Showing papers on "Linear-fractional programming published in 2015"

Mixed Integer Linear Programming Formulation Techniques

Abstract: A wide range of problems can be modeled as Mixed Integer Linear Programming (MIP) problems using standard formulation techniques. However, in some cases the resulting MIP can be either too weak or too large to be effectively solved by state of the art solvers. In this survey we review advanced MIP formulation techniques that result in stronger and/or smaller formulations for a wide class of problems.

Global optimization method for network design problem with stochastic user equilibrium

Abstract: In this paper, we consider the continuous road network design problem with stochastic user equilibrium constraint that aims to optimize the network performance via road capacity expansion. The network flow pattern is subject to stochastic user equilibrium, specifically, the logit route choice model. The resulting formulation, a nonlinear nonconvex programming problem, is firstly transformed into a nonlinear program with only logarithmic functions as nonlinear terms, for which a tight linear programming relaxation is derived by using an outer-approximation technique. The linear programming relaxation is then embedded within a global optimization solution algorithm based on range reduction technique, and the proposed approach is proved to converge to a global optimum.

Deciphering and handling uncertainty in shale gas supply chain design and optimization: Novel modeling framework and computationally efficient solution algorithm

Abstract: The optimal design and operations of shale gas supply chains under uncertainty of estimated ultimate recovery (EUR) is addressed. A two-stage stochastic mixed-integer linear fractional programming (SMILFP) model is developed to optimize the levelized cost of energy generated from shale gas. In this model, both design and planning decisions are considered with respect to shale well drilling, shale gas production, processing, multiple end-uses, and transportation. To reduce the model size and number of scenarios, we apply a sample average approximation method to generate scenarios based on the real-world EUR data. In addition, a novel solution algorithm integrating the parametric approach and the L-shaped method is proposed for solving the resulting SMILFP problem within a reasonable computational time. The proposed model and algorithm are illustrated through a case study based on the Marcellus shale play, and a deterministic model is considered for comparison. © 2015 American Institute of Chemical Engineers AIChE J, 61: 3739–3755, 2015

A new approach on solving intuitionistic fuzzy linear programming problem

Abstract: In this paper, we propose a new approach for solving Intuitionistic Fuzzy Linear Programming Problems (IFLPP) involving triangular intuitionistic fuzzy numbers (TIFN). We introduce a new algorithm for the solution of an Intuitionistic Fuzzy Linear Programming Problem without converting in to one or more classical Linear Programming Problems. Numerical examples are provided to show the efficiency of the proposed algorithm.

Linear and Mixed Integer Programming for Portfolio Optimization

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A polynomial projection algorithm for linear feasibility problems

Abstract: We propose a polynomial algorithm for linear feasibility problems. The algorithm represents a linear problem in the form of a system of linear equations and non-negativity constraints. Then it uses a procedure which either finds a solution for the respective homogeneous system or provides the information based on which the algorithm rescales the homogeneous system so that its feasible solutions in the unit cube get closer to the vector of all ones. In a polynomial number of calls to the procedure the algorithm either proves that the original system is infeasible or finds a solution in the relative interior of the feasible set.

Water Resources Management Models Based on Two-Level Linear Fractional Programming Method under Uncertainty

Abstract: A two-level linear fractional water management (TLFWM) model based on interactive fuzzy programming is developed. The model can solve multi-objective problems quantitatively, particularly for the ratio multi-objective problems (e.g., benefit per unit of water in water resources management system). Furthermore, it takes the cooperation relationship between decision makers into consideration. Considering the stochastic features of runoff, chance-constrained programming is integrated into the TLFWM model framework. A stochastic two-level linear fractional chance-constrained water management (STLFCWM) model is thus proposed with different flow levels involved in the STLFCWM model in the form of probabilities as well. The developed two models are applied to a real case study to allocate the limited water resources to different water users. The obtained solutions can demonstrate the feasibility and suitability of the TLFWM model and STLFCWM model and thus help decision makers to identify desired water r...

