., 2021). AHP provides a structured multi-criteria decision-making framework through pairwise comparisons but struggles with uncertainty and the imprecise data typical in submarine propulsion evaluations (Saaty, 2008). Bayesian models offer a probabilistic approach by integrating prior knowledge with observed data, but require well-defined priors, which may introduce biases and limit adaptability in changing conditions (Koller & Friedman, 2009). Consequently, a fuzzy logic-based DSS was selected for its ability to manage uncertainty in expert assessments while ensuring flexible, interpretable decision-making (Zadeh, 1996). Fuzzy inference enables the use of qualitative inputs, adapting to complex AIP trade-offs and enhancing decision reliability while addressing imprecision in AIP propulsion evaluation.

In the submarine domain, besides some fuzzy logic approaches used to enhance depth control (Ranga et al., 2017), and the design of a fuzzy logic system for optimizing decision-making during submarine attack scenarios (Ranga et al., 2017), a fuzzy DSS for managing repair priorities of shipboard weapon engineering equipment during combat situations (Simões-Marques et al., 2000), or a fuzzy expert system developed to assist submarine commanders in damage control situations (Shi et al., 2009), no submarine or ship DSS has been established to support the selection of AIP systems or related equipment. Selecting the most suitable AIP system is challenging due to the need to balance operational effectiveness, economic cost-effectiveness, and environmental impact, which involve complex and multidimensional factors that are difficult to quantify or compare. The proposed DSS utilizes fuzzy logic to enhance decision-making in the selection of submarine propulsion systems, addressing trade-offs and uncertainties in AIP evaluation. This advancement offers a structured, flexible framework that addresses a crucial gap in decision-support methodologies, especially in the naval sector and, more specifically, submarines.

The article is structured as follows: After introducing the problem and describing the contributions and objectives in this section, Section 2 defines the problem and the adopted methodologies. Section 3 presents and describes the developed DSS. Section 4 describes the developed DSS application and discusses the obtained results. Finally, in Section 5, conclusions are provided, and further research work and development topics are suggested.

2. Problem Formulation & Methodologies

Defining the problem helps understand it and outline methodologies for applying the framework to other case studies. Section 2.1 illustrates the submarine technology and the importance of the AIP system to its operations, while Section 2.2 describes the method used to develop the DSS based on fuzzy reasoning.

2.1. Submarine Technology

Global navies’ use of submarines provides a significant strategic advantage during operations. The design and technical requirements of these submarines should align with their respective nations’ operational and technological needs, influenced by strategic and military demands (Hartley, 2024). Militarily, submarine operations depend heavily on their ability to operate stealthily and independently (Howard, 2011).

Compared to nuclear submarines, the main disadvantage of conventional diesel-electric submarines is their limited submerged endurance (Rains & Mitchell, 2000). They must surface or perform snorkeling to intake air for diesel generators, which makes them vulnerable to detection due to exhaust gases and noise. Advancements in anaerobic energy systems, like AIP, are essential for improving underwater endurance (Hawley et al., 1994).

The endurance limitation is a frequent challenge for traditional submarine operations, particularly for navies lacking nuclear-powered submarines. AIP technologies effectively address this limitation, enabling prolonged submerged missions without surfacing, thereby maintaining stealth and tactical advantages. For many countries, AIP offers a budget-friendly option to enhance underwater operational capabilities without the complexities and high costs associated with nuclear propulsion.

As previously described, AIP systems for submarines typically consist of FC, SE, and ST (Ahmadi et al., 2017; Kerros et al., 1994; Kordesch & Simader, 1996; Menon et al., 2020; Wang et al., 2023). Each of these power sources has its advantages and disadvantages. FC are known for their high efficiency and low acoustic and thermal signatures, providing a quiet power source with water and heat as byproducts. However, they require pure hydrogen, which leads to complex storage and high costs. SE use standard diesel fuel and have a low acoustic signature but face challenges related to size and the need for significant maintenance. Closed-cycle ST deliver high power output, enabling submarines to achieve higher speeds, but they are complex, inefficient, and have higher operational costs. Each AIP method offers unique capabilities, making the propulsion system vital in submarine design and mission success.

Choosing an appropriate AIP system for submarines is crucial for any Navy due to its strategic implications and various factors influenced by technology. National needs and capabilities are essential for successful selection. Selection criteria vary based on the Navy type, mission profile, or analyst, necessitating a careful assessment. This complex process can result in time-consuming and subjective decisions. A clear illustration of the inefficiencies arising from poor AIP system selection is the Spanish S-80 Plus submarine program, which faced substantial delays and redesigns due to difficulties in integrating its complex AIP bioethanol-reforming fuel cell system (Navantia, 2020; Revista Ejércitos, 2024). As described in the next section, fuzzy inference systems can simplify these challenges, reducing decision time and imprecision while aligning results with analyst expertise.

2.2. Fuzzy Reasoning

Fuzzy reasoning, typically consists of three stages (Mendel, 1995): Fuzzification (Section 2.2.1), inference mechanism (Section 2.2.2), and defuzzification (Section 2.2.3). First, input data becomes fuzzy sets through fuzzification, changing numbers to linguistic variables using membership functions. Each variable receives membership degrees from 0 to 1, reflecting truth in fuzzy logic. Next, the inference mechanism links input data to knowledge-based rules and merges multiple logical results. Finally, fuzzy values revert to real values based on earlier inferences in defuzzification.

2.2.1. Fuzzification

In the fuzzification stage, we will use a variant of Gaussian membership functions, specifically a membership function based on a mixture of two Gaussian functions, which can be defined as (Zhao et al., 2019):

μ_{g} (x; a, b, σ_{1}, σ_{2}) = {\begin{cases} e^{- \frac{1}{2} {(\frac{x - a}{σ_{1}})}^{2}}, & x < a, \\ 1, & a \leq x \leq b, \\ e^{- \frac{1}{2} {(\frac{x - b}{σ_{2}})}^{2}}, & x > b . \end{cases}

(1)

This function enables us to obtain a membership value of one within a specified interval. It is essential to note that if we assume

a = b

and

σ_{1} = σ_{2}

, the function

μ_{g}

reduces to the standard Gaussian membership function (Das et al., 2020). This representation enables the conversion of numerical values into linguistic variables.

2.2.2. Inference Mechanism

The Mamdani model (Dadios, 2012; Mamdani, 1976) utilizes max-min composition, where the minimum operation evaluates the degree of truth for antecedents within each rule as an intersection, and the maximum operation combines the consequents from different rules through a union to form a cohesive output (Sakti, 2014). After fuzzification, the inference process applies logical operators to combine input fuzzy sets, defining new subsets from the initial sets and firing any rule where this combination yields a non-zero membership degree. The max-min composition corresponds to the AND and OR logical operators used in this process. The implication method then models the fuzzy output set (consequent) for each rule, with the resulting membership degree truncating the consequent fuzzy set by applying the min function to the maximum defined for that set and the value from the antecedent using the logical AND operator. If a rule is not fired, the consequence is a constant membership degree of zero. All rule consequents aggregate in the final inference step, generating the fuzzy output set as the union of the consequent fuzzy subsets with the logical OR operator.

2.2.3. Defuzzification

Different methods can be utilized for defuzzification, with the centroid method, also referred to as the Center of Gravity (COG) method, being the most common (Takagi & Sugeno, 1985). The centroid method calculates the center of mass of the area under the membership function curve, where the desired value is the coordinate of this center, defined as (Takagi & Sugeno, 1985):

c = \frac{\int_{x} μ_{A} (x) \cdot x d x}{\int_{x} μ_{A} (x)}

(2)

where

x

is the input variable,

c

is the output variable (centroid), and

μ_{A} (x)

is the membership degree of the fuzzy set

A

corresponding to

x

ranging from

[0, 1]

Consider a simple triangular membership function, as illustrated in Figure 1, defined by:

μ (x) = {\begin{cases} \frac{x - 2}{2}, & 2 \leq x \leq 4 \\ \frac{6 - x}{2}, & 4 < x \leq 6 \\ 0, & otherwise \end{cases}

(3)

Figure 1.

Example of a Triangular Membership Function $μ (x)$ for Illustrating Centroid-based Defuzzification.

Applying the centroid method, we obtain:

c = \frac{\int_{2}^{4} \frac{x - 2}{2} x d x + \int_{4}^{6} \frac{6 - x}{2} x d x}{\int_{2}^{6} μ (x) d x} = \frac{8}{2} = 4

(4)

As expected, the centroid of this triangular membership function coincides precisely with its midpoint, $c = 4$ .

