Datadriven Fuzzy Multiple Criteria Decision Making and its Potential Applications
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Lina Dai, Shizhen Bai, "An Approach to Selection of Agricultural Product Supplier Using Pythagorean Fuzzy Sets", Mathematical Problems in Engineering, vol. 2020, Article ID 1816028, 7 pages, 2020. https://doi.org/10.1155/2020/1816028
An Approach to Selection of Agricultural Product Supplier Using Pythagorean Fuzzy Sets
Abstract
The selection of agricultural product supplier is an important link to optimize the supply chain management of agricultural products. Due to the uncertainty factors and the lack of decisionmakers’ cognition, the selection of agricultural products suppliers has become a very complex and difficult work. Therefore, in order to effectively deal with these problems, this study proposes an agricultural product supplier selection algorithm based on the Pythagorean fuzzy power Bonferroni mean operator under Pythagorean fuzzy environment. In this method, first, the power operator and Bonferroni mean operator are combined and embedded in the Pythagorean fuzzy sets to build a Pythagorean fuzzy power Bonferroni mean operator. Then, a multiattribute decisionmaking method based on the operator proposed in this paper is proposed. Finally, an analysis of examples of agricultural product supplier selection is given to verify the rationality and effectiveness of this method.
1. Introduction
The sources of safety problems of agricultural products mainly include excessive use of pesticides and chemical fertilizers in agricultural production, environmental pollution caused by “three wastes” in the processing of agricultural products, environmental degradation caused by water loss and soil erosion in the ecological environment, excessive use of additives in food production and processing, and food packaging materials that do not meet the requirements. In addition, primary agricultural and livestock products, such as fruits, vegetables, meat, eggs, and aquatic products, are perishable and obviously affected by geography and seasons, and there is a certain degree of timespace separation between supply and demand, which will also cause safety problems of agricultural products. Therefore, the supply of agricultural products is closely related to people’s life and production and has been concerned by various countries for a long time [1, 2]. At present, a large number of enterprises, such as supermarkets, catering enterprises, and agricultural products processing enterprises, faced with the problem of selection of agricultural products suppliers [3]. The purchase of agricultural products is the starting point and key link of the supply chain of agricultural products. The price, coldchain transportation, distance, production standards, and other aspects of agricultural products will ultimately affect the market competitiveness of terminal products of fresh agricultural products supply chain [4]. How to select a reasonable supplier has become one of the concerns of many agricultural productsrelated enterprises.
Many internal and external uncertainty factors of enterprise need to be considered in the selection process of agricultural products suppliers, which is also affected by the subjective preferences of decisionmakers. Therefore, the selection of agricultural products suppliers is essentially a fuzzy multiattribute decisionmaking problem. Fuzzy multiattribute decisionmaking problem has always been a hot topic in the field of decisionmaking and has attracted the attention of many scholars [5–10]. The fuzzy set theory proposed by Zadeh [11] lays the foundation for practical operation of fuzzy multiattribute decisionmaking problems. With the continuous research of fuzzy set theory, Atanassov put forward the intuitionistic fuzzy set theory [12]. After that, many scholars have extended the intuitionistic fuzzy set [13–19]. Among them, Yager proposed that the Pythagorean fuzzy set is one of the important research studies [18, 19]. Compared with the intuitionistic fuzzy set, the Pythagorean fuzzy set has stronger information representation ability and is more close to the practical problems. Once the Pythagorean fuzzy set is put forward, it has gained widespread attention in academic circles.
