A Strategy-Group Evolution Algorithm for Planning of Multi-Stage Activities in Modular Shipbuilding Considering Uncertainty Duration

Abstract

Modular shipbuilding, as a cutting-edge ship construction paradigm, enables parallel manufacturing across workshops and stages—a core advantage that significantly shortens the total shipbuilding cycle, making it pivotal for modern shipyards to enhance productivity. However, this mode decomposes the integrated shipbuilding project into a large number of interdependent sub-activities spanning three key stages (fabrication, logistics, and assembly). Further, the duration of these sub-activities is inherently uncertain, primarily due to the extensive manual operations, variable on-site conditions, and supply chain fluctuations inherent in shipbuilding. These characteristics collectively pose a formidable challenge to project planning that pursues both high efficiency and low cost. To address this challenge, this paper proposes a Strategy-Group Evolution algorithm. First, the modular shipbuilding process scheduling problem is mathematically formulated as a resource-constrained three-stage multi-objective optimization model, where triangular fuzzy numbers are employed to characterize the uncertain sub-activity durations. Second, a two-layered Strategy-Group Evolution algorithm is designed for solving this model: the inner layer comprises 12 practical priority rules tailored to modular shipbuilding’s multi-stage features, while the outer layer adopts a genetic algorithm-based evolution policy to schedule and optimize the assignment of inner-layer rules to activity groups. The core of the Strategy-Group Evolution algorithm lies in dynamically assigning suitable strategies to different activity groups and evolving these assignments toward optimality—this avoids the limitation of a single priority rule for all stages, thereby facilitating the search for global optimal solutions. Finally, validation tests on real cruise ship construction projects and benchmark datasets demonstrate the efficacy and superiority of the proposed Strategy-Group Evolution algorithm.

1. Introduction

Effective planning and organization of the project is the cornerstone of the modern shipbuilding industry, directly determining production efficiency, resource utilization, and, ultimately, economic performance [1,2].

For modular shipbuilding in particular, a well-optimized schedule ensures the seamless flow of prefabricated module components across workshops and assembly stages, minimizes idle time for high-cost shipyard resources—such as dry docks (critical for final module assembly) and gantry cranes (used for module lifting and transportation)—and guarantees on-time delivery of modular vessels [3]. Conversely, poor modular planning may result in severe project tardiness, significant cost overspends from prolonged resource occupation or rework of misaligned modules, and logistical chaos in intra-yard module transportation [4]. Therefore, the development of advanced and reliable planning methodologies is not merely an academic exercise but a pressing industrial necessity for enhancing the overall productivity and profitability of shipyards. The adoption of modular shipbuilding—an advanced construction mode where a ship is built from pre-assembled, standardized modules—introduces a new dimension of complexity to this classical planning problem [5].

In modular shipbuilding, this modular scheduling paradigm enables the parallel manufacturing of hull blocks across different workshops and project stages, significantly shortening the overall shipbuilding cycle time. However, at the same time, it also decomposes the integral shipbuilding project into a large number of interdependent sub-activities, such as the fabrication of standardized hull modules, the transportation of modules between different areas within the shipyard, and the final assembly of modules into a complete vessel [6]. These sub-activities are typically distributed across the core stages of modular shipbuilding: they span module fabrication workshops, in-yard logistics routes, and the main assembly dock, forming a complex network of precedence relationships (e.g., module outfitting must be completed before inter-module connection) [7]. Additionally, the inherent indeterminacy in the timeline of these sub-activities poses a significant challenge, which mainly stems from three shipbuilding-specific factors: extensive manual operations in the precision assembly of modules, variable weather conditions that disrupt outdoor module transportation and dock assembly, and potential supply chain disruptions for key modular components (e.g., pre-installed piping systems) [8].

Conventional project planning methods, including the classical critical path method (CPM) [9] or priority rule-based heuristics [10,11], are often inadequate for addressing these challenges. The CPM struggles with resource constraints and is fundamentally deterministic, unable to handle duration uncertainty effectively. While meta-heuristics are applied to resource-constrained project planning problems, they frequently rely on a single priority rule or a fixed heuristic to guide the search throughout the entire project. This “one-strategy-fits-all” approach is ill-suited for the heterogeneous and multi-stage nature of modular shipbuilding. The optimal priority for activities in the fabrication stage may be drastically different from those in the erection stage. Consequently, these methods can easily become trapped in local optima, failing to find a globally efficient schedule for the entire project.

To address the limitations of existing approaches, this paper proposes a novel Strategy-Group Evolution algorithm, specifically tailored for multi-stage activity planning in modular shipbuilding under uncertain durations. This algorithm avoids the drawback of applying a single global rule, enabling a more flexible and adaptive search that better matches the complex structure of shipbuilding projects. The work’s core contributions are twofold: first, mathematically formulating the problem as a resource-constrained three-stage multi-objective optimization model; second, developing an innovative two-layered evolution mechanism. The efficacy and superiority of the presented Strategy-Group Evolution algorithm are confirmed through a comprehensive test based on a real-world cruise ship construction project.

The remaining text is organized as follows: Section 2 summarizes the relevant literature outcomes from recent years. The mathematical model of the modular shipbuilding process for project planning is designed in Section 3. Section 4 designs the proposed two-layered Strategy-Group Evolution algorithm. Section 5 presents the designed numerical experiments, analyzing outcomes compared with existing methods to showcase the proposed algorithm’s effectiveness and efficiency. Section 6 applies the proposed method to a real-world cruise ship construction scenario for validation. Section 7 presents the conclusions.

2. Literature Review

Modular shipbuilding, a key advancement in modern ship construction, reduces production cycles via parallel manufacturing but complicates planning by decomposing projects into multi-stage sub-activities. Such planning is typically framed as a Resource-Constrained Project Scheduling Problem (RCPSP). The RCPSP has been a central topic in project optimization and decision-making research for several decades. Many studies have raised this problem. As early as 1969, Pritsker et al. [12] stated that optimizing project scheduling involves determining activity start times and allocating task resources to achieve specific objectives. This process operates under dual constraints: limited resource availability and mandatory activity sequencing. The RCPSP has spawned many extended models, attracting increased scholarly investigation in recent years. Van der Beek et al. [13,14] introduced a flexible project structure based on the RCPSP model and proposed an exact solution method based on constraint propagation and a hybrid differential evolution algorithm. Hu et al. [15] considered the characteristics of the segment assembly process in shipyards and proposed a project planning method with a multi-dimensional resource constraint. To tackle project scheduling under resource constraints with multiple execution modes and uncertain costs, Xie et al. [16] proposed an MINLP approach. To handle the computational burden of exact approaches to large-scale scenarios, they devised a construction heuristic incorporating a genetic algorithm.

The planning process for a cruise ship construction project is a tangled web of various objectives that sometimes clash, all of which must be fine-tuned at the same time. In the realm of multi-objective research concerning RCPSP, Gomes et al. [17] examined the optimization of project duration and the total weighted start time for activities. They devised a multi-objective variable neighborhood search combined with the greedy randomized adaptive search procedure, appropriately named GMOVNS. Zarei et al. [18] presented a mixed integer number linear scheduling framework designed to shorten project timelines, lower expenses, and improve task quality for railway bridge construction initiatives employing multi-skilled labor forces. Rodriguez-Ballesteros et al. [19] put six more multi-objective evolutionary algorithms (MOEAs) through their paces on a bi-objective RCPSP. They were trying to see how well each algorithm performed at minimizing both the total time and the operation cost of resource utilization. Arman Ghamginzadeh et al. [20] examined the multi-objective, multi-skill project scheduling problem under fuzzy time uncertainty with the objectives of minimizing project makespan and total labor allocation cost, adopted a multi-objective imperialist competitive algorithm (MOICA), and compared it with NSGA-II based on three indicators, and the results showed that MOICA had good performance. Basically, they were looking for the best trade-off between completing the job quickly and completing it cheaply.

