A Hybrid Approach for the Multi-Criteria-Based Optimization of Sequence-Dependent Setup-Based Flow Shop Scheduling

Abstract

A key challenge in production management and operational research is the flow shop scheduling problem, characterized by its complexity in manufacturing processes. Traditional models often assume deterministic conditions, overlooking real-world uncertainties like fluctuating demand, variable processing times, and equipment failures, significantly impacting productivity and efficiency. The increasing demand for more adaptive and robust scheduling frameworks that can handle these uncertainties effectively drives the need for research in this area. Existing methods do not adequately capture modern manufacturing environments’ dynamic and unpredictable nature, resulting in inefficiencies and higher operational costs; they do not employ a fuzzy approach to benefit from human intuition. This study successfully demonstrates the application of Hexagonal Type-2 Fuzzy Sets (HT2FS) for the accurate modeling of the importance of jobs, thereby advancing fuzzy logic applications in scheduling problems. Additionally, it employs a novel Multi-Criteria Decision-Making (MCDM) approach employing Proportional Picture Fuzzy AHP (PPF-AHP) for group decision-making in a flow shop scheduling context. The research outlines the methodology involving three stages: group weight assessment through a PPF-AHP for the objectives, weight determination using HT2FS for the jobs, and optimization via Genetic Algorithm (GA), a method that gave us the optimal solution. This study contributes significantly to operational research and production scheduling by proposing a sophisticated, hybrid model that adeptly navigates the complexities of flow shop scheduling. The integration of HT2FS and MCDM techniques, particularly PPF-AHP, offers a novel approach that enhances decision-making accuracy and paves the way for future advancements in manufacturing optimization.

1. Introduction

A significant challenge in operational research is the flow shop scheduling problem, which plays a role in the complex manufacturing and production management world. This issue, which arises when jobs need to be completed in a particular order through a sequence of machines or processes, is not just a theoretical one; it is a real-world challenge that businesses aiming for productivity and efficiency must deal with. Understanding the complex nature of production environments and the limitations with traditional deterministic models, this study explores a novel approach that considers these factors. The dynamic nature of flow shop scheduling, influenced by various uncertainties such as fluctuating demand, variable processing times, and equipment failures, necessitates a robust and adaptive solution framework. The problem is a Non-Deterministic Polynomial-Time (NP-hard) problem [1]. This complexity mandates the operation of research models employed to solve such problems. The computational complexity of the Flow Shop Scheduling Problem (FSSP) is essential since it directly affects the applicability and efficiency of scheduling solutions in real-world applications. The FSSP is a well-known issue in operations research due to its significant computational requirements. This section comprehensively analyzes the time and space complexity linked to FSSP.

The FSSP, which involves processing a group of jobs in a specific order on multiple machines, is recognized as NP-hard. The NP-hard complexity implies that no polynomial-time algorithm exists that can solve all instances of the problem optimally. The problem of an FSSP with n jobs and m machines can be solved using a brute-force approach, which entails examining all potential permutations of the jobs. This approach leads to a temporal complexity of factorial order O(n!) which renders it unfeasible for large cases. Some heuristics have been implemented to solve specific problems, such as Johnson’s Algorithm, but such heuristics focused on only a limited case [2]. The FSSP model involves assigning jobs to machines to optimize an objective function. The goal of the objective function can either be maximization or minimization, depending on the objective.

Recent studies assume that uncertainties are a part of real business. In return, there are three types of uncertainties: complete unknowns, suspicions about the future, and known uncertainties [3]. Complete unknowns are unpredictable events (e.g., a sudden strike) about which no advance information is available, and suspicions about the future arise from the intuition or experience of human schedulers, both of which are challenging to incorporate into scheduling algorithms and difficult to quantify. The expertise of professionals and their perceptions of the future can be valuable tools when properly managed and integrated into decision-making. Suspicions about the future arise from the intuition and experience of the human scheduler [4]. Fuzziness is prevalent in the business environment. Fuzzy logic is an emerging area that can be integrated to reflect this ambiguity, as it allows the integration of human intuition in a decision-making problem. Fuzzy set theory is essential in managing ambiguity [5].

Based on this assumption, the proposed study integrated fuzzy logic into the methodology to benefit from the merits of fuzzy logic. As the methodology section shows, fuzzy logic is developed in so-defined Type-1 Fuzzy Sets (T1FS). On the other hand, more complex situations may require different approaches [1]. Zadeh [6] proposed an extended version of T1FS called Type-2 Fuzzy Sets (T2FS). T2FS approaches have emerged and increased accuracy and performance.

Applying T2FS may contribute to the accurate representation of human intuition, encompassing suspicions about the future. Given this assumption, this paper aims to integrate a novel hexagonal type-2 fuzzy set approach in an asymmetric flow shop scheduling problem. In the past, T2FS was widely used in different areas, and multiple sub-areas were applied successfully in other regions [7,8]. As a result of such applications, the study aims to integrate a novel T2FS approach called Hexagonal Type-2 Fuzzy Sets (HT2FS), introduced to incorporate the fuzzy approach [9]. Such integration allows the model to represent human intuition accurately.

The objectives of a flow shop scheduling problem can be either single or multiple objectives. Single objectives aim to maximize or minimize a single objective. Some objectives widely used for optimization are makespan, maximum tardiness, maximum lateness, and machine utilization. Defining a single objective to optimize the model may be challenging while respecting all decision-makers’ priorities. MCDM is suggested to overcome such concerns. The subsections of MCDM problems are twofold. The first approach is to optimize multiple objectives simultaneously. The objective is to find feasible solutions in which each objective is optimized and represented as Pareto-optimal. The other approach is called multi-attribute decision-making (MADM). MADM aims to combine different objectives into a single objective function. In this sense, the weights are calculated for each corresponding objective and integrated into a single performance objective function. The proposed model employs the MADM approach; however, many MADM models exist.

The Analytical Hierarchy Process (AHP) is a reliable, rigorous, and robust method for eliciting and quantifying subjective judgments in MCDM problems. The AHP is one of the most popular MCDM methods, proposed by Saaty ([10], p. 201). AHP is the most widely used approach for decision-making [11]. Since its introduction, AHP and Fuzzy Analytic Hierarchy Process (FAHP) methods have been constantly evolving, with new extensions. Some of those extensions have focused on the auction of the complexity of the models by proposing a Stepwise Weight Assessment Ratio Analysis (SWARA)-integrated AHP called SWARA-AHP [12]. Similarly, novel methods benefit from a fuzzy approach with AHP. The proposed model employs such an extension, called PPF-AHP [11].

