In this study, intuitionistic fuzzy flow shop scheduling concerns with a single transport facility and different machine setup times were investigated. In manufacturing companies, several practical situations exist where separate setup times for processing units and transportation times are inevitable. For example, before the production of different types of dyes, the manufacturing machine must be assigned a legitimate assignment, and machines need to be set up each time for casting parts of different diameters. In the printing industry, the machine has to be set up each time depending on the orders. Over the past few decades, numerous researchers have investigated deterministic scheduling and proposed several heuristic procedures. However, there are certain practical situations in which deterministic scheduling and heuristic approaches fail to consider the inherent uncertainty of reality. Neglecting uncertainty overlooks the real world and our perceptions of it. Examples of such an uncertain nature include the change in processing time owing to machine breakdown, interruption in production owing to electricity failures, and partial information about the problem.

The concept of fuzzy logic was first introduced by Zadeh [1] in 1965 to model uncertain or imprecise parameters. It considers only the grade of the membership function and has no space for the degree of hesitancy because the degree of the non-membership function is only a complement of one. In 1986, Atanassov [2] introduced an intuitionistic fuzzy set, which is a natural tool for modeling preferences, as it provides space for membership and non-membership grades, as well as the degree of hesitancy. This shows that the grade of non-membership is not necessarily equal to one minus the grade of membership, despite some potential hesitation degree. An intuitionistic fuzzy set can communicate ill-known information more productively and flexibly than a fuzzy set. Since then, more problems in intuitionistic fuzzy set theory [3–11] has been investigated and developed. Therefore, in the current competitive business environment, efficient scheduling has become obligatory to obtain the minimum total completion time required to improve productivity and maximize profit. In practical situations, it is difficult to assess the processing and setup times. Hence, processing and transportation times are represented as generalized triangular intuitionistic fuzzy numbers [12–14] to address unreliable situations more effectively.

McCahon and Lee [15] discussed the concept of combining fuzzy concepts with job sequencing issues by considering uncertain handling times as triangular and trapezoidal fuzzy numbers. They modified the Johnson and Ignall-Schrage algorithms to account for the vague processing times of jobs. In 1992, McCahon and Lee [16] introduced a fuzzy approach to flow shop scheduling problems by modifying the job sequencing algorithm proposed by Campbell-Dudek-Smith (CDS) to consider trapezoidal fuzzy processing times. Jeet [17] used the grey wolf optimization algorithm to schedule a two-machine flow shop in a fuzzy environment and performed a comparison study with other existing heuristic and metaheuristic approaches. Sharma [18] considered the job block criteria in the n-job 2-machine fuzzy flow shop scheduling problem, including the setup time and single transport service. A heuristic algorithm was proposed to obtain the minimum makespan and idle time of the machines. Jeet et al. [19] used the multi-objective black hole algorithm to optimize the makespan and idle time of the n-job 2-machine fuzzy flow shop scheduling problem and compared their results with those of other existing heuristic and meta-heuristic approaches. Sathish and Ganesan [20] investigated the flow shop scheduling problems for three machines with a double transport facility in a fuzzy environment.

Gupta et al. [21] proposed a fuzzy version of the methodology for the n-job 2-machine flow shop scheduling problem subject to different setup times for machines with a single transporting service where processing, setup, transporting and return times are denoted as triangular fuzzy numbers. In the study, the minimum makespan was obtained. Uthra et al. [22] discussed n-job 3-machine generalized intuitionistic fuzzy flow shop scheduling problem with transportation facility, which transfers semi-processed goods from machine 1–2 and 2–3; however, return and setup times were not considered. Recently, Alharbi and Khalifa [23] proposed a novel approach to the flow shop scheduling problem concerning processing time as a pentagonal fuzzy number. Selvakumari and Santhi [24] proposed a novel division algorithm for the intuitionistic fuzzy flow shop scheduling problem and compared it with the existing Johnson algorithm.

