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Third International Conference on
Network Analysis 2013
May 20th – May 22nd, 2013
Center for Applied optimization (CAO),
University of Florida, USA
Laboratory of Algorithms and Technologies for Networks Analysis
(LATNA), Higher School of Economics, Russia
Monday, May 20th
Room 209 HSE, 136 Rodionova Str.
14:00-14:30 Panos M. Pardalos
Third International Conference on Network Analysis 2013
14:30-15:20 Steffen Rebennack
Policy Evaluation for Power Generation Expansion Planning with Demand Response: A Dynamic Benders' Decomposition Variant
15:20-15:40 Coffee Break
15:40-17:20 Session 1
Dinh The Luc
Multi-product supply demand networks with elementary flows
Lev Afraimovich
Approximation network schemes for multi-index decomposable problems
Ludmila Egorova
Pattern analysis in the study of science, education and innovative activity in Russian regions
Maksim Lukoyanov
Graph theoretical analysis of resting state electroencephalogram of left- and right-handed people
Olga Khvostova
Students partnership network: the case of Russian university
Tuesday, May 21st
Room 209 HSE, 136 Rodionova Str.
09:30-10:20 Athanasios Migdalas
Heuristic and metaheuristic routing
10:20-10:40 Coffee Break
10:40-11:30 Julius Zilinskas
Multi-objective combinatorial optimization
11:30-12:30 Session 2
Pavel Sukhov
Heuristic for the preemptive single machine scheduling problem with arbitrary processing times
Mikhail Batsyn
Lower and upper bounds for the preemptive single machine scheduling problem with equal processing times
Ilya Bychkov
An exact model for the cell formation problem
12:40-14:00 Lunch Break
14:00-14:50 Yuri Kochetov
The Stackelberg games in facility location
14:50-15:10 Coffee Break
15:10-16:10 Session 3
Anton Kocheturov
Analysis of cluster structures in financial market graphs
Arseniy Vizgunov
Analysis of the globalization process using the market graph model
Grigory Bautin
Comparative analysis of two similarity measures for the market graph construction
16:10-16:30 Coffee Break
16:30-17:30 Session 4
Vyacheslav Chistyakov
Concepts of Stability in Discrete Optimization Involving A-Operations
Dmitriy Malyshev
«Critical» graph classes for the edge list ranking problem
Dmitriy Gribanov
Integer program with bounded minors
Wednesday, May 22nd
Room 209 HSE, 136 Rodionova Str.
9:30-10:20 Sonia Cafieri
On exact methods for network clustering
10:20-10:40 Coffee Break
10:40-12:20 Session 5
Evgeny Maslov
An exact algorithm for the protein structure alignment problem
Alexander Porshnev
Using open-source software for visualization of citation networks: solutions for students and scholars
Dmitriy Mokeev
Konig graphs with respect to 4-paths
Victor Zamaraev
On P7-free bipartite graphs
Petr A. Koldanov
Efficiency analysis of branch network
Policy Evaluation for Power Generation Expansion Planning with Demand Response: A Dynamic Benders' Decomposition Variant
Steffen Rebennack and Timo Lohmann
Colorado School of Mines, Division of Economics and Business, USA
Demand response is the ability of load to respond to short-term changes in electricity prices as opposed to a completely inelastic demand. It plays an increasingly important role with the share of electricity coming from renewable energy sources increasing, but has not been taken into consideration in long-term power generation expansion planning problems in the academic literature. The combination of demand response, a long planning horizon, an hourly time resolution due to load patterns, and a large power system, are challenging to model and, in particular, to solve.
We present a long-term power generation expansion planning model that contains the above features. We describe assumptions for the demand response function that are necessary to solve the model using a new approach to Benders' Decomposition. Further, we present an extension of the model that includes start-up restrictions of coal and nuclear power plants. The Benders' Decomposition approach is modified to handle the resulting coupling of hours. We demonstrate the efficiency of the proposed algorithms with a case study of the Texas power system and its deregulated market environment.
The mathematical framework allows for the evaluation and comparison of different renewable policy standards.
Multi-product supply demand networks with elementary flows
Dinh The Luc
Avignon University, France
In this talk we present a multi-product multi-criteria supply demand network with capacity constraints. We analyze different concepts of equilibrium and establish some relationships between them. Particular attention is paid on elementary flows and on the construction of variational inequalities which are equivalent to network equilibrium problems. A discussion on numerical methods for finding equilibrium flows is also given.
