Improved entropy weight methods and their comparisons in evaluating the high-quality development of Qinghai, China

Li Chen; Xiangyu Wang; Yuanhui Wang; Peichao Gao

doi:10.1515/geo-2022-0570

Article Open Access

Improved entropy weight methods and their comparisons in evaluating the high-quality development of Qinghai, China

Li Chen , Xiangyu Wang , Yuanhui Wang and Peichao Gao

Published/Copyright: December 31, 2023

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From the journal Open Geosciences Volume 15 Issue 1

Abstract

“High-quality development” is an important objective for China’s current development projects. For the betterment of the current situation of the underdeveloped regions, high-quality development is necessary. The comprehensive index method is one of the most widely used methods for evaluating high-quality development works, and indicator weighting is its key and at the same time the most controversial step. Among many weighting methods, the entropy weighting method is widely applied due to its easier understandability and use. This article focuses on the problem of weight distortion when the entropy values of indicators are close to 1 and the improvements to this problem. To verify these improvements, this article deduces with mathematical analysis, the underlying assumptions of the problem and evaluates the level of high-quality development in Qinghai Province, which is assumed to be an appropriate example of the underdeveloped regions in western China. Our mathematical deductions demonstrate that these underlying assumptions do not theoretically exist. In addition, both the improved and the original methods have similar performances in practice or, more specifically, have shown an overall upward trend in the level of high-quality development in Qinghai Province. Because the “problem” is extremely stringent and difficult to reproduce, we conclude that further improvements on the entropy weighting method should not be a focus of future research. To address the subjectivity or objectivity issues of weighting, weightless methods may be of use and should be developed.

Keywords: “high-quality development”; improved entropy weight method; comprehensive evaluation

1 Introduction

During the Sustainable Development Summit by the United Nations, held in 2015, all 193 member states officially adopted the 17 Sustainable Development Goals [1]. These goals go beyond interests of national territories and take into account the holistic improvements of economy, environment, and people everywhere in the world. This indicates that the inclusive consideration of pursuing multiple objectives has become a common goal for countries across the world. In October 2017, the 19th National Congress of the Communist Party of China put forward the concept of shifting China’s economy from high-speed development to high-quality development focused on all-round improvement for the longer term. This signifies that China’s development path is undergoing transformation, aiming for balanced development objectives in both social economy and ecological environments [2], to address the pressing issues of unbalanced, uncoordinated, and unsustainable development occurred in the high-speed growth stage of the Chinese economy [3,4,5].

In recent years, research and policies related to high-quality development have been continuously advancing. Experts, scholars, and policymakers have gradually realized that “high-quality development” is an important concept guiding the stable development of China’s socioeconomic improvements during the period of the 14th Five-Year Plan and beyond. Currently, the primary focus of the research on high-quality development is on exploring the connotation of “high-quality development” [6], designing indicators [7], evaluating its spatiotemporal development status [8], analyzing further development strategies and paths [9], and so on. Among these steps taken, conducting a scientific assessment of China’s high-quality development status is of great significance in guiding regional development paths and naturally has become a subject of great concern among scholars.

In the current research, various methods for quantitatively evaluating the high-quality development status of China exist. Among these methods, the most widely used is the composite index method. This method is about combining numerous geographical indicators associated to the subject of evaluation, and examining them through a series of mathematical operations to obtain one or more dimensionless indices. Based on the changing characteristics of the indices, the operations analyze the high-quality development status of the region. Many composite indices exist that are internationally recognized by scholars and are being constantly updated, such as the sustainable society index [10] and the human development index [11]. The composite index method is divided into three major steps: selecting indicators, weighting indicators, and aggregating indicators. Scholars widely believe that weighting indicators is a crucial and controversial step among the others. This weighting is an intuitive work of measuring the importance of indicators and is done by experts, scholars, or government staff. The higher the indicator weight, the greater is its importance in high-quality development evaluation.

