Non-membership Functions Research Articles

A novel interactive intuitionistic fuzzy approach to find satisfactory Pareto-optimal solutions for general bilevel multi-followers programming

This paper focuses on the approach for solving the satisfactory Pareto-optimal solutions of the general form of bilevel multi-followers programming(BMFP), including the BMFP with partial shared variables among followers, the so-called referential-uncooperative BMFP, the uncooperative BMFP, and the cooperative BMFP. Without losing generality, the BMFP with partial shared variables among followers will be taken as a representative to explain the universal applicability of the proposed generic interactive intuitionistic fuzzy method. Based on the theory of intuitionistic fuzzy set, the new definition of satisfactory Pareto-optimal solutions is introduced by means of membership function, non-membership function, and score function. In order to obtain satisfactory Pareto-optimal solutions of BMFP with partial shared variables among followers, a prior minimization optimization problem and a series of corresponding auxiliary maximization optimization problems are formulated, and we strictly prove that the optimal solution of the prior minimization optimization problem is not necessarily the satisfactory Pareto-optimal solutions, but at least one satisfactory Pareto optimal solution can be derived by further solving these auxiliary maximization optimization problems. Then, a detailed interactive intuitionistic fuzzy algorithm framework that consider both the balance of satisfactory degree between the leader and the followers and the balance of satisfactory degree among all followers is proposed. In the part of numerical experiments, four examples(including a linear BMFP problem with partial shared variables among followers, a BMFP model of a road network problem, an uncooperative BMFP problem, and a nonlinear BMFP problem with partial shared variables among followers) are illustrated to demonstrate the applicability, effectiveness and superior performance of the proposed technique.

An Investigation of Linear Diophantine Fuzzy Nonlinear Fractional Programming Problems

The linear Diophantine fuzzy set notion is the main foundation of the interactive method of tackling nonlinear fractional programming problems that is presented in this research. When the decision maker (DM) defines the degree α of α level sets, the max-min problem is solved in this interactive technique using Zimmermann’s min operator method. By using the updating technique of degree α, we can solve DM from the set of α-cut optimal solutions based on the membership function and non-membership function. Fuzzy numbers based on α-cut analysis bestowing the degree α given by DM can first be used to classify fuzzy Diophantine inside the coefficients. After this, a crisp multi-objective non-linear fractional programming problem (MONLFPP) is created from a Diophantine fuzzy nonlinear programming problem (DFNLFPP). Additionally, the MONLFPP can be reduced to a single-objective nonlinear programming problem (NLPP) using the idea of fuzzy mathematical programming, which can then be solved using any suitable NLPP algorithm. The suggested approach is demonstrated using a numerical example.

A note on direct product of complex intuitionistic fuzzy subfield

This paper presents the concepts of a complex intuitionistic fuzzy subfield (CIFSF) and the direct product of a complex intuitionistic fuzzy subfield which is generalized from the concept of a complex fuzzy subfield by adding the notion of intuitionistic fuzzy into a complex fuzzy subfield. The main contribution and originality of this research are adding the non-membership term to the definition of a complex fuzzy subfield that assigns for any element a complex-valued grade. We expand the complex fuzzy subfield and obtain a new structure called CIFSF. This new concept is innovative in that it may attain a wider range of values for both membership and non-membership functions where these functions are expanded to the unit disc in the complex plane. Furthermore, we discuss that the direct product of two CIFSFs is CIFSF, and some related properties are investigated. In addition, we present the definition of necessity and possibility operators on the direct product of CIFSF, and some associated theorems are given. Finally, we propose the level subsets of the direct product of two complex intuitionistic fuzzy subsets of a field and prove that the level subset of the direct product of two CIFSFs is a subfield and discuss some related results.

Performance evaluation of bagasse-based cogeneration power generation plant utilizing IFLT, IF-FMEA and IF-TOPSIS approaches