The authors of [21] have studied flow shop scheduling problems considering setup, transporting, and return times under a fuzzy environment. Therefore, we considered the example of Gupta et al. [21] in a generalized intuitionistic fuzzy environment and compared our results with theirs by proposing a novel approach for the n-job 2-machine flow shop scheduling.

We further study the case where only a single transport facility is considered for all the jobs; thus, it delivers semi-processed job 1 from machine 1 to machine 2 and returns to machine 1 to move the following job. Here, the processing and transportation durations are described as generalized triangular intuitionistic fuzzy numbers to address unpredictable situations more productively, which may help decision-makers obtain more accurate results. This study aims to obtain optimal intuitionistic fuzzy total completion duration with less vagueness.

The remainder of this paper is organized as follows: Section 2 presents the basic concepts of generalized intuitionistic fuzzy sets and their arithmetic operations. In Section 3, the formulation of the generalized intuitionistic fuzzy n-job 2-machine flow shop problem is explained. Section 4 presents the basic assumptions of this study. In Section 5, we propose a novel algorithm to solve the n-job 2-machine generalized intuitionistic fuzzy flow shop problem by applying the proposed arithmetic operations and ranking functions. In Section 6, we present a numerical example demonstrating the significance of the proposed algorithm. Section 7 presents a comparative study of the proposed and existing algorithms. In Section 7, we discussed the limitations and motivations of the proposed work in detail. Finally, Section 8 summarizes the paper, as well as some directions for future research.

In the following section, the basic definitions and concepts of intuitionistic fuzzy sets, generalized triangular intuitionistic fuzzy number, intuitionistic fuzzy arithmetic operations, and intuitionistic fuzzy ranking functions are discussed to develop the proposed work.

Definition 1 ([1]). Let the set of ordered pairs ã = {(x, μ_ã(x))/x ∈ X} be fuzzy set (FS) in the universe of discourse X, where μ_ã: X → [0, 1] and μ_ã(x) are called membership function and grade of membership of x in ã, respectively.

Definition 2 ([2]). Let ã^I = {(x, μ_ã^I (x), γ_ã^I (x))/x ∈ X} be intuitionistic fuzzy set (IFS) in the universe of discourse X. For all x ∈ ã^I we have 0 ≤ μ_ã^I (x) + γ_ã^I (x) ≤ 1 where the function μ_ã^I : X → [0, 1] determines the grade of membership and the function γ_ã^I : X → [0, 1] determines the grade of non-membership of every element x ∈ ã^I.

Definition 3 ([2]). For every common fuzzy subset ã^I ∈ X, π_ã^I (x) = 1 – μ_ã^I (x) – γ_ã^I (x) denotes the intuitionistic fuzzy index of an element x ∈ ã^I. For every x ∈ ã^I_, 0 ≤ π_ã^I (x) ≤ 1 is called the degree of hesitancy or degree of uncertainty.

Definition 4: The belonging function or membership function and non-belonging function or non-membership function of a generalized triangular intuitionistic fuzzy number (GTIFN) ã^I is defined as follows:

and

The above equations are denoted by ã^I = ((a, b, c), w_ã, v_ã), 0 ≤ w_ã ≤ 1, 0 ≤ v_ã ≤ 1, and 0 ≤ w_ã + v_ã ≤ 1 (Figure 1).

Definition 5 (generalized intuitionistic fuzzy number [GIFN]). ã^I ∈ G(R) can also be represented as a pair ã^I = (a, ā, a′, ā′) of functions a(r), ā(r), a′ (r*), ā′ (r*) that satisfy the following requirements:

• 1. The left legs a(r) and a′ (r*) are bounded monotonic increasing left continuous functions for membership and non-membership functions, respectively.

• 2. The right leg ā(r) and ā′ (r*) are bounded monotonic decreasing left continuous functions for membership and non-membership functions, respectively.

• 3. a(r) ≤ ā(r), 0 ≤ r ≤ 1, a′ (r*) ≤ ā′ (r*), 0 ≤ r* ≤ 1.

