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Decision-making, briefly defined as choosing the best among the possible alternatives within the possibilities and conditions available, is a far more comprehensive process than instant. While in the decision-making process, there are often a lot of criteria as well as alternatives. In this case, methods referred to as Multicriteria Decision-Making (MCDM) are applied. The main purpose of the methods is to facilitate the decision-maker's job, to guide the decision-maker and help him to make the right decisions if there are too many options. In cases where there are many criteria, effective and useful decisions have been taken for granted at the beginning of the 1960s for the first time and supported by day-to-day work. A variety of methods have been developed for this purpose. The basis of some of these methods is based on distance measures. The most known method in the literature based on the concept of distance is, of course, a method called Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS). In this study, a new MCDM method that uses distance, similarity, and correlation measures has been proposed. This new method is shortly called DSC TOPSIS to include the initials of distance, similarity, and correlation words, respectively, prefix of TOPSIS name. In the method, Euclidean was used as distance measure, cosine was used as similarity measure, and Pearson correlation was used as relation measure. Using the positive ideal and negative-ideal values obtained from these measures, respectively, a common positive ideal value and a common negative-ideal value were obtained. Afterward DSC TOPSIS is discussed in terms of standardization and weighting. The study also proposed three different new ranking indexes from the ranking index used in the traditional TOPSIS method. The proposed method has been tested on the variables showing the development levels of the countries that have a very important place today. The results obtained were compared with the Human Development Index (HDI) value developed by the United Nations.

Decision-making with the simplest definition is the process of making choices from the available alternatives. Although it was expressed in different forms, basically a decision-making process involves: identification of the objective, selection of the criteria, selection of the alternatives, selection of the weighting methods, determination of aggregation method, and making decisions according to the results.

To be able to adapt to rapidly changing environmental conditions and make effective decisions in parallel with this change can only be possible by using scientific methods that can evaluate a large number of qualitative and quantitative factors in the decision-making process [

Taking more than one criterion, choosing the most appropriate one among the alternatives, or alternating sorting problems is called Multicriteria Decision-Making (MCDM) problems. The MCDM could have been appropriate for the purposes of evaluating the alternatives in a particular order, or alternatively, in order to determine the alternatives [

TOPSIS, one of the MCDM methods, developed by Hwang and Yoon [

The basis of this study is based on the report presented in the statistics conference (istkon 2017) held in Turkey in 2017 [

In this study, a new MCDM method based on the traditional TOPSIS method is proposed. In the proposed method, ideal positive solution approximation or ideal negative solution distance is calculated based on the distance, similarity, and correlation (DSC) measures, unlike the traditional TOPSIS method. For this reason, this new unit of measure proposed to rank alternatives is named as DSC and the new MCDM method developed is called DSC TOPSIS. The main advantage of the proposed unit of measurement is that it does not only rank the alternatives according to the concept of distance but also according to the concepts of similarity and correlation. In other words, the major advantage of the proposed method is that it proposes a stronger unit of measure by considering the three basic concepts that can be used to compare units: distance, similarity, and correlation. Another advantage of the proposed new MCDM method is that it suggests three new different methods that can be used to rank alternatives according to their importance levels. In addition, the impact of the proposed method on the cases where the decision matrix is dealt with by row, column, and double standardization methods is discussed in detail. These methods are applied to the results obtained by the proposed method. Thus, the traditional TOPSIS method and the sorting technique used in this method are compared with the proposed MCDM method and the proposed three new sorting methods. The functioning of the method has been tested to determine the order of development of countries. For this purpose, the indicators of Human Development Index (HDI) calculated by UN Development Programme (UNDP) have been utilized. The results were compared with HDI and traditional TOPSIS values.

There are no studies in the literature that use the concepts of distance, similarity, and correlation together. In this context, this study will be the first. The closest one to the proposed method in this study is Deng’s method which was published in the study entitled “A Similarity-Based Approach to Ranking Multicriteria” in 2007. With this study Deng presented a similarity-based approach to ranking multicriteria alternatives for solving discrete multicriteria problems. Then Safari and et al. [

In the following sections, the concepts of distance, similarity, and correlation will be mentioned first (Section

Since the new unit of measurement proposed in the study, which is developed as an alternative to the unit of measurement used in the traditional TOPSIS method, is based on the concepts of distance, similarity, and correlation, these and related concepts will be briefly explained in the following subsections. Thus, a better understanding of the proposed unit of measurement will be provided.

