# Multi-Dimensional Interval Number Decision Model Based on Mahalanobis-Taguchi System with Grey Entropy Method and Its Application in Reservoir Operation Scheme Selection

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{a}(b

^{N}), where a is the number of tests, b the number of levels of each factor and N the number of factors that can be arranged at most in the orthogonal table. The orthogonal table is a prepared set of standard tables, from which the suitable one is chosen in actual application according to the number of factors and the number of levels of each factor. The orthogonal table designs a small number of tests and obtains comprehensive information, which can effectively reduce the loss of information. The Mahalanobis distance [28], proposed by Indian statistician Mahalanobis, is a covariance distance that, compared with Euclidean distance, can better reflect the correlation between attributes. The concept of signal-to-noise ratio (SNR) [29] originates from signal transmission and is defined as the ratio of signal power to noise power. Taguchi G. redefined the SNR, regarding the square (μ

^{2}) and the variance (σ

^{2}) of the expected value of an index (non-negative and continuous) as the signal power and the noise power, respectively. The SNR can be divided into three types: nominal-the-better, smaller-the-better and larger-the-better. The first one means that the closer to the expected value when it is positive, the better; the second one means that the smaller the expected value when it is 0, the better; the third one means that the larger the expected value when it is $+\infty $, the better. The signal-to-noise ratio can be used to measure the volatility of indexes and thereby ensure the accuracy of decision results.

## 2. Multi-Dimensional Interval Number and Mahalanobis-Taguchi System

#### 2.1. Orthogonal Test of Multi-Dimensional Interval Number

^{L}, a

^{U}∈R and a

^{U}≥a

^{L}), where a

^{L}and a

^{U}are, respectively, the lower and upper bounds of the interval number. Real numbers can be regarded as interval numbers whose lower bounds are equal to their upper bounds. For the basic operation rules of any two interval numbers $\tilde{a}=\left[{a}^{L},{a}^{U}\right]$ and $\tilde{b}=\left[{b}^{L},{b}^{U}\right]$, see reference [30].

#### 2.2. Signal-to-Noise Ratio and Mahalanobis Distance

^{*}is calculated as follows [31]:

^{2}and σ

^{2}are the mean square and the variance of population D, respectively.

^{2}and mean square μ

^{2}of population D are as follows:

^{*}, then the NB signal-to-noise ratio η

^{NB}can be obtained:

^{2}and σ

^{2}are, the better, meaning that the smaller ${\mu}^{2}+{\sigma}^{2}=E({D}^{2})$ is, the better. The unbiased estimator of D

^{2}is ${\widehat{D}}^{2}=\frac{1}{a}{\displaystyle {\sum}_{g=1}^{a}{D}_{g}^{2}}$. Take ${({\widehat{D}}^{2})}^{-1}$ as the signal-to-noise ratio, and the SB SNR η

^{SB}is calculated as follows:

^{−1}stands for an SB index. From the above derivation process of the η

^{SB}formula, the LB SNR η

^{LB}can be obtained:

**x**to population Z is:

**x**and population Z are taken as the input of MTS, then the Mahalanobis distance d between the alternative scheme and the reference scheme is the responding output.

## 3. Improved Grey Entropy Method by Mahalanobis-Taguchi System

#### 3.1. Grey Entropy

#### 3.2. The Comparison of Grey Entropy and Information Entropy

- Similarities
- With the same form of calculation formula, grey entropy and information entropy share some characteristics, such as symmetry, non-negativity, additivity, convexity and extremum property.
- The physical meanings of grey entropy and information entropy are essentially the same. The former is to measure the fluctuation degree of a grey number, while the latter is to describe the uncertainty of a signal source.

- Differences
- Grey entropy is defined in a finite information space, whereas information entropy is defined in an infinite information space.
- Grey entropy is a kind of non-probability entropy with greyness, that is, ${q}_{i}$ is a possible value. On the contrary, information entropy is a type of probability entropy with certainty, that is, ${p}_{i}$ is a certain value.

