Non-stationary Probability Distribution Research Articles

Assessing the impacts of reservoirs on downstream hydrological frequency based on a general rainfall-reservoir index

When large reservoirs are built and put into operation, the downstream hydrological processes will be altered significantly, and ecology and agricultural irrigation water of the basin will be affected to some extent. The reservoir index (RI) and the sediment trapping efficiency (TE) of reservoirs are defined to quantify the reservoir impacts on the water flow and sediment by considering the static storage capacity. However, the regulating effect of reservoirs on hydrological variables is not only related to static storage capacity, but also to dynamic reservoir operation. Thus, in this paper, a general rainfall-reservoir index (GRRI) is developed by coupling reservoir regulation indicator (RR, including RI and TE) and effective rainfall affecting the dynamic operation of reservoirs, and the GRRI is used as the covariate to carry out the nonstationary frequency analysis of flood (Q) and annual sediment load (S) at Gaochang (GC) station in Min River, Wulong (WL) station in Wu River, Ankang (AK), Huangjiagang (HJG) and Huangzhuang (HZ) station in Han River, and Cuntan (CT) station on the main stream of the upper Yangtze River. It is found that Q and S at six stations have obvious changes induced by reservoirs, the mean of Q decreases by 22.8%–60.6%, and S drops by 47.7%–89.5% after the change-point of time series. The nonstationary probability distribution models with GRRI as the covariate have better fitting effects than nonstationary models with RR as the covariate. With the incorporation of the impacts of effective rainfall, the GRRI can more accurately capture the occurrence of nonstationarity in the downstream hydrological frequency. These results might be helpful for exploring the impact mechanism of the reservoir regulation on the downstream hydrological variables as well as ecological management of basin.

Nonstationary analysis of hydrological drought index in a coupled human-water system: Application of the GAMLSS with meteorological and anthropogenic covariates in the Wuding River basin, China

Significant nonstationarity of hydrological sequences, driven by the coupled influence of climate change and anthropogenic activities, has challenged traditional hydrological drought analysis under the stationarity assumption in some regions, and has questioned regional hydrological drought analysis under the changing environment. To evaluate hydrological drought characteristics, the novelty of this paper is the proposition of a GAMLSS (Generalized Additive Models for Location, Scale, and Shape)-based nonstationary standardized runoff index (NSRI), considering climate change and anthropogenic influence, for investigating the hydrological drought regime in the Wuding River basin. To that end, nonstationary test methods were performed on hydrological sequences, and data was segmented into three parts according to the results of the Pettitt test. A nonstationary probability distribution was fitted to runoff data with meteorological covariates (precipitation, temperature) and anthropogenic covariates (water consumption for social development demand, and water consumption triggered by water impounding by check dams). After selecting an optimal combination of covariates, the NSRI was calculated utilizing the GAMLSS model. A newly constructed evaluation threshold was proposed, based on the traditional SRI threshold. The performance of SRI and NSRI were compared at the Dingjiagou station in the Wuding River basin. Results of comparison between index identification and recorded drought events demonstrated that the NSRI better performed in drought event identification. Moreover, the NSRI identified more frequent severe droughts and extreme droughts. Therefore, the proposed NSRI provided a more accurate basis for the identification of drought events, which can offer valuable information for drought planning, preparedness, and mitigation.

Characterization of Bias during Meteorological Drought Calculation in Time Series Out-of-Sample Validation

The standardized precipitation index (SPI) is used for characterizing and predicting meteorological droughts on a range of time scales. However, in forecasting applications, when SPI is computed on the entire available dataset, prior to model-validation, significant biases are introduced, especially under changing climatic conditions. In this paper, we investigate the theoretical and numerical implications that arise when SPI is computed under stationary and non-stationary probability distributions. We demonstrate that both the stationary SPI and non-stationary SPI (NSPI) lead to increased information leakage to the training set with increased scales, which significantly affects the characterization of drought severity. The analysis is performed across about 36,500 basins in Sweden, and indicates that the stationary SPI is unable to capture the increased rainfall trend during the last decades and leads to systematic underestimation of wet events in the training set, affecting up to 22% of the drought events. NSPI captures the non-stationary characteristics of accumulated rainfall; however, it introduces biases to the training data affecting 19% of the drought events. The variability of NSPI bias has also been observed along the country’s climatic gradient with regions in snow climates strongly being affected. The findings propose that drought assessments under changing climatic conditions can be significantly influenced by the potential misuse of both SPI and NSPI, inducing bias in the characterization of drought events in the training data.

