A location-transportation problem concerns designing a company’s distribution network consisting of one central warehouse with ample stock and multiple local warehouses for a long but finite time horizon. The network is designed to satisfy the demands of geographically dispersed customers for multiple products within given delivery time targets. The company needs to first decide on the locations of local warehouses before the start of the time horizon. During the time horizon, the stocks at the local warehouses are repeatedly replenished, and the company has to decide how far the inventory levels are increased at those moments. Our problem is such that we can use time-independent base stock levels at all warehouses at those moments. Between any two replenishments, integer-valued demands are realized multiple times, and the company needs to satisfy them by shipments from the central and local warehouses to the customers.
In this paper, we follow an adjustable robust optimization approach for the design of the distribution network. We prove two main characteristics of our location-transportation problems, namely convexity and non-decreasingness of the optimal shipment cost function. Using these characteristics, we show for two commonly used uncertainty sets (hyper-box and budget uncertainty sets) that the optimal decisions on the location and the base stock levels of local warehouses can be made efficiently. For a general bounded uncertainty set, we propose a new method, called the Simplex-type method, to find a locally robust solution. The numerical experiments show the superiority of our method over using the integer-valued affine decision rules as well as an exact solution approach.