A Novel Integrated Fuzzy PIPRECIA–Interval Rough Saw Model: Green Supplier Selection

: A novel integrated fuzzy–rough Multi-Criteria Decision-Making (MCDM) model based on integration fuzzy and interval rough set theories is presented. The model integrates the Fuzzy PIvot Pairwise RElative Criteria Importance Assessment - fuzzy PIPRECIA and Interval Rough Simple Additive Weighting (SAW) methods. An illustrative example of the model demonstration is proposed, representing the evaluation and supplier selection based on nine environmental criteria. The fuzzy PIPRECIA method is used to determine the significance of the following seven criteria: C1 – the environmental image, C2 – recycling, C3 – pollution control, C4 – the environmental management system, C5 – environmentally friendly products, C6 – resource consumption, and C7 – green competencies. The interval rough SAW method is applied so as to evaluate four alternatives. The results show that the third criterion is most important, whereas the fourth alternative is the best solution.


Introduction
Green supplier selection is one of the most important tasks for the functioning of the whole supply chain, especially for production companies. In this paper, an innovative integrated fuzzy-rough MCDM model is proposed for the evaluation of suppliers, based on environmental criteria. MCDM is an important and powerful tool for solving such problems, as is confirmed by Stević et al. (2020): Multi-criteria decision-making is quite an applicable tool for analyzing complex real problems because of its ability to evaluate different alternatives by using certain criteria. There are a certain number of research studies of green supplier selection by using various MCDM methods. Büyüközkan & Çifçi (2012) used a combination of MCDM methods in order to evaluate green suppliers. Qin et al. (2017) solved the problem of making a decision on green supplier selection by using a combination of MCDM methods. Considering various environmental performance requirements and criteria, Yazdani et al. (2017) introduced a new model, i.e. an integrated approach to green supplier selection. Green supplier selection is carried out in various business areas. Zhao & Guo (2014) made the green supplier selection for a supplier of thermal power equipment by using MCDM methods. Banaeian et al. (2018) made green supplier selection in the agri-food industry, while Tsui & Tzeng used the MCDM approach to improve the performance of green suppliers in the TFT-LCD industry. Uppala et al. (2017) used the MCDM approach to green supplier selection in an electronics company, whereas Yu & Hou (2016) conducted green supplier selection in the automotive manufacturing industry. From the economic and environmental aspects, Chen et al. (2016) used the fuzzy MCDM approach to green supplier selection. The paper is aimed at taking the advantages of the implemented approaches and allowing for more accurate and balanced decision-making through their integration.
The rest of the paper is structured as follows: in the second section, the applied methods are presented, i.e. the fuzzy PIPRECIA and interval rough SAW methods, and some basic operations with interval rough numbers are also shown; in the third section, the results obtained are demonstrated in detail, and the section is divided into two subsections; in the fourth section, the conclusion of the paper is given, inclusive of an emphasis on the advantages offered by the proposed integrated model.

Fuzzy PIvot Pairwise RElative Criteria Importance Assessment -the Fuzzy PIPRECIA Method
The main advantage of the PIPRECIA (Stanujkić et al. 2017) method is that it allows the evaluation of criteria without sorting them first by significance, which is not the case with the SWARA method (Keršuliene et al. 2010;Vesković et al. 2018). Today, the largest number of multi-criteria decision-making problems are solved by applying group decision-making. In such cases, especially as the number of decision-makers involved in the fuzzy PIPRECIA model increases, achieves its benefits. The Fuzzy PIPRECIA method was developed by Stević et al. (2018). It consists of the 11 steps shown below.
Step 1. Forming the required benchmarking set of criteria and forming a team of decision-makers. Sorting the criteria according to the marks from the first to the last, which means they need to be sorted unclassified. Therefore, their significance is irrelevant in this step.
Step 2. In order to determine the relative importance of the criteria, each decisionmaker individually evaluates the presorted criteria by starting from the second criterion, Equation (1). In order to obtain the matrix j s , it is necessary to perform the averaging of the matrix r j s by using the geometric mean. The decision-makers evaluate the criteria by applying the defined scales shown in Tables 1 and 2. The second and third steps of the developed method are closely interdependence, and new fuzzy scales are defined so as to meet the second and third steps of the fuzzy PIPRECIA method. If the fact that the nature of fuzzy number operations and the fact that, in the third step, the values are subtracted from two, it is then required that these scales should be define. It is important to note that, by defining these scales, the appearance of the number two is avoided, which might cause difficulties and wrong results when the calculation is concerned. Therefore, no other previously used fuzzy scales could be used. Only the scales defined in this paper are applicable. When the criterion is of greater importance in relation to the previous one, an assessment is made by using the above-mentioned scale in Table 1. In order to make it easier for the decision-makers to evaluate the criteria, the table shows the defuzzified value (DFV) for each comparison. When the criterion is of lesser importance compared to the previous one, an assessment is made by using the above-mentioned scale in Table 2.
Step 3. Determining the coefficient (2) j s Step 4. Determining the fuzzy weight j (3) Step 5. Determining the relative weight of the criterion In the following steps, it is necessary to apply the inverse methodology of the fuzzy PIPRECIA method.
Step 6. The evaluation of the applicable scale defined above, this time starting from the penultimate criterion.
' r j s denotes the evaluation of the criteria by the decision-maker r.
It is again necessary to average the matrix r j s by applying the geometric mean.
Step 7. Determining the coefficient n denotes a total number of the criteria. Specifically, in this case, it means that the value of the last criterion is equal to the fuzzy number one.
Step 8. Determining the fuzzy weight Step 9. Determining the relative weight of the criterion Step 10. In order to determine the final weights of the criteria, it is first necessary to perform the defuzzification of the fuzzy values j w and ' Step 11. Checking the results obtained by applying the Spearman and Pearson correlation coefficients.

