Possibility Distributions Research Articles

Evolutionary Voluntary Prisoner’s Dilemma Game under Deterministic and Stochastic Dynamics

The voluntary prisoner’s dilemma (VPD) game has sparked interest from various fields since it was proposed as an effective mechanism to incentivize cooperative behavior. Current studies show that the inherent cyclic dominance of the strategies of the VPD game results in periodic oscillations in population. This paper investigated the influence of the level of individual rationality and the size of a population on the evolutionary dynamics of the VPD game. Different deterministic dynamics, such as the replicator dynamic, the Smith dynamic, the Brown-von Neumann-Nash (BNN) dynamic and the best response (BR) dynamic, for the evolutionary VPD game were modeled and simulated. The stochastic evolutionary dynamics based on quasi birth and death (QBD) process was proposed for the evolutionary VPD game and compared with deterministic dynamics. The results indicated that with the increase of the loners’ fixed payoff, the loner is more likely to remain in the stable state of a VPD game under any of the dynamics mentioned above. However, the different speeds of motion under the dynamics in the cycle dominance proved to be diverse under different evolutionary dynamics and also highly sensitive to the rationality of individuals in a population. Furthermore, in QBD stochastic dynamics, the size of the population has a remarkable effect on the possibility distribution. When the population size increases, the limited distribution of the QBD process will be in accordance with the results in the deterministic dynamics.

Multi-objective biogeography-based optimization for supply chain network design under uncertainty

This paper proposes a new two-stage optimization method for multi-objective supply chain network design (MO-SCND) problem with uncertain transportation costs and uncertain customer demands. On the basis of risk-neutral and risk-averse criteria, we develop two objectives for our SCND problem. We introduce two solution concepts for the proposed MO-SCND problem, and use them to define the multi-objective value of fuzzy solution (MOVFS). The value of the MOVFS measures the importance of uncertainties included in the model, and helps us to understand the necessity of solving the two-stage multi-objective optimization model. When the uncertain transportation costs and customer demands have joined continuous possibility distributions, we employ an approximation approach (AA) to compute the values of two objective functions. Using the AA, the original optimization problem becomes an approximating mixed-integer multi-objective programming model. To solve the hard approximating optimization problem, we design an improved multi-objective biogeography-based optimization (MO-BBO) algorithm integrated with LINGO software. We also compare the improved MO-BBO algorithm with the multi-objective genetic algorithm (MO-GA). Finally, a realistic dairy company example is provided to demonstrate that the improved MO-BBO algorithm achieves the better performance than MO-GA in terms of solution quality.

Possibilistic AIRS induction from uncertain data

This paper presents a new approach in machine learning, especially, in supervised classification and reasoning under uncertainty. For many classification problems, uncertainty is often inherent in modeling applications and should be treated carefully and not rejected to make better decisions. Artificial immune recognition system (AIRS) is a well-known classifier that has provided good results with certain data. However, this method is not able to cope with uncertainty. To overcome this limitation, we propose a new classification approach combining the AIRS and possibility theory. The new approach is allowing to deal with uncertain attribute and also class values of training instances. The uncertainty is expressed via possibility distributions. Experimentations on real datasets from the U.C.I machine learning repository show good performances of the proposed approach.

Identification of optimal strategies for agricultural nonpoint source management in Ulansuhai Nur watershed of Inner Mongolia, China

In this study, an environmental-friendly modeling system was developed and applied to an agriculture nonpoint source (AGNPS) management in Ulansuhai Nur watershed. In this system, water environmental capacity, credibility-based chance-constrained programming (CCCP), and AGNPS optimization models were integrated into a general modeling framework. It could be used to calculate water environmental capacity of total nitrogen and total phosphorus in Ulansuhai Nur watershed, which could consequentially provide input data for the developed AGNPS optimization model. Also, the inherent uncertainties in estimating water environmental capacities that can be expressed as possibilistic distributions were reflected and addressed based on computational results of three widely used methods. Such uncertainties were consequentially transferred to the proposed CCCP model based on the adoption of multiple credibility satisfactory levels, significantly facilitating objectivity reflection of decision alternatives. The developed modeling system was then applied to Ulansuhai Nur watershed of Inner Mongolia, a semi-arid river basin in northwestern China. Optimal strategies for AGNPS management in Ulansuhai Nur watershed were generated with consideration of the maximum total agricultural income under multiple policy scenarios. The results showed that the total agricultural income would increase with point source pollution being cut down, and would decrease with rising credibility levels, representing decreasing system violation risks. It was indicated that the higher of total nitrogen/phosphorus discharge being less than water environmental capacity of Ulansuhai Nur, the lower the total agriculture incomes. The proposed methods could help decision makers establish various production patterns with cost-effective agriculture nonpoint source management schemes in the basin of Ulansuhai Nur, and gain in-depth insights into the trade-offs between total agricultural incomes and system reliabilities.

