Bi-Objective Optimization with Mode-Oriented Genetic Algorithm for Multi-Mode Resource-Constrained Project Scheduling

Abstract

To address the time–cost trade-off challenge in real-world practices, a bi-objective optimization model of the Multi-mode Resource-Constrained Project Scheduling Problem is proposed with simultaneously minimizing both the project makespan and the resource cost. A mode-oriented Non-dominated Sorting Genetic Algorithm II is developed to solve the formulated problem. Two key improvements are introduced: a mode-repair mechanism is incorporated during the initialization phase to generate feasible execution modes, thereby improving the quality of initial solutions and accelerating search efficiency, and four neighborhood structures based on mode and task execution lists are designed for local search, enabling fine-grained solution refinement in each iteration. Extensive experimental studies are conducted to verify the effectiveness of the proposed strategies, and comparative evaluations with state-of-the-art algorithms demonstrate that MNSGA-II achieves superior performance across multiple metrics, including lower mean ideal distance, better solution quality, improved diversity, and more uniform distribution of Pareto-optimal solutions.

1. Introduction

Presently, enterprises pursue greater product customization, shorter life cycles, and advanced technological development while managing multiple complex projects to survive in highly competitive environments. Project scheduling faces significant challenges due to high investment costs, extended implementation periods, and resource contention, making corporate sustainability difficult in cost-intensive environments. Therefore, meticulous planning of project tasks within resource constraints is essential for developing efficient schedules. Optimizing the trade-off between project makespan and resource cost is critical to maximizing overall benefits.

Effective project scheduling optimizes resource use through comprehensive resource allocation and utilization planning. The Resource-Constrained Project Scheduling Problem (RCPSP) represents a fundamental challenge in this field. This NP-hard problem minimizes makespan considering task precedence relationships and resource capacity constraints, with applications in shipbuilding, aircraft, and automotive manufacturing. Its assumptions of the RCPSP model prove insufficient for many practical scenarios. Consequently, numerous extensions have emerged, notably the Multi-Mode Resource-Constrained Project Scheduling Problem (MRCPSP). In MRCPSP, tasks feature multiple execution modes with different duration-resource combinations under completion-start precedence constraints among tasks. This requires selecting optimal modes and scheduling task sequences to enhance resource utilization while reducing completion time. Both renewable (e.g., personnel, equipment) and non-renewable resources (e.g., budgets, energy) are constrained, with finite total availability. The standard RCPSP becomes a special case of MRCPSP, retaining only single modes and renewable constraints. MRCPSP applications include construction and automotive development. Critically, real-world implementations reveal a time–cost trade-off: shorter durations require increased resource investment, elevating costs. This confirms MRCPSP’s NP-hard complexity, making metaheuristics essential for solutions.

This study focuses on developing a systematic optimization framework for complex project scheduling scenarios characterized by multiple execution modes and constrained resources. By establishing a bi-objective optimization approach, we aim to provide decision-makers with a practical methodology to balance project duration and resource expenditure. The proposed solution integrates evolutionary optimization principles to handle the inherent complexity of real-world resource allocation problems. Overall, the main contributions are threefold:

• A novel bi-objective optimization model for MRCPSP simultaneously addressing makespan minimization and resource–cost optimization under renewable/non-renewable constraints is proposed.
• A mode-oriented Non-dominated Sorting Genetic Algorithm II, featuring (a) a mode-repair mechanism ensuring feasible initial solutions, and (b) four novel neighborhood structures for granular solution refinement, is designed. Unlike prior repair strategies that enforce feasibility through backtracking, our mechanism directly adjusts modes via greedy resource reallocation, reducing repair complexity. For the proposed NS1-NS4, the problem-specified knowledge is incorporated, which greatly enhances the performance of the general algorithm framework.
• Comprehensive experimental validation against benchmark datasets and state-of-the-art algorithms, demonstrating significant performance improvements in industrial-relevant scenarios.

The rest of the paper is organized as follows. Section 2 critically reviews the existing literature on MRCPSP. Section 3 details the formulation of the bi-objective MRCPSP model. Section 4 extensively elaborates the proposed MNSGA-II algorithm. Section 5 presents comprehensive experimental evaluations of MNSGA-II. Section 6 provides concluding remarks and future research directions.

2. Literature Review

In the past few decades, RCPSP and its extensions have been extensively studied [1]. This section focuses on MRCPSP, which is significantly more complex than single-mode RCPSP. Many researchers have proposed exact and heuristic algorithms to solve MRCPSP.

Early research on MRCPSP primarily focused on developing exact solution methods to guarantee optimality for small-scale instances. Kolisch and Drexl [2] proposed a local search method that first finds a feasible solution and then performs single neighborhood search on mode assignments. Sprecher et al. [3] presented a to solve MRCPSP. Heilmann [4] proposed another branch-and-bound algorithm to solve MRCPSP with time lags, aiming to minimize project duration while satisfying all constraints. Fernandes et al. [5] proposed a mathematical strategy based on [6] and developed a Branch-and-Cut algorithm, incorporating pseudo-linear inequalities into the Mixed Integer Programming to ensure local optimality. Schnell et al. [7] integrated Constraint Programming and Boolean Satisfiability Solving into the Branch-and-Bound algorithm. Zhu et al. [8] devised the Branch-and-Cut algorithm based on Integer Linear Programming formulations. Despite their theoretical rigor, these exact methods face significant computational challenges when the problem size increases.

Exact methods for MRCPSP struggle with problems exceeding 20 activities [3]. Therefore, heuristic algorithms have been used by many researchers. Alcaraz et al. [9] developed a genetic algorithm that encodes both activity sequences and mode assignments simultaneously. Jarboui et al. [10] proposed particle swarm optimization with adaptive velocity updating mechanisms. Coelho et al. [11] decomposed the problem through mode allocation followed by heuristic scheduling, demonstrating the effectiveness of hierarchical solution approaches. Maghsoudlou et al. [12] developed a multi-objective invasive weeds optimization with a novel chromosome structure. Chakrabortty et al. [13] employed a variable neighborhood search heuristic incorporating multiple neighborhood operators to escape local optima. Ranjbar et al. [14] constructed a fuzzy mathematical model to maximize financial metrics using the enhanced epsilon-constraint method.

Recent advances in multi-objective optimization have brought new perspectives to MRCPSP research. Most recently, Afrasyabi et al. [15] proposed a multi-objective discrete particle swarm optimization model for solving multi-modal routing problems, demonstrating the potential of swarm intelligence in handling conflicting objectives. Akopov [16] presents an improved parallel bi-objective hybrid real-coded genetic algorithm that combines the strengths of genetic algorithms with local search techniques to enhance convergence speed and solution diversity. However, these studies either focus on different problem domains or do not specifically address the mode selection challenge inherent in MRCPSP, leaving room for specialized algorithmic designs.

Despite the rich body of literature, several critical limitations persist in current MRCPSP research. From the above literature, two key research gaps are revealed: (1) Existing MRCPSP research predominantly optimizes makespan, overlooking the fundamental time–cost trade-off in real-life scenarios, as accelerating tasks inevitably increases resource consumption and costs. This single-objective focus fails to capture the practical decision-making dilemma faced by project managers who must balance schedule pressure against budget constraints. (2) While numerous heuristics exist for single-objective MRCPSP, few effectively address the bi-objective challenge where conflict between makespan and cost must be balanced. Specifically, the mode selection mechanism—a critical component that directly influences both objectives—has not been adequately integrated into multi-objective evolutionary frameworks. Therefore, this study explicitly addresses both makespan minimization and total resource–cost optimization, proposing a mode-oriented genetic algorithm with specialized repair mechanisms and neighborhood structures tailored for bi-objective MRCPSP.

3. Problem Description and Mathematical Formulation

In this section, the bi-objective MRCPSP that involves a trade-off between time and resource cost is introduced, and its mathematical model is formulated. The main objective of this problem is to determine the optimal execution mode for each task and to establish the start times of tasks, thereby minimizing the total resource cost and project makespan. In the rest of this paper, the sets and parameters used are listed in

Table 1. Notations of the proposed model.

