Iranian Journal of Fuzzy Systems
https://ijfs.usb.ac.ir/
Iranian Journal of Fuzzy Systemsendaily1Fri, 01 Oct 2021 00:00:00 +0330Fri, 01 Oct 2021 00:00:00 +0330cover Vol 18-No 5-october 2021
https://ijfs.usb.ac.ir/article_6259.html
A bibliometric analysis of Iranian Journal of Fuzzy Systems (2007-2020)
https://ijfs.usb.ac.ir/article_6252.html
Iranian Journal of Fuzzy Systems (IJFS) is an international journal, based on the WoS core collection, this paper uses bibliometric methods and tools (VOSviewer and Bibliometrix) to analyze the publications of IJFS from 2007 to 2020. A total of 716 publications are analyzed. Firstly, according to the basic information of the publications, including the type of publications, the number of publications, the number of citations, the average number of citations and the h-index, we conduct a general statistical analysis to explore the basic characteristics and internal structure of the publications. Afterwards, by studying the citation structure of publications, this paper analyzes which groups and fields the journal is more concerned about. For the journal itself, we combine the number of citations and the number of publications to identify the most popular publications, the most productive and influential authors/institutes/countries/regions. In the collaboration network analysis, we map the collaboration network of countries/regions, institutes and authors to explore the different levels of collaboration. The co-citation network of the journal is studied to explore the relationship between IJFS and other international academic journals. In the analysis of keywords, we discuss the development trend and hot topics of keywords over time through co-occurrence analysis, word growth graph and thematic evolution analysis. Finally, based on the above analysis, we discuss the problems and challenges that the journal may face, and some main conclusions are drawn. This study has certain reference value for the further development of the related researchers�� scientific research and the journal.Multidimensional interval type 2 epistemic fuzzy arithmetic
https://ijfs.usb.ac.ir/article_6253.html
The article presents a new, multidimensional arithmetic of type 2 fuzzy numbers (M-IT2-F arithmetic) in which the result is a multidimensional fuzzy set. This arithmetic increases the accuracy of calculations and the scope of problems solved in relation to the currently used interval type 2 standard fuzzy arithmetic (IT2-SF arithmetic). The proposed M-IT2-F arithmetic has mathematical properties that IT2-SF arithmetic does not have. Thanks to these properties, it provides accurate calculation results that are not over- or under-estimated in terms of uncertainty. The paper contains comparisons of both types of arithmetic in application to two problems. Fuzzy arithmetic is not a finished work and is in a phase of continuous improvement and development. M-IT2-F arithmetic is a higher form of M-IT2 (non-fuzzy) arithmetic.Approximating credibilistic constraints by robust counterparts of uncertain linear inequality
https://ijfs.usb.ac.ir/article_6254.html
This paper studies a class of credibilistic optimization (CO) problems, in which a convex&nbsp; objective is minimized subject to ambiguous credibilistic constraints. The considered CO problem is usually computational intractable. Our purpose in this paper is to&nbsp; discuss the robust counterpart approximations of ambiguous credibilistic constraints. Under mild assumptions, the closed property about the feasible region of credibilistic constraint is discussed. Using the obtained results, this paper deals with the robust counterpart approximations of credibilistic constraints under two types of ambiguity sets of possibility distributions. The first type is exponential function-based ambiguity set of possibility distribution, while the second type of ambiguity set is a particular case of the first one, and it is based on range and expectation information of fuzzy variables. The developed approximation techniques are capable to utilize the knowledge of ambiguity sets of possibility distributions when building distribution uncertainty-immunized solutions. As a result, the obtained safe approximations of ambiguous credibilistic&nbsp; constraints are computationally tractable convex/linear constraints.&nbsp; To apply the proposed approximation approach, a portfolio optimization problem is addressed, in which the investor is to find a portfolio to maximize the value-at-risk of his total return under the support and expectation information of uncertain returns. We use two types of robust counterpart approximations to credibilistic constraints.&nbsp; The computational results support our arguments.Idempotent uninorms and nullnorms on bounded posets
https://ijfs.usb.ac.ir/article_6255.html
The paper deals with uninorms and nullnorms as basic semi-group&nbsp; operations which are commutative and monotone (increasing). These operations were first introduced on the unit interval and later generalized to bounded lattices. In [Kalina 2019], they were introduced on bounded posets. This contribution is a generalization and extension of the results in [Kalina 2019]. Some necessary and some sufficient conditions for the existence of idempotent uninorms and idempotent nullnorms&nbsp; on bounded posets are studied. Finally, some application examples are provided.A fuzzy generalized predictive controller to optimal drug dosage therapy of mathematical modeling of HIV
