The power-law memory effect is taken into consideration in a generalisation of the economic model of natural growth. The memory effect refers to a process's reliance on its current state and its history of previous changes. However, the study that focuses on natural growth in economics considering the memory effect with fractional order-linear differential equation model is still limited. The current investigation seeks to solve the natural growth with memory effect in the economics model and decide the best model using fractional differential equation (FDE), namely Adomian Decomposition and Variational Iteration Methods. Also, this study assumes the level of consumer loss memory during a certain time interval denoted by a parameter (α). This study showed the model of loss memory effect with 0 < α ≤ 1 given a slowdown in output growth compared to a model without memory effect. Besides that, this study also found that output Y(t) is growing faster with the Variational Iteration method compared to the Adomian decomposition method. Also, using graphical simulation, this study found the output Y(t) is closer to the exact solution with α=0.4 and α=0.9. In conclusion, this study successfully solved natural growth with memory effect in economics and decided the best model between FDE, namely Adomian decomposition and Variational iterative methods using numerical analysis.

Internet Shopping Optimization Problem (ISOP) is the application of optimization to online shopping activities of all complexity. The ISOP is useful for consumers in minimizing the cost of purchasing goods. This paper presents a bibliometric analysis of peer-reviewed papers based on ISOP topics by utilizing the R application program in the mapping. Overall, 101 papers (233 authors) in the Scopus database have used ISOP topics with research growth of 11.61% annually. The researcher presents a network of citations from productive authors, the impact of research, trends in terms that have been used, and shows a collaborative network of citations. Finally, the researcher presents the thematic analysis of the papers that apply the ISOP as a research topic and shows how the research forms clusters based on analytical solutions and numerical simulations that generate suggestions in finding the latest topics in the ISOP study. Another target for this paper is to produce review analysis results through Preferred Reporting Items for Systematic reviews and Meta Analyses (PRISMA). Through bibliometric and PRISMA analysis, it was found that the latest method in completing ISOP optimization is the ARC method. The ARC method in the ISOP is still little published among researchers in the world.