Minimizing makespan in a multimode multiprocessor shop scheduling problem

We study the problem of multimode scheduling tasks on dedicated processors, with the objective of minimizing the maximum completion time. Each task can be undertaken in one among a set of predefined alternative modes, where each mode specifies a required set of dedicated processors and a processing time. At any time each processor can be used by a single task at most. General precedence constraints exist among tasks, and task preemption is not allowed. The problem consists of assigning a mode and a starting time to each task, respecting processor and precedence constraints, to minimize the time required to complete all tasks. The problem is NP-hard in several particular cases. In previous works, we studied algorithms in which a solution was obtained by means of an iterative procedure that combines mode assignment and sequencing phases separately. In this paper, we present some new heuristics where the decision on the mode assignment is taken on the basis of a partial schedule. Then, for each task, the mode selection and the starting time are chosen simultaneously considering the current processor usage. Different lower bounds are derived from a mathematical formulation of the problem and from a graph representation of a particular relaxed version of the problem. Heuristic solutions and lower bounds are evaluated on randomly generated test problems. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 893–911, 1999     

Scheduling of a single machine to minimize total weighted completion time subject to release dates

In this paper the n/1/r_j Σ_j w_j C_j problem under the assumptions of nonpreemptive sequencing and sequence independent processing times is investigated. After pointing out the fundamental properties, some dominance sufficient conditions among sequences are obtained and a branch and bound algorithm is proposed. Computational results are reported and discussed.     

Scheduling tasks with sequence-dependent processing times

In this article we consider the problem of minimizing the maximum completion time of a sequence of n jobs on a single machine. Nonzero ready times and sequence-dependent processing times are allowed. Upper bounds, lower bounds, and dominance criteria are proposed and exploited in a branch-and-bound algorithm. Computational results are given.     

Nonpreemptive scheduling of independent tasks with prespecified processor allocations

In this article we study the problem of scheduling independent tasks, each of which requires the simultaneous availability of a set of prespecified processors, with the objective of minimizing the maximum completion time. We propose a graph-theoretical approach and identify a class of polynomial instances, corresponding to comparability graphs. We show that the scheduling problem is polynomially equivalent to the problem of extending a graph to a comparability graph whose maximum weighted clique has minimum weight. Using this formulation we show that in some cases it is possible to decompose the problem according to the canonical decomposition of the graph. Finally, a general solution procedure is given that includes a branch-and-bound algorithm for the solution of subproblems which can be neither decomposed nor solved in polynomial time. Some examples and computational results are presented. © 1994 John Wiley & Sons, Inc.