Hypersoft set theory is a most advanced form of soft set theory and an innovative mathematical tool for dealing with unclear complications. Pythagorean fuzzy hypersoft set (PFHSS) is the most influential and capable leeway of the hypersoft set (HSS) and Pythagorean fuzzy soft set (PFSS). It is also a general form of the intuitionistic fuzzy hypersoft set (IFHSS), which provides a better and more perfect assessment of the decision-making (DM) process. The fundamental objective of this work is to enrich the precision of decision-making. A novel mixed aggregation operator called Pythagorean fuzzy hypersoft Einstein weighted geometric (PFHSEWG) based on Einstein’s operational laws has been developed. Some necessary properties, such as idempotency, boundedness, and homogeneity, have been presented for the anticipated PFHSEWG operator. Multi-criteria decision-making (MCDM) plays an active role in dealing with the complications of manufacturing design for material selection. However, conventional methods of MCDM usually produce inconsistent results. Based on the proposed PFHSEWG operator, a robust MCDM procedure for material selection in manufacturing design is planned to address these inconveniences. The expected MCDM method for material selection (MS) of cryogenic storing vessels has been established in the real world. Significantly, the planned model for handling inaccurate data based on PFHSS is more operative and consistent.

The solution to the problems in our daily lives is based on the classification of information, data, the collection of facts, etc. The critical question in decision analysis is the absence of accurate facts. This statistical difference is bridged by taking a scientific model and applying the appropriate DM. DM’s ideas can support the manufacturing enterprise, assemble, and categorize multiple priorities from best to worst viable alternative. As a result, it is a tool to help us select, categorize, and establish our prospects and comprehensively evaluate alternatives. MS is intense in enterprise and product development. The material chosen affects the manufacturer’s success and affordability [

MCDM has deliberated the best applicable procedure for the verdict and the best adequate alternative from all possible choices, ensuing criteria, or attributes. In real-life circumstances, most decisions are taken when the objectives and limitations are usually indefinite or ambiguous. To overcome such ambiguities and anxieties, Zadeh offered the idea of the fuzzy set (FS) [

The models mentioned above have been well-recognized by specialists. Still, the existing IFS cannot handle the inappropriate and vague data because it is deliberate to envision the linear inequality concerning the membership and non-membership grades. For example, if decision-makers choose membership and non-membership values 0.9 and 0.6, respectively, then

Peng et al. [

All the methods mentioned above have too many applications in many fields. But due to their inefficiency, these methods have many limitations in terms of parameterization tools. In presenting the solution to obscurity and ambiguity, Molodtsov [

Smarandache [

The PFHSS competently deals the complex issues considering the multi sub-attributes of the considered parameters in the DM process. To keep this advantage in mind, we establish the AO for PFHSS.

The Einstein operator is a well-known charming guesstimate AO. It is noticed that the prevailing Einstein AOs look unenthusiastic in marking the exact judgment through the DM procedure in some circumstances. To overwhelm these particular difficulties, these AOs need to be modified. We demonstrate advanced operational laws based on Einstein norms for Pythagorean fuzzy hypersoft numbers (PFHSNs).

Establish the PFHSEWG operator using the above-mentioned Einstein operational laws with fundamental properties.

A novel MCDM technique was established based on the proposed PFHSEWG to cope with DM issues under the PFHSS environment.

MS is a significant aspect of engineering as it sees the practical standards of all constituents. MS is a time-consuming but significant step in the enterprise procedure. The industrialist’s productivity, effectiveness, and character will suffer as an outcome deprived of material selection.

Comparative analysis of the developed MCDM technique is proposed with current approaches to deliberate the practicality and supremacy of the planned model.

This study is systematized as follows: Basic knowledge of some important notions like SS, HSS, IFHSS, PFHSS, and Einstein norms have been deliberate in

This section remembers some essential concepts such as SS, HSS, IFHSS, and PFHSS.

Also, it can be defined as follows:

It is also defined as

It is also defined as

It is also defined as

A Pythagorean fuzzy hypersoft number (PFHSN) can be stated as

For readers’ aptness, the PFHSN

But, in some cases, the above-defined score function cannot handle the scenario. For example, if we consider the two PFHSNs, such as

The consequent comparative laws will be used

If

If

If

If

Under the Pythagorean fuzzy environment, Einstein sum

This section will introduce a novel Einstein-weighted AO such as the PFHSEWG operator for PFHSNs with essential properties.

Proof: We will use mathematical induction to demonstrate the above result.

