Right-hand Side Coefficients Research Articles

Fully Pythagorean fuzzy linear programming problems with equality constraints

There are several daily life problems where we have to deal with the uncertainties and we are forced to solve the uncertain linear programming models. Certain methods have been presented for dealing with linear programming problems based on fuzzy sets and intuitionistic fuzzy sets which are characterized by membership degree, membership and non-membership degrees, respectively. In this study, we first extend the concept of crisp linear programming problem in Pythagorean fuzzy environment based on triangular Pythagorean fuzzy numbers. The profit/cost coefficients in objective function, input/output coefficients and right-hand side coefficients and decision variables of a linear programming problem are considered as triangular Pythagorean fuzzy numbers. Further, we present methods for solving fully Pythagorean fuzzy linear programming problems for non-negative and unrestricted triangular Pythagorean fuzzy numbers with equality constraints. We also apply the proposed technique to solve practical models.

Benders decomposition with adaptive oracles for large scale optimization

This paper proposes an algorithm to efficiently solve large optimization problems which exhibit a column bounded block-diagonal structure, where subproblems differ in right-hand side and cost coefficients. Similar problems are often tackled using cutting-plane algorithms, which allow for an iterative and decomposed solution of the problem. When solving subproblems is computationally expensive and the set of subproblems is large, cutting-plane algorithms may slow down severely. In this context we propose two novel adaptive oracles that yield inexact information, potentially much faster than solving the subproblem. The first adaptive oracle is used to generate inexact but valid cutting planes, and the second adaptive oracle gives a valid upper bound of the true optimal objective. These two oracles progressively “adapt” towards the true exact oracle if provided with an increasing number of exact solutions, stored throughout the iterations. These adaptive oracles are embedded within a Benders-type algorithm able to handle inexact information. We compare the Benders with adaptive oracles against a standard Benders algorithm on a stochastic investment planning problem. The proposed algorithm shows the capability to substantially reduce the computational effort to obtain an epsilon -optimal solution: an illustrative case is 31.9 times faster for a 1.00% convergence tolerance and 15.4 times faster for a 0.01% tolerance.

An Approach to Solve the Linear Programming Problem Using Single Valued Trapezoidal Neutrosophic Number

While making a decision, the neutrosophic set theory includes the uncertainty part beside certainty part (i.e., Yes or No). In the present uncertain socio-economic fashion, this pattern is highly acceptable and hence, the limitations of fuzzy set and intuitionistic fuzzy set are overcome with neutrosophic set theory. The present study provides a modified structure of linear programming problem (LP-problem) and its solution approach in neutrosophic sense. A special type of neutrosophic set defined over the set of real number, viz., single valued trapezoidal neutrosophic number (SVTN-number) is adopted here as the coefficients of the objective function, right-hand side coefficients and decision variables itself of an LP-problem. In order to solve such problem, a parameter based ranking function of SVTN-number is newly constructed from the geometrical configuration of the trapezium. It plays a key role in the development of the solution algorithm. An LP-problem is normally solved under the asset of some given constraints. Besides that, there may be some hidden parameters (e.g., awareness level of nearer society for the smooth run of a clinical pharmacy, ruined structure of road to be met a profit from a bus, etc) of an LP-problem and these affect the solution badly when experts ignore them. This study makes an attempt to solve an LP-problem by giving importance to all these to attain a fair outcome. The efficiency of the proposed concept is illustrated in a real field. A real example is stated and is solved numerically under the present view.

Robust stability of linear time-varying implicit dynamic equations: a general consideration

In this paper, the problem of robust stability for linear time-varying implicit dynamic equations is generally studied. We consider the effect of uncertain structured perturbations on all coefficient matrices of equations. A formula of stability radius with respect to dynamic structured perturbations acting on the right-hand side coefficients is obtained. In case where structured perturbations affect on both derivative and the right-hand side, the lower bounds for the stability radius are derived. The results are novel and extend many previous results about robust stability for time-varying ordinary differential/difference equations, time-varying differential algebraic equations and time-varying implicit difference equations.

Two-stage robust optimization, state-space representable uncertainty and applications

The present paper addresses the class of two-stage robust optimization problems which can be formulated as mathematical programs with uncertainty on the right-hand side coefficients (RHS uncertainty). The wide variety of applications and the fact that many problems in the class have been shown to be NP-hard, motivates the search for efficiently solvable special cases. Accordingly, the first objective of the paper is to provide an overview of the most important applications and of various polynomial or pseudo-polynomial special cases identified so far. The second objective is to introduce a new subclass of polynomially solvable robust optimization problems with RHS uncertainty based on the concept of state-space representable uncertainty sets . A typical application to a multi period energy production problem under uncertain customer load requirements is described into details, and computational results including a comparison between optimal two-stage solutions and exact optimal multistage strategies are discussed.

