Deterministic Values Research Articles

Meso-mechanical investigations on the overall elastic properties of multi-phase construction materials using finite element method

Many construction materials are multi-phase composites in mesoscale, and therefore their overall mechanical properties strongly depend upon those of each component. Elastic properties are critical parameters in the overall performance and analysis of composites, and they are also of practical importance in design. Owning to randomness in the particle locations of each component, the overall performance of a multi-phase material also is not a deterministic value. As a result, prediction of the elastic properties of a multi-phase material is a challenging task. In the present study, a numerical simulation algorithm was proposed to investigate fully the overall elastic properties of composite samples. Finite element models, coupled with Monte Carlo simulations, subsequently were used to predict the overall modulus and Poisson’s ratio. Specifically, two typical types of construction materials were considered; soil-rock mixtures and concrete. For validation, the numerical simulation results of elastic moduli were compared with experimental test results and the comparison gave good agreement between numerical and the test results. Moreover, the influence of parameters, such as the elastic modulus of each phase, particle shape, size distribution and porosity, on the overall elastic modulus, were considered via parametric studies. Finally, the proposed numerical simulation algorithm was compared with some existing empirical models and the comparison results underscored the importance of the numerical model developed in this study.

Probabilistic bearing capacity of strip footing on reinforced soil slope

The probabilistic bearing capacity of the strip footing placed on the edge of a reinforced cohesionless soil slope is computed by combining lower bound finite element limit analysis method with anisotropic random field modelling and Monte Carlo simulation technique. Friction and dilation angle of purely cohesionless soil are considered as log-normally distributed random variables. The mean value of the bearing capacity factor Nγ reduces with the increasing randomness of the soil friction angle for particular values of correlation length in the horizontal and vertical directions. At smaller correlation lengths, the mean Nγ values of unreinforced and reinforced slopes are always lower than the deterministic values. However, the difference reduces with the increasing values of correlation lengths. At a higher correlation length, the deterministic and probabilistic mean values become almost equal. With the increasing value of correlation lengths and factor of safety, the failure probability of the bearing capacity factor associated with the reinforced slope reduces. Failure mechanisms of unreinforced and reinforced slopes obtained from the deterministic and probabilistic analysis are illustrated.

Effect of Geotechnical Parameter Variability on Soil-Pipeline Interaction

Soil properties are stochastic in nature, and at a particular location, the soil properties will have an associated mean and probability distribution. However, in soil-pipeline interaction studies on oil and gas pipelines where the maximum response is associated with stronger pipe-soil spring properties, the soil properties used to determine the maximum soil resistance and soil spring stiffness are usually taken as deterministic, assuming no variation in soil properties. Alternatively, for soil-pipeline interaction scenarios where the maximum response is associated with weaker pipe-soil spring properties, the geotechnical engineer may assign deterministic lower-bound values in an attempt to apply conservatism. This paper examines the effect of variability of soil properties on the maximum soil resistances and soil stiffness values for both drained and undrained soil loading conditions. Then, the resulting probabilistic distributions are applied to two example soil-pipeline interaction problems common in pipeline design, a longitudinal landslide and a strike-slip earthquake fault displacement. The results show that the stresses and strains in the pipeline resulting from drained soil loading are relatively insensitive to the probabilistic distribution of soil properties, and for the most part, the use of average values may be adequate. However, for some soil-pipeline interaction scenarios and loading conditions, and given the large natural variation in undrained soil properties, the use of single deterministic average soil properties under undrained loading conditions may underestimate important pipeline strains or stresses. The use of probabilistic soil properties for some pipeline loading scenarios is recommended.

Uncertainties in dynamic response of buildings with non-linear base-isolators

Dynamic response of base-isolated buildings under uni-directional sinusoidal base excitation is numerically investigated considering uncertainties in the isolation and excitation parameters. The buildings are idealized as single degree of freedom (SDOF) system and multi-degrees of freedom (MDOF) system with one lateral degree of freedom at each floor level. The isolation system is modeled using two different mathematical models such as: (i) code-recommended equivalent linear elastic-viscous damping model and (ii) bi-linear hysteretic model. The uncertain parameters of the isolator considered are time period, damping ratio, and yield displacement. Moreover, the amplitude and frequency of the sinusoidal base excitation function are considered uncertain. The uncertainty propagation is investigated using generalized polynomial chaos (gPC) expansion technique. The unknown gPC expansion coefficients are obtained by non-intrusive sparse grid collocation scheme. Efficiency of the technique is compared with the sampling method of Monte Carlo (MC) simulation. The stochastic response quantities of interest considered are bearing displacement and top floor acceleration of the building. Effects of individual uncertain parameters on the building response are quantified using sensitivity analyses. Effect of various uncertainty levels of the input parameters on the dynamic response of the building is also investigated. The peak bearing displacement and top floor acceleration are more influenced by the amplitude and frequency of the sinusoidal base excitation function. The effective time period of the isolation system also produces a considerable influence. However, in the presence of similar uncertainty level in the time period, amplitude and frequency of the sinusoidal forcing function, the effect of uncertainties in the other parameters of the isolator (e.g., damping ratio and yield displacement) is comparatively less. Interestingly, the mean values of the response quantities are found to be higher than the deterministic values in several instances, indicating the need of conducting stochastic analysis. The gPC expansion technique presented here is found to be a computationally efficient yet accurate alternative to the MC simulation for numerically modeling the uncertainty propagation in the dynamic response analyses of the base-isolated buildings.

Scenario uncertainties assessment within whole building LCA

Uncertainties evaluation is increasingly gaining traction within life cycle assessment (LCA), due to its key role as environmental decision support tool. When applied at whole-building scale, the large variety of materials, subjective choices and long lifespans introduce parameter, scenario and model uncertainties throughout the life cycle. Since normative choices are unavoidable within whole-building LCA (wbLCA), in this article we carried out a so-called ‘scenario’ uncertainty assessment for one illustrative case study. First, three uncertainty sources were selected, to include the two drivers most frequently cited in literature (reference service life and end of life management alternatives) and material wastage, a relevant issue to factor in variable construction optimization levels in contexts like Brazil. Cumulative energy demand (CED) and CML 2001 v.2.05 methods were used for calculating deterministic values of non-renewable embodied energy and global warming potential in SimaPro 7.3 The uncertainty assessment combined scenario analysis, stochastic modelling (Monte Carlo simulation of triangular probability distributions for the uncertainty drivers investigated) and global sensitivity analysis (GSA). The GSA confirmed the dominant contribution of the operational phase - strongly influenced by components replacement rate - to of life cycle non-renewable embodied energy and global warming potential result variance, whilst construction and end of life stages showed no correlation with life cycle results. Findings from this research also highlight the strategic importance of gathering service life information adherent to the assessed context. Building components replacement rates induced by the Brazilian standard are overestimated relatively to international figures used in LCAs worldwide.

Stochastic natural frequency analysis of skewed sandwich plates

PurposeThe purpose of this paper is to investigate the first three stochastic natural frequencies of skewed sandwich plates, considering uncertain system parameters. To conduct the sensitivity analysis for checking the criticality of input parameters.Design/methodology/approachThe theoretical formulation is developed based on higher-order-zigzag theory in accordance with the radial basis function (RBF) and stochastic finite element (FE) model. A cubic function is considered for in-plane displacement over thickness while a quadratic function is considered for transverse displacement within the core and remains constant in the facesheet. RBF is used as a surrogate model to achieve computational efficiency and accuracy. In the present study, the individual and combined effect of ply-orientation angle, skew angle, number of lamina, core thickness and material properties are considered for natural frequency analysis of sandwich plates.FindingsResults presented in this paper illustrates that the skewness in the sandwich plate significantly affects the global dynamic behaviour of the structure. RBF surrogate model coupled with stochastic FE approach significantly reduced the computational time (more than 1/18 times) compared to direct Monte Carlo simulation approach.Originality/valueThe stochastic results for dynamic stability of sandwich plates show that the inevitable source uncertainties present in the input parameters result in significant variation from the deterministic value demonstrates the need for inclusive design paradigm considering stochastic effects. The present paper comprehensively establishes a generalized new RBF-based FE approach for efficient stochastic analysis, which can be applicable to other complex structures too.

Analysis of Basic Failure Scenarios of a Truss Tower in a Probabilistic Approach

This paper concerns the system reliability analysis of steel truss towers. Due to failures of towers, the assessment of their reliability seems to be a very important problem. In the analysis, two cases are examined: when the buckling coefficient is a deterministic value and when it is a random variable. The impact of failures of single elements on the structure reliability was investigated. Calculations of the standard deviation of the capacity and reliability indexes were made using author-developed programs in the Mathematica environment.

PROBABILISTIC INFERENCE FOR INTERVAL PROBABILITIES IN DECISION-MAKING PROCESSES

The present paper considers one approach to Bayes’ formula based probabilistic inference under interval values of relevant probabilities; the necessity of it is caused by the impossibility to obtain reliable deterministic values of the required probabilistic evaluations. The paper shows that the approach proves to be the best from the viewpoint of the required amount of calculations and visual representation of the results. The execution of the algorithm of probabilistic inference is illustrated using a classical task of decision making related to oil mining. For visualisation purposes, the state of initial and target information is modelled using probability trees.

Development of an oral reference dose for the perfluorinated compound GenX

Ammonium 2,3,3,3‐tetrafluoro‐2‐(heptafluoropropoxy)‐propanoate, also known as GenX, is a processing aid used in the manufacture of fluoropolymers. GenX is one of several chemistries developed as an alternative to long‐chain poly‐fluoroalkyl substances, which tend to have long clearance half‐lives and are environmentally persistent. Unlike poly‐fluoroalkyl substances, GenX has more rapid clearance, but has been detected in US and international water sources. There are currently no federal drinking water standards for GenX in the USA; therefore, we developed a non‐cancer oral reference dose (RfD) for GenX based on available repeated dose studies. The review of the available data indicate that GenX is unlikely to be genotoxic. A combination of traditional frequentist benchmark dose models and Bayesian benchmark dose models were used derive relevant points of departure from mammalian toxicity studies. In addition, deterministic and probabilistic RfD values were developed using available tools and regulatory guidance. The two approaches resulted in a narrow range of RfD values for liver lesions observed in a 2‐year bioassay in rats (0.01–0.02 mg/kg/day). The probabilistic approach resulted in the lower, i.e., more conservative RfD. The probabilistic RfD of 0.01 mg/kg/day results in a maximum contaminant level goal of 70 ppb. It is anticipated that these values, along with the hazard identification and dose‐response modeling described herein, should be informative for risk assessors and regulators interested in setting health‐protective drinking water guideline values for GenX.

EFFICIENT COMPUTATION OF SOBOL’ QUASI-RANDOM GENERATOR

Purpose of the study: The quasi-Monte Carlo method is an essential tool for modeling and analyzing various complex problems in engineering, physical sciences, finance, and business. The crucial element of the method is a sequence of deterministic quasi-random values, which is often obtained by using the Sobol’ quasi-random generator. The purpose of this study is to consider the time complexity of generating the Sobol’ sequence.
 Methodology: Algorithms for determining the Sobol' sequence have been studied. The algorithms have been implemented in the Python programming language.
 Main Findings: It is established that this sequence can be generated in the linear time provided that generated numbers are based on 32-bit or 64-bit integers. The main result of the paper is the algorithm, which enables this time-bound.
 Applications of this study: The study can be applied in engineering, physical sciences, finance, and business.
 Novelty/Originality of this study: It is shown that Sobol’ sequence can be generated in linear time.

A Nonlinear Inexact Two-Stage Management Model for Agricultural Water Allocation under Uncertainty Based on the Heihe River Water Diversion Plan

In this study, a nonlinear inexact two-stage management (NITM) model is proposed for optimal agricultural irrigation water management problems under uncertainty conditions. The model is derived from incorporating interval parameter programming (IPP), two-stage stochastic programming (TSP) and quadratic programming (QP) within the agricultural water management model. This model simultaneously handles uncertainties not only in discrete intervals, but also in probability distributions, as well as nonlinearity in the objective function. A concept of the law of diminishing marginal utility is introduced to reflect the relationship between unit benefits and allocated water, which can overcome the limitation of general TSP framework with a linear objective function. Moreover, these inexact linear functions of allocated water can be obtained by an interval regression analysis method. The model is applied to a real-world case study for optimal irrigation water allocation in midstream area of the Heihe River Basin in northwest China. Two Heihe River ecological water diversion plans, i.e., the original plan and an improved plan, will be used to determine the surface water availabilities under different inflow levels. Four scenarios associated with different irrigation target settings are examined. The results show that the entire study system can arrive at a minimum marginal utility and obtain maximum system benefits when optimal irrigation water allocations are the deterministic values. Under the same inflow level, the improved plan leads to a lower water shortage level than that of the original plan, and thus leads to less system-failure risk level. Moreover, the growth rate of the upper bound of economic benefits between each of two scenarios based on the improved plan are greater than that from the original plan. Therefore, these obtained solutions can provide the basis of decision-making for agricultural water allocation under uncertainty.

The concept of pavement skid resistance state

The Concept of Skid Resistance State for highway pavements is introduced in this paper. The Concept states that at a given point in time, each unit length of a pavement has a fixed deterministic skid resistance value under a given set of pavement, environmental and vehicle operating conditions. A set of the operating conditions is defined by the following parameters: Vehicle tyre geometric and structural properties, wheel load, tyre pressure, vehicle speed, tyre slip ratio, tyre slip angle, pavement microtexture and macrotexture properties, ambient temperature, and pavement surface water film thickness. It is demonstrated in this study that, according to the proposed Concept, the skid resistance characteristics of a pavement can be fully described by a computer simulation model that analyses tyre–pavement–fluid interaction based on solid mechanics and fluid dynamics theories. Once the computer skid resistance model of a pavement section has been established, its skid resistance value under any vehicle–pavement–environmental operating condition can be calculated. Currently, there are already a number of computer skid resistance models developed by pavement researchers. Using one of these models for illustration, this paper explains how the Concept of Skid Resistance State is applied. The rationale, calibration process of the model, and input parameters are described; followed by a numerical demonstration of the establishment of the Skid Resistance State of an example pavement section. The significance and potential applications of the Concept on functional pavement mix design, pavement friction maintenance management, and harmonising of skid resistance measurements by different devices are also described.

Performance of large-scale polling systems with branching-type and limited service

Motivated by emerging Internet-of-Things (IoT) applications and smart building environments, we analyze the performance of large-scale symmetric polling systems where the number of queues grows large. We consider a scenario in which the total arrival rate is kept fixed and the individual switch-over time and service time distributions remain the same. This asymptotic regime leads to cycles of infinite length and queue lengths with non-trivial distributions. We show that for most traditional service policies the scaled cycle times converge to a deterministic value in the limit, which in turn implies that the queue lengths at the various nodes become asymptotically independent. Using these insights, we find that the behavior of individual queues simplifies to that of a discrete-time bulk service queue in the limit, so that the marginal queue length and waiting-time distributions become considerably easier to analyze. Additionally, we propose a new flexible k-limited service discipline aimed at striking a good balance between short mean queue lengths and predictable cycle times for deadline-critical applications.

Effectiveness of seismic strengthening to repeated earthquakes in historic urban contexts

PurposeThe purpose of this paper is to discuss the dynamics of the evolution of damage to the residential buildings within the city walls of Norcia during the six-month seismic swarm that hit Central Italy in the period 24th August 2016 to end of January 2019. This is accomplished by comparing the damage recorded by the Italian Civil Protection usability form (AEDES form) during this period after each event.Design/methodology/approachFirst, these outputs are compared with a qualitative assessment conducted by means of omnidirectional camera (ODC) imagery collected on site by the authors, to explore the ability of this technology to support post-earthquake damage assessment. The damage level attributed with these two techniques is then further compared with the output of the analytical vulnerability assessment method FaMIVE, which allows to correlate damage to vulnerability. Specifically, the objective is to investigate the efficacy and performance of historic and recent strengthening interventions.FindingsResults show that there is a good correspondence between AeDES and ODC assessments for low to medium damage grades (DG). Discrepancies in higher DGs are discussed in light of the different levels of information that can be recorded by using the two tools. The efficacy of strengthening is also well captured by the FaMIVE method. The procedure estimates a decrease of almost 40, 25 and 20 per cent of the total number of buildings failing out-of-plane, respectively, for the three seismic events considered, when restraining elements are in action.Research limitations/implicationsThe analysis conducted in this work make use of deterministic values of Norcia’s masonry fabric characteristics that have been found in literature, thus implying that neither the probabilistic aspects nor the related uncertainties have been properly investigated and addressed. However, this limitation is to be considered within the more general context of the legislation for the preservation of historic buildings which limits substantially any type of semi-destructive tests, hence limiting the reliability of the values available in literature. This in turn affects the decisions informing the design and implementation of strengthening interventions which can be confidently considered reliable and effective.Originality/valueThe paper addresses for the first time a systematic investigation of damage progression in historic masonry structures, part of urban aggregates in heritage cities. The current urban fabric is discussed in view of historic building codes as the basis for determining the present seismic vulnerability of the historic city centre of Norcia. The study provides new data sets for the city of Norcia and develops a statistical correlation between cumulative damage and analytical vulnerability functions for heritage buildings exposed to a swarm of earthquakes. The analytical assessment of the effect of historic strengthening is totally novel.

Analisys of organizational and technological reliability at the level of the schedule time parameters determination

Problem statement. One of the main tasks of the construction organization is to comply with the terms of construction of the object according to the design decisions. Organizational and technological reliability of construction determines the ability of a construction organization to achieve its purposes at given input parameters. When developing a time schedule, there is a contradiction between the probabilistic nature of real processes and the determined methods of their description. When the calculations are of a predictable nature, it is necessary to include into the calculation the probabilistic component, which determines the probability of a certain event occurring in the future. This is especially true of scheduling because of the prediction of indicators of the duration of production processes can be carried out only with a certain level of probability. The study is devoted to solve a scientific and applied problem of increasing the reliability of planning at the stage of determining the time parameters of construction processes as part of a schedule. The factors affecting the planned time of construction and installation work in the schedule are considered. The approach was used, according to which the time of completion of a certain amount of work is depended from the performer’s output and the intensity of the work. The duration of the work is not a deterministic value, but a range (interval) of possible values that can be described using the normal distribution, β-distribution and α-distribution. The purpose of the article is to determine the rational level of organizational and technological reliability of schedule time parameters. Conclusions. The reliability of the construction schedule completely depends on the accuracy of time parameters. The dependence between the output of performers and the reliability of work in the planning period is established. The rational level of organizational and technological reliability of the construction schedule at the stage of its time parameters modeling is determined.

Confidence intervals for resistance factors in geotechnical LRFD applications

Load and resistance factor design (LRFD) calibration studies in geotechnical engineering typically report resistance factors (ϕ) as fixed deterministic values. However, ϕ is based on a sample of measured and predicted resistances, so in reality the reported; i.e., “nominal” ϕ from a calibration study is an estimate of some underlying “true” ϕ, and as with any other statistic, ϕ is subject to variability arising from both intrinsic randomness and measurement uncertainty. The statistical uncertainty associated with finite sample size can be acknowledged by reporting confidence intervals for the parameter. In this study, confidence intervals for ϕ are calculated using both parametric and nonparametric bootstrap methodologies. These confidence intervals allow for a specific definition of precision for ϕ expressed in terms of the confidence interval coverage and the relative confidence interval width, and these parameters are directly related to sample size. By way of application, a set of charts for coverage and confidence interval width are provided, and the use of these charts is illustrated with selected data from published deep foundation reliability studies. The variability associated with ϕ goes to the very heart of reliability analysis and is significant for at least three reasons: (1) the variability may be linked to the sample size of the calibration study from which ϕ is derived, (2) the magnitude of variability provides a way to judge the quality of ϕ, and (3) by implication, the quality of ϕ affects interpretation of the probability of failure for an LRFD application. Thus, consideration of the confidence interval associated with ϕ provides insight into the probability of failure, which is the proper object of an LRFD resistance factor calibration study.

A method of uncertainty analysis for whole-life embodied carbon emissions (CO2-e) of building materials of a net-zero energy building in Australia

The construction of new buildings requires the use of a substantial amount of materials, which have an associated embodied energy for manufacturing, transport, construction and end-of-life disposal. A number of inventories have been developed to collate the typical embodied energy or carbon emissions associated with different building materials and activities, and these can be used to quantify the environmental impacts of different construction methods. However, uncertainty exists in the estimation of embodied CO2-e emissions and other environmental impact results, due to i) inconsistencies in typical embodied carbon emissions values in inventories; ii) errors in estimations of material quantities; iii) assumptions regarding building lifetimes, and iv) errors in estimations of transport distances.This current study quantified the uncertainties associated with the calculation of lifetime CO2-e emissions in a case study net-zero, in terms of operational energy, educational building. This study examined the lifetime impacts of building materials for the building based on a detailed Life Cycle Assessment (LCA) that had been previously undertaken for this site. The study considered the 19 building materials which most heavily influenced the total, transport and recurring embodied carbon footprint of the building and a probability distribution was generated to represent the variability for each of the following uncertain parameters: Lifetime, Embodied CO2-e and transport distance over the building's life. Random sampling was used to generate input variables (1000 samples) based on a probability distribution of each uncertain parameter relative to the building materials. Through the use of a Monte Carlo simulation, the environmental impact for each construction material for a 50-year building lifetime was predicted. Unlike the conventional LCA approach, which provides a single deterministic value, cumulative Monte Carlo distribution curves were used to provide a range of embodied CO2-e emissions for each construction material, and the whole building, through the lifetime of the building. The obtained results revealed a distribution of the total embodied CO2-e of a building which ranged from 2951 tCO2-e to 5254 tCO2-e. This variation in the life cycle carbon emissions highlights the importance of considering an uncertainty analysis in the LCA analysis.

RELIABILITY-BASED MULTI-OBJECTIVE OPTIMIZATION APPLIED TO CHEMICAL ENGINEERING DESIGN

ABSTRACT Chemical engineering optimization represents a significant challenge due to the complexity of the mathematical models that are frequently required in this area. These models are normally associated with nonlinear equations that represent mass, energy, and momentum balances, which are submitted to physical, constitutive, environmental, and design limitations. The design of chemical systems is generally carried out by considering the model, the vector of design variables, and system parameters as deterministic values, i.e., small variations in these quantities do not affect the objective function. In this contribution, a new methodology based on a double loop iteration process to evaluate the influence of uncertainties on chemical engineering design is proposed. The inner optimization loop is used to find the solution associated with the highest probability value by using the so-called Inverse Reliability Analysis and the outer loop is the regular optimization loop used to determine the vector of design variables. For this aim, the Multi-Objective Optimization Water Cycle Algorithm is improved, adopting a mechanism of neighborhood exploration. For illustration purposes, the proposed methodology is applied to mathematical functions and to chemical engineering design. The obtained results demonstrate that the proposed strategy represents an interesting alternative to reliability design in chemical engineering.

Empirical Bayes estimators for high-dimensional sparse vectors

The problem of estimating a high-dimensional sparse vector $\theta \in \mathbb{R}^n$ from an observation in i.i.d. Gaussian noise is considered. The performance is measured using squared-error loss. An empirical Bayes shrinkage estimator, derived using a Bernoulli-Gaussian prior, is analyzed and compared with the well-known soft-thresholding estimator. We obtain concentration inequalities for the Stein's unbiased risk estimate and the loss function of both estimators. The results show that for large $n$, both the risk estimate and the loss function concentrate on deterministic values close to the true risk. Depending on the underlying $\theta$, either the proposed empirical Bayes (eBayes) estimator or soft-thresholding may have smaller loss. We consider a hybrid estimator that attempts to pick the better of the soft-thresholding estimator and the eBayes estimator by comparing their risk estimates. It is shown that: i) the loss of the hybrid estimator concentrates on the minimum of the losses of the two competing estimators, and ii) the risk of the hybrid estimator is within order $\frac{1}{\sqrt{n}}$ of the minimum of the two risks. Simulation results are provided to support the theoretical results. Finally, we use the eBayes and hybrid estimators as denoisers in the approximate message passing (AMP) algorithm for compressed sensing, and show that their performance is superior to the soft-thresholding denoiser in a wide range of settings.

Risk-averse real driving emissions optimization considering stochastic influences

ABSTRACTOptimization of vehicle powertrains is usually based on specific drive cycles and is performed on testbeds under reproducible conditions. However, in real-world operation, energy consumption and emissions differ significantly from the values obtained in testbed environments, which also implies breaching legislative thresholds. Therefore, in order to close the gap between testbed and real world, it is necessary to take random effects, like varying road and ambient conditions or various traffic situations, into account during the engine calibration process. In this article a stochastic optimization approach based on risk measures, that quantify the prevalent uncertainties, is presented. Rather than optimizing a deterministic value for one specific scenario described by a drive cycle, the distribution of possible outcomes is shaped in a way that it reflects the risk aversion and preferences of the decision maker. Simulation results show that incorporating randomness in the optimization process yields substantially more robust and reliable results.