GA is a well-recognized and effective metaheuristic that is inspired by the process of natural selection. It is commonly applied to resolve complicated optimization problems, such as scheduling jobs in manufacturing and production environments. Recent studies have demonstrated the superiority of GAs in handling the intricacies and variabilities inherent in such problems. For instance, a study showed that GAs significantly improved job shop scheduling by optimizing machine usage and reducing idle times [1]. Similarly, Wang and Zhu [13] employed a GA for optimizing flexible job shop scheduling, achieving substantial reductions in makespan and total completion time.

Moreover, a recent study highlighted the robustness of GAs in dynamic scheduling environments, effectively adapting to changes in job priorities and processing times [14]. The adaptability of GAs makes them particularly suitable for real-world applications, where uncertainties and disruptions are typical. In another recent study, Zhang et al. [15] integrated GAs using a multi-criteria approach that demonstrates the versatility and evolving capabilities of GAs. These examples underscore the genetic algorithm’s relevance and effectiveness in optimizing scheduling problems, validating its application in our research to address the complexities of the flow shop scheduling problem. A systematic review also underlined the effective application of GA for scheduling problems. A vital review showed that the hybrid application of GA with other methods is becoming more common for better outcomes [16].

The main objectives and contributions of the paper are as follows:

⮚

To integrate HT2FS to model human intuition and uncertainties in real-world manufacturing environments accurately. This aspect is vital, as human intuition and experience may be integrated into the models to deal with completely unknown or semi-unknown situations.

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To utilize PPF-AHP to facilitate group decision-making, ensuring a balanced and comprehensive consideration of multiple scheduling criteria. The novel model integrates human intuition to generate outputs using expert inputs.

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To implement a GA to optimize the scheduling process, aiming to achieve minimal makespan, reduce maximum tardiness, and maximize machine utilization. Using metaheuristics and optimization in an acceptable timeframe is crucial to adapting to changes in the fast business environment.

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To provide a robust and adaptive scheduling framework capable of handling dynamic and unpredictable manufacturing conditions, improving overall operational efficiency, and reducing costs. The application will help in general integration with more extensive areas for business.

The rest of the paper is organized as follows. Section 2 reviews the literature on the relevant topics of the study. The proposed scheduling model is given in detail in Section 3. In Section 4, applications are performed and the results are compared and discussed, and conclusions, limitations, and future work are given in Section 5, in the Conclusions.

2. Literature Review

The following subsections provide a literature review of fuzzy sets, multi-attribute decision-making, flow shop scheduling problems, and their hybrid approaches.

2.1. Fuzzy Sets and Applications

Fuzzy logic is a method that is used to convert human intuition into decision-making. Fuzzy logic was introduced in 1965 [5] to integrate human thinking into decision-making. The proposed model is T1FS, which involves no uncertainty in human thinking. Although it initially drew attention in academic studies, it was determined that it lacks uncertainty in decision-making. As a result, a new type of fuzzy logic is proposed to cover the ambiguity in decision-making. The latest models are called T2FS. T2FS is proposed as an extended version of T1FS. In the T2FS, the degree of membership is a T1FS [6].

Another essential aspect of fuzzy logic is converting fuzzy numbers to crisp values. It is vital to integrate fuzzy approaches in models accurately. The methods used for such conversions are called defuzzification. The defuzzifications used are proposed based on the introduction of the fuzzy approaches. Recent studies have been proposed for different fuzzy types and fuzzy numbers to cover ambiguity in a broader range of decision-making scenarios, providing more comprehensive and flexible approaches to handle uncertainties in various applications [17]. Some well-known methods are proposed in the literature [18,19].

On the other hand, recent studies also propose fast and accurate methods for achieving results for Center of Gravity (COG) approaches [20,21]. The proposed approach uses the fast and precise novel method proposed by Naimi [20]. A detailed literature review of T2FS is performed by De et al. [17]. In the mentioned study, no study was found using fuzzy sets for flow shop scheduling. This outcome provides underlying information regarding the contribution of the proposed paper.

2.2. Multi-Attribute Decision-Making

Making decisions is inherent to human beings in all their activities. On many occasions, decisions need to be made more and more quickly in a globalized world and in increasingly complex contexts, such as environmental and social sustainability, legislation, political and economic crises, and pandemics, among other scenarios that allow for optimistic or pessimistic alternatives, with the urgency of their respective direction, as Basilio et al. assert [22]. Decision-making is one of the fundamental tasks of management as a way of achieving organizational goals, correctly expressing objectives, determining the different forms of solutions, analyzing their feasibility and consequences, and seeking to resolve problems by implementing the most favorable solution. In organizations, strategic, tactical, or operational decisions can be made. Decision-makers seek to make a decision that fulfills a single or several objectives, often conflicting. A decision-maker is expected to have several goals that can be converted into objectives. Basilio et al. [23], in their research into the evolution of multi-criteria methods over the last four decades, revealed that numerous methods had been developed to support decision-making in organizations, as summarized in Figure 1. One of the results shows that the five multi-criteria methods most used by experts are AHP, the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS), VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR), the Preference Ranking Organization Method for Enrichment of Evaluations (PROMETHEE), and Analytic Network Process (ANP). Another point highlighted in the survey [22] concerns the emergence of hybrid methods and their interface with new technologies and machine learning.

Table 1. Characteristics of the methods most used by researchers.

N	Method	Publication Time	Recorded Count	Research Areas	Publication Time (Integrated/Hybrid Model)	Hybrid Model	New Technologies (Machine Learning)
1	AHP	1990–2021	6.835	Engineering (2.329)	1995–2021	1.388	38
2	TOPSIS	1991–2021	4.907	Computer science (1.797)	2003–2021	1.024	47
3	VIKOR	2002–2021	1.475	Computer science (519)	2009–2021	416	5
4	PROMETHEE	1989–2021	1.382	Engineering (445)	2001–2021	202	16
5	ANP	2000–2021	1.262	Engineering (428)	2006–2021	488	10
6	ELECTRE	1991–2021	1.005	Computer science (331)	2003–2021	120	6
7	DEMATEL	2007–2021	888	Computer science (289)	2007–2021	476	5
8	GOAL PROGRAMMING	1983–2021	553	Operations research (202)	1993–2021	147	3
9	SAW	1997–2021	403	Engineering (137)	2007–2021	67	5
10	TODIM	1999–2021	306	Computer science (171)	2013–2021	56	2
11	COPRAS	2006–2021	294	Business economics (83)	2011–2021	100	2
12	WASPAS	2012–2021	214	Engineering (68)	2013–2020	67	0
13	MULTIMOORA	2011–2021	198	Computer science (75)	2011–2021	43	0
14	SWARA	2011–2021	181	Business economics (46)	2011–2021	90	1
15	MAUT	1984–2021	164	Engineering (56)	2007–2021	19	0
16	MACBETH	1999–2021	162	Computer science (47)	1999–2021	27	0
17	WSM	1994–2021	87	Engineering (29)	2014–2021	17	2
18	DRSA	2002–2021	85	Computer science (51)	2012–2021	20	4
19	WPM	1997–2021	57	Computer science (23)	2014–2021	7	0
20	CBR	1996–2021	40	Computer science (25)	2006–2020	10	1
21	CONDORCET	1999–2021	35	Business economics (9)	-	0	1
22	FITRADEOFF	2016–2021	29	Computer science (14)	-	0	0
23	UTADIS	1998–2020	27	Operations research (14)	2005–2016	2	0
24	SMART	1996–2021	22	Engineering (9)	2021	2	0
25	PAPRIKA	2014–2021	12	Computer science (4)	2020	1	0
26	THOR	2008–2021	5	Engineering (2)	-	0	0

Note: data used based on research by Basilio et al. [23].

shows this evolution and records the areas where each method is most widely used.

