Research Article ReliabilityandSensitivityAnalysisofaRepairableSystemwith WarrantyandAdministrativeDelayinRepair MohamedS.El-Sherbeny 1,2 andZienabM.Hussien 1 1 Department of Mathematics, Faculty of Sciences and Arts-Rabigh Campus, King Abdulaziz University, Jeddah, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt Correspondence should be addressed to Mohamed S. El-Sherbeny; m_el_sherbeny@yahoo.com Received 18 April 2021; Revised 24 May 2021; Accepted 31 May 2021; Published 10 June 2021 Academic Editor: Niansheng Tang Copyright © 2021 Mohamed S. El-Sherbeny and Zienab M. Hussien. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e purpose of this study is to analyze the behavior of some industrial systems in light of the cost-free warranty policy. According to this policy, we assume that the repairman is not always present in the system. When the active unit fails, the repairman will be called to visit the system; however, administrative procedures may delay the visit for some time. Once on the system, the repairman first inspects whether the fault is caused by the user or not and whether it is repairable or not. According to product warranty laws, the repairman carries out the repair or replacement of the faulty unit. e failure time, administrative delay time, inspection time, and repair time are assumed taken as a negative exponential distribution. e system model is analyzed by the supplementary variable technique and Laplace transform, as various performance metrics of system efficiency have been obtained. e sensitivity and relative sensitivity analyses for the system parameters have also been performed. Finally, an illustrative example is taken to illustrate the efficiency of the system. 1.Introduction In the present time, warranty is a necessary condition for selling many industrial products. Warranty is defined as the mandatory contract afforded by a manufacturer with regard to the sale of a product. In other words, the warranty is a legal responsibility in the following cases: early failure of the product or the disability of the product to perform its function. When selling a product, the warranty covers either free repair or free replacement. Rules of the warranty are classified as follows: (1) the manufacturer is not legally re- sponsible for the product in case of failure due to incorrect and careless use or deterioration of the product by the user; this makes the customer responsible for the cost of the product repair; (2) the customer is not legally responsible for the product in the following cases: (a) manufacturing defects and (b) errors in software; this makes the manufacturer re- sponsible for free repair or free replacement of the product. More recently, users have distinguished single-unit systems due to their ability to afford the cost and ingrained reliability. erefore, reliability models of single-unit sys- tems have been examined by researchers, such as Kumar et al. [1], Du et al. [2], Goel and Mumtaz [3], Ram and Kadyan [4], Attia et al. [5], and Malhotra and Taneja [6], under different conditions from failure and repair policies without interest paying much attention to any warranty policies and service agreement. e only way to ensure the efficiency of the product sold is after-sales services related to warranty policy. erefore, the longer warranty leads to best reliability and provides convenience for the users. Kadyan and Ramniwas [7] calculated the cost benefit of the single- unit system and derived some reliability measures when applying warranty policy on the system. e profit function of industrial system with the presence of a cost-free warranty policy is obtained by Ram and Garg [8]. Effect warranty policies on remanufactured products are discussed by Alqahtani and Gupta [9]. Ram et al. [10] study system consists of a single unit with inspection for the feasibility of repair beyond the warranty. Ram et al. [11] presented the concept of preventive maintenance (PM) beyond warranty Hindawi Journal of Mathematics Volume 2021, Article ID 9424215, 9 pages https://doi.org/10.1155/2021/9424215