GREEN SUPPLIER SELECTION BASED ON CODAS METHOD IN PROBABILISTIC UNCERTAIN LINGUISTIC ENVIRONMENT

Probabilistic uncertain linguistic sets (PULTSs) have widely been used in MADM or MAGDM. The CODAS method, which is a novel MADM or MAGDM tool, aims to acquire the optimal choice which have the largest Euclidean & Hamming distances from the NIS. This paper designs the probabilistic uncertain linguistic CODAS (PUL-CODAS) method with sine entropy weight. Finally, a numerical example for green supplier selection is given and the obtained results are compared with some existing models.


Introduction
The CODAS method was firstly designed by Keshavarz Ghorabaee (2016). It is a novel and useful model used to solve MADM problems with aid of deriving the Euclidean distance and Hamming distances to select the best alternative. Ghorabaee, Amiri, Zavadskas, Hooshmand, and Antuchevičienė (2018) defined the fuzzy CODAS method to select suppliers. Panchal et al. (2017) applied fuzzy CODAS to tackle the maintenance decision issue.  employed CODAS method to select the optimal desalination plant location in Libya. Yeni and Ozcelik (2019) defined the CODAS method for MAGDM under IVIFSs. Peng and Li (2019) designed the hesitant fuzzy soft CODAS method. Karasan, Bolturk, and Kahraman (2019) proposed neutrosophic CODAS method. Pamucar, Badi, Sanja, and Obradovic (2018) introduced linguistic neutrosophic CODAS method.
Due to certain complexity, experts couldn't depict their preferences through real numbers (Liao & Xu, 2014a, 2014b, 2014c, thus with help of other mathematical qualitative tool (Beg et al., 2019;Wang, 2019;Wu et al., 2019aWu et al., , 2019b. For example, the DMs could employ the linguistic terms to depict satisficing degree of a car (Herrera & Martinez, 2000b). In order to give qualitative assessment, Herrera and Martinez (2000a) designed the 2TLTSs for calculating along with words. Sohaib, Naderpour, Hussain, and Martinez (2019) defined 2-tuple linguistic TOPSIS for MAGDM issues. Furthermore, Rodriguez, Martinez, and Herrera (2012) designed the HFLTSs which depicts some possible linguistic values. Wei (2019a) defined the GDSM under HFLTSs. Liao, Xu, and Zeng (2015) developed VIKOR model under HFLTSs. Liao, Yang, and Xu (2018b) gave the ELECTRE II model with HFLTS and gave two ELECTRE II model based on the score-deviation and positive and negative ideal.
In certain situations, some DMs may depict their preferences through ULTSs (Xu, 2004). Inspired by PLTSs (Pang et al., 2016) and ULTSs (Xu, 2004), Lin, Xu, Zhai, and Yao (2018) defined probabilistic ULTSs (PULTSs). Xie, Ren, Xu, and Wang (2018) depicted some preference relation under PULTSs and designed the distance and similarity. But there are no recent existing literatures to use CODAS method to solve PUL-MAGDM. Therefore, it is very necessary to investigate such issue. The other remaining section of such paper is given. Section 1 reviews the definition of PULTSs. In Section 2, the CODAS method is defined for PUL-MAGDM along with sine entropy weight. In Section 3, a detailed example is developed and some comparative analysis is given. This paper finishes with conclusions in last Section.

Preliminaries
In such section, some basic mathematical definitions are simply reviewed.
Definition 1 (Gou et al., 2017). Let { } a = a = θ − − θ -, , 2, 1,0,1,2, L l   be an LTS, the l a could depict the corresponding information with b which is defined by using g: , 0,1 , , 2 g l l g l (1) b could also depicts the equivalent assessing information for l a which is defined with g -1 : Definition 2 (Pang et al., 2016). Given an LTS where l (f) (p (f) ) is the fth l (f) along with corresponding probability values (p (f) ), and #L(p) denotes the number of L(p). The l (f) in L(p) are listed with ascending order. Furthermore, Lin et al. (2018) defined the PULTSs based on ULTSs (Xu, 2004) and PLTSs (Pang et al., 2016).
Definition 3 (Lin et al., 2018). The PULTS is defined: where Definition 4 (Lin et al., 2018) Then, the sine entropy of PULTS is defined to get unknown attribute weights in MAGDM issue based on the idea of simplified Neutrosophic sine entropy (Cui & Ye, 2018).

