Two-stage Stochastic Linear Programming Problem Research Articles

Multicut Benders decomposition algorithm for process supply chain planning under uncertainty

In this paper, we present a multicut version of the Benders decomposition method for solving two-stage stochastic linear programming problems, including stochastic mixed-integer programs with only continuous recourse (two-stage) variables. The main idea is to add one cut per realization of uncertainty to the master problem in each iteration, that is, as many Benders cuts as the number of scenarios added to the master problem in each iteration. Two examples are presented to illustrate the application of the proposed algorithm. One involves production-transportation planning under demand uncertainty, and the other one involves multiperiod planning of global, multiproduct chemical supply chains under demand and freight rate uncertainty. Computational studies show that while both the standard and the multicut versions of the Benders decomposition method can solve large-scale stochastic programming problems with reasonable computational effort, significant savings in CPU time can be achieved by using the proposed multicut algorithm.

Risk-Averse Two-Stage Stochastic Linear Programming: Modeling and Decomposition

We formulate a risk-averse two-stage stochastic linear programming problem in which unresolved uncertainty remains after the second stage. The objective function is formulated as a composition of conditional risk measures. We analyze properties of the problem and derive necessary and sufficient optimality conditions. Next, we construct a new decomposition method for solving the problem that exploits the composite structure of the objective function. We illustrate its performance on a portfolio optimization problem.

Long-term scheduling of hydro-thermal power systems using scenario fans

This paper presents a method for solving the long-term multireservoir hydro-thermal scheduling problem based on stochastic linear programming. Decisions for each week are determined by solving a two-stage stochastic linear programming problem, where reservoir inflow stochasticity for future weeks is represented as a fan of historical inflow records. The overall scheduling problem is then solved as a sequence of such problems. The method is tested on a data set for the Nordic power system, and its performance—in terms of socio-economic surplus, reservoir handling and computation time—is compared with a well established model for solving the identical problem.

Optimization of multi-product batch plant design under uncertainty with environmental considerations

This paper presents a framework for the design and optimization of multi-product batch processes under uncertainty with environmental considerations. The uncertainties and environmental impacts are discussed. The profit and environmental impacts are considered as bi-objectives for batch plant design. The problem, thus, is formulated as a multi-objective stochastic programming problem. It can be converted into a single-objective two-stage stochastic linear programming problem using the weighted aggregation method. To solve the two-stage stochastic programming, we introduce both Monte Carlo sampling for the entire domain of the distribution function and the feasibility cut method based on dual theory in Benders’ decomposition. The detailed algorithm for problem-solving is presented. A numerical example is presented to illustrate the proposed framework.

A heuristical approach for Farmer's problem with uniform continuous random yields

The L-shaped method is useful for solving two-stage stochastic linear programming problems which have the form of a master problem and several subproblems represented by the side model. In order to make this approach possible, the random vector ξ, which is bringing the uncertainty into the model, must have finite support. With finite support we may write the deterministic equivalent program in the extensive form which is solvable by decomposition. To avoid the problems arising from the random elements that have continuous probability distributions, an approximate discretisation is needed. In this paper, the uniform continuous probability distributions of the yields in the Farmer's problem are discretised by using one of the low-discrepancy sequences, called Faure sequence, and based on the obtained approximate discrete distribution a heuristic is proposed to approximate the solution to the problem.

A new approach to solving stochastic programming problems with recourse

A new numerical approach to the solution of two-stage stochastic linear programming problems is described and evaluated. The approach avoids the solution of the first-stage problem and uses the underlying deterministic problem to generate a sequence of values of the first-stage variables which lead to successive improvements of the objective function towards the optimal policy. The model is evaluated using an example in which randomness is described by two correlated factors. The dynamics of these factors are described by stochastic processes simulated using lattice techniques. In this way, discrete distributions of the random parameters are assembled. The solutions obtained with the new iterative procedure are compared with solutions obtained with a deterministic equivalent linear programming problem. It is concluded that they are almost identical. However, the computational effort required for the new approach is negligible compared with that needed for the deterministic equivalent problem.

