A New Method to Support Decision-Making in an Uncertain Environment Based on Normalized Interval-Valued Triangular Fuzzy Numbers and COMET Technique

: Multi-criteria decision-making (MCDM) plays a vibrant role in decision-making, and the characteristic object method (COMET) acts as a powerful tool for decision-making of complex problems. COMET technique allows using both symmetrical and asymmetrical triangular fuzzy numbers. The COMET technique is immune to the pivotal challenge of rank reversal paradox and is proﬁcient at handling vagueness and hesitancy. Classical COMET is not designed for handling uncertainty data when the expert has a problem with the identiﬁcation of the membership function. In this paper, symmetrical and asymmetrical normalized interval-valued triangular fuzzy numbers (NIVTFNs) are used for decision-making as the solution of the identiﬁed challenge. A new MCDM method based on the COMET method is developed by using the concept of NIVTFNs. A simple problem of MCDM in the form of an illustrative example is given to demonstrate the calculation procedure and accuracy of the proposed approach. Furthermore, we compare the solution of the proposed method, as interval preference, with the results obtained in the Technique for Order of Preference by Similarity to Ideal solution (TOPSIS) method (a certain preference number).


Introduction
Decision-making is the most critical and fundamental tool in which decision-makers use to compare and rank different objects and alternatives based on a few particular criteria to make the best possible decision. Our daily life is full of different experiences and exposures, which lead us to numerous problems and situations where we need to follow the basics principles of operational research. It is a discipline that deals with the applications of advanced analytical methods of decision-making, which help make better decisions than any other technique. The fuzzy set theory [1] is the important field of mathematics, which provides a platform for multi-criteria decision-making (MCDM) to make decisions of such problems of daily life in complex situations. This theory was introduced by Zadeh [1] in 1965, which opened new corridors for decision-making. Bellman and Zadeh [2] used fuzzy logic for the decision-making process for the first time, and then it became extension of the fuzzy set to industrial decisional problems in soft computing based on the multi-criteria group assessment method. Lee et al. [58] used IVFNs for supplier selection, which represents the application of IVFNs with a different range of variety. The theory of normalized interval-valued triangular fuzzy numbers (NIVTFNs) is critical in dealing with the environment in which DMs feel hesitation in providing their assessments in a discrete structure.
In this paper, we propose a new approach that combines the advantages of NIVFNs and the COMET method. Previously, obtained COMET extensions were provided to solve decisional problems under uncertainty using hesitant fuzzy sets (HFS), where the source of uncertainty was that expert known a set of possible values of the membership for one element. The main contribution of this work is dealing with another source of uncertainty. The main difficulty of establishing the membership function is because the data from an expert can have a margin of error, or the chosen shape of the function is not entirely adequate. In the COMET method, we should identify the membership function the best as we can. Therefore, by using NIVFNs, the expert can provide more safety guarantee that NIVFNs will cover the right membership function than by simple TFN. It is easy to prove because we can simplify and say that a NIVFN is a TFN with an added error margin. It is worth noticing that this connection eliminates dangerous paradoxes in decision-making areas and a new source of uncertainty.
The rest of this paper is organized as follows. Some crucial definitions and basic concepts related to TFNs, IVTFNs, NIVTFNs with some basic operations are discussed in Section 2. In Section 3, the COMET methodology in the context of NIVTFNs is developed to deal with vague and uncertain environments in the MCDM problems. A simple example is given in Section 4 to demonstrate the practical feasibility study of the proposed approach. The paper is ended by Section 5 with some conclusions related to research.

Preliminaries
In this section, we will focus on some important concepts which can play pivotal role in understanding the proposed study. Definition 1. Basic operations on two intervals [8,60].

For any two intervals
and λ ∈ R, the following basic operations on A and B can be defined as Definition 2. Triangular Fuzzy Number [8,10,11,59].
A fuzzy numberÃ(a, m, b) over the set of real numbers R is called a TFN if its membership function is represented by The membership function µÃ(x) satisfies the following characteristics: If m − a = m − b then it is a symmetrical TFN otherwise we call it asymmetrical TFN.
An interval-valued fuzzy number (IVFN) can be expressed as in the following form and µÃ L ≤ µÃ U where µÃ L (x) and µÃ U (x) are known as the lower and upper degrees of membership function µÃ( Definition 4. Interval-valued triangular fuzzy number [12,21].
The numbers w LÃ and w Ũ A are called the heights ofÃ L x andÃ U x respectively.
A NIVTFN number is an IVTFN with the following two characteristics: . The graphs of the IVTFNÃ and NIVTFN A can be seen in Figures 1 and 2 respectively.

