A Generalized Multiobjective Reliability Redundancy Allocation With Uncertainties

Multiobjective reliability-redundancy allocation problem (MORRAP) needs to maximize system reliability and minimize cost, weight, and volume with underlining constraints. In the systems’ design and analysis phase, uncertainties can occur from various sources, such as manufacturing variability, environmental conditions, user behavior, etc. To deal with this, we present a generalization of the traditional MORRAP under multiple empirical and ambiguous circumstances, named interval type-2 fuzzy multiobjective reliability redundancy allocation problem (IT2FMORRAP). The newly formulated IT2FMORRAP considers optimizing goals as reliability, cost, and weight for a series-parallel system with interval type-2 fuzzy number. The mathematical formulation is established under which the proposed IT2FMORRAP model reduces to T1FMORRAP (type-1 fuzzy MORRAP), IVMORRAP (interval-valued MORRAP), and classical MORRAP. An Enhanced Karnik-Mendel and NSGA-II algorithm-based solving strategy is developed for the proposed IT2FMORRAP. The real-world dataset is considered to demonstrate the efficacy of the solution method for the proposed problem. A K-mean clustering technique identifies the best solution sets from the knee region of the generated Pareto fronts. An experimental study on commonly used performance metrics reveals that IT2FMORRAP performs significantly better than T1FMORRAP and crisp MORRAP. Further, the statistical analysis also confirms the hypothesis established in the empirical research. Finally, a comparative performance study has been conducted with notable state-of-the-art papers from the literature to encounter an appropriate establishment for the proposed work in the domain.


I. INTRODUCTION
Reliability optimization is a prominent investigation issue in the design and engineering discipline that has earned massive attention over the last several decades. Reliability refers to its propensity to perform accurately throughout a particular time. The most well-known method of enhancing system reliability is redundancy which concerns the surplus of extra components in the system. Component redundancies, therefore, are connected to unnecessary expenditures of rise The associate editor coordinating the review of this manuscript and approving it for publication was Gustavo Olague . in cost, weight, volume, etc., from the perspective of model development. As a result, systems designers put tremendous effort into developing solutions that smack the ideal balance of redundancies and reliability. Researchers have used a variety of methodologies over the decades to determine the optimal reliability and redundancy trade-off, and the issue has come to be acknowledged as the reliability redundancy allocation problem (RRAP) [1], [2].
MORRAP (multiobjective RRAP) seeks to optimize total reliability, weight, and cost with optimal reliability and redundancies, were one goal conflicts with another [2]. Although different system configurations were taken into VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ consideration by researchers to examine MORRAP, recently, research studies have primarily focused on the m states series parallel design as shown in Fig. 1. An m states series-parallel configuration has independent m subsystems in series, and each subsystem has n i (i ∈ 1, 2, . . . m) parallelly engineered components, as depicted in Fig. 1. A subsystem is in an active state even if some of its components are not functioning since they provide the same functionality.
The members of subsystems are frequently non-repairable, with two conceivable conditions, functional or nonfunctional. The operational conditions of the component's reliability are understood, defined, and autonomous. That implies the loss of an individual component does not harm the functional capabilities of the corresponding subsystem or the entire functioning of the system. Nevertheless, there are several empirical uncertainties built into the component reliability of the system that needs to be addressed [3]. Sources of the uncertainties in the subsystem are due to the facts listed as follows: • The reliability of a device is strongly affected by the environment under which the system is operating. Therefore, it is almost impossible to estimate the exact quantitative value representing the system's reliability, cost, and weight.
• Possibly, the redundant modules may be of various models (materials used by manufacturers) and lack critical information regarding the reliability, cost, and weight needed to evaluate component parameters.
• During the design process, it is difficult to identify the number of redundant modules that can be chosen in a subsystem to ensure superior outcomes for the system.
• System designers have no definite idea about the parameters, viz., cost, volume, and weight, during design. Consequently, they have only guessed and used the estimated values for designing decisions. These estimations are typically based on imprecise, incomplete, and insufficient knowledge.
Therefore, it is essential to use suitable techniques for managing such uncertainties and improving system modeling. These uncertainties are a rudimentary aspect of the modeling that has been unseen lately. Thus, the realistic construction of MORRAP necessitates the deliberation of such empirical uncertainties associated with parameters as they have a serious impact on the modeling framework. These uncertainties in the component parameters of the system are modeled as fuzzy quantities [4] by several researchers [5], [6], [7]. Because these inconsistencies in the component characteristics affect the goals, such as reliability, cost, weight, volume, and so on, the realistic construction of MORRAP demands that these uncertainties be taken into consideration. Several studies [5], [6], [7] represent these uncertainties as fuzzy quantities. However, the type-1 fuzzy numbers (T1FNs) are an inefficient approach to managing the uncertainty from the multiple sources of the system. It is because T1FNs have severe interpretability issues and inaccuracies, and the membership values of T1FNs are crisp. So, they cannot be used to model the higher-order uncertainties [8], [9], [10]. Type 2 Fuzzy Numbers (T2FNs) are an extension of T1FNs that allow for a more precise representation of uncertainty [11]. A T1FN is defined by a membership function that assigns a degree of membership between 0 and 1 to each element of the universe of discourse, whereas a T2FN is defined by a primary membership function and a secondary membership function that assign degrees of membership, respectively, between 0 and 1 to each element of the universe of discourse. It is a fact that the researcher intends to use another variety of T2FS called interval type-2 fuzzy set (IT2FS) due to the high computational effort of T2FS. A T2FS is converted to the IT2FS whenever the level of secondary association functions is equivalent to one. Several other researchers have carried out significant contributions [8], [12], [13], [14], [15] and made this concept a magnificent field of study. Interval type-2 fuzzy numbers (IT2FNs) can discourse the weakness of T1FNs while the degree of belongingness is also demarcated with a type-1 fuzzy illustration [3], [16], [17], [18], [19], [20], [21]. Therefore, the main reasons for taking IT2FNs into account for modeling the component characteristics in MORRAP may be summed up as follows: a) Subsystems may include components from several suppliers and be constructed using raw resources of mixed qualities. So, the cost, weight, and reliability parameters of the connected components used in separate subsystems may differ. b) Additionally, design engineers of a specific system could only have a limited understanding of the potential attributes for such component attribute values throughout the designing process. They must provide exact values for the characteristics of the parameters, which they rarely do: thus, they used only approximated values. Therefore, IT2 fuzzy quantities are the best option for modeling them. c) The degree of belongingness of the inherent uncertainties in the costs, weight, and reliability components may modify according to the perceptions of decisionmakers. So, IT2FNs become the most appropriate to model the scenario In this paper, an IT2FMORRAP is planned to exploit reliability and simultaneously diminish the cost and weight of a series-parallel configuration . With intention  parameters as IT2FNs, we highlighted circumstances wherever the theoretical IT2FMORRAP paradigm transforms  to T1FMORRAP with T1FNs, IVMORRAP with intervalvalue (IV) numbers, and MORRAP with crisp or real  numbers. Further, IT2FMORRAP, T1FMORRAP, and crisp MORRAP have indeed been addressed by using NSGA-IIbased solvent procedures. The dataset of a pharmaceutical plant is being used to examine the solution strategy's effectiveness for perceived cases. A k-mean classification representation is also used to locate the highest quality solution sets from the produced Pareto optimal solutions. The major contribution and motivations of this work are as follows: 1) Proposed a novel Multi-Objective Reliability Redundancy Allocation Problem (MORRAP) using Interval Type-2 Fuzzy Numbers (IT2FN) for modeling uncertainties namely, IT2FMORRAP.
2) The IT2FMORRAP optimizes three objectives simultaneously, such as reliability, cost, and weight of a series-parallel system. 3) The underlining situations are established under which the proposed model abridges to T1FMORRAP, which consider with IVMORRAP, which finally eases to classical MORRAP. 4) A novel solution approach is proposed using the Enhance Karnik-Mendel algorithm and NSGA-II. 5) Experimental simulations are conducted, and results demonstrate that the proposed IT2FMORRAP is superior to that of T1FMORRAP and crisp MORRAP. 6) A k-mean clustering technique is used to recognize the most suited solution region of the optimal Pareto fronts, and indicators, namely, the number of solutions, spacing, spread, diversity, hypervolume, and normalized hypervolume performance, are applied to compare the formulations. 7) Statistical analysis also confirms the hypothesis established in the experimental study. 8) A simulation study along with comparative performance analysis with some other state of art methods from the literature has been conducted.
The remaining work is prepared as follows: An related study of the research work has been given in Section II. Section III presents the mathematical preliminaries. Section IV describes the formulation of the proposed model and its exceptional cases. Section V explains the problem-solving strategy. Section VI has examined the simulation findings and comparative analyses are presented in Section VII. Section VIII is where the paper is concluded.