Lower Semicontinuity of the Feasible Set Mapping of Linear Systems Relative to Their Domains

This paper deals with stability properties of the feasible set of linear inequality systems having a finite number of variables and an arbitrary number of constraints. Several types of perturbations preserving consistency are considered, affecting respectively, all of the data, the left-hand side data, or the right-hand side coefficients.

PENDEKATAN ALGORITMA GENETIKA DALAM MENYELESAIKAN PERMASALAHAN FUZZY LINEAR PROGRAMMING

Fuzzy linear programming is one of the linear programming developments which able to accommodate uncertainty in the real world. Genetic algorithm approach in solving linear programming problems with fuzzy constraints has been introduced by Lin (2008) by providing a case which consists of two decision variables and three constraint functions. Other linear programming problem arise with the presence of some coefficients which are fuzzy in linear programming problems, such as the coefficient of the objective function, the coefficient of constraint functions, and right-hand side coefficients constraint functions. In this study, the problem studied is to explain the genetic algorithm approach to solve linear programming problems where the objective function coefficients and right-hand sides are fuzzy constraint functions.PT Dakota Furniture study case provides a linear programming formulation with a given objective function coefficients and right-hand side coefficients are fuzzy constraint functions. This study describes the use of genetic algorithm approach to solve the problem of linear programming of PT Dakota to maximize the mean income. The genetic algorithm approach is done by simulate every fuzzy number and each fuzzy numbers by distributing them on certain partition points. Then genetic algorithm is used to evaluate the value for each partition point. As a result, the Final Value represents the coefficient of fuzzy number. Fitness function is done by calculating the value of the objective function of linear programming problems. Empirical results indicated that the genetic algorithm approach can provide a very good solution by giving some limitations on each fuzzy coefficient.Genetic algorithm approach can be extended not only to resolve the case of PT Dakota Furniture, but can also be used to solve other linear programming case with some coefficients in the objective function and constraint functions are fuzzy.Keywords : Genetic Algorithm, Fuzzy Linear Programming, Linear Programming, Two-Phase Simplex Method

Tolerance analysis in linear systems and linear programming

In many practical linear programming problems, it is often important to know how different optimality criteria (optimal solution, optimal basis, optimal partition, etc.) change under input data perturbations. Our aim is to compute tolerances (intervals) for the objective function and the right-hand side coefficients such that these coefficients can independently and simultaneously vary inside their tolerances while preserving the corresponding optimality criterion. We put tolerance analysis in a unified framework that is convenient for algorithmic processing and that is applicable not only in linear programming but for other linear systems as well. We survey the known results (pioneered by R.E. Wendell) and propose an improvement that is optimal in some sense (the resulting tolerances are maximal and they take into account proportionality). We apply our approach to several optimality invariancies: optimal basis, support set and optimal partition invariancy. Thus, the approach is useful not only for simplex method solvers, but for the interior points methods, too. We also discuss time complexity and show that it is NP-hard to determine the maximal tolerances.

Economics and the Refinery's CO2Emissions Allocation Problem

The establishment of a market for CO2 emission rights in Europe leads oil-refining companies to add a cost associated with carbon emissions to the objective function of linear programming models used to manage refineries. These models may be used to compute the marginal contribution of each finished product to the CO2 emissions of the refinery. Babusiaux (Oil. Gas Sci. Technol. , 58, 2003, 685-692) has shown that, under some conditions, this marginal contribution is a relevant means of allocating the carbon emissions of the refinery. Thus, it can be used in a well-to-wheel Life Cycle Assessment. In fact, this result holds if the demand equations are the only binding constraints with a non-zero right-hand side coefficient. This is not the case for short-run models with fixed capacity. Then, allocating CO2 emissions on a marginal basis tends to overvalue (or undervalue) the total volume of emissions. In order to extend the existing methodology, we discuss two distinct solutions to this problem, inspired by economic theory: adapting either the Aumann-Shapley cost sharing method (Values of non-atomic games, 1974, Princeton University Press) or the Ramsey pricing formula (Econ. J. , 37, 1927, 47-61; J. Econ. Theory , 3, 1971, 219-240). We compare these two solutions, with a strong argument in favour of Ramsey prices, based on the determination of the optimal environmental tax rate to which imported finished products should be subject.

Modelling and allocation of CO 2 emissions in a multiproduct industry: The case of oil refining

As oil refining is a multiproduct industrial activity, there are innumerable ways to allocate a refinery’s CO 2 emissions among the various refined products. The linear-programming models used to manage refineries may serve to compute the marginal contribution of each finished product to the CO 2 emissions of the refinery. We show that, under some conditions, this marginal contribution is a relevant means of allocating all the refinery’s CO 2 emissions. The application of this allocation rule leads to interesting results which can be used in a well-to-wheel life cycle assessment. In fact, this allocation rule holds rigorously if the demand equations are the only binding constraints with a non-zero right-hand side coefficient. This is certainly not the case for short-run models with fixed capacity of processing units. To extend the application field of our approach, we therefore suggest three distinct solutions, inspired by economic theory: applying the Aumann–Shapley cost-sharing method, or adapting the Ramsey pricing-formula, or using proportionally-adjusted marginal contributions. A numerical application to a simplified refining model is presented.

