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Internet technology has provided an indescribable new way for businesses to attract new customers, track their behaviour, customise services, products, and advertising. Internet technology and the new trend of online shopping have resulted in the establishment of numerous websites to sell products on a daily basis. Products compete to be displayed on the limited pages of a website in online shopping because it has a significant impact on sales. Website designers carefully select which products to display on a page in order to influence the customers’ purchasing decisions. However, concerns regarding appropriate decision making have not been fully addressed. As a result, this study conducts a comprehensive comparative analysis of the performance of ten different metaheuristics. The ant lion optimiser (ALO), Dragonfly algorithm (DA), Grasshopper optimisation algorithm (GOA), Harris hawks optimisation (HHO), Moth-flame optimisation algorithm (MFO), Multi-verse optimiser (MVO), sine cosine algorithm (SCA), Salp Swarm Algorithm (SSA), The whale optimisation algorithm (WOA), and Grey wolf optimiser (GWO) are some of the recent algorithms that were chosen for this study. The results show that the MFO outperforms the other methods in all sizes. MFO has an average normalised objective function of 81%, while ALO has a normalised objective function of 77%. In contrast, HHO has the worst performance of 16%. The study’s findings add new theoretical and practical insights to the growing body of knowledge about e-commerce environments and have implications for planners, policymakers, and managers, particularly in companies where an unplanned advertisement wastes the budget.

We generalize the results obtained by Jun and Min (2009) and use fixed point method to obtain the stability of the functional equationf(x+σ(y))=F[f(x),f(y)], for a class of functions of a vector space into a Banach space whereσis an involution. Then we obtain the stability of the differential equations of the formy′=F[q(x),P(x)y(x)].

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