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In this paper, we investigate Bullen-type inequalities applicable to functions that are twice-differentiable. To explore these advanced inequalities, we utilize generalized convexity and Riemann-type fractional integrals. A comparative analysis is provided to highlight the more refined inequalities from among the explored results. By exploring the limiting cases, a relation with existing literature is established. Several examples are also presented to illustrate the outcomes and their accuracy is validated through graphical analysis. Additionally, applications in generalized means are also discussed., Competing Interests: The authors declare that they have no competing interests., (© 2024 The Author(s).)

In today's business world, choosing a logistics supplier is a critical factor for companies to improve operational efficiency and reduce business costs. With the development of market economy, it is very difficult for companies to choose a suitable logistics provider according to specific rules. Therefore, this study proposes a new three-way decision making (TWD) technique for supplier selection in logistics service value creation. For this, we first develop a new concept called intuitionistic double hierarchy linguistic term set (IDHLTSs) that can describe uncertainty and ambiguity in a more flexible way. Some Hamacher aggregation operators for collecting IDHLTSs information and its basic aspect are proposed. The unknown weight vector for decision experts and criteria is determined by using entropy measures. In addition, the conditional probability is determined using TOPSIS which makes the decision making process more rational. And the decision result is conducted according to minimum loss principle. Finally, an example of 3 PL supplier selection in the logistics service value co-creation environment and comparison is given to validate and demonstrate the effectiveness of the developed method., Competing Interests: The author declare that they have no known competing financial interest or personal relationship that could have appeared to influence the work repeated in this paper., (© 2023 Published by Elsevier Ltd.)