Enhancing Manufacturing Efficiency Through Symmetry-Aware Adaptive Ant Colony Optimization Algorithm for Integrated Process Planning and Scheduling

Abstract

Integrated process planning and scheduling (IPPS) is an intricate and vital issue in smart manufacturing, requiring the coordinated optimization of both process plans and production schedules under multiple resource and precedence constraints. This paper presents a novel optimization framework, symmetry-aware adaptive Ant Colony Optimization (SA-AACO), designed to resolve key limitations in existing metaheuristic approaches. The proposed method introduces three core innovations: (1) a symmetry-awareness mechanism to eliminate redundant solutions arising from symmetrically equivalent configurations; (2) an adaptive pheromone-updating strategy that dynamically balances exploration and exploitation; and (3) a dynamic idle time penalty system, integrated with time window-based machine selection. Benchmarked across ten IPPS scenarios, SA-AACO achieves a superior makespan in 9/10 cases (e.g., 29.1% improvement over CCGA in Problem 1) and executes 18-part processing within 30 min. While MMCO marginally outperforms SA-AACO in Problem 10 (makespan: 427 vs. 483), SA-AACO’s consistent dominance across diverse scales underscores the feasibility of its application in industry to balance quality and efficiency. By unifying symmetry handling and adaptive learning, this work advances the reconfigurability of IPPS solutions for dynamic industrial environments.

1. Introduction

Process planning (PP) is a vital research area in the manufacturing industry, playing a significant role in modern flexible manufacturing systems (FMSs) [1], whereas in traditional manufacturing systems, it is typically carried out in a static manner [2]. More precisely, a process plan is the systematic arrangement of consecutive manufacturing procedures that outlines a methodology for economically and competitively producing a product. Conversely, scheduling involves the allocation of resources to distinct activities within a defined timeframe, ensuring that each operational step is executed in the most efficient order to adhere to production deadlines and reduce delays. The scheduling process considers resource availability, task dependencies, and the time necessary for each activity [3]. However, in traditional production systems, process planning and scheduling are executed independently and sequentially, as the outcome of process planning serves as the input for scheduling. Moreover, if process planning and scheduling are conducted without considering the other’s constraints and objectives, the feasibility of the process plans cannot be guaranteed if the job shop conditions alter. In addition, conflicts between two activities will necessitate adjustments to process plans based on shop-floor conditions, leading to inefficient production. Therefore, to address the difficulty, mitigate scheduling conflicts, and achieve objectives, the integration of process planning and scheduling (IPPS) is recommended. Consequently, due to the enhanced adaptability of process plans and components, IPPS is essential for contemporary and versatile production systems [4].

Integrated process planning and scheduling (IPPS) signifies a vital transformation in modern manufacturing, moving from the conventional sequential methodology of distinct process planning and production scheduling to the use of integration strategies. Hence, the IPPS seeks to simultaneously optimize both processes, considering their interdependencies, and it has proven to be essential for improving production efficiency and adaptability in more complex and competitive situations. These integration problems are categorized as NP-Hard. To address them, it is necessary to complete the application of various metaheuristic algorithms, including genetic algorithms (GAs) [5], Particle Swarm Optimization (PSO) [6], evolutionary algorithms (EAs) [7], Honey Bee Optimization (HBO) [8], Simulated Annealing (SA) [9], Ant Colony Optimization (ACO) [10], Tabu Search (TS) [11], and several hybrid methodologies. Ant Colony Optimization (ACO) is a prevalent and recognized metaheuristic approach, being extensively employed to address integrated investigations; it has been chosen for this research due to its decentralized search mechanism, positive feedback loop, and robustness in handling discrete combinatorial optimization problems such as IPPS. Compared to other techniques like GA, PSO, and SA, ACO offers superior capabilities in adapting to dynamic environments and incorporating local heuristic information through pheromone trails. This makes it highly appropriate for IPPS problems, where machine availability, task dependencies, and process flexibility demand a balance between global exploration and local exploitation. Furthermore, ACO is inherently flexible and can be enhanced with additional features such as symmetry awareness and adaptive parameters, which are explored in this study.

Numerous studies in recent years have demonstrated the effectiveness of Ant Colony Optimization (ACO) in addressing various forms of the IPPS problem. Reference [12] utilized a Multi-Agent System structured with Ant Colony Optimization to address the IPPS problem, aiming to minimize makespan in a manufacturing environment. Reference [13] investigated a graph-based ACO, utilizing AND/OR structures to address the same issue. The authors aimed to optimize the same target as [12]. Initially, the authors constructed an AND/OR graph to depict the IPPS. They subsequently employed Ant Colony Optimization (ACO) to address the IPPS, utilizing the graph. Reference [14] examined the integration of process planning and scheduling by Ant Colony Optimization to reduce the overall completion time (makespan). Reference [15] proposed an improved Ant Colony Optimization to address the IPPS problem within a workshop environment, where they attempted to reduce makespan, job flow time, and computational time. In conclusion, they determined that the enhanced version of ACO outperforms other metaheuristic algorithms. Reference [16] implemented an Ant Colony Optimization approach to address the flexible process planning issue through the dynamic modification of viable alternative machining operations. Reference [17] employed the genetic algorithm (GA) method and Ant Colony Optimization (ACO) to address the dynamic integrated process planning, scheduling, and due date assignment problem (DIPPSDDA) with stochastic work arrivals. The research attempted to reduce earliness and tardiness and optimize due dates by identifying the optimal combination of dispatching rules, due date allocation, and work pathways. Consequently, experimental findings indicated that ACO surpasses GA, and combined methodologies yield superior overall manufacturing efficiency compared to distinct techniques. Moreover, hybrid studies have been suggested for the purpose of addressing the IPPS issue.

Within IPPS systems, symmetry presents a significant challenge for optimization algorithms. Nonetheless, symmetry arises when multiple process sequences yield identical or near-identical objective function values, despite representing entirely different schedules. Therefore, these symmetrical solutions create large flat regions in the solution space, where conventional optimization techniques struggle to differentiate among alternatives, often leading to premature convergence or stagnation. This challenge is particularly pronounced in manufacturing systems, where structural symmetry is introduced through duplicated machinery, repetitive operations, and mirrored processing flows. Such structural redundancies inflate the solution space and increase computational complexity. To address this, the proposed symmetry-aware Ant Colony Optimization (SA-ACO) algorithm integrates symmetry detection mechanisms directly into the search process, enhancing both exploration and exploitation. While traditional approaches rely on symmetry-breaking constraints to eliminate redundant scheduling options and improve search efficiency [18], more recent studies have demonstrated the value of symmetry-aware learning in metaheuristics and integer programming. For example, frameworks that incorporate symmetry considerations into integer linear programming have achieved superior solution quality at lower computational costs [19]. Building on these insights, SA-ACO introduces multi-dimensional evaluation criteria and adaptive mechanisms, explicitly designed to navigate symmetrical solution landscapes more effectively, thereby improving convergence behavior and overall scheduling performance.

Integrated process planning and scheduling presents a formidable combinatorial challenge, further complicated by real-world limitations like dynamic task contexts, machine–tool linkages, and precedence interactions. Conventional optimization methods sometimes exhibit suboptimal performance in achieving quality schedules because of early convergence, excessive centralization of use on a singular solution, and insufficient adaptability to system changes. This paper offers numerous essential modifications to the structure of the symmetry-aware adaptive Ant Colony Optimization (SA-AACO) framework designed for high-efficiency IPPS.

The primary issue addressed is the ineffective settlement of issues relating to machine assignment and scheduling. The issue is mitigated by the algorithm’s implementation of an optimal machine selection schedule, which ensures that operations are allocated during unimpeded time windows, hence improving machine usage and schedule viability.

A further issue examined is excessive idle time in production, which is mitigated using a dynamic penalty structure that calibrates fines based on the system’s present status. This adaptive penalty system enhances responsiveness to resource conditions, therefore minimizing non-operational machine time.

A primary contribution is a dynamic pheromone-updating technique that adjusts pheromone levels based on the optimality of the solution. This directly addresses the problem of early convergence by ensuring a balance between exploration and exploitation within the search space. In addition, an adaptive Q parameter independently influences the reliability of tailoring the reinforcement intensity of convergence across diverse problem instances, improving the consistency of convergence dependability amid varied scenarios.