Any function

If

If

In mathematics, distance and metric expressions are used in the same sense [

Any function_{ij} is defined as “dissimilarity” if the first three of the metric axioms described above are satisfied. Thus, dissimilarity is more general concept. The upper and lower bound of the most dissimilarity functions are 1 and 0, respectively (0 ≤_{ij} ≤ 1).

The most common measure used to compare two cases is similarity. The most important reason why similarity is more preferable than distance and dissimilarity is that it is easier for people to find similar aspects when comparing two things. Similar to dissimilarity, the concept of similarity varies between 0 and 1 (0 ≤ s_{ij} ≤ 1). Similarity is the complement of the dissimilarity. This relationship between the two concepts is shown in

The transformation given in (

The concepts of distance, dissimilarity, and similarity can be interpreted in geometrical terms because they express the relative positions of points in multidimensional space. On the other hand, the association and correlation concepts reveal the relations between the axes of the same space based on the coordinates of the points.

Except for covariance, most of association and correlation coefficients measure the strength of the relationship in the interval of

The

The

The

According to this, the

The “global distance” is a measure in which the result of combining the various distance measures is called “local distance” with different methods. The most common of these methods used to combine local distances are “total sum, weighted sum, and weighted average (e.g., geometric mean)” [

MCDM problem is a problem in which the decision-maker intends to choose one out of several alternatives on the basis of a set of criteria. MCDM constitutes a set of techniques which can be used for comparing and evaluating the alternatives in terms of a number of qualitative and/or quantitative criteria with different measurement units for the purpose of selecting or ranking [

MCDM problems are divided into Multiattribute Decision-Making (MADM) and Multiobjective Decision-Making (MODM) problems. The MADM problems have a predetermined number of alternatives and the aim is to determine the success levels of each of these alternatives. Decisions in the MADM problems are made by comparing the qualities that exist for each alternative. On the other hand, in the MODM problems, the number of alternatives cannot be determined in advance and the aim of the model is to determine the “best” alternative [

Regardless of the type of decision-making problem, the decision-making process generally consists of the following four basic steps:

determination of criteria and alternatives,

assignment of numerical measures of relative importance to criteria,

assigning numerical measures to alternatives according to each criterion,

numerical values for sorting alternatives.

The MCDM methods have been developed to effectively carry out the fourth stage of this process. There are different methods used in the literature for the solution of MCDM problems. The differences between the methods are due to the approaches that they recommend to make decisions.

In this study, a new MCDM method based on the traditional TOPSIS method is proposed. In the proposed method, ideal positive solution approximation or ideal negative solution distance is calculated based on the distance, similarity, and correlation measures, unlike the traditional TOPSIS method. For this reason, the method is called DSC TOPSIS. Three different new methods are also proposed in order to rank the alternatives according to their importance levels in DSC TOPSIS or similar MCDM methods. There are several studies in the literature that modify the TOPSIS method. Some of them can be listed as follows. Hepu Deng [^{+} D^{-} plane and constructing the P value to evaluate quality of alternative in their study titled “

Since the proposed method is based on the TOPSIS method, the steps of the traditional TOPSIS method will first be explained in the following subsections. And then the details of the proposed MCDM method will be discussed.

Determining the decision matrix:

Determining the weighting vector as follows:_{j} with respect to the overall objective of the problem is represented as_{j}.

Normalizing the decision matrix through Euclidean normalization:

Calculating the performance matrix:

The weighted performance matrix which reflects the performance of each alternative with respect to each criterion is determined by multiplying the normalized decision matrix (

Determining the PIS and the NIS:

The positive-ideal solution (PIS) and the negative-ideal solution (NIS) consist of the best or worst criteria values attainable from all the alternatives. Deng [

For PIS (

Calculating the degree of distance of the alternatives between each alternative and the PIS and the NIS:

The D^{+} and the

Calculating the overall performance index for each alternative across all criteria: _{i} value indicates the absolute closeness of the ideal solution. If _{i} is the PIS; if _{i} is the NIS.