An Interval Two‐Stage Classified‐Allocation Model for Regional Water Management under Nonstationary Condition

AbstractDue to climate change and human activities, the assumption of the stationarity of hydrologic features will no longer hold. Moreover, uncertainties, such as the apparent randomness of hydrologic elements, and complexities, such as the existence of various water users with different characteristics, also introduce huge challenges for water managers. To address nonstationarity, uncertainty, and complexity, a new approach is proposed for the optimal allocation of regional water resources. This objective was achieved via two steps: First, the generalized additive model was chosen to analyze the nonstationary probability distribution of the hydrologic dataset; then, an interval two‐stage classified‐allocation model is formulated by incorporating two‐stage stochastic programming, interval parameter programming and classification thought. The model can not only address uncertainties, which were expressed as interval parameters and probability distributions, but can also handle complexities by classifying the water users into agricultural and nonagricultural users. The approach was applied to the Zhanghe Irrigation District to optimize available water allocation for municipality, industry, hydropower, and agriculture in two planning years (namely 2010 and 2015). The annual inflow of the Zhanghe Reservoir is found to be nonstationary and can be well fitted by Gamma distribution with one location parameter based on a nonlinear function of time. Moreover, the difference in output between the two years with different inflow probability distributions indicates the need for nonstationary analysis. Comparison to the inexact two‐stage water management model that did not consider the variation of agricultural water requirement shows the meaning of classification. From the results, municipality and industry are more competitive than agriculture and then hydropower. For agriculture, winter rape and cotton have higher priority than rice. These solutions of the optimal targets and optimal water allocation for different water users can help managers to accurately develop allocation plans under uncertain and nonstationary conditions.

Nonstationary Analyses of the Maximum and Minimum Streamflow in Tamsui River Basin, Taiwan

This study aims to detect non-stationarity of the maximum and minimum streamflow regime in Tamsui River basin, northern Taiwan. Seven streamflow gauge stations, with at least 27-year daily records, are used to characterize annual maximum 1- and 2-day flows and annual minimum 1-, 7-, and 30-day flows. The generalized additive models for location, scale, and shape (GAMLSS) are used to dynamically detect evolution of probability distributions of the maximum and minimum flow indices with time. Results of time-covariate models indicate that stationarity is only noted in the 4 maximum flow indices out of 35 indices. This phenomenon indicates that the minimum flow indices are vulnerable to changing environments. A 16-category distributional-change scheme is employed to classify distributional changes of flow indices. A probabilistic distribution with complex variations of mean and variance is prevalent in the Tamsui River basin since approximate one third of flow indices (34.3%) belong to this category. To evaluate impacts of dams on streamflow regime, a dimensionless index called the reservoir index (RI) serves as an alternative covariate to model nonstationary probability distribution. Results of RI-covariate models indicate that 7 out of 15 flow indices are independent of RI and 80% of the best-fitted RI-covariate models are generally worse than the time-covariate models. This fact reveals that the dam is not the only factor in altering the streamflow regime in the Tamsui River, which is a significant alteration, especially the minimum flow indices. The obtained distributional changes of flow indices clearly indicate changes in probability distributions with time. Non-stationarity in the Tamsui River is induced by climate change and complex anthropogenic interferences.

Stochastic fluctuations as a driving force to dissipative non-equilibrium states

Most natural complex systems exhibit fluctuations-driven processes, which work at far from equilibrium states, and are generally dissipative processes, for instance living cells. We studied this phenomenon within the stochastic framework by taking a set of nonequilibrium, bimolecular, autocatalytic reactions, originally proposed by Nicolis (1972). We also extended this model to incorporate the concept of time delay. Firstly, for both non-delay and delay cases, we calculated the exact non-stationary probability distribution solutions of the corresponding Master equations, which are found to deviate from the Maxwell–Boltzmann distribution. The analytically calculated probability distribution P of an autocatalyst X in the chemically reacting model system is found to follow some universal class of probability distributions at different situations. At the thermodynamic limit with a large population, P obeys Normal distribution. Again, we showed that one of the causes of this peculiar behaviour is the fluctuations in the reacting system. The analytical result of the Fano factor F in the non-delay case predicted a noise-enhanced process for our dynamical stochastic system which could probably drive the system far from equilibrium. For the delay case, the analytically calculated F was found to depend on the time delay function, which predicts that time delay could play an important role in regulating the system dynamics. These analytical predictions were then verified using numerical experiments with the stochastic simulation algorithm (SSA) and delay stochastic simulation algorithm (DSSA). Indeed, numerical results from SSA and DSSA confirmed noise-enhanced processes which are far from equilibrium and dissipative in nature.