https://ijfs.usb.ac.ir/article_6256.html
This paper proposes a fuzzy-GPC based on a mathematical model of human immunodeficiency virus (HIV) to determine the drug dosage and control the progression of the illness. For this purpose, a Takagi-Sugeno (TS) fuzzy model is generated to identify the nonlinear behavior of HIV. The parameters of HIV are estimated by the least square error (LSE) estimation method. Moreover, three scenarios are proposed to control HIV. In scenario 1, according to TS fuzzy model, generalized Predictive Control (GPC) is designed for a daily base drug therapy. Scenario 2 and 3 are more practical. In scenario 2, since the biological behavior of patients are different, the variation in the patients biology is taken into account by generating data according to a group of patients with varying parameters in their mathematical model. In senario3, since daily diagnosis of patient&rsquo;s health is costly, it is assumed that a patient information is available every month, and drug dosage is determined each month. As a result of which, the sample time of the measurement increases to 30 make it a multi-rate system. The result shows that the TS fuzzy models&nbsp; the mathematical model of HIV very well, and in all scenarios, the proposed controller has a good performance and the number of healthy cells are controlled in acceptable amount.Hierarchical Chinese postman problem with fuzzy travel times
https://ijfs.usb.ac.ir/article_6257.html
The hierarchical Chinese postman problem (HCPP), one of the kinds of Chinese postman problem (CPP), aims to visit at least once the arcs are classified according to their precedence relations and to find the shortest tour or tours. In HCPP, if the travel time values between two nodes constituting the cost are not crisp, this problem type can be said to be the hierarchical Chinese postman problem with fuzzy travel times (HCPP-FTT). In real-life problems, as the travel time between the nodes is not fixed due to various factors, in this study triangular fuzzy numbers are used at objective function coefficients of a small-size sample problem. Different ranking methods are used to defuzzify the fuzzy numbers and the acquired results are compared. The mathematical model formed is resolved using CPLEX solver GAMS 24.2.3 software. Besides, in this study, a large-size real-life problem was discussed. The objective function coefficients representing the travel time between the two nodes in the case study discussed were considered as trapezoidal fuzzy numbers. Due to the NP-hard nature of the problem, the mathematical model cannot find solutions in large size problems in the appropriate time interval. For this reason, in this study, the heuristic method based on the Greedy Search (GS) algorithm and the meta-heuristic method based on ant colony optimization (ACO) is proposed for the solution of the real-life problem. The proposed algorithms are coded using the Matlab-2019b programming language. Using the developed algorithms, the results of different ranking methods and solution times were compared.Multi-criteria decision making based on q-rung orthopair fuzzy promethee approach
https://ijfs.usb.ac.ir/article_6258.html
The preference ranking organization method for enrichment of evaluations (PROMETHEE) {constitutes a family of outranking} multiple-attribute decision-making (MADM) methods {that has been adopted by researchers from many areas during} the last years.It provides reliable and clear results {thanks to the} advantages of different types of preference functions.{In this paper, we incorporate the benefits of $q$-rung orthopair fuzzy set (for short, $q$-ROFS) in this strategy of solution.} {This model, $q$-ROFS, is a generalized form of Pythagorean fuzzy set (PFS)}, as it {broadens} the space of acceptable orthopairs and {has} an ability to deal with more elaborate and vague information.The technique of {our extension of the} PROMETHEE method {uses} $q$-rung orthopair fuzzy numbers to {render} the ratings of alternatives, {which allows us} to express {uncertain and vague information more accurately}.The usual criterion preference function {has} been used to measure the preferences of {the} alternatives.A partial ordering of alternatives is obtained by considering the outgoing and incoming flows of alternatives, which is known as PROMETHEE I.Furthermore, a complete ordering is accomplished by taking into account the procedure of PROMETHEE II.As a numerical {exercise, we consider the selection of a contractor for a construction project}.{A full analysis} is performed {to illustrate the application of the technique that stems from our approach}.{Then we compare the results that we obtain with the results from existing approaches, including $q$-rung orthopair fuzzy ELECTRE, $q$-rung orthopair fuzzy TOPSIS, $q$-rung orthopair fuzzy VIKOR and $q$-rung orthopair fuzzy aggregation operators.In this way the accuracy and effectiveness of the presented work is conclusively validated}.Robust stabilization of uncertain rectangular singular fractional order T-S fuzzy systems with the fractional order 0 < α < 1