II. LITERATURE REVIEW
This section will briefly address the research works on the approaches described in this paper. Two subsections cover current IT2FS and reliability optimization studies.

A. INTERVAL TYPE-2 FUZZY SET AND APPLICATIONS
Many situations arise in daily life in which there are more than two choices needed to be considered while making a decision. Solving problems of such kind requires considering more than two possible truth values. Thus, binary yes/no is insufficient for these circumstances, and complex representations are needed. In 1965, Zadeh presented the fundamental concept of fuzzy sets (T1FSs) [4]. The T1FS's uncertainties make it difficult to calculate the precise degree of belongingness. To get around this, Zadeh devised the idea of T2FSs. After that, an IT2FS was described as a unique mathematical formulation of T2FS by assuming a uniform secondary degree [8], [13], [22]. The higher-order IT2FSs expand the range of uncertainties that can be addressed in real-time and broaden the applications. IT2FSs have been used in a variety of applications, including control systems, decision-making, inventory system, pattern recognition, and image processing [16], [23], [24]. IT2FSs are more effective than T1FNs in many cases, particularly in situations where the uncertainty is high or the data is imprecise.
Recently, Ashraf et al. [16] developed an interval type-2 fuzzy logic-based image steganographic system. Ashraf et al. [25] proposed a non-linear system, IT2 vendormanaged inventory system, and solve it with EKM with particle swarm optimization algorithms. IT2 fuzzy neural networks and grasshopper optimization algorithms were suggested by Amirkhani et al. [26] for the development of a vehicle antilock braking system. A robust single fuzzifier IT2 fuzzy C-means clustering method to adopt the interval-valued numbers for the application of land cover segmentation was presented by Wu and Gao [27]. Ashraf and shahid [28] established the multiobjective vender managed model with IT2FNs for demand and order quantity. Javanmard et al. [29] demonstrated a fuzzy solution to a linear programming problem where all coefficients are understood by IT2 FNs. The closest interval approximation is the foundation of the suggested approach.