This work performs comprehensive calibrations for the selection of suitable values of ACO parameters, including the effects of pheromone (α), heuristic weight (β), evaporation rate, and ant population, to attain optimal performance over a broader spectrum of industrial applications. These modifications enable the implementation of the algorithm across diverse and complex scenarios while conforming to constraints like operational sequence, machine–tool nominal compatibility, and process viability. Moreover, integrating symmetry-awareness functions into the modeling and search method improves scheduling performance and solution quality. Hence, the SA-AACO algorithm presents a novel, adaptive, and symmetry-conscious methodology that enhances the applicability of ACO to complex real-world IPPS systems.

The subsequent sections of this work are structured as follows: Section 2 examines the relevant research. Section 3 provides a problem description. Section 4 illustrates the proposed ACO algorithm. Section 5 presents the experiments and explanations of the outcomes. Section 6 summarizes the findings and indicates future directions.

2. Literature Review

2.1. Introduction to Integrated Process Planning and Scheduling (IPPS)

In manufacturing systems, the simultaneous coordination of process planning and scheduling tasks is known as integrated process planning and scheduling, or IPPS. Due to its potential to increase production systems’ flexibility and efficiency, the integrated approach has received a lot of attention. Hence, process planning and scheduling are frequently handled as distinct processes in conventional manufacturing settings, which results in less-than-ideal outcomes. The integration of process planning and scheduling is recommended in order to overcome difficulty, prevent scheduling stage conflicts, and achieve desired goals [4]. However, the complexity of contemporary industrial systems, dynamic limitations, and shifting demand conditions make it difficult to solve the IPPS problem. As a result, creating reliable optimization methods for IPPS has emerged as a pivotal research topic. Therefore, recent advancements in optimization techniques, particularly enhanced algorithms, have become crucial for adapting to the dynamic and evolving demands of intelligent manufacturing environments [20,21].

2.2. Classical Methods for IPPS

Conventional approaches to addressing the IPPS problem have mainly depended on mathematical programming and heuristic methodologies. Mathematical programming methods, like linear and mixed-integer programming, provide exact formulations of the scheduling problem but frequently encounter challenges with extensive combinatorial complexity. Ref. [22] presented a survey related to the use of different heuristics for solving optimization problems in IPPS. Ref. [23] established MILP models with network representations for IPPS. Ref. [24] employed Benders’ decomposition based on logic to iteratively solve subproblems; however, these methods have trouble handling large-scale issues. Ref. [25] recommended a graph-based constraint programming (GCP) method for addressing the IPPS problem with AND/OR graphs. Ref. [26] suggested a method based on improved constraint programming to address the overall issue, using sophisticated algorithms to increase the efficiency of the search and improve the constraint propagation. Ref. [27] introduced an effective mixed-integer linear programming (MILP) model and a type-2 priority-based heuristic algorithm (PBHA II) for addressing the integrated process planning and scheduling (IPPS) problem with a makespan objective. A mixed-integer linear programming model was developed for process planning and scheduling in reconfigurable manufacturing systems, validating significant cost reductions and improving demand fulfillment compared to conventional systems through detailed sensitivity and ANOVA analyses [28]. These techniques seek optimal solutions but face challenges with scalability and managing dynamic scheduling in extensive IPPS problems. Moreover, MILP models involve considerable computational expenses as the problem size increases. On the other hand, metaheuristic methods offer a more practical solution, but at the cost of exact optimality.

2.3. Metaheuristic Approaches for IPPS

Metaheuristic algorithms and their hybrid forms have received significant attention for use tackling the IPPS problem because of their capacity to handle complicated, large-scale, and dynamic scheduling contexts. Ref. [11] examined the integration of process planning, scheduling, and due date assignment to enhance manufacturing performance. Diverse methodologies, such as random search, semi-tabu, evolutionary techniques, and their hybrid forms, were evaluated across various job shop dimensions. Ref. [29] presented a layered methodology for addressing the single-objective IPPS problem, integrating harmony search (HS) for sub-route identification and a genetic algorithm (GA) for machine allocation and operation sequencing. The methodology encompasses both static and dynamic scoring techniques, together with a selection operator designed to enhance population diversity. Ref. [30] developed an adaptive multi-strategy artificial bee colony (AMSABC) method to address the IPPS problem, including both exploration and exploitation techniques. Ref. [31] employed the gray wolf optimization (GWO) technique to minimize makespan, contrasting integrated and non-integrated scenarios. Numerous optimization methods, especially AI-driven algorithms, have been investigated to tackle the intricate and combinatorial characteristics of IPPS [32]. Ref. [33] proposed an enhanced genetic algorithm, utilizing multi-layer encoding (IGA-ML) to resolve the IPPS problem, focusing on its interaction and multi-flexibility attributes. Ref. [34] designed a Mountain Gazelle Optimizer (MGO) to solve the IPPS problem. Ref. [35] implemented a hybrid fuzzy logic-based genetic algorithm (fuzzy-GA) to tackle process planning and scheduling in a distributed flexible job shop. The fuzzy-GA framework dynamically modified the mutation and crossover rates, using fuzzy inference to improve the algorithm’s ability to explore and exploit. A dynamic energy-efficient IPPS (EIPPS) approach based on digital twin technology was proposed, embedding event-based virtual–real mapping and game theory to handle shop-floor disruptions while minimizing makespan and energy consumption [36]. A surrogate-assisted hyper-heuristic algorithm was presented for dynamic IPPS with reconfigurable manufacturing cells, achieving significant improvements in computational efficiency while effectively handling dynamic scheduling and reconfiguration challenges [37]. Several studies have been conducted to observe the effectiveness of these metaheuristics and curiously, no specific algorithm has been found consistently superior to others in all circumstances, but hybrid methodologies that leverage strengths of different algorithms have been proven to yield better results.

2.4. Improvements in ACO for Minimizing Makespan

In the last decade, research efforts have concentrated on enhancing the efficacy of Ant Colony Optimization (ACO), together with its hybrid forms featuring other heuristics and metaheuristics, to minimize makespan in scheduling. Ref. [10] examined the application of constructive metaheuristics to address IPPS issues, with the objective of enhancing dynamic responsiveness. A paradigm utilizing AND/OR graphs was proposed, accompanied by a general framework for the implementation of diverse constructive search algorithms. This is exemplified by Ant Colony Optimization (ACO). Experimental results confirmed the efficacy of the suggested approach in reducing makespan on benchmark tasks. Ref. [38] introduced a hybrid ant colony algorithm incorporating crossover and mutation processes for no-wait flow shop, scheduling aimed at minimizing maximum completion time. The algorithm was evaluated on 192 benchmark instances and compared with the adaptive learning approach and genetic heuristic approach. The experimental findings indicated that the suggested hybrid ant colony algorithm surpasses the alternative techniques. Ref. [39] developed a novel Ant Colony Optimization (ACO) algorithm for IPPS that concurrently optimized process planning and scheduling, taking into account sequence-dependent setups and tool-related restrictions. As a result, the suggested ACO enhanced solution quality and computational efficiency by integrating individual and cooperative evolution, utilizing a streamlined pheromone update mechanism and standardized transition probability. Hence, the experimental findings indicate that the proposed Ant Colony Optimization (ACO) surpassed contemporary metaheuristics. Ref. [40] established a hybrid genetic algorithm-Ant Colony Optimization (hAG) method for addressing IPPS issues, wherein the superior algorithm was chosen in real-time. For this purpose, hAG circumvented convergence by resetting parameters and incorporated a local search approach, utilizing the interior point method for enhanced solution accuracy. Consequently, experimental findings indicated that hAG surpassed AIS, GA, and ACO across multiple benchmark scenarios. These enhancements significantly improved ACO’s ability to minimize makespan and provide better solutions for real-world scheduling problems.

In summary, while classical and metaheuristic approaches have contributed significantly to addressing various facets of integrated process planning and scheduling, several critical challenges remain insufficiently explored, most notably the treatment of symmetry and adaptability to real-world production. Furthermore, many existing algorithms neglect the structural symmetries inherent in modern manufacturing systems or handle them unrealistically, resulting in redundant solution evaluations and compromised scheduling efficiency. The literature on the use of Ant Colony Optimization (ACO) for integrated process planning and scheduling does not directly address the impact of symmetry or propose symmetry-aware ACO strategies. Likewise, traditional Ant Colony Optimization methods often face issues such as premature convergence, static pheromone control, and limited responsiveness to changing shop-floor conditions. These gaps underscore the need for a more robust, intelligent solution capable of navigating symmetrical solution spaces while adapting effectively to dynamic industrial constraints. Addressing this need, the present work introduces a symmetry-aware adaptive Ant Colony Optimization (SA-AACO) algorithm, which incorporates symmetry-breaking strategies, adaptive pheromone updating, time-sensitive machine selection, and dynamic penalty mechanisms. By evaluating the algorithm using enriched benchmark scenarios that better reflect real-world variability, this study aims to bridge the disconnect between theoretical models and practical scheduling demands, advancing the role of ACO in intelligent, industry-ready IPPS solutions.

