1 Introduction

Determination of the weights of the criteria in the multi-criteria decision-making (MCDM) problems is incredibly significant since it remarkably affects the results. In decision problems, failure to obtain reliable criterion weights can create uncertainty about the accuracy of the solution of the decision problem (Jessop 2004). In cases of uncertainty, subjective methods can be used by taking opinions of decision makers. However, there is also the possibility of error in human judgment. Due to such possible errors, it is difficult to ensure accurate evaluation of criterion weights (Pala 2023). For this reason, using objective weighting methods and determining criterion weights reliably in decision problems is particularly important. Depending on this situation, new MCDM methods are developed to be used in the solution of decision problems. Three of them are the method based on the removal effects of criteria (MEREC) method introduced by Keshavarz-Ghorabaee et al. (2021), the logarithmic percentage change-driven objective weighting (LOPCOW) Method proposed by Ecer and Pamucar (2022), and modified preference selection index (MPSI) Method suggested by Gligori´c et al. (2022). Since they are newly introduced to the literature and being objective methods, the difference between them has been a matter of curiosity for researchers. Therefore, it is aimed to examine the impact of MEREC, LOPCOW, and MPSI weighting methods to criteria weights and the ranking results in this study. In addition, the applicability of PSI method instead of MPSI method has been questioned, since MPSI is a modified version of PSI method. In this context, the quality of life index (QLI) values are used as data set. For this purpose, MEREC, LOPCOW, MPSI and multi-attributive ideal-real comparative analysis (MAIRCA) methods are used as a combined method. The criteria weights for the QLI were calculated with the MEREC, LOPCOW and MPSI methods, and the capitals of 30 Asian countries were ranked according to the QLI using the MAIRCA method.

The QLI values to be used in the study are published by Numbeo every 6 months and the general QLI of the cities is estimated based on eight different indexes (Numbeo 2022). By calculating the QLI for selected cities around the world, these cities are ranked by the effects of certain criteria. Since the Numbeo database contains both the data of the criteria and the ranking results of the cities, it allows testing the developed MCDM methods.

The indexes in the QLI can be considered as criteria in MCDM problems. Among these criteria, purchasing power, safety, health services, and climate indexes are the benefit criteria (maximization oriented); pollution, property price/income ratio, cost of living and traffic indices are cost criteria (minimization oriented). It is preferred that the criteria with the maximization direction be large, and the criteria with the minimization direction be small. The eight indexes used in this study are explained in Numbeo web site (Numbeo 2022).

The aim of this study is to determine criteria weights with MEREC, LOPCOW, and MPSI weighting methods, which are new and objective methods in the literature, to compare them and to reveal their effects on ranking. The competence of these methods in decision models with a large number of alternatives was also evaluated. The results that will solve the problems that other scientists who will use the method may encounter have been reached. In the application part of the study, MAIRCA was combined for the first time with the weighting methods mentioned. The novelty of the study is explained below.

1.1 Highlights

  • It is aimed to examine the effects of MEREC, LOPCOW, and MPSI weighting methods, which are new and objective methods in the literature, on criteria weights and ranking results.

  • A new usage area for the MEREC method has been added to the literature.

  • The applicability of the Preference Selection Index (PSI) method, which is a modified version of this method instead of the MPSI method, was investigated and it was found that the PSI method was inadequate in calculating the criteria weights in decision-making problems with a large number of alternatives. This result is an important problem solver for other studies. Therefore, MPSI method was preferred instead of PSI method.

  • MEREC, LOPCOW, MPSI, and MAIRCA methods were used as a combined method. With the MPSI method, methods that have not been encountered in the literature have been added to the combined MCDM methods and have been guiding the way for new areas of use by comparing them.

  • The usage of MAIRCA method with the mentioned weighting methods was encountered only in this study. The rest of the article consists of the literature review in the second section, methods in the third section, results in the fourth section, discussion in the fifth section, and conclusion in the final section.

2 Literature review

Although MEREC and LOPCOW methods have recently been added to the literature as weighting methods, they have been used in various studies alone or integrating with other methods.

Table 1 shows the studies in which MEREC, LOPCOW, and MAIRCA methods are preferred for weighting and ranking the criteria in various decision-making problems.

Table 1 MEREC, LOPCOW, and MAIRCA methods areas of use
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Unlike MEREC and LOPCOW methods, MPSI method has been appeared in very few studies which are the support system selection in an underground mine (Gligori´c et al. 2022), the anti-fatigue lightweight design of heavy tractor frame (Zhang et al. 2022) and the bank selection problem (Yılmaz 2023).

