# A Novel Multi-Criteria Decision-Making Model for Building Material Supplier Selection Based on Entropy-AHP Weighted TOPSIS

## Abstract

**:**

## 1. Introduction

## 2. Literature Review and Methodology

#### 2.1. Literature Review

#### 2.1.1. Literature on the Application of the Entropy, AHP, and/or TOPSIS Method

#### 2.1.2. Rank Reversals in Decision-Making

- Multi-attribute utility theory (MAUT).
- The TOPSIS method.
- The analytic hierarchy process (AHP) and some of its variants.
- The ELECTRE (outranking) method and its variants.
- The PROMETHEE (outranking) method.

#### 2.2. Entropy Weighted Method

#### 2.2.1. Entropy Weight Principle

#### 2.2.2. Significance and Nature of Entropy Weight Method

- If the values of elements in a column are the same, the maximum entropy is 1, and the entropy weight is 0. On behalf of an indicator, if the data of each evaluation object are the same, the indicator does not contain any valuable information.
- The greater the difference between the values of elements in a column, the smaller the entropy value of the elements in this column and the larger the entropy weight value. It indicates that the indicator has valuable information. Conversely, if the indicator’s entropy value is larger, the smaller its entropy weight and the less important this indicator is.

#### 2.3. AHP Method

#### 2.3.1. The Meaning of AHP

#### 2.3.2. Application of AHP

#### 2.3.3. Steps of AHP Method

#### 2.4. Combination Weighting Method

#### 2.5. Weights for Multi-Criteria Decision Making

## 3. Construction Steps of Entropy-AHP Weighted TOPSIS

## 4. Numerical Execution Example of Building Material Supplier Selection

_{1}: the rate of qualified products (%), C

_{2}: the product price (thousand dollars), C

_{3}: the rate of product market share (%), C

_{4}: the supply capacity (kg/time), C

_{5}: the new product development rate (%), C

_{6}: the delivery time (days), and C

_{7}: the delivery on time ratio (%). The analytic hierarchy diagrams of the two levels and the seven criteria are shown in Figure 3.

_{j}.

_{1}as an example, the proportion calculation formula of ${\mathrm{e}}_{1}$ is as follows:

_{1}= 1 − e

_{1}= 1 − 0.7276 = 0.2724; b

_{2}= 1 − e

_{2}= 1 − 0.8137 = 0.1863; b

_{3}= 1 − e

_{3}= 1 − 0.7533 = 0.2467

## 5. Results and Discussion

## 6. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 5.**Sensitivity analysis of the facet weight to the outcome of the alternatives. ntropy-AHP TOPSIS vs. AHP-based TOPSIS.

R-level | ${R}_{1}$ | ${R}_{2}$ | $\cdots $ | ${R}_{n}$ | S-level element combination weight | ||

Weight | ${r}_{1}$ | ${r}_{2}$ | $\cdots $ | ${r}_{n}$ | |||

S-level | |||||||

${S}_{1}$ | ${s}_{1}^{1}$ | ${s}_{1}^{2}$ | $\cdots $ | ${s}_{1}^{n}$ | ${\mathrm{s}}_{1}={\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{r}}_{\mathrm{i}}{\mathrm{s}}_{1}^{\mathrm{i}}$ | ||

${S}_{2}$ | ${s}_{2}^{1}$ | ${s}_{2}^{2}$ | ⋯ | ${s}_{2}^{n}$ | ${\mathrm{s}}_{2}={\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{r}}_{\mathrm{i}}{\mathrm{s}}_{2}^{\mathrm{i}}$ | ||

$\vdots $ | ⋮ | $\vdots $ | $\vdots $ | $\vdots $ | |||

${S}_{4}$ | ${s}_{m}^{1}$ | ${s}_{m}^{2}$ | $\cdots $ | ${s}_{m}^{n}$ | ${\mathrm{s}}_{\mathrm{m}}={\displaystyle {\displaystyle \sum}_{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{r}}_{\mathrm{i}}{\mathrm{s}}_{\mathrm{m}}^{\mathrm{i}}$ |

**Table 2.**Weights of various facets and criteria for building material supplier selection evaluated with the entropy method.

