Scheduling Resumable Deteriorating Jobs

We consider the following single-machine time-dependent scheduling problem with constraints on the machine availability. There is given a set of independent and deteriorating jobs Jj , 1 ≤ j ≤ n, to be processed on a single machine, starting from time t0 > 0. The processing time pj of job Jj is a proportional function of the starting time Sj of this job, pj = αjSj, where Sj ≥ t0 and deterioration rate αj > 0 for 1 ≤ j ≤ n. The machine is not continuously available and there are h disjoint periods of the machine non-availability. These periods are described by time intervals [Wi,1,Wi,2], where t0 < W1,1 and Wi,1 < Wi,2 for 1 ≤ i ≤ h. Since in real-world applications the number of non-availability periods is usually small, we will assume that h < n. Moreover, since any such non-availability period can interrupt the processing of any job, we will assume that the jobs are resumable, i.e. if a job has been interrupted by the start time of a non-availability period, then this job does not need to be restarted and can be completed after the machine becomes available again. The applied criterion of schedule optimality is the maximum completion time Cmax = max 1≤j≤n {C[j]}, where C[j] denotes the completion time of the j-th job in schedule. For simplicity of presentation, the above problem will be denoted in short by RDP (h). The problem RDP (h), introduced by Wu and Lee [7] for h = 1, is a combination of timedependent scheduling with scheduling on a machine with non-availability periods. Time-dependent scheduling has numerous practical applications, e.g. in financial management and modelling service operations. Scheduling on machines with non-availability periods is applied in manufacturing environments in which machines have conservation or maintenance breaks, caused by production or service reasons. We refer the reader to reviews by Cheng et al. [1] and Lee [5] for more details on time-dependent scheduling and scheduling on machines with non-availability periods, respectively. In this talk, first we extend some of the results obtained by Gawiejnowicz [2] and Ji et al. [4], where a counterpart of the problem RDP (1) with nonresumable jobs has been considered. In particular, we prove the ordinary NP-hardness of the RDP (1) problem and thus we give an answer to the open problem stated in [4]. Next, we present a pseudopolynomial dynamic programming algorithm, DP, for the problem RDP (h) in the case when h is a fixed number. After that we show that the algorithm DP allows us to construct an FPTAS for the RDP (1) problem. Finally, we prove that for the problem RDP (h) with h ≥ 2 there does not exist a polynomial-time approximation algorithm with a constant worst-case ratio, unless P = NP .