Capacitated assortment planning of a multi-location system under transshipments

We analyze a joint assortment optimization problem for multiple locations of a firm, where each of those locations has an assortment capacity. When one location does not keep a product in its assortment, it transships the requested product from another location when needed. If this is not possible, then demand may be fulfilled by a substitutable (second choice) product from the current location or another demand location through transshipment. We show that the resulting assortment planning problem is NP-complete. We provide structural properties of the optimal assortments that would simplify and speed up the search for the optimal solution. We specifically obtain the optimal product assortment in polynomial time when no substitutions are present. When both transshipments and substitutions are considered, the products adopted by all locations form a common assortment that includes the most popular products among customers, i.e., a popular set, which may not be valid for the individual assortment of each location. We derive an upper limit on customers' substitution probability for each location, such that at optimality, the location's capacity is utilized in full when the substitution probability is below this threshold value. Moreover, we show that when all locations have the same assortment capacity, utilizing theirs to the maximum reduces costly transshipment opportunities. Extensive numerical analyses are also conducted that illustrate the sensitivity of the optimal assortments to system characteristics and the benefits of allowing transshipment and demand substitution in multi-location assortment planning.