Picture Fuzzy Soft Robust VIKOR Method and its Applications in Decision-Making

This paper introduces the Euclidean, Hamming, and the generalized distance measures for picture fuzzy soft sets and discusses their properties. The numerical examples of decision-making and pattern recognition are focused. We also develop a robust VIKOR method for PFSSs. The relative and precise ideal picture fuzzy values (PFVs), robust factors and ranking indexes are defined. Different algorithmic procedures of robust VIKOR based on the relative and precise ideal PFVs, relative and precise robust factor, precise and picture fuzzy weights and relative and precise ranking indexes are proposed. In the end, the investment problem is solved by using the proposed method.


Introduction
To measure the similarity between any form of data is an important topic. The measures used to find the resemblance between data is called similarity measure. It has different applications in classification, pattern recognition, medical diagnosis, data mining, clustering, decision-making and in image processing.
Fuzziness, as developed in [1], is a kind of uncertainty which appears often in human decision-making problems. The fuzzy set theory deals with daily life uncertainties successfully. The membership degree is assigned to each element in a fuzzy set. The membership degrees can effectively be taken by fuzzy sets. But in real-life situations, the non-membership degrees should be considered in many cases as well, and it is not necessary that the non-membership degree be equal to the one minus the membership degree. Thus, Atanassov [2] introduced the concept of intuitionistic fuzzy set (IFS) that considers both membership and non-membership degrees. Here the non-membership The positive ideal (PI) solution is the best available solution in multi-criteria decisionmaking (MCDM) problems. The compromise solution by VIKOR approach is closest to the PI solution [35]. Thus basic idea in VIKOR method is to find the compromise solution which is closest to the PI solution and any improvement in VIKOR approach that not fulfil the basic idea is not reliable. Recently, Khan et al. [36] discussed the theoretical justifications of empirically successful VIKOR method.
Motivated from the Yang's model of PFSS [8] and remoteness-based VIKOR method for Pythagorean fuzzy sets by Chen [37]. We diversify this technique to PFSS and apply to MADM problems. We develop robust VIKOR method for PFSS to select priority area for investment. To measure the difference and similarity between PFSSs, different distance and similarity measures for PFSSs are presented.
The aim of this paper is to discuss the priority area for investment for an underdeveloping country using the robust VIKOR method for PFSS. Also, to define the distance and similarity measures for measuring the difference and similarity between PFSSs. Additionally, to discuss the different algorithmic procedures that incorporate the precise and picture fuzzy weights and precise and relative ideal values.
The paper has following contributions: (1) The Hamming, Euclidean and generalised distance measures are defined for PFSSs and strategic decision-making and pattern recognition problems are debated. (2) The relative and precise ideal values are defined to reach best available solution and avoid worst solution. (3) The relative and precise robust factors are defined for PFSSs. (4) Precise and picture fuzzy (PF) weights are introduced. (5) Different relative and precise ranking indexes are defined for PFSSs to cope with precise and PF weights. (6) The algorithmic procedures of robust VIKOR method are proposed. (7) The problem of selecting a priority area for investment is solved with proposed methods.
The remaining paper is written as follows: Section 2 discuss the basic definitions. The distance and similarity measures and their properties are focused in Section 3. The relative and precise ideals, robust factors, ranking indexes, precise and PF weights, and algorithmic procedures are expounded in Section 4. The application of the propose method in selection of priority area for investment is discussed in Section 5. The comparison and concluding remarks are focused in Sections 6 and 7, respectively.

Preliminaries
This segment contains the basic notions of soft set, IFS, PFS and PFSS. LetŶ = { 1 , 2 , . . . , m } represents the universal set throughout the paper which is discrete, finite, non-void discourse set and contains the alternatives, whileÊ = {j 1 , j 2 , . . . , j n } represents the characteristics or attributes of criteria and called criteria space.
A novel idea of soft set was proposed by Molodtsov [38], where uncertainty deals successfully in the light of parametric point of view. Each member in soft set can be viewed by some characteristic or attributes (criteria).
Atanassov defined the generalisation of fuzzy set by considering the non-membership function. The uncertainty model's more effectively in IFS.

Definition 2.2 ([2]):
The membership function (ξ R ) and non-membership function (ν R ) from universal set to unit interval, with a condition ξ R ( ) + ν R ( ) ≤ 1, define the IFS R over a universal setŶ as follows The hesitancy index of the element ∈Ŷ is defined as Coung defined the generalisation of IFS by including the neutral membership function and called the PFS. This model is important for the situations involves yes, abstain, no and refusal. Voting is a good example for PFS.

