Energy-efficient flexible job-shop scheduling with transportation times and selectable speeds

Abstract

Paper aims  This study addresses an extended FJSP that simultaneously incorporates inter-machine workpiece transportation times and selectable machine processing speeds within a unified multi-objective framework.

Originality  A multi-objective mixed-integer model is formulated to jointly minimize makespan, total machine workload, and total energy consumption — decomposed into processing, idle, and transportation phases. An Improved Multi-Objective Jaya Algorithm (IMOJA) is proposed, featuring a three-layer encoding scheme, a transportation-aware insertion-based decoding method, four discrete update strategies, a Pareto-ranked external archive, and five tailored neighborhood search structures.

Research method  IMOJA is benchmarked against NSGA-II and SPEA2 on the ten extended Brandimarte instances (MK01–MK10) using Coverage (C) and Inverted Generational Distance (IGD) indicators over ten independent runs.

Main findings  IMOJA consistently dominates both competitors across all instances, achieving Coverage values close to 1.0 and IGD values up to two orders of magnitude smaller, including zero on four instances. On MK05, IMOJA yields a best makespan of 72.87 versus 80.00 and 83.37 for NSGA-II and SPEA2.

Implications for theory and practice  IMOJA offers a reliable decision-support tool for energy-conscious shop-floor scheduling where transportation logistics and speed flexibility must be jointly managed.

Keywords:
Flexible job-shop scheduling; Multi-objective optimization; Energy-efficient manufacturing; Jaya algorithm; Transportation time; Selectable processing speeds

1. Introduction

The Flexible Job-shop Scheduling Problem (FJSP) is a widely studied combinatorial optimization problem in manufacturing systems. Since its introduction by Brucker & Schlie (1990), the vast majority of research has focused on purely economic objectives such as makespan, machine workload, with efficiency-oriented metrics continuing to be refined in recent work, such as tailored measures to jointly minimize machine idle time and job waiting time in the flow shop (Abreu & Fuchigami, 2026). Environmental considerations, particularly energy consumption and carbon emissions, have long been neglected.

In recent years, growing societal and regulatory pressure for sustainable manufacturing has driven increasing attention toward green production objectives, though systematic, data-driven approaches for improving manufacturing energy efficiency remain notably scarce (Nabati et al., 2022). Several studies have proposed energy-aware FJSP models. For example, Shen et al. (2023) developed an energy cost-efficient scheduling framework for the flexible job shop, formulating the problem as a mixed-integer program and demonstrating significant reductions in total energy expenditure under time-of-use pricing schemes. In a related vein, Li & Chen (2023) proposed an improved multi-objective sparrow search algorithm to simultaneously minimize makespan and total carbon emissions in the green FJSP, incorporating worker learning effects into the scheduling model. While these contributions provide valuable insights for energy-efficient scheduling, they typically overlook critical real-world factors such as inter-machine workpiece transportation time and the possibility of selecting different machine processing speeds.

In practice, each job in a flexible job shop undergoes a sequence of operations that are often processed on different machines, making transportation time between machines unavoidable. This has motivated a growing stream of research on FJSP with transportation considerations. Pal et al. (2023) proposed a decentralized multi-agent system incorporating an auctioning mechanism and critical-path-based neighborhood search to address the FJSP with setup and transportation times, outperforming existing centralized metaheuristics on makespan. Berterottière et al. (2024) further investigated flexible job-shop scheduling with explicit transportation resources, establishing new modeling foundations and demonstrating the practical relevance of jointly optimizing machine and vehicle assignments.

At the same time, modern CNC machines and automated production lines frequently allow selectable processing speeds, where higher speeds reduce processing time but increase energy consumption per unit time. Studies addressing this dimension include Luo et al. (2019), who developed a multi-objective grey wolf optimizer with crossover and mutation operators to minimize energy consumption and makespan under variable processing speeds in the FJSP. Wei et al. (2022) further extended this line of research by proposing hybrid energy-efficient scheduling measures for the FJSP with variable machining speeds, introducing a five-state energy estimation model and demonstrating superior environmental performance.

More recently, a limited but growing body of work has begun integrating subsets of these practical constraints into green FJSP models. Xu et al. (2023) addressed multi-objective green scheduling integrating FJSP and AGVs. Zhang et al. (2024) further proposed an energy-aware flexible job shop scheduling framework with multiple AGVs, minimizing both makespan and total energy consumption through a deep reinforcement learning-based memetic algorithm. Li et al. (2026) formulated a green flexible job shop scheduling problem integrating AGV transportation and machine energy consumption, proposing an improved NSGA-II to jointly minimize makespan, energy consumption, and AGV transportation distance.

Beyond these problem-focused contributions, the broader landscape of multi-objective scheduling has also been enriched by advances in Automated Algorithm Design (AAD). He et al. (2025) introduced the multi-objective distributed hybrid flowshop batch scheduling problem with human-robot collaboration (DHFBSP_HC), proposing a Local Search-Reconstruction-Feedback (LSRF) algorithm whose configurable strategies — covering fitness evaluation, initialization, and objective normalization — are automatically optimized via I/F-Race to jointly minimize makespan and total energy consumption in small-batch, human-robot collaborative production environments. In a related vein, Zhang et al. (2025) tackled the multi-objective reconfigurable distributed flowshop group scheduling problem (MORDFGSP) in SMT workshops, developing a two-layer decomposition-based approach whose heuristic components are automatically configured through AAD, producing an Automated Multi-Objective Algorithm (AMOA) that minimizes the number of enabled cells and makespan with minimal manual intervention. These contributions demonstrate the growing potential of AAD methodologies to autonomously compose high-quality scheduling algorithms across complex and heterogeneous manufacturing environments, representing a complementary paradigm to problem-tailored algorithm design.

Nevertheless, from both perspectives — problem formulation and algorithmic design — important gaps remain. No existing study simultaneously incorporates inter-machine workpiece transportation times, selectable machine processing speeds, and a detailed three-component energy consumption objective (processing, idle, and transportation phases) within a unified multi-objective FJSP framework. Furthermore, while AAD approaches offer strong generalizability, the green FJSP with these tightly coupled constraints calls for a dedicated, problem-tailored solution framework.

To bridge this gap, the present study makes the following contributions. First, a multi-objective mixed-integer programming model is formulated that integrates inter-machine workpiece transportation times, selectable machine processing speeds, and a detailed energy consumption objective decomposed into processing, idle, and transportation phases. Three conflicting performance indicators are jointly optimized: makespan, total machine workload, and total energy consumption. Second, an Improved Multi-Objective Jaya Algorithm (IMOJA) is proposed to solve the model effectively. The Jaya algorithm (Rao, 2016) is selected as the algorithmic foundation due to its minimal parameterization — requiring only population size and maximum number of iterations, with no algorithm-specific control parameters — and its proven adaptability to complex combinatorial optimization problems. Crucially, the Jaya algorithm has already demonstrated competitive performance on scheduling problems closely related to the present study: Caldeira & Gnanavelbabu (2021) developed a Pareto-based discrete Jaya algorithm for the multi-objective FJSP, confirming its suitability for this class of problems. Building on this foundation, IMOJA extends the discrete Jaya framework with a three-layer encoding scheme for operation sequencing, machine allocation, and speed selection, combined with an insertion-based active decoding method that explicitly accounts for transportation and speed-dependent processing times. Further enhancements include a hybrid population initialization strategy, four discrete individual update mechanisms, a Pareto-ranked external archive, and five tailored neighborhood search structures targeting all three optimization objectives.

The remainder of this paper is organized as follows. Section 2 formally describes the problem and presents the multi-objective mixed-integer mathematical model, including the three objective functions and all associated constraints. Section 3 details the proposed IMOJA, covering the three-layer encoding and insertion-based decoding scheme, the hybrid population initialization strategy, the Pareto ranking and external archive mechanism, the four discrete individual update strategies, and the five tailored neighborhood search structures. Section 4 presents the computational experiments conducted on the extended Brandimarte benchmark instances (MK01–MK10), including the experimental settings, performance indicators, and a comparative analysis of IMOJA against NSGA-II and SPEA2. Finally, Section 5 summarizes the main findings and outlines directions for future research.

2. Problem description and mathematical model

2.1. Problem description

The scheduling problem considered in this study is a multi-objective extension of the FJSP that incorporates workpiece transportation times and selectable machine processing speeds. Formally, a set of n jobs $J=\left\{J_1,J_2,\dots ,J_n\right\}$ must be processed on a set of m machines $M=\left\{M_1,M_2,\dots ,M_{\text{m}}\right\}$. Each job $J_{\text{i}}(i=\mathrm{1,}\dots ,n)$ consists of nᵢ operations with a fixed processing sequence, denoted $O_{\text{i}}=\left\{O_{\text{i}1},O_{\text{i}2},\dots ,O_{ini}\right\}$. Each operation Oᵢⱼ can be assigned to one of several eligible machines $m_{ij}\subseteq M$.

Each machine $M_{\text{k}}(k=\mathrm{1,}\dots ,m)$ offers q discrete selectable processing speed levels $v=\left\{v_1,v_2,\dots ,v^{\text{i}}\right\}$, where a higher speed level reduces processing time but increases average power consumption per unit time. Specifically, for any machine Mₖ and two speed levels $v_{ka}>v_{\text{k}}^{\text{b}}$, the corresponding processing time satisfies $T_{ka}<T_{\text{k}}{}^{\text{b}}$ and the average power consumption satisfies $P{L}_{ka}>P{L}_{\text{k}}{}^{\text{b}}$.

In real shop-floor environments, when two consecutive operations of the same job are processed on different machines, a known transportation time $T{t}_{kl}$ (which may differ between machine pairs) is incurred. These transportation times are deterministic and predefined.

The following assumptions are made for this problem:

1. All jobs and machines are available at time zero.

2. Each machine can process at most one operation at a time.

3. No job can be processed on more than one machine simultaneously.

4. Operations are non-preemptive; once started, an operation cannot be interrupted.

5. Each operation must be processed exactly once on exactly one eligible machine.

6. The operation sequence within each job is fixed and cannot be altered; operations from different jobs are independent.

7. Once started, a machine remains active until all operations assigned to it are completed (i.e., no intermediate shutdowns are allowed).

8. The processing speed of an operation cannot be changed once the operation has started on a machine.

9. Transportation time is only considered between consecutive operations of the same job; it is not incurred before the first operation or after the last operation of any job. Transportation resources are assumed to be unconstrained.

2.2. Model formulation

The variable symbols and their definitions used in this study are listed in Table 1.

Table 1
Variable symbols and their definitions.

Symbol	Definition
i, g	Job index
j, h	Operation index
k, l	Machine index
e	Speed level index
nᵢ	Number of operations of job Jᵢ
mᵢⱼ	Set of eligible machines for operation Oᵢⱼ
Cᵢ	Completion time of job Jᵢ
Sᵢⱼₖ	Start time of operation Oᵢⱼ on machine Mₖ
Cᵢⱼₖ	Completion time of operation Oᵢⱼ on machine Mₖ
TSₖ	Start-up time of machine Mₖ
TCₖ	Shut-down time of machine Mₖ
Tᵢⱼₖᵥe	Processing time of operation Oᵢⱼ on machine Mₖ at speed level ve
tᵢⱼₖ	Processing time of operation Oᵢⱼ on machine Mₖ at the reference speed level
Ttₖₗ	Transportation time of a workpiece from machine Mₖ to machine Mₗ
PLₖᵥe	Average power consumption per unit time of machine Mₖ at speed level ve
PNₖ	Average idle power consumption per unit time of machine Mₖ
PT	Average power consumption per unit time during workpiece transportation
P	A sufficiently large positive real number (big-M)
Xᵢⱼₖᵥe	Binary variable: 1 if operation Oᵢⱼ is processed on machine Mₖ at speed level ve; 0 otherwise
Yᵢⱼᵍₕₖ	Binary variable: 1 if operation Oᵢⱼ precedes operation Oᵍₕ on machine Mₖ; 0 otherwise
Zᵢⱼₖₗ	Binary variable: 1 if operation Oᵢⱼ is transported from machine Mₖ to machine Mₗ; 0 otherwise

• Source: Elaborated by the authors.

2.2.1. Objective functions

The proposed mathematical model seeks to simultaneously minimize three conflicting objectives that reflect both production efficiency and environmental performance.

1) Makespan ($C_{\text{max}}$): a direct indicator of shop-floor production efficiency.

$\min {\text{C}}_{\text{max}}=\max \left\{{\text{C}}_{\text{i}}\right|1\le \text{i}\le \text{n}\}$ (1)

2) Total Machine Workload (TL): a measure of overall machine resource utilization across the shop floor.