The centroid method computes the defuzzified output by identifying the geometric center of the aggregated fuzzy set, resulting in balanced and stable outcomes that demonstrate robustness (Takagi & Sugeno, 1985). In contrast, the bisector method divides the area of the fuzzy set into two halves but can be sensitive to asymmetric membership functions (Ross, 2005). Methods such as the Smallest Of Maximum (SOM) and Largest Of Maximum (LOM) select extreme points linked to maximum membership values, potentially causing abrupt shifts in the output. Conversely, the Mean Of Maximum (MOM) averages all points of maximum membership, which can sometimes produce unintuitive results (Mendel, 2017). Overall, centroid defuzzification typically provides smoother and more reliable outputs, making it particularly beneficial in control and decision-making applications (Mendel, 2017; Ross, 2005).

A two-stage FMM, using fuzzy reasoning, serves as a DSS. First, we evaluate the operational, economic, and environmental criteria with fuzzy rules to generate subcategory scores. Then, these scores are aggregated using higher-level fuzzy rules to provide an overall recommendation. This DSS assigns values to the analyzed AIP systems. The choice of economic, operational, and environmental criteria as the primary focus stems from the necessity of a thorough and well-rounded assessment of AIP systems. Economic criteria encompass the acquisition, maintenance, and operational costs, which are crucial for long-term adoption and budgetary considerations. Operational criteria measure efficiency, power, and acoustic signature, significantly affecting the systems’ tactical and strategic importance. Environmental criteria ensure the consideration of thermal emissions, byproducts, and their respective size. Collectively, these aspects provide a comprehensive perspective that enables informed, responsible, and mission-oriented decisions within defense frameworks. The following sections detail the methodology and analysis.

3. Decision Support System (DSS)

To develop the DSS, a theoretical study of the various systems that typically constitute an AIP was first conducted, as described in Section 2.1, to identify the key variables that support the decision-making process, as illustrated in Figure 2. After defining and organizing these variables into subcategories, a recommendation level is determined based on rule evaluation to identify the system that best complies.

Figure 2.

Defining Key Variables, Organizing them into Subcategories, and Determining the Recommendation Process Level.

After defining the variables, subcategories, and recommendation level, fuzzy sets were created by converting numerical data into linguistic variables and establishing the rules. This process involved analyzing related work and leveraging PoN expertise in submarine systems.

The DSS will explore the relationships between characteristics and subcategories, as well as between subcategories and recommendation levels, to provide a comprehensive profile of each system across three distinct fuzzy subcategories: operational, economic, and environmental. The definitions of variables and subcategories will be outlined in Section 3.1, and the recommendation level will be discussed in Section 3.2.

3.1. Variables & Subcategories

The systems studied can be installed in modern submarines, with any challenges, like structural stability, expected to be managed by naval engineering and therefore not addressed here. Key characteristics enable a clear comparison among the systems. While safety, autonomy, and consumption might seem non-essential, they are crucial in modern designs. Autonomy largely depends on the system and platform, while consumption reflects operational costs. The selected variables are grouped into three subcategories: operational, economic, and environmental, providing a comprehensive evaluation of technical characteristics, costs, and potential environmental impacts. These include variables like efficiency, power, acoustic signature, acquisition costs, maintenance costs, operating costs, thermal emissions, expelled byproducts, and size, as shown in Figure 3.

Figure 3.

The Subcategories and Respective Variables Chosen for Evaluation.

Due to the classified nature of the military systems discussed, access to data is limited. As a result, unavailable data is estimated using qualitative methods, drawing on theoretical studies and insights from PoN submarine experts to establish logical rules. The system’s logical rules were developed based on responses from a questionnaire administered to 21 PoN militaries with submarine specialization. The system is designed to adapt to real-world situations, with the Navy working alongside manufacturers and suppliers to gather precise data. Once obtained, this data modifies the existing system or standardizes new variables based on defined fuzzy set values.

3.1.1. Operational Subcategory

The operational subcategory objectively enhances submarine performance while meeting essential operational requirements. It comprises three variables: efficiency, power, and acoustic signature. Efficiency is the energy or heat input ratio to the work output produced by a system. Power refers to the rate at which energy is converted into work. The acoustic signature indicates the equipment’s noise level at full capacity.

A system with high efficiency offers significant operational advantages, allowing energy supplies to last much longer. In contrast, a system with low efficiency consumes energy faster, resulting in a shorter duration of energy supply. For this variable, four fuzzy sets have been defined based on concrete values ranging from 0 to 100%: {Very Poor (VP), Poor (P), Good (G), Very Good (VG)}. The membership functions for these fuzzy sets are determined using a mixture of two Gaussian functions, as described by Equation 1 and illustrated in Figure 4.

Figure 4.

Operation Membership Functions - Efficiency.

Table 1 presents the defined fuzzy sets for efficiency, along with their corresponding mean values $(a, b)$ and standard deviations $(σ_{1}, σ_{2})$ for each membership function. Specific standard deviation values are omitted where irrelevant to the membership function. The values $a$ , $b$ , $σ_{1}$ , and $σ_{2}$ used in this analysis, as well as those for other variables with obtainable concrete values, are based on insights from a theoretical study and PoN experts in submarine systems. This empirical evaluation aims to distinguish reliably between the values of different systems, particularly in the absence of a universal assessment scale. The primary objective was to categorize the variables into four distinct fuzzy sets, each with the same range. These divisions were then optimized for the fuzzy system, with an ideal intersection of two consecutive fuzzy sets, such as VP and P in the efficiency case, occurring at a membership degree of 0.5, ensuring that the maximum of one set aligns with the minimum of the next.

Table 1.

Parameters for Each Efficiency Membership Function.

Fuzzy Sets	$a$	$σ_{1}$	$b$	$σ_{2}$
Very Poor	0	–	8	8
Poor	27	8	42	8
Good	60	7	73	7
Very Good	90	7	100	–

A system with high power output is advantageous for operational purposes as it enables rapid battery charging. Conversely, a system with a relatively low power output significantly reduces its capacity for fast charging. Four fuzzy sets were defined for the power variable: {VP, P, G, VG}. The membership functions for these sets were established using a mixture of two Gaussian functions, as described in Equation 1 and illustrated in Figure 5. Table 2 presents the defined fuzzy sets and their corresponding parameters $a$ , $b$ , $σ_{1}$ , and $σ_{2}$ for each membership function.

Figure 5.

Operation Membership Functions - Power.

Table 2.

Parameters for Each Power Membership Function.

Fuzzy Sets	$a$	$σ_{1}$	$b$	$σ_{2}$
Very Poor	0	–	0.8	0.8
Poor	2.7	0.8	4.2	0.8
Good	6	0.7	7.3	0.7
Very Good	9	0.7	10	–

For the acoustic signature variable, four fuzzy sets have been defined: {Very Low (VL), Low (L), High (H), Very High (VH)}. The membership functions for these sets are identical to those for the power variable, as outlined in Equation 1 and illustrated in Figure 6. Table 3 presents the defined fuzzy sets along with their respective parameters $a$ , $b$ , and $σ_{1}$ , $σ_{2}$ for each membership function.

By defining the three previously mentioned variables, efficiency, power, and acoustic signature, we established the first subcategory, the operational. For this subcategory, three fuzzy sets were defined: {Low (L), Medium (M), High (H)}. The membership functions for these sets were created using a combination of two Gaussian functions, as outlined in Equation 1 and illustrated in Figure 7. Table 4 lists the specific parameters $a$ , $b$ , $σ_{1}$ , and $σ_{2}$ for each membership function. The values $(a, b, σ_{1}, σ_{2})$ were chosen to divide the operational, economic, and environmental subcategories into three distinct sets, each with the same range. These divisions were then further refined to optimize the fuzzy system for its intended purposes.

Figure 6.

Operation Membership Functions - Acoustic Signature.

As previously stated, the inference method will use the FMM, as outlined in Section 2. With the operational subcategory and its defining variables fully established, creating the rule base is necessary to obtain the required inferences.

3.1.2. Economic Subcategory

The economic subcategory refers to the costs associated with a system, encompassing several essential variables: acquisition, maintenance, and operational costs. The acquisition cost includes all expenses for obtaining the system and any auxiliary systems. Maintenance cost covers the expenses required to keep these systems and their auxiliaries in good working condition. Operational cost pertains to the costs associated with operating the systems, such as consumption and refueling, among other expenses. These three variables are defined by four sets: {VL, L, H, VH}, with respective membership functions aligned with the previously described variables. These membership functions are represented by a mixture of two Gaussian functions, as shown in Equation 1 and illustrated in Figures 8.

Table 3.
Parameters for Each Acoustic Signature Membership Function.