Numerical investigation on the power of parametric and nonparametric tests for trend detection in annual maximum series

Abstract. The need to fit time series characterized by the presence of a trend or change points has generated increased interest in the investigation of nonstationary probability distributions in recent years. Considering that the available hydrological time series can be recognized as the observable part of a stochastic process with a definite probability distribution, two main topics can be tackled in this context: the first is related to the definition of an objective criterion for choosing whether the stationary hypothesis can be adopted, whereas the second regards the effects of nonstationarity on the estimation of distribution parameters and quantiles for an assigned return period and flood risk evaluation. Although the time series trend or change points are usually detected using nonparametric tests available in the literature (e.g., Mann–Kendall or CUSUM test), the correct selection of the stationary or nonstationary probability distribution is still required for design purposes. In this light, the focus is shifted toward model selection criteria; this implies the use of parametric methods, including all of the issues related to parameter estimation. The aim of this study is to compare the performance of parametric and nonparametric methods for trend detection, analyzing their power and focusing on the use of traditional model selection tools (e.g., the Akaike information criterion and the likelihood ratio test) within this context. The power and efficiency of parameter estimation, including the trend coefficient, were investigated via Monte Carlo simulations using the generalized extreme value distribution as the parent with selected parameter sets.

A risk-based analytical framework for quantifying non-stationary flood risks and establishing infrastructure design standards in a changing environment

In a rapidly changing environment, analysis of risks associated with non-stationary hydroclimatic extremes has many important implications for resilient and sustainable water resources management, including the evaluation of risk for existing systems and the design of new infrastructure. This study develops a new risk-based analytical framework called the Non-Stationary Monte-Carlo (NSMC) to better address various problems associated with non-stationarity of hydrologic extremes. Current approaches in the literature evaluating non-stationary Probability Distribution Functions (PDFs) of extremes events commonly use trend extension or multivariate analysis based on observed data, which often fail to account for larger changes in the future. To avoid these problems, NSMC explicitly accounts for the projected changes in hydroclimatic extremes by analyzing the changing PDFs of extremes for each future year based on statistically downscaled climate projections and hydrologic simulations. Using Monte Carlo techniques, NSMC generates a Super Ensemble (SE) of extremes, the statistics of which can be readily applied to various problems in non-stationary flood frequency analysis. For example, we show that the estimation of design standards based on Design Life Level (DLL) or Average Risk of Failure (ARF) metrics can be reduced to a simple look-up process of quantiles in the SE of extremes. A case study analyzing extreme high streamflow and a hypothetical levee design for the Wabash basin (IN, USA) demonstrates the applicability of NSMC to real-world flood risk problems. Furthermore, this study also shows an example case of using NSMC to identify cost-effective design standards for new infrastructure combining future changing risk of failure, project design lifespan, and present value of future replacement costs.

Online Resource Allocation with Limited Flexibility

We consider a class of online resource-allocation problems in which there are n types of resources with limited initial inventory and n demand classes. The resources are flexible in that each type of resource can serve more than one demand class. In this paper, we focus on a special class of structures with limited flexibility, the long-chain design, which was proposed by Jordan and Graves [Jordan WC, Graves SC (1995) Principles on the benefits of manufacturing process flexibility. Management Sci. 41(4):577–594.] and has been an important concept in the design of sparse flexible processes. We study the long-chain design in an online stochastic environment in which the requests are drawn repeatedly and independently from a nonstationary probability distribution over the different demand classes. Also, the decision on how to address each request must be made immediately upon its arrival. We show the effectiveness of the long-chain design in mitigating supply–demand mismatch under a simple myopic online allocation policy. In particular, we provide an upper bound on the expected total number of lost sales that is irrespective of how large the market size is. This paper was accepted by Yinyu Ye, optimization.

