Inventory control as an identification problem based on fuzzy logic

An approach is proposed to solving inventory control problems using information available on current demand and stock. The approach is based on identification of nonlinear dependences using fuzzy knowledge bases. By tuning a fuzzy model against a learning sample, model control actions can be made very close to an expert’s decision. This approach can further be developed by creating adaptive (neuro-fuzzy) inventory control models for enterprises and trading companies. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 123–133, May–June 2006.     

A model of inventory control

A model of inventory control is considered. It is described by a semi-Markov random walk with a negative drift at an angle of 0 < α < 90°, with positive random jumps, a delay, an absorbing screen at zero, and a reflecting screen for a > 0 at an angle α. The Laplace transformation is found for the distribution of the first moment storehouse exhaustion, and the first and the second moments are explicitly obtained. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 50–59, July–August, 1999.     

Some Continuous Models of Inventory Control

The problem of optimal control of a manufacturing process is investigated. The dynamics of the process is described by a stochastic differential equation. Two methods of solution of this problem are considered. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 175–178, May–June 2005.     

Solving a continuous nonlinear problem of optimal set partition with arrangement of subset centers in the case of a convex objective functional

A continuous nonlinear single-commodity problem of optimal partition of a set Ω in an n-measurable Euclidean space into disjoint subsets with arrangement of their centers is analyzed using equality and inequality constraints in the case of a convex objective functional. A method and algorithm are proposed to solve this problem. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 134–152, March–April 2008.     

A Continuous problem of optimal partition with a nondifferentiable functional

A continuous problem of decomposition of an optimal set into disjoint subsets is considered. This problem is characterized by a non-differentiable performance criterion, the availability of a boundary-value problem for a parabolic type equation, and dependence of the initial condition function on the partition of some flat set. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 115–125, July–August, 2000.     

Synthesis of optimal control of dynamic systems with infinite aftereffect,a small parameter,and poisson perturbations

A methodology is obtained that makes it possible to synthesize an optimal control for stochastic systems of functional-differential equations with the entire prehistory and a small parameter and takes into account continuous Wiener-type and discontinuous Poisson-type perturbations. It is proved that the sought-for control can be found as an optimal control of some auxiliary problem. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 181–186, May–June 2007.     

Some Approaches to Solving Inventory Control Problems

Several approaches to the solution of inventory control problems are considered. These approaches are designed to find optimal strategies, control parameters for a given strategy, and probability estimates of resource supply. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 101–106, July–August 2005.     

Linear variational principle for convex vector maximization

For infinitely dimensional convex vector maximization problems, it is proved that there exists, under certain conditions, an arbitrarily small additive perturbation of the performance criterion by a linear continuous operator such that the perturbed problem has efficient solutions. The study was sponsored by the Ukrainian State Fund for Basic Research __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 105–114, March–April 2007.     

Partition of a metric space into continuous components

The fact that each discrete metric space is uniquely decomposed into continuous components is elegantly proved. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 185–186, September–October, 1999.     

Pricing Multi-Asset American Options: A Finite Element Method-of-Lines with Smooth Penalty

This paper studies the problem of pricing multi-asset American-style options in the Black–Scholes–Merton framework. The value function of an option contract is known to satisfy a partial differential variational inequality (PDVI) when early exercise is permitted. We develop a computational method for the valuation of multi-asset American-style options based on approximating the PDVI by a non-linear penalized PDE with a penalty term with continuous Jacobian. We convert the non-linear PDE to a variational (weak) form, discretize the weak formulation spatially by a Galerkin finite element method to obtain a system of ODEs, and integrate the resulting system of ODEs in time with an adaptive variable order and variable step size solver SUNDIALS. Numerical results demonstrate that employing a penalty term with continuous Jacobian in contrast to the penalty terms with discontinuous Jacobians in use in the literature improves computational performance of the adaptive temporal integrator. In our framework we are able to price American-style options with payoffs dependent on up to six assets on a PC. This is in contrast to the existing literature on the pricing of American options by PDE methods, that has so far been limited to at most three-dimensional problems. Our results open avenues for further applications to multi-dimensional problems, such as pricing convertible bonds in multi-factor models, that will be explored in future work. This research was supported by the National Science Foundation under grants DMI–0422937 and DMI–0422985.     

Stability of continuous hybrid systems

The stationary state of a continuous hybrid system is analyzed for stability using the method of Lyapunov functions. Sufficient conditions of stability and instability are established. These conditions are constructive and can easily be calculated. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 123–128, March–April 2007.     

Neuro-Adaptive Output Feedback Control for a Class of Nonlinear Non-Minimum Phase Systems

This paper presents an adaptive output-feedback control method for non-affine nonlinear non-minimum phase systems that have partially known Lipschitz continuous functions in their arguments. The proposed controller is comprised of a linear, a neuro-adaptive and an adaptive robustifying control term. The adaptation law for the neural network weights is obtained using the Lyapunov’s direct method. One of the main advantageous of the proposed method is that the control law does not depend on the state estimation. This task is accomplished by introducing a strictly positive-real augmented error dynamic and using the Leftshetz–Kalman–Yakobuvich lemma. The ultimate boundedness of the error signals will be shown analytically using the extension of Lyapunov theory. The effectiveness of the proposed scheme will be shown in simulations for the benchmark problem Translational Oscillator/Rotational Actuator (TORA) system.     

