Abstract: The coordination of production scheduling and transportation has recently received a lot of attention in logistics and supply chain management. We study a coordinated scheduling problem in which the jobs are transported by $m$ vehicles from a holding area to a single serial batching machine for further processing, each batch of jobs to be processed  occurs a processing cost. The objective is minimizing the sum of the jobs' total completion time and the total processing cost. Under the condition of the job processing times are equal, if the job assignment to the vehicles is predetermined, we provide a polynomial time dynamic programming algorithm, for the general problem,  prove that it is {NP}-hard. When the returning time of vehicle is $0$, we present the approximation algorithm and prove that the worst case ratio of the algorithm is  $2-\frac{1}{m}$.

Key words: supply chain scheduling, dynamic programming algorithm, complexity, worst case analysis