In actual ship project activities, there is usually uncertainty in the operation time of the activities due to multiple types of factors. Uncertain project timelines have garnered considerable focus. Typically, in the perspective of the project scheduling problem, this uncertainty duration can be described by fuzzy sets. This approach employs the three-point estimation technique, modeling activity durations as random variables that can assume any value within a range specified by minimum, maximum, and most plausible values. In addressing activity duration uncertainty in prefabricated building (PB) projects, Yuan et al. [21] employed triangular fuzzy numbers and developed a multi-objective RCPSP framework. Separately, Birjandi and Mousavi [22] established a mathematical formulation for the fuzzy RCPSP incorporating multiple execution paths, for which a three-phase heuristic was proposed to tackle large-scale instances. Their model captures uncertainties in both duration and resources through trapezoidal fuzzy numbers, utilizing an enhanced lexicographic technique for defuzzification.

In summary, the Resource-Constrained Project Scheduling Problem (RCPSP) has accumulated a solid research foundation, with three primary categories of techniques dominating its optimization landscape, as supported by relevant studies:

First, exact algorithms (for finding globally optimal solutions) have been advanced. Bold et al. [23] proposed a faster exact method for the robust multi-mode RCPSP, enhancing computational efficiency while ensuring solution optimality under uncertainty.

Second, constructive methods (incrementally building feasible schedules) remain fundamental. Hartmann et al. [24] conducted a systematic evaluation of state-of-the-art constructive heuristics, establishing benchmark standards for the RCPSP.

Third, meta-heuristics (balancing exploration and exploitation for high-quality solutions) are widely used for complex variants. Etminaniesfahani et al. [25] introduced an efficient relax-and-solve meta-heuristic for the multi-mode RCPSP, reducing computational complexity.

Despite the substantial body of research on shipbuilding scheduling, it remains a gap for modular shipbuilding projects. Most existing approaches rely on a single priority rule or a fixed heuristic throughout the entire project, which are ill-suited for the heterogeneous and multi-stage nature of modular shipbuilding. The optimal priority for activities in the fabrication stage may differ substantially from those in the transportation stage.

Unlike methods that rely on a single priority rule, our Strategy-Group Evolution algorithm incorporates multiple planning strategies that can be assigned for different groups of activities, better accommodating the heterogeneous nature of multi-stage shipbuilding projects.

3. Modeling of Modular Shipbuilding Processes for Project Planning

3.1. Problem Description

The modular cruise ship construction process works by breaking the ship down into individual modules (including blocks, cabin units, power systems, outfitting equipment, etc.), which are then fabricated and assembled ashore to form the complete ship. These modules have been produced in specific workshops or manufacturers and are then transported to the dock for complete ship assembly. The modular shipbuilding project encompasses three distinct phases: fabrication, logistics, and assembly, as depicted in Figure 1. The modular shipbuilding project offers advantages such as off-site prefabrication, leading to enhanced construction productivity and a shortened project timeline.

Due to the modular manufacturing method, various types of intermediate products of the modular shipbuilding project will be manufactured by specific departments in the shipyard or by project subcontractors according to demand, which is a parallel process. Each stage of the modular shipbuilding project is seemingly independent, but inherently related to the others. In addition, each stage consists of different construction or transportation activities with operation duration and resources requirements. Therefore, the project scheduling problem for the modular shipbuilding project can be modeled as a multi-stage RCPSP, as shown in Figure 2.

The modular shipbuilding project is essentially a spin-off from the RCPSP’s foundational structure. The project encompasses a series of interconnected tasks, each with a predefined processing duration and specific resource needs. Under the conventional RCPSP framework, the lengths of these activities are set in stone, meaning they remain a steadfast, unchanging figure. However, in the actual RCPSP, the production activities of shipyards are affected by various factors such as weather, manpower, equipment, etc., leading to uncertainty in the duration of activities. Fuzzy set theory is often used to describe time uncertainty in scheduling problems, which is applicable to the characteristics of the modular shipbuilding project scheduling problem. Therefore, the project scheduling of the modular shipbuilding project will be modeled as a multi-stage fuzzy multi-objective RCPSP (M-F-MORCPSP).

3.2. M-F-MORCPSP Modeling

In the M-F-MORCPSP model, we define i as the stage number, j as the task number, and r as the resource number, and the tasks have a fuzzy operation duration ${ \tilde{\mathrm{p}}}_{\mathrm{sj}}$. In analyzing the M-F-MORCPSP model, we must consider the precedence relationships between each stage, where one stage immediately follows another.

The M-F-MORCPSP model hinges on several key settings:

• A project is broken down into a series of tasks, each with a pre-determined baseline completion time, which is uncertain in practice.
• The exact duration of any given task is subject to variability.
• The initiation of any given task is contingent upon fulfilling its preconditions, which are defined by the logical relationships and completion status of other tasks in the project network.
• Resources are finite and cannot be replenished during the project’s lifecycle.
• There is no substitution between resources.
• Once an activity is underway, it cannot be interrupted.
• Each task has only one mode that can be executed.
• The manufacturing, transportation, and assembly phases are all connected and work as a cohesive unit.
• Once a phase is underway, it must be seen through to the end before jumping to another.
• The overarching optimization goal is the concurrent minimization of the project timeline and operational expenditures.

Based on the above settings, a mathematical description for the M-F-MORCPSP is developed.

Table 1. Information about M-F-MORCPSP.

Indices
$\mathrm{i}$	Stage number, $\mathrm{i}=1,2,\dots ,\mathrm{I}$
$\mathrm{j}$	Number of the task in stage $\mathrm{s}$, $\mathrm{j}=1,2,\dots ,\mathrm{J}$
$\mathrm{r}$	Number of resource, $\mathrm{r}=1,2,\dots ,\mathrm{R}$
$\mathrm{t}$	Time
Parameters
$\widetilde{{\mathrm{t}}_{\mathrm{ij}}^{\mathrm{S}}}$	Fuzzy start time of task $\mathrm{j}$ in stage $\mathrm{i}$.
$\widetilde{{\mathrm{t}}_{\mathrm{ij}}^{\mathrm{F}}}$	Fuzzy end time of task $\mathrm{j}$ in stage $\mathrm{i}$.
$\widetilde{{\mathrm{p}}_{\mathrm{ij}}}$	Fuzzy operation time of task $\mathrm{j}$ in stage $\mathrm{i}$.
${\mathrm{c}}_{\mathrm{ij}}$	Operational cost per time unit of activity $\mathrm{j}$ in stage $\mathrm{i}$.
${\mathrm{Pre}}_{\mathrm{i}}^{\mathrm{st}}$	Set of preorder stages of stage $\mathrm{i}$.
${\mathrm{Pre}}_{\mathrm{ij}}^{\mathrm{act}}$	Set of preorder tasks of task $\mathrm{j}$ in stage $\mathrm{i}$.
${\mathrm{N}}_{\mathrm{r}}$	Maximum amount of existing resource $\mathrm{r}$.
${\mathrm{T}}_{\max }$	Expected maximum timeline of the event.
$\mathrm{C}$	Total expected expenditures of the event.
Decision variables
${\mathrm{x}}_{\mathrm{ijt}}$	Decision variable, where ${\mathrm{x}}_{\mathrm{ijt}}=1$ if task $\mathrm{j}$ of stage $\mathrm{i}$ is scheduled at time $\mathrm{t}$, or else ${\mathrm{x}}_{\mathrm{ijt}}=0$.
${\mathrm{x}}_{\mathrm{ijr}}$	Decision variable, where ${\mathrm{x}}_{\mathrm{ijr}}=1$ if resource $\mathrm{r}$ is selected for task $\mathrm{j}$ in stage $\mathrm{i}$, or else ${\mathrm{x}}_{\mathrm{ijr}}=0$.