3. Problem Description

The IPPS problem involved processing I parts, each with a different number of features, using various machines. Each feature of a part could be processed through multiple potential operations (POs), where each PO consists of a combination of the machine, tool, and corresponding processing time. The selection of the most suitable PO is governed by several assumptions:

• Each part has several operations, and there is a total of I parts to be processed.
• Once a task started, it cannot be suspended or interrupted. This assumption streamlines the scheduling problem by eliminating intricate preemption logic and concentrating on uninterrupted job execution.
• Among all possible POs of a part, only one (the most suitable) is selected for processing.
• The precedence constraints among the selected processes must be strictly followed.
• Once a PO is allocated to a machine, it remains on that machine for the duration of its processing longevity. The reallocation of processes among machines is prohibited.
• All machines and parts are assumed to be accessible at time zero, and the scheduling procedure begins with all resources prepared for utilization.
• Upon completion of work using a machine, it is promptly conveyed to the subsequent machine in the process, with the transmission delay considered insignificant.
• The duration of every task is predetermined and established beforehand. The algorithm presumes that the processing time for each task is fixed and remains constant throughout execution.
• The cost associated with processing each task on a specific machine is constant and established in advance.
• Each machine is chosen according to a static time window that considers its availability and the processing duration of the preceding task.

Table 1. List of abbreviations and symbols.

Notation	Description
I	Parts set.
OSij	Operations set for part i.
PO	Potential operation set consisting of potential machines and tools for all the operations.
POij	Potential operation set for operation OSij.
N	Total number of POs for all parts I.
i, i′	Parts, 1 ≤ i ≤ │I│, where i′ is the next part.
j, j′	Operations, 1 ≤ j ≤ │Ji│, where j′ is the next operation.
k, k′	Potential operations, 1 ≤ k ≤ │Kij│, where k′ is the next PO.
im	Index of machine m (1 ≤ m ≤ │10│).
POijk	kth PO for operation OSij of the part i.
mijk	Index of the machine chosen for kth PO, POijk.
tijk	Index of the tool chosen for PO, POijk.
mtijk	Machine time of PO, POijk.
α	Influence of pheromone information.
β	Influence of heuristic information.
ρ	Pheromone evaporation rate.
Qmax	Upper bound for Q (pheromone deposition constant).
Qmin	Lower bound for Q (pheromone deposition constant).
τmax	The maximum pheromone level.
τmin	The minimum pheromone level.
Widle	Weight factor for idle time penalty.
λ	Scaling factor that regulates the impact of idle time on the total cost.
uijk	If the kth potential operation for jth operation of part i is selected, 1; 0, otherwise;
vijkj′k′	If the kth potential operation for jth operation is machined before the k′ th potential operation for j′th operation selected 1; 0, otherwise.
wijki′j′k′ ITidlem	1, the kth potential operation for jth operation of part i is machined before the k′th potential operation for j′th operation of part i’ and mijk = mi′j′k′; Idle time for machine im for all the parts.
MNim	The makespan of the im machine.
MNmax	The maximum makespan of all machines.
OMS	Overall or cumulative makespan of all machines.
Cij	The machine cost for operation j of part i;.
Ctool,i	The tool cost for operation j of part i.
Csetup,i	The setup cost for operation j of part i.
Cidle	Idle time cost.
Ctotal	Total cost.
ηijki′j′k′	Heuristic value.

shows the different abbreviations and symbols used in the paper.

The selection of potential processes in the proposed scheduling framework guarantees the optimal allocation of operations while adhering to problem-specific limitations. It is important to note that every process is defined by a combination of part number, feature number, machine, tool, TAD (Tool Feed Direction), and the corresponding processing time. During each cycle, feasible processes are identified by applying filters based on operational restrictions, such as machine availability, precedence relationships among features, and tool compatibility. Subsequently, the selection step is carried out by the proposed method, which considers both pheromone concentrations and key problem attributes, such as cost penalties and time inefficiencies, to guide the decision-making process more effectively. Next, it is incorporated into the schedule and the system. Then, we adjust the machine availability and apply the part restrictions conforming to the finished operation. This technique ensures that all the operations are correctly sequenced in terms of efficiency and constraint requirements and responds to changes as necessary in the most effective manner.

Prior to scheduling, the process scheme must be provided initially. It is presumed that there are I parts requiring machining. Let OS_ij denote the operation set for part i, defined as follows:

$$O{S}_{ij}=\left\{O{S}_{i1},O{S}_{i2,}O{S}_{i3},\dots \dots \dots ,O{S}_{i{J}_i}\right\}$$

(1)

An operation can be performed using a selection of suitable machines and tools. The combination of the potential machines and tools forms the potential operation (PO) set for a given operation. Let PO_ij denote the PO set for operation $O{S}_{ij}\left(j=1,2,3,\dots ,J_i\right)$ of part i, defined as follows:

$$P{O}_{ij}=\left\{P{O}_{ij1},P{O}_{ij2},\dots ,P{O}_{ijk},\dots ,P{O}_{ij{K}_{ij}}\right\}$$

(2)

where the PO_ijk is the kth PO for operation OS_ij of the part i.

k^th PO can be defined as follows:

$$P{O}_{ijk}=\left\{m_{ijk},t_{ijk},m{t}_{ijk}\right\}$$

(3)

where m_ijk is the index of the machine chosen, t_ijk is the index of the tool chosen, and mt_ijk is the machine time of kth PO, PO_ijk.

For operation OS_ij, K_ij POs must be chosen in the actual machining or computing process, but only one PO may be selected. Regarding part i with J_i operations, the total number of POs is as follows:

$$n_i={\displaystyle \sum }_{j=1}^{J_i}{K}_{ij}$$

(4)

The total number of POs for the I components is as follows:

$$N={\displaystyle \sum }_{i=1}^I{\displaystyle \sum }_{j=1}^{J_i}{K}_{ij}={\displaystyle \sum }_{i=1}^I{n}_i$$

(5)

3.1. Main Constraints

In the scheduling framework, only a single PO may be designated for each operation, as outlined below:

$${\displaystyle \sum }_{k=1}^{K_{ij}}{u}_{ijk}=1$$

(6)

For a part, the total number of selected POs is as follows:

$${\displaystyle \sum }_{j=1}^{J_i}{\displaystyle \sum }_{k=1}^{K_{ij}}{u}_{ijk}=J_i$$

(7)

The overall number of selected POs for the entire scheduling system is as follows:

$${\displaystyle \sum }_{i=1}^I{\displaystyle \sum }_{j=1}^{J_i}{\displaystyle \sum }_{k=1}^{K_{ij}}{u}_{ijk}={\displaystyle \sum }_{i=1}^I{J}_i$$

(8)

In part i for J_i operations, the number of connections between POs should be J_i − 1, as outlined below:

$${\displaystyle \sum }_{j=1}^{J_i}{\displaystyle \sum }_{k=1}^{K_{ij}}{\displaystyle \sum }_{j^{\prime }=1}^{{J_i}^{\prime }}{\displaystyle \sum }_{k^{\prime }=1}^{K_{i^{\prime }{j}^{\prime }}}{v}_{jk{j}^{\prime }{k}^{\prime }}^i=J_i-1$$

(9)

where

$$j\ne j^{\prime }$$

For a PO, there should be only one PO before it or after it, as follows:

$${\displaystyle \sum }_{j=1}^{J_i}{\displaystyle \sum }_{k=1}^{K_{ij}}{v}_{jk{j}^{\prime }{k}^{\prime }}^i\le 1$$

(10)

$${\displaystyle \sum }_{j^{\prime }=1}^{{J_i}^{\prime }}{\displaystyle \sum }_{k^{\prime }=1}^{K_{i^{\prime }{j}^{\prime }}}{v}_{jk{j}^{\prime }{k}^{\prime }}^i\le 1$$

(11)

The end of the machining process for the preceding PO must occur prior to the commencement of the machining process for the subsequent PO, as outlined below:

$$B{T}_{ijk}+m{t}_{ijk}\le B{T}_{i{j}^{\prime }{k}^{\prime }}$$

(12)

BT_ijk denotes the start or beginning time of the $k^{th}$ PO for the $j^{th}$ operation of part i, provided it is chosen for machining, and the POs, PO_ijk, and PO_i′j′k′ pertain to the same part i.