2.1 Parametric Representation of Generalized Triangular Intuitionistic Fuzzy Numbers

The modal value (location index) of membership and non-membership functions of any GTIFN ã^I are represented as a0=(a_(1)+a¯(1)2) and a0′=(a_′(0)+a¯′(0)2), respectively. The non-decreasing left continuous functions a_* = (a₀– a) and a′*=(a¯′-a0′) represent the left and right fuzziness index functions of the membership and non-membership functions, respectively. Similarly, the non-increasing left continuous functions a^* = (ā – a₀) and a*′=(a0′-a_′) represent the right and left fuzziness index functions of membership and non-membership functions, respectively. Hence, every GTIFN ã^I can also be represented by a˜I=(⟨a0,a*,a*⟩,⟨a0′,a*′,a′*⟩;wa˜,νa˜).

2.2 Arithmetic Operations for Generalized Intuitionistic Fuzzy Numbers

We proposed an improved generalized triangular intuitionistic fuzzy arithmetic operation that was developed and applied in this study based on the proposed arithmetic operation of Ma et al. [25] These improved arithmetic operations on generalized triangular intuitionistic fuzzy quantities satisfy certain beneficial properties, such as commutative, associative, and sub-distributive, and preserve the shape of membership and non-membership functions of GIFNs. The proposed arithmetic operation is expressed based on the parametric form of a GTIFN. This is expressed in terms of the location, right fuzziness, and left fuzziness indices of both membership and non-membership functions. The location index numbers assume the ordinary arithmetic, whereas the fuzziness index functions follow the lattice rule, which is the least upper bound in lattice L. That is, for all a, b ∈ L, a ∨ b = max{a, b} and a ∧ b = min{a, b}.

That is, for any arbitrary two GTIFNs a˜I={(a0,a*,a*),(a0′,a*′,a′*);wa˜,νa˜} and b˜I={(b0,b*,b*),(b0′,b*′,b′*);wa˜,νa˜}∈GR and * ∈ {+, −, ×, ÷}, the arithmetic operations on them are defined by where m = min{w_ã, w_b̃}, n = max{ν_ã, ν_b̃}. Particularly for a˜I={(a0,a*,a*),(a0′,a*′,a′*);wa˜,νa˜} and b˜I={(b0,b*,b*),(b0′,b*′,b′*);wa˜,νa˜}, we define

Addition:

where m = min{w_ã, w_b̃}, n = max{ν_ã, ν_b̃}.

Subtraction:

where m = min{w_ã, w_b̃}, n = max{ν_ã, ν_b̃}.

2.3 Ranking Function

Although intuitionistic fuzzy sets handle ill-known quantities more flexibly, the ranking or ordering of these numbers becomes difficult when numerical values are represented as GIFNs. In this regard, researchers have extensively studied ranking methods of intuitionistic fuzzy numbers. Li and his colleagues [26, 27] developed a novel ranking methodology based on value and ambiguity indices for GTIFNs and applied it to multi-attribute decision-making problems. To compare intuitionistic fuzzy numbers, Wang and Zhang [28] proposed a ranking method based on a score function, an accuracy function, and the expected values of intuitionistic fuzzy numbers. Arun Prakash et al. [29] discussed the ranking of intuitionistic fuzzy numbers using the centroid concept. We extend this ranking index to compare the parametric representation of GIFNs [30]. To compare any two GTFINs, its centroid point is considered. Each GTIFN is reduced to its corresponding crisp value by the centroid index that uses the geometric center corresponding to x(ã^I) value on the horizontal axis and y(ã^I) value on the vertical axis. These values are as follows:

The ranking function of any GTIFN ã^I is defined as

which denotes the Euclidean distance. For any two GTIFNs a˜I={(a0, a*, a*), (a0′, a*′, a′*); wa˜, νa˜} and b˜I={(b0, b*, b*), (b0′, b*′,b′*); wa˜, νa˜} in G(R), we define the ranking of ã^I and b̃^I by comparing the R(ã^I) and R(b̃^I) on R as follows:

• 1. If R(ã^I) < R(b̃^I), then ã^I ≺ b̃^I

• 2. If R(ã^I) > R(b̃^I), then ã^I ≻ b̃^I

• 3. If R(ã^I) = R(b̃^I), then ã^I ≈ b̃^I

Let us consider i jobs for i = 1, 2, . . ., n prepared by machines M₁ and M₂ in the sequence M₁M₂ to restrict passing. Each machine is set up before starting the processing of each job and is denoted by S˜i,1I and S˜i,2I. The handling periods for each task or job i on machines M₁ and M₂ are represented as p˜i,1I and p˜i,2I, respectively. Let t˜i,1→2I be the duration that the task or job is transported from machine 1 to machine 2 and r˜iI the return time of the transport agent, which transports semi-processed job from the 1st machine to 2nd machine, returning empty to M₁ to transport the next job to M₂. This process continues until all the processed jobs are taken to machine M₂. The parameters are represented as GTIFNs. The mathematical model for the problem is presented in Table 1.

• 1A single job cannot be processed simultaneously by more than one machine.

• 2. The setup, processing, transportation, and return times of each job are known.

• 3. Jobs are processed as soon as they are available.

• 4. The second machine takes up the job only after it is processed at the first machine.

• 5. Each job must be finished once it is started.

• 6. No passing is allowed.

• 7. There is only one machine of each type in the shop.

• 8. A machine is not kept idle if it can take up a job.

• 9. The storage space is available and the cost of holding inventory for each job is either the same or negligible.

• 10. Processing time intervals are independent of the order in which operations are conducted.

Definition 6. The total completion time of task or job i on the machine M_j is defined as

where p˜i,jI and S˜i,jI denote the intuitionistic fuzzy handling time and intuitionistic fuzzy setup time, respectively, for i-th job on the j-th machine, and t˜i,1→2I is the time at which the job is transported from machine 1 to 2. Moreover,

Theorem 1 ([21]). The optimal schedule for n jobs with processing time p˜i,jI:i=1,2,…,n; j = 1, 2 and setup time S˜i,jI:i=1,2,…,n; j = 1, 2 for two machines M₁ and M₂ in the intuitionistic fuzzy environment, including the transportation time t˜i,1→2I and with r˜iI as time taken by transporting agent to return to M₁ for the next job, is obtained by sequencing the jobs (i – 1), i and (i + 1) such that

where R˜i-1I=t˜i-1,1→2I+R˜i-1I-p˜i,1I and t˜i-1,1→2I+R˜i-1I-p˜i,1I>0 and R˜i-1I=0 if otherwise.

• Step 1: Express the given processing, setup, transportation, and return times into parametric form a˜I={(a0,a*,a*),(a0′,a*′,a′*);wa˜,νa˜}.

• Step 2: Calculate R˜i-1I using proposed arithmetic operations, where

  R˜i-1I={t˜i-1,1→2I+r˜i-1I-p˜i,1I,if t˜i-1,1→2I+r˜i-1I-p˜i,1I>0,0,otherwise.

• Step 3: It is assumed that the two fictitious machines with handling times G˜iI and H˜iI are expressed as

  G˜iI=R˜i-1I+t˜i,1→2I+p˜i,1I-S˜i,2I,

  and

  H˜iI=R˜i-1I+t˜i,1→2I+p˜i,2I-S˜i,1I.

• Step 4: Applying the proposed centroid-based ranking method, determine the corresponding crisp value for each G˜iI and H˜iI for i = 1, 2, . . ., n.

• Step 5: Using Johnson’s rule to the fictitious machine times G˜iI and H˜iI, determine the sequence S_q having minimum total elapsed time.

• Step 6: Prepare S_q an in-out table and measure the idle time for machines and transporting agents.

