Multi-Criteria Decision-Making Algorithm Selection and Adaptation for Performance Improvement of Two Stroke Marine Diesel Engines

Abstract

Selecting an appropriate Multi-Criteria Decision-Making (MCDM) algorithm for optimizing marine diesel engine operation presents a complex challenge due to the diversity in mathematical formulations, normalization schemes, and trade-off resolutions across methods. This study systematically evaluates fourteen MCDM algorithms, which are grouped into five primary methodological categories: Scoring-Based, Distance-Based, Pairwise Comparison, Outranking, and Hybrid/Intelligent System-Based methods. The goal is to identify the most suitable algorithm for real-time performance optimization of two stroke marine diesel engines. Using Diesel-RK software, calibrated for marine diesel applications, simulations were performed on a variant of the MAN-B&W-S60-MC-C8-8 engine. A refined five-dimensional parameter space was constructed by systematically varying five key control variables: Start of Injection (SOI), Dwell Time, Fuel Mass Fraction, Fuel Rail Pressure, and Exhaust Valve Timing. A subset of 4454 high-potential alternatives was systematically evaluated according to three equally important criteria: Specific Fuel Consumption (SFC), Nitrogen Oxides (NO_x), and Particulate Matter (PM). The MCDM algorithms were evaluated based on ranking consistency and stability. Among them, Proximity Indexed Value (PIV), Integrated Simple Weighted Sum Product (WISP), and TriMetric Fusion (TMF) emerged as the most stable and consistently aligned with the overall consensus. These methods reliably identified optimal engine control strategies with minimal sensitivity to normalization, making them the most suitable candidates for integration into automated marine engine decision-support systems. The results underscore the importance of algorithm selection and provide a rigorous basis for establishing MCDM in emission-constrained maritime environments. This study is the first comprehensive, simulation-based evaluation of fourteen MCDM algorithms applied specifically to the optimization of two stroke marine diesel engines using Diesel-RK software.

1. Introduction

Marine diesel engines, particularly two stroke variants like the MAN B&W L/S60 series and Wärtsilä Flex series engines, are widely used for global shipping but face escalating environmental and operational challenges. Tier III regulations, set by the International Maritime Organization (IMO), impose strict limits on nitrogen oxide (NOx) emissions from marine engines. In restricted Emission Control Areas (ECAs), such as the Baltic Sea, North Sea, and starting in 2026, the Canadian Arctic and Norwegian Sea, engines must emit no more than 3.4 g/kWh of NOx_x [1]. These standards are particularly relevant for large, low speed marine engines and aim to significantly reduce air pollution from shipping in environmentally sensitive regions [2]. Global fuel sulfur limits now require marine fuels to contain no more than 0.50% sulfur by mass (m/m, meaning mass of sulfur per mass of fuel) worldwide. Even stricter limits apply in restricted\Sulfur Emission Control Areas (SECAs), where the maximum sulfur content drops to 0.10% m/m. These include the Mediterranean Sea (effective May 2025) and the Canadian Arctic (effective March 2027) [2]. These measures target the industry’s disproportionate environmental impact; for example, maritime transport contributes 15–35% of global NO emissions [2]. Traditional performance improvement methods, such as manual and pre-defined calibration of injection timing (SOI) or exhaust valve timing (EVT), are inadequate for resolving the complex and changing trade-offs between emissions (NOx, PM, CO) and performance metrics, such as specific fuel consumption (SFC) [3,4,5,6,7,8].

Multi-Criteria Decision-Making algorithms provide a systematic framework to resolve these conflicts by evaluating multiple objectives simultaneously. For instance, Lamas et al. [9] employed three MCDM approaches in systematic study was conducted on the Wärtsilä 6 L 46 four stroke marine diesel engine using three well-established MCDM methods: Weighted Sum Method (WSM), which aggregates criteria through a linear combination of weighted values; Weighted Product Method (WPM), which applies a multiplicative approach to combine weighted criteria; and the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), which ranks alternatives based on their relative distance from ideal and non-ideal solutions. These methods were applied to evaluate various pre-injection strategies based on Performance and emission trade-offs. It was found that linear max, linear sum, vector, and logarithmic normalization methods provided the same optimal pre-injection configuration. This configuration achieved a 34.7% reduction in nitrogen oxides (NOx) emissions, with only a 6.7% increase in specific fuel consumption, demonstrating a favorable emissions-performance trade-off. TOPSIS was preferred for its ability to rank alternatives based on geometric proximity to ideal/anti-ideal solutions, which is crucial for balancing emission-performance trade-offs [9].

Modern marine diesel engines face complex challenges in meeting performance and environmental expectations. These engines are critical for global maritime logistics but are also significant contributors to NOx and SOx emissions [2]. As shown in [10] a CFD simulations (validated with experimental data on Wärtsilä 6 L 46, a 6-cylinder, four stroke, turbocharged marine diesel engine) was analyzed and validated with engine sea trial data. One hundred twenty-five injection scenarios were evaluated [10]. MCDM weighting methods compared to identifying the optimal strategy. Pre-injection strategies (rate, duration, timing) in the Wärtsilä 6 L 46 marine engine. It found that NOx reductions up to 24% were achieved with pre-injection, but this increased CO (7.6%) and HC emissions and raised specific fuel consumption (SFC) by 23.5% due to lower peak pressures [10]. Multi-Criteria Decision-Making methods are increasingly adopted to address this multi-dimensional optimization challenge [5]. These techniques help identify optimal engine operating strategies by evaluating multiple conflicting objectives such as minimizing specific fuel consumption (SFC), particulate matter (PM), carbon monoxide (CO), and nitrogen oxides (NOx) under varying operational conditions [10,11]. Prior work shows that MCDM has been applied at both subsystem and propulsion-architecture levels in marine powertrains. Interval type-2 fuzzy VIKOR has ranked alternatives for auxiliaries of a ship’s main diesel engine under uncertain, linguistic criteria [12]. At the plant/architecture level, a multi-criteria framework compared diesel-mechanical and diesel-electric options across cost, environmental impact, and risk dimensions [13]. Complementarily, some performance and monitoring studies provide calibrated performance/monitoring models for two stroke engines that inform criteria definition, weighting, and constraint setting in MCDM pipelines [14,15,16].

