Experts use Pythagorean fuzzy hypersoft sets (PFHSS) in their investigations to resolve the indeterminate and imprecise information in the decision-making process. Aggregation operators (AOs) perform a leading role in perceptivity among two circulations of prospect and pull out concerns from that perception. In this paper, we extend the concept of PFHSS to interval-valued PFHSS (IVPFHSS), which is the generalized form of interval-valued intuitionistic fuzzy soft set. The IVPFHSS competently deals with uncertain and ambagious information compared to the existing interval-valued Pythagorean fuzzy soft set. It is the most potent method for amplifying fuzzy data in the decision-making (DM) practice. Some operational laws for IVPFHSS have been proposed. Based on offered operational laws, two inventive AOs have been established: interval-valued Pythagorean fuzzy hypersoft weighted average (IVPFHSWA) and interval-valued Pythagorean fuzzy hypersoft weighted geometric (IVPFHSWG) operators with their essential properties. Multi-criteria group decision-making (MCGDM) shows an active part in contracts with the difficulties in industrial enterprise for material selection. But, the prevalent MCGDM approaches consistently carry irreconcilable consequences. Based on the anticipated AOs, a robust MCGDM technique is deliberate for material selection in industrial enterprises to accommodate this shortcoming. A real-world application of the projected MCGDM method for material selection (MS) of cryogenic storing vessels is presented. The impacts show that the intended model is more effective and reliable in handling imprecise data based on IVPFHSS.

MCGDM is deliberated as the most suitable method for a verdict on the adequate alternative from all possible choices, following criteria or attributes. Most decisions are taken when the intentions and confines are usually unspecified or unclear in real-life circumstances. Zadeh offered the idea of the fuzzy set (FS) [

The models mentioned above have been well-recognized by the specialists. Still, the existing IFS cannot handle the inappropriate and vague data because it envisions the linear inequality among the MD and NMD. For example, if decision-makers choose MD and NMD 0.6 and 0.7, respectively, then the IFS, as mentioned earlier, cannot deal with it because 0.6 + 0.7 ≥ 1. Yager [

All of the above techniques have broad applications, but these theories have some limitations on parametric chemistry due to their ineffectiveness. Molodtsov [

Samarandche [

The IVPFHSS capably contracts the multifaceted concerns seeing the multi sub-attributes of the deliberated parameters in the DM procedure. To preserve this benefit in concentration, we prolong the PFHSS to IVPFHSS and establish the AOs for IVPFHSS.

The AOs for IVPFHSS are well-known attractive estimate AOs. It has been observed that the prevalent AOs aspect is unresponsive to scratch the precise finding over the DM process in some situations. To overcome these specific complications, these AOs necessary to be revised. We determine innovative operational laws for interval-valued Pythagorean fuzzy hypersoft numbers (IVPFHSNs).

Interval-valued Pythagorean fuzzy hypersoft weighted average and geometric operators have been introduced with their necessary properties using developed operational laws.

A novel algorithm based on the planned operators to resolve the DM problem is established to resolve MCGDM issues under the IVPFHSS scenario.

Material selection is an imperative feature of manufacturing as it realizes the concrete conditions for all ingredients. MS is an arduous but significant step in professional development. The manufacturer’s efficiency, productivity, and eccentric will suffer due to the absence of material selections.

A comparative analysis of advanced MCGDM technique and current methods has been presented to consider utility and superiority.

The organization of this paper is assumed as follows: the second section of this paper involves some basic notions that support us in developing the structure of the subsequent study. In

This section contains some basic definitions that will structure the following work.

Also, it can be defined as follows:

It is also defined as

It is also defined as

(

It is also defined as

(

The PFHSN is stated as

In this section, we will extend the idea of IVPFSS to interval-valued Pythagorean fuzzy hypersoft sets (IVPFHSS) with some fundamental notions and introduce the operational laws for IVPFHSNs. We propose interval-valued Pythagorean fuzzy hypersoft weighted average (IVPFHSWA) and interval-valued Pythagorean fuzzy hypersoft geometric (IVPFHSWG) operators using the developed operational laws.