At present, the weighting methods can be classified into two categories [12]: subjective weighting methods and objective weighting methods. The former is based on the subjective judgment of experts’ experience, such as Delphi method and Analytic hierarchy process; the latter calculates weights based on the changing characteristics of indicator values, such as entropy weight method and ranked weight method. Different indicator weighting methods have different drawbacks, and the various weight results obtained through these methods are not interpreted equally and thus, they do not satisfy everyone [13]. To mitigate this problem, many experts also use equal-weighted or imprecise weighting methods in their research [14,15,16,17,18].

Among numerous weighting methods, the entropy weight method is widely used due to its ease of use and objective results. Due to the high use value, this method is frequently applied to a wide range of fields, such as engineering [19,20], environment [21,22,23], medicine [24], transport [25], and socioeconomic development [26,27,28], among others. In specific applications, many experts and scholars combine the entropy weight method with other weighting methods. For example, they integrate the entropy weight method with the analytic hierarchy process [29], the principal component analysis method [13], and the grey correlation model [30] to enhance the accuracy of their evaluations. These combinations do not change the principle of the entropy weight method. However, some scholars believe that the principle of the entropy weight method has inherent problems and have proposed improvements for this method. The purpose of this article is to provide an in-depth analysis of the rationale behind the improved entropy weight method and explore the practical necessity of its improvement. In empirical research, we evaluate the high-quality development in Qinghai Province to compare the application of the entropy weight method and its improvements. Qinghai Province is an important region in western China [31]. The issue of regional imbalances is an important factor in achieving the goal of region-specific high-quality development. The eastern coastal regions of China were rapidly developed where a large population enjoys the benefits of excellent transportation, land reclamation, and other facilities [32,33,34]. This significant difference in development between the western inland regions and the eastern coastal regions is an important point for consideration, and the inland regions are more in need of these high-quality development programs.

In summary, this article has the following purposes: (1) Evaluate the theoretical rationality of the improved entropy weight method by analyzing its improvement principle; (2) Assess the necessity of improving the entropy weight method by applying the evaluation example of high-quality development in Qinghai Province; and (3) Establish an evaluation indicator system for the high-quality socioeconomic development of Qinghai Province and assess the changes in the level of high-quality development in this region.

2 A review of entropy weight method and its improvements

2.1 The principle of entropy weight method

The entropy weight method is an objective weighting evaluation method that evaluates complex systems with multiple indicators. When applied to regional development quality evaluation, it is usually based on years of continuous development quality evaluation indicator data. The weights of each indicator are obtained through entropy calculation, and finally, the composite index is formed by summarizing the indicator values based on their weights. Assuming that there are n indicators, each with m years of continuous data, we can obtain a Matrix X :

X = x 11 … x 1 n ⋮ ⋱ ⋮ x n 1 … x nn

The specific implementation of the entropy weight method includes the following four main steps:

2.1.1 Standardization

To standardize the raw data using the following formula:

Profitability indicators (the larger, the better):

(1) r ij = x ij − min ( x ij ) max ( x ij ) − min ( x ij )

Cost performance indicators (the smaller, the better):

(2) r ij = max ( x ij ) − x ij max ( x ij ) − min ( x ij ) ,

where i represents the indicator ( 1 ≤ i ≤ n ), j represents the year ( 1 ≤ j ≤ m ), x ij represents the numerical value of the ith indicator in the jth year, max ( x ij ) represents the maximum value of the ith indicator in all years, min ( x ij ) represents the minimum value of the ith indicator in all years, and r ij represents the standardized value of the ith indicator in the jth year ( 0 ≤ r ij ≤ 1 ) .

2.1.2 Calculate the value of information entropy

The main purpose of this step is to use the standardized values of indicators to quantify the degree of fluctuation (also known as information content) of each indicator through information entropy (also known as Shannon entropy). The main calculation formula is as follows:

(3) E i = − 1 ln n ∑ j = 1 n p ij ln p ij ,

(4) p ij = r ij ∑ j = 1 n r ij ,

where p ij represents the proportion of an individual indicator value in relation to the sum of indicator values. E i represents the entropy value of the ith indicator. Furthermore, when r ij = 0 and p i ln p i = 0 . It needs to be noted that the information entropy formula used here is different from its common usage.