Many scholars mainly focus on the study of the extended form of the Pythagorean fuzzy set, including the study of the information integration operator of the Pythagorean fuzzy set and the fuzzy multiattribute decisionmaking based on Pythagoras. For example, Garg [20] improved the score function, which can better compare the sizes of Pythagorean fuzzy numbers. He et al. [21] studied the Pythagorean hesitant fuzzy integration operator and its decisionmaking application. In view of the important role of the integration operator in the Pythagorean hesitant fuzzy multiattribute decisionmaking and the imperfection of the integration operator, He et al. [21] systematically studied the Pythagorean hesitant fuzzy integration operator. Gou et al. studied the continuity and differentiation of Pythagorean fuzzy numbers. Peng et al. [22] combined the properties of Pythagorean fuzzy set and parameterization of the soft set, constructed the Pythagorean fuzzy soft set, introduced the properties of the Pythagorean fuzzy soft set and discussed its decisionmaking application, and then discussed its DeMorgan’s law. Zhao and Wang [23] proposed a multiattribute decisionmaking method based on the Hamacher operator in view of the multiattribute decisionmaking problem of dual hesitant Pythagorean fuzzy uncertain linguistic information. Then, based on the Hamacher operator, Zhao and Wang [23] defined the operation rule between dual hesitant Pythagorean fuzzy uncertain linguistic variables. He et al. [24] applied the power average operator to the Pythagorean fuzzy decisionmaking environment, defined the average operator of the Pythagorean fuzzy power, orderly weighted the average operator of Pythagorean fuzzy power, geometric operator of Pythagorean fuzzy power and orderly weighted geometric operator of Pythagorean fuzzy power, and then studied their properties, respectively. In the latest research progress, Chen [25] published a paper on proposing a new Pythagorean Chebyshev distance measure and established a practical method for eliminating and selecting the conversion based on Chebyshev distance measure. Shakeel et al. [26] defined the Einstein operator on the Pythagorean trapezoid fuzzy set and expanded it into two average aggregation operators. Ullah et al. [27] proposed the concept of the complex Pythagorean fuzzy set (CPFS) in view of the limitation of the complex fuzzy set and complex intuitionistic fuzzy set. Yang and Chang [28] defined the new concept of the intervalvalued Pythagorean normal fuzzy set and developed a series of aggregation operators for addressing the intervalvalued Pythagorean normal fuzzy information. Harish [29] presented a Pythagorean fuzzy neutrality aggregation operator. Han et al. [30] developed an intervalvalued Pythagorean prioritized operator from the perspective of game theory.
It can be seen from the above literature that up to now, the research of the Pythagorean fuzzy set has made some achievements, but it still has room to expand in the field of the information aggregation operator and its application. Therefore, this study proposes the average operator of Pythagorean fuzzy power Bonferroni, which will be applied to the study of agricultural product supplier selection, so as to build a decisionmaking support framework for agricultural product supplier selection based on the Pythagoras power Bonferroni average operator. The main contributions of this study are as follows:(1)The average operator of Pythagorean fuzzy power Bonferroni was proposed.(2)A decisionmaking algorithm for agricultural product supplier selection was proposed based on the Pythagorean fuzzy power Bonferroni average (PFPBA) operator.
The other contents of this paper are as follows: part 2 reviews the basic concepts, algorithms, and distance measures of Pythagorean fuzzy. Part 3 puts forward the Pythagorean fuzzy power Bonferroni mean operator and analyzes its properties. Part 4 constructs a multiattribute decisionmaking method based on the average operator of the Pythagorean fuzzy power Bonferroni mean operator. Part 5 gives an example to verify the validity and rationality of the method. Part 6 summarizes some conclusions.
2. Pythagorean Fuzzy Sets
Definition 1. (see [18, 19]). Assuming that is a nonempty general set, the expression of the Pythagorean fuzzy set A defined on iswhere : and : represent the membership function and nonmembership function of A and . In addition, the hesitancy degree is defined as .
For simplicity, is called a Pythagorean fuzzy number. Assuming that , is three Pythagorean fuzzy numbers, and is any real number greater than or equal to 0, then the following operation rules are specified:(1)(2)(3) (4) Therefore, the following conclusions can be drawn:(1)(2) (3)(4) (5)(6)
Defintion 2. (see [1819]). Assuming is a Pythagorean fuzzy number, the score function of is defined as , and the exact function of is defined as . For any two Pythagorean fuzzy numbers and , the definitions are as follows:(1)If , then (2)If , then when ; when
Definition 3. (see [18, 19]). Assuming that , and are any two Pythagorean fuzzy numbers, then the standard hamming distance of and can be defined aswhere and .
3. PFPBA Operator
Definition 4. (see [31]). Assuming s and t are nonnegative real numbers that are not both zero and is a series of nonnegative real numbers, ifThen, is the average operator of power Bonferroni, in which , and represents the supporting degree of a_{i} and a_{j} and meets the following conditions:(1)(2)(3) if and only if (4)If , then , in which is the standard hamming distance of a_{i} and a_{j}.
Definition 5. Assuming that s and t are nonnegative real numbers that are not both zero, is the set of a group of Pythagorean fuzzy numbers, and , and . Then, the Pythagorean fuzzy power Bonferroni mean (PFPBA) operator can be defined aswhere , and indicates the supporting degree of a_{i} and a_{j} and meets the following conditions:(1)(2)(3) if and only if (4)If , then , in which is the standard hamming distance of the two Pythagorean fuzzy numbers.