Bringing decision-making closer to the context of this research, we can exemplify the decision-making context of a production planner who aims to minimize the backlog and make a list of jobs. Multi-objective decision-making and multi-attribute decision-making approaches are used to solve these problems. Some examples of applying Multi-Objective-Decision-Making (MODM) to flow shop scheduling exist in the literature [24,25,26]. The proposed hybrid approach uses the MADM approach to solve flow shop problems. The difference between MODM and MADM is the number of objectives. The MADM uses a single objective based on several objectives with variable weights [27]. There are few studies on multi-criteria methods associated with flow shop scheduling problems. On 15 June 2024, the Scopus database was consulted using the search key TITLE-ABS-KEY ((“flow shop scheduling problem”) AND (“multi-criteria” OR “MCDA” OR “MCDM”))) AND (LIMIT-TO (DOCTYPE, “ar”)) AND (LIMIT-TO (LANGUAGE, “English”)), where MCDA represents Multi-Criteria-Decision-Analysis. Thirteen articles were found, only four of which were related to the topic in question, and they are summarized in

Table 2. Detailed table with research on FSSP and MCDA.

Authors	Methods	Brief Description
Makki et al. [28]	Ranking the Alternatives using the Trace to Median Index (RATMI)/PFSP	The researchers used the multi-criteria method, Ranking the Alternatives using the Trace to Median Index (RATMI), to evaluate the performance ranking of eight heuristic techniques for solving the permutation flow shop scheduling problem (PFSP).
Karacan et al. [29]	Random Key Genetic Algorithm (RKGA)/TOPSIS/FSSP	The researchers integrated the Random Key Genetic Algorithm (RKGA) and TOPSIS to calculate the makespan and solve the FSSP.
Gonzalez-Neira et al. [30]	GRASP/PAES/AHP	The experts proposed a hybridized metaheuristic approach that combines a Greedy Random Adaptive Search Procedure (GRASP), a Monte Carlo simulation, a Pareto Archival Evolution Strategy (PAES), and an AHP to solve a multi-criteria stochastic permutation flow shop problem with stochastic processing times and stochastic sequence-dependent setup times.
Giannopoulos et al. [31]	RVNS/GA/FSSP	In their article, this work presented a new hybrid metaheuristic for solving the FSSP problem. The metaheuristic approach combines a GA to evolve solutions and a reduced variable neighborhood search (RVNS) technique to achieve rapid improvement.

.

The AHP is the most used MCDM model in the literature [11]. Since its introduction, many novel methods have been proposed as extensions of AHP. A recent study proposed PPF-AHP as an AHP extension. The study is new in the literature, and the model is applied to a facility location selection problem [11]. The proposed model aims to integrate PPF-AHP into the model. To the best of our research knowledge, the application of this novel model is new to a production scheduling problem. Group decision-making is essential to integrate different approaches using multiple expert opinions into a single output. Many studies focus on group decision-making. Some recent studies regarding group decision-making focus on advanced statistics for application [32]. Some approaches use aggregation operators to combine group decision-making [33]. Based on the literature review, the methods may be complicated and computationally intensive. A recent study proposed a novel model for group decision-making using an ordinal priority approach. The proposed model performs better than alternative models and is easy to employ due to optimization [34].

2.3. Flow Shop Scheduling

Flow shop scheduling aims to maximize or minimize an objective or multiple objectives. To name a few of those objectives, the delay in the time of jobs, which is crucial for customer satisfaction and service level, was used in a recent study. Machine utilization is another objective that needs to be maximized for an efficient schedule [35]. Makespan and overall energy efficiency are some widely used objectives for flow shop scheduling. A recent study employed these objectives in the multi-objective optimization of flow shop scheduling [36].

An increase in machine utilization may lead to increased tardiness, and a reduction in work-in-process inventory (WIP) may lead to lower machine utilization and, eventually, customer satisfaction due to shortages and delays. Recent studies aim to overcome these problems by setting multiple objectives that must be satisfied simultaneously. Such models are called MODM. A particle swarm optimization (PSO), a well-established metaheuristic model, is applied to solve the multi-objective job shop problem [37]. A detailed literature review is given regarding multi-objective job scheduling problems [38].

The literature review shows detailed studies on single or multi-objective models that have been performed, as illustrated in

Table 3. Comparison of problems regarding criteria and objectives.

Scheduling Problem	Reference	Objective(s)	Minimization Criteria
			Mono-Criterion	Bi-Criteria	Multi-Criteria
FFSPw	Tyagi et al. [39]	Makespan Total tardiness
	Seidgar et al. [40]	Makespan Total tardiness
	Pargar et al. [41]	Makespan Total tardiness
	Marichelvam et al. [42]	Makespan Total flowtime
DFSP-BD	Li, Li et al. [43]	Makespan Energy consumptions
DPFSP	Bargaoui et al. [44]	Makespan
	Fernandez-Viagas and Framinan [45]	Total flowtime
DNWFSP	Lin and Ying [46]	Makespan
	Allali et al. [47]	Makespan Total tardiness
DNWFFSP	Shao et al. [48]	Makespan
DAPFSP	Ochi and Driss [49]	Makespan
	Song et al. [50]	Makespan
FSP	Bai et al. [51]	Maximum lateness
DCPFSP	Xiong et al. [52]	TWET
EFFSPw	Gong et al. [53]	Makespan total worker cost Green production indicator
DPFSP	Miyata and Nagano [54]	Makespan

Note: adapted from the study by Mraihi et al. [55].

. To the best of our research knowledge, the use of MADM combining multiple objectives into a single objective using weights has not been found in the literature. Also, the studies did not conduct novel methods using AHP extensions, the most widely used model. This study aims to fill this literature gap by proposing a novel approach, PPF-AHP-based FSSP.

