Multi-polar Information Research Articles

An efficient approach to study multi-polar fuzzy ideals of semirings

The multi polar fuzzy (m-PF) set has an extensive range of implementations in real world problems related to the multi-polar information, multi-index and multi-attributes data. This paper introduces innovative extensions to algebraic structures. We present the definitions and some important results of m-polar fuzzy subsemirings (m-PFSSs), m-polar fuzzy ideals (m-PFIs), m-polar fuzzy generalized bi-ideals (m-PFGBIs), m-polar fuzzy bi-ideals (m-PFBIs) and m-polar fuzzy quasi-ideals (m-PFQIs) in semirings. The main contributions of the paper include the derivation and proof of key theorems that shed light on the algebraic interplay and computational aspects of m-polar fuzzy ideals (m-PFIs), m-polar fuzzy generalized bi-ideals (m-PFGBIs), m-polar fuzzy bi-ideals (m-PFBIs) and m-polar fuzzy quasi-ideals (m-PFQIs) in semirings along with examples. Moreover, this paper deals with several important properties of m-PFIs and characterizes regular and intra-regular semirings by the properties of these ideals.

Measuring efficiency of retrieval algorithms with Schweizer-Sklar information aggregation

The task of an information retrieval system (IRS) is to retrieve relevant information from large datasets efficiently. However, selecting the best algorithm for an IRS is a complex decision support system (DSS) problem. The cubic m-polar fuzzy set (CmPFS) model is proposed to address the multi-polarity issue during algorithm selection and to express multi-polar information under m-fuzzy intervals. In this study, we introduce the Schweizer-Sklar (SS) aggregation operator based on CmPFS and develop various new aggregation operators using CmPFS and SS t-norm and t-conorm. New operations such as addition, usual multiplication, and power of two or more Cubic m-polar fuzzy Numbers (CmPFNs) based on SS t-norm and t-conorm are also defined. We also define score and accuracy functions to determine the priorities of alternatives. Four operators are proposed as follows: Cubic m-polar fuzzy Schweizer-Sklar weighted average (CmPFSSWA), Cubic m-polar fuzzy Schweizer-Sklar ordered weighted average (CmPOWA), Cubic m-polar fuzzy Schweizer-Sklar weighted geometric (CmPFSSWG), and Cubic m-polar fuzzy Schweizer-Sklar ordered weighted geometric (CmPFSSOWG). The desired qualities of these operators are explored and used to solve the DSS problem based on CmPFS. Finally, a practical application of an IRS is presented to demonstrate the effectiveness and reliability of the proposed operators.

Vectorial-Holography metasurface empowered by Orthogonality-Simplified Machine learning

Metasurfaces can provide unprecedented degree of freedom in manipulating electromagnetic waves and have been introduced to holography. Aiming to explore the full capability for information presentation, vectorial metasurface holography is sprung up, which exhibits mesmerizing capability in carrying information compared with scalar counterpart. However, the more flexible modulation means the more complex design dimension, which hinders the development of vectorial metasurface holography. In this work, we propose an orthogonal I-shaped structure embedded with resistors to synthesize arbitrary polarization with less crosstalk. Benefiting from the independent control of phase and amplitude on orthogonal base, the orthogonality-simplified machine learning framework is employed to assist vectorial metasurface holography design. As a proof-of-concept, a vectorial metasurface holography carrying multi-polarization information was designed, simulated and measured. All the results exhibit a high degree of consistency, which fully demonstrates the effectiveness of our design. Encouragingly, our method paves a new route to simplify machine learning framework based on physical significance, which can be handily extended to more functional structures.

Combination of Fuzzy-Weighted Zero-Inconsistency and Fuzzy Decision by Opinion Score Methods in Pythagorean m-Polar Fuzzy Environment: A Case Study of Sign Language Recognition Systems

