Linear programming with interval right hand sides

Linear program under changes in the system matrix coefficients has proved to be more complex than changes of the coefficients in objective functions and right hand sides. The most of the previous studies deals with problems where only one coefficient, a row (column), or few rows (columns) are linear functions of a parameter. This work considers a more general case, where all the coefficients are polynomial (in the particular case linear) functions of the parameter tT??R. For such problems, assuming that some non-singularity conditions hold and an optimal base matrix is known for some particular value t of the parameter, corresponding explicit optimal basic solution in the neighborhood of t is determined by solving an augmented LP problem with real system matrix coefficients. Parametric LP can be utilized for example to model the production problem where, technology, resources, costs and similar categories vary with time. .

Linear bilevel programming with interval coefficients

In recent years, many researchers in the field of fuzzy mathematics and operations research in the world are interested to solve the interval type-2 fuzzy linear programming (IT2FLP) problems. In IT2FLP problems all parameters, including coefficients in objective function, coefficients in the constraints and right hand sides of the constraints are interval type-2 fuzzy numbers (IT2FN). Hence, the IT2FLP model is one of the most important and the most appropriate models to interpret and determine the optimal decision. In this paper, a new ranking function method is proposed for solving these problems. Moreover, an example is used to study and describe the proposed method.

The design and optimization of a large-scale systems are the most difficalt problems. A large-scale system consists of a number of subsystems. For example, in a harvest for harvesting one can separate the following subsystems: the frame, driver's cab, platform, engine, transmission, and steering system. Different departments of the design office engaged in creating a machine optimize their ‘own’ subsystems, while ignoring others. A machine assembled from ‘autonomously optimal’ subsystems turns out to be far from perfect. A machine is a single whole. When improving one of its subsystems, we can unwittingly worsen others. This implies that it is not always possible to solve optimization problems directly even for determination of the feasible solution set. The correct determination of the feasible solution set was a major challenge in engineering optimization problems. In order to construct the feasible solution set, a method called the Parameter Space Investigation (PSI) has been created and successfully integrated into various fields of industry, science, and technology. The methods of approximation of the feasible solution and Pareto optimal sets and the regularization of the Pareto optimal set are described in our paper. These methods are applied to solving the multicriteria optimization problems of large- scale systems. For example, they were applied in an agricultural engineering to a harvester for harvesting design.

Linear programming is an important branch of operations research. The model is mature in theory and widely used in real life. However, various complex realistic scenarios involve fuzzy information. In this paper, we consider a fuzzy linear programming (FLP) model in which all parameters are trapezoidal interval type-2 fuzzy numbers (IT2FNs) and propose a solution method based on the nearest interval approximation and the best-worst cases (BWC) method. We prove the nearest interval approximation interval of trapezoidal IT2FNs, then the trapezoidal IT2FNs in the model are transformed into interval numbers which both upper and lower limits are interval numbers. With the help of best-worst cases (BWC) method, the sub-models of the transformed interval linear programming model are proposed, and four sub-solutions with different specific meanings can be obtained by solving them respectively. Finally, an application example is presented to show the rationality and practical significance of the method.

Modern socio-political-economic problems often represent situations in which the traditional numerical problem models fall far short of being adequate for determining formal solutions. The basic reason for this inadequacy is the fact that the objective functions in such problems contain abstract criteria. It, therefore, is difficult, if not impossible, to determine how much one alternative in the problem is preferred to another alternative. When such information is not available in a problem, a numerical model will serve to fabricate this information and, therefore, reduce the degree of isomorphism between the problem and the problem model. A linear program that uses a relations set as its objective function can maintain isomorphism. In such models, the single optimal solution may not be determinable. However, if the set of feasible solutions is ordered, the optimal solution can be determined. The same ordered approach also has some useful applications to games.

In constrast to “ordinary” sensitivity analysis in linear programming, the tolerance approach considers simultaneous and independent changes in the objective function coefficients and in the right-hand side terms. This approach yields a maximum tolerance percentage such that, as long as selected coefficients or terms are accurate to within that percentage of their estimated values, the same basis is optimal. In particular, if the objective function coefficients are accurate to within the maximum tolerance percentage of their specified values, then the same solution is optimal.

The traditional linear programming model is deterministic. The way that uncertainty is handled is to compute the range of optimality. After the optimal solution is obtained, typically by the simplex method, one considers the effect of varying each objective function coefficient, one at a time. This yields the range of optimality within which the decision variables remain constant. This sensitivity analysis is useful for helping the analyst get a sense for the problem. However, it is unrealistic because objective function coefficients tend not to stand still. They are typically profit contributions from products sold and are subject to randomly varying selling prices. In this paper, a realistic linear program is created for simultaneously randomizing the coefficients from any probability distribution. Furthermore, we present a novel approach for designing a copula of random objective function coefficients according to a specified rank correlation. The corresponding distribution of objective function values is created. This distribution is examined directly for central tendency, spread, skewness and extreme values for the purpose of risk analysis. This enables risk analysis and business analytics, emerging topics in education and preparation for the knowledge economy.

