Abstract: Subcontracting capacity to third parties has become a prevalent practice for improving an enterprise's competitive power and agility. In the problem,each order placed in a period is immediately shipped by some containers in the period and the total freight cost is proportional to the number of containers used. If the number of enterprise's containers is insufficient,then the enterprise needs to contract additional transportation capacity on the spot market,where the price of single container is higher and time-varying. The simultaneous determination of the lot sizes and the transportation policy that minimizes the total costs,including ordering cost,inventory holding cost and freight cost,is dealt with.As the problem is NP-hard,a lagrangian-based heuristic algorithm is proposed to compute lower and upper bounds,of which the comprehensive computational experiments show the compelling performance in terms of quality and speed.

Key words: subco ntracting, lot sizing, lagrangian relaxation, transportation costs, heuristic algorithm