In the fuzzy multicriteria decision-making approach, a committee of decision-makers is usually involved in the assessment of the suitability of different alternatives based on the evaluation criteria by using linguistic terms and their equivalent fuzzy numbers. In this context, researchers have developed the Pythagorean fuzzy set (PFS) to overcome the limitation of intuitionistic fuzzy set in the description of decision-maker information such as imposing restrictions on the representation of membership and nonmembership grades. On the one hand, PFS still does not have sufficient ability and flexibility to deal with such issues. On the other hand, multipolar technology is used to operate large-scale systems in real-life situations, especially in dealing with dissatisfaction and indeterminacy grades for the alternatives of the reference set. Thus, m-polar fuzzy set is utilized and applied with other fuzzy sets because of its remarkable ability as a tool for depicting fuzziness and uncertainty under multipolar information in many circumstances. With the practical features of m-polar fuzzy set in combination with PFS, this paper employs it to extend two considerable MCDM methods, namely, fuzzy decision by opinion score method and fuzzy-weighted zero inconsistency. Such extensions, called Pythagorean m-polar fuzzy-weighted zero-inconsistency (Pm-PFWZIC) method and Pythagorean m-polar fuzzy decision by opinion score method (Pm-PFDOSM), are formulated to weight the evaluation criteria followed by alternative ranking progressively. The research methodology is presented as follows. Firstly, the mechanisms of Pm-PFWZIC and Pm-PFDOSM are formulated and integrated into the development phase. Secondly, the description of the real-world case study of the evaluation and benchmarking of the sign language recognition systems is adapted and presented. The result of Pm-PFWZIC shows that the criterion of ‘finger movements’ has the highest weight amongst the rest of the criteria, whereas ‘misclassification error’ has the lowest weight. In the ranking results, a variation of ranking is scored by each expert, and group decision-making is applied to solve the individual ranking variety. The robustness of the formulated methods is evaluated using systematic ranking, sensitivity analysis and comparison analysis.

A study on weighted aggregation operators for q‐ rung orthopair m‐ polar fuzzy set with utility to multistage decision analysis

Modeling uncertainties with multipolar information is an important tool in computational intelligence to address complexities in real-world circumstances. An m-polar fuzzy set (mPFS) is the strong model to express multipolarity with m $m$ membership grades (MGs) in the unit closed interval [ 0 , 1 ] $[0,1]$ . A q-rung orthopair fuzzy set (qROFS) is the strong model to express vague and uncertain information with MGs and nonmembership grades (NMGs). The notion of q-rung orthopair m-polar fuzzy set is a new hybrid extension of both mPFS and qROFS. An ROmPFS is a generalized concept that has the ability to deal with multipolarity with m $m$ ordered pairs of MGs and NMGs. Motivated by these robust concepts, in this article, various aggregation operators (AOs) for the aggregation of q-rung orthopair m-polar fuzzy numbers are proposed, including q-rung orthopair m-polar fuzzy weighted averaging operator, symmetric q-rung orthopair m-polar fuzzy weighted averaging operator, q-rung orthopair m-polar fuzzy weighted geometric operator, symmetric q-rung orthopair m-polar fuzzy weighted geometric operator, and q $q$ -rung orthopair m-polar fuzzy Maclaurin symmetric mean operator. On the basis of proposed AOs, a robust multicriteria decision-making approach is proposed. An application of proposed AOs is presented to address economic crises during COVID-19. Furthermore, the comparison analysis is designed to discuss the validity and rationality of proposed AOs.

A new method to evaluate regular ternary semigroups in multi-polar fuzzy environment

<abstract> <p>Theory of $m$-polar fuzzy set deals with multi-polar information. It is used when data comes from $m$ factors $\left({m \ge 2} \right)$. The primary objective of this work is to explore a generalized form of $m$-polar fuzzy subsemigroups, which is $m$-polar fuzzy ternary subsemigroups. There are many algebraic structures which are not closed under binary multiplication that is a reason to study ternary operation of multiplication such as the set of negative integer is closed under the operation of ternary multiplication but not closed for the binary multiplication. This paper, presents several significant results related to the notions of $m$-polar fuzzy ternary subsemigroups, $m$-polar fuzzy ideals, $m$-polar fuzzy generalized bi-ideals, $m$-polar fuzzy bi-ideals, $m$-polar fuzzy quasi-ideals and $m$-polar fuzzy interior ideals in ternary semigroups. Also, it is proved that every $m$- polar fuzzy bi-ideal of ternary semigroup is an $m$-polar fuzzy generalized bi-ideal of ternary semigroup but converse is not true in general. Moreover, this paper characterizes regular and intra-regular ternary semigroups by the properties of $m$-polar fuzzy ideals, $m$-polar fuzzy bi-ideals.</p> </abstract>

A SAR-GMTI Approach Aided by Online Knowledge With an Airborne Multichannel Quad-Pol Radar System