Definition 2.4 ([4]):
For any two PFSs R and S inŶ, the following operations are defined as follows: To understand the construction of PFSS, we consider an example of selecting the class representative (CR). Let the class in the university need to select a CR and they agree for voting method. Let three candidates are available for CR and we represent as The candidates are evaluated based on their negotiation, communication, leadership, problem-solving and team-working skills and represented as j 1 , j 2 , j 3 , j 4 and j 5 , respectively. Each student of the class give their preference for the candidates against each attributes in the form of yes (ξ ), abstain (η) and no (ν). If the number of students of the class are L, then the evaluation of each candidate i against each criterion j j is calculated as b ij = Evaluation of all students for a ith candidate against jth criterion L where each b ij should follow the condition of Definition 2.3. Thus for criteria setÊ = {j 1 , j 2 , . . . , j 5 }, the mappingF :Ê → PF(Ŷ) can be define and for each criterion j j , theF(j j ) is a PFS as followsF ThisF(j j ), j ∈ {1, 2, . . . , 5} constitute the PFSS (F,Ê) and represented in tabular form in Table 1.

Distance and Similarity Measures
This section contains the Hamming, Euclidean and generalised distance measures for PFSSs. Additional properties and their applications in decision-making and pattern recognition are discussed here.

Definition 3.3:
For two PFSSs 1 = (F,Â) and 2 = (Ĝ,B) inŶ, the Hamming distances between 1 and 2 are defined as follows: Definition 3.4: Let 1 = (F,Â) and 2 = (Ĝ,B) be two PFSSs inŶ, the Euclidean distances between 1 and 2 are defined as follows: Definition 3.5: For two PFSSs 1 = (F,Â) and 2 = (Ĝ,B) inŶ, the generalised distance measures between 1 and 2 are defined as follows: Remark 3.1: The generalised distance measures D p and D • p are reduced to Hamming distances D h and D • h , respectively, for p = 1. Also, the Euclidean distances D e and D • e are obtained from D p and D • p , respectively, for p = 2.

Application in Strategic Decision-Making and Pattern Recognition
In this section, we solve the strategic decision-making and pattern recognition problem adopted from [32,39]. Assume that a firm wants to allocate a plant for making new products. The firm has to decide the standard of new products to obtain the highest benefits and optimal production strategy. After the review of the market, the firm consider four alternatives. LetŶ = { 1 , 2 , 3 , 4 } represents the four alternative, where i (1 ≤ i ≤ 4), stands for: product for upper class, upper middle class, lower middle class and for working class, respectively. To make the process of decision-making beneficial and effective, the firm hire the experts from the different fields and constitute a committee to make recommendations for choosing the potential alternative wisely. The committee set up the criteria (attribute) to evaluate the above-mentioned alternatives. LetÊ = {j 1 , j 2 , j 3 , j 4 , j 5 , j 6 } represents the six attributes, where j j (1 ≤ j ≤ 6), stands for: short-term benefits, mid-term benefits, long-term benefits, production strategy risk, potential market and market risk, and industrialisation infrastructure, human resources and financial conditions, respectively. The committee make assessment of four alternatives against the six attributes and give their preferences in the form of PFSS. The assessment of alternatives is presented in Table 2. Further, the committee decide the unknown production strategy y, with data as listed in Table 2. The distance between the each alternatives i and unknown production strategy y have calculated on the basis of above proposed distance measures.
From Table 3, we have seen that the distance between 4 and y is minimum, which shows that the unknown production strategy y belongs to alternative 4 . We consider the distance measures that includes the hesitancy index in their calculations. There is the slight difference in the ranking for higher values of the parameter p in the distance measures ( Table 3). The obtain ranking coincide with the ranking obtained by Wei [32].

The Robust VIKOR Method for PFSSs
In this section, the relative and precise ideal values are defined to reach best available solution and avoid worst solution, respectively. Based on the ideal values, the relative and precise robust factors are defined for PFNs. The relative and precise ideal values and robust factors are helpful to define ranking indexes. The algorithmic procedures of robust VIKOR method are proposed.