The selection of the appropriate MCDM method is not trivial. The literature distinguishes between Scoring-Based, Distance-Based, Pairwise Comparison, Outranking, and Hybrid/Intelligent System-Based categories. Depending on how they handle trade-offs, aggregate criteria, and derive rankings [17,18,19,20,21,22,23,24,25]. These categories differ in their mathematical foundations, assumptions, and practical implications for decision-making [17,18,19,20,21,22,23,24,25]. The following overview outlines each algorithm’s core principle and mathematical foundation, forming the basis for comparative evaluation in later sections.

Table 1. Categorization of implemented MCDM algorithms with mathematical justification [17,26,27,28,29,30,31,32,33,34,35,36,37,38].

Category	Method	Logical & Mathematical Justification	Contributor
Scoring-Based Methods	SAW Simple Additive Weighting	Classic linear function: Si = Σ wj·xij Each alternative is scored by summing the products of criterion weights (wj) and normalized performance values (xij). Higher scores indicate better Performance.	Kenneth R. MacCrimmon [26]
	MARCOS Measurement of Alternatives and Ranking according to Compromise Solution	Derives utility function Ki = Si/Sref; where Si is the weighted normalized score of an alternative, and Sref is a reference score (ideal or not ideal). It evaluates how close each option is to the reference.	Željko Stević, Dragan Pamučar, Adis Puška, Prasenjit Chatterjee [27]
	WASPAS Weighted Aggregated Sum Product Assessment	Combines additive and multiplicative utility: Qi = λ·Σ wj·xij + (1 − λ)·Π xij^wj; where it combines the additive (SAW) and multiplicative models for more flexibility. λ is a balance factor between 0 and 1.	EK Zavadskas, Z Turskis, J Antucheviciene, A Zakarevicius [28]
	MAIRCA Multi-Attribute Ideal–Real Comparative Analysis	Calculates theoretical vs. real preference difference: Di = Σ |Tij − Rij|; where Di measures deviation between theoretical preferences (Tij) and actual ratings (Rij). Lower Di means a better match to the ideal.	Gigović, Ljubomir, Dragan Pamučar, Zoran Bajić and Milić Milićević [29]
	MARA Magnitude of the Area for the Ranking of Alternatives	Calculates the geometric area formed by their normalized criteria values, interpreting each as a point in n-dimensional space. The larger the enclosed area (via polygon or hypervolume formulas), the better the overall, balanced performance.	Gligorić, Miloš, Zoran Gligorić, Suzana Lutovac, Milanka Negovanović and Zlatko Langović [30]
	ORESTE (Organisation Rangement Et SynThèsE de données relationnelles)	Uses ordinal regression and pseudo-criteria to translate qualitative ranks into utility scores.	Marc Roubens [31]
Distance-Based Methods	TOPSIS Technique for Order Preference by Similarity to Ideal Solution	Uses Euclidean distances to ideal/anti-ideal solutions: Ci = D−/(D+ + D−); calculates closeness to the ideal solution (D+) and distance from the worst (D−) using Euclidean distance.	Ching-Lai Hwang and Kwang Yoon [32]
	MABAC Multi-Attributive Border Approximation Area Comparison	Calculates how far each alternative is from a neutral or border zone; larger deviations from the border (toward ideal) are better.	Dragan Pamučar and Goran Ćirović [33]
	OCRA Operational Competitiveness Rating Analysis	Constructs a weighted normalized matrix and calculates Performance based on deviation from a reference vector.	Celik Parkan [34]
	VIKOR VIseKriterijumska Optimizacija Kompromisno Resenje	Qi = v·(Si − S*)/(S− − S*) + (1 − v)·(Ri − R*)/(R− − R*); where Si is the total gap of alternative i from the ideal (group utility). Ri is the worst performance of alternative i among all criteria (individual regret). S* and S− are the best and worst values of S among all alternatives. R* and R− are the best and worst values of R among all alternatives. v is weight, indicating the importance of group utility vs. individual regret.	Serafim Opricović [35]
Pairwise Comparison Methods	PIV Proximity Indexed Value	PIV ranks alternatives based on their relative closeness to both the ideal and anti-ideal solutions. PIVi = D−/(D+ + D−); where D+ is the distance to the ideal solution, and D− is the distance to the worst (anti-ideal) solution	Mufazzal, Sameera and S. M. Muzakkir. [36]
	WISP Integrated Simple Weighted Sum Product	WISP is a hybrid method that integrates both weighted sum (SAW) and weighted product (WPM) techniques. WISPi = λ · Σ(wj · xij) + (1 − λ) · Π(xij ^ wj); where λ ∈ [0, 1] is the control parameter that adjusts the influence of SAW vs. WPM.	Stanujkic, Dragisa, Gabrijela Popovic, Darjan Karabasevic, Ieva Meidute-Kavaliauskiene and Alptekin Ulutaş. [37]
Outranking Methods	PROMETHEE II Preference Ranking Organization METHod for Enrichment of Evaluations	Uses preference flows to compare alternatives on how much they outrank or are outranked by others.	Jean-Pierre Brans and Bertrand Mareschal [38]
Hybrid Method	Tri Metric Fusion	CAWI (Criteria Aggregated Weighted Index), BECI (Balanced Extreme Criteria Index), and Euclidean distance to synthesize rankings with high robustness and low sensitivity.	Hammoud [17]

provides a structured overview of the implemented MCDM algorithms, grouped by their methodological category: scoring-based, distance-based, pairwise comparison, outranking, and hybrid methods. The logical and core mathematical formulation for each algorithm is briefly summarized. This categorization clarifies the diversity of approaches within the MCDM landscape. It lays the foundation for comparative analysis regarding ranking behavior, sensitivity to weighting and normalization, and robustness across scenarios.

