This paper presents an inventory model with a price‐dependent stochastic demand and finite replenishment rate mode. It establishes a production‐inventory system with a controllable deterioration rate, preservation technology, different types of rework process, promotional effort, and carbon emission reduction. This study finds the optimal solution and determines the most efficient production and safety technology investments to maximize the manufacturer’s total profit over the planning horizon. Also, this inventory model balances sustainability and profitability to maximize the profit and to determine the optimal cycle length. In addition, the research highlights the most advantageous investments in production and preservation technology, emphasizing their critical role in increasing profitability and identifying the most effective duration of the production cycle. The study presents a comprehensive approach to inventory optimization that tackles practical and strategic challenges in manufacturing contexts. This research refers to cases and numerical examples, providing management insights and sensitivity analysis to determine the optimal course of action to change the parameters of the model.

In this paper, a two-warehouse inventory model with polynomial time-dependent demand for noninstantaneous deteriorating items is taken into consideration. As advertisement plays a vital role in business, more advertisements are to be made by the retailer to increase demand in the competitive market. In order to meet the high demand, it is necessary to maintain two warehouses to store the goods in inventory: one owned warehouse with limited capacity and one rented warehouse with unlimited capacity. In order to increase sales, the retailer anticipates a discount from the manufacturer, and in turn, the retailer is equipped to pay a portion of the purchase price in advance. The model aims to optimize the overall profit by determining the appropriate order quantity and cycle length. Furthermore, numerical examples are provided for the validation of the proposed model.

Two vertices of a graph \(G\) are said to be radial to each other if the distance between them is equal to the radius of the graph. The radial graph of a graph \(G\), denoted by \(R(G)\), has the vertex set as in \(G\) and two vertices are adjacent in \(R(G)\) if and only if they are radial to each other in \(G\). If \(G\) is disconnected, then the two vertices are adjacent in \(R(G)\) if they belong to different components of \(G\). The main objective of this paper is to determine the radial graphs of some families of product graphs.