2.1.3 Calculate indicator weights

Indicator weights are calculated based on the following formula:

(5) W i = 1 − E i ∑ i = 1 n ( 1 − E i ) ,

where W i represents the weight of the ith indicator and E i represents the entropy value of the ith indicator.

2.1.4 Calculate composite index

Based on the aforementioned weights and known indicator values, the composite index is calculated using a weighted sum method:

(6) S j = ∑ i = 1 n W i × r ij ,

where S j represents the composite index value in year j .

Through the aforementioned calculation steps of the entropy weight method, it becomes evident that this method determines the weight of indicators by considering the magnitude of data fluctuations. The greater the fluctuation and statistical dispersion of indicator data, the more sensitive the indicator is to external influences. Consequently, the indicator will more accurately prioritize the development-requiring areas and thus have a greater contribution to regional development. As a result, the weight assigned to the indicator will be higher.

2.2 A review of improved methods

The entropy weight method has been widely applied but was also marked for its lack of accuracy. Some scholars argue that the magnitude of fluctuations in indicator data does not necessarily reflect the actual importance of the indicators [35]. Compared to the entropy weight method, subjective weighting methods or combination weighting methods are recommended [36]. Furthermore, researchers have found distortions in the entropy weight method when the entropy values of indicators approach 1. Specifically, when the entropy values of all indicators are close to 1, even a slight difference in entropy values between two indicators can bring significant differences in weights. However, these significant differences in weights and the minor differences in entropy values do not exist in ordinary situations where the entropy values of the indicators are not uniformly close to 1. In response to this problem, several scholars have proposed three improvement methods, which will be detailed in the following discussion.

For the sake of brevity and without sacrificing generality, let us consider an indicator system consisting of three indicators. In extreme cases, assume that the entropy values of the three indicators are 0.9999, 0.9998, and 0.9997. At this point, it can be seen that all entropy values are close to 1, and the difference between these values is relatively small. According to the entropy weight method, the weights assigned to each indicator would be 0.1667, 0.3333, and 0.5000, respectively. Indeed, it can be observed that when the entropy value of an indicator changes from 0.9999 to 0.9998, the difference in value is only a decrease of 0.01%, but the corresponding weight increases significantly by 99.94%. In ordinary cases, let us assume that the entropy values of the three indicators are calculated as 0.3333, 0.6789, and 0.8579, respectively. According to the entropy weight method, the weights assigned to each indicator would be 0.5901, 0.2842, and 0.1258, respectively. It can be observed that when the entropy value of an indicator decreases by 50.91% from 0.6789 to 0.3333, the corresponding weight only increases by 51.84%.

In response to the aforementioned “problems,” Hui-Cheng et al. [37], Yinghai and Jianzhong [38], and Sen and Yili [39] have respectively proposed an improved weight calculation formula based on entropy (Table 1). As a result, three improved versions of the entropy weight method were introduced, referred to as Improvement 1, 2, and 3 respectively.

Table 1

The core formulas of the original entropy weight method and its improvements

Method	Core formulas	Annotation
Original	W i = 1 ‒ E i ∑ i = 1 n ( 1 ‒ E i )	E i is the entropy value of indicator i; H ˆ is the mean of the entropy values of each indicator with a non-1 entropy
Improvement 1	W i = ∑ k = 1 n E k + 1 ‒ 2 E i ∑ i = 1 n ∑ k = 1 n E k + 1 ‒ 2 E i
Improvement 2	W i = ( 1 ‒ H ˆ ) w 1 + H ˆ w 2 E i ≠ 1 0 E i = 1	w 1 = 1 ‒ E i ∑ i = 1 n ( 1 ‒ E i )
Improvement 3	W i = ( 1 ‒ H ˆ 35.35 ) w 1 + H ˆ 35.35 w 2 E i ≠ 1 0 E i = 1	w 2 = 1 ‒ E i + H ˆ ∑ i = 1 n ( 1 ‒ E i + H ˆ )

Improvement 1 is the earliest version in the series of improvements. It posits that the significant fluctuation in weight occurs primarily when the entropy values are relatively high, particularly when they approach 1. This phenomenon is attributed to the small value of ( 1 − E i ) presented in formula (5), which governs the relationship between entropy and weight. Therefore, the strategy proposed in Improvement 1 is to avoid the occurrence of this direct cause by adding ( ∑ k = 1 n E k − E i ) to the value of ( 1 − E i ) each time it is used. In other words, use ∑ k = 1 n E k + 1 − 2 E i to replace ( 1 − E i ) .

