Multi Criteria Decision Making Using Normalized 3 – Polar ELECTRE I and Fuzzy 3 – Polar Dombi Arithmatic AOs (Case Study of Determining Manufacture Location)

: The m-Polar fuzzy set is a set that not only overcomes data ambiguity, but can also handle multi-polar, multi-attribute, and multi-criteria information. The m-Polar fuzzy set is useful in describing uncertainty in multi-attribute decision-making. One of the techniques used in decision-making is the ELECTRE I method. The ELECTRE I method plays a role in conducting pairwise comparisons between alternatives given by the decision-maker, where alternatives, criteria, and weights are given by the decision-maker. Furthermore, the ranking results from the ELECTRE I method will be compared with the mF Dombi Weighted Averaging (m-FDWA) aggregation operator with the help of the arithmetic operator. The purpose of this study was to compare the ranking results of the mF ELECTRE I, and the normalized and non-normalized m-FDWA arithmatic methods. The data used is secondary data related to site selection for global manufacturing with 20 alternative countries (country) and 8 criteria. The results showed that the best alternative to the normalized mF ELECTRE I and m-FDWA methods was country 14. While the m-FDWA arithmetic method without normalization resulted in country 3 as the best alternative. The effectiveness test was applied to m-FDWA arithmetic method, both normalized and without normalization to test the validity of the model so that it can be seen that normalization does not affect the validity of the model.


Introduction
Multi Criteria Decision Making (MCDM) is a decision-making method that determines the best alternative based on criteria or several rules. The technic of MCDM is related with designing and evaluating decision structure and planning problem that involving several criteria [1]. MCDM has widely used to solve problem in several topics like in economy, business, technology information, and medical health. The solving of MCDM case can be done by considering problem representation, fuzzy set evaluation, choosing optimal alternative, data source, and type of data [2].
One of the method in MCDM, especially MADM is ELECTRE method [2]. ELECTRE was introduced by Benayon, et. al [3] to show the option between several alternatives by double-comparing those alternatives through outranking relation [4]. It then explained further by Roy [5] and defined as ELECTRE I. The fundamental concept of this method is determining the concordance and discordance of the set that each represents the positive and negative of every alternative so that it resulted in the best alternative [6,7].
ELECTRE I method is known as the appropriate outranking approach to be used in all type of information and desired to choose a group of preferred alternative but didn't produce option that exceed outranking alternative [1,8,9]. Aside of ELECTRE I, there are other ELECTRE methods such as ELECTRE II, ELECTRE III, ELECTRE IV, ELECTRE IS, and ELECTRE TRI that are another type from ELECTRE method [1,6], 9,10]. The positive of ELECTRE method is it has less input to solve a problem with moderately high number of alternative and criteria [10].
MADM fuzzy method is proposed to solve the problem that involving fuzzy data. Bellman dan Zadeh [11] are the first individuals that connect the fuzzy set theory with decision-making problem. Zadeh introduced fuzzy set that is able to handle ambiguous information, in which the membership degree falls between the interval of [0,1] [12]. In general, ordinary fuzzy set is limited. Several extension and generalization of fuzzy set has been introduced [13], where it includes Hesitant Fuzzy Sets (HFSs) [14], Bipolar Fuzzy Sets (BFSs) [15], dan m -Polar Fuzzy Sets (mF sets) [16].
In 2014, m-Polar set theory was introduced by Chen [16] where it was defined as generalization of bipolar fuzzy set. The set of m-Polar on X set is a mapping of ∶ → [0,1]. The concept of m-Polar information is happening because the real-world data comes from a lot of characters and understanding. The set of m -Polar shows a better representation from unclear set of data, that results in significantly better study in the parameter of equality, incompleteness, and data relation. [13].
Multipolar information holds an important role in many situations. Knowing the fact that the set of m-Polar has efficient power in handling unclear data that shows up in real-world problem, the need of aggregation operators (AOs) in combining information is critical [17][18]. AOs have important role in solving MCDM problem and combining data into one shape of data (single form) [19][20][21].
Several prior studies done by Hatami -Marbini [22] uses ELECTRE method to do a ranking in evaluation of health safety in Hazardous Waste Recycling (HWR) facility. Neha Waseem did a new approach in ELECTRE I method by using m-Polar fuzzy set to register the criteria and the alternative which later will be compared using fuzzy ELECTRE I [23]. Muhammad Akram [20] did a study with an approach of using Dombi aggregation operator in decision-making under m-Polar fuzzy set.
This study will explain about the comparison of decision making using ELECTRE I method and both normalized and unnormalized method of m-Polar Dombi Weighted Averaging (m-FDWA) Arithmetic AOs which will result in the rank of alternative and criteria given by decision maker. Then, by using m-Polar fuzzy set will do alternative input and criteria from decision maker. The result of this study is to show the best and optimal alternative which will be chosen alternative.

