New Extended Topsis Method Based On The Entropy Measure And Picture Fuzzy Rough Set Information And Their Application In Decision Support System

. In this article, we shall introduce a novel technique for order pref- erence by similarity to ideal solution (TOPSIS)-based methodology to resolve multicriteria group decision-making problems within picture fuzzy environ- ment, where the weights information of both the decision makers (DMs) and criteria are completely unknown. First, we brie(cid:135)y review the de(cid:133)nition of picture fuzzy sets (PFS), score function and accuracy function of PFRSs and their basic operational laws. In addition, de(cid:133)ned the generalized distance measure for PFRSs based on picture fuzzy rough entropy measure to compute the unknown weights information. Secondly, the picture fuzzy information- based decision-making technique for multiple attribute group decision making (MAGDM) is established and all computing steps are simply depicted. In our presented model, it(cid:146)s more accuracy and e⁄ective for considering the con(cid:135)ict-ing attributes. Finally, an illustrative example with robot selection is provided to demonstrate the e⁄ectiveness of the proposed picture fuzzy decision support approaches, together with comparison results discussion, proving that its results are feasible and credible.


Introduction
Multi-attribute group decision making (MAGDM) has made a signi…cant contribution to decision supprot system since its beginings [1,2,3,4,5,6].Robot selection for manufacturing industries is a multi-functional group decision-making problem that is frequently addressed by employing unprogrammed decision-making methodologies and including the company's massive contract.Various decision makers/analysis, such like development, research,engineering, and economics, are covered in a decision group. In reality, a single decision maker's interests may not be the same.When using the group decision maker (GDM) approach, the priority degree of each decision maker can have a large impact on the …nal result. The increased involvement of multi-functional teams in robot selection and estimating has a signi…cant impact on the buying …rm's e¢ciency.The representation of attribute value is a major issue in decision methods. Crisp numbers cause a problem in decision-making. Because in some circumstances, proving an attribute with crisp set can be challenging. As a result, decision-makers have the ability to make decisions at a good extent. The fuzzy set theory has been utilised to tackle collective decision-making challenges in a number of aspects, including management, engineering, and social sciences. Fuzzy set theory's application to decision-making has a substantial impact.
Due to uncertainties, there are numerous issues that occure in decision-making. For this, In1965, Zadeh was the …rst who give the idea of fuzzy set (FS) [7] which consist only the membership grade function based on [0,1]. The Zadeh theory of fuzzy sets has alot of interesting aspects. In the area of the fuzzy set theory, the challeng of making decisions that classify the elements in the given universe into more then one relevent place hass been examined. According to Atanassov, there are numerous ‡aws with FS. He saw that the idea of negative membership grade might be there as well, which is an important issue to consider when assembling the completely suggested structure and impacts of the di¢culties. As an alternative for proper values, the intuitionistic fuzzy (IF) set appropriatly introduces this type of grade. The element of Atanassov's IF set [8] are provided in ordered pair that consists of positive and negative membership grade characteristics that follow constraint that the sum of both given functions lie between [0,1].
After more outcomes Pawlak [9] is credited with being the …rst to explore the prevalent idea of rough sets (RS) theories. The classical set theory, that deals with inexact and inaccurate details, is generalised by this theory. Rough set analysis has advanced signi…cantly in recent years, both in terms of practical applications and theorectical understanding. The idea of rough sets has been expanded in di¤erent ways by many researchers. By utilizing fuzzy relations instead of crisp binary relations, Dubois and Prade [10] invented the idea of fuzzy rough collection. Cornelis et al. [11] established a combined analysis of IF rough set applying the hybrid notion of IFS and rough set as link between all these two theories (IFRS). By using IFR approximation operators (AOs), Zhou and Wu [12] established a constrictive and axiomatic analysis. By introducing the idea of crisp and fuzzy approximation space Zhou and Wu [13] created the idea of IFRS and rough IFS and presenting their constraining and axiomatic analysis in depth. The IF relation was established by Bustince and Burillo [14]. Zhang et al. [15] utilized general IF relations to examine the general structure of IFRS based on the principle of two universes. Yun and Lee [16] used topology to establish some properties of intutionistic fuzzy rough approximation operator (IFRAO) based on intuitionistic fuzzy relation. Many number of IFRS extensions are examined See [17,18,19,20] for more information. In addition, Mehmood et al. [21] invented the interaction between the rough and IFS and also discussed its various aggregation operators.