B. RELIABILITY OPTIMIZATION
Finding the best component redundancy distribution while maximizing reliability is one of the fundamental issues in reliability theory. The early studies on reliability optimization, a dynamic allocation technique [30], and a heuristic technique [31] were presented to determine the best reliability allocation. In a landmark study, M. S. Chern [32] showed that the reliability with redundancies optimization problem is essentially an NP-hard. It motivated the computational intelligence community to compare heuristic-based techniques for RRAP to conventional methodologies such as genetic algorithm (GA) [30] and [33], enhanced GA [34], particle swarm optimization (PSO) [35], [36], bee colony algorithm [37], cuckoo search algorithm with GA [38], imperialist competitive algorithm [39], co-evolutionary differential method with harmony search [40], Gradient-based optimizer [41] and so on. VOLUME 11, 2023 To solve MORRAP, Coit et al. [42] combined multiobjective optimization problems into a single objective problem. Sheikhalishahi et al. [43] employed a mix of two algorithms, the GA and PSO methods, to enhance reliability by decreasing the cost, weight, and volume under nonlinear constraints. The NSGA strategy was used by Taboada et al. to solve MORRAP [44]. With the universal moment generating function approach for reliability or availability indices, Taboada et al. [45] addressed the multiple objective multi-state RPAP. Under constraints, reliability and cost by NSGA-II are the two goals of Wang et al. [46] reliability optimization model. A multipleobjective evolutionary method for addressing conditions was used by Salazar et al. [47] to solve MORRAP. A knowledgebased simulated annealing approach to solve MORRAP was published by Zaretalab et al. [48]. Cao et al. created the decomposition-based technique in [49] to solve the MORRAP.
To address the issue of uncertain reliability allocation, Gupta et al. [50] used the interval-valued reliability of the components. Under many restrictions, Sahoo et al. [51] solved MORRAP with interval-valued component reliability. An interval-based MORRAP was developed by Zhang et al. [52] and solved using MOPSO. To model the MORRAP and provide a solution, Roy et al. [53] well-thought-out the interval number for the reliability parameters. A fuzzy MORRAP for series-parallel systems was developed by Garg and Sharma [6] utilizing linear and non-linear membership functions, and it was then solved using PSO. When examining the MORRAP, Ebrahimipour and Sheikhalishahi [54] considered a triangular T1FN. Jiansheng et al. [55] addressed MORRAP with unknown parameters like repair rate, failure rate, and other related coefficients in the repairable mode of series-parallel systems.
A fuzzy MORRAP in a type-2 fuzzy environment was first developed by Ashraf et al. [56]. Ashraf et al. [57] presented a PSO-based solution method to solve the type-2 fuzzy MORRAP model. Muhuri et al. [21] constructed an interval type-2 fuzzy reliability of the component-based model for MORRAP, which was solved using the KM and NSGA-II methods. Chebouba et al. [58] solve the multiobjective system reliability of fuzzy quantities using the non-sorting genetic algorithms (NSGA-III). To describe the series-parallel and parallel-series systems, Ashraf et al. [59] presented an IT2 Fuzzy membership function, EKM and PSO were used to solve the formulated interval type-2 fuzzy MORRAPs, and the outcomes were compared to GA.

III. MATHEMATICAL PRELIMINARIES
This section provides descriptions of the essential fundamental terms and techniques [9], [13], [15] used in our proposed problem formulation.
1) A type-2 fuzzy set (T2 FS)Ã is distinguished by the second-ordered grade of belongingness µ˜Ã (x, u) and mathematically interpreted as follows.
Here, x represents the principal component considered from the discourse of universe X , and u represents the secondary component function, that is, u ∈ J x ⊆ [0, 1]. For each element x∈X , J x describes the primary degree of belongingness of x and µ˜Ã(x, u) provides the secondary order degree of belongingness also called the type-2 membership function (T2 MF), such that 0 ≤ µ˜Ã (x, u) ≤ 1.
2) An interval type-2 fuzzy set (IT2 FS)Ã, is a T2 FS with the secondary order degree of belongingness µ˜Ã (x, u) is equality and expressed as: A closed area under IT2 MFs of an IT2 FSÃ represents the amount of uncertainty known as the footprint of uncertainty (FOU) and is defined The inferior and superior of FOU Ã are defined as follows.

IV. PROBLEM FORMULATION
This section explains the mathematical framework. The new IT2FMORRAP will be designed. Additionally, it is proved that T1FMORRAP, IVMORRAP, and MORRAP are all particular circumstances of IT2FMORRAP. Table 1 exhibits symbols and abbreviations. Muhuri et al. [21], Kuo et al. [60], Coit and Konak. [42], and Ashraf et al. [57], describe the formulations for the traditional series parallel m− stage framework. In [21], Muhuri et al. considered a structure function χ over a n−dimensional space, such that . . , x n ) . Now, the nature of (x 1 , x 2 , . . . , x n ) decides the nature of structure function (χ), whether it is crisp numbers, interval-value numbers, T1FNs, or IT2FNs. For a m−stage series-parallel configuration, as presented in Fig. 1, the chrematistics of each elements are assumed to be defined and autonomous. Thus, the total reliability functionR S corresponding to IT2 fuzzy reliabilities r 1 ,r 2 , . . . ,r n , can be constructed by using the structural function as mentioned above presented in Eq. (6): Here,r i are represented with IT2FNs. An IT2FN has been represented by a triangular IT2MF (µ˜r i ) which is actually a set value mapping with respect to IT2 fuzzy reliabilitỹ R s (x i ) = {1−µ f t (r i n i )}r i , can be defined as follows: where r up left and r up right are left and right extreme points of the upper membership function (UMF). Similarly, r lo left , and r lo right are left and right extreme points of lower membership function (LMF) (see Fig. 3). So, overall system reliabilitỹ R S r , n , IT2 fuzzy costC S r , n , IT2 fuzzy volumẽ V S w , n and IT2 fuzzy weight andW S w , n of m−stage system configuration [21] is mathematically represented as: Thus, IT2FMORRAP may be formulated as:  Minimize W s r , n In special cases, generalized IT2FMORRAP presented by Eq. (13) has been reduced to T1FMORRAP, IVMORRAP, and MORRAP, as discussed in the following cases.  a T1MF (µr i ) as follows: Thus, overall T1 fuzzy reliabilityR S (r, n), costC S (r, n), weightW S (w, n) and volumeṼ S (w, n) can be stated as: Thus, we formulate T1FMORRAP as: ] therefore, T1FN will express as an interval-valued number as illustrated in Fig. 5. Thus, the IVNORRAP model [52] with total intervalvalued reliability [R left , R right ] can be expressed follows: (19) Similarly, total interval-valued cost [C left , C right ], intervalvalued weight [W left , W right ] and interval-valued volume V left , V right are expressed as follows: Thus, the interval-valued MORRAP is:

C. CASE 3: CRISP MORRAP
Finally, the T1FMORRAP will present a MORRAP when r left = r right = r s , as the classical number is shown in Fig. 6 because of the degree of belonging µr i (x) will be present by the characteristic function ζ r i (x) : Therefore, MORRAP [60] is: As per the aforementioned discussion, we finally concluded that the formulation of IT2FMORRAP is a generalized framework for other MORRAP models, Viz. T1FMORRAP [6], IVMORRAP [52], MORRAP [60].  The stated argument comes under the cope with Zadeh's detection [10] of the excellent capabilities of fuzzy sets to generalize mathematical models. It is also genuinely agreeable with notable judgments [61], [62], which show crisp sets, interval-valued sets, and type-1 fuzzy sets are special cases of type-2 fuzzy sets. Fig. 7 shows the visual hierarchy among the IT2FMORRAP, T1FMORRAP, IVMORRAP, and crisp MORRAP [21].

