The product replacement graph Γk(G) is the graph on the generating k-tuples of a group G, with edges corresponding to Nielsen moves. We prove the exponential growth of product replacement graphs Γk(Gω) of Grigorchuk groups, for k ≥ 5.

Plastics are the most utilized materials for product packaging in manufacturing industries, from electronics to food and fashion accessories. However, numerous challenges surround plastics because of their non-biodegradability, which poses a severe threat environment. This study has uncovered possibilities replacing discouraging use products. A few scholarly articles have successfully proven th...

‎this paper proposes a multi-supplier multi-product inventory model in which the suppliers have unlimited production capacity, allow delayed payment, and offer either an all-unit or incremental discount. the retailer can delay payment until after they have sold all the units of the purchased product. the retailer’s warehouse is limited, but the surplus can be stored in a rented warehouse at a...

Product mix problem (PMP) is one of the most important and complicated problems in production systems. Different approaches have been applied to solve this problem, among them, theory of constraints (TOC) has been widely considered since 1990s. This paper develops a distinguished algorithm to solve product mix prob-lems that is efficient both in single and multi-bottleneck problems. At first, t...

One of the most important items for better file system performance is efficient buffering of disk blocks in main memory. Efficient buffering helps to reduce the widespeed gap between main memory and hard disks. In this buffering system, the block replacement policy is one of the most important design decisions that determines which disk block should be replaced when the buffer is full. To o...