Energy-eﬃcient Flexible Job Shop Scheduling Problem Considering Preventive Maintenance and Transportation Times

Abstract

The ﬂexible job shop scheduling problem (FJSP) has been extensively researched over the past decade. However, the integration of preventive maintenance (PM), transportation times, and energy eﬃciency remains under-explored. This paper addresses the energy-eﬃcient ﬂexible job shop scheduling problem with preventive maintenance and transportation times (EFJSP-PMT). To minimize both makespan and total energy consumption, a multi-population optimization algorithm with adaptive strategy and competition (MPOA-ASC) is proposed, featuring distinct early and later evolution phases. In the early evolution phase, we develop a hybrid heuristic initialization method, a new population division strategy, an adaptive mechanism for crossover and mutation probabilities, and three specialized local search operators. In the later phase, a novel inter-population competition mechanism is introduced to optimize the allocation of computational resources and enhance convergence performance. Extensive experimental results demonstrate that the proposed MPOA-ASC signiﬁcantly outperforms state-of-the-art algorithms in solving the EFJSP-PMT.

Keywords

energy-efficient flexible job shop scheduling problem preventive maintenance transportation times multi-population algorithm makespan

1. Introduction

The flexible job shop scheduling problem (FJSP) is an extension of the classic job shop scheduling problem, with the core feature that operations can be processed on multiple optional machines with different processing times. FJSP requires simultaneous decision-making for machine allocation and operation sequencing, making it an NP-hard problem. Effective scheduling is a key factor in improving the efficiency of manufacturing systems.

Furthermore, excessive energy consumption has led to increased costs, low resource utilization, and environmental pollution, prompting enterprises to focus on energy efficiency in the production process. As a core component of manufacturing, effective scheduling can directly improve production efficiency and reduce energy consumption. Related research is no longer restricted to traditional economic objectives—such as machine load, completion time, and tardiness—but increasingly focuses on green indicators, including energy consumption and carbon emission. With the introduction of these factors, the complexity and difficulty of scheduling are greatly increased.

Given the NP-hard nature of EFJSP, intelligent optimization algorithms have emerged as a key methodology. Numerous studies have addressed standard and distributed EFJSP by optimizing makespan ( $C_{max}$ ) and total energy consumption (TEC) using methods such as genetic algorithms (GA) (Hao et al., 2025; Ren et al., 2021; Wei et al., 2022), evolutionary algorithms (Qu et al., 2024; Wang et al., 2024a; 2026a; Yu et al., 2024; Zhang et al., 2024b, 2025b), and other metaheuristics like immune and memetic algorithms (Burmeister et al., 2024; Chen et al., 2023a, 2023b; Du et al., 2023; Gong et al., 2021, 2024; Hu et al., 2024; Li et al., 2020a, 2020b, 2022b; Tang et al., 2024; Wang & Zhao, 2025; Yang et al., 2023; Zhou et al., 2024; Zhu et al., 2024). Furthermore, EFJSP with dynamic events (e.g., machine breakdowns, rush orders) has also received significant attention, with researchers developing knowledge-guided and reinforcement learning-assisted algorithms (Akram et al., 2024; Feng et al., 2025; Li et al., 2025b; Peng et al., 2025; Zhu et al., 2023, 2025) to minimize $C_{max}$ and TEC.

Regarding EFJSP with transportation constraints, existing research has increasingly integrated logistics into scheduling. Approaches include quality-diversity algorithms (Qin et al., 2025), deep reinforcement learning (Chen & Wu, 2026; Cheng et al., 2025), and multi-population co-evolutionary algorithms (MCEA) (Liu et al., 2024) to minimize $C_{max}$ and TEC. Building on these foundations, the complexities of automated guided vehicles (AGVs) in real-world scenarios have become a prominent focus. For instance, recent studies have developed knowledge region selection enhanced quality-diversity algorithms tailored specifically for FJSP with AGV transportation (Qin et al., 2026), as well as advanced scheduling models that explicitly consider limited AGV transportation resources (Zhang et al., 2025a). Various transportation modes, such as multi-AGVs and cranes, along with setup times, have been widely modeled (Dai et al., 2019; Du et al., 2014; Jiang et al., 2022; Li & Lei, 2021; Li et al., 2022a, 2025a; Liu et al., 2024; Qin et al., 2026; Wang et al., 2024b; Zhang et al., 2021, 2023, 2024d, 2025a; Zhou et al., 2025).

Similarly, FJSP considering preventive maintenance (PM) has been extensively investigated. Researchers have developed hybrid meta-heuristics, collaborative variable neighborhood search algorithms, and co-evolutionary frameworks to jointly optimize job scheduling and maintenance activities (An et al., 2023, 2024; Khoukhi et al., 2017; Wang et al., 2024c; Wocker et al., 2024; Wu et al., 2025; Yan et al., 2025; Zhang et al., 2024a; Zhao et al., 2025a, 2025b).

As stated above, existing research has considered various constraints such as setup times, machine eligibility and order cancellation. However, preventive maintenance and transportation have rarely been combined with energy efficiency. Energy, transportation, and preventive maintenance are common factors that frequently co-occur in real-world flexible job shops. Considering these three conditions in the FJSP yields highly practical scheduling solutions and represents an important research direction. Hence, we focus on EFJSP-PMT in this study.

Multi-population optimization algorithms (MPOAs) feature multiple parallel sub-populations that can adopt differentiated search processes. This architecture prevents the homogenization of search strategies and non-independent evolution, thereby maintaining high population diversity and enhancing the algorithm's search capability. MPOA has been extensively applied with promising results to various production scheduling problems (Jia et al., 2024; Wang et al., 2026b; Zhang et al., 2025c). It is a potential method for solving the EFJSP-PMT.

The contributions of this study are summarized below. (1) MPOA-ASC is developed to address EFJSP-PMT, which consists of early evolution and later evolution. (2) A new population division strategy is designed, an adaptive strategy for adjusting crossover and mutation probabilities is given, and three local search operators are developed. (3) A competition mechanism between sub-populations is introduced to assign computation resources reasonably and improve convergence performance.

The remaining parts of the paper are organized as follows. Section 2 describes the problem. MPOA-ASC for EFJSP-PMT is designed in Section 3. The experimental results are presented in Section 4. Conclusions and directions for future research are provided in the final section.

2. Problem Description and Mathematical Model

Before formulating the mathematical model for the EFJSP-PMT, it is necessary to establish a clear mathematical notation system. The relevant symbols, including the sets, indices, given parameters, and decision variables used throughout this study, are systematically defined and summarized in Table 1.

Table 1.
Related Parameters and Their Definitions.

Symbol Definition

Sets and indices:

$i, h$ indices for jobs, $i, h \in {1, 2, \dots, n}$

$j, g$ indices for operations, $j, g \in {1, 2, \dots, n_{i}}$

$k, p$ indices for machines, $k, p \in {1, 2, \dots, m}$

$l$ index for the adjustable speed levels of machine k when processing $O_{i j}$ , $l \in {1, 2, \dots, L_{i j k}}$

$O_{i j}$ the $j^{t h}$ operation of the job $J_{i}$

$M_{i j}$ the set of candidate machines for $O_{i j}$

$P M_{k t}$ The $t^{t h}$ preventive maintenance task on machine $M_{k}$

Parameters:

$S_{i j}$ start time of $O_{i j}$

$C_{i j}$ completion time of $O_{i j}$

$C_{k}^{max}$ the ﬁnal completion on $M_{k}$

$S_{k}^{min}$ the initial start times on $M_{k}$

${\tilde{P}}_{i j k}$ /basic processing time of $O_{i j}$ on machine $M_{k}$

$E_{i j k l}$ processing power per unit time of $M_{k}$ for $O_{i j}$ at speed l

$S E_{k}$ standby (idle) power per unit time of $M_{k}$

$T_{k p}$ transportation time from $M_{k}$ to $M_{p} (T_{k k} = 0)$

$T E$ power consumption per unit time of the transportation equipment

$[S_{k t}^{P M}, E_{k t}^{P M}]$ start and end time of the $t^{t h}$ maintenance on $M_{k}$

$L$ a suﬃciently large positive constant

Decision variables:

$x_{i j k l} \in {0, 1}$ 1 if $O_{i j}$ is assigned to $M_{k}$ at speed l; 0 otherwise

$y_{i j, h g, k l} \in {0, 1}$ 1 if $O_{i j}$ precedes $O_{h g}$ on $M_{k}$ at speed l; 0 otherwise

$f_{i j, k t} \in {0, 1}$ 1 if $O_{i j}$ is scheduled before $M_{k}$ ; 0 otherwise

