# Mathematical Problems in Engineering Decision-Making Based on Improved Entropy Weighting Method: An Example of Passenger Comfort in a Smart Cockpit of a Car

Authors: Jianjun Yang; Shanshan Xing; Ruizhi Qiu; Yimeng Chen; Chunrong Hua; Dawei Dong
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The entropy method is used in many decisions on indicator weights and has been proven to be effective. The traditional entropy method has two shortcomings in calculating indicator weights. On the one hand, when the developed indicator is considered to have no influence, it will lead to the meaningless calculation formula of the entropy method, and on the other hand, when the entropy method is dealing with data with entropy values close to 1, there will be entropy jumps leading to data distortion. To solve the above shortcomings, two correction coefficients are introduced in this paper, and the improved entropy weight method is applied to the evaluation model of the passenger comfort of the intelligent cockpit of the car, and the effectiveness of the improved entropy weight method is proved.


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## Body

## 1. Introduction

Entropy is a thermodynamic concept, which was first introduced into information theory by Shannon, and is now widely used in engineering, economics, and other fields [1–3]. The entropy value indicates the relative importance of a parameter and is closely related to multiple sample statistics such as mean, standard deviation (SD), and coefficient of variation [4, 5]. If the information entropy of an indicator is smaller, it indicates that the variability of an indicator among evaluation objects is greater, which means that this indicator provides a lot of useful information [6]. Conversely, if the information entropy of an indicator is larger, it indicates that for this indicator, the variability of values among evaluation objects is low, and this indicator plays a less important role in the comprehensive evaluation [7].The current determination of indicator weights is broadly divided into subjective and objective assignment methods [8, 9]. However, subjective assignment methods, such as analytic hierarchy process (AHP) and least squares, end up with weights that are too much influenced by subjective preferences in decision making [10, 11], and compared with various subjective assignment models, the greatest advantage of entropy weight method (EWM) is that it avoids the interference of human factors on indicator weights, reflects and reveals the relevance of the intrinsic characteristics of thing indicators, and improves the scientificity of indicators [12, 13]. At the same time, compared with the pull-out grade method that highlights overall differences, the entropy weighting method highlights only local differences, which means that the greater the difference in levels among the assessment objects, the greater the weight the indicator receives and the greater the impact on the assessment results [14, 15].However, there are two main concerns when calculating using the traditional entropy method: (1) when experts believe that a certain indicator has little or no influence on the decision, it can lead to the meaningless formula of the traditional entropy method; (2) with entropy values close to 1 for each indicator, small differences between entropy values may cause the entropy weight to change exponentially [16, 17].In this paper, the entropy weight method is improved for the above deficiencies, and the improved entropy weight method is applied to the evaluation model of the passenger comfort of the intelligent cockpit of the car, and the weight sizes of different influence indicators are obtained.


## 2. Methods and Materials

### 2.1. Traditional Entropy Weight Method

When the entropy weight is used for evaluation, the score of thej th evaluation index in the ith expert is recorded as xij.In the evaluation using entropy weights, assuming that there are m evaluation indicators and n evaluation objects, the original indicator evaluation matrix X=xijm×n can be formed [18–20].(1)X=x11x12…x21x22…⋰⋰⋱x1nx2n⋰xm1xm2…xmn.The first step is the standardization of evaluation values. The weight of the first evaluation indicator in the first expert’s rating value is denoted aspij, which is calculated as(2)pij=xij∑i=1nxij.The entropy valueHj of the jth indicator in the entropy weights method is defined as(3)Hj=−1lnn∑i=1npijlnpij.Calculate the weightωj of the jth indicator:(4)ωj=1−Hj∑i=1n1−Hj.