https://ijfs.usb.ac.ir/article_6260.html
This paper presents a novel method to investigate the robust stabilization problem of uncertain rectangular singular fractional order Takagi-Sugeno (T-S) fuzzy systems with the fractional order 0 &lt; &alpha; &lt; 1. Firstly, the uncertain rectangular singular fractional order T-S fuzzy system is transformed into an augmented uncertain square singular fractional order T-S fuzzy system by designing a new T-S fuzzy dynamic compensator. Secondly, a sufficient condition in the form of linear matrix inequalities (LMI) is obtained for the robust stabilization of the uncertain rectangular singular fractional order T-S fuzzy system. Finally, a numerical example is given to verify the effectiveness of the results proposed.Topological structures induced by L-fuzzifying approximation operators
https://ijfs.usb.ac.ir/article_6261.html
This paper further studies topological structures induced by L-fuzzifying approximation operators, where L denotes a completely distributive De Morgan algebra. Firstly, the Alexandrov topologies induced by L-fuzzy relations are investigated with respect to L-fuzzifying approximation operators. Especially, the relationships among those Alexandrov topologies are discussed. Secondly, pseudo-similarity sets of L-fuzzy relations are proposed based on the Alexandrov topologies induced by upper and lower L-fuzzifying approximation operators. Meanwhile, the properties of pseudo-similarity sets are discussed, where some examples are presented to show the differences between similarity set of fuzzy relations and pseudo-similarity set of L-fuzzy relations.A novel defuzzification approach of type-2 fuzzy variable to solving matrix games: An application to plastic ban problem
https://ijfs.usb.ac.ir/article_6262.html
The novelty of the type-2 fuzzy variable (T2FV) is its fuzzy secondary possibility distribution. Here, we aim to formulate a new defuzzification model of triangular type-2 fuzzy variables (TT2FVs) based on type reduction. We also show its superiority compared to other existing models by providing some numerical examples. Later, we develop a matrix game problem in the type-2 fuzzy environment to show the applicability of the proposed defuzzification method in a real-world situation. Based on this defuzzification method, two crisp linear programming models are derived, which are subsequently solved by the simplex method using LINGO 17.0 software. Also, the expected pay-off for the maximizing player is calculated as TT2FV, which is desirable. Finally, to show the validity and applicability, the proposed methodology is illustrated with the plastic ban problem.On homogeneity of the smallest semicopula-based universal integral in the class of pseudo-continuous semicopulas
https://ijfs.usb.ac.ir/article_6263.html
In this paper, we introduce the concept of pseudo-continuous semicopula. We show its relationship with continuity in the second variable and provide a characterization of all semicopulas S such that the smallest semicopula-based universal integral is S-homogeneous. This completely solves Open problem 2.29 proposed by J. Borzov&acute;a-Moln&acute;arov&acute;a et al. in the paper [2].A novel method for multi-objective design optimization based on fuzzy systems
https://ijfs.usb.ac.ir/article_6264.html
A novel strategy to design optimization is expressed using the fuzzy preference function concept. This method efficiently uses the designer&rsquo;s experiences by preference functions and it is also able to transform a constrained multi-objective optimization problem into an unconstrained single-objective optimization problem. These two issues are the most important features of the proposed method which using them, you can achieve a more practical solution in less time. To implement the proposed method, two design optimizations of an unmanned aerial vehicle are considered which are: deterministic and non-deterministic optimizations. The optimization problem in this paper is a constrained multiobjective problem that with attention to the ability of genetic algorithm, this algorithm is selected as the optimizer. Uncertainties are considered and the Monte Carlo simulation (MCS) method is used for uncertainties modeling. The obtained results show a good performance of this technique in achieving optimal and robust solutions.Persian-translation Vol 18-No 5-october 2021
https://ijfs.usb.ac.ir/article_6265.html
(2012-6329) Adaptive control design for fixed-time synchronization of fuzzy stochastic cellular neural networks with discrete and distributed delay
https://ijfs.usb.ac.ir/article_6120.html
This paper studies the fixed-time synchronization problem of fuzzy stochastic cellular neural networks (FSCNNs) with discrete and distributed delay. Compared with the finite-time synchronization in the existing literature, the fixed-time synchronization of FSCNNs is studied for the first time, and the convergence time obtained does not depend on the upper bound of the initial value of the system. In addition, two kinds of control are designed, one is feedback control and the other is adaptive control. Besides, it is the first time to achieve fixed-time synchronization of FSCNNs via adaptive control. Finally, two numerical examples are also proposed to illustrate the practicability and validity of the results we proposed.(2008-6104) Calculation of centroid of high dimensional fuzzy number and application
