Showing papers on "Linear programming published in 2009"
TL;DR: A simple costless modification to iterative thresholding is introduced making the sparsity–undersampling tradeoff of the new algorithms equivalent to that of the corresponding convex optimization procedures, inspired by belief propagation in graphical models.
Abstract: Compressed sensing aims to undersample certain high-dimensional signals yet accurately reconstruct them by exploiting signal characteristics. Accurate reconstruction is possible when the object to be recovered is sufficiently sparse in a known basis. Currently, the best known sparsity–undersampling tradeoff is achieved when reconstructing by convex optimization, which is expensive in important large-scale applications. Fast iterative thresholding algorithms have been intensively studied as alternatives to convex optimization for large-scale problems. Unfortunately known fast algorithms offer substantially worse sparsity–undersampling tradeoffs than convex optimization. We introduce a simple costless modification to iterative thresholding making the sparsity–undersampling tradeoff of the new algorithms equivalent to that of the corresponding convex optimization procedures. The new iterative-thresholding algorithms are inspired by belief propagation in graphical models. Our empirical measurements of the sparsity–undersampling tradeoff for the new algorithms agree with theoretical calculations. We show that a state evolution formalism correctly derives the true sparsity–undersampling tradeoff. There is a surprising agreement between earlier calculations based on random convex polytopes and this apparently very different theoretical formalism.
2,412 citations
TL;DR: A fast algorithm for overcomplete sparse decomposition, called SL0, is proposed, which tries to directly minimize the l 1 norm.
Abstract: In this paper, a fast algorithm for overcomplete sparse decomposition, called SL0, is proposed. The algorithm is essentially a method for obtaining sparse solutions of underdetermined systems of linear equations, and its applications include underdetermined sparse component analysis (SCA), atomic decomposition on overcomplete dictionaries, compressed sensing, and decoding real field codes. Contrary to previous methods, which usually solve this problem by minimizing the l 1 norm using linear programming (LP) techniques, our algorithm tries to directly minimize the l 1 norm. It is experimentally shown that the proposed algorithm is about two to three orders of magnitude faster than the state-of-the-art interior-point LP solvers, while providing the same (or better) accuracy.
1,033 citations
Book•
17 Aug 2009
TL;DR: The author did not name the algorithm, but it is likely to be VAR Optimization by Evolutionary Algorithm, which is a very simple and straightforward way to go about solving the problem of how to Optimize Power Dispatch.
Abstract: Preface. 1 Introduction. 1.1 Conventional Methods. 1.2 Intelligent Search Methods. 1.3 Application of Fuzzy Set Theory. 2 Power Flow Analysis. 2.1 Mathematical Model of Power Flow. 2.2 Newton-Raphson Method. 2.3 Gauss-Seidel Method. 2.4 P-Q decoupling Method. 2.5 DC Power Flow. 3 Sensitivity Calculation. 3.1 Introduction. 3.2 Loss Sensitivity Calculation. 3.3 Calculation of Constrained Shift Sensitivity Factors. 3.4 Perturbation Method for Sensitivity Analysis. 3.5 Voltage Sensitivity Analysis. 3.6 Real-Time Application of Sensitivity Factors. 3.7 Simulation Results. 3.8 Conclusion. 4 Classic Economic Dispatch. 4.1 Introduction. 4.2 Input-Output Characteristic of Generator Units. 4.3 Thermal System Economic Dispatch Neglecting Network Losses. 4.4 Calculation of Incremental Power Losses. 4.5 Thermal System Economic Dispatch with Network Losses. 4.6 Hydrothermal System Economic Dispatch. 4.7 Economic Dispatch by Gradient Method. 4.8 Classic Economic Dispatch by Genetic Algorithm. 4.9 Classic Economic Dispatch by Hopfi eld Neural Network. 