#### 3.3. Fundamentals of Grey Entropy Method

**X**is converted into $\mathit{C}={\left[{c}_{ij}\right]}_{m\times n}$, and the alternative scheme is ${c}_{i}=({c}_{i1},{c}_{i2},\cdots ,{c}_{in})$. Given the reference scheme ${p}_{k}=({p}_{k1},{p}_{k2},\cdots ,{p}_{kn})$ (k = 1,2,…,z), the correlation degree ${G}_{ki}$ between the alternative scheme and the reference scheme is calculated as follows:

## 4. Multi-Dimensional Interval Number Decision Model Based on MTS-GEM

#### 4.1. Development of the Weighted Standardized Decision Matrix

#### 4.2. Orthogonal Test of Schemes and Calculation of Derivative Indicators

#### 4.3. Scheme Decision-Making

**Y**is constructed as shown in Equation (23), in which the benefit indicator is ${\eta}_{-i}$ and ${\gamma}_{+i}$, while the cost indicator is ${\eta}_{+i}$ and ${\gamma}_{-i}$:

- The n-dimensional real numbers are regarded as the points A
_{1},…,A_{i},…,A_{m}in an n-dimensional space with O as the origin, and then we can get the vectors ${\mathit{a}}_{1}={\overrightarrow{OA}}_{1}$,…,${\mathit{a}}_{i}={\overrightarrow{OA}}_{i}$,…,${\mathit{a}}_{m}={\overrightarrow{OA}}_{m}$. - Assuming that the reference scheme vector is $\mathit{p}$, $\left|{\mathit{a}}_{i}\right|$ and $\left|\mathit{p}\right|$ are the modules of vector
**a**_{i}and vector $\mathit{p}$, respectively. Between vectors**a**_{i}and $\mathit{p}$, calculate their angle ${\theta}_{i}=(\widehat{{\mathit{a}}_{i},\mathit{p}})=\mathrm{arccos}\left(\frac{{\mathit{a}}_{i}\cdot \mathit{p}}{\left|{\mathit{a}}_{i}\right|\left|\mathit{p}\right|}\right)$ (where ${\mathit{a}}_{i}\cdot \mathit{p}$ is their product), as well as their mapping distance ${\mathrm{MD}}_{i}=\left|{\mathit{a}}_{i}\right|\mathrm{Sin}{\theta}_{i}$. - The set of mapping distance $\mathrm{MD}=\left\{{\mathrm{MD}}_{i}|i=1,2,\cdots ,m\right\}$ can be obtained. According to the principle that the smaller ${\mathrm{MD}}_{i}$ is, the closer vector
**a**_{i}is to vector $\mathit{p}$, the scheme that satisfies the objective $\underset{i}{\mathrm{min}}{\mathrm{MD}}_{i}$ is selected as the optimal scheme.

## 5. Case Study

#### 5.1. Initial Interval Number Decision Matrix and Its Weighted Standardization

#### 5.2. Orthogonal Test of the Schemes

**C**

_{i}(i=1,2,…,6), the positive ideal scheme’s layout matrix ${\mathit{P}}_{+}$ and the negative ideal scheme’s layout matrix ${\mathit{P}}_{-}$ are shown in Table 8.

#### 5.3. Scheme Decision-Making and Result Evaluation

**C**

_{i}to the positive ideal scheme’s layout matrix ${\mathit{P}}_{+}$ and the negative ideal scheme’s layout matrix ${\mathit{P}}_{-}$ is worked out, as shown in Table 9.

**X**is derived, as shown in Table 10.

**X**is standardized and with the weight of each index being 0.25, the weighted standardized decision matrix

**C**is obtained, as shown in Table 11.

## 6. Conclusions

- MTS-GEM can effectively reduce the uncertainty created by interval numbers. In the model, the bounded uncertain n-dimensional interval number is expressed quantitatively as a hypercube in the n-dimensional space. Meanwhile, the alternative and reference schemes are all transformed into finite vertices of the hypercube, which realizes the transformation from an interval number decision vector to a real number decision vector.
- MTS-GEM can produce markedly distinctive decision results, which demonstrates the sufficiency of decision information contained in the model. The model not only considers the output response strength and the degree of balance and approach between alternative and reference schemes, but also uses the idea of further approaching the reference scheme by mapping distance, which elevates the accuracy and reliability of the decision results.
- The case study of selecting the optimal scheme of controlling the Pankou reservoir’s water level in flood season shows that the proposed method can pick out the best scheme that better coordinates risk and benefit, which further proves the comprehensive and excellent decision-making performance of the model.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Flow chart of multi-dimensional interval number decision-making based on Mahalanobis-Taguchi System with grey entropy method.