For n = 1, we get

For m = 1, we get

So,

Assume that the equation grasps for

Now we show the

So, it is true for

As we know that

Proof: As we know that

Again

Let PFHSWG

Then, inequalities

So,

If

If

Then,

From inequalities

Hence, from Examples 3.1 and 3.2, it is proved that

Proof: As we know that

Proof: Let

Let

Again, let

Let

Let

If

If

If

So proved that

Proof: Let

So,

This section proposes a DM method to address the difficulties of MCDM based on the planned PFHSEWG operator with a numerical example.

Consider

Step 1: Obtain decision matrices for each alternative in the form of PFHSNs

Step 2: Convert the cost type attributes to benefit type using the normalization rule.

Step 3: Use the settled PFHSEWG operator to collective the PFHSNs

Step 4: Use

Step 5: Pick the alternative with the highest score and check the ranking.

In this section, a practical MCDM problem comprises a decisive adequate material selection model to confirm that the conventional approach is pertinent and reasonable.

Consequently, energy scarcities resolve instantaneously if the world completely alters to a hydrogen budget that eradicates fossil fuel feasting. This component delivers significant tasks in verdict an appropriate power source. Though, this investigation will not insurance this issue. As mortality is impending, the ‘end of low-priced oil’ eras,” with complete compromise in science and power engineering that essential discover new energy exporters. Severe reduction procedure across nations exposed hydrogen will be the eventual optimal. Hydrogen, conceivable as complementary energy in cars, influences industrial innovations such as hydrogen fuel cells to deliver manufacturing deprived of producing any

It kinds intelligence to renovate fossil discarded energies such as natural gas is earliest transformed into hydrogen. In conclusion, while fossil fuels develop excessively and prospective unlawfully for worldwide warming, renewable, most important energy will originate into the depiction for financial or environmental causes. In expressions of power, the recently formed hydrogen fuel is dissimilar to the frequently used ones in gratified weight and volume. This hydrogen is frivolously associated with its energy capability is the top prominent feature. The energy content of hydrogen per kilogram is 120 MJ. Hydrogen has a little volumetric energy compactness related to its exceptional gravimetric density. The compactness of hydrogen is committed by its accumulation state. Unfluctuating densities up to 700 bar are not massive sufficient belongings of hydrocarbons similar to gasoline and diesel. Only fluid hydrogen can influence a reasonable amount, still less than a quarter of the amount of gasoline. So, hydrogen containers for motorized solicitations will conquer more than fluid hydrocarbons formerly used containers [

Fracture toughness: The steaming point of melted nitrogen is around −196°C, whereas the steaming point of liquefied nitrogen is around. The temperature of hydrogen gas is approximate −253°C. The substantial cannot find ductility and converts hard. So, the considerable requirement is robust and sufficient to endure Inelastic crack. Face-centered cube metal webs are suitable since they are impervious to low temperatures. All nickel-copper compounds, aluminum, its compounds, and austenitic stainless steels contain an extra 7% nickel to construct a storing cryogenic vessel [

Material assortment in any manufacturing arena is a very significant enterprise phase. Manufacturing enterprise is prepared by enactment, budget, ecological compassion objectives, And commonly inadequate by the material. The most acceptable product strategy selects the best appropriate material design criteria by providing an extreme presentation at the lowermost probable budget. Material selection is By seeing numerous contradictory DM procedures. AO shows a vital part in DM. The existing Einstein AO has originated as a DM procedure in this circumstance. These AOs must be modernized to talk about these definite concerns. We intend some novel operations and escorting AO for aggregating innumerable PFHSN. Our projected ideal outclasses other models. Conferring to the clarification stated above and DM perception, all structures can be categorized. The case study was shown in a motorized portions engineering corporation in Malaysia, and a motorized constituent, cryogenic storing, accompanied the study. As part of applying the concept of sustainability, companies must choose suitable materials for produced parts. It focuses first on cryogenic storage containers and then on other factors input of weights for gathering parameters and materials from DM. PFHSN theory and proposed AO are used to overcome complexity and indecision human judgment. MS with three remember the essential pillar of sustainability: materials must be reasonable, ecologically pleasant, and beneficial to humanity.

The most imperative aspects (parameters) to consider when selecting a substantial dashboard DM. Choose the procedure starts with an initial screening of material used for dashboards, captivating into justification structures intrinsic to the application. In the screening process, identify the fabrics that may be appropriate. It is serious about deciding the material that can be used initial MS for the instrument board process. Four materials are selected, subsequently examining the abilities: ^{T}. Let

Step 1: According to the expert’s opinion, Pythagorean fuzzy hypersoft decision matrices for all alternatives are given in

Step 2: All parameters are of the same type. So, no need to normalize.