Approximation network schemes for multi-index decomposable problems
Afraimovich L.G.
Lobachevsky State University of Nizhny Novgorod, Russia
We investigate integer multi-index transport-type problem, known to be NP-hard already in the three-index case. Earlier we proposed polynomial algorithm for multi-index problems with special decomposable structure. The algorithm is based on constructing corresponding min-cost flow problem and computing the solution of the multi-index problem as a decomposition of the optimal flow into simple cycles. In this paper we describe several approximation algorithms for special NP-hard classes of multi-index problems, which are based on the described above flow algorithm.
Pattern analysis in the study of science, education and innovative activity in Russian regions
Aleskerov F., Egorova L., Gokhberg L., Myachin A., Sagieva G.
National Research University Higher School of Economics, Moscow, Russia
We describe the method of pattern analysis and the results of its application to the problem of analyzing the development of science, education and the success of innovative activity in the regions of the Russian Federation. We examine characteristics of the regions of Russia such as the level of socio-economic conditions and the potential and efficiency of science, education and innovative activity from 2007 to 2010. Also we obtain a classification of regions by the similarity of the internal structure of these indicators, construct trajectories of regional development over time, and find groups of regions carrying out similar strategies.
Graph theoretical analysis of resting state electroencephalogram of left- and right-handed people
Maksim Lukoyanov
National Research University Higher School of Economics, Nizhny Novgorod, Russia
Analysis of network properties with graph theory is a perspective way in modern neuroscience. We used graph theory to examine differences in resting state electroencephalogram (EEG) in left- and right-handed people. EEG was recorded during passive perception of white screen and screen with simple objects (line with different slope) before and after memory tests. 19 channels were used for recording. 41 participants at age from 18 to 25 divided into two groups according their handedness (19 left-handed: 11 female, 22 right-handed: 15 female) took part in the experiment. Magnitude-squared coherence and imaginary part of the coherency was calculated in 6 different frequency bands and between each pair of electrodes to obtain graphs from which minimum spanning trees were constructed. From these trees, several parameters were calculated to characterize differences in network organization of left- and right- handers. Two patterns of differences were obtained: diameter and radius of MST form the first one, leaf number and tree hierarchy measure form the second. Additional investigating need to clarify neuronal sources of these results.
Students partnership network: the case of Russian university
Olga Khvostova
National Research University Higher School of Economics, Nizhny Novgorod, Russia
In this talk we present the analysis of social ties between students. One group of 20 students at the Business-informatics and applied mathematics faculty, NRU HSE ‑ Nizhny Novgorod is chosen. We apply graph theory to examine social ties in network of friendship and network of assistance in studying. We discuss differences between friendship and partnership relations. It is shown that network of friendship and network of assistance intersect. Particular attention is paid to students characteristics both in studying and financial and social status. We discuss what characteristics lead to friendship ties using the idea of similarity.
The author is partially supported by LATNA Laboratory, NRU HSE, RF government grant, ag. 11.G34.31.0057.
Heuristic and metaheuristic routing
Yannis Marinakis1 & Athanasios Migdalas2
1Technical University of Crete, Dept of Production Engineering and Management, University Campus, 73100 Chania, Crete, Greece, marinakis@ergasya.tuc.gr
2Aristotle University of Thessalonike, School of Engineering, Dept of Mathematical and Physical Sciences, Egnatia University Campus, 54124 Thessalonike, Macedonia, Greece, samig@gen.auth.gr
Routing has always been of immense importance in communication networks due to its impact on the network performance, and the significance of scalable and adaptive routing has sky-rocked during the last decade as a consequence of the ever increasing demand for Internet and mobile communications. A routing algorithm selects one or more paths over which devices communicate with each other. The selection of these paths is based on the status of the network and optimization criteria aiming at maximizing network performance by pursuing e.g. optimization of resource utilization, or/and minimization of congestion (packet delay), or/and minimization of packet loss, etc. Due to these facts, routing problems are combinatorial optimization problems which differ with respect to type, i.e., a routing may be of unicast, multicast or anycast type depending on the number of communicating devices, with respect to the network topology which may be of fixed infrastructure or of ad-hoc wireless infrastructure. This research topic has been very actively studied during the last years and a considerably number of specialised routing problems have been analyzed resulting in new algorithms of mainely heuristic and metaheuristic type. Particularly, population based metaheuristics have attracted considerable attention as they are capable to provide adaptivity and scalability absent in conventional protocols such as OSPF and RIP. We survey the research topic and also propose a new adaptive routing strategy based on a particle swarm optimization (PSO) algorithm. PSO simulates the social behavior of social organisms by using the physical movements of the individuals in the swarm. Its mechanism enhances and adapts to the global and local exploration. Preliminary computational tests are presented.
