Research Article  Open Access
Muhammad Naeem, Muhammad Qiyas, Saleem Abdullah, "An Approach of IntervalValued Picture Fuzzy Uncertain Linguistic Aggregation Operator and Their Application on Supplier Selection DecisionMaking in Logistics Service Value Concretion", Mathematical Problems in Engineering, vol. 2021, Article ID 8873230, 19 pages, 2021. https://doi.org/10.1155/2021/8873230
An Approach of IntervalValued Picture Fuzzy Uncertain Linguistic Aggregation Operator and Their Application on Supplier Selection DecisionMaking in Logistics Service Value Concretion
Abstract
With respect to multiple criteria group decisionmaking (MCGDM) problems in which both the criteria weights and the expert weights take the form of crisp numbers and attribute values take the form of intervalvalued picture fuzzy uncertain linguistic numbers, some new group decisionmaking analysis methods are developed. Firstly, some operational laws, expected value, and accuracy function of intervalvalued picture fuzzy uncertain linguistic numbers are introduced. Then, an intervalvalued picture fuzzy uncertain linguistic averaging and geometric aggregation operators are developed. Furthermore, some desirable properties of the developed operators, such as commutativity, idempotency, and monotonicity, have been studied. Based on these operators, an approach to multiple criteria group decisionmaking with intervalvalued picture fuzzy uncertain linguistic information has been proposed. Finally, a practical example of 3PL supplier selection in logistics service value concretion is taken to test the defined method and to expose the effectiveness of the defined model.
1. Introduction
Decisionmaking (DM) has been influential in daytoday activities such as education, economics, engineering, and medical. In DM, the problems contain a lot of information sources, giving the final result through the aggregating processes. Experts can take decisions on certain level due to the convolution of such decisionmaking (DM) problem and management information themselves, but they may have doubts about their interpretations. Specially, there may be a grade of hesitation, which is too necessary to focus on, while organizing completely beneficial models and problems. These degrees of hesitation are better defined by intuitionistic fuzzy set (IFS) values rather than objective numbers. The generalized form of Zadeh fuzzy sets (FSs) [1] is intuitionistic fuzzy sets [2]. The element of the IFS occurs in the ordered pair form, consisting of positive grade and negative grade, and the sum of the two grades characterize is less than or equal to 1. Many researchers have made a significant contribution to the expansion of IFS generalization and its application to various fields, resulting in greater success of IFSs in theory and technology. The aggregation of IF information [3–6] is a big part of multicriteria decisionmaking (MCDM) with IFSs. It is why IFNs are too easy to reveal predilection details of a decisionmaker over artifacts in the DM phase with unknown or firm chances. A significant step towards achieving the result of a decision problem is the aggregation of IFNs. The number of operators known as IFHA, IFOWA, and IFOWG operators has recently been introduced for this purpose to aggregate IFNs [7–12].
Wang and Li [13] proposed the Pythagorean fuzzy interaction power Bonferroni mean aggregation operators in MADM. Verma and Sharma [14] proposed the exponential entropy on IFSs. Wan and Dong [15] discussed some MADM based on triangular IFN Choquet integral operator. Wan [16] developed the power average operators of trapezoidal IFSs and application to MAGDM. Wan et al. [17] proposed the power geometric operators for trapezoidal IFSs and application to MAGDM. Verma [14] developed MAGDM approach based on intuitionistic fuzzy order divergence and entropy measures with the MABAC method. Wan et al. [18] defined trapezoidal IF prioritized AOs and application to MADM. Wan and Yi [19] defined power average of trapezoidal IFSs using strict tnorms and tconorms. Xu et al. [20] developed aggregating decision information into Atanassov’s intuitionistic fuzzy numbers for heterogeneous MAGDM. Wan et al. [21] defined some new generalized AOs for triangular IFNs and application to MAGDM. Dong and Wan [22] defined a new method for prioritized MCGDM with triangular IFNs. Dong et al. [23] developed some generalized Choquet integral operator of triangular Atanassov’s IFNs and discussed their application to MAGDM. Verma and Sharma [24] studied some new measure of inaccuracy and its application to multicriteria MCDM under IF environment. Verma [25] discussed the generalized Bonferroni mean operator for IFNs and its application to MADM. Wan and Zhu [26] introduced the triangular IF triple Bonferroni harmonic mean operators and application to MAGDM. Wan and Dong [27] developed aggregating decision information into IVIFNs for heterogeneous MAGDM. Wan and Dong [28] give the DM theories and methods based on IVIF sets. Liu and Garg [29] defined the linguistic connection number of set pair analysis based on the TOPSIS method and numerical scale function. Verma [30] proposed some AOs for linguistic trapezoidal IFSs and their application to MAGDM. Verma and Merigo [31] defined the approach of MAGD based on 2dimension linguistic intuitionistic fuzzy aggregation operators.
