The multi-weighted Steiner tree problem

We formulate and investigate the Multi-Weighted Steiner Problem (MWS), a generalization of the Steiner problem in graphs, involving more than one weight function. As a special case, it contains the hierarchical network design problem. With the notion of "bottleneck length/distance", a min-max measure, we analyze the interaction between differently weighted edges in a solution. Combining the results with known methods for the Steiner problem in graphs and the hierarchical network design problem, two heuristics for the MWS are developed, one based on weight modifications and the other on exchanging edges. Both are of time complexityO(kv ²), withv the number of nodes andk the number of special nodes in the graph. The first is also suited for thedirected MWS; the second is expected to perform better on the undirected version. Before actually solving the Steiner problem in graphs and the hierarchical network design problem, preprocessing techniques exploiting tests to reduce the problem graphs have proven to be valuable. We adapt three prominent tests for use in the MWS.     

The epistemic,the cognitive,the human: a commentary on the mathematical working space approach

ZDM – Mathematics Education - This article is a commentary on the mathematical working space (MWS) approach and draws on the articles contained in this ZDM issue. The article is divided into...     

Mathematical Working Spaces in schooling: an introduction

ZDM – Mathematics Education - The theoretical and methodological model of Mathematical Working Space (MWS) is introduced in this paper. For over 10 years, the model has been the...     

Mathematical working spaces through networking lens

ZDM – Mathematics Education - This issue of ZDM collects research works sharing a common reference to the theoretical framework of Mathematical Working Spaces (MWS), a construction which...     

Some observations on maximum weight stable sets in certain P5-free graphs

The maximum weight stable set problem (MWS) is the weighted version of the maximum stable set problem (MS), which is NP-hard. The class of P₅-free graphs – i.e., graphs with no induced path of five vertices – is the unique minimal class, defined by forbidding a single connected subgraph, for which the computational complexity of MS is an open question. At the same time, it is known that MS can be efficiently solved for (_P5,F)$(P_5\text{,}F)$-free graphs, where F is any graph of five vertices different to a C₅. In this paper we introduce some observations on P₅-free graphs, and apply them to introduce certain subclasses of such graphs for which one can efficiently solve MWS. That extends or improves some known results, and implies – together with other known results – that MWS can be efficiently solved for (_P5,F)$(P_5\text{,}F)$-free graphs where F is any graph of five vertices different to a C₅.     

Sensitivity and stability analysis in data envelopment analysis

One of the topics of interest in data envelopment analysis (DEA) is sensitivity and stability and stability analysis of the specific decision making unit (DMU), which is under evaluation. In DEA, efficient DMUs are of primary importance as they define the efficient frontier. In this paper, we develop a new sensitivity analysis approach for the CCR, BCC and Additive models, when variations in the data are considered for a specific efficient DMU and the data for the remaining DMUs are assumed fixed.     

LP-metric sensitivity analysis for single and multi-attribute decision analysis

Analyzing the sensitivity of decisions to probability estimation error in single and multi-attribute problems and to errors in estimating additive multi-attribute value models in multi-attribute problems is an integral part of decision analysis. This paper presents an intuitive and tractable approach to this sensitivity analysis. Here a decision is considered insensitive if: 1) the probabilities or multi-attribute weights required for any other alternative to become preferred are not close to the original estimated probabilities and weights, and 2) the rank order of states implied by the probabilities or the rank order of attributes implied by the additive multi-attribute weights must change for any other alternative to become preferred. The sensitivity analysis is conducted using straight forward linear programming models. An example is used to demonstrate their application.     

Sensitivity and stability analysis in fuzzy data envelopment analysis

As a useful management and decision tool, data envelopment analysis (DEA) has become a pop area of research. One of the topics of interests in DEA is sensitivity and stability analysis of decision making units (DMUs). Due to the uncertainty of the data in real life, this paper will give some DEA models in fuzzy environment. It is followed by a series analysis of sensitivity and stability for all DMUs. Finally a numerical example will be presented to give an illustration of the sensitivity and stability analysis.     