**Table 3.**Weights of various facets and the criterion of building material supplier selection evaluated with the analytic hierarchy process (AHP) method.

Main Target | Facet for the First Layer | Facet Weight $({}_{}{}^{1}\mathbf{w}{}_{\mathbf{h}\mathbf{i}})$ | Criterion (Indicator) for the Second Layer | Dimension | Indicator Weight $({}_{}{}^{2}\mathbf{w}{}_{\mathbf{h}\mathbf{j}})$ | Total Weight $({\mathbf{w}}_{\mathbf{h}\mathbf{j}=}{}_{}{}^{1}\mathbf{w}{}_{\mathbf{h}\mathbf{i}}\times {}_{}{}^{2}\mathbf{w}{}_{\mathbf{h}\mathbf{j}})$ |
---|---|---|---|---|---|---|

Suitable supplier selection | Product satisfaction (A) | 0.3916 | A1. Rate of qualified products | positive | 0.4125 | 0.1615 |

A2.Product price (thousand dollars) | negative | 0.3759 | 0.1472 | |||

A3.Rate of Product market share | positive | 0.2116 | 0.0829 | |||

Subtotal | 1 | --- | ||||

Supply innovation capability (B) | 0.2815 | B1.Supply capacity (kg/time) | positive | 0.5293 | 0.1490 | |

B2.New product development rate (%) | positive | 0.4707 | 0.1325 | |||

Subtotal | 1 | --- | ||||

Service level (C) | 0.3269 | C1. Delivery time (days) | negative | 0.3917 | 0.1280 | |

C2. Delivery on time ratio (%) | positive | 0.6083 | 0.1989 | |||

Subtotal | 1 | --- |

**Table 4.**Facet weights of building material supplier selection evaluated with the combination weighting method.

Weight Item | Product Satisfaction (A) | Supply Innovation Capability (B) | Service Level (C) |
---|---|---|---|

Entropy weight (${}_{}{}^{1}\mathrm{w}{}_{\mathrm{ei}}$) | 0.4426 | 0.2592 | 0.2982 |

AHP weight (${}_{}{}^{1}\mathrm{w}{}_{\mathrm{hi}}$) | 0.3916 | 0.2815 | 0.3269 |

Combination weight (${}_{}{}^{1}\mathrm{w}{}_{\mathrm{ci}}=\frac{{}_{}{}^{1}\mathrm{w}{}_{\mathrm{ei}}\times {}_{}{}^{1}\mathrm{w}{}_{\mathrm{hi}}}{{{\displaystyle \sum}}_{i=1}^{3}{}_{}{}^{1}\mathrm{w}{}_{\mathrm{ei}}\times {}_{}{}^{1}\mathrm{w}{}_{\mathrm{hi}}}$) | 0.5042 | 0.2122 | 0.2836 |

**Table 5.**Criterion weights of building material supplier selection evaluated by the combination weighting method.

Weight Item | Rate of Qualified Products (A1) | Product Price (Thousand Dollars) (A2) | Rate of Product Market Share (A3) | Supply Capacity (kg/ time) (B1) | New Product Development Rate (%) (B2) | Delivery Time (days) (C1) | Delivery on Time Ratio (%) (C2) |
---|---|---|---|---|---|---|---|

Entropy weight (${}_{}{}^{2}\mathrm{w}{}_{\mathrm{ej}}$) | 0.3862 | 0.2641 | 0.3497 | 0.4658 | 0.5342 | 0.4168 | 0.5832 |

AHP weight (${}_{}{}^{2}\mathrm{w}{}_{\mathrm{hj}})$ | 0.4125 | 0.3759 | 0.2116 | 0.5293 | 0.4707 | 0.3917 | 0.6083 |

Combination weight (${}_{}{}^{2}\mathrm{w}{}_{\mathrm{cj}}=\frac{{}_{}{}^{2}\mathrm{w}{}_{\mathrm{ej}}\times {}_{}{}^{2}\mathrm{w}{}_{\mathrm{hj}}}{{{\displaystyle \sum}}_{j=1}^{7}{}_{}{}^{2}\mathrm{w}{}_{\mathrm{ej}}\times {}_{}{}^{2}\mathrm{w}{}_{\mathrm{hj}}})$ | 0.4789 | 0.2985 | 0.2225 | 0.4951 | 0.5049 | 0.3152 | 0.6848 |