While there are studies in the literature where MEREC and LOPCOW methods are used together (Bektaş, 2022; Yalman et al. 2023; Yaşar and Ünlü, 2023; Yürüyen et al. 2023), only one study was found in which MEREC, LOPCOW, and PSI methods were used (Ulutaş et al. 2023). However, no study comparing MEREC, LOPCOW, and MPSI methods has yet been found.

In the literature, it is seen that the studies in which MEREC and MAIRCA methods are used together are few and mostly in mechanical engineering (Trung and Thinh 2021; Nguyen et al. 2022; Sapkota et al. 2022). The literature examples of studies which uses MEREC and MAIRCA methods in social sciences were mostly based on financial performance of BIST insurance index (Çilek 2022) and on banking sector (Özcan 2022). Few studies have been found in which MCDM methods are used for the ranking of the QLI. Some studies using MCDM methods for QLI ranking are given in Table 2.

Table 2 Some studies using MCDM methods for QLI ranking
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De Vicente Oliva and Romero-Ania (2022) aimed to develop a multidimensional QLI which can replace the current methodology designed by Eurostat and implemented by the national statistical institutes of the European Union member states using ELECTRE III method.

As a result of the literature review, any study comparing the difference of MEREC, LOPCOW, and MPSI methods or considering MAIRCA, LOPCOW, MPSI, and MEREC methods together has not been found. Furthermore, the studies which the mentioned weighting methods and MAIRCA method are used have never been focused on the field of QLI. With this study, the use of weight determination in the ranking of countries according to QoL is also added to the used areas of the MEREC, LOPCOW, MPSI, and MAIRCA methods. The studies containing QLI-MCDM have not focused on the Asian countries. Based on the literature review, no studies have been conducted to measure and analyze the impact of using different weighting methods in ranking and prioritizing Asian countries according to QLI. Therefore, the research in question will contribute significantly to the literature. In addition to all these, it is thought that examining the applicability of the PSI method in decision problems with a large number of alternatives will contribute significantly to the literature. For this purpose, first, the application steps of MEREC, LOPCOW, MPSI, PSI, and MAIRCA methods will be introduced. Then the weights of the criteria (quality of life indicators) will be calculated with the MEREC, LOPCOW, and MPSI methods. According to these weights, the Asian countries will be ranked with the MAIRCA method. Finally, the ranking results will be compared with the ranking on the Numbeo website.

3 Method

In this study, rankings were carried out for Asian countries using QLI between 2020 and 2023. While the indices were weighted by the MEREC, LOPCOW, and MPSI methods, rankings are computed by the MAIRCA method, and these methods were combined in the study. Eight index values, including purchasing power (including rent) index, pollution index, property prices to income ratio index, cost of living index, safety index, climate index, health care index, and traffic index were chosen as criteria and the values for the capitals of Asian countries were used. The data were pulled from numbeo.com for the years 2020, 2021, 2022, and 2023. Since 2023 data have not yet been completed, mid-year data were used as index values in the analysis. Concerning a convenient comparison with the other years, mid-year index values were also used for each year. The ranking results for quality of life by years are also available on this site. The ranking results obtained in this study were compared with the Numbeo ranking results by calculating correlation coefficients and Euclidean distances. During the analysis, EXCEL and SPSS programs were used.

The QLI available in the Numbeo website content (in Java) is calculated as (URL1):

index.main = Math.max (0,100 + purchasingPowerInclRentIndex/2.5-(housePriceToIncomeRatio × 1.0) − costOfLivingIndex/10 + safetyIndex/2.0 + healthIndex/2.5 − trafficTimeIndex/2.0 – pollutionIndex × 2.0/3.0 + climateIndex/3.0).

Since the index data of some Asian countries or their capitals could not be reached, the evaluation was made over 29, 28, 28, and 25 countries for years 2020, 2021, 2022, and 2023, respectively. The data sets are given at Tables 9, 10, 11 and 12 in Appendix A.

3.1 Method based on the removal effects of criteria (MEREC)

This method is added to MCDM techniques in 2021 by Keshavarz-Ghorabaee et al. The method called MEREC is used to determine the weights of the criteria and has been tested by its developers and found to give consistent and reliable results with other criteria weight calculation methods. For the criteria weights, subjective weighting methods, objective weighting methods, and hybrid weighting methods are used (Keshavarz-Ghorabaee et al 2021). There are places where each method is superior to the others. MEREC stands as one of the objective weighting methods.

