Dealing with uncertainty in constrained optimisation using decision theory

We consider constrained optimisation problems with a real-valued, bounded objective function on an arbitrary space. The constraints are expressed as a relation between the optimisation variable and the problem parameters. There is uncertainty about these problem parameters, which turns the optimisation problem into an ill-specified problem

Optimisation under uncertainty applied to a bridge collision problem

We consider the problem of modelling the load on a bridge pillar when hit by a vehicle. This load depends on a number of uncertain variables, such as the mass of the vehicle and its speed on impact. The objective of our study is to analyse their effect on the load. More specifically, we are interested in finding the minimum distance of the pillar to the side of the road passing under the bridge such that a given constraint on the load is satisfied in 99% of impact cases, i.e., such that the probability of satisfying the constraint is 0.99. In addition, we look for solutions to the following optimisation problem: find the distance that minimises a given cost function while still satisfying a given constraint on the load. This optimisation problem under uncertain constraints is not a well-posed problem, so we turn it into a decision problem under uncertainty. For both problems, we consider two typical cases. In the first, so-called precise-probability case, all uncertain variables involved are modelled using probability distributions, and in the second, so-called imprecise-probability case, the uncertainty for at least some of the variables (in casu the mass) is modelled by an interval of possible values, which is a special imprecise-probabilistic model. In the first case, we compute the joint distribution using simple Monte Carlo simulation, and in the second case, we combine Monte Carlo simulation with newly developed techniques in the field of imprecise probabilities. For the optimisation problem with uncertain constraints, this leads to two distinct approaches with different optimality criteria, namely maximality and maximinity, which we discuss and compare

Reliability analysis in vehicle collision with bridge pier

Much work has been done in bridge design specification via a set of structural design standards called Eurocodes to cover the design of all types of structures. We analyse the accidental force on a bridge pier when it is hit by vehicles in order to assess the reliability of a bridge. The force that comes from a vehicle—called vehicle impact force—is not deterministic and it depends on some uncertain parameters, such as the mass of the vehicle and its speed on impact. All the data and uncertainty models for the parameters are given by Eurocode 1. In this paper we analyse the force that is affected by these parameters. For doing that we consider two kinds of problems where in the both problems this force is a function on a distance—the distance between the bridge pier and the side of a road passing under the bridge. One of the problems proposes a design force as a function of the distance—called reliable distance—using a strength condition, the condition on the design forces and the other one suggests a tool for obtaining an economical optimum distance—called cost-optimal distance—by taking into account the optimum economical costs—the cost of bulding and repairing the bridge and human life. In both problems, we consider the safety of the distance where affects dynamic and static design forces and the impact force of vehicle which is not a constant. We show how reliable are Eurocodes by comparing these two distances calculated in two different problems. In other words, through these two problems/criteria we show the danger of using the data represented via Eurocodes for the parameters. In addition, we found linear functions on the distance and the (dynamic and static) design forces of the bridge

Day-ahead energy and reserve dispatch problem under non-probabilistic uncertainty

The current energy transition and the underlying growth in variable and uncertain renewable-based energy generation challenge the proper operation of power systems. Classical probabilistic uncertainty models, e.g., stochastic programming or robust optimisation, have been used widely to solve problems such as the day-ahead energy and reserve dispatch problem to enhance the day-ahead decisions with a probabilistic insight of renewable energy generation in real-time. By doing so, the scheduling of the power system becomes, production and consumption of electric power, more reliable (i.e., more robust because of potential deviations) while minimising the social costs given potential balancing actions. Nevertheless, these classical models are not valid when the uncertainty is imprecise, meaning that the system operator may not rely on a unique distribution function to describe the uncertainty. Given the Distributionally Robust Optimisation method, our approach can be implemented for any non-probabilistic, e.g., interval models rather than only sets of distribution functions (ambiguity set of probability distributions). In this paper, the aim is to apply two advanced non-probabilistic uncertainty models: Interval and ϵ-contamination, where the imprecision and in-determinism in the uncertainty (uncertain parameters) are considered. We propose two kinds of theoretical solutions under two decision criteria—Maximinity and Maximality. For an illustration of our solutions, we apply our proposed approach to a case study inspired by the 24-node IEEE reliability test system

Implementation of maximin and maximal solutions for linear programming problems under uncertainty

We present a software implementation of the methods for solving linear programming problems under uncertainty from previous work. Uncertainties about constraint parameters can be expressed as intervals or trapezoidal possibility distributions. The software computes the solutions for the optimality criteria maximin and maximality. For maximality with possibility distributions, only an approximate solution is obtained