A model for emergency supply management under extended EDAS method and spherical hesitant fuzzy soft aggregation information

Due to the frequent occurrence of numerous emergency events that have significantly damaged society and the economy, the need for emergency decision-making has been manifest recently. It assumes a controllable function when it is critical to limit property and personal catastrophes and lessen their negative consequences on the natural and social course of events. In emergency decision-making problems, the aggregation method is crucial, especially when there are more competing criteria. Based on these factors, we first introduced some basic concepts about SHFSS, and then we introduced some new aggregation operators such as the spherical hesitant fuzzy soft weighted average, spherical hesitant fuzzy soft ordered weighted average, spherical hesitant fuzzy weighted geometric aggregation, spherical hesitant fuzzy soft ordered weighted geometric aggregation, spherical hesitant fuzzy soft hybrid average, and spherical hesitant fuzzy soft hybrid geometric aggregation operator. The characteristics of these operators are also thoroughly covered. Also, an algorithm is developed within the spherical hesitant fuzzy soft environment. Furthermore, we extend our investigation to the Evaluation based on the Distance from Average Solution method in multiple attribute group decision-making with spherical hesitant fuzzy soft averaging operators. And a numerical illustration for “supply of emergency aid in post-flooding the situation” is given to show the accuracy of the mentioned work. Then a comparison between these operators and the EDAS method is also established in order to further highlight the superiority of the established work.


Spherical hesitant fuzzy soft average aggregation operators
Operators are necessary to develop a robust framework for decision-making in a spherical hesitant fuzzy soft environment, where uncertainty and hesitancy are inherent in the data. So, some of the averaging aggregated operators of SHFSS are given as: Spherical hesitant fuzzy soft weighted average (SHFSWA) aggregation operators. This operator is used to calculate the weighted average of a set of SHFSS, where the weights are represented by SHFSS. This operator is essential because it allows for the aggregation of multiple SHFSS with different degrees of uncertainty, which is a common scenario in decision-making problems.
Proof This conclusion has to be supported by mathematical induction.
For r = 2; By using the operational law, we have: . www.nature.com/scientificreports/ Thus, the results are true for r = 2 . Assume the results also hold for p = z.
Proof The proof is similar to above Theorem4.2.
Moreover, similarity to the SHFSWA operator, the SHFSOWA operator has some important properties, such as idempotency, boundedness, monotonicity.
Spherical hesitant fuzzy soft hybrid average (SHFSHA) operator. According to Definition 4.1 and 4.4, SHFSWA operators only weight the spherical hesitant fuzzy soft number itself, whereas SHFSOWA operators weight the ordered ranks of the spherical hesitant fuzzy soft number rather than the arguments themselves. Therefore, in both the SHFSWA and SHFSOWA operators, the weights represent two distinct aspects. However, merely one of them is taken into account by either operator. In the paragraphs that follow, we'll suggest using the spherical hesitant fuzzy soft hybrid average (SHFSHA) operator to tackle this issue.
Proof The proof is directly analogous to above Theorem-4.2.
Moreover, similarity to the SHFSWA operator, the SHFSHA operator has some important properties, such as idempotency, boundedness, monotonicity.

Spherical hesitant fuzzy soft weighted geometric aggregation (SHFSWGA) operator. This
operator is used to calculate the geometric average of a set of SHFSS. The use of this operator is important because it takes into account both the degree of MG, nMG, NMG, along with the degree of hesitancy of the elements in the SHFSS. This provides a more balanced consideration of all the elements in the set, instead of just focusing on the most dominant ones. www.nature.com/scientificreports/ operator is the mapping defined as SHFSWGA : ϒ p → ϒ , where ( ϒ is the family of all SHFSNs) such that SHFSWGA Ŵ(j r ) = (φ Ŵ(j r ) , χ Ŵ(j r ) , ψ Ŵ(j r ) ), (r = 1, 2, 3, .., p).