V. PROPOSED SOLUTION APPROACH
This section explains the solution approach, which uses the well-known non-dominated sorting genetic algorithm (NSGA-II) suggested by Deb et al. [63] and interval type-2 fuzzy system (IT2FS) [9]. Fig. 8 illustrates the nature of our planned solution strategy. The major processes involved in the approach are discussed as follows.

A. IT2 FUZZIFICATION
Multiple experts opinions on ambiguous decision-making variables leads complexity in the process, leaving scope of IT2 fuzzy systems [15]. IT2 fuzzy systems are employed with IT2 fuzzification to generate IT2FNs), adopted from [21]. Generally, the estimations of decision-makers are a specified range or interval (say I) represented via IT2FNs and may be in any one form i.e. Gaussian, trapezoidal, and triangular functions. Therefore, it requires some logical reasoning to choose best suited IT2MF µ˜Ã. As discussed earlier, reliability defines the likelihood of an operational device that will function correctly without any faults, and the time to failure (t) is the most promising value provided by the experts [21], [59]. Therefore, the IT2MF of the reliability at this particular time instance (t) is equivalent to one. In other terms, for r(t), the IT2MF (µ˜r ) should only be one with only one highest value, which also implies that IT2MF is normal. And, for any system that will operate over an endless period is also not feasible. Therefore, an extensive tail with more non-zero values is insignificant after a limit. Hence, neither trapezoidal nor Gaussian functions are suitable choices for modeling IT2FMORRAP. Thus, the triangular function is the best option to model IT2 fuzzy reliability, and weight parameters. It may also be possible that the endpoints of the interval of uncertainty (I), that is, left endpoint (I left ) and right endpoint (I right ), are uncertain due the multiple range provided by experts. Then, each I left and I right will also form the interval with lower bound and upper bounds. Specifically, for component reliability , r, the µ˜r will have r = [r left ,r right ] and each endpoint in interval will have lower and upper bounds, i.e., r left = [r up left , r lo left ] and r right = [r lo right , r up right ], respectively. The IT2FN of reliability (r) with the triangular IT2MF is presented in Eq. (7) and pictorially demonstrated in Fig. 3. A detailed algorithm for the generation of IT2FN with the parameters can be seen in [21].

B. IT2 FUZZY TYPE-REDUCTION AND DEFUZZIFICATION
The type-reduction procedure changes the IT2MFs into T1MFs. T1MFs transform into an output number by using the procedure of defuzzification. Several methods have been developed to perform the type-reduction and defuzzification of IT2 FN. Among them, Enhanced Karnik-Mendel (EKM) algorithm [13] is one of the most commonly used methods for the purpose. EKM algorithm is used for the typereduction process. The iteration number is kept low to offer an improvement in initializing and terminating situations. A centroid C˜Ã(x) is the combination of the centroids of all its embedded T1FSs. That is, The EKM algorithm is divided into two parts: in the first part, the computation of the c l is performed and in other parts c r is calculated. Eq. (27) and Eq. (28) are used for computing c l and c r .
The procedure of the EKM algorithm is explained in detail in [9], [16], and [59]; here, the first part is for the algorithm to compute c l and second one is for evaluating c r . The defuzzification will provide the ultimate output number (y d ) corresponding to IT2FN and is the average of the c l and c r , as follows:  Many scientists widely adopted NSGA-II [63], a multiobjective evolutionary optimization approach to solve various decision-making models with multiple goals. The pictorial representation of NSGA-II is given in Fig. 9. The subsequent sections explain the various sub-procedures of NSGA-II, including three goals, namely, (i) maximize the total  reliability of the system, (ii) minimize the overall cost of the system (iii) minimize the total weight of the system, under fixed volume and limiting constraints.

1) CHROMOSOME STRUCTURE AND OBJECTIVES EVALUATIONS
A chromosome/genome is an array of genomic sequences represented by floating-points. A genome is a uniformly distributed random solution that lies inside a predefined search space. In IT2FMORRAP, components' reliability (r i ), components' weight (w i ) and the redundant components (n i ) of the i − th subsystem are genes of the chromosome. The variables parameters r i , w i , and n i are of range [0.5, 0.99999], [7.0, 12.0], and [1], [5], respectively. A chromosome structure has been depicted in Fig. 10 to maximize the reliability (R S ), minimize the costs (C S ) and minimize the weight (W S ) of the system. The initialization of the initial population is similar to the original NSGA-II.

2) NON-DOMINATED SORT
The population generated initially has been sorted by a fast non-domination sorting algorithm introduced by Dev et al. [63]. In non-dominated sorting, solution X is supposed to dominate Y if and only if the objectives of X are no worse than the objective of Y and there must exist at least one objective of X , which is better than that of Y [64]. At first, we consider a set S containing all the N individual solutions of the population without losing the generality.
The sorting algorithm will then select the non-dominated solutions from S and assign them to rank-1 and call them Front-1 of the Pareto-front. Again, the left-out solutions of S are sorted, and the non-dominating ones are assigned as rank-2 forming the Front-2. This process repeats until there is no solution left in S to dominate others, and these solutions will create the kth rank Pareto-front as Front −k. The Pareto-front for IT2FMORRAP shaped via the nondominated sorting process with the three objectives is shown in Fig. 11. After sorting and assigning the rank to the fronts, the crowding distance of each front is determined.

3) CROWDING DISTANCE
The difference between individual solutions on the front is measured by the crowding distance depending on the objective function values, as shown in Fig. 12. In a Pareto-Front F i , the crowding distance (D jm ), for each individual j = 2, 3, . . . n i −1 and m = 1, 2, . . . , M number of objectives are arranged in increasing order is calculated as follows: The overall crowding distance of an individual j is D j = M m=1 D jm .

