1. Introduction
Human Development Index (HDI) is a social-economic indicator that shows the status of the countries in terms of progress and human development [
1]. This index was proposed by Mahub ul Haq [
1], and functions as a social-economic indicator that allows us to observe the status of countries in terms of progress and human development. However, before this index, economist Amartya Sen X presented a more inclusive method that defines it as development as freedom. This index takes an approach related to the inequality of capabilities, which prioritizes people as the real wealth of a nation and their capabilities for assessing its development [
2]. Although the two proposals may seem distant, they have a common element of aggregating information and obtaining averages that allow for ranking of countries. As the HDI is accepted by the international community, it emphasizes that people, their capabilities and expending human choices should be the ultimate criteria for assessing the development of a country, not only economic progress (UNIDP, 2020)
HDI is composed of three key dimensions; human development, long and healthy life, knowledge, and a decent standard of living, which consider several indicators and are measured between maximum and minimum values, and it is calculated through the geometric mean of normalized indices for each of the three key dimensions of human development. This index is apparently reliable but still needs to be improved, as inequalities and development are different from one nation to another. Thus, the index is susceptible to improvement as there are components that change over time, where dominant positions are reinforced, and new inequalities are emerging. This becomes a challenge to generate new ways of assessing the changes, inequality and inequity faced in the new context.
One of the problems that can be seen in the formulation of the HDI is in the way the index considers each key dimension. The results of each country are based on a geometric average where the relative importance of each component is the same (). Due to that, the analysis that has been conducted with the data is limited. Based on that, there is a possibility of generating different analyses using the same information but considering a different relative importance for each of the components, based on the realities experienced by each of the countries or the expectations of different decision makers or experts. The purpose is to be able to visualize how the ranking behaves using different data analysis, allowing us, in turn, to generate new positions in the countries and thus be able to confirm whether they have a high or low human development.
Based on the above, with the development of information science and mathematics, new ways of making closer approximations to reality have emerged. Of these advances, aggregation operators stand out, which allow information from different sources and types to be aggregated to obtain a single significant value [
3]. Of all the methodologies proposed, the Ordered Weighted Average Operator (OWA) stands out [
4]. This operator has the characteristic of taking into account the attitude of the decision maker (subjective information) [
5]. Among its developments are the following operators: OWA Induced [
6], OWA Heavy [
7], OWA Prioritized [
8], OWA Probabilistic [
9], OWA Bonferroni [
10], OWA Logarithmic [
11], OWA Pythagorean [
12], etc. Applications have also been developed in entrepreneurship [
13], finance [
14,
15,
16,
17], management [
18,
19,
20], as well as proposals for indices such as transparency [
21,
22]. In this sense, using these new methodologies can improve the index by being able to include more variables to generate a better assessment.
The main aim is to propose a new method for estimating the Human Development Index using ordered weighted average. For this purpose, we study the characteristics and properties of the following aggregation operators: ordered weighted geometric average (OWGA), and induced OWGA prioritized OWA (POWA) operator. In combination of the studied operators and the formula for determining the Human Development Index, a new method is proposed, called the prioritized induced ordered weighted geometric average (PIOWGA) operator. Using this new method, a mathematical application is made to estimate the Human Development Index and compare it with the traditional method and other existing methods. Finally, it is noted that decision makers have an influence on the order given in the ranking by its attitude and criterion, since they prioritize the three elements according to different resources, different developments and more or less stable political and economic systems throughout the subjective information prioritized by them.
This work is structured as follows:
Section 2 presents in a concrete way the theoretical concepts related to the Human Development Index.
Section 3 reviews the aggregation operators that will be useful to construct the proposed method.
Section 4 presents the mathematical application of the HDI. Finally, the conclusions are presented in
Section 5.
4. The Human Development Index by Using PIOWGA Operator
Having presented the arguments on the Human Development Index and using the aggregation operators for the development of the new methodological proposal for its calculation called prioritized induced ordered weighted geometric average, we proceed to apply it to the calculation of the HDI. To carry out the application of the new HDI estimation method, a 4-step process is established, which is explained below:
Step 1. Initially taken into account is the information of
and
, which is obtained by UNDP [
46], then the experts must be selected. In this step, the experts will be determined based on their experience in the topic. It is recommended that they have worked in different governmental organizations that work in the different fields that the HDI considers, that is, health, education, and income, and that also have experience in public policy making. These two aspects are very important because in that way they will know at first-hand how the different elements of how HDI works and how public policies are enacted to improve in that field.
Step 2. In this step, the weighting vector, induced and prioritized values are established. For the weighting vector, for expert 1 , expert 2 and expert 3 . In the case of the induced values, for expert 1 , expert 2 and expert 3 . Finally, the prioritized importance of each experts is . These last values are determined based on the expertise and research time in the field.
Step 3. The calculations are done for each country following the same process explained in
Section 3.3. All the results for the OWGA, IOWGA, POWGA and PIOWGA operator are found in
Appendix A.