Theorem 4.9
Let Ŵ(j r ) = (φ Ŵ(j r ) , χ Ŵ(j r ) , ψ Ŵ(j r ) ), (r = 1, 2, 3, .., p), be an SHFSNs, the aggregated data by SHFSWGA operator is also an SHFSNs, and given by Proof This conclusion has to be supported by mathematical induction. For r = 2; By using the operational law, we have: Thus, the results are true for r = 2 . Assume the results also hold for p = z. www.nature.com/scientificreports/ Further, suppose that the results are true for p = z + 1 , So combined the above two conditions, we have the following form; It is obvious from the expression above that aggregated value is also SHFSN. Consequently, the outcome is valid for all n.
which presents the proof.
Proof The proof is directly analogous to Theorem-4.9.
Moreover, similarity to the SHFSWGA operator, the SHFSOWGA operator has some important properties, such as idempotency, boundedness, monotonicity.
Spherical hesitant fuzzy soft hybrid geometric (SHFSHG) operator. According to Definition 4.8 and 4.11, SHFSWG operators only weight the spherical hesitant fuzzy soft number itself, whereas SHFSOWG operators weight the ordered ranks of the spherical hesitant fuzzy soft number rather than the arguments themselves. Therefore, in both the SHFSWG and SHFSOWG operators, the weights depict two distinct aspects. However, merely one of them is taken into account by either operator. In the paragraphs that follow, we'll suggest using the spherical hesitant fuzzy soft hybrid geometric (SHFSHG) operator to tackle this issue.
Moreover, similarity to the SHFSWGA operator, the SHFSHG aggregation operator has some important properties, such as idempotency, boundedness, monotonicity.