Multi-Criteria Decision-Making frameworks offer a mathematically structured means of navigating complex decision environments. MCDM does not seek to produce a single “correct” answer. Instead, it provides a formal structure for aggregating diverse economic, environmental, and operational criteria while acknowledging that preferences, trade-offs, and uncertainties must be explicitly modeled rather than assumed. However, the multiplicity of available MCDM algorithms introduces a practical challenge: different methods can lead to different rankings for the same dataset. This divergence is not due to computational error but from each algorithm’s underlying assumptions regarding distance metrics, normalization, compensatory behavior, and preference aggregation. As such, algorithm selection must be aligned with the structure of the decision environment and the nature of the trade-offs involved.

Although most prior studies focused on four stroke engines or applied only a few MCDM methods without a structured comparison. This paper fills a gap by analyzing two stroke marine engines, which are critical for shipping but are less studied [39].

The main research contribution is the systematic evaluation and comparison of fourteen Multi-Criteria Decision-Making algorithms, categorized as scoring-based, distance-based, pairwise, outranking, and hybrid methods for selecting optimal control strategies in two stroke marine diesel engines, under real-world emission and performance constraints.

This paper aims to compare 14 different MCDM algorithms, including SAW, TOPSIS, TMF, and others, in the context of optimizing control parameters (e.g., start of injection (SOI), fuel delay, exhaust valve timing (EVT), and injection pressure) for two stroke marine diesel engine, such as MAN B&W S60-MC-C8-8. Simulation data were generated using Diesel-RK software (a physics-based, cycle-resolved engine simulation software), calibrated for marine diesel applications will be used to construct a decision matrix, and the algorithms will be critically evaluated for ranking consistency, result reliability, and their potential for integration into real-time engine management systems.

The structure of the article is organized into five main sections. Following the introduction and the literature review in Section 1, we illustrate the key structure of the targeted marine diesel engines study, Diesel-RK simulation software, and MCDM algorithms in Section 2. After that, Section 3 presents the experimental framework and input parameter selection. Then, Section 4 demonstrates the results of the case study and the application of MCDM algorithms to a two stroke marine diesel engine. Section 5 presents the paper’s findings and conclusions.

2. Key Structure of the Targeted Marine Diesel Engines Study

The study focuses on a two-stroke, low-speed marine diesel engine similar to the MAN B&W S60-MC-C8-8 series. This engine is a benchmark design for ocean vessels due to its combination of high efficiency, operational flexibility, and suitability for advanced research. The key characteristics include:

• High thermal efficiency: Exceeding 50% at optimal load, which is crucial for reducing fuel consumption and operational costs.
• Superior scavenging control: The uniflow scavenging process offers improved gas exchange efficiency and controllability compared to other methods (e.g., loop scavenging), leading to better cylinder charging and lower emissions.
• Fuel flexibility: Compatibility with a wide range of fuels, including Heavy Fuel Oil (HFO), Marine Diesel Oil (MDO), and Liquefied Natural Gas (LNG) in diesel dual-fuel modes, allowing for adaptation to fuel availability and environmental regulations.
• Advanced combustion control: The integration of electronically controlled, common-rail fuel injection and low-speed operation (85–105 RPM) provides a stable and extended time window for combustion, making this engine an ideal platform for parametric studies. Specifically, it allows for detailed investigation of:
  Fuel injection strategies, such as timing, pressure, and split injection, which are critical for optimizing performance and meeting emission standards.
  Exhaust valve timing, including the impact of Exhaust Valve Opening (EVO) and Closing (EVC) on scavenging efficiency, combustion phasing, and pollutant formation.

2.1. Diesel-RK Simulation Software for Multi-Parameter Marine Engine Simulation

The thermodynamic simulation suite Diesel-RK software was selected as the primary modeling platform for this study due to its proven accuracy in compression-ignition engine modeling and its specialized capabilities tailored to large-bore, low-speed marine diesel engines. Its core strength lies in its suite of validated, physics-based combustion and emission sub-models, which include multi-zone combustion modeling to resolve spatial heterogeneities in temperature and equivalence ratio critical for accurately predicting localized flame quenching, thermal NOx formation via the Extended Zeldovich mechanism and particulate matter PM behavior. For PM prediction, the Moss-Brookes model is employed for soot nucleation [40] and the Razleytsev model for oxidation kinetics, achieving a mean absolute error of less than 5% when benchmarked against experimental data from MAN B&W test facilities [7].

The Diesel-RK software has been extensively validated through numerous academic and engineering studies, establishing its reliability for evaluating engine performance and emissions under a wide range of fuels and operating conditions [41,42,43,44,45].

In addition to its fidelity, Diesel-RK offers exceptional parametric flexibility, enabling precise control over key injection parameters such as timing, duration, fuel fraction, and rail pressure, including dynamic adjustment of Dwell Time with ±0.1° crank angle resolution. It also supports advanced valve timing configurations, including simulation of Exhaust Valve Opening and Closing (EVO/EVC) strategies and variable valve actuation (VVA) scenarios. The platform accommodates a wide range of fuel types, from Heavy Fuel Oil to LNG-diesel dual-fuel blends, allowing realistic modeling across regulatory and operational contexts.

This flexibility is critical for generating high-fidelity response surfaces needed in Multi-Criteria Decision-Making analysis, particularly when evaluating NOx, SFC, and PM trade-offs under IMO Tier III constraints. Furthermore, Diesel-RK integrates seamlessly into the MCDM workflow through its batch-processing application programming interface, enabling automated Design of Experiments (DoE) across a five-dimensional parameter space, such as SOI, fuel delay, rail pressure, and valve timing, while extracting structured decision matrices populated with key performance indicators (SFC, NOx, PM).