Triangular Membership Function
Membership function can be represented in several ways, one of it is function approach. The representation of triangular curve will be used to change crisp data into fuzzy data. In principle, the triangular curve is a combination of 2 linear as shown in Figure 1.  Membership function: Prior to solving MCDM problem, the explanation of m-Polar ELECTRE I method is needed. If = { 1 , 2 , 3 , … , } is a set of alternative and = { 1 , 2 , 3 , … , } is the set of criteria, the initial step of m-Polar ELECTRE I fuzzy method is to form a decision matrix filled with alternative that is suitable with the criteria, in which the decision matrix are symbolized by = ( ), with: After the decision matrix is formed, the next step is to determine the weight of decision maker by complying to normality condition, that is, The formed decision matrix then have to went through normalization using equation as follows: After normalization, the continuing step is the normalized decision matrix being multiplied with the weight so that it resulted in m-Polar fuzzy decision matrix weighted by = ( ), in which The next step is to calculate the level of concordance fuzzy m -Polar ̅ and discordance fuzzy m -Polar ̅ which is defined as the average of the index by using equation below: so the dominant matrix of both concordance fuzzy m -Polar and discordance fuzzy m -Polar can be formed in matrix showed below: peer to peer multiplication of entry and are done to make the aggregation dominant matrix showed as follows: The last step is to rank the alternatives based on outranking value from matrix , where there is trending side (represented in trending graph) from entry to if and only if the value of = 1. This resulted in three cases, that are:


There is trending side from to , which means is more preferred than .  There is trending side from to and to , which means there is no difference between the two.  No side in between to , or unable to be compared.

Dombi Aggregation Operator
On the principle, the sum of Dombi operation and Dombi product are t -norm and t -conorm, which are defined as follows.

Results and Discussions
On this research, there is an addition of step that is normalizing decision matrix on m-FDWA Arithmetic AOs which then will be compared with both m-FDWA Arithmetic AOs with no normalization method and mF ELECTRE I method to produce the best alternative of determining global manufacture location.
The data used is a secondary data extracted from online site (https://dc.uwm.edu/cgi/viewcontent.cgi? article=1238&context=etd). The alternative of the criteria is inputted in the form om multicriteria information so the set that is used is fuzzy m -Polar set, in which every membership degree represents sub criteria. In determining manufacture location, the developer, or the contractor in this case, need to consider several things related to determining the location of construction [24]. The specialists on the subject are already being asked as decision maker to determine the alternative and criteria of choosing the location. They will evaluate twenty alternatives or candidates ( = 1, 2, … , 20) , with criteria such as 1 : Cost, 2 : Labor, 3 : Infrastructure, 4 : Market, 5 : Other Locations, 6 : Economic, 7 : Quality of Life, and 8 : Political. The weight of the criteria is also applied by decision maker in = (0.07, 0.19, 0.06, 0.17, 0.12, 0.19, 0.08, 0.12) which is obtained from random values using Python software. Data from decision makers are shown in Table 1.