This section examines the problems discussed by Gupta et al. [21] using a generalized intuitionistic fuzzy version that considers membership and non-membership grades with setup time processing, transport, and return times as GTIFNs. We aim to achieve an optimal sequence with the minimum total elapsed time.

Example 1: Consider a 5-job 2-machine flow shop scheduling concern involving GTIFNs, which is shown in Table 2.

Table 3 lists GTIFNs in their parametric form.

Table 4 defines two fictitious machines G˜iI=R˜i-1I+t˜i,1→2I+p˜i,1I-S˜i,2I and H˜iI=R˜i-1I+t˜i,1→2I+p˜i,1I-S˜i,2I with intuitionistic fuzzy processing duration p˜i,1I and p˜i,2I.

Using Johnson’s procedure, the optimal sequence obtained is

Hence, the obtained minimum triangular intuitionistic fuzzy total elapsed time is

The idle time of machine M₁ is the difference between the total elapsed time and the completion time of the last job on the first machine

In contrast, the idle time of machine M₂

Lastly, the idle time of the transporting agent

From Table 5, we obtain the generalized triangular intuitionistic fuzzy total completion time of the production as ((73, 75, 77); 0.5, 0.5), whereas Gupta et al. [21] obtained a fuzzy total completion duration of (56, 67, 78). Although Gupta et al. also worked on the n-job 2-machine fuzzy flow shop scheduling problem, the membership functions of our obtained result produces narrower and less vague results than those of Gupta et al. [21]. This demonstrates the novelty and significance of the proposed method in a generalized intuitionistic fuzzy environment.

In this study, we proposed a simple method for obtaining the optimal and vagueness-reduced intuitionistic fuzzy completion duration of the n-job 2-machine flow shop scheduling problem. To the best of our knowledge, only a few authors have studied n-job m-machine flow shop scheduling problem in an intuitionistic fuzzy environment. Authors, such as Uthra et al. [22] discussed the n-job 3-machine generalized intuitionistic fuzzy flow shop scheduling problem along with transportation time. Selvakumari and Santhi [24] proposed a novel algorithm for the n-job 4-machine intuitionistic fuzzy flow shop scheduling problem. The aforementioned authors have defuzzified the given intuitionistic fuzzy flow shop problem into its corresponding crisp problem to avoid computational complexity and considered the rental policy for the machines; however, they did not consider the setup time for the machines and return time of the transport agent. This motivated us to work on the n-job m-machine generalized intuitionistic fuzzy flow shop scheduling problem considering these unaccounted variables. Hence, basing on the study of Gupta et al. [21] we considered an example in a generalized intuitionistic fuzzy environment and compared our results with theirs. This was accomplished by proposing a novel approach for n-job 2-machine flow shop scheduling.

This study discusses the n-job 2-machine flow shop scheduling problem considering different setup times for each machine to commence jobs, the transportation time for transporting semi-finished jobs from machine M₁ to machine M₂, and the return time of the transporting agent, which returns empty to the first machine to transport the next job to the second machine. Parameters such as setup, processing, transporting, and return times are interpreted as GTIFNs, which are of the form ã^I = ((a, b, c), w_ã, v_ã). To facilitate the computational complexity of the given problem, the processing, setup, transportation, and return times are represented in their corresponding parametric forms, and the proposed arithmetic operations and ranking functions are applied to obtain the optimal sequence of the scheduling problem. The comparative study clearly demonstrates that the proposed method effectively satisfies its primary objective while minimizing vagueness and preserving the intuitionistic fuzzy nature of the problem throughout the computation. The proposed method preserves the intuitionistic fuzzy nature of a given problem without converting it into an equivalent crisp problem. An illustration is provided to demonstrate the ease of this concept. This concept can be extended to a picture fuzzy environment, which is a generalization of fuzzy and intuitionistic fuzzy sets.

Table 1. Conceptual model of the problem in matrix form.

Table 2. Handling durations as GTIFNs.

Table 3. Parametric form of processing durations.

Table 4. Representation of two fictitious machines.

Table 5. In-out table.