Homotopy perturbation method for linear programming problems

In this paper, He’s homotopy perturbation method (HPM) is applied for solving linear programming (LP) problems. This paper shows that some recent findings about this topic cannot be applied for all cases. Furthermore, we provide the correct application of HPM for LP problems. The proposed method has a simple and graceful structure. Finally, a numerical example is displayed to illustrate the proposed method.     

Inference of statistical bounds for multistage stochastic programming problems

We discuss in this paper statistical inference of sample average approximations of multistage stochastic programming problems. We show that any random sampling scheme provides a valid statistical lower bound for the optimal (minimum) value of the true problem. However, in order for such lower bound to be consistent one needs to employ the conditional sampling procedure. We also indicate that fixing a feasible first-stage solution and then solving the sampling approximation of the corresponding (T–1)-stage problem, does not give a valid statistical upper bound for the optimal value of the true problem.Supported, in part, by the National Science Foundation under grant DMS-0073770.     

Reminiscences about the origins of linear programming

The author recalls the early days of linear programming, the contributions of von Neumann, Leontief, Koopmans and others.Linear Programming is viewed as a revolutionary development giving us the ability for the first time to state general objectives and to find, by means of the simplex method, optimal policy decisions for a broad class of practical decision problems of great complexity.     

Eliminating columns in the simplex method for linear programming

We propose a column-eliminating technique for the simplex method of linear programming. A pricing criterion is developed for checking whether a dual hyperplane corresponding to a column intersects a simplex containing all of the optimal dual feasible solutions. If the dual hyperplane has no intersection with this simplex, we can eliminate the corresponding column from further computation during the course of the simplex method.The author is grateful for many discussions with Professor G. B. Dantzig, Stanford University, and for his valuable suggestions about this work. The author also gratefully acknowledges the editor and two referees for their very helpful comments, corrections, and remarks.     

Parallel decomposition of multistage stochastic programming problems

A new decomposition method for multistage stochastic linear programming problems is proposed. A multistage stochastic problem is represented in a tree-like form and with each node of the decision tree a certain linear or quadratic subproblem is associated. The subproblems generate proposals for their successors and some backward information for their predecessors. The subproblems can be solved in parallel and exchange information in an asynchronous way through special buffers. After a finite time the method either finds an optimal solution to the problem or discovers its inconsistency. An analytical illustrative example shows that parallelization can speed up computation over every sequential method. Computational experiments indicate that for large problems we can obtain substantial gains in efficiency with moderate numbers of processors.This work was partly supported by the International Institute for Applied Systems Analysis, Laxenburg, Austria.     

PROMISE: A DSS for multiple objective stochastic linear programming problems

Most of the multiple objective linear programming (MOLP) methods which have been proposed in the last fifteen years suppose deterministic contexts, but because many real problems imply uncertainty, some methods have been recently developed to deal with MOLP problems in stochastic contexts. In order to help the decision maker (DM) who is placed before such stochastic MOLP problems, we have built a Decision Support System called PROMISE. On the one hand, our DSS enables the DM to identify many current stochastic contexts: risky situations and also situations of partial uncertainty. On the other hand, according to the nature of the uncertainty, our DSS enables the DM to choose the most appropriate interactive stochastic MOLP method among the available methods, if such a method exists, and to solve his problem via the chosen method.     

Confidence intervals in the solution of stochastic integer linear programming problems

A method is proposed to estimate confidence intervals for the solution of integer linear programming (ILP) problems where the technological coefficients matrix and the resource vector are made up of random variables whose distribution laws are unknown and only a sample of their values is available. This method, based on the theory of order statistics, only requires knowledge of the solution of the relaxed integer linear programming (RILP) problems which correspond to the sampled random parameters. The confidence intervals obtained in this way have proved to be more accurate than those estimated by the current methods which use the integer solutions of the sampled ILP problems.This research was partially supported by the Italian National Research Council contract no. 82.001 14.93 (P.F. Trasporti).     

A redundancy detection algorithm for fuzzy stochastic multi-objective linear fractional programming problems

The computational complexity of linear and nonlinear programming problems depends on the number of objective functions and constraints involved and solving a large problem often becomes a difficult task. Redundancy detection and elimination provides a suitable tool for reducing this complexity and simplifying a linear or nonlinear programming problem while maintaining the essential properties of the original system. Although a large number of redundancy detection methods have been proposed to simplify linear and nonlinear stochastic programming problems, very little research has been developed for fuzzy stochastic (FS) fractional programming problems. We propose an algorithm that allows to simultaneously detect both redundant objective function(s) and redundant constraint(s) in FS multi-objective linear fractional programming problems. More precisely, our algorithm reduces the number of linear fuzzy fractional objective functions by transforming them in probabilistic–possibilistic constraints characterized by predetermined confidence levels. We present two numerical examples to demonstrate the applicability of the proposed algorithm and exhibit its efficacy.     

Advanced basis construction in linear programming

This paper considers basis construction in a linear program when the number of activities with basic status is not equal to the number of rows in the particular scenario. This arises when starting with an ‘advanced basis’, obtained from a solution to another scenario. The goal here is to provide a triangular-seeking algorithm that takes advantage of structural properties during the construction of a basis agenda. For completeness, some facts, which are known but have not been published, are given about choosing an advanced basis and about spikes.     

