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Online optimization with gradual look-ahead

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Abstract

We extend the setting of online optimization with look-ahead to online optimization with gradual look-ahead. While look-ahead as considered so far refers to a deterministic outlook on future data, gradual look-ahead only allows for an uncertain outlook on future data which becomes more and more precise as an input element’s release time is approached. After a discussion of related concepts, we formally introduce the class of online optimization problems with gradual look-ahead. Since the course of look-ahead information of a single input element is tied to a corresponding uncertainty set trajectory, we examine how different forecasting methods and different algorithmic approaches for dealing with gradual look-ahead can be instantiated and compared to each other with respect to the optimization output. We exemplify the introduced concepts by numerical experiments for the two applications of lot sizing and vehicle routing under gradual look-ahead.

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Notes

  1. Besides the discussed information-related aspects, it is necessary to discuss the processing-related aspects that are relevant for algorithmic processing. Dunke (2014) and Dunke and Nickel (2016) have introduced the so-called rule set for this purpose; the rule set describes what is considered to be feasible in terms of processing \(\sigma _i\) once it became known at time \(n_i\).

  2. Under gradual look-ahead, we have that \(|U_i(t)| > 1\) for \(t \in [n_i, r_i)\). Thus, any description in the instance revelation rule of how \(U_i(t)\) evolves clearly needs to refer to random elements as otherwise \(\sigma _i\) could already be reconstructed before time \(r_i\) is reached according to a deterministic backtracking.

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Authors and Affiliations

  1. Institute of Operations Research, Discrete Optimization and Logistics, Karlsruhe Institute of Technology, Kaiserstr. 12, 76131, Karlsruhe, Germany

    Fabian Dunke & Stefan Nickel

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  1. Fabian Dunke
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  2. Stefan Nickel
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Correspondence to Fabian Dunke.

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Appendix: Computational results

Appendix: Computational results

Results in each table are cumulated over all parameter and algorithm configurations apart from the ones indicated by the row and column labels.

1.1 Online lot sizing with gradual look-ahead

100 input sequences with 100 demands have been tested with all combinations of the following parameterizations:

Look-ahead levels: 0, 1, 2, 5, 10, forecasting accuracies: 1, 2, 5, 10, 25, forecasting misestimation probabilities: 0, 0.1, 0.25, 0.5, step sizes of input data updates: 1, 2, 5, 10, reoptimization algorithms: MIP, Silver-Meal heuristic, Groff heuristic, Part-period heuristic (Table 7).

Table 7 Average results (costs) for the online lot sizing problem with gradual look-ahead
Full size table

1.2 Online vehicle routing with gradual look-ahead

100 input sequences with 125 customers have been tested with all combinations of the following parameterizations:

Look-ahead levels: 0, 1, 2, 5, 10, forecasting accuracies: 0.1, 0.25, 0.5, 1, 2.5 in both dimensions, forecasting misestimation probabilities: 0, 0.1, 0.25, 0.5, step sizes of input data updates: 1, 2, 5, 10, reoptimization algorithms: MIP, heuristic (Table 8).

Table 8 Average results (number of vehicles/travel distance) for the online vehicle routing problem with gradual look-ahead
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Dunke, F., Nickel, S. Online optimization with gradual look-ahead. Oper Res Int J 21, 2489–2523 (2021). https://doi.org/10.1007/s12351-019-00506-z

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