# A Fuzzy Parameterized Multiattribute Decision-Making Framework for Supplier Chain Management Based on Picture Fuzzy Soft Information

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## Abstract

**:**

## 1. Introduction

#### 1.1. Literature Review

#### 1.2. Research Gap and Motivation

- Uncertainty and vagueness appeared while choosing the appropriate parameters.
- Flexible opinions of the DMS in terms of truth, falsity, and neutrality grades.
- Approximate function for the assessment of alternatives.

**Hypothesis**

**1 (H1).**

**Hypothesis**

**2 (H2).**

**Hypothesis**

**3 (H3).**

#### 1.3. Salient Questions and Contributions

- a.
- How may the DMS’s uncertainty over the selection of parameters be addressed?
- b.
- How can the DMS’s impartiality in estimating the options be effectively controlled?
- c.
- What part does the approximate function play in the evaluation of different options?

- A novel mathematical context, i.e., FpPiFSS, is characterized as the combination of three significant concepts: FPara idea, picture fuzzy set, and SOS. Such a combination is trustworthy to address the limitations of the published literature (this is a theoretical aspect linked with the above-described research questions).
- In order to assess the vague nature of parameters, their respective fuzzy parameterized grades (FPGs) are determined by using the assigned weights of DMS (this is specifically linked with the first research question).
- Based on the set-theoretic properties of FpPiFSS, an intelligent decision framework is established, accompanied by an algorithm for the evaluation of timber suppliers (this is particularly linked with the third research question).

## 2. Fundamental Knowledge

**Definition**

**1**

**.**Let ${\tilde{\zeta}}_{T}$, ${\tilde{\zeta}}_{F}$, and ${\tilde{\zeta}}_{N}$ be membership, nonmembership, and neutral membership functions, respectively, with the nonempty set $\widehat{\u25b5}$ as their domain and [0, 1] as their range; then, PiFS A is defined as

**Definition**

**2**

**.**Let ${P}^{\widehat{\u25b5}}$ and $\widehat{\Xi}$ be the set of all subsets of $\widehat{\u25b5}$ and the set of attributes, respectively; then, the SoS B is defined as

**Definition**

**3**

**.**Let ${P}^{\widehat{\u25b5}}$ and $\widehat{\Xi}$ be the set of all subsets of $\widehat{\u25b5}$ and the set of attributes, respectively. Let ${\tilde{\zeta}}_{T}$, ${\tilde{\zeta}}_{F}$, and ${\tilde{\zeta}}_{N}$ be membership, nonmembership, and neutral membership functions, respectively, with universal set $\widehat{\u25b5}$ as their domain and [0, 1] as their range; then, PiFSS C is characterized by an approximate mapping $\tilde{\psi}:\widehat{\Xi}\to {\Omega}_{PiFS}$ and defined as

## 3. Materials and Methods

#### 3.1. Characterization of Proposed Structure, i.e., FpPiFSS

**Definition**

**4.**

**Example**

**1.**

- ${\tilde{\psi}}_{F}\left(\frac{e}{0.2}\right)=\left\{\frac{{\tilde{x}}_{1}}{\langle 0.21,0.33,0.32\rangle},\frac{{\tilde{x}}_{2}}{\langle 0.23,0.35,0.34\rangle},\frac{{\tilde{x}}_{3}}{\langle 0.25,0.37,0.36\rangle},\frac{{\tilde{x}}_{4}}{\langle 0.27,0.39,0.38\rangle}\right\}$,
- ${\tilde{\psi}}_{F}\left(\frac{e}{0.5}\right)=\left\{\frac{{\tilde{x}}_{1}}{\langle 0.31,0.21,0.11\rangle},\frac{{\tilde{x}}_{2}}{\langle 0.33,0.23,0.13\rangle},\frac{{\tilde{x}}_{3}}{\langle 0.35,0.25,0.15\rangle},\frac{{\tilde{x}}_{4}}{\langle 0.37,0.27,0.17\rangle}\right\}$,
- ${\tilde{\psi}}_{F}\left(\frac{e}{0.3}\right)=\left\{\frac{{\tilde{x}}_{1}}{\langle 0.11,0.31,0.21\rangle},\frac{{\tilde{x}}_{2}}{\langle 0.13,0.33,0.23\rangle},\frac{{\tilde{x}}_{3}}{\langle 0.15,0.35,0.25\rangle},\frac{{\tilde{x}}_{4}}{\langle 0.17,0.37,0.27\rangle}\right\}$,
- ${\tilde{\psi}}_{F}\left(\frac{e}{0.6}\right)=\left\{\frac{{\tilde{x}}_{1}}{\langle 0.41,0.11,0.31\rangle},\frac{{\tilde{x}}_{2}}{\langle 0.43,0.13,0.33\rangle},\frac{{\tilde{x}}_{3}}{\langle 0.45,0.15,0.35\rangle},\frac{{\tilde{x}}_{4}}{\langle 0.47,0.17,0.37\rangle}\right\}$.

