Microbial collectives -- communities of microbes of the same or different species -- can carry out functions that are often not achievable by individual microbes in isolation. Examples of collective functions include waste degradation, food fermentation, and vitamin production that are beneficial to humans. A human-desired collective function might be achieved by artificial selection on collectives. However, such selection on collectives is challenging due to the lack of theoretical understanding. Here, we examine a simple microbial system to develop a theoretical approach for understanding the conditions for successful artificial selection on collectives. In each selection cycle, multiple low-density "Newborn" collectives with wild and mutant cells are allowed to "mature" during which cells grow and possibly mutate. At the end of a cycle, "Adult" collective with the highest function is selected and is split into multiple Newborns for the next cycle. Repeating selection cycles, we aim to obtain the target composition that gives an optimal collective function. Investigating the possibility to get closer the target composition through artificial selection, we predict whether the selection is successful at given experimental setups. This approach allows us to find which experimental parameters are important for successful selection. Our results show that small bottleneck sizes play an important role while the effect of increasing collective numbers is not always significant.