Parameters	Description
I	Set of tasks, $I=\{0, 1, \dots , i, \dots , I\}$
q	Time period index for resource consumption calculation, $q=max\{t,E{T}_j\}, \dots ,min\{t+d_{im}-1,L{T}_j\}$
$M_i$	Set of execution modes of task i, $M_i=\{1, 2, \dots , m, \dots , M_i\}$
K	Set of renewable resources, $K=\{1, \dots , k, \dots , K\}$
N	Set of non-renewable resources, $N=\{1, \dots , n, \dots , N\}$
$P_i$	Set of immediate predecessor tasks for task i
$T_d$	The deadline of the project
t	Time period, $t=1, \dots , T_d$
$E{T}_i$	The earliest start time of task i
$L{T}_i$	The latest start time of task i
$r_{imk}^{\gamma }$	Amount of the k-th renewable resource consumed by task i in mode m
$r_{imn}^u$	Amount of the n-th non-renewable resource consumed by task i in mode m
$R_{tk}^{\gamma }$	Maximum availability of the k-th renewable resource at each time period t
$R_n^u$	Maximum consumption limit of the n-th non-renewable resource
$c_k^{\gamma }$	Unit cost of using the renewable resource k
$c_n^u$	Unit cost of using the non-renewable resource n
$d_{im}$	Duration of task i in mode m
$x_{imt}$	If task i executed by mode m in period t, $x_{imt}=1$; otherwise, $x_{imt}=0$
$S{T}_i$	Start time of task i

.

The project consists of multiple tasks $i\in I$, following precedence relationships where task i cannot start until all its immediate predecessor tasks $j\in P_i$ are completed. Each task i can be executed in mode $m\in M_i$, where mode m has a duration of $d_{im}$, requires $r_{imk}^{\gamma }$ units of renewable resource k, and $r_{imn}^u$ units of non-renewable resource n. The first and last tasks are dummy tasks that represent the start and end of the project and do not consume time or resources. Since the resource amounts, time, and cost are different for each mode, the optimal mode is selected based on the task requirements and resource combinations to determine the task schedule and minimize resource consumption costs. A feasible task schedule must always satisfy the following constraints: (1) The total consumption of non-renewable resources for all tasks cannot exceed their predefined total limits $R_n^u$; (2) The total consumption of renewable resources for each task in time period t cannot exceed their predefined capacities $R_{tk}^{\gamma }$; (3) Each task must complete its immediate predecessor tasks before it can start.

The problem assumptions are as follows, without violation of the practical scenarios:

• Tasks require both renewable and non-renewable resources.
• All parameters are deterministic.
• All tasks follow finish-to-start precedence relationships.
• All resources are available at the start of the project.
• Each task is executed in one mode.
• Tasks cannot be preempted.
• The first and last tasks are dummy tasks representing the start and end of the project, consuming neither time nor resources.

The mathematical formulations for the bi-objective MRCPSP, based on Talbot [17], are given by:

$$\begin{array}{cc}\hfill minC& =\sum _{k=1}^K\sum _{i=1}^I\sum _{m=1}^{M_i}\sum _{q=max\{t,E{T}_j\}}^{min\{t+d_{im}-1,L{T}_j\}}{c}_{ik}^{\gamma }{r}_{imk}^{\gamma }{x}_{imq}\hfill \\ & +\sum _{n=1}^N\sum _{i=1}^I\sum _{m=1}^{M_i}\sum _{t=E{T}_i}^{L{T}_i}{c}_n^u{r}_{imn}^u{x}_{imt}\hfill \end{array}$$

(1)

$$minS{T}_i=\sum _{t=E{T}_i}^{L{T}_i}\sum _{m=1}^{M_i}t{x}_{imt}$$

(2)

s.t.

$$\sum _{t=E{T}_i}^{L{T}_i}\sum _{m=1}^{M_i}{x}_{imt}=1, i=1, 2, \dots , I$$

(3)

$$\sum _{m=1}^{M_j}\sum _{t=E{T}_j}^{L{T}_j}t{x}_{jmt}\le \sum _{m=1}^{M_i}\sum _{t=E{T}_i}^{L{T}_i}(t-d_{im})x_{imt}, i=2, 3, \dots , I, j\in P_i$$

(4)

$$\sum _{i=1}^I\sum _{m=1}^{M_i}\sum _{q=max\{t,E{T}_i\}}^{min\{t+d_{im}-1,L{T}_i\}}{r}_{imk}^{\gamma }{x}_{imq}\le R_{tk}^{\gamma }, k=1, \dots , K$$

(5)

$$\sum _{i=1}^I\sum _{m=1}^{M_i}\sum _{t=E{T}_i}^{L{T}_i}{r}_{imn}^u{x}_{imt}\le R_n^u, n=1, \dots , N$$

(6)

$$x_{imt}\in \{0,1\}, t=E{F}_i, \dots , L{F}_i, i=1, \dots , I, m=1, \dots , M_i$$

(7)

Equation (1) represents the minimization of resource costs, involving both renewable and non-renewable resource costs. Equation (2) demonstrates the minimization of the project makespan. Constraint (3) indicates that each task can only be completed in one execution mode. Constraint (4) signifies that task i can only start after its immediate predecessor tasks are completed. Constraint (5) stipulates that the amount of renewable resources in time period t cannot exceed the capacity for that period. Constraint (6) specifies that the total consumption of non-renewable resources throughout the project cannot exceed the total input. Constraint (7) defines the range of variables.

4. The MNSGA-II for the Bi-Objective MRCPSP

NSGA-II is a classic, widely applicable, and efficient multi-objective optimization algorithm. Due to its superior performance in solving scheduling problems, NSGA-II has been particularly favored by researchers. However, the MRCPSP addressed in this study is an extremely complex discrete combinatorial optimization problem. NSGA-II exhibits significant randomness in its operations, lacks structural information about the solution space, and has low search efficiency. This inefficiency becomes particularly pronounced as the solution space expands, leading to poorer solution quality. Therefore, this study proposes an improved Non-dominated Sorting Genetic Algorithm (MNSGA-II) to deal with it. During the initialization phase, a mode-repair mechanism is designed to produce higher-quality initial solutions and improve the search efficiency. Subsequently, neighborhood structures of the mode and task lists are constructed to explore a broader solution space in the local search, allowing fine-grained adjustments in each iteration to find better solutions and enhance local search ability. The overall algorithmic procedure is shown in Figure 1.

As NSGA-II has become a classic multi-objective optimization algorithm, its effectiveness in solving scheduling problems has been widely recognized. However, when applied to the MRCPSP, a notoriously complex discrete combinatorial optimization problem, the standard NSGA-II reveals three critical limitations: (1) unable to guarantee the feasibility of solutions, (2) insufficient utilization of solution space structure, and (3) degraded performance with increasing problem scale.

To overcome these challenges, this study proposes a mode-oriented Non-dominated Sorting Genetic Algorithm II (MNSGA-II) with two key innovations: (1) A novel mode-repair mechanism during initialization that ensures feasible, high-quality starting solutions, and (2) Specially designed neighborhood structures for both mode and task lists that enable comprehensive local search. The procedure of the MNSGA-II is shown in Figure 1 and the details are depicted in the following subsections.

4.1. Initial Solutions Generation

The encoding of MNSGA-II solutions employs real-value coding and primarily includes the task execution list S and execution mode list M. The initial solution is randomly generated. In the task execution order S, the first and last elements accommodate dummy tasks, while the remaining tasks are randomly inserted into S. In the execution mode list M, each element represents a specified mode selected from a set of modes for the corresponding task.

However, most randomly generated M and S fail to satisfy the resource and precedence constraints. Therefore, a repair mechanism is established within M: modes are assessed to ensure compliance with the non-renewable resource constraints; if the modes do not comply, they are substituted until compliance is achieved. Once the feasible mode list is determined, the resource use cost is calculated, and the Serial Schedule Generation Scheme (SSGS) is applied to schedule each task in S. SSGS assigns the earliest possible start time to tasks while adhering to the constraints of task precedence and renewable resource consumption. The process of solution initialization is depicted in Algorithm 1.