After more outcomes BC Cuong developed the picture fuzzy set (PFS) [22] by adding the neutral membersip grade satisfying the condition that the sum of the positive, neutral and negative membership grades lie between [0,1], which is the generalization of FSs, and IFSs. PFSs are clearly better suited to dealing with fuzziness and ambiguity than IFSs. Under the picture fuzzy environment, H. Garg [23] introduced the picture fuzzy weighted averaging operator, picture fuzzy ordered weighted and hybrid averging operator. The correlation coe¢cient of PFS was given by P. Singh [24] in 2015. Wei [25,26] developed a decision-making methodology based on the fuzzy weighted cross-entropy of picture, which is used to distinguish the alternative.Wang, X.Zhou, H. Tu, and S. Tao (2017) researched multiple attribute choice problems based on the picture fuzzy setting, as well as constructed and discribed various picture fuzzy geometric operators. due to lack of understanding NEW EXTENDED TOPSIS M ETHOD 3 about the area of the problem and time constraints, DM sometimes uses picture fuzzy information, and knowledge about is weight is uncertain.
Numerous decision-making models have been presented in the research over the decades, with the methodology for order preference by similarity to ideal solution (TOPSIS) being one of the most often utilised and useful. The TOPSIS method was suggested by H wang and Yoon [27] to handle with multi-attribute DMPs. In DMPs, the best alternative is the one with the lowest distance from the positive ideal solution (PIS) and the greatest distance from the negative ideal solution (NIS). In the …rst part [28] Chen demonstrated how to solve DMPs using TOPSIS in the FS context. In recent years, many people have become interesting in TOPSIS and have applied it real-world DMPS using various extended structures of FS [29,30,31,32,33,34,35,36,37,38,39,40,41] in the …eld of decision sciencesIn the …rst part, Chen demonstrated how to solve DMPs using TOPSIS in the FS context. In recent years, many people have become interested in TOPSIS and have applied it to realworld DMPs using various extended structures of FS in the domains of decision sciences [41,42,43].It's also worth noting that the present TOPSIS procedure [29,30,31,32,33,34,35,36,37,38,39,40,41] have the issue of requiring either DMs weight [37] or criteria weights [33,40] or both [29,31,40,41,42,43] to solve DMPs. [44,45] in which criteria weights are availble, some authors assigned uncertain weight information regarding DMs. In MCGDM di¢culties, some method utilizes uncertain weight information for criterion with known weight knowledge for DMs. According to researchers, there appears to be no strategy in the …eld for handling MCGDM problems using PFR data where the information regarding DM weights and criteria is completely unknown.
In this reseach, a novel modi…ed TOPSIS-based approach is suggested to deal with the situation of unknown weight data for both DMs and criteria weights, as well as to handle the MAGDM issue once all the weights have been computed. It is necessary to choose the group decision/ ideal opinion that is closest to each DMs decision matrix when solving the DMPs. Ideal opinion is determined using the PFR average method in the proposed process. To evaluat the di¤erences between two PFRSs, a generalised distance measure is constructed. To …nd out the criteria weights using PFR fact in the given picture fuzzy rough, a generalised distance measure-based entropy measure is presented. In this research, TOPSIS is used to solve MAGDM problem. The main thought for calculating criteria weights using the entropy measure is that the lower the entropy measure of a criterian among alternatives, the higher the weight should be placed on that criterion, and instead, the lower weight should be placed on that critrion. A newly entropy measure for PFRSs is described as keeping in mind the priority of its membership grades to better evaluate the fuzziness of PFRSs and used to determine the criteria weights in addressing MAGDM problems with entirely unknown weight information using the PFRs entropy weight.
(i) Decision-making is provided by PFRNs, which means that DMs have more ability to share information about the alternative that match the criteria, i.e., DMs can easily apply their inaccurate judgement in the MAGDM process.
(ii)All DM weights and criteria are hidden bihind the judgement values produced by DMs for alternatives that statisfying the criteria. As a result, all uncertain weights are generated and evaluated during the procedure using all of the decisionmaking.
(iii) In order to solve MSGDM challenges with fully unknown weight information, a novelTOPSIS-based methodology is presented to calculate the weights of DMs (iv) The entropy weight approach is used to calculate the criteria weight in MAGDM situations with entirely uncertain weight information, using a newly established entropy measure for PFRSs.
(v) TOPSIS is further performed foe each DM separately after analysing the weights of criteria, in order to aviod the loss of aggregate information.
(vi) Since aggregation is conducted in the …nal phase utalising determined weight of the DMs, there is no risk of lossing collective information during the process.
The following is a summary of the paper's structure. The second section goes through some PFS and rough set concepts. The iiovative cocept of picture fuzzy rough entropy measure was introduced in section third.The modi…ed TOPSIS method was established in Section forth to solve the uncertainties in MAGDM situations. In section …ve, an illustrated example of the planned MAGDM technique for robot selection for manufacturing units is shown, along with a comparison to existing decision-making approaches. InSection seven, the paper's study is concluded.

