Column generation for minimizing total completion time in a parallel-batching environment

Abstract

This paper deals with the $$1|{p-\text {batch}, s_j\le b}|\sum C_j$$ 1 | p - batch , s j ≤ b | ∑ C j scheduling problem, where jobs are scheduled in batches on a single machine in order to minimize the total completion time. A size is given for each job, such that the total size of each batch cannot exceed a fixed capacity b. A graph-based model is proposed for computing a very effective lower bound based on linear programming; the model, with an exponential number of variables, is solved by column generation and embedded into both a heuristic price and branch algorithm and an exact branch and price algorithm. The same model is able to handle parallel-machine problems like $$Pm|{p-\text {batch}, s_j\le b}|\sum C_j$$ P m | p - batch , s j ≤ b | ∑ C j very efficiently. Computational results show that the new lower bound strongly dominates the bounds currently available in the literature, and the proposed heuristic algorithm is able to achieve high-quality solutions on large problems in a reasonable computation time. For the single-machine case, the exact branch and price algorithm is able to solve all the tested instances with 30 jobs and a good amount of 40-job examples.

Scheduling a single parallel-batching machine with non-identical job sizes and incompatible job families

“…More recently, a number of integer-programming-based methods have been developed for related batch scheduling problems that work under different assumptions regarding the scheduling environments or the objective function compared to our setting (Alfieri et al, 2021;Ozturk, 2020Ozturk, , 2022Trindade et al, 2021). For a more detailed overview, we refer to the surveys of Potts and Kovalyov (2000), Mathirajan and Sivakumar (2006), Mönch et al (2011), andMönch (2021).…”