Definition 6. Geometric mean
, ..., I n = (a U n , a L n , b n , c L n , c U n ) be n NIVTFNs. Then, the geometric mean of I 1 , I 2 , ..., I n can be defined as (1)

COMET Method with NIVTFNs
This section is devoted to a theoretical description of the proposed approach to solving MCDM problems with the use of NIVTFNs and COMET. The whole procedure has been divided into five colliding steps described below. Let A j (j = 1, 2, . . . .m) be a set of alternatives and C i (i = 1, 2, 3 . . . n) be the set of criteria. The whole decision-making process by using the COMET method and NIVTFNs is presented in Figure 3.  Step 1: Define the space of the problem Let C be the family of all NIVTFNs and N i = {N i1 , N i2 , ..., N ic i } be a collection of some NIVTFNs which are selected for each criterion C i (i = 1, 2, ..., n). As a result, the following families of NIVTFNs can be obtained for each criterion as follows:

MCDM problem
Now, we need to find the core of each NIVTFNs selected for each criterion C i (i = 1, 2, ..., n). Afterwards, the core of each criterion is obtained which can be described as the core of each NIVTFN involved in the families as mentioned above, i.e.
Step 2: Generate the COs The all possible COs can be obtained by taking the Cartesian product of all C(C i )(i = 1, 2, ..., n) as follow: As the result of this, the following ordered sets are obtained containing all the cores of respective NIVTFNs as: c i is the count of all the COs.
Step 3: Rank and evaluate the COs Collect the opinion of expert on the importance of all the COs via pairwise comparisons as represented by square matrix called the matrix of expert judgment (MEJ). The experts are requested to provide their assessments about CO l (1 ≤ l ≤ s) by using the pre-defined linguistic scales in the form of NIVTFNs which can express the relative importance of one CO over another. The MEJ = [I ij ] s×s can expressed as Each I ij is NIVTFN which denotes the degree to which CO i is preferred to CO j .
Step 4: Preference values of COs In this step, we will find two vectors known as SJ and P. The vector SJ called the vector of summed judgments is found by calculating the geometric mean of the corresponding elements in the form of NIVTFNs from the MEJ. This is represented by SJ = [v 1 , v 2 , ... v s ], where each v l = (a l , b l , c l , d l , e l ) is NIVTFN and is obtained by taking the geometric mean G(I l1 , I l2 , · · · , I ls ) of I l1 , I l2 , · · · , I ls (1 ≤ l ≤ s) as discussed in Equation (1). The next vector P = [P 1 , P 2 , ..., P s ] which actually contains the preference values of all the COs can be computed by the following formula where (3) Step 5: Inference in a fuzzy model and final ranking As every alternative can be represented with a set of crisp numbers such as where the following conditions must be satisfied for each element of A j (j = 1, 2, ..., m).
The number of COs are obviously 2 n where 1 ≤ 2 n ≤ s. Let p 1 , p 2 , ..., p 2 n be the approximate values of preference of the activated rules (COs) which were already calculated in Step 4. We denote N i (a ij ) = {N ik i (a ij ) | a ij ∈ A j , k i = 1, 2...(c i − 1)} the value of each family of NIVTFNs at a ij ∈ A j where i = 1, 2, . . . .n and j = 1, 2, . . . .m. It should be noted that each member of this family is an interval of the form [N ik i (a ij ), N ik i (a ij )] where N ik i (a ij ) ≤ N ik i (a ij ) for each i = 1, 2, . . . .n and j = 1, 2, . . . .m.
By using Definition 1, the preference value of each alternative A j (j = 1, 2, ..., m) in the form of interval can be computed as sum of the product of the preference values of all the COs and the fulfillment degrees of corresponding elements of A j , i.e.
The final preference value Pr(A j )(j = 1, 2, ..., m) of each alternative A j (1 ≤ j ≤ m) can be found by calculating the mean value of the corresponding preference interval [I j , I j ], i.e.
Finally, the final ranking of alternatives is obtained by sorting the final preference values of alternatives. The greater the preference value, the better the alternative A j (1 ≤ j ≤ m).