Scheduling under demand uncertainty using a new multiparametric programming approach

Previous article Next article Complementarity Theorems for Linear ProgrammingA. C. WilliamsA. C. Williamshttps://doi.org/10.1137/1012015PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Frank Eugene Clark, Mathematical Notes: Remark on the Constraint Sets in Linear Programming, Amer. Math. Monthly, 68 (1961), 351–352 MR1531192 0109.38204 CrossrefISIGoogle Scholar[2] A. J. Goldman, Resolution and separation theorems for polyhedral convex setsLinear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N. J., 1956, 41–51, in [4] MR0089113 0072.37505 Google Scholar[3] A. J. Goldman and , A. W. Tucker, Theory of linear programmingLinear inequalities and related systems, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N.J., 1956, 53–97, in [4] MR0101826 0072.37601 Google Scholar[4] H. W. Kuhn and , A. W. Tucker, Linear Inequalities and Related Systems, Princeton University Press, Princeton, 1956 0072.37502 Google Scholar[5] A. C. Williams, Boundedness relations for linear constraint sets, Linear Algebra and Appl., 3 (1970), 129–141 10.1016/0024-3795(70)90009-1 MR0266612 0201.22003 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Sparse solutions to an underdetermined system of linear equations via penalized Huber loss6 November 2020 | Optimization and Engineering, Vol. 22, No. 3 Cross Ref Dynamic Non-diagonal Regularization in Interior Point Methods for Linear and Convex Quadratic Programming26 February 2019 | Journal of Optimization Theory and Applications, Vol. 181, No. 3 Cross Ref Necessary and Sufficient Conditions for Noiseless Sparse Recovery via Convex Quadratic SplinesMustafa Ç Pinar12 February 2019 | SIAM Journal on Matrix Analysis and Applications, Vol. 40, No. 1AbstractPDF (416 KB)A labeling algorithm for the sensitivity ranges of the assignment problemApplied Mathematical Modelling, Vol. 35, No. 10 Cross Ref Bibliography15 August 2011 Cross Ref Sensitivity Analysis and Dynamic Programming15 February 2011 Cross Ref Bibliography Cross Ref Condition measures and properties of the central trajectory of a linear programMathematical Programming, Vol. 83, No. 1-3 Cross Ref New characterizations of ℓ1 solutions to overdetermined systems of linear equationsOperations Research Letters, Vol. 16, No. 3 Cross Ref Limiting behavior of weighted central paths in linear programmingMathematical Programming, Vol. 65, No. 1-3 Cross Ref Stability of linearly constrained convex quadratic programsJournal of Optimization Theory and Applications, Vol. 64, No. 1 Cross Ref Marginal values in mixed integer linear programmingMathematical Programming, Vol. 44, No. 1-3 Cross Ref A theory of linear inequality systemsLinear Algebra and its Applications, Vol. 106 Cross Ref Boundedness relations in linear semi-infinite programmingAdvances in Applied Mathematics, Vol. 8, No. 1 Cross Ref A Variable-Complexity Norm Maximization ProblemO. L. Mangasarian and T. -H. Shiau17 July 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 7, No. 3AbstractPDF (698 KB)Simple computable bounds for solutions of linear complementarity problems and linear programs26 February 2009 Cross Ref On polyhedral extension of some LP theoremsMathematical Programming, Vol. 30, No. 1 Cross Ref Polyhedral extensions of some theorems of linear programmingMathematical Programming, Vol. 24, No. 1 Cross Ref Optimal simplex tableau characterization of unique and bounded solutions of linear programsJournal of Optimization Theory and Applications, Vol. 35, No. 1 Cross Ref Projection and Restriction Methods in Geometric Programming and Related Problems Cross Ref Representation of Convex Sets Cross Ref Projection and restriction methods in geometric programming and related problemsJournal of Optimization Theory and Applications, Vol. 26, No. 1 Cross Ref The complementary unboundedness of dual feasible solution sets in convex programmingMathematical Programming, Vol. 12, No. 1 Cross Ref Theorems on the dimension of convex setsLinear Algebra and its Applications, Vol. 12, No. 1 Cross Ref On the primal and dual constraint sets in geometric programmingJournal of Mathematical Analysis and Applications, Vol. 32, No. 3 Cross Ref Volume 12, Issue 1| 1970SIAM Review History Submitted:23 December 1968Published online:18 July 2006 InformationCopyright © 1970 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1012015Article page range:pp. 135-137ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

