An iterative exact algorithm for the weighted fair sequences problem

Abstract

In this work, we present a new iterative exact solution algorithm for the weighted fair sequences problem, which is a recently introduced NP-hard sequencing problem with applications in diverse areas such as TV advertisement scheduling, periodic machine maintenance and real-time scheduling. In the problem we are given an upper bound on the allowed solution sequence length and a list of symbols. For each symbols, there is a positive weight and a number, which gives the minimum times the symbol has to occur in a feasible solution sequence. The goal is to find a feasible sequence, which minimizes the maximum weight-distance product, which is calculated for each consecutive appearance of each symbol in the sequence, including the last and first appearance in the sequence, i.e., the sequence is considered to be circular for the calculation of the objective function.Our proposed solution algorithm is based on a new mixed-integer programming model for the problem for a fixed sequence length. The model is enhanced with valid inequalities and variable fixings. We also develop an extended model, which allows the definition of an additional set of valid inequalities and present additional results which can allow us to skip the solution of the mixed-integer program for some sequence lengths.We conduct a computational study on the instances from literature to assess the efficiency of our newly proposed solution approach. Our approach manages to solve 404 of 440 instances to optimality within the given timelimit, most of them within five minutes. The previous best existing solution approach for the problem only managed to solve 229 of these instances, and its exactness depends on an unproven conjecture. Moreover, our approach is up to two magnitudes faster compared to this best existing solution approach.

An efficient implementation of a VNS heuristic for the weighted fair sequences problem

“…A series of breaking symmetry cuts was included in Pessoa et al (2018) to get rid of them. More recently, Sinnl (2021) proposed to solve the WFSP by solving a sequence of subproblems in which the sequence length is fixed. From that variable fixing, the author demonstrates a series of valid inequalities for those subproblems, allowing his algorithm to increase the number of WFSP benchmark instances solved to optimality.…”

“…As done by Sinnl (2021), the general outline of our heuristic, presented in Algorithm 1, decomposes the WFSP into a series of subproblems of fixed sequence length T = n, . .…”

“…Our heuristic methods, Iterative and Iterative-2, are compared against the state-of-the-art method of Sinnl (2021) in the 440 instances of Pessoa et al (2018) available at https://sites.google. com/site/weightedfairsequencesproblem/instances.…”

“…In addition to proposing a mathematical optimization formulation for the WFSP based on mixed-integer linear programming, Pessoa et al (2018) proved that the WFSP is NP-hard and proposed an heuristic method that iteratively solves a constrained version of the WFSP. More recently, Sinnl (2021) developed an iterative exact method that decomposes the WFSP in a series of constrained subproblems. By means of a series of new valid inequalities, the author was able to considerably increase the number of WFSP instances solved to optimality.…”

“…By means of a series of new valid inequalities, the author was able to considerably increase the number of WFSP instances solved to optimality. Our work proposes the first metaheuristic solution approach for the WFSP and compares its performance with the method of Sinnl (2021), both in terms of solution quality and efficiency.…”