Ranking the alternatives in the descending order of the performance index value.

The approaches used in the steps of “normalizing the decision matrix,” “calculating the distance of positive and negative-ideal solutions,” and “calculating the overall performance index for each alternative across all criteria” of the traditional TOPSIS method are open to interpretation and can be examined, improved, or modified. Approaches that can be recommended in these steps are briefly summarized below.

_{2}), which is used to calculate the distance of PIS and NIS, is also possible to use many different distance measures such as linear [

To offer solutions to the weaknesses listed above, in this study instead of

“normalization approach” applied in Step 3 of the traditional TOPSIS method,

“euclidean distance” used in Step 6,

“simple ratio” used in Step 7,

This method, which could be an alternative to MCDM methods especially TOPSIS, due to the new measure proposed is called DSC TOPSIS. The main objective of

using different standardization methods is to emphasize what can be done for different situations that can be encountered in real life problems;

developing a new measure for the traditional TOPSIS method in this study is to improve the approach of “evaluating the two alternatives only based on the Euclidean distance to the PIS and NIS values” and make it more valid;

developing new sorting methods is to criticize the method used in traditional TOPSIS because it is based on simple rate calculation.

At this point, an important issue mentioned earlier must be remembered again. That is, the fact that none of the MCDM methods is superior to the other. Therefore, the new MCDM method proposed in this study will certainly not provide a full advantage over other methods. But a different method will be given to the literature.

Basically, there are three approaches used to compare vectors. These are distance, similarity, and correlation. The concept of distance is the oldest known comparison approach and is the basis of many sorting or clustering algorithms. On the other hand both similarity and correlation are two other important concepts that should be used in vector comparisons. According to Deng [

In order to express an MCDM problem in “_{i} vector and PIS and NIS can be represented by _{i} and _{i} and

The degree of conflict.

In the light of the above-mentioned explanations and causes, the three approaches of “

Unlike the traditional TOPSIS method, this step has been developed on the basis of standardization. The differences of column and row-standardization were emphasized.

All parameters should have the same scale for a fair comparison between them when using the Euclidean distance and similar methods. Two methods are usually well known for rescaling data. These are “normalization” and “standardization.”

Standardization can be performed in three ways:

A row or column effect can be removed with a transformation that sets row or column means (totals) to equal values. Centering or standardization of row or column variables results in a partial removal of the main effects, by setting one set of means to zero. Double centering and double standardization remove both sets of means. Removal of the “magnitude” or “popularity” dimension allows the researcher to examine the data for patterns of “interactive” structure [

Centering or standardization within rows removes differences in row means but allows differences in column mean to remain. Thus the “consensus” pattern among rows, characterized by differences in the column means, remains relatively unaffected. Column-standardization also removes only one set of means; column means are set to zero. Centering of data, usually performed on column or row variables, is analogous to analyzing a covariance matrix. Data are sometimes double-centered to remove the magnitude or popularity dimension (Green, 1973).

These are approaches that can be used in sorting and clustering alternatives. According to the structure of the decision problem, it is necessary to determine which standardization method should be applied. That is,

In the context described above and in Section

As mentioned before, the methods that can be used to determine the differences between vectors are distance, correlation, and similarity. In (

Below column-standardization and row-standardization cases of this proposed measure were discussed, respectively. In the literature, the effects of row-standardization on the TOPSIS method have not been addressed at all. For this reason, the effects of row-standardization on the TOPSIS method in this study were also investigated.

Thus, the degree of distance of the alternatives between each alternative and the PIS and the NIS for column-standardized data are as follows:^{+} and D^{-} values and, for correlation similarity and cosine similarity, the S^{+} and S^{-} values are, respectively, as follows.

Let^{+},^{-} and^{+},^{-} values for square Euclidean distance, correlation similarity, cosine similarity, correlation distance, and cosine distance are as follows. Differently here the reduced states of the correlation and cosine formulas are used (for column-standardization was explained at ^{+} and^{-} formulas are equal to the formulas of cosine^{+} and^{-}, respectively.

