Warm Starting Integer Programming for the Internet SHopping Optimization Problem with Multiple Item Units (ISHOP-U)

In this discrete optimization work, we deal with Internet purchases that have become very popular and increased yearly. In this chapter, we study the Internet SHopping Optimization Problem with multiple item Units (ISHOP-U), a combinatorial NP-hard variant of the original ISHOP that considers purchasing one or more units of a product in a set of products. We performed a warm start in an Integer programming model in CPLEX using as a starting point the best-found solutions from two evolutionary algorithms: A Cellular Genetic Algorithm (CGA), A Genetic Algorithm (GA), and a nature-inspired Water Cycle Algorithm (WCA) present in the state-of-the-art. The test setup is on 15 synthetic instances in the literature, where unit costs and delivery prices adhere to a random uniform distribution. The results were as follows: for instances with 10 products and 25 stores, the average difference in CPU ticks between a cold start and a warm start is 0.222 CPU ticks when using solutions constructed by a GA and a CGA. Additionally, the difference for solutions built by a WCA is 0.022 CPU ticks.