Multi-objective combinatorial optimization
Julius Zilinskas
Vilnius University, Lithuania
Many applied optimization problems have more than one conflicting objectives. The aim of such multi-objective problems is to find Pareto set of non-dominated solutions. In this talk we consider multi-objective branch-and-bound to find the exact Pareto-optimal set and meta-heuristics for approximation of the Pareto-optimal front. We discuss some problems of multi-objective combinatorial optimization like cell formation, aesthetic visualization of business process diagrams, and competitive facility location. Cell formation problem aims at identification of groups of machines to form manufacturing cells, what may include conflicting objectives like the number of inter-cell moves and within-cell load variation. Visualization of business process diagrams involves multiple objectives like the length of connectors and the compatibility of the sequence flows with the favorable top-down left-right direction. The aim of competitive facility location aims at maximization of the market share captured by the new facilities and minimization of the lost market share of the old facilities caused by the entering of the new facilities in the market.
Heuristic for the preemptive single machine scheduling problem with arbitrary processing times
Pavel Sukhov
National Research University Higher School of Economics, Nizhny Novgorod, Russia
The preemptive single machine scheduling problem of minimizing the total weighted completion time with arbitrary processing times and release dates is an important NP-hard problem in scheduling theory. In this work we present a fast and high quality heuristic for this problem based on the WSRPT (Weighted Shortest Remaining Processing Time) rule. The running time increases only as a square of the number of jobs. Our computational study shows that very large size instances might be treated within extremely small CPU times.
Lower and upper bounds for the preemptive single machine scheduling problem with equal processing times
Mikhail Batsyn
National Research University Higher School of Economics, Nizhny Novgorod, Russia
The preemptive single machine scheduling problem of minimizing the total weighted completion time with equal processing times and arbitrary release dates is one of the four single machine scheduling problems with an open computational complexity status. In this talk we present lower and upper bounds for the exact solution of this problem based on the assignment problem. We also investigate properties of these bounds and worst-case behavior.
An exact model for the cell formation problem
Ilya Bychkov
National Research University Higher School of Economics, Nizhny Novgorod, Russia
Currently there are no exact approaches (known to the authors) suggested for solving the cell formation problem (CFP) with the grouping efficacy objective. The only exact model which solves the CFP in a restricted formulation is due to Elbenani & Ferland (2012). The restriction consists in fixing the number of production cells. The main difficulty of the CFP is the fractional objective function - the grouping efficacy. In this paper we address this issue for the CFP in its common formulation with a variable number of cells. Our computational experiments are made for the most popular set of 35 benchmark instances. For the 13 of these instances using CPLEX software we prove that the best known solutions are exact global optimums.
The Stackelberg games in facility location
Yuri Kochetov
Sobolev Institute of Mathematics, Novosibirsk, Russia
In the Stackelberg games, there are two decision makers which we refer to as a leader and a follower. In the case of facility location, they compete to provide customers from a given market by opening the certain number of facilities. The decision makers open facilities in turn. At first, the leader decides where to locate facilities taking into account the follower reaction. Later on, the follower opens own facilities. We assume that the customer’s preferences among the opened facilities are based on the distances to these facilities and the quality of service provided by the decision makers. We consider the binary preference rules (all or none), where each customer chooses the closest opened facility, and the gravity preference rules, where each customer uses all opened facilities as suppliers. Each customer has a demand (purchasing power or weight). We assume that the demands are essential, that is goods must be consumed. The weight of each customer is fixed and does not depend on how far or close the customer is to the facilities. The leader and the follower obtain a profit from serving the customer which coincides with its weight. Each decision maker maximizes its profit. The problem is to define the facilities for the leader which maximize his profit.
In this talk, we present exact mathematical formulations for these Stackelberg games as the bi-level mixed integer programming problems. Complexity status, the min-max reformulations, math-heuristics and exact methods for these games are reviewed. Finally, we present some applications of these molecommunications.