**Definition**

**5.**

- 1.
- The FpPiFHSS ${\widehat{\Pi}}_{3}=\left\{\left(\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{3}\left({\tilde{e}}_{i}\right)},{\tilde{\psi}}_{F}^{3}(\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{3}({\tilde{e}}_{i})})\right):\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{3}({\tilde{e}}_{i})}\in {\widehat{F}}_{\widehat{\Xi}}^{3},{\tilde{\psi}}_{F}^{3}(\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{3}\left({\tilde{e}}_{i}\right)})\subseteq {\Omega}_{PiFS}\right\}$ is their union such that$${\tilde{\psi}}_{F}^{3}(\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{3}\left({\tilde{e}}_{i}\right)})=\left\{\begin{array}{cc}\begin{array}{c}{\tilde{\psi}}_{F}^{1}(\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{1}\left({\tilde{e}}_{i}\right)})\\ {\tilde{\psi}}_{F}^{2}(\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{2}\left({\tilde{e}}_{i}\right)})\\ {\tilde{\psi}}_{F}^{1}(\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{1}\left({\tilde{e}}_{i}\right)})\cup {\tilde{\psi}}_{F}^{2}(\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{2}\left({\tilde{e}}_{i}\right)})\end{array}& \begin{array}{c};\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{3}\left({\tilde{e}}_{i}\right)}\in {\widehat{F}}_{\widehat{\Xi}}^{1}\setminus {\widehat{F}}_{\widehat{\Xi}}^{2}\\ ;\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{3}\left({\tilde{e}}_{i}\right)}\in {\widehat{F}}_{\widehat{\Xi}}^{2}\setminus {\widehat{F}}_{\widehat{\Xi}}^{1}\\ ;\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{3}\left({\tilde{e}}_{i}\right)}\in {\widehat{F}}_{\widehat{\Xi}}^{1}\cap {\widehat{F}}_{\widehat{\Xi}}^{2}\end{array}\end{array}\right.$$
- 2.
- The FpPiFHSS ${\widehat{\Pi}}_{4}=\left\{\left(\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{4}\left({\tilde{e}}_{i}\right)},{\tilde{\psi}}_{F}^{4}(\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{4}\left({\tilde{e}}_{i}\right)})\right):\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{4}\left({\tilde{e}}_{i}\right)}\in {\widehat{F}}_{\widehat{\Xi}}^{4},{\tilde{\psi}}_{F}^{4}(\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{4}\left({\tilde{e}}_{i}\right)})\subseteq {\Omega}_{PiFS}\right\}$ is their union such that ${\tilde{\psi}}_{F}^{4}(\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{4}\left({\tilde{e}}_{i}\right)})={\tilde{\psi}}_{F}^{1}(\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{1}\left({\tilde{e}}_{i}\right)})\cap {\tilde{\psi}}_{F}^{2}(\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{2}\left({\tilde{e}}_{i}\right)})$ when $\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{4}\left({\tilde{e}}_{i}\right)}\in {\widehat{F}}_{\widehat{\Xi}}^{1}\cap {\widehat{F}}_{\widehat{\Xi}}^{2}$ where ${\tilde{\psi}}_{F}^{1}(\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{1}\left({\tilde{e}}_{i}\right)})\cap {\tilde{\psi}}_{F}^{2}(\frac{{\tilde{e}}_{i}}{{\mu}_{T}^{2}\left({\tilde{e}}_{i}\right)})=\left\{\begin{array}{c}{\tilde{x}}_{j}/\langle min\phantom{\rule{0.166667em}{0ex}}\{{\zeta}_{T}^{1}\left({\tilde{x}}_{j}\right),{\zeta}_{T}^{2}\left({\tilde{x}}_{j}\right)\},max\phantom{\rule{0.166667em}{0ex}}\{{\zeta}_{F}^{1}\left({\tilde{x}}_{j}\right),{\zeta}_{F}^{2}\left({\tilde{x}}_{j}\right)\},max\phantom{\rule{0.166667em}{0ex}}\{{\zeta}_{N}^{1}\left({\tilde{x}}_{j}\right),{\zeta}_{N}^{2}\left({\tilde{x}}_{j}\right)\}\rangle :\\ {\tilde{x}}_{j}\in \widehat{\u25b5},{\zeta}_{T}^{k}\left({\tilde{x}}_{j}\right),{\zeta}_{F}^{k}\left({\tilde{x}}_{j}\right),{\zeta}_{N}^{k}\left({\tilde{x}}_{j}\right)\in [0,1]\end{array}\right\}.$

#### 3.2. Criteria for Selection of Parameters

#### 3.3. Determination of Fuzzy Parameterized Grades

#### 3.4. Decision Support Framework

#### Problem Scenario

#### 3.5. Proposed Algorithm

Algorithm 1: This algorithm consists of three stages: (1) Input, (2) Construction and Computations, (3) Output. These stages are explained as below: |

1. Input 1.1 Consider a set $\mathbb{Z}=\{{E}_{1},{E}_{2},{E}_{3},\dots ,{E}_{p}\}$ consisting of experts (DMS) hired for the evaluation process. 1.2 Assume a set of alternatives $\widehat{\u25b5}=\{{\tilde{x}}_{1},{\tilde{x}}_{2},{\tilde{x}}_{3},\dots ,{\tilde{x}}_{m}\}$ consisting of suppliers short listed by DMS through initial screening. 1.3 Assume a set $\widehat{\Xi}=\{{\tilde{e}}_{1},{\tilde{e}}_{2},{\tilde{e}}_{3},\dots ,{\tilde{e}}_{n}\}$ consisting of parameters selected by decision makers with mutual consensus. 1.4 Collect preferential weights ${\varpi}_{i,j},(i=1,2,3,\dots ,n;j=1,2,3,\dots ,p)$ from decision makers for each parameter. |

2. Construction and Computations 2.1 Determine FPGs for each parameter by using Equation (1). 2.2 Construct an FpPiFSS $\widehat{\Pi}$ based on the opinions provided by DMS for the approximation of alternatives based on fuzzy parameters and represent it in tabular form. 2.3 Convert each picture fuzzy value into fuzzy value by using the criterion $|{\tilde{\zeta}}_{T}\left(\tilde{x}\right)-\phantom{\rule{0.166667em}{0ex}}{\tilde{\zeta}}_{F}\left(\tilde{x}\right)-{\tilde{\zeta}}_{N}\left(\tilde{x}\right)|$ and represent them in matrix ${\mathbb{M}}_{1}$. 2.4 Construct decision matrix ${\mathbb{M}}_{2}$ by multiplying each row entry with its respective FPG. 2.5 Compute the score values $\mathbb{S}\left({\tilde{x}}_{m}\right)$ of each alternative by taking the sum of respective entries of the alternative column and represent them in matrix ${\mathbb{M}}_{3}$. |

