http://repub.eur.nl/res/aut/32/ List of Publications en http://repub.eur.nl/static-eur/img/logo.png http://repub.eur.nl http://repub.eur.nl/res/pub/37785/ 2012-09-01T00:00:00Z We consider a consumer electronics manufacturer's problem of controlling the inventory of spare parts in the final phase of the service life cycle. The final phase starts when the part production is terminated and continues until the last service contract or warranty period expires. Placing final orders for service parts is considered to be a popular tactic to satisfy demand during this period and to mitigate the effect of part obsolescence at the end of the service life cycle. Previous research focuses on repairing defective products by replacing the defective parts with properly functioning spare ones. However, for consumer electronic products there typically is considerable price erosion while repair costs stay steady over time. As a consequence, there might be a point in time at which the unit price of the product drops below the repair costs. If so, it is more cost effective to adopt an alternative policy to meet service demands toward the end of the final phase, such as offering customers a new product of the similar type or a discount on a next generation product. This study examines the cost trade-offs of implementing alternative policies for the repair policy and develops an exact expression for the expected total cost function. Using this expression, the optimal final order quantity and switching time from repair to an alternative policy can be determined simultaneously. Numerical analysis of a real world case sheds light on the cost benefits of these policies and also yields insights into the quantitative importance of the various cost parameters. http://repub.eur.nl/res/pub/18332/ 2010-03-02T00:00:00Z We consider a consumer electronics (CE) manufacturer’s problem of controlling the inventory of spare parts in the final phase of the service life cycle. The final phase starts when the part production is terminated and continues until the last service contract or warranty period expires. Placing final orders for service parts is considered to be a popular tactic to satisfy demand during this period and to mitigate the effect of part obsolescence at the end of the service life cycle. To satisfy demand for service in the final phase, previous research focuses on repairing defective products by replacing the defective parts with properly functioning spare ones. However, for consumer electronic products there is a remarkable price erosion while repair costs may stay steady over time. As a consequence, this introduces the idea that there might be a point in time at which the unit price of the product is lower than repair associated costs. Therefore, it would be more cost effective to adopt an alternative policy to meet demands for service such as offering customers a replacement of the defective product with a new one or giving a discount on the next generation of the product. This paper examines the cost trade-offs of implementing alternative policies for the repair policy and develops an exact formulation for the expected total cost function. Based on this developed cost function we propose policies to simultaneously find the optimal final order quantity and the time to switch from the repair to an alternative replacement policy. Numerical analysis of a real world case study sheds light over the effectiveness and advantage of these policies in terms of cost reduction and also yields insights into the quantitative importance of the various cost parameters. 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/18266/ 2009-05-01T00:00:00Z Steel structures such as bridges, tanks and pylons are exposed to outdoor weathering conditions. In order to prevent them from corrosion they are protected by an organic coating system. Unfortunately, the coating system itself is also subject to deterioration. Imperfect maintenance actions such as spot repair and repainting can be done to extend the lifetime of the coating. This paper considers the problem of finding the set of actions that minimizes the expected (discounted) maintenance costs over both a finite horizon and an infinite horizon. To this end the size of the area affected by corrosion is modelled by a non-stationary gamma process. An imperfect maintenance action is to be done as soon as a fixed threshold is exceeded. The direct effect of such an action on the condition of the coating is assumed to be random. On the other hand, due to maintenance the parameters of the gamma deterioration process may also change. It is shown that the optimal maintenance decisions related to this problem are a solution of a continuous-time renewal-type dynamic programming equation. To solve this equation time is discretized and it is verified theoretically that this discretization induces only a small error. Finally, the model is illustrated with numerical examples. 