The Relative and Precise Ideal Picture Fuzzy Values
The relative positive ideal PFV (rpi-PFV) and relative negative ideal PFV (rni-PFV) are the best available and worst avoidable values, respectively. The rpi-PFV and rni-PFV are based on the available data provided by the decision-makers. If the decision-maker change his/her preferences, then rpi-PFV and rni-PFV influenced. Instead of relative ideal values, the precise ideal values are fixed and not influence by the preferences of the decision-makers. The precise positive ideal PFV (ppi-PFV) and precise negative ideal (pni-PFV) are the best available and worst avoidable values in the domain. These ideal values are useful to obtain the best suitable alternative and to keep away from worst alternative.
As we know that there are two types of criteria, that is, benefit and cost criteria. LetÊ = {j 1 , j 2 , . . . , j n } be the criteria space andÊ b andÊ c are benefit and cost criterion, respectively. The rpi-PFV and rni-PFV for the PFV decision matrix are defined as follows.
The ppi-PFV and pni-PFV for the PFV decision matrix are defined as follows.

The Relative and Precise Robust Factors
As we discuss earlier, the ideal values are useful to obtain the best suitable alternative and to keep away from worst alternative. If the separation between each assessed value (b ij ) and rpi-PFV (b +j ) (represented as D(b ij , b +j )) reduces then the affirmative of b ij with ideal value surges. Since the rpi-PFV is based on the preferences of decision-makers and thus frequently changed among attributes. The separation between rpi-PFV and rni-PFV provides the upper bound of D(b ij , b +j ). So, we consider the ratio of . But for precise ideal values, the problem of an upper bound is insignificant due to the fact that the separation between ppi-PFV and pni-PFV is one, that is D(b+ j , b− j ) = 1. Now, we define the relative robust factor (RI d ) as follows.

Definition 4.3:
For a distance measure D, the relative robust factor RI d of assessed value (b ij ) is defined as:  (2) We calculate the Euclidean distance between two PFVs by using Equation (5) as follows: because D(b− j , b+ j ) = 1 for ppi-PFV and pni-PFVs.

Decision-Making Process
The aim of decision-making (DM) process is to choose the favourite option based on experts defined attributes. In DM process, letŶ = { 1 , 2 , . . .
The attributes in real-life scenario are not of equal significance. Some of them have more significance then others. The weights of criteria tackle this issue in DM process. There are two types of weights discuss in this paper, that is, the precise weights and PF weights. The single value of the criterion is assign as precise weight and represented as ω = {ω 1 , ω 1 , . . . , ω n } T such that n j=1 ω j = 1. While the PF weights contains the importance, neutralness and unimportance degrees of the attribute and represented by = { 1 , 1 , . . . , n } T , where The robust VIKOR method incorporate both type of weights. The relative and precise ideal values and robust factors, and precise and PF weights are used to define the ranking indexes. First we consider the relative robust factors and precise weights-based ranking indexes.

Definition 4.5:
The relative robust factor-based group utility indexŜ d of an alternative i is described as follows:Ŝ where ω j are the precise weights. The relative robust factor-based individual regret indexR d of i is defined as follows: The relative robust factor-based compromise indexQ d of i is defined as follows: where λ ∈ [0, 1] is the decision mechanism coefficient. Now, we consider the precise robust factors and precise weights-based ranking indexes.

Definition 4.6:
The precise robust factor-based group utility indexŜ f of an alternative i is described as follows:Ŝ where ω j are the precise weights. The precise robust factor-based individual regret indexR f of i is defined as follows: The precise robust factor-based compromise indexQ f of i is defined as follows: (27) where λ ∈ [0, 1] is the decision mechanism coefficient. Now, we propose two algorithmic procedures for robust VIKOR method. These algorithmic procedures for PFVs are based on the precise and relative ideals, precise weights, robust factors and ranking indexes.  (14) and (15)  The PF weights and relative robust factor-based ranking indexes are defined for alternatives.

