## Lianhui Li* , Guanying Xu* and Hongguang Wang**## |

Supplier | C_{1} | C_{2}/ full score is 17 | C_{3}/ full score is 9 | C_{4}/ full score is 1 | |||
---|---|---|---|---|---|---|---|

C_{11}/¥ | C_{12}/ error value (mm) | C_{13} | C_{14} | ||||

x_{1} | 6.4×10^{1} | 0.01 | Good | Very bad | 15 | [8.5, 9] | 0.9848 |

x_{2} | 1.9×10^{1} | 0.01 | Very good | Good | [15, 16] | [8.5, 9] | 1 |

x_{3} | 2.8×10^{1} | 0.03 | Very bad | Good | [5, 9] | / | 1 |

Corresponding to the remark level [TeX:] $$\left\{G_{1}, G_{2}, G_{3}, G_{4}, G_{5}\right\}$$, the reference values of the index belonging to quantitative type are as follows: [TeX:] $$G\left(C_{11}\right)=\left\{10^{5}, 10^{4}, 10^{3}, 10^{2}, 10^{1}\right\}, G\left(C_{12}\right)=\{0.05,0.04,0.03,0.02,0.01\}$$, [TeX:] $$G\left(C_{2}\right)=\{17,13,9,5,1\}, G\left(C_{3}\right)=\{1,3,5,7,9\} \text { and } G\left(C_{4}\right)=\{0,0.25,0.5,0.75,1\}$$.

Then, the membership degree of initial index value to every remark level is obtained. The data in Table 1 is translated into the membership degree form corresponding to remark grade. As shown in Table 2, the tendency degree form of initial index value is obtained.

Table 2.

Supplier | C_{11} | C_{12} | C_{13} | C_{14} | C_{2} | C_{3} | C_{4} |
---|---|---|---|---|---|---|---|

>x_{1} | 0.8500 | 1.0000 | 0.7500 | 0 | 0.8750 | 0.9688 | 0.9848 |

>x_{2} | 0.4750 | 1.0000 | 1.0000 | 0.7500 | 0.8813 | 0.9688 | 1.0000 |

>x_{3} | 0.2000 | 0.3000 | 0 | 0.7500 | 0.3750 | / | 1.0000 |

We define the set of candidate suppliers as the D-S theory identification framework: [TeX:] $$\Theta=\left\{x_{1}, x_{2}, x_{3}\right\}$$, Here, x_{1},x_{2} and x_{3} represent bearing-cage suppliers 1, 2, and 3, respectively.

For four indexes C_{11},C_{12},C_{13} and C_{14} and three criterions C_{2},C_{3}, and C_{4}, the weighted BPA values of all focal elements are obtained according to the tendency degree shown in Table 2 and the weight vectors [TeX:] $$\left(\omega\left(C_{11}\right), \omega\left(C_{12}\right), \omega\left(C_{13}\right), \omega\left(C_{14}\right)\right)=(0.30,0.23,0.25,0.22), \text { and }\left(\omega\left(C_{2}\right), \omega\left(C_{3}\right), \omega\left(C_{4}\right)\right)=(0.18,0.26,09)$$. The calculation result is as follows:

(1) C_{11}: [TeX:] $$\tilde{m}_{11}\left(x_{1}\right)=0.1672, \quad \tilde{m}_{11}\left(x_{2}\right)=0.0934, \quad \tilde{m}_{11}\left(x_{3}\right)=0.0393, \quad \tilde{m}_{11}(\Theta)=0.7000$$

(2) C_{12}: [TeX:] $$\tilde{m}_{12}\left(x_{1}\right)=0.0920, \quad \tilde{m}_{12}\left(x_{2}\right)=0.0920, \quad \tilde{m}_{12}\left(x_{3}\right)=0.0460, \quad \tilde{m}_{12}(\Theta)=0.7700$$

(3) C_{13}: [TeX:] $$\tilde{m}_{13}\left(x_{1}\right)=0.1071, \tilde{m}_{13}\left(x_{2}\right)=0.1429, \quad \tilde{m}_{13}\left(x_{3}\right)=0, \quad \tilde{m}_{13}(\Theta)=0.7500$$