5 Security-Constrained Economic Dispatch. 5.1 Introduction. 5.2 Linear Programming Method. 5.3 Quadratic Programming Method. 5.4 Network Flow Programming Method. 5.5 Nonlinear Convex Network Flow Programming Method. 5.6 Two-Stage Economic Dispatch Approach. 5.7 Security-Constrained ED by Genetic Algorithms. 6 Multiarea System Economic Dispatch. 6.1 Introduction. 6.2 Economy of Multiarea Interconnection. 6.3 Wheeling. 6.4 Multiarea Wheeling. 6.5 MAED Solved by Nonlinear Convex Network Flow Programming. 6.6 Nonlinear Optimization Neural Network Approach. 6.7 Total Transfer Capability Computation in Multiareas. 7 Unit Commitment. 7.1 Introduction. 7.2 Priority Method. 7.3 Dynamic Programming Method. 7.4 Lagrange Relaxation Method. 7.5 Evolutionary Programming-Based Tabu Search Method. 7.6 Particle Swarm Optimization for Unit Commitment. 7.7 Analytic Hierarchy Process. 8 Optimal Power Flow. 8.1 Introduction. 8.2 Newton Method. 8.3 Gradient Method. 8.4 Linear Programming OPF. 8.5 Modifi ed Interior Point OPF. 8.6 OPF with Phase Shifter. 8.7 Multiple-Objectives OPF. 8.8 Particle Swarm Optimization for OPF. 9 Steady-State Security Regions. 9.1 Introduction. 9.2 Security Corridors. 9.3 Traditional Expansion Method. 9.4 Enhanced Expansion Method. 9.5 Fuzzy Set and Linear Programming. 10 Reactive Power Optimization. 10.1 Introduction. 10.2 Classic Method for Reactive Power Dispatch. 10.3 Linear Programming Method of VAR Optimization. 10.4 Interior Point Method for VAR Optimization Problem. 10.5 NLONN Approach. 10.6 VAR Optimization by Evolutionary Algorithm. 10.7 VAR Optimization by Particle Swarm Optimization Algorithm. 10.8 Reactive Power Pricing Calculation. 11 Optimal Load Shedding. 11.1 Introduction. 11.2 Conventional Load Shedding. 11.3 Intelligent Load Shedding. 11.4 Formulation of Optimal Load Shedding. 11.5 Optimal Load Shedding with Network Constraints. 11.6 Optimal Load Shedding without Network Constraints. 11.7 Distributed Interruptible Load Shedding. 11.8 Undervoltage Load Shedding. 11.9 Congestion Management. 12 Optimal Reconfi guration of Electrical Distribution Network. 12.1 Introduction. 12.2 Mathematical Model of DNRC. 12.3 Heuristic Methods. 12.4 Rule-Based Comprehensive Approach. 12.5 Mixed-Integer Linear Programming Approach. 12.6 Application of GA to DNRC. 12.7 Multiobjective Evolution Programming to DNRC. 12.8 Genetic Algorithm Based on Matroid Theory. 13 Uncertainty Analysis in Power Systems. 13.1 Introduction. 13.2 Defi nition of Uncertainty. 13.3 Uncertainty Load Analysis. 13.4 Uncertainty Power Flow Analysis. 13.5 Economic Dispatch with Uncertainties. 13.6 Hydrothermal System Operation with Uncertainty. 13.7 Unit Commitment with Uncertainties. 13.8 VAR Optimization with Uncertain Reactive Load. 13.9 Probabilistic Optimal Power Flow. 13.10 Comparison of Deterministic and Probabilistic Methods. Author Biography. Index.
623 citations
Book•
26 Mar 2009
TL;DR: This chapter discusses the foundations of optimization, and some of the methods for unconstrained optimization, as well as topics from linear algebra, including the simplex method and other fundamentals.
Abstract: Preface Part I. Basics: 1. Optimization models 2. Fundamentals of optimization 3. Representation of linear constraints Part II. Linear Programming: 4. Geometry of linear programming 5. The simplex method 6. Duality and sensitivity 7. Enhancements of the simplex method 8. Network problems 9. Computational complexity of linear programming 10. Interior-point methods of linear programming Part III. Unconstrained Optimization: 11. Basics of unconstrained optimization 12. Methods for unconstrained optimization 13. Low-storage methods for unconstrained problems Part IV. Nonlinear Optimization: 14. Optimality conditions for constrained problems 15. Feasible-point methods 16. Penalty and barrier methods Part V. Appendices: Appendix A. Topics from linear algebra Appendix B. Other fundamentals Appendix C. Software Bibliography Index.