2.2. Decision Matrix Formation for MCDM Analysis

Decision-making, particularly in complex engineering and operational contexts, is fundamentally a matter of structure, not prediction. In contrast, most modern Artificial Intelligence (AI) systems are designed to generate forecasts or estimates with results in a process where modeling is mistaken for thinking. Current decision support systems often optimize isolated components without investing these improvements within a coherent, adaptive decision framework [8]. Yet operation support decisions, particularly in systems such as marine diesel engines, are made iteratively and evolve with operating conditions. In this context, intelligence is not static output but continuously updates that integrate outcomes into future decisions. Thus, the future of decision intelligence lies not in increasingly sophisticated AI models but in the rigorous structuring of decision systems that can close the loop from data to action to observed results and back to refinement.

In Multi-Criteria Decision-Making, the decision matrix is the foundational framework for evaluating and comparing alternatives against multiple, often conflicting, criteria. This structured representation enables systematic decision analysis, particularly in complex engineering systems such as marine diesel engine operation, where trade-offs between Performance, emissions, and efficiency must be carefully balanced.

Each element $x_{ij}$ in the matrix quantifies the value of the alternative $A_i$ to criterion $C_j$, derived from simulation data, empirical measurements, or expert assessments. The main elements of MCDM are:

• Decision alternatives $(A_1,A_2,\dots ,A_k)$ represent the different options available for evaluation. Each alternative is assessed based on how it meets the various criteria.
• Evaluation Criteria $(C_1,C_2,\dots ,C_m$) are the factors that must be considered when making the decision. Each criterion can be quantitative or qualitative.
• The decision-making problem is structured using a Decision Matrix, X [20]. This matrix systematically captures the performance of each alternative against each criterion. It is formulated as:

  $$\underset{\_}{\underset{\_}{X}}=\left(\begin{array}{ccc}{x}_{11}{x}_{12}& \dots & x_{1m}\\ ⋮& \ddots & ⋮\\ x_{k1}{x}_{k2}& \dots & x_{km}\end{array}\right)$$

  (1)

  where $x_{ij}$ is the performance score or measurable value of the i-th alternative (A_i) with respect to the j-th criterion (C_j). In this matrix, each row corresponds to an alternative, and each column corresponds to a criterion.

Each element $x_{ij}$ in the decision matrix represents the quantified evaluation of alternative $A_i$ concerning criterion $C_j$

2.3. Overview of the Implemented MCDM Algorithms

The primary objective of this study is to identify the most suitable Multi-Criteria Decision-Making algorithm for marine applications, focusing on enhancing the Performance of two stroke marine diesel engines through optimal control parameter selection. Given the complexity of marine engine optimization, which involves balancing multiple, often conflicting, performance indicators such as NOx, PM, and SFC. In addition to dynamic factors like peak cylinder pressure and injection duration, a structured decision-making framework is essential.

The study systematically evaluates representative MCDM algorithms across the major methodological categories identified in the literature [17,18,19,20,21,22,23,24,25]:

• Scoring-Based Methods.
• Distance-Based Methods.
• Pairwise Comparison-Based Methods.
• Outranking Methods.
• Hybrid or Intelligent System-Based Methods.

Each category offers distinct strengths and operational characteristics. For example, value-based methods such as (SAW) and Weighted Aggregated Sum Product Assessment (WASPAS) are computationally simple and suitable for real-time decision loops, while distance-based methods like TOPSIS and VIKOR handle trade-offs in performance-versus-emission scenarios with geometric clarity. Outranking methods like PROMETHEE II build preference functions and use outranking flows. Pairwise comparison techniques (e.g., PIV, WISP) enable expert-driven decisions based on hierarchical structures or network dependencies. Hybrid methods, such as TriMetric Fusion (TMF) [17], integrate complementary mathematical strategies to enhance ranking robustness. These methods draw upon recent trends in combining optimization, machine learning, and hierarchical logic to improve decision accuracy in engineering systems [40,46,47,48].

Different algorithms from each category have been selected for implementation and comparative testing based on their maturity, adaptability, and relevance to marine engine optimization problems. The simulation data used to evaluate these methods is generated using Diesel-RK software, modeling a variant of a large two stroke engine. The aim is to observe each algorithm’s consistency and ranking behavior and assess their practicality for onboard integration or automated control environments in marine operation systems.

The diverse set of fourteen MCDM algorithms was selected to capture the range of ranking behaviors encountered in engine optimization tasks.

By understanding the underlying principles of each algorithm, a more application appropriate method selection becomes possible. This foundation supports the comparative analysis presented in the following sections.

3. Simulation Framework and Input Parameters Selection

The optimization methodology applied in this study is grounded in a high-fidelity simulation framework. The assessment of engine performance and emissions for various control parameter combinations was conducted using the Diesel-RK software, a validated thermodynamic simulation tool. This simulation-based approach was selected for its ability to efficiently and systematically explore a vast multi-dimensional parameter space, comprising 241,920 potential configurations, which would be prohibitively expensive, time consuming, and impractical to investigate through physical experimentation. The control parameters varied within the simulations were selected based on their practical adjustability in real engine operation. These parameters are the following:

• Engine Control Unit (ECU) Calibration Simplicity: Variables such as the Start Of Injection (SOI) for the pre-injection phase and common-rail fuel pressure were selected because they can be modified via existing Engine Control Unit Maps without requiring hardware changes or complex recalibration [7,8].
• User Accessibility: Parameters such as Exhaust Valve Opening, Exhaust Valve Closing, and fuel mass fraction were included because they align with standard crew-adjustable controls in marine engines, ensuring practical applicability [49].
• High Impact on Performance and Emissions: The selected parameters directly affect critical outcomes such as NO_x emissions, primarily through their influence on in-cylinder temperature, which governs the Zeldovich thermal NO_x formation pathway [50,51], and particulate matter PM, which is modeled using the Moss-Brookes soot formation model. This model estimates soot mass based on key combustion parameters like local equivalence ratio, temperature, and pressure, accounting for both nucleation (soot particle formation from hydrocarbon precursors) and subsequent oxidation. These relationships have been validated in previous studies on low-speed marine engines, highlighting the importance of these control variables in managing emission-performance trade-offs [52,53].

