Engineers design systems to be reliable and work to fulfil their missions without failure for a specific period. However, the system components deteriorate with time and lead to its failures. A frequent system failure increases the management costs, hence posing a challenge to decision-makers. Therefore, for the avoidance of frequent system failures, preventive maintenance is necessary. The objective of any manufacturing firm is to maximize profit and minimize costs. The interval for preventive maintenance can be optimized if the system’s availability is maximized and its cost function minimized. This study evaluates the availability and cost function for a continuous operating series-parallel system under a fixed time environment. A multiobjective model is formulated to maximize the availability and minimize the cost function of the system. The study illustrated a numerical example and solved using goal programming (GP), fuzzy goal programming (FGP), genetic algorithm (GA), and particle swarm optimization (PSO) techniques. The results are compared using a robust statistical test and, the PSO proves to be better. A simulation study was carried out further to evaluate the availability and cost function using R and MATLAB packages.

This note introduces a new class of preinvexity called $({\mathsf{h}}_1,{\mathsf{h}}_2)$-Godunova-Levin preinvex functions that generalize earlier findings. Based on these notions, we developed Hermite-Hadamard, weighted Fejér, and trapezium type inequalities. Furthermore, we constructed some non-trivial examples in order to verify all the developed results. In addition, we discussed some applications related to the trapezoidal formula, probability density functions, special functions and special means. Lastly, we discussed the importance of order relations and left two open problems for future research. As an additional benefit, we believe that the present work can provide a strong catalyst for enhancing similar existing literature.

In this paper, a new approach for deriving continuous probability distributions is developed by incorporating an extra parameter to the existing distributions. Frechet distribution is used as a submodel for an illustration to have a new continuous probability model, termed as modified Frechet (MF) distribution. Several important statistical properties such as moments, order statistics, quantile function, stress-strength parameter, mean residual life function, and mode have been derived for the proposed distribution. In order to estimate the parameters of MF distribution, the maximum likelihood estimation (MLE) method is used. To evaluate the performance of the proposed model, two real datasets are considered. Simulation studies have been carried out to investigate the performance of the parameters’ estimates. The results based on the real datasets and simulation studies provide evidence of better performance of the suggested distribution.

Background A consistent referral system and patient patronage are the primary connections between the three tiers in the healthcare delivery system. Patients were scheduled to visit primary care clinics for the first time. Subsequently, patients are moved to more distinguished healthcare facilities to receive additional care. Objectives This study aims to investigate the reasons behind patients' transfer from primary and secondary care to tertiary care, as well as the critical factors that influence these referrals. Methods This study employs a mixed-methods approach to explore factors determining referral service provision across primary, secondary, and tertiary healthcare levels. Along with conducting semi-structured interviews with healthcare professionals, we systematically examined a wealth of retrospective data on 1331 referred patients from 130 health facilities, including patient records, demographics, referral status, and clinical presentation. All statistical analysis was processed in R, and Corel Draw 12 was also used for graphical illustration. Results Healthcare facilities referred most emergencies to several departments, including the cardiac care unit, medical, urology, intensive care unit/emergency, obstetrics and gynaecology, children's, orthopaedic, and psychiatry. The percentage of all cases referred is displayed ward-by-ward; the intensive care unit/emergency wards have a high referral ratio of 65.51%, while the obstetrics and gynaecology wards have an 18.40% referral ratio. Conclusion There is a need for increased government investments to strengthen the capacity, human resources, and availability of equipment in primary, secondary, and tertiary public health facilities to deliver quality services in order to reduce the patient referral ratio.

The novel coronavirus disease (COVID‐19) pandemic has had devastating effects on healthcare systems and the global economy. Moreover, coronavirus has been found in human feces, sewage, and in wastewater treatment plants. In this paper, we highlight the transmission behavior, occurrence, and persistence of the virus in sewage and wastewater treatment plants. Our approach follows the process of identifying a coronavirus hotspot through existing wastewater plants in major cities of Saudi Arabia. The mathematical distributions, including the log‐normal distribution, Gaussian model, and susceptible‐exposed‐infected‐recovered (SEIR) model, are adopted to predict the coronavirus load in wastewater plants. We highlight not only the potential virus removal techniques from wastewater treatment plants but also methods of tracing SARS‐CoV‐2 in humans through wastewater treatment plants. The results indicate that our modeling approach may facilitate the analysis of SARS‐CoV‐2 loads in wastewater for early prediction of the epidemic outbreak and provide significant implications to the public health system.