**Table 6.**The entropy-AHP weight (${\mathrm{w}}_{\mathrm{cj}}$) calculated by the combination weighting method.

Main Target | Facet for the First Layer | Facet Weight $({}_{}{}^{1}\mathbf{w}{}_{\mathbf{c}\mathbf{i}})$ | Criterion (indicator) for the Second Layer | Dimension | Indicator Weight $({}_{}{}^{2}\mathbf{w}{}_{\mathbf{c}\mathbf{j}})$ | Total Weight (Entropy-AHP ${\mathbf{w}}_{\mathbf{c}\mathbf{j}=}{}_{}{}^{1}\mathbf{w}{}_{\mathbf{c}\mathbf{i}}\times {}_{}{}^{2}\mathbf{w}{}_{\mathbf{c}\mathbf{j}}$) |
---|---|---|---|---|---|---|

Suitable supplier selection | Product satisfaction (A) | 0.5042 | A1.Rate of qualified products | positive | 0.4790 | 0.2415 |

A2.Product price (thousand dollars) | negative | 0.2985 | 0.1505 | |||

A3.Rate of Product market share | positive | 0.2225 | 0.1122 | |||

Subtotal | 1 | --- | ||||

Supply innovation capability (B) | 0.2122 | B1.Supply capacity (kg/time) | positive | 0.4951 | 0.1051 | |

B2.New product development rate (%) | positive | 0.5049 | 0.1071 | |||

Subtotal | 1 | --- | ||||

Service level © | 0.2836 | C1. Delivery time (days) | negative | 0.3152 | 0.0894 | |

C2. Delivery on time ratio (%) | positive | 0.6848 | 0.1942 | |||

Subtotal | 1 | --- |

Alternatives | ${\mathbf{A}}_{1}$ | ${\mathbf{A}}_{2}$ | ${\mathbf{A}}_{3}$ | ${\mathbf{A}}_{4}$ | ${\mathbf{A}}_{5}$ |
---|---|---|---|---|---|

${\mathrm{S}}^{+}$ | 0.0069 | 0.0149 | 0.0071 | 0.0101 | 0.0138 |

Alternatives | ${\mathbf{A}}_{1}$ | ${\mathbf{A}}_{2}$ | ${\mathbf{A}}_{3}$ | ${\mathbf{A}}_{4}$ | ${\mathbf{A}}_{5}$ |
---|---|---|---|---|---|

${\mathrm{S}}^{-}$ | 0.0123 | 0.0045 | 0.0105 | 0.0116 | 0.0061 |

Alternatives | ${\mathbf{A}}_{1}$ | ${\mathbf{A}}_{2}$ | ${\mathbf{A}}_{3}$ | ${\mathbf{A}}_{4}$ | ${\mathbf{A}}_{5}$ |
---|---|---|---|---|---|

${\mathsf{\phi}}_{\mathrm{i}}$ | 0.6395 | 0.2326 | 0.5946 | 0.5350 | 0.3074 |

Options | ${\mathbf{A}}_{1}$ | ${\mathbf{A}}_{2}$ | ${\mathbf{A}}_{3}$ | ${\mathbf{A}}_{4}$ | ${\mathbf{A}}_{5}$ |
---|---|---|---|---|---|

Rank | 1 | 5 | 2 | 3 | 4 |

**Table 11.**Facet weights of AHP-based technique for order preference by similarity to an ideal solution (TOPSIS) and entropy-AHP TOPSIS.

