Bicriterion Approach to a Single Machine Time-Dependent Scheduling Problem

Abstract

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 Hölder’s norm l _p , 1 ≤ p ≤ ∞. The main result of the paper is theorem saying that there exist numbers λ₁ > λ_o > 0 such that for all λ ∈ [0, λ_o] 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.