3. Output 3.1 Select the alternative with maximum score. |

#### 3.6. Validation of Algorithm 1

**Example**

**3.**

## 4. Discussion and Comparison Analysis

- Procurement has drawn a lot of attention because it has become crucial in determining the durability and efficacy of production teams. Purchaser–dealer correlations based solely on cost are insufficient to any further extent, as already discussed by Sarkis and Talluri [60]. Companies are being forced to reevaluate their strategies related to purchasing and evaluation as an effective procuring assessment directly depends on choosing the “right” supplier due to the increasing significance of supplier selection decisions.
- The SuSP is an MCDM problem, as previously mentioned in the literature review section, and it is simple to see that the key component of each MCDM is the bias displayed by specialists for the objects under observation with reference to each decisive element. It is also possible to examine the fact that the primary source of study in many studies is the opinions of experts. However, the computational process may be impacted if the opinions of experts show any flaws. Roughness in the computation and information is seen to be relevant in this situation.
- The works from investigators Xiao et al. [46], Liu et al. [61], Mukherjee et al. [62], Tan et al. [49], Liao et al. [50], and Quan et al. [51] are regarded as the most significant and pertinent to the recommended strategy for SuSP when the aforementioned discussion is taken into consideration. In order to deal with ambiguous information and imperfect expert opinions, these approaches overlooked soft settings, the consideration of three-dimensional membership values $\langle {\tilde{\zeta}}_{T}\left(\tilde{x}\right),{\tilde{\zeta}}_{F}\left(\tilde{x}\right),{\tilde{\zeta}}_{N}\left(\tilde{x}\right)\rangle $, and the concept of FPara. The suggested strategy can manage all of the aforementioned factors simultaneously.
- For the purpose of a favorable assessment, Table 14 and Table 15 elaborate on its computation and structural comparison with the aforementioned methods. The subsequent assessment criteria are taken into account in this regard:
- (i).
- Three-dimensional membership-based opinions (3DMO) (i.e., provision of opinions based on dependent positive, negative, and neutral membership grades).
- (ii).
- Soft settings (SoS) (i.e., parameterization mode: the inclusion of parameters for the approximation of alternatives. This kind of setting provides an approximate function to accomplish this task).
- (iii).
- Fuzzy parameterization idea (FPI) (i.e., provision of fuzzy parameterized parameters to handle the uncertainties of DMS regarding the selection of parameters).
- (iv).
- Consideration of categorical criteria (CCC) (i.e., parameters with their relevant categories of criteria).