A Conceptual Framework for the Scale‐Specific Stochastic Modeling of Transitions in Tropical Cyclone Intensities

At any given time, a tropical cyclone (TC) vortex has multiple intensity pathways that are possible. We conceptualize this problem as a scenario where each of the TC's intensity pathways is a distinct attractor basin, and a combination of several external and internal factors across multiple scales dictates as to which of the many pathways the TC vortex actually takes. As with any complex system, it is difficult to know the details of the multiscale processes that cause or initiate the tipping of the TC vortex into an attractor basin. A stochastic shock arising from any of the various scales within a TC vortex and the subsequent cross‐scale energy transactions may rapidly increase the probability of the vortex intensifying or weakening. To address this problem and apply our conceptual framework to actual TC case studies, we formulate a novel scale‐specific stochastic model that examines the multiscale energetics at and across individual wave numbers within the TC vortex. The stochastic term is modeled in a realistic manner in that the lower and higher wave numbers are treated differently. High‐resolution Hurricane Weather and Research Forecast model outputs of two Bay of Bengal TCs, Phailin (intensifying) and Lehar (weakening), are used as case studies. An ensemble of intensity pathways is generated, and the nonstationary probability distributions of the intensity transitions at each time are examined. Our approach is another step toward an improved understanding of the stochastic dynamics of multiscale transitions of a TC vortex.

Synesthetes perseverate in implicit learning: Evidence from a non-stationary statistical learning task.

Synesthetes automatically and consistently experience additional sensory or cognitive perceptions in response to particular environmental stimuli. Recent evidence suggests that the propensity to develop synesthesia is genetic while the particular associations experienced by a given synesthete are influenced by learning. Despite the potential role of implicit learning in the formation of synesthetic associations, there has been minimal investigation of synesthetes' implicit learning abilities. In this study, we examine linguistic-colour synesthetes' ability to implicitly learn from and adjust to non-stationary statistics in a domain unrelated to their particular form of synesthesia. Engaging participants in a computer game Whack-the-mole, we utilise the online measure of reaction time to assess the time course of learning. Participants are exposed to "worlds" of probabilities that, unbeknownst to them, undergo unannounced changes, creating unpredictable statistical shifts devoid of accompanying cues. The same small set of probability worlds are repeated throughout the experiment to investigate participants' ability to retain and learn from this repetitive probabilistic information. The reaction time data provide evidence that synesthetes require more information than nonsynesthetes to benefit from the non-stationary probability distributions. These findings demonstrate that linguistic-colour synesthetes' implicit learning abilities-in a domain far from their synesthetic experiences-differ from those of nonsynesthetes.

A non-stationary cost-benefit analysis approach for extreme flood estimation to explore the nexus of ‘Risk, Cost and Non-stationarity’

Cost-benefit analysis is commonly used for engineering planning and design problems in practice. However, previous cost-benefit based design flood estimation is based on stationary assumption. This study develops a non-stationary cost-benefit based design flood estimation approach. This approach integrates a non-stationary probability distribution function into cost-benefit analysis, and influence of non-stationarity on expected total cost (including flood damage and construction costs) and design flood estimation can be quantified. To facilitate design flood selections, a ‘Risk-Cost’ analysis approach is developed, which reveals the nexus of extreme flood risk, expected total cost and design life periods. Two basins, with 54-year and 104-year flood data respectively, are utilized to illustrate the application. It is found that the developed approach can effectively reveal changes of expected total cost and extreme floods in different design life periods. In addition, trade-offs are found between extreme flood risk and expected total cost, which reflect increases in cost to mitigate risk. Comparing with stationary approaches which generate only one expected total cost curve and therefore only one design flood estimation, the proposed new approach generate design flood estimation intervals and the ‘Risk-Cost’ approach selects a design flood value from the intervals based on the trade-offs between extreme flood risk and expected total cost. This study provides a new approach towards a better understanding of the influence of non-stationarity on expected total cost and design floods, and could be beneficial to cost-benefit based non-stationary design flood estimation across the world.

