Intuitionistic hesitant fuzzy set (IHFS) is a mixture of two separated notions called intuitionistic fuzzy set (IFS) and hesitant fuzzy set (HFS), as an important technique to cope with uncertain and awkward information in realistic decision issues. IHFS contains the grades of truth and falsity in the form of the subset of the unit interval. The notion of IHFS was defined by many scholars with different conditions, which contain several weaknesses. Here, keeping in view the problems of already defined IHFSs, we will define IHFS in another way so that it becomes compatible with other existing notions. To examine the interrelationship between any numbers of IHFSs, we combined the notions of power averaging (PA) operators and power geometric (PG) operators with IHFSs to present the idea of intuitionistic hesitant fuzzy PA (IHFPA) operators, intuitionistic hesitant fuzzy PG (IHFPG) operators, intuitionistic hesitant fuzzy power weighted average (IHFPWA) operators, intuitionistic hesitant fuzzy power ordered weighted average (IHFPOWA) operators, intuitionistic hesitant fuzzy power ordered weighted geometric (IHFPOWG) operators, intuitionistic hesitant fuzzy power hybrid average (IHFPHA) operators, intuitionistic hesitant fuzzy power hybrid geometric (IHFPHG) operators and examined as well their fundamental properties. Some special cases of the explored work are also discovered. Additionally, the similarity measures based on IHFSs are presented and their advantages are discussed along examples. Furthermore, we initiated a new approach to multiple attribute decision making (MADM) problem applying suggested operators and a mathematical model is solved to develop an approach and to establish its common sense and adequacy. Advantages, comparative analysis, and graphical representation of the presented work are elaborated to show the reliability and effectiveness of the presented works.

In modern decision science, multi-attribute decision making (MADM) is a vital investigation area on how to choose the correct option corresponding to many prominent attributes [

The total of the grade of MS and the grade of NMS of each ordered pair set is smaller than or equivalent to 1 and greater than or equivalent to 0. The IFS has been receiving more consideration since its arrival [

An HFS allows the MS taking a set of conceivable values for example, in order to obtain a sensible decision outcome, a decision association, containing many DMs, which is approved to assess the grade that an alternative should fulfill a criterion. Consider there are three situations, few DMs offer 0.2, few offer 0.4, and the rest offer 0.9, and these units may not convince one another, thus the grade that the alternative should fulfill the criterion can be signified by an HF

Mostly, this is noted that one fuzzy framework is not enough to deal with practical problems. There is a common trend of combining two or more fuzzy frameworks. Therefore, by mixing IFS and HFS established the theory of IHFS. IHFS is also described by the grade of MS and the grade of NMS, whose summation is smaller than or equivalent to 1 and greater than or equal to 0. IHFS has emerged as a powerful instrument for illustrating vagueness of the MADM difficulties. The determination of the article is to present the idea of power aggregation operators based on IHFS by combining the theory IFS and HFS. We found that two different definitions of IHFS which were proposed by Beg et al. [

Enlarge several PA operators, as IHFPA operator, IHFPWA operator, IHFPOWA operator, IHFPHA operator, IHFPG operator, IHFPWG operator, IHFPOWG operator, IHFPHG operator and check their characteristics.

Explore the similarity measures based on IHFSs and justified with the help of numerical example.

Describe a new DM method consists over the proposal operations.

Provide some numerical to demonstrate the reliability and supremacy of described techniques.

The making of article is followed as in portion 2, it gives few fundamental notions as well as in this section we reviewed the definition of IHFS which are established by Beg et al. [

In this study, we review the idea of IHFS which was established by Beg et al. [

where

where

where

satisfying

where

where

and _{1} from _{2}, which meets the given properties:

The support (

Based upon intuitionistic hesitant fuzzy PA operators and PG, we will describe few IHFPG aggregation operators. Next, we will establish few intuitionistic hesitant fuzzy power arithmetic aggregation operators.

The purpose of this section is to establish few novel aggregation operators for IHFSs which are IHFPA, IHFPG, IHFPWA, IHFPWG, IHFPOWA, IHFPOWG, IHFPHA, IHFPHG operators and verify their fundamental properties. The mentioned operators are not only developed in this section but also their characteristics have been studied and their fitness is established using induction phenomenon.

where

and _{j}_{i}

If we will choose the grade of truth and falsity in the form of singleton sets then the

where

If we will choose the grade of truth and falsity in the form of singleton sets then the

_{j}

where

If we will choose the grade of truth and falsity in the form of singleton sets then the

_{j}_{j}

_{j}

_{j}

Further, we give an IHFPOWA operator as follows:

where

where

where

_{j}_{j}

_{j}

_{j}

_{j}

where _{j}

where

where

where

where

where _{j}_{i}

_{j}_{i}_{s}_{t}

If we will choose the grade of truth and falsity in the form of singleton sets then the

where

_{j}

where

If we will choose the grade of truth and falsity in the form of singleton sets then the

_{j}_{j}

_{j}

_{j}

Further, we give an IHFPOWG operator below:

Such that

where

where

_{j}_{j}

_{j}

_{j}

_{j}

where _{j}

where _{j}

where

where

The VSM is one of the important tools for the similarity degree between objects. We straightforwardly utilized Jaccard, Dice and Cosine SM. Presently in this segment we characterize VSMs and weighted VSMs (WVSMs) for IHFSs.