Fuzzy Sets $a$ $σ_{1}$ $b$ $σ_{2}$

Very Low 0 – 0.8 0.8

Low 2.7 0.8 4.2 0.8

High 6 0.7 7.3 0.7

Very High 9 0.7 10 –

Figure 7.

Subcategory Operation Membership Functions.

Table 4.

Parameters for Each Operational Subcategory Membership Function.

Fuzzy Sets	$a$	$σ_{1}$	$b$	$σ_{2}$
Low	0	–	15	10
Medium	38	8	62	8
High	85	10	100	–

Figure 8.

Subcategory Economic: The Same Membership Functions Represent Acquisition, Maintenance, and Operation Costs.

The fuzzy sets defined, along with the parameters $a$ , $b$ , $σ_{1}$ , and $σ_{2}$ for each membership function, are indicated in Table 5. Given that the economic subcategory consists of the same three fuzzy sets as the operational subcategory, only Figure 9 is presented, where its membership functions can be visualized.

Figure 9.

Subcategory Economic Membership Functions.

Table 5.

Parameters for Each Economic Subcategory Membership Function.

Fuzzy Sets	$a$	$σ_{1}$	$b$	$σ_{2}$
Very Low	0	–	0.8	0.8
Low	2.7	0.8	4.2	0.8
High	6	0.7	7.3	0.7
Very High	9	0.7	10	–

3.1.3. Environmental Subcategory

The environmental subcategory evaluates the system’s impact on the environment in which it operates. It includes variables such as thermal emission, by-products, and size. Thermal emission reflects the temperatures resulting from the operation of various systems. The byproduct variable refers to the type of byproduct expelled by the systems when operating, indicating whether it is polluting. The size variable pertains to the space or volume occupied by the system.

Obtaining these specific values through theoretical study was possible for the thermal emission variable. For this variable, four fuzzy sets were defined: {VL, L, H, VH}, with respective membership functions represented by a mixture of two Gaussian functions, as shown in Equation 1 and illustrated in Figure 10. Table 6 also indicates the defined fuzzy sets, parameters $a$ , $b$ , and the standard deviations $σ_{1}, σ_{2}$ for each membership function.

Figure 10.

Environmental Membership Functions - Thermal Emissions.

Table 6.

Parameters for Each Thermal Emission Membership Function.

Fuzzy Sets	$a$	$σ_{1}$	$b$	$σ_{2}$
Very Low	0	–	60	80
Low	250	80	350	80
High	530	70	630	70
Very High	800	70	900	–

Given the inherent subjectivity in assessing the number of byproducts generated by the operation of various systems, this variable is represented by four fuzzy sets: {Low Pollutant (LP), Pollutant (P), Highly Pollutant (HP), Extremely Pollutant (EP)}. The membership functions for these sets are defined by a mixture of two Gaussian functions, as shown in Equation 1 and illustrated in Figure 11. Table 7 lists the parameters $a$ , $b$ , $σ_{1}$ , and $σ_{2}$ for each membership function.

Figure 11.

Environmental Membership Functions - Byproducts.

Table 7.

Parameters $a$ , $b$ , $σ_{1}$ , and $σ_{2}$ for Each Byproducts Membership Function.

Fuzzy Sets	$a$	$σ_{1}$	$b$	$σ_{2}$
Low Pollutant	0	–	0.8	0.8
Pollutant	2.7	0.8	4.2	0.8
Highly Pollutant	6	0.7	7.3	0.7
Extremely Pollutant	9	0.7	10	–

As with many other variables, obtaining concrete values for size was impossible, making the fuzzification purely qualitative. This variable is again defined by four fuzzy sets: {VL, L, H, VH}, with respective membership functions defined by a mixture of two Gaussian functions, as shown in Equation 1 and illustrated in Figure 12. Table 8 indicates the fuzzy sets and their respective parameters $a$ , $b$ , and standard deviations $σ_{1}, σ_{2}$ for each membership function.

Figure 12.

Environmental Membership Functions - Size.

Table 8.

Parameters for Each Size Membership Function.

Fuzzy Sets	$a$	$σ_{1}$	$b$	$σ_{2}$
Very Low	0	–	0.8	0.8
Low	2.7	0.8	4.2	0.8
High	6	0.7	7.3	0.7
Very High	9	0.7	10	–

Since the environmental subcategory variable is also composed of the same three fuzzy sets as the operational and economic subcategories {L, M, H}, only Figure 13 is presented, where its membership functions can be visualized. The mean values $a$ , $b$ , and the standard deviations $σ_{1}$ and $σ_{2}$ are identical to those in Table 4.

Figure 13.

Subcategory Environmental Membership Functions.

3.2. Recommendation Level

The recommendation level is derived as a system output variable by combining values obtained from different subcategories and after obtaining the center of mass using the centroid method described in Equation 2. This process allows us to establish the system’s recommendation level. As previously mentioned, the recommendation level is based on rules stemming from insights provided by PoN experts in submarine systems.

To implement these inferences, it is essential to define the output variable. This variable is categorized into five fuzzy sets: {VL, L, M, H, VH}, with respective membership functions defined by a mixture of two Gaussian functions, as shown in Equation 1 and illustrated in Figure 14. Table 9 indicates the fuzzy sets and their respective parameters $a$ , $b$ , and standard deviations $σ_{1}, σ_{2}$ for each membership function.

Figure 14.

Recommendation Level Membership Functions.

Table 9.

Parameters for Each Recommendation Level Membership Function.

Fuzzy Sets	$a$	$σ_{1}$	$b$	$σ_{2}$
Very Low	0	–	10	5
Low	23	5	33	5
Medium	45	5	55	5
High	68	5	78	5
Very High	90	5	100	–

To improve the clarity of the final value obtained at this stage, it would be beneficial for the system to provide an output between 0% and 100%. It is important to note that due to the specific nature of the centroid defuzzification method used, normalization was applied to the output values using the following expression:

r = \frac{n_{i} - n_{m i n}}{n_{m a x} - n_{m i n}} \times 100

(5)

where

r

represents a value from 0% to 100%,

n_{i}

is the value resulting from the final phase of the adopted model,

n_{m a x}

is the maximum value achievable by the model, and

n_{m i n}

is the minimum value.

While the transformation may not be essential since the original values already serve as a ranking, with higher values indicating better recommendations, normalization enhances the clarity of results by presenting them as percentages. In decision-making, users of this system should not only focus on the final value but also compare values across the three subcategories to ensure optimal outcomes. The final value should not be the only determining factor for choosing the best system.

4. System Application & Evaluation

This section presents and justifies the values chosen for each system, based on the variables discussed in Section 4.1, the experimental setup in Section 4.2, and the discussion and analysis of the results obtained, as outlined in Section 4.3.

4.1. Data Application

For characteristics for which concrete values could not be obtained, such as power, acoustic signature, acquisition costs, maintenance costs, operation costs, byproducts, and size, a scale ranging from 0 to 10 was established. The values assigned to different systems will be qualitative and based on theoretical study and the expertise of PoN experts in submarine systems. Once again, feedback from the PoN experts was obtained from the 21 questionnaires completed by PoN militaries with submarine specialization.

It is important to note that the scales are designed to reflect real-world conditions, facilitating the system’s future adaptation with specific values. For example, when considering the acoustic signature, the values will be measured in decibels (dB), with lower values indicating better performance. This principle applies to all other variables except power, where higher levels are preferred for better outcomes. Additionally, we will adopt the worst-case scenario values for each input variable when analyzing the different systems. This approach establishes a coherent method for comparison.

4.1.1. Data Application: Efficiency

For the variable of efficiency, FC can achieve efficiencies exceeding 70% (Kordesch & Simader, 1996; Milliken & Ruhl, 2002), which is relatively high compared to the other three systems. SE have an efficiency of about 40% (Menon et al., 2020), ST achieve efficiencies ranging from 25% to 30% (Lus, 2001), and DE have efficiencies around 30% to 35% (Lus, 2001). Therefore, the values to be considered for efficiency are as follows: (i) FC – 70%, (ii) SE – 40%, (iii) ST – 25%, and (iv) DE – 30%.

4.1.2. Data Application: Power

A theoretical analysis of the variable power revealed that the steam turbine system delivers the highest power output among the four systems. In contrast, SE exhibit the lowest power. When comparing the DE and FC systems, it is concluded that DE have a greater power output than FC. Therefore, we can establish the following power hierarchy: $P (Turbines) > P (Diesel) > P (Cells) > P (Stirling)$ . Consequently, the following values will be used for this comparison: (i) FC – 3, (ii) SE – 2, (iii) ST – 8, and (iv) DE – 6.

