A scenario generation-based lower bounding approach for stochastic scheduling problems

Sampling-Based Approximation Schemes for Capacitated Stochastic Inventory Control Models

Various stochastic programming problems can be formulated as problems of optimization of an expected value function. Quite often the corresponding expectation function cannot be computed exactly and should be approximated, say by Monte Carlo sampling methods. In fact, in many practical applications, Monte Carlo simulation is the only reasonable way of estimating the expectation function. We discuss converges properties of the sample average approximation (SAA) approach to stochastic programming. We argue that the SAA method is easily implementable and can be surprisingly efficient for some classes of stochastic programming problems.

The sample average approximation (SAA) approach is a widely used technique, based on Monte‐Carlo simulation, often applied to large‐scale stochastic optimization problems. In this approach, a set of sample average problems with multiple copies of sampled scenarios are generated and solved exactly. In other words, there is an implicit assumption that the sample average problems are solvable to optimality. In some instances, however, the sample average problems might be NP‐hard problems, often difficult or impractical to solve to optimality. In this article, we broaden the scope of the SAA approach and show that even without solving the sample problems to optimality, by combining a heuristic and a lower bounding approach, high‐quality solutions with tight confidence bounds on the optimal solution value can be obtained. We demonstrate this “inexact SAA approach” on two problems. First, we apply it to the Stochastic Connected Facility Location (SConFL) problem, the motivating application for this article, that arises in the design of telecommunications networks. As an additional application, we also use it for the Stochastic Uncapacitated Facility Location (SUFL) problem. Our computational results demonstrate the effectiveness of the inexact SAA approach. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(1), 19–33 2017

We provide a refined convergence analysis for the SAA (sample average approximation) method applied to stochastic optimization problems with either single or mixed CVaR (conditional value-at-risk) measures. Under certain regularity conditions, it is shown that any accumulation point of the weak GKKT (generalized Karush-Kuhn-Tucker) points produced by the SAA method is almost surely a weak stationary point of the original CVaR or mixed CVaR optimization problems. In addition, it is shown that, as the sample size increases, the difference of the optimal values between the SAA problems and the original problem tends to zero with probability approaching one exponentially fast.

Sample average approximation (SAA) method has recently been applied to solve stochastic programs with second order stochastic dominance (SSD) constraints. In particular, Hu et al. (Math Program 133:171–201, 2012) presented a detailed convergence analysis of \(\epsilon \)-optimal values and \(\epsilon \)-optimal solutions of sample average approximated stochastic programs with polyhedral SSD constraints. In this paper, we complement the existing research by presenting convergence analysis of stationary points when SAA is applied to a class of stochastic minimization problems with SSD constraints. Specifically, under some moderate conditions we prove that optimal solutions and stationary points obtained from solving sample average approximated problems converge with probability one to their true counterparts. Moreover, by exploiting some recent results on large deviation of random functions and sensitivity analysis of generalized equations, we derive exponential rate of convergence of stationary points.

This paper aims to optimise healthcare resource pre-positioning, patient scheduling, and patient transferring under uncertain demands and stochastic resource consumption. We propose a two-stage stochastic programming model that formulates the patient scheduling problem as a Markov decision process. To address complexities and uncertainties, we use Artificial Intelligence (AI) techniques to improve both model formulation and algorithmic performance. To tackle the limited data challenge, we introduce a Wasserstein distance-based ambiguity set and propose a two-stage distributionally robust optimisation (DRO) model, which derives a deterministic equivalent using the Lagrangian dual of non-anticipativity constraints. The solution process is accelerated with a scenario decomposition approach and the K-means clustering method. Both theoretical and numerical results demonstrate the consistency of the two-stage DRO model with the sample average approximation (SAA) method. The potential of AI to improve the model's performance is evident through the significant reduction in computation time achieved with the K-means clustering approach without compromising solution quality. Compared to the SAA method, the DRO model exhibits a considerable reduction in the waiting penalty cost for out-of-sample cases, ranging from 43.06 % to 81.23 % . Numerical results show that the proposed algorithm outperforms SAA methods and several benchmark policies in terms of computational efficiency and solutions.

This paper applies sample average approximation (SAA) method based on ‐space decomposition theory to solve stochastic convex minimax problems. Under some moderate conditions, the SAA solution converges to its true counterpart with probability approaching one and convergence is exponentially fast with the increase of sample size. Based on the ‐theory, a superlinear convergent ‐algorithm frame is designed to solve the SAA problem.

