Dynamic lot sizing with batch ordering and truckload discounts

This paper studies two important variants of the dynamic economic lot-sizing problem that are applicable to a wide range of real-world situations. In the first model, production in each time period is restricted to a multiple of a constant batch size, where backlogging is allowed and all cost parameters are time varying. Several properties of the optimal solution are discussed. Based on these properties, an efficient dynamic programming algorithm is developed. The efficiency of the dynamic program is further improved through the use of Monge matrices. Using the results developed for the first model, an O(n³log n) algorithm is developed to solve the second model, which has a general form of product acquisition cost structure, including a fixed charge for each acquisition, a variable unit production cost, and a freight cost with a truckload discount. This algorithm can also be used to solve a more general problem with concave cost functions.