Decision making model under shfs aggregation information
The flow chart of the the proposed model is shown in Fig. 1.
Here, an MCDM technique for solving MCDM issues that arise in the context of SHFSS is examined, that is based on SHFSWA, SHFSOWA, SHFSWGA, SHFSOWGA, SHFSHG, and SHFSHA aggregation operators.
Step 1 Arrange all expert assessment information for every alternative to their respective parameters to create the decision matrix.
Step 2 Ordered the overall decision matrix.
Step 3 Utilize the SHFSS decision matrix because of the grading,i.e., membership, neutral, and non-membership grade.
Step 5 Calculate the score values for each alternative according to the following formula; Step-6 Rank the outcomes for each alternative Ć = {e 1 , e 2 , e 3 , ..., e m } and select the most effective one.  decision-makers to solve because they are complicated, time-consuming, lack data, and have an impact on mental processes. With the use of membership, neutral, and non-membership values, SHFSS is more adaptable in illustrating the judgment of a group of "decision-makers" in EmDMPs. SHFSS enables decision-makers to choose an unbiased subset of attributes based on their intuition. In order to demonstrate the value of the existing work, we will give a thorough overview of the above-mentioned method to MADM in this part, using an illustrated example Case study 5.1. Supply of emergency aid for post-flooding situation. Natural disasters and global warming exert a serious threat to Pakistan. For years, society has been plagued by catastrophic events like earthquakes, typhoons, flooding, and drought, which frequently destroy the basics on which the existence of huge numbers of families is built. It has been seen that the community's response determines whether a disaster becomes a catastrophe. Pakistani people face numerous difficulties and require assistance in extreme and devastating weather events, which are becoming more frequent, such as the historic floods of 2022. That puts people's lives, well-being, and assets in danger. Therefore, the government has established a number of measures in the face of uncontainable natural disasters so that after the disaster hits, the disaster-stricken citizens can be swiftly saved and their lives and production plans can be restored. The use of SHFSS can provide a powerful and flexible tool for modeiling and analyzing emergency supply management information and allowing decision-makers to more effectively manage the complex and uncertain situations that arise in emergency scenarios. Building the emergency shelters is one of them, and it's very important. According to personal observation, three criteria are typically taken into account when establishing emergency shelters: In particular, three kinds of (alternatives) are taken to provide aid in emergency situations. such as; Availability ( e 1 ), Convenience ( e 2 ), Safety ( e 3 ).
In order to provide Emergency Aid as soon as possible, the emergency command department invited three decision-makers ER = {ER 1 , ER 2 , ER 3 } from government officials, experts in emergency decision-making, experts from international rescue organizations and local residents to participate in the emergency decision-making. These decision-makers evaluated the three alternatives according to three criteria, and have been displayed in Table 1. The three alternatives for emergency aid are listed in detail below.
Availability(e 1 ) Availability refers to the extent to which the supply of emergency aid is accessible and can be obtained in sufficient quantities to meet the needs of those affected by the post-flooding situation. It is the probability that an item will operate competently when it is used to restore emergency conditions in an ideal support environment. The service will be deemed unavailable if the people who have been impacted by the flood are unable to access the service. So, the availability of emergency aid to flood-infected people is most important. In the context of emergency aid, availability can be affected by various factors, such as the location and severity of the flooding, the availability of transportation and communication infrastructure, and the capacity of aid providers.
Convenience(e 2 ) The efficiency of being accessible, simple to use, beneficial, or helpful is known as convenience. Convenience, when compared to the availability of emergency aid due to one's ease of comfort and suitability becomes effective and also remains during post-flooding periods, which are commonly used for emergency aid. In the context of emergency aid, convenience can be affected by factors such as the accessibility of aid delivery points, the speed of aid delivery, and the suitability of aid for the needs of the recipients. We could save people due to the convenience of emergency aid to the post-flooding areas at the right time.
Safety (e 3 ) The safest zone for the supply of emergency aid is one of the important factors. It refers to the protection of both aid providers and recipients from harm or danger while delivering or receiving emergency aid. It is a condition in which, consequences and situations that can cause damage to physical, psychological, or assets are managed to protect people's health, property, and well-being. In the context of emergency aid, safety can be affected by various factors, such as the nature and quantity of the aid being delivered, the safety of transportation and communication infrastructure, and the safety of the recipients and aid providers in the affected areas. www.nature.com/scientificreports/ Safety measures are important to ensure the wellbeing of both aid providers and recipients, as well as the success of the aid delivery process. Table 1 depicts the spherical hesitant fuzzy soft decision matrix as Ŵ(j r ) 3×3 = (φ Ŵ(j r ) , χ Ŵ(j r ) , ψ Ŵ(j r ) ) 3×3 . And in this problem, by using the score function, we transformed the Spherical Hesitant fuzzy soft decision matrix to an ordered matrix, presented in Table 2 Table 2.
now, we apply SHFSWA Operator to find out the aggregated decision values, the outcomes are shown in Table 3a-d. Now, we apply SHFSWGA Operator to find out the aggregated decision values, the outcomes are shown in Table 4a-d. Now, we apply SHFSOWA Operator to find out the aggregated decision values, the outcomes are shown in Table 5a-d. Now, we apply SHFSOWGA Operator to find out the aggregated decision values, the outcomes are shown in Table 6a-d. Now, we find the weighted matrix shown in Table 7 to utilized in hybrid aggregation operators. Now, we apply SHFSHA Operator to find out the aggregated decision values, the outcomes are shown in Table 8a-d. Now, we apply SHFSHG Operator to find out the aggregated decision values, the outcomes are shown in Table 9a-d.
The comparison between the proposed operators is given below. The graphical representation of alternative ranking is shown in Fig. 2: Table 10 makes it clear that the overall rating values of the alternatives differ when different operators are used, but the ranking orders of the alternatives are not changed. As a result, the safest alternative is e 3 .So, the decision makers choose the third alternative with medication, food and safety shelters as emergency aid.  www.nature.com/scientificreports/  www.nature.com/scientificreports/ Step 4 Based on computed AvS, determine PDAS and NDAS by utilizing the below formula: Step 5 Further calculate the positive weight distance (SP i ) and negative weight distance (SN i ) Step 6 Normalized the SP i and SN i by using the below formula:  www.nature.com/scientificreports/ Step 7 Compute the appraisal score AS: Step 8 Sort the values in a particular way based on the value of AS i to achieve the superior rank.
The flow chart of the EDAS methodology is shown in Fig. 3.
Illustrative example based on EDAS method. With the same data as previously mentioned in Table 1, we present a real-world MCDM example to demonstrate the effectiveness and supremacy of the analyzed approach. Normalized collective data of experts is given in Table 11 as follows. Now the score value for the normalized collective data of experts is given in Table 12.
The results of average solution is given in Table 13:  www.nature.com/scientificreports/   Table 14. PDAS matrix and NDAS matrix.    Table 17.
The graphical representation of EDAS method is shown in Fig. 4.