Improvement 2 is based on the concept that Improvement 1 has indeed solved the aforementioned “problem,” but has raised two new issues. Although Improvement 1 has made progress in certain aspects, it has also introduced some negative effects. The goal of Improvement 2 is to find a better method to address these issues and enhance the accuracy and reliability of the entropy weight method. The first new problem (denoted as “New Problem 1”) is that when the entropy value of a single indicator is 1, the corresponding weight is no longer 0, as shown in the “Special Case” in Table 2. The New Problem 1 actually changes the important properties of the original entropy weight method. The New Problem 2 is an improvement to solve the problem by using ∑ k = 1 n E k + 1 − 2 E i to replace ( 1 − E i ) , which also affects the weight of “ordinary cases.” As shown in the “ordinary case” in Table 2, the weights obtained using the original entropy weight method are 0.5901, 0.2842, and 0.1258, respectively. However, the weights changed to 0.4525, 0.3105, and 0.2370 when Improvement 1 is used. To solve the new problems brought by Improvement 1, Improvement 2 has modified the calculation formula of Improvement 1 to a piecewise function, listed “when the entropy value of a single indicator is 1” as a special case for processing, and modified the specific expression of the functional expression in nonspecial cases. The example shows (Table 2) that to an extent, Improvement 2 has solved the new problems brought about by Improvement 1. Specifically, in the “special case” in Table 2, when the entropy value of a single indicator is 1, the weights given to the indicator by the Improvement 2 and original entropy weight methods are both 0; In the “ordinary case” of Table 2, the calculation results of Improvement 2 are closer to the results of the original entropy weight method than Improvement 1 (but they are still different).

Table 2

Entropy calculation weight under different conditions

Cases	Entropy values	Original	Improvement 1	Improvement 2	Improvement 3
Extreme	0.9999, 0.9998, 0.9997	0.1667, 0.3333, 0.5000	0.3333, 0.3333, 0.3334	0.3333, 0.3333, 0.3334	0.3321, 0.3333, 0.3346
Special	1.0000, 0.9998, 0.9997	0.0000, 0.4000, 0.6000	0.3333, 0.3333, 0.3334	0.0000, 0.4999, 0.5001	0.0000, 0.4991, 0.5009
Normal	0.3333, 0.6789, 0.8579	0.5901, 0.2842, 0.1258	0.4525, 0.3105, 0.2370	0.4903, 0.3033, 0.2064	0.5901, 0.2842, 0.1258

Improvement 3 optimized Improvement 2 and perfectly solved the New Problem 2 on the basis of Improvement 2 [39]. Specifically, Improvement 3 retains the segmented function form of Improvement 2, but provides specific values for the exponent through function fitting. The example proves that in ordinary cases, the weight calculated by Improvement 3 is highly consistent with the original entropy weight method.

3 Reflection on the improved entropy weight method: Theoretical rationality

From the previous section, it can be seen that Improvements 1, 2, and 3 are progressing layer by layer. They perfectly solve the problem of distortion in the original entropy weight method in extreme cases where the entropy values of each indicator are close to 1. However, this study suggests that the problem perceived is not an actual problem.

From a mathematical perspective, the “problem” is more thoroughly expressed in the following lines: the original entropy weight method changes the ratio of weight difference to entropy difference in “extreme cases” and “special cases.” In other words, the expectations for Improvements in 1, 2, and 3 are as follows: the ratio of weight difference to entropy difference in the original entropy weight method should remain unchanged in all cases.