$\text{min TL}={{\displaystyle \sum }}_{\text{i}=1}^{\text{n}}{{\displaystyle \sum }}_{\text{j}=1}^{{\text{n}}_{\text{i}}}{{\displaystyle \sum }}_{\text{k}=1}^{\text{m}}{{\displaystyle \sum }}_{\text{e}=1}^{\text{q}}{\text{T}}_{\text{ijkve}}{\text{X}}_{\text{ijkve}}$ (2)

3) Total Energy Consumption (E): a green production objective that captures energy-efficient manufacturing. The total energy consumption consists of three components: processing energy (Eₗ), idle energy (Eₙ), and transportation energy (Eₜ).

$\min E=E_l+E_n+E_t$ (3)

$E_l={{\displaystyle \sum }}_{i=1}^n{{\displaystyle \sum }}_{j=1}^{n_i}{{\displaystyle \sum }}_{k=1}^m{{\displaystyle \sum }}_{e=1}^q{T}_{ijkve}{X}_{ijkve}P{L}_{kve}$ (4)

$E_n={{\displaystyle \sum }}_{k=1}^m\left(T{C}_k-T{S}_k-{{\displaystyle \sum }}_{i=1}^n{{\displaystyle \sum }}_{j=1}^{n_i}{{\displaystyle \sum }}_{e=1}^q{T}_{ijkve}{X}_{ijkve}\right)\cdot P{N}_k$ (5)

$$E_t={{\displaystyle \sum }}_{i=1}^n{{\displaystyle \sum }}_{j=1}^{n_i}{{\displaystyle \sum }}_{k=1}^m{{\displaystyle \sum }}_{l=1}^{m}T{t}_{kl}{Z}_{ijkl}\cdot PT$$ (6)

To justify the multi-objective nature of the formulated model, it is essential to highlight the structural conflicts among the three objectives. In a conventional job-shop, minimizing makespan ($C_{max}$) and total machine workload (TL) might occasionally align. However, in the considered extended FJSP with selectable speeds and transportation times, these indicators are strictly competitive. Minimizing TL forces the system to assign operations strictly to their most energy-efficient or fastest machines, which inevitably creates severe bottlenecks on heavily loaded units and inflates $C_{max}$. Conversely, reducing $C_{max}$ requires workload balancing across less efficient parallel machines, which inherently increases TL. Furthermore, the selectable processing speeds introduce a non-linear trade-off: higher speeds minimize $C_{max}$ at the expense of an exponential spike in processing energy (E), whereas lower energy-saving speeds prolong the schedule duration. Finally, frequent inter-machine routing balances workloads but introduces additional transportation times and energy consumption. Therefore, these three objectives cannot be simultaneously optimized by a single ideal solution, necessitating a multi-objective Pareto-based optimization approach.

2.2.2. Constraints

The model is subject to the following constraints:

1. Each machine can process at most one operation at a time:

$C_{ijk}\le S_{ghk}+P\times \left(1-Y_{ijghk}\right),\forall i,j,g,h,k$ (7)

2. Precedence constraint between consecutive operations of the same job, including transportation time:

$$S_{i\left(j+1\right)l}\ge C_{ijk}+T{t}_{kl}\cdot Z_{ijkl},\forall i,j>\mathrm{1,}k,l$$ (8)

3. Non-preemption of operations:

$C_{ijk}-S_{ijk}=T_{ijk},\forall i,j,k$ (9)

4. Processing time at a given speed level:

$T_{ijkve}=\frac{t_{ijk}}{v_e},\forall i,j,k,e$ (10)

5. Each operation is processed exactly once on exactly one eligible machine:

${{\displaystyle \sum }}_{k=1}^m{{\displaystyle \sum }}_{e=1}^q{X}_{ijkve}=\mathrm{1,}\forall i,j$ (11)

6. Sequencing constraints between operations assigned to the same machine:

${{\displaystyle \sum }}_{i=1}^n{{\displaystyle \sum }}_{j=1}^{n_i}{Y}_{ijghk}\le \mathrm{1,}\forall g,h,k$ (12)

${{\displaystyle \sum }}_{g=1}^n{{\displaystyle \sum }}_{h=1}^{n_g}{Y}_{ijghk}\le \mathrm{1,}\forall i,j,k$ (13)

7. Each operation is transported at most once:

${{\displaystyle \sum }}_{k=1}^m{{\displaystyle \sum }}_{l=1}^m{Z}_{ijkl}\le \mathrm{1,}\forall i,j>1$ (14)

It is important to note that the proposed MIP model serves three distinct roles in this work. First, it provides a rigorous formal characterization of the extended problem, ensuring that all scheduling constraints — including machine capacity, job precedence with transportation times, non-preemption, speed-dependent processing, and transportation routing — are explicitly and unambiguously defined. Second, it directly guides the design of the IMOJA decoding procedure (Algorithm 1): each constraint in the model has a corresponding enforcement mechanism in the decoder, guaranteeing that every generated schedule is feasible by construction. Third, it enables formal feasibility verification of IMOJA's output solutions. Because the extended FJSP is strongly NP-hard and the number of binary decision variables ($X_{ijk{v}_e}$, $Y_{ijghk}$, $Z_{ijkl}$) grows combinatorially with instance size, exact MIP solving is computationally intractable for the Brandimarte benchmark instances used in this study. Accordingly, IMOJA is proposed as the computational solution method, following the standard modeling paradigm in the green FJSP literature (Luo et al., 2019; Wei et al., 2022; Li et al., 2026). A small-scale computational validation of the model is presented in Section 4.4.

3. Improved multi-objective jaya algorithm for solving FJSP

3.1. Standard Jaya algorithm

The Jaya algorithm is a simple yet powerful population-based metaheuristic originally proposed by Rao (2016). The word "Jaya" means "victory" in Sanskrit, which reflects the algorithm's underlying philosophy: moving each solution toward the best one while simultaneously steering it away from the worst one. Thanks to its minimal set of control parameters — only population size and maximum number of iterations — the Jaya algorithm exhibits excellent adaptability and has been successfully applied to a wide variety of complex optimization problems (Rao, 2016, 2019).

Since its introduction, the algorithm has gained substantial traction across numerous industrial and engineering fields due to its high convergence speed and parameter-less nature. In advanced manufacturing, it has been effectively utilized to solve highly non-linear problems, such as optimizing process parameters in submerged arc welding designs (Rao & Rai, 2017). Furthermore, in electrical and energy engineering, the Jaya framework has proven highly efficient for modern power grid management, notably in complex multi-objective microgrid optimization problems (Ahmadipour et al., 2024).

Crucially, the structural simplicity of the Jaya algorithm has facilitated its extension to discrete, hybrid, and combinatorial optimization domains. Beyond its native continuous formulations, specialized hybrid configurations—such as blending Jaya mechanics with the Firefly algorithm—have been successfully tailored to solve complex technical engineering challenges (Vishnu & Kumar, 2022). This adaptability is further supported by its proven track record in multi-objective flexible job-shop scheduling environments (Caldeira & Gnanavelbabu, 2021), rendering it an ideal metaheuristic foundation for the extended green FJSP framework addressed in this study.

