### Article

## The Mixed Chinese Postman Problem

The routing problems are important for logistic and transport sphere. Basically, the routing problems related to determining the optimal set of routes in the multigraph. The Chinese postman problem (CPP) is a special case of the routing problem, which has many potential applications. We propose to solve the MCPP (special NP-hard case of CPP, which defined on mixed multigraph) using the reduction of the original problem into General Travelling Salesman Problem (GTSP). The variants of CPP are pointed out. The mathematical formulations of some problems are presented. The algorithm for reduction the MCPP in multigraph into GTSP is shown. The experimental results of solving MCPP in multigraph through the reduction into GTSP are presented.

The routing problems are important for logistic and transport sphere. Basically, the routing problems related to determining the optimal set of routes in the multigraph. The Chinese postman problem (CPP) is a special case of the routing problem, which has many potential applications. We propose to solve the MCPP (special NP-hard case of CPP, which defined on mixed multigraph) using the reduction of the original problem into General Travelling Salesman Problem (GTSP). The variants of CPP are pointed out. The mathematical formulations of some problems are presented. The algorithm for reduction the MCPP in multigraph into GTSP is shown. The experimental results of solving MCPP in multigraph through the reduction into GTSP are presented.

Motivated by applications in manufacturing industry, we consider a supply chain scheduling problem, where each job is characterised by non-identical sizes, different release times and unequal processing times. The objective is to minimise the makespan by making batching and sequencing decisions. The problem is formalised as a mixed integer programming model and proved to be strongly NP-hard. Some structural properties are presented for both the general case and a special case. Based on these properties, a lower bound is derived, and a novel two-phase heuristic (TP-H) is developed to solve the problem, which guarantees to obtain a worst case performance ratio of . Computational experiments with a set of different sizes of random instances are conducted to evaluate the proposed approach TP-H, which is superior to another two heuristics proposed in the literature. Furthermore, the experimental results indicate that TP-H can effectively and efficiently solve large-size problems in a reasonable time.

The resource efficiency of different implementations of the branch-and-bound method for the classical traveling salesman problem depends, inter alia, on ways to organize a search decision tree generated by this method. The classic «time-memory» dilemma is realized herein either by an option of storing reduced matrices at the points of the decision tree, which leads to reduction in the complexity with additional capacity cost, or matrix recalculation for the current node, which leads to an increase in complexity while saving memory. The subject of this paper is an experimental study of temporal characteristics of solving the traveling salesman problem by the branch-and-bound method to identify a real reduction of span time using additional memory in a selected structure of a decision tree. The ultimate objective of the research is to formulate recommendations for implementing the method in practical problems encountered in logistics and business informatics. On the basis of experimental data, this paper shows that both considered options of the classic algorithm for the traveling salesman problem by the branch-and-bound method generate software implementations with an exponential dependence on the execution time of the input length. The experimental results permit us to suggest that the applicability of an additional memory capacity of no more than 1 GB results in a significant (up to five times) reduction of the time span. The estimate of the resulting trend makes it possible to recommend practical application of the software implementation of the branch-and-bound algorithm with storage of matrices - with a really available 16 GB random-access memory and with limitation of the expected average computation time of about one minute on modern personal computers whereby problems having a dimension no more than 70 can be solved exactly.

The Travelling Salesman Problem (TSP) is a fundamental task in combinatorial optimization. A special case of the TSP is Metric TSP, where the triangle inequality holds. Solutions of the TSP are generally used for costs minimization, such as finding the best tour for round-the-world trip or construction of very large-scale integration schemes. Since the TSP is NP-hard, heuristic algorithms providing near optimal solutions will be considered. The objective of this article is to find a group of Pareto optimal heuristic algorithms for Metric TSP under criteria of run time efficiency and qualitative performance as a part of the experimental study. Classification of algorithms for Metric TSP is presented. Feasible heuristic algorithms and their prior estimates are described. The data structure and the details of the research methodology are provided. Finally, results and prospective research are discussed.

The theory of single upper and lower tolerances for combinatorial minimization problems has been formalized in 2005 for the three types of cost functions sum, product and maximum, and since then shown to be rather useful in creating heuristics and exact algorithms for the Traveling Salesman Problem and related problems. In this paper for these three types of cost functions we extend this theory from single to set tolerances and the related reverse set tolerances. In particular, we characterize specific values of (reverse) set upper and lower tolerances as positive and infinite, and we present a criterion for the uniqueness of an optimal solution to a combinatorial minimization problem. Furthermore, we present formulas or bounds for computing (reverse) set upper and lower tolerances using the relation to their corresponding single tolerance counterparts. Finally, we give formulas for the minimum and maximum (reverse) set upper and lower tolerances using again their corresponding single tolerance counterparts.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.