4) GENETIC OPERATIONS
Selection operation: The tournament selection approach is used in the selection process. The population's solutions are adopted by a crowded-comparison operator [65]. The comparative operator guides the selection method using ranks measured in the sorting process and the crowding distance.
For a particular individual, if an individual has a lower or the equivalent rank and higher crowding distance, it is said that the comparative operator is more dominant than the other. Solutions are selected to create a mating pool to perform the crossover and mutation procedures to generate the new offspring just after the selection process. Crossover: Simulated binary crossover strategies generate child individuals for each parent individual, as shown in Fig. 13. The binary crossover procedure is as follows.
Step-1: Create a randomly generated number u from a range between 0 and 1.
Step-2: Find the spread factor ρ as: where, η is a probability index also known as crossover probability.
Step-3: Suppose par 1,j ,par 2,j be two parents. Now the two children are generated as follows: Mutation: the polynomial mutation procedure is used to restore unexpected solutions to prevent getting stuck into the local optimum, resulting in exploration ability and diversity in the population. as demonstrated in fig. 14, a parent par j with upper bounds (par u j ) and lower bounds par l j , child ch j is calculated as: in eq. (34), δ j is evaluated using eq. (35) where u represents a random number and η m is mutation probability.

5) RECOMBINATION
In the recombination procedure, the older population and newly-generated offspring are mixed to ensure elitism in the NSGA-II. The current population is then shifted over to non-domination sorting to form the fronts. Suppose, the population size of solutions is more than N number of individuals on the Pareto-front. Then, individuals are sorted using the non-dominated sorting and arranged in decreasing order based on crowding distance to pick the first N solutions. The same is repeated until the stopping criteria are met. The typical illustration of the NSGA-II is pictorially demonstrated in Fig. 15. Finally, NSGA-II effectively establishes the Pareto-front solutions and offers a comprehensive knowledge of multiple optimal solutions over the search space. The proposed solution model is accomplished by two procedures, namely IT2FS and NSGA-II. The IT2FS consist of three sub-procedures, such as, IT2 fuzzification, type reduction and defuzzification. The time complexity of these procedures for each system components, reliability, cost and weight, are O (n), O (n) and O (1), respectively. Hence, the combined time complexity of IT2FS= O (n) +O (n) +O (1) ≃ O (n). Now, the time complexity of NSGAII is O (m × n × n) , for m objective and n population size. So, the worst-case time complexity for the proposed solution approach is O (n) +O(m × n × n) ≃ O(mn 2 ).

VI. EXPERIMENTAL SIMULATIONS
To demonstrate the proposed solution for IT2FMORRAP, we have taken a real application data set of a pharmaceutical factory, presented in Table 2 [6]. The experimental simulations have been conducted on MATLAB 2015b on a processor 3.40 GHz Xeon with 16 GB RAM and a Windows 10 operating system. The IT2FMORRAP formulations in Section IV are mixed-integer MOOPs with nonlinear objectives and constrained functions. Now, the proposed NSGA-II solution approach is applied to solve the given problem as in Table 2. Table 2, the formulations of the IT2FMORRAP, T1FMORRAP, and MORRAOP are as given below:
The problems IT2FMORRAP, T1FMORRAP, and crisp MORRAP are solved using the proposed approach, as mentioned in Fig. 8. Specifically, the IT2FMORRAP will be VOLUME 11, 2023   solved using the IT2 fuzzy system with the component parameters, reliability, and weight as IT2FNs. The initial population passes through the IT2 fuzzy system, and the IT2 fuzzification, type-reduction, and defuzzification process take place, after that the NSGA-II procedures are applied to find the Pareto front for the IT2 fuzzy reliability, IT2 fuzzy cost, and IT2 fuzzy weight objectives. On the other hand, for solving the T1FMORRAP model, the proposed approach will utilize the T1 fuzzy system (T1 fuzzification and defuzzification process) for the generation of T1FNs corresponding to the reliability and weight components. Then, NSGA-II is applied to generate the Pareto front for the T1 fuzzy reliability, T1 fuzzy cost, and T1 fuzzy weight objectives. While, for solving the MORPAP model, the NSGA-II will directly apply as a multi-objective evolutionary optimization approach because there is no fuzziness involved in this scenario. It is mentioned that the reliability, weight, and number of redundant components are the decision variables in all three problems. The redundancy of a subsystem must always be an integer, so, at the time of evaluation, it has to be converted into the nearest integer value.

B. RESULTS AND DISCUSSION
The results are presented in the form of an optimal Pareto optimal set for the MORRAP, T1FMORRAP, and IT2FMORRAP corresponding to the different number of iterations. We have performed 50 runs for all the various instances of iterations and presented the suitable optimal set of solutions. It provides the support to justify the relative convergence and performances amongst the solution algorithms. The fitness function evaluation was conducted approximately 280 times, and the infeasibility for the individual solutions (for which the feasibility was not fulfilled). Moreover, with the initial population set to 100, approximately 200 individuals are created using genetic operations to form the mating pool. These individuals were sorted according to non-domination to have 100 final individuals ultimately.
The optimal Pareto-Fronts corresponding to the three approaches is shown in Fig. 16 for (a) 50, (b) 100, (c) 200, (d) 400, (e) 500, and (f) 1000 instances, respectively. Fig. 16 evidence that the Pareto optimal solutions for models converge as the iteration rises. From Fig. 16, we see that as the system's total reliability goes higher, its total cost and weight of the system become higher. For example, in iteration numbers 50 and 100, in Fig. 16 (a) and (b), respectively, there is a clear trade-off among the nondominated outcomes represented by the IT2FMORRAP front than other approaches. So, the proposed IT2FMORRAP model has higher reliability and lower cost and weight function values. Further, there are tiny variations among the fronts when the numbers of iterations are 500 or more; see Fig. 16 (e) and (f). Thus, the IT2FMORRAP achieves a better front in a lesser iteration number. It means that the statistics of the Pareto optimal solutions are precisely noticeable at the lesser iteration number for IT2FMORRAP. Table 3 illustrates the extreme and most minor boundaries of the generated optimal set of solutions for the IT2FMORRAP, T1FMORRAP, and MORRAP models. The lower (min) and upper (max) ranges of the total reliability, cost, and weight functions in the approaches are [0.168070, 0.99501], [50.300058, 701.9070], and [32.10064, 386.6790], respectively. The range and the non-dominated solutions in all three approaches are acceptable. As mentioned, all approaches were run 50 times; we have also said the mean of each objective value of the front in Table 3. From Table 3, the average of the total reliability function values of the IT2FMORRAP approach is higher than the other two approaches, and the cost values are lower for the smaller iteration number. However, the weight of the system is higher. Similarly, the IT2FMORRAP values are comparatively better for the other iteration number. It is to be noted that there is very little difference between the function values when the number of iterations is more significant. Therefore, the IT2MORRAP can be used as a conventional foundation to distinguish a single run's performance. We need to reduce the size of the solution sets to further deeply analyze the Pareto optimal solution sets for the three approaches. Since the decision-maker might have their own choices over selecting a particular individual solution, we have to use the data clustering approach to prune the optimal solutions. Moreover, we have used famous comparison metrics and statistical testing the compare the results.