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/19268/ 2007-08-01T00:00:00Z In this paper, we consider first the most important classes of cone convexlike vector-valued functions and give a dual characterization for some of these classes. It turns out that these characterizations are strongly related to the closely convexlike and Ky Fan convex bifunctions occurring within minimax problems. Applying the Lagrangian perturbation approach, we show that some of these classes of cone convexlike vector-valued functions show up naturally in verifying strong Lagrangian duality for finite-dimensional optimization problems. This is achieved by extending classical convexity results for biconjugate functions to the class of so-called almost convex functions. In particular, for a general class of finite-dimensional optimization problems, strong Lagrangian duality holds if some vector-valued function related to this optimization problem is closely K-convexlike and satisfies some additional regularity assumptions. For K a full-dimensional convex cone, it turns out that the conditions for strong Lagrangian duality simplify. Finally, we compare the results obtained by the Lagrangian perturbation approach worked out in this paper with the results achieved by the so-called image space approach initiated by Giannessi. http://repub.eur.nl/res/pub/10455/ 2007-07-03T00:00:00Z Steel structures such as bridges, tanks and pylons are exposed to outdoor weathering conditions. In order to prevent them from corrosion they are protected by an organic coating system. Unfortunately, the coating system itself is also subject to deterioration. Imperfect maintenance actions such as spot repair and repainting can be done to extend the lifetime of the coating. In this paper we consider the problem of finding the set of actions that minimizes the expected maintenance costs over a bounded horizon. To this end we model the size of the area affected by corrosion by a non-stationary gamma process. An imperfect maintenance action is to be done as soon as a fixed threshold is exceeded. The direct effect of such an action on the condition of the coating is assumed to be random. On the other hand, maintenance may also change the parameters of the gamma deterioration process. It is shown that the optimal maintenance decisions related to this problem are a solution of a continuous-time renewal-type dynamic programming equation. To solve this equation time is discretized and it is verified theoretically that this discretization induces only a small error. Finally, the model is illustrated with a numerical example. http://repub.eur.nl/res/pub/10437/ 2007-07-02T00:00:00Z Steel structures such as bridges, tanks and pylons are exposed to outdoor weathering conditions. In order to prevent them from corrosion they are protected by an organic coating system. Unfortunately, the coating system itself is also subject to deterioration. Imperfect maintenance actions such as spot repair and repainting can be done to extend the lifetime of the coating. In this paper we consider the problem of finding the set of actions that minimizes the expected maintenance costs over a bounded horizon. To this end we model the size of the area affected by corrosion by a non-stationary gamma process. An imperfect maintenance action is to be done as soon as a fixed threshold is exceeded. The direct effect of such an action on the condition of the coating is assumed to be random. On the other hand, maintenance may also change the parameters of the gamma deterioration process. It is shown that the optimal maintenance decisions related to this problem are a solution of a continuous-time renewal-type dynamic programming equation. To solve this equation time is discretized and it is verified theoretically that this discretization induces only a small error. Finally, the model is illustrated with a numerical example. 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/10149/ 2007-05-24T00:00:00Z The non-stationary gamma process is a non-decreasing stochastic process with independent increments. By this monotonic behavior this stochastic process serves as a natural candidate for modelling time-dependent phenomena such as degradation. In condition-based maintenance the first time such a process exceeds a random threshold is used as a model for the lifetime of a device or for the random time between two successive imperfect maintenance actions. Therefore there is a need to investigate in detail the cumulative distribution function (cdf) of this so-called randomized hitting time. We first relate the cdf of the (randomized) hitting time of a non-stationary gamma process to the cdf of a related hitting time of a stationary gamma process. Even for a stationary gamma process this cdf has in general no elementary formula and its evaluation is time-consuming. Hence two approximations are proposed in this paper and both have a clear probabilistic interpretation. Numerical experiments show that these approximations are easy to evaluate and their accuracy depends on the scale parameter of the non-stationary gamma process. Finally, we also consider some special cases of randomized hitting times for which it is possible to give an elementary formula for its cdf. 