## 5. Conclusions

- The DMS are sometimes faced with such situations that it becomes difficult for them to determine which parameters to select and which to reject, which to give more importance, and which to give less importance. In other words, they face some degree of uncertainty and ambiguity in selecting, testing, and evaluating features.
- The DMS sometimes need a DMG environment that not only reinforces their positive and negative opinions but also takes into account their impartiality to evaluate alternatives on the basis of parameters.
- A suitable mode of settings for approximating the alternatives based on parameters.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Bhutta, M.K.S. Supplier selection problem: Methodology literature review. J. Int. Inf. Manag.
**2003**, 12, 53–71. [Google Scholar] - Ware, N.; Sing, S.; Banwet, D. Supplier selection problem: A state-of-the-art review. Manag. Sci. Lett.
**2012**, 2, 1465–1490. [Google Scholar] [CrossRef] - Lambert, D.M.; Cooper, M.C. Issues in supply chain management. Ind. Mark. Manag.
**2000**, 29, 65–83. [Google Scholar] [CrossRef] - Mentzer, J.T.; DeWitt, W.; Keebler, J.S.; Min, S.; Nix, N.W.; Smith, C.D.; Zacharia, Z.G. Defining supply chain management. J. Bus. Logist.
**2001**, 22, 1–25. [Google Scholar] [CrossRef] - Tan, K.C. A framework of supply chain management literature. Eur. J. Purch. Supply Manag.
**2001**, 7, 39–48. [Google Scholar] [CrossRef] - Power, D. Supply chain management integration and implementation: A literature review. Supply Chain. Manag.
**2005**, 10, 252–263. [Google Scholar] [CrossRef] - Ng, W.L. An efficient and simple model for multiple criteria supplier selection problem. Eur. J. Oper. Res.
**2008**, 186, 1059–1067. [Google Scholar] [CrossRef] - Li, L.; Zabinsky, Z.B. Incorporating uncertainty into a supplier selection problem. Int. J. Prod. Econ.
**2011**, 134, 344–356. [Google Scholar] [CrossRef] - Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] - Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1986**, 20, 87–96. [Google Scholar] [CrossRef] - Cuong, B.C.; Kreinovich, V. Picture fuzzy sets. J. Comput. Sci. Cybern.
**2014**, 30, 409–420. [Google Scholar] [CrossRef] - Molodtsov, D. Soft set theory-first results. Comput. Math. Appl.
**1999**, 37, 19–31. [Google Scholar] [CrossRef] - Maji, P.K.; Biswas, R.; Roy, A.R. Soft set theory. Comput. Math. Appl.
**2003**, 45, 555–562. [Google Scholar] [CrossRef] - Ali, M.I.; Feng, F.; Liu, X.; Min, W.K.; Sabir, M. On some new operations in soft set theory. Comput. Math. Appl.
**2009**, 57, 1547–1553. [Google Scholar] [CrossRef] - Babitha, K.V.; Sunil, J.J. Soft set relations and functions. Comput. Math. Appl.
**2010**, 60, 1840–1849. [Google Scholar] [CrossRef] - Babitha, K.V.; Sunil, J.J. Transitive closure and ordering in soft set. Comput. Math. Appl.
**2011**, 61, 2235–2239. [Google Scholar] [CrossRef] - Sezgin, A.; Atagün, A.O. On operations of soft sets. Comput. Math. Appl.
**2011**, 61, 1457–1467. [Google Scholar] [CrossRef] - Maji, P.K.; Biswas, R.; Roy, A.R. An application of soft sets in a decision making problem. Comput. Math. Appl.
**2002**, 44, 1077–1083. [Google Scholar] [CrossRef] - Maji, P.K.; Biswas, R.; Roy, A.R. Fuzzy soft sets. J. Fuzzy Math.
**2001**, 9, 589–602. [Google Scholar] - Cağman, N.; Enginoğlu, S.; Citak, F. Fuzzy soft set theory and its applications. Iran. J. Fuzzy Syst.
**2011**, 8, 137–147. [Google Scholar] [CrossRef] - Maji, P.K.; Biswas, R.; Roy, A.R. Intuitionistic fuzzy soft sets. J. Fuzzy Math.
**2001**, 9, 677–692. [Google Scholar] - Cuong, B.C.; Kreinovich, V. An extended soft set model: Picture fuzzy soft set. In Seminar Neuro-Fuzzy Systems with Applications; Preprint 04/2014; Institute of Mathematics: Hanoi, Vietnam, 2014. [Google Scholar]
- Yang, Y.; Liang, C.; Ji, S.; Liu, T. Adjustable soft discernibility matrix based on picture fuzzy soft sets and its applications in decision making. J. Intell. Fuzzy Syst.
**2015**, 29, 1711–1722. [Google Scholar] [CrossRef] - Memiş, S. Another view on picture fuzzy soft sets and their product operations with soft decision-making. J. New Theory
**2022**, 2022, 1–13. [Google Scholar] [CrossRef] - Khan, M.J.; Kumam, P.; Ashraf, S.; Kumam, W. Generalized picture fuzzy soft sets and their application in decision support systems. Symmetry
**2019**, 11, 415. [Google Scholar] [CrossRef] - Khan, M.J.; Phiangsungnoen, S.; Kumam, W. Applications of generalized picture fuzzy soft set in concept selection. Thai J. Math.
**2020**, 18, 296–314. [Google Scholar] - Arshad, M.; Rahman, A.U.; Saeed, M. An abstract approach to convex and concave sets under refined neutrosophic set environment. Neutrosophic Sets Syst.
**2023**, 53, 274–296. [Google Scholar] - Rahman, A.U.; Arshad, M.; Saeed, M. A conceptual framework of convex and concave sets under refined intuitionistic fuzzy set environment. J. Prime Res. Math.
**2021**, 17, 122–137. [Google Scholar] - Ihsan, M.; Saeed, M.; Khan, K.A.; Nosheen, A. An algebraic approach to the variants of convexity for soft expert approximate function with intuitionistic fuzzy setting. J. Taibah Univ. Sci.
**2023**, 17, 2182144. [Google Scholar] [CrossRef] - Asghar, A.; Khan, K.A.; Albahar, M.A.; Alammari, A. An optimized multi-attribute decision-making approach to construction supply chain management by using complex picture fuzzy soft set. PeerJ Comput. Sci.
**2023**, 9, e1540. [Google Scholar] [CrossRef] - Vimala, J.; Mahalakshmi, P.; Rahman, A.U.; Saeed, M. A customized TOPSIS method to rank the best airlines to fly during COVID-19 pandemic with q-rung orthopair multi-fuzzy soft information. Soft Comput.
**2023**, 27, 14571–14584. [Google Scholar] [CrossRef] - Rajesh, R.; Ravi, V. Supplier selection in resilient supply chains: A grey relational analysis approach. J. Clean. Prod.
**2015**, 86, 343–359. [Google Scholar] - ÓBrien, W.J.; Formoso, C.T.; Vrijhoef, R.; London, K.A. Construction Supply Chain Management Handbook. Constr. Manag. Econ.