^{+} and^{-} values of the square Euclidean distance for row-standardized data are equal to formulas of (^{+} and^{-} values of the square Euclidean distance for column-standardized data).

For both cases the PIS and NIS vectors must be standardized on a row basis.

In this study three different performance indexes were proposed. The significance of these three new different_{i} values were tried to emphasize by taking into account “the_{i} value suggested by the traditional TOPSIS method” (_{i} value proposed by Ren et al. [_{i} values using the Euclidean distance. For this reason, this value will be called the

^{+}^{-} plane is established with^{+} at the^{−} at the

Explanation of the Euclidean distance.

_{i} value detailed above.

In this study alternative to the Euclidean_{i} value (_{i} value (_{q} distance family, just like the_{q} family. The Euclidean, Manhattan, and Chebyshev distances are detailed in Section

Manhattan distance.

Chebyshev distance.

According to the standardization method used (Structure of the Performance Matrix) and whether the criteria are weighted or not the distance and similarity values mentioned above can be calculated by means of which formulas (Proposed Method Formulas) are shown in Table

Implementation of the proposed method.

Cases | Structure of the Performance Matrix | Proposed Method Formulas | |||||||
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Column- | Row- | Weighting | Sq_Euc | Corr_s | Cos_s | Cos_d | Cos_d | Proposed | |

Case 1 | Used | Unused | Unused | Equations ( | Equations ( | Equations ( | Equations ( | Equations ( | Equations ( |

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Case 2 | Used | Unused | Used | Equations | Equations | Equations | Equations | Equations | Equations ( |

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Case 7 | Used | | Unused | Equations | Equations | Equations | Equations | Equations | Equations |

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Case 8 | Used | | Used | Equations | Equations | Equations | Equations | Equations | Equations |

Explanations about the cases in Table

Case 1, Case 2, Case 3, and Case 4 are the cases where only one of the column or row-standardization methods are applied, while the remaining last four cases, Case 5, Case 6, Case 7, and Case 8, showed the conditions under which “double standardization” are applied.

In Case 1, Case 3, Case 5, and Case 7 the “standardized matrix” will be also “performance matrix” because any weighting method is not used.

In Case 3, Case 4, Case 5, and Case 6 after the row-standardization the “PIS and NIS values” should be also standardized according to “row-standardization.” This correction is needed since the PIS and NIS are considered as a single value in the column-standardization and as a vector in row-standardization.

The application types that can be used for the cases detailed in Table

In this section all of the proposed methods were applied on the HDI data. Therefore, in the following subsections, information about HDI will be given first, and then implementation steps of the proposed method will show on criteria and alternatives that constitute HDI. Finally, the results obtained will be compared with the HDI value.

Human development, or the human development approach, is about expanding the richness of human life, rather than simply the richness of the economy in which human beings live [

It has been pointed out that the opportunities and choices of people have a decisive role in the measurement. These are to live long and healthy, to be educated and to have access to resources for a decent standard of living [

The first Human Development Report introduced the HDI as a measure of achievement in the basic dimensions of human development across countries. This somewhat crude measure of human development remains a simple unweighted average of a nation’s longevity, education, and income and is widely accepted in development discourse. Before 2010 these indicators are used to measure HDI. Over the years, some modifications and refinements have been made to the index. In HDI 20th anniversary edition in 2010, the indicators calculating the index were changed. Figure

The dimensions and indicators of HDI.

As can be seen from Figure

In order to calculate the HDI first these three core dimensions are put on a common (0, 1) scale. For this purpose the following equation is used:

In this section the degree of similarity of the alternatives between each alternative and the PIS and the NIS is calculated with proposed ^{-} value at (_{i} values at (

From this point of view, in order to operate the DSC TOPSIS method proposed in this study, the development level of the preferred countries and the problem of deciding to create a new HDI were discussed through these three dimensions. Since 188 countries were considered in the HDR report published in 2016 (prepared with 2015 values), the initial decision matrix created in the implementation part of the study was formed from 188 alternatives. Income Index, Life Index, and Education Index are considered as criteria so that the initial decision matrix was formed from 188 alternatives and 3 criteria. For the tables not to take up too much space, the data and the results of the first five countries in alphabetical order from 188 countries are given. In terms of giving an idea before the steps of the DSC TOPSIS method are implemented, the first and last five countries with the highest and lowest Income Index, Life Index, and Education Index values in addition to HDI value are given in Table