Interval Rough Numbers
The process of group decision-making is accompanied by a large amount of uncertainty and subjectivity, so decision-makers often have dilemmas when assigning certain values to decision attributes. In this paper, a new approach in rough sets theory based on interval rough numbers (IRN) is applied so as to process uncertainty contained in data in group decision-making. Suppose that one decision attribute should be assigned a value represented by a qualitative scale, whose values range from 1 to 5. The first decision-maker (DM) may consider that the decision attribute should have a value ranging between 3 and 4, the second DM may consider that a value between 4 and 5 should be assigned, whereas the third DM has no dilemma about the value of the decision attribute and assigns it the value 4. The presented dilemmas are extremely common in the group decision-making process. In such situations, one of the solutions is to geometrically average two values, which individual decision-makers are in doubt which one to assign. In such situations, however, the uncertainty (ambiguity) that prevailed in the decision-making process would be lost, and a further calculation would be reduced to crisp values. On the other hand, the use of fuzzy or grey techniques would entail predicting the existence of uncertainty and subjectively defining the interval which such uncertainty is exploited by. Subjectively defined intervals in further data processing may significantly influence the final decision (Duntsch et al., 1997), which should definitely be avoided if impartial decision-making is aimed at.
On the contrary, the approach based on interval rough numbers includes the exploitation of the uncertainty contained in the obtained data. By applying the arithmetic operations explained in the following section, the values of the attributes that fully describe the specified uncertainties without subjectively affecting their values are obtained. Thus, the uncertainties of the first DM can be described by an interval rough number IRN = [(3,3.67), (4,4.33)], of the second DM by IRN = [(3.67,4), (4.33,5)], while those of the third DM can be described by IRN = [(3.67,4), (4,4.33)]. The detailed procedure for the determination of an IRN is explained in the following section.
the DM's preferences, The upper approximations of    (16) and (17), as follows: As can be seen, each class of the objects is defined by its lower and upper limits, which represent the interval rough number defined as The IRN determination procedure will be explained by the example of the determination of the weight coefficient of the criterion wi, which is participated in by four experts. The experts evaluated the criteria by using the scale that includes integer values, ranging within the following 1-5 intervals: 1 -a very small impact, 2 -a small impact, 3 -a medium impact, 4 -a large impact, and 5 -a very large impact. The experts' evaluations are shown in Table 3. The experts' evaluations in Table 3 are presented in the form of ordered pairs (ai;bi), where ai and bi are the values assigned by the experts to the criteria from the 1-5 scale. The experts who cannot confidently opt for one of the values in the scale enter both values they have a dilemma of (E1, E2 and E3). In our example, only the expert E4 had no dilemma and chose a unique value from the scale.
These uncertainties can be represented by trapezoidal fuzzy numbers of the form A=(a1, a2, a3, a4), where a2 and a3 represent the values in which the membership function reaches its maximum value, whereas a1 and a4 represent the left and the right limits of a fuzzy set, respectively. In our example (Table 3), the four trapezoidal fuzzy numbers A (E1) = (1,2,3,4), A (E2) = (2,3,4,5), A (E3) = (3. 4,5,5) and A (E4) = (4,5,5,5) were obtained. The trapezoidal fuzzy numbers are graphically shown in Figure 1, where the darker nuance indicates the values in which the membership function reaches its maximum value (a2 and a3), whereas the light nuance indicates the elements of the set more or less belonging to the fuzzy set (a1 and a4). In addition to the fuzzy approach, the uncertainties described can also be presented by interval rough numbers, since it was defined in the previous section (the equations (10)-(21)) that an IRN consists of two rough sequences and the two classes of the objects wi and w ' i:  (20) and (21) are formed for each class of the objects. For the first class of the objects, the following was obtained: (2) 2 Lim  , , (26) and (27): (1) The addition of interval rough numbers, "+", (2) the subtraction of interval rough numbers, "-", (3) the multiplication of interval rough numbers, "×",