Analysis of cluster structures in financial market graphs
Anton Kocheturov
National Research University Higher School of Economics, Nizhny Novgorod, Russia
Analyzing of the financial markets is a developing sphere of research. A number of network-based approaches can be applied to study stocks and their behavior. In this paper we analyze how cluster structures derived from a market graph by means of the p-median model change while the number of clusters is increasing. We have studied these changes for different time periods and found different behavior during the crises and non-crises periods. Also we have discovered some interesting properties which are typical for financial markets.
Analysis of the globalization process using the market graph model
Arseniy Vizgunov
National Research University Higher School of Economics, Nizhny Novgorod, Russia
All countries stock markets are influenced by the globalization process and it is important to have models describing this influence. We use the network representation of the stock market referred to as market graph. The basis of the model is the market network constructed using correlation as a measure of similarity between stocks. In the market graph each vertex represents a stock. Two vertices are connected by an edge if the corresponding correlation coefficient is larger than or equal to the specified threshold. The influence of a stock is measured by volume of trades. We understand globalization process in two ways. The first one is the tendency of the stock markets graphs of the different countries to have the similar dynamics of the structural properties. The second one is existence of the stable cluster of stocks with disproportionate influence on the market. Our computational results show that Brazilian, Russian and Indian stock markets are more influenced by the globalization process than US stock market.
Comparative analysis of two similarity measures for the market graph construction
Grigory A. Bautin, Valery A. Kalyagin, and Alexander P. Koldanov.
National Research University Higher School of Economics, Nizhny Novgorod, Russia
Market graph is built on the basis of some similarity measure for financial asset returns. The paper considers two similarity measures: classic Pearson correlation and sign correlation. We study the associated market graphs and compare the conditional risk of the market graph construction for these two measures of similarity. Our main finding is that the conditional risk for the sign correlation is much better than for the Pearson correlation for larger values of threshold for several probabilistic models. In addition, we show that for some model the conditional risk for sign correlation dominates over the conditional risk for Pearson correlation for all values of threshold. These properties make sign correlation a more appropriate measure for the maximum clique analysis.
Concepts of Stability in Discrete Optimization Involving A-Operations
Vyacheslav Chistyakov
National Research University Higher School of Economics, Nizhny Novgorod, Russia
The talk addresses the tolerance approach to the sensitivity analysis of optimal solutions to the nonlinear optimization problem of the form
over S,
where S is a collection of nonempty subsets of a finite set X such that and , C is a cost (or weight) function from X into R or , and is a continuous, associative, commutative, nondecreasing and unbounded binary operation of generalized addition on R , called an A-operation. Simple examples of A-operations are the usual addition operation , the operation of taking the maximum , and the usual operation of multiplication , (). We evaluate and present sharp estimates for upper and lower bounds of costs of elements from X, for which an optimal solution to the above problem remains stable. These bounds extend in a unified way most known results in the sensitivity analysis. We define an invariant of the optimization problem – the tolerance function, which is independent of optimal solutions, and establish its basic properties, among which we mention a characterization of the set of all optimal solutions, the uniqueness of optimal solutions and the extremal values of the tolerance function.
This is a joint work with Panos M. Pardalos.
Supported by LANTA Laboratory, NRU HSE, Russian Federation government grant, ag. 11.G34.31.0057.
Integer program with bounded minors
Dmitriy Vladimirovich Gribanov
National Research University Higher School of Economics,
National Research University Lobachevsky State University of Nizhniy Novgorod
The work consists of several sections:
· In the first section we show that the integer program with almost unimodular matrix can be solved in polynomial time.
· In the second section we will provide a polynomial algoritm to find some integer solution of the integer program with bounded minors on the assumption that the polyhedron has sufficient width (see [6]).
· In the third section we show that some corner polyhedron of integer program with bimodular matrix can have an exponential number of edges (part. uses results from [3]).
· Finally, we consider the polynomially solvable program with a bimodular matrix.
This work significantly uses results from paper “S.I. Veselov, A.J. Chirkov Integer program with bimodular matrix” [1].
Also significant results were obtained if the matrix is a {0, 1}-matrix in paper [5].
The author is partially supported by LATNA Laboratory, NRU HSE, RF government grant, ag. 11.G34.31.0057; and by FTP in support program for scientific research that is conducted by teams of research and education centers for the scientific direction, grant № 2012-1.1-12-000-1005-870.
