The comparison of MCDM Methods including AHP, TOPSIS and MAUT with an Application on Gender Inequality Index

Gender Inequality Index is a major indicator presenting level of development of the countries as Human Development Index, which is calculated regularly every year by UN. In this study, an alternative calculation has been proposed for measuring gender inequality index which is an important barrier for the human development. Each indicator in the index integrated as MAUT-AHP and also AHP-TOPSIS and these methods carried out again for the alternative ranking member and candidate countries of the European Union. The main objective here is to represent that the indicators form gender inequality index can be reclassified with different weights for each indicator


136
In this study the methods commonly used in the literature TOPSIS&AHP and AHP&MAUT are integrated and proposed as an alternative methods doing fair classification for the indicators form gender inequality index.

Methodology:
In this section we give brief explanations about the methods used in this study.

AHP:
The Analytic Hierarchy Process (AHP) introduced by Saaty is a multi-criteria decision-making technique to solve complex decision problems (1977 and 1994). This method uses a multi-level hierarchical structure of objectives, criteria, sub criteria, and alternatives ( Figure 1). AHP is a preferable model due to its easy to use has been extensively studied and is used in a wide variety of decision situations by many researchers, in fields such as, business, industry, healthcare etc. AHP methodology can be implemented in three steps. Each step needs to be performed to be resolved in a decisionmaking problem with AHP are described below. In the following m refers to the alternative numbers and n refers to the criteria numbers.
Step 1: It can be stated objective (goal) and in turn defined the criteria picked the alternatives.
Step 2: In this step firstly, elements can be compared to one another, two at a time, with respect to their importance on an element above them in the hierarchy and then structured the comparison matrix (a square matrix of size n×n). All   The Standard Preference Scale used in the AHP method is provided in Table 1 as follows. In the AHP method, the scale range 1-9 is assumed sufficiently representing human beings' perception.

Preference Level Numerical Value
Equally Preferred 1 Equally to Moderately Preferred 2

Moderately Preferred 3
Moderately to Strong Preferred 4

Strongly Preferred 5
Strongly to Very Strongly Preferred 6

Very Strongly Preferred 7
Very Strongly to Extremely Preferred 8 Extremely Preferred 9 Step 3: It has been normalized each matrix element by the sum of elements in each column and we calculate the sum for each row. B column vectors are utilized in the calculation of the equation (2). Priority vector which is specified below by W column vector is obtained by forming the arithmetic average of the each line of the B matrix.
Measuring consistency of the judgements, Saaty(1980) proposed Consistency Index (CI), which is a measure consistency of the subjective judgements. It is calculated given following formula below; The consistency ratio (CR) is obtained by comparing CI with the set of numbers called random consistency index (RI) with the following formula given below.
If Consistency Ratio is greater than 10%, test results are inconsistent (CR ≥ 10%), then the result from the AHP method will be of no use in decision making. The higher consistency ratio, the assessment result becomes more inconsistent.
TOPSIS Method: The TOPSIS method was initially presented by Yoon and Hwang (Yoon and Hwang, 1981) and Lai, Liu and Hwang (Lai, Liu and Hwang, 1994). This method is a process of finding the best solution among all practical alternatives. TOPSIS is based on that the chosen alternative should have the shortest geometric distance from the positive ideal solution (PIS) (Assari, A. , Mahesh, T. , Assari, E. , 2012) and the longest geometric distance from the negative ideal solution (NIS). The TOPSIS method is expressed with six steps as follows: Step 1: Firstly create an evaluation matrix consisting of m alternatives and n criteria, with the intersection of each alternative and criteria given as aij, therefore a matrix in form (aij)m×n    Step 4: Determine the ideal (A * ) and negative ideal (A¯) solutions.
Step 5: Calculate the separation measures using the m-dimensional Euclidean distance. Determine the worst alternative and the best alternative, respectively, are as follows: Step 6: Calculate the relative closeness to the ideal solution. (12) Step 7: Rank the alternatives according to siw (i=1, 2…………. . m)

MAUT (Multi Attribute Utility Theory):
Utility is a measure of desirability and gives to a uniform scale to compare and/or combine tangible and intangible criteria (Ang, Tang, 1984). Utility function is a device which quantifies the preferences of a decision-maker by assigning a numerical index to varying levels of satisfaction of a criterion (Mustafa, Ryan, 1990). For a single criterion ( X ), the utility of satisfaction of a consequence x' is denoted by ( ) ' ux . The utility is generally calculated as the sum of the marginal utilities that each criteria assigns to the considered action (Figueira, Greco, Ehrgott, 2005). Multi Attribute Utility Theory takes into consideration the decision maker's preferences in the form of the utility function which is defined over a set of attribute (Pohekar, Ramachandran, 2004). In this method both quantitative and qualitative criteria can be used. The most common method of multicriteria utility function is the additive model (Keeney, Raiffa, 1993). Step 6: Rank the alternatives, Choose an alternative which gain the most utility.

Findings:
In this article, we studied on Gender Inequality Index (GII) Indicators for the Candidate and Member countries of European Union. This index measures reflecting inequality in achievements between women and men in three dimensions: reproductive health, empowerment and the labor market as seen Table 2 given below.
142 When examined GII calculations, it can be seen that all of the indicator's importance is in the same level. However, it has criticisms from some scholars and policy makers about indicators since they are not equal each other, as in the human development index (Safari, Ebrahimi, 2014). By thinking these critics, it has been created as an alternative method of ranking countries in terms of gender inequality index.
This study is compromised two important stages. Firstly by using analytical hierarchical process method, it can be achieved the comparing elements (indicators) to one another, two at a time, with respect to their importance with in the hierarchy and structured the comparison matrix (a square matrix of size n×n). Weights given below in Table 3 have been created randomly in order to set an assignment for the criteria.  Table 4 represents normalized values for each element of the comparison matrix. The last column of the Table 4 called Priority vector (Criteria Weights) obtained by forming arithmetic average of each line.

. 03) has been calculated by using formula (5), which represents that AHP is reasonable for the analysis. Further, countries are listed with TOPSIS and MAUT method after defining weights with AHP.
In the TOPSIS method, initially evaluation matrix is formed consisting of 32 alternative countries and 7 criteria. Table 5 given below represents evaluation matrix for TOPSIS method partially.  Table 6 represents weighted normalized evaluation matrix, which is calculated by multiplying criteria weights with each column of the Table 5.  Table 7 represents the ideal (A + ) and negative ideal (A¯) solutions of weighted (with AHP) normalized decision matrix.  Table 8. After getting the ranking with TOPSIS, it has been performed MAUT method. Marginal Utility Scores, which is the identification of best and worst values in the MAUT method, is given as follows. Total utility values have been calculated for each country after normalized values are obtained by multiplying with AHP coefficients (Table 10).