The aforementioned expectations are unreasonable, have not been achieved in Improvements 1–3, and are unnecessary. If this expectation proves to be achievable, it would be possible to calculate the weights of different indicators by directly using fixed ratios in all probable scenarios. Consequently, the specific calculation process of the entropy weight method would no longer be required to determine indicator weights and will render the entropy weight method as irrelevant. Meanwhile, the aforementioned expectations have not been achieved in Improvements 1–3, as evidenced by the following:

Let the entropy value of indicator i be E i . Its corresponding weight is W i . Assuming Improvements 1–3 meet the aforementioned expectations, we obtain the following:

(7) k = W 1 − W 2 E 1 − E 2 , and k is a constant .

By substituting the weight calculation formulas for Improvements 1–3 into formula (7), it can be concluded that:

(8) Improvement 1 : μ 1 = − 2 ∑ i = 1 n ∑ k = 1 n E k + 1 − 2 E j .

(9) Improvement 2 : μ 2 = H ˆ − 1 ∑ ( 1 − E j ) − H ˆ ∑ ( 1 − E j + H ˆ ) .

(10) Improvement 3 : μ 3 = H ˆ 35.35 − 1 ∑ ( 1 − E j ) − H ˆ 35.35 ∑ ( 1 − E j + H ˆ ) .

From formulas (8–10), it can be seen that, μ 1 , μ 2 , and μ 3 all vary with the entropy value of the indicator and are not constants. Therefore, the aforementioned expectations have not been achieved in Improvements 1–3.

More importantly, the aforementioned expectations are unrealistic and not essential in real-world examples. Any case that uses the entropy weight method only involves a single case of the distribution of indicator entropy values (whether it is the extreme case, special case, or ordinary case in Table 2). In a single case, meeting the expectation of unchanged ratio of weight difference and entropy difference in any case is not necessary. Meanwhile, it needs to be noted that in a single case, the ratio of weight difference to entropy difference remains unchanged. The proof is as follows:

Substituting the weight calculation formula of the original entropy weight method into formula (7) yields:

(11) μ 4 = − 1 ∑ j ( 1 − E j ) .

As E j does not change, μ 4 is a constant.

4 Comparison of improved entropy weighting methods: Empirical necessity

In this section, we have demonstrated through practical cases that not only the so-called extreme situation is least likely to occur in actual cases but also the calculation results of improved 1 and 2 vary from the original entropy weight method when the “extreme situation” does not occur. Specifically, we selected Qinghai Province as the research sample and constructed an evaluation index system for high-quality economic and social development in the region. Qinghai Province is an area in the western inland region of China, which is a good example of the country’s underdeveloped regions that are in greater need of progress (Figure 1). Despite the extensive policy support given to the western regions by the Chinese government, the per capita GDP in Qinghai Province has consistently been much lower than the national average in China (Figure 2). Although per capita GDP is only one aspect of measuring economic development, it does reflect the imbalanced state of economic development in the country and the significant development potential in western regions represented by Qinghai Province. At the same time, the Qinghai Province also has ecologically fragile areas [40,41], so the socioeconomic development programs need to be balanced with the region’s environmental protection needs.

Figure 1

Location of Qinghai province in China and its elevation.

Figure 2

Per capita GDP comparison (in ten thousand Yuan) between Qinghai Province and China.

4.1 Construction of index system

High-quality development is a transformation and a reflection of China’s economic development reaching anew height. This transformation is no longer solely focuses on high-speed economic growth, but takes into account the environmental and social impacts, for a comprehensive developmental process. It means that the assessment of high-quality development requires the consideration of its multiple dimensions, making it a vast and complex system. When evaluating complex systems, it is necessary to select appropriate indicators based on different dimensions [42]. For example, Chen and Huo evaluated the overall level of China’s high-quality economic development from five dimensions: innovation, coordination, green development, openness, and sharing [43]. Meanwhile, Wang et al. assess the high-quality development of China’s energy system by decomposing it into five subsystems: energy innovation, energy coordination, green energy, energy openness, and energy sharing [44].