### 2.2. Evaluation Index of Passenger Comfort of the Intelligent Cockpit of the Car

With the gradual penetration of electrification, intelligence, and networking, the degree of car intelligence is increases and will be gradually transformed into the third space [21]. As the place where people interact directly with the car, the study of the comfort level of the intelligent cockpit is very important. Some scholars have found that acoustic environment, optical environment, thermal environment, human-computer interaction, and cabin air quality all have an impact on passenger comfort [22–27]. Therefore, in this paper, when using the entropy weight method to study the passenger comfort model of the automotive intelligent cockpit, the acoustic environment, optical environment, thermal environment, human-computer interaction, and cockpit air quality are used as the first-level indicators, and in order to more comprehensively evaluate the comfort of intelligent cabin passengers, a comprehensive comfort evaluation hierarchy model of the intelligent cockpit of the car was established, as shown in Figure 1.Figure 1
Intelligent cockpit passenger comfort hierarchy evaluation system.When dividing the human comfort-taking intervals, this paper draws on the indoor human comfort evaluation model, uses a 10-point scale to score, and divides the car intelligence cockpit passenger comfort into five intervals, as shown in Table1 [28, 29].Table 1
Automotive intelligent cockpit passenger comfort classification standards.
Comfort evaluation levelScoring rangeIntolerable0,2Very uncomfortable2,4Uncomfortable4,6Slightly uncomfortable6,8Comfortable8,10


## 2.1. Traditional Entropy Weight Method

When the entropy weight is used for evaluation, the score of thej th evaluation index in the ith expert is recorded as xij.In the evaluation using entropy weights, assuming that there are m evaluation indicators and n evaluation objects, the original indicator evaluation matrix X=xijm×n can be formed [18–20].(1)X=x11x12…x21x22…⋰⋰⋱x1nx2n⋰xm1xm2…xmn.The first step is the standardization of evaluation values. The weight of the first evaluation indicator in the first expert’s rating value is denoted aspij, which is calculated as(2)pij=xij∑i=1nxij.The entropy valueHj of the jth indicator in the entropy weights method is defined as(3)Hj=−1lnn∑i=1npijlnpij.Calculate the weightωj of the jth indicator:(4)ωj=1−Hj∑i=1n1−Hj.


## 2.2. Evaluation Index of Passenger Comfort of the Intelligent Cockpit of the Car

With the gradual penetration of electrification, intelligence, and networking, the degree of car intelligence is increases and will be gradually transformed into the third space [21]. As the place where people interact directly with the car, the study of the comfort level of the intelligent cockpit is very important. Some scholars have found that acoustic environment, optical environment, thermal environment, human-computer interaction, and cabin air quality all have an impact on passenger comfort [22–27]. Therefore, in this paper, when using the entropy weight method to study the passenger comfort model of the automotive intelligent cockpit, the acoustic environment, optical environment, thermal environment, human-computer interaction, and cockpit air quality are used as the first-level indicators, and in order to more comprehensively evaluate the comfort of intelligent cabin passengers, a comprehensive comfort evaluation hierarchy model of the intelligent cockpit of the car was established, as shown in Figure 1.Figure 1
Intelligent cockpit passenger comfort hierarchy evaluation system.When dividing the human comfort-taking intervals, this paper draws on the indoor human comfort evaluation model, uses a 10-point scale to score, and divides the car intelligence cockpit passenger comfort into five intervals, as shown in Table1 [28, 29].Table 1
Automotive intelligent cockpit passenger comfort classification standards.
Comfort evaluation levelScoring rangeIntolerable0,2Very uncomfortable2,4Uncomfortable4,6Slightly uncomfortable6,8Comfortable8,10


## 3. Experiment

### 3.1. Expert Scoring

The NIO ES8 is equipped with a pure electric powertrain and intelligent driving features such as HUD head-up digital display, automatic parking, and fatigue driving reminder, making it a better all-around intelligent vehicle, as shown in Figure2. For the evaluation of the vehicles, we invited five experts in the automotive field, selected with gender in mind, two of whom were women. All five experts are PhDs, professors, or engineers who have been working in the field of smart cars for a long time. The five experts rated the NIO ES8 vehicle based on five evaluation indicators, and the results are shown in Table 2.Figure 2
The cockpit of the NIO ES8 car.Table 2
Expert scoring table.
Acoustic environmentOptical environmentThermal environmentHuman-computer interactionCockpit air qualityExpert I8.38.28.38.67.7Expert II7.67.47.77.97.8Expert III8.48.28.39.18.5Expert IV8.27.988.88.3Expert V8.48.28.58.68.3