Projective pseudodifferential analysis and harmonic analysis

We consider pseudodifferential operators on functions on Rn+1 which commute with the Euler operator, and can thus be restricted to spaces of functions homogeneous of some given degree. The symbols of such restrictions can be regarded as functions on a reduced phase space, isomorphic to the homogeneous space Gn/Hn=SL(n+1,R)/GL(n,R), and the resulting calculus is a pseudodifferential analysis of operators acting on spaces of appropriate sections of line bundles over the projective space Pn(R): these spaces are the representation spaces of the maximal degenerate series (πiλ,ε) of Gn. This new approach to the quantization of Gn/Hn, already considered by other authors, has several advantages: as an example, it makes it possible to give a very explicit version of the continuous part from the decomposition of L²(Gn/Hn) under the quasiregular action of Gn. We also consider interesting special symbols, which arise from the consideration of the resolvents of certain infinitesimal operators of the representation πiλ,ε.     

Efficiency measurement using independent component analysis and data envelopment analysis

Efficiency measurement is an important issue for any firm or organization. Efficiency measurement allows organizations to compare their performance with their competitors’ and then develop corresponding plans to improve performance. Various efficiency measurement tools, such as conventional statistical methods and non-parametric methods, have been successfully developed in the literature. Among these tools, the data envelopment analysis (DEA) approach is one of the most widely discussed. However, problems of discrimination between efficient and inefficient decision-making units also exist in the DEA context (Adler and Yazhemsky, 2010). In this paper, a two-stage approach of integrating independent component analysis (ICA) and data envelopment analysis (DEA) is proposed to overcome this issue. We suggest using ICA first to extract the input variables for generating independent components, then selecting the ICs representing the independent sources of input variables, and finally, inputting the selected ICs as new variables in the DEA model. A simulated dataset and a hospital dataset provided by the Office of Statistics in Taiwan’s Department of Health are used to demonstrate the validity of the proposed two-stage approach. The results show that the proposed method can not only separate performance differences between the DMUs but also improve the discriminatory capability of the DEA’s efficiency measurement.     

Policy analysis and organizational politics

It is widely recognized that the formulation and implementation of policy is a ‘political’ process. Yet this aspect of policy and policy making remains rarely explicity included in the analysis and models of the professional policy analyst. Where political aspects have been considered the emphasis has tended to be upon the legitimated political arena rather than internal organizational politics. It is our concentention that to ignore the internal political dimension is to decrease significantly the effectiveness of the policy analyst in aiding and influencing the policy-making process. The purpose of this article is to explore this concentention and specifically to discuss a particular approach to policy analysis derived from it. It is one which involves the construction of models for the policy maker as client that are importantly different in nature from typical rational-scientific models of operational research. The models deliberately attempt to successfully capture his attention and commitment because they reflect the ramifications of his political organizational environment as well as the more legitimate technical aspects of his problem. The conceptual framework and methodology that underlie this approach use concepts and methods drawn from ideas in cognitive psychology and sociology and political science, as well as operational research. The paper discusses these perspectives and points to potential uses of the models we have constructed by providing examples derived specifically from work we are currently undertaking in the public sector.     

Computable analysis and Blaschke products

We show that if a Blaschke product defines a computable function, then it has a computable sequence of zeros in which the number of times each zero is repeated is its multiplicity. We then show that the converse is not true. We finally show that every computable, radial, interpolating sequence yields a computable Blaschke product.

     

Towards integrated design and analysis

A systems analysis technique is used to investigate and better understand the processes involved in finite element modelling. Two different starting points are considered: engineering drawings and solid models. The breakdown of activities involved in finite element analysis from a solid model is used as a requirements definition and functional specification for software tools to facilitate automation of finite modelling and eventual integration of design and analysis.     

Convex analysis and financial equilibrium

Convexity has long had an important role in economic theory, but some recent developments have featured it all the more in problems of equilibrium. Here the tools of convex analysis are applied to a basic model of incomplete financial markets in which assets are traded and money can be lent or borrowed between the present and future. The existence of an equilibrium is established with techniques that include bounds derived from the duals to problems of utility maximization. Composite variational inequalities furnish the modeling platform. Models with and without short-selling are handled, moreover in the absence of any requirement that agents must initially have a positive amount of every asset, as is typical in equilibrium work in economics.     