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/10095/ 2007-05-10T00:00:00Z The non-stationary gamma process is a non-decreasing stochastic process with independent increments. By this monotonic behavior this stochastic process serves as a natural candidate for modelling time-dependent phenomena such as degradation. In condition-based maintenance the first time such a process exceeds a random threshold is used as a model for the lifetime of a device or for the random time between two successive imperfect maintenance actions. Therefore there is a need to investigate in detail the cumulative distribution function (cdf) of this so-called randomized hitting time. We first relate the cdf of the (randomized) hitting time of a non-stationary gamma process to the cdf of a related hitting time of a stationary gamma process. Even for a stationary gamma process this cdf has in general no elementary formula and its evaluation is time-consuming. Hence two approximations are proposed in this paper and both have a clear probabilistic interpretation. Numerical experiments show that these approximations are easy to evaluate and their accuracy depends on the scale parameter of the non-stationary gamma process. Finally, we also consider some special cases of randomized hitting times for which it is possible to give an elementary formula for its cdf. 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/19261/ 2007-03-01T00:00:00Z In this paper we discuss necessary and sufficient conditions for different minimax results to hold using only linear programming duality and the finite intersection property for compact sets. It turns out that these necessary and sufficient conditions have a clear interpretation within zero-sum game theory. We apply these results to derive necessary and sufficient conditions for strong duality for a general class of optimization problems. 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/7809/ 2006-06-10T00:00:00Z In this chapter we give an overview on the theory of noncooperative games. In the first part we consider in detail for zero-sum (and constant-sum) noncooperative games under which necessary and sufficient conditions on the payoff function and different (extended) strategy sets for both players an equilibrium saddlepoint exists. This is done by using the most elementary proofs. One proof uses the separation result for disjoint convex sets, while the other proof uses linear programming duality and some elementary properties of compact sets. Also, for the most famous saddlepoint result given by Sion's minmax theorem an elementary proof using only the definition of connectedness is given. In the final part we consider n-person nonzero-sum noncooperative games and show by a simple application of the KKM lemma that a so-called Nash equilibrium point exists for compact strategy sets and concavity conditions on the payoff functions. 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/7756/ 2006-05-11T00:00:00Z In this chapter we give an overview on the theory of noncooperative games. In the first part we consider in detail for zero-sum (and constant-sum) noncooperative games under which necessary and sufficient conditions on the payoff function and different (extended) strategy sets for both players an equilibrium saddlepoint exists. This is done by using the most elementary proofs. One proof uses the separation result for disjoint convex sets, while the other proof uses linear programming duality and some elementary properties of compact sets. Also, for the most famous saddlepoint result given by Sion's minmax theorem an elementary proof using only the definition of connectedness is given. In the final part we consider n-person nonzero-sum noncooperative games and show by a simple application of the KKM lemma that a so-called Nash equilibrium point exists for compact strategy sets and concavity conditions on the payoff functions. 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/14695/ 2006-01-01T00:00:00Z Steel structures such as bridges, tanks and pylons are exposed to outdoor weathering conditions. In order to prevent them from corrosion they are protected by organic coating systems. Unfortunately, the coating system itself is also subject to deterioration. In this paper we use the gamma process to model the coating deterioration in continuous time. The condition of the coating is represented by a continuous quantity. As soon as the condition exceeds the intervention level, maintenance is to be done. However, maintenance of coating systems is usually imperfect. Most maintenance optimization models do not take this into account. We incorporate imperfect maintenance in our deterioration model by assuming that maintenance actions have a random effect on the condition of the coating. This random effect makes that the time between two maintenance actions is a randomized hitting time. We consider a number of probability distributions for the random effect and we will derive (explicit) expressions for the distribution and the expected value of the time between two maintenance actions. It is shown that the time between two maintenance actions is Weibull distributed if the random reduction in condition is exponentially distributed and the expected deterioration follows a power law