This research explores the Location Routing Problem (LRP), a complex NP-hard problem that combines facility location and vehicle routing to minimize operational costs. The focus is on a variant of LRP with covering constraints. The primary objective is to investigate the impacts of different single-objective functions in LRPs, contrasting the traditional approach of minimizing combined opening and transportation costs with objectives prioritizing either the minimization of the number of facilities or the total distance.

A Mixed-Integer Programming (MIP) model is developed to analyze the effects of these objectives on facility location and routing efficiency. The methodology involves running the MIP model under various objective functions and observing the changes in total distance and number of facilities, including the use of preemptive goal programming.

Results from small and large problem instances reveal that focusing on minimizing the number of facilities significantly reduces their count but increases the total travel distance. On the other hand, prioritizing distance minimization shows minimal distance reduction but a slight increase in the number of facilities compared with the traditional approach of minimizing combined opening and transportation costs, but this may be due to the proneness to the scale of data. A Pareto analysis shows the trade-offs between these objectives.

This research contributes to a deeper understanding of the importance of objective selection in LRPs, offering valuable insights for decision-makers in adapting LRP strategies to specific operational priorities.