4.1.3. Data Application: Acoustic Signature

According to research conducted and insights provided by experts in PoN submarine systems, it was concluded that the least noisy system is the fuel cell system, which operates solely through electrochemical reactions and produces almost no noise. In contrast, the other three systems generate significant noise due to their moving parts. DE are particularly loud because of the explosions within them. ST also produce considerable noise from the rotation of the turbine itself. However, they are much quieter than DE. Finally, SE generate some noise from their moving parts, mainly the pistons. Still, this noise level is significantly lower than that of ST. Based on this evaluation, the following values will be adopted: (i) FC – 1, (ii) SE – 3, (iii) ST – 7, and (iv) DE – 9.

4.1.4. Data Application: Acquisition Cost

When analyzing the acquisition cost variable, we conclude that FC and ST are complex systems requiring highly specialized infrastructures. Consequently, their acquisition costs are significantly higher, comparable to each other, than those of the other two systems DE and SE. DE are regarded as much less complex than SE, resulting in lower acquisition costs. Based on this assessment, we will adopt the following values: (i) FC – 8, (ii) SE – 5, (iii) ST – 8, and (iv) DE – 3.

4.1.5. Data Application: Maintenance Cost

Based on the findings from the study, it was noted that the maintenance routines for FC are significantly less extensive than those for DE, SE, and ST. However, FC tend to have a shorter lifespan than the other systems, leading to the need for more frequent replacement of various internal components, which can be relatively costly. As a result, fuel cell maintenance costs are comparable to DE. In contrast, the maintenance cycles for DE, SE, and ST are much longer due to exposure to corrosion and contamination. However, ST have a distinct advantage over the other engine types. Additionally, SE, due to their complexity, incur higher maintenance costs than DE. Thus, to compare this variable, the following values will be adopted: (i) FC – 5, (ii) SE – 6, (iii) ST – 8, and (iv) DE – 5.

4.1.6. Data Application: Operating Cost

Regarding operating costs, FC are significantly more expensive than DE and SE. This is primarily due to their need for two essential components, oxygen ( $O_{2}$ ) and hydrogen ( $H_{2}$ ), to generate energy. The hydrogen component’s operational, storage, and refueling costs are considerably higher than diesel’s. Additionally, ST that use methanol or ethanol have high consumption rates, placing their operating costs on par with FC. Based on this assessment, the following values will be used: (i) FC – 7, (ii) SE – 3, (iii) ST – 7, and (iv) DE – 3.

4.1.7. Data Application: Thermal Emission

The theory behind the operation of the different systems made it possible to obtain concrete temperature data. FC operating temperatures range from 50°C to 1000°C (Lus, 2001). However, these values correspond to the Proton Exchange Membrane (PEM) FC installed in the PoN Trident class submarines and will be used for comparison. Assuming this, these cells achieve operating temperatures around 80°C. Regarding SE, they reach operating temperatures around 500°C (Sripakagorn & Srikam, 2011), ST achieve temperatures around 700°C (Lus, 2019), and DE operate around 500°C. Thus, the following values will be adopted: (i) FC – 80°C, (ii) SE – 500°C, (iii) ST – 700°C, and (iv) DE – 500°C.

4.1.8. Data Application: Byproducts

When discussing FC, it is essential to note that the byproduct of their chemical reactions is solely distilled water. This is advantageous for maintaining stability, as it helps regulate weight by compensating for the fuel consumption of hydrogen ( $H_{2}$ ) and oxygen ( $O_{2}$ ). The distilled water produced helps make up for the loss of reactants. However, this purified water needs to be expelled during the process. While expelling the water does have some environmental impact, it is minimal and can be considered negligible.

In contrast, the other three systems primarily produce carbon dioxide ( $C O_{2}$ ), one of the most polluting gases. This gas must be released into the sea, which poses a risk to the surrounding environment. Based on this evaluation, the following values will be adopted: (i) FC – 1, (ii) SE – 8, (iii) ST – 8, and (iv) DE – 8.

4.1.9. Data Application: Size

Both FC and ST are complex systems that must be implemented appropriately. ST complexity primarily arises from their size, while FC involve additional systems contributing to their complexity. SE also have some complexity, which is lessened by incorporating a specialized module into the submarine’s design. In contrast, DE are the easiest to implement among these options. Based on this assessment, the following values will be adopted: (i) FC – 7, (ii) SE – 5, (iii) ST – 7, and (iv) DE – 3.

4.1.10. Data Application: Variables Resume

Table 10 summarizes the values to be adopted for each variable of the systems mentioned. The accuracy of this definition will directly impact the reliability of the developed DSS. While significant technological advances may gradually influence the values of these variables, the inherent nature of the systems makes abrupt or significant changes unlikely. Nonetheless, a new propulsion system can be integrated into the analysis, ensuring the DSS remains adaptable and up-to-date.

Table 10.
Variables Values in the Considered Subcategories - Summary.

Operational Subcategory

AIP Systems Efficiency (%) Power Acoustic Signature

Fuel Cells 70 3 1

Stirling Engines 40 2 3

Steam Turbines 25 8 7

Diesel Engines 30 6 9

Economic Subcategory

AIP Systems Acquisition Cost Maintenance Cost Operating Cost

Fuel Cells 8 5 7

Stirling Engines 5 6 3

Steam Turbines 8 8 7

Diesel Engines 3 5 3

Environmental Subcategory

AIP Systems Thermal Emission (°C) Byproducts Size

Fuel Cells 80 1 7

Stirling Engines 500 8 5

Steam Turbines 700 8 7

Diesel Engines 500 8 3

	Operational Subcategory
Fuel Cells	70	3	1
Stirling Engines	40	2	3
Steam Turbines	25	8	7
Diesel Engines	30	6	9
	Economic Subcategory
AIP Systems	Acquisition Cost	Maintenance Cost	Operating Cost
Fuel Cells	8	5	7
Stirling Engines	5	6	3
Steam Turbines	8	8	7
Diesel Engines	3	5	3
	Environmental Subcategory
AIP Systems	Thermal Emission (°C)	Byproducts	Size
Fuel Cells	80	1	7
Stirling Engines	500	8	5
Steam Turbines	700	8	7
Diesel Engines	500	8	3

4.2. Experimental Setup

After defining the variables and ensuring the reproducibility of the results, this subsection details the experimental setup for the proposed DSS, as illustrated in Figure 15. Section 4.2.1 will describe the computational tools used, while Section 4.2.2 will outline the fuzzy categories considered based on the specified variables. Section 4.2.3 will summarize the adopted inference framework, and Section 4.2.4 will discuss the defuzzification process and normalization.

Figure 15.

Experimental Setup - Simplified Illustration.

4.2.1. Computational Tools

The DSS was developed entirely in Python because of its widespread use, user-friendly nature, and strong support for scientific computing and fuzzy logic. The following Python libraries were utilized:

–
Scikit-Fuzzy (skfuzzy): Applied for fuzzy modeling and inference, providing the needed tools for the fuzzy logic applications;
–
Numerical Python (NumPy): Utilized for mathematical operations and data manipulation;
–
Matplotlib: Employed for visualizing results and membership function graphs.

4.2.2. Definition of Variables

As outlined in Section 3.1, the analyzed variables were categorized into three fuzzy subcategories:

–
Operational: Efficiency, power, and acoustic signature;
–
Economic: Acquisition, maintenance, and operational costs;
–
Environmental: Thermal emissions, pollutants, and system size.

Each variable was modeled using a combination of two Gaussian functions, as detailed in Equation 1. The values assigned to each system were determined following the methodology presented in the preceding subsections.
4.2.3. Fuzzy Inference Methodology

As described previously, the developed DSS is based on a two-stage FMM, as illustrated in Figure 15, which can be summarized as follows:

First Stage: The individual input variables, as outlined in Section 3.1, are processed through fuzzy rules to generate operational, economic, and environmental scores;

Second Stage: The three subcategory scores are combined using fuzzy inference rules to determine the final recommendation score.

The fuzzy inference rules used in both stages are adequately defined in Section 3.

4.2.4. Defuzzification and Normalization

The centroid defuzzification method outlined in Equation 4 converted fuzzy values into numerical outputs in both stages. The final recommendation score was normalized to a percentage scale (0%–100%), as noted in Equation 5, which helps interpret and compare recommendation values.

4.3. Discussion of Results

The fuzzy system described in Section 3 has been implemented. In this section, we will analyze the results by presenting the membership degrees of each input across subcategories and their classifications. We will also include the outcomes from both system phases. For a detailed analysis of output values, see Appendix A for the activated rules.