In the complicated geographical environment, there will be a seriously deleterious effect to the performance of synthetic aperture radar (SAR)-ground moving target indication (SAR-GMTI) system, because it is difficult to obtain the homogeneous training samples to accurately estimate the clutter covariance matrix (CCM) without prior information of the observed scene. To this end, this paper proposes a SAR-GMTI approach aided by online knowledge with an airborne multichannel quadrature-polarimetric (quad-pol) radar system. Generally, this paper can be divided into two parts: online knowledge acquisition and polarization knowledge-aided (Pol-KA) SAR-GMTI processing. Firstly, based on the similarity of pixels from the multichannel and multi-polarization information, a weighed estimation method of polarimetric coherency matrix is proposed, which can overcome the over-smoothing problem and increase the estimation accuracy of coherency matrix. Further, a hybrid weighted local K-means based on geodesic distance (GD-HWLKM) clustering algorithm is proposed to achieve the aim of unsupervised classification. Here, geodesic distance (GD) is exploited to measure the distance between multi-feature region covariance matrixes (MFRCMs) and a hybrid weight from different scales (including local cover class distribution, region and pixel) is calculated to automatically update the cluster centroid, which can make full use of the local spatial information by taking the interclass samples' similarity and the diversity of different classes into consideration. Secondly, with the assistance of the previous polarization SAR (PolSAR) image classification result, a Pol-KA SAR-GMTI method is developed. For each ground cover category, an accurate CCM can be estimated with the independent and identically distributed (IID) training samples. Then, the multichannel clutter suppression and preliminary constant false alarm rate (CFAR) detection are performed. Finally, with an airborne multichannel quad-pol radar system, the experimental results on real measured data demonstrate that the proposed method can efficiently improve the clutter suppression preformation and moving-target detection preformation.

POLSAR Target Recognition Using a Feature Fusion Framework Based on Monogenic Signal and Complex-Valued Nonlocal Network

With the continuous development of synthetic aperture radar (SAR) systems, multi-polarization information has been increasingly applied to numerous fields, and automatic target recognition (ATR) in polarimetric SAR (POLSAR) has been recognized as vital problem. The SAR recognition methods can primarily fall into handcrafted feature-based algorithms and deep learning algorithms. The former exhibits excellent interpretability but insufficient generalization; the latter achieves stronger representational ability but relies on a considerable number of samples. To solve above problems, a feature fusion framework is proposed in this paper based on monogenic signal and complex-valued non-local network (CVNLNet) for POLSAR target recognition. The proposed feature fusion framework effectively uses the complementarity of handcrafted features and deep features, while making up for the disadvantages of single feature-based methods. First, a Mono-BOVW model is proposed based on monogenic signal and bag-of-visual-words (BOVW) model to extract handcrafted features, which can more fully mine the information covered in POLSAR data in multi-scale space. Moreover, CVNLNet is built for deep feature extraction to use both the amplitude and phase covered in POLSAR data. Next, a kernel discrimination correlation analysis algorithm (KDCA) is proposed to jointly analyze and transform the two features, so as to remove redundant information while retaining effective and discriminative information. Experiments on the MSTAR dataset and the GOTCHA dataset show that the proposed framework has superior performance on single polarimetric and fully polarimetric datasets.

The Orientation Estimation of Elongated Underground Objects via Multipolarization Aggregation and Selection Neural Network

The horizontal orientation angle and the vertical inclination angle of an elongated subsurface object are key parameters for object identification and imaging in ground-penetrating radar (GPR) applications. Conventional methods can only extract the horizontal orientation angle or estimate both angles in narrow ranges due to limited polarimetric information and detection capability. To address these issues, this letter, for the first time, explores the possibility of leveraging neural networks with multipolarimetric GPR data to estimate both angles of an elongated subsurface object in the entire spatial range. Based on the polarization-sensitive characteristic of an elongated object, we propose a multipolarization aggregation and selection network (MASNet), which takes the multipolarimetric radargrams as inputs, integrates their characteristics in the feature space, and selects discriminative features of reflected signal patterns for accurate orientation estimation. Numerical results show that our proposed MASNet achieves high estimation accuracy with an angle estimation error of less than 5°. The promising results obtained by the proposed method encourage one to think of new solutions for GPR-related tasks by integrating multipolarization information with deep learning techniques. The data and code implemented in the letter can be found at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://haihan-sun.github.io/GPR.html</uri> .