PurposeTo meet energy demand and tackle the challenges posed by global warming, Bagasse-based Cogeneration Power Generation (BCPG) plant in sugar mills have tremendous potential due to large-scale supply of renewable fuel called bagasse. To meet this goal, an integrated framework has been proposed for analyzing performance issues of BCPG.Design/methodology/approachIntuitionistic Fuzzy Lambda-Tau (IFLT) approach was implemented to compute various reliability parameters. Intuitionistic Fuzzy Failure Mode and Effect Analysis (IF-FMEA) approach has been implemented for studying risk issues results in decrease in plant's availability. Moreover, IF- Technique for Order Performance by Similarity to Ideal Solution (IF-TOPSIS) is implemented to verify accuracy of IF-FMEA approach.FindingsFor membership and non-membership functions, availability decreases to 0.0006% and 0.0020% respectively for spread ±15% to ±30%, and further decreases to 0.0127% and 0.0221% for spread ±30% to ±45%. Under risk assessment failure causes namely Storage tank (ST3), Valve (VL6), Transfer pump (TF8), Deaerator tank (DT11), High pressure heater and economiser (HP15), Boiler drum and super heater (BS22), Forced draft and Secondary air fan (FS25), Air preheater (AH29) and Furnace (FR31) with Intuitionistic Fuzzy Hybrid Weighted Euclidean Distance (IFHWED) based output scores – 0.8988, 0.9752, 0.9400, 0.8988, 0.9267, 1.1131, 1.0039, 0.8185, 1.0604 were identified as the most critical failure causes.Research limitations/implicationsReliability and risk analysis results derived from IFLT and IF-FMEA approaches respectively, to address the performance issues of BCPG is based on the quantitative and qualitative data collected from the industrial experts and maintenance log book. Moreover, to take care of hesitation in expert's knowledge, IF theory-based concept is incorporated so as to achieve more accuracy in analysis results. Reliability and risk analysis results together will be helpful in analyzing the performance characteristics and diagnosis of critical failure causes, which will minimize frequent failure in BCPG.Practical implicationsThe framework will help plant managers to frame optimal maintenance policy in order to enhance the operational aspects of the considered unit. Moreover, the accurate and early detection of failure causes will also help managers to take prudent decision for smooth operation of plant.Social implicationsThe results obtained ensure continuous operation of plant by utilizing the bagasse as fuel in boiler and also mitigate the wastages of fuel. If this bagasse (green fuel) is not properly utilized, there remains a dependency on coal-based power plants to meet the power demand. The results obtained are useful for decreasing dependency on coal, and promoting bagasse as the green, and alternative fuel, the emission by burning of these fuels are not harmful for environment and thereby contribute in preventing the environment from harmful effect of GHGs gases.Originality/valueIFLT approach has been implemented to develop reliability modeling equations of the BCPG unit, and furthermore to compute various reliability parameters for both membership and non-membership function. The ranking results of IF-FMEA are compared to IF-TOPSIS approach. Sensitivity analysis is done to check stability of proposed framework.

Intuitionistic SIR Technique with Double Parameters to Detect the Operative Vaccine of COVID-19

Since 2019, COVID disease threats the world. It originated in China, and due to its easy spread characteristics, it spreads quickly among all the countries. As the world faced a new kind of virus, the government could not control its spread, and the fatality rate had also hugely increased. In 2021, the vaccine for the virus was introduced by various countries, which aids in controlling the spread and mortality rate. The mutant form of this virus exhibits high efficiency. So, this study analyzes the effective of COVID vaccines released after 2020 against all COVID variants through the fuzzy superiority and inferiority ranking (SIR) decision-making method. The vaccines and variants are analyzed by the [Formula: see text] matrix pattern under the fuzzy SIR-integrated weight method. The disease variants are taken as criteria and that are assigned integrated weight based on subjective, objective weight approaches. The intuitionistic fuzzy set-double parameter (IFS-DP) handles the vagueness by strictly including only membership and non-membership functions with the condition [Formula: see text]. As a result, the vaccines Moderna and Covishield are attained top rank and have a high likelihood to combat all the COVID variants compared to other vaccines. The reached outcomes are corroborated through sensitivity and comparative analysis. The sensitivity and comparative studies are measured by increasing the order of IFS-DP and extant fuzzy MCDM methods, respectively. The comparison of these results with those obtained from the proposed method is similar to the sensitivity study, along with minor variations observed in the comparative study. Therefore, the proposed method generates an efficient outcome. Moreover, the testing hypothesis technique is applied to check the efficiency of proposed method.

