Abstract
Linear programming (LP)-based relaxations have proven to be useful in enumerative solution procedures for \(\mathbf {NP}\)-hard min–sum scheduling problems. We take a dual viewpoint of the time-indexed integer linear programming (ILP) formulation for these problems. Previously proposed Lagrangian relaxation methods and a time decomposition method are interpreted and synthesized under this view. Our new results aim to find optimal or near-optimal dual solutions to the LP relaxation of the time-indexed formulation, as recent advancements made in solving this ILP problem indicate the utility of dual information. Specifically, we develop a procedure to compute optimal dual solutions using the solution information from Dantzig–Wolfe decomposition and column generation methods, whose solutions are generally nonbasic. As a byproduct, we also obtain, in some sense, a crossover method that produces optimal basic primal solutions. Furthermore, the dual view naturally leads us to propose a new polynomial-sized relaxation that is applicable to both integer and real-valued problems. The obtained dual solutions are incorporated in branch-and-bound for solving the total weighted tardiness scheduling problem, and their efficacy is evaluated and compared through computational experiments involving test problems from OR-Library.
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The second author was supported by the National Science Foundation of China [Grant 71422003 and Grant 71201003].
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Pan, Y., Liang, Z. Dual relaxations of the time-indexed ILP formulation for min–sum scheduling problems. Ann Oper Res 249, 197–213 (2017). https://doi.org/10.1007/s10479-014-1776-2
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DOI: https://doi.org/10.1007/s10479-014-1776-2