Geostatistical seismic inversion for frontier exploration

Numerical 3D high-resolution models of subsurface petroelastic properties are key tools for exploration and production stages. Stochastic seismic inversion techniques are often used to infer the spatial distribution of the properties of interest by integrating simultaneously seismic reflection and well-log data also allowing accessing the spatial uncertainty of the retrieved models. In frontier exploration areas, the available data set is often composed exclusively of seismic reflection data due to the lack of drilled wells and are therefore of high uncertainty. In these cases, subsurface models are usually retrieved by deterministic seismic inversion methodologies based exclusively on the existing seismic reflection data and an a priori elastic model. The resulting models are smooth representations of the real complex geology and do not allow assessing the uncertainty. To overcome these limitations, we have developed a geostatistical framework that allows inverting seismic reflection data without the need of experimental data (i.e., well-log data) within the inversion area. This iterative geostatistical seismic inversion methodology simultaneously integrates the available seismic reflection data and information from geologic analogs (nearby wells and/or analog fields) allowing retrieving acoustic impedance models. The model parameter space is perturbed by a stochastic sequential simulation methodology that handles the nonstationary probability distribution function. Convergence from iteration to iteration is ensured by a genetic algorithm driven by the trace-by-trace mismatch between real and synthetic seismic reflection data. The method was successfully applied to a frontier basin offshore southwest Europe, where no well has been drilled yet. Geologic information about the expected impedance distribution was retrieved from nearby wells and integrated within the inversion procedure. The resulting acoustic impedance models are geologically consistent with the available information and data, and the match between the inverted and the real seismic data ranges from 85% to 90% in some regions.

Appropriate model selection methods for nonstationary generalized extreme value models

Several evidences of hydrologic data series being nonstationary in nature have been found to date. This has resulted in the conduct of many studies in the area of nonstationary frequency analysis. Nonstationary probability distribution models involve parameters that vary over time. Therefore, it is not a straightforward process to apply conventional goodness-of-fit tests to the selection of an appropriate nonstationary probability distribution model. Tests that are generally recommended for such a selection include the Akaike’s information criterion (AIC), corrected Akaike’s information criterion (AICc), Bayesian information criterion (BIC), and likelihood ratio test (LRT). In this study, the Monte Carlo simulation was performed to compare the performances of these four tests, with regard to nonstationary as well as stationary generalized extreme value (GEV) distributions. Proper model selection ratios and sample sizes were taken into account to evaluate the performances of all the four tests. The BIC demonstrated the best performance with regard to stationary GEV models. In case of nonstationary GEV models, the AIC proved to be better than the other three methods, when relatively small sample sizes were considered. With larger sample sizes, the AIC, BIC, and LRT presented the best performances for GEV models which have nonstationary location and/or scale parameters, respectively. Simulation results were then evaluated by applying all four tests to annual maximum rainfall data of selected sites, as observed by the Korea Meteorological Administration.

Nonstationary decision model for flood risk decision scaling

AbstractHydroclimatic stationarity is increasingly questioned as a default assumption in flood risk management (FRM), but successor methods are not yet established. Some potential successors depend on estimates of future flood quantiles, but methods for estimating future design storms are subject to high levels of uncertainty. Here we apply a Nonstationary Decision Model (NDM) to flood risk planning within the decision scaling framework. The NDM combines a nonstationary probability distribution of annual peak flow with optimal selection of flood management alternatives using robustness measures. The NDM incorporates structural and nonstructural FRM interventions and valuation of flows supporting ecosystem services to calculate expected cost of a given FRM strategy. A search for the minimum‐cost strategy under incrementally varied representative scenarios extending across the plausible range of flood trend and value of the natural flow regime discovers candidate FRM strategies that are evaluated and compared through a decision scaling analysis (DSA). The DSA selects a management strategy that is optimal or close to optimal across the broadest range of scenarios or across the set of scenarios deemed most likely to occur according to estimates of future flood hazard. We illustrate the decision framework using a stylized example flood management decision based on the Iowa City flood management system, which has experienced recent unprecedented high flow episodes. The DSA indicates a preference for combining infrastructural and nonstructural adaptation measures to manage flood risk and makes clear that options‐based approaches cannot be assumed to be “no” or “low regret.”

Local Quasi-equilibrium Description of Multiscale Systems

Abstract Systems whose dynamics result from the existence of a wide variety of time and length scales frequently exhibit slow relaxation behavior, manifested through the aging compartment of the correlations and the nonexponential decay of the response function. Experiments performed in systems such as amorphous polymers and supercooled liquids and glasses seem to indicate that these systems undergo, in general, non-Markovian and nonstationary dynamics. Hence, in this contribution, we present a dynamical description of slow relaxation systems based on a generalization of Onsager’s theory to nonequilibrium aging states. By assuming the existence of a local quasi-equilibrium state characterized by a nonstationary probability distribution the entropy of the system is expressed in terms of the conditional probability density by means of the Gibbs entropy postulate. Thus, by taking into account probability conservation and the rules of nonequilibrium thermodynamics, the generalized Fokker–Planck equation is derived.