MCDM Method | Product Satisfaction (A) | Supply Innovation Capability (B) | Service Level (C) |
---|---|---|---|

AHP-based TOPSIS | 0.3916 | 0.2815 | 0.3269 |

Entropy-AHP TOPSIS | 0.5042 | 0.2122 | 0.2836 |

MCDM Method | Rate of Qualified Products (A1) | Product Price (Thousand Dollars) (A2) | Rate of Product Market Share (A3) | Supply Capacity (kg/ time) (B1) | New Product Development Rate (%) (B2) | Delivery Time (days) (C1) | Delivery on Time Ratio (%) (C2) |
---|---|---|---|---|---|---|---|

AHP-based TOPSIS | 0.4125 | 0.3759 | 0.2116 | 0.5293 | 0.4707 | 0.3917 | 0.6083 |

Entropy-AHP TOPSIS | 0.4790 | 0.2985 | 0.2225 | 0.4951 | 0.5049 | 0.3152 | 0.6848 |

**Table 13.**Sensitivity analysis of the facet A weight (${}_{}{}^{1}\mathrm{w}{}_{\mathrm{h}1}\mathrm{in}{}_{}{}^{1}\mathrm{w}{}_{\mathrm{c}1})$ to the outcome of the alternatives. in entropy-AHP TOPSIS.

${}_{}{}^{1}\mathbf{w}{}_{\mathbf{h}1}=$$-50\mathbf{\%}$ | ${}_{}{}^{1}\mathbf{w}{}_{\mathbf{h}1}=$$-40\mathbf{\%}$ | ${}_{}{}^{1}\mathbf{w}{}_{\mathbf{h}1}=$$-30\mathbf{\%}$ | ${}_{}{}^{1}\mathbf{w}{}_{\mathbf{h}1}=-20\mathbf{\%}$ | ${}_{}{}^{1}\mathbf{w}{}_{\mathbf{h}1}=$$-10\mathbf{\%}$ | ${}_{}{}^{1}\mathbf{w}{}_{\mathbf{h}1}=$$0$ | ${}_{}{}^{1}\mathbf{w}{}_{\mathbf{h}1}=10\mathbf{\%}$ | ${}_{}{}^{1}\mathbf{w}{}_{\mathbf{h}1}=$$20\mathbf{\%}$ | ${}_{}{}^{1}\mathbf{w}{}_{\mathbf{h}1}=$$30\mathbf{\%}$ | ${}_{}{}^{1}\mathbf{w}{}_{\mathbf{h}1}=$$40\mathbf{\%}$ | ${}_{}{}^{1}\mathbf{w}{}_{\mathbf{h}1}=$$50\mathbf{\%}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathrm{A}}_{1}$ | 0.6406 | 0.6402 | 0.6400 | 0.6398 | 0.6396 | 0.6395 | 0.6394 | 0.6393 | 0.6392 | 0.6392 | 0.6406 |

${\mathrm{A}}_{2}$ | 0.2423 | 0.2391 | 0.2367 | 0.2350 | 0.2336 | 0.2326 | 0.2317 | 0.2310 | 0.2304 | 0.2299 | 0.2423 |

${\mathrm{A}}_{3}$ | 0.6000 | 0.5982 | 0.5969 | 0.5959 | 0.5952 | 0.5946 | 0.5941 | 0.5937 | 0.5934 | 0.5931 | 0.6000 |

${\mathrm{A}}_{4}$ | 0.5247 | 0.5283 | 0.5308 | 0.5326 | 0.5339 | 0.5350 | 0.5359 | 0.5366 | 0.5371 | 0.5376 | 0.5247 |

${\mathrm{A}}_{5}$ | 0.3664 | 0.3483 | 0.3343 | 0.3234 | 0.3146 | 0.3074 | 0.3014 | 0.2964 | 0.2921 | 0.2884 | 0.3664 |

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**MDPI and ACS Style**

Chen, C.-H.
A Novel Multi-Criteria Decision-Making Model for Building Material Supplier Selection Based on Entropy-AHP Weighted TOPSIS. *Entropy* **2020**, *22*, 259.
https://doi.org/10.3390/e22020259

**AMA Style**

Chen C-H.
A Novel Multi-Criteria Decision-Making Model for Building Material Supplier Selection Based on Entropy-AHP Weighted TOPSIS. *Entropy*. 2020; 22(2):259.
https://doi.org/10.3390/e22020259

**Chicago/Turabian Style**

Chen, Chun-Ho.
2020. "A Novel Multi-Criteria Decision-Making Model for Building Material Supplier Selection Based on Entropy-AHP Weighted TOPSIS" *Entropy* 22, no. 2: 259.
https://doi.org/10.3390/e22020259