https://ijfs.usb.ac.ir/article_6161.html
In this paper, the conception of centroid of n-dimensional fuzzy number is introduced viaregarding its membership function as the density function on its support set, and some properties of it are obtained. Compared with the mean of the multi dimensional fuzzy number, the centroid takes into account the overall relationship between the edge membership functions of the membership function of the multi dimensional fuzzy number. Therefore, it can approximate (characterize) the fuzzy number more objectively and reasonably than using the mean of the multi dimensional fuzzy number. The most important work of this paper is that for two special kinds of multi dimensional fuzzy numbers (fuzzy n-cell numbers and fuzzy n-ellipsoid numbers), we respectively give calculation formulas, which can be used conveniently in application since the formulas are based on a definite integral of the level set functions of the multi dimensional fuzzy number on the unit interval [0,1], rather than the multiple integral of the membership function of the multi dimensional fuzzy number itself on its support set. Then, by using the calculation formulas, we obtain another special property of the centroid for fuzzy n-cell number and fuzzy n-ellipsoid number. Finally, as an example of application, by using the centroid of multi dimensional fuzzy number, we define a fuzzy order on n-dimensional fuzzy number space, which can be used to rank uncertain or imprecise multichannel digital information.(2008-6077) Idempotent uninorms and nullnorms on bounded posets
https://ijfs.usb.ac.ir/article_6162.html
The paper deals with uninorms and nullnorms as basic semi-group&nbsp; operations which are commutative and monotone (increasing). These operations were first introduced on the unit interval and later generalized to bounded lattices. In [Kalina 2019] they were introduced on bounded posets. This contribution is a generalization and extension of the results in [Kalina 2019]. Some necessary and some sufficient conditions for the existence of idempotent uninorms and idempotent nullnorms&nbsp; on bounded posets are studied. Finally, some application examples are provided.(2007-6024) Ranking of generalized fuzzy numbers based on accuracy of comparison
https://ijfs.usb.ac.ir/article_6164.html
Ranking generalized fuzzy numbers plays an important role in many applied models and, in particular, decision-making procedures. In ranking process of two generalized fuzzy numbers, it is natural to compare the sets of values in support of two the generalised fuzzy numbers. Accordingly, the comparison of a real number and a generalised fuzzy number as well as two generalised fuzzy numbers have to be considered. On the other hand, it is seen that a definitive process of comparison of a real number and a generalised fuzzy number, as well as two generalised fuzzy numbers, is not possible. So in this study, a method for comparing a real number and a generalised fuzzy number with a degree of accuracy (between a zero and one) is defined and then the method is generalized to compare two generalised fuzzy numbers. In general, an index to rank a real number and generalised fuzzy number is constructed. Eventually, this index is extended to rank two generalised fuzzy numbers based on the concept of accuracy of comparison. The advantage of our method is that it can compare two generalised fuzzy numbers with an accuracy of comparison. Also, a definition is introduced to make a definitive comparison. Finally, the proposed method is illustrated by some numerical examples.(2011-6313) End-point linear functions
https://ijfs.usb.ac.ir/article_6165.html
Positive homogeneity is represented as a constraint 0-homogeneity and generalized into z-homogeneity, called also z-end point linearity. Several special z-homogeneous aggregation functions are studied, in particular semicopulas, quasi-copulas, copulas, overlap functions, etc.(1912-5600) Type 2 adaptive fuzzy control approach applied to variable speed DFIG based wind turbines with MPPT algorithm
https://ijfs.usb.ac.ir/article_6166.html
In this research, a Type 2 adaptive fuzzy controller approach is formulated and designed to be applied to variable speed doubly fed induction generator-based wind turbines directly connected to the grid. It brings this study to evaluate the whole operation of the system to capture the highest rate of power in the wind turbines. The controlling approach is considered to keep the stator reactive power to the ideal value. In contrast to the other researches, here the controlling technique is developed through the nonlinear systems.&nbsp; By the aim of making progress in system operation, in contrast with the Type 1 adaptive fuzzy system, type two adaptive fuzzy theory is proposed to approximate a large number of uncertainties and the dynamic nonlinearities, exists in tracking errors which may limit the system performance. Feedback linearization control approach helps us to algebraically alter the system into a linearized plant. Thanks to the Lyapunov theorem, the introduced type two adaptive fuzzy approach is proved to meet the uniformly ultimately boundness (UUB) property. On the other hand, it results better tracking function. The simulation outputs represent that the proposed technique is robust enough in presence of parameter variations and unstructured uncertainties.(2011-6279) Structural topology in a category