In the MEREC method, the effects of each criterion on the overall performance of the alternatives are examined for criterion weights. The effect of removing a criterion on the overall performance of the alternatives is examined. If the total performance is more affected by the removal of which criterion, that criterion should have a greater weight. Thus, both the weights can be given to the criteria and the criteria deemed unnecessary by the expert can be removed. To obtain objective criteria weights, MEREC uses the perspective of subtraction rather than inclusion, which is the basis of other objective weighting methods (Keshavarz-Ghorabaee 2021). In this method, criteria and alternatives are evaluated simultaneously. The method is based on maximizing the overall performance of alternatives with variation within and between criteria, measured using standard deviation and correlation (Sabaghian et al. 2021). The application steps of the method are as follows (Keshavarz-Ghorabaee 2021; Keshavarz-Ghorabaee et al 2021):

Step 1. Creating decision matrix: a decision matrix is created showing the ratings of each alternative or the values of each alternative according to each criterion. The elements of this matrix are denoted by \({x}_{ij}\), and these elements must be greater than zero. If there is negative data, it should be converted to positive using appropriate methods. A decision matrix \(X=\left[{x}_{ij}\right]\) of size \(m\times n\) can be represented as follows, where \(m\) represents the number of alternatives and \(n\) the number of criteria:

$$X=\left[\begin{array}{c}{x}_{11} {x}_{12} \cdots {x}_{1j} \cdots {x}_{1n}\\ {x}_{21} {x}_{22} \cdots {x}_{2j} \cdots {x}_{2n}\\ \vdots \vdots \ddots \vdots \ddots \vdots \\ {x}_{i1} {x}_{i2} \cdots {x}_{ij} \cdots {x}_{in}\\ \vdots \vdots \ddots \vdots \ddots \vdots \\ {x}_{m1} {x}_{m2} \cdots {x}_{mj} \cdots {x}_{mn}\end{array}\right]$$

Step 2. The normalized matrix \({M}^{x}=\left[{m}_{ij}^{x}\right]\) is obtained using the following appropriate equations.

$$m_{ij}^{x} = \frac{{\mathop {\min }\limits_{i} x_{ij} }}{{x_{ij} }}{\text{if the j}}{\text{. criterion is maximization oriented (When high is preferred)}}$$
(1)
$$m_{ij}^{x} = \frac{{x_{ij} }}{{\mathop {\max }\limits_{i} x_{ij} }}{\text{if the j}}{\text{. criterion is minimization oriented (When low is preferred)}}$$
(2)

Step 3. In this step, the performances of the alternatives are calculated with the help of a logarithmic measure with equal criterion weights.

$$S_{i} = \ln \left[ {1 + \left( {\frac{1}{n}\mathop {\sum\nolimits_{j}^{{}} {\left| {\ln \left( {m_{ij}^{x} } \right)} \right|} } } \right)} \right]$$
(3)

Step 4. In this step, the logarithmic measure is used similarly to the previous step. The difference from Step 3 is that the performances of the alternatives are determined based on the removal of each criterion separately. By removing each criterion one by one, the performances of the alternatives are calculated. Let \(S_{ij}{\prime}\) be the overall performance of the \({i}^{th}\) alternative computed with the removal of the \({j}^{th}\) criterion, then the value for each alternative is calculated as below:

$$S_{ij}{\prime} = \ln \left[ {1 + \left( {\frac{1}{n}\sum\nolimits_{k,k \ne j}^{{}} {\left| {\ln \left( {m_{ik}^{x} } \right)} \right|} } \right)} \right]$$
(4)

Step 5. The effect of removing the criterion is calculated. For this, the results of Step 3 and Step 4 are used. Thus, the sum of the absolute deviations is found and the effect of removing the criterion is computed. Let \({E}_{j}\) be the removal effect. \({E}_{j}\) is found with the help of the following equation:

$$E_{j} = \mathop \sum \limits_{i} \left| {S_{ij}{\prime} - S_{i} } \right|$$
(5)

Step 6. The final weights of the criteria are determined. In this step, the objective weight of each criterion is calculated using the removal effects (\({E}_{j}\)) calculated in Step 5. Let \({w}_{j}\) represent the weight of the \({j}^{th}\) criterion:

$$w_{j} = \frac{{E_{j} }}{{\mathop \sum \nolimits_{k} E_{k} }}$$
(6)

Here the sum of the weights of the criteria is equal to 1 (\(\sum_{j}{w}_{j}=1\)).