C. CLUSTERING OF PARETO-FRONTS: K -MEAN CLUSTERING ALGORITHM
The next step is to prune solution sets after achieving acceptable Pareto-optimal solutions. One of the many significant purposes of shrinking the optimal solutions' size is to select meaningful solutions by decision-makers. Two strategies are designed to do the same, i.e., deciding based on the importance of the objective components with professional decision-makers and the data clustering technique. The first strategy is less fruitful than the decision-maker would hardly anticipate the analytical preference of the system's goals. However, the second approach of clustering to classify solutions in alternative domains and then decide the best solutions is relevant if there is an emphasis on their categorization. The use of the data clustering approach for reducing the size of an optimal set is presented in more detail in [45] and [66].
The k-mean identifies the best solution region of the Pareto-front from the three models. It offers excellent assistance for system designers investigating the variances between the three models. We used the silhouette plot strategy to evaluate the optimal number of clusters. This method calculates the allocation of objects to clusters to see if the groups are about the same quality (e.g., an equal number of objects in every collection). For example, Fig. 17 illustrates the clusters of Pareto optimal sets generated by IT2FMORRAP. There are two methods for identifying the optimal solution among these five clusters. First, the cluster center (centroid) is calculated and considered the best representative solution. Second, the ''knee'' cluster is discovered since it includes the significantly superior Pareto solutions for trade-offs between the goals. The knee cluster is composed of the most impressive outcomes of the Pareto fronts, solutions where a minute increase in one goal would direct a considerable decline in at least one other objective. However, locating the knee area is challenging due to the unsystematic nature of the Pareto-fronts; hence, we have grouped all Pareto-solutions produced via various iterations. VOLUME 11, 2023  The cluster in the knee region provides the individuals responsible for making decisions with a list of recommended alternatives from which they may choose the most appropriate solution. Fig. 18 demonstrates the cluster center, and centroids of the Pareto-Front found for various iterations (50,100,200, 400, 500, and 1000) concerning the IT2MORRAP, T1FMORRAP, and MORRAP. Fig. 18 demonstrates the clusters point along with the centroids of the Pareto Front for various iterations; 50, 100, 200, 400, 500, and 1000 for the IT2MORRAP, T1FMORRAP, and MORRAP. Moreover, the five representative ranges of solutions (corresponding to different clusters) are shown in Table 4 and Table 5. From Table 4 and Table 6, the expert chooses a solution region and analyzes the extreme objective function values. Once the clustering is done, we can further investigate the system objectives from Fig. 18 and Tables 4-5 as follows: • For iteration number 50, as we can see from Fig. 18   The results mentioned above demonstrate that the designed IT2FMORRAP formulation archives better knee region solutions than the T1FMORRAP and MORRAP models for 50, 100, and 200 instances of iterations. The obtained amount of solutions of Cluster#4 for 50 instances, Cluster#5 for 100 instances, and Cluster#5 for 200 instances for IT2FMORRAP consisted of a higher number of solutions as achieved a more comprehensive range of objective values in comparison to the other two approaches. For 400, 500, and 1000 iterations, Cluster#3, Cluster#5, and Cluster#3 for IT2FMORRAP, respectively, are the most prominent solutions for decision-makers. When the numbers of instances are 500 or higher than it, the differences between the three approaches are minimal (see Table 5). This will again justify our argument that the IT2 fuzzy modeling of reliability optimization is more realistic and faster [21].  Finally, the above results are plotted pairwise (a) reliability vs. cost and (b) reliability vs. weight objectives for illustration in Figs. 19-24 for 50, 100, 200, 400, 500, and 1000 iterations. This interpretation of the optimal solution sets can be useful to the decision-maker since it would have found suitable tradeoffs in the knee region.
In Fig. 19(a)-(b) for iteration 50, Clusters#1, Cluster#5, and Cluster#4 are considered for MORRAP, T1FMORRAP, and IT2FMORRAP, respectively, because they possess complete reliability vs. cost and reliability vs. weight of the system compared to the remaining clusters. Also, the IT2FMORRAP delivers the lowest values for cost and weight among the three clusters. Whereas in Fig. 20(a)-(b), Clusters#1 for MORRAP and Cluster#4 for T1FMORRAP have high reliability than Cluster#5 of IT2FMORRAP. However, it is accomplished at a relatively expensive cost and weight value. As mentioned earlier, in a similar context, solution clusters at the knee region in Fig. 21 to Fig. 24 reliability vs. cost and reliability vs. weight of the system compared. It can be easily observed that the IT2FMORRAP  cluster solutions outperform both T1FMORRAP and Crisp MORRAOP models by producing better Pareto solutions.
Although the trade-off solutions in the decision-making can only be confirmed as the most acceptable solution with knowing the decision preference, it is undoubtedly fascinating to point out the apparent and straightforward trade-off solution to three objectives higher than others and the final choice. Hence, the selection of one individual depends on the prospect of the decision-making and understanding of the system's needs, responsibilities, and expectations of the anticipated customer/users.