**2009**, 27, 1265–1266. [Google Scholar] - Aretoulis, G.N.; Kalfakakou, G.P.; Striagka, F.Z. Construction material supplier selection under multiple criteria. Oper. Res.
**2010**, 10, 209–230. [Google Scholar] [CrossRef] - Safa, M.; Shahi, A.; Haas, C.T.; Hipel, K.W. Supplier selection process in an integrated construction materials management model. Autom. Constr.
**2014**, 48, 64–73. [Google Scholar] [CrossRef] - Stević, Ž.; Pamučar, D.; Vasiljeviéć, M.; Stojić, G.; Korica, S. Novel integrated multi-criteria model for supplier selection: Case study construction company. Symmetry
**2017**, 9, 279. [Google Scholar] [CrossRef] - Yin, S.; Li, B.; Dong, H.; Xing, Z. A New Dynamic Multicriteria Decision-Making Approach for Green Supplier Selection in Construction Projects under Time Sequence. Math. Probl. Eng.
**2017**, 2017, 7954784. [Google Scholar] [CrossRef] - Cengiz, A.E.; Aytekin, O.; Ozdemir, I.; Kusan, H.; Çabuk, A. A multi-criteria decision model for construction material supplier selection. Procedia Eng.
**2017**, 196, 294–301. [Google Scholar] [CrossRef] - Yazdani, M.; Wen, Z.; Liao, H.; Banaitis, A.; Turskis, Z. A grey combined compromise solution (CoCoSo-G) method for supplier selection in construction management. J. Civ. Eng. Manag.
**2019**, 25, 858–874. [Google Scholar] [CrossRef] - Hoseini, S.A.; Fallahpour, A.; Wong, K.Y.; Mahdiyar, A.; Saberi, M.; Durdyev, S. Sustainable supplier selection in construction industry through hybrid fuzzy-based approaches. Sustainability
**2021**, 13, 1413. [Google Scholar] [CrossRef] - Matić, B.; Jovanović, S.; Das, D.K.; Zavadskas, E.K.; Stević, Ž.; Sremac, S.; Marinković, M. A new hybrid MCDM model: Sustainable supplier selection in a construction company. Symmetry
**2019**, 11, 353. [Google Scholar] [CrossRef] - Eshtehardian, E.; Ghodousi, P.; Bejanpour, A. Using ANP and AHP for the supplier selection in the construction and civil engineering companies; Case study of Iranian company. KSCE J. Civ. Eng.
**2013**, 17, 262–270. [Google Scholar] [CrossRef] - Yazdani, M.; Chatterjee, P.; Pamucar, D.; Abad, M.D. A risk-based integrated decision-making model for green supplier selection: A case study of a construction company in Spain. Kybernetes
**2020**, 49, 1229–1252. [Google Scholar] [CrossRef] - Wang, T.K.; Zhang, Q.; Chong, H.Y.; Wang, X. Integrated supplier selection framework in a resilient construction supply chain: An approach via analytic hierarchy process (AHP) and grey relational analysis (GRA). Sustainability
**2017**, 9, 289. [Google Scholar] [CrossRef] - Kannan, D.; Khodaverdi, R.; Olfat, L.; Jafarian, A.; Diabat, A. Integrated fuzzy multi criteria decision making method and multi-objective programming approach for supplier selection and order allocation in a green supply chain. J. Clean. Prod.
**2013**, 47, 355–367. [Google Scholar] - Xiao, Z.; Chen, W.; Li, L. An integrated FCM and fuzzy soft set for supplier selection problem based on risk evaluation. Appl. Math. Model.
**2012**, 36, 1444–1454. [Google Scholar] [CrossRef] - Patra, K.; Mondal, S.K. A supplier selection model with fuzzy risk analysis using the balanced solution technique with a soft set. Pac. Sci. Rev. Nat. Sci. Eng.
**2016**, 18, 162–168. [Google Scholar] [CrossRef] - Chang, K.H. Enhanced assessment of a supplier selection problem by integration of soft sets and hesitant fuzzy linguistic term set. Proc. Inst. Mech. Eng. Part B J. Eng. Manuf.
**2015**, 229, 1635–1644. [Google Scholar] [CrossRef] - Tan, K.C.; Kannan, V.R.; Handfield, R.B. Supply chain management: Supplier performance and firm performance. Int. J. Purch. Mater. Manag.
**1998**, 34, 1–23. [Google Scholar] - Liao, C.N.; Fu, Y.K.; Wu, L.C. Integrated FAHP, ARAS-F and MSGP methods for green supplier evaluation and selection. Technol. Econ. Dev. Econ.
**2016**, 22, 651–669. [Google Scholar] [CrossRef] - Quan, M.Y.; Wang, Z.L.; Liu, H.C.; Shi, H. A hybrid MCDM approach for large group green supplier selection with uncertain linguistic information. IEEE Access
**2018**, 6, 50372–50383. [Google Scholar] [CrossRef] - Çağman, N.; Çıtak, F.; Enginoğlu, S. Fuzzy parameterized fuzzy soft set theory and its applications. Turk. J. Fuzzy Syst.
**2010**, 1, 21–35. [Google Scholar] - Zhu, K.; Zhan, J. Fuzzy parameterized fuzzy soft sets and decision making. Int. J. Mach. Learn. Cybern.
**2016**, 7, 1207–1212. [Google Scholar] [CrossRef] - Bashir, M.; Salleh, A.R. Fuzzy parameterized soft expert set. Abstr. Appl. Anal.
**2022**, 2022, 258361. [Google Scholar] [CrossRef] - Sulukan, E.; Çağman, N.; Aydin, T. Fuzzy parameterized intuitionistic fuzzy soft sets and their application to a performance-based value assignment problem. J. New Theory
**2019**, 2019, 79–88. [Google Scholar] - Riaz, M.; Hashmi, M.R. Certain applications of fuzzy parameterized fuzzy soft sets in decision-making problems. Int. J. Algebra Stat.
**2016**, 5, 135–146. [Google Scholar] [CrossRef] - Hazaymeh, A.; Abdullah, I.B.; Balkhi, Z.; Ibrahim, R. Fuzzy Parameterized Fuzzy Soft Expert Set. Appl. Math. Sci.
**2012**, 6, 5547–5564. [Google Scholar] - Thammajitr, K.; Visit, P.; Suebsan, P. Fuzzy parameterized relative fuzzy soft sets in decisionmaking problems. Int. J. Innov. Comput. Inf. Control
**2022**, 18, 867–881. [Google Scholar] - Rotjanasom, C.; Inbunleu, C.; Suebsan, P. Applications of fuzzy parameterized relative soft sets in decision-making problems. IAENG Int. J. Appl. Math.
**2021**, 51, 607–612. [Google Scholar] - Sarkis, J.; Talluri, S. A model for strategic supplier selection. J. Supply Chain Manag.
**2002**, 38, 18–29. [Google Scholar] [CrossRef] - Liu, F.H.F.; Hai, H.L. The voting analytic hierarchy process method for selecting supplier. Int. J. Prod. Econ.
**2005**, 97, 308–317. [Google Scholar] [CrossRef] - Mukherjee, S.; Kar, S. Multi attribute decision making based on fuzzy logic and its application in supplier selection problem. Oper. Supply Chain Manag. Int. J.
**2014**, 5, 76–83. [Google Scholar] [CrossRef]

Abbreviations | Stand for | Abbreviations | Stand for |
---|---|---|---|

SuCM | supply chain management | MADM | multiattribute decision making |

MCDM | multicriteria decision making | FuS | fuzzy sets |

InFS | intuitionistic fuzzy sets | PiFS | picture fuzzy set |

SoS | soft set | DMG | decision making |

FuSS | fuzzy soft set | InFSS | intuitionistic fuzzy soft set |

PiFSS | picture fuzzy soft set | SuSP | supplier selection problem |

CSuCM | construction supply chain management | CSuC | construction supply chains |

DMS | decision makers | FpPiFSS | fuzzy parameterized picture fuzzy soft set |

FPara | fuzzy parameterization | FPGs | fuzzy parameterized grades |

$\widehat{\Pi}$ | ${\tilde{\mathit{x}}}_{1}$ | ${\tilde{\mathit{x}}}_{2}$ | ${\tilde{\mathit{x}}}_{3}$ | ${\tilde{\mathit{x}}}_{4}$ |
---|---|---|---|---|

$\frac{{\tilde{e}}_{1}}{0.2}$ | $\langle 0.21,0.33,0.32\rangle $ | $\langle 0.23,0.35,0.34\rangle $ | $\langle 0.25,0.37,0.36\rangle $ | $\langle 0.27,0.39,0.38\rangle $ |

$\frac{{\tilde{e}}_{2}}{0.5}$ | $\langle 0.31,0.21,0.11\rangle $ | $\langle 0.33,0.23,0.13\rangle $ | $\langle 0.35,0.25,0.15\rangle $ | $\langle 0.37,0.27,0.17\rangle $ |

$\frac{{\tilde{e}}_{3}}{0.3}$ | $\langle 0.11,0.31,0.21\rangle $ | $\langle 0.13,0.33,0.23\rangle $ | $\langle 0.15,0.35,0.25\rangle $ | $\langle 0.17,0.37,0.27\rangle $ |

$\frac{{\tilde{e}}_{4}}{0.6}$ | $\langle 0.41,0.11,0.31\rangle $ | $\langle 0.43,0.13,0.33\rangle $ | $\langle 0.45,0.15,0.35\rangle $ | $\langle 0.47,0.17,0.37\rangle $ |

${\widehat{\Pi}}_{1}$ | ${\tilde{\mathit{x}}}_{1}$ | ${\tilde{\mathit{x}}}_{2}$ | ${\tilde{\mathit{x}}}_{3}$ | ${\tilde{\mathit{x}}}_{4}$ |
---|---|---|---|---|

$\frac{{\tilde{e}}_{1}}{0.2}$ | $\langle 0.21,0.33,0.32\rangle $ | $\langle 0.23,0.35,0.34\rangle $ | $\langle 0.25,0.37,0.36\rangle $ | $\langle 0.27,0.39,0.38\rangle $ |

$\frac{{\tilde{e}}_{2}}{0.5}$ | $\langle 0.31,0.21,0.11\rangle $ | $\langle 0.33,0.23,0.13\rangle $ | $\langle 0.35,0.25,0.15\rangle $ | $\langle 0.37,0.27,0.17\rangle $ |

$\frac{{\tilde{e}}_{3}}{0.3}$ | $\langle 0.11,0.31,0.21\rangle $ | $\langle 0.13,0.33,0.23\rangle $ | $\langle 0.15,0.35,0.25\rangle $ | $\langle 0.17,0.37,0.27\rangle $ |

$\frac{{\tilde{e}}_{4}}{0.6}$ | $\langle 0.41,0.11,0.31\rangle $ | $\langle 0.43,0.13,0.33\rangle $ | $\langle 0.45,0.15,0.35\rangle $ | $\langle 0.47,0.17,0.37\rangle $ |

${\widehat{\Pi}}_{2}$ | ${\tilde{\mathit{x}}}_{1}$ | ${\tilde{\mathit{x}}}_{2}$ | ${\tilde{\mathit{x}}}_{3}$ | ${\tilde{\mathit{x}}}_{4}$ |
---|---|---|---|---|

$\frac{{\tilde{e}}_{1}}{0.2}$ | $\langle 0.22,0.32,0.31\rangle $ | $\langle 0.24,0.34,0.33\rangle $ | $\langle 0.26,0.36,0.35\rangle $ | $\langle 0.28,0.38,0.37\rangle $ |

$\frac{{\tilde{e}}_{2}}{0.5}$ | $\langle 0.32,0.20,0.10\rangle $ | $\langle 0.34,0.22,0.12\rangle $ | $\langle 0.36,0.24,0.14\rangle $ | $\langle 0.38,0.26,0.16\rangle $ |

$\frac{{\tilde{e}}_{3}}{0.3}$ | $\langle 0.12,0.30,0.20\rangle $ | $\langle 0.14,0.32,0.22\rangle $ | $\langle 0.16,0.34,0.24\rangle $ | $\langle 0.18,0.36,0.26\rangle $ |

$\frac{{\tilde{e}}_{4}}{0.6}$ | $\langle 0.42,0.10,0.30\rangle $ | $\langle 0.44,0.12,0.32\rangle $ | $\langle 0.46,0.14,0.34\rangle $ | $\langle 0.48,0.16,0.36\rangle $ |

${\widehat{\Pi}}_{3}$ | ${\tilde{\mathit{x}}}_{1}$ | ${\tilde{\mathit{x}}}_{2}$ | ${\tilde{\mathit{x}}}_{3}$ | ${\tilde{\mathit{x}}}_{4}$ |
---|---|---|---|---|

$\frac{{\tilde{e}}_{1}}{0.2}$ | $\langle 0.22,0.32,0.31\rangle $ | $\langle 0.24,0.34,0.33\rangle $ | $\langle 0.26,0.36,0.35\rangle $ | $\langle 0.28,0.38,0.37\rangle $ |

$\frac{{\tilde{e}}_{2}}{0.5}$ | $\langle 0.32,0.20,0.10\rangle $ | $\langle 0.34,0.22,0.12\rangle $ | $\langle 0.36,0.24,0.14\rangle $ | $\langle 0.38,0.26,0.16\rangle $ |

$\frac{{\tilde{e}}_{3}}{0.3}$ | $\langle 0.12,0.30,0.20\rangle $ | $\langle 0.14,0.32,0.22\rangle $ | $\langle 0.16,0.34,0.24\rangle $ | $\langle 0.18,0.36,0.26\rangle $ |

$\frac{{\tilde{e}}_{4}}{0.6}$ | $\langle 0.42,0.10,0.30\rangle $ | $\langle 0.44,0.12,0.32\rangle $ | $\langle 0.46,0.14,0.34\rangle $ | $\langle 0.48,0.16,0.36\rangle $ |

${\widehat{\Pi}}_{4}$ | ${\tilde{\mathit{x}}}_{1}$ | ${\tilde{\mathit{x}}}_{2}$ | ${\tilde{\mathit{x}}}_{3}$ | ${\tilde{\mathit{x}}}_{4}$ |
---|---|---|---|---|

