Bicriterion Approach to a Single Machine Time-Dependent Scheduling Problem

In the paper a single machine time-dependent scheduling problem with simultaneous minimization of two criteria is considered. Processing time P j of each task is described by a linear function of the starting time t of the task, p j ; = 1 + α j t, where α j > 0 for j = 0, 1, 2,... , n. The criterion of optimality of a schedule is a convex combination of norms ‖·‖i and ‖·‖∞ in the form of λ‖·‖1+(1 − λ)‖·‖∞, where λ ∈ [0,1] is an arbitrary real number and ‖ · ‖ p is Holder’s norm l p , 1 ≤ p ≤ ∞. The main result of the paper is theorem saying that there exist numbers λ1 > λo > 0 such that for all λ ∈ [0, λo] the problem is solvable in O(n log n) time and for all λ ∈ [λ1, 1] the optimal schedule for the problem has V-shape.