This research proposes multicriteria decision-making (MCDM)-based real-time Mesenchymal stem cells (MSC) transfusion framework. The testing phase of the methodology denotes the ability to stick to plastic surfaces, the upregulation and downregulation of certain surface protein markers, and lastly, the ability to differentiate into various cell types. First, two scenarios of an enhanced dataset based on a medical perspective were created in the development phase to produce varying levels of emergency. Second, for real-time monitoring of COVID-19 patients with different emergency levels (i.e., mild, moderate, severe, and critical), an automated triage algorithm based on a formal medical guideline is proposed, taking into account the improvement and deterioration procedures from one level to the next. For this strategy, Einstein aggregation information under the Pythagorean probabilistic hesitant fuzzy environment (PyPHFE) is developed. Einstein operations on PyPHFE such as Einstein sum, product, scalar multiplication, and their properties are investigated. Then, several Pythagorean probabilistic hesitant fuzzy Einstein aggregation operators, namely the Pythagorean probabilistic hesitant fuzzy weighted average (PyPHFWA) operator, Pythagorean probabilistic hesitant fuzzy Einstein weighted geometric (PyPHFEWG) operator, Pythagorean probabilistic hesitant fuzzy Einstein ordered weighted average (PyPHFEOWA) operator, Pythagorean probabilistic hesitant fuzzy Einstein ordered weighted geometric (PyPHFEOWG) operator, Pythagorean probabilistic hesitant fuzzy Einstein hybrid average (PyPHFEHA) operator and Pythagorean probabilistic hesitant fuzzy Einstein hybrid geometric (PyPHFEHG) operator are investigated. All the above-mentioned operators are helpful in design the algorithm to tackle uncertainty in decision making problems. In last, a numerical case study of decision making is presented to demonstrate the applicability and validity of the proposed technique. Besides, the comparison of the existing and the proposed technique is established to show the effectiveness and validity of the established technique.

Coronavirus disease 2019 (COVID-19) is a highly contagious sickness that can range from mild to severe. Fever, cough, and shortness of breath are common symptoms of COVID-19, which can escalate to acute respiratory distress syndrome, which causes breathing problems, a low ratio of the partial pressure arterial oxygen to the fraction of inspired oxygen (PaO2/FiO2), and multi-organ failure [

In fact, a system’s complexity grows more and more, making it impossible for DMs to select the best option among a wide range of viable options. It is difficult to put into words how challenging achieving a single objective is, yet it is not unattainable. Several corporations are faced with encouraging workers, identifying aims, and creating viewpoint. As a result, whether affecting individuals or panels, organizational decisions must consider a number of goals simultaneously. In this context, each DM should be restricted to finding the optimum solution for each factor associated with real problems, based on criteria that can be handled consciously. For decision-makers, establishing suitable ways for deciding on the appropriate option has become increasingly vital.

Classic or crisp procedures may not necessarily be the most successful when dealing with unstructured situations in decision-making circumstances. Zadeh [

Atanassov [

Wherever the MG and NMG numbers are more than one, the DMs are frequently confronted with certain qualities. In that circumstance, IFSs have little chance of producing a satisfactory result. To handle such a circumstance, Yager [

The concept of Pythagorean HFS (PyHFS) was suggested by Khan et al. [

Different priority levels are frequently assigned to each criterion in each emergency situation, adding to the task’s complexity. Ultimately, a COVID-19 patient prioritization process inside each emergency/triage situation necessitates a synchronized evaluation of the inverse relationship among the criteria, resulting in a trade-off. To address the first issue, an accurate automated triage guideline must be developed and implemented into the proposed MSC transfusion system. By prioritizing COVID-19 patients within each triage level, MCDM approaches are critical for overcoming the aforementioned problems. MCDM is a multi-objective decision model that is an extension of decision theory. It can address multi-criteria decision-making problems by constructing a decision matrix based on the intersection of each triage level’s evaluation criteria. The fundamental purpose of the MCDM is to rank/prioritize a group of alternatives based on a variety of evaluation criteria. For the MCDM technique Pythagorean Probabilistic Hesitant Fuzzy Information Aggregation when dealing with difficulties like this, Einstein Operations can be quite useful.

In this research work, we administered the Einstein aggregation operators (AOs) to PyPHFS environment, i.e., PyPHFEWA, PyPHFEOWA, PyPHFEHWA, PyPHFEWG, PyPHFEOWG and PyPHFEHWG operator. Idempotency, boundedness, and monotonicity are among the properties of the recommended operators that are established. The PyPHF AOs are taken into consideration by such operators in probabilistic scenarios, which is their main benefit. In the case of probabilistic material, the lack of PyPHFE AOs could lead to a scarcity of probabilistic information.

