Intuitionistic Fuzzy Rough TOPSIS Method for Robot Selection using Einstein operators

. Rough set and intuitionistic fuzzy set are very vital role in the decision making method for handling the uncertain and imprecise data of decision makers. The technique for order preference by similarity to ideal solution (TOPSIS) is very attractive method for solving the ranking and multi-criteria decision making (MCDM) problem. The primary goal of this paper is to intro- duce the Extended TOPSIS for industrial robot selection under intuitionistic fuzzy rough (IFR) information, where the weights of both, decision makers (DMs) and criteria are not-known. First, we develop Intuitionistic fuzzy rough (IFR) aggregation operators based on Einstein T-norm and T-conom, For this (cid:133)rstly we give the idea of intuitionistic fuzzy rough Einstein weighted averaging (IFREWA), intuitionistic fuzzy rough Einstein hybrid averaging (IFREHA) and intuitionistic fuzzy rough ordered weighted averaging (IFREOWA) aggregation operators. The fundamental properties of the proposed operators are described in detail. Furthermore to determine the unknown weights, a generalized distance measure are de(cid:133)ned for IFRSs based on intuitionistic fuzzy rough entropy measure. Following that, the intuitionistic fuzzy rough information-based decision-making technique for multi-criteria group decision making (MCGDM) is developed, with all computing steps depicted in sim-plest form. For considering the con(cid:135)icting attributes, our proposed model is more accurate and e⁄ective. Finally, an example of e¢ cient industrial robot selection is presented to illustrate the feasibility of the proposed intuitionistic fuzzy rough decision support approaches, as well as a discussion of comparative outcomes, demonstrating that the results are feasible and reliable.


