Fakultät für Mathematik
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Item On the stochastic vehicle routing problem with time windows, correlated travel times, and time dependency(2021-03-12) Bomboi, Federica; Buchheim, Christoph; Pruente, JonasMost state-of-the-art algorithms for the Vehicle Routing Problem, such as Branch-and-Price algorithms or meta heuristics, rely on a fast feasibility test for a given route. We devise the first approach to approximately check feasibility in the Stochastic Vehicle Routing Problem with time windows, where travel times are correlated and depend on the time of the day. Assuming jointly normally distributed travel times, we use a chance constraint approach to model feasibility, where two different application scenarios are considered, depending on whether missing a customer makes the rest of the route infeasible or not. The former case may arise, e.g., in drayage applications or in the pickup-and-delivery VRP. In addition, we present an adaptive sampling algorithm that is tailored for our setting and is much faster than standard sampling techniques. We use a case study for both scenarios, based on instances with realistic travel times, to illustrate that taking correlations and time dependencies into account significantly improves the quality of the solutions, i.e., the precision of the feasibility decision. In particular, the nonconsideration of correlations often leads to solutions containing infeasible routes.Item Laplace contour integrals and linear differential equations(2021-07-17) Steinmetz, NorbertThe purpose of this paper is to determine the main properties of Laplace contour integrals Λ(z)=12πi∫Cϕ(t)e−ztdt that solve linear differential equations L[w](z):=w(n)+∑j=0n−1(aj+bjz)w(j)=0. This concerns, in particular, the order of growth, asymptotic expansions, the Phragmén–Lindelöf indicator, the distribution of zeros, the existence of sub-normal and polynomial solutions, and the corresponding Nevanlinna functions.Item Malmquist-type theorems for cubic Hamiltonians(2021-02-06) Steinmetz, NorbertThe aim of this paper is to classify the cubic polynomials H(z,x,y)=∑j+k≤3ajk(z)xjyk over the field of algebraic functions such that the corresponding Hamiltonian system x′=Hy, y′=−Hx has at least one transcendental algebroid solution. Ignoring trivial subcases, the investigations essentially lead to several non-trivial Hamiltonians which are closely related to Painlevé’s equations PI, PII, P34, and PIV . Up to normalisation of the leading coefficients, common Hamiltonians are HI:HII/34:HIV:−2y3+12x2−zyx2y−12y2+12zy+κxx2y+xy2+2zxy+2κx+2λy13(x3+y3)+zxy+κx+λy, but the zoo of non-equivalent Hamiltonians turns out to be much larger.