Planning Problems in Public Transit

Borndörfer, Ralf; Grötschel, Martin; Jäger, Ulrich

doi:10.1007/978-3-642-11248-5_6

Cited by 14 publications

(7 citation statements)

References 12 publications

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“…Perumal et al 2020) or by means of incorporating issues regarding disturbances and robustness (cf. Borndörfer et al 2010;Ge et al 2020). Fairness settings regarding workload balancing are considered (e.g., by Xie and Suhl 2015;Er-Rbib et al 2021a).…”

Section: Integrated Vehicle and Crew Schedulingmentioning

confidence: 99%

Revisiting the richness of integrated vehicle and crew scheduling

Kliewer²,

Nourmohammadzadeh

et al. 2022

Public Transp

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The last decades have seen a considerable move forward regarding integrated vehicle and crew scheduling in various realms (airline industry, public transport). With the continuous improvement of information and communication technology as well as general solvers it has become possible to formulate more and more rich versions of these problems. In public transport, issues like rostering, delay propagation or days-off patterns have become part of these integrated problems. In this paper we aim to revisit an earlier formulation incorporating days-off patterns and investigate whether solvability with standard solvers has now become possible and to which extent the incorporation of other aspects can make the problem setting more rich and still keep the possible solvability in mind. This includes especially issues like delay propagation where in public transport delay propagation usually refers to secondary delays following a (primary) disturbance. Moreover, we investigate a robust version to support the claim that added richness is possible. Numerical results are provided to underline the envisaged advances.

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Section: Integrated Vehicle and Crew Schedulingmentioning

confidence: 99%

Revisiting the richness of integrated vehicle and crew scheduling

Kliewer²,

Nourmohammadzadeh

et al. 2022

Public Transp

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“…be a matrix in which we collect all of the above defined distance matrices. Let t be the number of columns of the matrix D. For each configuration C ∈ C we introduce a vector of variables x C of length t whose entries we index x C,σ ; we set the box constrains to x C,σ = n C ∀C ∈ C 4 The model is taken from [23].…”

Section: Closest Stringmentioning

confidence: 99%

“…with A an integer m × n matrix, b ∈ Z m , w ∈ Z n , l, u ∈ (Z ∪ {±∞}) n . It is well known to be strongly NP-hard, but models many important problems in combinatorial optimization such as planning [30], scheduling [14], and transportation [4] and thus powerful generic solvers have been developed for it [27]. Still, theory is motivated to search for tractable special cases.…”

Section: Introductionmentioning

confidence: 99%

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Evaluating and Tuning n -fold Integer Programming

Altmanová

Knop

Koutecký

2019

ACM J. Exp. Algorithmics

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In recent years, algorithmic breakthroughs in stringology, computational social choice, scheduling, etc., were achieved by applying the theory of so-called n-fold integer programming. An n-fold integer program (IP) has a highly uniform block structured constraint matrix. Hemmecke, Onn, and Romanchuk [Math. Programming, 2013] showed an algorithm with runtime ∆ O(rst+r 2 s) n 3 , where ∆ is the largest coefficient, r, s, and t are dimensions of blocks of the constraint matrix and n is the total dimension of the IP; thus, an algorithm efficient if the blocks are of small size and with small coefficients. The algorithm works by iteratively improving a feasible solution with augmenting steps, and n-fold IPs have the special property that augmenting steps are guaranteed to exist in a not-too-large neighborhood. However, this algorithm has never been implemented and evaluated.We have implemented the algorithm and learned the following along the way. The original algorithm is practically unusable, but we discover a series of improvements which make its evaluation possible. Crucially, we observe that a certain constant in the algorithm can be treated as a tuning parameter, which yields an efficient heuristic (essentially searching in a smaller-than-guaranteed neighborhood). Furthermore, the algorithm uses an overly expensive strategy to find a "best" step, while finding only an "approximately best" step is much cheaper, yet sufficient for quick convergence. Using this insight, we improve the asymptotic dependence on n from n 3 to n 2 log n.Finally, we tested the behavior of the algorithm with various values of the tuning parameter and different strategies of finding improving steps. First, we show that decreasing the tuning parameter initially leads to an increased number of iterations needed for convergence and eventually to getting stuck in local optima, as expected. However, surprisingly small values of the parameter already exhibit good behavior while significantly lowering the time the algorithm spends per single iteration. Second, our new strategy for finding "approximately best" steps wildly outperforms the original construction. ACM Subject ClassificationTheory of computation → Parameterized complexity and exact algorithms; Mathematics of computing → Solvers; Theory of computation → Discrete optimization Keywords and phrases n-fold integer programming, integer programming, analysis of algorithms, primal heuristic, local search Supplement Material https://github.com/katealtmanova/nfoldexperiment

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“…In the remainder of this paper, we denote by P(c, A, b, , u, I) a MIP of form (1) in dependence of the provided data. This allows to model many real-world optimization problems from various fields like production planning [28], scheduling [20], transportation [13], or telecommunication networks [24]. On the other hand, the strict specifications for the problem statement make it possible to solve arising optimization problems for all these applications using the same algorithm.…”

Section: Introductionmentioning

confidence: 99%

Structure-Based Primal Heuristics for Mixed Integer Programming

Gamrath

Berthold

Stefan

et al. 2015

Optimization in the Real World

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Primal heuristics play an important role in the solving of mixed integer programs (MIPs). They help to reach optimality faster and provide good feasible solutions early in the solving process. In this paper, we present two new primal heuristics which take into account global structures available within MIP solvers to construct feasible solutions at the beginning of the solving process. These heuristics follow a large neighborhood search (LNS) approach and use global structures to define a neighborhood that is with high probability significantly easier to process while (hopefully) still containing good feasible solutions. The definition of the neighborhood is done by iteratively fixing variables and propagating these fixings. Thereby, fixings are determined based on the predicted impact they have on the subsequent domain propagation. The neighborhood is solved as a sub-MIP and solutions are transferred back to the original problem. Our computational experiments on standard MIP test sets show that the proposed heuristics find solutions for about every third instance and therewith help to improve the average solving time.

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Planning Problems in Public Transit

Cited by 14 publications

References 12 publications

Revisiting the richness of integrated vehicle and crew scheduling

Revisiting the richness of integrated vehicle and crew scheduling

Evaluating and Tuning n -fold Integer Programming

Structure-Based Primal Heuristics for Mixed Integer Programming

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