Definition 6. Assuming that is the Pythagorean fuzzy number and s and t are nonnegative real numbers that are not both zero, then the integration value of these Pythagorean fuzzy numbers obtained by using the operator is still a Pythagorean fuzzy number, andAccording to the Definition 3, it is easy to prove that the above formula is true by mathematical induction.
Some basic properties of the average operators of Pythagorean fuzzy power Bonferroni are discussed as follows:(1)(Power equitability) Assuming that is the Pythagorean fuzzy number and , then(2)(Permutation invariance) Assuming that is the number and is any permutation and combination of , then(3)(Boundedness) Assuming that is the Pythagorean fuzzy number, , , and , then
4. Multiattribute DecisionMaking Model Based on the PFPBA Operator
In the problem of Pythagorean fuzzy information multiattribute decisionmaking, assume that there are n alternative schemes and m decision attributes . The experts provide the evaluation information of Pythagorean fuzzy and set the attribute value of the scheme under the attribute of to be . In which is the Pythagorean fuzzy number, and then, the decision matrix of Pythagorean fuzzy can be obtained. and represent the value of membership degree and nonmembership degree of the attribute j of the alternative scheme i. A decision method based on Pythagorean fuzzy information is proposed as follows in combination with the Pythagorean fuzzy power Bonferroni operator, and the specific steps are illustrated in Figure 1: Step 1: use the PFPBA operator to integrate the attribute values of schemes Step 2: calculate the score function of each scheme according to Definition 2 Step 3: sort the schemes according to the score function Step 4: select the best scheme according to the scheme ranking
5. Example Analysis of Agricultural Product Supplier Selection
5.1. Calculation Process
The safety of agricultural products is a major livelihood issue. Countries have raised their standards for the production and processing of agricultural products, and ordinary people have also raised their awareness of consumption of agricultural products. In order to adapt to the rapidly changing market demand of agricultural products, reduce the operating costs of agricultural production enterprises, and improve the core competitiveness of the company, enterprises need to select appropriate agricultural products suppliers for cooperation. When selecting suppliers, enterprises often need to define their own needs and select appropriate suppliers according to their own needs. In addition, as the concept of green agricultural products has been strongly advocated and gradually gained popularity in recent years, enterprises should also consider environmental protection and implement the strategy of sustainable development when selecting suppliers for cooperation. An agricultural product enterprise in a city intends to select a supplier as a stable source of supply for agricultural products. After preliminary market research, it selects suppliers from multiple perspectives and selects four suppliers with core competitiveness, which are represented with , respectively. The decisionmaker evaluates four suppliers from 4 aspects: C_{1} green technology level (including pollution control level and environmental planning ability), C_{2} product advantage (including product price, product quality, and the product warranty period), C_{3} risk bearing capacity (including the company’s capital scale and enterprise prospect), and C_{4} enterprise production ability (including the company’s equipment quantity and production efficiency) to select the best supplier. After consultation, the expert group gives the following decision matrix as shown in Table 1:

5.2. Comparison with Different Methods
(1)When the Pythagorean fuzzy power average operator is used to integrate these attribute values: Step 1: Establish the Pythagorean fuzzy decision matrix according to actual situation, as shown in Table 1. Step 2: Use the Pythagorean fuzzy power average operator to calculate the comprehensive attribute value of each scheme. Then, the comprehensive attribute value of each scheme can be calculated: Step 3: The score value of each scheme can be calculated as , respectively. Step 4: After comparison of score values, it can be known according to that the scheme is the best one.(2)When the average operator of Pythagorean fuzzy Bonferroni mean is used to integrate these attribute values: Step 1: establish the Pythagorean fuzzy decision matrix according to actual situation, as shown in Table 1 Step 2: use the Pythagorean fuzzy Bonferroni mean operator to calculate the comprehensive attribute value of each scheme with as follows: Step 3: the score value of each scheme can be calculated as , respectively Step 4: after comparison of score values, it can be known that the scheme is the best one
From the above results, it can be known that the results of the method proposed in this paper are the same as the optimal scheme based on the Pythagorean fuzzy power average operator, but their total order is not the same, and it is also different from the sorting results based on the Pythagorean fuzzy Bonferroni mean operator. The reason for this difference is that the two methods do not consider the heterogeneity between attributes and their impact on the evaluation results. Therefore, the method proposed in this paper has strong advantages.