Présentation d'une propriété nouvelle des multiplicateurs de Lagrange et application à la fonction de coût d'une firme avec produits liés

Dans le cadre d’un problème classique d’optimisation sous contraintes, nous mettons en évidence une propriété nouvelle des multiplicateurs de Lagrange: à l’optimum, le produit de la valeur de la fonction objectif par son élasticité d’échelle est égal à la somme des produits de chaque multiplicateur de Lagrange par le second membre et par l’élasticité d’échelle de la contrainte correspondante. En d’autres termes, lorsque la fonction valeur d’un problème d’optimisation sous contraintes est exprimée en fonction des seconds membres de ces contraintes, une forme générale de la propriété d’Euler est vérifiée. Nous appliquons cette propriété à la détermination de la fonction de coût d’une firme multiproduits, lorsque le vecteur des quantités de biens obtenues à partir d’une combinaison donnée de facteurs peut être représenté par une fonction de production vectorielle. Notre approche conduit alors à proposer une clé de répartition du coût particulièrement pertinente. Le coût alloué à un produit donné correspond, sous certaines hypothèses, au coût moyen d’une forme décentralisée de production de ce seul produit.

A fuzzy method to repair infeasibility in linearly constrained problems

In this paper we introduce a fuzzy method to deal with infeasibility in linearly constrained programs. Given an infeasible instance, we determine how much we should perturb the right-hand side coefficients in order to attain feasibility and propose a ‘feasible reformulation’ of the problem. Although we prove that our algorithm always finds such a reformulation the convenience of using it can be decided by the analyst. By this, we mean that the method also provides a simple way to compute lower bounds on the changes on every right-hand side coefficient, and if the decision maker considers that some of the magnitudes are unacceptable, he or she simply stops at this step. We think that it will be specially useful for those situations in which the cause of the infeasibility is in the requirement of specifying exact values for the parameters in the mathematical programs formulation.

Chance constrained fuzzy goal programming with right-hand side uniform random variable coefficients

We present a basic idea about stochastic goal programming and the approach of chance constrained linear goal programming (CCLGP). We developed the method to transform probabilistic goal programming into deterministic goal programming and consider the stochastic fuzzy goal programming problem when the right-hand side coefficients are random variables distributed according to uniform distribution. A chance constrained goal programming in the case of having uncertain aspiration levels is considered. However, these aspiration levels are random variables with known probability distribution functions. The equivalent deterministic goal program is developed and an illustrative example is given.

A Branch-and-Bound Approach Using Polyhedral Results for a Clustering Problem

This article describes a variation of the branch and bound method for solving a clustering problem stated as a partitioning problem on edge-weighted graphs. The key features of the approach are two. First, it employs the transformation of a subproblem according to some heuristic solution. For this, two clustering heuristics, constructive and iterative improvement, are adopted. Second, the lower bound computation is based on the use of the inequalities with zero right-hand side coefficient, each defining a facet of the polytope related to the transformed subproblem. A procedure is described for covering negative edges of the subproblem graph by cycles and paths inducing such inequalities. The objective function is modified each time by adding the left-hand side of the selected inequality to it. The lower bound is obtained by summing the weights of uncovered negative edges. The algorithm can be used for solving clustering problems in the area of qualitative data analysis. Computational results on both real and randomly generated clustering problems demonstrate the efficiency of the algorithm. in particular, all 10 real problems from the literature were solved in a total of less than 9 seconds on an IBM PC.

A Method for the Solution of the Distribution Problem of Stochastic Linear Programming

A method is proposed for finding a closed form expression for the cumulative distribution function (CDF) of the maximum value of the objective function in a stochastic linear programming problem in the case where either the objective function coefficients or the right-hand side coefficients are given by known probability distributions. If the objective function coefficients are random, the CDF is obtained by integrating over the space for which a given feasible basis remains optimal. A transformation is presented using the Jacobian theorem which simplifies the regions of integration and often simplifies the integration itself. A similar analysis is presented in the case where the right-hand side coefficients are random. An example is given to illustrate the technique.

On Chance Constrained Programming Problems with Joint Constraints

In this paper we consider chance constrained programming problems with joint constraints shown in the literature to be equivalent deterministic nonlinear programming problems. Since most existing computational methods for solution require that the constraints of the equivalent deterministic problem be concave, we obtain a simple condition for which the concavity assumption holds when the right-hand side coefficients are independent random variables. We show that it holds for most probability distributions of practical importance. For the case where the random vector has a multivariate normal distribution, nonexistence of any efficient numerical methods for evaluating multivariate normal integrals necessitates the use of lower bound approximations. We propose an approximation for the case of positively correlated normal random variables.