We will defuzzify the fuzzy variables using the centroid calculation method as shown in Equation 2 to complete the process and obtain the final recommendation scores. Alongside the final recommendation score for each system, we will also present its normalization using Equation 5, yielding a score between 0% and 100%.

4.3.1. Discussion of Results: Fuel Cells

Regarding the operational aspects of FC, the data presented in Table 11 reveals that the system rated the efficiency as Good, the power as Poor, and the acoustic signature primarily as Very Low. However, there is also a slight possibility of it being Low, with a weak membership of 0.1.

Table 11.
Degree of Membership and Classification - Operational - FC.

Variable Degree of Membership Classification

Efficiency 1.0 Good

Power 1.0 Poor

Acoustic Signature 1.0 Very Low

0.1 Low

Variable	Degree of Membership	Classification
Efficiency	1.0	Good
Power	1.0	Poor
Acoustic Signature	1.0	Very Low
	0.1	Low

These classifications triggered the rules presented in Appendix A, resulting in the area shown in Figure 16. Thus, it can be observed that the operational subcategory scored 79.

Figure 16.

Final Fuzzy set of the Operational Subcategory - FC.

Regarding the economic subcategory, the data presented in Table 12 reveals that the system classified the acquisition cost as High and Very High, the maintenance cost as Low and High, and the operating cost as High.

Table 12.

Degree of Membership and Classification - Economic - FC.

Variable	Degree of Membership	Classification
Acquisition cost	0.6	High
	0.4	Very High
Maintenance cost	0.6	Low
	0.4	High
Operating cost	1.0	High

Again, after the rules presented in Appendix A were activated, a resulting colored area was obtained, as illustrated in Figure 17, showing that the economic subcategory achieved a score of 42.

Figure 17.

Final Fuzzy Set of the Economic Subcategory - FC.

Regarding the environmental subcategory, the data presented in Table 13 reveals that the system classified the thermal emission as primarily Very Low. Still, it could also be considered Low, with a meager membership degree of 0.1. The byproduct is predominantly a Low Pollutant but could also be regarded as a Pollutant, with a membership degree of 0.1 and a High size.

Table 13.

Degree of Membership and Classification - Environmental - FC.

Variable	Degree of Membership	Classification
Thermal Emission	1.0	Very Low
	0.1	Low
Byproduct	1.0	Low Pollutant
	0.1	Pollutant
Size	1.0	High

Again, after the rules presented in Appendix A were activated, a resulting area was obtained, as illustrated in Figure 18. It can be observed that the environmental subcategory achieved a score of 79.

Figure 18.

Final Fuzzy Set of the Environmental Subcategory - FC.

Considering the scores achieved in each subcategory mentioned before {79, 42, 79}, it is possible to ascertain that each of them received the classifications described in Table 14.

Table 14.

Degree of Membership and Classification - Recommendation - FC.

Subcategory	Degree of Membership	Classification
Operational	0.8	High
	0.1	Medium
Economic	1.0	Medium
Environmental	0.8	High
	0.1	Medium

Again, after the rules presented in Appendix A were activated, resulting in the area illustrated in Figure 19, where the recommendation level for FC reached a High level, with a score of 80.

Figure 19.

Final Fuzzy Set of the Recommendation Level - FC.

To better understand the final score of 80, this value was normalized using Equation 5, from which it can be concluded that the fuel cell system achieved a recommendation level of 86%.

4.3.2. Discussion of Results: Stirling Engines

Regarding the operational subcategory of SE, based on the data presented in Table 15, it can be observed that the system classified the efficiency as Poor, the power as Poor and also Very Poor, and the acoustic signature as Low.

Table 15.
Degree of Membership and Classification - Operational - SE.

Variable Degree of Membership Classification

Efficiency 1.0 Poor

Power 0.7 Poor

0.3 Very Poor

Acoustic Signature 1.0 Low

Variable	Degree of Membership	Classification
Efficiency	1.0	Poor
Power	0.7	Poor
	0.3	Very Poor
Acoustic Signature	1.0	Low

Again, after the rules presented in Appendix A were activated, a resulting colored area was obtained in orange, as illustrated in Figure 20, where it is observed that the operational subcategory achieved a score of 50.

Figure 20.

Final Fuzzy Set of the Operational Subcategory - SE.

Regarding the economic subcategory, the data presented in Table 16 reveals that the system classified the acquisition cost as both High and Low. The maintenance cost was predominantly High, but it could also be considered Low, with a meager membership of 0.1. The operating cost was classified as Low.

Table 16.

Degree of Membership and Classification - Economic - SE.

Variable	Degree of Membership	Classification
Acquisition Cost	0.6	Low
	0.4	High
Maintenance Cost	1.0	High
	0.1	Low
Operating Cost	1.0	Low

Again, after the rules presented in Appendix A were activated, the resulting colored area was obtained, as illustrated in Figure 21, where the economic subcategory scored 50.

Figure 21.

Final Fuzzy Set of the Economic Subcategory - SE.

Regarding the environmental subcategory, the data presented in Table 17 reveals that the system classified the thermal emission as predominantly High. Still, it could also be considered Low, with a meager membership of 0.2. Classifies the byproducts as Highly and Extremely Polluting and the size as Low.

Table 17.

Degree of Membership and Classification - Environmental - SE.

Variable	Degree of Membership	Classification
Thermal Emission	0.9	High
	0.2	Low
Byproduct	0.6	Highly Polluting
	0.4	Extremely Polluting
Size	1.0	Low

Again, after the rules presented in Appendix A were activated, the resulting area was obtained, as illustrated in Figure 22, where the environmental subcategory scored 42.

Figure 22.

Final Fuzzy Set of the Environmental Subcategory - SE.

Considering the scores achieved in each of the dimensions above {50, 50, 42}, it is possible to ascertain that each received the classifications described in Table 18.

Table 18.

Degree of Membership and Classification - Recommendation - SE.

Subcategory	Degree of Membership	Classification
Operational	1.0	Medium
Economic	1.0	Medium
Environmental	1.0	Medium

Again, after the rules presented in Appendix A were activated, resulting in the area illustrated in Figure 23, where the recommendation level for SE reached a Medium level, with a score of 49. Through normalization, the system achieved a recommendation level of 49%.

Figure 23.

Final Fuzzy Set of the Recommendation Level - SE.

4.3.3. Discussion of Results: Steam Turbines

Regarding the operational subcategory of ST, based on the data presented in Table 19, it can be observed that the system classified the efficiency as predominantly Poor, but also as Very Poor, with a relatively low degree of membership of 0.1, the power as Good and also Very Good, and the acoustic signature as High.

Table 19.
Degree of Membership and Classification - Operational - ST.

Variable Degree of Membership Classification

Efficiency 1.0 Poor

0.1 Very Poor

Power 0.6 Good

0.4 Very Good

Acoustic Signature 1.0 High

Variable	Degree of Membership	Classification
Efficiency	1.0	Poor
	0.1	Very Poor
Power	0.6	Good
	0.4	Very Good
Acoustic Signature	1.0	High

Again, after the rules presented in Appendix A were activated, the resulting colored area was obtained, as illustrated in Figure 24, where it can be observed that the operational subcategory achieved a score of 48.

Figure 24.

Final Fuzzy Set of the Operational Subcategory - ST.

Regarding the economic subcategory, the data presented in Table 20 reveals that the system classified the acquisition cost as High and Very High, the maintenance cost as High and Very High, and the operating cost as High.

Table 20.

Degree of Membership and Classification - Economic - ST.

Variable	Degree of Membership	Classification
Acquisition Cost	0.6	High
	0.4	Very High
Maintenance Cost	0.6	High
	0.4	Very High
Operating Cost	1.0	High

Again, after the rules presented in Appendix A were activated, a resulting colored area was obtained, as illustrated in Figure 25, allowing us to observe that the economic subcategory scored 42.

Figure 25.

Final Fuzzy Set of the Economic Subcategory - ST.

Regarding the environmental subcategory, the data presented in Table 21 reveals that the system classified the thermal emission as High and Very High, the byproduct as Highly Polluting and Extremely Polluting, and the size as High.

Table 21.

Degree of Membership and Classification - Environmental - ST.

Variables	Degree of Membership	Classification
Thermal Emission	0.6	High
	0.4	Very High
Byproduct	0.6	Highly Polluting
	0.4	Extremely Polluting
Size	1.0	High

Again, after the rules presented in Appendix A were activated, a resulting area was obtained, as illustrated in Figure 26, where the environmental subcategory scored 16.

Figure 26.

Final Fuzzy Set of the Environmental Subcategory - ST.