The decision variables employed to establish the constraints for the POs handled by the same machine, represented by binary values, are specified as follows:

$${\displaystyle \sum }_{i=1}^I{\displaystyle \sum }_{j=1}^{J_i}{\displaystyle \sum }_{k=1}^{K_{ij}}{\displaystyle \sum }_{i^{\prime }=1}^I{\displaystyle \sum }_{j^{\prime }=1}^{{J_i}^{\prime }}{\displaystyle \sum }_{k^{\prime }=1}^{K_{i^{\prime }{j}^{\prime }}}{w}_{ijk{i}^{\prime }{j}^{\prime }{k}^{\prime }}=M-m_{used}$$

(13)

For the POs which are machined by the same machine, there should be only one PO before it or after it, as follows:

$${\displaystyle \sum }_{i=1}^I{\displaystyle \sum }_{j=1}^{J_i}{\displaystyle \sum }_{k=1}^{K_{ij}}{w}_{ijk{i}^{\prime }{j}^{\prime }{k}^{\prime }}\le 1$$

(14)

where

• $i^{\prime }=1,2,3,\dots ,I$, $j^{\prime }=1,2,3,\dots ,{J_i}^{\prime }$, $k^{\prime }=1,2,3,\dots ,K_{i^{\prime }{j}^{\prime }}$

$${\displaystyle \sum }_{i^{\prime }=1}^I{\displaystyle \sum }_{j^{\prime }=1}^{{J_i}^{\prime }}{\displaystyle \sum }_{k^{\prime }=1}^{K_{i^{\prime }{j}^{\prime }}}{w}_{ijk{i}^{\prime }{j}^{\prime }{k}^{\prime }}\le 1$$

(15)

where

• $i=1,2,3,\dots ,I$, $j=1,2,3,\dots ,J_i$, $k=1,2,3,\dots ,K_{ij}$

If PO_ijk is the preceding PO of PO_i′j′k′ and both are processed by the same machine, the completion time of the machining process for PO_ijk must precede the initiation time of the machining process for PO_i′j′k′ as stated below:

$$B{T}_{ijk}+m{t}_{ijk}\le B{T}_{i^{\prime }{j}^{\prime }{k}^{\prime }}$$

(16)

where

$$m_{ijk}=m_{i^{\prime }{j}^{\prime }{k}^{\prime }}$$

Let MS_i^m be the makespan of the machine i^m, where it denotes the sum of the time necessary to complete all tasks allocated to that machine, encompassing both processing time and idle time. So, the MS_i^m is as follows:

$$M{S}_{i^m}={\displaystyle \sum }_{i=1}^I{\displaystyle \sum }_{j=1}^{J_i}{\displaystyle \sum }_{k=1}^{K_{ij}}{u}_{ijk}.m{t}_{ijk}.\Omega 1\left(i^m,m_{ijk}\right)+{\displaystyle \sum }_{i=1}^{I_{i^m}^{idle}}I{T}_{idlem}$$

(17)

where Ω1(i^m,m_ijk) is a function that checks if a task is assigned to the correct machine. It returns 1 if the task is on the right machine and 0 if not.

Set MN_max as the maximum makespan over all machines, as described below:

$$MN\mathit{max}={\left(M{N}_{i^m}\right)}_{max}$$

(18)

Take OMS as the overall or cumulative makespan of all machines, as detailed below:

$$OMS={\displaystyle \sum }_{i^m=1}^{I^m}M{N}_{i^m}$$

(19)

The cost of the machine is linked to the specific machine chosen for a given operational process. For each operation j of part i on the corresponding machine, the cost of the machine is specified as follows:

$$C_{ij}=w_1\times C_{mac}\left(m_{ijk}\right)$$

(20)

The cost of the tool is based upon the specific tool chosen for each operation and varies according to the type of tool employed.

The tool cost for operation j of part i is determined as follows:

$$C_{tool,i}=w_2\times C_{tool}\left(t_{ijk}\right)$$

(21)

The setup cost is the fixed expense associated with preparing a machine for a particular task. The cost of this arrangement may vary based on the machine utilized.

$$C_{setup,i}=w_3\times C_{setup}$$

(22)

The idle time cost pertains to the inefficiency incurred during periods when a machine is not in operation. The expense is dependent upon the duration of the machine’s idleness.

The idle time penalty cost for machine i^m is as follows:

$$C_{idle,i^m}=W_{idle}\times I{T}_{idlem}\left(i^m\right)\times \lambda$$

(23)

The total cost for operation j of part i on a machine m_ijk comprises the machine cost, tool cost, setup cost, and idle time cost. The overall expense for operation j is specified as follows:

$$C_{total,i}=C_{ij}+C_{tool,i}+C_{setup,i}+C_{idle,i^m}$$

(24)

The total system cost is the sum of the costs associated with all the operations of all the parts in the scheduling procedure. The overall system cost is determined by aggregating the total costs of all operations J_i of all parts I. This is defined as follows:

$$C_{total}={\displaystyle \sum }_{i=1}^I{\displaystyle \sum }_{j=1}^{J_i}\left(C_{ij}+C_{tool,i}+C_{setup,i}+C_{idle,i^m}\right)$$

(25)

3.2. Objective

The objective is to minimize the overall completion time. The classical MILP model of a given IPPS problem can be found in the literature [14]. The formulation is as follows:

$$\mathit{min}f=OMS$$

(26)

4. Proposed Symmetry-Aware Adaptive Ant Colony Optimization (SA-AACO) Algorithm

Traditional ACO excels in combinatorial optimization but struggles with complex scheduling tasks due to inefficiency and slow convergence under constraints like machine–tool allocations. In contrast, enhanced versions of ACO that incorporate dynamic components have demonstrated a more balanced exploration of the solution space, leading to improved convergence toward optimal or near-optimal solutions. As a result, such adaptive strategies are particularly well-suited for addressing the complexities of modern manufacturing systems, offering tangible benefits in minimizing total completion time, reducing production costs, and avoiding idle time penalties.

4.1. The Framework of Proposed SA-AACO

The proposed symmetry-aware adaptive Ant Colony Optimization (SA-AACO) framework (as shown in Figure 1) is designed to efficiently address integrated process planning and scheduling (IPPS) by dynamically constructing solutions while managing symmetry and adapting exploration intensity. The process is detailed in the following steps:

Step 1: The algorithm initiates by activating the optimization workflow and preparing the system for execution.

Step 2: Relevant process data such as part features, machine capabilities, operation sequences, tools, and precedence constraints are gathered from the predefined dataset or input files. This information serves as the foundational layer for subsequent decision-making.

Step 3: A cost matrix is constructed by evaluating various performance metrics including processing time, machine setup costs, and tool-switching penalties. These costs influence the heuristic desirability of operation-to-resource assignments during solution construction.

Step 4: Before the search begins, the parameters are set for the ant colony system. These include the number of ants, the pheromone influence factor, heuristic information weight, pheromone evaporation rate, and the adaptive parameter Q, which is adapted based on solution quality over time.

Step 5: The algorithm enters its main iterative loop, which runs for a predefined number of iterations. Each iteration simulates a collective decision-making process among the ants.

Step 6: At the beginning of each iteration, ants are initialized. Each ant represents a candidate solution constructor that begins from a blank slate and builds a schedule through a series of informed decisions.

Step 7: The status of all jobs, machines, and tools is initialized. This includes tracking job completion progress, machine occupancy, and tool usage, ensuring that ants make decisions based on the current state of the system.

Step 8: A sequence of feasibility checks is applied to identify operations that are eligible for scheduling. These filters consider machine availability, operation precedence, tool readiness, and temporal constraints. Multiple filters are used in tandem to ensure that only valid options are passed forward for construction.

Step 9: Each ant incrementally constructs a solution by selecting feasible operations and assigning them to resources. The selection process is guided by pheromone trails, heuristic cost values, and real-time feasibility checks. Symmetry-disrupting strategies are incorporated here to diversify the paths taken by ants, thus reducing redundancy and improving exploration of the solution space.

Step 10: Once a solution is built, its performance is evaluated using a cost function. This function aggregates the total processing time, machine idle time, switching costs, and overall makespan, which allows the algorithm to compare the quality of different solutions.