A fuzzy portfolio selection model with background risk

In financial markets, the presence of background risk may affect investors’ investments. This article develops a fuzzy portfolio selection model with background risk, based on the definitions of the possibilistic return and possibilistic risk. For the returns of assets obey LR-type possibility distribution, we propose a specific portfolio selection model with background risk. Then, a numerical study is carried out by using the data concerning some stocks. Based on the data, we obtain the efficient frontier of the possibilistic portfolio with background risk, and compare it with the efficient frontier of the portfolio without background risk. Finally, we conclude that the background risk can better reflect the investment risk of the real economy environment which make the investors choose a more suitable portfolio to them.

Possibility Distribution of Wind Power Fluctuation

Possibility Distribution of Wind Power Fluctuation

Credibility-Based Biobjective Fuzzy Optimization for Supplier Selection Problem with Disruption

This paper addresses a supplier selection problem in which a buyer procures multiple products from multiple suppliers under disruption risk. The problem is formulated as a new credibility-based biobjective fuzzy optimization model. In the proposed model, cost, capacity, and demand are characterized by fuzzy variables with known possibility distributions. The objectives of our model are to maximize the total quality of purchased products and minimize the expected total cost. Two credibility constraints are used to guarantee that the chance about the supplier capacity and buyer demand can satisfy the predetermined levels. The main concern in solving the optimization model is to calculate the expected value of the objective function and the credibility in the constraints. When the key parameters are mutually independent triangular fuzzy variables, the expected cost objective and credibility constraints can be transformed into their equivalent forms. Taking advantage of the structural characteristics of the equivalent model, the goal programming method is employed to solve the supplier selection model, which can be solved by conventional optimization method. At last, some numerical experiments have been performed to illustrate the effectiveness of the proposed model and solution strategy.

Two-Stage Fuzzy Portfolio Selection Problem with Transaction Costs

This paper studies a two-period portfolio selection problem. The problem is formulated as a two-stage fuzzy portfolio selection model with transaction costs, in which the future returns of risky security are characterized by possibility distributions. The objective of the proposed model is to achieve the maximum utility in terms of the expected value and variance of the final wealth. Given the first-stage decision vector and a realization of fuzzy return, the optimal value expression of the second-stage programming problem is derived. As a result, the proposed two-stage model is equivalent to a single-stage model, and the analytical optimal solution of the two-stage model is obtained, which helps us to discuss the properties of the optimal solution. Finally, some numerical experiments are performed to demonstrate the new modeling idea and the effectiveness. The computational results provided by the proposed model show that the more risk-averse investor will invest more wealth in the risk-free security. They also show that the optimal invested amount in risky security increases as the risk-free return decreases and the optimal utility increases as the risk-free return increases, whereas the optimal utility increases as the transaction costs decrease. In most instances the utilities provided by the proposed two-stage model are larger than those provided by the single-stage model.

Modelling of Equipment Failure Rate Accounting for the Uncertainty

Abstract A fuzzy model for failure rate with the consideration of the effects of uncertain factors in distribution reliability evaluation is presented. The possibility and credibility distribution analyzed on the basis of sample datum are used for quantifying effects of the uncertainty done to failure rate. Mathematically, the failure rate can be obtained in the interval integration. Moreover, aiming to make the calculating quantity of system reliability evaluation simple and easy, the fuzzy clustering analysis of equipment is adopted. The technique proposed has been implemented in an example distribution system for illustration and the results obtained have been compared with those obtained with average model.