Integrated decision-making methods based on 2-tuple linguistic $ m $-polar fuzzy information

&lt;abstract&gt;&lt;p&gt;The 2-tuple linguistic $ m $-polar fuzzy sets (2TL$ m $FSs) are acknowledged to represent the multi-polar information owing to the practical structure of $ m $-polar fuzzy sets with the help of linguistic terms. The TOPSIS and ELECTRE series are efficient and widely used methods for solving multi-attribute decision-making problems. This paper aim to augment the literature on multi-attribute group decision making focusing on the the strategic approaches of TOPSIS and ELECTRE-I methods for the 2TL$ m $FSs. In the 2TL$ m $F-TOPSIS method, the relative closeness index is used to rank the alternatives. For the construction of concordance and discordance sets, the superiority and inferiority of alternatives over each other are accessed by using the score and accuracy functions. In the 2TL$ m $F ELECTRE-I, selection of the best alternative is made by the means of an outranking decision graph. At the final step of the 2TL$ m $F ELECTRE-I method, a supplementary approach is developed for the linear ranking of alternatives based on the concordance and discordance outranking indices. The structure of the proposed techniques are illustrated by using a system flow diagram. Finally, two case studies are used to demonstrate the correctness, transparency, and effectiveness of the proposed methods for selecting highway construction project manager and the best textile industry.&lt;/p&gt;&lt;/abstract&gt;

Novel multiple criteria decision-making analysis under m-polar fuzzy aggregation operators with application.

Aggregation is a very efficient indispensable tool in which several input values are transformed into a single output value that further supports dealing with different decision-making situations. Additionally, note that the theory of m-polar fuzzy (mF) sets is proposed to tackle multipolar information in decision-making problems. To date, several aggregation tools have been widely investigated to tackle multiple criteria decision-making (MCDM) problems in an m-polar fuzzy environment, including m-polar fuzzy Dombi and Hamacher aggregation operators (AOs). However, the aggregation tool to deal with m-polar information under Yager's operations (that is, Yager's t-norm and t-conorm) is missing in the literature. Due to these reasons, this study is devoted to investigating some novel averaging and geometric AOs in an mF information environment through the use of Yager's operations. Our proposed AOs are named as the mF Yager weighted averaging (mFYWA) operator, mF Yager ordered weighted averaging operator, mF Yager hybrid averaging operator, mF Yager weighted geometric (mFYWG) operator, mF Yager ordered weighted geometric operator and mF Yager hybrid geometric operator. The initiated averaging and geometric AOs are explained via illustrative examples and some of their basic properties, including boundedness, monotonicity, idempotency and commutativity are also studied. Further, to deal with different MCDM situations containing mF information, an innovative algorithm for MCDM is established under the under the condition of mFYWA and mFYWG operators. After that, a real-life application (that is, selecting a suitable site for an oil refinery) is explored under the conditions of developed AOs. Moreover, the initiated mF Yager AOs are compared with existing mF Hamacher and Dombi AOs through a numerical example. Finally, the effectiveness and reliability of the presented AOs are checked with the help of some existing validity tests.

Correlation Measures for Cubic m-Polar Fuzzy Sets with Applications

A cubic m -polar fuzzy set (CmPFS) is a new hybrid extension of m -polar fuzzy set and cubic set. A CmPFS is a robust model to express multipolar information in terms of m fuzzy intervals representing membership grades and m fuzzy numbers representing nonmembership grades. In this article, we explore some new operational laws of CmPFSs, produce some related results, and discuss their consequences. We propose relative informational coefficients and relative noninformational coefficients for CmPFSs. These coefficients are analyzed to investigate further properties of CmPFSs. Based on these coefficients, we introduce new correlation measures and their weighted versions for CmPFSs. The value of proposed correlation measures is symmetrical and lies between −1 and 1. Moreover, the applications of the proposed correlation in pattern recognition and medical diagnosis are developed. The feasibility and efficiency of suggested correlation measures is determined by respective illustrative examples.