Step 11: If an ant constructs a better solution than the current best option, it replaces the best-so-far solution. The algorithm checks for improvement trends or stagnation based on solution quality across iterations.

Step 12: Based on the observed performance, the Q parameter is adaptively tuned. If improvement is detected, Q is increased to amplify the reinforcement effect; otherwise, it is decreased to reduce the influence of stagnated paths and encourage exploration.

Step 13: Pheromone levels are updated to reflect recent search experiences. First, evaporation reduces the influence of older pheromone trails. Then, reinforcement increases pheromone levels on the paths chosen by the most successful ants, making them more attractive in the future.

Step 14: The pheromone matrix is revised based on the adjusted Q values and solution fitness, effectively encoding the collective learning of the ant population up to the current iteration.

Step 15: Intermediate results, such as Gantt charts, operation sequences, or performance metrics, are visualized to provide insights into the evolution of the search process and the current solution status.

Step 16: At the end of each iteration, the algorithm checks whether the stopping criterion has been met. If not, it returns to initializing the ants for the next iteration. If so, the loop terminates.

Step 17: Once terminated, the algorithm outputs the best solution found. This includes detailed scheduling information, performance metrics, and any visual representations of the final schedule.

Step 18: The algorithm concludes its execution, completing the optimization process.

4.1.1. Heuristic Information

Within the framework of this proposed scheduling algorithm, a crucial element is the heuristic value η_{ijki′j′k′}, which assesses the attractiveness of choosing a certain potential operation (PO) PO_i′j′k′, as the subsequent operation in the schedule. The processing efficiency of a particular machine defines the heuristic value through inverse machine or processing time calculations. As a result, each PO follows feasibility conditions that determine eligibility for scheduling by using the binary variable w_{ijki′j′k′}. This equation defines how to compute the heuristic value:

$${\eta }_{ijk{i}^{\prime }{j}^{\prime }{k}^{\prime }}={\displaystyle \frac{1}{m{t}_{i^{\prime }{j}^{\prime }{k}^{\prime }}}} \mathrm{when}, w_{ijk{i}^{\prime }{j}^{\prime }{k}^{\prime }}=1$$

(27)

The value of ${\eta }_{ijk{i}^{\prime }{j}^{\prime }{k}^{\prime }}$ will be equal to zero if $w_{ijk{i}^{\prime }{j}^{\prime }{k}^{\prime }}=$0.

4.1.2. Time Window Mechanism for Machine Selection

To implement the proposed SA-AACO algorithm, a static time window mechanism is utilized during the scheduling phase to facilitate machine selection, as illustrated in Figure 2. This static time window defines a fixed-duration interval during which machine availability is assessed for the purpose of assigning tasks. This serves as a baseline structure that ensures tasks are scheduled for machines without overlap or conflict, thereby maintaining order and feasibility in the scheduling process. Machines that are either busy or otherwise unavailable during the defined interval automatically exclude themselves from consideration, ensuring only eligible machines are evaluated. This approach contributes to efficient scheduling by reducing execution delays and minimizing machine idleness.

However, while the time window itself remains static during a single scheduling cycle, the overall process incorporates dynamic elements that adapt in real time to improve decision-making. For instance, the algorithm dynamically adjusts idle time penalties based on machine utilization patterns, encouraging balanced workloads across resources. Additionally, pheromone levels are updated adaptively to reflect the success of previous scheduling decisions, guiding the ants toward more promising machine task assignments in subsequent iterations. These dynamic mechanisms complement the static time window by introducing responsiveness to changing system conditions, leading to improved scheduling outcomes over time.

The time window-based approach to machine candidate selection further enhances symmetry management. This mechanism provides flexibility in process assignment while maintaining efficiency, allowing the algorithm to consider multiple symmetrical scheduling options within a controlled time window. This approach helps prevent myopic scheduling decisions that might arise from strictly greedy selection methods.

4.1.3. Pheromone-Based Learning in SA-AACO Algorithm

In the SA-AACO framework, ants are guided toward more promising solutions through pheromone-based learning. Two key stages—adaptive Q adjustment and pheromone update (evaporation and reinforcement)—work in tandem to balance exploration and exploitation. These mechanisms are central to effective convergence and symmetry disruption, and their order of execution in the algorithm is crucial. After ants construct solutions and the best solution is updated, the algorithm enters the “Adjust Q Parameters” step. This is executed before any pheromone updates are made, ensuring that the latest Q value reflects the current state of the search and is then applied during reinforcement.

The first step adapts the Q parameter, which determines the amount of pheromone deposited during the reinforcement. The value of Q is dynamically updated based on the success of the current iteration as follows:

• If an improvement is found, Q rises by Q_{step_increase}, although it is constrained by Q_max.

$$Q\left(t+1\right)=\mathit{min}\left(Q\left(t\right)+Q_{step\_increase},Q_{max}()\right)$$

(28)

2.

If no enhancement is observed after a certain number of iterations (monitored by no_improvement_count), Q is reduced by Q_{step_decrease}, with a lower limit of Q_min.

$$Q\left(t+1\right)=\mathit{max}\left(Q\left(t\right)-Q_{step\_decrease},Q_{min}()\right)$$

(29)

where

• $Q(t)$ is the pheromone magnitude at iteration t;
• $Q_{step\_increase}$ is the rate at which Q increases when improvements are found;
• $Q_{step\_decrease}$ is the rate at which Q decreases when no improvements are found;
• $Q_{\mathit{max}}$ and $Q_{\mathit{min}}$ are the upper and lower bounds for Q, respectively.

The final adaptive rule for Q can be compactly represented as follows:

$$Q\left(t+1\right)=\{\begin{array}{c}\min \left(Q\left(t\right)+Q_{step\_increase,}{Q}_{max}()\right)\\ \max \left(Q\left(t\right)-Q_{step\_decrease,}{Q}_{min}()\right)\end{array}$$

(30)

This adaptive mechanism dynamically adjusts the pheromone deposit rate (Q) based on solution improvement patterns. When improvements are found, Q increases to intensify the exploitation of promising regions. After periods without improvement, often indicating the algorithm has encountered a symmetrical plateau, Q decreases to promote the exploration of alternative solution paths. This balance between intensification and diversification is crucial for effectively handling symmetrical regions of the solution space.

This adaptive Q mechanism directly influences the next stage, pheromone reinforcement, by determining how strongly newly found solutions should be emphasized in the pheromone trail.

Once Q is adjusted, it is used during pheromone updating, which consists of two key sub-steps:

• Pheromone Evaporation:

When simulating the natural decay of pheromones, all trail values are reduced by a factor of (1 − ρ), where ρ ∈ (0,1). This avoids the over-reinforcement of early paths and encourages exploration by weakening outdated solutions.

2

Pheromone Reinforcement:

Using the updated Q from the previous step, pheromones are deposited on the paths used by high-quality solutions:

$${\tau }_{ijk{i}^{\prime }{j}^{\prime }{k}^{\prime }}\left(t+1\right)=\max \left({\tau }_{min}{\left(\left(1-\rho \right)·{\tau }_{ijk{i}^{\prime }{j}^{\prime }{k}^{\prime }}\left(t\right)+{\displaystyle \frac{Q}{\cos t}},{\tau }_{max}()\right)}_{min}\right)$$

(31)

where

• Evaporation reduces the pheromone by a factor of (1 − ρ);
• Reinforcement increases the pheromone based on the inverse of the solution’s cost;
• τ_min and τ_max ensure that pheromone values remain bounded, preventing early convergence and maintaining search diversity.

The implementation of minimum and maximum pheromone bounds adds another layer of symmetry management. These bounds prevent pheromone stagnation in symmetrical regions, ensuring that even when strong pheromone trails develop along certain paths, alternative paths maintain minimum exploration potential, allowing the algorithm to escape local optima in symmetrical plateaus.

This pheromone update mechanism explicitly breaks symmetry by differentiating between solutions with similar costs but different idle time distributions. Solutions that minimize idle time receive stronger pheromone reinforcement, creating a gradient within otherwise symmetrical regions of the solution space. This encourages ants to favor more time-efficient sequences, even when traditional cost metrics would suggest equivalence.

4.1.4. Dynamic Idle Time Penalty Adjustment

The Dynamic Idle Time Penalty Adjustment mechanism is implemented in the SA-AACO algorithm to reduce the ineffectiveness that results from idle time within a schedule by adjusting the penalty factor based on existing data on machine availability and progress in the process.