This approach ensures the findings are actionable for operators while avoiding parameters (e.g., combustion chamber geometry) that would necessitate risky ECU reprogramming or mechanical modifications.

Five independent variables were selected based on their influence on combustion phasing, fuel-air mixing, and exhaust emissions:

• Start of Injection (SOI); Measured in Crank Angle Degrees Before Top Dead Center [°CA-BTDC], this parameter varied between 1° and 8°. It defines when fuel injection begins during the engine cycle. Advancing or delaying SOI influences ignition delay and the degree of premixed combustion, which in turn affects NO_x emissions.
• Pilot Injection Fuel Mass Fraction [%]: Ranging from 1% to 12% of the total fuel mass, this parameter defines the proportion of fuel allocated to the pilot injection event. It was varied to study how splitting the fuel dose between a small pilot and a larger main injection impacts combustion characteristics, including the formation of soot (particulate matter) and nitrogen oxides. The subsequent parameter, Dwell Time, defines the crank angle interval between this pilot and the main injection.
• Dwell Time [°CA]; This is the time interval between the pilot and main injection events, measured in crank angle degrees [°CA]. Values from 1° to 5° were tested to optimize the combustion process in terms of efficiency and emissions, particularly under kinetically controlled combustion conditions.
• Exhaust Valve Timing (EVT); This includes Exhaust Valve Opening (EVO) and Exhaust Valve Closing (EVC), both measured in Crank Angle Degrees relative to Bottom Dead Center [BDC]. Adjusting these values changes the effective compression ratio and the engine’s scavenging efficiency, affecting residual gas levels and emission behavior.
  • Exhaust Valve Opening (EVO): 60° to 74° before bottom dead center [°CA-BBDC].
  • Exhaust Valve Closing (EVC): 100° to 130° after bottom dead center [°CA-ABDC].
• Common-Rail Fuel Pressure [bar]; Tested between 600 and 1000 bar, this parameter controls how forcefully fuel is injected into the combustion chamber. Higher pressures improve atomization (breaking the fuel into finer droplets), which enhances combustion efficiency and reduces particulate formation.

To establish a computationally tractable yet comprehensive parameter space for optimization, the selected ranges for the key operational and design parameters were defined based on simulation feasibility and engine design limits observed during model calibration using Diesel-RK [33].

Parameter bounds were rigorously constrained using:

• Manufacturer’s operational limits (MAN Diesel Engines Technical Guides) [52,54].
• International Maritime Organization’s compliance thresholds [1,2].
• Prior experimental validations [39].

The design space refers to the full set of simulation scenarios generated by varying five key engine control parameters. Each parameter was discretized based on realistic operating constraints and simulation feasibility.

Table 2. Design space for the Multi-Criteria Decision-Making.

Parameter	Range	Step Size	Number of Steps (N)
Start of Injection (SOI)	1° to 8° [°CA-BTDC]	1°	8
Fraction per Injection Event (F)	1 to 12 [%]	1 [%]	12
Dwell Time (D)	1° to 5°	1°	5
Common-Rail Fuel Pressure (Pfuel)	600 to 1000 [bar]	50 [bar]	9
Exhaust Valve Opening (EVO)	60° to 74° [°CA-BBDC]	2°	8
Exhaust Valve Closing (EVC)	100° to 130° [°CA-ABDC]	5°	6

summarizes the parameter ranges, discretization steps, and the resulting number of values (N) for each.

A full factorial exploration of this space results in a total of: the total configurations of input parameters are 8·12·5·9·8·6 = 241,920 unique engine configurations.

A full exploration of this design space would yield 241,920 unique combinations of input parameters, presenting a significant computational demand. To thoroughly but efficiently explore the design space, we used the Latin Hypercube Sampling (LHS) method, Shields and Jiaxin [55], generating 500 representative cases. Unlike random sampling, LHS is a statistical method that ensures balanced coverage by dividing each parameter into equal intervals and selecting one sample per interval. This approach minimizes clustering while capturing the full range of possible scenarios, making our analysis both computationally efficient and statistically robust [55].

Mathematically, the input range of each parameter is divided into N non-overlapping intervals of equal probability. One value is randomly selected from each interval. This method reduces variance and improves coverage efficiency. The sampled 500 configuration results were analyzed in alignment with a set of regulatory, operational, and combustion stability constraints, including:

First, the analysis process considers the following primary performance criteria:

• Specific Fuel Consumption (SFC): SFC ≤ 0.2 [kg/kWh]

(Objective: minimize fuel consumption per unit power output)

2.

NO_x Emissions: NO_x ≤ 3.4 [g/kWh]

(To comply with IMO Tier III regulations)

3.

Particulate Matter (PM) Emissions: PM ≤ 0.15 [g/kWh]

(To ensure soot and particulate emission control)

Second, to ensure physically realistic and mechanically safe operating conditions, the following constraints are imposed during optimization:

4.

Combustion Duration: Must remain within 0.35° to 45° crank angle [°CA]

5.

Peak Cylinder Pressure (P_max): P_max ≤ 170 [bar]

6.

Maximum Pressure Rise Rate (dp/dθ): dp/dθ ≤ 4 [bar/°CA]

7.