Improving convergence of the stochastic decomposition algorithm by using an efficient sampling technique

This work focuses on the basic stochastic decomposition (SD) algorithm of Higle and Sen [J.L. Higle, S. Sen, Stochastic Decomposition, Kluwer Academic Publishers, 1996] for two-stage stochastic linear programming problems with complete recourse. The algorithm uses sampling when the random variables are represented by continuous distribution functions. Traditionally, this method has been applied by using Monte Carlo (MC) sampling to generate the samples of the stochastic variables. However, Monte Carlo methods can result in large error bounds and variance. Hence, some other approaches use importance sampling to reduce variance and achieving convergence faster that the method based on the Monte Carlo sampling technique. This work proposes an improvement on this respect. Hence, we propose to replace the use of the Monte Carlo sampling technique or the importance sampling in the SD algorithm by the use of the novel Hammersley sequence sampling (HSS) technique. Recently, such a technique has proved to provide better uniformity properties than other sampling techniques and, as a consequence, the variance and the number of samples required for convergence are reduced. Also, we use a fractal dimension approach to characterize the error of the estimation of the recourse function based on sampling. The computational implementation of the algorithm involves a framework that integrates the GAMS modeling environment, the HSS sampling code (FORTRAN) and a C++ program which generates appropriate LP problems for each SD iteration. The algorithm has been tested with several case studies representing chemical engineering applications and the results are discussed.

The augmented system variant of IPMs in two-stage stochastic linear programming computation

The application of interior point methods (IPM) to solve the deterministic equivalent of two-stage stochastic linear programming problems is a known and natural idea. Experiments have proved that among the interior point methods, the augmented system approach gives the best performance on these problems. However, most of their implementations encounter numerical difficulties in certain cases, which can result in loss of efficiency. We present a new approach for the decomposition of the augmented system, which ‘automatically’ exploits the special behavior of the problems. We show that the suggested approach can be implemented in a fast and numerically robust way by solving a number of large-scale two-stage stochastic linear programming problems. The comparison of our solver with fo1aug, which is considered as a state-of-the-art augmented system implementation of interior point methods, is also given.

A regularized stochastic decomposition algorithm for two-stage stochastic linear programs

In this paper a regularized stochastic decomposition algorithm with master programs of finite size is described for solving two-stage stochastic linear programming problems with recourse. In a deterministic setting cut dropping schemes in decomposition based algorithms have been used routinely. However, when only estimates of the objective function are available such schemes can only be properly justified if convergence results are not sacrificed. It is shown that almost surely every accumulation point in an identified subsequence of iterates produced by the algorithm, which includes a cut dropping scheme, is an optimal solution. The results are obtained by including a quadratic proximal term in the master program. In addition to the cut dropping scheme, other enhancements to the existing methodology are described. These include (i) a new updating rule for the retained cuts and (ii) an adaptive rule to determine when additional reestimation of the cut associated with the current solution is needed. The algorithm is tested on problems from the literature assuming both descrete and continuous random variables.

Computational methods for solving two-stage stochastic linear programming problems

Approximating a given continuous probability distribution of the data of a linear program by a discrete one yields solution methods for the stochastic linear programming problem with complete fixed recourse. For a procedure along the lines of [8], the reduction of the computational amount of work compared to the usual revised simplex method is figured out. Furthermore, an alternative method is proposed, where by refining particular discrete distributions the optimal value is approximated.

Stochastic Scheduling of Linear Construction Sites

Essential basic data of scheduling linear construction sites are subject to stochastically variable influence factors, e.g., weather conditions. To evaluate economical low-cost alternatives which determine the production sequence of segments of a linear site through a projected construction time, a mathematical model is developed which is formulated as a two-stage stochastic linear programming problem and can be solved by standard linear programming methods. Criteria for interpretation and evaluation of model solutions in terms of a time-velocity diagram are derived and illustrated with a simple example problem. By means of stability analysis, risk factors and probability distributions of project duration and cost can be obtained for different construction alternatives to prepare a final decision. It is emphasized that the methodology presented is not seen as an approach to determine “the optimal decision,” but rather as a tool to facilitate decision making, since this ultimately relies on practical engineering judgment.