For multi-attribute decision-making problems with interval numbers whose attribute weights are completely unknown, it is difficult to compare the value of interval numbers. So this paper proposes a method based on TOPSIS and the weighted parameter to deal with it. Firstly, this paper transforms the interval number matrix into two exact number matrices which reduces sorting complexity. Secondly, a parameter is given when determining the weights with entropy weight in order to reflect all the information of the interval numbers. Then TOPSIS is used to determine the order of each scheme. In addition, the average value of the ranking number is used to reflect the actual situation better. Finally, the analysis of the car purchase case shows the proposed method is feasible and practical. And the comparative analysis with the other method based on practical application dataset demonstrates the proposed method is stable and effective.
Due to the complex objective world and its uncertainty, it is difficult to describe the relevant attributes in decision-making problems with exact numbers. To make the model closer to the facts, interval numbers without clear preference information are often used when processing data, thus causing interval number multi-attribute decision-making problems, such as .
At present, the research on such problems has gained attention from scholars at home and abroad. The research focuses on the following four aspects. The first is to sort the interval numbers according to possibility degree, relative superiority degree. Li Z W  used relative superiority degree and defined a new sorting vector to rank interval numbers. Li D Q  found the ranking method of possibility degree may obtain contrary results to the meaning of possibility degree. So a revised ranking by Boolean matrix was discussed. But the scientificity of the new method needs to be further verified. Yao N  defined an interval number ranking method considering symmetry axis compensation which consider multiple attitudes of decision makers with different risk appetites. Firozja, M  proposed a new interval distance of two fuzzy numbers that satisfy on metric properties. The second is to determine the weight by studying the application of entropy weight method , . Dong P Y  proposed a combined weight method. Yue Z L  developed a determining weights method for group decision-making problems which each individual decision information is interval numbers. The third is to sort the schemes with the help of projection, grey relation analysis, etc , , , , , , , , , . The forth is to propose the methods for group decision-making problems , , .
Conclusion and promotion
It is difficult to compare the value of two intervals. In this paper, a novel method is proposed to solve multi-attribute decision-making problems with interval numbers. Two types of exact numbers which are left end point and length are used to replace an interval number which facilitates the determination of attribute weights. At the same time, when determining weights, a variety of possible rankings are considered, and the information brought by interval numbers is fully utilized to ensure the accuracy of results.
When extracting exact numbers from interval numbers, other methods can also be applied, that is, interval numbers can be replaced with the right end point and length, or the left end point and the midpoint, or the right end point and the midpoint. And the other steps are the same as the corresponding content in this paper. All of them can reduce the difficulty of the problem, and the optimal solution obtained is the same as the method presented in this paper.
Finally, the feasibility and effectiveness of the proposed method have been demonstrated by car purchasing. The method can also be used to solve the group decision-making problems with intervals.