The author wishes to express special thanks for the invaluable assistance to A.J. Chirkov, S.I. Veselov, D.S. Malyshev, V.N. Shevchenko and S.V. Sorochan.
Bibliography
1. S.I. Veselov, A.J. Chirkov; Integer program with bimodular matrix // Discrete Optimization. Volume 6. Issue 2. May 2009. Pages 220–222.
2. G. Cornuejols, L.F. Zuluaga; On Padberg's conjecture about almost totally unimodular matrices // Operations Research Letters. Vol. 27. Issue 3. October 2000. P. 97‑99.
3. J.W. Grossman, D.M. Kilkarni, I.E. Schochetman; On the minors of an incidence matrix and its Smith normal form // Linear Algebra Appl. 1995. Vol.218. P. 213-224.
4. V.E. Alekseev, D.V. Zaharova; Independent sets in graphs with bounded minors of the extended incidence matrix // Discrete Analysis and Operations Research. 2010. Vol. 17. Num 1. P. 3-10. (in Russian)
5. D.S. Malyshev, D.V. Gribanov, P.M. Pardalos; About the complexity of integer linear programming with bounded minors of the restrictions matrix. //Preprint submitted in Discrete Optimization and Operations Research. (in Russian)
6. M.E. Houle, G.T. Toussaint; Computing the width of a set // IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 10, no. 5, pp. 761-765, Sept. 1988.
7. V.N. Shevchenko, V.V. Chumakov; About some quantitative characteristics of the minors of integral matrices // Bulletin of the Nizhny Novgorod Lobachevsky State University. Vol 1(2). 2004. P. 215-219. (in russian)
8. V.N. Shevchenko; Qualitative Topics in Integer Linear Programming // AMS. October 15, 1996. -ISBN-10: 0821805355. -ISBN-13: 978-0821805350.
9. A. Schrijver; Theory of Linear and Integer Programming // WileyInterscience series in discrete mathematics. John Wiley & Sons, 1998.
On exact methods for network clustering
Sonia Cafieri
The French University of Civil Aviation, Toulouse, France
Clustering on networks, or graphs, is attracting a growing attention in a wide variety of domains. Networks are an extremely useful representation of many complex systems, that often consist of entities linked together in some way, and clusters correspond to subgroups of entities among which strong relations exist.
The presently most used clustering criterion is modularity, due to Newman and Girvan. The modularity maximization problem has been extensively studied both from the algorithmic and from the applications viewpoints. Many heuristics, based on various concepts, have been proposed and yield rapidly near optimal solution or sometimes optimal ones but without a guarantee of optimality. In contrast, papers proposing exact algorithms or using mathematical programming are rare. We present recent advances on exact methods, used either to solve the whole underlying optimization problem or, locally, subproblems arising in hierarchical heuristics, or to refine solutions previously obtained by other methods.
An exact algorithm for the protein structure alignment problem
Evgeny Maslov
National Research University Higher School of Economics, Nizhny Novgorod, Russia
The protein structure alignment problem consists in finding the best alignment of two proteins defined by their primary structures. It finds the longest sequence of amino acids of one protein which matches (aligns with) a sequence of amino acids of another protein. This problem is polynomially reducible to the maximum clique problem (MCP) for the protein alignment graph. In this paper we present an efficient exact algorithm for the protein structure alignment problem based on the MCS algorithm by Tomita et al. (2010) for the MCP. To reduce the alignment problem to the MCP we follow the DAST approach introduced by Malod-Dognin et al. (2010). Our main contributions include: applying the ILS heuristic by Andrade et al. (2012), alignment graph preprocessing, preallocating of dynamic memory, and bit implementation of the adjacency matrix. The computational results are provided for the popular Skolnick test set of 40 proteins. The computations show that the suggested algorithm is faster than the MCS algorithm on all the considered instances and much faster than the Ostergard's (2002) algorithm.