2.4. Hybrid Approaches for Flow Shop Scheduling

Due to the general practical applications of flow shop scheduling problems, hybrid approaches are used to solve different types of flow shop problems with different objectives. The studies used multiple methods to model the flow shop problems simultaneously. A cooperative coevolution algorithm similar to the genetic algorithm was applied to solve a distributed hybrid flow shop problem. The goal was to explore and exploit the proposed model’s capabilities simultaneously [56]. A similar approach for optimization was performed in an energy-aware distributed hybrid flow shop scheduling process. The results showed the proposed model’s higher performance compared to existing models [57]. Hyperheuristics are widely used approaches for optimization. Hyperheuristics employ different metaheuristics for optimization. A recent study employed hyperheuristics for the multi-objective optimization of blocking flow shop scheduling, with the goal of energy efficiency [58].

The literature review underlines the following insights. Fuzzy approaches are widely used to convert human inputs from fuzzy values to crisp values. Weights of all jobs reflect the importance of each job. Using T2FS, suspicions about the future and known uncertainties proposed by Wu et al. [59] can be accurately modeled. Incorporating HT2FS in Stage 2 significantly advances fuzzy logic applications for scheduling problems. By accurately modeling uncertainties and facilitating more informed decision-making, this approach enhances the robustness and flexibility of the scheduling system, paving the way for more efficient and reliable production processes. Different objectives are widely used to prioritize other goals, and the proposed approach is simultaneously a novel PPF-AHP approach with an ordinal priority approach for group decision-making in a hybrid approach. The application of these two methods is easy to comprehend and apply. As a result, due to these advantages, PPF-AHP and group decision-making models are used in a hybrid model [11,33].

3. Materials and Methods

Flow shop scheduling problems involve multiple possible objectives. The proposed methodology aims to integrate different separate models. As a result, the proposed models integrate different approaches for consecutive steps. Figure 2 shows the workflow of the proposed approach. The notation used for modeling is given in

Table 4. Problem description parameter set.

Parameter	Parameter Description
n	Total number of jobs
i, l	Job index, i, l = [1,2, …,n]
wi	Weights of job i
m	Total number of machines
Mk	kth machine
k, m	Machine index k, k = [1,2, …,m]
wo	oth weight of the objective
fp	pth objective
t	Objective index p, p = [1,2, …,t]
wsik	Waste at the for job i at the kth machine
Ji	ith job
j	Operations index
Oij	jth operation of Ji
ni	Number of operations in Ji
t	Current scheduling time

and

Table 5. Mathematical description parameter set.

Parameter	Parameter Description
${\mathrm{t}}_{\mathrm{i}{\mathrm{M}}_{\mathrm{k}}}$	Processing time of Ji on Mk
${\mathrm{c}}_{\mathrm{i}{\mathrm{M}}_{\mathrm{k}}}$	Processing completion time of Ji on Mk
${\mathrm{a}}_{\mathrm{i}{\mathrm{M}}_{\mathrm{h}}{\mathrm{M}}_{\mathrm{k}}}$	When Mh is processes Ji before Mk = 1; otherwise Mk = 0
${\mathrm{b}}_{\mathrm{i}\mathrm{l}{\mathrm{M}}_{\mathrm{k}}}$	When Ji is processed on the machine before J1 = 1; otherwise J1 = 0
${\mathrm{C}}_{\mathrm{i}}$	Estimated processing time for Ji

.

3.1. Stage 1 Proportional PicDoneture Analytic Hierarchy Process for Group Weight Assessment

The first phase employs an integration of two distinct approaches. Initially, expert opinions are translated into criteria weights, reflecting the importance of each criterion. This phase utilized an integrated PPF-AHP method, as thoroughly detailed in Kahraman’s [11] study. Multiple experts provided input, which was essential for the robustness of the model. Figure 3 illustrates the flowchart of the PPF-AHP process implemented in this phase. The inputs, which are criteria comparisons, culminate in the determination of weights. The study’s primary objective is to synthesize various experts’ outputs into a cohesive model.

Consequently, each expert’s weights are aggregated using a geometric mean approach to ensure a comprehensive and balanced outcome. The application of PPF-AHP generates outputs for the second stage. The objectives’ weights allow us to convert experts’ opinions into weights of all objectives essential in an MCDM problem. Therefore, converting multiple objectives into a single objective that reflects the priorities is possible.

3.2. Stage-2 Type-2 Hexagonal Fuzzy Inputs for Weight Calculations

In the advancement of flow shop scheduling problems, incorporating uncertainties and ambiguities inherent in real-world scenarios is paramount. Stage 2 of our proposed methodology introduces HT2FS for weight calculations, leveraging their capability to handle the multidimensional nature of uncertainty more effectively than traditional fuzzy sets. The weight of each job is not equal in real-life cases. Different weights allow us to better represent each job among other alternatives. In Stage 2, such generation using fuzzy application enables the use of the weights in the optimization phase. HT2FS represents a novel approach in fuzzy logic, where each element’s membership degree is a fuzzy set, allowing for an additional dimension of uncertainty. This area particularly applies to the modeling of human intuition and suspicions about future events in scheduling problems. Given its geometrically balanced structure, the hexagonal representation of these fuzzy sets facilitates an enhanced depiction of uncertainty, providing a more natural and intuitive way of handling ambiguities. The weight calculation process using HT2FS involves several key steps:

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Fuzzification of Input Variables: The first step is converting crisp inputs into HT2FS. The fuzzification is achieved by mapping each input parameter to a hexagonal fuzzy number based on expert assessments and historical data, considering both the degree of certainty and the range of possible values.

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Construction of Hexagonal Fuzzy Decision Matrix: With inputs fuzzified, a decision matrix is constructed, encapsulating the relationships between various scheduling criteria and their respective importance, all represented as HT2FS.

⮚

Application of Fuzzy Arithmetic Operations: These fuzzy values are then combined and adjusted using fuzzy arithmetic operations, which makes it easier to determine composite weights for every criterion. This step is essential to fully capture various scheduling parameters’ interdependencies and relative importance under uncertainty.

⮚

Defuzzification: The final stage involves converting the hexagonal type-2 fuzzy numbers into crisp values, providing actionable weights for each criterion. The defuzzification process is carefully designed to preserve the nuanced information captured by the HT2FS. The weights are aggregated by summing each expert result and then normalizing it. The details of the proposed HT2FS can be found in the study proposed by Naimi et al. [20].

This step ensures that the complexities and uncertainties of real-world scheduling problems are accurately represented, leading to more effective and reliable scheduling solutions.

3.3. Stage-3 Multi-Criteria Optimization Using a Genetic Algorithm with Fuzzy Inputs

The third stage of our proposed methodology involves the application of Multi-criteria Optimization using GA to solve flow shop scheduling problems effectively. This stage capitalizes on the weighted criteria derived from applying HT2FS in Stage 2, aiming to find a scheduling solution that optimizes multiple objectives concurrently. The purpose of the proposed model is to use a multi-criteria approach using PPF-AHP. The weights are results of the first stage, given as W = {w1, w2, …, wn}.

In a flow shop environment containing several machines, M = {M1, M2, …, Mm}, there are a number of jobs, J = {J1, J2, …, Ji, …, Jn}; each job, say Ji, contains a serial of operations Oi = {Oi1, Oi2, …, Oij, …, Oini} which need to be processed in a predefined technological sequence [60]. The performance of the model is described as F = {$f_1,f_2,f_3,\dots f_n\},$ where $f_n$ denotes the result of the solution from the criterion n.

Based on these inputs, the objective function can be defined as

$$\mathrm{M}\mathrm{i}\mathrm{n} {\mathrm{w}}_1{\mathrm{f}}_1+{\mathrm{w}}_2{\mathrm{f}}_2+{\mathrm{w}}_3{\mathrm{f}}_3\dots {\mathrm{w}}_{\mathrm{n}}{\mathrm{f}}_{\mathrm{t}}$$

(1)

$f_1$ represents the normalized Minimization of Makespan (C_max). The makespan represents the time to complete all jobs. The formulation is given as Equation (2).

Min. C_max = max{C_j}

(2)

$f_2$ represents the normalized Minimization of Maximum Weighted Tardiness (T_max). Weighted tardiness is the tardiness of a job weighted by its importance. The goal is to minimize the maximum weighted tardiness across all jobs. The formulation is given as Equation (3).

Min. T_max = max{w_o * max{0, C_j − D_j}}

(3)

$f_3$ represents the normalized Minimization of Total Waste Due to Setups (W_total). Setup waste refers to wasted material during the setup or changeover between jobs. The objective is to minimize the total waste across all setups, given in Equation (4).

$$\mathrm{Min}.{\mathrm{W}}_{\mathrm{total}}={\sum }_{i,j}{ws}_{ik}$$

(4)

$f_4$ represents the normalized Minimization of Total Time Lost Due to Setups (S_total). This objective represents the total time spent on setups or changeovers between jobs, which does not contribute to productive work. The goal is to minimize this total setup time. The formulation is given as Equation (5).

$$\mathrm{Min}.{\mathrm{S}}_{\mathrm{total}}={\sum }_{i,j}{s}_{ik}$$

(5)

Normalization can be performed by calculating each objective’s standard score (z-score). As a preprocessing step, we randomly generate R combinations (e.g., 500 combinations), and from these, we calculate the mean and standard deviation. Since the normalized values can be negative and positive, this might interfere with the weighted combinations in the objective function. We add a constant to prevent this, to ensure that all values are positive. We chose a constant of 9, corresponding to 3 standard deviations, to ensure that the values remain positive without affecting the overall calculations.

Flow shop scheduling has multiple constraints [60]. Operation Assignment: Every operation must be assigned to exactly one machine. The relevant constraint is given in Equation (6).

$${\sum }_{k\in M}{x}_{ijk}=1\forall j\in J,\forall i\in O_j$$

(6)

Operation Sequence: The operations for each job must follow a specified sequence, as given in Equation (7).

$$C_{i-1,j}\le C_{ij},\forall j\in J,\forall i\in O_j\backslash {\{O}_{j1}\}$$

(7)

Machine Non-Overlap: A machine can handle only one operation at a time. The Machine Non-Overlap constraint is given in Equation (8).

$$C_{ij}\le C_{kl},\forall j\in J,\forall i\in O_j\backslash {\{O}_{j1}\}$$

(8)

Non-negativity and Completeness: Equations (9)–(11) give the relevant constraint equations, respectively.

$$C_{ij}\ge 0,\forall j\in J,\forall i\in O_j$$

(9)

$$x_{ijk}\in \left\{0,1\right\},\forall j\in J,\forall i\in O_j,\forall k\in M$$

(10)

$$y_{ijkl}\in \left\{0,1\right\},\forall i,k\in O_j,\forall j,l\in J$$

(11)

After modeling the multi-criteria flow shop problem, the GA method is used to solve the problem. A brief description of the genetic algorithm is as follows. A detailed survey gives detailed information about the GA application [61]. The GA workflow is presented in Figure 4. The GA is a metaheuristic based on Darwinian principles of genetics, natural selection, and evolution. The GA uses mutation and crossover of the genes, primarily the candidate solution. The objective function allows the changes in the candidate solution using mutation, crossover, new gene entry, or hybrid approaches to these problems to generate better solutions. Each generation aims to improve the objective function to arrive at a better fitness solution that represents the goal of the model. Like the research in optimization and modeling, there are constant general models or problem-specific approaches to the GAs. The main differences between GAs and mathematical optimization techniques can be summarized into three topics. First, GAs operate on the encoded string of the control variables rather than the actual control variables of the problem. Secondly, GAs are population-based optimization techniques and use more candidates instead of a single point in their search. Lastly, GAs do not require any prior knowledge regarding the objective function. The only requirement is the calculation of the FF, which assigns the quality value to each solution (and presupposes knowledge of the status of the system’s non-controllable elements, which is a given). The details of the GA are also provided in Papazoglu and Biskas [61] for further reference.

4. Numerical Application

This section presents a detailed case study of a flow shop scheduling problem, considering ten jobs and nine machines and optimizing across four objectives: Makespan, Weighted Tardiness, Total Waste, and Total Setup Time. This case study and its results can be accessed through this link: https://bit.ly/3wVfmzq (accessed on 30 May 2024), and the source code is available at https://bit.ly/3KmQy6s (accessed on 30 May 2024). Our numerical example demonstrates that our approach is practical in solving complex scheduling problems. The parameters of the numerical application of the proposed model can be classified into three classes. The first stage covers the objective weight determination. Experts’ opinions are used in this stage. The PPF-AHP has no additional parameter, as the MCDM method converts experts’ opinions to weights using comparison tables among objectives. The four objectives in Section 1 are the most widely used in scheduling problems, as presented in the literature review. The goal of such a selection is to use objectives not only profit-cost-wise but also sustainability-wise. In the second stage, the weights of the jobs are assessed. Again, a range of weights is used to represent the importance of each job. This aspect is crucial, and HT2FS converts experts’ opinions into weights. The parameters regarding the HT2FS are mainly associated with the methodology. In the last stage, the numerical analysis is performed for modeling and optimization using GA. The numerical analysis is performed based on ten jobs and nine machine cases. Random values are generated to show the applicability of the model. The optimum range is impossible when job numbers are over 10, as the potential candidate number becomes over 10!. The results given in the model show that such parameter selection proved the applicability of the proposed model.