### 3.2. Improvement of Entropy Weight

Substituting the expert scores into (1), the original evaluation matrix for the passenger comfort of the car’s intelligent cockpit can be obtained as follows:(5)X=8.38.27.67.48.48.28.38.67.77.77.97.88.39.18.58.27.98.48.288.88.38.58.68.3.When substituting the original matrixX into (2), it is found that since 0≤pij≤1, 0≤−∑k=1npijlnpij≤1,0≤Hj≤1. However, when experts believe that a certain indicator has a very small or even irrelevant effect on the comfort of the car’s intelligent cockpit, it occurs that pij=0. In this case, lnpij is meaningless. When using the entropy weights method, lnpij=0 for the case where pij=0. However, it has been pointed out that the standardized result of the entropy weights method is easily distorted when there are too many zero values in the evaluation value. Also, this result will lead to overweighting of the index [30]. Therefore, this paper introduces a correction factor K1, which modifies the weight of indicator evaluation value pij to(6)pij′=pij+K1∑k=1mpij+K1,where the correction factor K1 is a constant whose value should at least satisfy(7)pij+K1>0.At the same time, it is necessary to ensure thatlnpij is mathematically meaningful while keeping its effect on the entropy value Hj within a reasonable range [31].Therefore, in this paper,K1 is taken as 10−5,10−4,and10−3, respectively, and the entropy weights of each index are obtained by substituting the original evaluation matrix X into equations 2∼5, and the improved entropy weights are compared with the original entropy weights, as shown in Table 3.Table 3
The effect ofK1 on the weight of each indicator of intelligent cockpit.
Indicators/methodsTraditional entropy weight methodK1=10−5K1=10−4K1=10−3Acoustic environment0.176290.176290.176290.17629Optical environment0.201450.201450.201450.20144Thermal environment0.153480.153480.153480.15348Human-computer interaction0.276320.276320.276320.27632Cockpit air quality0.192460.192460.192460.19247From Table3, it can be learned that when the correction coefficient K1 takes the values of 10−5 and 10−4, the entropy weights of the five indicators of acoustic environment, optical environment, thermal environment, human-computer interaction, and cockpit air quality change after the fifth decimal compared with the traditional entropy weight method. When the correction coefficient K1 takes the value of 10−3, the changes of each entropy weight obtained compared with the traditional entropy weight method are all after the fourth decimal place. It means that all three values of the correction coefficient have a small effect on the magnitude of the entropy weights. Finally, after comprehensive consideration, K1 is taken as 10−4, and then the formula is modified as(8)pij′=pij+10−4∑k=1mpij+10−4.In this case, the entropy valueHj of the j th indicator is then corrected to(9)Hj=−1lnn∑i=1npij′lnpij′.There is an inherent deficiency in the traditional entropy formula, that is when experts believe that the importance of each indicator is very similar, the entropy value of the indicator is close to 1 (the small difference between each other will cause the corresponding entropy weight to change exponentially, the phenomenon of entropy jump, making the calculated indicator weights have a large error. If the numerator and denominator are added with a larger positive correction factor, the above entropy jump phenomenon can be corrected. In this case, the entropy formula will be changed as follows:(10)ωj=1−Hj+K2∑i=1m1−Hj+K2K2>0.Regarding the correction coefficientK2, the smaller the change in entropy weight due to the change in the numerator denominator, the smaller the correction effect is made. Some scholars believe that the value K2 can be taken as 1/m∑j=1m1−Hj, and m is usually taken as 3∼10 . A set of Hj⟶1's entropy values 0.999,0.9998,0.9997 is also selected to justify the introduction of the correction coefficient K2, as shown in Table 4.Table 4
Comparison of entropy weights obtained by different entropy methods when the entropy valueHj is close to 1.
Entropy value/methodTraditional entropy weight methodm=3m=7m=100.99990.16670.25000.21670.20510.99980.33330.33330.33330.33330.99970.50000.41670.45000.4615From Table4, it can be seen that, when the entropy value Hj is close to 1, the indicator weights obtained using the traditional entropy method are unreasonable, a different value of m will make the entropy distribution results of the improved entropy method more objective. At the same time, in order to verify the impact on the indicator weights brought by adding the correction coefficient when the entropy value of the indicator is normal. The values of m are taken to be 3, 5, and 10, respectively, while the entropy values are made to compare the entropy weights Hi= obtained by different entropy weighting methods, as shown in Table 5. Also, the entropy weights obtained by various entropy weighting methods are put in the graph for comparison, as shown in Figure 3. From Figure 3, it can be seen that the difference between the improved entropy weight and the entropy weight obtained by the traditional entropy weight method is smaller when m=10 when the entropy value is general, thus proving the effectiveness of the improved entropy weight. At this point, the entropy weight of each index is obtained by substituting the original evaluation matrix X into formulas 2∼6, as shown in Figure 4. Moreover, entropy weights after the introduction of different correction coefficients K2 are compared with the traditional entropy weights, as shown in Table 6.Table 5
The effect ofK2 takes on the entropy weight of the normal entropy value.
Entropy value/methodTraditional entropy weight methodm=3m=7m=100.20.40000.36670.38000.38460.30.35000.34170.34500.34150.50.25000.29170.27500.2692Figure 3
The effect ofK2 takes on the entropy weight of normal entropy value.Figure 4
Impact ofK2 on the weighting of intelligent cockpit indicators.Table 6
Impact ofK2 on the weighting of intelligent cockpit indicators.
Indicators/methodsTraditional entropy weight methodm=3m=7m=10Acoustic environment0.176290.191110.186170.18419Optical environment0.201450.200540.200840.20096Thermal environment0.153480.182560.172860.16899Human-machine interaction0.276320.228620.244520.25088Cabin air quality0.192460.197180.195610.19498From Figure4, it can be seen that when m=10, the difference between the entropy weight of the indicator derived from the improved entropy weight method and the original entropy weight is the smallest when calculating the weight of each influence indicator of the passenger comfort of the NIO ES8 car intelligent cockpit. At this time, K2=1/10∑j=1m1−Hj, so this paper corrects the formula as(11)ωi=1−Hj+1/10∑j=1m1−Hj∑j=1m1−Hj+1/10∑j=1m1−Hj.Finally, by substituting the original evaluation matrixX into formulas (1)–(9), the indicator entropy weights of the five indicators affecting the passenger comfort of the NIO ES8 smart cabin were obtained: 0.18419, 0.20096, 0.16899, 0.25088, and 0.19498 for the acoustic environment, optical environment, thermal environment, human-computer interaction, and cabin air quality.