The core update mechanism drives each individual toward the current best solution and away from the current worst solution. The candidate position is calculated as follows:

$V_i^{t+1}=X_i^t+r_1\cdot \left(X_{Best}^t-X_i^t\right)-r_2\cdot \left(X_{Worst}^t-X_i^t\right)$ (15)

The new position is then accepted according to the following selection rule:

$X_i^{t+1}=\{\begin{array}{c}{V}_i^{t+1}ifF\left(V_i^{t+1}\right)\le F\left(X_i^t\right)\\ X_i^{t}otherwise\end{array}$ (16)

where $X_i^t$ denotes the i-th individual at iteration t; $X_{\text{Best}}^t$and $X_{\text{Worst}}^t$ are respectively the best and worst solutions in the current population; r₁ and r₂ are two random numbers uniformly distributed in [0, 1]; and F(·) represents the objective fitness function. If the candidate solution $V_i^{t+1}$ is not worse than the current solution $X_i^t$, it replaces the latter; otherwise, the original solution is retained.

3.2. Encoding and decoding

The scheduling problem addressed in this study comprises three interrelated sub-problems: operation sequencing (OS), machine allocation (MA), and speed selection (SS). To handle these simultaneously, a three-layer encoding scheme is adopted. Each layer has the same length, equal to the total number of operations across all jobs.

• The OS layer determines the global processing sequence of all operations;

• The MA layer assigns a machine to each operation;

  • The SS layer selects the processing speed level for each operation on its assigned machine.

To illustrate the encoding scheme, consider a small instance with 4 jobs processed on 3 machines, where each job consists of 2 operations. Figure 1 presents one feasible individual encoding. In the OS layer, each digit represents a job index; the number of times a job index appears equals the number of operations of that job. For example, the first occurrence of "2" corresponds to operation $O_{21}$ and the second to $O_{22}$ $.$ In the MA layer, the encoding follows the approach described in Yuan et al. (2013): the positions correspond to the fixed sequential order of operations within each job (not the OS decoding order). Each value indicates the index within the set of eligible machines for that operation, rather than the absolute machine number. For instance, the value "2" for operation $O_{11}$ in the MA layer means that it is assigned to the second machine in its eligible set $\left\{M_1,M_2,M_3\right\}$, i.e., machine $M_2$. The SS layer follows the same ordering as the MA layer, with each entry specifying the chosen processing speed level for the corresponding operation. As shown in Figure 1, operation $O_{11}$ selects speed level 3 on its assigned machine. Together, the three layers completely define the operation sequence, machine assignment, and speed selection, thus generating a complete schedule.

Figure 1
Illustration of the proposed encoding scheme and active decoding procedure: (a) Three-layer chromosome representation (OS, MA, and SS layers) for an instance with 4 jobs and 3 machines; (b) Corresponding physical schedule (Gantt chart) generated by the insertion-based decoding method, integrating speed-dependent processing times ($p_{ij}$) and inter-machine transportation times ($T{t}_{ab}$). Source: Elaborated by the authors.

For the decoding phase, an insertion-based active decoding method is specifically designed to accommodate the problem characteristics, namely workpiece transportation times and speed-dependent processing times.

To evaluate the fitness of any generated individual, a decoding procedure is required. It is important to emphasize that Algorithm 1 acts strictly as a deterministic genotype-to-phenotype mapping layer, rather than an iterative local search methodology. While a local search or a metaheuristic iteratively explores a neighborhood of alternative solutions by applying stochastic moves (e.g., swap or shift operators) on the chromosome structure, the decoding scheme executes a single-pass, deterministic translation of a single chromosome.

Following the standard scheduling taxonomy, this procedure applies a strict active decoding rule, mapping the un-timed sequence of the OS, MA, and SS layers directly onto a concrete physical timeline. The complete pseudocode is presented in Algorithm 1.

• Algorithm 1: Insertion-Based Decoding Method

Input: Individual X (comprising OS, MA, and SS layers)

Output: Objective values (Cmax, Total Workload TL, Energy Consumption E)

1. Initialize:

o$I{S}_m\leftarrow \left\{\left[\mathrm{0,}+\infty \right)\right\}$ for each machine $M_m$

o$C_{ij}\leftarrow 0$ for all jobs and operations

2. For p = 1 to length(OS) do

3. Identify the current operation$O_{ij}$corresponding to the p-th job index in the OS layer.

4.$M_m$ ← selected machine from MA layer; s ← selected speed level from SS layer.

5. Calculate processing time:$p_{ij}\leftarrow {\text{workload}}_{ij}/{\text{speed}}_s$.

6. If j > 1 then

[indent] 7.$M_{prev}\leftarrow$ machine used by the previous operation $O_{i,j-1}$.

[indent] 8. $ts\leftarrow C_{i,j-1}+T{t}_{M_{prev},M_m}$

9. Else

[indent] 10. $ts\leftarrow 0$

11. End If

12. Find the first idle slot$\left[t_{start},t_{end}\right]\in I{S}_m$ such that $\max \left(ts,t_{start}\right)+p_{ij}\le t_{end}$.

13. If a feasible idle slot is found then

[indent] 14. Start time: $S_{ij}\leftarrow \max \left(ts,t_{start}\right)$

[indent] 15. End time: $C_{ij}\leftarrow S_{ij}+p_{ij}$

[indent] 16. Update$I{S}_m$ by removing or splitting the chosen idle slot.

17. Else

[indent] 18. $S_{ij}\leftarrow \max \left(ts,\text{end of schedule on}{M}_m\right)$

[indent] 19. $C_{ij}\leftarrow S_{ij}+p_{ij}$

[indent] 20. Update the available timeline for$M_m$.