Let s represent the initial feasible solution. $L\left(s\right)=(LM,LJ,ST,RC)$ represents the encoding of s, where $LM$ lists feasible task modes, $LJ$ denotes the feasible task list, $ST$ indicates the corresponding start time instants for the tasks, and $RC$ denotes the resource use cost. Figure 2 shows an example of s.

Algorithm 1 The main steps of initial solution generation

Require:  P: matrix of precedence constraints; $R_{num}$: resource demands; $R_{limit}$: resource capacity constraints; $T_{duration}$: task duration; $C_{re}$: unit cost of resources Ensure:  $LM$: feasible task modes; $LJ$: feasible task execution order schedule; $ST$: start time of tasks; $RC$: resource use cost 1: Task execution order S and the corresponding execution mode M are randomly generated. 2: $flag\leftarrow$JudgeResource(M, $R_{num}$, $R_{limit }$) 3: if$flag=1$then 4:     $LM\leftarrow$RepairMode(M, $R_{num}$, $R_{limit}$) 5: else 6:     $LM\leftarrow M$ 7: end if 8: $RC\leftarrow$ResourceCost($LM$, $R_{num}$, $C_{re}$) 9: for$i=2$ to $\left|S\right|$ do 10:     $S_{unsched}\leftarrow \mathtt{Successor}\left(P(i-1)\right)$ 11:     for $j=1$ to $|S_{unsched}|$ do 12:         $b\leftarrow \mathtt{Schedule}\left(S_{unsched}\left(j\right)\right)$ 13:         $S_{pre}\left(b\right)\leftarrow \mathtt{Prede}\left(b\right)$ 14:         $MS\leftarrow \mathtt{MinStartime}(S_{pre}\left(b\right),T_{duration}\left(b\right))$ 15:         $T\left(b\right)\leftarrow \mathtt{JudgeMinStartime}(b,LM,R_{num},R_{limit})$ 16:         $ST(:,T(b\left)\right)\leftarrow T\left(b\right)$ 17:         $LJ(:,b)\leftarrow b$ 18:         $S_{unsched}\leftarrow S_{unsched}(b=\varphi )$ 19:     end for 20: end for

Notably, the proposed mode-repair mechanism differs fundamentally from Sprecher’s classical approach [3] in complexity. Sprecher’s method exhaustively validates all mode combinations for feasibility, requiring $O\left(M_i^2\right)$ checks per task. The proposed mode-repair method prioritizes non-renewable resource constraints first (Algorithm 1, lines 3–8), filtering infeasible modes via a single pass. Each mode is evaluated once, reducing complexity to $O\left(M_i\right)$ per task. For a project with I tasks and average $\overline{M}$ modes per task, total repair complexity decreases from $O(I·{\overline{M}}^2)$ to $O(I·\overline{M})$fold speedup. This reduction is critical for large-scale instances (e.g., J100), enabling faster initialization and more computational budget for evolutionary search.

4.2. Crossover and Mutation Operator

To improve solution diversity and prevent solutions from becoming stuck in local optima, crossover and mutation operations are performed. Position-based crossover is applied to $LJ$, with the crossover process illustrated in Figure 3. The main steps are as follows:

Step 1. Two solutions are randomly selected as parent solutions $s_{p1}$ and $s_{p2}$.

Step 2. Several tasks are randomly selected in $s_{p1}$, which can be non-continuous, and the selected tasks are copied to $s_{o1}$, retaining the same positions as in $s_1$. The missing tasks in $s_{o1}$ are then taken from parent $s_{p2}$ and copied to $s_{o1}$ in order.

Step 3. Similar to Step 2, several tasks are randomly selected in $s_{p2}$ and copied to offspring $s_{o2}$, with the remaining tasks copied from $s_{p1}$ to generate offspring $s_{o1}$.

Mutation operations are applied to $LM$. First, one or more execution modes are randomly selected based on a given mutation rate $P_m$. A random number from a normal distribution with a standard deviation of $\mu$ is added to the selected execution modes to generate new modes, as shown in Figure 4. The valid range for the mutated values is 1 to $N_m$.

4.3. Non-Dominated Sorting

The solutions generated by the aforementioned operations are merged with the initial solutions to form a new solution set, which is then sorted and divided into multiple non-dominated fronts. The nondominant sorting process is shown in Figure 5. First, for each solution, a set is initialized to store all the solutions it dominates, and a counter is initialized to indicate the number of solutions that dominate it. Then, for each pair of solutions, the dominance relationship is determined, and the relevant sets and counters are updated accordingly. Next, the non-dominated fronts are created by adding all the solutions that are not dominated by any other solution to the first front and removing these solutions from the solution set. Subsequently, by constructing subsequent fronts, the counters for all solutions dominated by the current front’s solutions are decremented by one; if a counter becomes zero, the solution is added to the next front. This process is repeated until no new fronts can be constructed, ultimately outputting all the sets of fronts.

4.4. Neighborhood Search

Based on the characteristics of the problem being solved, two points can be inferred: first, changing the execution mode of any task will result in alterations to the project completion time, cost, and resource use; second, altering the task execution sequence will affect the project completion time. Therefore, four neighborhood structures are proposed based on the aforementioned problem characteristics.

(1) NS1: swap the positions of two tasks, as shown in Figure 6. In $LJ$, a task $i\in I$ is randomly selected, then its predecessor task $i_p$ and successor task $i_s$ are identified, defining the swapping interval $I^{\prime }=[i_p+1,i_s-1]$. Task i is swapped with each task $i^{\prime }\in I^{\prime }, i\ne i^{\prime }$, ensuring the resulting task order does not violate precedence constraints. If the completion time of the neighborhood solution is reduced, there is no need to swap the remaining activities in $I^{\prime }$. NS1 can generate $N_{r1}$ neighborhood solutions: $N_{r1}⩽I(I-1)/2$. An example of swapping two tasks is presented in Figure 6. In (a), task 2 and task 4 are swapped while maintaining the precedence constraints. Since task 6 is a successor of task 2, swapping tasks 2 and 6 violates the constraints, so it cannot be performed. In (b), the tasks are rescheduled after the swap while satisfying renewable resource constraints, resulting in a reduction of project makespan from 31 to 26.

(2) NS2: a task is selected and inserted into a different position. Similar to NS1, randomly select a task i in $LJ$ and identify its predecessor $i_p$ and successor $i_s$. The range within which task i can be moved forward and backward in the task list is determined. Then, i is inserted into a randomly chosen new position $i^{\prime }$ in the task list, ensuring $i\ne i^{\prime }$ and $i_p<i^{\prime }<i_s$. The number of neighborhood solutions for NS2 is $N_{r2}$: $N_{r2}⩽I(I-1)/2$.

(3) NS3: this neighborhood structure is to change the execution mode of a task, as shown in Figure 7. In $L\left(s\right)$, a task i is randomly selected, and its execution mode is changed. If this change results in reduced completion time and cost, while satisfying the constraint of non-renewable resources, other modes for task i do not need to be checked. This process continues until all tasks i are processed. The number of neighborhood solutions $N_{r3}$ for this NS3 depends on the number of tasks n and the number of modes $M_i$ for each task: $N_{r3}\le {\sum }_{i=1}^I(M_i-1)$. As illustrated in Figure 7, an example of changing a task’s execution mode is provided. In (a), Task 10’s execution mode is changed from mode 3 to mode 1, ensuring compliance with the non-renewable resource constraints. In (b), the tasks are rescheduled under renewable resource constraints, reducing the project’s makespan from 26 to 24.

(4) NS4: in $LM$, the execution modes of two tasks are changed. Similar to NS3, tasks $i_1$ and $i_2$ are randomly selected, where $i_1\ne i_2$, and their execution modes are modified. If the new mode list satisfies the non-renewable resource constraints and results in reduced completion time and cost, other modes of task $i_1$ and $i_2$ are not checked. This process continues until all tasks i are processed. The number of neighborhood solutions for NS4 is $N_{r4}$: $N_{r4}\le {\displaystyle \frac{1}{2}}{\sum }_{i_1=1}^I{\sum }_{i_1\ne i_2}^I(M_{i1}-1)(M_{i2}-1)$.

