A variable sized bucket indexed formulation for nonpreemptive single machine scheduling problems

The single machine scheduling problem (SMSP) is a classic problem in optimisation which has been extensively studied over the past 60 years. To solve an instance of the problem a set of jobs must be scheduled on a single machine, so that at any time the machine is either idle or processing exactly one job. We consider a nonpreemptive version of this problem which requires that the processing of a job continue uninterrupted for the duration of its processing time. Several mixed integer linear programs exist for the SMSP, with the classic time indexed (TI) model being the most common formulation. The TI formulation can be applied to a range of SMSP variations, with all standard min-sum scheduling criteria capable of being expressed as linear functions of the TI variables. Many complex production planning problems are modelled using TI variables, and so the TI formulation often serves as natural basis for designing mixed integer linear programs to solve these problems. To formulate the TI model the problem data is assumed to be integer and a suffciently large planning horizon is discretised into time periods of unit length. The length of the planning horizon can be no smaller than the sum of all processing times, and hence grows pseudopolynomially with the size of the problem input. The TI formulation is known to have a strong linear relaxation compared to alternative formulations, however for instances where the sum of processing times is large the resulting model may be intractable due to the large number of constraints and variables. The authors recently proposed a mixed integer linear program, named the bucket indexed (BI) formulation, for which the time horizon is discretised into periods of the same length and no larger than the processing time of the shortest job. The BI model generalises the TI model to one in which either at most two or three jobs can be processing in each period. In this paper we present a model, named the variable sized bucket indexed (BI-VAR) formulation, in which the lengths of the periods are not required to be identical. This model generalises the BI model to one in which each period is characterised as either permitting at most two, three, or an arbitrary number of jobs to be processed within it. In addition we present necessary conditions for a partition of the time horizon to be valid for the BI-VAR model