Typically, the current algorithm examines the idle time with regard to each machine, which indicates the time that the machine remained inactive for between consecutive operations. The time period is then used to arrive at a penalty, which is then taken to be added to the cost function of the scheduling solution. The penalty is then computed with the help of a weighted function, the idle weight, the total idle time, and an adaptive parameter λ. It should be noted that the cost of this penalty is given in (23).

$$Idl{e}_{penalty}=W_{idle}\ast I{T}_{idlem}\left(i^m\right)\ast \lambda$$

(32)

A significant role of the λ parameter allows the algorithm to adjust dynamically based on scheduling process changes. The modification of this parameter allows the algorithm to manage idle time penalties effectively, which supports improved resource usage throughout optimization runs.

The flexible idle time penalty mechanism of the algorithm manages the exploration vs. exploitation balance during the optimization period. Hence, a changeable algorithm structure adapts to changes in machine availability and production constraints, leading to improvements in scheduling quality together with better resource utilization. Thus, the established penalty amount becomes part of the scheduling solution’s final cost along with other costs regarding machinery, tooling, and setup. The system repeats this procedure multiple times until it reaches the best possible scheduling solution, which optimizes machine efficiency and production capability.

The idle time penalty cost is included in the final cost to mitigate idle time, which is crucial for guiding the algorithm toward optimal machine usage by discouraging decisions that lead to idle periods, hence preventing inefficient schedules. Figure 3 illustrates the procedure for determining the overall cost, including the idle time penalty.

The many cost elements, including machine costs, tool costs, and setup changes in the symmetry-aware adaptive ACO algorithm, influence both the quality of solutions and the search dynamics. These aspects establish a multi-faceted framework for optimization that directs the algorithm towards effective manufacturing scheduling through several methods. The cost matrix directly influences transition probabilities through the heuristic factor (η), as transitions with lower costs are more likely to be chosen. Secondly, the weight parameters effectively equilibrate conflicting priorities within the manufacturing processes, enabling the algorithm to adapt to diverse production environments by modifying the operational cost penalty associated with changeover. Third, the understanding of symmetry arises from a holistic perspective that considers all costs, thus preventing asymmetrical optimizations—solutions that enhance one aspect while neglecting others. The base costs (machine, tool) dictate direct resource selection, while the change costs (machine, setup, tool alterations) promote continuity, and the penalties for idle time serve to dynamically enhance load balance. This comprehensive technique is enhanced by the adaptive Q parameter, which modifies pheromone reinforcement based on the improvement of a solution, thus increasing or decreasing the influence of cost components throughout various stages of the search.

These factors collectively create an accurate representation of the limitations and economics of the manufacturing environment, ensuring the scheduling algorithm attains a reasonable equilibrium among resource utilization, changeover minimization, and production expenses while considering practical aspects.

4.1.5. Probability for Random Selection of Processes

The random selection, executed by ant r, to determine the subsequent operation PO, PO_i′j′k′, dependent upon the prior selection of PO, PO_ijk, can be expressed probabilistically as shown in (33). The likelihood that ant r chooses the subsequent PO, PO_i′j′k′, is dependent upon the pheromone concentrations τ_{ijki′j′k′} and the heuristic values η_{ijki′j′k′}, which signify the appeal of the operation and the machine, respectively. The formula includes parameters α and β, which regulate the relative significance of pheromone and heuristic values in the decision-making process. Furthermore, J_r denotes the collection of unselected potential operations available to ant r, whereas S_{ri′j′k′} serves as a scaling factor for each potential selection, and is dynamically modified according to the current state of the search as the ants advance. This component is generally elevated for solutions that demonstrated superior performance in prior iterations, hence reinforcing successful trajectories within the solution space. Consequently, ants have a tendency to choose actions that yield superior solutions, thus enhancing the pursuit of optimal routes.

The probability P^r_{ijki′j′k′} of ant r selecting PO_{ik″jk″kk″} from the remaining potential operations J_r(ijk), accounting for the idle penalty, is expressed as follows:

$$P_{ijk{i}^{\prime }{j}^{\prime }{k}^{\prime }}^r=\{\begin{array}{c}{\displaystyle \frac{{\left({\tau }_{ijk{i}^{\prime }{j}^{\prime }{k}^{\prime }}\right)}^{\alpha }·{\left({\eta }_{ijk{i}^{\prime }{j}^{\prime }{k}^{\prime }}\right)}^{\beta }·S_{r{i}^{\prime }{j}^{\prime }{k}^{\prime }}}{{{\displaystyle \sum }}_{i^{″}{j}^{″}{k}^{″}\in J_r\left(ijk\right)}\left({\left({\tau }_{ijk{i}^{″}{j}^{″}{k}^{″}}\right)}^{\alpha }·{\left({\eta }_{ijk{i}^{″}{j}^{″}{k}^{″}}\right)}^{\beta }+Idl{e}_{penalty}\right)}}\\ 0\end{array}$$

(33)

It selects the $P{O}_{i^{″}{j}^{″}{k}^{″}}$ from the remaining operations; otherwise, it equals zero, indicating that no operations remain among the potential operations.

During the selection of the subsequent operation, ant r assesses all viable potential operations pending scheduling, modifying the selection probability of each depending on pheromone concentrations, heuristic attractiveness, and the scaling factor. For this purpose, the equation above illustrates the equilibrium between the exploitation of high-quality solutions, indicated by pheromone intensity, and the investigation of new prospective solutions, derived from heuristic knowledge. Furthermore, by integrating the idle penalty, the program guarantees that machines are utilized efficiently, avoiding potential inefficiencies. Therefore, the likelihood of selecting an operation is modified to decrease the probability of choosing solutions with excessive idle time, hence enhancing overall scheduling efficiency. Consequently, this methodology promotes a more equitable search that evolves with the algorithm, prioritizing high-quality solutions while preserving the necessary diversity to investigate the complete solution space. The following section shows the proposed SA-AACO algorithm results combined with a comparison analysis of current methods and proof of the successful application of proposed changes.

5. Experiments and Discussions

Prior to its application to the IPPS problem, the algorithm was validated against Brandimarte’s widely recognized flexible job shop scheduling (FJSSP) benchmarks [41], demonstrating competitive performance, including a 10–15% faster convergence rate compared to genetic algorithm (GA)-based approaches. After that, multiple test problems consisting of different manufacturing situations formed part of the experimental objects. The test problems consisted of different part quantities that needed to process through specific machine groups. The benchmark problem tests varied in complexity, ranging from scenarios with 6 parts to more challenging cases involving up to 18 parts [14,42].

As is pointed out above, every test problem has many parts, and each part may have different characteristics concerning the type of machine that is expected to handle the problem, and the amount of time expected to complete the job’s processing. The makespan of all the parts is to be minimized, and these parts are indexed by the dataset in order to develop the proposed ACO algorithm for constructing the schedule.

Table 2. Test problems.

Problem	Number of Parts	Index of Parts
1	6	1 4 8 12 15 17
2	6	2 6 7 10 14 18
3	6	1 4 7 10 13 16
4	9	1 4 5 7 8 10 13 14 16
5	12	4 5 6 7 8 9 13 14 15 16 17 18
6	12	1 2 4 5 7 8 10 11 13 14 16 17
7	12	2 3 5 6 8 9 11 12 14 15 17 18
8	15	2 3 4 5 6 8 9 10 11 12 13 14 16 17 18
9	15	1 4 5 6 7 8 9 11 12 13 14 15 16 17 18
10	18	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

presents a summary of the test problems. The quantity of parts and their corresponding indices are enumerated for each problem case. These test cases encompass a range of problem sizes and configurations, rendering them indicative of standard scheduling challenges in manufacturing.

The calculations of the proposed Ant Colony Optimization (ACO) algorithm were performed on a personal computer which had an Intel i5 CPU and 8GB RAM. MATLAB version (R2022a), with corresponding supporting tools that allow comprehensive mathematical modeling and computational optimization, was installed in the system. Windows 10 was used as an operating system for a stable and compliant environment to run the MATLAB code.