JSMs fulfill the following axioms:

If we will choose the grade of truth and falsity in the form of singleton sets then the

WJSMs fulfill the following axioms:

where

DSMs fulfills the following axioms:

If we will choose the grade of truth and falsity in the form of singleton sets then the

WDSMs fulfills the following axioms

where

In the portion, we use IHF power aggregation operators to multiple attribute DM through intuitionistic hesitant fuzzy data. Following hypotheses or concepts are utilized to signify the multiple attribute DM difficulties for possible calculation of developing technology commercialization with intuitionistic hesitant fuzzy data. Consider

where justify _{ij}_{ik}

where

_{j}_{ij}_{ij}_{ik}

where calculated weight _{ij}

where

or

to receive the total preference objects _{i}

_{i}_{i}_{i}_{i}_{j}_{i}_{j}_{i}_{j}_{i}_{j}_{i}_{j}

_{i}_{i}

_{1} is the technical development (_{2} is the potential market and market risk; (_{3} is the industrialized structure, human resource management, and economic circumstances (_{4} is the job creation and the development of science and technology. The five possible developing technology enterprises _{i}

Next, we use the method established to indicate potential evaluation of developing technology commercialization of four possible developing technology enterprises, see

_{i}

By using the formula of score value, we examine the score values of the aggregated values of

0.2112 | 0.1897 | |

−0.0498 | −0.1476 | |

0.3706 | 0.196 | |

0.118 | 0.1008 | |

0.156 | 0.1196 |

Further, we examine ranking results of the score values.

Methods | Ranking |
---|---|

IHFPWA | _{3} > _{1} > _{5} > _{4} > _{2} |

IHFPWG | _{3} > _{1} > _{5} > _{4} > _{2} |

From above analysis, we get as best option alternative

By using the

Ranking values of the above measures are summarized as follows:

Methods | Results | Ranking |
---|---|---|

Peng et al. [ |
||

Beg et al. [ |
||

Explored measures | ||

The best option is _{2}. Additionally, if we choose the intuitionistic hesitant fuzzy types of information’s with existing conditions that are the sum of the maximum (also for minimum) of the truth grade and minimum (also for maximum) of the falsity grade cannot exceed from unit interval, and the sum of the maximum of the truth grade and falsity grade exceeds the unit interval, then it is very difficult to cope with such types of issues. But, when we choose the condition as in this explorative study then the sum of the maximum of the truth grade and falsity grade cannot exceed from the unit interval. The theories of intuitionistic fuzzy set and hesitant fuzzy set describe the foundation of the intuitionistic hesitant fuzzy set. When we choose the intuitionistic fuzzy types of information’s or hesitant fuzzy types of information then the explored approach easily copes with it. But, if we choose the intuitionistic hesitant fuzzy types of information’s, then the existing types of theories are cannot able to cope with it.

The comparative analysis of the explored measures with selected existing measures are summarized in

From above analysis, the three different measures above share the same ranking values and the best option is _{2}. The graphical representation for the information of

From the above analysis, the explored measures and operators based on IHFSs are more perfect and more proficient then existing methods and measures.

We explored the improved intuitionistic hesitant fuzzy set with a new condition that is the sum of the maximum of the truth grade and maximum of the falsity grade which cannot exceed from the unit interval. Additionally, we examine the multi-attribute decision making challenge built upon power aggregating operators with IHF data. So, inspired from the model of power aggregating operators, we established few power aggregation operators for aggregating IHF information: IHFPA operator, IHFPG operator, IHFPWA operator, IHFPWG operator, IHFPOWA operator, IHFPOWG operator, IHFPHA operator, and IHFPHG operator. Additionally, some similarity measures based on IHFSs are also explored and their special cases discussed. Outstanding feature of these recommended operators are examined. So, we used operators to establish few methods to resolve the IHF multi attribute DM difficulties. A helpful example is presented to confirm the established methodology and to determine its practicability and efficiency. The advantages, comparative analysis, and geometrical representation of the presented works are also discussed in detailed.

Notably, that the article results of the article can be expanded to the IvIHF situation and further fuzzy situations. In superior study, it is enough to get the implementation of these operators to resolve the actual DM drawbacks as fuzzy investigation, unsure programming and image recognition, etc. We must also deal with few new operators for the foundation of PHFNs for example, modify them to complex q-rung fuzzy aggregation operators [