524 citations
01 Aug 2009
TL;DR: In this paper, the authors present a process discovery algorithm using concepts taken from the language-based theory of regions, a well-known Petri net research area and identify a number of shortcomings of this theory from the process discovery perspective, and provide solutions based on integer linear programming.
Abstract: The research domain of process discovery aims at constructing a process model (e.g. a Petri net) which is an abstract representation of an execution log. Such a model should (1) be able to reproduce the log under consideration and (2) be independent of the number of cases in the log. In this paper, we present a process discovery algorithm where we use concepts taken from the language-based theory of regions, a well-known Petri net research area. We identify a number of shortcomings of this theory from the process discovery perspective, and we provide solutions based on integer linear programming.
361 citations
TL;DR: Computational results indicate that by using the strengthened formulations of PCLP, instances that are considerably larger than have been considered before can be solved to optimality.
Abstract: Linear programs with joint probabilistic constraints (PCLP) are difficult to solve because the feasible region is not convex. We consider a special case of PCLP in which only the right-hand side is random and this random vector has a finite distribution. We give a mixed-integer programming formulation for this special case and study the relaxation corresponding to a single row of the probabilistic constraint. We obtain two strengthened formulations. As a byproduct of this analysis, we obtain new results for the previously studied mixing set, subject to an additional knapsack inequality. We present computational results which indicate that by using our strengthened formulations, instances that are considerably larger than have been considered before can be solved to optimality.
327 citations
TL;DR: This paper focuses on detecting critical nodes, or nodes whose deletion results in the minimum pair-wise connectivity among the remaining nodes, and proposes a heuristic for the problem which exploits the combinatorial structure of the graph.
325 citations
TL;DR: To solve the CDLP for real-size networks, it is proved that the associated column generation subproblem is indeed NP-hard and a simple, greedy heuristic is proposed to overcome the complexity of an exact algorithm.
Abstract: During the past few years, there has been a trend to enrich traditional revenue management models built upon the independent demand paradigm by accounting for customer choice behavior. This extension involves both modeling and computational challenges. One way to describe choice behavior is to assume that each customer belongs to a segment, which is characterized by a consideration set, i.e., a subset of the products provided by the firm that a customer views as options. Customers choose a particular product according to a multinomial-logit criterion, a model widely used in the marketing literature.
In this paper, we consider the choice-based, deterministic, linear programming model (CDLP) of Gallego et al. (2004) [Gallego, G., G. Iyengar, R. Phillips, A. Dubey. 2004. Managing flexible products on a network. Technical Report CORC TR-2004-01, Department of Industrial Engineering and Operations Research, Columbia University, New York], and the follow-up dynamic programming decomposition heuristic of van Ryzin and Liu (2008) [van Ryzin, G. J., Q. Liu. 2008. On the choice-based linear programming model for network revenue management. Manufacturing Service Oper. Management10(2) 288--310]. We focus on the more general version of these models, where customers belong to overlapping segments. To solve the CDLP for real-size networks, we need to develop a column generation algorithm. We prove that the associated column generation subproblem is indeed NP-hard and propose a simple, greedy heuristic to overcome the complexity of an exact algorithm. Our computational results show that the heuristic is quite effective and that the overall approach leads to high-quality, practical solutions.
303 citations
TL;DR: An approach to multiattribute decision making with incomplete attribute weight information where individual assessments are provided as interval-valued intuitionistic fuzzy numbers (IVIFNs) is proposed by employing a series of optimization models.
300 citations
TL;DR: Recoverable robustness combines the flexibility of stochastic programming with the tractability and performances guarantee of the classical robust approach and is exemplified in delay resistant, periodic and aperiodic timetabling problems, and train platforming.
Abstract: We present a new concept for optimization under uncertainty: recoverable robustness A solution is recovery robust if it can be recovered by limited means in all likely scenarios Specializing the general concept to linear programming we can show that recoverable robustness combines the flexibility of stochastic programming with the tractability and performances guarantee of the classical robust approach We exemplify recoverable robustness in delay resistant, periodic and aperiodic timetabling problems, and train platforming
289 citations
TL;DR: In this article, a bilevel model for transmission expansion planning within a market environment, where producers and consumers trade freely electric energy through a pool, is presented, where the target of the transmission planner, modeled through the upper-level problem, is to minimize network investment cost while facilitating energy trading.