Method | Advantage | Shortage |
---|---|---|

TOPSIS | Alternative schemes are evaluated by both how close they are to the positive ideal scheme and how far away from the negative ideal scheme. | The Euclidean distance does not consider the correlation between indexes, producing indistinct decision result, “reverse order” and other problems. |

SPA | Alternative schemes are evaluated by their identical-discrepant-contrary degree to the optimal scheme, preventing the "reverse order" problem of TOPSIS. | The uncertainty of the discrepancy coefficient may lead to decision risk. |

GRA | Alternative schemes are evaluated by their geometric proximity to the optimal scheme. | The Euclidean distance is used to calculate the correlation coefficient, which neglects the correlation between indexes; larger correlation coefficients determine the correlation degree, which causes information loss. |

GTM | Alternative schemes are evaluated by their distance from the optimal scheme (off-target distance). | The single bull’s-eye may cause the decision result to be indistinct; the off-target distance is calculated using the Euclidean distance, which neglects the correlation between indexes, and index weight is not considered. |

GEM | Alternative schemes are evaluated by how evenly close they are to the optimal scheme, which remedies GRA in some way. | The Euclidean distance is used to calculate the correlation coefficient, which neglects the correlation between indexes; the calculation of correlation degree does not consider index weight. |

Test | Factor | ||
---|---|---|---|

${\tilde{\mathit{x}}}_{1}$ | ${\tilde{\mathit{x}}}_{2}$ | ${\tilde{\mathit{x}}}_{3}$ | |

One | 1 | 1 | 1 |

Two | 1 | 2 | 2 |

Three | 2 | 1 | 2 |

Four | 2 | 2 | 1 |

Item | Unit | Pankou |
---|---|---|

Dead water level | m | 330 |

Flood control limit water level | m | 347.6 |

Normal water level | m | 355 |

Flood control high water level | m | 358.4 |

Design flood water level | m | 357.14 |

Spillway flood water level | m | 360.82 |

Minimum storage capacity | 10^{8} m^{3} | 8.5 |

Regulating storage capacity | 10^{8} m^{3} | 11.2 |

Total storage capacity | 10^{8} m^{3} | 23.38 |

Regulating performance | – | Annual regulating |

Installed power capacity | MW | 500 |

Average annual generated power | 10^{8} kW·h | 10.474 |

Scheme i | $\mathbf{Flood}\mathbf{Control}\mathbf{Risk}\mathbf{Rate}{\tilde{\mathit{x}}}_{\mathit{i}1}/\%$ | $\mathbf{Annual}\mathbf{Generated}\mathbf{Power}{\tilde{\mathit{x}}}_{\mathit{i}2}/{10}^{8}\mathbf{kW}\xb7\mathbf{h}$ | $\mathbf{Water}\mathbf{Storage}\mathbf{at}\mathbf{the}\mathbf{End}\mathbf{of}\mathbf{Flood}\mathbf{Season}{\tilde{\mathit{x}}}_{\mathit{i}3}/{10}^{8}{\mathbf{m}}^{3}$ |
---|---|---|---|

1 | [1.812,4.371] | [6.531,14.392] | [4.887,11.200] |

2 | [1.882,4.408] | [7.086,14.983] | [5.411,11.200] |

3 | [1.993,4.421] | [7.687,15.639] | [6.033,11.200] |

4 | [2.211,4.689] | [8.275,16.136] | [6.751,11.200] |

5 | [2.534,5.028] | [8.699,16.641] | [7.523,11.200] |

6 | [2.977,5.594] | [9.320,17.343] | [8.431,11.200] |

Scheme i | $\mathbf{Flood}\mathbf{Control}\mathbf{Risk}\mathbf{Rate}{\tilde{\mathit{b}}}_{\mathit{i}1}$ | $\mathbf{Annual}\mathbf{Generated}\mathbf{Power}{\tilde{\mathit{b}}}_{\mathit{i}2}$ | $\mathbf{Water}\mathbf{Storage}\mathbf{at}\mathbf{the}\mathbf{End}\mathbf{of}\mathbf{Flood}\mathbf{Season}{\tilde{\mathit{b}}}_{\mathit{i}3}$ |
---|---|---|---|