A duality approach for solving bounded linear programming problems with fuzzy variables based on ranking functions and its application in bounded transportation problems

Abstract: There are several methods, in the literature, for solving fuzzy variable linear programming problems fuzzy linear programming in which the right-hand-side vectors and decision variables are represented by trapezoidal fuzzy numbers. In this paper, the shortcomings of some existing methods are pointed out and to overcome these shortcomings a new method based on the bounded dual simplex method is proposed to determine the fuzzy optimal solution of that kind of fuzzy variable linear programming problems in which some or all variables are restricted to lie within lower and upper bounds. To illustrate the proposed method, an application example is solved and the obtained results are given. The advantages of the proposed method over existing methods are discussed. Also, one application of this algorithm in solving bounded transportation problems with fuzzy supplies and demands is dealt with. The proposed method is easy to understand and to apply for determining the fuzzy optimal solution of bounded fuzzy variable linear programming problems occurring in real-life situations.

A Fuzzy Programming Approach for Solving Intuitionistic Fuzzy Linear Fractional Programming Problem

Abstract: This paper develops an approach for solving intuitionistic fuzzy linear fractional programming problem (IFLFPP). The cost of the objective function, the resources, and the technological coefficients are taken to be triangular intuitionistic fuzzy numbers. Here, the IFLFP problem is transformed into an equivalent crisp multi-objective linear fractional programming problem (MOLFPP). By using fuzzy mathematical programming approach, the transformed MOLFPP is reduced into a single objective linear programming problem (LPP) which can be solved easily by suitable LPP algorithm. The proposed procedure is illustrated by a numerical example.

Two-Stage Chance-Constrained Fractional Programming for Sustainable Water Quality Management under Uncertainty

Abstract: In this study, a two-stage chance-constrained fractional programming (TCFP) method is developed for dealing with water quality management problems associated with stochastic inputs. Two-stage chance-constrained fractional programming is a hybrid of stochastic linear fractional programming (SLFP) and two-stage stochastic programming (TSP) methods. It can not only balance objectives of two aspects through converting a bi-objective problem into a ratio one but can also analyze various policy scenarios when the promised production targets are violated. For demonstrating its advantages, the proposed TCFP method is applied to a case study of water quality management where managers have to consider conflicting objectives between economic development and environmental conservation, as well as stochastic features expressed as probability distributions. The obtained solutions under different significance levels can help managers to identify desired policies under various environmental, economic, and constra...

Linear optimal tracking control: An adaptive dynamic programming approach

Abstract: This paper addresses the optimal output regulation problem of linear systems with unknown system dynamics. The exogenous signal is presumed to be generated by a continuous-time linear exosystem. Firstly, we formulate the linear optimal output regulation problem (LOORP). Then, we give an offline solution of LOORP to design the optimal static state-feedback servoregulator by solving an algebraic Riccati equation (ARE) and a regulator equation. Instead of solving these two equations directly, by using state, input and exogenous signals collected online, we employ an approximate/adaptive dynamic programming (ADP) technique to seek online approximations of above equations whereby we get the approximated optimal servoregulator. Rigorous stability analysis shows that the closed-loop linear system is exponentially stable. Also, the output of the plant asymptotically tracks the given reference. Simulation results demonstrate the effectiveness of the proposed approach.

Linear Programming Method for Solving Semi-Latticized Fuzzy Relation Geometric Programming with Max-Min Composition

Abstract: Motivated by application in BitTorrent-like Peer-to-Peer resource sharing system, we introduce the maximization and minimization semi-latticized fuzzy relation geometric programming problems with max-min composition in this paper. The objective function in the proposed problem in nonlinear and the feasible domain is non-convex. The maximization problem is converted into a fuzzy relation monomial geometric programming and then solved. However, this approach is not effective for the minimization one. The linear programming method is applied to deal with the minimization problem. A step-to-step algorithm is develop to carried out the linear programming method and tow illustrative examples are provided at last.