(4) C_{14}: [TeX:] $$\tilde{m}_{14}\left(x_{1}\right)=0, \quad \tilde{m}_{14}\left(x_{2}\right)=0.1100, \tilde{m}_{14}\left(x_{3}\right)=0.1100, \tilde{m}_{14}(\Theta)=0.7800$$

(5) C_{2}: [TeX:] $$\tilde{m}_{2}\left(x_{1}\right)=0.0739, \quad \tilde{m}_{2}\left(x_{2}\right)=0.0744, \quad \tilde{m}_{2}\left(x_{3}\right)=0.0317, \quad \tilde{m}_{2}(\Theta)=0.8200$$

(6) C_{3}: [TeX:] $$\tilde{m}_{3}\left(x_{1}\right)=0.1300, \quad \tilde{m}_{3}\left(x_{2}\right)=0.1300, \quad \tilde{m}_{3}(\Theta)=0.7400$$

(7) C_{4}: [TeX:] $$\tilde{m}_{4}\left(x_{1}\right)=0.0297, \tilde{m}_{4}\left(x_{2}\right)=0.0302, \tilde{m}_{4}\left(x_{3}\right)=0.0302, \tilde{m}_{4}(\Theta)=0.9100$$

After that, we take [TeX:] $$\tilde{m}_{11}\left(x_{i}\right), \tilde{m}_{12}\left(x_{i}\right), \tilde{m}_{13}\left(x_{i}\right), \text { and } \tilde{m}_{14}\left(x_{i}\right)$$ as the evidence input and implement the first evidence fusion. The BPA values of all focal elements are obtained as follows: [TeX:] $$m_{1}\left(x_{1}\right)=0.1001,m_{1}\left(x_{2}\right)=0.7815, m_{1}\left(x_{3}\right)=0.0772, m_{1}\left(x_{1}, x_{2}\right)=0.0102, m_{1}\left(x_{2}, x_{3}\right)=0.0201, m_{1}\left(x_{1}, x_{3}\right)=0.0098, \text { and } m_{1}(\Theta)=0.0011.$$

We normalize BPA values [TeX:] $$m_{1}\left(A_{i}\right)$$ of the suppliers to be evaluated and on index C_{1}. With the consideration of [TeX:] $$\omega\left(C_{1}\right)$$, the weighted BPA values are obtained as follows: [TeX:] $$\tilde{m}_{1}\left(x_{1}\right)=0.0571, \tilde{m}_{1}\left(x_{2}\right)=0.4455, \tilde{m}_{1}\left(x_{3}\right)=0.0440, \quad \tilde{m}_{1}\left(x_{1}, x_{2}\right) 0.0058, \quad \tilde{m}_{1}\left(x_{2}, x_{3}\right)=0.0115, \tilde{m}_{1}\left(x_{1}, x_{3}\right)=0.0056, \text { and } \tilde{m}_{1}(\Theta)=0.0006$$.

Then, we take [TeX:] $$\tilde{m}_{1}\left(A_{i}\right), \tilde{m}_{2}\left(A_{i}\right), \tilde{m}_{3}\left(A_{i}\right) \text { and } \tilde{m}_{4}\left(A_{i}\right)$$ as the evidence input and implement the second evidence fusion. The comprehensive BPA values of all focal elements are obtained as follows: [TeX:] $$m\left(x_{1}\right)=0.1255, m\left(x_{2}\right)=0.7088, m\left(x_{3}\right)=0.0102, m\left(x_{1}, x_{2}\right)=0.0999, m\left(x_{2}, x_{3}\right)=0.0032, m\left(x_{1}, x_{3}\right)=0.0506\ and\ m(\Theta)=0.0018_{\circ}$$

[TeX:] $$\operatorname{Bel}\left(A_{i}\right) \text { and } \operatorname{Pl}\left(A_{i}\right)$$ of all suppliers are calculated. Then the trust intervals of all suppliers are obtained as follows:

(1) [TeX:] $$x_{1} :[0.1255,0.2778]$$

(2) [TeX:] $$x_{2} :[0.7088,0.8137]$$

(3) [TeX:] $$x_{3} :[0.0102,0.0658]$$

On the basis of the D-S theory decision regulations, the result is as follows:

(1) [TeX:] $$P\left(x_{1}>x_{2}\right)=0,$$ so x_{1} x_{2}.