function. Obviously, the (expected) time between two maintenance actions plays an important role in evaluating the cost of a maintenance strategy. A number of (imperfect) maintenance actions are available to improve the condition of a coating: spot repair, repainting and replacement. Each action has a different (random) effect on the coating. We derive an expression for the expected average maintenance cost per time unit for a given sequence of maintenance actions. Subsequently, we consider the problem of finding the set of maintenance actions that minimizes the maintenance costs over a bounded horizon. This problem is formulated as a stochastic dynamic programming model. http://repub.eur.nl/res/pub/19273/ 2006-01-01T00:00:00Z In this paper we discuss the level set method of Joó and how to use it to give an elementary proof of the well-known minimax theorem of Sion. Although this proof technique was initiated by Joó and based on the intersection of upper level sets and a clever use of the topological notion of connectedness, it is not very well known and accessible for researchers in optimization. At the same time we simplify the original proof of Joó and give a more elementary proof of the celebrated minimax theorem of Sion. 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/19414/ 2005-06-01T00:00:00Z In this note we review some known minimax theorems with applications in game theory and show that these results form an equivalent chain which includes the strong separation result in finite dimensional spaces between two disjoint closed convex sets of which one is compact. By simplifying the proofs we intend to make the results more accessible to researchers not familiar with minimax or noncooperative game theory. http://repub.eur.nl/res/pub/6558/ 2005-06-01T00:00:00Z In this note we review some known minimax theorems with applications in game theory and show that these results form an equivalent chain which includes the strong separation result in finite dimensional spaces between two disjoint closed convex sets of which one is compact. By simplifying the proofs we intend to make the results more accessible to researchers not familiar with minimax or noncooperative game theory. http://repub.eur.nl/res/pub/1928/ 2005-04-03T00:00:00Z In this technical note we give a short proof based on standard results in convex analysis of some important characterization results listed in Theorem 3 and 4 of [1]. Actually our result is slightly general since we do not specify the convex set X. For clarity we use the same notation for the different equivalent optimization problems as done in [1]. http://repub.eur.nl/res/pub/1931/ 2005-04-03T00:00:00Z In this paper we will show that the closely K-convexlike vector-valued functions with K Rm a nonempty convex cone and related classes of vector-valued functions discussed in the literature arise naturally within the theory of biconjugate functions applied to the Lagrangian perturbation scheme in finite dimensional optimization. For these classes of vectorvalued functions an equivalent characterization of the dual objective function associated with the Lagrangian is derived by means of a dual representation of the relative interior of a convex cone. It turns out that these characterizations are strongly related to the closely convexlike and Ky-Fan convex bifunctions occurring within minimax problems. Also it is shown for a general class of finite dimensional optimization problems that strong Lagrangian duality holds in case a vector-valued function related to the functions in this optimization problem is closely K-convexlike and satisfies some additional regularity condition. http://repub.eur.nl/res/pub/1927/ 2005-04-01T00:00:00Z In this note we show that the strong duality theorem of an unconstrained (generalized) geometric programming problem as defined by Peterson (cf.[1]) is actually a special case of a Lagrangian duality result. Contrary to [1] we also consider the case that the set C is compact and convex and in this case we do not need to assume the standard regularity condition. http://repub.eur.nl/res/pub/19412/ 2005-04-01T00:00:00Z In this note we show that the strong duality theorem of an unconstrained (generalized) geometric programming problem as defined by Peterson (cf.[1]) is actually a special case of a Lagrangian duality result. Contrary to [1] we also consider the case that the set C is compact and convex and in this case we do not need to assume the standard regularity condition. http://repub.eur.nl/res/pub/19410/ 2005-01-23T00:00:00Z In this technical note we give a short proof based on standard results in convex analysis of some important characterization results listed in Theorem 3 and 4 of [1]. Actually our result is slightly general since we do not specify the convex set X. For clarity we use the same notation for the different equivalent optimization problems as done in [1]. http://repub.eur.nl/res/pub/1610/ 2004-09-17T00:00:00Z Single-ratio and multi-ratio fractional programs in applications are often generalized convex programs. We begin with a survey of applications of single-ratio fractional programs, min-max fractional programs