$\frac{{\tilde{e}}_{1}}{0.2}$ | $\langle 0.21,0.33,0.32\rangle $ | $\langle 0.23,0.35,0.34\rangle $ | $\langle 0.25,0.37,0.36\rangle $ | $\langle 0.27,0.39,0.38\rangle $ |

$\frac{{\tilde{e}}_{2}}{0.5}$ | $\langle 0.31,0.21,0.11\rangle $ | $\langle 0.33,0.23,0.13\rangle $ | $\langle 0.35,0.25,0.15\rangle $ | $\langle 0.37,0.27,0.17\rangle $ |

$\frac{{\tilde{e}}_{3}}{0.3}$ | $\langle 0.11,0.31,0.21\rangle $ | $\langle 0.13,0.33,0.23\rangle $ | $\langle 0.15,0.35,0.25\rangle $ | $\langle 0.17,0.37,0.27\rangle $ |

$\frac{{\tilde{e}}_{4}}{0.6}$ | $\langle 0.41,0.11,0.31\rangle $ | $\langle 0.43,0.13,0.33\rangle $ | $\langle 0.45,0.15,0.35\rangle $ | $\langle 0.47,0.17,0.37\rangle $ |

Sr. No. | Category | Parameter | Adoptation |
---|---|---|---|

1 | Reputation | Purchase Cost | Valid |

2 | Certifications | Product Quality | Valid |

3 | Financial Health | Capacity | Valid |

4 | Collaboration | Delivery Level | Valid |

5 | Product Development | Lead Time | Valid |

6 | Customer Base | Location | Valid |

7 | Social Responsibility | Flexibility | Valid |

8 | Sustainability | Green Degree | Valid |

Parameters | DM ${\mathit{E}}_{1}$ | DM ${\mathit{E}}_{2}$ | DM ${\mathit{E}}_{3}$ | DM ${\mathit{E}}_{4}$ |
---|---|---|---|---|

${\tilde{e}}_{1}$ | 0.21 | 0.32 | 0.29 | 0.18 |

${\tilde{e}}_{2}$ | 0.11 | 0.42 | 0.39 | 0.08 |

${\tilde{e}}_{3}$ | 0.15 | 0.35 | 0.25 | 0.25 |

${\tilde{e}}_{4}$ | 0.28 | 0.22 | 0.23 | 0.27 |

${\tilde{e}}_{5}$ | 0.41 | 0.19 | 0.26 | 0.14 |

${\tilde{e}}_{6}$ | 0.42 | 0.18 | 0.22 | 0.18 |

${\tilde{e}}_{7}$ | 0.22 | 0.22 | 0.16 | 0.40 |

${\tilde{e}}_{8}$ | 0.14 | 0.16 | 0.25 | 0.45 |

**Table 9.**The FPGs $\tilde{\mu}\left({\tilde{e}}_{i}\right)$ for selected parameters ${\tilde{e}}_{i},i=1,2,3,4$.

${\tilde{\mathit{e}}}_{\mathit{i}}$ | $\tilde{\mathit{\mu}}\left({\tilde{\mathit{e}}}_{\mathit{i}}\right)$ | ${\tilde{\mathit{e}}}_{\mathit{i}}$ | $\tilde{\mathit{\mu}}\left({\tilde{\mathit{e}}}_{\mathit{i}}\right)$ |
---|---|---|---|

${\tilde{e}}_{1}$ | $\frac{0.2500+0.2434+0.2368}{3}=0.2434$ | ${\tilde{e}}_{5}$ | $\frac{0.2500+0.2308+0.2140}{3}=0.2316$ |

${\tilde{e}}_{2}$ | $\frac{0.2500+0.1948+0.1507}{3}=0.1985$ | ${\tilde{e}}_{6}$ | $\frac{0.2500+0.2339+0.2218}{3}=0.2352$ |

${\tilde{e}}_{3}$ | $\frac{0.2500+0.2393+0.2283}{3}=0.1559$ | ${\tilde{e}}_{7}$ | $\frac{0.2500+0.2359+0.2242}{3}=0.2367$ |

${\tilde{e}}_{4}$ | $\frac{0.2500+0.2487+0.2474}{3}=0.2487$ | ${\tilde{e}}_{8}$ | $\frac{0.2500+0.2241+0.2039}{3}=0.2260$ |

$\widehat{\Pi}$ | ${\tilde{\mathit{x}}}_{1}$ | ${\tilde{\mathit{x}}}_{2}$ | ${\tilde{\mathit{x}}}_{3}$ | ${\tilde{\mathit{x}}}_{4}$ |
---|---|---|---|---|

$\frac{{\tilde{e}}_{1}}{0.2434}$ | $\langle 0.27,0.39,0.38\rangle $ | $\langle 0.25,0.37,0.36\rangle $ | $\langle 0.23,0.35,0.34\rangle $ | $\langle 0.21,0.33,0.32\rangle $ |

$\frac{{\tilde{e}}_{2}}{0.1985}$ | $\langle 0.37,0.27,0.17\rangle $ | $\langle 0.35,0.25,0.15\rangle $ | $\langle 0.33,0.23,0.13\rangle $ | $\langle 0.31,0.21,0.11\rangle $ |

$\frac{{\tilde{e}}_{3}}{0.1559}$ | $\langle 0.17,0.37,0.27\rangle $ | $\langle 0.15,0.35,0.25\rangle $ | $\langle 0.13,0.33,0.23\rangle $ | $\langle 0.11,0.31,0.21\rangle $ |

$\frac{{\tilde{e}}_{4}}{0.2487}$ | $\langle 0.47,0.17,0.37\rangle $ | $\langle 0.45,0.15,0.35\rangle $ | $\langle 0.43,0.13,0.33\rangle $ | $\langle 0.41,0.11,0.31\rangle $ |

$\frac{{\tilde{e}}_{5}}{0.2316}$ | $\langle 0.46,0.18,0.38\rangle $ | $\langle 0.35,0.25,0.25\rangle $ | $\langle 0.41,0.11,0.31\rangle $ | $\langle 0.43,0.13,0.33\rangle $ |