mF ELECTRE I
Based on Error! Reference source not found., the initial step of mF ELECTRE I method approach in solving the problem of determining global manufacture location is to do a fuzzification on crisp data showed in Table 1 using triangular function so it will be changed into fuzzy. The fuzzification result is shown in Table 2, in which those data is the data of 3 polar fuzzy numbers where every value represents the value in sub criteria.
The normalization of decision matrix is done next in accordance with equation (2), with the result written as follows: After the weighed normalized decision matrix is obtained, the next step is to form the index of concordance and discordance, each are shown in Table 3 and Table 4. The continuing step is to determine the level of 3F concordance and level 3F discordance which will be used as threshold parameter in determining the dominant matrix of concordance and discordance. Based EKSAKTA journal.uii.ac.id/eksakta February 2022, Volume 3, Issue 1, 39-49 on equation (5) and (6), the level value of 3F concordance ̅ = 0.5055 and level 3F discordance ̅ = 0.85734. So, the dominant matrix of concordance and discordance can be written as follows: After the dominant matrix of concordance and discordance are obtained, the next step is to determine the aggregate dominant matrix of 3F which obtained from peer to peer multiplication of entry and , so the aggregate dominant matrix of is showed by matrix below: The last step is to represent the result of aggregate dominant matrix by using outranking relation into trending graph which is shown in Figure 2. From figure 2, the red color shows A14 which dominates other alternatives as evidenced by the edge of the graph that comes out of the A14 vertex and leads to the top of the vertex in the graph above, this means that A14 has superior criteria compared to other alternatives. It can be concluded that A14 or country 14 is the best location for global manufacturing.
Normalized mFDWA Method By using normalized matrix data shown in equation (8), the next step is to calculate the value of ̂ in determining the location of global manufacturing with = 3 in accordance with equation (7), so that the result is obtained as follows The next step is to decide the score value of (̂) from 3FNs ̂, which is the average value of ̂, so that the result is obtained as follows So, according to the score value of (̂), ( = 1, 2, … ). Based on Figure 2 it is obtained where 14 > 15 > 16 > 1 > 8 > 3 > 2 > 11 > 5 > 7 > 20 > 17 > 9 > 18 > 4 > 19 > 6 > 10 > 13 > 12 . Because 14 has the highest score, it can be concluded that 14 or country 14 is the best alternative for the location of global manufacturing.

Effectiveness Test Using Normalized Mfdwa Method
To ensure the validity of both, normalized and non-normalized, mFDWA algorithm method, the criteria test developed by Wang [25] is needed. Criteria test are done to change the value on the least optimal alternative, in this case alternative A12 and A10. The exchange value of alternative A12 and A10 each are shown in Table 5 and Table 6. The calculation then are being redone using mFDWA method but there was no different in the rank result where A14 still have the highest score and are the best alternatives. Since there was no difference detected in the rank of optimal alternative after going through effectiveness test, it can be concluded that the two methods of mFDWA are valid.

Conclusion
It can be concluded from this research that m-Polar fuzzy set has a role in managing multicriteria information given by decision maker, which is information in the form of criteria in determining the location of global manufacturing with 8 criteria with each is made of 3 sub criteria. According to the calculation, the ranking of the methods used in this research (3-Polar ELECTRE I fuzzy method and normalized 3-Polar Dombi Arithmetic AOs fuzzy method) showed the same alternative. By using effectiveness test, it then revealed that adding normalization step in mF Dombi Arithmatic AOs didn't affect the method validity. Even, by having a normalization, the triangular fuzzy calculation is helped in the case of in which the result is undefined value or 1/0. The next research can focus on the other ELECTRE method in wider perspective in considering the suitability level and unsuitability of information given. Other type of fuzzification are also open to be explained further in decision-making method with the output as m-Polar fuzzy set, hesitant fuzzy, fuzzy intuition, etc.