Multi-view intuitionistic fuzzy support vector machines with insensitive pinball loss for classification of noisy data

Multi-view support vector machines (MvSVMs) have been widely used to solve multi-view classification problems. However, the conventional MvSVMs often overlook the presence of noise and outliers that commonly exist in the original data. In this paper, we propose two novel multi-view intuitionistic fuzzy support vector machines with insensitive pinball loss that can not only handle the general multi-view classification problems but also be robust to noisy data. In the proposed convex optimization models, the pinball loss is incorporated into the multi-view learning, which enables the maximization of the quantization distance. Moreover, to utilize multi-view information more effectively, the intuitionistic fuzzy score is introduced to assign a weight to each multi-view sample. The intuitionistic fuzzy score combines the membership and non-membership functions, which provides an efficient mechanism to assign weights to the multi-view samples. Further, we provide a discussion of the proposed models with the state-of-the-art technologies. Experiments are conducted on a series of datasets, and the results show that the proposed convex models outperform several state-of-the-art models. The code is available at 1.

Intuitionistic Trapezoidal Fuzzy based Aggregation Operator: Applications in Medical Diagnosis

Since the inception of fuzzy sets given by Zadeh, uncertainty arises due to partial information or imprecise information has been measured. The generalized version of fuzzy sets has been introduced by Atanassov, known as intuitionistic fuzzy sets (IFSs), which have wide applications in decision making processes and consider both membership and non-membership functions. A generalized fuzzy based decision making method with aggregation operator has been discussed with application in medical diagnosis. In this paper, IF based method have been discussed for the application of the diagnosis of the type of child cancer.

A new ranking method for trapezoidal intuitionistic fuzzy numbers and its application to multi-criteria decision making

The ranking of intuitionistic fuzzy numbers is paramount in the decision making process in a fuzzy and uncertain environment. In this paper, a new ranking function is defined, which is based on Robust’s ranking index of the membership function and the non-membership function of trapezoidal intuitionistic fuzzy numbers. The mentioned function also incorporates a parameter for the attitude of the decision factors. The given method is illustrated through several numerical examples and is studied in comparison to other already-existent methods. Starting from the new classification method, an algorithm for solving fuzzy multi-criteria decision-making (MCDM) problems is proposed. The application of said algorithm implies accepting the subjectivity of the deciding factors, and offers a clear perspective on the way in which the subjective attitude influences the decision-making process. Finally, a MCDM problem is solved to outline the advantages of the algorithm proposed in this paper.

On fundamental algebraic characterizations of complex intuitionistic <i>Q</i>-fuzzy subfield

<abstract> <p>The main objective of this study is to propose a new notion of a complex intuitionistic $ Q $-fuzzy subfield of a field $ F $ that is developed from the concept of a complex fuzzy subfield of a field $ F $ by adding the notion of intuitionistic $ Q $-fuzzy into a complex fuzzy subfield. We establish a new structure of complex fuzzy subfields which is called complex intuitionistic $ Q $-fuzzy subfield. The most significant advantage of this addition appears to be that it broadens the scope of the investigation from membership function values to membership and non-membership function values. The range of complex fuzzy subfields is expanded to the unit disc in the complex plane for both membership and non-membership functions. Some fundamental operations, especially the intersection, union, and complement of complex intuitionistic $ Q $-fuzzy subfields are studied. We define the necessity and possibility operators on a complex intuitionistic $ Q $-fuzzy subfield. Moreover, we show that each complex intuitionistic $ Q $-fuzzy subfield generates two intuitionistic $ Q $-fuzzy subfields. Subsequently, several related theorems are proven.</p> </abstract>

Application of Intuitionistic Z-Numbers in Supplier Selection

Intuitionistic fuzzy numbers incorporate the membership and non-membership degrees. In contrast, Z-numbers consist of restriction components, with the existence of a reliability component describing the degree of certainty for the restriction. The combination of intuitionistic fuzzy numbers and Z-numbers produce a new type of fuzzy numbers, namely intuitionistic Z-numbers (IZN). The strength of IZN is their capability of better handling the uncertainty compared to Zadeh's Z-numbers since both components of Z-numbers are characterized by the membership and non-membership functions, exhibiting the degree of the hesitancy of decision-makers. This paper presents the application of such numbers in fuzzy multi-criteria decision-making problems. A decision-making model is proposed using the trapezoidal intuitionistic fuzzy power ordered weighted average as the aggregation function and the ranking function to rank the alternatives. The proposed model is then implemented in a supplier selection problem. The obtained ranking is compared to the existing models based on Z-numbers. The results show that the ranking order is slightly different from the existing models. Sensitivity analysis is performed to validate the obtained ranking. The sensitivity analysis result shows that the best supplier is obtained using the proposed model with 80% to 100% consistency despite the drastic change of criteria weights. Intuitionistic Z-numbers play a very important role in describing the uncertainty in the decision makers’ opinions in solving decision-making problems.

