The main aim of this paper is to find an optimal interval control law for quadratic linear problems under interval uncertainty. For this purpose, using Bellman's optimality principle and interval Hamilton-Jacobi-Bellman inequalities, the interval optimal control problem is transformed into a system of interval differential inequalities. These inequalities are called Riccati's differential inequalities, which is a result of the dynamic programming method. To solve this system of inequalities, we use inclusion relations and interval arithmetic. By this method, we can obtain the upper and lower bounds of the solutions. We also use Hokuhara's generalized difference to reduce errors in the interval arithmetic. We apply the presented method for solving some interval quadratic linear optimal control problems by using MATLAB software. The obtained results show the efficiency of the proposed method.

This paper addresses a bi-objective mixed integer optimization model under uncertainty for population partitioning problem. The objective functions are to minimize the number of communications between partitions and to balance their population. The main constraints are defined for creating contiguous and compact partitions as well as assigning uniquely each basic unit to one partition. To deal with the uncertainty of parameters, a robust programming method is proposed that causes the uncertainty parameters lie between the interval of bestcase (the deterministic mode) and worst-case (the highest uncertainty level for all parameters). As the suggested method is NP-Hard, three meta-heuristic algorithms NSGAII, PESA, and SPEA are developed and, to evaluate the efficiency of the algorithms, 10 small-size examples, 10 medium-size examples and, 10 large-size examples are generated and solved. According to computational results, the SPEA has the best performance. The method is examined for a real-world application, as a case study in Iran.

Earthquake as the most devastating natural disaster in urban areas causes huge physical and human damages worldwide. One way to assist reducing the impact of the earthquake on people and infrastructures is to produce a reliable seismic vulnerability map. The physical seismic vulnerability of a region as a multi criteria problem is concerned with seismic intensity, land slope, the number of building floors, building age and quality. Among the most important sources of uncertainty in determining the vulnerability of each urban statistical unit, is the uncertainty related to the conflicts in expert opinions concerning the level of severity of the seismic vulnerability. The main objective of this paper is to manage uncertainty considering different vulnerability classes allocated by the experts in integration of the concerned parameters. In this model, to reduce the uncertainty in the decision making process related to the expert opinions on allocating a seismic physical vulnerability class to each urban statistical unit, interval mathematics, genetic algorithm and granular computing methods are used. The physical seismic vulnerability map has been produced for Tehran on the basis of activation of North Tehran fault. Among 3174 urban statistical units, 150 randomly selected samples have been selected by 5 experts in related geoscience fields. The experts are asked to fill a questionnaire for allocating the physical seismic vulnerability of the samples. Due to the disaggregation in the experts’ knowledge on the physical seismic vulnerability of each statistical unit, their opinions have been integrated using interval mathematics. For the conflict resolution among the experts, genetic algorithm is used. Granular computing has been applied to manage the uncertainty caused by the large amount of information achieved from the parameters affecting the physical vulnerability to assess the seismic physical vulnerability. The relations among the input parameters and the vulnerability classes are presented in a decision table. The rules with a minimum conflict from the decision table are extracted. The vulnerability classes have been sorted from 1 to 5 considering 1 as the least vulnerable class and 5 as the most vulnerable class. According to the results, most of the statistical units in Tehran fall within interval class vulnerabilities of [3 4] and [5 4]. To compare the similarity between the results of the model and those of the previous research by Khamespnah in the same study area, who used an integrated model of granular computing and rough set theory, Spearman rank correlation coefficient was employed. The value of this coefficient was 0.47 that shows some similarities between the results. The accuracy of 76% was achieved in this research using Kappa index verifying the importance of managing uncertainty using interval mathematics.

Dempster Shafer theory and entropy are two methods for representation and quantitative measurement of the uncertainty in information systems. Deextended later by Shafer. Entropy is a measure of uncertainty as a basic concept in the information theory. Entropy can be applied as an uncertainty measurement of the systems in specific situation. In this paper, a new method has been proposed for measurement of the uncertainty upper and lower bound with the combination of mathematical models of entropy and Dempster Shafer theory. According to this analysis maximum and minimum of the uncertainty are calculated.