Step 4. All results are obtained. To present the results, three different tables were produced, organized into the Top 10 countries (evaluation from position 1 to 10, see
Table 3), Middle 10 countries (evaluation from position 91 to 100, see
Table 4) and Worst 10 (evaluation from position 180 to 189, see
Table 5). As can be seen, with the analysis made from Top 10, Middle 10, and Worst 10 is possible to visualize how much the ranking can change depending in the weights assigned to each dimension of the HDI calculation.
The first analysis is performed based on Top 10 countries. With this information, it is possible to detect that Norway is the country with the highest score even when different aggregation operators were used. After that, Switzerland and Singapore are usually between number 2 and 3 in many of the operators. An interesting case can be Germany because in the traditional formulation, its position is number 5 and within the it is number 2. Additionally, Liechtenstein is not considered in the traditional Top 10 but can be in the ranking according to the , , or operators. Hence, it is possible to visualize that even when the weights of the dimension are changed, Norway is always number 1, but after that, some important changes in the ranking can be seen.
In the Middle 10 countries, it is possible to find some notorious changes, for example, Fiji is presented in the traditional and IPOWGA operator only. Botswana presents important changes, being in position 95 in the traditional score and number 91 in the POWGA operator. Another example can be Jamaica, which is just presented in 4 of the 9 operators that are being compared. The ranking with more changes is Middle 10. This is because the ranking in that section is sensitive to changes, and because of that, if a dimension has more weight than another one, the results can change drastically. Additionally, we observe the ultimate idea that the aggregation operators, such as the IPOWGA operator, can help to generate new scenarios.
Finally, the Worst 10 countries is the ranking that present fewer changes. In the case of position 188–189, there is always Niger or Central Africa; Chad goes from rank 187–186, South Sudan from 186–185 and in some cases, for example, Congo can be in or out of the ranking. The Worst 10 presented the least changes in comparison to another two rankings because even when the weights to each dimension are changed, they are usually the lowest score in all of them. In this sense, it is a must that the politics that are assigned to these countries and the help that they receive from other ones improve, in a general way, the three dimensions.
The following figures show a graphical comparison of the results obtained from the traditional ranking (T) and the rankings generated by the proposed POWGA and IPOWGA operators (see
Figure 2,
Figure 3 and
Figure 4). Initially, the radials for all methods show a spiral shape and their desiccant sequence is clockwise. Secondly, it is observed that the results for the proposed operators have two main changes; the order of the ranking where they change position and enter and leave countries according to their results; the distance of the results as their distances are wider and the spirals are steeper. The results for the POWGA and IPOWGA operators show that the countries within the ranking vary according to the traditional one. The shape of the graph is easier to see these changes, as it differs at different points, showing lower results in a more concave manner and higher results in a more convex manner, while the traditional graph has straighter aspects. Likewise, the lowest scores are even more pronounced in the 10 worst-ranked countries. These changes are due to the amount of information that is aggregated. Here, in the proposed methods, the shape of the radial marks a more pronounced spiral, which indicates that the distances between each of the countries’ results. Hence, the influence that the different criteria of those evaluated can have on the levels of development of the countries is evident.
These results lead us to address aspects related to the notion of average. The mean is most commonly understood as the average value of a set of numerical data, which is calculated by dividing the sum of the set of values by the amount of data. For the topic under discussion, the geometric mean is used, which is the calculation of the
n-th root of a joint product, and its characteristic is to have positive numbers. For our proposal, we take into account the ordered weighted average (OWA) as an operator that provides a parameterized family of aggregation operators between maximum and minimum values, allowing us to over or underestimate the information according to the attitude of the decision maker in the studied problem [
4,
5,
47]. This operator has the versatility to allow aggregation of those degrees of truth or optimism in a formal method, i.e., it has the ability to model in a linguistic way through instructions that aggregate subjective information to obtain a single representative value [
4].
In this sense, averaging and geometric averaging allow you to efficiently measure the data you aggregate in your operation, although it is quite complicated to give meaning to a broad mass valuation to a mere numerical value. With the OWA operator being able to follow subjective instructions for the aggregation of arguments, it has a greater versatility in responding to situations in the environment that link individual’s reasoning and situations that deserve to be explained in terms of their meaning [
48]. Likewise, it allows us to propose wider extensions that help to take into account data that is not possible with traditional methods. Comparing the three methods, it is evident that the OWA operator has more advantages by being able to add in its formulation soft aspects (approximate reasoning and degrees of truth) that have an influence on the final result. Thus, the two proposed methods contain the characteristics of the mean, geometric mean and OWA operator and additionally, the characteristics of the induced and prioritized variables. The latter two aspects are closely related to the judgement and subjectivity of the individuals. Finally, the figures presented show those nuances, some more prominent than others, but which can help to give meaning to the information by the origin of the data, i.e., one can always turn to the expert or individual who provided the initial judgement (soft information).
Based on the above, the main idea of the HDI is to generate a vision to each country and that in results can improve and generate new politics, but not all the countries have the same amount of resources to make changes in all three dimensions at the same time and, because of that, a ranking based on dimensions that have different weights will make them visualize if the politics that are being implemented are changing their position in the ranking and, also, not all the countries have the same problems and, because of that, the prioritization of one dimension above another is something that has to be taken into account.