INTRODUCTION
Multi-criteria group decision making (MCGDM) has become extremely valuable in last few decades in the study of decision support systems [1,2,3,4,5,6]. The complexity of decision-making (DM) problems increases time to time in this world of global competition as the socio-economic framework becomes more dynamic. Therefore, making wise and successful decisions in this situation is very complicated for a single decision expert. In this real world, group DM models are often used to combine the opinions of team of experienced experts in order to generate highly accurate and optimal values. Hence, MCGDM have an excellent ability and systematic method for improving and evaluating various competing criteria in all aspect of DM in order to achieve more appropriate and possible DM outcomes. In DM issues, the knowledge based about even a fact is often unclear, making the decision-making task more complicated and ambiguous. To resolve this limitation, Zadeh [7] in 1965 …rst established the idea of fuzzy set (FS) by considering the membership degree. This concept has been investigated in a various real-world problems, including clustering analysis [8] decision-making problems (DMPs) [9] and medical diagnosis [10]. Atanassov [11] developed the Intuitionistic Fuzzy Set (IFS) as an extension of the FS, and its constraint is that the sum of the degree of membership and the degree of non-membership will be less than or equal to one. IFS has been a hot research topic for researchers who examined its hybrid structure in a variety of ways. The idea of intuitionistic fuzzy weighted averaging (IFWA) aggregation operators was …rst investigated by Xu [12]. Xu and Yager [13] developed the de…nition of intuitionistic fuzzy weighted geometric (IFWG) aggregation operators. The graphical techniques for rating score and accuracy functions was created by Ali et al., [14]. He et al., [15] explored the concept of intuitionistic fuzzy neutral averaging operators (IFNAO). He et al., [16] invented the notions of geometric relationship averaging operator and proposed its implementation in DM. Also Atanassov [17] presented preference relation based on IVIF. Further Wan and Don [18] work on extension of best-worst method based on IFS. Zaho et al., [19] were the …rst to apply the generalized intuitionistic fuzzy weighted averaging (GIFWA), generalized intuitionistic fuzzy order weighted averaging (IFOWA), and generalized intuitionistic fuzzy hybrid averaging (GIFHA) operators to DM. However, it was discovered that these aggregation process was performed by using Archimedean Tnorms and T-conorm by operators. As an equivalent to algebraic product and sum, Einstein-based T-norm and T-conorm provides the best estimate for product and sum of intuitionistic fuzzy numbers (IFNs). Wang and Liu [20,21] provided IF Einstein weighted averaging (IFEWA) and IF Einstein weighted geometric (IFEWG) operators by using the concept of Einstein operation.
After more outcomes," Pawlak [22] is credited with being the …rst to explore the prevalent idea of rough sets (RS) theories. This theory expanded on the classical set theory, which deals with ambiguous and imprecise information. Rough set analysis has advanced signi…cantly in recent years, both in terms of practical applications and theoretical understanding. The idea of rough sets has been expanded in di¤erent ways by many researchers. The idea of fuzzy rough (FR) collection was …rst invented by Dubois and Prade [23] by utilizing fuzzy relations instead of crisp binary relation. Cornelis et al., [24] established a combined analysis of intuitionistic fuzzy rough set (IFRSs) applying the hybrid concept of IFS and RS as link between all these two theories. By using IFR approximation operators, Zhou and Wu [25] established a constrictive and axiomatic analysis. Zhou and Wu [26] created the idea of IFRS, rough intuitionistic fuzzy set and presented their constraining and axiomatic analysis in detail by introducing the idea of crisp and fuzzy approximation space. The IF relation was established by Bustince and Burillo [27]. Zhang et al., [28] utilized general IF relations to examine the general structure of IFRS based on the principle of two universes. Yun and Lee [29] used topology to establish some properties of IFR approximation operator based on IF relation. Many number of IFRSs extensions are examined, see [30,31,32,33] for more information. In addition, Mehmood et al., [34] invented the interaction between the rough and IFS and also discussed its various aggregation operators.
Several decision-making processes are being proposed in the literature over the years, with techique for order preference by similarity to ideal solution (TOPSIS) being one of the most widely and e¢ ciently used. To handle with MCGDM problems, Hwang and Yoon [35] addressed the TOPSIS. In DM problems, the best alternative is the one with the lowest distance from the positive ideal solution (PIS) and the highest distance from negative ideal solution (NIS). In [36] Chen discussed how to solve DM problems using TOPSIS under the FS. In recent decades, several researchers have become interested in TOPSIS and have applied it to real-world DM problems using various extended structures of FS [37] to [49] in the …elds of decision sciences [49,50,51]. It's also worth noting that the current TOPSIS procedures [37] to [49] have the downside of requiring either DMs weights [45] or criteria weights [41,48] or both [37,39,48,49,50,51] to be known when solving DMPs. Some scholars assigned unknown weight details regarding DMs in which the parameters weights are known [52,53].
Motivation: As for our information and based from the above review, no implementation of the MCGDM approach to a combination review of intuitionistic fuzzy set and rough sets using intuitionistic fuzzy Einstein averaging aggregation operators in intuitionistic fuzzy Rough setting in which the knowledge about DMs weights and criteria are completely unknown. MCGDM is used to demonstrate the e¢ ciency of the established IFR Extended TOPSIS framework, which is focused on IFR Einstein averaging operators. We proposed the various average aggregation operators like intuitionistic fuzzy rough Einstein weighted averaging (IFREWA), intuitionistic fuzzy rough Einstein order weighted averaging (IFREOWA) and intuitionistic fuzzy rough Einstein hybrid averaging (IFREHA), by using the idea of Einstein T-norm and T-conorm for solving MCGDM. The main contribution of the proposed work are as following: 1) Using the Einstein T-norm and T-conorm to develop Einstein-aggregation operators for intuitionistic fuzzy rough information 2) We develop a new extended version of TOPSIS method by using the intuitionistic fuzzy rough Einstein aggregation information 3) The proposed techniques applied to industrial robot selection problems 4) The proposed decision techniques compared with previous decision making models and the comparative analysis show that the proposed method has more reliable and e¤ective for ranking and decision making problems.
Throughout in this article, a novel Extended TOPSIS-based approach is developed to deal a problem with uncertain weight knowledge for both DMs and criteria weights, as well as to resolve the MCGDM problem even with all the weights have been computed. Ideal opinion (IO) which is closed to every DMs should be chosen for solving DM problems. Under proposed IFR Einstein average method, the ideal opinion is nominated in the presented procedure. To …nd the di¤erences between two IFRSs, a generalized distance measure is de…ned. Generalized distance measures-based entropy measure is implemented in the presented IFR-Extended TOPSIS approach for solving MCGDM problems to …nd the criteria weights in the form of IFR information which are explained in this article. The fundamental principle for calculating criteria weights using the entropy measure is that the lower the entropy measure of a criterion between alternatives, the larger the weight should be applied on that criterion, but instead, the smaller weight should be applied on that criterion. A newer entropy measure for IFRSs is described as taking into consideration the value of its membership, non-membership grade and hesitancy degree in order to better measure the fuzziness of IFRSs, which is used to achieve the criteria weights in solving MCGDM problems with completely unknown weight information using the IFR entropy weight.
The rest of this article is arranged as follows: Section 2, includes the essential principles of IFRSs that will be useful in following section. Section 3, exists the Einstein operation laws for IFRSs and concepts of Einstein averaging operators for IFRSs such as IFREWA, IFREOWA, (IFREHA), also discussed their properties. Section 4, here in this section we proposed entropy measure and distance measures using IFRs information. Section 5, utilizing all these developed ideas we have presented the proposed Extended TOPSIS to resolve the uncertainty in MCGDM problems in this section. Section 6, exists numerical example for robot selection using the built MCGDM technique. Section 7 present comparison to other DM approaches. Section 8, draws a conclusion to the article.
where successor neighborhood is denoted by (p ) of an object p w.r.t. . Then the pair ( ; ) as known as crisp approximation space. The lower and upper approximation for any $ w.r.t. approximation space denoted by ( ; ); are de…ned as follows: Then the rough set is represented by the pair ( ($); ($)) and ($); ($) :  ($); which are de…ned as follows:

Einstein operations for intuitionistic fuzzy rough sets
Here we present the Einstein operations and fundamental properties of IFRSs by taking the ideas of [56]. If T-norm and T-conorm are represented by T and S. Then Einstein product and sum are expressed by T & andŜ & respectively: The Einstein sum and product are equivalent to the generalized union and intersection of two IFRSs, which are de…ned as follows: Further more, we can derive the following forms: (2) (3) Also, certain fundamental properties will be debated. IF REW A( ($ 1 ); ($ 2 ); ::: Theorem 1 shows the aggregated results for the IFREWA operator based on the above de…nition.

Methodological Development of Intuitionistic Fuzzy Rough Entropy Measure
Based on the distance model [54,55], this part presented the generalized distance and weighted generalized distance measures using intuitionistic fuzzy rough information to determine the di¤erences between the two IFRSs, After that, we presented a novel entropy measure for IFRSs to measure its fuzziness, based on generalized distance measures. Then for any > 0 (2 R) the generalized distance measure between any two IFRSs are de…ned as follows:     (l) mn It should be noted that in the decision-making process, all details about the weights of (DM's) and attributes (criteria) is completely unknown.

INTUITIONISTIC FUZZY ROUGH EXTENDED TOPSIS METHOD.
The method is divided into …ve parts. In the …rst part, a TOPSIS-based method for computing decision-makers (DMs) weights is proposed. Second part consists of computing the weights of attributes (criteria) with help of new proposed entropy measure. With PIS and NIS, the …nal aspect is a rating system based on degree of similarity to the ideal solution.
Step-(2c): Here, we use equation (4.1) to compute the distance of decision matrices N (l) ij to G D IŜ; G RĨD and G LĨD; and symbolically represented by DG DĨ S; DG RĨD and DG LĨD respectively, which are mentioned below: for i = 1; 2; :::; m and l = 1; 2; :::; e: Step-(2d): Following the model proposed [53], We measure the closeness indices (CIs) as follows: For l = 1; 2; :::; e: Step-(2e): In this step, by using following formula to determined the DMs weights: Step-(3a): Using the proposed IFRVs entropy measure to …nd the weights of attributes (criteria), and the revised group decision (RG DĨ S) was calculated as follows: Step-(3b): The entropy measure for each attribute in IFRV is determined by using Equation (4.4): Ec j = E R Ĩ S 1j ; R Ĩ S 2j ; :::; R Ĩ S mj ; j = 1; 2; :::; n: Step-(3c): The following formula is used to measure attribute weights: Ec j ; j = 1; 2; :::; n: Step-(4a): The weighted normalized decision matrices by using criteria weight vector are determined as follows: for each (l = 1; 2; :::; e) : Step-(4b): Determine P IS (l) and N IS (l) for each DM's by using weighted normalized decision matrices DM (N ) Step-(4c): By using equation 4.2 the WGDM from DM (N ) (l) to P IS (l) and N IS (l) ; are computed as follows: and for each i = 1; 2; :::; m: Step-(4d): For each DM's the revised closeness indices denoted by (R _ C I i ) are calculated as follows: Step-(5): Using the DMs weights, compute the …nal revised closeness indices (F R _ C Is) as follows: In ascending order, rank the measured F R _ C Is values; the alternative with the greatest value is our best choice.