To assess the level of high-quality development in Qinghai Province of China, this article has constructed an evaluation index system. On the basis of the widely accepted connotation of high-quality development, we have identified seven of its dimensions, namely, economic quality, innovation, openness, social civilization, ecological civilization, people’s livelihood, and security. Fulfillment of these seven dimensions appears to form an acceptable measure of high-quality development. On the basis of these seven dimensions, we have established 17 specific development goals. Following principles such as “objectivity of evaluation,” “comparability of evaluation objects,” “systematic composition of evaluation indicators,” and “consistency with policies,” we have selected indicators and further screened them according to their availability. Finally, our 17 development goals were presented with 50 indicators to form an evaluation index system for high-quality development in Qinghai Province (shown in Table 3). Data sources used for the evaluation process include the “Statistical Yearbook of Qinghai Province,” “China Social Statistical Yearbook,” “China Environmental Statistical Yearbook,” “China Rural Statistical Yearbook,” etc., covering the period of 1999–2019. The time resolution is annual, and the spatial resolution is at the provincial level.

Table 3

Evaluation index system of high-quality development in Qinghai Province

System	Dimension	Target	Indicator	Attribute
High-quality regional development	Economic quality	Economic level	(1) Labor productivity of the whole society	+
		Economic level	(2) Per capita GDP	+
		Structural optimization	(3) Proportion of nonagricultural industries	+
		Structural optimization	(4) Turnover of high-tech industries as a percentage of GDP	+
		Environmental protection	(5) Energy consumption of 10,000 Yuan of GDP	−
			(6) Water consumption of 10,000 Yuan of GDP (at the current price)	−
			(7) Human-made CO₂ emissions per unit of GDP	−
	Driven by innovation	Investment in innovation	(8) Research and development (R&D) expenditure as a percentage of GDP	+
			(9) Full-time equivalent per 10,000 R&D personnel	+
			(10) The number of participants in science popularization activities per 10,000 people	+
			(11) The amount of foreign technology import contracts per 10,000 people	+
		Innovation output	(12) Number of patents granted per 10,000 people	+
			(13) Technology market turnover as a percentage of GDP	+
			(14) Number of patent applications per 10,000 people	+
	Opening	Exchange of ideas	(15) Number of international tourists	+
		Exchange of ideas	(16) The number of people engaged in labor service cooperation abroad per 10,000 people	+
		Foreign trade	(17) Total import and export of goods per capita	+
			(18) Per capita amount of foreign funds actually used	+
			(19) Per capita total imports and exports of foreign investment enterprises	+
			(20) Turnover of external contracting projects per 10,000 people	+
			(21) International tourism revenue per 10,000 population	+
	Social civilization	Cultural construction	(22) Public library collections per 10,000 people	+
			(23) Number of intangible cultural heritage of China	+
			(24) Mass cultural facilities for every 10,000 people	+
			(25) The number of cultural and artistic activities organized by art centers and cultural centers per 10,000 people	+
		Education	(26) Education expenditure as a percentage of GDP	+
			(27) Student–teacher ratio of ordinary colleges and universities (teachers = 1)	−
			(28) Student–teacher ratio in compulsory education (teacher = 1)	−
	Ecological civilization	Ecological protection	(29) The proportion of nature reserves in the land area	+
			(30) Degree of soil and water conservation	+
			(31) The proportion of investment completed in industrial pollution control to GDP	+
		Environmental quality	(32) Excellent proportion of surface water quality	+
			(33) Surface PM2.5 concentration	−
			(34) Fertilizer application rate (cultivated land)	−
			(35) Vegetation restoration potential index	−
	People’s well-being	Health and hygiene	(36) Health technicians per 10,000 population	+
			(37) Number of healthcare beds per 10,000 population	+
			(38) Health expenditure as a share of GDP	+
			(39) Access to rural sanitation, including latrines	+
		People’s well-being	(40) Per capita green park space	+
			(41) The proportion of urban and rural residents participating in basic old-age insurance	+
		Regional coordination	(42) Urban-rural income ratio (rural = 1)	−
		Regional coordination	(43) Density of transport infrastructure networks (railways and high-grade roads)	+
	Security guarantees	Rule of law security	(44) Number of lawyers per 10,000 people	+
		Rule of law security	(45) Government transparency	+
		Food security	(46) Pesticide use per cultivated land	−
			(47) Food production per capita	+
			(48) Food consumer price index (previous year = 100)	−
		Disaster losses	(49) Direct economic losses from disasters (natural and geological) as a percentage of GDP	−
		Disaster losses	(50) Number of people affected by disasters (natural and geological) as a proportion of total population	−

“+” is profitability indicator; “−” is cost performance indicator.