Preliminaries
Here we sorts out the essential knowledge about, fuzzy set, intuitionistic FS, picture FS and rough set. De…nition 1. [7] Assume that ½ U be a …xed set. A fuzzy set (FS) = in the universe ½ U is a set having the form; where the value = (~) 2 [0; 1] is called membership grade of~: Assume that ½ U be a …xed set . An intutionistic FS = in the universe ½ U is a set having the form; where the values = (~) 2 [0; 1],ñ = (~) 2 [0; 1] are called positive and negative membership grades of~and = (~) +ñ = (~) 1; 8~2 ½ U: for each~2 ½ U. The degree of hesitancy of IFS is determined by = (~) = 1 ( (~) +ñ (~)) : Then, the operational rules are as follows: De…nition 3. [22] Assume that ½ U be a …xed set. An picture FS = in the universe ½ U is a set having the form; where Then, the operational rules are as follows: Assume that ½ U be a …xed set, and $ 2 ½ U ½ U be a crisp relation.

Development of the PFR Entropy Measure methodology
In order to calculate the di¤erences between the two PFRVs, this segment developed generalized and weighted generalized distance measures incorporating PFR information based on the distance model [46,47]. In order to measure the fuzziness of PFRVs, we propose entropy measures for PFRS based on the developed distance operators.
: Then, generalized distance measure (GDM) is described for any > 0 (2 R) as : Then, generalized distance measure (GDM) is described for any > 0 (2 R) as in which g (g = 1; 2; :::; n) the weights satisfying condition g 0 and P n g=1 g = 1: Remark 1. (1) The above mentioned distance measure are said to be Hamming distance, if = 1: (2) The above mentioned distance measure are said to be Euclidean distance, if = 2: the GDM de…ned in 9 reduced as follows , > 0 (2 R) : 3.2. PFR Entropy Measure. In this part we invente a novel entropy measure for PFRSs, by using the notion of Guo and Song [46]. N) : Then, PFR entropy measure is described as  N) : Then, PFR entropy measure holds the following propoerties: The expert evalution matrix is described as: are the PF rough values. NOTE: All data about the weights of DMs and criteria is compeletly hidden in the decision-making setting.

PF TOPSIS method.
The method is divided into …ve parts.The procedure contains …ve crucial parts. In the …rst section, a TOPSIS-based method for computing DM weights is proposed.In the …rst part, TOPSIS-based technique for compute the weights of DMs is proposed.The proposed entropy calculation is used to determine the weights of the criteria in the second section. The second part is about calculate the weights of the criteria using the proposed entropy measure.With PIM and NIM, the …nal aspect is a rating system based on degree of similarity to the ideal matrix. The last section is ranking approach using PIM and NIM tha is founded on the degree of similarity to the ideal matrix.
The following steps are implemented to solve the PFRS MAGDM problem using a TOPSIS-based procedure: Step-1(a): Construct the experts evaluation matrices (E) k . 2 where k represents the number of expert.
Step-1(b): Evaluate normalized experts matices N k ; that is Step-2(a): The expert ideal matrix (EIM) is calculated using PFRWA operator, which is closer to each expert information.  Step-2(c): Using De…netion 9 evalute the distance of N (k) ij to EIM; ERIM and ELIM as follows DEIM; DERIM and DELIM respectively.
Ej ($)+Ej ($) 2 ; j = 1; 2; :::; n: Step-4(a): The following is how the weighted normalised experts matices are calculated using the attribute weight vector: for each k = 1; 2; :::; e: Step-4(b): P IM (k) and N IM (k) for each EM s can be calculated by using weighted normalized experts matices EM ( N ) ij ; which are de…ned as follows: Step-4(c):ŴĜ µ DM are evaluated by using the de…nition 10 from EM ( N ) (k) to P IM (k) and P IM (k) which is represented and de…ned as follows: or each i = 1; 2; :::; m: Step-4(d): The following is how each EM's revised closeness indices (RCIs) are calculated: Step-5: Through using EMs weights, compute the …nal revised closeness indices (FRCIs): Rank the obtained FRCIs values in decreasing order; the alternative with the highest value is our best option.