Definition 4.7:
The relative robust factor-based group utility index S d of an alternative i with a set of PF weights j = ( ξ j , η j , ν j ) for all j j ∈Ê is defined as follows: Algorithm 2 Scenario 2: precise ideals, precise weights and PF decision matrix 1. Steps 1 and 2 are same as Algorithm 1.
3. Equations (16) and (17) are used to formulate the ppi-PFV and pni-PFV for each attribute, respectively. 4. The separations between each assessed value b ij and ppi-PFV are calculated and represented as D(b ij , b+ j ).
5. The precise robust factor of each PFV b ij , that is, RI d (b ij ) is calculated by Equation (20).
6. The precise robust factor-based group utility indexŜ f and individual regret indexR f are formulated by Equations (25) and (26), respectively. Then the precise robust factorbased compromise indexQ f for each alternative is computed by Equation (27) where δ and b ij ∈ [b ij ] m×n are score function and PFVs, respectively. Additionally, the multiplication of PFV with scalar in Equation (28) is defined in Definition 2.4. The relative robust factor and PF weights-based individual regret index R d of i is defined as follows: The relative robust factor and PF weights compromise index Q d of i is defined as follows: The PF weights and precise robust factor-based ranking indexes are defined for alternatives.

Definition 4.8:
The precise robust factor-based group utility index S f of an alternative i with a set of PF weights j = ( ξ j , η j , ν j ) for all j j ∈Ê is defined as follows: where δ and b ij ∈ [b ij ] m×n are score function and PFVs, respectively. Additionally, the multiplication of PFV with scalar in Equation (31) has defined in Definition 2.4. The precise robust factor and PF weights-based individual regret index R f of i is defined as follows: The precise robust factor and PF weights-based compromise index Q f of i is defined as follows: Again we propose two algorithmic procedures for VIKOR method. These algorithmic procedures represent the robust VIKOR methods for PFVs based on PF weights. This VIKOR method is based on precise and relative ideals, precise and relative indexes, PF weights and precise and relative ranking indexes.

Selectio of Priority Area for Investment in Under-Developing Countries
Mostly under-developing countries are facing the problems of corruption. Therefore, the economic situations of many countries are going down day by day. Some countries are taking some decisions against the corrupt elements. The economy, environment and budget should be focused while making an investments by under-developing countries. The shortand long-term benefits, operational costs, job creation, maintenance, revenue generated, yield, reliability and minimum effect on environment and peoples are important parameters. A good sector should focus on job opportunity for the peoples. Therefore, the suitable area for investment for under-developing country should be choosen wisely.
In this part, we study the problems of selecting an area for investment for underdeveloping countries. LetŶ = { 1 , 2 , 3 , 4 , 5 , 6 } represents the set of different sectors or areas for investment (alternative), where 1 , 2 , 3 , 4 , 5 and 6 stands for food Algorithm 3 Scenario 3: relative ideals, PF weights and PF decision matrix 1. Let the options (alternatives) set and attribute set are represented byŶ = { 1 , 2 , ..., m } andÊ = {j 1 , j 2 , ..., j n }, respectively. 2. The PF decision matrix b = [b ij ] m×n is obtained when each option i , 1 ≤ i ≤ m is assessed against each attribute j j , 1 ≤ j ≤ n. The PF weights can be acquired by decision-makers or choosing the suitable linguistics variables. 3. Steps 3-5 are same as Algorithm 1. 6. The relative robust factor-based group utility index S d and individual regret index R d are formulated by Equations (28) and (29), respectively. Then the relative robust factorbased compromise index Q d for each alternative is computed by Equation (30). 7. The computations of S d , R d and Q d provide the three ranking lists of the alternatives. 8. The minimum value in the ranking list of Q d serves as the compromise solution if the following standards holds: C1. Acceptable advantage: where is the second minimum alternative in the ranking list of Q d . C2. Acceptable stability: The ranking lists of S d and R d also have the minimum value for alternative . The violation of any above-mentioned standards will lead to the set of the ultimate compromise solution, which consists of: a. The violation of second standard C2 leads to the compromise solution that contains and . b. While the violation of the first standard C1 leads to the compromise solution that contains , , ..., p , where p is the maximum value for which processing, textile, logistics, automobiles, IT & ITes, power sector, respectively. LetÊ = {j 1 , j 2 , j 3 , j 4 , j 5 , j 6 } represents the set of criteria (attributes), where j 1 , j 2 , j 3 , j 4 , j 5 and j 6 stands for short term benefits, long-term benefits, operational costs, job creation, revenue generated and reliability, respectively. The shortlisted areas are evaluated against the six parameters (criteria).
To solve this problem, a committee of different personals established that consist of economists, decision-makers, managers, governments servants and some other policy makers. The committee is responsible for assessment of the available alternatives against predefined criteria. It is possible for a committee to change the criteria or characteristics to assess the options. So, the committee evaluate the alternatives against criteria and propose their preferences in the form of PFV. Their preferences generates the PF decision matrix of six rows and six columns and represented as b = [b ij ] 6×6 , where b ij shows the evaluation of i th alternative against j th criterion. The PF decision matrix for this problem is displayed in Table 4.
The PF decision matrix is normalised, because j 3 is the cost criterion, by following equation:

Formulation by Algorithm 1
The normalised PF decision matrix is already generated in Table 5. Equations (14) and (15) are used to formulate the rpi-PFVs b +j and rni-PFVs b −j as follows: Equation (19) is used to calculate the separation between rpi-PFVs and rni-PFVs. We use this equation to formulate the separation between two PFVs. The results summarised in Equation (37).
The separation between each PFVs b ij and the rpi-PFVs b +j is calculated by Equation (19) and represented as D • e (b ij , b +j ) = c ij . The results are displaced in Equation (38).  (39).
Equations (22), (23) and (24) Thus both conditions (standards C1 & C2) in step 8 of Algorithm 1 are satisfied for textile sector. Therefore, textile sector is the compromise solution for investment problem. The order of the investment options is 2 5 6 4 3 1 .

Formulation by Algorithm 2
The PFVs (1, 0, 0) and (0, 0, 1) serve as the ppi-PFV and the pni-PFV for all criteria due to normalisation in Table 5. The separation between b ij and b+ j is formulated by Equation (19) and represented as D • e (b ij , b+ j ) = C ij . The separation results are presented in Equation (42).
The precise robust factor of b ij is equal to the distance between b ij and b+ j , that is,  (43).
By sorting theŜ f ( i ),R f ( i ) andQ f ( i ) values in ascending order, the three ranking lists 1 , respectively, are obtained from Equation (45). All ranking lists mark IT & ITes sector is the best among different options and the standard C1 holds well. Moreover, Thus the first standard (acceptable advantage) in Step 8 of Algorithm 2 is not satisfied. Therefore, the ultimate compromise solution is proposed. The IT & ITes and textile are the ultimate compromise solutions of the problem. The order of the investment problem is { 5 , 2 } 3 6 4 1 .

Formulation by Algorithm 3
Algorithm 3 is employed to solve this problem by PF weights. The ideal values rpi-PFVs b+ j and rni-PFVs b− j (j = {1, 2, . . . , 6}) have calculated in Equations (35) and (36) The relative robust factor-based group utility index S d ( i ), individual regret index R d ( i ) and the compromise index Q d ( i ) with a set of PF importance weights j = ( ξ j , η j , ν j ) are calculated by using Equations (28) For acceptable stability, the optimal alternative should have the minimum value for ranking indexes. But, the alternative 2 have minimum value for S d and 5 have minimum values for R d and Q d . Hence the condition of acceptable stability is not satisfied. The condition of acceptable advantage is satisfied, i.e.

Formulation by Algorithm 4
Algorithm 4 is employed to solve this problem by PF weights. The PFVs (1, 0, 0) and (0, 0, 1) serve as the ppi-PFV and pni-PFV for all criteria due to normalisation in Table 5. The separation between b ij and b+ j is formulated by Equation (42). The precise robust factor of b ij is equal to the separations between b ij and b+ j , that is, RI f (b ij ) = D • e (b ij , b+ j ) (Due to Definition 4.4).
The precise robust factor and PF weights-based group utility index S f , individual regret index R f and the compromise index Q f are calculating by Equations (31), (32) and (33), respectively, and the results are presented in Equation (48).

Conclusion
The Euclidean, Hamming and the generalised distance measures for PFSSs have introduced. Additional properties of the distance measures and their applications in decision-making and pattern recognition have focused. We have developed the robust VIKOR method for PFSSs. The relative and precise ideal PFVs, relative and precise robust factors, and relative and precise ranking indexes have defined. Different algorithmic procedures of robust VIKOR based on the relative and precise ideal PFVs, the relative and precise robust factor, precise and PF weights and relative and precise ranking indexes have proposed. In the end, investment problems have solved by using different algorithmic procedures of the robust VIKOR method. In the future, we will find other distance and similarity measures for PFSSs and use them to define the MCDM methods. Additionally, we will focus on distance and similarity measures for GPFSSs [16], linear Diophantine fuzzy set [43,44], bipolar fuzzy sets [45,46] and temporal IFSs [47].

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi (RMUTT). Her research interests include fuzzy optimization, fuzzy regression, fuzzy nonlinear mappings, least squares method, optimization problems, and image processing.