http://repub.eur.nl/res/aut/2350/ List of Publications en http://repub.eur.nl/static-eur/img/logo.png http://repub.eur.nl http://repub.eur.nl/res/pub/16675/ 2009-09-09T00:00:00Z We consider an economic order quantity type model with unit out-of-pocket holding costs, unit opportunity costs of holding, fixed ordering costs and general transportation costs. For these models, we analyze the associated optimization problem and derive an easy procedure for determining a bounded interval containing the optimal cycle length. Also for a special class of transportation functions, like the carload discount schedule, we specialize these results and give fast and easy algorithms to calculate the optimal lot size and the corresponding optimal order-up-to-level. http://repub.eur.nl/res/pub/16323/ 2009-01-01T00:00:00Z In this paper, we introduce robust versions of the classical static and dynamic single-leg seat allocation models. These robust models take into account the inaccurate estimates of the underlying probability distributions. As observed by simulation experiments, it turns out that for these robust versions the variability compared to their classical counterparts is considerably reduced with a negligible decrease in average revenue. http://repub.eur.nl/res/pub/13050/ 2008-08-21T00:00:00Z Several approaches exist to model decision making under risk, where risk can be broadly defined as the effect of variability of random outcomes. One of the main approaches in the practice of decision making under risk uses mean-risk models; one such well-known is the classical Markowitz model, where variance is used as risk measure. Along this line, we consider a portfolio selection problem, where the asset returns have an elliptical distribution. We mainly focus on portfolio optimization models constructing portfolios with minimal risk, provided that a prescribed expected return level is attained. In particular, we model the risk by using Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). After reviewing the main properties of VaR and CVaR, we present short proofs to some of the well-known results. Finally, we describe a computationally efficient solution algorithm and present numerical results. http://repub.eur.nl/res/pub/19255/ 2007-07-01T00:00:00Z In this note we give an elementary proof of the Fritz-John and Karush–Kuhn–Tucker conditions for nonlinear finite dimensional programming problems with equality and/or inequality constraints. The proof avoids the implicit function theorem usually applied when dealing with equality constraints and uses a generalization of Farkas lemma and the Bolzano-Weierstrass property for compact sets. http://repub.eur.nl/res/pub/10151/ 2007-05-24T00:00:00Z In this paper portfolio problems with linear loss functions and multivariate elliptical distributed returns are studied. We consider two risk measures, Value-at-Risk and Conditional-Value-at-Risk, and two types of decision makers, risk neutral and risk averse. For Value-at-Risk, we show that the optimal solution does not change with the type of decision maker. However, this observation is not true for Conditional-Value-at-Risk. We then show for Conditional-Value-at-Risk that the objective function can be approximated by Monte Carlo simulation using only a univariate distribution. To solve the equivalent Markowitz model, we modify and implement a finite step algorithm. Finally, a numerical study is conducted. http://repub.eur.nl/res/pub/19256/ 2007-05-16T00:00:00Z We present a thorough analysis of the economic production quantity model with shortages under a general inventory cost rate function and piecewise linear concave production costs. Consequently, an effective solution procedure, particularly useful for an approximation scheme, is proposed. A computational study is appended to illustrate the performance of the proposed solution procedure. http://repub.eur.nl/res/pub/9412/ 2007-03-28T00:00:00Z We discuss a class of risk measures for portfolio optimization with linear loss functions, where the random returns of financial instruments have a multivariate elliptical distribution. Under this setting we pay special attention to two risk measures, Value-at-Risk and Conditional-Value-at-Risk and differentiate between risk neutral and risk averse decision makers. When the so-called disutility function is taken as the identity function, the optimization problem is solved for a risk neutral investor. In this case, the optimal solutions of the two portfolio problems using the Value-at-Risk and Conditional-Value-at-Risk measures are the same as the solution of the classical Markowitz model. We adapt an existing less known finite algorithm to solve the Markowitz model. Its application within finance seems to be new and outperforms the standard quadratic programming procedure quadprog within MATLAB. When the disutility function is taken as a convex increasing function, the problem at hand is associated with a risk averse investor. If the Value-at-Risk is the choice of measure we show that the optimal solution does not differ from the risk neutral case. However, when Conditional-Value-at-Risk is preferred for the risk averse decision