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On integration of Fuchs-class equations with four singular points in connection with a conformal mapping problem
Published in C Constanda, J Saranen, S Seikkala, Integral methods in science and engineering, 2020
In this paper we consider Fuchs-class equation with four real-valued singular points and the associated conformal mapping on to a circular quadrangle. A method for the integration of the equation and for constructing the mapping proposed in [11] is presented. The method includes: 1) the determination of the auxiliary parameters; 2) representations for the solution of the Fuchs-class equation in the form of Burmann-Lagrange series by combinations of exponentials and elliptic functions; 3) the associated representation for the direct and inverse conformal mappings; 4) an exact algorithm for obtaining the coefficients of these representations; 5) the determination of the monodromy group and transfer group. The method has been numerically realised and applied to some specific problems [11].
Graph Algorithms I
Published in R. Balakrishnan, Sriraman Sridharan, Discrete Mathematics, 2019
R. Balakrishnan, Sriraman Sridharan
An algorithm A is said to be an approximate algorithmapproximate algorithm for the traveling salesman problem (TSP), if for any instance I of the TSP (an instance of a problem is obtained by specifying particular values of the parameters of the problem, for example, the graph of Figure 1.38 is an instance of TSP) if the ratio r(A)=A(I)/E(I) is bounded above by a constant c, where A(I) is the value found by the algorithm A on the input I and E(I) is the exact value of the instance. Theoretically, we may obtain E(I) by executing the brute-force algorithm on the instance I. Note that c ≥ 1. If the constant c is 1, then A is an exact algorithmexact algorithm.
Methodology
Published in Tolga Bektaş, Freight Transport and Distribution, 2017
Matheuristics are hybrid methods that combine mathematical programming, or in general optimisation methods, with heuristic algorithms. The aim is to take advantage of the power and capabilities of both classes of methods. Hybridisation can be done in at least two ways, as described here: Embed mathematical programming within a heuristic or a metaheuristic algorithm, in which one or several optimisation problems are solved optimally. This might be the case, for example, when integrated problems are solved by using a heuristic, but where each individual and smaller problem forming the integrated problem is solved optimally.Use heuristics within the framework of an exact algorithm, in which smaller or subproblems can be solved using a heuristic algorithm. One example to this approach would be to solve the subproblems arising in Benders decomposition or Lagrangean relaxation by using a heuristic or metaheuristic algorithm. A benefit of this approach would be a speed-up in the running time of the algorithm, but this would be at the expense of loss of optimality, or a guarantee thereof. In other words, the bounds provided by such an algorithm would not necessarily be valid.
Robust identical parallel machine scheduling with two-stage time-of-use tariff and not-all-machine option
Published in International Journal of Production Research, 2023
Future research can be extended in a few directions. First, because the proposed IR algorithm is shown to be computationally expensive in the numerical experiment, some approaches to drop inactive regret cuts can be adopted to accelerate the convergence of the IR algorithm. Other solution methods such as branch-and-bound procedure can also be used to develop a more efficient exact algorithm. Another weakness of our studied model is the assumption on the two-stage TOU tariff. Hence, it would be interesting to develop a more general model with cyclical and multi-stage TOU tariffs. Besides, it is also important to extend the robust scheduling problem into the hybrid flow shop, which is more complicated than the parallel machine scheduling environment.