A note on the influence of missing operations on scheduling problems

Abstract

This paper attempts to resolve the existing confusion concerning missing operations. Scheduling problems are classified in two groups: (i) nullcontinuous (NC) -comprising the problems where an optimal schedule remains optimal on replacement of arbitrarily small processing times (existing operations) with zeros (missing operations); (ii) null-discontinuous (NDC) -comprising those problems which are not null-continuous.A "zero processing time" of an operation refers to either of the two following contingencies: (i) An actual operation whose processing time tends to zero. Thus, if ijj denotes the processing time of operation Oj in job Ji, then for a sufficiently small positive number E, scheduling problems with ijj = E and tjj = 0 have the same optimal schedules (algorithms), and E may reasonably be replaced with zero to facilitate calculations, and (ii) A nonexisting (missing) operation.Since an infinitesimal (arbitrarily small) processing time operation (i) has a starting time while a missing operation (ii) has not, the two types have a different effect on a scheduling problem and must be differentiated to prevent ambiguity. Accordingly, we propose to designate an infinitesimal processing time operation as e , and a missing operation as a zero.Scheduling problems involving operations with arbitrarily small processing times (existing operations where, for all practical purposes, the length may be considered as zero, but have starting times) are basically the same as those with the usual strictly-positive processing times and do not merit separate consideration.Whether a discipline (flowshop or openshop) allows missing operations or not depends on its definition, without clear preference for one definition over another. However, for the sake of understanding, the definition (whatever it is) must be known and accepted. We propose to define flowshop and openshop disciplines allowing missing operations. 535NAVAL RESEARCH LOGISTICS QUARTERLY

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