In this study, we consider a perishable inventory system that has an (s, Q) ordering policy, along with a finite waiting hall. The single server, which provides an item to the customer after completing the required service performance for that item, only begins serving after N customers have arrived. Impatient demand is assumed in that the customers waiting to be served lose patience and leave the system if the server’s idle time overextends or if the arriving customers find the system to be full and will not enter the system. This article analyzes the impatient demands caused by the N-policy server to an inventory system. In the steadystate, we obtain the joint probability distribution of the level of inventory and the number of customers in the system. We analyze some measures of system performance and get the total expected cost rate in the steadystate. We present a beneficial cost function and confer the numerical illustration that describes the impact of impatient customers caused by N-policy on the inventory system’s total expected cost rate.

Consider the Diophantine equation $y^n=x+x(x+1)+\cdots +x(x+1)\cdots (x+k)$, where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality $n=$ 19,736 to obtain all solutions $(x,y,n)$ of the equation for the fixed positive integers $k\le 10$. In this paper, we improve the bound as $n\le$ 10,000 for the same case $k\le 10$, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

Over the past few decades, extensive attention has been given by researchers and practitioners to the development and application of multi-criteria decision-making (MCDM) methods within intuitionistic fuzzy environments across a wide range of fields and disciplines. This challenging research area has emerged as one of the most prominent topics, and its importance and popularity are expected to continue growing in the future. The elliptic intuitionistic fuzzy set (EIFS) addresses complex, multidimensional, non-symmetrical vagueness and uncertainty more effectively than other traditional intuitionistic fuzzy sets (IFSs). Sustainable renewable energy source selection is a critical decision-making (DM) process aiming to identify the most suitable energy alternative. The process of selecting sustainable renewable energy sources necessitates a comprehensive assessment of numerous criteria, which encompass environmental ramifications, economic feasibility, and societal acceptance. Contemporary research suggests novel methodologies to enhance this selection process, highlighting the need for an MCDM framework that integrates a variety of factors. This study presents an innovative DM framework for sustainable renewable energy source selection based on EIFS and a newly developed aggregation operator, the Elliptic Intuitionistic Fuzzy Weighted Muirhead Mean Aggregation (EIFWMMA) operator. These mechanisms expand upon conventional intuitionistic fuzzy frameworks by employing an elliptical portrayal of membership and non-membership degrees, facilitating a more accurate and lifelike representation of uncertainty and hesitation in evaluations by experts. To enhance computational efficiency, the framework weaves together machine learning-driven dimensionality reduction and weight optimization strategies of principal component analysis (PCA) for DM. The suggested operators are employed in an MCDM scenario centered around the selection of sustainable renewable energy sources, where the hierarchy of alternatives is established through score values derived from EIFWMMA. A comparative exploration of Circular Intuitionistic Fuzzy Sets (C-IFSs) and Interval-Valued Intuitionistic Fuzzy Sets (IVIFSs) uncovers that the elliptical formulation yields consistently reliable, precise, and geometrically comprehensible results. The findings affirm that EIFS-based operators offer a resilient, adaptable, and broadly applicable strategy for tackling MCDM challenges amidst uncertainty. The Min–Max normalization method is employed to validate our proposed methodology for identifying alternatives within the MCDM paradigm. It also improves accuracy, stability, and scalability in comparison to conventional approaches.