To reflect uncertain data in practical problems, stochastic versions of the mathematical program with complementarity constraints (MPCC) have drawn much attention in the recent literature. Our concern is the detailed analysis of convergence properties of a regularization sample average approximation (SAA) method for solving a stochastic mathematical program with complementarity constraints (SMPCC). The analysis of this regularization method is carried out in three steps: First, the almost sure convergence of optimal solutions of the regularized SAA problem to that of the true problem is established by the notion of epiconvergence in variational analysis. Second, under MPCC-MFCQ, which is weaker than MPCC-LICQ, we show that any accumulation point of Karash-Kuhn-Tucker points of the regularized SAA problem is almost surely a kind of stationary point of SMPCC as the sample size tends to infinity. Finally, some numerical results are reported to show the efficiency of the method proposed.

A scenario-based planning for the pickup and delivery problem with time windows, scheduled lines and stochastic demands

A note on the sample average approximation method for stochastic mathematical programs with complementarity constraints

In this paper, we combine two approaches to handling uncertainty: we use techniques for finding optimal solutions in the expected sense to solve combinatorial optimization problems in an online setting.

Motivated by the real-world problem of a logistics company, this paper proposes a novel distribution-free prescriptive analytics approach—termed kernelized empirical risk minimization (kernelized ERM)—to solve a complex two-stage capacity planning problem with multivariate demand and vector-valued capacity decisions and compares this approach both theoretically and numerically with an extension of the well-known sample average approximation (SAA) approach termed weighted SAA. Both approaches use integrated machine learning algorithms to prescribe capacities directly from historical demand and numerous features (covariates) without having to make assumptions about the underlying multivariate demand distribution. We provide extensive analytical insights into both approaches. Most important, we prove the universal approximation property for the kernelized ERM approach when using a universal (data-independent) kernel and show how out-of-sample guarantees can be derived for various kernels. We demonstrate the applicability of both approaches to a real-world planning problem and evaluate their performance relative to traditional parametric approaches that first estimate a multivariate demand distribution and then solve a stochastic optimization problem and a nonparametric approach (SAA). Our results suggest that the two prescriptive analytics approaches can result in substantial performance improvements of up to 58% compared with traditional approaches. Additional numerical analyses shed light on the behavior and performance drivers of the various approaches and demonstrate that in our case, the prescriptive approaches are much more robust to variations of exogenous cost parameters than traditional approaches. This paper was accepted by J. George Shanthikumar, Management Science Special Section on Data-Driven Prescriptive Analytics.

The drone swarm that performs missions in an autonomous and intelligent overall collaborative manner is an important operational factor in the complex flight space. However, the complex and changing battlefield environment greatly affects the mission execution and survivability of drone swarm. In this paper, we formulated the problem of formation reconfiguration to avoid collision with obstacles with minimum cost in a complex flight space as a two-stage stochastic programming with considering uncertain moving obstacles, where we first calculate the optimal formation parameters of the drone swarm that can avoid certain fixed obstacles with minimum reconfiguration cost in the first-stage. For the uncertain moving obstacles, we calculate the extra cost for avoiding in the second-stage after the distribution of moving obstacles is known, and add this part of the cost to the first-stage to ensure the overall reconfiguration cost is minimized. Moreover, the sample average approximation (SAA) method is exploited to solve this stochastic programming problem. Extensive experimental results from the OMNeT ++ simulation environment indicate both the feasibility and effectiveness of our proposed two-stage stochastic programming approach and the better performs, compared with the existing approach.

Sample average approximation method is one of the well-behaved methods in the stochastic optimization.This paper presents a sample average approximation method based on a D-gap function for stochastic variational inequality problems.An unconstrained optimization reformulation is proposed for the expected-value formulation of stochastic variational inequality problems based on the D-gap function. An implementable sample average approximation method for the reformulation is established andit is proven that the optimal values and the optimal solutions of the approximation problems converge to their true counterpart with probability one as the sample size increases under some moderate assumptions.Finally, the preliminary numerical results for some test examples are reported, which show that the proposed method is promising.

Studies have been being carried out to make production faster and more organised in the shipbuilding industry, as in other industry. The fact that automation-based works are limited in the shipbuilding industry is one of the biggest challenges encountered in block production as in other stages of shipbuilding. The blocks are time-consuming and difficult components to produce in the shipbuilding process. They are the structures formed by joining the cut metal sheets, profiles and other components. These activities are carried out at the different stations of shipyards. Labour planning is one of the crucial issues in shipbuilding. In this study, the allocation of the required capacity during the pre-production stations of the block production, namely C and D, is examined stochastically. The amount of work, revisions and worker performance under uncertainty factors to be experienced in the production process are included in the problem. A two-stage stochastic mathematical recourse model was established to determine the amount of workforce capacity requirement (man*day) of the planning period at the pre-production station depending on the factors. Scenarios are determined randomly and the near-optimum solution was tried to be obtained by the Sample Average Approximation (SAA) approach.