, which synthesizes the required indices, definitions, and notations for all parameters and decision variables, is constructed through on-site investigations and specifically integrated into this study.

Based on the above assumptions and notation definitions, the multi-objective M-F-MORCPSP model for the modular shipbuilding project is constructed as follows:

According to the above description, we can define the objective functions as:

$$\min {\mathrm{T}}_{\max }=\max \left\{\mathrm{E}\left(\widetilde{{\mathrm{t}}_{\mathrm{ij}}^{\mathrm{F}}}\right)\right\}$$

(1)

$$\min \mathrm{C}={{\displaystyle \sum }}_{\mathrm{i},\mathrm{j}}\mathrm{E}\left({\mathrm{c}}_{\mathrm{ij}}·\widetilde{{\mathrm{p}}_{\mathrm{ij}}}·{\mathrm{x}}_{\mathrm{ijt}}\right)$$

(2)

Equation (1) represents the maximum timeline of the project, representing the efficiency of the project. Equation (2) represents the total expenditures of the project, which are obtained by adding the expenditures of all tasks.

M-F-MORCPSP constraints are specified below:

$${\displaystyle \sum }_{\mathrm{i},\mathrm{j}}{\mathrm{x}}_{\mathrm{ijt}}=1, \forall t$$

(3)

$${\displaystyle \sum }_{\mathrm{i},\mathrm{j},\mathrm{r}}{\mathrm{x}}_{\mathrm{ijr}}=1, r=1,2,\dots ,R$$

(4)

$$\widetilde{{\mathrm{t}}_{\mathrm{ij}}^{\mathrm{F}}}=\widetilde{{\mathrm{t}}_{\mathrm{ij}}^{\mathrm{S}}}+{{\displaystyle \sum }}_{\mathrm{t}}\widetilde{{\mathrm{p}}_{\mathrm{ij}}}·{\mathrm{x}}_{\mathrm{ijt}}, \mathrm{i}=1,2,\dots ,\mathrm{I}, \mathrm{j}=1,2,\dots ,\mathrm{J},\mathrm{t}\ge \widetilde{{\mathrm{t}}_{\mathrm{ij}}^{\mathrm{S}}}$$

(5)

$$\widetilde{{\mathrm{t}}_{\mathrm{ij}}^{\mathrm{F}}}-\tilde{{\mathrm{t}}_{\mathrm{i} {\mathrm{j}}^{\prime }}^{\mathrm{F}}}\ge {{\displaystyle \sum }}_{\mathrm{t}}\widetilde{{\mathrm{p}}_{\mathrm{ij}}}·{\mathrm{x}}_{\mathrm{ijt}}, \mathrm{i}=1,2,\dots ,\mathrm{I}, {\mathrm{j}}^{\prime }\in {\mathrm{Pre}}_{\mathrm{ij}}^{\mathrm{act}},\mathrm{t}\ge \tilde{{\mathrm{t}}_{\mathrm{i} {\mathrm{j}}^{\prime }}^{\mathrm{F}}}$$

(6)

$$\widetilde{{\mathrm{t}}_{\mathrm{i}1}^{\mathrm{F}}}-\tilde{{\mathrm{t}}_{ {\mathrm{i}}^{\prime } \mathrm{J}}^{\mathrm{F}}}\ge {{\displaystyle \sum }}_{\mathrm{t}}\widetilde{{\mathrm{p}}_{\mathrm{i}1}}·{\mathrm{x}}_{\mathrm{i}1\mathrm{t}}, {\mathrm{i}}^{\prime }\in {\mathrm{Pre}}_{\mathrm{i}}^{\mathrm{st}},\mathrm{t}\ge \tilde{{\mathrm{t}}_{ {\mathrm{i}}^{\prime } \mathrm{J}}^{\mathrm{F}}}$$

(7)

$${\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{I}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{J}}{\displaystyle \sum }_{\mathrm{r}=1}^{\mathrm{R}}{\mathrm{x}}_{\mathrm{ijt}}·\mathrm{r}\le {\mathrm{N}}_{\mathrm{r}}, \mathrm{i}=1,2,\dots \mathrm{I}, \mathrm{j}=1,2,\dots ,\mathrm{J}$$

(8)

$${\mathrm{x}}_{\mathrm{ijt}}=\left\{0,1\right\}$$

(9)

$${\mathrm{x}}_{\mathrm{ijr}}=\left\{0,1\right\}$$

(10)

The constraints at Equations (3) and (4) make certain that each task $\mathrm{j}$ of stage $\mathrm{i}$ is conducted by resource $\mathrm{r}$. Equation (5) indicates the relationship between the beginning and ending time of task $\mathrm{j}$ in stage $\mathrm{i}$. Equation (6) makes absolutely certain that activities within a single stage happen in the correct order. Equation (7) establishes the necessary order between the multiple stages of the entire event. Equation (8) constrains the maximum amount of resources. Equations (9) and (10) limit the acceptable scope for the decision-variable values.

Having defined the core constraints of the scheduling model—covering resource allocation, task time logic, inter- and intra-stage sequencing, and decision-variable bounds—the model as currently formulated assumes deterministic processing times for all activities. However, in practical modular shipbuilding project scheduling, time uncertainty is a prevalent and non-negligible factor that directly impacts the model’s applicability and the reliability of scheduling results. To address this gap and enhance the model’s alignment with real-world scenarios, the next section will focus on explicitly representing and quantifying the fuzzy processing times of activities, laying the groundwork for a more robust scheduling framework.

3.3. Representation of Fuzzy Processing Time of Activities

Based on the above problem analysis, time uncertainty exists in the project scheduling of a modular shipbuilding project. This ambiguity is quantified using fuzzy set theory, and while fuzziness can be represented in various ways, triangular fuzzy numbers (TFNs) outperform probabilistic methods when modeling different types of uncertainties in engineering and management scenarios. Probabilistic methods rely on sufficient historical data to fit fixed probability distributions—a requirement that is often unmet for such uncertainties. In contrast, triangular fuzzy numbers only require three intuitive values (pessimistic, most likely, and optimistic values) derived from expert experience or practical insights, making them more adaptable to data-scarce situations. Additionally, these uncertainties are mostly characterized as “cognitive fuzziness” rather than “objective randomness.” TFNs directly quantify this fuzziness without imposing forced random assumptions, thus avoiding model distortion caused by improper distribution settings; in contrast, probabilistic methods may misalign with the fuzzy nature of such uncertainties, resulting in inaccurate modeling outcomes.

As early as 2006, Xing et al. [26] modeled the imprecise processing times in job-shop scheduling problems as triangular fuzzy numbers to address uncertainties such as variable setup times and transportation times caused by human operations. Soheil Sadi Nezhad et al. [27] addressed the issue of triangular fuzzy number shape distortion caused by the maximum operator in fuzzy scheduling, proposed an approximation method based on the preference ratio, and developed a fuzzy CDS algorithm accordingly.