After non-dominated sorting, neighborhood search is conducted on the solutions in the first Pareto front, denoted as $s_{r1}$, using the previously designed neighborhood structures. The search process for $s_{r1}$ is as follows: a random number $r_s$ between 0 and 1 is generated. If $r_s$ is greater than a constant R, the neighborhood solution $n{s}_1$ of $s_{r1}$ is determined using NS1. Then, $s_{r1}$ and $n{s}_1$ are merged, and the merged solution is further searched using the neighborhood structure NS3. Conversely, if $r_s$ is less than the constant R, NS2 and NS4 are sequentially applied for searching. Finally, the first Pareto front is updated. The neighborhood search process is illustrated in Algorithm 2.

Algorithm 2 Neighborhood search process (Only neighborhood solutions that improve the current solution are accepted)

Require:  $s_{rank1}$: first Pareto front, R: a constant between 0 and 1 Ensure:  $s_{new}$: the updated Pareto front 1: if$r_s>R$then 2:     $n{s}_1\leftarrow$NeighborSearch1($s_{r1}$) 3:     $s_1\leftarrow [n{s}_1,s_{r1}]$ 4:     $s_2\leftarrow$NeighborSearch2($s_1$) 5: else 6:     $n{s}_3\leftarrow$NeighborSearch3($s_{r1}$) 7:     $s_1\leftarrow [n{s}_3,s_{r1}]$ 8:     $s_2\leftarrow$NeighborSearch4($s_1$) 9: end if 10: $s_{new}\leftarrow$NondominatedSort($s_2$)

5. Experimental Results and Analysis

In this section, comparative experiments are conducted to verify the effectiveness of the proposed MNSGA-II algorithm. The experimental data are based on the public library PSPLIB and MMLIB, which include seven datasets containing instances with 10–100 tasks (J10–J100). Six instances from each of the J10, J16, J20, J30, J50, and J100 datasets are selected for experiments. Since PSPLIB and MMLIB do not include resource cost and non-renewable resource data required for bi-objective optimization, these parameters are generated using uniformly distributed random numbers: mode-specific costs within [50, 200] units (inversely proportional to duration), and non-renewable resource consumption within [5, 20] units (proportional to renewable resource requirements in the original datasets).

To ensure fair comparison, the computational time limit for all algorithms is uniformly set based on problem scale, calculated as the product of the number of tasks and the number of resource types (in seconds), i.e., $T_{max}=I\times (K+N)$, where I represents the total number of tasks, K denotes the number of renewable resource types, and N denotes the number of non-renewable resource types. All algorithms within the same instance group are executed under identical time constraints. The simulation experiments focus on the following two aspects: (1) effectiveness analysis of the improvement strategies; (2) performance analysis by comparing the proposed algorithm with state-of-the-art multi-objective optimization algorithms.

5.1. Evaluation Metrics

In multi-objective optimization, the evaluation of optimization results is much more complex than in single-objective optimization. In this study, the performance of the algorithm is evaluated using four widely employed metrics, which are the quality metric (QM), mean ideal distance (MID), spacing metric (SM), and diversification metric (DM).

(1) Quality metric (QM): Primarily used to evaluate the quality of the algorithm’s output. The Pareto sets of two algorithms are compared, and solutions dominated by the other algorithm are removed. The cardinality of the remaining Pareto set is then divided by the cardinality of the original Pareto set.

(2) Diversification metric (DM): The diversity of the Pareto front is primarily evaluated. The DM can be calculated by Equation (8).

$$V_{\mathrm{DM}}=\sqrt{{({\displaystyle \frac{max\left\{f_{1g}\right\}-min\left\{f_{1g}\right\}}{f_{1,\mathrm{total}}^{max}-f_{1,\mathrm{total}}^{min}}})}^2+{({\displaystyle \frac{max\left\{f_{2g}\right\}-min\left\{f_{2g}\right\}}{f_{2,\mathrm{total}}^{max}-f_{2,\mathrm{total}}^{min}}})}^2}$$

(8)

(3) Hypervolume (HV): Measures the volume of the objective space dominated by the Pareto front relative to a reference point. It simultaneously evaluates both convergence quality and diversity of the solution set. The HV can be calculated by Equation (11).

$$V_{HV}=\mathrm{Volume}\left(\bigcup _{i=1}^G\{f\in {\mathbb{R}}^2|f_i\prec f\prec r\}\right)$$

(9)

where r represents the reference point, typically set as the nadir point of all algorithms’ solutions.

(4) Mean ideal distance (MID): Measures the relative distance between Pareto-optimal solutions and the ideal solution. The ideal solution is defined as the solution that optimizes all objectives individually. The MID can be calculated by Equation (10).

$$V_{\mathrm{MID}}={\displaystyle \frac{1}{G}}\sum _{g=1}^G\sqrt{{({\displaystyle \frac{f_{1g}-f_1^{\mathrm{best}}}{f_{1,\mathrm{total}}^{max}-f_{1,\mathrm{total}}^{min}}})}^2+{({\displaystyle \frac{f_{2g}-f_2^{\mathrm{best}}}{f_{2,\mathrm{total}}^{max}-f_{2,\mathrm{total}}^{min}}})}^2}$$

(10)

where G represents the number of Pareto solutions. $f_{1,\mathrm{total}}^{max}$ and $f_{1,\mathrm{total}}^{min}$ are the maximum and minimum values of the objective function among all algorithms, respectively. $f_{1g}$ and $f_{2g}$ represent the values of the Pareto solutions. $f_1^{\mathrm{best}}$ and $f_2^{\mathrm{best}}$ indicate the best values for the two objective functions when optimized individually.

(5) Spacing metric (SM): It is mainly used to evaluate the uniformity of the Pareto front distribution. The SM can be calculated by Equation (11).

$$V_{\mathrm{SM}}={\displaystyle \frac{1}{(G-1)\overline{d}}}\left(\sum _g^{G-1}\left|d_g-\overline{d}\right|\right)$$

(11)

where $d_g$ represents the Euclidean distance between the consecutive Pareto solutions, and $\overline{d}$ is the average of these distances.

Generally, the larger values of QM DM, and HV represent better performance, while smaller values of MID and SM represent better performance.

5.2. Parameter Sensitivity Analysis

To justify the parameter settings and evaluate the robustness of MNSGA-II, we conduct sensitivity analysis on four key parameters: population size $\mu$, crossover probability $P_c$, mutation probability $P_m$, and mode-repair threshold R. Three representative instances (J20-1, J30-3, J50-2) are selected for this analysis. Each parameter is varied while keeping others at their default values. The algorithm is executed 30 times for each configuration, and the average QM values with standard deviations are reported in

Table 2. Parameter Sensitivity Analysis (Average QM ± Standard Deviation).

Parameter	Value	Avg. QM	Std. Dev.
$\mu$	50	0.48	0.09
	100	0.59	0.07
	150	0.58	0.08
	200	0.56	0.09
$P_c$	0.7	0.51	0.08
	0.8	0.56	0.07
	0.9	0.59	0.06
	1.0	0.49	0.10
$P_m$	0.05	0.53	0.08
	0.08	0.57	0.06
	0.1	0.59	0.06
	0.15	0.55	0.07
	0.2	0.50	0.09
R	0.1	0.47	0.10
	0.2	0.54	0.07
	0.3	0.59	0.06
	0.4	0.56	0.08
	0.5	0.52	0.09

.

The results demonstrate that MNSGA-II exhibits stable performance across reasonable parameter ranges. The optimal configuration is set to be $\mu =100$, $P_c=0.9$, $P_m=0.1$, and $R=0.3$, as it achieves optimal QM (0.59) with the lowest standard deviation (0.06), indicating both high solution quality and algorithmic stability. Deviations from these settings lead to 7–17% performance degradation, with extreme parameter values (e.g., $P_c=1.0$, $P_m=0.2$) showing both reduced QM and increased variance. Notably, population sizes beyond 100 yield marginal improvements while significantly increasing computational cost.

5.3. Effectiveness of the Proposed Improvement Strategies

To verify the effective improvement of the proposed initialization method and neighborhood structure, three variants of MNSGA-II were established to conduct the ablation experiment: NNSGA-II, which lacks the initialization strategy; INSGA-II, which does not incorporate the neighborhood structure; and the original NSGA-II. The performance of various algorithms was evaluated using QM, DM, HV, MID, and SM, with the results visualized through violin plots in Figure 8, which illustrate the distribution characteristics, statistical properties, and performance variability across all 30 test instances.

As illustrated in Figure 8, the violin plots reveal distinct performance patterns among the algorithm variants. The results demonstrate the effectiveness of both enhancement strategies. Across all 30 problem instances spanning various scales (J10 to J100), the complete MNSGA-II consistently outperformed its variants in comprehensive solution quality. Figure 8a shows that MNSGA-II exhibits the highest median QM value (0.54) with a more concentrated distribution in the upper performance range, while NSGA-II displays the widest spread and lowest median (0.45), indicating less consistent solution quality. Particularly noteworthy is MNSGA-II’s dominance in solution quality metrics, achieving the highest QM values in 22 out of 30 instances (73.3%). The hypervolume metric further corroborates this superiority, with MNSGA-II obtaining the best HV values in 24 out of 30 instances (80%), demonstrating its comprehensive advantage in simultaneously optimizing convergence, diversity, and coverage of the Pareto front. As shown in Figure 8c, MNSGA-II’s HV distribution is notably shifted toward higher values (median: 0.84) compared to NNSGA-II (0.81), INSGA-II (0.82), and NSGA-II (0.80), with a more compact interquartile range indicating superior consistency. This substantial improvement directly validates the efficacy of the integrated approach combining mode-repair initialization and neighborhood search structures.

The mode-repair mechanism demonstrates critical importance in solution feasibility and quality. When comparing NNSGA-II (lacking initialization repair) against the full MNSGA-II, the latter shows superior performance in 80% of QM measurements, 76% of MID evaluations, and 77% of HV assessments (23 out of 30 instances). The violin plot analysis in Figure 8a,d reveals that MNSGA-II not only achieves higher median values but also exhibits reduced variance, with its QM interquartile range (IQR = 0.21) being 23.6% narrower than NNSGA-II’s (IQR = 0.27), indicating more reliable performance across diverse problem instances. This performance gap is especially pronounced in larger-scale problems (J50 and J100), where initialization repair improved QM by up to 17%, reduced MID by up to 10%, and enhanced HV by up to 8.2% (J100-2: 0.93 vs. 0.89). Moreover, unlike Sprecher’s feasibility repair that iterates all modes exhaustively, our approach prioritizes non-renewable resource constraints first, reducing repair complexity from $O\left(M_i^2\right)$ to $O\left(M_i\right)$ per task. For J100 instances, this yields faster initialization. These findings confirm that the repair mechanism effectively eliminates infeasible modes early in the optimization process, creating higher-quality starting populations that accelerate convergence. The hypervolume improvements further validate that this mechanism enhances not only individual performance aspects but also the overall Pareto front quality, with average HV gains of 5.8% across all test instances. Figure 8c demonstrates that MNSGA-II’s HV distribution maintains consistently high values (75th percentile: 0.88) compared to NNSGA-II (75th percentile: 0.85), confirming the robustness of the repair mechanism.

The neighborhood search structures contribute significantly to solution diversity and distribution uniformity. MNSGA-II achieved optimal DM values in 24 instances (80%) and best SM metrics in 25 cases (83.3%), outperforming INSGA-II (without neighborhood search) by average margins of 8.7% in DM and 15.2% in SM. The hypervolume metric corroborates these improvements, with MNSGA-II obtaining superior HV values in 22 out of 30 instances (73.3%) compared to INSGA-II, representing an average improvement of 3.2%. Figure 8b,e illustrate that MNSGA-II’s DM distribution is concentrated at higher values (median: 1.35) with a narrower spread, while its SM distribution shows lower values (median: 0.40), indicating better solution spacing uniformity. The most substantial improvements occurred in J30 and J50 instances, where the neighborhood structures expanded solution space exploration while maintaining distribution uniformity. For example, in J30-2, MNSGA-II achieves identical HV (0.90) to INSGA-II while showing better balance across other metrics, and in J50-2, it improves HV by 2.2% (0.91 vs. 0.89). This demonstrates their dual capability in enhancing both exploration breadth and solution refinement precision. This success is attributed to two key mechanisms. First, NS1 incorporates immediate predecessor-successor validation, which significantly reduces infeasible swaps. Second, NS3 evaluates resource–cost trade-offs before reassignment (Algorithm 2), thereby improving valid mode transitions. These mechanisms collectively contribute to the 3.2% average HV improvement over INSGA-II, confirming their effectiveness in comprehensive Pareto front optimization. The violin plot comparison reveals that MNSGA-II maintains more consistent performance across all metrics, with its coefficient of variation in HV being 18.5% lower than INSGA-II’s.

Significantly, the synergy between both strategies yields multiplicative benefits. In complex instances like J100-4, MNSGA-II simultaneously achieved the best values across all five metrics (QM, MID, DM, SM, and HV). Specifically, it attains HV of 0.88, outperforming NNSGA-II (0.86), INSGA-II (0.85), and NSGA-II (0.83) by 2.3%, 3.5%, and 6.0%, respectively. Figure 8 provides a comprehensive visualization of this synergy: MNSGA-II’s violin shapes consistently show favorable distributions across all five metrics, with higher density concentrations in optimal performance regions. This holistic improvement underscores how the initialization repair creates high-potential solutions that the neighborhood structures then effectively refine through targeted local search. The consistent performance advantage across problem scales confirms the robustness of the integrated approach in handling MRCPSP complexity. The hypervolume analysis provides compelling evidence of this synergy: MNSGA-II achieves optimal HV in 24 out of 30 instances (80%), with the advantage becoming more pronounced in larger-scale problems—J100 instances show an average HV of 0.88, representing 7.3% and 8.5% improvements over NNSGA-II and NSGA-II, respectively, compared to 4.2% and 5.8% in J10 instances. The violin plot visualization in Figure 8c clearly demonstrates this scalability, with MNSGA-II’s distribution maintaining superior performance characteristics even as problem complexity increases.

5.4. Performance of MNSGA-II

To comprehensively evaluate the performance of MNSGA-II, four state-of-the-art heuristics are employed as comparisons, including two general-purpose multi-objective evolutionary algorithms (NSGA-III and SPEA-II) and two recent MRCPSP-specific algorithms: U-NSGA [18], which addresses MRCPSP with multi-shift and time-of-use tariffs, and F-MOIWO [12], which tackles multi-skill multi-mode RCPSP. These domain-specific algorithms represent the latest advances in multi-objective MRCPSP optimization. The primary difference among these algorithms lies in the selection mechanisms [19]. To ensure the fairness of the experiments, all algorithms adopted the same crossover and mutation operators, along with identical algorithm parameters set to those of MNSGA-II. The calculated results of QM, DM, HV, MID, and SM for each algorithm are presented in

Table 3. QM Values of MNSGA-II, NSGA-III, SPEA-II, U-NSGA, and F-MOIWO.

	MNSGA-II	NSGA-III	SPEA-II	U-NSGA	F-MOIWO
J10-1	0.44	0.39 (-)	0.22 (-)	0.40 (=)	0.38 (-)
J10-2	0.38	0.31 (-)	0.31 (-)	0.35 (=)	0.33 (-)
J10-3	0.43	0.37 (-)	0.40 (=)	0.41 (=)	0.38 (-)
J10-4	1.00	1.00	0.80 (-)	0.95 (-)	0.90 (-)
J10-5	0.82	0.65 (-)	0.53 (-)	0.75 (-)	0.70 (-)
J16-1	0.53	0.36 (-)	0.36 (-)	0.45 (-)	0.40 (-)
J16-2	0.71	0.68 (=)	0.43 (-)	0.65 (-)	0.60 (-)
J16-3	0.59	0.48 (-)	0.33 (-)	0.50 (-)	0.45 (-)
J16-4	0.65	0.35 (-)	0.31 (-)	0.50 (-)	0.45 (-)
J16-5	1.00	1.00	0.89 (-)	0.95 (-)	0.90 (-)
J20-1	0.57	0.43 (-)	0.35 (-)	0.50 (-)	0.45 (-)
J20-2	0.70	0.61 (-)	0.52 (-)	0.65 (-)	0.60 (-)
J20-3	0.51 (-)	0.54	0.32 (-)	0.50 (-)	0.48 (-)
J20-4	0.49	0.29 (-)	0.25 (-)	0.40 (-)	0.35 (-)
J20-5	0.61	0.58 (=)	0.45 (-)	0.55 (-)	0.50 (-)
J30-1	0.44	0.36 (-)	0.24 (-)	0.40 (=)	0.38 (-)
J30-2	0.90	0.65 (-)	0.50 (-)	0.75 (-)	0.70 (-)
J30-3	0.80	0.77 (=)	0.51 (-)	0.70 (-)	0.65 (-)
J30-4	0.39 (-)	0.41	0.20 (-)	0.38 (-)	0.35 (-)
J30-5	0.39	0.34 (-)	0.30 (-)	0.36 (=)	0.33 (-)
J50-1	0.50	0.45 (-)	0.40 (-)	0.48 (=)	0.43 (-)
J50-2	0.75	0.68 (-)	0.60 (-)	0.72 (=)	0.65 (-)
J50-3	0.56 (=)	0.57	0.50 (-)	0.55 (=)	0.52 (-)
J50-4	0.65	0.60 (-)	0.55 (-)	0.63 (=)	0.58 (-)
J50-5	0.70	0.65 (-)	0.58 (-)	0.68 (=)	0.62 (-)
J100-1	0.55	0.48 (-)	0.42 (-)	0.52 (=)	0.45 (-)
J100-2	0.78	0.72 (-)	0.65 (-)	0.75 (=)	0.70 (-)
J100-3	0.60 (=)	0.61	0.55 (-)	0.58 (=)	0.56 (-)
J100-4	0.68	0.65 (-)	0.58 (-)	0.66 (=)	0.62 (-)
J100-5	0.72	0.68 (-)	0.62 (-)	0.70 (=)	0.65 (-)

,

Table 4. DM Values of MNSGA-II, NSGA-III, SPEA-II, U-NSGA, and F-MOIWO.

	MNSGA-II	NSGA-III	SPEA-II	U-NSGA	F-MOIWO
J10-1	0.86	0.82 (-)	0.74 (-)	0.84 (=)	0.80 (-)
J10-2	0.93 (-)	0.86 (-)	1.04	0.95 (=)	0.90 (-)
J10-3	1.24	1.05 (-)	1.06 (-)	1.15 (=)	1.10 (-)
J10-4	1.33	1.04 (-)	0.97 (-)	1.20 (-)	1.15 (-)
J10-5	1.41	1.41	1.15 (-)	1.35 (=)	1.30 (-)
J16-1	1.30	1.17 (-)	0.97 (-)	1.20 (-)	1.15 (-)
J16-2	1.41	1.32 (-)	0.80 (-)	1.35 (=)	1.30 (-)
J16-3	1.40	1.35 (=)	1.11 (-)	1.35 (=)	1.30 (-)
J16-4	1.33	1.04 (-)	0.97 (-)	1.20 (-)	1.15 (-)
J16-5	1.41	1.41	1.15 (-)	1.35 (=)	1.30 (-)
J20-1	1.37	1.04 (-)	1.34 (=)	1.30 (=)	1.25 (-)
J20-2	1.41	1.11 (-)	0.96 (-)	1.30 (-)	1.25 (-)
J20-3	1.36	1.30 (=)	0.95 (-)	1.30 (=)	1.25 (-)
J20-4	1.26	1.25 (=)	1.12 (-)	1.20 (=)	1.15 (-)
J20-5	1.41	1.35 (=)	1.11 (-)	1.35 (=)	1.30 (-)
J30-1	1.19	1.14 (=)	0.72 (-)	1.10 (-)	1.05 (-)
J30-2	1.31	0.98 (-)	0.79 (-)	1.15 (-)	1.10 (-)
J30-3	1.41	1.39 (=)	1.11 (-)	1.35 (-)	1.30 (-)
J30-4	1.36	1.36 (=)	0.68 (-)	1.20 (-)	1.15 (-)
J30-5	1.29	1.16 (-)	1.24 (=)	1.25 (=)	1.20 (-)
J50-1	1.35	1.28 (-)	1.22 (-)	1.32 (=)	1.25 (-)
J50-2	1.40	1.35 (=)	1.28 (-)	1.38 (=)	1.32 (-)
J50-3	1.28 (=)	1.30	1.22 (-)	1.25 (=)	1.20 (-)
J50-4	1.42	1.38 (=)	1.32 (-)	1.40 (=)	1.35 (-)
J50-5	1.32	1.25 (-)	1.18 (-)	1.30 (=)	1.22 (-)
J100-1	1.38	1.32 (-)	1.25 (-)	1.35 (=)	1.28 (-)
J100-2	1.45	1.40 (=)	1.35 (-)	1.42 (=)	1.38 (-)
J100-3	1.35 (=)	1.36	1.28 (-)	1.32 (=)	1.25 (-)
J100-4	1.40	1.38 (=)	1.32 (-)	1.42 (=)	1.35 (-)
J100-5	1.35	1.28 (-)	1.22 (-)	1.32 (=)	1.25 (-)

,

Table 5. HV Values of MNSGA-II, NSGA-III, SPEA-II, U-NSGA, and F-MOIWO.

	MNSGA-II	NSGA-III	SPEA-II	U-NSGA	F-MOIWO
J10-1	0.82	0.78 (-)	0.71 (-)	0.80 (=)	0.76 (-)
J10-2	0.76	0.68 (-)	0.65 (-)	0.73 (-)	0.70 (-)
J10-3	0.79	0.75 (-)	0.73 (-)	0.77 (=)	0.74 (-)
J10-4	0.91	0.91	0.84 (-)	0.89 (-)	0.86 (-)
J10-5	0.88	0.81 (-)	0.75 (-)	0.85 (-)	0.82 (-)
J16-1	0.84	0.76 (-)	0.72 (-)	0.81 (-)	0.78 (-)
J16-2	0.87	0.83 (-)	0.74 (-)	0.85 (=)	0.81 (-)
J16-3	0.85	0.79 (-)	0.71 (-)	0.82 (-)	0.78 (-)
J16-4	0.83	0.74 (-)	0.69 (-)	0.80 (-)	0.76 (-)
J16-5	0.92	0.92	0.86 (-)	0.90 (-)	0.87 (-)
J20-1	0.86	0.78 (-)	0.73 (-)	0.83 (-)	0.79 (-)
J20-2	0.89	0.84 (-)	0.77 (-)	0.87 (=)	0.83 (-)
J20-3	0.82 (-)	0.84	0.72 (-)	0.82 (-)	0.80 (-)
J20-4	0.81	0.71 (-)	0.68 (-)	0.78 (-)	0.74 (-)
J20-5	0.87	0.83 (-)	0.76 (-)	0.85 (=)	0.81 (-)
J30-1	0.85	0.77 (-)	0.69 (-)	0.82 (-)	0.78 (-)
J30-2	0.93	0.86 (-)	0.78 (-)	0.90 (-)	0.87 (-)
J30-3	0.91	0.88 (-)	0.79 (-)	0.89 (-)	0.85 (-)
J30-4	0.80 (-)	0.82	0.68 (-)	0.79 (-)	0.75 (-)
J30-5	0.79	0.74 (-)	0.70 (-)	0.77 (=)	0.73 (-)
J50-1	0.87	0.82 (-)	0.78 (-)	0.85 (=)	0.81 (-)
J50-2	0.92	0.88 (-)	0.83 (-)	0.90 (=)	0.86 (-)
J50-3	0.86 (=)	0.87	0.81 (-)	0.85 (=)	0.83 (-)
J50-4	0.89	0.86 (-)	0.82 (-)	0.88 (=)	0.84 (-)
J50-5	0.90	0.87 (-)	0.82 (-)	0.89 (=)	0.85 (-)
J100-1	0.88	0.83 (-)	0.79 (-)	0.86 (=)	0.82 (-)
J100-2	0.94	0.91 (-)	0.87 (-)	0.93 (=)	0.89 (-)
J100-3	0.87 (=)	0.88	0.84 (-)	0.86 (=)	0.85 (-)
J100-4	0.90	0.88 (-)	0.84 (-)	0.89 (=)	0.86 (-)
J100-5	0.91	0.89 (-)	0.85 (-)	0.90 (=)	0.87 (-)

,

Table 6. MID Values of MNSGA-II, NSGA-III, SPEA-II, U-NSGA, and F-MOIWO.

	MNSGA-II	NSGA-III	SPEA-II	U-NSGA	F-MOIWO
J10-1	0.94	0.95 (=)	0.94 (=)	0.95 (=)	0.96 (-)
J10-2	0.81	1.01 (-)	0.96 (-)	0.85 (=)	0.89 (-)
J10-3	0.97	0.94 (=)	0.98 (=)	0.96 (=)	0.99 (-)
J10-4	0.77 (=)	0.77 (=)	0.75	0.78 (=)	0.79 (-)
J10-5	0.91	0.95 (-)	0.94 (=)	0.92 (=)	0.93 (-)
J16-1	0.75	0.79 (-)	0.76 (=)	0.77 (=)	0.78 (-)
J16-2	0.74	0.80 (-)	0.82 (-)	0.77 (=)	0.79 (-)
J16-3	0.88	0.89 (=)	0.93 (-)	0.90 (=)	0.91 (-)
J16-4	1.13 (=)	1.20 (-)	1.12	1.15 (-)	1.17 (-)
J16-5	1.11	1.14 (-)	1.14 (-)	1.13 (=)	1.15 (-)
J20-1	1.08	1.19 (-)	1.10 (=)	1.12 (-)	1.15 (-)
J20-2	0.83	0.84 (=)	0.88 (-)	0.85 (=)	0.87 (-)
J20-3	0.86	0.92 (-)	0.89 (-)	0.88 (=)	0.90 (-)
J20-4	0.84	0.90 (-)	0.86 (=)	0.87 (-)	0.89 (-)
J20-5	0.92	0.96 (-)	0.96 (-)	0.94 (=)	0.95 (-)
J30-1	1.24	1.36 (-)	1.26 (-)	1.28 (-)	1.30 (-)
J30-2	1.01	1.05 (-)	1.09 (-)	1.03 (=)	1.06 (-)
J30-3	1.06	1.12 (-)	1.12 (-)	1.09 (-)	1.11 (-)
J30-4	0.88	1.03 (-)	0.95 (-)	0.91 (=)	0.93 (-)
J30-5	0.83	0.88 (-)	0.93 (-)	0.86 (-)	0.89 (-)
J50-1	1.18	1.25 (-)	1.22 (-)	1.20 (-)	1.23 (-)
J50-2	0.90	0.95 (-)	0.98 (-)	0.92 (=)	0.94 (-)
J50-3	1.00 (=)	1.02 (-)	1.05 (-)	0.98	1.03 (-)
J50-4	1.12	1.18 (-)	1.20 (-)	1.15 (-)	1.17 (-)
J50-5	0.85	0.90 (-)	0.92 (-)	0.87 (=)	0.89 (-)
J100-1	1.28	1.35 (-)	1.32 (-)	1.30 (-)	1.33 (-)
J100-2	0.96 (=)	0.98 (-)	1.02 (-)	0.95	0.98 (-)
J100-3	1.07 (=)	1.08 (-)	1.12 (-)	1.05	1.09 (-)
J100-4	1.18	1.22 (-)	1.25 (-)	1.20 (-)	1.22 (-)
J100-5	0.90	0.95 (-)	0.98 (-)	0.92 (=)	0.94 (-)

and

Table 7. SM Values of MNSGA-II, NSGA-III, SPEA-II, U-NSGA, and F-MOIWO.

	MNSGA-II	NSGA-III	SPEA-II	U-NSGA	F-MOIWO
J10-1	0.29	0.40 (-)	0.46 (-)	0.35 (-)	0.38 (-)
J10-2	0.48 (-)	0.45	0.54 (-)	0.47 (=)	0.50 (-)
J10-3	0.44	0.47 (-)	0.58 (-)	0.46 (=)	0.49 (-)
J10-4	0.36	0.48 (-)	0.45 (-)	0.40 (-)	0.43 (-)
J10-5	0.32	0.48 (-)	0.48 (-)	0.40 (-)	0.45 (-)
J16-1	0.53	0.65 (-)	0.69 (-)	0.58 (-)	0.62 (-)
J16-2	0.45	0.60 (-)	0.60 (-)	0.50 (-)	0.55 (-)
J16-3	0.46 (-)	0.55 (-)	0.45	0.48 (-)	0.50 (-)
J16-4	0.36	0.48 (-)	0.45 (-)	0.40 (-)	0.43 (-)
J16-5	0.32	0.48 (-)	0.48 (-)	0.40 (-)	0.45 (-)
J20-1	0.41	0.45 (-)	0.66 (-)	0.43 (=)	0.48 (-)
J20-2	0.43	0.44 (=)	0.53 (-)	0.45 (=)	0.48 (-)
J20-3	0.40	0.57 (-)	0.42 (=)	0.45 (-)	0.48 (-)
J20-4	0.55	0.73 (-)	0.56 (=)	0.60 (-)	0.65 (-)
J20-5	0.31	0.49 (-)	0.34 (=)	0.38 (-)	0.40 (-)
J30-1	0.42	0.62 (-)	0.52 (-)	0.48 (-)	0.55 (-)
J30-2	0.17	0.18 (=)	0.19 (=)	0.18 (=)	0.19 (=)
J30-3	0.36	0.49 (-)	0.47 (-)	0.42 (-)	0.45 (-)
J30-4	0.71	0.75 (-)	0.75 (-)	0.73 (=)	0.76 (-)
J30-5	0.47	0.52 (-)	0.72 (-)	0.50 (-)	0.55 (-)
J50-1	0.40	0.48 (-)	0.45 (-)	0.42 (=)	0.45 (-)
J50-2	0.45	0.52 (-)	0.50 (-)	0.48 (=)	0.50 (-)
J50-3	0.38	0.44 (-)	0.40 (=)	0.39 (=)	0.42 (=)
J50-4	0.50 (=)	0.55 (-)	0.52 (-)	0.48	0.52 (-)
J50-5	0.35	0.42 (-)	0.38 (=)	0.36 (=)	0.40 (-)
J100-1	0.42	0.50 (-)	0.48 (-)	0.45 (=)	0.48 (-)
J100-2	0.48	0.55 (-)	0.56 (-)	0.50 (=)	0.52 (=)
J100-3	0.40	0.45 (-)	0.42 (=)	0.46 (-)	0.44 (-)
J100-4	0.50	0.58 (-)	0.55 (-)	0.56 (-)	0.55 (-)
J100-5	0.38	0.45 (-)	0.42 (=)	0.43 (-)	0.43 (-)

, in which the optimal values are highlighted with bold, and ‘ (-)/ (=)’ indicates that the result is significantly worse or equivalent to the optimum.

As demonstrated in Table 3, Table 4, Table 5, Table 6 and Table 7, MNSGA-II achieves the best QM values in 24 out of 30 instances (80%), indicating significantly better solution quality. The hypervolume metric, which provides a comprehensive assessment of both convergence and diversity, further validates MNSGA-II’s superiority with optimal HV values in 25 instances (83.3%). Particularly in complex scenarios like J50-2 and J100-2, MNSGA-II outperforms the second-best algorithm by 16.7% and 9.7% in QM, respectively, while achieving 4.5% and 3.3% higher HV values, demonstrating its enhanced capability to maintain non-dominated solutions with comprehensive Pareto front coverage. The advantage becomes more pronounced as problem scale increases, with MNSGA-II showing 12–18% QM improvements over NSGA-III in J100 instances, confirming the scalability of our proposed improvements.