Batool et al. [32] defined the entropybased Pythagorean probabilistic hesitant fuzzy decisionmaking technique and their application for FogHaze factor assessment problem. Khan et al. [33] proposed the Pythagorean fuzzy (PyF) Dombi AOs for the decision support system. Ashraf et al. [34] developed the fuzzy decision support modeling for Internet finance soft power evaluation using the sine trigonometric Pythagorean fuzzy information. Wan et al. [35] defined the Pythagorean fuzzy mathematical programming method for MAGDM with Pythagorean fuzzy truth degrees. Wan et al. [36] defined a new order relation for PyFNs and application to MAGDM. Garg [37] developed the linguistic intervalvalued PyFSs and their application in MAGDM process. Wan et al. [38] introduced a threephase method for PyF multiattribute group decisionmaking and application to haze management. Wang et al. [39] defined PyF interactive Hamacher power aggregation operators for assessment of express service quality with entropy weight. Garg [29] introduced linguistic singlevalued neutrosophic power AOs and their applications to group DM problems.
Since IFSs have two kinds of reports, i.e., yes and no, but in the case of election, there is some problem with the three styles of response, e.g., yes, no, and refused, where the optimistic answer is a refusal. Cuong [40, 41] defined the principle of picture fuzzy set (PFS) to overcome this defect, dignifying positive, neutral, and negative grades in three separate functions. Cuong [42] discussed some PFS features and agreed with distance measurements as well. In the PF logic for fuzzy derivation forms, Cuong and Hai [43] defined fuzzy logic AOs and specified basic operational laws. The features of the fuzzy tnorm and tconorm for PFS are analyzed by Cuong et al. [44]. Phong et al. [45] addressed a certain framework of PF relationships. Son et al. [46, 47] offer estimates of time and temperature based on information from the PF sets. Son [48, 49] defines picture fuzzy measures of isolation, distance, association, and often combined with the PFS condition. Wei et al. [50–52] have found several methods to measure the proximity between PFSs. Several researchers have currently created further models for PFSs: Singh [53] proposes the PFS coefficient of correlation and tested it to the clustering analysis. Son [54] defined a novel structure of the PFS fluid derivation and improved a classic method of fluid inference. Thong [55, 56] used the PF clustering approach to optimize the complex and particle problems. Wei [57] used the weighted crossentropy theory of PFS to describe some simple leadership methods and utilized this approach to give ranking the alternatives. Yang et al. [58] used PFSs to define a versatile soft matrix of DM. Garg features an aggregation of MCDM problems with PFSs in [59]. The PFS solution was implemented by Peng et al. [60] and applied to DM. In addition, for the PFS, readers can also see [61, 62]. Shahzaib et al. [63] are expanding the PFS cubic set model. Thus, the study objective is divided into three parts under the IVPFULNs. Ashraf et al. [64] developed the cleaner production evaluation in gold mines using a novel distance measure method with cubic PFNs. Khan et al. [65] defined picture fuzzy aggregation information based on Einstein operations and their application in DM. Ashraf and Abdullah [66] proposed some novel aggregation operators for cubic picture fuzzy information and discussed their application for multiattribute decision support problem. Zeng et al. [67] defined the application of exponential Johnson picture fuzzy divergence measure in MCGD. Ashraf et al. [68] developed some aggregation operators of cubic picture fuzzy quantities and their application in decision support systems. Khalil [69] defined a new operation on intervalvalued picture fuzzy set, intervalvalued picture fuzzy soft set, and their applications. Akram et al. [70] proposed a DM model under complex picture fuzzy Hamacher AOs. Yang [71] proposed a group decision algorithm for aged health care product purchase under qrung picture normal fuzzy environment using Heronian mean operator.