Therefore, we use TFNs to describe the processing time of activities in the RCPSP. Figure 3 illustrates $\widetilde{{\mathrm{p}}_{\mathrm{ij}}}$, a fuzzy processing time of task $\mathrm{j}$ in stage $\mathrm{i}$. The affiliation function of $\widetilde{{\mathrm{p}}_{\mathrm{ij}}}$ is indicated in Equation (11).

$$\mathrm{A}\left(\mathrm{x}\right)=\left\{\begin{array}{c}0 \mathrm{i}\mathrm{f} x\le {\mathrm{p}}_1\\ {\displaystyle \frac{\mathrm{x}-{\mathrm{p}}_1}{{\mathrm{p}}_2-{\mathrm{p}}_1}} \mathrm{i}\mathrm{f} x\in \left({\mathrm{p}}_1,{\mathrm{p}}_2\right]\\ {\displaystyle \frac{{\mathrm{p}}_3-\mathrm{x}}{{\mathrm{p}}_3-{\mathrm{p}}_2}} \mathrm{i}\mathrm{f} x\in \left({\mathrm{p}}_2,{\mathrm{p}}_3\right)\\ 0 \mathrm{i}\mathrm{f} x\ge {\mathrm{p}}_3\end{array}\right.$$

(11)

In the proposed model, the operation time of a task j at stage i is characterized via a triangular fuzzy number (TFN) defined by three values, p1, p2, and p3, which correspond to the minimum, most likely, and maximum possible durations, respectively. Consequently, solving the model involves the manipulation of and comparison between TFNs. The specific arithmetic operations and ordering principles for fuzzy numbers adhere to the methodology presented in

Table 2. The operation rules of TFNs.

Operations	Rules
$+\left(\mathrm{sum}\right)$	${\tilde{\mathrm{p}}}^1+{ \tilde{\mathrm{p}}}^2=\left({\mathrm{p}}_1^1+{\mathrm{p}}_1^2,{\mathrm{p}}_2^1+{\mathrm{p}}_2^2,{\mathrm{p}}_3^1+{\mathrm{p}}_3^2\right)$
$\vee \left(\max \right)$	${ \tilde{\mathrm{p}}}^1 \vee { \tilde{\mathrm{p}}}^2=\left({\mathrm{p}}_1^1 \vee { \mathrm{p}}_1^2,{\mathrm{p}}_2^1 \vee { \mathrm{p}}_2^2,{\mathrm{p}}_3^1 \vee { \mathrm{p}}_3^2\right)$
Ranking	Criterion 1. ${\mathrm{C}}_1\left(\tilde{\mathrm{p}}\right)= ({\mathrm{p}}_1+2{\mathrm{p}}_2+{\mathrm{p}}_3)/4$
	Criterion 2. ${\mathrm{C}}_2\left(\tilde{\mathrm{p}}\right)={ \mathrm{p}}_2$
	Criterion 3. ${\mathrm{C}}_3\left(\tilde{\mathrm{p}}\right)={\mathrm{p}}_3-{\mathrm{p}}_1$

[28].

Here are two examples from Table 2. If ${\tilde{\mathrm{p}}}^1=\left(1, 2, 3\right)$ and ${\tilde{\mathrm{p}}}^2=\left(2, 3, 4\right)$, their sum is (3, 5, 7). For ${\tilde{\mathrm{p}}}^1=\left(1, 3, 5\right)$ and ${\tilde{\mathrm{p}}}^2=\left(2, 2, 6\right)$, the max is (2, 3, 6).

In Table 2, When comparing two TFNs ${\tilde{\mathrm{p}}}^{\mathrm{A}}$ and ${\tilde{\mathrm{p}}}^{\mathrm{B}}$, the ordering mechanism follows a hierarchical comparison procedure:

Step 1. Compare ${\mathrm{C}}_1\left({\tilde{\mathrm{p}}}^{\mathrm{A}}\right)$ and ${\mathrm{C}}_1\left({\tilde{\mathrm{p}}}^{\mathrm{B}}\right)$. If ${\mathrm{C}}_1\left({\tilde{\mathrm{p}}}^{\mathrm{A}}\right)$> ${\mathrm{C}}_1\left({\tilde{\mathrm{p}}}^{\mathrm{B}}\right)$, ${\tilde{\mathrm{p}}}^{\mathrm{A}}$ ranks higher; if equal, proceed to Step 2.

Step 2. Compare ${\mathrm{C}}_2\left({\tilde{\mathrm{p}}}^{\mathrm{A}}\right)$ and ${\mathrm{C}}_2\left({\tilde{\mathrm{p}}}^{\mathrm{B}}\right)$. If ${\mathrm{C}}_2\left({\tilde{\mathrm{p}}}^{\mathrm{A}}\right)$ > ${\mathrm{C}}_2\left({\tilde{\mathrm{p}}}^{\mathrm{B}}\right)$, ${\tilde{\mathrm{p}}}^{\mathrm{A}}$ ranks higher; if equal, proceed to Step 3.

Step 3. Compare ${\mathrm{C}}_3\left({\tilde{\mathrm{p}}}^{\mathrm{A}}\right)$ and ${\mathrm{C}}_3\left({\tilde{\mathrm{p}}}^{\mathrm{B}}\right)$. A smaller ${\mathrm{C}}_3$ means a higher rank.

In addition, the TFN expectation calculation will be referred to Criterion 1, i.e., $\mathrm{E}\left(\tilde{\mathrm{p}}\right)=({\mathrm{p}}_1+2{\mathrm{p}}_2+{\mathrm{p}}_3)/4$.

4. Strategy-Group Evolution Algorithm

The Strategy-Group Evolution architecture boils down to two pivotal layers: (1) The inner layer embodies a portfolio of diverse planning strategies, each representing a different priority rule within the M-F-MORCPSP framework to inform its scheduling choices; (2) The outer layer intelligently assigns the most suitable strategies to different groups of activities across various stages and evolves these assignments toward the optimal state. Specifically, the inner-layer strategies are designed as 12 priority rules that are commonly used in the modular shipbuilding project scheduling, while the outer-layer evolution policy is designed based on the genetic algorithm (GA) to govern the inner-layer strategies, because of the natural suitability of its coding and decoding mechanisms for high-dimensional discrete scheduling policy solution spaces, by which it is efficiently possible to search for a suitable dynamic scheduling policy.

4.1. Inner-Layer Strategies

Since the shipbuilding project is divided into a large number of activities in a modular manner, it is complex and tedious to manipulate the real starting and finishing time solutions of all these activities during the evolution, and it is not appropriate to apply one rule for all activities with different characteristics. By doing this, the proposed method transforms the solution space of the M-F-MORCPSP problem to the priority rule space of the inner-layer strategy. The outer layer manipulates the inner-layer rules and applies them to activity groups. Thus, the complexity of the searching process can be reduced to a large extent, without sacrificing solution diversity.

The inner-layer strategy is composed of 12 commonly used priority rules that match the characteristics of the M-F-MORCPSP problem, and they are often adopted in the shipbuilding project planning. The 12 priority rules are shown in

Table 3. Priority rules utilized as the inner-layer strategies of the proposed Strategy-Group Evolution algorithm.

No.	Inner-Layer Strategy	Priority Rule	Description
1	${\mathrm{st}}_1$	SOT	Shortest operation time
2	${\mathrm{st}}_2$	LOT	Longest operation time
3	${\mathrm{st}}_3$	EBT	Earliest beginning time
4	${\mathrm{st}}_4$	EET	Earliest ending time
5	${\mathrm{st}}_5$	LBT	Latest beginning time
6	${\mathrm{st}}_6$	LET	Latest ending time
7	${\mathrm{st}}_7$	GRR	Greatest resource required
8	${\mathrm{st}}_8$	GCRR	Greatest cumulative resource required
9	${\mathrm{st}}_9$	MIS	Most immediate amount of successors
10	${\mathrm{st}}_{10}$	MTS	Most total amount of successors
11	${\mathrm{st}}_{11}$	GRPW	Greatest rank positional weight
12	${\mathrm{st}}_{12}$	Random	Random priority

. It is noteworthy that the rule set can be adapted or modified as needed to suit specific application contexts, without altering the core framework of the proposed Strategy-Group Evolution algorithm. The strategy group of priority rule sequentially incorporates activities into the schedule, ensuring that each is fully booked before moving on to the next strategy group. Different strategy groups with corresponding priority rules work together to form an integral schedule of all activities.