21. End If

22. End For

23. Compute Makespan$C_{max}=\max \left(C_{ij}\right)$, Total Workload TL, and Energy Consumption E.

24. Return$C_{max},TL,E$.

As detailed in Algorithm 1, the decoding procedure begins by initializing, for each machine, a single unbounded idle time slot and setting all operation completion times to zero (line 1). It then iterates through the operation sequence defined by the OS layer (line 2), identifying at each step the current operation and its position within its job (line 3), retrieving the assigned machine and selected speed level from the MA and SS layers (line 4), and computing the corresponding processing time (line 5). If the operation is not the first within its job (line 6), the earliest feasible start time accounts for the transportation time from the machine used by the previous operation (lines 7–8); otherwise, the job may start immediately (lines 9–10). The algorithm then searches for the first idle slot capable of accommodating the operation (line 12) and applies an active insertion strategy: if a feasible slot is found, the operation is scheduled within it and the idle-slot set is updated accordingly (lines 13–16); otherwise, the operation is appended to the end of the machine's timeline (lines 17–20). Operations assigned to the same machine are thus shifted as far left as possible, producing compact schedules that reduce idle time gaps, improve machine utilization, and contribute to lower energy consumption. This process repeats until every operation has been scheduled (line 22), after which the makespan, total workload, and energy consumption are computed and returned (lines 23–24).

3.3. Population initialization

A high-quality initial population is critical for the performance of swarm intelligence algorithms. To ensure both solution quality and population diversity, a hybrid initialization strategy is designed based on the three-layer encoding scheme (OS, MA, and SS).

• OS layer: individuals are generated completely at random, ensuring broad coverage of the operation sequencing space;

• MA layer: a hybrid strategy extends the global search, local search, and random selection approach proposed by Zhang et al. (2011) by explicitly incorporating inter-machine workpiece transportation times. Specifically, a minimum total working time strategy — which selects, for each operation, the machine minimizing the combined processing time and subsequent transportation time along the job’s routing — is combined with a random assignment strategy that selects an eligible machine uniformly at random, injecting diversity into machine assignments;

• SS layer: three strategies are employed in combination: (i) low-speed selection, which favors reduced machine energy consumption; (ii) high-speed selection, which shortens processing times and thereby contributes to smaller makespan values; and (iii) random speed selection, which promotes diversity across the speed dimension.

Together, these strategies ensure that the initial population covers a broad region of the search space and reflects trade-offs among all three optimization objectives from the outset.

3.4. Pareto ranking and external archive

3.4.1. Pareto ranking

In contrast to single-objective optimization, where solution quality can be assessed by direct fitness comparison, multi-objective problems require Pareto ranking to determine the non-domination level of each individual in IMOJA. A solution x is said to dominate a solution y (written $x\succ y$) if and only if x is no worse than y on all objectives and is strictly better on at least one objective.

By evaluating pairwise dominance relationships, the population is partitioned into successive non-dominated fronts. Individuals with a lower front rank are considered superior. All solutions in the first front collectively constitute the current Pareto optimal set. Within the same front, relative quality is assessed using crowding distance, which promotes diversity: a larger crowding distance indicates a more isolated — and therefore preferred — solution. Boundary solutions are assigned an infinite crowding distance to ensure they are always retained. The crowding distance of solution i is calculated as:

$\frac{D\left(i\right)={{\displaystyle \sum }}_{n=1}^N\left(f_n\left(i+1\right)-f_n\left(i-1\right)\right)}{f_n^{\text{max}}-f_n^{\text{min}}}$ (17)

where N is the number of objectives; $f_n^{\text{max}}$ and $f_n^{\text{min}}$ are the maximum and minimum values of the n-th objective over the current front; and $f_n\left(i+1\right)$and $f_n\left(i-1\right)$are the n-th objective values of the two neighboring solutions of solution i in the sorted front.

3.4.2. External archive

To avoid losing high-quality non-dominated solutions during the search, a fixed-size external archive of capacity nArchive (N_A) is maintained throughout the optimization. The archive is updated as follows:

• Initialization: after decoding and evaluating the initial population, all rank-1 (non-dominated) solutions are added to the archive.

• Iteration update: at each iteration, the non-dominated solutions of the current generation are merged with all solutions in the archive. The combined set is re-ranked using Pareto dominance, and the resulting new non-dominated set replaces the archive content. If the archive exceeds N_A, solutions with the smallest crowding distance are removed first, retaining the most widely spread solutions to maximize diversity.

3.5. Individual update

To adapt the standard Jaya algorithm to the discrete nature of the FJSP, four update strategies are designed to generate new candidate solutions while respecting the three-layer encoding. These strategies are inspired by the Jaya principle of attraction toward the best and repulsion from the worst, while additionally incorporating crossover with the external archive to enhance population diversity.

1. Worst-removal update (producing$X_i^t$). Following Caldeira & Gnanavelbabu (2021), subsequences of $X_i^t$ that are identical to those of $X_{\text{Worst}}^t$ are replaced by the corresponding subsequences from $X_{\text{Best}}^t$, producing $X_{i\mathrm{,1}}^t$. This moves the individual away from the worst while pulling it toward the best. The OS-layer update is illustrated in Figure 2, and the MA- and SS-layer updates in Figure 3.

   

   Figure 2
   OS encoding update in method (1). Source: Elaborated by the authors.
   
   

   Figure 3
   MA and SS encoding update in method (1). Source: Elaborated by the authors.
   

2. Best-preservation update (producing$X_{i\mathrm{,2}}^t$). Subsequences of $X_i^t$ identical to those of $X_{\text{Best}}^t$ are retained, while the differing subsequences are replaced by the corresponding subsequences from $X_{\text{Worst}}^t$ that also differ from both, producing $X_{i\mathrm{,2}}^t$ This preserves components shared with the best solution while diversifying remaining positions. The OS-layer update is illustrated in Figure 4, and the MA- and SS-layer updates in Figure 5.

   

   Figure 4
   OS encoding update in method (2). Source: Elaborated by the authors.
   
   

   Figure 5
   MA and SS encoding update in method (2). Source: Elaborated by the authors.
   

3. Archive crossover (producing $X_{i\mathrm{,3}}^t$ and $X_{i\mathrm{,4}}^t$). To increase solution diversity, $X_i^t$ is crossed with a randomly selected solution X_Archive^t from the external archive, generating two offspring. The OS layer uses Precedence Operation Crossover (POX), which preserves job precedence constraints, while the MA and SS layers use Machine Precedence Crossover (MPX). The OS-layer update is illustrated in Figure 6, and the MA- and SS-layer updates in Figure 7.

   

   Figure 6
   OS encoding update using POX crossover. Source: Elaborated by the authors.
   
   

   Figure 7
   MA and SS encoding update using MPX crossover. Source: Elaborated by the authors.
   

If $X_i^t$ is identical to $X_{\text{Best}}^t$ or $X_{\text{Worst}}^t$, the resulting candidate may be a duplicate, risking premature convergence. In such cases, the candidate is re-initialized. Finally, among the four candidate solutions ${\text{X}}_{\text{i}\mathrm{,1}}^{\text{t}}$, ${\text{X}}_{\text{i}\mathrm{,2}}^{\text{t}}$, ${\text{X}}_{\text{i}\mathrm{,3}}^{\text{t}}$, ${\text{X}}_{\text{i}\mathrm{,4}}^{\text{t}}$, and the original $X_i^t$, the best one according to Pareto ranking and crowding distance becomes the next-generation individual $X_i^{t+1}$.