Abstract: We present a bilevel model for transmission expansion planning within a market environment, where producers and consumers trade freely electric energy through a pool. The target of the transmission planner, modeled through the upper-level problem, is to minimize network investment cost while facilitating energy trading. This upper-level problem is constrained by a collection of lower-level market clearing problems representing pool trading, and whose individual objective functions correspond to social welfare. Using the duality theory the proposed bilevel model is recast as a mixed-integer linear programming problem, which is solvable using branch-and-cut solvers. Detailed results from an illustrative example and a case study are presented and discussed. Finally, some relevant conclusions are drawn.
TL;DR: In this article, a Branch-and-Bound bounding procedure was proposed for finding exact solutions of Max-Cut, the problem of finding a cut of maximum weight in a weighted graph.
Abstract: We present a method for finding exact solutions of Max-Cut, the problem of finding a cut of maximum weight in a weighted graph. We use a Branch-and-Bound setting that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a “nearly optimal” solution of the basic semidefinite Max-Cut relaxation, strengthened by triangle inequalities. The expensive part of our bounding procedure is solving the basic semidefinite relaxation of the Max-Cut problem, which has to be done several times during the bounding process. We review other solution approaches and compare the numerical results with our method. We also extend our experiments to instances of unconstrained quadratic 0–1 optimization and to instances of the graph equipartition problem. The experiments show that our method nearly always outperforms all other approaches. In particular, for dense graphs, where linear programming-based methods fail, our method performs very well. Exact solutions are obtained in a reasonable time for any instance of size up to n = 100, independent of the density. For some problems of special structure we can solve even larger problem classes. We could prove optimality for several problems of the literature where, to the best of our knowledge, no other method is able to do so.
TL;DR: This paper proposes an algorithm, MAPEL, which globally converges to a global optimal solution of the WTM problem in the general SINR regime and provides an important benchmark for performance evaluation of other heuristic algorithms targeting the same problem.
Abstract: Achieving weighted throughput maximization (WTM) through power control has been a long standing open problem in interference-limited wireless networks. The complicated coupling between the mutual interferences of links gives rise to a non-convex optimization problem. Previous work has considered the WTM problem in the high signal to interference-and-noise ratio (SINR) regime, where the problem can be approximated and transformed into a convex optimization problem through proper change of variables. In the general SINR regime, however, the approximation and transformation approach does not work. This paper proposes an algorithm, MAPEL, which globally converges to a global optimal solution of the WTM problem in the general SINR regime. The MAPEL algorithm is designed based on three key observations of the WTM problem: (1) the objective function is monotonically increasing in SINR, (2) the objective function can be transformed into a product of exponentiated linear fraction functions, and (3) the feasible set of the equivalent transformed problem is always ldquonormalrdquo, although not necessarily convex. The MAPEL algorithm finds the desired optimal power control solution by constructing a series of polyblocks that approximate the feasible SINR region in an increasing precision. Furthermore, by tuning the approximation factor in MAPEL, we could engineer a desirable tradeoff between optimality and convergence time. MAPEL provides an important benchmark for performance evaluation of other heuristic algorithms targeting the same problem. With the help of MAPEL, we evaluate the performance of several existing algorithms through extensive simulations.
TL;DR: In this paper, a fuzzy-random interval programming (FRIP) model is proposed to identify optimal strategies in the planning of energy management systems under multiple uncertainties through the development of a FRIP model, which is based on an integration of the existing interval linear programming, superiority-inferiority-based fuzzy-stochastic programming (SI-FSP) and mixed integer linear programming (MILP).
TL;DR: The proposed lower bound is superior to the ones presented in the literature and that the exact algorithm can solve, for the first time ever, several test instances of all problem types considered.