## 3.1. Expert Scoring

The NIO ES8 is equipped with a pure electric powertrain and intelligent driving features such as HUD head-up digital display, automatic parking, and fatigue driving reminder, making it a better all-around intelligent vehicle, as shown in Figure2. For the evaluation of the vehicles, we invited five experts in the automotive field, selected with gender in mind, two of whom were women. All five experts are PhDs, professors, or engineers who have been working in the field of smart cars for a long time. The five experts rated the NIO ES8 vehicle based on five evaluation indicators, and the results are shown in Table 2.Figure 2
The cockpit of the NIO ES8 car.Table 2
Expert scoring table.
Acoustic environmentOptical environmentThermal environmentHuman-computer interactionCockpit air qualityExpert I8.38.28.38.67.7Expert II7.67.47.77.97.8Expert III8.48.28.39.18.5Expert IV8.27.988.88.3Expert V8.48.28.58.68.3


## 3.2. Improvement of Entropy Weight

Substituting the expert scores into (1), the original evaluation matrix for the passenger comfort of the car’s intelligent cockpit can be obtained as follows:(5)X=8.38.27.67.48.48.28.38.67.77.77.97.88.39.18.58.27.98.48.288.88.38.58.68.3.When substituting the original matrixX into (2), it is found that since 0≤pij≤1, 0≤−∑k=1npijlnpij≤1,0≤Hj≤1. However, when experts believe that a certain indicator has a very small or even irrelevant effect on the comfort of the car’s intelligent cockpit, it occurs that pij=0. In this case, lnpij is meaningless. When using the entropy weights method, lnpij=0 for the case where pij=0. However, it has been pointed out that the standardized result of the entropy weights method is easily distorted when there are too many zero values in the evaluation value. Also, this result will lead to overweighting of the index [30]. Therefore, this paper introduces a correction factor K1, which modifies the weight of indicator evaluation value pij to(6)pij′=pij+K1∑k=1mpij+K1,where the correction factor K1 is a constant whose value should at least satisfy(7)pij+K1>0.At the same time, it is necessary to ensure thatlnpij is mathematically meaningful while keeping its effect on the entropy value Hj within a reasonable range [31].Therefore, in this paper,K1 is taken as 10−5,10−4,and10−3, respectively, and the entropy weights of each index are obtained by substituting the original evaluation matrix X into equations 2∼5, and the improved entropy weights are compared with the original entropy weights, as shown in Table 3.Table 3
The effect ofK1 on the weight of each indicator of intelligent cockpit.
Indicators/methodsTraditional entropy weight methodK1=10−5K1=10−4K1=10−3Acoustic environment0.176290.176290.176290.17629Optical environment0.201450.201450.201450.20144Thermal environment0.153480.153480.153480.15348Human-computer interaction0.276320.276320.276320.27632Cockpit air quality0.192460.192460.192460.19247From Table3, it can be learned that when the correction coefficient K1 takes the values of 10−5 and 10−4, the entropy weights of the five indicators of acoustic environment, optical environment, thermal environment, human-computer interaction, and cockpit air quality change after the fifth decimal compared with the traditional entropy weight method. When the correction coefficient K1 takes the value of 10−3, the changes of each entropy weight obtained compared with the traditional entropy weight method are all after the fourth decimal place. It means that all three values of the correction coefficient have a small effect on the magnitude of the entropy weights. Finally, after comprehensive consideration, K1 is taken as 10−4, and then the formula is modified as(8)pij′=pij+10−4∑k=1mpij+10−4.In this case, the entropy valueHj of the j th indicator is then corrected to(9)Hj=−1lnn∑i=1npij′lnpij′.There is an inherent deficiency in the traditional entropy formula, that is when experts believe that the importance of each indicator is very similar, the entropy value of the indicator is close to 1 (the small