The first and last five countries with high HDI,

No | HDI | | | |
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1 | Norway | Hong Kong, China (SAR) | Australia | Liechtenstein |

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2 | Switzerland | Japan | Denmark | Singapore |

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3 | Australia | Italy | New Zealand | Qatar |

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4 | Germany | Singapore | Norway | Kuwait |

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5 | Singapore | Switzerland | Germany | Brunei Darussalam |

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184 | Eritrea | Guinea-Bissau | Guinea | Madagascar |

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185 | Sierra Leone | Mozambique | Ethiopia | Togo |

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186 | South Sudan | Nigeria | Sudan | Mozambique |

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187 | Mozambique | Angola | Mali | Malawi |

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188 | Guinea | Chad | Djibouti | Guinea |

In this matrix 188 countries, which have been investigated in the 2016 HDR, are considered to be alternatives and the three HD dimension formed the criteria. The decision matrix is shown at Table

The original data for the first five countries: Decision matrix.

Country | | | |
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Afghanistan | 0.626 | 0.398 | 0.442 |

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Albania | 0.892 | 0.715 | 0.699 |

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Algeria | 0.847 | 0.658 | 0.741 |

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Andorra | 0.946 | 0.718 | 0.933 |

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Angola | 0.503 | 0.482 | 0.626 |

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| 0.790 | 0.639 | 0.687 |

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| 0.127 | 0.174 | 0.180 |

Since the purpose is to explain the operation of the model rather than the weighting, the criteria are handled without any weighting (

“Equal weighting” is a method of giving equal importance to the criteria being considered. Since three criteria are considered in the study, the weight of each criterion is determined as “

Before ranking the countries, to put data on a common scale standardization method are used column-standardization and row-standardization. It was performed with (

In the column-standardization part of Table

Standardized data (or performance matrix of unweighting data) for the first five countries.

Country | Column-Standardization | Row-Standardization | ||||||||
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| | | Mean | St. Dev. | | | | Mean | St. Dev. | |

Afghanistan | -1.289 | -1.386 | -1.363 | | | 1.391 | -0.918 | -0.473 | | |

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Albania | 0.801 | 0.439 | 0.069 | | | 1.410 | -0.614 | -0.797 | | |

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Algeria | 0.448 | 0.111 | 0.303 | | | 1.271 | -1.172 | -0.099 | | |

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Andorra | 1.226 | 0.456 | 1.372 | | | 0.768 | -1.412 | 0.644 | | |

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Angola | -2.255 | -0.902 | -0.338 | | | -0.535 | -0.866 | 1.401 | | |

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The performance matrix is obtained by multiplying the weight vector by the matrix of the standardized values (Table

It should be reminded again at this point that if the criterion weighting is not performed, then the matrix obtained as a result of column or row-standardization (standardized data, Table

Performance matrix for the first five countries.

Country | Column-Standardization | Row-Standardization | ||||||||
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| | | Mean | St. Dev. | | | | Mean | St. Dev. | |

Afghanistan | -0.430 | -0.462 | -0.454 | -0.449 | 0.014 | 0.464 | -0.306 | -0.158 | | |

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Albania | 0.267 | 0.146 | 0.023 | 0.145 | 0.100 | 0.470 | -0.205 | -0.266 | | |

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Algeria | 0.149 | 0.037 | 0.101 | 0.096 | 0.046 | 0.424 | -0.391 | -0.033 | | |

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Andorra | 0.409 | 0.152 | 0.457 | 0.339 | 0.134 | 0.256 | -0.471 | 0.215 | | |

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Angola | -0.752 | -0.301 | -0.113 | -0.388 | 0.268 | -0.178 | -0.289 | 0.467 | | |

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The PIS and the NIS are attainable from all the alternatives (188 countries) across all three criteria according to (

PIS (I^{+}) and NIS (I^{-}) value.

