Some properties of Skorokhod metric on fuzzy sets

This paper discusses the properties of Skorokhod metric on normal and upper semi-continuous fuzzy sets on metric space. All fuzzy sets mentioned below refer to this type of fuzzy sets. We confirm that the Skorokhod metric and the enhanced-type Skorokhod metric are equivalent on compact fuzzy sets. However, the Skorokhod metric and the enhanced-type Skorokhod metric are not necessarily equivalent on L-p-integrable fuzzy sets, which include compact fuzzy sets. We point out that the L-p-type dpmetric, p >= 1, is well-defined in common cases but the d(p) metric is not always well-defined on all fuzzy sets. We introduce the d(p)(*) metric which is an expansion of the dpmetric, and write d(p)(*) as dpin the sequel. Then, we investigate the relationship between these two Skorokhod-type metrics and the dpmetric. We show that the relationship can be divided into three cases. On compact fuzzy sets, the Skorokhod metric is stronger than the dpmetric. On L-p-integrable fuzzy sets, the Skorokhod metric is not necessarily stronger than the dpmetric, but the enhanced-type Skorokhod metric is still stronger than the dpmetric. On all fuzzy sets, even the enhanced-type Skorokhod metric is not necessarily stronger than the dpmetric. We also show that the Skorokhod metric is stronger than the sendograph metric. At last, we give a simple example to answer some recent questions involved the Skorokhod metric. (C) 2021 Elsevier B.V. All rights reserved.