5.1. Parameter Settings

Extensive experimentation was carried out to optimize the values of critical parameters in the experimental setup, after which specific critical parameters were chosen for use in the proposed SA-ACO method in the experimental setup. The iteration count (T = 100) was determined through preliminary experiments and aligned with established practices in the ACO literature [10,14,39]. Empirical observations confirmed that SA-AACO achieves stable convergence (no improvement in makespan for 20 consecutive iterations) by T = 100 across problem scales, while maintaining practical execution times (e.g., ≤28.6 min for 18 parts). Prior studies in IPPS optimization [14,39] similarly adopt 50–150 iterations, balancing solution quality and computational effort. SA-AACO’s adaptive mechanisms (e.g., symmetry awareness, dynamic pheromone updates) further reduce reliance on prolonged iterations by efficiently navigating symmetrical search spaces. This parameter ensures robust performance without compromising real-world applicability.

The quantity of ants utilized in the method was established at 20. An increased population of ants generally improves the diversity of solutions investigated, leading to superior overall outcomes. Nonetheless, augmenting the ant population escalates computational requirements, so 20 ants were selected as a suitable balance. The impact of pheromone (α) on route selection was established at 2, prompting the ants to choose pathways with more potent pheromone trails. Simultaneously, the heuristic factor (β) was established at 4, prompting the ants to depend more significantly on problem-specific knowledge included in the heuristic factor. The pheromone evaporation rate, ρ, was established at 0.20, regulating the decay rate of pheromone intensity over time. An increased evaporation rate mitigates stagnation by facilitating the rapid dissipation of pheromone trails on suboptimal courses, hence promoting the investigation of other routes.

The method utilized an adaptive Q parameter, first established at a value of 80. The Q_min and Q_max parameters provide the minimum and maximum thresholds for Q, ensuring it remains within an acceptable range. If the algorithm decreases the cost, Q is elevated by a step size of 10, thus facilitating exploration. If no enhancement is observed over 5 successive iterations, Q is diminished by a decrement of 3, facilitating more exploration and preventing premature convergence on suboptimal solutions. The adaptive traits of Q promote equilibrium between exploration and exploitation, allowing the algorithm to adapt to different stages of the search process.

To increase scheduling efficiency, a dynamic idle time penalty mechanism was introduced in this study. This was suitable for adjusting the penalty weight by iteration progress and total idle time. The SA-AACO algorithm adapts the weight instead of using a fixed idle penalty to strike a balance between exploration and exploitation. Initially, a more relaxed form of penalty (W_idle * 0.3) was used to explore the solution space to a greater extent. Later on, the penalty became more stringent (W_idle * 1.0) to stop at a better solution as the objective was to minimize machine idle time. Hence, the dynamic scaling of the idle penalty factor, λ, was set between 0.3 ≤ λ ≤ 1.0, guaranteeing that longer idle durations are greater contributions to total cost function. Thus, the proposed method adapted penalty values non-linearly to aid in the development of a penalty-based, fair, stable, and efficient optimization strategy.

5.2. Analysis of Experimental Results

This section provides a thorough examination of the proposed SA-AACO algorithm for the IPPS issue. The effectiveness of the algorithm can be seen from its performance on ten IPPS evaluation problems. To make simulations more accurate, these instances were handled using a log-normal distribution for processing time. The SA-AACO was assessed in comparison to classical and mixed metaheuristics, for example, CCGA, SEA, HA, ALGA, TGA, ICA, standard ACO, GA-VNS, MMCO, and MSCOA [14,43,44,45]. The outcomes are derived from ten benchmark tasks, with the makespan values presented in

Table 3. Optimization results for Makespan.

Prob.	CCGA	SEA	HA	ALGA	TGA	ICA	ACO	GA VNS	MMCO	MSCOA	Proposed Algorithm
1	378	372	372	372	372	372	283	372	372	372	268
2	363	343	343	347	352	343	295	343	343	339	270
3	312	306	306	306	306	306	293	306	306	306	265
4	360	328	327	327	321	327	306	319	318	322	268
5	466	431	423	423	414	443	392	362	344	384	324
6	396	379	377	377	363	384	380	349	318	349	309
7	535	490	476	474	468	490	362	427	427	445	338
8	567	534	518	513	493	529	480	433	427	481	412
9	531	498	470	470	456	495	461	388	372	440	354
10	611	587	544	548	523	577	487	446	427	522	483

.

Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 illustrate the Gantt charts for the scheduling outcomes related to the problem situations discussed in this study. The figures illustrate the allocation of operations to machines over time, showcasing the efficacy of the proposed SA-AACO algorithm in addressing IPPS challenges. Each phase is represented by a colored rectangular block identified as O_a,b, where ‘a’ signifies the part number and ‘b’ indicates the operation index of that part. The annotation adjacent to the charts indicates the color assigned to each part, facilitating the identification and differentiation of the elements within the overall timetable.

The processing time on the x-axis of each Gantt chart is represented in uniform units, while the machine number assigned to processes is indicated on the y-axis. This structure facilitates the clear comprehension of the operational sequence, task distribution among machines, and resource use. Charts ensure that precedence restrictions are upheld by arranging the actions of the same part in the required sequence, preventing overlap on a single machine at any given moment.

Furthermore, the charts illustrate that the proposed strategy effectively accommodates the flexible nature of job shop scheduling. In Figure 4 (pertaining to a smaller-sized problem consisting of 6 parts), the operation blocks are distributed at considerable intervals, signifying reducing system congestion and expediting makespan convergence. Progressing to Figure 5 and Figure 6 (containing 9 and 12 parts, respectively) introduces additional parts, resulting in increased complexity in scheduling. The method demonstrates adaptable job allocation, since the machines maintain balanced workloads and avoid idle periods through strategic sequencing. Moreover, Figure 7 and Figure 8, representing the problems with 15 and 18 parts, respectively, further illustrate how the algorithm’s robustness enables it to manage large-scale scenarios that simultaneously account for machine availability and operational precedence. The seamless and harmonious distribution of operations exemplifies the efficacy of the implemented symmetry-aware improvements in the algorithm.

These Gantt charts act as both a visual depiction of the schedule and as proof of the algorithm’s capacity to optimize makespan, equilibrate machine workloads, and adapt to differing system scales. They offer an exhaustive perspective on the scheduling results, which is crucial for evaluating the effectiveness and relevance of the proposed SA-AACO framework in practical flexible manufacturing systems.

Table 3 presents the comparative makespan results for ten benchmark instances evaluated across multiple optimization algorithms, including CCGA, SEA, HA, ALGA, TGA, ICA, ACO, GA VNS, MMCO, MSCOA, and the proposed SA-AACO. The performance of the SA-AACO algorithm is demonstrably superior, achieving the lowest makespan in 9 out of the 10 tested instances. This consistent performance highlights the robustness and effectiveness of the proposed algorithm in addressing complex scheduling problems within the IPPS domain. Notably, SA-AACO shows significant improvement in several challenging scenarios. For instance, in Problem 1, it reduced the makespan to 268, compared to 372 in most other methods and 283 in traditional ACO, reflecting a fair improvement over standard technique. Similar substantial reductions are evident in Problem 5 (from a maximum of 466 to 324), Problem 7 (from a peak of 535 to 338), and Problem 9 (from 531 to 354), indicating the proposed method’s strength in handling larger and more symmetrical scheduling problems where conventional algorithms tend to stagnate. The only instance where SA-AACO did not outperform all other methods was Problem 10, where MMCO achieved the lowest makespan of 427, whereas SA-AACO delivered one of 483. However, even in this case, the proposed method’s performance remains competitive, showing that while SA-AACO excels in the majority of cases, it may exhibit room for fine-tuning under certain problem structures or parameter settings.

Table 4. Best makespan values and CPU execution times for each problem instance.

Problem	Best Makespan Value	Algorithm	CPU Time (minutes)
1	268	Proposed Algorithm	7.8
2	270	Proposed Algorithm	8.5
3	265	Proposed Algorithm	6.2
4	268	Proposed Algorithm	12.4
5	324	Proposed Algorithm	18.5
6	309	Proposed Algorithm	16.1
7	338	Proposed Algorithm	21.7
8	412	Proposed Algorithm	24.3
9	354	Proposed Algorithm	23.5
10	427	MMCO	28.6

displays the best makespan values achieved for each problem by the corresponding algorithms. Furthermore, to address the computational efficiency of SA-AACO, we measured CPU execution times for all problem instances. As shown in Table 4, SA-AACO solves smaller problems (e.g., Problems 1–3 with 6 parts) in 6.2–8.5 min, while scaling efficiently to larger instances like Problem 8 (15 parts, 24.3 min) and Problem 9 (15 parts, 23.5 min). For the most complex case (Problem 10, 18 parts), the MMCO algorithm achieved a marginally better makespan (427 vs. SA-AACO’s 483) but required a comparable runtime of 28.6 min, demonstrating SA-AACO’s practical feasibility in real-world scenarios.