1 | [0.323,1.000] | [0.000,0.727] | [0.000,1.000] |

2 | [0.314,0.981] | [0.051,0.782] | [0.083,1.000] |

3 | [0.310,0.952] | [0.107,0.842] | [0.182,1.000] |

4 | [0.239,0.895] | [0.161,0.888] | [0.295,1.000] |

5 | [0.150,0.809] | [0.201,0.935] | [0.418,1.000] |

6 | [0.000,0.692] | [0.258,1.000] | [0.561,1.000] |

Index | $\mathbf{Flood}\mathbf{Control}\mathbf{Risk}\mathbf{Rate}{\tilde{\mathit{b}}}_{1}$ | $\mathbf{Annual}\mathbf{Generated}\mathbf{Power}{\tilde{\mathit{b}}}_{2}$ | $\mathbf{Water}\mathbf{Storage}\mathbf{at}\mathbf{the}\mathbf{End}\mathbf{of}\mathbf{Flood}\mathbf{Season}{\tilde{\mathit{b}}}_{3}$ |
---|---|---|---|

Flood control risk rate ${\tilde{b}}_{1}$ | 1.000 | 0.613 | 0.573 |

Annual generated power ${\tilde{b}}_{2}$ | 0.613 | 1.000 | 0.971 |

Water storage at the end of flood season ${\tilde{b}}_{3}$ | 0.573 | 0.971 | 1.000 |

Scheme i | $\mathbf{Flood}\mathbf{Control}\mathbf{Risk}\mathbf{Rate}{\tilde{\mathit{c}}}_{\mathit{i}1}$ | $\mathbf{Annual}\mathbf{Generated}\mathbf{Power}{\tilde{\mathit{c}}}_{\mathit{i}2}$ | $\mathbf{Water}\mathbf{Sstorage}\mathbf{at}\mathbf{the}\mathbf{End}\mathbf{of}\mathbf{Flood}\mathbf{Season}{\tilde{\mathit{c}}}_{\mathit{i}3}$ |
---|---|---|---|

1 | [0.092, 0.485] | [0.000, 0.286] | [0.000, 0.351] |

2 | [0.090, 0.476] | [0.016, 0.308] | [0.010, 0.351] |

3 | [0.089, 0.462] | [0.034, 0.332] | [0.021, 0.351] |

4 | [0.068, 0.434] | [0.052, 0.350] | [0.035, 0.351] |

5 | [0.043, 0.392] | [0.064, 0.368] | [0.049, 0.351] |

6 | [0.000, 0.336] | [0.083, 0.394] | [0.066, 0.351] |

Matrix | Flood Control Risk Rate | Annual Generated Power | Water Storage at the End of Flood Season | Matrix | Flood Control Risk Rate | Annual Generated Power | Water Storage at the End of Flood Season |
---|---|---|---|---|---|---|---|

C_{1} | 0.092 | 0.000 | 0.000 | C_{5} | 0.043 | 0.064 | 0.049 |

0.092 | 0.286 | 0.351 | 0.043 | 0.368 | 0.351 | ||

0.485 | 0.000 | 0.351 | 0.392 | 0.064 | 0.351 | ||

0.485 | 0.286 | 0.000 | 0.392 | 0.368 | 0.049 | ||

C_{2} | 0.090 | 0.016 | 0.010 | C_{6} | 0.000 | 0.083 | 0.066 |

0.090 | 0.308 | 0.351 | 0.000 | 0.394 | 0.351 | ||

0.476 | 0.016 | 0.351 | 0.336 | 0.083 | 0.351 | ||

0.476 | 0.308 | 0.010 | 0.336 | 0.394 | 0.066 | ||

C_{3} | 0.089 | 0.034 | 0.021 | ${\mathit{P}}_{+}$ | 0.092 | 0.083 | 0.066 |

0.089 | 0.332 | 0.351 | 0.092 | 0.394 | 0.351 | ||

0.462 | 0.034 | 0.351 | 0.485 | 0.083 | 0.351 | ||

0.462 | 0.332 | 0.021 | 0.485 | 0.394 | 0.066 | ||

C_{4} | 0.068 | 0.052 | 0.035 | ${\mathit{P}}_{-}$ | 0.000 | 0.000 | 0.000 |

0.068 | 0.350 | 0.351 | 0.000 | 0.286 | 0.351 | ||

0.434 | 0.052 | 0.351 | 0.336 | 0.000 | 0.351 | ||

0.434 | 0.350 | 0.035 | 0.336 | 0.286 | 0.000 |

Scheme i | ${\mathit{d}}^{2}\left[{\mathit{c}}_{\mathit{i}}^{(\mathit{g})},{\mathit{P}}_{+}\right]$ | ${\mathit{d}}^{2}\left[{\mathit{c}}_{\mathit{i}}^{(\mathit{g})},{\mathit{P}}_{-}\right]$ | ||||||
---|---|---|---|---|---|---|---|---|

