Abstract
This paper addresses a single-machine scheduling problem with periodic preventive maintenance activities that are predeterministic so that the machine is not available all the time, and jobs have to be processed between two consecutive maintenance periods. We propose a mixed integer programming (MIP) model and two heuristics to minimize the makespan. With more constraints in our model, the model is more efficient than the recent model of Perez-Gonzalez and Framinan , and our model could solve problems with up to fifty jobs. Two heuristic algorithms, namely, H (MW) and H (LB∗), are also proposed, in which two bin-packing policies of the minimum waste and minimum lower bound are used, respectively. Furthermore, we also proposed an improvement procedure. The results showed that the heuristic H (MW) outperformed other heuristics of the paper, indicating that the bin-packing policy of the minimum waste is more effective than well-known ones such as full batch and best fit. Additionally, all the heuristic algorithms addressed in this paper combined with the improvement procedure could achieve a similar and high quality of solutions with a very tiny increase in computational expense.
Highlights
The Calculation of Lower BoundFor an NP-hard scheduling problem, lower bounds used to measure the performance of heuristic algorithms are commonly encountered in the literature because optimal solutions are hard to obtain
Mathematical Problems in Engineering the maintenance time is a nondecreasing function of its starting time, i.e., the later the maintenance activity begins, the longer its duration
Based on the studies of Perez-Gonzalez and Framinan [16] and Ji et al [9], the LPT rule for generating job sequences is better than other dispatching rules in the first phase; in this paper, the jobs first were sorted in nonincreasing order of their processing time, i.e., p[1] ≥ p[2] ≥ · · · ≥ p[n], and we developed two binpacking policies of the minimum waste and the minimum lower bound that are different from the well-known ones such as full batch and best fit that are prevalently encountered in the literature for solving the scheduling problem with machine maintenance
Summary
For an NP-hard scheduling problem, lower bounds used to measure the performance of heuristic algorithms are commonly encountered in the literature because optimal solutions are hard to obtain. nj 1 B2min pj/T, (nj 1 which is the same as that pj + Wmax)/T, where Wmax of is the maximum waste and will be explained in the following; and (3) B3min is obtained by the integer programming model for bin-packing problems provided by Martello et al [39]. To obtain the value of Wmax and B2min, we developed a maximum waste (Max_W) method. Regarding the estimation of the lower bound, we modified formula (19) based on the properties as follows: LB∗. Because B1min is dominated by B2min and the Max_W method is much more efficient than the integer programming model of Martello et al [39], we adopted B2min to obtain the estimated number of batches in this paper
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