Complex Linear Diophantine Fuzzy Sets over AG-Groupoids with Applications in Civil Engineering

Intuitionistic fuzzy sets (IFS), Pythagorean fuzzy sets (PFS), and q-rung orthopair fuzzy sets (q-ROFS) are among those concepts which are widely used in real-world applications. However, these theories have their own limitations in terms of membership and non-membership functions, as they cannot be obtained from the whole unit plane. To overcome these restrictions, we developed the concept of a complex linear Diophantine fuzzy set (CLDFS) by generalizing the notion of a linear Diophantine fuzzy set (LDFS). This concept can be applied to real-world decision-making problems involving complex uncertain information. The main motivation behind this paper is to study the applications of CLDFS in a non-associative algebraic structure (AG-groupoid), which has received less attention as compared to associative structures. We characterize a strongly regular AG-groupoid in terms of newly developed CLDF-score left (right) ideals and CLDF-score (0,2)-ideals. Finally, we construct a novel approach to decision-making problems based on the proposed CLDF-score ideals, and some practical examples from civil engineering are considered to demonstrate the flexibility and clarity of the initiated CLDF-score ideals.

Testing the Difference of Means of Populations With Respect to Intuitionistic Fuzzy Sets

In this study, the tests of statistical hypotheses with crisp data using small samples are extended to with the membership function and the non-membership function of the intuitionistic fuzzy set. The test procedure of statistical hypotheses for means of two normally distributed populations with respect to any intuitionistic fuzzy set is proposed.

A mathematical model using temporal intuitionistic fuzzy sets

Krassimir T. Atanassov’s intuitionistic fuzzy sets (IFS), one of the extensions of fuzzy sets, have shown to be one of the most effective ways to handle ambiguity. John N. Mordeson and Davender S. Malik developed the idea of a fuzzy finite state machine. Intuitionistic fuzzy finite state machines were created by Jun as a generalisation of fuzzy finite state machines. In order to increase the uncertainty and lower the periodic functions in intuitionistic fuzzy finite state automata, new membership and non-membership functions based on transitions were introduced in this study. Also, temporal intuitionistic fuzzy automata (TIFA) were defined and used to model a pattern.

A Study on Intuitionistic Fuzzy Critical Path Problems Through Centroid Based Ranking Method

In this study the intuitionistic fuzzy version of the critical path method has been proposed to solve networking problems with uncertain activity durations. Intuitionistic fuzzy set [1] is an extension of fuzzy set theory [2] unlike fuzzy set, it focuses on degree of belonging, the degree of nonbelonging or non-membership function and the degree of hesitancy which helps the decision maker to adopt the best among the worst cases. Trapezoidal and the triangular intuitionistic fuzzy numbers are utilized to describe the uncertain activity or task durations of the project network. Here trapezoidal and triangular intuitionistic fuzzy numbers are converted into their corresponding parametric form and applying the proposed intuitionistic fuzzy arithmetic operations and a new method of ranking based on the parametric form of intuitionistic fuzzy numbers, the intuitionistic fuzzy critical path with vagueness reduced intuitionistic fuzzy completion duration of the project has been obtained. The authentication of the proposed method can be checked by comparing the obtained results with the results available in pieces of literature.

A circular intuitionistic fuzzy evaluation method based on distances from the average solution to support multiple criteria intelligent decisions involving uncertainty