The paper is arranged in the prescribed manner.

In this section, let’s review the fundamentals of fuzzy sets, intuitionistic fuzzy sets, and Pythagorean fuzzy sets. These concepts will be used here when they have been reviewed.

In what follows, we symbolize by

(1)

(2)

(3)

(4)

(1)

(2)

(3)

(1)

(2)

(3)

(1)

(2)

(3)

(1)

(2)

(3)

(4)

(1) If

(2) If

(1) the PyPHFWA operator can be described as

(2) the PyPHFWG operator can be described as

(1) PyPHFOWA operator can be described as

(2) PyPHFOWG operator can be described as

(1)

(2)

The application of t-norms in FS theory at the intersection of two FSs is widely recognized. T-conorms are being used to model disjunction or union. These are a straightforward explanation of the conjunction and disjunction in mathematical fuzzy logic syntax, and they are utilized in MCDM to combine criteria.

The Einstein sum (

Based on the above Einstein operations, we give the following new operations on PyPHFEs.

(1)

(2)

(3)

(4)

In this section, we develop several new Einstein operators for PyPHFNs, namely the Pythagorean probabilistic hesitant fuzzy Einstein weighted averaging (PyPHFEWA) operator, the Pythagorean probabilistic hesitant fuzzy Einstein ordered weighted averaging (PyPHFEOWA) operator, the Pythagorean probabilistic hesitant fuzzy Einstein weighted geometric (PyPHFEWG) operator, and the Pythagorean probabilistic hesitant fuzzy Einstein ordered weighted geometric (PyPHFEOWG) operator.

Since both

Then

Thus, the result holds for

Assume that the results holds for

Now we will prove for

Thus

There are some properties which are fulfilled by the PyPHFEWA as follows:

Since

Thus,

Consider

Then

Let,

and

Let

There are some properties which are fulfilled by the PyPHFEWG as follows:

We developed the following ordered weighted operators based on PyPHFNs.

(2) Let

Flow chart of the proposed decision making methodology is given in

The Emergency Decision Making for COVID-19’s emergency circumstance is discussed in this part. Several researchers made decisions to minimize the influence of COVID-19 [

To combat NCoViD-2019, there are four emergency options:

Similarly for other alternatives

Therefore

The best choice is

Therefore

The best choice is

To show the applicability and appropriateness of the established Pythagorean probabilistic hesitant fuzzy Einstein operators-based methodology, we compared the approach based on the suggested operator in accordance with the existing decision assistance approach to validate the advantages and stability of the developed methodology.

Batool et al. [

Operator | Scores | Ranking | |||
---|---|---|---|---|---|

PyPHFWA [ |
0.0965 | 0.4777 | 0.0740 | 0.2172 | |

PyPHFWG [ |
0.4432 | 0.5644 | 0.1701 | 0.3827 | |

PyPHFOWA [ |
0.0935 | 0.4671 | 0.0735 | 0.1954 | |

PyPHFOWG [ |
0.4315 | 0.5511 | 0.1609 | 0.3842 | |

PyPHFHWA [ |
0.0978 | 0.4606 | 0.0717 | 0.2156 | |

PyPHFHWG [ |
0.4301 | 0.5511 | 0.1672 | 0.3725 | |

PyPHFEWA (Proposed) | 0.0496 | 0.3103 | 0.0449 | 0.2133 | |

PyPHFEWG (Proposed) | −0.0498 | 0.1185 | −0.036 | 0.1030 |

In a nutshell, our suggested solutions are expected to become more extensive and flexible than a few existing strategies for addressing MADM problems. Collected specialist data [

Given weight vector is

Comparative studies with collected specialist data by [

We conducted a thorough investigation into PyPHF aggregation operators in this paper. First, we have defined certain operational guidelines based on Einstein operations for PyPHFS. The weighted averaging and geometric Einstein aggregation operators have also been intended to deal with PyPHF data, such as Pythagorean probabilistic hesitant fuzzy averaging (PyPHFA) and Pythagorean probabilistic hesitant fuzzy geometric (PyPHFG) aggregation data in decision-making issues, due to the importance of operating procedures and a number of useful characteristics have been shown on them. To explain the usage, adaptability, and effectiveness of the procedure, we created a Healthcare framework and employed specified aggregation material to solve this problem.

We will examine the conceptual framework of Pythagorean probabilistic hesitant fuzzy sets for Einstein operations in future work using innovative decision-making approaches like as TOPSIS, VIKOR, EDAS and how these techniques are used in domains including analytical thinking, intelligent systems, soft computing, etc.

The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work by Grant Code: 22UQU4310396DSR32.

The authors received no specific funding for this study.

The authors declare that they have no conflicts of interest to report regarding the present study.