Fuzzy Responses and Bifurcations of a Forced Duffing Oscillator with a Triple-Well Potential

Responses and bifurcations of a forced triple-well potential system with fuzzy uncertainty are studied by means of the Fuzzy Generalized Cell Mapping (FGCM) method. A rigorous mathematical foundation of the FGCM is established as a discrete representation of the fuzzy master equation for the possibility transition of continuous fuzzy processes. The FGCM offers a very effective approach for solutions to the fuzzy master equation based on the min–max operator of fuzzy logic. A fuzzy response is characterized by its topology in the state space and its possibility measure of membership distribution functions (MDFs). A fuzzy bifurcation implies a sudden change both in the topology and in the MDFs. The response topology is obtained based on the qualitative analysis of the FGCM involving the Boolean operation of 0 and 1. The MDFs are determined by the quantitative analysis of the FGCM with the min–max calculations. With an increase of the intensity of fuzzy noise, noise-induced escape from each of the potential wells defines two types of bifurcations, namely catastrophe and explosion. This paper focuses on the evolution of transient and steady-state MDFs of the fuzzy response. As the intensity of fuzzy noise increases, steady-state MDFs cover a bigger area in the state space with higher membership values spreading out to a larger area. The previous conjectures are further confirmed that steady-state MDFs are dependent on initial possibility distributions due to the nonsmooth and nonlinear nature of the min–max operation. It is found that as time goes on, transient MDFs spread around three potential wells. The evolutionary orientation of transient MDFs aligns with unstable invariant manifolds leading to stable invariant sets. Two examples of additive and multiplicative fuzzy noise are given.

An economic production quantity problem with backorders and fuzzy cycle times

Optimization models combining Economic Production Quantity models and fuzzy set theory are important to the process industry as they are capable of modeling the numerous uncertainties inherent in this context. In this paper, we incorporate backorders i.e the inventory to go below zero in a fuzzy Economic Production Quantity EPQ model. The uncertainties in the backorders for different products are modeled using triangular possibility distributions. We present an example describing a typical decision making problem in the paper industry to illustrate our model.

Extreme points of the credal sets generated by comparative probabilities

When using convex probability sets (or, equivalently, lower previsions) as uncertainty models, identifying extreme points can help simplifying various computations or the use of some algorithms. In general, sets induced by specific models such as possibility distributions, linear vacuous mixtures or 2-monotone measures may have extreme points easier to compute than generic convex sets. In this paper, we study extreme points of another specific model: comparative probability orderings between the singletons of a finite space. We characterize these extreme points by means of a graphical representation of the comparative model, and use them to study the properties of the lower probability induced by this set. By doing so, we show that 2-monotone capacities are not informative enough to handle this type of comparisons without a loss of information. In addition, we connect comparative probabilities with other uncertainty models, such as imprecise probability masses.

Jeffrey’s rule of conditioning with various forms for uncertainty

We first introduce Jeffrey’s rule of conditioning and explain how it allows us to determine the probability of an event related to one variable from information about a collection of conditional probabilities of that event conditioned on the state another variable. We note that in the original Jeffrey paradigm we have the uncertainty about the state of the conditioning variable expressed as a probability distribution. Here we extend this by allowing alternative formulations for the uncertainty about the conditioning variable. We first consider the case where our uncertainty is expressed in terms of a measure. This allows us to consider the case where our uncertainty is a possibility distribution. We next consider the case where our uncertainty about the conditioning variable is expressed in terms of a Dempster–Shafer belief structure. Finally we consider the case where we are ignorant about the underlying distribution and must use the decision maker’s subjective attitude about the nature of uncertainty to provide the necessary information to use in the Jeffrey rule.

Introducing parametric uncertainty into a nonlinear friction model

AbstractWe introduce parametric uncertainty into a nonlinear model for static and kinetic friction of solids. The uncertain parameters are modeled by possibility distributions and an analytical approach is used to propagate the uncertainties through the computations. As a result, we obtain a symbolic expression for the whole spectrum of possible calculation results. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The Mathematical Theory of Evidence and Measurement Uncertainty - Comparison of Measurement Results Expressed in Terms of Random-Fuzzy Variables

In the previous papers, it was shown how possibility distributions can be effectively employed to represent and propagate uncertainty in measurements. In particular, it was shown how the Random-Fuzzy variables (RFVs) can be effectively employed to represent a measurement result. The effects of the systematic and random contributions to uncertainty can be well identified in the RFV, and all confidence intervals at all confidence levels are provided, so that complete information about the measurement result is given. Moreover, this distinction also allows one to model the propagation of the systematic and the random contributions in two different ways, according to their different nature and different behavior when they combine.