It is also defined as

(

The IVPFHSN can be stated as

The score and accuracy functions have been presented to compute the alternative ranking for IVPFHSS can be stated as, if

And

For

For

So, the above theorem is proved for

Assume that for

For

Hence, it holds for

Let

If

As

Let

Similarly,

Let

Using the score function, we have

By order relation between two IVPFHSNs, we have

Let

So,

Prove that

So,

For

For

So, for

Now, for

For

So, it is proved the for

By using the above theorem, we have

If

As

Let

Similarly,

If

Using the score function,

By order relation between two IVPFHSNs, we have

Let

So,

Prove that

So,

A decision-making method has been present to resolve the MCGDM obstacles to authenticate the implication of the planned AOs. Also, a statistical illustration has been offered to confirm the pragmatism of the developed methodology.

Consider

Step-1: Obtain a decision matrix in IVPFHSNs for each alternative according to the expert’s opinion.

Step-2: Convert the cost type attributes to benefit type using the normalization rule and establish the normalized decision matrices.

Step-3: Calculate the aggregated values for each alternative using developed IVPFHSWA and IVPFHSWG.

Step-4: Calculate the score values for each alternative.

Step-5: Examine the ranking of the alternatives.

It is an intelligent transformation of fossil waste energy, such as natural gas first converted into hydrogen. In inference, despite the overdevelopment of fossil fuels and the potential for global warming, the most important renewable energy sources will originate from the description of financial or environmental reasons. The recently formed hydrogen fuel is different in weight and volume from the commonly used hydrogen fuel in power performance. This hydrogen, irrelevant to its energy capacity, is the most prominent feature. The energy content per kilogram of hydrogen is 120 MJ. The advantage of methanol is an extraordinary six times [

The most significant features (parameters) to deliberate when electing a materiality dashboard DM. The assortment method initiates with a preliminary screening of the material used for the dashboard and is captivated by the validation configuration in-built into the application. Throughout the airing progression, potentially proper materials are acknowledged. Defining the ingredients that can be used by the preliminary MS of the dashboard fashioning is serious. Then select from four material assessment abilities:

Step-1 and Step-2 are similar to

Subsequently, the material assessment wonders at the theoretical level through the depiction phase of the strategy; there is more possibility to the extent of the correctness of the specific materials. Face-centered cube materials are typically used at minor temperatures −163°C and

A comparison among the projected model and prevalent approaches is planned to validate the efficacy of the offered technique in the subsequent section.

The intended method is proficient and realistic; in the IVPFHSS setting, we construct an inventive MCGDM model on the IVPFHSWA and PFHSEWG operators. Our planned model is more talented than prevalent techniques and can produce the most subtle implications in MCGDM difficulties. The cooperative model is multipurpose and conversant, adjusting to evolving instability, commitment, and output. Different models have particular ranking processes, so there is a straight modification among the rankings of the anticipated methods conferring to their expectations. This systematic study and assessment determined that the outcomes attained from prevailing procedures are irregularly equated to hybrid structures. Also, due to some favorable situations, many mixed IVFS, IVIFS, IVPFS, IVIFSS, and IVPFSS grow into special in IVPFHSS. It is easy to syndicate insufficient and ambiguous data in DM procedures. Imprecise and anxious facts are mixed in the DM procedure. Hence, our scheduled method will be more proficient, crucial, superior, and better than numerous mixed FS structures.