3.2 LOgarithmic percentage change-driven objective weighting (LOPCOW)

The LOPCOW method was introduced by Ecer and Pamucar in 2022 as an objective weighting method. The distinguishing features of the LOPCOW method are that it eliminates the difference (gap) caused by the size of the data by expressing it as a percentage of the standard deviations of the mean square values of the series and it provides the opportunity to work with negative values in the data set (Bektaş, 2022). Therefore, the distribution of criterion weights is less affected by changes in the performance values of the alternatives and provides a reasonably accurate result (Biswas et al. 2022). In addition, it is not affected by the influence of a large number of alternatives and criteria (Bektaş, 2022; Biswas et al. 2022). The steps of the LOPCOW method are as follows (Ecer and Pamucar 2022):

Step 1. A decision matrix is created with the ratings of each alternative or the values of each alternative according to each criterion. The elements of this matrix are denoted by \({x}_{ij}\), and these elements can be negative unlike the MEREC method. A decision matrix \(X=\left[{x}_{ij}\right]\) of size \(m\times n\) can be formed as in MEREC method, where \(m\) represents the number of alternatives and \(n\) the number of criteria.

Step 2. The normalized decision matrix \({M}^{x}=\left[{m}_{ij}^{x}\right]\) can be computed by Eqs. 7 and 8 in accordance with the characteristics of the criterion.

$$m_{ij}^{x} = \frac{{x_{ij} - x_{i}^{ - } }}{{x_{i}^{ + } - x_{i}^{ - } }}{\text{if the j}}{\text{. criterion is maximization oriented (When low is preferred)}}$$
(7)
$$m_{ij}^{x} = \frac{{x_{i}^{ + } - x_{ij} }}{{x_{i}^{ + } - x_{i}^{ - } }}{\text{if the j}}{\text{. criterion is minimization oriented (When low is preferred)}}$$
(8)

In equalities above, \({x}_{ij}\) is the value of the ith alternative at jth criterion, \({x}_{i}^{-}\) denotes the lowest value of the criterion gets from the alternative, and \({x}_{i}^{+}\) denotes the highest value of criterion takes from the alternative.

Step 3. Calculating the percentage values (PV) of the criteria: in this step, to eliminate the difference (gap) caused by the data size, the mean square value as a percentage of the standard deviations of each criterion is obtained with the help of Eq. 9.

$$PV_{ij} = \left| {\ln \left( {\frac{{\sqrt {\frac{{\mathop \sum \nolimits_{i = 1}^{m} (m_{ij}^{x} )^{2} }}{m}} }}{\sigma }} \right).100} \right|$$
(9)

where \(\sigma\) is the standard deviation and \(m\) represents the number of alternatives.

Step 4. Objective weights for each criterion are computed by Eq. 10.

$$w_{j} = \frac{{PV_{ij} }}{{\mathop \sum \nolimits_{i = 1}^{n} PV_{ij} }}$$
(10)

where the sum of the weights of the criteria is 1 \((\sum_{j}{w}_{j}=1)\).

3.3 Modified preference selection index (MPSI)

Preference selection index (PSI) method introduced by Mania and Bhatt (2010) eliminates the need to determine the relative importance of the criteria and is used both for finding criterion weights and for ranking purposes (Mufazzal and Muzakkir 2017), which makes it preferred in the literature. However, in decision problems with several alternatives as in this study, it is not possible to use the PSI method since the addition of numerous normalized values between 0 and 1 can be greater than 2 and as a result, overall preference value becomes negative. In addition, Zhang et al. (2022) argued in their study that the PSI method “prefers” certain criteria, and this leads to injustice in making decisions. Due to such disadvantages, different normalization techniques have been tried for the PSI method in the literature (Do and Nguyen 2023) or the PSI method has been modified (Gligorić et al. 2022; Zhang et al. 2022). The MPSI method is a modification of the traditional PSI method, achieved only by eliminating a certain step from the original PSI method. This small change increases the final values of the weighting coefficients, bringing them closer to the values obtained with other objective weighting methods. MPSI method is one of them and the application steps are explained below (Gligorić et al. 2022):

Step 1. A decision matrix is formed with the ratings of each alternative or the values of each alternative according to each criterion same as LOPCOW and MEREC. A decision matrix \(X=\left[{x}_{ij}\right]\) of size \(m\times n\) is formed, where \(m\) represents the number of alternatives and \(n\) the number of criteria.