and sum-of-ratios fractional programs. Given the limited advances for the latter class of problems, we focus on an analysis of min-max fractional programs. A parametric approach is employed to develop both theoretical and algorithmic results. http://repub.eur.nl/res/pub/1611/ 2004-09-17T00:00:00Z In the first chapter of this book the basic results within convex and quasiconvex analysis are presented. In Section 2 we consider in detail the algebraic and topological properties of convex sets within Rn together with their primal and dual representations. In Section 3 we apply the results for convex sets to convex and quasiconvex functions and show how these results can be used to give primal and dual representations of the functions considered in this field. As such, most of the results are well-known with the exception of Subsection 3.4 dealing with dual representations of quasiconvex functions. In Section 3 we consider applications of convex analysis to noncooperative game and minimax theory, Lagrangian duality in optimization and the properties of positively homogeneous evenly quasiconvex functions. Among these result an elementary proof of the well-known Sion’s minimax theorem concerningquasiconvex-quasiconcave bifunctions is presented, thereby avoiding the less elementary fixed point arguments. Most of the results are proved in detail and the authors have tried to make these proofs as transparent as possible. Remember that convex analysis deals with the study of convex cones and convex sets and these objects are generalizations of linear subspaces and affine sets, thereby extending the field of linear algebra. Although some of the proofs are technical, it is possible to give a clear geometrical interpretation of the main ideas of convex analysis. Finally in Section 5 we list a short and probably incomplete overview on the history of convex and quasiconvex analysis. 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/1537/ 2004-08-30T00:00:00Z In this paper we discuss the level set method of Joó and how to use it to give an elementary proof of the well-known Sion’s minimax result. Although this proof technique is initiated by Joó and based on the inter-section of upper level sets and a clever use of the topological notion of connectedness, it is not very well known and accessible for researchers in optimization. At the same time we simplified the original proof of Joó and give a more elementary proof of the celebrated Sion’s minimax theorem. 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/1219/ 2004-04-14T00:00:00Z In this paper we discuss necessary and sufficient conditions for different minimax results to hold using only linear programming duality and the finite intersection property of compact sets. It turns out that these necessary and sufficient conditions have a clear interpretation within zero-sum game theory. In the last section we apply these results to derive necessary and sufficient conditions for strong Lagrangean duality for a large class of optimization problems. 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/240/ 2002-10-22T00:00:00Z In this paper the well-known minimax theorems of Wald, Ville and Von Neumann are generalized under weaker topological conditions onthe payoff function ƒ and/or extended to the larger set of the Borel probabilitymeasures instead of the set of mixed strategies. http://repub.eur.nl/res/pub/158/ 2002-01-24T00:00:00Z In this paper we review known minimax results with applications in game theory and show that these results are easy consequences of the first minimax result for a two person zero sum game with finite strategy sets published by von Neumann in 1928: Among these results are the well known minimax theorems of Wald, Ville and Kneser and their generalizations due to Kakutani, Ky-Fan, König, Neumann and Gwinner-Oettli. Actually it is shown that these results form an equivalent chain and this chain includes the strong separation result in finite dimensional spaces between two disjoint closed convex sets of which one is compact. To show these implications the authors only use simple properties of compact sets and the well-known Weierstrass Lebesgue lemma. http://repub.eur.nl/res/pub/6845/ 2001-08-27T00:00:00Z In this paper which will appear as a chapter in the Handbook of Generalized Convexity we discuss the basic ideas of convex and quasiconvex analysis in finite dimensional Euclidean spaces. To illustrate the usefulness of this branch of mathematics also applications to optimization theory and noncooperative game theory are considered. http://repub.eur.nl/res/pub/15346/ 1999-10-16T00:00:00Z In most multi-item inventory systems, the ordering costs consist of a major cost and a minor cost for each item included. Applying for every individual item a cyclic inventory policy, where the cycle length is a multiple of some basic cycle time, reduces the major ordering costs. An efficient algorithm to determine the optimal policy of this type is discussed in this paper. It is shown that this algorithm can be used for deterministic multi-item inventory problems, with general cost rate functions and possibly service level constraints, of