Synthetic Emission Index (SE): Computed using Diesel-RK’s internal formulation to capture the NO_x–PM trade-off during combustion [51].

$$SE=max{\left(1,{\displaystyle \frac{{NO}_X}{{NO}_{X_0}}}\right)}^{m_1}+max{\left(1,{\displaystyle \frac{PM}{{PM}_0}}\right)}^{m_2}+{\displaystyle \frac{SFC}{{SFC}_0}}$$

(2)

where

NOx: Simulated nitrogen oxide emission level [g/kWh]

NO_x0: Target (reference) NO_x emission level [g/kWh]

PM: Simulated particulate matter emission level [g/kWh]

PM₀: Target (reference) PM emission level [g/kWh]

m₁: Exponent controlling the weight of NO_x penalty

m₂: Exponent controlling the weight of PM penalty

SFC: Simulated specific fuel consumption [g/kWh]

SFC₀: Reference average or target SFC [g/kWh]

The max(1, …) function is applied only to NO_x and PM to act as penalty functions. This ensures that if emissions are below their respective targets (i.e., NO_x < NO_x0 or PM < PM₀), they do not reduce the overall SE value. The penalty activates only when emissions exceed the targets, increasing nonlinearly based on the exponents m₁ and m₂.

SFC is not placed inside a max() function because it contributes to the index linearly regardless of its value. Fuel consumption is always considered in the SE, even if it is better than the reference level, which aligns with operational goals that continuously prefer lower SFC values and only consider PM and NO_x when they are over compliance.

The feasible cases were analyzed following the initial simulation to identify refined parameter ranges that maintained engineering relevance and regulatory compliance while reducing the decision space for subsequent high-resolution optimization within the MCDM framework.

Table 3. Control parameter ranges.

Parameter	Original Range	Adjusted Range
SOI	1–8	1–5
F	0.01–0.12	0.01–0.08
D	1–5	2–5
Pfuel	600–1000	700–950
EVO	60–74	62–68
EVC	100–130	110–125

summarizes the refinement process applied to the control parameters used in this study. The ranges were adjusted to ensure regulatory compliance, combustion stability, and physical realism within a two-stroke marine diesel engine context. The adjustments were guided by specific constraints, such as avoiding excessive peak cylinder pressure (P_max), maintaining efficient scavenging, and minimizing emissions (NO_x, PM). The final narrowed ranges reduce computational complexity and preserve the optimization potential within the Multi-Criteria Decision-Making framework.

Based on the refined parameter ranges, the total number of possible combinations of input parameters becomes 15,360 feasible cases, which reflects the new design space, and after applying the limitation conditions to the latest ranges, the total number of possible combinations of input parameters becomes 4454 feasible cases significantly reducing computational complexity while maintaining physical and regulatory relevance.

Table 4. Decision matrix.

Alternative Ai	SOI [°CA-BTDC]	F [%]	D [°CA]	Pfuel [bar]	EVO [°CA-BBDC]	EVC [°CA- ABDC]	Criterion Cj
							SFC [kg/kWh]	PM [g/kWh]	NOx [g/kWh]
1	1	0.04	3	700	70	120	0.1937	0.10194	2.2259
2	8	0.06	5	1000	72	110	0.17763	0.0802	3.3741
3	4	0.09	5	950	62	120	0.18834	0.13495	2.6197
4	5	0.12	4	600	66	125	0.20071	0.18008	2.0606
5	1	0.09	2	900	68	125	0.19281	0.13468	2.3362
.	.	.	.	.	.	.	.	.
.	.	.	.	.	.	.	.	.	.
4454	.	.	.	.	.	.	.	.	.

presents the decision matrix constructed by Diesel-RK simulation results for the 4454 feasible cases. Each row represents a distinct alternative (i.e., a unique combination of control parameters), while the columns correspond to either an input variable or an output performance criterion. The decision variables include the Start of Injection (SOI), fuel mass fraction (F), Dwell Time (D), rail pressure (P_fuel), Exhaust Valve Opening (EVO), and Exhaust Valve Closing (EVC). These were varied systematically to generate representative engine operating scenarios within the refined parameter space.

The input parameters Specific Fuel Consumption (SFC), Particulate Matter (PM), and Nitrogen Oxides (NO_x) serve as the decision criteria (C_j) for subsequent Multi-Criteria Decision-Making analysis. The dataset captures the multi-objective trade-offs inherent to marine diesel engine operation, where improvements in fuel efficiency often compete with stricter emission constraints. This matrix forms the quantitative foundation upon which various MCDM algorithms (e.g., SAW, TOPSIS, TMF) are applied to rank and identify optimal configurations.

The decision matrix provides a structured and quantifiable basis for evaluating engine control configurations across multiple performance criteria. It bridges simulation data with multi-criteria reasoning, enabling trade-off analysis among emissions, fuel efficiency, and operational constraints. This foundation is essential for applying MCDM algorithms to identify optimal marine diesel engine operation strategies.

4. Case Study—Application of MCDM Algorithms to a Two Stroke Marine Diesel Engine

To accurately characterize and optimize the operational Performance of a two stroke marine diesel engine, this study used a validated thermodynamic model of the MAN B&W S60-MC-C8-8 engine constructed in the Diesel-RK v.4.189 software version. The simulation setup was calibrated to full-load conditions, producing a piston engine power of 19,003 kW at 105 RPM, with a brake mean effective pressure (P_mep) of 20.0 bar and a specific fuel consumption (SFC) of 181.9 g/kWh, in line with published test data for ISO reference conditions.

To represent the fuel injection process in this simulation, an approximate injection profile was created by combining data from three prototype injector systems, labeled System A, B, and C in Figure 1. These are not official injectors from MAN B&W, but external sources used to approximate realistic injection behavior. The noted experimental diagram is typical, and it was used at combustion model calibration in the simulation of spatial interaction of fuel sprays in a combustion chamber of two stroke marine diesel engine [56,57].