Step 3: Apply the proposed PFHSEWG operator to the obtained data (

Step 4: Use

Step 5: Compute the ranking of the alternatives

Since the material estimation surprises at the theoretic phase through the enactment stage of the plan, there is extra scope to area the appropriateness of the particular materials. Face-centered cube materials are used at small temperatures of −163°C. Austenitic steel

To demonstrate the efficiency of the anticipated approach, a comparison with some standing methods under the IFS, IFSS, IFHSS, PFS, PFSS, and proposed PFHSS model.

The planned methodology is competent and realistic; we have established an innovative MCDM model under the PFHSS setting over the PFHSEWG operator. Our projecting model is more talented than prevalent methods and can produce the most delicate significance in MCDM problems. The collective model is multipurpose and familiar, adapting to budding volatility, engagement, and productivity. Different models have specific ranking procedures, so there is an immediate difference between the rankings of the proposed techniques to be feasible according to their assumptions. This scientific study and evaluation conclude that results obtained from existing methods are unpredictable compared to hybrid structures. Furthermore, many hybrid FS, IFSS, IFHSS, and PFSS become uncommon in PFHSS due to some fortunate circumstances. It is easy to combine incomplete and uncertain facts in DM techniques. They were mixing inaccurate and insecure data in the DM process. Thus, our intended methodology will be more skilled, imperative, superior, and restored than various hybrid-structured FS.

Fuzzy information | Aggregated parameters information | Einstein aggregated parameters information | Multi sub-attributes information of each attribute | |
---|---|---|---|---|

IFEWG [ |
✓ | × | ✓ | × |

IFWG [ |
✓ | × | × | × |

IFSWG [ |
✓ | ✓ | × | × |

IFHSWG [ |
✓ | ✓ | × | ✓ |

PFSWG [ |
✓ | ✓ | × | × |

PFEWG [ |
✓ | ✓ | ✓ | × |

PFSEOWG [ |
✓ | ✓ | ✓ | × |

PFHSWG [ |
✓ | ✓ | × | ✓ |

Proposed operator | ✓ | ✓ | ✓ | ✓ |

To endorse the usefulness of the planned technique, we compare the attained outcomes with some state-of-the-arts in the PFSS setting are concise in

Authors | AO | Alternatives ranking | Optimal material |
---|---|---|---|

Wang et al. [ |
IFWG | ||

Wang et al. [ |
IFOWG | ||

Arora et al. [ |
IFSWG | ||

Wang et al. [ |
IFEWG | ||

Wei et al. [ |
PFWG | ||

Rahman et al. [ |
PFEWG | ||

Zulqarnain et al. [ |
PFSWG | ||

Zulqarnain et al. [ |
PFSEWG | ||

Zulqarnain et al. [ |
PFSEOWG | ||

Zulqarnain et al. [ |
IFHSWG | ||

Siddique et al. [ |
PFHSWG | ||

Proposed | PFHSEWG |

It is also an appropriate tool for dealing with contemptible inaccuracies and misrepresentation in DM plans. The advantage of expecting skill and associated dealings compared to existing methods is to avoid inspirations based on abominations. Hence, it is a proper tool for integrating erroneous and vague data in DM.

In engineering, the subtle stability of designing is impartial; authentic materials and manufacture comprise wide-ranging matters. Mathematical modeling in manufacturing enterprise establishments utilizes all capitals while combining design objectives under financial, superior, and security constraints. Questions must be defined for the most acceptable decision, conferring to judgment necessities. In actual DM, the assessment of alternative facts delivered by the expert is habitually imprecise, rough, and unpredictable; thus, PFHSNs can be used to conduct this indeterminate info. The core goal of this research is to use Einstein’s norms to develop some operational laws for PFHSS. Then, a new operator, such as PFHSEWG, developed according to the designed operational laws. Some fundamental properties have been presented using our developed PFHSEWG operator. Also, a DM approach is established to address MCDM problems based on the endorsed operator. To certify the robustness of the settled approach, we provide an inclusive mathematical illustration for MS in the manufacturing industry. A comparative analysis has been presented to ensure the practicality of the planned model. Lastly, based on the outcomes attained, it is determined that the technique projected in this study is the most practical and effective way to solve the problem of MCDM. In the future, several other hybrid AOs for PFHSS will be introduced with their decision-making techniques. Furthermore, the developed AOs can be extended to T-spherical fuzzy hypersoft, and q-rung orthopair fuzzy hypersoft settings with decision-making approaches.

The authors extend their appreciation to Deanship of Scientific Research at King Khalid University, for funding this work through General Research Project under Grant No. GRP/93/43.