Using open-source software for visualization of citation networks: solutions for students and scholars
Alexander Porshnev
National Research University Higher School of Economics
The modern university culture with its orientation toward scientific research and motto: “publish or perish” challenges students and scholars with growing demand for more quantity of high quality papers and a fact that authors are encouraged to write about new and interdisciplinary topics make this challenge more and more complex. The one of the possible ways how to establish better understanding of a research area is to visualize its topics, authors and keywords by network analysis (Bergström & Whitehead Jr, 2006). Network analysis and semantic technologies are technologies which could help in mapping a research area, creating first impression and collecting information about most important authors and works. Visualization could also show some black holes in literature review. In our paper we discuss possible steps of visualizing scientific area based on two examples from behavioral finance and psychology. We propose three steps model: 1. collecting information about scientific area: searching for keywords, usage of co-occurrence network (Dunning 1993) and Semantic networks (Sowa 1983) to enlarge to initial set of keywords; 2. gathering information from e-libraries for citations analysis; 3. applying PageRank (Brin and Page 1998), Hyperlinked-Induced Topic Search (Kleinberg 1999) algorithms to visualize information from created network. We discuss possibility to use existing free and open-source solutions like WordNet (Miller 1995) project for semantic networks, visualizing patterns and trends in scientific literature with CiteSpace (Chen, C. et al. 2010), CircleView (Bergström & Whitehead Jr, 2006), Cytoscape (Michael Smoot et al. 2010), and etc. As well we provide solutions and recommendations for students and scholars for application these technologies for academic purposes.
On P7-free bipartite graphs
Victor Zamaraev1, Andrew Collins2, Vadim Lozin2
1National Research University Higher School of Economics, Nizhny Novgorod, Russia
2DIMAP and Mathematics Institute, University of Warwick
A class of graphs is an infinite set of graphs closed under isomorphism. For a class X we write Xn for the set of graphs in X with vertex set 1,2, …, n. The cardinality of Xn is known as the speed of class X.
A class is called hereditary if it is closed under taking induced subgraphs. A hereditary class X is factorial if for some positive constants c1 and c2. Hereditary classes with the speed slower than factorial are well structured. The situation with factorial classes is more complicated and less explored, although this family includes many classes of theoretical and practical importance, such as planar graphs or graphs of bounded vertex degree. To simplify the study of factorial classes, V. Lozin proposed the following conjecture [2]: the speed of a hereditary class X is factorial if and only if the fastest of the following three properties is factorial: bipartite graphs in X, co-bipartite graphs in X and split graphs in X. From the point of view of this conjecture it makes sense to investigate the factorial subclasses of bipartite graphs. In [1] P. Allen studied the speed of subclasses of bipartite graph defined by a single forbidden bipartite induced subgraph. He classified almost all these classes with respect to being or not being factorial. The only class for which this answer is still unknown is the class of P7-free bipartite graphs. In this talk, in order to understand what makes this case difficult, we consider some subclasses of P7-free bipartite graphs.
1. P. Allen, Forbidden induced bipartite graphs, J. Graph Theory, 60 (2009) 219-241.
2. V. Lozin, C. Mayhill and V. Zamaraev, A note on the speed of hereditary properties, Electronic J. Combinatorics, 18 (2011) Research paper 157.
Efficiency analysis of branch network
Petr A. Koldanov
National Research University Higher School of Economics, Nizhny Novgorod, Russia
Consider the activity of an organization having branch network. Let these branches operate in different regional centers, under control of one common center and follow common strategy. One of the problems of control consists in comparison of several branches activity efficiency. This information is necessary, in particular, to select a rational method of resources distribution, namely, into which branches resources should be allocated.
Solution of this problem can be based on comparison of numerical indicators of these branches efficiency. In the report such indicator is the quantity of sales on corresponding product in different branches. We assume, that the quantities of sales are random variables, distribution of which depends on some unknown parameters. Let the comparison between parameters define the preferences of branches in efficiency. The main problem now is to construct a rule of ranking of branches on the base of set of samples of a small volume.
We consider such problem from mathematical statistic point of view. Our solution is based on the Lehmann's theory of multiple decision statistical procedure. This theory is based on three points: choice of the generating hypothesis, compatibility of the family of tests for the generating hypothesis with decision space for the original hypothesis, and additivity of the loss function.
In the report the problem of comparison of branch activity efficiency as a multiple decision problem is formulated. Then the choice of generating hypothesis and compatibility condition is discussed, and the condition of additivity of the loss function and condition of unbiasedness is investigated. The multiple decision statistical procedure of ranking of branches is constructed. It is shown that this multiple decision statistical procedure is optimal in the class of unbiased multiple decision statistical procedures.
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