A fuzzy closeness approach to fuzzy multi-attribute decision making

The aim of this paper is to develop a new fuzzy closeness (FC) methodology for multi-attribute decision making (MADM) in fuzzy environments, which is an important research field in decision science and operations research. The TOPSIS method based on an aggregating function representing “closeness to the ideal solution” is one of the well-known MADM methods. However, while the highest ranked alternative by the TOPSIS method is the best in terms of its ranking index, this does not mean that it is always the closest to the ideal solution. Furthermore, the TOPSIS method presumes crisp data while fuzziness is inherent in decision data and decision making processes, so that fuzzy ratings using linguistic variables are better suited for assessing decision alternatives. In this paper, a new FC method for MADM under fuzzy environments is developed by introducing a multi-attribute ranking index based on the particular measure of closeness to the ideal solution, which is developed from the fuzzy weighted Minkowski distance used as an aggregating function in a compromise programming method. The FC method of compromise ranking determines a compromise solution, providing a maximum “group utility” for the “majority” and a minimum individual regret for the “opponent”. A real example of a personnel selection problem is examined to demonstrate the implementation process of the method proposed in this paper.     

A New Method for Finding the Characteristic Roots of E<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>/E<Subscript><Emphasis Type="Italic">m</Emphasis></Subscript>/1 Queues

In 1953, Smith (Proc Camb Philos Soc 49:449–461, 1953), and, following him, Syski (1960) suggested a method to find the waiting time distribution for one server queues with Erlang-n arrivals and Erlang-m service times by using characteristic roots. Syski shows that these roots can be determined from a very simple equation, but an equation of degree n + m. Syski also shows that almost all of the characteristic roots are complex. In this paper, we derive a set of equations, one for each complex root, which can be solved by Newton’s method using real arithmetic. This method simplifies the programming logic because it avoids deflation and the subsequent polishing of the roots. Using the waiting time distribution, Syski then derived the distribution of the number in the system after a departure. E _n /E _m /1 queues can also formulated as quasi birth-death (QBD) processes, and in this case, the characteristic roots discussed by Syski are closely related to the eigenvalues of the QBD process. The QBD process provides information about the number in system at random times, but they are much more difficult to formulate and solve.     

A portfolio selection model using fuzzy returns

We study a static portfolio selection problem, in which future returns of securities are given as fuzzy sets. In contrast to traditional analysis, we assume that investment decisions are not based on statistical expectation values, but rather on maximal and minimal potential returns resulting from the so-called α-cuts of these fuzzy sets. By aggregating over all α-cuts and assigning weights for both best and worst possible cases we get a new objective function to derive an optimal portfolio. Allowing for short sales and modelling α-cuts in ellipsoidal shape, we obtain the optimal portfolio as the unique solution of a simple optimization problem. Since our model does not include any stochastic assumptions, we present a procedure, which turns the data of observable returns as well as experts’ expectations into fuzzy sets in order to quantify the potential future returns and the investment risk.     

A continuum mechanics approach for modelling arbitrary degradation and erosion processes in polymers

Composed of rather simple components some polymers show the ability to decompose from complex structures into substances that can be easily absorbed by the surrounding environment. The responsible processes behind chain shortening and eventually the material remove are degradation and erosion. The aim of this work is to model the chemo-mechanical condition of the polymer material based on a discretised network model. Further, the influence of erosion processes on the morphology is investigated. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Organizing product releases in time-driven development processes: a probabilistic analysis

In this paper, the problem is addressed of how to organizea new product development process that is fast on the one handand that provides good-quality results on the other hand. Severalplanning techniques, like PERT and CPM, are available to analysethe completion time or product release time of such complexprocesses. Although these techniques are all well known andare widely used in project management, they do not address theiterative mode of operation that is characteristic for suchuncertain processes. Neither do they offer a tool or guidelinesto ‘design’ a composition of iterative processesbecause they are analytical. In this paper, a quantitative concept is presented for modellingthe release time of a single uncertain iterative activity, asa random variable to deal with the probabilistic aspect in asimple way. From this simple model, the complexity is extendedsystematically to model fundamentally different configurations,that are on the one hand simple enough to be studied analyticallyand on the other hand exhibit their fundamentally differentrelease characteristics, as experienced in real life. From theanalysis, guidelines are formulated for organizing or (re)configuringa complex process configuration. It is demonstrated that organizing uncertain processes fora fast product release requires a balance between the exploitationof the principles of concurrent engineering and the risk ofoverrunning time targets. An important factor in this balanceis the decision structure for the release of intermediate results.The ‘empowerment’ structure, a structure where decisionsare not clustered as milestones but are made without any delay,offers great opportunity in terms of small mean release timesand small variances thereof.     

Simplicity and reality in computational modeling of politics

Modeling a polity based on viable scientific concepts and theoretical understanding has been a challenge in computational social science and social simulation in general and political science in particular. This paper presents a computational model of a polity (political system) in progressive versions from simple to more realistic. The model, called SimPol to highlight the fundamental structures and processes of politics in a generic society, is developed using the combined methodologies of object-based modeling (OOM), the Unified Modeling Language (UML), and the methodology of Lakatos’ research programs. SimPol demonstrates that computational models of entire political systems are methodologically feasible and scientifically viable; they can also build on and progress beyond previous theory and research to advance our understanding of how polities operate across a variety of domains (simple vs. complex) and levels of analysis (local, national, international). Both simple and realistic models are necessary, for theoretical and empirical purposes, respectively.

Thoughts on modeling complexity

The simplest and probably the most familiar model of statistical processes in the physical sciences is the random walk. This simple model has been applied to all manner of phenomena, ranging from DNA sequences to the firing of neurons. Herein we extend the random walk model beyond that of mimicking simple statistics to include long‐time memory in the dynamics of complex phenomena. We show that complexity can give rise to fractional‐difference stochastic processes whose continuum limit is a fractional Langevin equation, that is, a fractional differential equation driven by random fluctuations. Furthermore, the index of the inverse power‐law spectrum in many complex processes can be related to the fractional derivative index in the fractional Langevin equation. This fractional stochastic model suggests that a scaling process guides the dynamics of many complex phenomena. The alternative to the fractional Langevin equation is a fractional diffusion equation describing the evolution of the probability density for certain kinds of anomalous diffusion. © 2006 Wiley Periodicals, Inc. Complexity 11: 33–43, 2006     

Graph-Theoretic Approach to Stochastic Integrals with Clifford Algebras

Given a fixed probability space (Ω,ℱ,ℙ) and m≥1, let X(t) be an L²(Ω) process satisfying necessary regularity conditions for existence of the mth iterated stochastic integral. For real-valued processes, these existence conditions are known from the work of D. Engel. Engel’s work is extended here to L²(Ω) processes defined on Clifford algebras of arbitrary signature (p,q), which reduce to the real case when p=q=0. These include as special cases processes on the complex numbers, quaternion algebra, finite fermion algebras, fermion Fock spaces, space-time algebra, the algebra of physical space, and the hypercube. Next, a graph-theoretic approach to stochastic integrals is developed in which the mth iterated stochastic integral corresponds to the limit in mean of a collection of weighted closed m-step walks on a growing sequence of graphs. Combinatorial properties of the Clifford geometric product are then used to create adjacency matrices for these graphs in which the appropriate weighted walks are recovered naturally from traces of matrix powers. Given real-valued L²(Ω) processes, Hermite and Poisson-Charlier polynomials are recovered in this manner.     

An extension of the basic Palm measure correspondence

The notion of Palm measure provides a one-to-one correspondence between the distributions of simple, stationary point processes on ℝ with finite intensity and those of stationary sequences of positive random variables with finite mean. Here we extend this result to any stationary random measures on ℝ, using a new notion of spacing measure based on an elementary measure inversion. Our approach is symmetric, in the sense that an iteration of the spacing measure construction leads back to the original distribution. The classical results for point processes arise as simple special cases. Received: 30 September 1998 / Revised version: 9 June 1999 / Published online: 30 March 2000     

<Emphasis Type="Italic">f</Emphasis>-Vectors of barycentric subdivisions

For a simplicial complex or more generally Boolean cell complex Δ we study the behavior of the f- and h-vector under barycentric subdivision. We show that if Δ has a non-negative h-vector then the h-polynomial of its barycentric subdivision has only simple and real zeros. As a consequence this implies a strong version of the Charney–Davis conjecture for spheres that are the subdivision of a Boolean cell complex or the subdivision of the boundary complex of a simple polytope. For a general (d − 1)-dimensional simplicial complex Δ the h-polynomial of its n-th iterated subdivision shows convergent behavior. More precisely, we show that among the zeros of this h-polynomial there is one converging to infinity and the other d − 1 converge to a set of d − 1 real numbers which only depends on d. F. Brenti and V. Welker are partially supported by EU Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272 and the program on “Algebraic Combinatorics” at the Mittag-Leffler Institut in Spring 2005.     

The importance of systems thinking in ethical and sustainable decision-making

Ethical values beyond the satisfaction of basic human needs are specific to each society at a given time. Modern societies are confronted to the challenges of disappearing natural resources, fierce competition on global markets, and climate change. In this paper we define ‘good’, and at the same time ‘ethical‘ decisions in the 21st century as being in accordance with the principles of Sustainable Development. Operations Research can assist sustainable decision-making in two ways: (1) through the tools of systems thinking, in particular system dynamics and soft system modelling, to provide insight into the way complex non-linear living systems and human societies function, and help making sustainable decisions, and (2) through educating and training young people in systems thinking. The paper presents examples of simple models that could serve in classrooms.     

Time series aggregation,disaggregation, and long memory

We study the aggregation/disaggregation problem of random parameter AR(1) processes and its relation to the long-memory phenomenon. We give a characterization of a subclass of aggregated processes which can be obtained from simpler, “elementary”, cases. In particular cases of the mixture densities, the structure of the aggregated process is investigated. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 466–481, October–December, 2007.     

On the ranks of principal submatrices of diagonalizable matrices

As is well known, the rank of a diagonalizable complex matrix can be characterized as the maximum order of the nonzero principal minors of this matrix. The standard proof of this fact is based on representing the coefficients of the characteristic polynomial as the (alternating) sums of all the principal minors of appropriate order. We show that in the case of normal matrices, one can give a simple direct proof, not relying on those representations. Bibliography: 2 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 42–44.     

Simple sheaves along dihedral Lagrangians

We prove the unicity of a complex of sheavesF whose microsupport is carried by a “dihedral” Lagrangian Λ ofT ^* X (X=a real manifold) and which is simple with a prescribed shift at a regular point of Λ. Our method consists in reducing Λ, by a real contact transformation, to the conormal bundle to aC ¹-hypersuface, and then in using [K-S 1, Prop. 6.2.1] in the variant of [D'A-Z 1]. This is similar to [Z 2] but more general, since complex contact transformations and calculations of shifts are not required. We then consider the case of a complex manifoldX, and obtain some vanishing theorems for the complex of “microfunctions along Λ” similar to those of [A-G], [A-H], [K-S 1] (cf. also [D'A-Z 3 5], [Z 2]).     

The zeta function of a simplicial complex

Given a simplicial complex δ on vertices {1, …,n} and a fieldF we consider the subvariety of projective (n−1)-space overF consisting of points whose homogeneous coordinates have support in δ. We give a simple rational expression for the zeta function of this singular projective variety overF _q and show a close connection with the Betti numbers of the corresponding variety over ℂ. This connection is particularly simple in the case when Δ is Cohen-Macaulay.     