Step 2. The normalized matrix \({M}^{x}=\left[{m}_{ij}^{x}\right]\) is obtained using Eqs. 1 and 2.

Step 3. Calculate the preference variation value \({p}_{j}\) as follows:

$$p_{j} = \sum\nolimits_{i = 1}^{m} {\left( {m_{ij}^{x} - \overline{m}_{j} } \right)^{2} }$$
(11)

where \({\overline{m} }_{j}\) is the mean normalized value given in Eq. 12.

$$\overline{m}_{j} = \frac{1}{n}\sum\nolimits_{i = 1}^{n} {m_{ij}^{x} }$$
(12)

Step 4. The weights of criteria \({w}_{j}\) are computed by Eq. 13.

$$w_{j} = \frac{{p_{j} }}{{\mathop \sum \nolimits_{j = 1}^{n} p_{j} }}$$
(13)

where the sum of the weights of the criteria is 1 \((\sum_{j}{w}_{j}=1)\). PSI method involves an extra step that computes PSI by \(q_{j} = 1 - p_{j}\) and then calculate the weights according to \({w}_{j}=\frac{{q}_{j}}{\sum_{j=1}^{n}{q}_{j}}.\)

3.4 Multi-attributive ideal-real comparative analysis (MAIRCA)

Pamucar et al. introduced the MAIRCA method, which was developed at the Center for Defense Logistics Studies at the University of Belgrade, at a conference in 2014 (Pamucar et al. 2014). The method is based on the logic of defining the difference between theoretical and real results. Thanks to the linear normalization algorithm used, very reliable results can be attained (Ecer 2022). It can also be used easily with other MCDM or alternative solution methods. There have been many studies for ranking aim where it is used alone or with other solution methods. The advantages of this method have been considered in choosing the MAIRCA method. These advantages are (Ecer 2022) listed below:

  • It can be used in problems with many evaluation criteria and alternatives.

  • It can solve decision problems with both qualitative and quantitative evaluation criteria.

  • It is easy to understand and apply.

  • It produces consistent solutions thanks to its own algorithm.

The method consists of six steps (Pamucar et al. 2014):

Step 1. Creating the decision matrix: in this step, an initial matrix with alternatives and criteria is built. A decision matrix \(X=\left[{x}_{ij}\right]\) is formed with \(m\times n\) dimension, where \(m\) represents the number of alternatives and \(n\) the number of criteria.

Step 2: determination of preference according to alternatives: at this stage, there is no priority calculation for the decision maker to choose between alternatives. The decision maker gives equal priority to alternatives. Since there will be equal priority among \(m\) alternatives, the preference ratio (\({P}_{{A}_{i}}\)) among the alternatives is calculated as follows:

$$P_{{A_{i} }} = \frac{1}{m}; \sum\nolimits_{i = 1}^{m} {P_{{A_{i} }} = 1, i = 1,2, \ldots ,m}$$
(14)

Step 3: determination of theoretically considered matrix elements: this matrix is denoted by (\({T}_{p}\)), and the elements of the matrix are determined as the product of the preference ratio of the preferred alternatives (\({P}_{{A}_{i}}\)) and the criterion weighting coefficients (\({w}_{n}\)). Let \(n\) be the number of criteria:

$${w}_{1} {w}_{2} \dots {w}_{n}$$
$$T_{p} = P_{{A_{i} }} \left[ {P_{{A_{i} }} {*}w_{1} { }P_{{A_{i} }} {*}w_{2} { } \ldots { }P_{{A_{i} }} {*}w_{n} } \right]$$
(15)

Step 4: calculation of real rating matrix (\({T}_{r}\)) elements: here, \(n\) is the number of criteria and \(m\) is the total number of alternatives:

$${C}_{1} {C}_{2} \dots {C}_{n}$$
$${T}_{r}=\begin{array}{c}{A}_{1}\\ \begin{array}{c}{A}_{2}\\ \begin{array}{c}\dots \\ {A}_{m}\end{array}\end{array}\end{array}\left[\begin{array}{ccc}{t}_{r11}& {t}_{r12}& \begin{array}{cc} \dots & {t}_{r1n}\end{array}\\ {t}_{r21}& {t}_{r22}& \begin{array}{cc}\dots & {t}_{r2n}\end{array}\\ \begin{array}{c}\dots \\ {t}_{rm1}\end{array}& \begin{array}{c}\dots \\ {t}_{rm2}\end{array}& \begin{array}{c}\begin{array}{cc}\dots & \dots \end{array}\\ \begin{array}{cc} \dots & {t}_{rmn}\end{array}\end{array}\end{array}\right]$$