6. Conclusion
Considering the uncertain information faced in the selection of agricultural product suppliers, this study proposes a multiattribute decisionmaking method based on the Pythagorean fuzzy power Bonferroni mean operator. In this method, the heterogeneous relationship between attributes and the abnormal value of evaluation information are fully considered, and the power operator and Bonferroni average operator are integrated and introduced into the Pythagorean fuzzy information environment to construct an agricultural product supplier selection algorithm based on the Pythagorean fuzzy power Bonferroni mean operator. This algorithm has a strong ability in dealing with uncertain information, and at the same time, it considers the influence of the internal relationship between attributes on the decision results and avoids the adverse effect of the extreme value in the evaluation information on the ranking results through the PA operator. In the future, this study can be combined with other operators, such as Muirhead mean operator and Hamy mean operator, to propose more extensive Pythagorean fuzzy information aggregation operators. At the same time, the operators proposed in this paper can also be applied in other practical decisionmaking fields, such as traffic route selection, enterprise performance evaluation, and human resource management.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors acknowledge that this work was supported by the Natural Science Foundation of China (No. 71671054) and Heilongjiang Social Science Foundation of China (No. 18JYB147).
References
 Y. L. Liu and T. T. Jin, “Application of Raman spectroscopy technique to agricultural products quality and safety determination,” Spectroscopy and Special Analysis, vol. 35, no. 9, pp. 2567–2572, 2015. View at: Google Scholar
 C. Ma, D. Wang, D. Wang et al., “Considerations of constructing quality, health and safety management system for agricultural products sold via ecommerce,” International Journal of Agricultural and Biological Engineering, vol. 11, no. 1, pp. 31–39, 2018. View at: Publisher Site  Google Scholar
 B. Yan, C. Yan, C. Ke, and X. Tan, “Information sharing in supply chain of agricultural products based on the internet of things,” Industrial Management & Data Systems, vol. 116, no. 7, pp. 1397–1416, 2016. View at: Publisher Site  Google Scholar
 G. Chen, “Production decision of agricultural products: A game model based on negative exponential utility function,” Journal of Intelligent & Fuzzy Systems, vol. 37, no. 5, pp. 6139–6149, 2019. View at: Publisher Site  Google Scholar
 Z. Yang, T. Ouyang, X. Fu, and X. Peng, “A decision‐making algorithm for online shopping using deep‐learningbased opinion pairs mining and q ‐rung orthopair fuzzy interaction Heronian mean operators,” International Journal of Intelligent Systems, vol. 35, no. 5, pp. 783–825, 2020. View at: Publisher Site  Google Scholar
 Z. L. Yang, H. Garg, J. Li, G. Srivastava, and Z. Cao, “Investigation of multiple heterogeneous relationships using a qrung orthopair fuzzy multicriteria decision algorithm,” Neural Computing and Applications, pp. 1–22, 2020. View at: Publisher Site  Google Scholar
 X. Zhang and Z. Xu, “A new method for ranking intuitionistic fuzzy values and its application in multiattribute decision making,” Fuzzy Optimization and Decision Making, vol. 11, no. 2, pp. 135–146, 2012. View at: Publisher Site  Google Scholar
 G. Yu, “An algorithm for multiattribute decision making based on soft rough sets,” Journal of Computational Analysis and Applications, vol. 20, no. 7, pp. 1248–1258, 2016. View at: Google Scholar
 Z. Yang, X. Li, H. Garg, and M. Qi, “Decision support algorithm for selecting an antivirus mask over COVID19 pandemic under spherical normal fuzzy environment,” International Journal of Environmental Research and Public Health, vol. 17, no. 10, p. 3407, 2020. View at: Publisher Site  Google Scholar
 Z. L. Yang, X. Li, Z. H. Cao, and J. Q. Li, “Qrung orthopair normal fuzzy aggregation operators and their application in multiattribute decisionmaking,” Mathematics, vol. 7, no. 12, Article ID 1142, 2019. View at: Publisher Site  Google Scholar
 L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965. View at: Publisher Site  Google Scholar
 K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87–96, 1986. View at: Publisher Site  Google Scholar