Symbol	Definition
Sets and indices:
$i, h$	indices for jobs, $i, h \in {1, 2, \dots, n}$
$j, g$	indices for operations, $j, g \in {1, 2, \dots, n_{i}}$
$k, p$	indices for machines, $k, p \in {1, 2, \dots, m}$
$l$	index for the adjustable speed levels of machine k when processing $O_{i j}$ , $l \in {1, 2, \dots, L_{i j k}}$
$O_{i j}$	the $j^{t h}$ operation of the job $J_{i}$
$M_{i j}$	the set of candidate machines for $O_{i j}$
$P M_{k t}$	The $t^{t h}$ preventive maintenance task on machine $M_{k}$
Parameters:
$S_{i j}$	start time of $O_{i j}$
$C_{i j}$	completion time of $O_{i j}$
$C_{k}^{max}$	the ﬁnal completion on $M_{k}$
$S_{k}^{min}$	the initial start times on $M_{k}$
${\tilde{P}}_{i j k}$	/basic processing time of $O_{i j}$ on machine $M_{k}$
$E_{i j k l}$	processing power per unit time of $M_{k}$ for $O_{i j}$ at speed l
$S E_{k}$	standby (idle) power per unit time of $M_{k}$
$T_{k p}$	transportation time from $M_{k}$ to $M_{p} (T_{k k} = 0)$
$T E$	power consumption per unit time of the transportation equipment
$[S_{k t}^{P M}, E_{k t}^{P M}]$	start and end time of the $t^{t h}$ maintenance on $M_{k}$
$L$	a suﬃciently large positive constant
Decision variables:
$x_{i j k l} \in {0, 1}$	1 if $O_{i j}$ is assigned to $M_{k}$ at speed l; 0 otherwise
$y_{i j, h g, k l} \in {0, 1}$	1 if $O_{i j}$ precedes $O_{h g}$ on $M_{k}$ at speed l; 0 otherwise
$f_{i j, k t} \in {0, 1}$	1 if $O_{i j}$ is scheduled before $M_{k}$ ; 0 otherwise

EFJSP-PMT is described as follows. There are n jobs $J = {J_{1}, J_{2}, \dots, J_{n}}$ to be processed on m machines. $M = {M_{1}, M_{2}, \dots, M_{m}}$ . Each job $J_{i}$ consists of $n_{i}$ operations. $O_{i j}$ can be processed on any compatible machines $M_{i j}$ , $M_{i j} \subseteq M$ . $P_{i j k l} = {\tilde{P}}_{i j k} / l$ . All machines operate in one of three modes: processing, idle, or preventive maintenance. Jobs can only be processed between two consecutive maintenance windows. $u_{k}$ represents the time length between two consecutive maintenance intervals and $w_{k}$ is the duration of each maintenance on $M_{k}$ . The beginning time of the $q^{t h}$ maintenance is $q \times u_{k}$ . Besides, the transportation time from $M_{k}$ to $M_{p}$ is $T_{k p}$ .

In addition, the following assumptions are adopted. (1) Each machine can process at most one operation at any given time. (2) A job cannot be processed on multiple machines simultaneously. (3) All operations are non-preemptive, i.e., they can not be interrupted once started. (4) There is an infinite capacity of transportation resources (e.g., AGVs or robots). Complex logistical factors such as specific routing paths, collision avoidance, and fleet scheduling are not considered in this model; only the deterministic transportation time between machines is integrated into the scheduling constraints.

The objectives of EFJSP-PMT are makespan and TEC which consists of processing energy ( $T E C_{p r o c}$ ), idle energy ( $T E C_{i d l e}$ ), and transportation energy ( $T E C_{t r a n s}$ ), formulated as follows.

\begin{aligned} min f_{1} & = C_{max} \end{aligned}

(1)

\begin{aligned} min f_{2} & = T E C = T E C_{p r o c} + T E C_{i d l e} + T E C_{t r a n s} \end{aligned}

(2)

\begin{aligned} T E C_{p r o c} & = \sum_{i = 1}^{n} \sum_{j = 1}^{n_{i}} \sum_{k \in M_{i j}} \sum_{l = 1}^{L_{i j k}} x_{i j k} \times P_{i j k l} \times E_{i j k l} \end{aligned}

(3)

\begin{aligned} T E C_{i d l e} & = \sum_{k = 1}^{m} S E_{k} \times [(C_{k}^{max} - S_{k}^{min}) - \sum_{i = 1}^{n} \sum_{j = 1}^{n_{i}} \sum_{l = 1}^{L_{i j k}} x_{i j k} \times P_{i j k l} - \sum_{t} (E_{k t}^{P M} - S_{k t}^{P M})] \end{aligned}

(4)

\begin{aligned} T E C_{t r a n s} & = \sum_{i = 1}^{n} \sum_{j = 1}^{n_{i} - 1} \sum_{k \in M_{i j}} \sum_{p \in M_{i, j + 1}} \sum_{l = 1}^{L_{i j k}} (x_{i j k l} \times x_{i, j + 1, p} \times T_{k p} \times T E) \end{aligned}

(5)

subject to:

\begin{aligned} \sum_{k \in M_{i j}} \sum_{l = 1}^{L_{i j k}} x_{i j k l} = 1, \forall i \in {1, 2, \dots n}, j \in {1, 2, \dots n_{i}} \end{aligned}

(6)

\begin{aligned} S_{i, j + 1} \geq S_{i j} + \sum_{k \in M_{i j}} \sum_{l = 1}^{L_{i j k}} x_{i j k l} \times P_{i j k l} + \sum_{k \in M_{i j}} \sum_{p \in M_{i, j + 1}} \sum_{l = 1}^{L_{i j k}} (x_{i j k l} \times x_{i, j + 1, p l} \times T_{k p}), \forall i, j < n_{i} \end{aligned}

(7)

\begin{aligned} {\begin{cases} S_{h g} + L \times (1 - y_{i j, h g, k}) \geq S_{i j} + \sum_{k \in M_{i j}} \sum_{l = 1}^{L_{i j k}} x_{i j k l} \times P_{i j k l} \\ S_{i j} + L \times y_{i j, h g, k l} \geq S_{h g} + \sum_{k \in M_{h g}} \sum_{l = 1}^{L_{i j k}} x_{h g k l} \times P_{h g k l} \end{cases} \forall (i, j), (h, g), k \end{aligned}

(8)

\begin{aligned} {\begin{cases} S_{i j} + \sum_{k \in M_{i j}} \sum_{l = 1}^{L_{i j k}} x_{i j k l} \times P_{i j k} \leq S_{k t}^{P M} + L \times (1 - f_{i j, k t} + (1 - x_{i j k l})) \\ S_{i j} \geq E_{k t}^{P M} - L \times (f_{i j, k t} + (1 - x_{i j k l})) \end{cases} \forall i, j, k, t \end{aligned}

(9)

\begin{aligned} C_{max} \geq S_{i, n_{i}} + \sum_{k \in M_{i, n_{i}}} x_{i, n_{i}, k} \times P_{i, n_{i}, k}, \forall i \in {1, 2, \dots, n} \end{aligned}

(10)

Constraint (6) stipulates that each operation must be assigned to exactly one compatible machine, and simultaneously, exactly one speed l must be selected from its available speed levels. This ensures the integration of machine allocation and speed selection. Constraint (7) enforces operation precedence while accounting for transportation delays, ensuring that a subsequent operation begins only after its predecessor is completed and the job has been transferred to the destination machine. Constraint (8) represents the machine capacity constraint, which precludes the simultaneous processing of multiple operations on a single machine. Constraint (9) restricts any processing activities during the scheduled maintenance windows of machine

M_{k}

. Finally, constraint (10) deﬁnes

C_{max}

by ensuring it serves as the upper bound for the completion times of all jobs.

3. MPOA-ASC for EFJSP-PMT

By partitioning the global population into several sub-populations and facilitating parallel exploration across diverse regions of the solution space, MPOAs eﬀectively maintain population diversity and mitigate the risk of premature convergence. However, despite their importance, the dynamic adaptation of search strategies guided by evolutionary states and the competitive interactions among sub-populations remain largely under-explored. To address these gaps, this paper develops a MPOA-ASC. As illustrated in Figure 1, the proposed framework incorporates distinct early and later evolutionary stages speciﬁcally tailored to solve the EFJSP-PMT.

Unlike generic metaheuristic frameworks, the design of MPOA-ASC is deeply driven by the inherent physical characteristics of the EFJSP-PMT. The integration of preventive maintenance (PM) and transportation times constraints creates a highly discontinuous scheduling landscape with severe bottlenecks and fragmented machine idle times. To effectively utilize these specific problem characteristics, MPOA-ASC incorporates problem-specific initialization rules, critical-path-based local search operators, and an adaptive competition mechanism to meticulously navigate the trade-off between makespan and TEC.

To clarify the algorithm flow depicted in Figure 1, the MPOA-ASC framework structurally begins with a hybrid triple-layer initialization. This is followed by a rank-based population division that assigns elite individuals to anchor distinct sub-populations. During the early evolution phase, each sub-population independently executes crossover and mutation operations, with their parameters adaptively controlled by a real-time ‘evolution quality’ metric. Subsequently, the algorithm enters the later evolution phase, where a global competition mechanism evaluates the normalized total fitness of all sub-populations, dynamically redistributing computational resources (sub-population sizes) to prevent stagnation before iterating until the termination criterion is met.

To clearly articulate the innovations of the proposed MPOA-ASC and differentiate it from existing multi-population frameworks, a structured methodological comparison is presented in Table 2. Unlike traditional methods that rely on static resource allocation and random population division, MPOA-ASC introduces a stage-based evolutionary architecture. It dynamically adjusts search strategies based on real-time evolution quality and employs a dynamic competition mechanism to optimize computational resource allocation.

Figure 1.

Framework Diagram of MPOA-ASC.

Table 2.

Methodological Comparison Between Traditional MPOAs and the Proposed MPOA-ASC.