4.2 Comparison of empirical results

This article uses annual data from various indicators from 1999 to 2019 to calculate the entropy values of each indicator after standardization. The results (see Table 4) indicate that the entropy distribution of the 50 indicators is scattered, with the largest being 0.9834 (the proportion of disaster affected population to the total population) and the smallest being 0.6289 (the contract amount for foreign technology introduction per 10,000 people). The distribution histogram of entropy values for these indicators is shown in Figure 3. According to the graph, no extreme situation where the entropy values of each indicator are close to 1.

Table 4

Entropy of each index

No.	Entropy value	No.	Entropy value	No.	Entropy value	No.	Entropy value	No.	Entropy value
(1)	0.8620	(11)	0.6289	(21)	0.9024	(31)	0.8512	(41)	0.8273
(2)	0.8651	(12)	0.7420	(22)	0.9013	(32)	0.9773	(42)	0.8666
(3)	0.9721	(13)	0.8329	(23)	0.8110	(33)	0.9554	(43)	0.8945
(4)	0.8447	(14)	0.7554	(24)	0.7624	(34)	0.9405	(44)	0.8219
(5)	0.9422	(15)	0.9181	(25)	0.8358	(35)	0.9487	(45)	0.9078
(6)	0.9522	(16)	0.9291	(26)	0.9108	(36)	0.8291	(46)	0.9223
(7)	0.9547	(17)	0.9009	(27)	0.8788	(37)	0.8191	(47)	0.9260
(8)	0.9461	(18)	0.8885	(28)	0.8893	(38)	0.8955	(48)	0.9690
(9)	0.9271	(19)	0.7724	(29)	0.9687	(39)	0.9416	(49)	0.9626
(10)	0.6936	(20)	0.6645	(30)	0.8713	(40)	0.9385	(50)	0.9834

Figure 3

Histogram of entropy.

Even in theory, this extreme situation is difficult to occur when the indicator system includes numerous statistical pointers. Indicators with an entropy value close to 1 for a single indicator often have a sudden increase in data for only 1 year, exceeding far from that of other years. This is improbable to occur in reality, and even if it does occur, it is difficult to see such data values in the entire indicator system; On the contrary, indicators with entropy values not close to 1 often exhibit multiyear volatility, appearing much more frequently in real-world situations.

This article takes the evaluation index system for high-quality development in Qinghai Province as an example, and applies the original entropy weight method, Improvement 1, Improvement 2, and Improvement 3 to the high-quality development status of Qinghai Province from 1999 to 2019. The composite index of high-quality development in Qinghai Province calculated by these four methods in the period of 1999–2019 is shown in Figure 4. The results showed that Improvement 3 is able to help the “problem” of the original entropy weight method better than Improvements 1 and 2, which was almost consistent with the evaluation results of the original entropy weight method. In this example of evaluating the development of Qinghai, the entropy value of the indicator is not entirely close to the extreme case of 1, and a significant difference is found between the evaluation results of Improvements 1 and 2 and the original entropy weight method. The aforementioned theoretical and empirical analysis shows that the problems addressed by Improvements 1, 2, and 3 are difficult to solve in practical situations, and Improvements 1 and 2 bring errors to the entropy weight method, thus becoming new problems.

Figure 4

The high-quality development level in Qinghai Province from 1999 to 2019.

5 Discussion

The entropy weight method and its improvements are discussed in terms of empirical research and its theoretical significance. The results indicate that the “problem” does not exist, and it is difficult to reproduce it in reality due to stringent conditions. In addition, Improvements 1 and 2 introduce errors due to “corrections.” This article provides a comprehensive evaluation of the high-quality development situation in Qinghai Province through empirical research and found that the development has increased in recent years.

