### Article

## Two-Station Single Track Scheduling Problem

Single track segments are common in various railway networks, in particular in various supply chains. For such a segment, connecting two stations, the trains form two groups, depending on what station is the initial station for the journey between these two stations. Within a group the trains differ by their cost functions. It is assumed that the single track is sufficiently long so several trains can travel in the same direction simultaneously. The paper presents polynomial-time algorithms for different versions of this two-station train scheduling problem with a single railway track. The considered models differ from each other by their objective functions.

The volume is dedicated to Boris Mirkin on the occasion of his 70th birthday. In addition to his startling PhD results in abstract automata theory, Mirkin’s ground breaking contributions in various fields of decision making and data analysis have marked the fourth quarter of the 20th century and beyond. Mirkin has done pioneering work in group choice, clustering, data mining and knowledge discovery aimed at finding and describing non-trivial or hidden structures—first of all, clusters, orderings, and hierarchies—in multivariate and/or network data.

This volume contains a collection of papers reflecting recent developments rooted in Mirkin's fundamental contribution to the state-of-the-art in group choice, ordering, clustering, data mining, and knowledge discovery. Researchers, students, and software engineers will benefit from new knowledge discovery techniques and application directions.

We consider a problem of the astronaut training scheduling. Each astronaut has his own set of tasks which should be performed with respect to resource and time constraints. The problem is to determine start moments for all considered tasks. For this issue a mathematical model based on integer linear programming is proposed. Computational results of the implemented model and experiments on real data are presented.

Financial Decision Making Using Computational Intelligence covers all the recent developments in complex financial decision making through computational intelligence approaches. Computational intelligence has evolved rapidly in recent years and it is now one of the most active fields in operations research and computer science. The increasing complexity of financial problems and the enormous volume of financial data often make it difficult to apply traditional modeling and algorithmic procedures. In this context, the field of computational intelligence provides a wide range of useful techniques, including new modeling tools for decision making under risk and uncertainty, data mining techniques for analyzing complex data bases, and powerful algorithms for complex optimization problems.

This book constitutes the proceedings of the 9th International Conference on Discrete Optimization and Operations Research, DOOR 2016, held in Vladivostok, Russia, in September 2016.

The 39 full papers presented in this volume were carefully reviewed and selected from 181 submissions. They were organized in topical sections named: discrete optimization; scheduling problems; facility location; mathematical programming; mathematical economics and games; applications of operational research; and short communications.

The computerized system of statistical operations research of pulsed optical tomography of spherical microobjects is under consideration. The system serves for creating new methods of dynamic space-time reconstruction of microobjects by their pulse discrete projection images and allows to perform dynamic measurements of main geometric miroobject properties. On this basis, original methods of on-line control and sorting by geometric dimensions and shape of nuclear fuel microobjects have been developed. These methods use the statistical reconstruction of the size (*D*) and shape (*K*) of each microobject on the base of their outline overall dimensions of three mutually orthogonal two-dimensional pulsed discrete projection images of each microobject. The processing speed of these methods are nearly 100 MO/s in the diameter range of 400–1500 μm.

On the one hand, the relevance of this research is determined by an attempt of solving the problem of optimal inventory allocation, which can open the possibilities for increase in stock turnover. On the other hand, there was an attempt to extend the list of problems which can be solved by operations research methods. The potential of application of operations research methods (transport model as a specific case of linear programming, in particular) is underestimated. According to Taha [2011], a transport model is a problem of finding optimal allocation of homogeneous objects from accumulators (a i ) to receivers (b i ) with minimizations of costs on displacement or movement. In our opinion, the canonical form of a transport model represents accumulators as points of departures, receivers as clients and cost on displacement as transport costs. The paradigm of using this model is constrained by using the latter in transport logistics only. In fact we can apply this model in much more problems (micro, meso or macro level). This study shows that objects and variables from the canonical transport model can be represented as objects from different fields (beyond logistics) thus helping to find an optimal solution to a certain problem. Our study represents accumulators as nominal cells where work-in-process (WIP) product is in the warehouse, receivers - as production lines and costs on displacements as mileage of loaders. Thus, the cost function Z that we want to optimize is the function of mileage of loaders. Minimizing function Z will enable us to find the optimal allocation of WIP products to production lines (next production stage)

Wildfires are a naturally occurring phenomenon in many places of 4 the world. While they perform a number of important ecological functions, the 5 proximity of human activities to forest landscapes requires a measure of control/pre- 6 paredness to address safety concerns and mitigate damage. An important technique 7 utilized by forest managers is that of wildfire fuel management, in which a portion 8 of the available combustible material in the forest is disposed of through a variety 9 of fuel treatment activities. A number of operations research approaches have been 10 applied to locate and schedule these fuel treatment activities, and herein we review 11 and discuss the various models and approaches in the literature

The cosmonauts training planning problem is a problem of construc- tion of cosmonauts training timetable. Each cosmonaut has his own set of tasks which should be performed with respect to resource and time con- straints. The problem is to determine start moments for all considered tasks. This problem is a generalization of the resource-constrained project scheduling problem with “time windows”. In addition, the investigated problem is extended with restrictions of different kinds. Previously, for solving this problem the authors proposed an approach based on methods of integer linear programming. However, this approach turned out to be ineffective for high-dimensional problems. A new heuristic method based on constraint programming is developed. The effectiveness of the method is verified on real data.

The system is designed to create new methods of dynamic spatial reconstruction of the size and shape of microobjects on the base of their pulsed discrete projection images. These methods provide high-performance monitoring and sorting of microobjects in real time.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.