Topological Structure of Single-Valued Neutrosophic Hesitant Fuzzy Sets and Data Analysis for Uncertain Supply Chains
Abstract
:1. Introduction
2. Preliminaries
- If , then is superior to , designated by .
- If and , then is superior to , designated by .
- If and , then is superior to , designated by .
- If and , then is equal to designated by .
- Complement: The complements of SVNHFEs and can be expressed as follows:
- Inclusion: and for each .
- The union of two SVNHFEs is defined as follows:.
- The intersection of two SVNHFEs is defined as follows: .
- .
- .
- .
- .
3. SVNHF Topology
- .
- For each , , .
- For any , .
- and are SVNHF open sets;
- is an SVNHF open set, where each is an SVNHF open set;
- is an SVNHF open set, where each is an SVNHF open set.
- From the definition of SVNHF topology τ, , . Hence, and are SVNHF open sets.
- Let be SVNHF open sets. Then, . By the definition of τHence, is SVNHF open set.
- Let be SVNHF open sets. Then, by the definition of τ,Hence, is an SVNHF open set.
- are SVNHF closed sets over ;
- is an SVNHF closed set over , where each is an SVNHF closed set;
- is an SVNHF closed set over , for any SVNHF closed sets and .
- is an SVNHF open set
- if
- This is obvious from the definition of the SVNHF interior.
- Since is an SVNHF open set and it is also the biggest SVNHF open subset of itself, .
- If is an SVNHF open subset, then will be an SVNHF interior of itself since it is the largest SVNHF open subset. Conversely, if , then is an SVNHF open set because is SVNHF open.
- Since , from part , . is an SVNHF open subset of and so, by the definition of , we have
- From part ,andandso thatFurthermore, since , so that is an SVNHF open subset of . Hence,Thus,.
- Fromwe haveso that, because is SVNHF open,.
- and
- is an SVNHF closed set
- if
- By definition, . Since is an SVNHF closed superset of itself, . Thus, . Similarly, .
- By definition, , because is the intersection of all SVNHF closed supersets of .
- The proof is obvious.
- Since is an SVNHF closed set, by we have .
- Suppose as . Therefore, . This means that is an SVNHF closed superset of . Thus, .
- As we know that and , by using part , and ..Conversely, suppose that and .Thus, .Since is s SVNHF closed superset of .Therefore, .Thus, .
- If and , then and . Thus, .
- contains the largest SVNHF open set .
- is SVNHF open .
- .
- .
- .
- If is SVNHF open, then .
- By the definition of the SVNHF frontier, .
- Since , by taking the SVNHF complement on both sides, we obtain .by Theorem 5.
- Let be an SVNHF open set; this yields that is SVNHF closed. Utilizing , , and by , we obtain .
- .
- .
- For any subset in , is open if, and only if, is a null SVNHFS.
- For any subset in , is closed if, and only if, .
- For any subset in , is both open and closed if, and only if, is a null SVNHFS.
4. Extension of SIR Method for SVNHF Information
Algorithm 1: (SIR method for SVNHFSs) |
Let be an assemblage of substitutes/alternatives and is the collection of accredits/attributes. Assume that be the collection of experts with weight vectors . Suppose is the SVNHF decision matrix, where designates the accredits value that substitutes and persuades the accredits designated by expert . The accredits weighted decision matrix is , where designates the weight value of the accredits designated by expert . A novel approach based on SVNHF-SIR is addressed below: Step 1: Calculate the discrete/individual measure degree via the weights of experts, which take the form of SVNHFEs. The relative closeness coefficient is procured as follows: Step 2: To make the sum into a unit, normalize the and obtain as follows: We obtain the vector of real numbers that have been normalized as discrete/individual measure degrees. Step 3: Employ the SVNHFWA operator to aggregate individual perspectives into group perspectives as follows:
From this step, the group-integrated decision matrix and the attribute weight vector are acquired. |
Step 4: Acquire the SVNHF superiority/inferiority matrix
Step 5: Calculate the superiority flow and inferiority flow as follows: -flow
-flow
By using Equation (1), we calculate the score function of the corresponding -flow and -flow , respectively. Hence, we obtain the -flow and -flow of alternative as . It seems that if the -flow is larger and the -flow is smaller, the alternative is preferable. Step 6: Superiority ranking rule (-Rule): . If and , then ; . If and , then ; . If and , then . |
Inferiority ranking rule (-Rule): . If and , then ; . If and , then ; . If and , then . Step 7: By incorporating the -Rule and the -Rule, we can achieve the best alternative . A flow chart of the SIR method for supplier selection is shown in Figure 1. |
5. Extension of CV Method for SVNHF Information