which the well-known joint replenishment problem is a special case. Some useful results in determining the optimal control parameters are derived, and worked out for piecewise linear cost rate functions. Numerical results for this case show that the algorithm significantly outperforms other solution methods, both in the quality of the solution and in the running time. http://repub.eur.nl/res/pub/1600/ 1999-09-15T00:00:00Z In this paper we will discuss a general framework for single item inventory models based on the theory of regenerative processes. After presenting without proof the main theorems for regenerative processes we analyze in detail how the different single item models can be embedded within this general theory. This facilitates to write down the expressions for the average cost associated with an arbitrary costrate function ƒ, and some of the service measures, which appear most frequently in the literature. http://repub.eur.nl/res/pub/1601/ 1999-09-15T00:00:00Z This article is a continuation of the paper "Inventory Control and Regenerative processes: Theory" (Bazsa et al.; 1998) and presents closed form expressions for Laplace transforms associated with the cost functions of the classical single item inventory models with indivisible items, a fixed lead time and backlogging. For given instances of these inventory control models these Laplace transforms will be used as input functions in a newly developed Laplace transform inversion algorithm (Iseger; A new method for inverting Laplace transforms; 1998) which yields highly accurate computational results of the associated cost functions. http://repub.eur.nl/res/pub/11536/ 1999-08-01T00:00:00Z In this paper, we introduce several classes of generalized convex functions already discussed in the literature and show the relation between these classes. Moreover, a Gordan–Farkas type theorem is proved for all these classes and it is shown how these theorems can be used to verify strong Lagrangian duality results in finite-dimensional optimization. http://repub.eur.nl/res/pub/1527/ 1998-12-31T00:00:00Z In this paper we will discuss a general framework for single item inventory models based on the theory of regenerative processes. After resenting without proof the main theorems for regenerative processes we analyze in detail how the different single item models can be embedded within this general theory. This facilitates to write down the expressions for the average cost associated with an arbitrary costrate function f. Since these expressions are still complicated, involving convolutions, we use a recently developed numerically stable Laplace inversion algorithm to compute these objective functions in MATLAB. This enables us to compute the exact costs instead of using approximations. http://repub.eur.nl/res/pub/1528/ 1998-11-26T00:00:00Z In this paper we review and unify some of the classes of generalized convex functions introduced by different authors to prove minimax results in infinite dimensional spaces and show the relations between those classes. We also list for the most general class already introduced by Jeyakumar an elementary proof of a minimax result. The proof of this result only uses a finite dimensional separation theorem and although this minimax result is already presented by Neumann and independently Jeyakumar we believe that the present proof is shorter and more transparent. http://repub.eur.nl/res/pub/15259/ 1997-10-01T00:00:00Z In this paper we discuss a general framework for single component replacement models. This framework is based on the regenerative structure of these models and by using results from renewal theory a unified presentation of the discounted and average finite and infinite horizon cost models is given. Finally, some well-known replacement models are discussed, and making use of the previous results an easy derivation of their cost functions is presented. http://repub.eur.nl/res/pub/15258/ 1997-09-01T00:00:00Z In this paper we investigate the conditions under which the marginal cost approach of Refs. 1–3 holds. As observed in Ref. 4, the validity of the marginal cost approach gives rise to a useful framework of single-component maintenance optimization models, which covers almost all models used in practice. For the class of unimodal finite-valued marginal cost functions, we show that these optimization models are easy to solve. http://repub.eur.nl/res/pub/15306/ 1997-06-01T00:00:00Z During the last two decades, many heuristic procedures for the joint replenishment problem have appeared in the literature. The only available optimal solution procedure was based on an enumerative approach and was computationally prohibitive. In this paper we present an alternative optimal approach based on global optimisation theory. By applying Lipschitz optimisation one can find a solution with an arbitrarily small deviation from an optimal value. An efficient procedure is presented which uses a dynamic Lipschitz constant and generates a solution in little time. The running time of this procedure grows only linearly in the number of items. http://repub.eur.nl/res/pub/2228/ 1997-06-01T00:00:00Z In