Based on the above, the problem-solving approach operates at a meta-level, distinct from direct solution construction. In the proposed Strategy-Group Evolution algorithm, the chromosomes are encoded for representing the dynamic evolution policy of the M-F-MORCPSP, instead of the solution to the actual problem. In detail, the chromosomes are encoded in a two-vector matrix manner. Each scheduling policy, representing an individual ${\mathrm{P}}_{\mathrm{k}}$ in the population of evolution process, consists of two vectors, ${\mathrm{H}}_{\mathrm{k}}$ and ${\mathrm{A}}_{\mathrm{k}}$. The two vectors have the same length $\mathrm{D}$, but each with a different meaning. The vector ${\mathrm{H}}_{\mathrm{k}}=\left({\mathrm{h}}_{1,\mathrm{k}},{ \mathrm{h}}_{2,\mathrm{k}},\dots ,{ \mathrm{h}}_{\mathrm{D},\mathrm{k}}\right)$ contains integer numbers corresponding to strategies expressed by the priority rules in Table 3. The vector ${\mathrm{h}}_{\mathrm{d},\mathrm{k}}$ denotes the priority rule adopted at position $\mathrm{d}$. The vector ${\mathrm{H}}_{\mathrm{k}}$ denotes the order of application for the priority-based schedule construction rules. The vector ${\mathrm{A}}_{\mathrm{k}}=\left({\mathrm{a}}_{1,\mathrm{k}},{ \mathrm{a}}_{2,\mathrm{k}},\dots ,{ \mathrm{a}}_{\mathrm{D},\mathrm{k}}\right)$ represents a sequence of integer numbers ${\mathrm{a}}_{\mathrm{d},\mathrm{k}}$, which represents the number of activities scheduled by the priority rule ${\mathrm{h}}_{\mathrm{d},\mathrm{k}}$ at position $\mathrm{d}$, and it must be satisfied that $\sum _{\mathrm{d}=1}^{\mathrm{D}}{\mathrm{a}}_{\mathrm{d},\mathrm{k}}=\mathrm{J}$. The two vectors ${\mathrm{H}}_{\mathrm{k}}$ and ${\mathrm{A}}_{\mathrm{k}}$ construct a series of strategy groups (SGs), and the SGs evolve by the outer-layer operation.

Figure 4 shows an example of an individual Strategy-Group Evolution algorithm. The length of vectors ${\mathrm{H}}_{\mathrm{k}}$ and ${\mathrm{A}}_{\mathrm{k}}$ $\mathrm{D}=6$ for a problem with 12 non-dummy activities. A total of 12 non-dummy activities will determine their own order of activities according to the respective priority rules adopted, thus enabling the generation of M-F-MORCPSP problem solutions. The sequence of priority rules will be obtained based on the priority rules corresponding to each position of the individual and the corresponding number of activities.

After generating M-F-MORCPSP solutions through the encoded individuals, these solutions need to be further evaluated and managed to ensure the evolutionary process converges toward high-quality, non-dominated outcomes. Specifically, each encoded individual Pk—defined by its ${\mathrm{H}}_{\mathrm{k}}$ and ${\mathrm{A}}_{\mathrm{k}}$ vectors—corresponds to a unique scheduling policy, and the solution generated by this policy must first undergo performance evaluation. Once evaluated, these solutions are not discarded immediately; instead, they are fed into an external repository designed to preserve non-dominated solutions, which serves as a core component for guiding subsequent evolutionary operations.

The presented Strategy-Group Evolution algorithm incorporates an external repository for preserving the set of non-dominated solutions. This reference set was used for calculating RN(Sj) and AD(Sj) for all algorithms in the comparison. The repository can accommodate up to ${\mathrm{N}}_{\mathrm{rep}}^{\max }$ items. Initially, it starts off as a blank slate. As the evolution process churns along, it keeps meticulously updating and safeguarding the repository. Once the repository reaches its capacity of ${\mathrm{N}}_{\mathrm{rep}}^{\max }$ non-dominated solutions, it is crucial to manage the entries based on their sorting and crowding distances to make certain that only the cream of the crop remains. The evaluation procedure of the solutions is below.

Step 1: Perform non-dominated sorting on the current population pop to assign Pareto ranks. All non-dominated solutions (rank 1) are inserted into the external register rep.

Step 2: Conduct another non-dominated sort within rep. Only individuals with rank 1 are preserved; all others are removed.

Step 3: Check whether the size of rep exceeds its predefined capacity. If so, advance to Step 4; or else, advance to Step 5.

Step 4: Hold only the top elite solutions in rep based on a diversity maintenance metric.

Step 5: Conclude the maintenance process of the external archive.

4.2. Outer-Layer Evolution Policy

In this study, the outer-layer evolution policy is designed based on a genetic algorithm (GA), which comprises four key components: initialization of the population, selection, crossover, and mutation.

• Population initialization

The population initialization process is shown in Algorithm 1. Each individual of the proposed Strategy-Group Evolution algorithm is made of two vectors, ${\mathrm{H}}_{\mathrm{k}}$ and ${\mathrm{A}}_{\mathrm{k}}$. For ${\mathrm{H}}_{\mathrm{k}}$ generation, we introduce a tabu list $\mathrm{TL}$ to enhance the diversity of the priority rules’ sequence. The specific operations are as follows: First we construct the set of priority rules. Priority rules are randomly assigned to all locations when performing individual initialization and selected priority rules are added to the $\mathrm{TL}$ until an ${\mathrm{H}}_{\mathrm{k}}$ completes generation to empty the $\mathrm{TL}$. For generating vector ${\mathrm{A}}_{\mathrm{k}}$, each priority rule within ${\mathrm{H}}_{\mathrm{k}}$ receives a random integer ${\mathrm{a}}_{\mathrm{d},\mathrm{k}}$ selected from $\left[0,\mathrm{m}\right]$, where $\mathrm{m}$ is a positive integer. Note that the ${\mathrm{a}}_{\mathrm{d},\mathrm{k}}$ needs to meet the constraints $\sum _{\mathrm{d}=1}^{\mathrm{D}}{\mathrm{a}}_{\mathrm{d},\mathrm{k}}=\mathrm{J}$.

Algorithm 1: Population initialization

Input:

$ps$: population size, $D$: length of vectors, $S_h$: set of all priority rules in Table 3, $TL$: tabu list, $J$: number of non-dummy activities

Output:

Initial population

Begin

$k=1$

$S_0=S_h$

while $k\le ps$ do

for $d=1$ to $D$ do

$h_r\leftarrow$ Randomly select a priority rule from $S_h$

$h_{d,k}=s$

$TL=TL\cup \left\{h_s\right\}$

$S_h$ remove $h_s$

$A_k=\left[rand\left[0,1\right]*m\right]$

end for

$S_h=S_0$

$\theta =J-sum\left(A_k\right)$

if $\theta >0$ do

Select a random set of $\theta$ elements whose values are less than m and augment them by 1.

else if $\theta <0$

Select a random set of $\left|\theta \right|$ elements whose values are less than m and augment them by $-1$.

end if

end while

• Selection

The principle of “survival of the fittest” is central to GA selection. Chromosomes demonstrating superior fitness suggest a better adaptation to the problem and increase the likelihood that their corresponding scheduling scheme will be chosen. However, a Strategy-Group Evolution algorithm solution cannot directly calculate selection probability from a single objective function since it is a multi-objective model. Consequently, a stochastic tournament model (STM) is employed during the selection phase. The STM bypasses explicit fitness evaluation, focusing instead on a comparative analysis of individual strengths and weaknesses within the population. In our Strategy-Group Evolution algorithm, we implement a stochastic league selection process as follows:

Step 1: We randomly select N individuals from the parent population to create a league set, denoted as S.

Step 2: The individuals within league set S are then ranked according to non-domination, and the individual exhibiting the highest level of dominance is chosen to advance into the offspring population.

Step 3: These steps are repeated until the offspring population reaches the desired size, $\mathrm{ps}$.

• Crossover

Population diversity is ensured through crossover operators, which are typically tailored to specific problem types. In the proposed Strategy-Group Evolution algorithm, the offspring individual ${\mathrm{P}}_{\mathrm{k}}\left(\mathrm{t}+1\right)$ is gained from different individuals, ${\mathrm{P}}_{{\mathrm{k}}_1}\left(\mathrm{t}\right)$ and ${\mathrm{P}}_{{\mathrm{k}}_2}\left(\mathrm{t}\right)$. The following equations are performed on ${\mathrm{P}}_{{\mathrm{k}}_1}\left(\mathrm{t}\right)$ and ${\mathrm{P}}_{{\mathrm{k}}_2}\left(\mathrm{t}\right)$ for generating new individuals.

$${\mathrm{H}}_{\mathrm{k}}\left(\mathrm{t}+1\right)={\mathrm{H}}_{{\mathrm{k}}_1}\left(\mathrm{t}\right)+\mathsf{\eta }·\left({\mathrm{H}}_{{\mathrm{k}}_1}\left(\mathrm{t}\right)-{\mathrm{H}}_{{\mathrm{k}}_2}\left(\mathrm{t}\right)\right)$$

(12)

$${\mathrm{A}}_{\mathrm{k}}\left(\mathrm{t}+1\right)={\mathrm{A}}_{{\mathrm{k}}_1}\left(\mathrm{t}\right)+\mathsf{\eta }·\left({\mathrm{A}}_{{\mathrm{k}}_1}\left(\mathrm{t}\right)-{\mathrm{A}}_{{\mathrm{k}}_2}\left(\mathrm{t}\right)\right)$$

(13)

The scaling factor $\mathsf{\eta }$ for every individual in the population is drawn from a normal distribution with $\mathrm{N}\left(0.5,0.3\right)$ in the range $\left(0, 2\right]$. The obtained ${\mathrm{H}}_{\mathrm{k}}\left(\mathrm{t}+1\right)$ and ${\mathrm{A}}_{\mathrm{k}}\left(\mathrm{t}+1\right)$ values are real, and they are converted to integers via these equations:

$${\mathrm{h}}_{\mathrm{d},\mathrm{k}}\left(\mathrm{t}+1\right)=\left\{\begin{array}{c}1 \mathrm{i}\mathrm{f} {\mathrm{h}}_{\mathrm{d}}\left(\mathrm{t}+1\right)<1\\ {\mathrm{N}}_{\mathrm{h}} \mathrm{i}\mathrm{f} {\mathrm{h}}_{\mathrm{d}}\left(\mathrm{t}+1\right)>{\mathrm{N}}_{\mathrm{h}} \\ ⌊{\mathrm{h}}_{\mathrm{d},\mathrm{k}}\left(\mathrm{t}+1\right)⌋ \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\right.$$

(14)

$${\mathrm{a}}_{\mathrm{d},\mathrm{k}}\left(\mathrm{t}+1\right)=\left\{\begin{array}{c}0 \mathrm{i}\mathrm{f} {\mathrm{a}}_{\mathrm{d},\mathrm{k}}\left(\mathrm{t}+1\right)<0\\ \mathrm{J} \mathrm{i}\mathrm{f} {\mathrm{a}}_{\mathrm{d},\mathrm{k}}\left(\mathrm{t}+1\right)>\mathrm{J}\\ ⌊{\mathrm{a}}_{\mathrm{d},\mathrm{k}}\left(\mathrm{t}+1\right)⌋ \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\right.$$

(15)

where $⌊*⌋$ represents the floor value of the given real number.

It is important to clarify that the arithmetic operations on vectors ${\mathrm{H}}_{\mathrm{k}}$ and ${\mathrm{A}}_{\mathrm{k}}$ in Equations (12)–(15) are symbolic–numerical hybrid operations, where the integer values (representing discrete priority rules or activity counts) are treated as numerical proxies for their symbolic meanings during evolution, while retaining their interpretability as rule indices.

The subsequent conversion of real values back to integers is validated by two key considerations: (1) The normal distribution of η limits large deviations from original integer values, preserving the symbolic relevance of the resulting indices/counts; (2) The clamping operations ensure the transformed values remain semantically valid. This hybrid approach balances the flexibility of continuous optimization with the interpretability of discrete symbolic representations, a trade-off widely adopted in evolutionary algorithms for combinatorial problems with discrete decision variables.

The vector ${\mathrm{A}}_{\mathrm{k}}\left(\mathrm{t}+1\right)$ obtained from Equations (12)–(15) might violate the constraint $\sum _{\mathrm{d}=1}^{\mathrm{D}}{\mathrm{a}}_{\mathrm{d},\mathrm{k}}=\mathrm{J}$. Given $\mathsf{\theta }=\mathrm{J}-\sum _{\mathrm{d}=1}^{\mathrm{D}}{\mathrm{a}}_{\mathrm{d},\mathrm{k}}$, if $\mathsf{\theta }>0$, $\mathsf{\theta }$ elements in ${\mathrm{A}}_{\mathrm{k}}\left(\mathrm{t}+1\right)$ with values below $\mathrm{J}$ are chosen and incremented by 1. Instead, if $\mathsf{\theta }<0$, $\left|\mathsf{\theta }\right|$ elements greater than zero are selected and reduced by 1. An example of a crossover operator is visualized in Figure 5.

• Mutation

The mutation operator tweaks individuals in the current population by introducing minor, random alterations. This shot in the arm of fresh genetic material spices things up, boosting the population’s diversity and sharpening the algorithm’s global search prowess. In this study, we are rolling with the swap mutation (SM) operator, which randomly picks two spots and then flips the genes residing there. An example of SM is visualized in Figure 6.

In detail, the frame and flow path for the proposed Strategy-Group Evolution algorithm is summarized in Figure 7, and the procedure is as follows.

(1) Initialize the strategy-group population by the algorithm in Algorithm 1.

(2) Run into the inner layer.

(2.1) For every given individual of the population, obtain its SG matrix.

(2.2) Generate the priority-rule sequence according to the SG matrix and the 12 rules in the strategy pool, as shown in Figure 4.

(2.3) According to the priority-rule sequence, select the activities in the sequence, thus obtaining the activity planning sequence.

(2.4) Generate the planning result by applying fuzzy durations under the resource constraint.

(2.5) Evaluate the total duration and the operation cost fitness for every planning result.

(3) Run back to the outer layer.

(3.1) Judge whether the maximum evolution time is reached. If no, go to (3.2); else, go to (3.5).

(3.2) Select parents based on the Pareto fitness of the planning results.

(3.3) Perform crossover and mutation on the strategy-group matrix.

(3.4) Generate a new offspring population, and run back to the inner layer of step (2).

(3.5) Output the final evolved Pareto-optimal solutions and end the algorithm.

5. Verification Tests

This section details quantitative evaluations assessing the efficacy of the proposed Strategy-Group Evolution algorithm. All experimental verification programs were run on a PC with a 3.00 GHz core and 16 G of running memory.

5.1. Data Setup

Lacking a standard problem set for the M-F-MORCPSP, we have built our own using the J30, J60, and J90 benchmark problems from the PSPLIB as a foundation, tweaking them by incorporating fuzzy activity durations. As outlined in

Table 4. Information about basic utilized datasets.