The Proposed Improved TOPSIS Method's Numerical Application
The developed MAGDM method is inially demonstrated in this section with a numerical application involving robot selection. Then, in order to demonstrate the characteristics and advantages of the proposed technique, a comparison ia made between it and aother decision-making techniques that use picture fuzzy rough information.

5.1.
Example. 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 MAGDM 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¤erents types of industrial robots S 1 , S 2 , S 3 and S 4 with di¤erent features and there are three professional experts D i (i = 1; 2; 3) having unknown weights vectors . The experts assesed these four robots concerning the …ve criteria ff 1 ; f 2 ; f 3 ; f ,f 5 g with unknown weight vector 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 behaviour.
(2) Instrumentation and Control Systems These are the 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.
(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) Robotic Architecture The geometry and movements needed to push around the robot's surroundings are important in the design and analysis of the robot.
(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 installationand/or operation.
The invited decision makers are divided into three expert panels where each expert panel is required to provide uni…ed evaluation results in the form of picture fuzzy rough values with unkown expert and criteria weight information.
The expert evaluation information in the form of picture fuzzy rough values is enclosed in Table-1-3:   Table-   Step-1(b): According to the experts,the all attributes f 1 , f 2 , f 3 , f 4 and f 5 are bene…ts type, Step-2(a): The EIM is calculated in Table-4 to EIM; ERIM and ELIM as follows in Table-     Step-2(d): The closeness indices is calculated as follows: CI (1) CI (2) CI (3) 0:742781 0:726000 0:659046 Step-2(e): Expert weight information is calculated as follows: 0:349 0:341 0:310 Step-3(a): The revised expert ideal matrix is calculated in Table-10:  Step-3(c): The weights of the attributes are computed as follows:  Step-4(c): Distance are computed as follows:  Step-4(d): The revised closeness indices for each EM are determined as follows:  Step-5: When using EM s weights, the …nal revised closeness indicess (RCI) are calculated as follows: Hence,Ŝ 2 is our best alteenative.

Comparison Analysis
The bene…ts of the created methodology are demonstrated in this part by comparing the characteristics of the proposed upgraded TOPSIS method and the designed MAGDM approach. This comparision is made by analyzing the prpperties of several decision-making techniques found in the letrature, The EDAS approach for intutionistic fuzzy rough aggregation operators is described in the technique [21]. Table- Expert weight information is calculated as follows: (1) (2) 0:336 0:347 0:317 Attributes weightes are computed as follows: With the help of EM s weights thr …nal closeness indices (F RCIs) are evaluated as follows: Table-23: Alternatives S 1 S 2 S 3 S 4 z R C Is 0:5783 0:6681 0:5878 0:3935 As a resultŜ 2 is our best option.
6.1. Results and Discussion. The details are given by the decision maker in the form of picture fuzzy rough sets.We utilised the picture improved TOPSIS plan to resolve the information in the comparison section, considering the neutral term to be zero.In terms of obtaining outcomes, S2 is the best option, which is the same as the one stated in [21]. As a result, the suggested methodology seems to be more realistic, practicable, useful, and generalised for solving MAGDM problems with uncertain information between EMs and criteria.

Conclusion
PFRS are new and e¤ective generalised procedures that have been chosen as operable opportunity to manage the uncertainties and ambiguity associated with MAGDM di¢culties, and so DMs feel much more comfortable using PFRS information in their judgement than IFS, IFRS, and PFRS. In this work, an unique improved TOPSIS-based decision-making strategy is developed to address with MAGDM problems in a PFRS system with entirely unknown DMs and criteria weights. To construct the PFRS entropy weight structure for evaluating the criteria weights over PFRS data, a GDM-based unique PFRS entropy measure is provided. Aggregation is conducted in the last steps, utilising the determined DMs weights to create a …nal ranking of alternatives, to prevent the failure of collective information during the process.Finally, numerical examples are illustrated to present the applicability and advantage of the introduced technique. Furthermore, the suggested method can be enhanced for future studies by adding other existing fuzzy sets and applying them to various MCGDM issues involving undetermined DM and criteria weights.
Authorship Contributions: All autors have equally contributed. Declaration Con ‡icts of interest: The authors declare that they have no con ‡ict of interest.