Considering the scores achieved in each of the dimensions above {48, 42, 16}, it is possible to ascertain that each received the classifications described in Table 22.

Table 22.

Degree of Membership and Classification - Recommendation - ST

Variables	Degree of Membership	Classification
Operational Subcategory	1.0	Medium
Economic Subcategory	1.0	Medium
Environmental Subcategory	1.0	Low

Again, after the rules presented in Appendix A were activated, resulting in the area illustrated in Figure 27, where the recommendation level for ST reached a Low level, with a score of 29. The system achieved a recommendation level of 24% after normalization.

Figure 27.

Final Fuzzy Set of the Recommendation Level - ST.

4.3.4. Discussion of Results: Diesel Engines

Regarding the operational aspects of DE, the data presented in Table 23, it can be observed that the system classified the efficiency as Poor, the power as predominantly Good, but also as Poor, with a meager degree of membership of 0.1, and the acoustic signature as predominantly Very High, but also High, with a meager degree of membership of 0.2.

Table 23.
Degree of Membership and Classification - Operational - DE.

Variable Degree of Membership Classification

Efficiency 1.0 Poor

Power 1.0 Good

0.1 Poor

Acoustic Signature 1.0 Very High

0.2 High

Variable	Degree of Membership	Classification
Efficiency	1.0	Poor
Power	1.0	Good
	0.1	Poor
Acoustic Signature	1.0	Very High
	0.2	High

Again, after the rules presented in Appendix A were activated, a resulting colored area was obtained, as illustrated in Figure 28, where the operational subcategory scored 18.

Figure 28.

Final Fuzzy Set of the Operational Subcategory - DE.

Regarding the economic subcategory, the data presented in Table 24 reveals that the system classified the acquisition cost as Low, the maintenance cost as Low and High, and the operating cost as Low.

Table 24.

Degree of Membership and Classification - Economic - DE.

Variable	Degree of Membership	Classification
Acquisition Cost	1.0	Low
Maintenance Cost	0.6	Low
	0.4	High
Operating Cost	1.0	Low

Again, after the rules presented in Appendix A were activated, a resulting colored area was obtained, as illustrated in Figure 29, where the economic subcategory scored 50.

Figure 29.

Final Fuzzy Set of the Economic Subcategory - DE.

Regarding the environmental subcategory, the data presented in Table 25 reveals that the system classified the thermal emission as predominantly High. Still, it could also be considered Low, with a meager membership of 0.2, the byproduct being Highly and Extremely Pollutant, and the size being Low.

Table 25.

Degree of Membership and Classification - Environmental - DE.

Variable	Degree of Membership	Classification
Thermal Emission	0.9	High
	0.2	Low
Byproduct	0.6	Highly Pollutant
	0.4	Extremely Pollutant
Size	1.0	Low

Again, after the rules presented in Appendix A were activated, a resulting area was obtained, as illustrated in Figure 30, where the environmental subcategory scored 50.

Figure 30.

Final Fuzzy Set of the Environmental Subcategory - DE.

Considering the scores achieved in each of the dimensions as mentioned above {18, 50, 50}, it is possible to ascertain that each received the classifications described in Table 26.

Table 26.

Degree of Membership and Classification - Recommendation - DE.

Subcategory	Degree of Membership	Classification
Operational	1.0	Low
Economic	1.0	Medium
Environmental	1.0	Low

Again, after the rules presented in Appendix A were activated, resulting in the area illustrated in Figure 31, the recommendation level for DE reached a Low level, with a score of 29. After normalization, the system achieved a recommendation level of 24.

Figure 31.

Final Fuzzy Set of the Recommendation Level - DE.

4.3.5. Results & Discussion

To summarize all the scores across the dimensions and levels of recommendation achieved by the different systems, along with their respective normalization, Table 27 has been created.

Table 27.
Final Obtained Results - Resume.

System Operational Economic Environmental Recommendation Level %

Fuel Cells 79 42 79 80 86

Stirling Engines 50 50 42 49 49

Steam Turbines 48 42 16 29 24

Diesel Engines 18 50 50 29 24

System	Operational	Economic	Environmental	Recommendation Level	%
Fuel Cells	79	42	79	80	86
Stirling Engines	50	50	42	49	49
Steam Turbines	48	42	16	29	24
Diesel Engines	18	50	50	29	24

The data shown in Figure 32 indicates that FC outperform other systems. They exhibit significantly higher operational efficiency, produce virtually no noise during operation, and have a minimal environmental impact, generating nearly zero pollution.

Figure 32.

Starplot Illustrating the Final Obtained Results.

The final recommendation value is not the only factor in this DSS. Comparing dimension values is essential for decision-makers. Although FC systems are preferred, this analysis is critical as new systems can emerge, complicating decisions. A two-phase system provides a recommendation value while evaluating systems operationally, economically, and environmentally for thorough comparison.

An equation was applied that assigned different weights based on the opinions of PoN submarine systems experts to compare values from the fuzzy system with those from linear and non-fuzzy applications. Normalized values from the first phase were used for the operational subcategory ( $x$ ), economic subcategory ( $y$ ), and environmental subcategory ( $z$ ) as follows:

R = \frac{4.81 x + 3.43 y + 3.19 z}{11.43}

(6)

where the obtained results are described in Table 28. By comparing the two sets of results, it is evident that they are not significantly different.

Table 28.

Final Recommendation Level Analysis.

System	DSS (%)	Confirmation (%)
Fuel Cells	86	80
Stirling Engines	49	46
Steam Turbines	24	28
Diesel Engines	24	27

These results are further validated by analyzing the most recently developed systems. The FC AIP systems have been integrated into a new generation of non-nuclear submarines, including the Republic of Singapore Navy’s Type 218SG, commissioned in 2020 (Naval News, 2020), the German–Norwegian Type 212CD, which concluded its design phase in 2024 (Army Recognition, 2024a; Naval News, 2024b), and the Turkish Navy’s Type 214TN, with its lead ship commissioned in 2024 (Naval News, 2024c).

This study employed a fuzzy model instead of a linear one, offering the DSS greater flexibility and variability by assigning unlimited membership degrees. It accommodates vague human assessments, providing an effective problem-solving method and intuitively enhancing understanding.

5. Conclusions & Future Work

AIP systems boost submarines’ stealth and underwater power. They enable non-nuclear submarines to compete with nuclear ones by reducing the need for battery recharge. Their compact size and low noise make AIP systems strategic assets for countries without a large nuclear fleet. This study confirms the superiority of fuel cells among AIP systems, highlighting the DSS as a key decision-making tool for AIP implementation.

A two-phase FMM approach was developed to create a DSS for selecting the best submarine AIP system. It evaluated four systems: (i) FC, (ii) SE, (iii) closed-cycle ST, and (iv) closed-loop DE. FC received the highest recommendation at about 86%, followed by SE at 49%, and both closed-cycle ST and closed-loop DE at 24%. This systematic method provided clear recommendations, allowing for comparison across key operational, economic, and environmental categories.

The implementation faced challenges in accessing technical data due to military confidentiality. Nevertheless, insights from PoN experts and theoretical studies significantly alleviated this issue. The system’s flexible nature allows for future adjustments as concrete data becomes available through collaboration with manufacturers and suppliers, enabling the expansion of this study to other systems as well. Although the DSS has been validated with naval experts, its applicability remains limited by the availability of high-fidelity operational data, potential refinements in fuzzy rule definitions, and the necessity for integration with real-time propulsion performance monitoring systems.

This method introduced an application of fuzzy logic to AIP selection, filling a void not addressed by the existing DSS frameworks. By incorporating fuzzy logic, the system effectively manages uncertainties and imprecision in expert evaluations, providing a flexible and adaptable solution for assessing AIP systems.

Recent advancements in battery technology yield higher capacities and smaller sizes than traditional lead-acid batteries used in submarines. Future research should focus on meeting Navy requirements through AIP systems and batteries, bypassing traditional generators. This could lead to more efficient submarine designs that align with modern technology and operational needs. Additionally, we should focus on enhancing the system’s adaptability to evolving propulsion technologies and exploring hybrid decision-making techniques to optimize performance further. Integrating machine learning for self-improving decision rules could further enhance the DSS, allowing it to learn from historical data and refine its recommendations over time, thereby improving decision-making accuracy and adaptability in more dynamic environments.

This DSS can provide navies with a robust, data-driven framework for evaluating and selecting AIP technologies. It aligns choices with operational, economic, and environmental demands and priorities while minimizing the risk of technical mismatches, reducing long-term costs, and enhancing the overall effectiveness of submarine missions.