4.1. PPF-AHP

In this stage, expert opinions are input for the proposed model. The first stage’s goal is to convert expert opinions on the importance of each criterion combined for a single objective. The experts are from the production planning area and have over ten years of experience in production planning and MCDM approaches. Their familiarity with spreadsheet tools is also essential to avoid mistakes during data entry. Proper documentation is shared with the experts, and guidelines are given for the documents. Their experience in production planning and information about the meanings of objectives are also vital for accurate data entry. Their expertise and background in industrial engineering also helped them analyze PPF-AHP. The details of each criterion are given in

Table 6. Objectives’ definitions.

Objectives	Definitions
f1	Makespan
f2	Weighted Tardiness
f3	Total Waste
f4	Total Setup Times

and are shared with experts for the PPF-AHP stage. The model does not employ any stage with a consensus or removal of outliers. The model uses a consistency check for each separate data input. As a result, the consistency of each expert is validated.

The final weights for the objectives were derived by aggregating the calculated values from the PPF-AHP method using the geometric mean of all expert opinions, as illustrated in

Table 7. Weights of each objective after PPF-AHP.

	Expert-1	Expert-2	Expert-3	Geometric Mean
f1	0.0840	0.0297	0.2644	0.0871
f2	0.1774	0.3443	0.2541	0.2495
f3	0.1439	0.4671	0.2011	0.2382
f4	0.5947	0.1589	0.2804	0.2981

.

4.2. T2HFS Approach for the Weight Determination of Jobs

Fuzzy approaches are widely preferred to convert human intuition to quantitative values. The input is used to convert H2TFS values for the assessment of weights. The weights are calculated based on random values that use a nine-point Likert scale. Each job weight is shown in

Table 8. Weight of jobs using HT2FS.

i	wi
1	0.1001
2	0.1032
3	0.1068
4	0.0944
5	0.1047
6	0.1005
7	0.0957
8	0.1015
9	0.0951
10	0.0979

.

4.3. Dataset and Optimization

To illustrate our case study, we examined a flow shop scheduling problem involving ten jobs and nine machines.

Table 9. Flow shop dataset.

Jobs	Sequence	Due Dates
J0	(0, 3), (3, 5), (2, 2), (4, 4), (1, 3), (5, 2), (7, 2), (8, 3), (6, 1)	25
J1	(0, 4), (3, 3), (2, 6), (4, 2), (1, 5), (5, 4), (7, 2), (8, 3), (6, 1)	30
J2	(0, 5), (3, 4), (2, 3), (4, 2), (1, 1), (5, 5), (7, 2), (8, 3), (6, 4)	18
J3	(0, 2), (3, 6), (2, 4), (4, 3), (1, 5), (5, 1), (7, 4), (8, 8), (6, 5)	27
J4	(0, 3), (3, 2), (2, 5), (4, 4), (1, 3), (5, 2), (7, 3), (8, 4), (6, 4)	40
J5	(0, 5), (3, 2), (2, 3), (4, 3), (1, 5), (5, 1), (7, 4), (8, 8), (6, 5)	10
J6	(0, 4), (3, 1), (2, 3), (4, 2), (1, 2), (5, 1), (7, 2), (8, 1), (6, 1)	33
J7	(0, 2), (3, 2), (2, 2), (4, 2), (1, 2), (5, 3), (7, 2), (8, 3), (6, 4)	58
J8	(0, 5), (3, 5), (2, 2), (4, 2), (1, 2), (5, 2), (7, 2), (8, 3), (6, 2)	12
J9	(0, 4), (3, 2), (2, 5), (4, 2), (1, 3), (5, 4), (7, 3), (8, 4), (6, 1)	29

Note: Data that has been entered into the MCDM Scheduler code available at Supplementary Materials: https://bit.ly/3wVfmzq (accessed on 30 May 2024) and https://bit.ly/3KmQy6s (accessed on 30 May 2024).

provides a detailed dataset for this problem, where each job must pass through all machines in a specified sequence with predefined processing times. The goal is to minimize the total processing time while adhering to due dates and minimizing waste.

Table 9 presents the dataset, where each job (J0 to J9) follows a distinct route through the machines. Each entry in the Sequence column indicates the machine and the processing time in time units. For instance, Job 0 (J0) follows this processing order: Machine 0 (three time units); Machine 3 (five time units); Machine 2 (two time units); Machine 4 (four time units); Machine 1 (three time units); Machine 5 (two time units); Machine 7 (two time units); Machine 8 (three time units); and Machine 6 (one time unit).

The Due Dates column specifies the due date for each job, representing the latest allowable completion time for each job to meet customer demands or production schedules.

Flow shop operations often incur waste and setup times when switching between jobs. These factors are crucial in scheduling, impacting the total processing time and efficiency.

Table 10. Flow shop waste.

Waste
	J0	J1	J2	J3	J4	J5	J6	J7	J8	J9
J0	0	3	2	5	4	2	5	4	7	6
J1	3	0	9	1	7	9	2	4	9	7
J2	2	9	0	3	2	8	1	9	9	6
J3	5	1	3	0	6	4	5	6	1	8
J4	4	7	2	6	0	9	6	5	8	2
J5	2	9	8	4	9	0	9	2	7	3
J6	5	2	1	5	6	9	0	2	1	4
J7	4	4	9	6	5	2	2	0	8	5
J8	7	9	9	1	8	7	1	8	0	2
J9	6	7	6	8	2	3	4	5	2	0

Note: Data that has been entered into the MCDM Scheduler code available at Supplementary Materials: https://bit.ly/3wVfmzq (accessed on 30 May 2024) and https://bit.ly/3KmQy6s (accessed on 30 May 2024).

lists the waste material generated during the setup or changeover between jobs. The values represent the waste (in arbitrary units) incurred when transitioning from one job to another.

Table 11. Flow shop setup.

Waste
	J0	J1	J2	J3	J4	J5	J6	J7	J8	J9
J0	0	1	2	4	7	5	4	4	4	4
J1	1	0	5	3	2	3	4	1	1	7
J2	2	5	0	1	4	2	8	9	2	1
J3	4	3	1	0	8	5	5	5	3	3
J4	7	2	4	8	0	7	6	5	2	2
J5	5	3	2	5	7	0	1	2	3	4
J6	4	4	8	5	6	1	0	2	5	8
J7	4	1	9	5	5	2	2	0	6	3
J8	4	1	2	3	2	3	5	6	0	1
J9	4	7	1	3	2	4	8	3	1	0

shows the total time (in time units) spent on setups or changeovers between jobs. Minimizing these setup times is essential to optimize the flow shop schedule.