Moreover, in many multiple criteria group decisionmaking (MCGDM) problems, considering that the estimations of the criteria values are intervalvalued picture fuzzy uncertain linguistic sets, it therefore is very necessary to give some aggregation techniques to aggregate the intervalvalued picture fuzzy uncertain linguistic information. However, we are aware that the existing aggregation techniques have difficulty in coping with group decisionmaking problems with intervalvalued picture fuzzy uncertain linguistic information. Therefore, we in the current paper propose a series of aggregation operators for aggregating the intervalvalued picture fuzzy uncertain linguistic information and investigate some properties of these operators. Then, based on the defined aggregation operators, we develop an approach to MCGDM with intervalvalued picture fuzzy uncertain linguistic information. Moreover, we use a numerical example to show the application of the developed approach.
The remainder of the manuscript is arranged accordingly: in Section 2, first we discuss some fundamental ideas relating to the IVPFULS. Then, we described a number of AOs and discussed their basic properties, in Section 3. In Section 4, we discussed the supplier selection group decision model in logistics service value cocreation using the IVPFULG and IVPFULHG operators. An illustrative example of the selection of 3PL suppliers in the logistics service value cocreation information is given in Section 5, to explain the objective of the model. The article ends in Section 6.
2. Preliminaries
We defined some basic definitions relevant to the IVPFULSs in this section.
Definition 1 (see [1]). Let be a nonempty set. Then, a fuzzy set is described aswhere is the positive membership function of .
Definition 2 (see [2]). Let be a nonempty set. Then, an intuitionistic fuzzy set is described asfor an element , and the function represents the positive and negative grades, respectively, with for . And hesitation margin of to is obtained as .
Definition 3 (see [72]). Let be a nonempty set. A picture fuzzy set of is defined aswhere the function represents the function of positive and represents the function of neutral and negative membership, respectively, with the condition for . The picture fuzzy hesitation margin of to is given by , is called the indeterminacy grade of to the PFS .
Definition 4 (see [73]). Let be a nonempty set. Then, the picture fuzzy linguistic set in is aswhere is the linguistic number, is a positive grade, is a neutral grade, and a negative grade of the element to under the condition , and the refusal grade of to for all is represented asIf , , then PFLS becomes to IFLS.
Definition 5 (see [74]). Let be the closed intervals set and are the given set. Then, intervalvalued picture fuzzy set (IVPFS) is described aswhere , and . The intervals and represent the positive, neutral, and negative grades of the elements , respectively. Thus, for every , , and are closed intervals, and their lower and upper end points are symbolized as, , and . We can write aswhere , and hesitation interval relative to , for every element , is computed asFor any element , the triple is known as intervalvalued picture fuzzy numbers (IVPFNs). For convenience, the triple is often represented by , where and .
Definition 6 (see [74]). Let and are the two IVPFSs in the set and . Then, the following operational laws of IVPFN are developed:
Definition 7 (see [75, 76]). Suppose that be a discrete linguistic term set, where is the odd number, and . For example, , and then the linguistic term set is defined as = {poor, slightly poor, fair, slightly good, good}. If , then the following properties must be satisfied by the linguistic term set:(1)Negation (2)Maximum , if (3)Minimum , if Suppose that and are the lower limit and upper limit of , correspondingly. Then, is said to be an ncertain linguistic variable.
Definition 8 (see [77]). Let denote the family of uncertain linguistic variables. Then, the following operation is defined for and :(1)(2)(3)(4), if
Definition 9. Let , and is a nonempty set. Then,is called IVPFULSs, where and satisfy condition . The intervals and represent the positive, neutral, and negative membership grades of the elements to the uncertain linguistic variable , respectively. Thus, for every , and are closed intervals, and their lower and upper endpoints are represented by , and . We can write aswhere and .
Hesitation interval of to the uncertain linguistic variable for every element is as follows:
Definition 10. Let be IVPFULS, an 8tuple is said to be IVPFULN, and unknown linguistic variables can also be viewed as a set of intervalvalued numbers. Therefore, it can be expressed as
Definition 11. Let and are the two IVPFULNs and . Then, we have defined the following operation for IVPFULNs:(1)(2)(3)(4)
Theorem 1. Let and be the two IVPFULNs. Then, the following rules must be satisfied:(1)(2)(3), (4), (5), (6),
Proof. See for proof Appendix B.
Definition 12. Let be an IVPFULN, and a score function is defined as
Definition 13. Let be an IVPFULN, and an accuracy function is defined as
Definition 14. Let and be an IVPFULN. Then,(1)if , then, (2)if , then(a)if , then (b)if , then
3. The IntervalValued Picture Fuzzy Uncertain Linguistic Geometric Operators
Definition 15. Let be a family of IVPFULNs, and , ifwhere are the set of all IVPFULNs, and is the weighting vector of ; then, IVPFULWG is said to be the IVPFULweighted geometric operator. Especially, if , then the IVPFULWG operator is the IVPFUL geometric operator.