On weak convergence of random fields

We suggest simple and easily verifiable, yet general, conditions under which multi-parameter stochastic processes converge weakly to a continuous stochastic process. Connections to, and extensions of, R. Dudley’s results play an important role in our considerations, and we therefore discuss them in detail. As an illustration of general results, we consider multi-parameter stochastic processes that can be decomposed into differences of two coordinate-wise non-decreasing processes, in which case the aforementioned conditions become even simpler. To illustrate how the herein developed general approach can be used in specific situations, we present a detailed analysis of a two-parameter sequential empirical process.     

Complex Lorentz rotations on vectors⊕pseudovectors

We study the complex structure of the space of vectors and pseudovectors Λ≡ ≡ Λ₁ ⊕ Λ₃ arising from the properties of the Hodge duality in 4-dimensional spacetimes. This structure appears naturally in the framework of its real Clifford geometric algebraCl _1,3 orCl _3,1, in which the Hodge duality is the simple multiplication by the volume unit ∈. Interpreting the linear combination of a scalar and a pseudoscalar α+β∈ as a complex number (Λ₀ ⊕ Λ₄ ⋍ ℂ), the odd subspace Λ_ ∈Cl is a left/right complex linear space, previously studied by Sobczyk. From the real metric definingCl, Λ_ inherits a natural complex bilinear normN(a). Corresponding to this norm there is the group of complex Lorentz rotations Spin_1,3 (ℂ), for which we find a formulation using exclusively the real algebraCl. We discuss its behaviour in some examples and find expressions for different decompositions into real and selfdual, real and anti-selfdual, and real and pure imaginary angle rotations. We finally apply our results to the implementation of the rotations corresponding to Euclidean and null signatures inside the real Lorentzian Clifford algebraCl.     

Generators of simple Lie algebras in arbitrary characteristics

In this paper we study the minimal number of generators for simple Lie algebras in characteristic 0 or p > 3. We show that any such algebra can be generated by 2 elements. We also examine the ‘one and a half generation’ property, i.e. when every non-zero element can be completed to a generating pair. We show that classical simple algebras have this property, and that the only simple Cartan type algebras of type W which have this property are the Zassenhaus algebras. The author was partially supported by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00287 (RTN Network “K-Theory, Algebraic Groups and Related Structures”) and a long-term research grant from the D.A.A.D.     

Gaudin magnet and off-mass-shell Bethe wave functions

The general Bethe equation (or off-shell Bethe ansatz equation) is proved for the Gaudin magnet for a broad class of simple complex Lie algebras. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 1, pp. 13–28, October, 1997.     

Complexity of networks I: The set‐complexity of binary graphs

The balance between symmetry and randomness as a property of networks can be viewed as a kind of “complexity.” We use here our previously defined “set complexity” measure (Galas et al., IEEE Trans Inf Theory 2010, 56), which was used to approach the problem of defining biological information, in the mathematical analysis of networks. This information theoretic measure is used to explore the complexity of binary, undirected graphs. The complexities, Ψ, of some specific classes of graphs can be calculated in closed form. Some simple graphs have a complexity value of zero, but graphs with significant values of Ψ are rare. We find that the most complex of the simple graphs are the complete bipartite graphs (CBGs). In this simple case, the complexity, Ψ, is a strong function of the size of the two node sets in these graphs. We find the maximum Ψ binary graphs as well. These graphs are distinct from, but similar to CBGs. Finally, we explore directed and stochastic processes for growing graphs (hill‐climbing and random duplication, respectively) and find that node duplication and partial node duplication conserve interesting graph properties. Partial duplication can grow extremely complex graphs, while full node duplication cannot do so. By examining the eigenvalue spectrum of the graph Laplacian we characterize the symmetry of the graphs and demonstrate that, in general, breaking specific symmetries of the binary graphs increases the set‐based complexity, Ψ. The implications of these results for more complex, multiparameter graphs, and for physical and biological networks and the processes of network evolution are discussed. © 2011 Wiley Periodicals, Inc. Complexity, 17,51–64, 2011