Knowledge that Transforms

N jobs are to be processed sequentially on a single machine. While waiting for processing, jobs deteriorate, causing the random processing requirement of each job to grow at a job-specific rate. Under such conditions, the actual processing times of the jobs are no longer exchangeable random variables and the expected makespan is no longer invariant under any scheduling strategy that disallows idleness. In this paper, we analyze the effects of different deterioration schemes and derive optimal scheduling policies that minimize the expected makespan, and, for some models, policies that minimize the variance of the makespan. We also allow for random setup and detaching times. Applications to optimal inventory issuing policies are discussed and extensions are considered.

Many interesting combinatorial problems can be optimized efficiently using recursive computations often termed discrete dynamic programming. In this paper, we develop a paradigm for a general class of such optimizations that yields a polyhedral description for each model in the class. The elementary concept of dynamic programs as shortest path problems in acyclic graphs is generalized to one seeking a least cost solution in a directed hypergraph. Sufficient conditions are then provided for binary integrality of the associated hyperflow problem. Given a polynomially solvable dynamic program, the result is a linear program, in polynomially many variables and constraints, having the solution vectors of the dynamic program as its extreme-point optima. That is, the linear program provides a succinct characterization of the solutions to the underlying optimization problem expressed through an appropriate change of variables. We also discuss projecting this formulation to recover constraints on the original variables and illustrate how some important dynamic programming solvable models fit easily into our paradigm. A classic multiechelon lot sizing problem in production and a host of optimization problems on recursively defined classes of graphs are included.

We show, using small examples, that two algorithms previously published for the Bilevel Linear Programming problem (BLP) may fail to find the optimal solution and thus must be considered to be heuristics. A proof is given that solving BLP problems is NP-hard, which makes it unlikely that there is a good, exact algorithm.