The real grading matrix (\({T}_{r}\)) is made by multiplying the elements of the theoretically considered (\({T}_{p}\)) matrix with the elements of the initial decision matrix (\(X\)) according to the formulas given below:

a) If the criterion is the benefit criterion, that is, if it is preferred to be maximum:

$$t_{rij} = t_{pij} *\left( {\frac{{x_{ij} - x_{i}^{ - } }}{{x_{i}^{ + } - x_{i}^{ - } }}} \right)$$
(16)

b) If the criterion is the cost criterion, that is, if it is preferred to be minimum:

$$t_{rij} = t_{pij} *\left( {\frac{{x_{ij} - x_{i}^{ + } }}{{x_{i}^{ - } - x_{i}^{ + } }}} \right)$$
(17)

In equalities above, \({x}_{ij}\) is the value of the ith alternative at jth criterion, \({x}_{i}^{-}\) denotes the lowest value of the criterion gets from the alternative, and \({x}_{i}^{+}\) denotes the highest value of criterion takes from the alternative.

Step 5: creating the total gap matrix. The elements of the total gap matrix (\(G\)) are obtained by calculating the difference between the theoretical considered matrix (\({T}_{p}\)) and the actual grading matrix (\({T}_{r}\)) created in Step 3.

Step 6: calculation of final values (\({Q}_{i}\)) of criterion functions according to alternatives: at this stage, the final values are obtained by summing the rows of the gap matrix (\(G\)) for each alternative.

4 Results

The descriptive statistics of data sets for the years are given at Table 3 where C1 is purchasing power index, C2 is safety index, C3 is health care index, C4 is cost of living index, C5 is property prices to income ratio index, C6 is traffic index, C7 is pollution index, and C8 is climate index.

Table 3 Descriptive statistics of criteria for mid points of 2020, 2021, 2022, and 2023
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Sd standard deviation, Min minimum, Max maximum, K kurtosis, S skewness

Table 1 shows the descriptive statistical values of the data sets used for the analyses. Mean, standard deviation, minimum, maximum, skewness, and kurtosis values are presented, and it is observed that the data set does not have extreme values. The following sub-sections include weighting method results and applicability results of PSI method, then ranking results using MAIRCA method.

4.1 Criteria weights with MEREC, LOPCOW, and MPSI methods

The steps of the MEREC, LOPCOW, and MPSI methods were applied to the four decision matrices given in Appendix A. The criteria C1, C2, C3, and C8 are maximum oriented criteria and C4, C5, C6, and C7 are minimum oriented criteria in the analysis. The removal effect (\({E}_{j}\)) values in MEREC method, the percentage values (\({PV}_{j}\)) in LOPCOW method, and preference variation values (\({p}_{j}\)) in MPSI method were calculated for each criterion and results are given in Table 4.

Table 4 Analysis results for MEREC, LOPCOW, and MPSI
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When the results for MPSI method are examined, it is seen that the preference variation values for a few criteria is higher than 1; namely, criterion C1, C5, and C8 in 2020 and 2022; C1, C4, C5, and C8 in 2021; C1, C5 in 2023. This situation caused to get negative values at the PSI values (\({q}_{j}=1-{p}_{j}\)) calculation step of PSI method. This occurs because as the number of alternatives increases, the sum of preference variation values increases and exceeds 1. For this reason, the PSI method becomes dysfunctional when the number of alternatives is large. Therefore, PSI method could not be applied in this study, MPSI method was preferred instead. The criteria weights attained by MEREC, LOPCOW, and MPSI methods are illustrated with Fig. 1.

Fig. 1
figure 1

Criteria weights according to MEREC, LOPCOW, and MPSI methods by years

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According to Fig. 1, while MEREC and MPSI methods gave the close and high values, LOPCOW gave very low value for criteria C1. The opposite situation was observed for the criteria C3 and C6. The highest weighted criterion was C8 in 2020, C5 in 2021 and 2022, and C4 in 2023 by MEREC method. The lowest weighted criterion was C7 in each year. In the results of LOPCOW method, the highest weighted criterion was found as C5 for 2020, 2021, and 2022, but C4 for 2023. The lowest weighted criterion was C8 at each year. The highest weighted criterion was found as C8 and lowest weighted criterion was C3 by MPSI method at each year.

