A comparison of deterministic, reliability-based and risk-based structural optimization under uncertainty

In this paper, the effects of uncertainty and expected costs of failure on optimum structural design are investigated, by comparing three distinct formulations of structural optimization problems. Deterministic Design Optimization (DDO) allows one the find the shape or configuration of a structure that is optimum in terms of mechanics, but the formulation grossly neglects parameter uncertainty and its effects on structural safety. Reliability-based Design Optimization (RBDO) has emerged as an alternative to properly model the safety-under-uncertainty part of the problem. With RBDO, one can ensure that a minimum (and measurable) level of safety is achieved by the optimum structure. However, results are dependent on the failure probabilities used as constraints in the analysis. Risk optimization (RO) increases the scope of the problem by addressing the compromising goals of economy and safety. This is accomplished by quantifying the monetary consequences of failure, as well as the costs associated with construction, operation and maintenance. RO yields the optimum topology and the optimum point of balance between economy and safety. Results are compared for some example problems. The broader RO solution is found first, and optimum results are used as constraints in DDO and RBDO. Results show that even when optimum safety coefficients are used as constraints in DDO, the formulation leads to configurations which respect these design constraints, reduce manufacturing costs but increase total expected costs (including expected costs of failure). When (optimum) system failure probability is used as a constraint in RBDO, this solution also reduces manufacturing costs but by increasing total expected costs. This happens when the costs associated with different failure modes are distinct. Hence, a general equivalence between the formulations cannot be established. Optimum structural design considering expected costs of failure cannot be controlled solely by safety factors nor by failure probability constraints, but will depend on actual structural configuration. (c) 2011 Elsevier Ltd. All rights reserved.

Robust scenario formulations for strategic supply chain optimization under uncertainty

Strategic supply chain optimization (SCO) problems are often modelled as a two-stage optimization problem, in which the first-stage variables represent decisions on the development of the supply chain and the second-stage variables represent decisions on the operations of the supply chain. When uncertainty is explicitly considered, the problem becomes an intractable infinite-dimensional optimization problem, which is usually solved approximately via a scenario or a robust approach. This paper proposes a novel synergy of the scenario and robust approaches for strategic SCO under uncertainty. Two formulations are developed, namely, naïve robust scenario formulation and affinely adjustable robust scenario formulation. It is shown that both formulations can be reformulated into tractable deterministic optimization problems if the uncertainty is bounded with the infinity-norm, and the uncertain equality constraints can be reformulated into deterministic constraints without assumption of the uncertainty region. Case studies of a classical farm planning problem and an energy and bioproduct SCO problem demonstrate the advantages of the proposed formulations over the classical scenario formulation. The proposed formulations not only can generate solutions with guaranteed feasibility or indicate infeasibility of a problem, but also can achieve optimal expected economic performance with smaller numbers of scenarios.

Risk management for mathematical optimization under uncertainty

We present a general multistage stochastic mixed 0-1 problem where the uncertainty appears everywhere in the objective function, constraints matrix and right-hand-side. The uncertainty is represented by a scenario tree that can be a symmetric or a nonsymmetric one. The stochastic model is converted in a mixed 0-1 Deterministic Equivalent Model in compact representation. Due to the difficulty of the problem, the solution offered by the stochastic model has been traditionally obtained by optimizing the objective function expected value (i.e., mean) over the scenarios, usually, along a time horizon. This approach (so named risk neutral) has the inconvenience of providing a solution that ignores the variance of the objective value of the scenarios and, so, the occurrence of scenarios with an objective value below the expected one. Alternatively, we present several approaches for risk averse management, namely, a scenario immunization strategy, the optimization of the well known Value-at-Risk (VaR) and several variants of the Conditional Value-at-Risk strategies, the optimization of the expected mean minus the weighted probability of having a "bad" scenario to occur for the given solution provided by the model, the optimization of the objective function expected value subject to stochastic dominance constraints (SDC) for a set of profiles given by the pairs of threshold objective values and either bounds on the probability of not reaching the thresholds or the expected shortfall over them, and the optimization of a mixture of the VaR and SDC strategies.

A risk-averse optimization model for trading wind energy in a market environment under uncertainty

In this paper, a stochastic programming approach is proposed for trading wind energy in a market environment under uncertainty. Uncertainty in the energy market prices is the main cause of high volatility of profits achieved by power producers. The volatile and intermittent nature of wind energy represents another source of uncertainty. Hence, each uncertain parameter is modeled by scenarios, where each scenario represents a plausible realization of the uncertain parameters with an associated occurrence probability. Also, an appropriate risk measurement is considered. The proposed approach is applied on a realistic case study, based on a wind farm in Portugal. Finally, conclusions are duly drawn. (C) 2011 Elsevier Ltd. All rights reserved.

Optimization methods for the operating room management under uncertainty: stochastic programming vs. decomposition approach

The operating theatres are the engine of the hospitals; proper management of the operating rooms and its staff represents a great challenge for managers and its results impact directly in the budget of the hospital. This work presents a MILP model for the efficient schedule of multiple surgeries in Operating Rooms (ORs) during a working day. This model considers multiple surgeons and ORs and different types of surgeries. Stochastic strategies are also implemented for taking into account the uncertain in surgery durations (pre-incision, incision, post-incision times). In addition, a heuristic-based methods and a MILP decomposition approach is proposed for solving large-scale ORs scheduling problems in computational efficient way. All these computer-aided strategies has been implemented in AIMMS, as an advanced modeling and optimization software, developing a user friendly solution tool for the operating room management under uncertainty.

Multi-objective optimization of environmentally conscious chemical supply chains under demand uncertainty

In this work, we analyze the effect of demand uncertainty on the multi-objective optimization of chemical supply chains (SC) considering simultaneously their economic and environmental performance. To this end, we present a stochastic multi-scenario mixed-integer linear program (MILP) with the unique feature of incorporating explicitly the demand uncertainty using scenarios with given probability of occurrence. The environmental performance is quantified following life cycle assessment (LCA) principles, which are represented in the model formulation through standard algebraic equations. The capabilities of our approach are illustrated through a case study. We show that the stochastic solution improves the economic performance of the SC in comparison with the deterministic one at any level of the environmental impact.

Efficient Methods for Robust Classification Under Uncertainty in Kernel Matrices

In this paper we study the problem of designing SVM classifiers when the kernel matrix, K, is affected by uncertainty. Specifically K is modeled as a positive affine combination of given positive semi definite kernels, with the coefficients ranging in a norm-bounded uncertainty set. We treat the problem using the Robust Optimization methodology. This reduces the uncertain SVM problem into a deterministic conic quadratic problem which can be solved in principle by a polynomial time Interior Point (IP) algorithm. However, for large-scale classification problems, IP methods become intractable and one has to resort to first-order gradient type methods. The strategy we use here is to reformulate the robust counterpart of the uncertain SVM problem as a saddle point problem and employ a special gradient scheme which works directly on the convex-concave saddle function. The algorithm is a simplified version of a general scheme due to Juditski and Nemirovski (2011). It achieves an O(1/T-2) reduction of the initial error after T iterations. A comprehensive empirical study on both synthetic data and real-world protein structure data sets show that the proposed formulations achieve the desired robustness, and the saddle point based algorithm outperforms the IP method significantly.

Valuing Expansions of the Electricity Transmission Network under Uncertainty: The Binodal Case

Transmission investments are currently needed to meet an increasing electricity demand, to address security of supply concerns, and to reach carbon-emissions targets. A key issue when assessing the benefits from an expanded grid concerns the valuation of the uncertain cash flows that result from the expansion. We propose a valuation model that accommodates both physical and economic uncertainties following the Real Options approach. It combines optimization techniques with Monte Carlo simulation. We illustrate the use of our model in a simplified, two-node grid and assess the decision whether to invest or not in a particular upgrade. The generation mix includes coal-and natural gas-fired stations that operate under carbon constraints. The underlying parameters are estimated from observed market data.

A soft robust model for optimization under ambiguity

In this paper, we propose a framework for robust optimization that relaxes the standard notion of robustness by allowing the decision maker to vary the protection level in a smooth way across the uncertainty set. We apply our approach to the problem of maximizing the expected value of a payoff function when the underlying distribution is ambiguous and therefore robustness is relevant. Our primary objective is to develop this framework and relate it to the standard notion of robustness, which deals with only a single guarantee across one uncertainty set. First, we show that our approach connects closely to the theory of convex risk measures. We show that the complexity of this approach is equivalent to that of solving a small number of standard robust problems. We then investigate the conservatism benefits and downside probability guarantees implied by this approach and compare to the standard robust approach. Finally, we illustrate theme thodology on an asset allocation example consisting of historical market data over a 25-year investment horizon and find in every case we explore that relaxing standard robustness with soft robustness yields a seemingly favorable risk-return trade-off: each case results in a higher out-of-sample expected return for a relatively minor degradation of out-of-sample downside performance. © 2010 INFORMS.

