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Optimization approaches for the modernization of buildings mostly consider one-time investments for the design decision. Also the uncertainty of boundary conditions is rarely taken into account. In this work, we propose a robust extension of a mixed-integer linear
program that determines modernization schedules considering multiple points in time for the design decision. An initially conducted sensitivity analysis of the original model reveals high influences of user behavior, emission factors and economic parameters on
the objectives and design decisions. By applying the method to a typical building, it is shown that robust differ from nominal solutions. The shown investigation of different degrees of robustness should facilitate the decision for a modernization path in the future.

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... With this MILP, modernization roadmaps of single buildings can be determined for observation periods of up to 30 years. Since most of the yearly parameters in this model underlie uncertainty by observing these long observation periods, a robust extension of this MILP was proposed in [16] to handle this uncertainty. In [17], the MILP was extended by the possibility of setting emission goals at specific points in time during a modernization roadmap. ...

... (2) Quantify resources in economic optimal unlimited roadmap. Basically, there are two options to limit resources res in the model: Either, the sum of a resource over the whole roadmap may be limited by Equation (16) where red res is the reduction parameter of a resource that can be set between 0 and 1. Thereby, roadmaps with different levels of a limited resource can be determined. ...

... It is possible to quantify the influences of these uncertainties with sensitivity analyses as proposed in [61]. Furthermore, stochastic (e.g., [12]) or robust (e.g., [16]) optimization approaches can handle the uncertainties inside the optimization. ...

Great potential for saving carbon emissions lies in modernizing European buildings. Multi-year modernization roadmaps can plan modernization measures in terms of time and are able to consider temporal interactions. Therefore, we have developed a mixed-integer program that determines modernization roadmaps. These roadmaps include changing the energy supply system, improving the envelope, and considering annually varying boundary conditions. High craftwork capacities are required to implement the necessary modernizations to meet climate goals. Unfortunately, studies showed that the current shortage of craftworkers will intensify in the next years. Other important limitations correspond to energy resources. Recent crises show that many energy systems need to handle these limitations. Therefore, we extended the mixed-integer program by a method to handle these limitations inside the roadmaps. By the use of data from 90 interviews with craftwork specialists about the time needed to realize modernization measures, the method is applied. The main purpose is to analyze how modernization strategies change under limited resources, especially in terms of craftwork capacities. Hence, the method is exemplified by a representative single-family dwelling. Within this use case, modernization roadmaps with different craftwork capacity levels were calculated. The results show that modernization roadmaps change comprehensively over these levels. Key findings are that costs and emissions rise with decreasing craftwork capacities. Furthermore, smaller storages and pv systems are implemented at low craftwork capacities. The electrification of the heat supply supported by medium insulation standards should also be implemented with limited craftwork capacities.

... In addition, in their case study, the myopic approach leads to higher overall costs of the energy system. On building level, multi-period approaches have been successfully applied to optimize building retrofit schedules [39,40] and investment pathways for building energy systems [41,42,30,43]. On district level, Wei et al. [44] optimized a microgrid (modeled as single node) and consider long-term trends of declining investments for batteries. ...

... However, especially for long-term design problems with long planning horizons, these parameters are subject to substantial uncertainties. Therefore, robust optimization approaches could be added to the model formulation, as presented in [24,41]. However, robust optimization models substantially increase the computational complexity and make results more difficult to communicate to decision makers in real-world design problems (compared to straight-forward scenario analyses). ...

... With this MILP, modernization roadmaps of single buildings can be determined for time horizons of about 30 years. Since most of the yearly parameters in this model underlie uncertainty by observing these long time horizons, a robust extension of this work was proposed in [12] to handle this uncertainty. In contrast to stochastic optimization which is often used to consider uncertainty, robust optimization does not have the need of probability distributions for parameters. ...

... The robust extension of the described MILP is described in detail in [12]. An uncertainty characterization of all parameters as well as a sensitivity analysis was done. ...