4.2 Ranking by MAIRCA method

In this section, Asian countries are ranked according to eight criteria weighted by the MEREC, LOPCOW, and MPSI methods for each year. The decision matrices are same as the matrices used for weighting and are given in Appendix A. In the next step, since there will be equal priority among the alternatives, the preference rate (\({P}_{{A}_{i}}\)) is calculated as in Table 5 using Eq. 14.

Table 5 Preference rates
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After second step, all calculations are made using the weight values determined by the MEREC, LOPCOW, and MPSI methods, separately. The ranking results by years and methods are summarized at the following Table 6.

Table 6 Summarized ranking results by MAIRCA method
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The results were tested by comparing the ranking results with the current ranking status on the Numbeo website. In the rankings, Oman is generally in the best condition among Asian countries according to the QLIs, while Philippines took the last place mostly. According to the results, there was no statistically significant difference between the rankings.

Spearman rank correlation results were analyzed to determine the correlations between the ranking scores. The Spearman correlation coefficient enables the comparison of rankings and the determination of the degree of relatedness between the results. The Spearman correlation coefficient approaching 1 indicates an increasing relationship between the two rankings. According to the results at Table 7, high correlation was observed between all multi-criteria method rankings at 0.01 level of significance. Among them, the results of MPSI and MAIRCA methods have the highest correlation coefficient with Numbeo ranking results, which means that MPSI weights give the most successful criteria weights compared to MEREC and LOPCOW methods.

Table 7 Correlation coefficients of ranking results
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Euclidean distance shows the distance between two points, and with these distances, it can be determined how far the sets of rankings are from each other. A small distance value indicates that the two compared clusters are closer to each other. The Euclidean distance between the ranking results and Numbeo ranking was computed and presented at Table 8, and the results implied that the closest ranking was MPSI + MAIRCA.

Table 8 Euclidean distances between the ranking results
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5 Discussion

The aim of the study was to compare the MEREC, LOPCOW, and MPSI methods and their effects to criteria weights and ranking results in the decision problems involving over 20 alternatives. From the mathematical point of view, MEREC method is based on logarithmic function and measure the removal effects of the criteria. LOPCOW method is based on a percentage of the standard deviations of the mean square values and allows negative numbers in the data sets, unlike MEREC and MPSI methods. MPSI method considers relative importance of the criteria.

In the normalization step, a simple linear normalization is used, and all the criteria are transformed to the minimization type criteria in MEREC method. LOPCOW uses Weitendorf’s Linear Normalization technique, and MPSI method involves 0–1 interval normalization using Max–Min values (Aytekin 2021). In terms of ease of application, it has been seen that the MPSI method is the most easily applicable method at normalization and calculation steps.

Based on the criteria weighting results given with Fig. 1, it can be said that the MPSI method gives more stable criteria weights and is not affected by small changes in the data set. Comparing MPSI, MEREC, and LOPCOW weighting values, it can be said that MEREC and MPSI gives close results, in contrast to the LOPCOW method. The weight results could not be criticized with the literature, since there has been no study that compared the MEREC and MPSI or LOPCOW and MPSI methods as far as known. However, MEREC and LOPCOW results can be compared to literature. It is seen in the literature that the LOPCOW method gives exactly the opposite weights compared to the MEREC method. Bektaş (2022), Yalman et al. (2023), Ulutaş et al. (2023), and Yaşar and Ünlü (2023) state in their study that since the criteria with higher weights in the MEREC method receive lower weights in the LOPCOW method, the results obtained from both objective criterion weighting methods are combined for a more stable weighting.

Upon examination of the change in criterion weights over time, it becomes evident that C1, C5, and C8 receive the most weight in the MEREC and MPSI methods. While the criteria that receive the highest weight as a result of the LOPCOW method vary from year to year, they include criteria C1, C5, and C8. The purchasing power index (C1) indicates the relative purchasing power of goods and services in a given city in comparison to the average net salary. The property price-to-income ratio (C5) is defined as the ratio of average apartment prices to the average disposable family income. In the climate index (C8), settlements with high quality of life have moderate temperatures and low humidity. The criteria weights imply that countries seeking to enhance the quality of life should prioritize indexes pertaining to purchasing power, property price-to-income ratio, and climate. In other words, the increase of the purchasing power and climate index values, in conjunction with a reduction in the property price-to-income ratio, significantly influences the ranking of countries. Consequently, while these countries endeavor to enhance climatic conditions, they should implement political measures to regulate purchasing power and property price-to-income ratios. Given their geographical location, countries are unable to alter air temperature and humidity, yet they can implement environmental policies that improve these conditions within the constraints of their own climatic conditions.