Feature / Evolutionary Stage	Traditional MPOA Approaches	Proposed MPOA-ASC
Population Division (Early Phase)	Sub-populations are typically divided randomly. This often leads to an uneven distribution of initial quality, causing some sub-populations to converge prematurely while others waste resources in non-promising regions.	Employs a novel division method based on non-dominated sorting. Elite solutions are strategically distributed to anchor sub-populations, ensuring a balanced starting front and guiding parallel exploration effectively.
Parameter Adaptation (Early Phase)	Crossover and mutation probabilities are either kept constant or adjusted simply based on the current generation number.	Dynamically adjusts crossover and mutation probabilities based on the real-time “evolution quality” of each specific sub-population. Well-adapted sub-populations maintain strategies, while underperforming ones adapt to balance exploration and exploitation.
Resource Allocation & Competition (Later Phase)	The numerical count and scale of sub-populations remain rigid and fixed throughout the evolutionary process.	Sub-population sizes and counts are fluid and adaptive. Superior sub-populations are rewarded with expanded sizes, while stagnant or underperforming ones are autonomously restructured or deleted to prevent resource waste.
Diversity Maintenance (Later Phase)	Often struggles to balance rapid convergence with diversity, risking premature convergence if one sub-population dominates.	Constructs a specific allocation vector Q during competition to prevent the optimal sub-population from monopolizing all computational resources. This sustains population diversity while accelerating global convergence.

3.1. Encoding and Decoding

EFJSP-PMT consists of three sub-problems: scheduling, machine assignment and speed selection. A three-layer encoding representation is adopted, as presented in Figure 2. For OSV, the $j^{t h}$ occurrence of each number represents the $j^{t h}$ operation of $J_{i}$ , namely $O_{i j}$ . For MSV and SSV, the numbers indicate machines and speeds determined to process $O_{i j}$ . Additionally, the lengths of the OSV, MSV, and SSV are all equal to the total number of operations. The decoding procedure is described in Algorithm 1.

Figure 2.

A Schematic Diagram of Encoding.

3.2. Initialization and Population Division

To eﬀectively balance global convergence and population diversity while addressing the trade-oﬀ between time eﬃciency and energy conservation, a hybrid heuristic initialization method is developed, as shown in Algorithm 2. The population Pop with N solutions is generated based on a speciﬁc probability distribution across the three encoding layers: OSV, MSV and SSV.

(1)
MSV Layer: Comprehensive Load Balancing Strategies

The MSV layer determines the allocation of resources. To prevent premature convergence and ensure a high-quality starting front, three strategies are employed:

Improved Global Selection (GSA, 60%): This strategy evaluates the comprehensive cost of each candidate machine. To account for the variable speed levels in the SSV layer, the selection is performed using standardized parameters (at a median or standard speed level):
$\begin{aligned} min_{k \in M_{i j}} {ω \times T_{i, j, k}^{s t d} + (1 - ω) \times \frac{P_{k}^{s t d} \times T_{i, j, k}^{s t d}}{E_{a v g}}} \end{aligned}$
(11)
Where $T_{i, j, k}^{s t d}$ and $P_{k}^{s t d}$ represent the processing time and power consumption of operation $O_{i j}$ on machine k at the standard speed, respectively. $ω \in [0, 1]$ is a random weight factor used to diversify the search toward either the time or energy objective. $ω = 0.5$ by experiments. $E_{a v g}$ serves as a normalization factor representing the average energy consumption across all compatible machines.

Local Selection Strategy (LSA, 30%): This focuses on machine workload balancing. For each operation, the machine with the minimum current cumulative workload is selected to minimize potential idle energy consumption.

Random Selection (RS, 10%): Machines are assigned randomly from the set of compatible machines $M_{i j}$ to preserve diversity within the initial population. (2)
SSV Layer: Energy-Eﬃciency Regulation Strategies

The SSV layer is initialized immediately following the MSV assignment. For a selected machine with speed levels $V \in {1, 2, \dots, v_{k}}$ , the following rules are applied:

Minimum Processing Time Rule (40%): The maximum speed level $v_{max}$ is selected for all operations to provide the algorithm with initial solutions focused on minimizing the makespan ( $f_{1}$ ).

Maximum Energy Eﬃciency Rule (40%): This strategy targets the minimization of $f_{2}$ by selecting the speed level that minimizes the direct processing energy:
$\begin{aligned} v * = \arg min_{v \in {1, 2, \cdot \cdot \cdot, v_{k}}} {P_{k, l} \times T_{i, j, k, l}} \end{aligned}$
(12)
Where $P_{k l}$ and $T_{i, j, k, l}$ are the power and duration associated with speed level l.

Random Speed Selection (20%): Speed levels are chosen randomly to explore the intermediary regions of the Pareto front. (3)
OSV Layer: Bottleneck-Aware Sequencing Strategies

The OSV layer deﬁnes the execution sequence of the assigned operations:

Most Work Remaining (MWKR, 20%): To mitigate the impact of bottleneck jobs on the critical path, the total remaining processing time for each job $J_{i}$ is calculated. Jobs with higher cumulative workloads are prioritized and placed earlier in the sequence.

Random Permutation (80%): The majority of the OSV strings are generated via random permutations of the job indices. This ensures that the precedence constraints are satisﬁed while allowing the MPOA-ASC to explore a vast permutation space.

The ﬁtness value of the solution $x_{i}$ is calculated by Equation (13).
$\begin{aligned} f i t (x_{i}) = max_{_{l \in N}} {r a n k_{l}} - r a n k_{i} + c r o w d_{i} / \sum_{u \in Θ_{r a n k_{i}}} c r o w d_{u} \end{aligned}$
(13)
where $r a n k_{i}$ indicates the rank value of $x_{i}$ by non-dominated sorting on Pop (Deb et al., 2002), $Θ_{r a n k_{i}}$ is a set of solutions with $r a n k_{i}$ and represents $c r o w d_{i}$ crowding distance of $x_{i}$ .

For MPOA, Pop is typically randomly divided into several sub-populations. Although simple, this method often leads to an uneven distribution of initial solution quality among sub-populations. Some sub-populations may coincidentally receive a concentration of high-quality individuals, causing premature convergence. In contrast, others might consist largely of low-quality individuals whose search direction deviates signiﬁcantly from promising regions, thereby wasting substantial computation resources. To overcome the above shortcomings, a new population division method is designed, shown in Algorithm 3, where $ρ$ is a coeﬃcient, $α$ and $β$ are an integer and determined by
$\begin{aligned} α & = {\begin{array}{lc} c e i l {(0.3 - 0.025 \times η) \times (1 - ρ \times N)}, & η < ρ \times N \\ c e i l {(1 - ρ) \times N / (ρ \times N)}, & η \geq ρ \times N \end{array} \end{aligned}$
(14)

$\begin{aligned} β & = {\begin{array}{lc} c e i l {(1 - ρ) \times N - η \times α}, & η < ρ \times N \\ 0, & η \geq ρ \times N \end{array} \end{aligned}$
(15)

Suppose $ρ \times N = 6$
$\begin{aligned} α = {\begin{array}{lc} c e i l {0.275 \times (1 - ρ) \times N}, & i f η = 1 \\ c e i l {0.25 \times (1 - ρ) \times N}, & i f η = 2 \\ c e i l {0.225 \times (1 - ρ) \times N}, & i f η = 3 \\ c e i l {0.20 \times (1 - ρ) \times N}, & i f η = 4 \\ c e i l {0.175 \times (1 - ρ) \times N}, & i f η = 5 \\ c e i l {(1 - ρ) \times N / 6}, & i f η \geq 6 \end{array} \end{aligned}$
(16)

The $γ_{i}$ solutions assigned to $s u b - p o p_{i}^{g e n}$ (i = 1,2,…, $ρ \times N$ ) are described in Algorithm 4, where $s u b - p o p_{i}^{g e n}$ indicates the $i^{t h}$ sub-population at $g e n$ generation.

3.3. Early Evolution

In MPOA-ASC, sub-populations exhibit distinct evolutionary states due to diﬀerences in initial distribution. The adoption of a search strategy should be aligned with its speciﬁc evolutionary state. In this paper, an adaptive adjustment strategy for crossover probability $p c_{i}^{g e n}$ and mutation probability $p m_{i}^{g e n}$ based on evolution quality of $s u b - p o p_{i}^{g e n}$ is proposed to dynamically and intelligently balance exploration and exploitation, shown in Algorithm 5.

The evolution quality $T_f i t_{i}^{g e n}$ of $s u b - p o p_{i}^{g e n}$ is computed by

\begin{aligned} T_f i t_{i}^{g e n} & = F_f i t_{i}^{g e n} + S_f i t_{i}^{g e n} \end{aligned}

(17)

\begin{aligned} F_f i t_{i}^{g e n} & = {\begin{cases} \sum_{j}^{| {Θ^{'}}_{r n a k_{1}} |} f i t (x_{j}), & i f | {Θ^{'}}_{r n a k_{1}} | > 0 \\ 0, & e l s e \end{cases} \end{aligned}

(18)

\begin{aligned} S_f i t_{i}^{g e n} & = {\begin{cases} 0.1 \times \frac{1}{D_{v}} \times \sum_{j}^{D_{v}} f i t (x_{j}), & i f D_{v} > 1 \\ 0, & e l s e \end{cases} \end{aligned}

(19)

where

D_{v} = | s u b - p o p_{i}^{g e n} | - | {Θ^{'}}_{r a n k_{1}} |

{Θ^{'}}_{r a n k_{1}}

is a set of solutions with

r a n k_{1}

s u b - p o p_{i}^{g e n}

Selecting appropriate crossover and mutation operators is essential to achieve global search and avoid local optima. In this paper, the precedence operation crossover (POX) (Wang et al., 2026a) is implemented for the OSV part, and universal crossover (UX) (Wang et al., 2026a) is applied to the MSV and SSV, presented in Figures 3 and 4.

Figure 3.

POX Operators.