maker, the corresponding portfolio problem has a different optimal solution. In this case the used objective function can be easily approximated by Monte Carlo simulation. For the actual solution of the Markowitz model, we modify and implement the less known finite step algorithm and explain its core idea. After that we present numerical results to illustrate the effects of two disutility functions as well as to examine the convergence behavior of the Monte Carlo estimation approach. http://repub.eur.nl/res/pub/7808/ 2006-06-10T00:00:00Z In this paper we introduce robust versions of the classical static and dynamic single leg seat allocation models as analyzed by Wollmer, and Lautenbacher and Stidham, respectively. These robust models take into account the inaccurate estimates of the underlying probability distributions. As observed by simulation experiments it turns out that for these robust versions the variability compared to their classical counter parts is considerably reduced with a negligible decrease of average revenue. http://repub.eur.nl/res/pub/7755/ 2006-05-11T00:00:00Z In this paper we introduce robust versions of the classical static and dynamic single leg seat allocation models as analyzed by Wollmer, and Lautenbacher and Stidham, respectively. These robust models take into account the inaccurate estimates of the underlying probability distributions. As observed by simulation experiments it turns out that for these robust versions the variability compared to their classical counter parts is considerably reduced with a negligible decrease of average revenue. http://repub.eur.nl/res/pub/19277/ 2006-03-16T00:00:00Z We consider equilibrium constrained optimization problems, which have a general formulation that encompasses well-known models such as mathematical programs with equilibrium constraints, bilevel programs, and generalized semi-infinite programming problems. Based on the celebrated KKM lemma, we prove the existence of feasible points for the equilibrium constraints. Moreover, we analyze the topological and analytical structure of the feasible set. Alternative formulations of an equilibrium constrained optimization problem (ECOP) that are suitable for numerical purposes are also given. As an important first step for developing efficient algorithms, we provide a genericity analysis for the feasible set of a particular ECOP, for which all the functions are assumed to be linear. http://repub.eur.nl/res/pub/7030/ 2005-11-07T00:00:00Z In this note we give an elementary proof of the Fritz-John and Karush-Kuhn-Tucker conditions for nonlinear finite dimensional programming problems with equality and/or inequality constraints.The proof avoids the implicit function theorem usually applied when dealing with equality constraints and uses a generalization of Farkas lemma and the Bolzano-Weierstrass property for compact sets. http://repub.eur.nl/res/pub/6992/ 2005-10-14T00:00:00Z In this note we give an elementary proof of the Fritz-John and Karush-Kuhn-Tucker conditions for nonlinear finite dimensional programming problems with equality and/or inequality constraints. The proof avoids the implicit function theorem usually applied when dealing with equality constraints and uses a generalization of Farkas lemma and the Bolzano-Weierstrass property for compact sets. http://repub.eur.nl/res/pub/1535/ 2004-08-30T00:00:00Z The present paper discusses an approach to solve the joint replenishment problem in a production environment with concave production cost functions. Under this environment, the model leads to a global optimization problem, which is investigated by using some standard results from convex analysis. Consequently, an effective solution procedure is proposed. The proposed procedure is guaranteed to return a solution with a predetermined quality in terms of the objective function value. A computational study is provided to illustrate the performance of the proposed solution procedure with respect to the running time. http://repub.eur.nl/res/pub/1536/ 2004-08-30T00:00:00Z We present a thorough analysis of the economic order quantity model with shortages under a general inventory cost rate function and concave production costs. By using some standard results from convex analysis, we show that the model exhibits a composite concave-convex structure. Consequently, an effective solution procedure, particularly useful for an approximation scheme, is proposed. A computational study is appended to illustrate the performance of the proposed solution procedure. http://repub.eur.nl/res/pub/1325/ 2004-06-21T00:00:00Z The present paper proposes a new approach to solve generalized fractional programming problems through user interaction. Capitalizing on two alternatives, we review the Dinkelbach-type methods and set forth the main difficulty in applying these methods. In order to cope with this difficulty, we propose an approximation approach that can be controlled by a predetermined parameter. The proposed approach is promising particularly when a decision maker is involved in the solution process and agrees upon finding an effective but nearoptimal value in an efficient manner. The decision maker