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Efficient generation of performance bounds for a class of traffic scheduling problems

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Abstract

This work seeks to develop (lower) performance bounds for a traffic scheduling problem that arises in many application contexts, ranging from industrial material handling and robotics to computer game animations and quantum computing. In a first approach, the sought bounds are obtained by applying the Lagrangian relaxation method to a MIP formulation of the considered scheduling problem that is based on a natural notion of “state” for the underlying traffic system and an analytical characterization of all the possible trajectories of this state over a predefined time horizon. But it is also shown that the corresponding “dual” problem that provides these bounds, can be transformed to a linear program (LP) with numbers of variables and constraints polynomially related to the size of the underlying traffic system and the employed time horizon in the MIP formulation. Furthermore, the derived LP formulation constitutes the LP relaxation of a second MIP formulation for the considered scheduling problem that can be obtained through an existing connection between this problem and the “integral multi-commodity flow” (IMCF) model of network optimization theory. Finally, the theoretical developments of the paper are complemented with a computational part that demonstrates the efficacy of the pursued methods in terms of the quality of the derived bounds, and their computational tractability.

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Notes

  1. The time that it is required by any feasible traffic schedule for the considered problem to move all the traveling agents from their current locations to their destinations, is known as the “makespan” of this schedule; hence, in more technical terms, our problem is a “min makespan” scheduling problem. We shall provide a detailed positioning of this problem in later parts of this work.

  2. We emphasize, however, that our results have been developed in the Ph.D. thesis program of the first author independently from, and in parallel to, the corresponding developments of Yu and LaValle (2016) and Ma et al. (2016).

  3. While facilitating the presentation of the subsequent developments, the presumed uniformity of the zone traversal times is not restrictive, since one can adapt the presented developments to the non-uniform case by using the greatest common divisor of the various zone traversal times as the discretizing time unit.

  4. As acknowledged in Yu and LaValle (2016), this is a typical requirement for most models of guidepath-based traffic systems. In the context of zone-controlled until-load MHS (Heragu 2008) and of the qubit transport systems that are employed by quantum computing (Daugherty 2017), this requiirement is motivated by the need to ensure collision-freedom for the traveling agents during the unobservable, transitional phase between time periods t and t + 1.

  5. Besides its practical relevance in many applications, the selection of the schedule makespan as the employed objective function also expresses our intention to address one of the most difficult variations of the considered scheduling problem, since this criterion is a nonlinear function of the dynamics of the underlying traffic.

  6. This constraint is dictated by the broader logic that defines the MPC scheme that provides context for the considered MIP formulation.

  7. In the statement of this constraint, we further assume that the constraint is observed by the problem data that specify the initial positions, sa, and the destinations, da, of the agents \(a\in {\mathcal A}\).

  8. We also notice, for completeness, that the MIP formulation of Eqs. (1)–(10) can be infeasible for some of its instantiations, and the assessment of the corresponding (in-)feasibility is a hard problem in itself. In general, the feasibility of the considered MIP will depend on (i) the topology of the underlying guidepath network and the relative positioning of the edges sa and da, \(a\in {\mathcal A}\), in this topology, as well as (ii) the selection of the parameter T. Determining feasibility w.r.t. the first of the above two elements is a hard “reachability” problem that should be addressed in the context of the “untimed” dynamics of the underlying traffic system, using, for instance, some “linguistic” modeling framework for these dynamics, like automata theory or Petri nets (Cassandras and Lafortune 2008). On the other hand, for feasible problem instances w.r.t. criterion (i), a pertinent T value that will not compromise this feasibility, can be obtained through the solution of the corresponding scheduling problem by a heuristic method.

  9. A complete definition of matrix A and vector β2 can be obtained from the parsing of the right-hand-side of Eq. (11) that defines the Lagrangian function employed in this work.

  10. A more exact characterization of the elements \(F_{a},\ \boldsymbol {\beta }^{1}_{a}\) and Qa that appear in Eqs. (41) and (42), is provided in Section 5.

  11. As remarked in the introductory section, in the general case, the bounds obtained from the LP relaxation of a MIP formulation might not be as tight as the corresponding bounds that are obtained through Lagrangian duality theory (Bertsekas 1999).

  12. In the graph of Fig. 1 the available zones are encoded by the graph nodes and not by its edges; but the translation of this structure to the corresponding model of Section 2 is pretty straightforward.

  13. In fact, in certain cases, we can actually infer that the observed gap is due primarily to the sub-optimality of the schedule that is utilized in the estimation of the corresponding gap. As a concrete example, we refer to the case of pair (D5, H5) in the line of Table 1 corresponding to 15 agents. Looking at the column for H5, we observe the sequence 〈17, 20, 18〉 for the rows corresponding to number of agents 12, 15 and 18. But the construction of the corresponding problem instances through the addition of three agents from each instance to the next, implies that the optimal makespans for these three problem instances should be monotonically increasing. Therefore, the actual optimal makespan for the fifth problem instance with 15 traveling agents is actually 17 or 18, and not 20.

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Authors and Affiliations

  1. Roadie, Atlanta, GA, USA

    Greyson Daugherty

  2. School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA, USA

    Spyros Reveliotis

  3. Georgia Tech Research Institute, Georgia Institute of Technology, Atlanta, GA, USA

    Greg Mohler

Authors
  1. Greyson Daugherty
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  2. Spyros Reveliotis
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  3. Greg Mohler
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Correspondence to Spyros Reveliotis.

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This work was partially supported by NSF grant ECCS-1707695.

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Daugherty, G., Reveliotis, S. & Mohler, G. Efficient generation of performance bounds for a class of traffic scheduling problems. Discrete Event Dyn Syst 29, 211–235 (2019). https://doi.org/10.1007/s10626-019-00284-y

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