A flow shop problem is an NP-complete problem [1]. Consequently, finding solutions within acceptable limits necessitates the use of metaheuristics. We employ GA to optimize our model, which provides a robust framework for solving combinatorial optimization problems. Our GA operates through the following steps:

• Initial Population Creation—The first step in the genetic algorithm involves creating an initial population of potential solutions. Our study set the population size to 10 individuals, each representing a unique sequence of job processing orders. The process of creating this initial population is as follows:

  ⮚

  Randomization: Each individual (solution) in the population is generated randomly. This randomness ensures a diverse set of initial solutions, covering various parts of the solution space. For example, a sequence might look like [J3, J1, J4, J0, J2, J5, J6, J7, J8, J9], where the jobs are ordered randomly.

  ⮚

  Representation: Each individual is represented as a permutation of job sequences. This representation allows the GA to manipulate and evolve solutions effectively.

2.

Selection Mechanism—The selection mechanism is crucial for guiding the evolutionary process by choosing parent solutions for reproduction. We use the roulette wheel method for selection, which operates as follows:

⮚

Fitness Evaluation: Each individual in the population is evaluated based on a fitness function, which, in our case, could be the total processing time, waste, and setup times.

⮚

Probability Assignment: The fitness values assign selection probabilities to individuals. Better-performing solutions (with lower processing times and waste) have higher probabilities of being selected.

⮚

Roulette Wheel Selection: Imagine a roulette wheel where each segment represents an individual, and the size of each segment is proportional to the individual’s fitness. The wheel is spun to select parents for the next generation, ensuring that fitter individuals are more likely to be chosen.

3.

Crossover Operator—The crossover operator combines two parent solutions to produce offspring, aiming to inherit the best characteristics from each parent. We use the Best Cost Route Crossover (BCRC) function, which operates as follows:

⮚

Parent Selection: Two-parent solutions are selected based on the roulette wheel method.

⮚

Segment Selection: BCRC selects segments from both parents to combine them into a new sequence. For example, it might take the first part of the sequence from one parent and the second from the other.

⮚

Offspring Creation: The segments are combined to create a new offspring solution. This method ensures that the offspring inherit beneficial traits from both parents, potentially improving the sequence.

4.

Mutation with Local Search—A mutation function is applied to introduce variability and prevent premature convergence. Additionally, a local search (two-opt method) is incorporated to refine solutions further:

⮚

Mutation: The mutation function randomly swaps two jobs in a sequence. For instance, if the sequence is [J0, J1, J2, J3, J4], a mutation might swap J1 and J4, resulting in [J0, J4, J2, J3, J1].

⮚

Two-opt Local Search: This method optimizes the sequence by iteratively reversing segments to find a shorter path. It evaluates the impact of reversing every possible pair of edges and keeps the best improvement. This local search helps achieve fast convergence and stability.

⮚

Mutation Rate: The mutation rate is set to 0.10, meaning that 10% of the population undergoes mutation in each generation, balancing exploration (new solutions) and exploitation (refinement of existing solutions).

5.

Tracking and Improving the Elite Solution—Throughout the evolutionary process, it is crucial to preserve and improve the best-performing solution, known as the elite solution:

⮚

Elite Preservation: The best solution from each generation is retained in the next generation, ensuring that the quality of solutions does not degrade over time.

⮚

Elite Improvement: The elite solution is continuously evaluated and improved if a better solution is found in subsequent generations. In our study, we maintain one elite member (the elite solution is set to 1).

By iteratively applying these steps across multiple generations, the GA explores and exploits the solution space, aiming to find an optimal or near-optimal solution to the flow shop scheduling problem. The process is carried out for 250 generations, allowing sufficient iterations to refine and improve the solutions effectively.

Hence, utilizing a population size of 10 and a mutation rate of 0.10, retaining one elite member, and executing the genetic algorithm over 250 generations, we could effectively solve the flow shop scheduling problem through this configuration, as illustrated in Figure 5.

The optimal sequence identified was Jobs 5, 3, 8, 9, 4, 2, 0, 1, 7, 6. This sequence yielded a Makespan of 67, a Weighted Tardiness of 3.7391, a Total Waste of 22, and a Total Setup Time of 21. To ensure the robustness of our model, we employed a brute-force approach, exhaustively testing all possible solutions. Given the ten jobs, there are precisely 3,628,800 possible combinations. The results from the genetic algorithm were optimal, as the same sequence was identified, validating the effectiveness of our model.

4.4. Sensitivity Analysis

A sensitivity analysis was performed to analyze how the results change when parameter changes occur. The sensitivity analysis is vital to validate the applicability of the proposed model. The main goal of the sensitivity analysis is to determine whether the results reflect the changes in the parameters of the model. The alternative scenarios are performed to observe how the model changes and analyze the behavior of the proposed model. The outcomes of different models are given in

Table 12. Sensitivity analysis results.

Variable Revision		Amendments	GA	OR
Weight Amendments	Base Results		45.57	45.47
	Scenario-1a	f2, f4 = +%20 f1, f3 = −%20	46.09	44.98
	Scenario-1b	f1, f3 = +%20 f2, f4 = −%20	45.88	45.88
Parameter Changes	Scenario-2a	Waste = [5 20]	66.22	66.16
	Scenario-2b	Due Time = [30 50]	42.92	42.92
	Scenario-2c	Processing Times = [5 20]	67.79	67.79
	Scenario-2d	Waste = [5 20]	88.77	88.65
		Due Time = [10 20]
		Processing Times = [5 20]

. The model is analyzed on the best results of a single case. The results are presented in Table 12, which compares the proposed model and optimum solution where m = 5 and n = 9.

The sensitivity analysis amended the parameters as given in the amendment column. The rest of the parameters were the same as the base results. As provided in Scenario-1a and Scenario-1b, the weight change did not change the results in an excessive case. This change is in line with the expectation, as the weights of the models were also close to each other. The proposed model performed well, achieving either the optimal solution in Scenario-1b or the close solution in Scenario-1a. The proposed model also demonstrated results very close to the optimal results. This aspect is crucial, as one of the main objectives of the proposed model is to achieve close-to-optimal results in an acceptable computation time.