Advanced basis construction in linear programming

This paper considers basis construction in a linear program when the number of activities with basic status is not equal to the number of rows in the particular scenario. This arises when starting with an advanced basis, obtained from a solution to another scenario. The goal here is to provide a triangular-seeking algorithm that takes advantage of structural properties during the construction of a basis agenda. For completeness, some facts, which are known but have not been published, are given about choosing an advanced basis and about spikes.     

On a time consistency concept in risk averse multistage stochastic programming

We discuss time consistency of multistage risk averse stochastic programming problems. The concept of time consistency is approached from an optimization point of view. That is, at each state of the system optimality of a decision policy should not involve states which cannot happen in the future.     

A compromise solution for the multiobjective stochastic linear programming under partial uncertainty

This paper solves the multiobjective stochastic linear program with partially known probability. We address the case where the probability distribution is defined by crisp inequalities. We propose a chance constrained approach and a compromise programming approach to transform the multiobjective stochastic linear program with linear partial information on probability distribution into its equivalent uniobjective problem. The resulting program is then solved using the modified L-shaped method. We illustrate our results by an example.     

Sensitivity analysis in interval-valued trapezoidal fuzzy number linear programming problems

The aim of this article is to introduce a formulation of fuzzy linear programming problems involving the level (h^L,h^U)$(h^L\text{,}{h}^U)$-interval-valued trapezoidal fuzzy numbers as parameters. Indeed, such a formulation is the general form of trapezoidal fuzzy number linear programming problems. Then, it is demonstrated that study of the sensitivity analysis for the level (h^L,h^U)$(h^L\text{,}{h}^U)$-interval-valued trapezoidal fuzzy number linear programming problems gives rise to the same expected results as those obtained for trapezoidal fuzzy number linear programming problems.     

Some fundamental issues of basic line search algorithm for linear programming problems

In this article we present the fundamental idea, concepts and theorems of a basic line search algorithm for solving linear programming problems which can be regarded as an extension of the simplex method. However, unlike the iteration of the simplex method from a basic point to an improved adjacent basic point via pivot operation, the basic line search algorithm, also by pivot operation, moves from a basic line which contains two basic feasible points to an improved basic line which also contains two basic feasible points whose objective values are no worse than that of the two basic feasible points on the previous basic line. The basic line search algorithm may skip some adjacent vertices so that it converges to an optimal solution faster than the simplex method. For example, for a 2-dimensional problem, the basic line search algorithm can find an optimal solution with only one iteration.     

Fuzzy stochastic goal programming problems

In this paper, we present a model to measure attainment value of fuzzy stochastic goals. Then, the new measure is used to de-randomize and de-fuzzify the fuzzy stochastic goal programming problem and obtain a standard linear program (LP). A numerical example is provided to illustrate the proposed method.     

The Expected loss in the discretization of multistage stochastic programming problems—estimation and convergence rate

In the present paper, the approximate computation of a multistage stochastic programming problem (MSSPP) is studied. First, the MSSPP and its discretization are defined. Second, the expected loss caused by the usage of the “approximate” solution instead of the “exact” one is studied. Third, new results concerning approximate computation of expectations are presented. Finally, the main results of the paper—an upper bound of the expected loss and an estimate of the convergence rate of the expected loss—are stated.     

An active-set strategy in an interior point method for linear programming

We will present a potential reduction method for linear programming where only the constraints with relatively small dual slacks—termed active constraints—will be taken into account to form the ellipsoid constraint at each iteration of the process. The algorithm converges to the optimal feasible solution in O( L) iterations with the same polynomial bound as in the full constraints case, wheren is the number of variables andL is the data length. If a small portion of the constraints is active near the optimal solution, the computational cost to find the next direction of movement in one iteration may be considerably reduced by the proposed strategy.This research was partially done in June 1990 while the author was visiting the Department of Mathematics, University of Pisa.     

Designing a majorization scheme for the recourse function in two-stage stochastic linear programming

We discuss issues pertaining to the domination from above of the second-stage recourse function of a stochastic linear program and we present a scheme to majorize this function using a simpler sublinear function. This majorization is constructed using special geometrical attributes of the recourse function. The result is a proper, simplicial function with a simple characterization which is well-suited for calculations of its expectation as required in the computation of stochastic programs. Experiments indicate that the majorizing function is well-behaved and stable.     

Stability in multistage stochastic programming

Multistage stochastic programs are regarded as mathematical programs in a Banach spaceX of summable functions. Relying on a result for parametric programs in Banach spaces, the paper presents conditions under which linearly constrained convex multistage problems behave stably when the (input) data process is subjected to (small) perturbations. In particular, we show the persistence of optimal solutions, the local Lipschitz continuity of the optimal value and the upper semicontinuity of optimal sets with respect to the weak topology inX. The linear case with deterministic first-stage decisions is studied in more detail.This research has been supported by the Schwerpunktprogramm Anwendungsbezogene Optimierung und Steuerung of the Deutsche Forschungsgemeinschaft.     

Airline network revenue management by multistage stochastic programming

A multistage stochastic programming approach to airline network revenue management is presented. The objective is to determine seat protection levels for all itineraries, fare classes, points of sale of the airline network and all dcps of the booking horizon such that the expected revenue is maximized. While the passenger demand and cancelation rate processes are the stochastic inputs of the model, the stochastic protection level process represents its output and allows to control the booking process. The stochastic passenger demand and cancelation rate processes are approximated by a finite number of tree structured scenarios. The scenario tree is generated from historical data using a stability-based recursive scenario reduction scheme. Numerical results for a small hub-and-spoke network are reported. This research is supported by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.