This paper studies the problem of tangible assets acquisition within the company by proposing a new hybrid model that uses linear programming and fuzzy numbers. Regarding linear programming, two methods were implemented in the model, namely: the graphical method and the primal simplex algorithm. This hybrid model is proposed for solving investment decision problems, based on decision variables, objective function coefficients, and a matrix of constraints, all of them presented in the form of triangular fuzzy numbers. Solving the primal simplex algorithm using fuzzy numbers and coefficients, allowed the results of the linear programming problem to also be in the form of fuzzy variables. The fuzzy variables compared to the crisp variables allow the determination of optimal intervals for which the objective function has values depending on the fuzzy variables. The major advantage of this model is that the results are presented as value ranges that intervene in the decision-making process. Thus, the company’s decision makers can select any of the result values as they satisfy two basic requirements namely: minimizing/maximizing the objective function and satisfying the basic requirements regarding the constraints resulting from the company’s activity. The paper is accompanied by a practical example.

The various robust linear programming models investigated so far in the literature essentially appear to be based either on what is referred to as rowwise uncertainty models or on columnwise uncertainty models (these basically assume that the rows, or the columns, of the constraint matrix are subject to changes within a well-specified uncertainty set). In this chapter, we discuss a special case of columnwise uncertainty, namely the subclass of robust linear programming (LP) models with uncertainty limited to the right hand side (RHS) only (this subclass does not appear to have been significantly investigated so far). In this context we introduce the concept of a ‘two-stage robust LP model’ as opposed to the standard case (which might be referred to as a ‘single-stage robust LP model’) and we address the question of whether LP duality can be used to convert a LP problem with RHS uncertainty into a robust LP problem with uncertainty on the objective function. We show how to derive both statements of (a) the dual to the robust model and (b) the robust version of the dual. The resulting expressions of the objective function to be optimized in both cases, appear to be clearly distinct. Moreover, from a complexity point of view, one appears to be efficiently solvable (it reduces to a convex optimization problem), whereas the other, as a nonconvex optimization problem, is expected to be computationally difficult in the general case. As an application of the two-stage robust LP model introduced here, we next investigate the PERT (program evaluation and review technique) scheduling problem, considering two possible natural ways of specifying the uncertainty set for the task durations: the case where the uncertainty set is a scaled ball with respect to the L ∞ norm; the case where the uncertainty set is a scaled Hamming ball of bounded radius (which, though leading to a quite different model, bears some resemblance to the well-known Bertsimas–Sim approach to robustness). We show that in both cases the resulting robust optimization problem can be efficiently solved in polynomial time.

Multiple attribute decision-making theory under uncertainty is an important part of decision making theory, in this paper, a grey multiple attribute decision-making approach is proposed to study multi-criteria decision-making problems under uncertainty in which criteria values are interval gray numbers. Using grey related analysis as a means to reflect uncertainty in multiple attribute models through interval numbers. According to related theory, a non-risk standard expected value decision-making matrix can be obtained. Though determining the ideal grey relation reference sequence and non-ideal grey relation reference sequence, we can calculate the distance-proximity and angle-proximity of each alternative, and then the best alternative can be obtained by optimal membership degree which is based on distance-proximity and angle-proximity. At last the numerical illustrations prove the new model to be effective and scientific.

This paper establishes a new model for portfolio selection in which the returns and risk of assets are treated as interval numbers respectively. We develop an algorithm for solving portfolio selection problems, which can be transformed to a classical linear programming. The method obtained is illustrated by a case study.

Differential evolution algorithm requires diversified population to solve multi- modalproblems. DE supports non-distributed population. Differential evolution algorithm (DE) versions include differential evolution algorithm with best selection method and species evolution (DESBS) and differential evolution algorithm with hierarchical fair competition modal (HFCDE). This article analyzed the efficiency of HFCDE and DESBS to solve the multi-modal s’ problems. HFCDE and DESBS support non-distributed population structured. HFCDE starts with set of feasible solution then it distributed them in the different hierarchy. HFCDE provides the fair competition. DESBS is another semi distributed, differential evolution algorithm.It starts with set of feasible solutions (population). Later it identifies the niches and create the sub-groups within population. Both HFCDE and DESBS have outperformed the other variant of state-of-art variants of DE. Here, the performance of DESBS and the HFCDE are rigorously tested on the multimodal problems. The success of DESBS over HFCDE in multi-modal difficulties managed to overcome the phenomena of elitism to resolve the complex problems, it has been observed that DESBS performs better than HFCDE in complex multi-modal scenarios.