Additional to the above inferences, the usability of the PSI method is also discussed in this study. In Table 2, the preference variation values of criteria such as C1, C5, and C8 are higher than 1. This situation caused to get negative values while computing the PSI values (\({q}_{j}=1-{p}_{j}\)) for PSI method. This occurs because as the number of alternatives increases, the sum of preference variation values increases, so that it exceeds 1. For this reason, the PSI method gives incorrect criteria weights, and it becomes useless as happened in this study, especially when the number of alternatives is large. Therefore, MPSI method was preferred instead of PSI method. This result will be an important guide when choosing the right MCDM method for other studies. Accordingly, when the results of the study are examined in terms of the number of alternatives, although PSI method is preferable, objective, and easy to apply, this study indicates that the PSI method is sensitive to the number of alternatives. However, the MEREC, LOPCOW, and MPSI methods are not affected by the number of alternatives.

When the ranking results were compared to Numbeo, there was no statistically significant difference between the rankings. Furthermore, it can be concluded that the MPSI method is more preferable compared to the MEREC and LOPCOW methods because the correlation between the MPSI + MAIRCA and Numbeo rankings is between 0.97 and 0.99, the closest ranking is again MPSI + MAIRCA with values between 6.16 and 10.68 based on the ranking distances, and the ease of application of MPSI.

6 Conclusion

In MCDM, subjective, objective, and combined methods can be used to determine the criteria weights. These combined methods can be created by hybridizing different subjective and objective weighting methods. The reason for using the combination method is that it is thought that more consistent results are obtained. In this study, the data reached at Numbeo website were analyzed by MEREC, LOPCOW, MPSI, and MAIRCA methods. The Numbeo QLI determines and calculates the quality of life of the world countries with eight different criteria. These data are shared in the database mentioned in certain periods of the year. This study presents the ranking of Asian countries in this database on the basis of selected QLIs by years with MCDM methods. Eight criteria and over twenty-five alternative countries were used to calculate the index values. MEREC, LOPCOW, and MPSI methods were used to determine the weights of the criteria, and MAIRCA method was used to find the rankings of the alternatives. MEREC + MAIRCA, LOPCOW + MAIRCA, and MPSI + MAIRCA values were compared with the results in Numbeo database.

The following results were obtained from the study:

  • MEREC, LOPCOW, and MPSI methods are newly added to the literature and have been compared for the first time and have been used for the first time to rank countries according to their QLIs and make significant contributions to the domestic and foreign literature. Among these weighting methods, MPSI weights gave the closest ranking results to Numbeo.

  • By combining the MPSI method with the MAIRCA method, methods that have not been encountered before in the literature have been added to the MCDM methods, thus this study has guided new areas of use.

  • This study also examines the applicability of PSI method to the decision problems containing more than 20 alternatives, which indicated that PSI method cannot be applicable at problems with high number of alternatives. This result is an important problem solver while choosing the right MCDM method for other studies. Thus, MPSI method was preferred instead of PSI method.

  • The MAIRCA method was used together with the mentioned weighting methods in only this study. Therefore, this study contributes to the field.

In future studies, the weighting methods can be combined to a single weight, hybridized with different MCDM techniques and expanded according to different fuzzy set bases. A method can be developed to obtain a single weight by integrating these objective weighting results. Analyses can be performed for different countries and different year intervals and thus the generalizability of the method used can be examined. In addition, possible reasons for changes in country rankings during the study period, such as economic or social changes, and how these factors may have affected quality of life indices can be discussed in future studies.

6.1 Limitations of study

In the research, QLI data in Asian countries taken from the Numbeo website were used. The limitation of this study is that data for every country could not be accessed for every year. For this reason, the number of alternatives in the data sets between 2020 and 2023 included in the study is different. For this reason, analysis results may vary depending on different countries and different years. This situation also makes it difficult to interpret the change in the quality of life in countries over the years. To overcome these limitations in future studies, the data for other continents can be used or the countries with missing data for some years may not be included in the study.