Figure 4.

Ux Operator.

Generic evolutionary operators often destroy the delicate machine-operation matching in highly constrained spaces. Therefore, to deeply utilize the inherent characteristics of the EFJSP-PMT, three specialized local search operators (Figure 5) are meticulously designed:

Figure 5.

The Designed Local Search Operators.

LS1 (Critical Path Machine Reassignment): PM tasks and transportation times often create severe bottlenecks on the critical path. LS1 identifies the last completed operation on this critical path and transfers it to the machine with the absolute shortest completion time, directly attacking the makespan bottleneck.

LS2 (Critical Path Workload Balancing): to further utilize the workload distribution characteristics, LS2 randomly selects a critical operation and reassigns it to a faster machine, effectively alleviating congestion caused by maintenance windows.

LS3 (Energy-Aware Speed Adjustment): deeply utilizing the multi-speed feature of the problem, LS3 targets a non-critical operation and lowers its machine's speed level. This intelligently absorbs the fragmented idle gaps caused by preceding transportation delays, minimizing energy consumption without extending the overall makespan.

Algorithm 6 describes procedures of crossover and mutation, where $L S_{g} (x_{u})$ indicates the neighborhood solutions of $x_{u}$ by using $L S_{g}$ .

After crossover and mutation are performed, $s u b - p o p_{i}^{g e n + 1}$ (i = 1,2,…, $ρ \times N$ )is produced. All sub-populations are sorted in descending order of $T_f i t_{i}^{g e n}$ . $ρ \times N$ sub-populations are categorized into two groups $G r o u p_{1}$ and $G r o u p_{2}$ , where the ﬁrst half of them is assigned to $G r o u p_{1}$ and the rest to $G r o u p_{2}$ . The crossover and mutation probabilities for sub-populations in group $G r o u p_{1}$ remain invariant. Conversely, for sub-populations in group $G r o u p_{2}$ , these probabilities are adaptively alternated between pre-defined upper and lower bounds. For sub-populations in $G r o u p_{1}$ , they have the potential to yield signiﬁcantly higher positive $T_f i t_{i}^{g e n + 1}$ . In other words, they are well-adapted, making the introduction of new probabilities unnecessary. Obviously, $p c_{i}^{g e n + 1}$ and $p m_{i}^{g e n + 1}$ are decided by the evolution quality.

3.4. Later Evolution

In the proposed MPOA-ASC, the competition mechanism facilitates the autonomous redistribution of computational resources. Superior sub-populations are prioritized with expanded population sizes or extended iteration cycles, whereas under-performing ones are either restructured or eliminated to prevent stagnation. This strategy eﬀectively mitigates resource expenditure on low-potential regions of the solution space, thereby accelerating global convergence. Furthermore, by preserving multiple successful evolutionary trajectories, the mechanism maintains high population diversity, prevents premature convergence, and sustains a robust dynamic balance between exploration and exploitation.

In this Section, this competitive framework is integrated into MPOA-ASC to ensure the rational allocation of computational resources. Unlike traditional static models, the numerical count and scale of sub-populations in our approach remain ﬂuid and adaptive. The speciﬁc implementation of this competitive process is detailed in Algorithm 7.

The normalized total ﬁtness $N T_f i t_{i}^{g e n}$ is computed by

\begin{aligned} N T_{-} f i t_{i}^{g e n} = T_{-} f i t_{i}^{g e n} / \sum_{j = 1}^{| s u b - p o p_{i}^{g e n} |} T_{-} f i t_{j}^{g e n} \end{aligned}

(20)

To prevent the optimal subpopulation from prematurely monopolizing all resources, which would lead to a sharp decline in population diversity and premature convergence, a vector Q is constructed.

\begin{aligned} Q = [N T_{-} f i t_{1}^{g e n} - ε_{1}, N T_{-} f i t_{2}^{g e n} - ε_{2}, \dots, N T_{-} f i t_{ρ \times N}^{g e n} - ε_{ρ \times N}] \end{aligned}

(21)

where

ε_{i} \in U [0, 1]

The number $t r_n u m_{w}^{g e n}$ allocated into $s u b - p o p_{w}^{g e n}$ is computed by

\begin{aligned} t r_n u m_{w}^{g e n} = {\begin{array}{ll} 0, & N T_{-} f i t_{w}^{g e n} / N T_{-} f i t_{t}^{g e n} \leq 1.05 \\ 0.05 \times | s u b - p o p_{t}^{g e n} |, & 1.05 < N T_{-} f i t_{w}^{g e n} / N T_{-} f i t_{t}^{g e n} \leq 1.1 \\ 0.1 \times | s u b - p o p_{t}^{g e n} |, & 1.1 < N T_{-} f i t_{w}^{g e n} / N T_{-} f i t_{t}^{g e n} \leq 1.15 \\ 0.15 \times | s u b - p o p_{t}^{g e n} |, & 1.15 < N T_{-} f i t_{w}^{g e n} / N T_{-} f i t_{t}^{g e n} \leq 1.2 \\ 0.2 \times | s u b - p o p_{t}^{g e n} |, & N T_{-} f i t_{w}^{g e n} / N T_{-} f i t_{t}^{g e n} > 1.2 \end{array} \end{aligned}

(22)

After the competition is ﬁnished, the crossover and mutation are executed as described in early evolution.

4. Computational Experiments and Results

All experiments are coded in Python and run on a PC with an Intel Core i7-1260 CPU (2.10 GHz), 16.0 GB RAM, and a 64-bit Windows 11 operating system.

4.1. Instances, Metrics and Comparative Algorithms

In this study, we choose mk01-15 (Brandimarte, 1993), 01a-18a (Dauzère-Pérès & Paulli, 1997) and 21 instances from (Barnes & Chambers, 1996), which are designed for testing FJSP and extended by adding energy consumption, preventive maintenance and transportation times information. $u_{k} \in [80, 120]$ , $w_{k} \in [5, 15]$ , $T_{k p} \in [5, 10]$ , $V = {1, 1.3, 1.55, 1.8, 2}$ , $E_{i j k} = ξ \times v_{i}^{2}$ , $ξ \in [2, 4]$ , $S E_{k} = 1$ , $T E = 3$ .

To quantitatively assess the algorithm performance, three widely recognized metrics are employed in this study: Hypervolume ( $H V$ ) (Zhang et al., 2024c), Inverted Generational Distance ( $I G D$ ) (Bosman & Thierens, 2003), and the Set Coverage ( $C$ -metric) (Zitzler & Thiele, 1999). These indicators have been extensively utilized to evaluate Pareto fronts in recent multi-objective scheduling research.

The $H V$ indicator computes the size of the objective space that is collectively dominated by the obtained Pareto approximation set S, bounded by a predefined reference coordinate $Z *$ . Mathematically, it is defined as:

\begin{aligned} H V (S) = λ (⋃_{x \in S} [f_{1} (x), Z_{1} *] \times [f_{2} (x), Z_{2} *]) \end{aligned}

(23)

Where

λ

denotes the Lebesgue measure. A larger

H V

signifies a superior Pareto front approximation, reflecting both excellent convergence towards the true front and a diverse distribution of solutions.

Z * = (1.1, 1.1)

As for the $I G D$ metric, it evaluates the average Euclidean distance from a uniformly distributed set of reference points $P *$ (sampled from the true or estimated Pareto front) to the closest solutions in the generated set S. It can be formulated as:

\begin{aligned} I G D (S, P *) = \frac{\sum_{v \in P *} d (v, S)}{| P * |} \end{aligned}

(24)

where

d (v, S)

represents the minimum Euclidean distance from a specific reference point v to the set S, and

| P * |

denotes the cardinality of the reference set. A smaller

I G D

value indicates that the obtained solutions are closer to the true Pareto front and exhibit a well-proportioned spread.

Finally, the $C$ -metric ( $C (A, B)$ ) is utilized to compare the relative coverage between two distinct non-dominated sets, A and B. It calculates the proportion of solutions in set B that are dominated by at least one solution in set A:

\begin{aligned} C (A, B) = \frac{| {u \in B : \exists v \in A, v ≻ u} |}{| B |} \end{aligned}

(25)

A higher $C (A, B)$ value implies that set A has a stronger dominance over set B.

In this paper, MPOA, QMOEA/D (Wang et al., 2026a), MCEA (Liu et al., 2024), EARL (Zhang et al., 2024d), IMOEA (Li et al., 2025a) are selected as comparative algorithms.

MPOA is designed as a baseline to demonstrate the effectiveness of the proposed strategies. In MPOA, the crossover and mutation probabilities for each sub-population remain fixed, and the competition mechanism is removed. The QMOEA/D is proposed by Wang et al. (Wang et al., 2026a) to solve low-carbon FJSP, aiming to minimize $C_{max}$ , total machine workload and total carbon emissions. After speed selection, preventive maintenance and transportation information are added, it can be directly applied to address EFJSP-PMT. Therefore, QMOEA/D is chosen as the ﬁrst comparative algorithm. The studies in (Li et al., 2025a; Liu et al., 2024; Zhang et al., 2024d) focused on addressing the EFJSP with transportation. The effectiveness of these algorithms has been validated in previous studies; therefore, we selected them as benchmarks for comparison.

4.2. Parameter Settings

The key parameters of MPOA-ASC are listed below: population size N, coeﬃcient $ρ$ , the termination condition of the early evolution Et₁, β₁, β₂, γ₁, γ₂. The Taguchi method is utilized to set the parameters. The levels of parameters are shown in Table 3. Table 4 gives the orthogonal array $L_{27} (3^{7})$ , where $I G D$ and $H V$ represent average value obtained by MPOA-ASC and each combination is run 20 times independently for instances mk10, 10a and setb4xxx. Figure 6 describes the main eﬀect plots of $I G D$ and HV.