In the statistical downscaling methods which is based on the relationship between AOGCMs data and ground based climatic variables (such as rain and temperature), the future period of those variables are simulated. Since in the simulation, all effective parameters cannot be modeled, estimated values suffers from be uncertainty. The outputs of downscaling models are used as inputs to agriculture and water resources models; therefore, identifying the models inputs’ error or uncertainty is essential to realize the reliability of obtained results. In this research, an attempt is made to investigate the uncertainty of Artificial Neural Network (ANN) as a downscaling model in a case study in the northwest of Iran. For this purpose, precipitation, minimum and maximum temperature variables were used in the designed ANN model, and the NCEP data was employed for its calibration and validation. The HadCM3 was the selected AOGCM in this study. Observed daily time series were gathered at all stations in the study period and on the basis of bootstrap method the 99% confidence interval was calculated for all the variables. In the next step, the simulated (downscaled) mean and variance of the variables by the ANN model, compared to the calculated confidence interval. To compare the results, the criterion of the number of station-month was used. The results showed that the average maximum temperature at 14 station-months were within the confidence interval. The results of monthly analysis showed that the accuracy of ANN model in summer was low and its uncertainty is more than the other seasons. In the simulation of minimum temperature based on this criterion, 18 station-months were within the confidence interval. The accuracy of ANN to estimate the minimum temperature in summer was low with high uncertainty in almost all the stations. Moreover, in June and August in any of the stations estimated values were not within the confidence interval. Due to the high variability of rainfall in relation to temperature, confidence range was very high, and in some stations was more than 50% of average monthly precipitation. Because of the high confidence rang of precipitation, in 53 Stations-month cases were within the confidence interval.

It is very necessary to consider the uncertainty in the data and how to deal with it when performance measurement using data envelopment analysis. Because a little deviation in the data can lead to a significant change in the performance results. However, in the real world and in many cases, the data is uncertain. interval data envelopment analysis is one of the most widely used approaches to deal with interval data uncertainty. The purpose of the present paper is to provide a robust model of interval data envelopment analysis in order to performance measurement under double uncertainty situations. In addition to the uncertainty caused by the interval of data, there is also uncertainty at the lower bound and upper bound of the interval for each data. Using the approach presented in this study can greatly increase the conservatism and validity of the efficiency results and ranking. Finally, it should be noted that the results of the proposed models are illustrated by using of a numerical example.

A motivation for using fuzzy systems stems in part from the fact that they are particularly suitable for processes when the physical systems or qualitative criteria are too complex to model and they have provided an efficient and effective way in the control of complex uncertain nonlinear systems. To realize a fuzzy model-based design for chaotic systems, it is mostly preferred to represent them by T–S fuzzy models. In this paper, a new fuzzy modeling method has been introduced for chaotic systems via the interval type-2 Takagi–Sugeno (IT2 T–S) fuzzy model. An IT2 fuzzy model is proposed to represent a chaotic system subjected to parametric uncertainty, covered by the lower and upper membership functions of the interval type-2 fuzzy sets. Investigating many well-known chaotic systems, it is obvious that nonlinear terms have a single common variable or they depend only on one variable. If it is taken as the premise variable of fuzzy rules and another premise variable is defined subject to parametric uncertainties, a simple IT2 T–S fuzzy dynamical model can be obtained and will represent many well-known chaotic systems. This IT2 T–S fuzzy model can be used for physical application, chaotic synchronization, etc. The proposed approach is numerically applied to the well-known Lorenz system and Rossler system in MATLAB environment.

Despite modern scientific knowledge and computational power in hydrology, the key to properly addressing hydrologic uncertainty remains a critical and challenging one. Here, we applied lumped HBV hydrological model to describe the uncertainty in runoff prediction in Chehl-Chay watershed in Golestan province. We applied a new framework for uncertainty analysis that is rooted on ideas from predicting model residual uncertainty. The uncertainty calculated by local Errors and Clustering (EEC) is compared with estimates from two non parametric methods (quantile regression (QR) and random forest (RF)) and a parametric method (GLUE). Firstly, the model parameters were optimized by Shuffled Complex Evolution approach and model residuals of test data were computed. Fuzzy clustering in EEC is carried out by the fuzzy c-means method and employs four clusters, predictive discharges, observed discharges, rainfall values and residuals in study basin. The results of this case study show that the uncertainty estimates obtained by EES which is trained by SVM gives wider uncertainty band and RF gives narrower uncertainty band. The best overall uncertainty estimates according to the PICP, MPI and ARIL indices were obtained with QR and then EEC. In comparison with non-parametric, with respct to all indices nonparametric methods had better performance than GLUE method.