this paper we adapt the well-known parametric approach from fractional programming to solve a class of fractional programs with a noncompact feasible region. Such fractional problems belong to an important class of single component preventive maintenance models. Moreover, for a special but important subclass we show that the subproblems occurring in this parametric approach http://repub.eur.nl/res/pub/11651/ 1997-05-01T00:00:00Z In this paper we analyze some variants of the classical uncapacitated facility location problem with a ratio as an objective function. Using basic concepts and results of fractional programming, we identify a class of one-level fractional location problems which can be solved in polynomial time in terms of the size of the problem. We also consider the fractional two-echelon location problem, which is a special case of the general two-level fractional location problem. For this two-level fractional location problem we identify cases for which its solution involves decomposing the problem into several one-level fractional location problems. http://repub.eur.nl/res/pub/1401/ 1997-01-01T00:00:00Z In this paper we introduce several classes of generalized convex functions already discussed in the literature and show the relation between those function classes. Moreover, for some of those function classes a Farkas-type theorem is proved. As such this paper unifies and extends results existing in the literature and shows how these results can be used to verify Farkas-type theorems and strong Lagrangian duality results in finite dimensional optimization. http://repub.eur.nl/res/pub/1402/ 1997-01-01T00:00:00Z In this paper we discuss a general framework for single item inventory control models. This framework is based on the regenerative structure of these models. Using results from the theory of regenerative processes a unified presentation of those models is presented. Although most of the results are already known for special cost structures this unified presentation yields us the possibility to show that the same techniques can be applied to each instance. http://repub.eur.nl/res/pub/1405/ 1997-01-01T00:00:00Z In this paper we analyse the effect of satisfying in a different way customers with an order larger than a prespecified cutoff transaction size, in a simple newsboy setting. For compound Poisson demand with discrete order sizes, we show how to determine the expected costs and the optimal cutoff transaction size. Moreover, by approximating the distribution of the total demand during a period by the normal distribution one can determine an expression for the average cost function that depends on the cutoff transaction size only. A main advantage of this approximation is that the computational effort is much less. The quality of using the normal approximation is evaluated through a number of numerical experiments, which show that the approximative results are satisfactory. http://repub.eur.nl/res/pub/1411/ 1997-01-01T00:00:00Z In this paper we consider stochastic purchase timing models used in marketing for low-involvement products and show that important characteristics of those models are easy to compute. As such these calculations are based on an elementary probabilistic argument and cover not only the well-known condensed negative binomial model but also stochastic purchase timing models with other interarrival and mixing distributions. http://repub.eur.nl/res/pub/1423/ 1997-01-01T00:00:00Z In most multi-item inventory systems, the ordering costs consist of a major cost and a minor cost for each item included. Applying for every individual item a cyclic inventory policy, where the cycle length is a multiple of some basic cycle time, reduces the major ordering costs. An efficient algorithm to determine the optimal policy of this type is discussed in this paper. It is shown that this algorithm can be used for deterministic multi-item inventory problems, with general cost rate functions and possibly service level constraints, of which the well-known joint replenishment problem is a special case. Some useful results in determining the optimal control parameters are derived, and worked out for piecewise linear cost rate functions. Numerical results for this case show that the algorithm significantly outperforms other solution methods, both in the quality of the solution as in the running time. http://repub.eur.nl/res/pub/11538/ 1996-04-01T00:00:00Z In this paper, a class of min-max continuous location problems is discussed. After giving a complete characterization of th stationary points, we propose a simple central and deep-cut ellipsoid algorithm to solve these problems for the quasiconvex case. Moreover, an elementary convergence proof of this algorithm and some computational results are presented. http://repub.eur.nl/res/pub/11539/ 1996-04-01T00:00:00Z This paper studies the problem of deciding whether the present iteration point of some algorithm applied to a planar singlefacility min-max location problem, with distances measured by either anl p -norm or a polyhedral gauge, is optimal or not. It turns out that this problem is equivalent to the decision problem