$\frac{{\tilde{e}}_{6}}{0.2352}$ | $\langle 0.48,0.19,0.39\rangle $ | $\langle 0.25,0.25,0.45\rangle $ | $\langle 0.43,0.15,0.35\rangle $ | $\langle 0.42,0.17,0.37\rangle $ |

$\frac{{\tilde{e}}_{7}}{0.2367}$ | $\langle 0.41,0.11,0.31\rangle $ | $\langle 0.45,0.25,0.15\rangle $ | $\langle 0.45,0.17,0.37\rangle $ | $\langle 0.47,0.19,0.36\rangle $ |

$\frac{{\tilde{e}}_{8}}{0.2260}$ | $\langle 0.45,0.12,0.32\rangle $ | $\langle 0.47,0.24,0.34\rangle $ | $\langle 0.41,0.11,0.31\rangle $ | $\langle 0.48,0.21,0.21\rangle $ |

${\mathbb{M}}_{1}$ | ${\tilde{\mathit{x}}}_{1}$ | ${\tilde{\mathit{x}}}_{2}$ | ${\tilde{\mathit{x}}}_{3}$ | ${\tilde{\mathit{x}}}_{4}$ |
---|---|---|---|---|

$\frac{{\tilde{e}}_{1}}{0.2434}$ | 0.50 | 0.48 | 0.46 | 0.44 |

$\frac{{\tilde{e}}_{2}}{0.1985}$ | 0.07 | 0.05 | 0.03 | 0.01 |

$\frac{{\tilde{e}}_{3}}{0.1559}$ | 0.47 | 0.45 | 0.43 | 0.41 |

$\frac{{\tilde{e}}_{4}}{0.2487}$ | 0.07 | 0.05 | 0.03 | 0.03 |

$\frac{{\tilde{e}}_{5}}{0.2316}$ | 0.10 | 0.15 | 0.01 | 0.03 |

$\frac{{\tilde{e}}_{6}}{0.2352}$ | 0.10 | 0.45 | 0.07 | 0.12 |

$\frac{{\tilde{e}}_{7}}{0.2367}$ | 0.01 | 0.05 | 0.09 | 0.08 |

$\frac{{\tilde{e}}_{8}}{0.2260}$ | 0.01 | 0.11 | 0.01 | 0.06 |

${\mathbb{M}}_{2}$ | ${\tilde{\mathit{x}}}_{1}$ | ${\tilde{\mathit{x}}}_{2}$ | ${\tilde{\mathit{x}}}_{3}$ | ${\tilde{\mathit{x}}}_{4}$ |
---|---|---|---|---|

${\tilde{e}}_{1}$ | 0.121700 | 0.116832 | 0.111964 | 0.107096 |

${\tilde{e}}_{2}$ | 0.013895 | 0.009925 | 0.005955 | 0.001985 |

${\tilde{e}}_{3}$ | 0.073273 | 0.070155 | 0.067037 | 0.063919 |

${\tilde{e}}_{4}$ | 0.017409 | 0.012435 | 0.007461 | 0.007461 |

${\tilde{e}}_{5}$ | 0.023160 | 0.034740 | 0.009264 | 0.006948 |

${\tilde{e}}_{6}$ | 0.023520 | 0.105840 | 0.016464 | 0.028224 |

${\tilde{e}}_{7}$ | 0.002367 | 0.011835 | 0.021303 | 0.018936 |

${\tilde{e}}_{8}$ | 0.002260 | 0.024860 | 0.002260 | 0.013560 |

${\mathbb{M}}_{3}$ | ${\tilde{\mathit{x}}}_{1}$ | ${\tilde{\mathit{x}}}_{2}$ | ${\tilde{\mathit{x}}}_{3}$ | ${\tilde{\mathit{x}}}_{4}$ |
---|---|---|---|---|

$\mathbb{S}\left({\tilde{x}}_{m}\right)$ | 0.277584 | 0.386622 | 0.241708 | 0.248129 |

References | Application | Ranking |
---|---|---|

Mukherjee et al. [62] | SuSP | ${\tilde{x}}_{3}>{\tilde{x}}_{2}>{\tilde{x}}_{1}>{\tilde{x}}_{4}$ |

Liao et al. [50] | SuSP | ${\tilde{x}}_{1}>{\tilde{x}}_{2}>{\tilde{x}}_{4}>{\tilde{x}}_{3}$ |

Quan et al. [51] | SuSP | ${\tilde{x}}_{2}>{\tilde{x}}_{1}>{\tilde{x}}_{3}>{\tilde{x}}_{4}$ |

Proposed Approach | SuSP | ${\tilde{x}}_{2}>{\tilde{x}}_{1}>{\tilde{x}}_{4}>{\tilde{x}}_{3}$ |

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## Share and Cite

**MDPI and ACS Style**

Rahman, A.U.; Alballa, T.; Alqahtani, H.; Khalifa, H.A.E.-W.
A Fuzzy Parameterized Multiattribute Decision-Making Framework for Supplier Chain Management Based on Picture Fuzzy Soft Information. *Symmetry* **2023**, *15*, 1872.
https://doi.org/10.3390/sym15101872

**AMA Style**

Rahman AU, Alballa T, Alqahtani H, Khalifa HAE-W.
A Fuzzy Parameterized Multiattribute Decision-Making Framework for Supplier Chain Management Based on Picture Fuzzy Soft Information. *Symmetry*. 2023; 15(10):1872.
https://doi.org/10.3390/sym15101872

**Chicago/Turabian Style**

Rahman, Atiqe Ur, Tmader Alballa, Haifa Alqahtani, and Hamiden Abd El-Wahed Khalifa.
2023. "A Fuzzy Parameterized Multiattribute Decision-Making Framework for Supplier Chain Management Based on Picture Fuzzy Soft Information" *Symmetry* 15, no. 10: 1872.
https://doi.org/10.3390/sym15101872