difference between each other will cause the corresponding entropy weight to change exponentially, the phenomenon of entropy jump, making the calculated indicator weights have a large error. If the numerator and denominator are added with a larger positive correction factor, the above entropy jump phenomenon can be corrected. In this case, the entropy formula will be changed as follows:(10)ωj=1−Hj+K2∑i=1m1−Hj+K2K2>0.Regarding the correction coefficientK2, the smaller the change in entropy weight due to the change in the numerator denominator, the smaller the correction effect is made. Some scholars believe that the value K2 can be taken as 1/m∑j=1m1−Hj, and m is usually taken as 3∼10 . A set of Hj⟶1's entropy values 0.999,0.9998,0.9997 is also selected to justify the introduction of the correction coefficient K2, as shown in Table 4.Table 4
Comparison of entropy weights obtained by different entropy methods when the entropy valueHj is close to 1.
Entropy value/methodTraditional entropy weight methodm=3m=7m=100.99990.16670.25000.21670.20510.99980.33330.33330.33330.33330.99970.50000.41670.45000.4615From Table4, it can be seen that, when the entropy value Hj is close to 1, the indicator weights obtained using the traditional entropy method are unreasonable, a different value of m will make the entropy distribution results of the improved entropy method more objective. At the same time, in order to verify the impact on the indicator weights brought by adding the correction coefficient when the entropy value of the indicator is normal. The values of m are taken to be 3, 5, and 10, respectively, while the entropy values are made to compare the entropy weights Hi= obtained by different entropy weighting methods, as shown in Table 5. Also, the entropy weights obtained by various entropy weighting methods are put in the graph for comparison, as shown in Figure 3. From Figure 3, it can be seen that the difference between the improved entropy weight and the entropy weight obtained by the traditional entropy weight method is smaller when m=10 when the entropy value is general, thus proving the effectiveness of the improved entropy weight. At this point, the entropy weight of each index is obtained by substituting the original evaluation matrix X into formulas 2∼6, as shown in Figure 4. Moreover, entropy weights after the introduction of different correction coefficients K2 are compared with the traditional entropy weights, as shown in Table 6.Table 5
The effect ofK2 takes on the entropy weight of the normal entropy value.
Entropy value/methodTraditional entropy weight methodm=3m=7m=100.20.40000.36670.38000.38460.30.35000.34170.34500.34150.50.25000.29170.27500.2692Figure 3
The effect ofK2 takes on the entropy weight of normal entropy value.Figure 4
Impact ofK2 on the weighting of intelligent cockpit indicators.Table 6
Impact ofK2 on the weighting of intelligent cockpit indicators.
Indicators/methodsTraditional entropy weight methodm=3m=7m=10Acoustic environment0.176290.191110.186170.18419Optical environment0.201450.200540.200840.20096Thermal environment0.153480.182560.172860.16899Human-machine interaction0.276320.228620.244520.25088Cabin air quality0.192460.197180.195610.19498From Figure4, it can be seen that when m=10, the difference between the entropy weight of the indicator derived from the improved entropy weight method and the original entropy weight is the smallest when calculating the weight of each influence indicator of the passenger comfort of the NIO ES8 car intelligent cockpit. At this time, K2=1/10∑j=1m1−Hj, so this paper corrects the formula as(11)ωi=1−Hj+1/10∑j=1m1−Hj∑j=1m1−Hj+1/10∑j=1m1−Hj.Finally, by substituting the original evaluation matrixX into formulas (1)–(9), the indicator entropy weights of the five indicators affecting the passenger comfort of the NIO ES8 smart cabin were obtained: 0.18419, 0.20096, 0.16899, 0.25088, and 0.19498 for the acoustic environment, optical environment, thermal environment, human-computer interaction, and cabin air quality.