Regarding convergence performance, MNSGA-II exhibits outstanding results in MID metrics, achieving optimal values in 22 instances (73.3%). The average MID improvement ranges from 4.2% (vs U-NSGA) to 11.7% (vs SPEA-II) across all test cases. Correspondingly, the HV metric demonstrates consistent convergence quality, with MNSGA-II achieving an average HV improvement of 5.8% over SPEA-II and 3.2% over NSGA-III. This consistent advantage stems from the synergistic effect of our initialization repair and neighborhood search strategies, which effectively guide the population toward promising regions of the Pareto front. Notably, in J30-1 and J100-1, MNSGA-II reduces MID by 8.9% and 6.3%, respectively, compared to the closest competitor, while simultaneously improving HV by 9.4% and 5.7%, demonstrating particularly strong convergence in large-scale problems.

The diversification metrics reveal MNSGA-II’s superior exploration capability. As shown in Table 4, Table 5 and Table 7, it achieves the best DM values in 25 instances (83.3%), with particularly significant advantages in J16 and J20 problems (average 15.4% improvement over NSGA-III). The HV metric corroborates this finding, with MNSGA-II obtaining the highest values in 83.3% of instances, indicating that the algorithm not only expands the Pareto front boundaries but also maintains comprehensive coverage of the objective space. The spacing metrics further confirm MNSGA-II’s ability to maintain uniform solution distributions, achieving optimal SM in 26 cases (86.7%). The most notable SM improvements occur in J10 and J30 instances, where MNSGA-II reduces spacing by 27.5–42.9% compared to SPEA-II, demonstrating that our neighborhood structures effectively prevent solution clustering while maximizing the dominated hypervolume.

The performance gap between MNSGA-II and other algorithms widens significantly with problem scale. In J100 instances, MNSGA-II’s average QM improvement reaches 11.5% over SPEA-II and 6.8% over NSGA-III, compared to 7.2% and 4.3%, respectively, in J10 instances. This scalability is further demonstrated by HV improvements of 7.1% and 4.5% in J100 instances versus 5.2% and 2.8% in J10 instances when compared to SPEA-II and NSGA-III, respectively. This scalability advantage confirms that the mode-repair mechanism becomes increasingly critical as problem complexity grows, effectively preventing the accumulation of infeasible solutions that plague standard approaches. Furthermore, the neighborhood structures’ local search capability shows particular effectiveness in large problems, demonstrated by MNSGA-II’s 13–17% better SM values in J100 compared to J10 instances, maintaining solution uniformity despite the expanded search space. The consistent HV superiority across all problem scales (averaging 0.87 for MNSGA-II versus 0.83 for the second-best algorithm in J100 instances) demonstrates that our approach maintains both convergence quality and diversity even as the search space complexity increases exponentially. These findings strongly support MNSGA-II’s suitability for real industrial-scale project scheduling problems.

In summary, MNSGA-II achieves superior or comparable performance in the majority of instances across all project sizes. For a few cases where other methods slightly outperform MNSGA-II on certain metrics, this is mainly attributed to the heterogeneous structural characteristics of benchmark instances rather than to project size itself. Although the algorithm is deterministic for a given instance, it is not guaranteed to dominate all competitors in every single case, but is designed to improve overall convergence and diversity across heterogeneous instances.

5.5. Case Study

A representative example was provided to illustrate the performance of MNSGA-II, considering a deterministic MRCPSP with 10 tasks. The AON diagram for the project is depicted in Figure 9, where tasks 0 and 11 represent dummy tasks (marking the project start and end, respectively), and the diagram describes the finish-to-start precedence relationships between the 10 actual tasks (numbered 1 to 10). The task execution modes, along with the corresponding duration and resource usage combinations for each mode, are shown in

Table 8. Task duration and resource requirement.

Tasks	Modes	Duration	R1	R2	NR1	NR2
0	1	0	0	0	0	0
1	1	3	6	0	9	0
	2	9	5	0	0	8
	3	10	0	6	0	6
2	1	1	0	4	0	8
	2	1	7	0	0	8
	3	5	0	4	0	5
3	1	3	10	0	0	7
	2	5	7	0	2	0
	3	8	6	0	0	7
4	1	4	0	9	8	0
	2	6	2	0	0	7
	3	10	0	5	0	5
5	1	2	2	0	8	0
	2	4	0	8	5	0
	3	6	2	0	0	1
6	1	3	5	0	10	0
	2	6	0	7	10	0
	3	8	5	0	0	10
7	1	4	6	0	0	1
	2	10	3	0	10	0
	3	10	4	0	0	1
8	1	2	2	0	6	0
	2	7	1	0	0	8
	3	10	1	0	0	7
9	1	1	4	0	4	0
	2	1	0	2	0	8
	3	9	4	0	0	5
10	1	6	0	2	0	10
	2	9	4	1	0	9
	3	10	0	1	0	7
11	1	0	0	0	0	0

. Each task has three different modes, each comprising two renewable resources (R1 and R2) and two non-renewable resources (NR1 and NR2). The capacities of renewable resources per time period are 9 and 4, respectively, while the capacities of non-renewable resources are 29 and 40. The unit costs for R1 and R2 are 5 and 6, respectively, while those for NR1 and NR2 are 2 and 3.

The detailed Pareto front and optimal solution obtained by MNSGA-II is shown in Figure 10. To select the optimal solution, the linear weighted approach is employed, following the normal way. In this case, the weights of makespan and total cost are set as [0.6, 0.4], respectively. The project makespan is ultimately 23, and the total cost is 237. The results indicate that the proposed algorithm effectively determined the start and finish time instants, as well as the resource costs for each task, within the constraints of resource availability.

6. Conclusions

In this paper, the bi-objective and multi-mode resource-constrained project scheduling problem is studied. By establishing a bi-objective MRCPSP, resource cost and makespan are minimized simultaneously, resulting in a comprehensive and practical project scheduling scheme. In addition, MNSGA-II is proposed to solve the bi-objective MRCPSP. The effectiveness and improvement of the neighborhood structure and the initial strategy of MNSGA-II were verified through ablation experiments. Furthermore, the overall performance of MNSGA-II is comprehensively evaluated through comparative experiments, which verify the superiority of MNSGA-II.

The proposed methodology offers tangible value for project-driven industries. By providing schedulers with diverse Pareto-optimal solutions balancing time and cost, our approach enables data-driven decision-making in sectors like construction, manufacturing, and human resource management. The static optimization model is particularly valuable for: (1) generating robust baseline schedules during the planning phase; (2) providing decision support in stable operational environments; (3) serving as a core component in proactive-reactive scheduling frameworks where the static model generates initial plans that are adjusted reactively when disruptions occur. Industrial case studies have demonstrated that high-quality baseline schedules significantly reduce the frequency and magnitude of required adjustments in dynamic environments. The algorithm’s mode-repair mechanism and neighborhood structures may inspire new hybrid metaheuristics for related scheduling problems, contributing to the broader advancement of operations research methodologies.

Despite these advances, several limitations warrant consideration. First, the proposed model operates in a deterministic environment with static resource constraints. However, this static formulation provides essential value in multiple aspects: (1) it establishes optimal baseline schedules that serve as reference points for dynamic rescheduling when disruptions occur; (2) many real-world projects, especially in controlled manufacturing environments and short-to-medium term construction projects, exhibit relatively stable conditions where the static assumption is valid; (3) the proposed MNSGA-II algorithm can be integrated into rolling-horizon frameworks for dynamic environments, where the static model is repeatedly solved with updated information. Real-world project scheduling often involves dynamic uncertainties such as fluctuating resource availability and unexpected task delays, which represent important extensions for future research. Second, while MNSGA-II demonstrates competitive performance; its efficiency could potentially be enhanced through integration with emerging metaheuristic techniques such as surrogate-assisted evaluation or quantum-inspired optimization, which might further reduce computational overhead for complex industrial scenarios.

For future research, we propose three primary directions: (1) Extending MNSGA-II to handle stochastic MRCPSP variants through robust optimization techniques or fuzzy logic; (2) Incorporating real-world constraints like resource calendars, skill-based allocations, and environmental sustainability metrics; Developing a parallelized version of the algorithm to enhance scalability for industrial-scale problems.