It is important to note that this type of operator has some limitations. First, the weighting vector is proposed by the experts or decision makers and, because it is an essential element for the calculation, if different weights are proposed, the results can vary. Additionally, this flexibility has a positive effect because the results can change depending on the attitude and expectations of the decision maker. Second, these operators are useful when the relative importance of each of the variables that compose the HDI is not the same, if the expert considers that each variable should weight there is no reason to use the different aggregation operators. Finally, with the use of these operators, different results can be obtained and, because of that, it is not possible to assume that only one result is correct, because each result is based on the weighting vector proposed, and if there is not a consensus, it is possible to obtain as many different results as there are decision makers, making it difficult to achieve only one result; however, the idea of this type of operator is that they produce different scenarios based on the aptitude, expectations and knowledge of the decision makers that can be considered in the decision making process, and, with that, there is a better understating and general vision of the phenomenon to study.
Another limitation of the paper is that the study is based on the OWA operator and its extensions, but there are many other operators whose formulation is interesting and can be useful in this type of analysis of information. Such is the case of the Hami means that are used to aggregate values, simultaneously including mutual correlation among multiple arguments [
49,
50], Bonferroni means that are an averaging aggregation function that allows capturing the interrelationship between arguments [
51,
52], the use of different Dombi operators [
53,
54,
55], among others.
5. Conclusions
This paper has reviewed the main aspects of the Human Development Index and its components. By revising the general formula for determining the HDI, a new method is proposed using the aggregation operators [
3]. The proposed method is called the prioritized induced ordered weighted geometric average (PIOWGA) operator. The main idea of this operator is that the reordering step will be performed based on induced variables determined by the decision maker, and allows for considering different values and to everyone to be included in the results. Each one of them are given a weight based on their experience and importance in the decision [
44,
45]. Thus, by having induced variables and decision-prioritizing aspects in a group of experts, it is possible to consider both the information derived from the data and the opinions of the decision makers in group decisions.
The mathematical application developed follows a 4-stage process that considers objective input information on education, health and standard of living and subjective information that allows different scenarios to be considered to observe the usefulness and applicability of the methodological proposal. In this sense, three different levels, Top 10, Middle 10 and Worst 10 are shown. For the Top 10, comparing the traditional method with IPOWGA, there are significant variations in general, and the ten countries only move a little in the ranking from position 5 to position 10. For Middle 10, changes are evident, such as Fiji being present in the traditional and IPOWGA operator only. Botswana presents important changes, being in position 95 in the traditional score and position 91 in the POWGA operator. Ranking in this section is sensitive to changes, because it is a dimension that has more weight than another one; therefore, the results can change drastically. For Worst 10, the rankings show no major alterations, only a change of positions in the country of Niger, Central Africa, Chad, and South Sudan. This smaller variation is because when the weight to each dimension is changed, they are usually the lowest score in all of them. In addition, looking at the results graphically, it can be seen that the proposed new method leads to changes in the rankings, the order of countries and the exit and entry of countries in the three boundaries presented. These changes are due to the amount of information that is aggregated. Hence, the influence that the different criteria of those evaluated can have on the levels of development of the countries is evident. Thus, with these results, it can be assumed that the valuation given by the different decision makers has an influence on the order given in the ranking, either by extreme positions of pessimism or optimism. This can occur because of how decision makers prioritize the three elements that make up the HDI, given that each country has different resources, different developments, and more or less stable political and economic systems. Hence, the proposed method brings us closer to the reality that each country has in terms of human development because of the importance given to the subjective information prioritized by decision makers.
Additionally, it is important to visualize the different applications that the IPOGWA operator can has, because any problem that is analyzed by the geometric average (GA) and where the relative importance of each variable that compose the GA is considered by the decision maker that should not be the same these operators can be done. For example, The OGWA operator can be used when only one decision maker is considered, and the weights are ordered based on maximum or minimums; the IOGWA operator is useful when the ordering between weights and attributes want to be induced by specific values and the POWGA operator can be used when more than one decision maker is considered and their importance in the results is not the same. Additionally, the IPOWGA operator is used when the complexity of the problem requires it, that is, in a group decision making problem where the problem is analyzed by a geometric average where the variables that compose it do not have the same relative importance and the reordering of the weights is based on induced values. Thus, the two proposed methods contain the characteristics of the mean, geometric mean and OWA operator and additionally, the characteristics of the induced and prioritized variables. The latter two aspects are closely related to the judgement and subjectivity of the individuals. Finally, the figures presented show those nuances, some more prominent than others, but which can help to give meaning to the information by the origin of the data, i.e., one can always turn to the expert or individual who provided the initial judgement (soft information).
Finally, the research provides a different vision of how to approach the evaluation of the different indices that measure and provide reliable information for decision making at the governmental level in each country. In addition, it shows the usefulness of aggregation operators to consider different types of information. For future lines of research, new proposals can be made with indices such as the happiness, GINI, or competitiveness indices. Likewise, in the development of methodological proposals, these tools can be improved by complementing them with Bonferroni means [
56], multi-person aggregation [
57], soft multi-set [
58], among others.