We present a framework for solving some types of 0-1 multi-stage scheduling/planning problems under uncertainty in the objective function coefficients and the right-hand-side. A scenario analysis scheme with full reco...
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We present a framework for solving some types of 0-1 multi-stage scheduling/planning problems under uncertainty in the objective function coefficients and the right-hand-side. A scenario analysis scheme with full recourse is used. The solution offered for each scenario group at each stage takes into account all scenarios but without subordinating to any of them. The constraints are modelled by a splitting variables representation via scenarios. So, a 0-1 model for each scenario is considered plus the non-anticipativity constraints that equate the 0-1 variables from the same group of scenarios in each stage. The mathematical representation of the model is very amenable for the proposed framework to deal with the 0-1 character of the variables. A branch-and-fix coordination approach is introduced for coordinating the selection of the branching nodes and branching variables in the scenario subproblems to be jointly optimized. Some computational experience is reported for different types of problems. (C) 2002 Elsevier B.V. All rights reserved.
We examine several methods for evaluating resource acquisition decisions under uncertainty. Traditional methods may underestimate equipment benefit when part of this benefit comes from decision flexibility. We develop...
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We examine several methods for evaluating resource acquisition decisions under uncertainty. Traditional methods may underestimate equipment benefit when part of this benefit comes from decision flexibility. We develop a new, practical method for resource planning under uncertainty, and show that this approach is more accurate than several commonly used methods. We successfully applied our approach to an investment problem faced by a major firm in the aviation information industry. Our recommendations were accepted and resulted in estimated annual savings in excess of $1 million (US). (C) 2003 Wiley Periodicals, Inc.
The problem of finding a truss of minimal weight subject to stress constraints and stochastic loading conditions is considered. We demonstrate that this problem is ill-posed by showing that the optimal solutions chanc...
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The problem of finding a truss of minimal weight subject to stress constraints and stochastic loading conditions is considered. We demonstrate that this problem is ill-posed by showing that the optimal solutions chance discontinuously as small changes in the modelling of uncertainty are introduced. We propose a relaxation of this problem that is stable with respect to such errors. We establish a classic E-perturbation result for the relaxed problem, and propose a solution scheme based on discretizations of the probability measure. Using Chebyshev's inequality we give an a priori estimation of the probability of stress constraint violations in terms of the relaxation parameter. The convergence of the relaxed optimal designs towards the original (non-relaxed) optimal designs, as the relaxation parameter decreases to zero, is established.
In this paper, we propose and analyze an SQP-type method for solving linearly constrained convex minimization problems where the objective functions are too complex to be evaluated exactly. Some basic results for glob...
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In this paper, we propose and analyze an SQP-type method for solving linearly constrained convex minimization problems where the objective functions are too complex to be evaluated exactly. Some basic results for global convergence and local superlinear convergence are obtained according to the properties of the approximation sequence. We illustrate the applicability of our approach by proposing a new method for solving two-stage stochastic programs with fixed recourse.
Capacitated location-allocation problem with stochastic demands is originally formulated as expected value model, chance-constrained programming and dependent-chance programming according to different criteria. For so...
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Capacitated location-allocation problem with stochastic demands is originally formulated as expected value model, chance-constrained programming and dependent-chance programming according to different criteria. For solving these stochastic models efficiently, the network simplex algorithm, stochastic simulation and genetic algorithm are integrated to produce a hybrid intelligent algorithm. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed algorithm. (C) 2003 Elsevier Science Ltd. All lights reserved.
We consider structural topology optimization problems, including unilateral constraints arising from, for example, non-penetration conditions in contact mechanics or non-compression conditions for elastic ropes. To co...
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We consider structural topology optimization problems, including unilateral constraints arising from, for example, non-penetration conditions in contact mechanics or non-compression conditions for elastic ropes. To construct more realistic models and to circumvent possible failures or inefficient behaviour of optimal structures, we allow parameters (for example, loads) defining the problem to be stochastic. The resulting non-smooth stochastic optimization problem is an instance of stochastic mathematical programs with equilibrium constraints (MPEC), or stochastic bilevel programs. We propose a solution scheme based first on the approximation of the given topology optimization problem by a sequence of simpler sizing optimization problems, and second on approximating the probability measure in the latter problems. For stress-constrained weight-minimization problems, an alternative to E-perturbation based on a new penalty function is proposed.
In the setting of stochastic recourse programs, we consider the problem of minimizing the probability of total costs exceeding a certain threshold value. The problem is referred to as the minimum risk problem and is p...
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In the setting of stochastic recourse programs, we consider the problem of minimizing the probability of total costs exceeding a certain threshold value. The problem is referred to as the minimum risk problem and is posed in order to obtain a more adequate description of risk aversion than that of the accustomed expected value problem. We establish continuity properties of the recourse function as a function of the first-stage decision, as well as of the underlying probability distribution of random parameters. This leads to stability results for the optimal solution of the minimum risk problem when the underlying probability distribution is subjected to perturbations. Furthermore, an algorithm for the minimum risk problem is elaborated and we present results of some preliminary computational experiments.
The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is...
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The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps. We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and first-stage integer variables. For each of the three problem classes, we use decomposition and branch-and-cut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 2(1694) scenarios to within an estimated 1.0% of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably near-optimal solutions to these difficult stochastic programs using only a moderate amount of computation time.
In stochastic programming models we always face the problem of how to represent the random variables. This is particularly difficult with multidimensional distributions. We present an algorithm that produces a discret...
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In stochastic programming models we always face the problem of how to represent the random variables. This is particularly difficult with multidimensional distributions. We present an algorithm that produces a discrete joint distribution consistent with specified values of the first four marginal moments and correlations. The joint distribution is constructed by decomposing the multivariate problem into univariate ones, and using an iterative procedure that combines simulation, Cholesky decomposition and various transformations to achieve the correct correlations without changing the marginal moments. With the algorithm, we can generate 1000 one-period scenarios for 12 random variables in 16 seconds, and for 20 random variables in 48 seconds, on a Pentium III machine.
The problem of optimizing the expected performance of a discrete-event, stochastic system was studied. The study proposed an optimization-via-simulation algorithm for solving the stochastic, discrete-event simulation ...
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The problem of optimizing the expected performance of a discrete-event, stochastic system was studied. The study proposed an optimization-via-simulation algorithm for solving the stochastic, discrete-event simulation problem. The decision variables of the system were such that they might be subjected to the deterministic linear integer constraints. The proposed approach, which consisted of a global guidance system, a selection-of-the-best procedure, and local improvement method, was globally convergent under mild conditions.
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