The hub location–allocation problem under uncertainty is a real-world task arising in the areas such as public and freight transportation and telecommunication systems. In many applications, the demand is considered as inexact because of the forecasting inaccuracies or human’s unpredictability. This study addresses the robust uncapacitated multiple allocation hub location problem with a set of demand scenarios. The problem is formulated as a nonlinear stochastic optimization problem to minimize the hub installation costs, expected transportation costs and expected absolute deviation of transportation costs. To eliminate the nonlinearity, the equivalent linear problem is introduced. The expected absolute deviation is the robustness measure to derive the solution close to each scenario. The robust hub location is assumed to deliver the least costs difference across the scenarios. The number of scenarios increases size and complexity of the task. Therefore, the classical and improved Benders decomposition algorithms are applied to achieve the best computational performance. The numerical experiment on CAB and AP dataset presents the difference of resulting hub networks in stochastic and robust formulations. Furthermore, performance of two Benders decomposition strategies in comparison with Gurobi solver is assessed and discussed.

This paper presents an algorithm of hub number p robustness estimation using specific simulations in the single allocation hub location problem. The simulation has to model the service demand trends in each origin node to each destination node. This idea is based on the hub network dependence on service demand forecasting, which is modeled by random values from random distribution with parameters reflecting the demand changes. The algorithm includes the mixed integer programming model which describes the hub location-allocation problem with single allocation (each node is connected exactly to one hub). The model chooses the optimal locations for the fixed number of hubs p from the fixed possible location set in the problem. The perturbed data simulate the changes in the service need and present the perspectives of the network changes, and the algorithm fixes these changes. The number of changes in the network is consolidated into the variety frequencies which describe the variabilities in the set of simulations. The algorithm is implemented on Python 3.5 and model optimization is fulfilled using Gurobi Optimizer 7.0.1 software. The results in the real dataset are illustrated and discussed. Refs 18. Fig. 1. Tables 3.

The article proposes a method for estimation of “correct” number of clusters. The stability methodology is used and quantative stability level assessment is introduced. This problem is important and common in cluster analysis and do not have unique criterion for all dataset types. The suggested approach solves one more problem. Often in socio-economics information analysis there are situations when initial numerical data contains various kinds of inaccuracies (measurement errors, intentional misrepresentations, errors of calculation, errors of mathematical model and other possible sources of errors). In this regard, it is important to choose those classifications of study objects which have the stability property with respect to random noise. The work is aimed to get a reliable and accurate clustering which is stable with respect to random perturbations and solves “ill posed” problems in clustering analysis, i.e. to find suggested number of clusters. In current paper the probabilistic approach to problem resolving is being offered. The variability frequency based on random perturbation is introduced and examined as a main metric for assessing clustering results. This estimation can be used for different clustering algorithms and their stability indices can be compared without additional procedures. The experiment on artificial data using the k-mean clustering approach is carried out.