1 | 4.139 | 1.567 | 3.263 | 2.428 | 1.658 | 1.649 | 4.166 | 4.175 |

2 | 3.774 | 1.662 | 2.966 | 2.290 | 1.424 | 1.906 | 3.858 | 4.190 |

3 | 3.386 | 1.790 | 2.630 | 2.155 | 1.187 | 2.222 | 3.478 | 4.192 |

4 | 3.152 | 2.076 | 2.238 | 1.909 | 1.054 | 2.583 | 2.930 | 3.936 |

5 | 3.071 | 2.436 | 1.901 | 1.668 | 1.037 | 3.018 | 2.308 | 3.583 |

6 | 3.132 | 3.112 | 1.542 | 1.543 | 1.177 | 3.806 | 1.628 | 3.355 |

Scheme i | ${\mathit{\eta}}_{+\mathit{i}}$ | ${\mathit{\eta}}_{-\mathit{i}}$ | ${\mathit{\gamma}}_{+\mathit{i}}$ | ${\mathit{\gamma}}_{-\mathit{i}}$ |
---|---|---|---|---|

1 | 2.998 | 3.010 | 0.753 | 0.652 |

2 | 2.829 | 2.894 | 0.775 | 0.660 |

3 | 2.508 | 2.784 | 0.801 | 0.669 |

4 | 1.869 | 2.633 | 0.824 | 0.686 |

5 | 2.429 | 2.446 | 0.841 | 0.705 |

6 | 2.984 | 3.010 | 0.838 | 0.710 |

Scheme i | ${\mathit{\eta}}_{+\mathit{i}}^{\ast}$ | ${\mathit{\eta}}_{-\mathit{i}}^{\ast}$ | ${\mathit{\gamma}}_{+\mathit{i}}^{\ast}$ | ${\mathit{\gamma}}_{-\mathit{i}}^{\ast}$ |
---|---|---|---|---|

1 | 0.000 | 0.250 | 0.000 | 0.250 |

2 | 0.037 | 0.199 | 0.063 | 0.216 |

3 | 0.109 | 0.150 | 0.136 | 0.177 |

4 | 0.250 | 0.083 | 0.202 | 0.103 |

5 | 0.126 | 0.000 | 0.250 | 0.022 |

6 | 0.003 | 0.250 | 0.241 | 0.000 |

Scheme i | Method 1 | Method 2 | This Study’s Method | |||
---|---|---|---|---|---|---|

${\mathit{\eta}}_{\mathit{i}}$ | Ranking | ${\mathit{\gamma}}_{\mathit{i}}$ | Ranking | ${\mathit{MD}}_{\mathit{i}}$ | Ranking | |

1 | 0.5010 | 6 | 0.5359 | 6 | 0.2500 | 6 |

2 | 0.5057 | 3 | 0.5401 | 5 | 0.1585 | 3 |

3 | 0.5260 | 2 | 0.5449 | 2 | 0.0492 | 1 |

4 | 0.5848 | 1 | 0.5457 | 1 | 0.1378 | 2 |

5 | 0.5018 | 5 | 0.5440 | 3 | 0.1983 | 4 |

6 | 0.5021 | 4 | 0.5413 | 4 | 0.2443 | 5 |

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**MDPI and ACS Style**

Ji, C.; Liang, X.; Peng, Y.; Zhang, Y.; Yan, X.; Wu, J.
Multi-Dimensional Interval Number Decision Model Based on Mahalanobis-Taguchi System with Grey Entropy Method and Its Application in Reservoir Operation Scheme Selection. *Water* **2020**, *12*, 685.
https://doi.org/10.3390/w12030685

**AMA Style**

Ji C, Liang X, Peng Y, Zhang Y, Yan X, Wu J.
Multi-Dimensional Interval Number Decision Model Based on Mahalanobis-Taguchi System with Grey Entropy Method and Its Application in Reservoir Operation Scheme Selection. *Water*. 2020; 12(3):685.
https://doi.org/10.3390/w12030685

**Chicago/Turabian Style**

Ji, Changming, Xiaoqing Liang, Yang Peng, Yanke Zhang, Xiaoran Yan, and Jiajie Wu.
2020. "Multi-Dimensional Interval Number Decision Model Based on Mahalanobis-Taguchi System with Grey Entropy Method and Its Application in Reservoir Operation Scheme Selection" *Water* 12, no. 3: 685.
https://doi.org/10.3390/w12030685