This study aims to construct an evaluation method based on distances from the average solution (EDAS) based on circular intuitionistic fuzzy (C-IF) sets to support intelligent decision-making involving multiple criteria and intricate uncertainty. C-IF sets possess a resilient circular structure that can model decision-makers’ hesitancy and indecision like intuitionistic fuzzy sets and can add ambiguity to fuzzy membership and nonmembership functions like type-2 fuzzy sets. Related to the existing literature on EDAS in uncertain environments, a new interpretation is needed so that the decision assistance procedure can exploit higher-order fuzzy membership functions as well as in type-2 fuzzy settings, and can address decision hesitancy as well as in IF settings. However, EDAS has not been studied to discuss its applicability in C-IF circumstances and make headway a suitable extension model, which forms a research question worth exploring. To address this issue, this study endeavors to construct a C-IF EDAS methodology and propound novel concepts and measurements that can improve the core techniques of traditional EDAS, enabling them to handle complicated C-IF information and treat highly equivocal and indistinct decision-making tasks. The C-IF analytical procedure for decision-making is mainly built upon two representations from semantic assessments, namely C-IF assessment ratings and C-IF importance weights. This study puts forth the concept of aggressive and conservative estimates to identify anchored datum-based measurements that provide new interpretations of positive distance and negative distance concerning the C-IF average solution; besides this study evolves an extended EDAS decision rule in C-IF circumstances. Based on an aggregated operation and a membership score function, anchored datum-based (normalized) weighted sums about positive/negative distances can be rendered to produce a joint appraisal score for a compromise ranking of available alternatives. To corroborate the applicability of the established notions and measurements, this study utilizes the C-IF EDAS model and decision procedure to tackle a realistic site selection issue for a pandemic hospital within C-IF environments. The suitability, robustness, and flexibility of the developed C-IF EDAS methodology are confirmed through some comparative analyses and experimental studies. First, this study verifies the correctness and trustworthiness of the executive outcomes from the viewpoints of membership, nonmembership, and vague score functions. Next, this study confirms the effectuality and rationality of the proposed C-IF EDAS by comprehensive comparison with several related methods. Furthermore, this study suggests a flexible way of setting appraisal and distance parameters in the C-IF EDAS to facilitate intelligent decision-making in reality. Finally, some conclusive discussions of the C-IF EDAS solution are systematically investigated to reveal its practicality and application merits in the realm of multiple criteria decision analysis with information uncertainty; it is also confirmed that the C-IF set is well suited as a fundamental of problem modeling within complex decision-making environments.

Topological analysis of intuitionistic fuzzy distance measures with applications in classification and clustering

The distance measure is a vital classifier used to solve classification and clustering problems in a metric space. In this paper, we discussed the types of distance measures and elaborated that they generate uniquely shaped balls. Due to their balls, they cannot guarantee a satisfactory classification outcome for a given dataset. To address this limitation, researchers used intuitionistic fuzzy sets (IFSs) to lose the boundary of belongingness of data points in a given metric space with respect to the chosen membership and non-membership functions and accomplished some novel intuitionistic fuzzy distance measures (IFDMs). IFDMs are successfully used in various applications; however, they also predict counterintuitive cases. To address this issue, in this paper, we conducted a topological analysis of intuitionistic fuzzy distance measures. To do that, we constructed three categories of distance measures; Type 1 distance measures, Type 2 distance measures, and Intuitionistic fuzzy distance measures (in this paper, we called them Intuitive Distance Measures (IDMs)). We further sub-categorized Intuitive Distance Measures into Type 1 Intuitive Distance Measures (T1IDMs) and Type 2 Intuitive Distance Measures (T2IDMs). We established a homeomorphic relation between the topological spaces generated by the T1IDMs and T2IDMs. Using the proposed homeomorphic relation and proving the necessary lemma and theorems, we presented the type 2 distance measures of well-known normable T1IDMs. We applied the proposed type 2 variants to solve some classification and clustering problems of machine learning. The result analysis shows that the proposed type 2 variants overcome the drawbacks of their type 1 counterparts.

Application of Fuzzy Neutrosophic Cone in Decision Making

Aims: This article deals with a new decision-making process under a neutrosophic fuzzy environment. First of all, we develop various types of neutrosophic set by means of neutrosophic cones. In fact, this set has been developed from the general equation of second degree in the field of classical geometry. Considering the neutrosophic components “true membership”, the “falsity membership” and the “indeterminacy” as the three variables of three-dimensional rectangular axes we develop various types of cones like structures of the traditional neutrosophic set and hence a new defuzzification method. Background: Fuzzy set has some limitations in its domain [0,1] to describe real-life decisionmaking problems. The problem of difficulties lies in the variation of lower and upper bound and also the single valued logic (membership function only) systems. In reality, three valued logics (membership function, non-membership function and indeterminacy) have been established in the name of Neutrosophic logic/sets, and two valued logics (membership and non-membership functions) have developed in the name of Intuitionistic fuzzy logic/sets. In three valued logic system, the concepts of negation are now a growing subject of any group decision making problems. However, to draw a clear estimation of a neutrosophic decision has not yet been studied by modern researchers. Objective: Various kinds of new establishments of the Neutrosophic set have been studied from the algebraic point of view, along with some polynomial structures. We have seen that; no finite geometric structures have been developed yet to qualify the real-world problems. Method: We consider the three components of a neutrosophic set as the variables of threedimensional geometry. Since, the decisions are compact and constructive, we may consider the convex neutrosophic cone for analyzing single/ multiple group decision making problems. Result: Various definitions are made over the cone- fundamentals using non-standard neutrosophic set in the domain [−1,1] × [−1,1] × [−1,1]. Then, we studied the constructions of several expressions/ functions of neutrosophic cones, such as reciprocal cone, and enveloping cone via a novel thinking process. Then using some examples, we have developed a new ranking method along with their geometric structures exclusively. Conclusion: In this changing world, the nature of decision-making behaviors is also changing rapidly. So, the need of establishing new concepts is an emerging area of research. However, more attention is required in discussing such vital issues in near future. The proposed approach may be applied to the decision-making problems of global issues also.