Weighting | PIS | Column-Standardization | Row-Standardization | ||||||||
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| | | Mean | St. Dev. | | | | Mean | St. Dev. | ||

| ^{ + } | 1.548 | 1.728 | 1.746 | | | 1.414 | 1.358 | 1.401 | | |

^{ - } | -2.711 | -2.491 | -2.338 | | | -1.404 | -1.414 | -1.414 | | | |

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| ^{ + } | 0.516 | 0.576 | 0.582 | | | 0.471 | 0.453 | 0.467 | | |

^{ - } | -0.904 | -0.830 | -0.779 | | | -0.468 | -0.471 | -0.471 | | |

After the column and row-standardizations, in this step first the degree of similarities and distances of the alternatives between each alternative and the PIS and the NIS are calculated. And then from these values the degree of

PIS (I^{+}) and NIS (I^{-}) value for Case 4.

Row-Standardization | ||||
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| | | Mean | St. Dev. |

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0.958 | -1.380 | 0.422 | | |

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1.414 | -0.705 | -0.710 | | |

The overall proposed performance index for each alternative across all three criteria is calculated based on (_{i} values (

Table

The application results of Case 2 were given in Table

Table

The application results of Case 4 were given in Table

The overall performance index _{i} values for the first five countries-Case 1.

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Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | |

Afghanistan | 0.479 | 169 | 0.281 | 172 | 0.061 | 182 | 7.035 | 185 | 9.938 | 185 | 5.197 | 172 | 7.136 | 186 |

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Albania | 0.764 | 75 | 0.697 | 70 | 0.787 | 78 | 1.520 | 32 | 2.119 | 40 | 1.241 | 27 | 1.587 | 32 |

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Algeria | 0.745 | 83 | 0.667 | 83 | 0.793 | 75 | 1.761 | 46 | 2.468 | 53 | 1.402 | 41 | 1.827 | 47 |

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Andorra | 0.858 | 32 | 0.819 | 33 | 0.879 | 52 | 1.496 | 29 | 1.948 | 32 | 1.386 | 40 | 1.593 | 33 |

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Angola | 0.533 | 150 | 0.338 | 155 | 0.166 | 153 | 5.396 | 138 | 6.304 | 128 | 5.303 | 176 | 5.650 | 142 |

The overall performance index _{i} value for the first five countries-Case 2.

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Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | |

Afghanistan | 0.479 | 169 | 0.281 | 172 | 0.061 | 182 | 3.382 | 185 | 4.778 | 185 | 2.499 | 172 | 3.431 | 186 |

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Albania | 0.764 | 75 | 0.697 | 70 | 0.787 | 78 | 0.731 | 32 | 1.019 | 40 | 0.597 | 27 | 0.763 | 32 |

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Algeria | 0.745 | 83 | 0.667 | 83 | 0.793 | 75 | 0.847 | 46 | 1.186 | 53 | 0.674 | 41 | 0.878 | 47 |

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Andorra | 0.858 | 32 | 0.819 | 33 | 0.879 | 52 | 0.719 | 29 | 0.937 | 32 | 0.667 | 40 | 0.766 | 33 |

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Angola | 0.533 | 150 | 0.338 | 155 | 0.166 | 153 | 2.594 | 138 | 3.031 | 128 | 2.549 | 176 | 2.716 | 142 |

The overall performance index _{i} value for the first five countries-Case 3.

| | | | | | | | |||||||
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Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | |

Afghanistan | 0.479 | 169 | 0.006 | 171 | 0.079 | 149 | 3.609 | 119 | 3.954 | 113 | 3.590 | 155 | 3.714 | 117 |

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Albania | 0.764 | 75 | 0.230 | 114 | 0.007 | 170 | 3.682 | 169 | 4.307 | 142 | 3.617 | 169 | 3.856 | 152 |

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Algeria | 0.745 | 83 | 0.086 | 141 | 0.597 | 71 | 3.438 | 75 | 3.561 | 75 | 3.436 | 100 | 3.478 | 75 |

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Andorra | 0.858 | 32 | 0.464 | 94 | 0.970 | 15 | 2.789 | 26 | 2.814 | 26 | 2.788 | 29 | 2.797 | 26 |

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Angola | 0.533 | 150 | 0.875 | 48 | 0.706 | 62 | 1.527 | 1 | 2.158 | 1 | 1.113 | 1 | 1.542 | 1 |

The overall performance index _{i} value for the first five countries-Case 4.