An Illustrative Example
In this section, we solve an illustrative example by using proposed approach. This example and presented calculations are intended to help the reader to understand the presented method. It will allow using the given technique to various types of problems by readers with a lower level of expertise in fuzzy sets and their extensions.
Let us consider the problem of selecting the new tank to buy by the government for the army. A tank is used as a primary armored fighting vehicle for front-line combat. The basic parameters providing good combat value and maneuverability are firepower, strong armor, good quality tracks, and a powerful engine. Let us say that we should analyze ten offers (alternatives). Each one was assessed separately in the three criteria, according to Firepower (FP), Battlefield Maneuverability (BM), and Engine Power (EP). The offers performance is presented in the form of a decision matrix with established three criteria and reference ranking by using expert knowledge, which can be seen in Table 1. The detailed calculation procedure will be presented in the next 5 steps. Step 1: Suppose that N 1 , N 2 , and N 3 represent the three families of subsets of C selected for the criteria C 1 , C 2 and C 3 respectively, where The graphical representations of the families N 1 , N 2 , and N 3 for each criterion C 1 , C 2 and C 3 can be seen in Figures 4-6 respectively. The cores for each family with respect to each criterion is determined as C(N 1 ) = {40, 70, 100}, C(N 2 ) = {0, 9.5} and C(N 3 ) = {0, 3, 5.5}.  Step 2: The solution of COMET is obtained for different numbers of COs which can be obtained by taking the Cartesian product of the sets C(N 1 ), C(N 2 ) and C(N 3 ). The list of all the COs with their set values are given as under: preferred CO will get linguistic variable "absolutely important", the largest weaker CO will get linguistic variable "weakly important" and the COs with same comparison will get the linguistic variable "equally important". Strongly Important (SI) (0.7, 0.7, 0.7, 0.8, 0.9) 5 Absolutely Important (AI) (0.8, 0.8, 0.8, 0.9, 1) As a result, the matrix MEJ = [I ij ] 18×18 is obtained which can be seen in Tables 3 and 4.    Step 5: The preference interval indicating the approximate preference value of the first alternative A 1 = {84, 4, 3} is computed by using Formula (6), which is given as follows: The final preference value of A 1 can be found by calculating the mean score value as Similarly, the final preference values of all the remaining alternatives are obtained which are presented in Table 5. However, to compare result of our proposed method, TOPSIS method is applied to solve the same problem. The positive ideal solution d + i (i = 1, ..., 10) and negative ideal solution d − i (i = 1, ..., 10) are determined, and the relative closeness coefficients RC i (i = 1, ..., 10) and final ranking are presented in Table 5.
The final ranking obtained by TOPSIS method is A 8 A 6 A 9 and the most desirable alternative is A 8 . However, by using the COMET method with NIVTFNs, the ranking of the alternatives is A 8 A 9 which adequately match as those with the ranking obtained in the TOPSIS method as well as the reference ranking In the proposed approach, the ranking has been determined as an average of the received intervals. The best and the worst alternatives have the same place in the ranking, and correlation is very high (ρ = 0.9636). However, some important differences are also observed in the ranking order, i.e., the alternatives A 4 , A 5 , A 7 and A 10 . The proposed approach takes into account the new type of uncertainty compared to the previous extension of COMET. The result obtained here is the interval number, the so-called preference interval. Based on uncertain data, it is not possible to obtain a certain solution. Let us look at the assessment of alternatives A 4 and A 5 (Figure 7). The new quality of our solution is justified and possible discrepancies. The interval measure for A 4 is greater, but only in 74.07% of the cases for random values from these orders, we will obtain such a dependence. The solutions obtained by TOPSIS are the certain numbers and the any difference in the ranking has not explanation. The example above shows how a complete decision-making process can be carried out in order to make the result more knowledgeable about alternatives than other methods. This approach ensures that the phenomenon of rank reversal is avoided.

Conclusions
The uncertainty and diversity of assessment information provided by the DMs can be well reflected and modeled using NIVTFNs. The NIVTFNs are very useful to express vagueness and uncertainty more accurately as compared to fuzzy sets. Therefore, we synthesis a new method based on the COMET method and NIVTFNs. In that way, we obtained a helpful technique for solving MCDM problems under uncertain environment. In this study, we observed the difference between the proposed approach and TOPSIS methods for decision-making under uncertainty. We show that using preference intervals is more accurate and can justify the difference between rankings in an uncertain environment. Results of this approach are still free of rank reversal paradox due to the application of COMET, and it also permits decision-making under uncertain environment, especially where imprecise pieces of evidence and information are main hurdles for the decision-maker.
The presented approach is also following actual research trends in terms of methodological backgrounds. This paper provides theoretical manipulations of the extended approach of MCDM. It establishes a degree of membership in the form of interval instead of a crisp number, which is more suitable to tackle uncertainty during decision-making processes. To illustrate the whole calculation procedure of the COMET method using NIVTFNs, we studied a simple example. Future work will be geared towards the formulation of a comprehensive COMET-based system to support decision-making with an awareness of practical relevance and utility. Moreover, further research direction on the use of this approach and how to compare different rankings in the uncertain environment. As the next future works, we research the COMET method and: