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Four lemmas, which constitute the theoretical foundation necessary todetermine optimal control strategies for a class of single machine familyscheduling problems, are presented in this technical report. The schedulingproblem is characterized by the presence of sequence-dependent batch setup andcontrollable processing times; moreover, the generalized due-date model isadopted in the problem. The lemmas are employed within a constructive procedure(proposed by the Author and based on the application of dynamic programming)that allows determining the decisions which optimally solve the schedulingproblem as functions of the system state. Two complete examples of singlemachine family scheduling problem are included in the technical report with theaim of illustrating the application of the fundamental lemmas in the proposedapproach.

Fundamental lemmas for the determination of optimal control strategies for a class of single machine family scheduling problems