Revealed Preference and Choice under Uncertainty

We employ the theory of rational choice to examine whether observable choices from feasible sets of prospects can be generated by the optimization of some underlying decision criterion under uncertainty. Rather than focusing on a specific theory of choice, our objective is to formulate a general approach that is designed to cover the various decision criteria that have been proposed in the literature. We use a mild dominance property to define a class of suitable choice criteria. In addition to rationalizability per se, we characterize transitive and Suzumura consistent rationalizability in the presence of dominance.

Robust linear semi-infinite programming duality under uncertainty

In this paper, we propose a duality theory for semi-infinite linear programming problems under uncertainty in the constraint functions, the objective function, or both, within the framework of robust optimization. We present robust duality by establishing strong duality between the robust counterpart of an uncertain semi-infinite linear program and the optimistic counterpart of its uncertain Lagrangian dual. We show that robust duality holds whenever a robust moment cone is closed and convex. We then establish that the closed-convex robust moment cone condition in the case of constraint-wise uncertainty is in fact necessary and sufficient for robust duality. In other words, the robust moment cone is closed and convex if and only if robust duality holds for every linear objective function of the program. In the case of uncertain problems with affinely parameterized data uncertainty, we establish that robust duality is easily satisfied under a Slater type constraint qualification. Consequently, we derive robust forms of the Farkas lemma for systems of uncertain semi-infinite linear inequalities.

Dynamic Models under Uncertainty for Large-Scale Enterprise Systems

Thesis (Ph.D.)--University of Washington, 2016-06

Parallel nonconvex generalized Benders decomposition for natural gas production network planning under uncertainty

A scenario-based two-stage stochastic programming model for gas production network planning under uncertainty is usually a large-scale nonconvex mixed-integer nonlinear programme (MINLP), which can be efficiently solved to global optimality with nonconvex generalized Benders decomposition (NGBD). This paper is concerned with the parallelization of NGBD to exploit multiple available computing resources. Three parallelization strategies are proposed, namely, naive scenario parallelization, adaptive scenario parallelization, and adaptive scenario and bounding parallelization. Case study of two industrial natural gas production network planning problems shows that, while the NGBD without parallelization is already faster than a state-of-the-art global optimization solver by an order of magnitude, the parallelization can improve the efficiency by several times on computers with multicore processors. The adaptive scenario and bounding parallelization achieves the best overall performance among the three proposed parallelization strategies.

Bayesian methodology incorporating expert judgment for ranking countermeasure effectiveness under uncertainty: Example applied to at grade railroad crossings in Korea

This paper describes the formalization and application of a methodology to evaluate the safety benefit of countermeasures in the face of uncertainty. To illustrate the methodology, 18 countermeasures for improving safety of at grade railroad crossings (AGRXs) in the Republic of Korea are considered. Akin to “stated preference” methods in travel survey research, the methodology applies random selection and laws of large numbers to derive accident modification factor (AMF) densities from expert opinions. In a full Bayesian analysis framework, the collective opinions in the form of AMF densities (data likelihood) are combined with prior knowledge (AMF density priors) for the 18 countermeasures to obtain ‘best’ estimates of AMFs (AMF posterior credible intervals). The countermeasures are then compared and recommended based on the largest safety returns with minimum risk (uncertainty). To the author's knowledge the complete methodology is new and has not previously been applied or reported in the literature. The results demonstrate that the methodology is able to discern anticipated safety benefit differences across candidate countermeasures. For the 18 at grade railroad crossings considered in this analysis, it was found that the top three performing countermeasures for reducing crashes are in-vehicle warning systems, obstacle detection systems, and constant warning time systems.

An approach to access control under uncertainty

In dynamic and uncertain environments such as healthcare, where the needs of security and information availability are difficult to balance, an access control approach based on a static policy will be suboptimal regardless of how comprehensive it is. The uncertainty stems from the unpredictability of users’ operational needs as well as their private incentives to misuse permissions. In Role Based Access Control (RBAC), a user’s legitimate access request may be denied because its need has not been anticipated by the security administrator. Alternatively, even when the policy is correctly specified an authorised user may accidentally or intentionally misuse the granted permission. This paper introduces a novel approach to access control under uncertainty and presents it in the context of RBAC. By taking insights from the field of economics, in particular the insurance literature, we propose a formal model where the value of resources are explicitly defined and an RBAC policy (entailing those predictable access needs) is only used as a reference point to determine the price each user has to pay for access, as opposed to representing hard and fast rules that are always rigidly applied.