Algorithm 2: (CV method for SVNHFSs) |
Step 1: Obtain the decision matrices from the decision makers, with alternative evaluated on the basis of criterion , given in Table 6. The aggregated decision matrix is obtained using step 1, step 2 and step . Step 2: Decision makers also give weight to the criteria with the condition that the sum of weights must be equal to 1. Then, the multiplication of decision matrix is computed with criteria weights, to obtain the matrix . Step 3: Find the score function of each SVNHFN. Step 4: Compute the ranking of the alternatives according to their score function values. |
Case Study
FSCM IT Systems
Instance | Firm | Network | Enhancement |
---|---|---|---|
Aramyan et al. [53] | Tomato firm | Performance | Efficiency and flexibility |
measurement system | have both improved, | ||
food quality has improved, | |||
and there is a faster response time. | |||
Bevilacqua et al. [54] | Tronto Valley | ARIS | Three types of costs |
are being reduced; | |||
improved traceability. | |||
Pagell and Wu [55] | Pizza restaurants | TQM Lean/JIT | Enhanced information sharing, |
superior quality, | |||
enhanced logistical efficiency. | |||
Tuncel and Alpan [56] | A medium size | Risk management | The percentage of orders completed |
on time has increased to , | |||
with risk reduction rising by . | |||
Zhu et al. [57] | A food manufacturer | Customer cooperation | Customer cooperation has improved; |
system | internal environment management | ||
has been improved. | |||
Jacxsens et al. [58] | A fresh producer | Food safety | Food of higher quality; |
management system | improved risk management ability. | ||
Friel et al. [59] | Agri-food supply chain | H&S food | A more nutritious diet, |
decision-making | with improved environmental | ||
system | sustainability. | ||
Savino et al. [60] | A chestnut | Value chain | Increased long-term viability, |
company | management | CO2 reduced emissions, | |
system | enhanced value chain. | ||
Banasik et al. [61] | A mushroom | Supply chain | Overall profitability increased by , |
manufacturer | management | with improved environmental | |
system | performance | ||
Sgarbossa and Russo [62] | 6 Firms | FSCM system | Conserving energy, |
costs of disposal avoided, | |||
enhanced productivity. |
6. MCDM Process
Comparative Analysis
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Fuzzy Sets | Fuzzy Numbers | Constraints |
---|---|---|
IFS [2] | , | |
PFS [3] | , | |
q-ROFS [4] | , | |
PiFS [5] | , | |
SFS [6,7,8] | , | |
NS [9] | , | |
SVNS [10] | , |
Researchers | Benchmarks | Applications |
---|---|---|
Tam et al. [20] | SIR method | Concrete pump selection |
Tom and Tong [21] | SIR method | Developments in the project concerning |
the location of the large-scale harbor | ||
Liu [22] | IF SIR method | Supply chain management |
Ma et al. [23] | HF SIR method | Selection of outstanding teachers from overseas |
Peng and Yang [24] | PF SIR method | Investment in internet stocks |
Rouhani [25] | Fuzzy SIR method | Software selection in IT field |
Chen [26] | PF PROMETHEE | Bridge construction |
method with superiority | ||
and inferiority PFNs | ||
Tavana et al. [27] | IFG SIR method | Solution of third-party reverse |
logistics problem | ||
Zhao et al. [28] | SIR method with HFL | Sustainable energy technology evaluation |
prioritized value | ||
Geetha and | PF SIR method | For investment selection of the internet |
Narayanamoorthy [29] | Stock marketing companies | |
Jie et al. [30] | IVIF SIR method | Engineering investment selection |
Union | ||||
---|---|---|---|---|
Intersection | ||||
---|---|---|---|---|
Experts | SVNHFEs |
---|---|
Alternatives | Score Values |
---|---|
0.8309 | |
0.8158 | |
0.8253 | |
0.8233 |
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Riaz, M.; Almalki, Y.; Batool, S.; Tanveer, S. Topological Structure of Single-Valued Neutrosophic Hesitant Fuzzy Sets and Data Analysis for Uncertain Supply Chains. Symmetry 2022, 14, 1382. https://doi.org/10.3390/sym14071382
Riaz M, Almalki Y, Batool S, Tanveer S. Topological Structure of Single-Valued Neutrosophic Hesitant Fuzzy Sets and Data Analysis for Uncertain Supply Chains. Symmetry. 2022; 14(7):1382. https://doi.org/10.3390/sym14071382
Chicago/Turabian StyleRiaz, Muhammad, Yahya Almalki, Sania Batool, and Shaista Tanveer. 2022. "Topological Structure of Single-Valued Neutrosophic Hesitant Fuzzy Sets and Data Analysis for Uncertain Supply Chains" Symmetry 14, no. 7: 1382. https://doi.org/10.3390/sym14071382
APA StyleRiaz, M., Almalki, Y., Batool, S., & Tanveer, S. (2022). Topological Structure of Single-Valued Neutrosophic Hesitant Fuzzy Sets and Data Analysis for Uncertain Supply Chains. Symmetry, 14(7), 1382. https://doi.org/10.3390/sym14071382