The paper considers the multi-criteria decision-making problem based on linguistic picture fuzzy information. Firstly, we propose the concept of linguistic picture fuzzy set(LPFS), where the positive-membership, the neutral-membership and the negative-membership are represented by linguistic variables, and its operation rules are also discussed. The linguistic picture fuzzy weighted averaging (LPFWA) operator and linguistic picture fuzzy weighted geometric (LPFWG) operator are developed based on the proposed operation rules. Secondly, we propose the generalized weighted distance measure, the generalized weighted Hausdorff distance measure, and the generalized hybrid weighted distance measure between LPFSs and discuss their properties. Thirdly, we extend the technique for order of preference by similarity to the ideal solution (TOPSIS) method and the TODIM (an acronym in Portuguese of interactive and multi-criteria decision-making) method to the proposed distance measure, and the linguistic picture fuzzy entropy method is proposed to calculate the weights of the criteria. Finally, an illustrative example is given to verify the feasibility and effectiveness of the proposed methods, the comparative analysis with other existing methods and sensitivity analysis of the proposed methods are also discussed.

中文翻译：

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语言图像模糊集及其在多准则决策中的应用：基于熵权的 TOPSIS 和 TODIM 方法的说明

论文考虑了基于语言图像模糊信息的多准则决策问题。首先，我们提出了语言图像模糊集（LPFS）的概念，其中正隶属、中性隶属和负隶属由语言变量表示，并讨论了其运算规则。语言图片模糊加权平均（LPFWA）算子和语言图片模糊加权几何（LPFWG）算子是基于提出的操作规则开发的。其次，我们提出了LPFS之间的广义加权距离测度、广义加权Hausdorff距离测度和广义混合加权距离测度，并讨论了它们的性质。第三，我们将通过相似性的偏好顺序技术扩展到理想解决方案 (TOPSIS) 方法和 TODIM（葡萄牙语中交互式和多标准决策的首字母缩写词）方法到所提出的距离度量，以及语言图片模糊熵方法建议计算标准的权重。最后，通过一个实例验证了所提出方法的可行性和有效性，并讨论了与其他现有方法的比较分析和所提出方法的敏感性分析。