We realized the need for establishing an evaluation index system by electing appropriate weight determination methods and index aggregation methods based on a clear understanding of the research questions. The evaluation of regional high-quality development often requires a scientifically established evaluation index system that works well with local conditions. Regional high-quality development programs aligned to sustainable development goals also need to maintain environment-conscious practices, and the Qinghai Province is particularly important in this regards as large areas of the province are ecologically fragile [40,41]. For this reason, while formulating the evaluation index system, we pay special attention to relevant environmental indicators. Studies have shown that vegetation has an important role in ecological protection of an area [45], and so, we have included the “vegetation restoration potential index” in the evaluation index system to better evaluate the environmental protection situation in Qinghai Province.

Regarding weight determination methods, objective weighting methods represented by entropy weight method are essentially data driven. This means that under the basic assumption of accepting objective weighting methods, the value of indicators will determine the weight of the indicators themselves. Moreover, although these methods have different initial assumptions (e.g., entropy weighting and coefficient of variation consider that indicators with greater volatility [46] are more important), they all extract initial data characteristics to construct statistics (e.g., entropy weighting uses Shannon entropy and coefficient of variation uses coefficient of variation) to further calculate weights. Therefore, even for the same indicator system, it is not possible to obtain a uniform and stable weight through objective weighting methods [47]. Objective weighting methods represented by the entropy weight method often lack generalizability, and the obtained weights cannot be applied well to research in general other than studies similar to this. Similarly, studies have shown that weight reversal may occur when the entropy weighting method is applied to the problem of dynamic changes in indicators. Some studies have also pointed out that the weights determined by the entropy weight method cannot be generalized for evaluations in different periods or in regions, and have other limitations as well [48]. In addition to these binding conditions, the principle of using the size of data fluctuations to measure the importance of indicators through the entropy weighting method has also been challenged by some scholars. They believe that if significant differences in the actual importance of indicators are found, the entropy weighting method should not be used to determine weights [49,50]. However, some studies have shown that objective weighting methods outperform subjective weighting methods when applied to certain problems [51,52]. The principle and scope of application for weighting methods need to be reviewed and be noted that the connotation of indicator importance needs to serve research questions, which is crucial for accurately characterizing indicator weights.

6 Conclusion

The entropy weight method, as a widely applied approach for indicator weighting, has been questioned by experts and scholars. Addressing the issue raised by scholars regarding the original method, which states that “tiny differences in entropy values among indicators can result in significant differences in their respective weights,” this article examines all the proposed improvements (Improvement 1, Improvement 2, and Improvement 3) as research cases, discussing them from both theoretical and practical perspectives.

The “problem” raised in the study could not be proved, and therefore, its “correction” is out of scope. From a practical perspective, we have constructed an evaluation indicator system for assessing the level of high-quality development in Qinghai Province through the entropy weight method and its improvements. The results show that all the evaluations using different methods have indicated a growth trend in the high-quality development in Qinghai Province and speak of the Chinese government’s efforts toward regional high-quality development.

Another important mention is that during the practical application process, it is found that the conditions under which the mentioned “problem” occurs are extremely stringent and difficult to reproduce. It is also discovered that Improvement 1 and Improvement 2 introduce errors due to the “correction.” From the aforementioned findings, we have deduced that the mentioned “problem” does not need to be “corrected.” We also believe that when conducting comprehensive evaluations, it is important to select appropriate evaluation indicator systems and weighting methods that are tailored to the specific problem at hand.

Funding information: This research work was funded by National Natural Science Foundation of China (Grant No. 42271418), Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDA23100303), Second Tibetan Plateau Scientific Expedition and Research Program (Grant No. 2019QZKK0608), National Natural Science Foundation of China (Grant No. 42230106), State Key Laboratory of Earth Surface Processes and Resource Ecology (Grants No. 2022-ZD-04), and State Key Laboratory of Earth Surface Processes and Resource Ecology (Grants No. 2023-WT-02). The authors gratefully acknowledge the support of the aforementioned funds.
Conflict of interest: The authors declare no conflict of interest.

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Received: 2023-08-18

Revised: 2023-10-10

Accepted: 2023-12-11

Published Online: 2023-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/geo-2022-0570

Keywords for this article

“high-quality development”; improved entropy weight method; comprehensive evaluation

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