Cubic M-polar Fuzzy Hybrid Aggregation Operators with Dombi’s T-norm and T-conorm with Application

A cubic m-polar fuzzy set (CmPFS) is a new hybrid extension of cubic set (CS) and m-polar fuzzy set (mPFS). A CS comprises two parts; one part consists of a fuzzy interval (may sometimes be a fuzzy number) acting as membership grade (MG), and the second part consists of a fuzzy number acting as non-membership grade (NMG). An mPFS assigns m number of MGs against each alternative in the universe of discourse. A CmPFS deals with single as well as multi-polar information in the cubic environment. In this article, we explore some new aspects and consequences of the CmPFS. We define score and accuracy functions to find the priorities of alternatives/objects in multi-criteria decision-making (MCDM). For this objective, some new operations, like addition, scalar/usual multiplication, and power, are defined under Dombi’s t-norm and t-conorm. We develop several new aggregation operators (AOs) using cubic m-polar fuzzy Dombi’s t-norm and t-conorm. We present certain properties of suggested operators like monotonicity, commutativity, idempotency, and boundedness. Additionally, to discuss the application of these AOs, we present an advanced superiority and inferiority ranking (SIR) technique to deal with the problem of conversion from a linear economy to a circular economy. Moreover, a comparison analysis of proposed methodology with some other existing methods is also given.

Novel concepts of $ m $-polar spherical fuzzy sets and new correlation measures with application to pattern recognition and medical diagnosis

&lt;abstract&gt;&lt;p&gt;In this paper, we introduce the notion of $ m $-polar spherical fuzzy set ($ m $-PSFS) which is a hybrid notion of $ m $-polar fuzzy set ($ m $-PFS) and spherical fuzzy set (SFS). The purpose of this hybrid structure is to express multipolar information in spherical fuzzy environment. An $ m $-PSFS is a new approach towards computational intelligence and multi-criteria decision-making (MCDM) problems. We introduce the novel concepts of correlation measures and weighted correlation measures of $ m $-PSFSs based on statistical notions of covariances and variances. Correlation measures estimate the linear relationship of the two quantitative objects. A correlation may be positive or negative depending on the direction of the relation between two objects and its value lies the interval $ [-1, 1] $. The same concept is carried out towards $ m $-polar spherical fuzzy ($ m $-PSF) information. We investigate certain properties of covariances and the correlation measures to analyze that these concepts are extension of crisp correlation measures. The main advantage of proposed correlation measures is that these notions deal with uncertainty in the real-life problems efficiently with the help of $ m $-PSF information. We discuss applications of $ m $-polar spherical fuzzy sets and their correlation measures in pattern recognition and medical diagnosis. To discuss the superiority and efficiency of proposed correlation measures, we give a comparison analysis of proposed concepts with some existing concepts.&lt;/p&gt;&lt;/abstract&gt;

On the Use of Multipolarization Satellite SAR Data for Coastline Extraction in Harsh Coastal Environments: The Case of Solway Firth

This study deals with coastline extraction using multipolarization spaceborne synthetic aperture radar (SAR) imagery acquired over coastal intertidal areas. The latter are very challenging environments where mud flats lead to a large variability of normalized radar cross section, which may trigger a significant number of false edges during the extraction process. The performance of SAR-based coastline extraction methods that rely on a joint combination of multipolarization information (either single- or dual-polarization metrics) and speckle filtering (either local and nonlocal approaches) are analyzed using global positioning system (GPS) samples and colocated SAR imagery collected under different incidence angles. Our test site is an intertidal zone with a wetland (i.e., salt marsh) in the Solway Firth, south-west along the Scottish-English border. Experimental results, obtained processing a pair of RadarSAT-2 full-polarimetric and a pair of Sentinel-1 dual-polarimetric SAR imagery augmented by colocated GPS samples, show that: first, the multipolarization information outperforms the single-polarization counterpart in terms of extraction accuracy; second, among the single-polarization channels, the cross-polarized one performs best; third, both single- and dual-polarization methods perform better when nonlocal speckle filtering is applied; fourth, the joint combination of nonlocal speckle filter and dual-polarization information provides the best accuracy; and finally, the incidence angle plays a role in the extraction accuracy with larger incidence angles resulting in the best performance when dual-polarization metric is used.