Optimal sampling-based Feedback Motion Trees among obstacles for controllable linear systems with linear constraints

Abstract: The RRT* algorithm has efficiently extended Rapidly-exploring Random Trees (RRTs) to endow it with asymptotic optimality. We propose Goal-Rooted Feedback Motion Trees (GR-FMTs) that honor state/input constraints and generate collision-free feedback policies. Given analytic solutions for optimal local steering, GR-FMTs obtain and realize safe, dynamically feasible, and asymptotically optimal trajectories toward goals. Second, for controllable linear systems with linear state/input constraints, we propose a fast method for local steering, based on polynomial basis functions and segmentation. GR-FMTs with the method obtain and realize trajectories that are collision-free, dynamically feasible under constraints, and asymptotically optimal within a set we define. The formulation includes linear or quadratic programming of small sizes, where constraints are identified by root-finding in low or medium order of polynomials and added progressively.

Linear programming and dynamics

Abstract: In a Hilbert space we consider the linear boundary value problem of optimal control based on the linear dynamics and the terminal linear programming problem at the right end of the time interval. There is provided a saddle-point method to solve it. Convergence of the method is proved.

Inverse parametric linear/quadratic programming problem for continuous PWA functions defined on polyhedral partitions of polyhedra

Abstract: Constructive solution to inverse parametric linear/quadratic programming problems has recently been investigated and shown to be solvable via convex liftings [15], [14]. These results were stated and solved starting from polytopic partitions of a polytope in the parameter space. Therefore, the case of polyhedral partitions of unbounded polyhedra, was not handled by this method and deserves a complete characterization to address the general inverse optimality problem. This paper has as main objective to overcome the unboundedness limitation of the given polyhedral partition and to extend the constructive solution put forward in [14] for this omitted case.

Any discontinuous PWA function is optimal solution to a parametric linear programming problem

Abstract: Recent studies have investigated the continuous functions in terms of inverse optimality. The continuity is a primordial structural property which is exploited in order to link a given piecewise affine (PWA) function to an optimization problem. The aim of this work is to deepen the study of the PWA functions in the inverse optimality context and specifically deal with the presence of discontinuities. First, it will be shown that a solution to the inverse optimality problem exists via a constructive argument. The loss of continuity will have an implication on the structure of the optimization problem which, albeit convex, turns to have a set-valued optimal solution. As a consequence, the original PWA function will represent an optimal solution but the uniqueness is lost. From the numerical point of view, we introduce an algorithm to construct an optimization problem that admits a given discontinuous PWA function as an optimal solution. This construction is shown to rely on convex liftings. A numerical example is considered to illustrate the proposal.

Logical analysis of multi-class data

Abstract: Logical Analysis of Data (LAD) is a two-class learning algorithm which integrates principles of combinatorics, optimization, and the theory of Boolean functions. This paper proposes an algorithm based on mixed integer linear programming to extend the LAD methodology to solve multi-class classification problems, where One-vs-All (OvA) learning models are efficiently constructed to classify observations in predefined classes. The utility of the proposed approach is demonstrated through experiments on multi-class benchmark datasets.

A single stage single constraints linear fractional programming problem: An approach

Abstract: In the present paper we present a new method for solving a class of single stage single constraints linear fractional programming (LFP) problem. The proposed method is based on transformation the objective value and the constraints also. After reducing the fractional program in to equivalent linear program with the help of transformation technique, after that we apply Simplex method to find objective value. Numerical examples are constructed to show the applicability of the above technique

Equivalence between polyhedral projection, multiple objective linear programming and vector linear programming

Abstract: Let a polyhedral convex set be given by a finite number of linear inequalities and consider the problem to project this set onto a subspace. This problem, called polyhedral projection problem, is shown to be equivalent to multiple objective linear programming. The number of objectives of the multiple objective linear program is by one higher than the dimension of the projected polyhedron. The result implies that an arbitrary vector linear program (with arbitrary polyhedral ordering cone) can be solved by solving a multiple objective linear program (i.e. a vector linear program with the standard ordering cone) with one additional objective space dimension.

Benchmarks for Current Linear and Mixed Integer Optimization Solvers

Abstract: Linear programming (LP) and mixed integer linear programming (MILP) problems belong among very important class of problems that find their applications in various managerial consequences. The aim of the paper is to discuss computational performance of current optimization packages for solving large scale LP and MILP optimization problems. Current market with LP and MILP solvers is quite extensive. Probably among the most powerful solvers GUROBI 6.0, IBM ILOG CPLEX 12.6.1, and XPRESS Optimizer 27.01 belong. Their attractiveness for academic research is given, except their computational performance, by their free availability for academic purposes. The solvers are tested on the set of selected problems from MIPLIB 2010 library that contains 361 test instances of different hardness (easy, hard, and not solved).