As shown in Figure 1, the graph plots the fuel injection rate (in milligrams per second) with the crankshaft rotation angle (in degrees). The injection event occurs around Top Dead Center (TDC), which is the point of maximum piston compression. Each colored line shows how a single injector delivers fuel over time (A, B, C) based on external data sources used to approximate realistic injector behavior. The variations between them represent differences in injector design, such as nozzle hole size or hydraulic flow characteristics. The black line (Sum) combines the effects of all three injectors and represents the total injection profile used in the simulation. While this composite profile was created to mimic a complex, real-world injection event that might involve multiple pilot or main injections, it introduces some margin of error in key simulation outputs, as it can slightly distort the prediction of the Heat Release Rate (HRR) peak. Despite these limitations, the approximation is sufficiently accurate for comparative studies and trend analysis within the MCDM framework. The use of a consistent, though approximated, injection profile across all simulations is crucial. It ensures that the performance differences between the alternatives are due solely to the changes in the control parameters and not from random variation in the injection model. This controlled input is what allows for a fair and valid comparison using MCDM methods.

This brings an error in calculating the average droplet diameter (Sauter Mean Diameter) and the combustion process. Figure 2 serves as a validation and error analysis chart, comparing the simulation’s predictive capability with real-world experimental data. The experimental Heat Release Rate (HRR) line represents empirical data measured from a physical engine test. It is the benchmark for the comparison. The simulated HRR line shows the heat release rate predicted by the Diesel-RK software when using the manually composed “Sum” injection profile from Figure 1. The discrepancy between simulation and experimental data highlights the limitations of the composite injection approximation.

Despite the simplified injection profile, the simulation results exhibited sufficient precision to provide a robust basis for parametric studies and multi-dimensional optimization workflows. The Diesel-RK platform’s batch simulation capabilities allowed automated evaluation across the five-dimensional input space, forming the foundation for the MCDM decision matrix used in subsequent algorithm comparisons.

4.1. Comparison of MCDM Algorithms Results

This section presents a comprehensive comparative analysis of the investigated fourteen MCDM algorithms applied to 4454 simulation-derived alternatives for two stroke marine diesel engine optimization. The goal is to assess algorithmic consistency, consensus, correlation, and stability using various rank-based evaluation techniques, including Borda count aggregation [58] and sensitivity analysis [59]. The fourteen representative methods were selected and categorized according to their underlying mathematical principles. These algorithms were implemented to rank decision alternatives derived from the simulation data. We will present our results from different perspectives.

4.1.1. Consensus Analysis of the MCDM Algorithms’ Results

To synthesize the diverse rankings produced by the 14 individual MCDM algorithms, we employed the Borda Count method [58]. The Borda Count is a consensus-based voting procedure used to generate a single, aggregate ranking from a set of multiple ordered lists. Its primary advantage is its ability to identify alternatives that are broadly favored across different evaluation perspectives, thereby producing a more robust and less biased final ranking [58].

In this study, the application of the Borda Count was performed as follows:

• Rank Assignment: Each of the 14 MCDM methods provided a complete ranking of the 4454 marine engine configurations.
• Point Allocation: For each of these 14 rankings, an alternative received points equal to its rank. For example, an alternative ranked 1st by an algorithm received 1 point, an alternative ranked 10th received 10 points, and an alternative ranked last (4454th) received 4454 points.
• Score Aggregation: For each of the 4454 alternatives, the points from all 14 MCDM algorithms were summed. This sum is the alternative’s final Borda Score.
• Final Ranking: The alternatives were then ranked based on their Borda Scores, from lowest to highest. A lower Borda Score signifies a more preferable alternative, as it indicates a consistently higher existence across the majority of the MCDM methods.

This process effectively identifies the engine configurations that represent the best compromise solution, mitigating the risk of selecting an alternative that performs exceptionally well by one algorithm’s logic but poorly by others.

Figure 3 provides a comparative visualization of the top 15 ranked alternatives across the 14 selected MCDM algorithms. The color scale ranges from Green (most consistent) to Red (least consistent), where alternatives closer to Green indicate higher agreement among the 14 sets of results. The Y-axis represents: Alternative ID: {SOI, Fraction, Delay, P_fuel, EVO, EVC} SE = Value.

The comparative analysis revealed clear variation in algorithmic agreement, as visually captured in the consistency matrix (Figure 3). Alternatives exhibiting green-coded consensus reflect strong algorithmic convergence and are thus ideal candidates for real-world implementation or further validation. Conversely, red-coded rankings signaled greater instability, suggesting sensitivity to the choice of decision-making methodology.

This integrative approach mitigates the limitations inherent in relying on a single MCDM method and reinforces the importance of ensemble evaluation in complex engineering decision contexts. By emphasizing consensus and stability, this framework supports the development of a more resilient engine performance optimization strategy.

4.1.2. Stability Analysis of the MCDM Algorithms’ Results

To ensure that the recommended engine alternatives are not just high-performing but also reliable, a stability analysis was conducted. This means we checked how consistently each engine setting was ranked across all 14 MCDM algorithms.

For every engine alternative, we calculated:

• Standard deviation (σ): This tells us how much the rankings for a given alternative varied across the 14 algorithms. A lower σ means the alternative was ranked similarly (and consistently) by most methods.
• Range: This is the difference between the highest and lowest ranks that an alternative received. A smaller range means the rankings were closely grouped.