Nonstationary Analysis of Annual Maximum Streamflow of Canada

Abstract Both natural climate change and anthropogenic impacts may cause nonstationarities in hydrological extremes. In this study, long-term annual maximum streamflow (AMS) records from 145 stations over Canada were used to investigate the nonstationary characteristics of AMS, which include abrupt changes and monotonic temporal trends. The nonparameteric Pettitt test was applied to detect abrupt changes, while temporal monotonic trend analysis in AMS series was conducted using the nonparameteric Mann–Kendall and Spearman tests, as well as a parametric Pearson test. Nonstationary frequency analysis of the AMS series was done using a group of nonstationary probability distributions. The nonstationary characteristics of Canadian AMS were further investigated in terms of the Hurst exponent (H), which represents the long-term persistence (LTP) of streamflow data. The results presented here indicate that for Canadian AMS data, abrupt changes are detected more frequently than monotonic trends, partly because many rivers began to be regulated in the twentieth century. Drainage basins that have experienced significant land-use changes are more likely to show temporal trends in AMS, compared to pristine basins with stable land-use conditions. The nonstationary characteristics of AMS were accounted for by fitting the data with probability distributions with time-varying parameters. Large H found in almost ⅔ of the Canadian AMS dataset indicates strong LTP, which may partly represent the presence of long-term memories in many Canadian river basins. Furthermore, H values of AMS data are positively correlated with the basin area of Canadian rivers. It seems that nonstationary frequency analysis, instead of the traditional stationary hydrologic frequency analysis, should be employed in the future.

Fluctuation theorems and inequalities generalizing the second law of thermodynamics out of equilibrium

We present a general framework for systems which are prepared in a nonstationary nonequilibrium state in the absence of any perturbation and which are then further driven through the application of a time-dependent perturbation. By assumption, the evolution of the system must be described by Markovian dynamics. We distinguish two different situations depending on the way the nonequilibrium state is prepared; either it is created by some driving or it results from a relaxation following some initial nonstationary conditions. Our approach is based on a recent generalization of the Hatano-Sasa relation for nonstationary probability distributions. We also investigate whether a form of the second law holds for separate parts of the entropy production and for any nonstationary reference process, a question motivated by the work of M. Esposito et al. [Phys. Rev. Lett. 104, 090601 (2010)]. We find that although the special structure of the theorems derived in this reference is not recovered in the general case, detailed fluctuation theorems still hold separately for parts of the entropy production. These detailed fluctuation theorems contain interesting generalizations of the second law of thermodynamics for nonequilibrium systems.

PARAMETRIC AND NON-PARAMETRIC METHODS FOR THE STUDY OF THE VARIABILITY OF WAVE DIRECTIONS: APPLICATION TO THE ATLANTIC URUGUAYAN COASTS

Wave direction is a fundamental variable in coastal engineering, whether one is interested in analyzing coastal processes or designing harbor structures. In this work a mixture of circular (wrapped) distributions is introduced for modeling the non-stationary probability distribution function of mean wave directions. The proposed distribution is able to accommodate seasonal and inter-annual variability, as well the influence of several climatic indices. This function was applied to the Atlantic Uruguayan coast, finding that both the Tropical South Atlantic index and the Antarctic Oscillation index have a significant influence on the variability of the wave directions.

A Global Chance-Constraint for Stochastic Inventory Systems Under Service Level Constraints

We consider a class of production/inventory control problems that has a single product and a single stocking location, for which a stochastic demand with a known non-stationary probability distribution is given. Under the widely-known replenishment cycle policy the problem of computing policy parameters under service level constraints has been modeled using various techniques. Tarim and Kingsman introduced a modeling strategy that constitutes the state-of-the-art approach for solving this problem. In this paper we identify two sources of approximation in Tarim and Kingsman's model and we propose an exact stochastic constraint programming approach. We build our approach on a novel concept, global chance-constraints, which we introduce in this paper. Solutions provided by our exact approach are employed to analyze the accuracy of the model developed by Tarim and Kingsman.