3.6. Neighborhood search

To further refine solution quality, a neighborhood search phase is applied at each iteration to the top 20% of individuals in the population, ranked by Pareto front and crowding distance. Five neighborhood structures are designed; if a generated neighbor is non-dominated with respect to the current individual (assessed by Pareto dominance and crowding distance), it replaces the latter. The five structures target specific dimensions of the three-layer encoding and optimization objectives:

• NS1 — OS reversal: randomly select a non-identical subsequence in the OS encoding and reverse it, exploring alternative operation processing sequences;

• NS2 — Machine reassignment (workload): randomly select one operation from the most heavily loaded machine and reassign it to another eligible machine, directly targeting the total machine workload objective;

• NS3 — Machine reassignment (transportation): randomly select two adjacent operations of the same job with a long transportation time and reassign the latter to a machine yielding a shorter transportation time, reducing transportation energy and potentially improving makespan;

• NS4 — Speed reduction: randomly select one operation currently processed at a high speed and reduce its speed level by one, decreasing machine processing energy consumption;

• NS5 — Speed increase: randomly select one operation currently processed at a low speed and increase its speed level by one, shortening the processing time and contributing to a reduced makespan.

Together, NS1–NS5 form a complementary set that collectively addresses all three objectives — makespan, total workload, and energy consumption — across the OS, MA, and SS encoding dimensions.

3.7. Overall procedure of IMOJA

The complete procedure of IMOJA, integrating all components described in Sections 3.1–3.6, is summarized in Algorithm 2.

• Algorithm 2: IMOJA Overall Procedure

Input: Population size n_Pop, maximum iterations MaxIt,

archive size N_A, job/machine data.

Output: Pareto optimal solution set PF.

1) Generate initial population POP (Section 3.3).

2) Decode individuals (Algorithm 1); apply Pareto ranking

(Section 3.4); add non-dominated solutions to ArchivePop.

3) For It = 1 to$MaxIt$

4) Identify $X_{Best}$ and $X_{Worst}$ in current population.

5) For i = 1 to nPop

6) Update individual using strategies in Section 3.5;

select best candidate as $X_i^{t+1}$.

7) Decode $X_i^{t+1}$; evaluate objective functions.

8) End For

9) Apply Pareto ranking; extract non-dominated set PF.

10) Apply neighborhood search NS1–NS5 (Section 3.6) to

top 20% of individuals; re-evaluate improved solutions.

11) Merge PF with ArchivePop; re-rank; update ArchivePop

(trim to N_A by crowding distance if needed).

12) End For

13) Return ArchivePop.

After generating and decoding the initial population (lines 1–2), the algorithm iteratively updates each individual using the strategies described in Section 3.5 and retains the best candidate (lines 4–7), applies Pareto ranking, and refines the top 20% of individuals through the neighborhood search structures NS1–NS5 (lines 9–10). The resulting non-dominated solutions are merged with the external archive and re-ranked at each iteration (line 11). This process repeats until the maximum number of iterations is reached (line 3), after which the final Pareto optimal archive is returned (line 13).

4. Simulation experiments and results analysis

To evaluate the performance and verify the effectiveness of the proposed Improved Multi-Objective Jaya Algorithm (IMOJA), comprehensive simulation experiments were conducted. All algorithms were implemented in MATLAB R2020b and executed on a computer equipped with a 3.20 GHz processor and 8.0 GB of RAM.To ensure a fair and reproducible experimental comparison, the computational parameters were standardized across all metaheuristics. Following standard conventions in the multi-objective flexible job-shop scheduling literature (e.g., Deb et al., 2002; Zitzler et al., 2001) and reflecting the broader adoption of Pareto-based multi-objective frameworks in AI-assisted production engineering applications (Camargo et al., 2025), these configurations were selected based on widely adopted empirical baselines rather than instance-specific algorithmic tuning, thereby avoiding any risk of parameter overfitting. Furthermore, it is worth noting that a key theoretical advantage of the Jaya framework (and the proposed IMOJA) is its parameter-less nature regarding evolutionary operators; unlike GA-based approaches, it does not require the tuning of crossover or mutation probabilities. Accordingly, the population size was set to 50 and the maximum number of iterations to 100 for all algorithms. The external archive size was fixed at 50 for both IMOJA and SPEA2. For the benchmark algorithms (NSGA-II and SPEA2), the crossover probability ($P_c$) and mutation probability ($P_m$) were set to 0.8 and 0.1, respectively. Each benchmark instance was solved independently 10 times, and the parameter settings remained strictly consistent across all trials.

4.1. Test instances

This study uses the well-known Brandimarte benchmark dataset (MK01–MK10), which has been extended to meet the requirements of the proposed problem formulation. Each machine is assigned five discrete selectable speed levels $v=\left\{\mathrm{1.0,}\mathrm{1.5,}\mathrm{2.0,}\mathrm{2.5,}3.0\right\}$, and the processing times provided in the original benchmark are taken as the normal processing times at the lowest speed level $\left(v^1=1.0\right)$.

The average power consumption per unit time of machine M_k at speed level $v^e$ is defined as:

$P{L}_{kve}=r\cdot v_e^2$

where r is drawn randomly from [3, 6] independently for each machine. The average idle power consumption per unit time is set to $P{N}_k=4r$. Inter-machine workpiece transportation times satisfy$T{t}_{kl}\in \left[\mathrm{1,}5\right]$, drawn uniformly at random for each machine pair. The unit energy consumption per unit transportation time is fixed at PT = 2. These parameter settings are consistent across all ten instances and all compared algorithms, ensuring a fair and reproducible experimental comparison.

4.2. Performance Indicators

To assess the performance of the proposed algorithm, two widely used multi-objective metrics are employed: the Coverage indicator C and the Inverted Generational Distance (IGD). Both metrics are computed on the Pareto solution sets produced by each algorithm at the end of the 10 independent runs.

The Coverage indicator between two Pareto solution sets S₁ and S₂ is defined as:

$C\left(S_1,S_2\right)=\frac{\left|(x\in S_2|\exists y\in S_1:y\succ x)\right|}{\left|S_2\right|}$ (18)

where |S₂| denotes the cardinality of S₂, and $y\succ x$ indicates that y dominates x. A larger value of C(S₁, S₂) implies that a greater proportion of solutions in S₂ are dominated by at least one solution from S₁, indicating the superiority of the algorithm that produced S₁. Note that C(S₁, S₂) and C(S₂, S₁) are generally not complementary and must both be reported.

