We consider the problem of optimally maintaining an offshore wind farm in which major components progressively degrade over time due to normal usage and exposure to a randomly varying environment. The turbines exhibit both economic and stochastic dependence due to shared maintenance setup costs and their common environment. Our aim is to identify optimal replacement policies that minimize the expected total discounted setup, replacement, and lost power production costs over an infinite horizon. The problem is formulated using a Markov decision process (MDP) model from which we establish monotonicity of the cost function jointly in the degradation level and environment state and characterize the structure of the optimal replacement policy. For the special case of a two-turbine farm, we prove that the replacement threshold of one turbine depends not only on its own state of degradation but also on the state of degradation of the other turbine in the farm. This result yields a complete characterization of the replacement policy of both turbines by a monotone curve. The policies characterized herein can be used to optimally prescribe timely replacements of major components and suggest when it is most beneficial to share costly maintenance resources.

The use of topology optimization in the design of fluid dynamics systems is still in its infancy. With the decreasing cost of additive manufacture, the application of topology optimization in the design of structural components has begun to increase. This paper provides a method for using topology optimization to reduce the power dissipation of fluid dynamics systems, with the novelty of it being the first application of stochastic mechanisms in the design of 3D fluid–solid geometrical interfaces. The optimization algorithm uses the continuous adjoint method for sensitivity analysis and is optimized against an objective function for fluid power dissipation. The paper details the methodology behind a vanilla gradient descent approach before introducing stochastic behavior through a minibatch-based system. Both algorithms are then applied to a novel case study for an internal combustion engine's piston cooling gallery before the performance of each algorithm's resulting geometry is analyzed and compared. The vanilla gradient descent algorithm achieves an 8.9% improvement in pressure loss through the case study, and this is surpassed by the stochastic descent algorithm which achieved a 9.9% improvement, however this improvement came with a large time cost. Both approaches produced similarly unintuitive geometry solutions to successfully improve the performance of the cooling gallery.

This work is devoted to a vast extension of Sanov’s theorem, in Laplace principle form, based on alternatives to the classical convex dual pair of relative entropy and cumulant generating functional. The abstract results give rise to a number of probabilistic limit theorems and asymptotics. For instance, widely applicable non-exponential large deviation upper bounds are derived for empirical distributions and averages of independent and identically distributed samples under minimal integrability assumptions, notably accommodating heavy-tailed distributions. Other interesting manifestations of the abstract results include new results on the rate of convergence of empirical measures in Wasserstein distance, uniform large deviation bounds, and variational problems involving optimal transport costs, as well as an application to error estimates for approximate solutions of stochastic optimization problems. The proofs build on the Dupuis–Ellis weak convergence approach to large deviations as well as the duality theory for convex risk measures.

We introduce a new parameter to discuss the behavior of a genetic algorithm. This parameter is the mean number of exact copies of the best-fit chromosomes from one generation to the next. We believe that the genetic algorithm operates best when this parameter is slightly larger than 1 and we prove two results supporting this belief. We consider the case of the simple genetic algorithm with the roulette wheel selection mechanism. We denote by ℓ the length of the chromosomes, m the population size, pC the crossover probability, and pM the mutation probability. Our results suggest that the mutation and crossover probabilities should be tuned so that, at each generation, the maximal fitness multiplied by (1 - pC)(1 - pM)ℓ is greater than the mean fitness.

This paper focuses on the problem of robust time-optimal trajectory planning of robotic manipulators to track a given path under a probabilistic limited actuation, that is, the probability for the actuation to be limited is no less than a given bound κ. We give a general and practical method to reduce the probabilistic constraints to a set of deterministic constraints and show that the deterministic constraints are equivalent to a set of linear constraints under certain conditions. As a result, the original problem is reduced to a linear optimal control problem which can be solved with linear programming in polynomial time. In the case of κ = 1, the original problem is proved to be equivalent to the linear optimal control problem. Overall, a very general, practical, and efficient algorithm is given to solve the above problem and numerical simulation results are used to show the effectiveness of the method.