Set Number	Number of Instances	Number of Tasks per Instance	Number of Resources
J30	50	30	4
J60	50	60	4
J90	50	90	4

, the basic RCPSP datasets include the following information: the number of instances (“Ins”), the number of tasks per instance (“Acts”), and the number of resources (“Res”).

We are operating under the setting that all activity durations are fuzzy normal variables, and our tests use their expectations, $\overline{{\mathsf{\mu }}_{{\mathrm{d}}_{\mathrm{i}}}}=\left({\mathrm{d}}_{\mathrm{i},1},{ \mathrm{d}}_{\mathrm{i},2},{ \mathrm{d}}_{\mathrm{i},3}\right)$. The duration of each task $\mathrm{i}$ is set via the following:

(1) Set an original ${\mathrm{d}}_{\mathrm{i},2}$.

(2) ${\mathrm{d}}_{\mathrm{i},1}={\mathrm{d}}_{\mathrm{i},2}-\mathsf{\alpha }$, and $\mathsf{\alpha }=$ Uniform $\left[0, \mathrm{d}\right]$.

(3) ${\mathrm{d}}_{\mathrm{i},3}={\mathrm{d}}_{\mathrm{i},2}+\mathsf{\beta }$, and $\mathsf{\beta }=$ Uniform $\left[0, \mathrm{d}\right]$.

(4) Dummy activity start/end values are projected as (0, 0, 0).

Finally, using established benchmark datasets, a new dataset incorporating activity-specific operational costs was developed. This dataset explicitly accounts for the relationship between these costs and the properties of the corresponding triangular fuzzy numbers ${\mathrm{c}}_{\mathrm{i}}$.

$${\mathrm{c}}_{\mathrm{i}}=\mathrm{Uniform}\left(\overline{\mathrm{c}}-\mathsf{\alpha },\overline{\mathrm{c}}+\mathsf{\alpha }\right)+\mathsf{\beta }·\mathrm{Unform}\left({\mathrm{d}}_{\mathrm{i},1},{ \mathrm{d}}_{\mathrm{i},3}\right)$$

(16)

5.2. Performance Metrics

The M-F-MORCPSP studied in this paper has two optimization objectives: the timeline and the total cost. Three metrics assessed algorithmic efficacy in addressing this multi-objective challenge.

• Number of solutions $\left|{\mathrm{S}}_{\mathrm{j}}\right|$. An excellent algorithm must avoid a local optimum solution and search for possible solutions as much as possible. Therefore, we hope the larger this indicator is, the better.
• Ratio of non-dominated solutions ${\mathrm{R}}_{\mathrm{N}}\left({\mathrm{S}}_{\mathrm{j}}\right)$, which is calculated as:

$${\mathrm{R}}_{\mathrm{N}}\left({\mathrm{S}}_{\mathrm{j}}\right)={\displaystyle \frac{\left|{\mathrm{S}}_{\mathrm{j}}-\{\mathrm{x}\in {\mathrm{S}}_{\mathrm{j}}|\exists \mathrm{r}\in {\mathrm{S}}^{*}:\mathrm{r}\prec \mathrm{x}\}\right|}{\left|{\mathrm{S}}_{\mathrm{j}}\right|}}$$

(17)

Above, ${\mathrm{S}}^{*}$ means the optimal Pareto solution set. Since we prefer non-dominated results in real application, this indicator is also better the larger it is.

• Average distance $\mathrm{AD}\left({\mathrm{S}}_{\mathrm{j}}\right)$, which is calculated as:

$$\mathrm{AD}\left({\mathrm{S}}_{\mathrm{j}}\right)=\frac{1}{\left|{\mathrm{S}}_{\mathrm{j}}\right|}{\displaystyle \sum }_{\mathrm{r}\in {\mathrm{S}}^{*}}\min \{{\mathrm{d}}_{\mathrm{rx}}|\mathrm{x}\in {\mathrm{S}}_{\mathrm{j}}\}$$

(18)

where ${\mathrm{d}}_{\mathrm{rx}}$ is computed as:

$${\mathrm{d}}_{\mathrm{rx}}=‖\left(\begin{array}{c}{\mathrm{f}}_1\left(\mathrm{r}\right)\\ {\mathrm{f}}_2\left(\mathrm{r}\right)\end{array}\right)-\left(\begin{array}{c}{\mathrm{f}}_1\left(\mathrm{x}\right)\\ {\mathrm{f}}_2\left(\mathrm{x}\right)\end{array}\right)‖$$

(19)

This indicator quantifies convergence quality by computing the mean Euclidean among each point inside the solution set and the nearest point on the reference Pareto-optimal front. Thus, a smaller GD corresponds to superior convergence performance.

To evaluate solution quality, we extended the model by incorporating two objectives—timeline and cost—using established benchmark datasets (J30, J60, and J90). Since the true Pareto-optimal set is unknown a priori, a reference set was constructed through the following procedure: all algorithms were executed using their optimally tuned parameters and run with a sufficiently extended computation time. The resulting non-dominated solutions from all runs were combined to form a non-dominated set, which serves as the reference Pareto front.

5.3. Testing Results

In the following, the proposed Strategy-Group Evolution algorithm is put to the test to see how well it performs. To appraise the advantages of the proposed Strategy-Group Evolution algorithm, the outcomes are taken in contrast with some of more established players in the multi-objective optimization game. Specifically, we are going head-to-head with SPEA2, NSGA-II, and MOPSO. To keep things fair, all algorithms were given a budget of 5000 solution evaluations, and we also made sure to appropriately adjust the parameters of each approach to their optimal settings. The parameter settings for each algorithm are presented in

Table 5. Parameter settings of different algorithms.

Algorithm	Strategy-Group Evolution	SPEA2	NSGA-II	MOPSO
Parameter Settings	crossover probability: 0.8 mutation probability: 0.2 population size: 30 max iterations: 150	crossover probability: 0.8 mutation probability: 0.2 population size: 30 max iterations: 150	crossover probability: 0.8 mutation probability: 0.2 population size: 30 max iterations: 150	inertia weight: 0.5 cognitive constant: 1 social constant: 2 n.g.r.i.d.: 30 max iterations: 150

. Each instance was run 50 times and the mean value of the three evaluation indicators were recorded, as shown in Figure 8, Figure 9 and Figure 10.

As expressed in Figure 8, Figure 9 and Figure 10, the best outcomes of $\left|{\mathrm{S}}_{\mathrm{j}}\right|$, ${\mathrm{R}}_{\mathrm{N}}\left({\mathrm{S}}_{\mathrm{j}}\right)$, and $\mathrm{AD}\left({\mathrm{S}}_{\mathrm{j}}\right)$ are the results of the proposed Strategy-Group Evolution algorithm. In terms of $\left|{\mathrm{S}}_{\mathrm{j}}\right|$, during the evolution process, the proposed Strategy-Group Evolution algorithm has a slight disadvantage at 500 evaluated solutions (e.g., Figure 8a and Figure 9a). Further from Figure 10a, the Strategy-Group Evolution algorithm is able to maintain a better optimization as the problem size increases. In terms of ${\mathrm{R}}_{\mathrm{N}}\left({\mathrm{S}}_{\mathrm{j}}\right)$, similar to $\left|{\mathrm{S}}_{\mathrm{j}}\right|$, the Strategy-Group Evolution algorithm shows obvious advantages after 2000 evaluated solutions. In terms of $\mathrm{AD}\left({\mathrm{S}}_{\mathrm{j}}\right)$, the proposed Strategy-Group Evolution algorithm demonstrates increasing efficacy as the evolutionary process advances.