Footnotes

ORCID iDs

Nuno Pessanha Santos

Ricardo Moura

Author Contributions

Nuno Pessanha Santos: Conceptualization, Methodology, Investigation, Formal analysis, Project administration, Supervision, Validation, Visualization, Writing – Original draft, Writing – Review & editing. Ricardo Moura: Conceptualization, Methodology, Investigation, Formal analysis, Software, Project administration, Resources, Supervision, Validation, Visualization, Writing – Original draft, Writing – Review & editing. Ana Reis Sintra: Conceptualization, Methodology, Investigation, Formal analysis, Software, Resources, Validation, Visualization.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by national funds through the Fundação para a Ciência e a Tecnologia (FCT). Nuno Pessanha Santos received support under the projects LA/P/0083/2020, UIDP/50009/2020, and UIDB/50009/2020 (DOI: 10.54499/LA/P/0083/2020, 10.54499/UIDP/50009/2020, and 10.54499/UIDB/50009/2020) – Laboratory of Robotics and Engineering Systems (LARSyS). Ricardo Moura received support under the projects UIDB/00297/2020 and UIDP/00297/2020 (DOI: 10.54499/UIDB/00297/2020 and 10.54499/UIDP/00297/2020) – Center for Mathematics and Applications.

Declaration of Conflicting Interests

The authors declare that there are no known competing financial interests or personal relationships that may have influenced the work reported in this paper.

Data Availability

The manuscript includes all data and materials supporting the presented conclusions. Full access to the implementation code is available in the GitHub repository at (accessed on March 15, 2025). For further inquiries, please contact the corresponding author.

Appendix

References

Ahmadi

M. H.

Ahmadi

M.-A.

Pourfayaz

(2017). Thermal models for analysis of performance of stirling engine: A review. Renewable and Sustainable Energy Reviews, 68, 168–184. https://doi.org/10.1016/j.rser.2016.09.033

Army Recognition. (2024a, August 23). German–Norwegian submarine program reaches key milestone with completion of design phase. Army Recognition. https://armyrecognition.com/news/navy-news/2024/german-norwegian-submarine-program-reaches-key-milestone-with-completion-of-design-phase

Avallone

L. E. M.

(2000). Submarine development prior to world war I. Naval Engineers Journal, 112(2), 19–25. https://doi.org/10.1111/j.1559-3584.2000.tb03280.x

Başhan

(2022). Comparative evaluation and selection of submarines with air-independent propulsion systems. International Journal of Hydrogen Energy, 47(86), 36659–36671. https://doi.org/10.1016/j.ijhydene.2022.08.191

Brighton

D. R.

Mart

P. L.

Clark

G. A.

Rowan

M. J. M.

(1994). The use of fuel cells to enhance the underwater performance of conventional diesel-electric submarines. Journal of Power Sources, 51(3), 375–389. https://doi.org/10.1016/0378-7753(94)80106-1

Chin

(2019). Technology, war and the state: Past, present and future. International Affairs, 95(4), 765–783. https://doi.org/10.1093/ia/iiz106

Chopra

(2023). Military Applications of Nuclear Power Plants. In The Global Nuclear Landscape (pp. 60–76). Routledge.

Dadios

(2012). Fuzzy logic: Controls, concepts, theories and applications. BoD–Books on Demand.

Das

Sen

Maulik

(2020). A survey on fuzzy deep neural networks. ACM Computing Surveys, 53(3), Article 54, 1–25. https://doi.org/10.114000000005/3369798

10.

Friedman-Jimenez

Kato

Factor-Litvak

Shore

(2022). Mortality of enlisted men who served on nuclear-powered submarines in the united states navy. Journal of Occupational and Environmental Medicine, 64(2), 131–139. https://doi.org/10.1097/JOM.0000000000002364

11.

Gabbay

D. M.

Woods

(2004). Greek, Indian and arabic logic. Elsevier.

12.

Gupta

Singh

Singla

(2022). Fuzzy Logic-based Systems for Medical Diagnosis – A Review. In Proceedings of the 2022 3rd International Conference on Electronics and Sustainable Communication Systems (ICESC) (pp. 1058–1062). https://doi.org/10.1109/ICESC54411.2022.9885338

13.

Haffa

R. P.

Patton

J. H.

(1991). Analogues of stealth: Submarines and aircraft. Comparative Strategy, 10(3), 257–271. https://doi.org/10.1080/01495939108402847

14.

Hartley

(2024). European defence policy: Prospects and challenges. Defence and Peace Economics, 35(4), 504–515. https://doi.org/10.1080/10242694.2023.2185425

15.

Hawley

J. G.

Ashcroft

S. J.

Patrick

M. A.

(1994). Advanced underwater power systems. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, 208(1), 37–45.

16.

Howard

C. Q.

(2011). Recent developments in submarine vibration isolation and noise control. In Proceedings of the 1st Submarine Science Technology and Engineering Conference (Vol. 17, pp. 1–7).

17.

Jones

G. M.

(2022). Improving the Submersible. In The Development of Nuclear Propulsion in the Royal Navy, 1946-1975 (pp. 23–45). Springer International Publishing. https://doi.org/10.1007/978-3-031-05129-6_2

18.

Kerros

Inizan

Grousset

(1994). MESMA: AIP system for submarines. In Proceedings of OCEANS’94 (Vol. 3, pp. III/457–III/466). https://doi.org/10.1109/OCEANS.1994.364242

19.

Koller

Friedman

(2009). Probabilistic graphical models: Principles and techniques. MIT Press.

20.

Kordesch

Simader

(1996). Fuel cells and their applications. Wiley-VCH, Germany.

21.

Lautenschläger

(1987). The submarine in naval warfare, 1901-2001. International Security, 11(3), 94–140. https://doi.org/10.2307/2538886

22.

Liadze

Macchiarelli

Mortimer-Lee

Sanchez Juanino

(2023). Economic costs of the Russia-Ukraine war. The World Economy, 46(4), 874–886. https://doi.org/10.1111/twec.13336

23.

Lus

(2001). Submarine hybrid propulsion systems. Journal of KONES, 8(1-2), 265–270.

24.

Lus

(2019). Waiting for breakthrough in conventional Submarine’s prime movers. Transactions on Maritime Science, 8(01), 37–45.

25.

Mamdani

E. H.

(1976). Advances in the linguistic synthesis of fuzzy controllers. International Journal of Man-Machine Studies, 8(6), 669–678. https://doi.org/10.1016/S0020-7373(76)80028-4

26.

Marx

R. F.

(1990). The history of underwater exploration. Courier Corporation.

27.

Mendel

J. M.

(1995). Fuzzy logic systems for engineering: A tutorial. Proceedings of the IEEE, 83(3), 345–377. https://doi.org/10.1109/5.364485

28.

Mendel

J. M.

(2017). Uncertain rule-based fuzzy systems: Introduction and new directions (2nd ed.). Springer. https://doi.org/10.1007/978-3-319-51370-6

29.

Menon

R. R.

Vijayakumar

Pandey

J. K.

(2020). Selection of optimal air independent propulsion system using forced decision matrix. Defence Science Journal, 70, 1. https://doi.org/10.14429/dsj.70.13678

30.

Milliken

C. E.

Ruhl

R. C.

(2002). Low cost, high efficiency reversible fuel cell systems. In Proceedings of the 2002 US DOE Hydrogen Program Review, NREL/CP-610-32405 (pp. 1–14).

31.

Moorhouse

(2015). A modern history of the manned submersible. Marine Technology Society Journal, 49(6), 65–78. https://doi.org/10.4031/MTSJ.49.6.9

32.

Naval News. (2020, September 25). Singapore Navy’s First Type 218SG Invincible-Class Submarine Starts Sea Trials. Naval News. https://www.navalnews.com/naval-news/2020/09/singapore-navys-first-type-218sg-invincible-class-submarine-started-sea-trials/

33.

Naval News. (2024b, September 4). Type 212CD AIP will change underwater game for Norwegian Navy, says submarine CO. Naval News. https://www.navalnews.com/naval-news/2024/06/type-212cd-aip-will-change-underwater-game-for-norwegian-navy-says-submarine-co/

34.

Naval News. (2024c, August 24). Turkish Navy commissions first Reis-class AIP submarine TCG Piri Reis. Naval News. https://www.navalnews.com/naval-news/2024/08/turkish-navy-commissions-first-reis-class-aip-submarine-tcg-piri-reis/

35.

Navantia. (2020). The S-80 Programme successfully completes the development of the Air-independent Propulsion System (AIP). Navantia News https://www.navantia.es/en/news/news-s-80/the-s-80-programme-successfully-completes-the-development-of-the-air-independent-propulsion-system-aip/

36.