Table 3.
Parameters of MPOA-ASC and Their Levels.

Factor Level

Parameter 1 2 3

$N$ 40 60 80

$ρ$ 0.05 0.1 0.15

Et ₁ 0.05 0.1 0.15

β ₁ 0.7 0.8 0.9

β ₂ 0.4 0.5 0.6

γ ₁ 0.1 0.2 0.3

γ ₂ 0.2 0.3 0.4

	Factor Level
$N$	40	60	80
$ρ$	0.05	0.1	0.15
Et ₁	0.05	0.1	0.15
β ₁	0.7	0.8	0.9
β ₂	0.4	0.5	0.6
γ ₁	0.1	0.2	0.3
γ ₂	0.2	0.3	0.4

Table 4.

The Orthogonal Array $L_{27} (3^{7})$ .

	Factor
No.	$N$	$ρ$	Et ₁	β ₁	β ₂	γ ₁	γ ₂	HV	IGD
1	1	1	1	1	1	1	1	0.059	0.026
2	1	1	1	1	2	2	2	0.046	0.491
3	1	1	1	1	3	3	3	0.058	0.162
4	1	2	2	2	1	1	1	0.061	0.408
5	1	2	2	2	2	2	2	0.010	0.634
6	1	2	2	2	3	3	3	0.025	0.698
7	1	3	3	3	1	1	1	0.020	0.335
8	1	3	3	3	2	2	2	0.051	0.619
9	1	3	3	3	3	3	3	0.068	0.068
10	2	1	2	3	1	2	3	0.025	0.403
11	2	1	2	3	2	3	1	0.026	0.600
12	2	1	2	3	3	1	2	0.015	0.415
13	2	2	3	1	1	2	3	0.010	0.506
14	2	2	3	1	2	3	1	0.030	0.378
15	2	2	3	1	3	1	2	0.016	0.678
16	2	3	1	2	1	2	3	0.036	0.478
17	2	3	1	2	2	3	1	0.004	0.648
18	2	3	1	2	3	1	2	0.048	0.513
19	3	1	3	2	1	3	2	0.054	0.149
20	3	1	3	2	2	1	3	0.012	0.278
21	3	1	3	2	3	2	1	0.055	0.179
22	3	2	1	3	1	3	2	0.046	0.116
23	3	2	1	3	2	1	3	0.059	0.267
24	3	2	1	3	3	2	1	0.014	0.539
25	3	3	2	1	1	3	2	0.025	0.119
26	3	3	2	1	2	1	3	0.045	0.072
27	3	3	2	1	3	2	1	0.065	0.229

Figure 6.

Main Effect Plots of IGD and HV.

The best setting of MPOA-ASC is $N = 60$ , $ρ = 0.1$ , β₁ = 0.8, β₂ = 0.5, γ₁ = 0.2, γ₂ = 0.3.

In addition, empirical observations show that MPOA, EARL, QMOEA/D, and MCEA can fully converge within this timeframe. Thus, the uniform stopping condition for all algorithms is set to $0.2 \times m \times n$ CPU seconds. All comparison algorithms adopt the same method to set parameters and the best parameter settings are listed below.

IMOEA: population size $N = 20$ , crossover probability $p c = 0.6$ .

EARL: population size $N = 120$ , penalty of being early $α_{s} = 1$ , penalty of being late $α_{ε} = 5$ , number of objective divisions $p = 7$ .

QMOEA/D: crossover rate $C r = 0.5$ mutation rate $M r = 0.2$ , discount factor $γ = 0.7$ , learning rate $α = 0.4$ , greedy factor $ε = 0.2$ .

MCEA: population size $N = 50$ .

To ensure a strictly fair and objective evaluation, special attention was given to standardizing the parameter tuning principles and establishing an equal computational budget across all evaluated algorithms. First, because differences in population sizes and internal operator complexities can significantly skew performance when measured by iteration counts, we discarded fixed generation limits in favor of a uniform maximum CPU time strategy. Specifically, the stopping condition for MPOA-ASC and all comparison algorithms (MPOA, EARL, IMOEA, QMOEA/D, and MCEA) was universally set to $0.2 \times m \times n$ seconds. This dynamic time limit guarantees that every algorithm receives the exact same computational resources tailored to the specific scale of the problem instance.

Second, regarding parameter configurations, we ensured that no algorithm was disadvantaged by sub-optimal settings. The Taguchi method, utilizing an orthogonal array, was employed to rigorously determine the optimal parameters for the proposed MPOA-ASC. To maintain strict fairness, all comparison algorithms adopted the exact same Taguchi method to independently tune and finalize their respective best parameter settings (such as population size, crossover rate, and mutation rate). This principle ensures that the comparative results reflect the peak performance of each algorithm under their individually optimized configurations.

4.3. Results and Analyses

To reduce the eﬀects of randomness, MPOA-ASC, MPOA, IMOEA, EARL, QMOEA/D and MCEA are executed independently 20 times for all test instances.

Since the true Pareto front $P *$ of the investigated problem is unknown, we construct a reference set $P *$ for EFJSP-PMT by combining the non-dominated solutions $P_{1} \cup P_{2} \cup P_{3} \cup P_{4} \cup P_{5} \cup P_{6}$ , where $P_{1}, P_{2}, P_{3}, P_{4}, P_{5}$ and $P_{6}$ represent the non-dominated solution sets obtained by MPOA-ASC, MPOA, IMOEA, EARL, QMOEA/D, and MCEA, respectively. Performance comparison of all algorithms via metrics $I G D$ , $H V$ and $C$ is summarized in Tables 5 –7, where M, M₁, I, E, Q and M₂ are abbreviations for MPOA-ASC, MPOA, IMOEA, EARL, QMOEA/D and MCEA, respectively.

Table 5.
Performance Comparison of all Algorithms via IGD-Metric.

