A Heuristic Solution Procedure to Minimize Makespan on a Single Machine with Non-linear Cost Functions

We consider the static single-facility scheduling problem where the processing times of jobs are a nondecreasing and differentiable function of their starting (waiting) times and the objective is to minimize the total elapsed time (makespan) in which all jobs complete their processing. We give a criterion for optimality of two jobs to be scheduled next to each other, and based on this criterion we propose a heuristic algorithm to solve the problem. The effectiveness of the algorithm is empirically evaluated for quadratic and exponential cost functions. In the quadratic case it is compared with the static heuristic algorithm proposed by Gupta and Gupta.