Pourjavad

Shahin

(2018). The application of mamdani fuzzy inference system in evaluating green supply chain management performance. International Journal of Fuzzy Systems, 20, 901–912. https://doi.org/10.1007/s40815-017-0378-y

37.

Rains

D. A.

Mitchell Jr

K. A.

(2000). Nuclear vs. Non-nuclear attack submarine powerplants. Naval Engineers Journal, 112(2), 39–46. https://doi.org/10.1111/j.1559-3584.2000.tb03283.x

38.

Ranga

Dahiya

Singh

A. K.

(2017). Depth control of submarine: A fuzzy logic approach. IOSR Journal of Electrical and Electronics Engineering (IOSR-JEEE), 12, 64–79. https://doi.org/10.9790/1676-1205026479

39.

Revista Ejércitos. (2024). S-80 Program - The multiple problems of the AIP. Revista Ejércitos https://www.revistaejercitos.com/en/articulos/programa-s-80-los-multiples-problemas-del-aip/

40.

Ross

T. J.

(2005). Fuzzy Logic with Engineering Applications. John Wiley & Sons.

41.

Saaty

T. L.

(2008). Decision making with the analytic hierarchy process. International Journal of Services Sciences, 1(1), 83–98. https://doi.org/10.1504/IJSSCI.2008.017590

42.

Sakti

(2014). Methodology of fuzzy logic with Mamdani fuzzy models applied to the microcontroller. In Proceedings of the 2014 The 1st International Conference on Information Technology, Computer, and Electrical Engineering (pp. 93–98). https://doi.org/10.1109/ICITACEE.2014.7065721

43.

Shafi

Ansari

Din

Jeon

Paul

(2021). Artificial neural networks as clinical decision support systems. Concurrency and Computation: Practice and Experience, 33(22), e6342. https://doi.org/10.1002/cpe.6342

44.

Shi

Wang

(2009). A fuzzy expert system for submarine damage control. In Proceedings of the 2009 International Conference on Mechatronics and Automation (pp. 638–643). IEEE. https://doi.org/10.1109/ICMA.2009.5246494

45.

Simões-Marques

Ribeiro

R. A.

Gameiro-Marques

(2000). A fuzzy decision support system for equipment repair under battle conditions. Fuzzy Sets and Systems, 115(1), 141–157. https://doi.org/10.1016/S0165-0114(99)00023-8

46.

Sripakagorn

Srikam

(2011). Design and performance of a moderate temperature difference Stirling engine. Renewable Energy, 36(6), 1728–1733. https://doi.org/10.1016/j.renene.2010.12.014

47.

Stelzer

Proll

John

R. I.

(2007). Fuzzy Logic Control System for Autonomous Sailboats. In Proceedings of the 2007 IEEE International Fuzzy Systems Conference (pp. 1–6). https://doi.org/10.1109/FUZZY.2007.4295347

48.

Takagi

Sugeno

(1985). Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man, and Cybernetics, SMC-15(1), 116–132. https://doi.org/10.1109/TSMC.1985.6313399

49.

Wang

Zhu

Han

(2023). Industrial development status and prospects of the marine fuel cell: A review. Journal of Marine Science and Engineering, 11(2), Article 238, 1–40. https://doi.org/10.3390/jmse11020238

50.

Zadeh

L. A.

(1965). Fuzzy sets. Information and Control, 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X

51.

Zadeh

L. A.

(1988). Fuzzy logic. Computer, 21(4), 83–93. https://doi.org/10.1109/2.53

52.

Zadeh

L. A.

(1996). Fuzzy logic = computing with words. IEEE Transactions on Fuzzy Systems, 4(2), 103–111. https://doi.org/10.1109/91.493904

53.

Zhao

Cao

(2019). Self-organising interval type-2 fuzzy neural network with asymmetric membership functions and its application. Soft Computing, 23, 7215–7228. https://doi.org/10.1007/s00500-018-3367-7

	Operational Subcategory
AIP Systems	Efficiency (%)	Power	Acoustic Signature
Fuel Cells	70	3	1
Stirling Engines	40	2	3
Steam Turbines	25	8	7
Diesel Engines	30	6	9
	Economic Subcategory
AIP Systems	Acquisition Cost	Maintenance Cost	Operating Cost
Fuel Cells	8	5	7
Stirling Engines	5	6	3
Steam Turbines	8	8	7
Diesel Engines	3	5	3
	Environmental Subcategory
AIP Systems	Thermal Emission (°C)	Byproducts	Size
Fuel Cells	80	1	7
Stirling Engines	500	8	5
Steam Turbines	700	8	7
Diesel Engines	500	8	3

Two-Stage Fuzzy Inference for Submarine Air-Independent Propulsion Selection

Abstract

Keywords

1. Introduction

2. Problem Formulation & Methodologies

2.1. Submarine Technology

2.2. Fuzzy Reasoning

2.2.1. Fuzzification

2.2.3. Defuzzification

Table 3. Parameters for Each Acoustic Signature Membership Function. Fuzzy Sets a σ 1 b σ 2 Very Low 0 – 0.8 0.8 Low 2.7 0.8 4.2 0.8 High 6 0.7 7.3 0.7 Very High 9 0.7 10 –

4.1. Data Application

4.1.1. Data Application: Efficiency

4.1.2. Data Application: Power

4.1.3. Data Application: Acoustic Signature

4.1.4. Data Application: Acquisition Cost

4.1.5. Data Application: Maintenance Cost

4.1.6. Data Application: Operating Cost

4.1.7. Data Application: Thermal Emission

4.1.8. Data Application: Byproducts

4.1.9. Data Application: Size

4.1.10. Data Application: Variables Resume

4.2.4. Defuzzification and Normalization

4.3. Discussion of Results

4.3.1. Discussion of Results: Fuel Cells

Table 11. Degree of Membership and Classification - Operational - FC. Variable Degree of Membership Classification Efficiency 1.0 Good Power 1.0 Poor Acoustic Signature 1.0 Very Low 0.1 Low

Table 15. Degree of Membership and Classification - Operational - SE. Variable Degree of Membership Classification Efficiency 1.0 Poor Power 0.7 Poor 0.3 Very Poor Acoustic Signature 1.0 Low

Table 19. Degree of Membership and Classification - Operational - ST. Variable Degree of Membership Classification Efficiency 1.0 Poor 0.1 Very Poor Power 0.6 Good 0.4 Very Good Acoustic Signature 1.0 High

Table 23. Degree of Membership and Classification - Operational - DE. Variable Degree of Membership Classification Efficiency 1.0 Poor Power 1.0 Good 0.1 Poor Acoustic Signature 1.0 Very High 0.2 High

Table 27. Final Obtained Results - Resume. System Operational Economic Environmental Recommendation Level % Fuel Cells 79 42 79 80 86 Stirling Engines 50 50 42 49 49 Steam Turbines 48 42 16 29 24 Diesel Engines 18 50 50 29 24

Footnotes

ORCID iDs

Author Contributions

Funding

Declaration of Conflicting Interests

Data Availability

Appendix

References

Table 3.
Parameters for Each Acoustic Signature Membership Function.

Fuzzy Sets $a$ $σ_{1}$ $b$ $σ_{2}$

Very Low 0 – 0.8 0.8

Low 2.7 0.8 4.2 0.8

High 6 0.7 7.3 0.7

Very High 9 0.7 10 –

Table 11.
Degree of Membership and Classification - Operational - FC.

Variable Degree of Membership Classification

Efficiency 1.0 Good

Power 1.0 Poor

Acoustic Signature 1.0 Very Low

0.1 Low

Table 15.
Degree of Membership and Classification - Operational - SE.

Variable Degree of Membership Classification

Efficiency 1.0 Poor

Power 0.7 Poor

0.3 Very Poor

Acoustic Signature 1.0 Low

Table 19.
Degree of Membership and Classification - Operational - ST.

Variable Degree of Membership Classification

Efficiency 1.0 Poor

0.1 Very Poor

Power 0.6 Good

0.4 Very Good

Acoustic Signature 1.0 High

Table 23.
Degree of Membership and Classification - Operational - DE.

Variable Degree of Membership Classification

Efficiency 1.0 Poor

Power 1.0 Good

0.1 Poor

Acoustic Signature 1.0 Very High

0.2 High

Table 27.
Final Obtained Results - Resume.

System Operational Economic Environmental Recommendation Level %

Fuel Cells 79 42 79 80 86

Stirling Engines 50 50 42 49 49

Steam Turbines 48 42 16 29 24

Diesel Engines 18 50 50 29 24