Convexity plays a crucial role in the development of fractional integral inequalities. Many fractional integral inequalities are derived based on convexity properties and techniques. These inequalities have several applications in different fields such as optimization, mathematical modeling and signal processing. The main goal of this article is to establish a novel and generalized identity for the Caputo–Fabrizio fractional operator. With the help of this specific developed identity, we derive new fractional integral inequalities via exponential convex functions. Furthermore, we give an application to some special means.

Order picking is referred as a critical process of selecting items requested by a customer in a warehouse. Meeting the demand of every customer is the main objective in this area. Large warehouses pose a challenge since the order-picking process is slowed considerably by the lengthy time it takes to transport items across the warehouse. Throughout the study, the system is hoped to develop proper procedures in the order-picking process. In handling this scenario, the decision-makers need to take any possible action to ensure the warehouses can keep operating and meeting the requirements and satisfaction of the customers. Due to this, the study’s main objective is to determine whether the Floyd–Warshall algorithm or the dynamic programming method will give the most accurate shortest path and minimum travel distance for order pickers. Two data sets (nine nodes and nineteen nodes) are used to determine the optimal path and minimum travel distance for the order picker to meet and satisfy customer orders for the warehouse. The two models were modified and applied to address real-world case studies from the automotive manufacturing company in Malaysia. The results show a big difference between the total distance by 113.48% for 19 nodes. Through this finding, the company may choose which method suits their preferences. Concurrently, this study may also contribute to problem-solving issues in any warehouse operation with a similar procedure.

In the classical Economic Order Quantity (EOQ) model, the common unrealistic assumptions are that all the purchased items are of perfect quality and the demand is constant. However, in a real-world environment, a portion of the purchased items might be damaged due to mishandling or an accident during the shipment process, and the demand rate may increase or decrease over time. Many companies are torn between repairing or replacing the imperfect items with new ones. The right decision on that options is crucial in order to guarantee that there is no shortage of stocks while at the same time not jeopardising the items’ quality and maximising the company’s profit. This paper investigates two EOQ models for imperfect quality items by assuming the demand rate varies with time. Under Policy 1, imperfect items are sent for repairs at an additional cost to the makeup margin; under Policy 2, imperfect items are replaced with equivalent quality items from a local supplier at a higher price. Two mathematical models are developed, and numerical examples along with sensitivity analyses are provided to illustrate these models. Our results reveal that Policy 1 is preferable to Policy 2 most of the time. However, Policy 2 outperforms Policy 1 if there is no minimum threshold on the purchased stock quantity. This research allows a company to discover solutions to previously identified inventory problems and make the inventory-patching process more controlled.

A two-warehouse inventory model with deteriorating items and rework process with time varying demand rate is presented. The Last-In-First-Out (LIFO) and First-In-First-Out (FIFO) policies are considered with the assumption that the holding cost is higher in the rented warehouse (RW) compared to the owned warehouse (OW). The aim of the proposed model is to determine the optimum values of time in a production cycle that will minimise the total relevant cost, TRC*. We have utilised Microsoft Excel Solver as a solution tool, in which the generalised reduced gradient (GRG Nonlinear) method has been chosen as the solving method. The result is further verified using the built-in function in the Mathematica software. We observed that given same changes are made to the parameters in both the LIFO and FIFO systems, a lower total relevant cost, TRC* is obtained in the LIFO system. This shall mean that the LIFO system is less expensive than the FIFO system, provided that the holding cost in RW is higher than the holding cost in OW. The flow of inventory in the LIFO system suggests that items stored last in the owned warehouse will be dispatched first. This is an important factor for manufacturers in ensuring that items are distributed at optimal freshness.

Fractional integral operators have been studied extensively in the last few decades, and many different types of fractional integral operators have been introduced by various mathematicians. In 1967 Michele Caputo introduced Caputo fractional derivatives, which defined one of these fractional operators, the Caputo Fabrizio fractional integral operator. The main aim of this article is to established the new integral equalities related to Caputo-Fabrizio fractional integral operator. By incorporating this identity and convexity theory to obtained a novel class of Simpson type inequality. In this paper, we present a novel generalization of Simpson type inequality via s-convex and quasi-convex functions. Then, utilizing this identity the bounds of classical Simpson type inequality is improved. Finally, we discussed some applications to Simps,on s quadrature rule.