Complexity analysis of dynamic noncooperative game models for closed-loop supply chain with product recovery

In this paper, we consider a closed-loop supply chain (CLSC) with product recovery, which is composed of one manufacturer and one retailer. The retailer is in charge of recollecting and the manufacturer is responsible for product recovery. The system can be regarded as a coupling dynamics of the forward and reverse supply chain. Under different decision criteria, two noncooperative game models: Stackelberg game model and peer-to-peer game model are developed. The dynamic phenomena, such as the bifurcation, chaos and sensitivity to initial values are analyzed through bifurcation diagrams and the largest Lyapunov exponent (LLE). The influences of decision parameters on the complex nonlinear dynamics behaviors of the two models are further analyzed by comparing parameter basin plots, and the results show that with the improvement of retailer’s competitive position, the CLSC system will be more easier to enter into chaos.     

Two lessons from fractals and chaos

We used to think that a good measurement is characterized by its mean and variance and that a good theory is characterized by its ability to predict the values measured in an experiment. The properties of nonlinear systems called fractals and chaos have now taught us that this isn't necessarily true. Data from fractal systems extend over many scales and so cannot be characterized by a single characteristic average number. Data from chaotic systems do not repeat the same time series of values, even if they are started very close to the same initial conditions. This means that a valid mathematical model will not be able to predict the values of the time series.     

Fractal dimension evolution and spatial replacement dynamics of urban growth

This paper presents a new perspective of looking at the relation between fractals and chaos by means of cities. Especially, a principle of space filling and spatial replacement is proposed to interpret the fractal dimension of urban form. The fractal dimension evolution of urban growth can be empirically modeled with Boltzmann’s equation. For the normalized data, Boltzmann’s equation is just equivalent to the logistic function. The logistic equation can be transformed into the well-known 1-dimensional logistic map, which is based on a 2-dimensional map suggesting spatial replacement dynamics of city development. The 2-dimensional recurrence relations can be employed to generate the nonlinear dynamical behaviors such as bifurcation and chaos. A discovery is thus made in this article that, for the fractal dimension growth following the logistic curve, the normalized dimension value is the ratio of space filling. If the rate of spatial replacement (urban growth) is too high, the periodic oscillations and chaos will arise. The spatial replacement dynamics can be extended to general replacement dynamics, and bifurcation and chaos mirror a process of complex replacement.     

Modelling transport in disordered media via diffusion on fractals

When viewed at an appropriate scale, a disordered medium can behave as if it is strictly less than three-dimensional. As fractals typically have noninteger dimensions, they are natural models for disordered media, and diffusion on fractals can be used to model transport in disordered media. In particular, such diffusion processes can be used to obtain bounds on the fundemantal solution to the heat equation on a fractal. In this paper, we review the work in this area and describe how bounds on branching processes lead to bounds on heat kernels.     

A stochastic process for fractal design

A special family of stochastic processes which allow the design of fractals is considered. We show that a process with arbitrary but continuous given functions of variance and mean value can be constructed. The paths of the process are fractals. They are computed by a very simple and fast algorithm.     

Constructing easily iterated functions with interesting properties

A number of schools have recently introduced new courses dealing with various aspects of iteration theory or at least have found ways of including topics such as chaos and fractals in existing courses. In this note, we will consider a family of functions whose members are especially well suited to illustrate many of the concepts involved in these topics. The main advantage of these functions is that the student can create and investigate his or her own examples using the approach given in this note.     

Analysis of decision-making in economic chaos control

In some economic chaotic systems, players are concerned about whether their performance is improved besides taking some methods to control chaos. In the face of chaos occurring in competition, whether one player takes controlling measures or not affects not only their own earning but also other opponents’ income. An output duopoly competing evolution model with bounded rationality is introduced in this paper. Using modern game theory, decision-making analyses about chaos control of the model are taken by taking aggregate profits as players’ payoff. It is found that the speed of players’ response to the market and whether the decisive parameters are in the stable region of the Nash equilibrium or not have a distinct influence on the results of the game. The impact of cost function’ type on results of the game is also found. The mechanism of influences is discovered by using numerical simulation.     

Comparative analysis of bacterial essential and nonessential genes with Hurst exponent based on chaos game representation

Essential genes are indispensable for the survival of an organism. Investigating features associated with gene essentiality is fundamental to the prediction and identification of essential genes with computational techniques. We use fractal theory approach to make comparative analysis of essential and nonessential genes in bacteria. The Hurst exponents of essential genes and nonessential genes available in the DEG database for 27 bacteria are calculated based on their gene chaos game representations. It is found that for most analyzed bacteria, weak negative correlation exists between Hurst exponent and gene length. Moreover, essential genes generally differ from nonessential genes in their Hurst exponent. For genes of similar length, the average Hurst exponent of essential genes is smaller than that of nonessential genes. The results of our work reveal that gene Hurst exponent is very probably useful gene feature for the algorithm predicting essential genes.     

Complex dynamic properties of Cournot duopoly games with convex and log-concave demand function

A Cournot duopoly game is proposed where the interdependence between firms depends on convex and log-concave demand function. In this paper, a model of two rational firms that are in competition and produce homogeneous commodities is introduced. The equilibrium points of this model are obtained and their dynamical characteristics such as stability, bifurcation and chaos are investigated. Furthermore, a multi-team Cournot game is introduced. Through simulation the dynamical characteristics of the equilibrium points of this game are illustrated.     

A new chaos optimization algorithm based on symmetrization and levelling approaches for global optimization

Chaos optimization algorithm is a recently developed method for global optimization based on chaos theory. It has many good features such as easy implementation, short execution time and robust mechanisms for escaping from local minima compared with existing stochastic searching algorithms. In the present paper, we propose a new chaos optimization algorithm (COA) approach called SLC (symmetric levelled chaos) based on new strategies including symmetrization and levelling: the proposed SLC method is, to our knowledge, the first chaos approach that can efficiently and successfully operates in higher-dimensional spaces. The proposed method is tested on a number of benchmark functions, and its performance comparisons are provided against previous COAs. The experiment results show that the proposed method has a marked improvement in performance over the classical COA approaches. Moreover, among all COA approaches, SLC is the only one to work efficiently in higher-dimensional spaces.     

Chaos game representation of the Dst index and prediction of geomagnetic storm events

This paper proposes a two-dimensional chaos game representation (CGR) for the D_st index. The CGR provides an effective method to characterize the multifractality of the D_st time series. The probability measure of this representation is then modeled as a recurrent iterated function system in fractal theory, which leads to an algorithm for prediction of a storm event. We present an analysis and modeling of the D_st time series over the period 1963–2003. The numerical results obtained indicate that the method is useful in predicting storm events one day ahead.     

Existence of self-similar energies on finitely ramified fractals

A self-similar energy on finitely ramified fractals can be constructed starting from an eigenform, i.e., an eigenvector of a special operator defined on the fractal. In this paper, we prove two existence results for regular eigenforms that consequently are existence results for self-similar energies on finitely ramified fractals. The first result proves the existence of a regular eigenform for suitable weights on fractals, assuming only that the boundary cells are separated and the union of the interior cells is connected. This result improves previous results and applies to many finitely ramified fractals usually considered. The second result proves the existence of a regular eigenform in the general case of finitely ramified fractals in a setting similar to that of P.C.F. self-similar sets considered, for example, by R. Strichartz in [11]. In this general case, however, the eigenform is not necessarily on the given structure, but is rather on only a suitable power of it. Nevertheless, as the fractal generated is the same as the original fractal, the result provides a regular self-similar energy on the given fractal.     

A finite step algorithm via a bimatrix game to a single controller non-zero sum stochastic game

Given a non-zero sum discounted stochastic game with finitely many states and actions one can form a bimatrix game whose pure strategies are the pure stationary strategies of the players and whose penalty payoffs consist of the total discounted costs over all states at any pure stationary pair. It is shown that any Nash equilibrium point of this bimatrix game can be used to find a Nash equilibrium point of the stochastic game whenever the law of motion is controlled by one player. The theorem is extended to undiscounted stochastic games with irreducible transitions when the law of motion is controlled by one player. Examples are worked out to illustrate the algorithm proposed.The work of this author was supported in part by the NSF grants DMS-9024408 and DMS 8802260.     

Learning algorithms for repeated bimatrix Nash games with incomplete information

The purpose of this paper is to study a particular recursive scheme for updating the actions of two players involved in a Nash game, who do not know the parameters of the game, so that the resulting costs and strategies converge to (or approach a neighborhood of) those that could be calculated in the known parameter case. We study this problem in the context of a matrix Nash game, where the elements of the matrices are unknown to both players. The essence of the contribution of this paper is twofold. On the one hand, it shows that learning algorithms which are known to work for zero-sum games or team problems can also perform well for Nash games. On the other hand, it shows that, if two players act without even knowing that they are involved in a game, but merely thinking that they try to maximize their output using the learning algorithm proposed, they end up being in Nash equilibrium.This research was supported in part by NSF Grant No. ECS-87-14777.     

A Globally Convergent Algorithm to Compute All Nash Equilibria for <Emphasis Type="Italic">n</Emphasis>-Person Games

In this paper we present an algorithm to compute all Nash equilibria for generic finite n-person games in normal form. The algorithm relies on decomposing the game by means of support-sets. For each support-set, the set of totally mixed equilibria of the support-restricted game can be characterized by a system of polynomial equations and inequalities. By finding all the solutions to those systems, all equilibria are found. The algorithm belongs to the class of homotopy-methods and can be easily implemented. Finally, several techniques to speed up computations are proposed.     

Can mathematics educate for peace?

In this paper we simulate a discussion with our students, in which we alternate specific mathematical issues (as the PERT (Program Evaluation and Review Technique) in 1.2, a discrete dynamical system in 2.1 and fractals in 2.2) that can be applied in situations such as the evolution of populations (2.1) or other well-known contexts such as the economic equilibrium theory or the ecological systems. The students are given occasions to deepen both the meaning and use of the studied model. We emphasize those cases where the model leads to analyses of behaviour depending on parameters whose values highlight stable states of dynamic systems (regions of peace) and regions where control over the system evolution is lost, and where the system passes from bifureations to chaos (regions of war).     

A new approach to superfractals

V-variable fractals and superfractals have recently been introduced by Barnsley [Barnsley Michael, Hutchinson John, Stenflo Örjan. A fractal valued random iteration algorithm and fractal hierarchy. Fractals 2005;13(2):111–46 [MR2151094 (2006b:28014)]] to the world of mathematics and computer graphics. In this paper, we introduce superior iterates to study the role of contractive and non-contractive operators in relation to superfractals. A modified algorithm along with details of computer implementation is also provided to compute V-variable fractals. A brief discussion about the various aspects of the computed figures indicates usefulness of the study.     

Potential theory for mean payoff games

The paper presents an O(mn2ⁿ log Z) deterministic algorithm for solving the mean payoff game problem, m and n being the numbers of arcs and vertices, respectively, in the game graph, and Z being the maximum weight (the weights are assumed to be integers). The theoretical basis for the algorithm is the potential theory for mean payoff games. This theory allows one to restate the problem in terms of solving systems of algebraic equations with minima and maxima. Also, in order to solve the mean payoff game problem, the arc reweighting technique is used. To this end, simple modifications, which do not change the set of winning strategies, are applied to the game graph; in the end, a trivial instance of the problem is obtained. It is shown that any game graph can be simplified by n reweightings. Bibliography: 16 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 340, 2006, pp. 61–75.     

Chaos gray-coded genetic algorithm and its application for pollution source identifications in convection–diffusion equation

In order to reduce the computational amount and improve computational precision for nonlinear optimizations and pollution source identification in convection–diffusion equation, a new algorithm, chaos gray-coded genetic algorithm (CGGA) is proposed, in which initial population are generated by chaos mapping, and new chaos mutation and Hooke–Jeeves evolution operation are used. With the shrinking of searching range, CGGA gradually directs to an optimal result with the excellent individuals obtained by gray-coded genetic algorithm. Its convergence is analyzed. It is very efficient in maintaining the population diversity during the evolution process of gray-coded genetic algorithm. This new algorithm overcomes any Hamming-cliff phenomena existing in other encoding genetic algorithm. Its efficiency is verified by application of 20 nonlinear test functions of 1–20 variables compared with standard binary-coded genetic algorithm and improved genetic algorithm. The position and intensity of pollution source are well found by CGGA. Compared with Gray-coded hybrid-accelerated genetic algorithm and pure random search algorithm, CGGA has rapider convergent speed and higher calculation precision.     

Sub-Riemannian vs. Euclidean dimension comparison and fractal geometry on Carnot groups

We solve Gromov's dimension comparison problem for Hausdorff and box counting dimension on Carnot groups equipped with a Carnot-Carathéodory metric and an adapted Euclidean metric. The proofs use sharp covering theorems relating optimal mutual coverings of Euclidean and Carnot-Carathéodory balls, and elements of sub-Riemannian fractal geometry associated to horizontal self-similar iterated function systems on Carnot groups. Inspired by Falconer's work on almost sure dimensions of Euclidean self-affine fractals we show that Carnot-Carathéodory self-similar fractals are almost surely horizontal. As a consequence we obtain explicit dimension formulae for invariant sets of Euclidean iterated function systems of polynomial type. Jet space Carnot groups provide a rich source of examples.