Instance MPOA-ASC MPOA IMOEA EARL QMOEA/D MCEA

mk01 0 . 040 0.115 0.201 0.299 0.692 0.486

mk02 0.015 0.198 0.046 0.106 0.569 0.480

mk03 0.035 0.999 0.138 0.190 0.706 0.783

mk04 0.043 0.242 0.051 0.315 0.951 0.879

mk05 0.033 0.142 0.085 0.185 0.072 0.094

mk06 0.017 1.259 0.208 0.053 0.361 0.340

mk07 0.038 0.172 0.052 0.075 0.390 0.409

mk08 0.063 0.236 0.026 0.145 0.578 0.356

mk09 0.026 0.463 0.051 0.103 0.604 0.449

mk10 0.047 0.503 0.084 0.078 0.589 0.469

mk11 0.053 0.196 0.056 0.080 0.229 0.199

mk12 0.047 0.193 0.058 0.197 0.750 0.359

mk13 0.018 0.782 0.112 0.206 1.072 0.986

mk14 0.016 0.241 0.051 0.272 1.284 0.889

mk15 0.094 0.338 0.153 0.175 0.493 0.430

01a 0.024 0.144 0.082 0.194 0.091 0.094

02a 0.093 0.408 0.146 0.061 0.126 0.098

03a 0.095 0.571 0.272 0.153 0.206 0.177

04a 0.109 0.568 0.212 0.125 0.128 0.102

05a 0.015 0.126 0.067 0.150 0.059 0.068

06a 0.023 0.151 0.070 0.062 0.049 0.043

07a 0.033 0.396 0.185 0.103 0.229 0.128

08a 0.020 0.315 0.136 0.070 0.127 0.111

09a 0.045 0.457 0.192 0.035 0.068 0.071

10a 0.024 0.306 0.102 0.080 0.157 0.097

11a 0.038 0.257 0.095 0.068 0.082 0.109

12a 0.023 0.804 0.233 0.037 0.277 0.186

13a 0.018 0.553 0.089 0.105 0.187 0.149

14a 0.028 0.323 0.091 0.055 0.096 0.109

15a 0.046 0.313 0.098 0.056 0.112 0.072

16a 0.037 0.409 0.076 0.085 0.172 0.061

17a 0.086 0.223 0.116 0.103 0.102 0.119

18a 0.086 0.909 0.241 0.111 0.163 0.139

mt10c1 0.000 0.177 0.123 0.455 0.121 0.173

mt10cc 0.023 0.228 0.044 0.251 0.323 0.205

mt10x 0.036 0.209 0.063 0.146 0.297 0.272

mt10xx 0.042 0.229 0.040 0.164 0.349 0.175

mt10xxx 0.068 0.225 0.105 0.112 0.233 0.206

mt10xy 0.023 0.209 0.074 0.168 0.420 0.252

mt10xyz 0.056 0.171 0.079 0.289 0.469 0.364

setb4c9 0.020 0.124 0.038 0.103 0.161 0.179

setb4cc 0.042 0.176 0.045 0.186 0.384 0.223

setb4x 0.064 0.125 0.040 0.091 0.229 0.115

setb4xx 0.084 0.112 0.179 0.151 0.192 0.131

setb4xxx 0.097 0.231 0.103 0.182 0.476 0.213

setb4xy 0.034 0.131 0.111 0.153 0.300 0.280

setb4xyz 0.053 0.139 0.105 0.140 0.226 0.179

seti5c12 0.065 0.193 0.164 0.186 0.153 0.115

seti5cc 0.048 0.333 0.700 0.137 0.377 0.277

seti5x 0.080 0.167 0.090 0.111 0.296 0.145

seti5xx 0.028 0.253 0.111 0.129 0.275 0.322

seti5xxx 0.046 0.298 0.054 0.250 0.380 0.319

seti5xy 0.011 0.395 0.145 0.115 0.405 0.195

seti5xyz 0.064 0.346 0.061 0.307 0.379 0.334

Instance	MPOA-ASC	MPOA	IMOEA	EARL	QMOEA/D	MCEA
mk01	0 . 040	0.115	0.201	0.299	0.692	0.486
mk02	0.015	0.198	0.046	0.106	0.569	0.480
mk03	0.035	0.999	0.138	0.190	0.706	0.783
mk04	0.043	0.242	0.051	0.315	0.951	0.879
mk05	0.033	0.142	0.085	0.185	0.072	0.094
mk06	0.017	1.259	0.208	0.053	0.361	0.340
mk07	0.038	0.172	0.052	0.075	0.390	0.409
mk08	0.063	0.236	0.026	0.145	0.578	0.356
mk09	0.026	0.463	0.051	0.103	0.604	0.449
mk10	0.047	0.503	0.084	0.078	0.589	0.469
mk11	0.053	0.196	0.056	0.080	0.229	0.199
mk12	0.047	0.193	0.058	0.197	0.750	0.359
mk13	0.018	0.782	0.112	0.206	1.072	0.986
mk14	0.016	0.241	0.051	0.272	1.284	0.889
mk15	0.094	0.338	0.153	0.175	0.493	0.430
01a	0.024	0.144	0.082	0.194	0.091	0.094
02a	0.093	0.408	0.146	0.061	0.126	0.098
03a	0.095	0.571	0.272	0.153	0.206	0.177
04a	0.109	0.568	0.212	0.125	0.128	0.102
05a	0.015	0.126	0.067	0.150	0.059	0.068
06a	0.023	0.151	0.070	0.062	0.049	0.043
07a	0.033	0.396	0.185	0.103	0.229	0.128
08a	0.020	0.315	0.136	0.070	0.127	0.111
09a	0.045	0.457	0.192	0.035	0.068	0.071
10a	0.024	0.306	0.102	0.080	0.157	0.097
11a	0.038	0.257	0.095	0.068	0.082	0.109
12a	0.023	0.804	0.233	0.037	0.277	0.186
13a	0.018	0.553	0.089	0.105	0.187	0.149
14a	0.028	0.323	0.091	0.055	0.096	0.109
15a	0.046	0.313	0.098	0.056	0.112	0.072
16a	0.037	0.409	0.076	0.085	0.172	0.061
17a	0.086	0.223	0.116	0.103	0.102	0.119
18a	0.086	0.909	0.241	0.111	0.163	0.139
mt10c1	0.000	0.177	0.123	0.455	0.121	0.173
mt10cc	0.023	0.228	0.044	0.251	0.323	0.205
mt10x	0.036	0.209	0.063	0.146	0.297	0.272
mt10xx	0.042	0.229	0.040	0.164	0.349	0.175
mt10xxx	0.068	0.225	0.105	0.112	0.233	0.206
mt10xy	0.023	0.209	0.074	0.168	0.420	0.252
mt10xyz	0.056	0.171	0.079	0.289	0.469	0.364
setb4c9	0.020	0.124	0.038	0.103	0.161	0.179
setb4cc	0.042	0.176	0.045	0.186	0.384	0.223
setb4x	0.064	0.125	0.040	0.091	0.229	0.115
setb4xx	0.084	0.112	0.179	0.151	0.192	0.131
setb4xxx	0.097	0.231	0.103	0.182	0.476	0.213
setb4xy	0.034	0.131	0.111	0.153	0.300	0.280
setb4xyz	0.053	0.139	0.105	0.140	0.226	0.179
seti5c12	0.065	0.193	0.164	0.186	0.153	0.115
seti5cc	0.048	0.333	0.700	0.137	0.377	0.277
seti5x	0.080	0.167	0.090	0.111	0.296	0.145
seti5xx	0.028	0.253	0.111	0.129	0.275	0.322
seti5xxx	0.046	0.298	0.054	0.250	0.380	0.319
seti5xy	0.011	0.395	0.145	0.115	0.405	0.195
seti5xyz	0.064	0.346	0.061	0.307	0.379	0.334

Table 6.

Performance Comparison of all Algorithms via HV-Metric.

Instance	MPOA-ASC	MPOA	IMOEA	EARL	QMOEA/D	MCEA
mk01	0.897	0.788	0 . 915	0.717	0.423	0.568
mk02	0.843	0.612	0.803	0.750	0.372	0.405
mk03	0.924	0.300	0.889	0.782	0.341	0.376
mk04	0.954	0.818	0.957	0.785	0.479	0.602
mk05	0.788	0.611	0.699	0.585	0.760	0.728
mk06	0.896	0.184	0.700	0.893	0.634	0.654
mk07	0.815	0.668	0.809	0.798	0.547	0.521
mk08	0.833	0.621	0.873	0.737	0.340	0.529
mk09	0.862	0.491	0.843	0.774	0.450	0.526
mk10	0.830	0.357	0.758	0.733	0.268	0.365
mk11	0.776	0.588	0.740	0.719	0.575	0.598
mk12	0.906	0.758	0.875	0.784	0.368	0.610
mk13	0.919	0.512	0.802	0.822	0.409	0.436
mk14	0.938	0.754	0.936	0.772	0.238	0.434
mk15	0.936	0.634	0.891	0.788	0.404	0.532
01a	0.975	0.821	0.851	0.680	0.864	0.866
02a	0.808	0.543	0.694	0.809	0.768	0.809
03a	0.795	0.437	0.635	0.750	0.688	0.756
04a	0.804	0.458	0.663	0.769	0.725	0.779
05a	1.022	0.842	0.946	0.789	0.965	0.948
06a	0.736	0.583	0.694	0.675	0.700	0.777
07a	0.764	0.468	0.613	0.748	0.524	0.623
08a	0.798	0.539	0.686	0.734	0.723	0.723
09a	0.777	0.412	0.603	0.795	0.704	0.718
10a	0.804	0.442	0.716	0.734	0.630	0.717
11a	0.815	0.561	0.729	0.780	0.762	0.755
12a	0.872	0.476	0.708	0.800	0.688	0.743
13a	0.904	0.505	0.788	0.796	0.724	0.756
14a	0.748	0.469	0.671	0.709	0.672	0.660
15a	0.778	0.406	0.703	0.745	0.661	0.718
16a	0.745	0.401	0.707	0.684	0.641	0.754
17a	0.680	0.535	0.602	0.667	0.643	0.679
18a	0.899	0.456	0.759	0.883	0.850	0.852
mt10c1	1.048	0.729	0.859	0.421	0.831	0.792
mt10cc	0.855	0.659	0.801	0.641	0.548	0.655
mt10x	0.832	0.599	0.774	0.682	0.519	0.534
mt10xx	0.877	0.690	0.818	0.768	0.593	0.717
mt10xxx	0.806	0.584	0.781	0.735	0.583	0.611
mt10xy	0.860	0.695	0.806	0.728	0.520	0.666
mt10xyz	0.834	0.724	0.811	0.619	0.475	0.573
setb4c9	0.813	0.693	0.811	0.738	0.684	0.654
setb4cc	0.760	0.610	0.709	0.635	0.464	0.620
setb4x	0.810	0.755	0.859	0.742	0.638	0.772
setb4xx	0.775	0.713	0.730	0.664	0.592	0.647
setb4xxx	0.820	0.682	0.843	0.705	0.514	0.702
setb4xy	0.833	0.791	0.811	0.719	0.585	0.610
setb4xyz	0.769	0.616	0.728	0.630	0.477	0.548
seti5c12	0.848	0.593	0.719	0.706	0.616	0.698
seti5cc	0.797	0.521	0.738	0.675	0.490	0.614
seti5x	0.728	0.663	0.679	0.693	0.550	0.722
seti5xx	0.825	0.573	0.672	0.684	0.569	0.544
seti5xxx	0.801	0.484	0.775	0.574	0.416	0.462
seti5xy	0.879	0.524	0.767	0.804	0.528	0.696
seti5xyz	0.840	0.650	0.801	0.701	0.597	0.692

Table 7.

Performance Comparison of all Algorithms via C-Metric.

Instance	$C (M, M_{1})$	$C (M_{1}, M)$	$C (M, I)$	$C (I, M)$	$C (M, E)$	$C (E, M)$	$C (M, Q)$	$C (Q, M)$	$C (M, M_{2})$	$C (M_{2}, M)$
mk01	0 . 727	0.080	0.130	0.680	1.000	0.000	1.000	0.000	1.000	0.000
mk02	1.000	0.000	0.321	0.500	1.000	0.000	1.000	0.000	1.000	0.000
mk03	1.000	0.000	0.800	0.231	0.923	0.000	1.000	0.000	1.000	0.000
mk04	1.000	0.000	0.286	0.500	0.875	0.000	1.000	0.000	1.000	0.000
mk05	0.895	0.044	0.846	0.000	0.722	0.022	0.581	0.111	0.405	0.133
mk06	1.000	0.000	1.000	0.000	0.125	0.273	0.875	0.000	1.000	0.000
mk07	0.909	0.038	0.208	0.731	0.700	0.038	0.917	0.000	1.000	0.000
mk08	1.000	0.000	0.111	0.417	1.000	0.000	1.000	0.000	0.800	0.000
mk09	1.000	0.000	0.625	0.000	0.929	0.000	1.000	0.000	1.000	0.000
mk10	1.000	0.000	0.600	0.059	0.500	0.000	1.000	0.000	1.000	0.000
mk11	0.850	0.000	0.758	0.105	0.885	0.000	0.800	0.000	0.708	0.000
mk12	1.000	0.000	0.071	0.700	0.929	0.000	1.000	0.000	0.917	0.000
mk13	1.000	0.000	0.615	0.250	1.000	0.000	1.000	0.000	1.000	0.000
mk14	0.842	0.043	0.571	0.391	1.000	0.000	1.000	0.000	1.000	0.000
mk15	1.000	0.000	0.933	0.000	1.000	0.000	0.905	0.000	1.000	0.000
01a	1.000	0.000	0.933	0.053	0.833	0.000	0.760	0.053	0.480	0.132
02a	1.000	0.000	0.667	0.083	0.462	0.583	0.824	0.000	0.796	0.333
03a	1.000	0.000	0.875	0.000	0.857	0.545	0.428	0.273	0.556	0.273
04a	0.938	0.000	1.000	0.000	0.700	0.000	0.600	0.286	0.571	0.286
05a	1.000	0.000	0.783	0.094	1.000	0.000	0.618	0.076	0.654	0.113
06a	1.000	0.000	0.643	0.000	0.563	0.235	0.706	0.118	0.500	0.294
07a	1.000	0.000	0.833	0.000	0.400	0.154	1.000	0.000	0.500	0.154
08a	1.000	0.000	1.000	0.000	0.796	0.071	0.846	0.000	0.857	0.000
09a	1.000	0.000	0.875	0.000	0.462	0.333	0.733	0.167	0.700	0.083
10a	1.000	0.000	0.929	0.000	0.692	0.105	1.000	0.000	0.706	0.000
11a	0.929	0.000	0.917	0.000	0.700	0.190	0.643	0.095	0.700	0.095
12a	1.000	0.000	1.000	0.000	0.278	0.077	0.778	0.000	0.846	0.000
13a	1.000	0.000	0.823	0.111	0.923	0.111	0.941	0.000	0.867	0.000
14a	0.944	0.000	0.600	0.154	0.538	0.077	0.882	0.000	0.813	0.077
15a	0.875	0.000	0.769	0.000	0.500	0.267	0.842	0.000	0.833	0.067
16a	1.000	0.000	0.750	0.000	0.938	0.143	0.941	0.000	0.600	0.000
17a	0.909	0.000	0.727	0.222	0.429	0.333	0.357	0.333	0.636	0.556
18a	1.000	0.000	1.000	0.000	0.750	0.125	0.667	0.000	0.667	0.000
mt10c1	0.895	0.019	1.000	0.000	1.000	0.000	1.000	0.000	1.000	0.000
mt10cc	1.000	0.000	0.267	0.214	1.000	0.000	1.000	0.000	0.909	0.000
mt10x	1.000	0.000	0.444	0.289	0.889	0.000	0.875	0.000	1.000	0.000
mt10xx	0.923	0.100	0.444	0.200	0.857	0.100	1.000	0.000	0.786	0.100
mt10xxx	0.786	0.000	0.500	0.154	0.667	0.000	0.833	0.000	0.833	0.000
mt10xy	1.000	0.000	0.333	0.467	1.000	0.000	1.000	0.000	1.000	0.000
mt10xyz	0.900	0.000	0.250	0.688	0.917	0.000	0.714	0.000	0.944	0.000
setb4c9	0.917	0.045	0.444	0.182	0.643	0.000	0.857	0.091	0.909	0.045
setb4cc	1.000	0.000	0.450	0.231	0.909	0.000	0.813	0.000	0.778	0.000
setb4x	0.700	0.000	0.333	0.563	0.700	0.063	0.750	0.000	0.600	0.000
setb4xx	0.500	0.071	0.600	0.429	0.692	0.143	0.857	0.000	0.625	0.000
setb4xxx	0.823	0.000	0.583	0.500	0.818	0.000	0.714	0.000	0.583	0.000
setb4xy	0.846	0.111	0.750	0.222	1.000	0.000	0.857	0.000	0.917	0.000
setb4xyz	0.714	0.000	0.875	0.167	0.846	0.083	0.714	0.083	1.000	0.000
seti5c12	0.643	0.143	0.833	0.000	1.000	0.000	0.714	0.143	0.786	0.143
seti5cc	0.875	0.000	0.800	0.000	0.824	0.000	0.688	0.000	0.900	0.000
seti5x	0.700	0.083	0.455	0.333	0.750	0.000	0.900	0.000	0.444	0.167
seti5xx	1.000	0.000	0.636	0.000	1.000	0.000	1.000	0.000	1.000	0.000
seti5xxx	1.000	0.000	0.500	0.611	1.000	0.000	1.000	0.000	0.938	0.000
seti5xy	0.909	0.000	0.917	0.000	0.933	0.000	0.800	0.000	0.818	0.000
seti5xyz	1.000	0.000	0.417	0.500	0.750	0.167	1.000	0.000	0.900	0.083

Figure 7.

Violin Plots of all Algorithms on Metrics IGD, HV and C.

Figure 8.

Non-dominated Solution Distributions on mk05, 01a, 05a and mt10c1.

Figure 9.

The Gantt Chart of a non-Dominated Solution Produced by MPOA-ASC on Instance 05a.

As shown in Tables 5 –7, MPOA-ASC achieves superior results on metrics $I G D$ , $H V$ and $C$ on most test instances. A detailed analysis reveals the following: the metric $I G D$ of MPOA-ASC performs better than those of MPOA, IMOEA, EARL, QMOEA/D and MCEA on 54, 50, 52, 54 and 52 test instances. The metric $H V$ of MPOA-ASC is superior to MPOA, IMOEA, EARL, QMOEA/D and MCEA on 54, 51, 52, 54, 51 test instances. MPOA-ASC outperforms all comparison algorithms over 90% instances on these two metrics. For the metric $C$ , the values obtained by MPOA-ASC outperform compared algorithms on most test instances. $C (M, M_{1})$ , $C (M, Q)$ , $C (M, M_{2})$ are both greater than $C (M_{1}, M)$ , $C (Q, M)$ , $C (M_{2}, M)$ on all instances. The values of $C (M, I)$ , $C (M, E)$ , $C (I, M)$ perform better than those of $C (E, M)$ regarding nearly 80% of test instances. Therefore, MPOA-ASC is superior to MPOA, IMOEA, EARL, QMOEA/D and MCEA on metrics $I G D$ , $H V$ and $C$ .

Figures 7 and 8 are violin plots of all algorithms on metrics $I G D$ , HV, $C$ and non-dominated solution distributions on mk05, 01a, 05a and mt10c1. Figure 9 shows the Gantt chart of a non-dominated solution produced by MPOA-ASC on instance 05a. As shown in Figure 7, for the metric $I G D$ ( $H V$ ), the average values of MPOA-ASC are lower (higher) than that of MPOA, IMOEA, EARL, QMOEA/D and MCEA. It indicates that MPOA-ASC is superior to MPOA, IMOEA, EARL, QMOEA/D and MCEA.

In addition, the non-dominated solution distributions in Figure 8 visually corroborate the statistical findings. While competitive algorithms, particularly QMOEA/D, exhibit comparable performance in certain localized and central regions of the Pareto front, MPOA-ASC demonstrates distinct superiorities upon closer examination. First, MPOA-ASC consistently discovers better extreme solutions at the boundaries of the Pareto front (i.e., schedules with the absolute minimum makespan or the absolute minimum TEC). Second, the solution front generated by MPOA-ASC is generally positioned closer to the theoretical ideal point (the lower-left origin) across all four depicted instances (mk05, 01a, 05a, and mt10c1). Finally, MPOA-ASC maintains a more uniform and continuous spread of solutions, avoiding the sparse gaps often observed in the distributions of QMOEA/D and MCEA. This visual superiority in convergence and diversity is mathematically validated by the massive effect sizes ( ${\hat{A}}_{12}$ ) reported in Section 4.4, fully demonstrating the overall superiority of the proposed method.

4.4. Statistical Analysis and Discussion

4.4.1. Friedman Test and Wilcoxon Signed-Rank Test

The Friedman and Wilcoxon signed-rank tests are employed to assess variability differences among related samples. Tables 8 and 9 present the results of the Friedman test, where CN represents the number of instances. In Tables 8 and 9, the p-value is equal to 0, which indicates there are signiﬁcant diﬀerences among all algorithms. Additionally, Figure 10 visually depicts notable variations among the algorithms. These results aﬃrm that MPOA-ASC outperforms comparison algorithms.

Figure 10.

Related-Samples Friedman's two-way Analysis of Variance by Ranks.

Table 8.

The IGD Results of the Friedman Test.

Algorithm	Rank	CN	Mean	Std.Deviation	Min	Max
MPOA-ASC	1.15	54	0.047	0.026	0.010	0.110
MPOA	4.72	54	0.341	0.241	0.110	1.260
IMOEA	2.91	54	0.120	0.101	0.030	0.700
EARL	2.98	54	0.138	0.073	0.040	0.320
QMOEA/D	5.17	54	0.346	0.257	0.050	1.280
MCEA	4.07	54	0.264	0.214	0.040	0.990
p-value	4.6026E-32

Table 9.

The HV Results of the Friedman Test.

Algorithm	Rank	CN	Mean	Std.Deviation	Min	Max
MPOA-ASC	1.24	54	0.824	0.061	0.680	0.950
MPOA	4.89	54	0.570	0.133	0.180	0.820
IMOEA	2.81	54	0.759	0.085	0.600	0.960
EARL	2.96	54	0.732	0.064	0.570	0.890
QMOEA/D	5.26	54	0.561	0.133	0.240	0.850
MCEA	3.96	54	0.638	0.115	0.370	0.850
p-value	4.544E-33

The Wilcoxon signed-rank test, a widely adopted non-parametric statistical method, is employed to assess the signiﬁcance of performance diﬀerences between MPOA-ASC and other comparison algorithms. The Wilcoxon test operates based on the ranks of diﬀerences between paired observations, allowing inferences about whether such diﬀerences occur by chance. The results of the Wilcoxon signed-rank test at a 95% conﬁdence level $(α = 0.05)$ are provided in Table 10, where “Yes” represents that significant diﬀerence exists between MPOA-ASC and comparison algorithm. In the results, $R^{+}$ denotes the sum of ranks where MPOA-ASC outperforms another algorithm, while $R^{-}$ represents the opposite. For each algorithm pair, the sum of $R^{+}$ and $R^{-}$ equals the total number of compared cases.

Table 10.

Results Achieved by the Wilcoxon Test.

Comparison	Metric	$R^{+}$	$R^{-}$	p-Value	$α = 0.05$
MPOA-ASC vs. MPOA	$I G D$	54	0	1.6229E-10	Yes
	$H V$	54	0	1.6254E-10	Yes
	$C$	54	0	6.5488E-11	Yes
MPOA-ASC vs. IMOEA	$I G D$	50	4	1.3795E-9	Yes
	$H V$	51	3	3.6745E-9	Yes
	$C$	43	11	4.4998E-7	Yes
MPOA-ASC vs. EARL	$I G D$	52	2	3.6468E-10	Yes
	$H V$	52	2	2.5401E-10	Yes
	$C$	52	2	2.0857E-10	Yes
MPOA-ASC vs. QMOEA/D	$I G D$	54	0	1.6251E-10	Yes
	$H V$	54	0	1.6254E-10	Yes
	$C$	54	0	1.2991E-10	Yes
MPOA-ASC vs. MCEA	$I G D$	53	1	1.8183E-10	Yes
	$H V$	52	2	3.4505E-10	Yes
	$C$	54	0	1.4232E-10	Yes

A signiﬁcance level of $α = 0.05$ is used in this study; thus, a p-value less than $α$ indicates a statistically signiﬁcant diﬀerence between the paired algorithms. As shown in this Table, all p-values are below 0.05, conﬁrming that MPOA-ASC exhibits significant diﬀerences compared to each of the other algorithms—namely MPOA, IMOEA, EARL, QMOEA/D and MCEA—in solving the EFJSP-PMT problem, demonstrating superior performance.

4.4.2. Discussion on Algorithm Behaviors

To rigorously measure the magnitude of the performance differences, Vargha and Delaney's ${\hat{A}}_{12}$ effect sizes for the IGD and HV metrics are calculated and summarized in Table 11. While the exact p-values and these effect sizes statistically validate the superiority of MPOA-ASC, it is essential to trace these performance differences back to the underlying algorithmic mechanisms. The significant improvements observed can be attributed to three core structural advantages of the proposed algorithm:

Table 11.
Vargha and Delaney's ${\hat{A}}_{12}$ Effect Sizes for IGD and HV Metrics.

Comparison ${\hat{A}}_{12}$ for IGD Effect Size (IGD) ${\hat{A}}_{12}$ for HV Effect Size (HV)

MPOA-ASC vs. MPOA 0.0000 Large 0.9546 Large

MPOA-ASC vs. IMOEA 0.1423 Large 0.7227 Large

MPOA-ASC vs. EARL 0.0706 Large 0.8772 Large

MPOA-ASC vs. QMOEA/D 0.0225 Large 0.9388 Large

MPOA-ASC vs. MCEA 0.0262 Large 0.9239 Large

Comparison	${\hat{A}}_{12}$ for IGD	Effect Size (IGD)	${\hat{A}}_{12}$ for HV	Effect Size (HV)
MPOA-ASC vs. MPOA	0.0000	Large	0.9546	Large
MPOA-ASC vs. IMOEA	0.1423	Large	0.7227	Large
MPOA-ASC vs. EARL	0.0706	Large	0.8772	Large
MPOA-ASC vs. QMOEA/D	0.0225	Large	0.9388	Large
MPOA-ASC vs. MCEA	0.0262	Large	0.9239	Large

The massive statistical gap between MPOA-ASC and the baseline MPOA ( ${\hat{A}}_{12} = 0.000$ for IGD) highlights the critical impact of the proposed competition mechanism. Standard multi-population approaches allocate resources statically, which inevitably wastes computational effort on sub-populations that become trapped in local optima. By dynamically evaluating the normalized total fitness and reallocating resources, MPOA-ASC successfully rewards high-potential sub-populations with expanded sizes or extended iterations, while forcefully restructuring underperforming ones. This dynamic routing directly contributes to the significantly lower IGD scores, indicating accelerated and more accurate convergence.

Furthermore, the distinct advantage of MPOA-ASC over generic algorithms such as QMOEA/D and MCEA can be attributed to its specialized handling of problem-specific constraints. The integration of preventive maintenance and transportation times creates a highly discontinuous and rugged fitness landscape. The three local search operators embedded in the early evolution phase explicitly target critical path operations and machine idle gaps, providing MPOA-ASC with a highly directed exploitation capability to minimize both makespan and TEC. Generic crossover and mutation operators employed by the comparison algorithms often disrupt these delicate machine-operation-speed combinations, leading to inferior solutions.

Finally, the robust HV metrics achieved by MPOA-ASC, with all ${\hat{A}}_{12} > 0.71$ , demonstrate the algorithm's ability to avoid premature convergence and maintain Pareto front diversity. This behavior is fundamentally driven by the allocation vector Q introduced during the later competition phase, which restricts the optimal sub-population from monopolizing all computational resources. Coupled with the adaptive adjustment strategy for crossover and mutation probabilities based on evolution quality, MPOA-ASC preserves multiple successful evolutionary trajectories, sustaining a dynamic and intelligent balance between global exploration and local exploitation.

5. Conclusions

This paper presents a multi-population optimization algorithm with adaptive strategy and competition (MPOA-ASC) to address EFJSP-PMT. Extensive experimental results demonstrate that the proposed MPOA-ASC signiﬁcantly outperforms state-of-the-art comparison algorithms across the majority of test instances.

The core contributions of this study are summarized as follows. We investigated the EFJSP-PMT, a widespread yet complex problem in modern industry that integrates energy eﬃciency with maintenance and logistics constraints. A dual-phase evolutionary framework (early and later evolution) was developed. The early phase features a novel population division strategy, adaptive parameter control for crossover and mutation, and three tailored local search operators. The later phase introduces a sub-population competition mechanism to dynamically allocate computational resources and accelerate convergence.

While the proposed approach achieves superior results in a static environment, certain limitations remain. Specifically, dynamic uncertainties inherent in real-world manufacturing—such as unexpected job delays and machine breakdowns—were not considered. Furthermore, as our model primarily addresses transportation times, it neglects the constraints of finite transportation resources, such as limited AGV fleet sizes, battery charging cycles, and complex routing paths. Future research will extend the EFJSP-PMT framework to include factors like rework, operation sequence flexibility, and the joint optimization of job scheduling with dynamic AGV routing.

Footnotes

Acknowledgment

This work is supported by the Anhui Provincial University Outstanding Youth Project in Humanities and Social Sciences (Grant No. 2024AH020022) and the Anhui Province Social Science Innovation and Development Research Project (Grant No. 2024CX067).

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Anhui Province Social Science Innovation and Development Research Project, Anhui Provincial University Outstanding Youth Project in Humanities and Social Sciences, (grant number 2024CX067, 2024AH020022).

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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	Factor Level
Parameter	1	2	3
$N$	40	60	80
$ρ$	0.05	0.1	0.15
Et ₁	0.05	0.1	0.15
β ₁	0.7	0.8	0.9
β ₂	0.4	0.5	0.6
γ ₁	0.1	0.2	0.3
γ ₂	0.2	0.3	0.4

Energy-eﬃcient Flexible Job Shop Scheduling Problem Considering Preventive Maintenance and Transportation Times

Abstract

Keywords

1. Introduction

2. Problem Description and Mathematical Model

4.1. Instances, Metrics and Comparative Algorithms

Table 3. Parameters of MPOA-ASC and Their Levels. Factor Level Parameter 1 2 3 N 40 60 80 ρ 0.05 0.1 0.15 Et 1 0.05 0.1 0.15 β 1 0.7 0.8 0.9 β 2 0.4 0.5 0.6 γ 1 0.1 0.2 0.3 γ 2 0.2 0.3 0.4

4.4.1. Friedman Test and Wilcoxon Signed-Rank Test

Footnotes

Acknowledgment

Funding

Declaration of Conflicting Interests

References

Table 3.
Parameters of MPOA-ASC and Their Levels.

Factor Level

Parameter 1 2 3

$N$ 40 60 80

$ρ$ 0.05 0.1 0.15

Et ₁ 0.05 0.1 0.15

β ₁ 0.7 0.8 0.9

β ₂ 0.4 0.5 0.6

γ ₁ 0.1 0.2 0.3

γ ₂ 0.2 0.3 0.4