Dual relaxations of the time-indexed ILP formulation for min–sum scheduling problems

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• 作者：Yunpeng Pan ; Zhe Liang
• 关键词：Min–sum scheduling ; Time ; indexed formulation ; Duality ; Polynomial size
• 刊名：Annals of Operations Research
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：249
• 期：1-2
• 页码：197-213
• 全文大小：
• 刊物类别：Business and Economics
• 刊物主题：Operation Research/Decision Theory; Combinatorics; Theory of Computation;
• 出版者：Springer US
• ISSN：1572-9338
• 卷排序：249

文摘

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.