 K. Atanassov and G. Gargov, “Interval valued intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 31, no. 3, pp. 343–349, 1989. View at: Publisher Site  Google Scholar
 P. Liu and S.M. Chen, “Multiattribute group decision making based on intuitionistic 2tuple linguistic information,” Information Sciences, vol. 430431, pp. 599–619, 2018. View at: Publisher Site  Google Scholar
 Z. Yang, G. Xiong, Z. Cao, Y. Li, and L. Huang, “A decision method for online purchases considering dynamic information preference based on sentiment orientation classification and discrete DIFWA operators,” IEEE Access, vol. 7, pp. 77008–77026, 2019. View at: Publisher Site  Google Scholar
 H. M. Nehi, “A new ranking method for intuitionistic fuzzy numbers,” International Journal of Fuzzy Systems, vol. 12, no. 1, pp. 80–86, 2010. View at: Google Scholar
 Z. Yang, J. Li, L. Huang, and Y. Shi, “Developing dynamic intuitionistic normal fuzzy aggregation operators for multiattribute decisionmaking with time sequence preference,” Expert Systems with Applications, vol. 82, pp. 344–356, 2017. View at: Publisher Site  Google Scholar
 R. R. Yager, “Pythagorean membership grades in multicriteria decision making,” IEEE Transactions on Fuzzy Systems, vol. 22, no. 4, pp. 958–965, 2014. View at: Publisher Site  Google Scholar
 R. R. Yager and A. M. Abbasov, “Pythagorean membership grades, complex numbers, and decision making,” International Journal of Intelligent Systems, vol. 28, no. 5, pp. 436–452, 2013. View at: Publisher Site  Google Scholar
 H. Garg, “New logarithmic operational laws and their aggregation operators for Pythagorean fuzzy set and their applications,” International Journal of Intelligent Systems, vol. 34, no. 1, pp. 82–106, 2019. View at: Publisher Site  Google Scholar
 X. He, W. F. Liu, and X. Y. Du, “Pythagorean hesitant fuzzy aggregation operators and their applications in decision making,” Application Research of Computers, vol. 37, no. 8, pp. 1–7, 2019. View at: Google Scholar
 X. Peng, Y. Yang, J. P. Song et al., “Pythagorean fuzzy soft set and its application,” Computer Engineering, vol. 7, pp. 224–229, 2015. View at: Google Scholar
 X. D. Zhao and F. Wang, “Dual hesitant Pythagorean fuzzy uncertain linguistic Hamacher aggregation operators in multiple attribute decision making,” Fuzzy Systems and Mathematics, vol. 33, no. 5, pp. 89–106, 2019. View at: Google Scholar
 X. He, Y. X. Du, and W. F. Liu, “Pythagorean fuzzy power average operators,” Fuzzy Systems and Mathematics, vol. 30, no. 6, pp. 116–124, 2016. View at: Google Scholar
 T.Y. Chen, “New Chebyshev distance measures for Pythagorean fuzzy sets with applications to multiple criteria decision analysis using an extended ELECTRE approach,” Expert Systems With Applications, vol. 147, Article ID 113164, 2020. View at: Publisher Site  Google Scholar
 M. Shakeel, S. Abdullah, M. Aslam, and M. Jamil, “Ranking methodology of induced Pythagorean trapezoidal fuzzy aggregation operators based on Einstein operations in group decision making,” Soft Computing, vol. 24, no. 10, pp. 7319–7334, 2020. View at: Publisher Site  Google Scholar
 K. Ullah, T. Mahmood, Z. Ali, and N. Jan, “On some distance measures of complex Pythagorean fuzzy sets and their applications in pattern recognition,” Complex & Intelligent Systems, vol. 6, no. 1, pp. 15–27, 2020. View at: Publisher Site  Google Scholar
 Z. Yang and J. Chang, “Intervalvalued Pythagorean normal fuzzy information aggregation operators for multiattribute decision making,” IEEE Access, vol. 8, pp. 51295–51314, 2020. View at: Publisher Site  Google Scholar
 G. Harish, “Neutrality operationsbased Pythagorean fuzzy aggregation operators and its applications to multiple attribute group decisionmaking process,” Journal of Ambient Intelligence and Humanized Computing, vol. 11, no. 7, pp. 3021–3041, 2020. View at: Publisher Site  Google Scholar
 Y. Han, Y. Deng, Z. Cao, and C.T. Lin, “An intervalvalued Pythagorean prioritized operatorbased game theoretical framework with its applications in multicriteria group decision making,” Neural Computing and Applications, vol. 32, no. 12, pp. 7641–7659, 2020. View at: Publisher Site  Google Scholar
 P. Liu and X. Liu, “Multiattribute group decision making methods based on linguistic Intuitionistic fuzzy power Bonferroni mean operators,” Complexity, vol. 2017, Article ID 3571459, pp. 1–15, 2017. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2020 Lina Dai and Shizhen Bai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.