Tabu Search Algorithms for Cyclic Machine Scheduling Problems

A general framework is developed to model a large variety of different cyclic machine scheduling problems with blocking with the objective to minimize the cycle time. For this framework tabu search algorithms are developed. Neighborhoods are derived which generalize the block-approach based neighborhoods which have been successfully applied to noncyclic job-shop problems and cyclic job-shop problems without blocking. keywords: Local Search, Machine Scheduling Cyclic scheduling is concerned with the planning of activities that have to be identically repeated at regular intervals. Such types of scheduling problems arise in different application areas like compiler design, manufacturing, digital signal processing, railway scheduling, timetabling, etc. Hanen [2] introduces a general cyclic scheduling problem which among others contains the cyclic job-shop problem as special case. Brucker and Kampmeyer [1] developed tabu-search algorithms for this general problem. Now this general model is extended to cover also blocking situations. This is done by using the alternative graph model [3]. This General Cyclic Scheduling Problem (GCSP) can be described as follows. Let T = {1, . . . , n} be a set of generic tasks or operations (in connection with shop scheduling problems). Task (operation) i has processing time pi > 0 and must be performed infinitely often. We denote by < i; k > the k-th occurrence of the generic task i. A schedule assigns a starting time t(i; k) to each occurrence < i; k >. A schedule is called periodic with cycle time α if t(i; k) = t(i; 0) + αk for all i ∈ T, k ∈ Z. We assume that ti := t(i; 0) ≥ 0 for all i ∈ T . A periodic schedule is defined by the vector (ti)i∈T and the cycle time α. Generalized precedence constraints of the form t(i; k) + Lij ≤ t(j; k +Hij) may be given for all arcs (i, j) ∈ E of a directed graph G = (T,E) with vertex set T . Lij is called (start-start) delay and Hij is called the height of the generalized precedence constraint. Delays are assumed to be nonnegative integers while heights may be arbitrary integers. We also postulate that t(i; k) + pi ≤ t(i; k + 1) is satisfied for all i ∈ T , k ∈ Z. Finally, associated with each task i there is a dedicated machine M(i) ∈ M = {1, . . . ,m}, on which each occurrence < i; k > of i must be processed. Consider two operations i and j which are processed on the same machine m. Assume, that both operations are blocking operations and operation i has a successor s(i) and j has a successor s(j). Then we get the following disjunctive constraints: t(s(i); l +Hi,s(i)) ≤ t(j; k) ∨ t(s(j); k +Hj,s(j)) ≤ t(i; l) (1) ∗supported by Cusanuswerk 1