is asked to decide the parameter and our analysis shows how good is the value found by the approximation corresponding to this parameter. In addition, we present several observations that may be suitable for boosting up the performance of the proposed approach. Finally, we support our discussion through extensive numerical experiments. http://repub.eur.nl/res/pub/1164/ 2004-02-18T00:00:00Z We consider a class of stochastic mathematical programs with complementarity constraints, in which both the objective and the constraints involve limit functions or expectations that need to be estimated or approximated. Such programs can be used for modeling \\average" or steady-state behavior of complex stochastic systems. Recently, simulation-based methods have been successfully used for solving challenging stochastic optimization problems and equilibrium models. Here we broaden the applicability of so-called the sample-path method to include the solution of certain stochastic mathematical programs with equilibrium constraints. The convergence analysis of sample-path methods rely heavily on stability conditions. We first review necessary sensitivity results, then describe the method, and provide sufficient conditions for its almost-sure convergence. Alongside we provide a complementary sensitivity result for the corresponding deterministic problems. In addition, we also provide a unifying discussion on alternative set of sufficient conditions, derive a complementary result regarding the analysis of stochastic variational inequalities, and prove the equivalence of two different regularity conditions. http://repub.eur.nl/res/pub/1068/ 2003-12-03T00:00:00Z We consider equilibrium constrained optimization problems, which have a general formulationthat encompasses well-known models such as mathematical programs with equilibrium constraints, bilevel programs, and generalized semi-infinite programming problems. Based on the celebrated K K M lemma, we prove the existence of feasible points for the equilibrium constraints. Moreover, we analyze the topological and analytical structure of the feasible set. Alternative formulations of an equilibrium constrained optimization problem (ECOP) that are suitable for numerical purposes are also given. As an important _rst step for developing ef_cient algorithms, we provide a genericity analysis for the feasible set of a particular ECOP, for which all the functions are assumed to be linear. http://repub.eur.nl/res/pub/267/ 2003-01-21T00:00:00Z An alternative smoothing method for the high dimensional max function has been studied. The proposed method is a recursive extension of the two dimensional smoothing functions. In order to analyze the proposed method, a theoretical framework related to smoothing methods has been discussed. Moreover, we support our discussion by considering some application areas. This is followed by a comparison with an alternative well-known smoothing method. http://repub.eur.nl/res/pub/525/ 2002-12-31T00:00:00Z A new smoothing approach based on entropic perturbation is proposed for solving mathematical programs with equilibrium constraints. Some of the desirable properties of the smoothing function are shown. The viability of the proposed approach is supported by a computational study on a set of well-known test problems. http://repub.eur.nl/res/pub/526/ 2002-12-31T00:00:00Z We study a variational inequality problem whose domain is defined by infinitely many linear inequalities. A discretization method and an analytic center based inexact cutting plane method are proposed. Under proper assumptions, the convergence results for both methods are given. We also provide numerical examples for the proposed methods. http://repub.eur.nl/res/pub/527/ 2002-12-31T00:00:00Z By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that the proposed algorithm finds an exact solution of VLCP in a finite number of iterations, under some conditions milder than those assumed in literature. Some computational results are included to illustrate the potential of this approach. http://repub.eur.nl/res/pub/225/ 2002-09-16T00:00:00Z By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that the proposed algorithm finds an exact solution of VLCP in a finite number of iterations, under some conditions milder than those assumed in literature. Some computational results are included to illustrate the potential of this approach. http://repub.eur.nl/res/pub/224/ 2002-09-13T00:00:00Z A new smoothing approach based on entropic perturbation is proposed for solving mathematical programs with equilibrium constraints. Some of the desirable properties of the smoothing function are shown. The viability of the proposed approach is supported by a computationalstudy on a set of well-known test problems. http://repub.eur.nl/res/pub/219/ 2002-08-06T00:00:00Z We study a variational inequality problem whose domain is defined by infinitely many linear inequalities. A discretization method and an analytic center based inexact cutting plane method are proposed. Under proper assumptions, the convergence results for both methods are given. We also provide numerical examples for the proposed methods.