D. COMPUTATIONAL ILLUSTRATIONS OF PERFORMANCE INDICATORS
We have measured the number of Pareto solutions, diversity, spacing, spread, hyper volume, and normalized hyper volume performance indices [64] to compare the Pareto-fronts obtained from our experiments.
1) The number of Pareto solutions (NPS): It states that algorithm-I is better than algorithm-II; if algo-I generates P results and algo-II Q, provided P > Q.   Using the mentioned indicators, we compare the solutions obtained from the three models: (1) IT2FMORRAP, (2) T1FMORRAP, and (3) Crisp MORRAP, at different instances of iterations. The measures were computed for three methods over the independent simulation of 50. Table 6 gives the numeric values of these indicators for 50, 100, 200, 400, 500, and 1000 instances. Fig. 25 pictorially illustrates the critical observation for all indicators. It can be inferred from Fig. 25(a) that the mean of NPS in the established IT2FMORRAP process is greater than the other two methods, indicating the amount of non-dominated alternatives for IT2FMORRAP is more enormous than the MORRAP and T1FMORRAP. The mean of the diversity measure in the suggested IT2FMORRAP seems to have more relevance to other methods (see Fig. 25(b)). It implies that the IT2FMORRAP process has VOLUME 11, 2023   fewer non-convergences strategies for experimental tests. The average calculation of the spacing indicator (from Fig. 25 (c)) in the developed model is significantly smaller than other methods. That implies that the relative difference between the sequential non-dominated resolutions in the IT2FMORRAP is small. However, the suggested method's mean spread value has a much more significant effect than the T1FMORRAP and MORRAP (see Fig. 25 (d)). Fig. 25(e)-(f) shows that IT2FMORRAP produces nondominated frameworks with significantly higher amounts for the hypervolume and normalized hypervolume measurements. Thus, the non-dominated results achieved mostly by the IT2FMORRAP approach are much more distributed equally than all those observed by other systems. Hence, IT2FMORRAP provides better representation in appearances of the NPS, diversity, spacing, HV, and NHV indicators compared with T1FMORRAP and crisp MORRAP.

E. STATISTICAL ANALYSIS
The statistical analysis has been given to evaluate the hypothesis about the results of the experimental investigation of the Pareto-fronts performance Viz. NPS, diversity, spacing, spread, HV, and NHV. Foremost, the data samples of performance metrics were checked to see if they were normal. After that, we compared the efficacy of IT2FMORRAP, T1FMORRAP, and Crisp MORRAP by analyzing the models with a 95% confidence level. Following are the assumptions considered for the normality, homogeneity, and comparison. • Normality (Hypothesis 1): Sample data is insignificantly variated and normally distributed.
• Homogeneity (Hypothesis 2): Samples data follows the identical distribution of points.
• Comparison (Hypothesis 3): No significant difference in IT2FMORRAP, T1FMORRAP, and MORRAP on X, where X represents the vector of objectives. Table 7 shows simulation samples of 50 runs for diversities of Pareto front generated by IT2FMORRAP, T1FMORRAP, and MORRAP, respectively. For the diversity data samples described in Table 7, Table 8 displays the results of a normality test directed employing the Kolmogorov-Smirnov and Shapiro-Wilk tests. In this test, we will use a sample size of 50 for each condition. The values considered significant (Sig.) are lower than 0.05. As a result, the normality test hypothesis (Hypothesis 1) is disproved. It denotes the diversity sample data deviation from the normal distribution. Table 9 shows the result of the Leven Test, which was used to test for homogeneity. Significant values, in this case, are more than 0.05. As a result, Hypothesis 2 is confirmed, pointing toward the samples being homogenous.
As earlier demonstrated, diverse data samples are typically not dispersed but relatively homogenous. As a result, the Kruskal-Wallis test is used to evaluate them. The statistical results for the Kruskal Wallis test are demonstrated in Table 11. In Table 12, diversity has a significant (Sig.) value of 0.000, less than 0.05, at a 5% significance threshold. As a result, rejecting the hypothesis (Hypothesis 3) demonstrates that samples vary greatly, showing the improved performance of the suggested IT2FMORRAP over T1FMORRAP and MORRAP for diversity. Table 11 contains the statistical results of the Kruskal-Wallis test for different performance measures, including spacing, spread, NPS, HV, and NHV. The pattern mirrors that of Table 10. Thus, from Table 9 to Table 10, we prove that the proposed IT2FMORRAP considerably varies from T1FMORRAP and MORRAP on the performance metrics considered. The findings of the hypothesis test, which are described in Table 11, support the validity of the inferences made from the experimental investigation.

F. CRITICAL ANALYSIS AND DISCUSSION
From the obtained results of the IT2FMORRAP, T1FMORRAP, and crisp MORRAP in Section VI (B-E), we found that IT2FMORRAP outperforms T1FMORRAP and crisp MORRAP in terms of the performance matrices. However, it is almost impossible to design and implement a high-performing system in a real-world scenario, as uncertainties are always present. Nowadays, the IT2FN is the most precise and renowned approach to epistemic uncertainties.
In summary, this work provides practical and indirect solutions to the decision-makers for finding the best optimal solution among the trade-off objectives of reliability, cost, and weight under the most suited environment. The K-mean clustering approach provides the best-suited region of the Pareto optimal set that helps the decision-makers to select an appropriate solution value of optimal reliability, cost, and weight according to their requirement. Further, the performance matrices, such as NPS, diversity, spacing, spread, HV, and NHV, express the superiority of the proposed IT2FMORRAP over the others.