As a new business form, the buy-online and pick-up-in-store (BOPS) mode allows consumers to pay for goods online and pick them up in a physical store. In this paper, an equilibrium model is constructed to formulate an optimal decision-making problem for online and offline retailers under the BOPS mode, where the online retailer determines the retail price of the goods and the consignment quantity in a physical store, while the offline retailer chooses the revenue share of profit by a consignment contract. Different to the existing models, the cost of overstocking and loss of understocking are incorporated into the profit function of the online retailer due to the randomness of demand. For the objective function of the offline retailer, the cross-sale quantity generated by the BOPS mode is taken into account. Then the game between the online and offline retailers is expressed as a stochastic Nash equilibrium model. Based on the analytic properties of the model, necessary conditions for the equilibrium solution are obtained. A case study and sensitivity analysis are employed to reveal the managerial implications of the model, which can provide a number of valuable suggestions on optimizing the strategies for the online and offline retailers under the BOPS mode.

A risk-averse irrigated corn producer would be better off choosing the more expensive subsurface drip irrigation (SDI) over center-pivot sprinkler (CPS), given limited aquifer life and swine effluent and urea fertilization. A stochastic optimization using EPIC data maximized expected utility of 100 years' worth of net revenues for a quarter section. Phosphorus accumulation was more likely with the CPS than with the SDI but soil nitrogen was constant under both systems. SDI conserves more water than CPS per acre but depletes the aquifer faster because a greater area is irrigated. These results were invariant in the sensitivity analysis.

In many markets, especially in energy markets, electricity markets for instance, the detention of the physical asset is quite difficult. This is also the case for crude oil as treated by Davis (2000). So one can identify a good proxy which is an asset (financial or physical) (one)whose the spot price is significantly correlated with the spot price of the underlying (e.g. electicity or crude oil). Generally, the market could become incomplete. We explicit exact hedging strategies for exponential utilities when the risk premium is bounded. Our result is based upon backward stochastic differential equation (BSDE) and a good choice of admissible strategies which allows us to solve our hedging problem.

In this paper we propose a new genetic algorithm specifically based on mutation and selection in order to maximize a fitness function. This mutation-selection algorithm behaves as a gradient algorithm which converges to local maxima. In order to obtain convergence to global maxima we propose a new algorithm which is built by randomly perturbing the selection operator of the gradient-like algorithm. The perturbation is controlled by only one parameter: that which allows the selection pressure to be governed. We use the Markov model of the perturbed algorithm to prove its convergence to global maxima. The arguments used in the proofs are based on Freidlin and Wentzell's (1984) theory and large deviation techniques also applied in simulated annealing. Our main results are that (i) when the population size is greater than a critical value, the control of the selection pressure ensures the convergence to the global maxima of the fitness function, and (ii) the convergence also occurs when the population is the smallest possible, i.e. 1.

We study an optimal investment problem in a continuous-time framework where the interest rates follow Cox-Ingersoll-Ross dynamics. Closed form formulae for the optimal investment strategy are obtained by assuming the completeness of financial markets and the CRRA utility function. In particular, we study the behaviour of the solution when time approaches the terminal date.

We present a method for using rational expectations in a linear-quadratic optimization framework. Following the approach put forward by Sims, we solve the model through a QZ decomposition, which is generally easier to implement than the more widely used Blanchard-Kahn method.

We study a markovian evolutionary process which encompasses the classical simple genetic algorithm. This process is obtained by randomly perturbing a very simple selection scheme. Using the Freidlin-Wentzell theory, we carry out a precise study of the asymptotic dynamics of the process as the perturbations disappear. We show how a delicate interaction between the perturbations and the selection pressure may force the convergence toward the global maxima of the fitness function. We put forward the existence of a critical population size, above which this kind of convergence can be achieved. We compute upper bounds of this critical population size for several examples. We derive several conditions to ensure convergence in the homogeneous case and these provide the first mathematically well-founded convergence results for genetic algorithms.

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons:

(i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means.

(ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means.

We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR.

Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

We consider the problem of scheduling n jobs non-preemptively on m parallel, non-identical processors to minimize a weighted expected cost function of job completion times, where the weights are associated with the jobs. The cost function is assumed to be increasing and concave but otherwise arbitrary. Processing times are IFR with different distributions for different processors. Jobs may be processed on any processor and there are no precedences. We show that the optimal policy orders the jobs in decreasing order of their weights and then uses the individually optimal policy for each job. In other words, processors are offered to jobs in order, and each job considers its own expected cost function for its completion time to decide whether to accept or reject a processor. Therefore, the optimal policy does not depend on the weights of the jobs except through their order. Special cases of our objective function are weighted expected flowtime, weighted discounted expected flowtime, and weighted expected number of tardy jobs.