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In this study, we investigate various goal programming models and highlight their problematic outcomes, offering our revised versions. While previous publications relied on numerical examples to determine the optimal alternative, we provide theoretical explanations for these examples to enhance researchers' understanding of goal programming models. Additionally, we employ a fast integration approach to solve a challenging calculus problem concerning the centroid of trapezoidal fuzzy numbers. Our findings aim to assist researchers in comprehending intricate theoretical concepts and facilitate the application of these results in their own research endeavors. [ABSTRACT FROM AUTHOR]

Researchers developed a method to apply the row geometric mean method to improve the consistent index of a given reciprocal comparison matrix. We provide a detailed examination of their approach to find out that sometimes their adjustments will violate the 1-9 scale rule proposed by the original restriction in Analytic Hierarchy Process. Hence, their development contained severely questionable results. We advise practitioners with caution when trying to apply their approach to adjust their comparison matrices. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a complete and simplified solution procedure for the inventory model under trapezoidal type demand, Weibull-distributed deterioration, and partial backlogging. A published paper already provide a solution approach for this kind of inventory models. However, we will show that his approach is too complicated that depending on the partition the domain of a trapezoidal type demand to result in three sub-optimal problems. However, this paper will show that the already exist approach is correct but tedious. Moreover, a generalization of this kind of inventory systems from a trapezoidal type demand to any positive demand under a condition of generalized backlogged rate is also developed by this paper. [ABSTRACT FROM AUTHOR]

We aim to address an unresolved question raised in a previously published paper regarding the 1-9 bounded scale and consistency index. The objective of this study is to demonstrate that researchers need not be concerned about the inconsistency phenomenon highlighted in the aforementioned article. This paper has a four-fold purpose. Firstly, we illustrate that by employing an alternative ordering approach for the entries of a comparison matrix, the issue of inconsistency can be eliminated. Secondly, we emphasize that, by accepting Murphy's assumptions, we can present an efficient method for deriving the priority vector without relying on the comparison matrix. This reveals that Murphy's assumptions are excessively stringent for ordinary researchers to accept. Additionally, we offer a simplified version to establish the condition for the optimal solution in an inventory model with a temporary price discount, which was previously proposed in another published paper. Consequently, an additional condition put forth by the aforementioned paper becomes redundant. We have made revisions to a related paper discussing local stability intervals in the Analytic Hierarchy Process. [ABSTRACT FROM AUTHOR]

We study the minimum problem proposed by a recently published paper to point out that their formulated optimal solution is incomplete since in their formulations variables are mingled together. For a reduced version of their minimum problem, we convert the objective function from six variables into a variable minimum problem. For a special case of the fuzzy controlling parameter, we derive an approximated formulated solution that attains the minimum value within 0.0208% average estimation error. In the direction of future research, we propose a minimum problem under symmetric expressions. [ABSTRACT FROM AUTHOR]