The analysis of the performance of the proposed algorithm highlights some key points. The superior performance of SA-AACO compared to standard ACO highlights how incorporating symmetry awareness and adaptive learning significantly improves the results. First, the algorithm can handle more difficult issues and still provide excellent results. Secondly, the adaptability in the algorithm helps it to stop early convergence, which is normally a big issue in basic ACO for very large or greatly similar problems. Besides its well-built base, the algorithm succeeds largely because of its special efforts to counteract symmetry problems that can slow things down. As an example, the time given to select a machine provides flexibility so that various groupings of jobs can be considered within a given time frame. As a result, the algorithm can study a variety of workable schedules rather than being caught up in poor or short-sighted decision chains.

SA-AACO is the most powerful option because its architecture takes symmetry into account. It is common for traditional metaheuristics to become stuck in symmetric areas of the solution space where many possible equal solutions block their advance. To solve this, SA-AACO applies a number of creative approaches for handling symmetry. Instead of checking many symmetric configurations, the algorithm uses time window-based selection, which increases the quality of the exploration significantly. Furthermore, the penalty assigned to inaction adjusts dynamically based on system requirements, reducing unnecessary equipment idle time and improving the overall production rate. Also, there are minimum and maximum bounds set for pheromone values, which ensure that paths with too much reinforcement do not slow down the process occurring away from them. As a result, backup plans can still be applied in very strong regions. The pheromone quantity (Q) adapts itself in real time based on how well the search is improved as a result of the dynamic pheromone deposit. Once the algorithm spots that a search seems stuck in the same place, which typically indicates that people are trapped symmetrically, it reduces discounting to push the exploration of new possibilities. On the other hand, in improving regions, the Q ratio goes up to encourage greater exploitation. Because of this balance, the algorithm can handle symmetric plateaus and does not become stuck in unfavorable local solutions. To make sure the system worked well on problems of different sizes and difficulties, each ACO parameter was carefully set. Only feasible solutions were accepted in the sequence of work, ensuring the problem remained relevant to the real world.

Improvements in makespan, scalability, and robustness suggest that SA-AACO is suitable for use in smart manufacturing where uncertainty, changing workloads, and limited resources exist. Its ability to handle a wide range of machine choices, reduce redundant solutions, and adapt in real time makes it highly suitable for Industry 4.0 scheduling applications. Moreover, the fact that the algorithm can deal with various constraints and goals means it may be suited for multi-objective optimization in upcoming research.

5.3. Statistical Validation

To rigorously validate the comparative performance of SA-AACO, we conducted comprehensive statistical analysis using non-parametric tests and effect size measures. Given the small sample size (10 problem instances) and non-normal distribution of makespan values, the Wilcoxon signed-rank test, a robust alternative to paired t-tests, was employed to compare SA-AACO to each benchmark algorithm. The null hypothesis (H₀) suggested no difference in median makespan between SA-AACO and the compared algorithm, while the alternative hypothesis (H₁) asserted the superiority of SA-AACO.

As shown in

Table 5. Statistical comparison of SA-AACO against benchmark algorithms.

Algorithm	p-Value	Cohen’s d	Mean Improvement (%)
CCGA	0.003	2.1	24.9
SEA	0.008	1.8	22.1
HA	0.005	1.7	21.3
ALGA	0.007	1.6	20.5
TGA	0.009	1.5	19.8
ICA	0.010	1.4	18.2
ACO	0.015	1.3	17.6
GA-VNS	0.018	1.1	15.9
MMCO	0.012	1.2	13.0
MSCOA	0.022	0.9	12.7

, the Wilcoxon test rejected H₀ for all benchmark algorithms at a significance level of α=0.05 (p<0.05), confirming that SA-AACO’s improvements are statistically significant in 9 out of 10 problem instances. The sole exception is Problem 10, where MMCO achieved a lower makespan (427 vs. 483), highlighting its effectiveness in extreme-scale scheduling scenarios. Despite this outlier, SA-AACO maintains dominance in the majority of cases.

In order to quantify the practical significance of these improvements, Cohen’s d effect sizes were calculated, measuring the standardized difference between SA-AACO and each benchmark algorithm’s mean makespan. All effect sizes exceeded 0.8 (Table 5), indicating large practical improvements according to Cohen’s conventions [46]. SA-AACO demonstrated the strongest advantage over CCGA (d = 2.1), while its smallest (but still substantial) edge was seen against MSCOA (d = 0.9).

The distribution of makespan values across algorithms, illustrated in Figure 9, further reinforces SA-AACO’s dominance. The box plots reveal that SA-AACO not only achieves the lowest median makespan but also exhibits the tightest interquartile range, highlighting its consistency in avoiding suboptimal solutions. These results validate the statistical significance (p < 0.05) and large practical improvements (Cohen’s d>0.8) reported in Table 5, confirming SA-AACO’s robustness in minimizing idle time and navigating symmetrical solution spaces.

This statistical validation underscores that SA-AACO’s performance gains are both statistically significant and practically meaningful, solidifying its suitability for real-world IPPS applications where reliability and efficiency are critical.

6. Conclusions and Future Work

6.1. Conclusions

This study introduces a novel optimization framework, symmetry-aware adaptive Ant Colony Optimization (SA-AACO), to address the integrated process planning and scheduling (IPPS) problem, recognized as one of the most intricate and computationally intensive challenges in modern manufacturing systems. The presented approach incorporates several modifications that signify enhancements in the Ant Colony Optimization (ACO) paradigm. The first modification is a symmetry-awareness mechanism that removes redundant solutions arising from symmetrically equivalent combinations of machine tools and operations. This approach decreases the effective size of the search space, hence enhancing the algorithm’s convergence by discarding useless computational efforts applied to functionally equivalent configurations. Secondly, the proposed algorithm incorporates the static time window which allows us to remove the strictness of machine selection while offering flexibility. In the proposed algorithm, the third innovation comprises dynamic idle time penalty, which ensures the maximum utilization of machines. The fourth improvement is the adaptive pheromone-updating technique that adjusts the concentrations of pheromones based on the quality and diversity of candidate solutions. This alteration guarantees equilibrium between exploration and exploitation.

The performance and robustness of SA-AACO were evaluated using ten benchmark IPPS instances. The results validate that the SA-AACO algorithm is reliably effective across diverse levels of issue complexity. It attained superior makespan performance in nine out of ten benchmark tasks, surpassing a diverse array of algorithms including CCGA, SEA, HA, ALGA, TGA, ICA, traditional ACO, GA-VNS, MMCO, and MSCOA. These results substantiate the assertion about the efficacy of the proposed modifications, which incorporated symmetry-awareness and adaptive learning strategies that enhanced convergence reliability. With the exception of issue 10, when GA-VNS and MMCO exhibited somewhat superior performance, SA-AACO demonstrated remarkable dominance and consistency throughout all other issue instances, especially in the intricate, high-dimensional scenarios.

The findings suggest that the algorithm is especially beneficial in real production settings due to its efficacy in the presence of stochastic processing time variability. In intelligent production systems marked by significant unpredictability and uncertainty, SA-AACO provides resilient, sophisticated scheduling that effectively accommodates evolving restrictions and fluctuating resource configurations. This advances optimization research towards real commercial applications, especially within the framework of Industry 4.0 and cyber–physical production systems.

Despite the promising results achieved by the proposed SA-AACO algorithm in various benchmark scenarios, there are a few limitations that should be acknowledged. First, the current framework has only been tested on static benchmark datasets with deterministic processing times; hence, its applicability to highly dynamic or real-time shop-floor environments remains to be explored. Second, the model does not account for machine breakdowns, setup changeover times between part types, or human operator constraints, which are critical in practical manufacturing settings. Third, the algorithm’s scalability for extremely large-scale industrial problems with hundreds of parts and machines has not been thoroughly validated.

6.2. Future Work

Though the SA-AACO algorithm stands out in dealing with the IPPS problem in stochastic environments, there are still some aspects that can be enhanced further. One of the most significant extensions is turning the current single-objective structure into a multi-objective optimization model, which could optimize makespan alongside its production cost, energy consumption, resource utilization, and other parameters simultaneously. Moreover, such an integration with real-time data environments empowered by digital twin systems or industrial IoT platforms could allow for dynamic remodeling and responsive restructuring and planning based on real-time shop-floor activities like machine failures and order modifications.