The top five most stable alternatives, those exhibiting the lowest standard deviation in their ranks across all algorithms:

• Alternative 2203|{3, 0.01, 3, 750, 66, 115}, SE = 2.9132: σ = 2.87, range = 7
• Alternative 2200|{3, 0.01, 3, 750, 64, 115}, SE = 2.9156: σ = 3.41, range = 11
• Alternative 1149|{2, 0.01, 3, 850, 64, 110}, SE = 2.907: σ = 4.10, range = 10
• Alternative 2197|{3, 0.01, 3, 750, 62, 115}, SE = 2.9157: σ = 4.31, range = 14
• Alternative 2206|{3, 0.01, 3, 750, 68, 115}, SE = 2.9161: σ = 5.57, range = 20

Among all alternatives, Alternative 2203 (with the configuration (SOI = 3, Fraction = 0.01, D = 3, P_fuel = 750, EVO = 66, EVC = 115) and SE = 2.9132), emerged as the most stable, with the lowest standard deviation (σ = 2.87), while also achieving 100% algorithmic consensus. This strong convergence across methods affirms its reliability and resilience to ranking bias. Notably, it was ranked within the top five by most scoring- and distance-based methods, with WASPAS, MABAC, MAIRCA, and PROMETHEE II showing near-perfect alignment.

4.1.3. Synthetic Emission Index (SE) Analysis of the MCDM Algorithms’ Results

The absolute best SE value was observed in Alternative 3216 (this alternative is out of the top 5), (SOI = 4, F = 0.01, D = 3, P_fuel = 750, EVO = 66, EVC = 115), with SE = 2.906. Although marginally better in the Synthetic Emission Index, this alternative exhibited lower consensus and stability across algorithms. The algorithms that ranked this configuration highest were PROMETHEE II, MABAC, and MAIRCA.

5. Discussion

The results of this analysis reveal a critical insight: the alternative with the lowest SE value does not necessarily represent the most robust or widely supported choice. This is clearly demonstrated by the case of Alternative 3216 (SOI = 4, Fraction = 0.01, Delay = 3, P_fuel = 750, EVO = 66, EVC = 115), which despite achieving the lowest SE index (2.906) across all 4454 simulated alternatives, exhibited poor ranking stability (σ = 12.19). Its significant fluctuation across the different MCDM algorithms suggests that its top performance is highly sensitive to the specific weighting and calculation method used, thereby limiting its reliability. In contrast, Alternatives 2203 and 2200, with slightly higher SE values, demonstrated exceptional ranking stability and appeared in the top ranks across all 14 algorithms, achieving high consensus as well. These results highlight that algorithmic consensus and ranking stability are critical robust indicators. For real-world applications such as marine engine tuning, where consistent and reliable decision-making is essential, this stability is arguably as important as the raw performance metric itself.

This divergence proves that no single MCDM method should be treated as universally optimal for marine applications. Therefore, the most dependable foundation for optimization in such complex, regulation-bound environments is provided by methods that successfully balance performance with stability, namely the WISP, PIV, and TMF methods.

Thus, while multiple alternatives exhibit strong performance, WISP, PIV, and TMF methods stand out for their high alignment with consensus rankings, low variability, and ability to identify stable and optimal SE configurations. These three algorithms offer the best overall balance of ranking accuracy, consensus alignment, and discriminatory sensitivity, making them the most suitable MCDM method for marine engine support systems under complex multi-criteria trade-offs.

These findings carry substantial practical relevance for marine engine operators, regulators, and control system developers. In real-world applications, consistency and robustness across decision-making frameworks are critical to ensure safe and optimized performance under varying operating conditions. By identifying alternatives that are both high-performing (in terms of emission index SE) and stable across diverse MCDM rankings, the study supports data-driven tuning of engine parameters, a valuable contribution to engine onboard decision support systems, and IMO compliance strategies. Particularly, the demonstrated effectiveness of WISP, PIV, and TMF methods in isolating both optimal and stable configurations offers a blueprint for real-time engine control systems where algorithmic consensus is essential to avoid oscillatory behavior or sub-optimal tuning.

6. Conclusions

This study systematically examined and compared fourteen MCDM algorithms to recommend operation strategies for two stroke marine diesel engines under stringent emission and performance constraints. Using Diesel-RK simulation data for a variant of the MAN B&W S60-MC-C8-8 engine, this study focused on the effects of Start of Injection, Dwell Time, Fuel Fraction, Fuel Rail Pressure, and Exhaust Valve Timing (EVO/EVC) on critical performance metrics, namely SFC, NO_x, and PM. A refined design space of 15,360 feasible cases was sampled, and a high-fidelity decision matrix was constructed for subsequent MCDM evaluation.

The results of the comparative analysis revealed significant variability in ranking behavior, stability, and consensus across different MCDM classes, scoring-based, distance-based, pairwise comparison, outranking, and hybrid methods. Despite these differences, a subset of alternatives, particularly Alternative 2203 (SOI = 3, F = 0.01, D = 3, P_fuel = 750, EVO = 66, EVC = 115), consistently appeared in the top rankings across all 14 algorithms, achieving the highest consensus (100%) and exhibiting the highest stability. This confirms the robustness and reliability of the identified optimal solutions, independent of the ranking method.

Among the algorithms, WISP, PIV, and TMF methods demonstrated superior overall consistency and adaptability among the evaluated algorithms, particularly in emission-constrained contexts. They are recommended as the preferred methods for real-time or semi-automated engine management systems targeting emission compliance and fuel efficiency optimization.

This paper presents the first comprehensive comparison of 14 MCDM algorithms applied to two-stroke marine diesel engines using Diesel-RK simulations in the literature. It evaluates 4454 optimized cases across five key engine control parameters. WISP, PIV, and TMF methods emerged as the most robust, and consensus aligned. WISP, PIV, and TMF methods showed high stability and minimal sensitivity to normalization, making them ideal for real-time applications. The study supports emissions-compliant engine optimization under IMO Tier III. Its methodology lays the foundation for intelligent marine engine decision support systems. Future research will expand on this foundation by integrating fuzzy logic and dynamic weighting schemes into WISP, PIV, and TMF methods to address uncertainty and evolving operational contexts. In addition, including transient operating conditions will enhance model reliability and ensure practical applicability for real-world marine support systems.