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Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | Value | Rank | |

Afghanistan | 0.479 | 169 | 0.006 | 171 | 0.441 | 149 | 1.203 | 119 | 1.318 | 113 | 1.197 | 155 | 1.238 | 117 |

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Albania | 0.764 | 75 | 0.230 | 114 | 0.390 | 170 | 1.227 | 169 | 1.436 | 142 | 1.206 | 169 | 1.285 | 152 |

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Algeria | 0.745 | 83 | 0.086 | 141 | 0.511 | 71 | 1.146 | 75 | 1.187 | 75 | 1.145 | 100 | 1.159 | 75 |

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Andorra | 0.858 | 32 | 0.464 | 94 | 0.623 | 15 | 0.930 | 26 | 0.938 | 26 | 0.929 | 29 | 0.932 | 26 |

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Angola | 0.533 | 150 | 0.875 | 48 | 0.622 | 62 | 0.509 | 1 | 0.719 | 1 | 0.371 | 1 | 0.514 | 1 |

The alternatives could be ranked in the descending order of the proposed_{i} indexes values. In Tables _{i} values are different for Case 1 and Case 2, the results are given together because the ranks are the same.

If Table

Figure

According to Table

In Figures

Top and bottom five countries with high and low level of development-Case 1 and Case 2.

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1 | Norway | Norway | Norway | France | Japan | Korea (Republic of) | France |

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2 | Australia | Switzerland | Finland | Korea (Republic of) | France | France | Korea (Republic of) |

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3 | Switzerland | Australia | Netherlands | Japan | Korea (Republic of) | Israel | Japan |

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4 | Germany | Netherlands | Switzerland | Israel | Israel | Japan | Israel |

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5 | Denmark | Germany | United States | Spain | Spain | Spain | Spain |

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184 | Burundi | Guinea-Bissau | Central African Republic | Burkina Faso | Burkina Faso | Chad | Burkina Faso |

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185 | Burkina Faso | Mozambique | Guyana | Afghanistan | Afghanistan | Benin | Gambia |

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186 | Chad | Sierra Leone | South Sudan | Gambia | Gambia | Guyana | Afghanistan |

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187 | Niger | Chad | Chad | Niger | Niger | Guinea-Bissau | Niger |

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188 | Central African Republic | Central African Republic | Congo | Guinea | Guinea | Congo | Guinea |

Top and bottom ten countries with high and low level of development-Case3 and Case 4.

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1 | Norway | Azerbaijan | Austria | Angola | Angola | Angola | Angola |

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2 | Australia | France | Hong Kong, | United States | United States | United States | United States |

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3 | Switzerland | Lesotho | Azerbaijan | Nigeria | Nigeria | Nigeria | Nigeria |

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4 | Germany | Finland | Libya | Botswana | Botswana | Botswana | Botswana |

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5 | Denmark | Spain | Chad | Equatorial | Equatorial Guinea | Equatorial Guinea | Equatorial Guinea |

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184 | Burundi | Niger | Israel | Malawi | Palau | Argentina | Slovenia |

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185 | Burkina Faso | El Salvador | Argentina | Bolivia | Latvia | Nepal | Armenia |

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186 | Chad | Cabo Verde | Nepal | Palestine, State of | Estonia | Ghana | Liberia |

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187 | Niger | Dominica | Ghana | Denmark | Fiji | Haiti | Burundi |

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188 | Central African Republic | Ethiopia | Haiti | Uganda | Ukraine | Lithuania | Hungary |

Comparison of the development levels of countries for all indices, Case 1 and Case 2.

Comparison of the development levels of countries for all indices, Case 3 and Case 4.

Findings obtained from the study: in DSC TOPSIS method, standardization, weighting, and performance indexes were found to be important in order to rank the alternatives. It is possible to interpret the results obtained in the application part of the study as follows.