CODAS method for PUL-MAGDM issue
In such part, the PUL-CODAS model for MAGDM is designed.
is named a group of given alternatives, is called a group of given attributes along with weight ( ) Then, the CODAS model is devised to deal with PUL-MAGDM issues. The calculating steps are given soon afterwards and the flowchart is given in Figure 1.
Step 1. Convert cost attribute into beneficial attribute. If cost attribute value is  Step 2. Switch the     , Step 3. Figure  Step 4. Figure up the attributes weight by sine entropy.
Since the uncertainty of one attribute increases, the attribute weight should decrease correspondingly. Thus, we may figure up unknown weights of each attribute based on the sine entropy measure formula Eq. (9). Firstly, the probabilistic uncertain linguistic sine entropy measure (PULSEM) of ( ) ij NPULDM p are designed as follows: Then, the attribute weights is: Step 5. Design the PULNIS: Step 6. Derive the probabilistic uncertain linguistic weighted Euclidean distance Step 7. Derive the PUL relative assessment matrix (PULRAM): where k∈{1, 2, …, m} and the threshold formula is given by Eq. (21): where the threshold parameter τ is between 0.01 and 0.05. In this paper, τ = 0.02 are always used to compute (Lin et al., 2018).
Step 8. Derive the PUL assessment score Step 9. Sort the alternatives with PULAS i , the optimal choice should have maximum value.

A numerical example
In today's world of resource shortage and increasingly serious environmental pollution, facing the strict environmental protection system, how to make the coordinated development of supply chain and environment while pursuing economic benefits will become an important means and decisive factor for enterprises to succeed in market competition (Tavana et al., 2017;Tong, 2017;Wang et al., 2017). Green supply chain management, as a new management mode of core enterprises under sustainable development, has been widely recognized and valued by the academic and business circles (Wei et al., 2020a(Wei et al., , 2020b(Wei et al., , 2020c(Wei et al., , 2020d. Green supplier management is the important part of green supply chain management through the coordination and cooperation with suppliers to achieve cost reduction to reduce resource consumption to improve the environment, and can make the enterprise faster response to market demand, improve the core competitiveness, establish corporate social image. Supplier selection is also an important link along with implementation of green supplier management. Green supplier selection is a very common decision issue Wei et al., 2020;2020d;Zavadskas et al., 2019;Zhang et al., 2020). Thus, in such section, an example about green supplier selection is given to proof the defined method. There are some green suppliers ( ) = 1,2,3,4,5 i GS i for experts to select according to four assessing attributes: ① Q 1 is the environmental competencies; ② Q 2 is transportation cost of suppliers; ③ Q 3 is environmental improvement quality; ④ Q 4 is financial conditions of suppliers. All these four attributes are adapted from Lei, Wei, Gao, Wu, and Wei (2020). The Q 2 is cost index and other indices are beneficial. These potential green suppliers    Then, we employ the PUL-CODAS model designed to choose the optimal green supplier.
Step 1. Convert cost index Q 2 into beneficial index (See Tables 6-10). For example, in Table 1, the ULTS [M, G] is given for alternative A 1 under G 2 by the first DM, the converted beneficial attribute value is [P, M].  Step 2. Convert the ULTSs into PULTSs (See Table 11).       Step 3. Compute the normalized PULTSs (Table 12).    Table 11 Alternatives Step 4. Derive the attributes weight from Eq. (10)-(11), the attributes weight is given in Table 13. Step 5. Obtain the PULNIS (Table 14).   Table 12 Step 6. Compute the ( )  Table 15).   Table 17). GS GS GS GS GS . That's to say, GS 4 is the optimal alternative.

Compared with ULWA
Firstly, we deal with such example by using the ULWA (Xu, 2004) with same weight to aggregate these ULTSs into a group matrix (See Table 18).  GS GS GS GS GS . Thus, we could obtain the optimal green supplier GS 4 .

Compared with PUL-TOPSIS
Then, the PUL-CODAS is compared with the PUL-TOPSIS model (Lin et al., 2018), then the derived result is obtained (Table 19). Thus, the best green supplier is GS 4 .

Compared with PULWA
Finally, the PUL-CODAS is compared with PULWA (Lin et al., 2018), the attributes weight is: = = = = Then, the score of these five alternatives are derived through Definition 9 (Lin, et al., 2018): GS GS GS GS GS and the optimal green supplier is GS 4 .

Conclusions
In such paper, we developed the CODAS model for MAGDM based on PULTSs and sine entropy weight. Firstly, the Euclidean and Hamming distance under PULTSs are introduced. Then, the CODAS method is proposed for PUL-MAGDM and its main merit is that it highlights Euclidean and Hamming distance from PULNIS. Finally, an example analysis about green supplier selection is utilized to show the defined algorithms and some detailed comparative analysis are used to elucidate the effectiveness in practical decision making. However, there still remains some unfinished work to be done. Since the computational process of the PULTSs is complicated, we need to further investigate the operations of PULTSs. Except that, the consensus analysis between different groups should be take into account. In future, we are also going to carry out researches on these two aspects and devote to apply the designed methods to other fields, such as pattern recognition, industrial engineering, E-commerce, and so on. At the same time, the corresponding application of the designed algorithms under PULTSs are studied through some other uncertain MADM and uncertain settings and the basic concept of PULTSs could be employed to expand some other fuzzy settings with help of their corresponding probability.