Comparative analysis
The proposed approaches is better than previously developed decision making techniques. Becaue in it we take into account the hesitant fuzzy sets along with the membership, neutral, and non-membership grades and with the parameterized structure. Firstly, SHFSS provides a more nuanced representation of the decision-making problem by incorporating spherical fuzzy sets. Spherical fuzzy sets allow for more flexibility in modeling the uncertainties and ambiguities of real-world decision-making problems. This can result in more accurate and effective decision-making outcomes. Secondly, SHFSS allows for the integration of both fuzzy sets and soft sets, which can be particularly useful when dealing with decision-making problems that involve both quantitative and qualitative information. The combination of these two approaches can help to balance the strengths and weaknesses of each, leading to more effective decision-making outcomes. Thirdly, the use of hesitant membership functions in SHFSS can help to capture the hesitant attitudes of decision-makers. This can be particularly useful when dealing with decision-making problems that involve multiple decision-makers with differing opinions or preferences. By incorporating hesitant membership functions, SHFSS can help to balance and integrate these different perspectives, leading to more equitable and effective decision-making outcomes. Finally, SHFSS has been shown to be effective in a wide range of applications. This versatility suggests that SHFSS may be a valuable tool in many different decision-making contexts.

Conclusion
While it is difficult to make definitive claims about the superiority of SHFSS over other decision-making techniques, the approach's ability to incorporate spherical fuzzy sets, integrate both fuzzy sets and soft sets, capture hesitant attitudes, and its wide range of applications suggest that it may be a valuable tool for decision-makers looking to tackle complex, uncertain problems. So, this study offers the latest numerical modelling of effective management through fuzzy decision support systems. For this purpose, we proposed a hybrid structure of aggregation operators, called spherical hesitant fuzzy soft aggregation operators, to aggregate SHFS data. Following that, we present an algorithm for dealing with SHFS MADM problems. The the numerical illustration along with aggregation operators and EDAS method was provided to validate the established strategy and shows its applicability and efficiency. Therefore, compared to other models currently in use, this new model is more accurate, realistic, and useful. In order to solve the problem of decision-making, this paper aims to establish a customizable soft dicision matrix. According to the study's findings, the proposed approach is more convenient and consistent with other existing selection processes. We are hopeful that this modified concept will be helpful in dealing with several problems related to uncertainty and will yield more convincing outcomes. www.nature.com/scientificreports/ Limitations of the proposed work. The theory of the aggregation operator and the EDAS approach based on the SHFSS are highly beneficial and dominant in assessing tricky and imprecise information in real-life issues, but they do not operate successfully in specific scenarios or instances due to their structure and requirements. When we came across information in the form of yes, abstain, no, and refusal with an expanded domain, the theories we had developed under the SHFSS information were ignored and could not be processed. In this regard, the sum of square of their data does not belong to the closed interval 0,1. So, here the fundamental criteria violated and we say that these conceptions are also limited.
Future work. In the future, we will be focusing on developing new operators that improve the accuracy and efficiency of SHFS decision-making methods. This theory can be extended to complex hesitant fuzzy soft sets for gernelized fuzzy set and for Aczel-Alsina aggragation operators. Another challenge in group decisionmaking with SHFSS is the difficulty of eliciting individual opinions in a consistent and reliable way. The process of assigning hesitancy degrees and determining the spherical shape of the set can be subjective and dependent on individual preferences, which can introduce bias and inconsistency in the decision-making process. To address these challenges, there is a need for further research and standardization in the methods used for group decision-making with SHFSS. This could include the development of standardized aggregation methods and the establishment of best practices for eliciting individual opinions in a consistent and reliable way. Additionally, some innovative approaches like LINMAP, TAOV for decision-making artificial intelligence and neural networks in the multi-parameter framework of SHFSS would be defined. The theory of yager aggregation operators can be adapted for SHFSS and MADM. Further we needed to create the theory of T-spherical hesitant fuzzy sets and complex T-spherical hesitant fuzzy sets.
Ethics approval. This article does not contain any studies with human participants or animals performed by any of the authors.