Numerical Application of the Proposed EXTENDED TOPSIS Method
Now a days we are facing many problems like selection of a robot for a particular industrial application has always been a crucial issue due to the market's availability of various types of industrial robots with varying capacities, functionality, facilities, and speci…cations. For this we will present MCGDM a practical example to …nd the best optimal solution for selecting the di¤erent types of industrial robots. Let us suppose there are four di¤erent types of industrial robots = fp 1 ; p 2 ; p 3 ; p 3 g with di¤erent features and there are three professional experts D g (g = 1; 2; 3) having unknown weights vectors [ (l) . The experts assessed these four robots concerning the …ve criteria C = fc 1 ; c 2 ; c 3 ; c 4 ; c 5 g with unknown weight vector [ cj which are given as under 1) Performance (Static and Dynamic) A robot's performance characteristics are divided into two categories: static and dynamic. The values given under steady state conditions are called static characteristics, while dynamic characteristics refer to the robot's time-dependent behavior.
2) Robotic Architecture The geometry and movements needed to push around the robot's surroundings are important in the design and analysis of the robot.
3) Operating Environment Robotic systems must operate in di¢ cult and unpredictable settings, so the ability to communicate with and cope with the environment, whether on land, underwater, in the air, underground, or in space, is a vital skill.

4) Instrumentation and Control Systems
These are characteristics that are in charge of making important decisions based on input values from sensors and transducers, as well as monitoring and calculating the quantities of controllable parameter values.

5) General and Physical
These characteristics are a mix of cost-e¤ective and desirable technological features relevant to a robot's e¢ ciency and quality that aren't needed to complete any task but are extremely useful during installation and/or operation. Now, in the form of IFRVs, the DM's assessed their assessment report for each p i in the form of IFRVs against the corresponding criteria are as follow in Table  1  Step-(2d): The following are the closeness indices (CIs): :7420 :7512 Step-(2e): In this step, the following equation is used to determine the weights of DMs: Step-(3a): Group revised decision RG DĨ S are computed as follows: Table- Step-(3b): The Intuitionistic fuzzy rough entropy measure for each attribute is calculated as follows: Step-3(c): Criteria weight are calculated as follows:     Step-(4d): For each DM's, revised closeness indices (R _ C Is) are calculated as follows: Step-(5): Using the DM's weights, the …nal revised closeness indices (F R _ C I i ) are calculated as follows: As a result, based on the attributes, P 1 is our best choice.

Comparative Study
In this part, we will discuss the advantages of the our new proposed method Extended TOPSIS which are demonstrated by comparing its characteristics with di¤erent MCGDM methods. The PIS and NIS are used in the Extended TOPSIS system. The best alternative is PIS because of its superior value and NIS because of its inferior value. A comparison of our investigated IFR TOPSIS method with some existing methods in background has been made to prove the supremacy of our investigative IFR TOPSIS method (see [12,34,57,58]) which are discussed in the following subsection. 7.1. Comparison with IFWA. In this subsection we can compare our new proposed work with the existing methods. As in existing methods Xu [12] take the values in the form IFSs, and de…ned the averaging aggregation operators such as IFWA and IFOWA, to aggregate the information with IFRVs, when applied to decision making based on intuitionistic fuzzy knowledge, this makes the decision results more precise and realistic. But here in our paper we take the data in the form of IFRSs and used basic concept of Einstein T-norm and T-conorm to introduced a new aggregation operators like IFREWA, IFREOWA and IFREHWA to aggregate the MCGDM issues. Further more we have used these aggregation operators for solving the MCGDM in form of IFRSs environment with all unknown information about the weights of criteria and DMs. From Table-20 it is clear that the existing aggregation operators are incapable to solve the formed illustrated example with IFRVs. So by comparing this proposed work with the existing work we can see that our proposed work is more accurate and more applicable than existing methods. In this subsection we also compare our proposed work with the another existing methods. As in existing methods M. R. Seikh et al. [57] give the idea of Dombi T-norm and T-conorm to aggregate intuitionistic fuzzy knowledge. Also discussed there basic operational laws in the form of IFNs environment. On the basis of these operational laws, in existing method the aggregation operators were de…ned like IFDWA, IFDOWA, and IFDHA operators for aggregating the MCGDM problems under an intuitionistic fuzzy environment. But here in our paper we take the data in the form of IFRSs and used basic concept of Einstein T-norm and T-conorm we introduced a new aggregation operators like IFREWA, IFREOWA and IFREHWA to aggregate the MCGDM issues. Moreover we have used these aggregation operators for solving the MCGDM in form of IFRSs environment with all unknown information about the weights of criteria and DMs. It is clear from Table 21 that the current aggregation operators are incapable to solve the formed illustrated example with IFVs. So by comparing this proposed work with the existing work we can see that our new work is more accurate and more applicable than existing methods.

