PICTURE FUZZY WEIGHTED DISTANCE MEASURES AND THEIR APPLICATION TO INVESTMENT SELECTION

The picture fuzzy set (PFS) is a powerful tool to collect and handle large amounts of uncertain assess information in a new light. In this study, we explore some distance measures for the PFSs and propose Picture fuzzy ordered weighted distance measure and Picture fuzzy hybrid weighted distance measure. Some of their properties are also mathematically explored. Moreover, we introduce a model for the aforesaid distance measures to solve multiple attribute group decision making (MAGDM) method in an updated way. And at the end of our paper a practical application of investment alternatives selection is provided to illustrate the validity and applicability of the presented work


Introduction
The selection of an appropriate investment scheme is a complicated multi-attribute group decision-making (MAGDM) problem, because it is necessary to take into account the multiple characteristics and attributes of the scheme during decision process.This kind of problems often includes some uncertain and vague information as it needs evaluate the schemes from different aspects, while the decision makers (DMs) may not be familiar with the characteristics of all schemes because of distinctive academic background.In this case, it is impossible for DMs to give evaluations on alternatives with crisp values.Recently, the picture fuzzy set (PFS) proposed by Cuong and Kreinovich (2013) has been proved to be a powerful tool that allows DMs to handle such uncertain and vague information with ease.Compared with the existing fuzzy tools, such as intuitionistic fuzzy set (IFS) (Atanassov, 1986) and Pythagorean fuzzy set (Yager, 2014), the main advantages of the PFS is that it is distinguished by three difffferent functions, namely the degree of membership, the degree of neutral membership and the degree of non-membership, which makes it fully consider the degree of acceptance, rejection conflicting and refusal during the analysis.Currently, some progress has been carried out in the research of the PFS condition: Cuong (2014) explored several properties of PFSs and studied a few distance measures between PFSs.Sing (2015) developed correlation coefficient of PFS and studied its application in clustering analysis.Wei (2016) studied a basic leadership method for weighted crossentropy of PFS and applied it in ranking the choices.Wei (2017) exhibited some aggregation operators for PFS and applied them to MADM for selecting EPR problems.Garg (2017) explored a few aggregation operations for PFSs and use them to MADM problems.Zhang et al. (2018) employed PFS to collect and describe uncertain assess information for offshore wind power station selection.Wang et al. (2018) developed a normalized projection-based method to evaluate construction project.Wei (2018a) extended the conventional TODIM model to solve MADM problems with Picture fuzzy information.Wei (2018b) developed Hamacher aggregation operators for PFS and studied their application to MADM.Wei et al. (2018c) established a projection method for PFSs and developed its application in evaluating emerging technology commercialization.Jana et al. (2019) initroduced the some Dombi aggregation operators for PFS and studied their usefulness in MADM problems.
In 2008, Xu and Chen introduced a new distance measure known as the order weighted distance (OWD) measure, which can underestimate or overestimate the influence of extremely large/small deviations on the aggregation results by adjusting the corresponding weights.For this advantage, the researches on the OWD has become a hot research issue and have achieved numerous accomplishments, for example, Zeng and Su (2011) studied the application of the OWD in IFS situation, and developed the IFOWD measure.Zeng et al. (2012) studied the fuzzy OWD and explored its application in MADM.Zhou et al. (2013) developed a MADGM model based on the continuous OWD measure.Cai et al. (2014) introduced the linguistic OWD (LOWD) measure, and they proposed a pattern recognition method based on the presented LOWD measure.Merigó et al. (2017) introduced a new extension of the OWD by using weighted averages and Bonferroni means.Qin et al. (2017) developed the Pythagorean fuzzy OWD measure and applied it to assess the service quality of airlines.Sahin et al. (2018) explored the usefulness of the OWD measure in simplified neutrosophic situation and used it to select investment company.Silva et al. (2018) used the OWD to aid the logistics problems.Zeng et al. (2016) explored the usefulness of OWD measure in Pythagorean fuzzy MAGDM.Zeng and Xiao (2018) studied the application of the OWD and TOPSIS method in hesitant fuzzy situations.
The previous instances and discussions shows that PFS is a powerful and effective tool to demonstrate uncertain and vague information in real-world issues.However, none of researches mentioned previously focuses on the application of the OWD measure in PFS environment.Therefore, the primary focus of this study is to explore the usefulness of the OWD measure mentioned earlier in PFS situation.With this aim in mind, we first present a Picture fuzzy ordered weighted distance (PFOWD) measure.The PFOWD is very suitable to handle the deviations between the PFSs.Moreover, it can reduce the influence of extremely large (or small) values on the final results by giving them low (or high) weights.Furthermore, it is a generalized model that includes diversity cases, such as the Picture fuzzy weighted distance (PFWD) measure, the Picture fuzzy ordered weighted Hamming distance (PFOWHD) measure, and so on.Then, based on the PFOWD measure, a Picture fuzzy hybrid weighted distance (PFHWD) measure is presented, whose merit is that it not only includes the ordered weights of individual distances, but also considers their importance.A MAGDM approach is further established based on the PFHWD measure.Finally, an illustrative example concerning investment selection problem is provided to demonstrate the usefulness of the developed method, and a comparative analysis with existing relative methods is also presented.

