Single processor scheduling with job values depending on their completion times

The paper deals with a single processor scheduling problem in which the sum of values of all jobs is maximized. The value of a job is characterized by a stepwise nonincreasing function with one or more moments at which the changes of job value occur. Establishing an order of processing of datagrams which are sent by router is a practical example of application of such problems. We prove that the special case of our problem, with a single moment of change of job values, is equivalent to the well-known, NP-hard in the ordinary sense, problem of minimizing weighted number of late jobs. Next, we show that, based on this equivalence, the existing algorithms for solving the latter problem can be adopted to solve special cases of our problem. Additionally, we construct a pseudopolynomial time algorithm based on the dynamic programming method, for the case with arbitrary number of common moments of job value changes. At the end of the paper, we generalize this algorithm to the corresponding case with parallel processors. Thus, we show that these two problems are also NP-hard in the ordinary sense. Moreover, we construct exact polynomial time algorithms for two further special cases of our problem. Finally, in order to solve the general version of the problem, we construct and experimentally test a number of heuristic algorithms.