## 4. Results and Discussion

In the traditional entropy weight method, the final entropy weight value derived often has a slight error. This is caused by two defects of the traditional entropy method. First, when there are too many zero values in the evaluation value, the entropy weight method will lead to errors in the final results due to the zeroing process. Secondly, the traditional entropy weighting method will have a jump in entropy weighting when the entropy valueHj⟶1, which makes the obtained index weights have large errors.To solve the shortcomings of the traditional entropy weight method, this paper introduces correction coefficientsK1 and K2. Also, in order to determine the best value of the correction coefficients, examples are introduced for the study, and it is found that at K1=10−4, K2=1/10∑i=1m1−Hi, the traditional entropy weight method is the improvement which is the best, and by comparing with the weights derived from the traditional entropy weight method, it is found that the improved entropy weight method solves the shortcomings of the traditional entropy weight method while having little effect on the change of weights. Finally, the improved entropy weight method was applied to the field of evaluation of passenger comfort in intelligent cockpits of automobiles, and it was found that among the influencing factors affecting passenger comfort in cockpits, the most influential was human-computer interaction, followed by optical environment, cockpit air quality, acoustic environment and thermal environment, respectively.Although the objective weights of the five indicators can be obtained by the improved entropy weight method, the disadvantage of the entropy weight method is that it is too objective and cannot fully reflect the influence of each indicator on comprehensive comfort. In the future, we can consider combining subjective methods such as the hierarchical analysis method to establish a comprehensive evaluation model of passenger comfort in intelligent cockpits of automobiles to make a more reasonable evaluation of passenger comfort in intelligent cockpits. The mathematical approach involved in the model can also be extended to the fields of road, rail, air, and ocean transportation.



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