Abstract: This paper presents a unified exact method for solving an extended model of the well-known Capacitated Vehicle Routing Problem (CVRP), called the Heterogenous Vehicle Routing Problem (HVRP), where a mixed fleet of vehicles having different capacities, routing and fixed costs is used to supply a set of customers. The HVRP model considered in this paper contains as special cases: the Single Depot CVRP, all variants of the HVRP presented in the literature, the Site-Dependent Vehicle Routing Problem (SDVRP) and the Multi-Depot Vehicle Routing Problem (MDVRP). This paper presents an exact algorithm for the HVRP based on the set partitioning formulation. The exact algorithm uses three types of bounding procedures based on the LP-relaxation and on the Lagrangean relaxation of the mathematical formulation. The bounding procedures allow to reduce the number of variables of the formulation so that the resulting problem can be solved by an integer linear programming solver. Extensive computational results over the main instances from the literature of the different variants of HVRPs, SDVRP and MDVRP show that the proposed lower bound is superior to the ones presented in the literature and that the exact algorithm can solve, for the first time ever, several test instances of all problem types considered.
TL;DR: In this article, an inexact community-scale energy model (ICS-EM) is developed for planning renewable energy management (REM) systems under uncertainty, which allows uncertainties presented as both probability distributions and interval values to be incorporated within a general optimization framework.
Abstract: In this study, an inexact community-scale energy model (ICS-EM) has been developed for planning renewable energy management (REM) systems under uncertainty. This method is based on an integration of the existing interval linear programming (ILP), chance-constrained programming (CCP) and mixed integer linear programming (MILP) techniques. ICS-EM allows uncertainties presented as both probability distributions and interval values to be incorporated within a general optimization framework. It can also facilitate capacity-expansion planning for energy-production facilities within a multi-period and multi-option context. Complexities in energy management systems can be systematically reflected, thus applicability of the modeling process can be highly enhanced. The developed method has then been applied to a case of long-term renewable energy management planning for three communities. Useful solutions for the planning of energy management systems have been generated. Interval solutions associated with different risk levels of constraint violation have been obtained. They can be used for generating decision alternatives and thus help decision makers identify desired policies under various economic and system-reliability constraints. The generated solutions can also provide desired energy resource/service allocation and capacity-expansion plans with a minimized system cost, a maximized system reliability and a maximized energy security. Tradeoffs between system costs and constraint-violation risks can also be tackled. Higher costs will increase system stability, while a desire for lower system costs will run into a risk of potential instability of the management system. They are helpful for supporting (a) adjustment or justification of allocation patterns of energy resources and services, (b) formulation of local policies regarding energy consumption, economic development and energy structure, and (c) analysis of interactions among economic cost, system reliability and energy-supply security.
TL;DR: In this paper, a retailer decides its level of involvement in the futures market and in the pool as well as the selling price offered to its potential clients with the goal of maximizing the expected profit at a given risk level.
Abstract: This paper presents a bilevel programming approach to solve the medium-term decision-making problem faced by a power retailer. A retailer decides its level of involvement in the futures market and in the pool as well as the selling price offered to its potential clients with the goal of maximizing the expected profit at a given risk level. Uncertainty on future pool prices, client demands, and rival-retailer prices is accounted for via stochastic programming. Unlike in previous approaches, client response to retail price and competition among rival retailers are both explicitly considered in the proposed bilevel model. The resulting nonlinear bilevel programming formulation is transformed into an equivalent single-level mixed-integer linear programming problem by replacing the lower-level optimization by its Karush-Kuhn-Tucker optimality conditions and converting a number of nonlinearities to linear equivalents using some well-known integer algebra results. A realistic case study is solved to illustrate the efficient performance of the proposed methodology.
TL;DR: This work develops a column generation algorithm to solve the problem for a multinomial logit choice model with disjoint consideration sets (MNLD), and derives a bound as a by-product of a decomposition heuristic.
Abstract: We consider a network revenue management problem where customers choose among open fare products according to some prespecified choice model. Starting with a Markov decision process (MDP) formulation, we approximate the value function with an affine function of the state vector. We show that the resulting problem provides a tighter bound for the MDP value than the choice-based linear program. We develop a column generation algorithm to solve the problem for a multinomial logit choice model with disjoint consideration sets (MNLD). We also derive a bound as a by-product of a decomposition heuristic. Our numerical study shows the policies from our solution approach can significantly outperform heuristics from the choice-based linear program.
TL;DR: In this paper, the authors proposed a method to approximate the nonlinear objective function of the problem by means of piecewise-linear functions, so that UC can be approximated by an mixed-integer linear program (MILP).
