CRITIC-Entropy based Fuzzy Decision Making Models: A Systematic Analysis

This research article presents a comprehensive analysis of weighting methods used in fuzzy multi attribute decision making (MADM) methods. These methods involve various criteria in order to evaluate alternatives and determining the weights of criteria is a significant problem that arises very often in many MADM problems. In this research paper, CRTITIC and Entropy weighting methods have been used for finding criteria’s weights like in many research works. Using these unsupervised methods of assigning criteria weights, seven fuzzy MADM methods are examined in the context of ranking the best company to invest in. From the results of these methods, ranking order of alternatives is obtained and are analysed for reliability.


Introduction
In some of the situations characterized by uncertainty and imprecision, Boolean logic fails to represent adequately. So in order to handle these uncertain situations, the fuzzy set theory [15] has been extended to different forms like Intuitionistic fuzzy sets (IFSs) [1], Pythagorean fuzzy sets (PFSs) [14], Neutrosophic sets (NSs) [10], Spherical fuzzy sets (SFSs) [8] and so on. It is evident from the literature that these concepts have many real time applications in multi attribute decision making (MADM) [5,6] methods. These MADM methods are widely used in decision making theory. They involve several decision factors or criteria to deal with in any decision making situation. At same time, the attribute weights are extremely important in decision-making process. Decision maker is always struggling to assign criteria ISSN: 2456-8686, 5(1), 2021:132-141 https://doi.org/10.26524/cm100 weights in selection problem. For this purpose, two unsupervised weighting methods, namely CRITIC [7] and Entropy [11], are widely used, and some of well known fuzzy decision making methods were utilized based on attribute weighting methods. Many researchers have undertaken such analysis and they differ in their selective approach. For example, CRITIC and Entropy based fuzzy TOPSIS methods have been implemented in supply chain risk assessment [4] and petrochemical industries [12]. Also, Many MADM methods have been developed based on entropy measure [3,9]. Moreover, Sahin has done a comprehensive analysis of six weighting and seven multi criteria methods [13].
The rest of this paper is organized as follows: Section 2 describes the preliminaries; Section 3 gives the methodology of weighting methods; Section 4 presents the combined fuzzy MCDM methods with a case study in order to test the applicability of the decision making tools. The comparative analysis is given in section 5. Finally section 6 gives conclusion.

Preliminaries
Definition 2.1 [15] A fuzzy setF is defined on a universe of discourse as the form: Here ξF(ṙ) denotes membership function to eachṙ. Definition 2.2 [1] An intuitionistic fuzzy setsĨ F is defined as a set of ordered pairs over a universal set given bỹ and with the condition ξĨ F (ṙ) + ψĨ F (ṙ) ≤ 1 for each elementṙ ∈ . Here the membership and non-membership functions are denoted as ξĨ F (ṙ) and ψĨ F (ṙ) respectively. Suppose that the condition ξ 2 (ṙ) ≤ 1 satisfies for each elementṙ ∈ , then it is called Pythagorean fuzzy sets [14].
CRITIC Method: The CRitria Importance Thorough Inter critria Correlation (CRITIC) method [7] is used to find criteria weight based on the contrast intensity and conflict evaluation of the decision problem. The steps of CRITIC method are given to determine the attribute weight under fuzzy decision matrix. Step 1. The decision matrix [DM ]∀i = 1, 2, 3, . . . , m, j = 1, 2, 3, . . . , n is formed: Step 2. Decision matrix is normalized using the following equation Step 3. Calculate the correlation coefficient of the attribute C j to C k by Eq.2 Step 4. Estimate the standard deviation for each attribute Step 5. Compute the deviation degree θ of criterion C j from the other criteria Step 6. Obtain the original weights of attributes (ii). Entropy Method: The entropy concept, which is a measure of uncertainty in information expressed in terms of probability theory, was introduced by Shannon [11]. Shannon's entropy method interprets the relative intensities of the criterion importance depending on the discrimination among data to evaluate the relative weights. These are the steps involved in the Shannon's entropy method: Step 1. Normalize the initial decision matrix Step 2. Calculate the entropy of each criterion Step 3. Calculate the variation coefficient of criterion Step 4. Determine the weight of each criterion C j by the following equation

MCDM methods
In this section, the CRITIC and Entropy weighting methods are used to analyze seven fuzzy decision making methods such as weighted sum method (WSM), weighted product method (WPM), the multiple criteria optimization compromise solution (VIKOR), the technique for order performance by similarity to the ideal solution (TOPSIS), the Weighted Aggregates Sum Product Assessment (WASPAS), COmbinative Distance-Based ASsessment (CODAS), and the Evaluation Based on Distance from Average Solution (EDAS). In order to make a comprehensive study the following case study has been taken.

Case study:
An illustrative example on the selection problem of investment alternatives is taken from [2] to analyse the mentioned MCDM methods with fuzzy decision matrix. A company wants a sum of money to be invested in an industry. Then the committee suggests the following four feasible alternatives: A 1 -is a textile company; A 2 -is an automobile company; A 3 -is a computer company; A 4 -is a software company. Suppose that three attributes namely, C 1 -is the risk; C 2 -is the growth; C 3 -is the environmental impact; are taken into the evaluation requirements of the alternatives. Then the expert or decision maker is asked to evaluate each alternative on attributes in the form of fuzzy numbers. Thus, the assessment data can be represented by fuzzy decision matrix D = [λ ij ] 4×3 and it is given by   Based on the finding results, we observe that A 2 is the best alternative in overall.

Comparative Analysis
To compare the results of this study, a pictorial representation is given in Figures.  1 and 2 for the better understanding.
From above table, we observe that the ranking order of the alternatives are the same for all fuzzy decision making methods. Hence we find that the objective weighting methods such as CRITIC and Entropy are efficient and more reliable to deal with all types of fuzzy MCDM methods.

Conclusion
In this study, two objective weighting methods and seven fuzzy MCDM methods have been used for comparative analysis of ranking of alternatives. The weighting vectors are involved in fuzzy decision making methods and they play a very important role in finding the best output. According to the evaluation and results of the methods, we conclude that the seven methods under analysis are more reliable in decision making as the ranking obtained are almost similar. This research could be ISSN: 2456-8686, 5(1), 2021:132-141 https://doi.org/10.26524/cm100 extended for the systematic analysis of decision making methods based on complex fuzzy sets.