The Inverted Generational Distance is defined as:

$IGD=\frac{{{\displaystyle \sum }}_{i=1}^{\left|S^{*}\right|}{D}_i}{\left|S^{*}\right|}$ (19)

where S is the Pareto solution set produced by the algorithm; S^* is the true Pareto front (or a reference approximation); and D_i is the shortest Euclidean distance from the i-th solution in S^* to the nearest solution in S. Because the three objectives differ in scale, all objective values are normalized prior to computing D_i as follows:

$F_i=\frac{f_i-f_{i,\text{min}}}{f_{i,\text{max}}-f_{i,\text{min}}},i=\mathrm{1,2,3}$

where $f_{i,\text{max}}$and $f_{i,\text{min}}$ are the maximum and minimum values of the i-th objective over S^*. A smaller IGD value indicates better convergence and diversity of the obtained solution set. Since the true Pareto front S^* is unknown in practice, it is approximated in this study by merging all non-dominated solutions produced by the compared algorithms across all runs.

4.3. Comparative experiments

The well-established NSGA-II and SPEA2 algorithms (Deb et al., 2002; Zitzler et al., 2001) are employed as benchmarks to validate the feasibility and effectiveness of the proposed IMOJA. The experimental results are reported as averages over the 10 independent runs for the IGD and C indicators, and as best values across runs for the objective comparison in Table 2. Table 2 presents the best objective values achieved by each algorithm across the ten extended Brandimarte instances, with the best results highlighted in bold. As shown in Table 2, IMOJA consistently obtains superior best values for all three objectives — makespan ($C_{\text{max}})$, total machine workload (TL), and total energy consumption (E) — on every instance compared with both NSGA-II and SPEA2. This demonstrates stronger convergence and higher solution quality across all optimization directions.

Table 2
Comparison of best objective values obtained by each algorithm.

Instance	IMOJA			NSGA-II			SPEA2
	Cmax	TL	E	Cmax	TL	E	Cmax	TL	E
MK01	22.23	57.70	1103.81	23.87	60.80	1184.34	25.17	63.33	1235.73
MK02	19.67	54.23	1040.16	20.83	57.40	1136.88	21.97	59.77	1054.35
MK03	77.83	344.10	6097.73	84.47	368.57	6409.36	85.83	367.03	6682.36
MK04	34.30	128.40	2356.49	36.43	138.40	2571.49	38.63	143.53	2553.58
MK05	72.87	262.23	4322.49	80.00	275.87	4515.53	83.37	284.83	4563.55
MK06	56.07	144.30	3584.13	60.17	163.17	3911.92	60.70	170.83	3741.23
MK07	63.50	268.10	4369.99	67.33	272.83	4835.83	77.03	292.43	4850.99
MK08	202.23	1009.33	18110.88	211.13	1027.67	18549.98	225.73	1100.43	19761.19
MK09	145.23	966.67	16127.57	156.17	1012.03	16993.20	165.90	1051.73	17673.48
MK10	120.13	809.00	14553.03	134.10	859.43	15325.38	138.13	882.33	16052.16

• Source: Elaborated by the authors.

The results for the Coverage (C) and Inverted Generational Distance (IGD) indicators are presented in Tables 3 and 4, respectively. In Table 3, values of exactly 1 and 0 are highlighted in bold; in Table 4, the best IGD value per instance is highlighted in bold. As shown in Table 3, C(IMOJA, NSGA-II) equals 1 on six instances and is close to 1 on the remaining four, while C(NSGA-II, IMOJA) equals 0 on seven instances and is close to 0 on the other three. This indicates that IMOJA's non-dominated solution set dominates the vast majority of NSGA-II's solutions, whereas NSGA-II is largely unable to dominate IMOJA's solutions. Similarly, C(IMOJA, SPEA2) equals 1 on four instances and exceeds 0.9 on all others, while C(SPEA2, IMOJA) equals 0 on seven instances and remains below 0.007 on the rest, confirming that IMOJA substantially dominates SPEA2's solutions. As shown in Table 4, IMOJA achieves smaller IGD values than both NSGA-II and SPEA2 on all instances, with values orders of magnitude smaller and equal to zero on four instances, demonstrating its superior convergence and diversity.

Table 3
Comparison of C indicator results for different algorithms.

Instance	C(IMOJA, NSGA-II)	C(NSGA-II, IMOJA)	C(IMOJA, SPEA2)	C(SPEA2, IMOJA)
MK01	0.9273	0.0167	0.9133	0.0067
MK02	1.0000	0.0000	0.9924	0.0000
MK03	0.9333	0.0035	0.9235	0.0000
MK04	0.9441	0.0059	0.9113	0.0059
MK05	0.9935	0.0000	1.0000	0.0000
MK06	1.0000	0.0000	1.0000	0.0000
MK07	1.0000	0.0000	0.9937	0.0000
MK08	1.0000	0.0000	0.9509	0.0033
MK09	1.0000	0.0000	1.0000	0.0000
MK10	1.0000	0.0000	1.0000	0.0000

• Source: Elaborated by the authors.

Table 4
Comparison of IGD indicator results for different algorithms.

Instance	IMOJA	NSGA-II	SPEA2
MK01	0.0004	0.0330	0.0452
MK02	0.0001	0.0459	0.0598
MK03	0.0008	0.0383	0.0465
MK04	0.0004	0.0307	0.0342
MK05	0.0000	0.0235	0.0315
MK06	0.0000	0.1905	0.3004
MK07	0.0002	0.0399	0.0474
MK08	0.0004	0.0236	0.0396
MK09	0.0000	0.0607	0.0706
MK10	0.0000	0.0598	0.0682

• Source: Elaborated by the authors.

Figure 8 presents the Pareto optimal fronts and three-view projections obtained by all algorithms on the MK09 instance, where red circles denote IMOJA’s solutions, blue squares denote NSGA-II’s solutions, and green triangles denote SPEA2’s solutions. IMOJA’s Pareto front dominates those of both competitors along all three objective axes, confirming its ability to obtain a superior Pareto front structure. Figures 9 to 11 show the Gantt charts corresponding to the schedule with the minimum makespan obtained by each algorithm on MK05 (15 jobs, 4 machines, 106 operations) in one representative run. IMOJA produces a noticeably more compact schedule with fewer idle gaps between operations on each machine. Although NSGA-II and SPEA2 use the same population initialization and insertion-based decoding methods proposed in this paper, their schedules exhibit larger idle gaps, resulting in higher makespan values. The best objective values found across the 10 independent runs for IMOJA, NSGA-II, and SPEA2 on MK05 are ($C_{max}$, TL, E) = (72.87, 262.23, 4322.49), (80.00, 275.87, 4515.53), and (83.37, 284.83, 4563.55), respectively, matching the aggregated experimental updates summarized in Table 2. In the visualized IMOJA schedule (Figure 9), the minimum makespan of 72.87 is achieved while simultaneously maintaining highly competitive values for workload and energy consumption, further demonstrating the practical evolutionary effectiveness of the proposed algorithm.

Figure 8
Pareto front diagrams and three-view projections for MK09. Source: Elaborated by the authors.

Figure 9
Gantt chart obtained by IMOJA for MK05. Source: Elaborated by the authors.

Figure 10
Gantt chart obtained by NSGA-II for MK05. Source: Elaborated by the authors.

Figure 11
Gantt chart obtained by SPEA2 for MK05. Source: Elaborated by the authors.

4.4. Small-scale model validation

To validate the correctness of the proposed MIP model and assess the solution quality of IMOJA, a small-scale custom instance was constructed: 3 jobs, 3 machines, 2 operations per job, and 3 selectable speed levels. Inter-machine transportation times were drawn uniformly from [1, 5], and power consumption parameters followed the same settings as in Section 4.1. This instance yields a total of 6 operations, making the MIP tractable for exact solution.

The MIP model was implemented in MATLAB R2020b and solved using the intlinprog solver with a time limit of 3,600 seconds and an optimality gap tolerance of 1%. IMOJA was applied to the same instance over 10 independent runs using the parameter settings of Section 4. Table 5 reports the best objective values obtained by each approach.

Table 5
Comparison of objective values obtained by the MIP exact solver (MATLAB intlinprog) and IMOJA on a small-scale validation instance (3 jobs, 3 machines, 2 operations per job, 5 speed levels). IMOJA results are reported as best and mean ± standard deviation over 10 independent runs. IMOJA Gap (%) measures the relative deviation of the best IMOJA solution from the MIP optimal.

Objective	MIP Optimal	Solver Gap (%)	IMOJA Best	IMOJA Mean ± Std
Cmax	13.33	0.00	13.33	13.58 ± 0.34
TL	19.87	0.00	19.87	20.24 ± 0.67
E	241.53	0.00	243.85	247.12 ± 5.43
CPU (s)	42.7	—	0.13	—

• Note: IMOJA Gap (%) = (IMOJA Best − MIP Optimal) / MIP Optimal × 100. Results averaged over 10 independent runs. MIP solved using MATLAB intlinprog with a 3,600-second time limit and 1% optimality gap tolerance.

The results confirm two key findings. First, the MIP solver produces feasible schedules that satisfy all constraints defined in Section 2.2.2, validating the correctness of the model formulation. Second, IMOJA matches the exact optimal values for makespan and total workload, and lies within 0.96% of the optimum for energy consumption, demonstrating that the proposed algorithm reliably recovers near-optimal solutions on tractable instances. For larger instances (MK01–MK10), the exact MIP becomes computationally intractable within any reasonable time limit, justifying the use of IMOJA as the solution method.

5. Conclusions

This study addressed an extended version of the FJSP that integrates two practically relevant yet frequently overlooked factors: inter-machine workpiece transportation times and selectable machine processing speeds. A multi-objective optimization model was formulated to simultaneously minimize three key performance indicators: makespan, total machine workload, and total energy consumption (including processing, idle, and transportation phases). To solve this model, an IMOJA was proposed. The algorithm employs a three-layer encoding scheme for operation sequencing, machine allocation, and speed selection, combined with a deterministic, insertion-based active decoding procedure that explicitly maps chromosomes onto physical schedules while accounting for transportation times and speed-dependent processing times. In addition, a hybrid population initialization strategy, multiple discrete update mechanisms, and five tailored neighborhood structures were designed to enhance the algorithm’s local search capability and solution diversity.

Computational experiments performed on extended Brandimarte benchmark instances (MK01–MK10) demonstrated the effectiveness of IMOJA through systematic comparisons with two well-established multi-objective evolutionary algorithms, NSGA-II and SPEA2. The results confirm that IMOJA consistently produces superior Pareto optimal solutions in terms of both convergence and diversity, as evidenced by dominant Coverage indicator values and significantly smaller IGD values across all instances. Crucially, these stable results were achieved through standard literature-based baselines rather than instance-tailored parameter tuning, highlighting the high generalization capability of the framework and its immunity to parameter overfitting. These findings demonstrate that IMOJA provides a highly robust and parameter-efficient guidance for energy-efficient shop-floor scheduling decisions in modern manufacturing environments.

Future research will focus on incorporating additional real-world production constraints, such as machine breakdowns, dynamic job arrivals, and limited transportation resources. The objective set will also be extended to include due-date-related measures and explicit production costs, further aligning the model with the complex requirements of sustainable and responsive manufacturing systems. Furthermore, the scalability of IMOJA on larger-scale instances and its integration with digital twin technologies for real-time adaptive scheduling represent promising directions for future investigation.

Data availability

The benchmark instances (extended Brandimarte MK01–MK10) are publicly available (Brandimarte, 1993); the extended parameters (speed levels, transportation times, power coefficients) are fully reported in Section 4.1. The IMOJA implementation is available from the corresponding author upon reasonable request.

How to cite this article:

Mili, K. A., & Argoubi, M. (2026). Energy-efficient flexible job-shop scheduling with transportation times and selectable speeds. Production, 36, e20260050. https://doi.org/10.14488/1980-5411.20260050

Financial Support

This work was supported by Lusail University.

Ethical Statement

This study was performed in accordance with the ethical standards laid down in the 2013 Declaration of Helsinki. Ethical approval was obtained from the Institutional Review Board of Lusail University in Qatar (IRB Protocol: LU/IRB/03-12/01, dated 3 December 2025).

Artificial Intelligence Use Statement

Claude (Anthropic) was used during the proofreading and correction of the English text of this manuscript. The authors reviewed the resulting text and take full responsibility for all content included in the manuscript.

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*

[e-mail]

Conflict of Interest

No potential conflict of interest was reported by the author(s).

Editor(s)

Adriana Leiras

Rodrigo Caiado

Publication Dates

• Publication in this collection
  18 Sept 2026
• Date of issue
  2026

History

• Received
  10 Apr 2026
• Accepted
  17 Aug 2026