On Some Extension of Intuitionistic Fuzzy Synthetic Measures for Two Reference Points and Entropy Weights.

In this paper, a novel Double Intuitionistic Fuzzy Synthetic Measure (DIFSM), based on intuitionistic fuzzy values for handling multi-criteria decision-making problems used to rank alternatives, is presented. In the studies, intuitionistic fuzzy sets (IFSs) represented uncertain, imprecise information or human judgment. The intuitionistic fuzzy sets can also reflect the approval, rejection, and hesitation of decision-makers. The degrees of satisfiability and non-satisfiability and uncertainty of each alternative with respect to a set of criteria are described by membership functions, non-membership functions, and hesitancy indexes, respectively. The aggregation algorithm DIFSM is inspired by Hellwig’s method based on two reference points: ideal point (pattern) and anti-ideal point (anti-pattern), measuring distances between the alternative and ideal point and distance between the ideal and anti-ideal point. The proposed methods take into consideration the entropy-based weights of criteria. An illustrative example is given to demonstrate the practicality and effectiveness of the proposed approach. Additionally, the comparative analysis results, using the DIFSM and the Intuitionistic Fuzzy TOPSIS-based framework, are presented.

Contribution to decision-making in the big data industry based on the multiparametric similarity measure for Pythagorean fuzzy sets

Abstract Big Data allows analysing and assessing all human production types with its 5Vs, which are Volume, Velocity, Variety, Veracity and Value. Big Data is useful to improve decision-making to adjust it better to market demand, specifically selection of supplier that is an important link to optimize the logistic chain of enterprises. In this case, leadership or decider is ahead one serious complex problem, inexact and fuzzy. Pythagorean fuzzy set (PFS) is disposing the indeterminacy data by the membership and the nonmembership functions; it is a generalization of the intuitionist fuzzy set when the last set is limited. First, some results for PFSs are displaying in this study as particular cases and generalization of some binary operations. After, an improved score function of Pythagorean fuzzy number is proposed to avoid the comparison problem in practice. In addition, an existing approach exploring the combined alternatives weight to settle Pythagorean fuzzy issue by multi-parametric similarity measure is applied with the new proposed score function to selection of supplier issue with five serious criteria as a Big Data industry decision-making problem in economic environment. Finally, a comparison of the presented method with some existing approaches has been executed in the light of counterintuitive phenomena for validating its advantages.

Solving Multi-Objective Multilevel Programming problems using two-phase Intuitionistic Fuzzy Goal Programming method

This paper presents a two-phase intuitionistic fuzzy goal programming (two-phase IFGP) algorithm to solve Multi-Objective Multilevel Programming (MO-MLP) problems. The coefficient of each objective and constraint function is assumed to be triangular intuitionistic fuzzy parameters and the crisp MO-MLP problems are obtained using the accuracy function method. To avoid decision lock, the top levels set tolerance limits for decision variables to control the lower levels. The problem is modeled in the intuitionistic fuzzy environment using membership and non-membership functions for each objective function at all levels and decision variables controlled by the top levels. Then, we proposed an IFGP algorithm to achieve the highest degree of each membership and non-membership goal by minimizing unwanted deviational variables and generating compensatory solutions for all decision-makers at all levels. Moreover, in the proposed approach, two-phase IFGP is modeled to yield a compromise solution that satisfies both the MN-Pareto optimal solution and the Pareto optimal solution at each level. Also, verification of the proposed method is discussed with numerical examples.