https://ijfs.usb.ac.ir/article_6167.html
Several fuzzy topologies are defined and studied by different authors. In this article, we unify five of the most common fuzzy topologies existing in the literature, as well as the standard topology. This is done by introducing the notion of structural topology on objects in a category and proving that topologies on a set as well as fuzzy topologies on fuzzy sets and fuzzy topologies on fuzzy subsets are all structural topologies. We also introduce the notion of structural continuity and we show that the fuzzy continuity defined in the literature in all the above mentioned cases, as well as the standard continuity are structural.(2012-6338) Design and Analysis of Acceptance Sampling Plans Based on Intuitionistic Fuzzy Linguistic Terms
https://ijfs.usb.ac.ir/article_6211.html
Acceptance sampling plans (ASPs) offer inspection of a small set of items from a lot within a predefined plan to procure a certain output quality level with minimum cost in terms of time, effort, and damage to the inspected items. Although traditional ASPs use crisp plan parameters, quality characteristics of the incoming items or human evaluations about inspection process may contain uncertainties and may not always be defined as crisp values in real life problems. The fuzzy set theory (FST) is one of the most popular techniques to model these uncertainties by defining plan parameters as fuzzy numbers. Despite the advantages, traditional fuzzy sets are not flexible enough to model all kind of uncertainties. For example, it has some disadvantages because of defining the status of any item based on defectiveness or non-defectiveness conditions and presuming the parts as non-defective whose defectiveness is not indeed determined. New extensions of FST can improve the quality of uncertainty modeling of ASPs. Intuitionistic Fuzzy Sets (IFSs) allow slackness for non-determination about the membership and give more sensitive modeling opportunity in human-related evaluations by the help of this ability. Since the inspection procedure of the ASPs depends on human-related judgements, IFSs have been used to define the defectiveness degree of the items in this study. ASPs based on interval-valued IFSs (IVIFSs) have also been designed and some characteristic functions of ASPs, such as acceptance probability ($P_a$), average sample number ($ASN$) and average total inspection ($ATI$) have been reformulated. Intuitionistic binomial and Poisson distributions have been defined to be able to formulate the ASPs. Additionally, the defectiveness of the items has been represented by using linguistic terms to overcome the difficulty of quantifying the verbal evaluation results as numerical measures. The $\alpha$-cut technique has been combined with the linguistic approach to allow defining with multiple $\alpha$ values for different product segments. Finally, some numerical examples have been presented to analyze the effectiveness of proposed ASPs and discuss the obtained results.( 2005-5897) Different classes ratio and Laplace summation operator based intuitionistic fuzzy rough attribute selection
https://ijfs.usb.ac.ir/article_6212.html
In real-world data deluge, due to insignificant information and high dimension, irrelevant and redundant attributes reduce the ability of experts both in predictive accuracy and speed, respectively. Attribute selection is the notion of selecting those attributes that are essential as well as enough to specify the target knowledge preferably. Fuzzy rough set-based approaches play a crucial role in selecting relevant and less redundant attributes from a high-dimensional dataset. Intuitionistic fuzzy set-based approaches can handle uncertainty as it gives an additional degree of freedom when compared to fuzzy approaches. So, it has a more flexible and practical ability to deal with vagueness and noise available in the information system. In this paper, we introduce two new robust approaches for attribute selection based on intuitionistic fuzzy rough set theory using the concepts of Different Classes ratio and Laplace Summation operator. Firstly, Different Classes ratio and Laplace Summation operator based lower andupper approximations are established based on intuitionistic fuzzy rough set concept. Moreover, we present algorithms and illustrative examples for a better understanding of our approaches. Finally, experimental analysis is performed on some real-valued datasets for attribute selection and classification accuracies.( 2008-6119 ) A fuzzy approach to review-based recommendation: Design and optimization of a fuzzy classification scheme based on implicit features of textual reviews
https://ijfs.usb.ac.ir/article_6222.html