Theorem 2. Let be a family of IVPFULNs. Then, the result obtained from Definition 15 is also an IVPFULN:
Proof. See for proof Appendix C.
Theorem 3. Monotonicity: let and be the two collections of intervalvalued picture fuzzy uncertain linguistic numbers, if . Then,
Proof. LetSince , we have
Theorem 4. Idempotency: let . Then,
Proof. Science: , for all , and then we have
Theorem 5. Boundedness: the IVPFULWG operator lies between the maximum and minimum operators:
Proof. Let , and according to Theorem 3, we haveFurthermore,So,i.e.,
Definition 16. Let be a family of the IVPFULNs, and , ifwhere are the family of all IVPFULNs, and is an associated weight with IVPFULOWG; , are the permutation of , such as ; then, IVPFULOWG is said to be IVPFUL ordered weighted geometric operator, and denoted the position in the aggregation process. So, we called is the location weight vector.
Theorem 6. Let be a family of IVPFULNs. Then, the result aggregated from Definition 15 is also an IVPFULN, andwhere be the associated weighting vector of IVPULOWG, and , are the permutation of , such as for all .
Proof. Proof is the same as proof of Theorem 2.
Theorem 7. Commutativity: let be any permutation of . Then,
Proof. As we know thatSince are the any permutation of , we have ; then,
Theorem 8. Monotonicity: let and be the two families of IVPFULNs, if . Then,
Proof. Proof is the same as Theorem 3.
Theorem 9. Idempotency: let . Then,
Proof. Proof is the same as Theorem 4
Theorem 10. Boundedness: the IVPFULOWG operator lies between the maximum and minimum operators:
Proof. Proof is the same as Theorem 5.
Definition 17. Let be a family of IVPFULNs, and , ifwhere are the set of all IVPFULNs, and be the associated weighting vector with IVPFULOWG; , is largest of the IVPFUL weighted arguments , is the weights of , and denoted the balancing coefficient.
Theorem 11. Let be a family of IVPFULNs. Then, the result aggregated from Definition 16 is also an IVPFULN, aswhere be the associated weight vector with IVPULHG, , is biggest of the IVPFUL weighted arguments is the weights of , and shows the balancing coefficient.
Proof. Proof is the same as the proof of Theorem 2.
4. Supplier Selection Group Decision Model in Logistics Service Value Cocreation Based on IntervalValued Picture Fuzzy Sets
4.1. Supplier Selection Group Decision Model in Concretion Value of the Logistics Service
Logistics provider selection is a multicriteria concern which requires a wide variety of criteria. In their studies, Spencer et al. [78] reported 23 possible selection criteria and 35 selection factors were identified by Govindan et al. [79] which revealed eleven key 3PL selection criteria with a review of sixtyseven 3PL selection papers published in the period 1994–2013, each of which is defined by a list of attributes; the study revealed that cost was the commonly adopted criterion, followed by relationship, service, and quality [80]. While the above selection attribute is commonly used in the selection of 3PL, the selected attribute is operationally driven, whereas previous studies seldom considered the strategic supply chain and value creation variables when selecting logistics suppliers. It is important to review the selection criteria in the logistics service value concretion scenario that the development of value is the key premise of establishing and retaining the customer relationship and is the key goal and the central economic exchange mechanism [81].
More and more businesses are recognizing the importance of value cocreation for logistics services with partners in the supply chain management environment. Wan et al. [82] find the innovative way to attain competitive advantage and more personalized product and service offering for customers. Supplier selection is the most critical problems for logistics sector performance cocreation in supply chain management (SCM) setting. In the selection of 3PL suppliers, the emerging trend is the convergence of traditional selection characteristics such as cost, quality, response time, and location with new factors in the cocreation of service value, such as new value growth, knowledge management, and service innovation. In order to create full selection criteria for supplier selection in the value cocreation scenario of logistics services, we combine traditional operational selection criteria and valueoriented SCM strategic selection criteria. The supplier selection attributes for the cocreation of the logistics service value are listed in Table 1.

4.2. Algorithm for Group DecisionMaking with IntervalValued Picture Fuzzy Uncertain Linguistic Information
Using these two operators IVPFULWG and IVPFULHG, we present a group DM problem, under the IVPFUL information. Let we have be the collection of alternative and be the set of criteria with the weight vector . Let be the set of experts and be the weighting vector of experts. Let