Abstract: The short-term unit commitment (UC) problem in hydrothermal power generation is a large-scale, mixed-integer nonlinear program, which is difficult to solve efficiently, especially for large-scale instances. It is possible to approximate the nonlinear objective function of the problem by means of piecewise-linear functions, so that UC can be approximated by an mixed-integer linear program (MILP); applying the available efficient general-purpose MILP solvers to the resulting formulations, good quality solutions can be obtained in a relatively short amount of time. We build on this approach, presenting a novel way to approximating the nonlinear objective function based on a recently developed class of valid inequalities for the problem, called ldquoperspective cuts.rdquo At least for many realistic instances of a general basic formulation of UC, an MILP-based heuristic obtains comparable or slightly better solutions in less time when employing the new approach rather than the standard piecewise linearizations, while being not more difficult to implement and use. Furthermore, ldquodynamicrdquo formulations, whereby the approximation is iteratively improved, provide even better results if the approximation is appropriately controlled.
TL;DR: In this survey, a classification of 24 asymmetric traveling salesman problem (ATSP) formulations is presented and the strength of their LP relaxations is discussed and known relationships from the literature are reviewed.
TL;DR: In this article, the authors present mixed-integer linear programming formulations of transaction costs, a constraint limiting the number of stocks that can be purchased and a limit on the total transaction cost incurred.
TL;DR: This work provides general deterministic primal-dual algorithms for online fractional covering and packing problems and also provides deterministic algorithms for several integral online covering andpacking problems.
Abstract: We study a wide range of online covering and packing optimization problems. In an online covering problem, a linear cost function is known in advance, but the linear constraints that define the feasible solution space are given one by one, in rounds. In an online packing problem, the profit function as well as the packing constraints are not known in advance. In each round additional information (i.e., a new variable) about the profit function and the constraints is revealed. An online algorithm needs to maintain a feasible solution in each round; in addition, the solutions generated over the different rounds need to satisfy a monotonicity property. We provide general deterministic primal-dual algorithms for online fractional covering and packing problems. We also provide deterministic algorithms for several integral online covering and packing problems. Our algorithms are designed via a novel online primal-dual technique and are evaluated via competitive analysis.
19 Apr 2009
TL;DR: This paper incorporates convex analysis and Craig's criterion to develop a minimum-volume enclosing simplex (MVES) formulation for hyperspectral unmixing, and provides a non-heuristic guarantee of the MVES problem formulation, where the existence of pure pixels is proved to be a sufficient condition for MVES to perfectly identify the true endmembers.
Abstract: Hyperspectral unmixing aims at identifying the hidden spectral signatures (or endmembers) and their corresponding proportions (or abundances) from an observed hyperspectral scene. Many existing approaches to hyperspectral unmixing rely on the pure-pixel assumption, which may be violated for highly mixed data. A heuristic unmixing criterion without requiring the pure-pixel assumption has been reported by Craig: The endmember estimates are determined by the vertices of a minimum-volume simplex enclosing all the observed pixels. In this paper, using convex analysis, we show that the hyperspectral unmixing by Craig's criterion can be formulated as an optimization problem of finding a minimum-volume enclosing simplex (MVES). An algorithm that cyclically solves the MVES problem via linear programs (LPs) is also proposed. Some Monte Carlo simulations are provided to demonstrate the efficacy of the proposed MVES algorithm.
02 Aug 2009
TL;DR: This formulation of the problem of non-projective dependency parsing as a polynomial-sized integer linear program is able to handle non-local output features in an efficient manner and is compatible with prior knowledge encoded as hard constraints, and can also learn soft constraints from data.
Abstract: We formulate the problem of non-projective dependency parsing as a polynomial-sized integer linear program. Our formulation is able to handle non-local output features in an efficient manner; not only is it compatible with prior knowledge encoded as hard constraints, it can also learn soft constraints from data. In particular, our model is able to learn correlations among neighboring arcs (siblings and grandparents), word valency, and tendencies toward nearly-projective parses. The model parameters are learned in a max-margin framework by employing a linear programming relaxation. We evaluate the performance of our parser on data in several natural languages, achieving improvements over existing state-of-the-art methods.