(2) [TeX:] $$P\left(x_{1}>x_{3}\right)=1,$$ so x_{3} x_{1}

Therefore, the evaluation result of three suppliers is x_{3} x_{1} x_{2} and supplier 2 is the optimal bearingcage supplier. Thus, the proposed adaptive weight D-S theory model can solve the supplier evaluation problem in GSC even the initial index value is uncertain and incomplete (See in Table 1, the initial values of x_{2} and x_{3} on index C_{2} are interval values and the initial value of x_{3} on index C_{3} is missing).

To verify the effectiveness of the proposed adaptive weight D-S theory model, we use traditional TOPSIS method [18,21] to make a comparison. Because the traditional TOPSIS method can only solve the evaluation problem with certain and complete index value, we replace the interval with its mid-value and ignore the index with missing index value (The initial values of x_{2} and x_{3} on index C_{2} are 15.5 and 7, respectively. Index C3 is ignored). The tendency degree method is still used to process the initial index value. Then, processing of the adaptive weights is executed on the basis of the hierarchical structure shown in Fig. 1 and the final weight vector of C_{11},C_{12},C_{13},C_{14},C_{2} and C_{4} is [TeX:] $$\omega=(0.15,0.11,0.12,0.09,0.18,0.09 )$$ in which index C_{3} is ignored. In Table 3, the weighted index value matrix is obtained.

Table 3.

Supplier | C_{11} | C_{12} | C_{13} | C_{14} | C_{2} | C_{4} |
---|---|---|---|---|---|---|

x_{1} | 0.1275 | 0.1100 | 0.0900 | 0 | 0.1575 | 0.0886 |

x_{2} | 0.0712 | 0.1100 | 0.1200 | 0.0675 | 0.1586 | 0.0900 |

x_{3} | 0.0300 | 0.0330 | 0 | 0.0675 | 0.0675 | 0.0900 |

From Table 3, the positive and negative ideal points are (0.1275, 0.1100, 0.1200, 0.0675, 0.1586, 0.0900) and (0.0300, 0.0330, 0, 0, 0.0675, 0.0886), respectively. So we can obtain the close degree to positive ideal point of each supplier is as follows:

(1) x_{1}:0.7065.

(2) x_{2}:0.7686.

(3) x_{3}:0.2569.

Therefore, the evaluation result of three candidate suppliers by traditional TOPSIS method is x_{3} x_{1} x_{2} and the bearing manufacturing enterprise should choose supplier 2 as the optimal bearing cage supplier. The evaluation results of the proposed adaptive weight D-S theory model and traditional TOPSIS method are consistent. This shows that the proposed adaptive weight D-S theory model is feasible and effective.

In this paper, an adaptive weight D-S theory model is proposed for the evaluation problem characterized by uncertainty and incompleteness and variable index weight in GSC. In addition, a fuzzyrough- sets-AHP approach is designed to obtain the adaptive index weight. The index framework is established considering of the main factors affecting the supplier evaluation in GSC, which can improve the scientific nature and rationality. The case study and the comparison with TOPSIS show that the optimal supplier of manufacturing enterprise can be correctly selected by the proposed adaptive weight D-S theory model.

This paper is supported by Key Scientific Research Projects in 2017 at North Minzu University (No. 2017KJ22), the Third Batch of Ningxia Youth Talents Supporting Program (No. TJGC2018048), Natural Science Foundation of Ningxia Province (No. NZ17113), and Ningxia First-class Discipline and Scientific Research Project: Electronic Science and Technology (No. NXYLXK2017A07).

He received his M.E. degree in Vehicle Engineering from Henan University of Science and Technology, China, in 2010 and his Ph.D. degree in Aeronautics and Astronautics Manufacturing Engineering from Northwestern Polytechnical University, China, in 2016. He is currently a lecturer at North Minzu University, China. His current research interests include CAD/CAM and logistics engineering.

He received his B.S. degree in Computer Science and Engineering from Anhui University of Technology, China, in 2017. He is currently a master’s degree candidate in Computer Science and Engineering at North Minzu University. His current research interests include uncertainty theory, fuzzy decision making, and data stream classification.

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