Comparison with IF-EDAS Method.
We can compare our new proposed work with the existing methods in this subsection. Ghorabaee et al. [58] proposed an advance method of Evaluation depend on Distance from Average solution EDAS by taking the data in the form of IFS for multi-criteria inventory classi…cation and also examined its step-wise algorithms. But here in this paper we can take the data in the form of IFRSs and used the idea of Einstein T-norm and T-conorm. Further we can also extend and proposed the new aggregation operators like IFREWA, IFREOWA and IFREHWA, to aggregate the MCGDM issues. Finally we have used these developed operators on new Extended TOPSIS method with all unknown information about weights of DMs and criteria for …nding best options. It is clear that IF-EDAS method, are clearly incapable to solve the established illustrated example in the form of IFR information, as shown in Table- 22. We can see that current methods lack IF knowledge, and this approach is incapable of solving and ranking the established example. As a result, our newly established system is more capable and reliable than current approaches which are shown as follows:

7.4.
Comparison with IFR-EDAS Method. We can compare our new proposed work with the existing methods in this subsection. In this existing methods Mehmood et [34] take the idea of IFRSs and de…ned its basic aggregation operators like IFRWA, IFROWA and IFRHWA also de…ned a new score and accuracy function. And applied these operators on new proposed method IFR-(EDAS) in form of IFRSs with all known informations about the criteria weights and DMs. The step-wise details were given in literature. But here in our paper we take the same data and used basic concept of Einstein t-norm and t-conorm to introduced a new aggregation operators like IFREWA, IFREOWA and IFREHWA to aggregate the MCGDM issues. Further more we used these aggregation operators on new proposed IFR-Extended TOPSIS method in form of IFRSs environments with unknown weights of criteria and DMs. From obtaining outcome shown in Table-23, p 1 is the best alternative, that's the same as given in [34]. Overall Ranking Table-24  Approaches Score values Ranking IFWA [12] IFDWA [57] IF-EDAS [58] IFRSs-EDAS [34] 0:66 0:52 0:37 0:34 p 1 > p 2 > p 3 > p 4 IFREWA (Extended TOPSIS) :6984 :4400 :3995 :4517 p 1 > p 4 > p 2 > p 3 7.5. Results and Discussion. As a results given in Tables-24, we conclude that current approache like IF-EDAS method [58], and several aggregation operators [12,57] are incapable to solve the established illustrated example in the form of IFR setting except the IFR-EDAS. In comparison with IFR-EDAS given in Table-24, the information is provided by the DM's in the form of IFRSs we applied Extended TOPSIS approach with all unknown details about the weights of criteria and DMs. As a result, p 1 is the best option, that is the same as mentioned in [34]. We can see most of the current methods lack rough details, and these are unable to solve or rank the established example.
Hence, the presented method is more reliable, advisable, e¢ cient, and generalized for solving MCGDM problems with totally unknown DM's and criteria knowledge.

Conclusion
The MCGDM has a massive potential and discipline process for improving and evaluating various con ‡icting criteria in all aspects of DM in order to obtain more appropriate and realistic DM outcome. In DM issues, the e¤ective learning regarding a particular fact is often unknown, making the decision-making task more complicated and dynamic. RSs and intuitionistic fuzzy sets are general mathematical method that can easily handle ambiguous and imprecise knowledge. When there are several con ‡ict criteria in MCGDM problems, the TOPSIS approach plays an important part in the success. In this article we established a new Extended TOPSIS approach in the form of IFR information which is based on Einstein operators, with all unknown information about weights of criteria and DM's. For this we give the idea IFR-Einstein aggregation operators like IFREWA, IFREOWA and IFREHA. To create the IFR entropy weight framework for calculating the criteria weights with in IFR data, a new GDM-based IFR entropy measure is presented. In last steps, aggregation is conducted by using determined DMs weights to achieve the …nal ranking, to prevent the loss of knowledge base in this method. Finally, examples are given to show the technique's potential application and advantage. Furthermore, the established approach can be expanded for future research by incorporating other existing fuzzy sets and applying them to various MCGDM problems involving unknown DM and criteria weights.