Coordinated scheduling on single serial batching machine with transportation considerations

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 each job is transported to a single serial batching machine for further processing, each batch to be processed occurs a processing cost, and the objective is minimizing the sum of the makespan and the total processing cost. Under the condition of the jobs' 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, we prove it is NP-hard. When the returning time of vehicle is zero, we present the approximation algorithm and prove that the worst case ratio of the algorithm is not greater than 1 2 m  , and the bound is tight.