The second sensitivity analysis performed different amendments to the model. This analysis did not change the weights, but modified the range of parameters. In Scenario-2a, the waste range increased. The outcomes were as expected: the objective function increased, whereas the proposed model achieved a close-to-optimal solution. The waste quantity increased in line with the parameters. In Scenario-2b, due time parameters increased. The expected result was a decrease in the tardiness of the model. The due dates reflect when tardiness occurs; after that period, delays are defined as tardiness. The increased due dates reflect flexibility in the model. As given in Table 12, the outputs of Scenario-2b also achieved the expected results. In Scenario-2c, similar results were achieved due to increased processing times, and the proposed model was used successfully to find the optimal solution. Also, the proposed model successfully found the optimal results using GA. Scenario-2c would increase the total processing times, and eventually makespan and possible tardiness. The results showed that the sensitivity analysis performed as expected, as the objective function results should show an increase due to possible tardiness and makespan. Finally, in Scenario-2d, waste, due times of jobs, and processing times increased. The increases would increase the total waste due to increased waste parameters, due times, and processing times, affecting the makespan. As expected, the objective function increased. In addition, the proposed model achieved very close results to the optimum solution. Based on the results in Table 12, we can conclude that the proposed model successfully achieved results under changing conditions and in line with the expectation of the sensitivity analysis.

5. Discussion and Conclusions

The proposed study in this manuscript investigates the dynamic sequence-dependent setup time flow shop scheduling problem with a novel approach that integrates HT2FS and MCDM techniques, particularly PPF-AHP, for group decision-making. This approach is novel in terms of applying a novel MCDM approach to an asymmetric optimization problem. As shown in the literature review, the proposed model represents a novel approach that integrates multiple stages to convert expert opinion parameters of the problem with asymmetric optimization.

Using GA for the optimization stages allows for optimal solutions quickly, and this performance shows the applicability of the proposed model when the number of jobs is increased. Although flow shop scheduling is a widely used area in manufacturing, the proposed hybrid approach can be used to integrate expert opinions into problems as a single entity or group. The fuzzy sets given as HT2FS are a novel approach, and as shown in the literature, there are a wide area of potential applications to convert opinions into perceptions. The proposed methodology stands out for its capacity to accurately represent manufacturing uncertainties and the complex dynamics of scheduling criteria, providing a robust framework for optimizing production processes.

The scientific contribution of the proposed study can be summarized in three separate areas. First, this study is among the first to apply Hexagonal Type-2 Fuzzy Sets in flow shop scheduling, providing a more flexible and realistic approach to dealing with production ambiguities. The second contribution is in applying a novel MCDM and extending the FAHP method. The use of PPF-AHP enhances the accuracy of decision-making, which is essential for achieving optimal scheduling results in complex manufacturing environments. Third and lastly, by combining HT2FS and PPF-AHP with GA, the study presents a robust framework that can adapt to and effectively manage the complexities of modern manufacturing environments, improving operational efficiency, reducing production costs, and enhancing delivery reliability.

While the model showcases significant advantages in adapting to and effectively managing the complexities of modern manufacturing environments through a representation of uncertainties and a balanced consideration of multiple objectives, it has limitations. The computational complexity and the need for high-level domain knowledge for setting fuzzy parameters and decision criteria weights may restrict its scalability and accessibility. Similarly, the proposed study uses expert opinions to represent the HT2FS approach for weight determination and the assessment of multi-criteria in the first stage. The opinions may be subject to subjective assessments. One limitation is a reliance on expert opinions to determine and assess weights in the HT2FS approach and the MCDM process. While this methodology ensures that the model captures a wide range of human insights and experiences, it also introduces subjectivity and potentially limits the scalability and accessibility of the approach. In addition, generally, fuzzy logic, and particularly novel HT2FS methods, is effective for generating outcomes from human intuition and experience. However, the application and relevant models can be too complicated for experts. As a result, proper training and effective user-friendly applications are essential for an effective model. As given in the numerical application section, group consensus is not a part of the model. The group consensus approach is vital, as it may limit the effect of outliers.

Future studies might focus on improving computational efficiency, potentially via novel algorithms or parallel processing, and investigating the incorporation of machine learning for automating the estimation of model parameters, therefore diminishing dependence on specialized inputs. Additional research on how well the model works in other industrial contexts may provide crucial new information about the approach’s impact and adaptability. The model offers managers a sophisticated tool for improving operational efficiency, reducing production costs, and enhancing delivery reliability by leveraging advanced techniques to handle uncertainties and optimize scheduling objectives. The application of HT2FS in flow shop scheduling is a notable advancement, presenting a flexible and realistic approach to dealing with production ambiguities. PPF-AHP enhances decision-making accuracy, which is essential for reaching the best scheduling results. By tackling the difficulties of modern manufacturing, this work not only improves the subject of operational research but also paves the way for future advancements in production scheduling. Also, a group consensus approach integration can be an essential improvement to overcome limitations regarding possible outliers in the expert opinions. Future research could investigate other directions to extend the proposed approach’s scope to encompass different varieties of flow shop scheduling challenges.

The approach could be customized for flexible flow shop situations by modifying the GA component to accommodate numerous machines at each stage. The extension of the accommodation of multiple machines entails the establishment of novel crossover and mutation operators that adhere to the restrictions of the flexible flow shop. Additionally, it requires an assessment of the effectiveness of HT2FS in accurately representing the increased intricacies of flexible configurations. Furthermore, the concept can also be applied to the batch processing of flow shops, which includes the processing of jobs in groups. Enhancing the applicability of the technique might be achieved by making adjustments in the GA for batch scheduling and integrating batch-specific criteria in PPF-AHP. Subsequent research could prioritize examining the difficulties of creating and arranging batches in this particular setting.

In addition, integrating stochastic aspects into the flow shop scheduling model could enhance its versatility. This integration entails incorporating random processing durations or machine failures into the HT2FS framework to better manage uncertainties in real-world scenarios. Developing robust optimization algorithms within the GA to effectively handle stochastic factors will be essential. This strategy can also be advantageous in hybrid flow shop environments, which incorporate characteristics of both job shop and flow shop settings. The suggested method can be customized to accommodate hybrid limitations and objectives by incorporating extensions to the HT2FS and PPF-AHP components, making it suitable for more intricate situations. The model can be tailored to suit specific sectors, like semiconductor manufacturing or textile production, by including industry-specific factors in the PPF-AHP and adjusting the GA to align with industry-specific limitations and goals. This alignment would require a cooperative endeavor with industry professionals to enhance the imprecise sets and decision standards. By implementing these extensions, the suggested approach can become adaptable and resilient, able to handle various flow shop scheduling difficulties across multiple production settings. Further investigations should prioritize improving computing efficiency by implementing parallel processing or innovative algorithms and integrating machine learning methods for parameter estimation, thus minimizing the dependence on expert inputs.

6. Patents

As a result of this research, the authors obtained Computer Program Registration Certificate No. BR512024001768-2 from the National Institute of Industrial Property of the Ministry of Development, Industry, Commerce and Services of the Federative Republic of Brazil for the tool developed in Python called the “MCDM Scheduler-A MCDM approach for Scheduling Problems”.