A Location-Transportation (LT) 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 items within given delivery time targets. The company needs to decide on the locations of local warehouses and their basestock levels while considering the optimal shipment policies from central or local warehouses to customers. In this paper, we deal with integer uncertain demands in LT problems to design a robust distribution network. We prove two main characteristics of our LT problems, namely convexity and nondecreasingness of the optimal shipment cost function. Using these characteristics, we show for two commonly used uncertainty sets (box and budget uncertainty sets) that the optimal decisions on the location and the basestock levels of local warehouses can be made by solving a polynomial number of deterministic problems. For a general uncertainty set, we propose a new method, called 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, which is the only available method for this class of problems.