VII. COMPARISON WITH THE WELL-KNOWN STATE OF ART MODELS
A comprehensive study of the proposed model with some well-known states of the art models is done here. The bibliographic literature on reliability optimization techniques is rich due to the popularity and applicability of the reliability issue. The researcher developed several single objectives and multi-objectives heuristics and meta-heuristics schemes to solve the problem. Moreover, there are different ways of managing the randomness and uncertainties: interval approach, probabilistic distributions, possibility theory, T1 fuzzy sets, intuitionistic fuzzy sets, and IT2 fuzzy set theories. Therefore, we have mainly focused on those related works in which the multiobjective optimization approaches are presented under the umbrella of uncertainties [3], [67], [68]. Table 12 shows a thorough comparative study of the proposed model with the well-known state-of-art models.
Wang et al. [46] addressed the MORRAP aiming the R S maximization and C S minimization with NSGA-II under the weight constraint concerning the component reliability and redundancies as crisp numbers. Khalil-Damghani et al. [69] presented a dynamically tuned multiobjective particle swarm approach for solving MORRAP with three objectives: R S maximization, C S minimization, and W S minimization by finding the best-suited component reliability and redundancies for the series-parallel system. Taboada et al. [45] wellthought-out objectives, R S maximization, C S minimization, and W S minimization in MORRAP and Pareto Front studies with the k-mean techniques. Zhang et al. [70] presented the barebones-based multiobjective PSO algorithm to address the three objectives (R S , C S , and W S ) MORRAP. Further, the authors have applied a sensitivity-based clustering technique to reduce the size of the optimal solution sets. In [45], [46], [69], and [70] deliberate the MORRAP, wherever indecision, hesitation, vagueness, and ambiguity in the featuring decision variables or parameters was not measured and therefore not appropriate to the practical-life systems. Sahoo et al. [51] solved IVMORRAP (intervalvalued MORRAP) to maximize interval-valued reliability and cost functions using an entropy-based region-reducing GA. Roy et al. [53] considered the interval-valued reliability [R Left , R right ], and interval-valued cost [C Left , C right ] and proposed the IVMORRAP of the series-parallel system, then solved the proposed model using the GA algorithm.
Zhang et al. [52] formulated an IVMORRAP for the interval-valued reliability and cost functions. The authors introduced a multiobjective PSO algorithm to solve a SCADA system for water resources. Unfortunately, this is far from being a legitimate basis since considering an entire range of numbers (also known as an interval) to have equivalent probability is an uncommon occurrence in the existing natural system. Garg and Sharma [6] formulated a T1 fuzzy MORRAP for series-parallel systems using linear and nonlinear membership functions corresponding to the system's reliability maximization and cost minimization objectives. The authors have used a de-fuzzified approach to establish a crisp model and solved it using a PSO algorithm. In [71], Garg et al. demonstrated intuitionistic fuzzy programming to design the reliability and cost functions of the system via the intuitionistic fuzzy membership functions with triangular interval data.
Recently, Muhuri et al. [21] presented the higher-order uncertainty associated with reliability components with IT2 fuzzy numbers and introduced an IT2FMORRAP model to two objectives: system reliability vs. cost and reliability vs. weight. The bi-objective IT2FMORRAP models are solved and compared with the T1 FN representations of parameters. In another work, Ashraf et al. [59] modeled the system reliability and cost functions with the interval-type-2 fuzzy membership functions by using Zadeh's extension principles and solving the formulated FMORRAP model using the EKM with particle swarm optimization algorithm. Therefore, no work has been considering the system's reliability, cost, and weight parameters with IT2 FNs to model the three objectives of IT2FMORRAP.
Therefore, we brought across our claims of rationales to prove those component parameters like reliability, cost, and weight, respectively, are much more practically formed with IT2 FNs than most other uncertainty designing methods, including T1 FNs or Interval-valued numbers [61], [62], [72]. Consequently, the work presented in this work is a general model of the MORRAP; almost all current MORRAP frameworks will be seen to be the exceptional circumstances of the presented IT2FMORRAP design.

VIII. CONCLUSION AND FUTURE WORKS
In this paper, an interval type-2 fuzzy multiobjective reliability redundancy allocation problem (IT2FMORRAP) under higher-order uncertainties in the system's reliability, cost, and weight parameters has been formulated. IT2FMORRAP considers three objectives: reliability, cost, and weight of a series-parallel system with parameters such as IT2FN. The underlining situations are established under which the proposed IT2FMORRAP model reduces to T1FMORRAP, IVMORRAP, and classical MORRAP. A novel solution approach is presented using the Enhance Karnik-Mendel algorithm and NSGA-II. Experimental simulations have been conducted using the pharmaceutical plant dataset to produce Pareto Fronts for all the considered models. The simulation reveals that the Pareto Fronts generated by IT2FMORRAP are superior to T1FMORRAP and crisp MORRAP. A k-mean clustering approach is employed to demonstrate the cluster center and centroids of the Pareto-Front found for various iterations 50, 100, 200, 400, 500, and 1000. Also, the five representative ranges of solutions corresponding to different clusters are shown. The most suited knee region cluster for IT2FMORRAP with 21  .8093] respectively. The clusters at the knee region of the Pareto-front for two objectives (reliability vs. cost and reliability vs. weight) and three (reliability vs. cost vs. weight) are better for the proposed model compared to the other models for the all-considered iteration set with faster convergence for NSGA-II.
Moreover, the performance indicators Viz. the number of Pareto solutions, diversity, spacing, spread, hypervolume, and normalized hypervolume, indicate that the IT2FMORRAP model surpasses both considered models. Statistical analysis has also been conducted to compare the samples from the different runs of performance matrices. Kruskal Walli's test is used to confirm the hypothesis established in the experimental study. A simulation study and comparative performance analysis with other state-of-the-art methods from the literature have been conducted to find a suitable place for the proposed work in the domain.
As a limitation, we can observe that it is almost impossible to design and implement an exact optimal system in a real-world scenario with uncertainties. Therefore, for any solution model, continuous improvement is needed. Moreover, the characterization of both the epistemic and aleatory uncertainties simultaneously is still computationally challenging the researcher in practical applications.
The future direction will be considered as the use of General type-2 fuzzy number, Intuitionistic fuzzy number, and Interval type-2 Intuitionistic fuzzy number to represent the reliability, cost, and weight parameters of the MORRAP system. Further, some recent multi-objective meta-heuristics optimization algorithms may be used to solve the formulated model.