Solving tri-level programming problems using a particle swarm optimization algorithm

Abstract: Tri-level programming, a special case of multilevel programming, arises to deal with decentralized decision-making problems that feature interacting decision entities distributed throughout three hierarchical levels. As tri-level programming problems are strongly NP-hard and the existing solution approaches lack universality in solving such problems, the purpose of this study is to propose an intelligence-based heuristic algorithm to solve tri-level programming problems involving linear and nonlinear versions. In this paper, we first propose a general tri-level programming problem and discuss related theoretical properties. A particle swarm optimization (PSO) algorithm is then developed to solve the tri-level programming problem. Lastly, a numerical example is adopted to illustrate the effectiveness of the proposed PSO algorithm.

A New Approach of Solving Linear Fractional Programming Problem (LFP) by Using Computer Algorithm

Abstract: In this paper, we study a new approach for solving linear fractional programming problem (LFP) by converting it into a single Linear Programming (LP) Problem, which can be solved by using any type of linear fractional programming technique. In the objective function of an LFP, if β is negative, the available methods are failed to solve, while our proposed method is capable of solving such problems. In the present paper, we propose a new method and develop FORTRAN programs to solve the problem. The optimal LFP solution procedure is illustrated with numerical examples and also by a computer program. We also compare our method with other available methods for solving LFP problems. Our proposed method of linear fractional programming (LFP) problem is very simple and easy to understand and apply.

An interval approach based on expectation optimization for fuzzy random bilevel linear programming problems

Abstract: This paper considers a class of bilevel linear programming problems in which the coefficients of both objective functions are fuzzy random variables. The main idea of this paper is to introduce the Pareto optimal solution in a multi-objective bilevel programming problem as a solution for a fuzzy random bilevel programming problem. To this end, a stochastic interval bilevel linear programming problem is first introduced in terms of α-cuts of fuzzy random variables. On the basis of an order relation of interval numbers and the expectation optimization model, the stochastic interval bilevel linear programming problem can be transformed into a multi-objective bilevel programming problem which is solved by means of weighted linear combination technique. In order to compare different optimal solutions depending on different cuts, two criterions are given to provide the preferable optimal solutions for the upper and lower level decision makers respectively. Finally, a production planning problem is given to demonstrate the feasibility of the proposed approach.

On Tinhofer’s Linear Programming Approach to Isomorphism Testing

Abstract: Exploring a linear programming approach to Graph Isomorphism, Tinhofer (1991) defined the notion of compact graphs: A graph is compact if the polytope of its fractional automorphisms is integral. Tinhofer noted that isomorphism testing for compact graphs can be done quite efficiently by linear programming. However, the problem of characterizing and recognizing compact graphs in polynomial time remains an open question. In this paper we make new progress in our understanding of compact graphs. Our results are summarized below: We show that all graphs G which are distinguishable from any non-isomorphic graph by the classical color-refinement procedure are compact. In other words, the applicability range for Tinhofer’s linear programming approach to isomorphism testing is at least as large as for the combinatorial approach based on color refinement. Exploring the relationship between color refinement and compactness further, we study related combinatorial and algebraic graph properties introduced by Tinhofer and Godsil. We show that the corresponding classes of graphs form a hierarchy and we prove that recognizing each of these graph classes is P-hard. In particular, this gives a first complexity lower bound for recognizing compact graphs.

Linear and Mixed Integer Programming

Abstract: Linear Programming (LP) is one of the most famous optimization techniques introduced independently by Kantarowitsch in 1939 and by Dantzig in 1949 (Kreko, 1973). LP is applicable in decision situations where quantities (variables) can take any real values only restricted by linear (in-) equalities, e. g. for representing capacity constraints. Still, LP has turned out to be very useful for many companies so far. LP is used in APS e. g. in Master Planning as well as in Distribution and Transport Planning. Very powerful solution algorithms have been developed (named solvers), solving LP models with thousands of variables and constraints within a few minutes on a personal computer.