Design decisions concerning future energy systems of existing buildings must be made to realize necessary building modernizations. Previous research has mainly used single-year investment approaches for these design decisions. This approach cannot consider parameter changes over time, such as energy price changes. The uncertainty of these parameters is also rarely respected. The main goal of this work is to consider time-varying effects of parameters and their uncertainty. Thus, in addition to deciding about optimal technologies, we propose a multi-year approach to optimize the time for the implementation of these technologies in a building. A robust extension of this approach allows finding optimal solutions under many possible parameter scenarios. We develop a robust mixed-integer linear program to determine modernization investment roadmaps that include yearly decisions concerning the energy supply system and envelope insulation of a building. Parameters and constraints of the model are also set separately for each year. By conducting a multi-objective optimization , total costs and emissions of a considered time horizon are minimized. In a second optimization, we set emission goals for specific years while minimizing costs. Furthermore, different levels of robustness are calculated. These levels indicate how much uncertainty of the parameters is considered. Based on developed characteristic values for the roadmaps, solutions are compared for typical buildings and over multiple levels of robustness. The results show that setting emission goals leaves large savings potentials unused compared to minimizing cumulative emissions of the time horizon considered. Another key finding is that higher degrees of robustness tend to lead to more investments in the energy supply system and less in the insulation of the envelope.

... A good overview is given by Ben-Tal et al. (2009) or more recently by Bertsimas and Hertog (2022), where both books also cover adjustable robustness. Robust optimization for energy supply design was for example done by Moret et al. (2020) and for building renovation by Richarz et al. (2021). ...

Optimizing a building's energy supply design is a task with multiple competing criteria, where not only monetary but also, for example, an environmental objective shall be taken into account. Moreover, when deciding which storages and heating and cooling units to purchase (here-and-now-decisions), there is uncertainty about future developments of prices for energy, e.g. electricity and gas. This can be accounted for later by operating the units accordingly (wait-and-see-decisions), once the uncertainty revealed itself. Therefore, the problem can be modeled as an adjustable robust optimization problem. We combine adjustable robustness and multicriteria optimization for the case of building energy supply design and solve the resulting problem using a column and constraint generation algorithm in combination with an $\varepsilon$-constraint approach. In the multicriteria adjustable robust problem, we simultaneously minimize worst-case cost regret and carbon emissions. We take into account future price uncertainties and consider the results in the light of information gap decision theory to find a trade-off between security against price fluctuations and over-conservatism. We present the model, a solution strategy and discuss different application scenarios for a case study building.

The energetic modernization of buildings has a high carbon emission savings potential. This potential can be leveraged through the strategic planning of modernization measures. In addition to the identification of measures, this planning should include the chronological sequence of measures and incorporate boundary conditions that change over time. Therefore, the core of this work is the development of an optimization model that determines both measures for the energy supply system and the building envelope, as well as their implementation times. For this purpose, physical effects of the building envelope as well as the operation of a building's energy supply system are modelled on an hourly level. Modernization measures for the entire building energy system are planned over decades on the basis of economic interrelationships to determine optimal modernization roadmaps. Changes in economic and ecological boundary conditions are incorporated into the model and a robust extension of the model integrates the uncertainties of these conditions into the optimization. Another optimization model uses pareto-optimal modernization roadmaps of single buildings as input to determine modernization roadmaps for building portfolios. The methods determine economically optimal modernization strategies for single buildings and building portfolios to save cumulative carbon emissions. Furthermore, the influence of emission target paths, uncertain boundary conditions, and the availability of craftwork capacities on these strategies are investigated. The methods are applied to typical German residential and non-residential buildings as well as to a building portfolio created from these. The results show that modernization measures should be coordinated in time. The electrification of the heat supply combined with building envelope measures of medium level depending on the age of the building form the basis of most of the strategies identified. Considering emission saving paths leads to economically inefficient strategies in single buildings, but can contribute to an accelerated decarbonization at the portfolio level. Compared to nominal strategies, robust strategies incur higher costs while decreasing emissions and increasing the share of renewable energy sources. Limited availability of craftwork capacities leads to higher costs and higher emissions. In individual buildings, this effect can be counteracted by larger thermal and electrical storage units, which increase the self-consumption of solar power and require low capacities for installation. Since autonomously calculated individual building roadmaps represent the input for the portfolio model, modernization measures in portfolio roadmaps often occur at the same time and the possibility of distributing craftwork capacities over time is severely restricted.

Optimization models for long-term energy planning often feature many uncertain inputs, which can be handled using robust optimization. However, uncertainty is seldom accounted for in the energy planning practice, and robust optimization applications in this field normally consider only a few uncertain parameters. A reason for this gap between energy practice and stochastic modeling is that large-scale energy models often present features - such as multiplied uncertain parameters in the objective and many uncertainties in the constraints - which make it difficult to develop generalized and tractable robust formulations.
In this paper, we address these limiting features to provide a complete robust optimization framework allowing the consideration of all uncertain parameters in energy models. We also introduce an original approach to make use of the obtained robust formulations for decision support and provide a case study of a national energy system for validation.

In this paper, a Mixed-Integer Linear Programming model is proposed for the design of the energy renovation of existing buildings, considering both Energy Supply Systems and the adoption of Energy Saving Measures to reduce the demand of buildings in retrofitting towards the nearly Zero Energy Building standard. The method is applied to an existing building located in Bilbao (northern Spain), getting the optimal design, i.e. lower annual net cost, for different limits of non-renewable primary energy consumption. The demand reduction produced by the Energy Saving Measures is included as an input from previously validated dynamic simulations and a simple method is presented for its specific distribution in reference days. This simple method, based on degree-days, allows reference days to be generated that, through an Energy Saving Measure based base temperature, consider the weather, the use and the thermal properties’ dependency on the distribution of the demand. The optimization method is used to provide the design selection and operation strategy of the renovation of buildings to meet different non-renewable primary energy consumption limits and to provide designs for different constraints: economic, space availability, etc.

Various countries and communities are defining or rethinking their energy strategy driven by concerns for climate change and security of energy supply. Energy models, often based on optimization, can support this decision-making process. In the current energy planning practice, most models are deterministic, i.e. they do not consider uncertainty and rely on long-term forecasts for important parameters. However, over the long time horizons of energy planning, forecasts often prove to be inaccurate, which can lead to overcapacity and underutilization of the installed technologies. Although this shows the need of considering uncertainty in energy planning, uncertainty is to date seldom integrated in energy models. The main barriers to a wider penetration of uncertainty are i) the complexity and computational expense of energy models; ii) the issue of quantifying input uncertainties and determining their nature; iii) the selection of appropriate methods for incorporating uncertainties in energy models. To overcome these limitations, this thesis answers the following research question
"How does uncertainty impact strategic energy planning and how can we facilitate the integration of uncertainty in the energy modeling practice?"
with four novel methodological contributions. First, a mixed-integer linear programming modeling framework for large-scale energy systems is presented. Given the energy demand, the efficiency and cost of energy conversion technologies, the availability and cost of resources, the model identifies the optimal investment and operation strategies to meet the demand and minimize the total annual cost or greenhouse gas emissions. The concise formulation and low computational time make it suitable for uncertainty applications. Second, a method is introduced to characterize input uncertainties in energy planning models. Third, the adoption of a two-stage global sensitivity analysis approach is proposed to deal with the large number of uncertain parameters in energy planning models. Fourth, a complete robust optimization framework is developed to incorporate uncertainty in optimization-based energy models, allowing to consider uncertainty both in the objective function and in the other constraints.
To evaluate the impact of uncertainty, the presentation of the methods is systematically associated to their validation on the real case study of the Swiss energy system. In this context, a novelty is represented by the consideration of all uncertain parameters in the analysis. The main finding is that uncertainty dramatically impacts energy planning decisions. The results reveal that uncertainty levels vary significantly for different parameters, and that the way in which uncertainty is characterized has a strong impact on the results. In the case study, economic parameters, such as the discount rate and the price of imported resources, are the most impacting inputs; also, parameters which are commonly considered as fixed assumptions in energy models emerge as critical factors, which shows that it is crucial to avoid an a priori exclusion of parameters from the analysis. The energy strategy drastically changes if uncertainty is considered. In particular, it is demonstrated that robust solutions, characterized by a higher penetration of renewables and of efficient technologies, can offer more reliability and stability compared to investment plans made without accounting for uncertainty, at the price of a marginally higher cost.

Old buildings, heterogeneity concerning building characteristics within districts and a high energy heating consumption characterize the building stock of many institutions. Hence, a high potential for reducing heating consumption is given by retrofitting buildings. In this context, it must be assumed that most of the institutions are not able to carry out a near-termed re-development of all their properties for financial as well as organizational reasons. This paper provides a method to deduce an optimized retrofit order of buildings by minimizing life-cycle costs, considering financial and organizational factors.

Various countries and communities are defining strategic energy plans driven by concerns for climate change and security of energy supply. Energy models can support this decision-making process. The long-term planning horizon requires uncertainty to be accounted for. To do this, the uncertainty of input parameters needs to be quantified. Classical approaches are based on the calculation of probability distributions for the inputs. In the context of strategic energy planning, this is often limited by the scarce quantity and quality of available data.
To overcome this limitation, we propose an application-driven method for uncertainty characterization, allowing the definition of ranges of variation for the uncertain parameters. To obtain a proof of concept, the method is applied to a representative mixed-integer linear programming national energy planning model in the context of a global sensitivity analysis (GSA) study. To deal with the large number of inputs, parameters are organized into different categories and uncertainty is characterized for one representative parameter per category. The obtained ranges serve as input to the GSA, which is performed in two stages to deal with the large problem size.
The application of the method generates uncertainty ranges for typical parameters in energy planning models. Uncertainty ranges vary significantly for different parameters, from [-2%, 2%] for electricity grid losses to [-47.3%, 89.9%] for the price of imported resources. The GSA results indicate that only few parameters are influential, that economic parameters (interest rates and price of imported resources) have the highest impact, and that it is crucial to avoid an arbitrary a priori exclusion of parameters from the analysis. Finally, we demonstrate that the obtained uncertainty characterization is relevant by comparing it with the assumption of equal levels of uncertainty for all input parameters, which results in a fundamentally different parameter ranking.

While the building sector has a significant thermodynamic improvement potential, exergy analysis has been shown to provide new insight for the optimisation of building energy systems. This paper presents an exergy-based multi-objective optimisation tool that aims to assess the impact of a diverse range of retrofit measures with a focus on non-domestic buildings. EnergyPlus was used as a dynamic calculation engine for first law analysis, while a Python add-on was developed to link dynamic exergy analysis and a Genetic Algorithm optimisation process with the aforementioned software. Two UK archetype case studies (an office and a primary school) were used to test the feasibility of the proposed framework. Different measures combinations based on retrofitting the envelope insulation levels and the application of different HVAC configurations were assessed. The objective functions in this study are annual energy use, occupants' thermal comfort, and total building exergy destructions. A large range of optimal solutions was achieved highlighting the framework capabilities. The model achieved improvements of 53% in annual energy use, 51% of exergy destructions and 66% of thermal comfort for the school building, and 50%, 33%, and 80% for the office building. This approach can be extended by using exergoeconomic optimisation.

We address two critical choices in Global Sensitivity Analysis (GSA): the choice of the sample size and of the threshold for the identification of insensitive input factors. Guidance to assist users with those two choices is still insufficient. We aim at filling this gap. Firstly, we define criteria to quantify the convergence of sensitivity indices, of ranking and of screening, based on a bootstrap approach. Secondly, we investigate the screening threshold with a quantitative validation procedure for screening results. We apply the proposed methodologies to three hydrological models with varying complexity utilizing three widely-used GSA methods (RSA, Morris, Sobol’). We demonstrate that convergence of screening and ranking can be reached before sensitivity estimates stabilize. Convergence dynamics appear to be case-dependent, which suggests that “fit-for-all” rules for sample sizes should not be used. Other modellers can easily adopt our criteria and procedures for a wide range of GSA methods and cases. Open access: http://www.sciencedirect.com/science/article/pii/S1364815216300251

The choice of sensitivity analysis methods for a model often relies on the behavior of model outputs. However, many building energy models are “black-box” functions whose behavior of simulated results is usually unknown or uncertain. This situation raises a question of how to correctly choose a sensitivity analysis method and its settings for building simulation. A performance comparison of nine sensitivity analysis methods has been carried out by means of computational experiments and building energy simulation. A comprehensive test procedure using three benchmark functions and two real-world building energy models was proposed. The degree of complexity was gradually increased by carefully-chosen test problems. Performance of these methods was compared through the ranking of variables’ importance, variables’ sensitivity indices, interaction among variables, and computational cost for each method. Test results show the consistency between the Fourier Amplitude Sensitivity Test (FAST) and the Sobol method. Some evidences found from the tests indicate that performance of other methods was unstable, especially with the non-monotonic test problems. © 2015, Tsinghua University Press and Springer-Verlag Berlin Heidelberg.

Uncertainty is often present in environmental and energy economics. Traditional approaches to optimization under uncertainty,
e.g., stochastic programming, chance-constrained programming or stochastic dynamic programming, encounter the most severe numerical difficulties because models in this area are large and complex, already in their deterministic
formulation. The goal of the present chapter is to introduce a relatively new field, known as robust optimization, as an alternative to traditional methods and formulations. Through an illustrative example, we suggest ways of putting robust
optimization at work in environmental and energy optimization models.

Optimal solutions of Linear Programming problems may become severely infeasible if the nominal data is slightly perturbed.
We demonstrate this phenomenon by studying 90 LPs from the well-known NETLIB collection. We then apply the Robust Optimization
methodology (Ben-Tal and Nemirovski [1–3]; El Ghaoui et al. [5, 6]) to produce “robust” solutions of the above LPs which are
in a sense immuned against uncertainty. Surprisingly, for the NETLIB problems these robust solutions nearly lose nothing in
optimality.

In 1991 Morris proposed an effective screening sensitivity measure to identify the few important factors in models with many factors. The method is based on computing for each input a number of incremental ratios, namely elementary effects, which are then averaged to assess the overall importance of the input. Despite its value, the method is still rarely used and instead local analyses varying one factor at a time around a baseline point are usually employed.In this piece of work we propose a revised version of the elementary effects method, improved in terms of both the definition of the measure and the sampling strategy. In the present form the method shares many of the positive qualities of the variance-based techniques, having the advantage of a lower computational cost, as demonstrated by the analytical examples.The method is employed to assess the sensitivity of a chemical reaction model for dimethylsulphide (DMS), a gas involved in climate change. Results of the sensitivity analysis open up the ground for model reconsideration: some model components may need a more thorough modelling effort while some others may need to be simplified.

A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. A concern with such an approach is that it might be too conservative. In this paper, we propose an approach that attempts to make this trade-off more attractive; that is, we investigate ways to decrease what we call the price of robustness. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations. An attractive aspect of our method is that the new robust formulation is also a linear optimization problem. Thus we naturally extend our methods to discrete optimization problems in a tractable way. We report numerical results for a portfolio optimization problem, a knapsack problem, and a problem from the Net Lib library. Subject classifications: Programming, stochastic: robust approach for solving LP/MIP with data uncertainties. Area of review: Financial Services. History: Received September 2001; revision received August 2002; accepted December 2002.

The sector of existing non-residential buildings accounts for a significant proportion of the final energy consumption of the entire German building stock. Thus improving these buildings can make a significant contribution to the declared emission reduction targets. Modernization measures are rarely realized in the life cycle of a building, but have long-term effects on its energetic performance. Therefore, it is essential to systematically identify optimal and future-oriented modernization measures and schedule the identified measures over the life cycle of a building. For this purpose, we present a mixed-integer linear program that identifies the optimal measures for the building energy system including envelope and supply system. Along with the optimal combination of modernization measures, the developed approach determines the respective optimal point of time when each measure shall be realized. A multi-objective optimization is conducted aiming of minimizing carbon emissions and equivalent annualized costs. We use a multi-zone building model that comprises different usage zones of a non-residential building. Therefore, hourly profiles of internal loads and user electricity caused by the presence and activity of people are modelled. Economic and ecological boundary conditions are specified separately for each prospective year of the schedule’s time horizon. Results show that the constellation of a building energy system underlies multiple changes over a modernization schedule. Determined Pareto efficient solutions reveal possibilities to save carbon emissions cost-efficiently. Furthermore, we detect dependencies between the modernization measures of different years.

Sustainable design of distributed energy supply systems involves multiple aims. Therefore, multi-objective optimization is the appropriate concept for sustainable design. However, input parameters are in general uncertain. If uncertainties are disregarded in the optimization, solutions usually become infeasible in practice. To incorporate uncertain parameters, we apply the concept of minmax robust multi-objective optimization for designing sustainable energy supply systems. We propose a mixed-integer linear problem formulation. The proposed formulation allows to identify robust sustainable designs easily guaranteeing security of energy supply. Energy systems are shown to typically exhibit objective-wise uncertainties. Thus, a Pareto front can still be derived.
In a real-world case study, robust designs are identified with a good trade-off between economic and ecologic criteria. The robust designs perform remarkably well in the nominal scenario. The presented problem formulation transfers the important theoretical concept of minmax robust multi-objective optimization into engineering practice for the design of sustainable energy systems.

The optimal design of buildings is a complex task involving energy systems as well as construction measures. Typically, in exact optimization models, only energy systems are considered, whereas envelope components are neglected. When considering both, heuristics are commonly used, which do not guarantee optimal or close to optimal results.
Thus, this paper presents the governing equations, validation and exemplary usage of a building model suitable for exact optimization problems. The developed model simultaneously considers energy systems and building envelopes. It is based on ISO 13790 and validated according to ASHRAE 140 and further compared to a more detailed model. The findings show that the developed model largely complies with the ASHRAE requirements and is able to assess buildings’ dynamic behavior regarding indoor air temperatures as well as hourly, peak load, and annual heating loads.
The simultaneous optimization of energy system and envelope is further demonstrated analyzing retrofitting options of a residential building. We consider solely installing additional PV units, modernizing the building envelope according to German regulations and an optimization without constraints regarding building envelope and energy system. The results indicate that installing additional PV units can moderately reduce total costs and CO2 emissions. The envelope modernization according to governmental regulations leads to largely increased costs at lower emissions, whereas the unconstrained optimization is able to simultaneously achieve significant cost and CO2 emission advantages.

Distributed energy systems (DES) are widely accepted as the future generation of the energy systems. The number of studies in all related fields corroborates the assertion that these systems are in their infancy and need to develop more in terms of efficiency and economizing. Admittedly, these systems are hardly lucrative and poor planning is one of many hurdles standing in the way of their profitability. Disregarding uncertainty as an innate characteristic of the real world seems one of the improper simplifications of this planning. To cover this gap, the paper is mainly focused on designing an energy system in a neighborhood including its pipeline network under demand uncertainty concerning data insufficiency. Therefore, a new model for planning in a neighborhood is presented and then reformulated to its robust counterpart. Various technologies like PV array, chillers, boiler, storage tank, and CHPs are considered in order to meet the cooling, heating and electrical demands. The probable consequences of the demand uncertainty are studied to the length. The outcomes reveal that the unit sizes and pipeline network are highly dependent on the decision maker's level of conservatism.

A distributed energy system (DES) is recognized as a local energy system. Over the past decade, number of studies in technological fields, economizing and its regulatory aspects corroborated that an increasing attention has been paid toward the planning of these systems. However, lack of a proper planning because of simplifications in the assumptions may overshadow the potentials of the DESs hitherto. Disregarding some intrinsic characteristics of a real world like uncertainties and noises may distort results and consequently undermine its efficiency. This paper mainly focused on designing a building's energy system under demand, costs (like carbon emission cost, primary energy saving…) and prices (like fuel tariff and electricity prices…) uncertainties concerning data insufficiency by means of robust optimization. Various sustainable technologies such as photovoltaic arrays were also considered as an alternative. In order to study the probable consequences, the proposed robust model was applied to a real-world problem and the outcomes were presented, analyzed and compared to the deterministic model's results.

We consider a robust shortest path problem when the cost coefficient is the product of two uncertain factors. We first show that the robust problem can be solved in polynomial time by a dual-variable enumeration with shortest path problems as subproblems. We also propose a path enumeration approach using a K -shortest paths finding algorithm that may be efficient in many real cases. An application in hazardous materials transportation is discussed, and the solution methods are illustrated by numerical examples. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013

Models and Sensitivity AnalysisMethods and Settings for Sensitivity Analysis – an IntroductionNonindependent Input FactorsPossible Pitfalls for a Sensitivity AnalysisConcluding RemarksExercisesAnswersAdditional ExercisesSolutions to Additional Exercises

A computational model is a representation of some physical or other system of interest, first expressed mathematically and then implemented in the form of a computer program; it may be viewed as a function of inputs that, when evaluated, produces outputs. Motivation for this article comes from computational models that are deterministic, complicated enough to make classical mathematical analysis impractical and that have a moderate-to-large number of inputs. The problem of designing computational experiments to determine which inputs have important effects on an output is considered. The proposed experimental plans are composed of individually randomized one-factor-at-a-time designs, and data analysis is based on the resulting random sample of observed elementary effects, those changes in an output due solely to changes in a particular input. Advantages of this approach include a lack of reliance on assumptions of relative sparsity of important inputs, monotonicity of outputs with respect to inputs, or adequacy of a low-order polynomial as an approximation to the computational model.

This note formulates a convex mathematical programming problem in which the usual definition of the feasible region is replaced by a significantly different strategy. Instead of specifying the feasible region by a set of convex inequalities, fi(x) ≦ bi, i = 1, 2, …, m, the feasible region is defined via set containment. Here n convex activity sets {Kj, j = 1, 2, …, n} and a convex resource set K are specified and the feasible region is given by $X =\{x \in R^{n}\mid x_{1}K_{1} + x_{2}K_{2} + \cdots + x_{n}K_{n} \subseteq K, x_{j}\geq 0\}$ where the binary operation + refers to addition of sets. The problem is then to find x̄ ∈ X that maximizes the linear function c · x. When the res...

This paper examines different levels of analytical sophistication in the treatment of uncertainties in risk analysis, and the possibility of transfer of experience across fields of application. First, this paper describes deterministic and probabilistic methods of treatment of risk and uncertainties, and the different viewpoints that shape these analyses. Second, six different levels of treatment of uncertainty are presented and discussed in the light of the evolution of the risk management philosophy in the US. Because an in-depth treatment of uncertainties can be complex and costly, this paper then discusses when and why a full (two-tier) uncertainty analysis is justified. In the treatment of epistemic uncertainty, an unavoidable and difficult problem is the encoding of probability distributions based on scientific evidence and expert judgments. The last sections include a description of different approaches to the aggregation of expert opinions and their use in risk analysis, and a recent example of methodology and application (in seismic hazard analysis) that can be transferred to other domains.

Europe's buildings under the microscope

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Stock and typology of heated non-residential buildings in West Germany

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