Probability methods and nonlinear analysis

The concepts of accretive and differentiable operator in a Banach space B are used to show that certain approximations to a solution of a nonlinear evolution equation converge. When B is a space of continuous functions it is shown that the approximations and the solution be represented as integrals with respect to a signed measure on a function space. As an example, a new proof is given for the existence and uniqueness of solutions to a nonlinear parabolic differential equations with coefficients dependent upon solutions. Integral representations of these solutions follow.     

Continuous convergence and functional analysis

The usual setting for Functional Analysis is the category LCS of locally convex topological vector spaces. There are, however, advantages in working in a larger setting, the category CVS of convergence vector spaces—even if one's interest is restricted to LCS. In CVS, one has access to a dual structure, continuous convergence, unavailable in LCS.

We show that theorems such as Grothendieck's completion theorem, Ptak's closed graph and open mapping theorems and the Banach-Steinhaus theorem are transformed from technical results in LCS to transparent and elegant results when examined in CVS with continuous convergence. In the theory of distributions, important bilinear mappings such as evaluations, multiplication and convolution, which are separately continuous when viewed in LCS, become jointly continuous in CVS.     

Data analysis and decision support

In this paper, we consider developmental lines of computer-assisted decision support (with consideration of knowledge-based approaches) for data analysis problems. First, we discuss some situations where it is obviously appropriate to apply computer-assisted decision support in connection with data analysis tasks. Then, a brief historical retrospect is given viewing the development of this area of research and its interfaces to knowledge-based approaches. Against this background we illustrate two prototypes of knowledge-based decision support systems for specific data-analysis problems related to fields of interest of our own. Finally, we indicate possible progress and future activities in this area.     

Super-efficiency and DEA sensitivity analysis

This paper discusses and reviews the use of super-efficiency approach in data envelopment analysis (DEA) sensitivity analyses. It is shown that super-efficiency score can be decomposed into two data perturbation components of a particular test frontier decision making unit (DMU) and the remaining DMUs. As a result, DEA sensitivity analysis can be done in (1) a general situation where data for a test DMU and data for the remaining DMUs are allowed to vary simultaneously and unequally and (2) the worst-case scenario where the efficiency of the test DMU is deteriorating while the efficiencies of the other DMUs are improving. The sensitivity analysis approach developed in this paper can be applied to DMUs on the entire frontier and to all basic DEA models. Necessary and sufficient conditions for preserving a DMU’s efficiency classification are developed when various data changes are applied to all DMUs. Possible infeasibility of super-efficiency DEA models is only associated with extreme-efficient DMUs and indicates efficiency stability to data perturbations in all DMUs.     

Intelligent data analysis and model interpretation with spectral analysis fuzzy symbolic modeling

This paper proposes fuzzy symbolic modeling as a framework for intelligent data analysis and model interpretation in classification and regression problems. The fuzzy symbolic modeling approach is based on the eigenstructure analysis of the data similarity matrix to define the number of fuzzy rules in the model. Each fuzzy rule is associated with a symbol and is defined by a Gaussian membership function. The prototypes for the rules are computed by a clustering algorithm, and the model output parameters are computed as the solutions of a bounded quadratic optimization problem. In classification problems, the rules’ parameters are interpreted as the rules’ confidence. In regression problems, the rules’ parameters are used to derive rules’ confidences for classes that represent ranges of output variable values. The resulting model is evaluated based on a set of benchmark datasets for classification and regression problems. Nonparametric statistical tests were performed on the benchmark results, showing that the proposed approach produces compact fuzzy models with accuracy comparable to models produced by the standard modeling approaches. The resulting model is also exploited from the interpretability point of view, showing how the rule weights provide additional information to help in data and model understanding, such that it can be used as a decision support tool for the prediction of new data.