Fuzzy information | Aggregated attributes information | Aggregated sub-attributes information of any attribute | Aggregated information in intervals form | |
---|---|---|---|---|

IVFS [ |
✓ | × | × | ✓ |

IVIFWA [ |
✓ | × | × | ✓ |

IVIFWG [ |
✓ | × | × | ✓ |

IVPFWA [ |
✓ | × | × | ✓ |

IVPFWG [ |
✓ | × | × | ✓ |

IFSWA [ |
✓ | ✓ | × | × |

IFSWG [ |
✓ | ✓ | × | × |

IVIFSWA [ |
✓ | ✓ | × | ✓ |

IVIFSWG [ |
✓ | ✓ | × | ✓ |

PFSWA [ |
✓ | ✓ | × | × |

PFSWG [ |
✓ | ✓ | × | × |

PFSIWA [ |
✓ | ✓ | × | × |

PFSIWG [ |
✓ | ✓ | × | × |

IVPFSWA [ |
✓ | ✓ | × | ✓ |

IVPFSWG [ |
✓ | ✓ | × | ✓ |

IFHSWA [ |
✓ | ✓ | ✓ | × |

IFHSWG [ |
✓ | ✓ | ✓ | × |

PFHSWA [ |
✓ | ✓ | ✓ | × |

PFHSWG [ |
✓ | ✓ | ✓ | × |

PFHSIWA [ |
✓ | ✓ | ✓ | × |

PFHSIWG [ |
✓ | ✓ | ✓ | × |

Proposed IVPFHSWA | ✓ | ✓ | ✓ | ✓ |

Proposed IVPFHSWG | ✓ | ✓ | ✓ | ✓ |

To prove the usefulness of the planned technique, we equate the attained consequences with some prevailing approaches under the setting of IVPFS, IVIFSS, and IVPFSS. A summary of outcomes is specified in

Authors | AO | Alternatives ranking | Optimal choice | ||||
---|---|---|---|---|---|---|---|

Wang et al. [ |
IVIFWA | 0.4573 | 0.3509 | 0.3681 | 0.2146 | ||

Xu et al. [ |
IVIFWG | 0.3952 | 0.3104 | 0.2914 | 0.2753 | ||

Peng et al. [ |
IVPFWA | 0.0251 | 0.0154 | 0.0198 | 0.0247 | ||

Rahman et al. [ |
IVPFWG | 0.0856 | 0.0475 | 0.0786 | 0.0302 | ||

Zulqarnain et al. [ |
IVIFSWA | 0.0723 | 0.0530 | 0.0584 | 0.0235 | ||

Zulqarnain et al. [ |
IVIFSWG | 0.7234 | 0.2365 | 0.5840 | 0.6525 | ||

Zulqarnain et al. [ |
IVPFSWA | 0.0834 | 0.0377 | 0.0121 | 0.0141 | ||

Zulqarnain et al. [ |
IVPFSWG | 0.0754 | 0.0524 | 0.0251 | 0.0114 | ||

Proposed | IVPFHSWA | 0.0599 | 0.0578 | 0.0266 | −0.0382 | ||

Proposed | IVPFHSWG | 0.0752 | 0.0654 | 0.0242 | 0.0114 |

The graphical demonstration of

In manufacturing, the refined solidity of manipulation is neutral; authentic materials and fabrication encompass wide-ranging materials. Mathematical demonstration in industrial inventiveness formations exploits all assets while merging design intentions under financial, superior, and safety limitations. Inquiries must be restricted for best judgment, consulting to decision requirements. In genuine DM, the valuation of alternative facts conveyed by the professional is consistently inaccurate, irregular, and impulsive, so IVPFHSNs can be used to comport this uncertain data. The principal objective of this work is to prolong the Pythagorean fuzzy hypersoft sets to interval-valued Pythagorean fuzzy hypersoft sets. Firstly, we introduce the operational laws for the interval-valued Pythagorean fuzzy hypersoft setting. Considering the developed operational laws, we presented the IVPFHSWA and IVPFHSWG operators for IVPFHSS with their desired properties. Also, a DM method has been planned to address MCGDM complications based on the validated operators. To state the stoutness of the developed methodology, we deliver a comprehensive mathematical illustration for MS in manufacturing engineering. A comprehensive analysis of some existing procedures is described to ensure the practicality of the developed approach. Lastly, based on the consequences achieved, it is determined that the method proposed in this study is the most practical and operative way to explain the problem of MCGDM.