In the design of recommender systems, it is believed that the set of reviews written by a user can somehow reveal his/her interests, and the content of an item can also be implied from its corresponding reviews. The present study attempts to model both the users and the items via extracting key information from the existing textual reviews. Based on this information, a fuzzy rule-based classifier is designed and tuned, which aims to predict whether a typical user will be interested in a typical item or not. For this purpose, the set of all reviews belonging to a user are mapped to a vector representing the user's interests. Similarly, the set of reviews written by different users over an item are merged and mapped to a vector representing the item. By conjoining these two vectors, a longer vector is obtained which will be used as the input of the classifier. To optimize the classifier, an adaptive approach is suggested and rule-weight learning is carried out, accordingly. The performance of the proposed fuzzy recommender system was evaluated on the Amazon dataset. Experimental results narrate from the promising classification ability of the proposed recommender system compared to state of the art.(2008-6132) A novel fuzzy sliding mode control approach for chaotic systems
https://ijfs.usb.ac.ir/article_6248.html
The purpose of this paper is to study the stabilization problem for a class of uncertain chaotic systems against unknown dynamics and disturbances, based on fuzzy sliding mode controller approaches. To fulfill this aim, the first- and the second-orders sliding mode controllers and an adaptive variable universe fuzzy sliding mode controller are combined to a set of linguistic rules, to design some novel approaches for improving the performance of the control action and eliminating the chattering issue. The stability analysis of the closed-loop system is proved via the Lyapunov stability theorem, and also the convergence of the tracking error to zero in finite-time is guaranteed. The new proposed control laws contribute the control actions to outperform the conventional one in terms of chattering reduction and elimination, along with lessening in the reaching time. Moreover, some numerical simulations are provided to depict that the proposed control laws are not only robust with respect to uncertainties and external disturbances, which lead the system to the desired state, but also can significantly eliminate the chattering effect.(2102-6503) Fuzzy arithmetic with product t-norm
https://ijfs.usb.ac.ir/article_6250.html
Fuzzy arithmetic performed with the product t-norm is the focus of this paper. The subject is handled from both practical and theoretical perspectives. Explicit formulas for product-sum and product-multiplication of triangular fuzzy numbers are obtained. These formulas can effectively replace the computational methods proposed so far. The issue that these operations are not shape preserving is solved by the presentation of appropriate approximations. Finally, the product arithmetic is compared in detail to the arithmetic performed with the boundary t-norms, namely the minimum and drastic sum.(2011-6272) Design and analysis of process capability indices cpm and cpmk by neutrosophic sets
https://ijfs.usb.ac.ir/article_6251.html
Process capability indices (PCIs) have been widely used to analyze capability of the process that measures how the customer expectations have been conformed. Two of the well-known PCIs, named indices $ C_{pm} $ and $ C_{pmk} $ have been developed to consider customers' ideal value that called target value ($ T $). Although, these indices have similar features of the well-known indices $ C_{p} $ and $ C_{pk} $, one of the most important differences is to consider \textit{T}. In real case problems, we need to add some uncertainties related with human's evaluations into process capability analysis (PCA). One of the uncertainty modelling methods called neutrosophic sets (NSs), have an important role in modeling uncertainty based on incomplete and inconsistent information. For this aim, the PCIs have been designed by using NSs to manage the uncertainties of systems and to increase sensitiveness, flexibility and to obtain more detailed results of PCA in this paper. For this aim, the indices $ C_{pm} $ and $ C_{pmk} $ have been performed and re-designed by using single valued neutrosophic numbers for the first time in the literature. Additionally, specification limits (SLs) have been re-considered by using NSs. The neutrosophic state of the SLs provide us to have more knowledge about the process and easily applied for engineering problems that includes uncertainty. Finally, the neutrosophic process capability indices (NPCIs) $ \widetilde{\dddot{C}}_{pm}$ and $ \widetilde{\dddot{C}}_{pmk} $ have been obtained and the main formulas of them have been produced. Additionally, the proposed $ \widetilde{\dddot{C}}_{pm} $ and $ \widetilde{\dddot{C}}_{pmk} $ have been applied on real case studies from manufacturing industry. The obtained results show that the indices $ \widetilde{\dddot{C}}_{pm} $ and $ \widetilde{\dddot{C}}_{pmk} $ include more informative and flexible results to evaluate capability of process.