Additionally, as the Strategy-Group Evolution algorithm generates scheduling schemes by traversing nearly all commonly used scheduling strategies, it also exhibits the highest computational load. It is necessary to determine whether the advantages of the obtained results outweigh the reduction in computational efficiency. To align with practical modular shipbuilding projects, the experimental group with the largest data volume was selected for analysis. The average computation time of the algorithm per run on the J90 instance (with each instance repeated 50 times) is presented in

Table 6. Average computation time per run of different algorithms.

Algorithm	Strategy-Group Evolution	SPEA2	NSGA-II	MOPSO
Average computation time per run	12.26 s	10.14 s	9.33 s	11.41 s

.

Analysis of the data in Table 6 reveals that the average computation time per run of different algorithms typically ranges from a few seconds to 12 s. Comparative tests show that the Strategy-Group Evolution algorithm has an average optimality gap of only 1.8%. Combined with Figure 10 and Table 6, when handling large-scale instances, compared with the mainstream algorithms SPEA2, NSGA-II, and MOPSO, the Strategy-Group Evolution algorithm achieves a 28% higher average computational benefit, while its average computational efficiency is only 14% lower. Although the Strategy-Group Evolution algorithm takes the longest time, the gap between it and other algorithms can be fully offset by the higher benefits of its results. Moreover, in practical engineering scenarios, although the gap in computational efficiency will be amplified, it still remains within an acceptable range.

In summary, a rigorous evaluation of the experimental outcomes confirms its superior optimization capabilities over established benchmark algorithms.

6. Real-Case Application

To conduct a rigorous assessment of the introduced Strategy-Group Evolution framework’s practical applicability for addressing real-world modular shipbuilding project challenges, a project example of a cruise ship manufacturing company is used for validation. In this section, the production, transport, embarkation and installation activities of prefabricated cabins, an important type of intermediate product in cruise ship construction projects, are selected as examples for validation. In an actual modular shipbuilding project, prefabricated cabins are often planned as a turn-key sub-project, requiring the shipyard to plan specific production activities. The logistical relationship between the production activities and the detailed activity information are shown in Figure 11 and

Table 7. Production activities of prefabricated cabin in the modular shipbuilding project.

Activity Node	Activity Name	Labor Resource	Equipment Resource	Fuzzy Duration
0	Start node (dummy)	-	-	-
1	#1 Wet unit positioning and installation	17	20	(1, 2, 4)
2	#1 Cabin prefabricate	9	8	(1, 2, 3)
3	#1 Cabin storage and transportation	21	15	(3, 5, 8)
4	#1 Cabin embarkation and pushing	12	5	(2, 4, 7)
5	#1 Cabin installation and commission	12	18	(2, 6, 9)
6	#2 Wet unit positioning and installation	17	20	(1, 2, 4)
7	#2 Cabin prefabricate	9	8	(1, 2, 3)
8	#1 Cabin storage and transportation	21	15	(3, 5, 8)
9	#2 Cabin embarkation and pushing	12	5	(2, 4, 7)
10	#2 Cabin installation and commission	12	18	(2, 6, 9)
11	#3 Wet unit positioning and installation	17	20	(1, 2, 4)
12	#3 Cabin prefabricate	9	8	(1, 2, 3)
13	#3 Cabin storage and transportation	21	15	(3, 5, 8)
14	#3 Cabin embarkation and pushing	12	5	(2, 4, 7)
15	#3 Cabin installation and commission	12	18	(2, 6, 9)
16	Finish node (dummy)	-	-	-

, respectively.

Notably, all data supporting these figures and tables are derived from in-depth field research at a large domestic cruise ship manufacturing enterprise. After conducting research and performing cleaning and processing on the abnormal data, the resulting data is presented in Table 7. This indicates that the experimental data is highly consistent with the actual project operation data.

Due to the need to protect enterprise-confidential information, this paper represents both labor resources and processing equipment resources using single numerical values. Specifically, the labor resource requirement for a certain activity can be understood as the number of operators needed, while the processing equipment resource requirement refers to the number of machines to be used. Fuzzy time is expressed using triangular fuzzy numbers, also in numerical form. This conversion method leverages systems such as working hour forecasting and expert estimation employed within the enterprise.

In this case, the upper limit of labor resources invested per day in the production of prefabricated chambers is 26, and the upper limit of equipment resources is 33. The unit cost of production activities follow a uniform distribution of $\left[120, 150\right]$. The limit of the total operational cost (i.e., non-renewable resource) is 140,000. The Strategy-Group Evolution algorithm proposed in this paper is applied. The population size number is 30. The crossover probability number is 0.8. The mutation probability number is 0.1. The maximum iterations number is $200$. The Pareto solution set solved by the Strategy-Group Evolution algorithm is shown in

Table 8. The Pareto solution set of the testing case.

Scheme No.	Fuzzy Duration	Operational Cost
1	(21, 42, 60)	184,680
2	(22, 48, 68)	152,760
3	(28, 51, 69)	136,460
4	(30, 62, 71)	129,160
5	(32, 65, 72)	122,340

and Figure 12.

Based on the results in Table 8 and Figure 12, it is indicated that each solution in the Pareto solution set does not dominate any other. Compared with other scheduling schemes, in the optimal scheduling scheme obtained by this method, the fuzzy makespan is shortened by an average of 16.2%, and the operational cost is reduced by an average of 8.3%. In addition, a reduction in the total fuzzy duration of the production activity represents more resources invested, leading to an increase in total operational cost, and the application of the most efficient scheduling scheme can lead to cost exceeding the limit. In actual modular shipbuilding projects, prefabricated cabin production managers in a shipyard usually expect to produce as efficiently as possible while maintaining cost input constraints. For balancing the efficiency and cost, scheme 3 can be selected as the final schedule. For saving cost, scheme 5 can be selected as the production implementation scheme for production. Figure 13 and Figure 14 illustrate two activities plans for implementing activities that meet the resources limitations (schemes 3 and 5), respectively.

7. Conclusions

This study focuses on solving the project planning problem in modular shipbuilding, a domain characterized by high uncertainty and multi-objective trade-offs. To address this challenge, we propose a novel multi-objective optimization model that advances conventional resource-constrained project scheduling formulations in two key ways:

• It explicitly models uncertain activity durations using triangular fuzzy numbers, capturing the ambiguity of manual operations and dynamic resource allocation in modular shipbuilding;
• It formulates a bi-objective optimization framework that simultaneously minimizes fuzzy makespan and total operational cost, aligning with real-world project management needs.

To solve the proposed model, we developed a Strategy-Group Evolution algorithm with a dual-layer structure: the inner layer embeds 12 heuristic scheduling rules tailored to the modular shipbuilding processes, while the outer layer optimizes strategy selection and assignment via an evolutionary policy. This design enhances the algorithm’s ability to balance exploration.

Validation is conducted through two complementary experiments:

• Grouped tests based on the PSPLIB benchmark instances show that, when handling large-scale instances, compared with the mainstream algorithms SPEA2, NSGA-II, and MOPSO, the Strategy-Group Evolution algorithm achieves a 28% higher average computational benefit, while its average computational efficiency is only 14% lower—verifying the superiority of its algorithmic performance;
• A real-world case study from a cruise ship prefabricated cabin project demonstrates that, compared to the shipyard’s existing manual scheduling method, the algorithm reduces the project’s fuzzy makespan by 16.2% and operational cost by 8.3%, confirming its practical viability.

Future work will focus on two directions:

• Incorporating dynamic events into the model to enhance its adaptability to real-time project disruptions;
• Extending the Strategy-Group Evolution algorithm’s strategy pool to include multi-site coordination rules, addressing the cross-yard transportation and assembly challenges in large-scale modular shipbuilding.