Necessary and sufficient conditions for the equality of the interactive and non-interactive sums of two fuzzy numbers

This study aims to answer the open question proposed by Carlsson et al. (2004) [4], which asks whether it is possible to find the properties needed by a joint possibility distribution function such that the interactive sum with respect to this function and the standard sum of two fuzzy numbers will coincide. We provide the necessary and sufficient conditions such that this equality would hold. In addition, we find the necessary and sufficient conditions such that the interactive and standard sums will coincide at some point in the domain. In order to obtain an interactive addition that does not coincide with the standard addition, we show that we need to search for a joint possibility distribution that coincides as little as possible with the independent joint possibility distribution throughout the so-called diagonal of the two fuzzy numbers. As simple consequences, we deduce similar results for triangular norm-based additions of fuzzy numbers.

Traffic Congestion Forecasting Based on Possibility Theory

Traffic congestion state identification is one of the most important tasks of ITS. Traffic flow is a nonlinear complicated system. Traffic congestion state is affected by many factors, such as road channelization, weather condition, drivers’ different driving behavior and so on. It is difficult to collect all necessary traffic information. Traffic congestion auto identification result based on incomplete traffic information exists uncertain. Little work has been done to analyze the uncertainty. Possibility theory introduced by Zadeh is an efficient means to present incomplete knowledge. Possibility distribution determination is an important task of possibility theory. In this paper, possibility theory is used to describe the uncertainty of traffic state. The possibility distribution of traffic state is determined according to the probability distribution of traffic flow parameters, such as volume, speed and occupancy and so on. The multi-variable distribution of traffic flow parameters is determined with large-scale traffic flow data. Large-scale traffic congestion samples are generated with parallel k-mean clustering method. Traffic congestion forecasting is based on the forecasting of traffic flow parameters with SVM (Support Vector Machines). At last, a practical example is analyzed with the proposed method.

Uncertainty analysis of the coefficients of friction during the tightening process of bolted joints

In this paper, we introduce closed-form symbolic expressions for the possibility distributions of the coefficients of friction during the tightening process of bolted joints. The parameters in the distribution functions are then identified by a standardized fastener testing system according to ISO 16047. An uncertainty analysis finally shows that the total amount of uncertainty in the coefficient of bearing friction is almost 40% larger than in the coefficient of thread friction. Furthermore, the real value of the coefficent of bearing friction is likely to be about 5% higher than the expected value.

A case-oriented approach to a lead/acid battery closed-loop supply chain network design under risk and uncertainty

A new scenario based stochastic and possibilistic mixed integer programming model for a multi-objective closed-loop supply chain network design problem by considering financial and collection risks is proposed. Uncertainties in the form of randomness and fuzziness are handled together for a better reflection of the problem. Different risk measures such as “variability index”, “downside risk” and “conditional value at risk” are integrated within the proposed model. Particularly, for the downside risk measure, target/threshold values are considered as fuzzy and described by their own possibility distribution. The proposed hybrid model is applied to an illustrative example inspired by the lead/acid industry in Turkey. Computational results of applying the different risk measures suggest that using downside risk model with fuzzy targets is found to be an appropriate choice for the stated closed-loop supply chain network design problem in terms of “computational efficiency”, “handling uncertainty” and “solution quality”

Possibilistic Bayesian inference based on fuzzy data

A Bayesian approach in the possibilistic environment is introduced, when the available data of the underlying statistical model are fuzzy. In this approach, based on a possibilistic model and a prior possibility distribution, the possibilistic posterior distribution is defined, when the available data are fuzzy. While the probabilistic Bayesian approach is suitable when we have stochastic uncertainty in the underlying model and available information, the proposed possibilistic Bayesian approach is proper when we come across the possibilistic uncertainty which is related to the notions of compatibility and consistency. A few numerical examples and a couple of applied examples in the field of concept learning are presented to illustrate the applicability of the proposed models.