01 Jan 2009
TL;DR: This survey examines ways to reformulate integer and mixed integer programs so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm.
Abstract: We examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover reformulations based on decomposition, such as Lagrangean relaxation, the Dantzig-Wolfe reformulation and the resulting column generation and branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.
TL;DR: In this paper, an ordinal optimisation (OO) method for specifying the locations and capacities of distributed generation (DG) such that a trade-off between loss minimisation and DG capacity maximisation is achieved.
Abstract: This study presents an ordinal optimisation (OO) method for specifying the locations and capacities of distributed generation (DG) such that a trade-off between loss minimisation and DG capacity maximisation is achieved. The OO approach consists of three main phases. First, the large search space of potential combinations of DG locations is represented by sampling a relatively small number of alternatives. Second, the objective function value for each of the sampled alternatives is evaluated using a crude but computationally efficient linear programming model. Third, the top-s alternatives from the crude model evaluation are simulated via an exact non-linear programming optimal power flow (OPF) programme to find the best DG locations and capacities. OO theory allows computing the size s of the selected subset such that it contains at least k designs from among the true top-g samples with a pre-specified alignment probability AP. This study discusses problem-specific approaches for sampling, crude model implementation and subset size selection. The approach is validated by comparing with recently published results of a hybrid genetic algorithm OPF applied to a 69-node distribution network operating under Ofgem (UK) financial incentives for distribution network operators.
TL;DR: Several integrated models for the optimization of airline scheduling are presented for the first time, and solved by applying an enhanced Benders decomposition method combined with accelerated column generation, which significantly reduces airline costs.
TL;DR: This paper model the network retrofit problem as a two-stage stochastic programming problem that optimizes a mean-risk objective of the system loss and develops an efficient algorithm to efficiently handle the binary integer variables in the first stage and the nonlinear recourse in the second stage of the model formulation.
TL;DR: An integrated fuzzy and linear programming approach to the supplier selection problem is presented and order quantities assigned to each supplier according to the linear programming model.
Abstract: In supply chain management process, the firm select best supplier takes the competitive advantage to other companies. Thus, supplier selection is an important issue and with the multiple criteria decision-making approach, the supplier selection problem includes both tangible and intangible factors. This paper is aimed to present an integrated fuzzy and linear programming approach to the problem. Firstly, linguistic values expressed in trapezoidal fuzzy numbers are applied to assess weights and ratings of supplier selection criteria. Then a hierarchy multiple model based on fuzzy set theory is expressed and fuzzy positive and negative ideal solutions are used to find each supplier's closeness coefficient. Finally, a linear programming model based on the coefficients of suppliers, buyer's budgeting, suppliers' quality and capacity constraints is developed and order quantities assigned to each supplier according to the linear programming model. The integrated model is illustrated by an example in a textile firm.
TL;DR: This work provides a linear bilevel programming formulation for this hazmat transportation network design problem that takes into account both total risk minimization and risk equity, and provides a heuristic algorithm for the bileVEL model able to always find a stable solution.
Abstract: In this work we consider the following hazmat transportation network design problem. A given set of hazmat shipments has to be shipped over a road transportation network in order to transport a given amount of hazardous materials from specific origin points to specific destination points, and we assume there are regional and local government authorities that want to regulate the hazmat transportations by imposing restrictions on the amount of hazmat traffic over the network links. In particular, the regional authority aims to minimize the total transport risk induced over the entire region in which the transportation network is embedded, while local authorities want the risk over their local jurisdictions to be the lowest possible, forcing the regional authority to assure also risk equity. We provide a linear bilevel programming formulation for this hazmat transportation network design problem that takes into account both total risk minimization and risk equity. We transform the bilevel model into a single-level mixed integer linear program by replacing the second level (follower) problem by its KKT conditions and by linearizing the complementary constraints, and then we solve the MIP problem with a commercial optimization solver. The optimal solution may not be stable, and we provide an approach for testing its stability and for evaluating the range of its solution values when it is not stable. Moreover, since the bilevel model is difficult to be solved optimally and its optimal solution may not be stable, we provide a heuristic algorithm for the bilevel model able to always find a stable solution. The proposed bilevel model and heuristic algorithm are experimented on real scenarios of an Italian regional network.