Abstract
In this paper, we describe new complexity results and approximation algorithms for single-machine scheduling problems with non-renewable resource constraints and the total weighted completion time objective. This problem is hardly studied in the literature. Beyond some complexity results, only a fully polynomial-time approximation scheme (FPTAS) is known for a special case. In this paper, we discuss some polynomially solvable special cases and also show that under very strong assumptions, such as the processing time, the resource consumption and the weight is the same for each job; minimizing the total weighted completion time is still NP-hard. In addition, we also propose a 2-approximation algorithm for this variant and a polynomial-time approximation scheme (PTAS) for the case when the processing time equals the weight for each job, while the resource consumptions are arbitrary.
Highlights
Non-renewable resources, such as raw material, energy or money, are used in all sectors of production, and depending on the stocking policy, they have varying impact on the preparation of daily and weekly production schedules
In all problems studied in this paper, there are a single machine, a non-renewable resource, and a finite set of jobs J
The constraint w j = p j means that for each job j, its weight equals its processing time, while w j = λ p j indicates that w j is proportional to p j, λ > 0 which is a common ratio
Summary
Non-renewable resources, such as raw material, energy or money, are used in all sectors of production, and depending on the stocking policy, they have varying impact on the preparation of daily and weekly production schedules. Consider for instance the preparation of the weekly schedule of a production line, where some of the raw materials built into the products arrive over the week, and the supplies constrain what can be produced and when. We consider single-machine scheduling problems with one additional non-renewable resource. In all problems studied in this paper, there are a single machine, a non-renewable resource, and a finite set of jobs J. Let S be a schedule specifying a start time S j for each job j. We mention that a feasible schedule exists only if j∈J a j ≤
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