Continuous procedure of stochastic approximation with singular perturbation under balance conditions

Sufficient conditions for the convergence of a continuous stochastic approximation procedure are established for the case where the regression function depends on a Markovian environment and has a singular perturbation that satisfies balance conditions. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 133–139, May–June 2006.     

A Nonoverlapping Domain Decomposition Method for Legendre Spectral Collocation Problems

We consider the Dirichlet boundary value problem for Poisson’s equation in an L-shaped region or a rectangle with a cross-point. In both cases, we approximate the Dirichlet problem using Legendre spectral collocation, that is, polynomial collocation at the Legendre–Gauss nodes. The L-shaped region is partitioned into three nonoverlapping rectangular subregions with two interfaces and the rectangle with the cross-point is partitioned into four rectangular subregions with four interfaces. In each rectangular subregion, the approximate solution is a polynomial tensor product that satisfies Poisson’s equation at the collocation points. The approximate solution is continuous on the entire domain and its normal derivatives are continuous at the collocation points on the interfaces, but continuity of the normal derivatives across the interfaces is not guaranteed. At the cross point, we require continuity of the normal derivative in the vertical direction. The solution of the collocation problem is first reduced to finding the approximate solution on the interfaces. The discrete Steklov–Poincaré operator corresponding to the interfaces is self-adjoint and positive definite with respect to the discrete inner product associated with the collocation points on the interfaces. The approximate solution on the interfaces is computed using the preconditioned conjugate gradient method. A preconditioner is obtained from the discrete Steklov–Poincaré operators corresponding to pairs of the adjacent rectangular subregions. Once the solution of the discrete Steklov–Poincaré equation is obtained, the collocation solution in each rectangular subregion is computed using a matrix decomposition method. The total cost of the algorithm is O(N ³), where the number of unknowns is proportional to N ².      

Polyhedral coherent risk measures and investment portfolio optimization

The paper studies a class of polyhedral coherent risk measures for risk-return portfolio optimization problems under partial uncertainty, with unknown scenario probabilities estimated by some polyhedron. Such portfolio problems are reduced to linear programming problems. As an example, continuous problems of optimal investment allocation under risk of catastrophic floods are described. The study was supported from the Ukrainian Science and Technology Center, Project G3127. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 120–133, March–April 2008.     

Necessary and sufficient optimality conditions of a design decision on the structure of a local computer network

A problem of nonlinear programming with Boolean variables is considered, for which the possibility of extension of the optimality conditions of solving a special problem of nonlinear programming (with continuous variables) is shown by statistical estimation of possible design decisions. This contributed to the solution of the initial problem by simpler methods. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 133–137, September–October, 1999.     

Continuous stochastic approximation with semi-Markov switchings in the diffusion approximation scheme

The sufficient conditions of convergence are obtained for a continuous stochastic approximation procedure in the diffusion approximation scheme under balance conditions imposed on singular perturbations of the regression function in a semi-Markov environment. To this end, a singular perturbation problem is solved for the asymptotic representation of a compensating operator of a Markov renewal process. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 170–178, July–August 2007.     

Boundedness analysis for open Chemical Reaction Networks with mass-action kinetics

This paper describes the working principles of an algorithm for boundedness analysis of open Chemical Reaction Networks endowed with mass-action kinetics. Such models can be thought of both as a special class of compartmental systems or a particular type of continuous Petri Nets, in which the firing rates of transitions are not constant or preassigned, but expressed as a function of the continuous marking of the network (function which in chemistry is referred to as the “kinetics”). The algorithm can be applied to a broad class of such open networks, and returns, as an outcome, a classification of the possible dynamical behaviors that are compatible with the network structure, by classifying each variable either as bounded, converging to 0 or diverging to ∞. This can be viewed as a qualitative study of Input–Output Stability for chemical networks, or more precisely, as a classification of its possible I–O instability patterns. Our goal is to analyze the system irrespectively of values of kinetic parameters. More precisely, we attempt to analyze it simultaneously for all possible values. Remarkably, tests on non-trivial examples (one of which is discussed in this paper) showed that, as the kinetic constants of the network are varied, all the compatible behaviors could be observed in simulations. Finally, we discuss and illustrate how the results relate to previous works on the qualitative dynamics of closed reaction networks.     

A catalytically powered electrokinetic lens: toward channelless microfluidics

A chemically powered channelless microfluidic device was designed, fabricated, and characterized. The device consists of an asymmetric silver–gold bimetallic catalytic junction fabricated on a silicon dioxide surface. The decomposition of hydrogen peroxide at the silver–gold interface generates a proton gradient and an associated electric field which in turn drives electroosmosis and electrophoresis when a charged particle is present in the vicinity of the field. By engineering an asymmetric device consisting of an isolated junction, continuous electroosmotic fluid flow across the device has been achieved. In addition, a new device geometry has been developed which is capable of focusing and directing negatively charged particles along a desired path without the need of microchannels. The efficiency and ease of the fabrication suggest the possibility of many versatile applications including biological molecule sorting and manipulation.     

Projection theorem for Banach and locally convex spaces

The problem of representing linear functionals and linear operators using a given bilinear mapping is considered. For linear continuous functionals and linear compact operators that act in Banach and locally convex spaces, theorems on their representability using a given bilinear operator are formulated. Our results generalize the classical Lax-Milgram theorem. The study was sponsored by the State Fund for Fundamental Research of Ukraine. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 109–116, September–October 2008.