Approximation schemes for single machine scheduling with non-renewable resource constraints

Abstract

In this paper we discuss exact and approximation algorithms for scheduling a single machine with additional non-renewable resource constraints. Given the initial stock levels of some non-renewable resources (e.g. raw materials, fuel, money), and time points along with replenishment quantities, a set of resource consuming jobs has to be scheduled on the machine such that there are enough resources for starting each job, and the makespan is minimized. We show that the problem admits a pseudo polynomial time algorithm when the number of replenishments is not part of the input, and also present an FPTAS when there is only a single resource, and it is replenished only once. We also describe a PTAS for the problem with a constant number of replenishments.

A PTAS for a resource scheduling problem with arbitrary number of parallel machines

“…These assumptions led to a polynomial time algorithm for minimizing the makespan, which is in a strong contrast to the NPhardness of the scheduling problem analyzed in this paper. Further results can be found in e.g., [16], [17], [7], [5], [8], [9], [10], [14], [11].…”

“…In [8], [9] and [10] there are several approximability results for the single machine variant of the problem. [11] provided PTAS-es for some parallel machine variant of the problem and showed that the problem with two resources and two supplies is APX-hard.…”

Approximability of scheduling problems with resource consuming jobs

“…In Györgyi and Kis (2014) a PTAS for scheduling resource consuming jobs with a single non-renewable resource and a constant number of supply dates was developed, and also an FPTAS was devised for the special case with q = 2 supply dates and one non-renewable resource only. In Györgyi and Kis (2015) it was shown, among other results, that there is no FPTAS for the problem of scheduling jobs on a single machine with two non-renewable resources and q = 2 supply dates, unless P = N P , which is in strong contrast with the existence of an FPTAS for the special case with one non-renewable resource only (Györgyi and Kis, 2014). In Györgyi and Kis (2015) and Györgyi and Kis (2014), variants of the knapsack problem are solved as a subproblem using combinatorial techniques like enumeration of feasible packings.…”

Approximability of total weighted completion time with resource consuming jobs

“…Further results can be found in e.g., [10], [2,3], [8], [11,12]. In particular, Grigoriev et al [10] studied the complexity of several variants and developed some constant ratio approximation algorithms.…”

“…Briskorn et al [2], and [3] studied scheduling problems with an initial inventory only, where some of the jobs produce, and other jobs consume the resources, the objective being the minimization of the inventory levels. In [11] a PTAS for scheduling resource consuming jobs with a single non-renewable resource and a constant number of supply dates was developed. Also, an FPTAS was devised for the special case with q = 2 supply dates.…”