Interval Rough SAW Method
The SAW method is a simple and easily applicable multi-criteria decision-making method. Using only crisp numbers, however, it is impossible to obtain the results that treat uncertainty and objectivity in an adequate way (Stević et al. 2017). The Rough SAW method was developed two years ago and presented in the study (Stević et al., 2017). The Interval Rough SAW method consists of the following steps : Step 1: Forming a multi-criteria decision-making model which consists of m alternatives and n criteria.
Step 2: Forming a team of r experts, who will make an assessment of alternatives according to all the criteria and sub-criteria.
Step 3: The transformation of individual matrices into a group interval rough matrix. In this step, it is necessary to transform each individual matrix of the experts r1,r2,...,rn into an interval rough group matrix by using the equations (10)-(21): where m denotes the number of alternatives and n denotes the number of criteria.
Step 4: The normalization of the initial interval rough group matrix (33) by using the equations (34) and (35) If the criterion belongs to the benefit group, then Equation (34) is used for the normalization process: , , , ( ) , , , max , , , whereas for the criteria belonging to the cost group, Equation (35) is applied: Equations (25) and (26) are further broken down into Equations (36) and (37): '' min min min min , , , , , , Step 5: Weighting the previously normalized matrix: Step 6: Summing up all of the values of the obtained alternatives (summing up by rows): Step 7: Ranking the alternatives in descending order, i.e. the highest value is the best alternative. In order to rank the potential solutions more easily, a rough number can be converted into a crisp number by using the average value.

Results
The selection of a green supplier depends on the precise determination and selection of adequate criteria and their evaluation. A novel integrated MCDM model is modified from  where supplier selection carried out 21 sustainable criteria. In this example we left only environmental criteria and made decision. The criteria for selecting a sustainable supplier are as follows: C1 -environmental image, C2 -recycling, C3 -pollution control, C4 -environmental management system, C5environmentally friendly products, C6 -resource consumption and C7 -green competencies.

Determining Criteria Weights by Using the Fuzzy PIPRECIA Method
The evaluation of the criteria was performed by using the linguistic scale that involves quantification into fuzzy triangle numbers. Table 4 shows the evaluation of the criteria for fuzzy PIPRECIA and Inverse fuzzy PIPRECIA carried out by the decision-makers.
Based on the evaluation of the criteria and Equation (1)  In order to determine the final weights of the criteria, it was necessary to apply Equations (5)-(9), or the methodology of the inverse fuzzy PIPRECIA method. Based on the evaluation performed by the decision-makers, the matrix sj' was obtained as follows:  The results of the applied methodology are presented in Table 5. Using Equation (9), the final weights of the criteria were obtained. Before applying this equation, it was necessary to defuzzify the values of the criteria obtained by applying the equations (1)-(9). Table 5 shows the complete previous calculation, and the last column shows the defuzzified values of the relative weights of the criteria.
The Spearman (Erceg et al., 2019) correlation coefficient for the obtained ranks is 0.964, which means that there is a minimum difference in these ranks. The first and the fifth criteria are replaced in the third and the fourth place, respectively. The Pearson (Stevic et al., 2018) correlation coefficient for the criterion weights (0.977) was also calculated. Table 6 presents the final weight results obtained by using the fuzzy PIPRECIA method. In Table 6, the criteria are ranked by significance. The most significant criterion is C3pollution control. The function value of this criterion is 0.247. The least significant criterion is C6resource consumption. The function value of this criterion is 0.090.    Table 10 shows the final results of the integrated fuzzy PIPRECIA-Interval Rough SAW approach. The ranking was performed in descending order, which means that the highest value was the best and the lowest value was the worst solution. The alternative 4 is the most acceptable solution according to the results obtained.

Conclusion
In this paper, the evaluation of green suppliers was carried out by applying an innovative fuzzy-rough MCDM model. The advantages of fuzzy PIPRECIA, which was used to determine the criteria weights, and the interval rough SAW method, applied for supplier evaluation, are demonstrated throughout the paper. The fuzzy PIPRECIA method allows for the evaluation of criteria without first sorting them by significance. Group decision-making is also an advantage of this method. Today, the largest number of multi-criteria decision-making problems are solved by applying group decisionmaking. In such cases, especially given the fact that the number of decision-makers involved in the fuzzy PIPRECIA model increases, benefits are achieved from it. The SAW method is a simple and easily applicable multi-criteria decision-making method. Using only crisp numbers, however, it is impossible to obtain the results that treat uncertainty and objectivity in an adequate manner. For that reason, the interval rough SAW method was implemented for supplier selection based on the environmental criteria. The obtained results show that the fourth supplier is the best solution.