Physically Based Object Contour Edge Display Using Adjustable Linear Polarization Ratio for Passive Millimeter-Wave Security Imaging

Passive millimeter-wave (PMMW) imaging has been used for several close-range applications such as personal security checks, scene monitoring, and so on. PMMW images contain a variety of types of image edges which represent intensity discontinuities. As a special edge, an object contour edge is an important feature for object detection and recognition, which denotes the external shape of the object. In this article, a physically based contour edge display method using adjustable linear polarization ratio (ALPR) by multipolarization imaging is proposed. Polarization brightness temperature (TB) models of the object and radiometer observation are built. According to the physical principle of edge generation, the ALPR properties of the contour edge are investigated. By fusing multipolarization information, the simple feature parameter ALPR is sensitive to the object contour edge. Then, the specific operation process of the proposed method based on multipolarization images is presented. Simulations and measurements of polarimetric imaging for personal security inspection scenes were conducted to verify the contour edge display performance. Finally, the applicability of the method is discussed and summarized.

Concealed object enhancement using multi-polarization information for passive millimeter and terahertz wave security screening.

Passive millimeter and terahertz wave imaging has become a significant potential technique for human security check and scene monitoring. Due to the small difference of the brightness temperatures between human body and concealed objects, the temperature sensitivity and spatial resolution of radiometers are always the key performance indexes which are difficult to improve. Therefore, when the hardware performance is given, improving detectivity becomes a pressing need. In this paper, a physically-based concealed object enhancement method using multi-polarization information is presented. The polarization model and polarization property of human body and concealed objects have been analyzed. By fusing multiple polarization images, we can obtain a complete polarization image in which the contrast between human body and concealed objects is enhanced and stable. The experimental results of simulation and measurement demonstrate the enhancement performance, and Differential Signal Noise Ratio (DSNR) is obviously improved by using the proposed method.

A hybrid decision-making framework using rough mF bipolar soft environment

In this article, we present the notions of two novel hybrid multi-attribute decision-making models, namely m-polar fuzzy bipolar soft set (mFBSS, for short) model and rough mFBSS model. These mathematical models allow us to handle multipolar information under fuzziness with bipolar soft sets. We use these proposed models to handle complicated problems in which the degree of membership of an object in a given set uses m fuzzy values, to rank all the objects, and to find a suitable option. We illustrate our proposed concepts with examples. Furthermore, we investigate some useful properties, including complement, restricted union and intersection, extended union and intersection, AND and OR. We discuss two practical applications of mFBSSs and rough mFBSSs, that is, selecting a suitable employee for promotion and place for house construction, respectively. We develop efficient algorithms to solve multi-attribute decision-making problems. We also discuss a comparison analysis of the proposed approaches with some existing models.

Soft rough Pythagorean m-polar fuzzy sets and Pythagorean m-polar fuzzy soft rough sets with application to decision-making

The aim of this paper is to present the notions of soft rough Pythagorean m-polar fuzzy sets (SRPMPFSs) and Pythagorean m-polar fuzzy soft rough sets (PMPFSRSs), by combining the prevailing ideas of soft sets, rough sets, Pythagorean sets and m-polar fuzzy sets. The suggested models of SRPMPFSs and PMPFSRSs are more efficient to discuss the roughness of the input data by means of crisp soft and Pythagorean m-polar fuzzy soft approximation spaces. We present some fundamental properties of Pythagorean m-polar fuzzy soft approximation spaces and their related examples. We discuss a case study of image denoising method in the field of image processing with the help of proposed models. We develop two novel algorithms for the selection of best image denoising method under crisp soft and Pythagorean m-polar fuzzy soft approximation spaces. We also demonstrate the feasibility and flexibility of the proposed models for solving multi-criteria decision-making problems involving parameterizations, Pythagorean grades and multi-polar information.

Unify Steiner Weiner Distance for Some Class of m-Polar Fuzzy Graphs

ABSTRACT Sometimes information in a network model is based on multi-agent, multi-attribute, multi-object, multi-polar information or uncertainty rather than a single bit. An m-polar fuzzy model is useful for such network models which gives more and more precision, flexibility, and comparability to the system as compared to the classical, fuzzy models. On the other, The Steiner tree problem in networks, and particularly in graphs, was formulated by Hakimi [25] and Levi [21] by definition minimal size connected tree sub graph that contains the vertices in S. Steiner trees have applications to multiprocessor computer networks. For example, it may be desired to connect a certain set of processors with a sub network that uses the fewest communication links. In this paper, we extend Steiner distance SWk (G) for m-polar fuzzy graphs and give this parameter for Join, composition and Cartesian product of two m-polar fuzzy graphs.