In order to sort and compare the alternatives, HDI values, the results of the traditional TOPSIS method and the results of the proposed DSC TOPSIS method were given together and a general comparison was made. With the DSC TOPSIS method, a new measure that is more sensitive and stronger than the measure used in the traditional TOPSIS method was proposed. The reason why the proposed unit of measurement is expressed as more sensitive and stronger is the use of similarity and correlation distances in addition to the Euclidean distance used in the traditional TOPSIS method in order to rank the alternatives. In other words, DSC TOPSIS uses three units that can be used to compare alternatives: distance, similarity, and correlation. From the results of the application it was observed that HDI, TOPSIS, and DSC TOPSIS results are not exactly the same and show some differences. This is the expected result because the measurement units used in each method are different. At this point, which method is preferred depends on the decision-maker's opinion.

In the DSC TOPSIS method, although the performance indexes for Case 1 and Case 2, which have the objective to bring the alternatives to the same unit for each criterion, have different values, the ranking values are the same. This is expected because the equal weighting method is actually equivalent to the situation where there is no weighting. Of course, if different weights are used for the criteria in Case 2, it is highly probable that both different performance index values and different rankings can be obtained.

Case 3 and Case 4, which bring the criteria to the same unit for each alternative, refer to situations that are performed without weighting and equal weighting, respectively. The situation described above is also valid here and the results of the indexes are the same.

Which standardization method is preferred depends on the decision-maker's opinion and the nature of the problem being addressed. The decision problem in this study was the calculation of HDI and the proposed solution method was DSC TOPSIS. When the structure of the decision problem is examined, it can be concluded that it is more reasonable to prefer Case 1 and Case 2 where column-standardization is used. Considering the advantage of the fact that different weighting methods can be tried as well as equal weighting in Case 2, it is obvious that the most reasonable solution would be Case 2.

Which performance index is preferred to rank the alternatives depends on the decision-maker. At this point, it is expected that^{5}, which is the average of all performance indexes proposed in the study, is preferred.

There are lots of efficient MCDM methods which are suitable for the purpose of ranking alternatives across a set of criteria. None of MCDM methods in the literature gives a complete superiority over the other. Each of them can give different results according to their main purposes. For this reason, the method to be applied should be decided according to the structure of the decision problem and the situation to be investigated. In some cases, the results obtained by trying different MCDM methods are compared and the most reasonable solution is selected.

Undoubtedly, the oldest and valid method used to rank the alternatives is distance. The concept of distance forms the basis of the TOPSIS method (it was called “traditional TOPSIS” in the study). In TOPSIS (the steps of implementation are given in detail in Section

As previously stated, “normalizing the decision matrix (Step 3),” “calculating the distance of positive and negative-ideal solutions (Step 6),” and “calculating the overall performance index for each alternative across all criteria (Step 7)” of the traditional TOPSIS method steps can be examined, improved, or modified. The aim of the study is to draw attention to the above-mentioned weaknesses of the traditional TOPSIS method and to propose new solutions and approaches. For this purpose

The DSC TOPSIS method is particularly remarkable in its

The formulation of the DSC TOPSIS method is detailed for the eight different cases defined (Table ^{5} performance index could be preferred when using DSC TOPSIS method in the calculation of HDI.

In conclusion, the effect of the proposed DSC TOPSIS method on the following concepts is discussed in all aspects:

different standardization methods,

different weighting methods,

different performance indexes.

The proposed DSC TOPSIS method does not have any limitations, and the method can be easily applied to all decision problems where traditional TOPSIS can be applied.

In the study, the standardization of the decision matrix in different ways is also emphasized. According to the preferred standardization type, it is emphasized that the decision matrix will have different meanings and different results will be obtained. The only issue to be considered in the application of the method is to decide the standardization method to be applied. The following topics can be given as examples for future applications.

establishing different versions of the new measurement proposed by the DSC TOPSIS method: e.g., combining the concepts of distance, similarity, and correlation with a different technique instead of a geometric mean or experimenting with different combinations of these concepts together (distance and similarity, distance and correlation, similarity and correlation, etc.),

suggesting different ranking index (_{i}) formulas to sort alternatives,

experiment of different weighting techniques in DSC TOPSIS method,

blurring the DSC TOPSIS method: fuzzy DSC TOPSIS.

Data are obtained from “http://hdr.undp.org/en/humandev, United Nations Development Programme Human Development Reports” website.

The author declares that they have no conflicts of interest.