Preliminaries
In this section, some essential concepts concerning PFS and OWD measure are reviewed, which will be used in the rest of this work.

Picture fuzzy set (PFS)
Definition 1 (Atanassov, 1986) where the function 0 ( ) 1 A Ez  and 0 ( ) 1 A Nz  are named as the degree of membership and the non-membership, respectively, and satisfy 0 (z)+ ( ) 1 Definition 2 (Cuong and Kreinovich, 2013) is called the degree of refusal membership of Z.
Obviously, If ( ) 0 P Iz  , then the PFS reduces to the Atanassov's IFS (Atanassov, 1989), which shows that the IFS is a special case of the PFS.A picture fuzzy number (PFN), is simply denoted as ) , , ( be two PFNs, then some of their operational laws are defined as follows (Wei, 2017):

The OWD measure
On the basis of the ordered weighted averaging (OWA) operator (Yager, 1988), Xu and Chen (2008)

 
(1), (2),..., ( ) n And if 2   , then the ordered weighted Euclidean distance (OWED) measure: The OWD measure has been extended to a variety of uncertain environments, except for PFS.In this study, we explore the application of OWD measure in PFS situation.

=2
 , then we can perform the PFOWD to calculate distance between P and Q : It can be seen from Example 1 that the PFOWD measure focuses on the importance of the ordered position of the individual distances instead of the arguments themselves, while the PFWD measure only considers the importance of given individual distances.Therefore, the weights vector in the PFWD and PFOWD measures denote different means.Next we develop a Picture fuzzy hybrid weighted distance (PFHWD) measure to combine the advantages of both the PFWD and PFOWD measures.
is called the Picture fuzzy hybrid weighted distance (PFHWD) measure between P and Q , where


is the j th largest of the weighted distance ( , ) From the Example 2 and Definition 8, we know that the operational rules of the PFHWD is that it first weights the input individual distances, and then re-ranks these weighted individual distances in descending order, and finally converts them into a collective one by the its weights and the parameter  .Therefore, The PFHWD measure unifies both advantages of the PFWD and PFOWD measure, and reflects the importance degrees of both the given individual distances and their ordered positions.
Theorem 1.The PFWD measure is a special case of the the PFHWD measure.

Proof
Theorem 2. The PFOWD measure is a special case of the PFHWD measure.

An approach based on the PFHWD measure for MAGDM under Picture fuzzy environment
In this section, we present a new MAGDM model based on the proposed PFHWD measure, whose process can be summarized as follows.
Step Step 2. Aggregate all individual decision information into a collective one and then form the decision matrix: where the PFN () Step 3: Compute the ideal levels for each attributes to form the ideal scheme (see Table no.1):  Step 5: Rank the alternatives and identify the best one(s) according to the results obtained from Step 4.

Illustrative example
In the following, we present a numerical example concerning investment selection to show the usefulness of the developed MAGDM model.Assume a investment form wants to application shows that the PFHWD is able to reflect the degrees of pessimism or optimism of multiple decision makers, as well as the importance of various attributes during the process of aggregation.Additionally, it can deal with more complex decision making problems with uncertain information evaluated with the PFNs.
For future research, we expect to propose some new methods and application of the PFHWD measure, such as induced aggregation and the probability information.
introduced the ordered weighted distance (OWD) measure: ordered weighted distance (OWD) between X and Y , where 0   , then the OWD measure reduces to the ordered weighted Hamming AE Picture Fuzzy Weighted Distance Measures and their Application to Investment Selection distance (OWHD) measure: Picture fuzzy weighted distance (PFWD) measure between P and Q , considers the importance of the individual distances, and then aggregates these distances together with their weights.Next, we define the Picture fuzzy OWD (PFOWD) measure, which takes the order weights into consideration.fuzzy OWD (PFOWD) measure between P and Q , then the PFOWD measure will generate the Picture fuzzy ordered weighted Hamming distance (PFOWHD) measure and Picture fuzzy ordered weighted Euclidean distance (PFOWED) measure, respectively: of the weights is n is the balancing coefficient, playing a role of balance.Example 2. (Example 1 continuation).In order to utilize the PFHWD, let Utilize the PFWHD to calculate the distances between the ideal scheme I and each alternative( 1, 2,..., )

Fuzzy Weighted Distance Measures and their Application to Investment Selection 690 Amfiteatru Economic
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