of whether 0 belongs to the convex hull of either a finite number of points in the plane or a finite number of differentl q -circles . Although both membership problems are theoretically solvable in polynomial time, the last problem is more difficult to solve in practice than the first one. Moreover, the second problem is solvable only in the weak sense, i.e., up to a predetermined accuracy. Unfortunately, these polynomial-time algorithms are not practical. Although this is a negative result, it is possible to construct an efficient and extremely simple linear-time algorithm to solve the first problem. Moreover, this paper describes an implementable procedure to reduce the second decision problem to the first with any desired precision. Finally, in the last section, some computational results for these algorithms are reported. http://repub.eur.nl/res/pub/11635/ 1996-01-01T00:00:00Z A new dual problem for convex generalized fractional programs with no duality gap is presented and it is shown how this dual problem can be efficiently solved using a parametric approach. The resulting algorithm can be seen as “dual” to the Dinkelbach-type algorithm for generalized fractional programs since it approximates the optimal objective value of the dual (primal) problem from below. Convergence results for this algorithm are derived and an easy condition to achieve superlinear convergence is also established. Moreover, under some additional assumptions the algorithm also recovers at the same time an optimal solution of the primal problem. We also consider a variant of this new algorithm, based on scaling the “dual” parametric function. The numerical results, in case of quadratic-linear ratios and linear constraints, show that the performance of the new algorithm and its scaled version is superior to that of the Dinkelbach-type algorithms. From the computational results it also appears that contrary to the primal approach, the “dual” approach is less influenced by scaling. http://repub.eur.nl/res/pub/1370/ 1996-01-01T00:00:00Z In this paper we discuss a general framework for single component replacement models. This framework is based on the regenerative structure of these models and by using results from renewal theory a unified presentation of the discounted and average finite and infinite horizon cost models is given. Moreover, we present sufficient conditions for the numerator of the cost function to have a vanishing singular part. Finally, some well-known replacement models are discussed, and making use of the previous results an easy derivation of their cost functions is presented. http://repub.eur.nl/res/pub/1371/ 1996-01-01T00:00:00Z In this paper we verify the conditions under which the marginal cost approach of Berg (1980, 1995) holds. As observed by Aven and Dekker (1996) the validity of the marginal cost approach gives rise to a useful framework of single component maintenance optimization models which covers almost all models used in practice. For the class of unimodal finite-valued marginal cost functions we show that these optimization models are easy to solve. http://repub.eur.nl/res/pub/1376/ 1996-01-01T00:00:00Z We propose in this paper an algorithm for solving linearly constrained nondifferentiable convex programming problems. This algorithm combines the ideas of the affine scaling method with the subgradient method. It is a generalization of the dual and interior point method for min-max problems proposed by Sturm and Zhang [J.F. Sturm and S. Zhang, A dual and interior point approach to solve convex min-max problems, in: D.-Z. Du and P.M. Pardalos eds., Minimax and Applications, (1995) 69-78, Kluwer]. In the new method, the search direction is obtained by projecting in a scaled space a subgradient of the objective function with a logarithmic barrier term. The stepsize choice is analogous to the stepsize choice in the usual subgradient method. Convergence of the method is established. http://repub.eur.nl/res/pub/1393/ 1996-01-01T00:00:00Z A number of optimization methods require as a first step the construction of a dominating set (a set containing an optimal solution) enjoying properties such as compactness or convexity. In this note we address the problem of constructing dominating sets for problems whose objective is a componentwise nondecreasing function of (possibly an infinite number of) convex functions, and we show how to obtain a convex dominating set in terms of dominating sets of simpler problems. The applicability of the results obtained is illustrated with the statement of new localization results in the fields of Linear Regression and Location. http://repub.eur.nl/res/pub/2224/ 1996-01-01T00:00:00Z http://repub.eur.nl/res/pub/11537/ 1995-10-01T00:00:00Z In this paper, we introduce a variant of a cutting plane algorithm and show that this algorithm reduces to the well-known Dinkelbach-type procedure of Crouzeix, Ferland, and Schaible if the optimization problem is a generalized fractional program. By this observation, an easy geometrical interpretation of one of the most important algorithms in generalized fractional programming is obtained. Moreover, it is shown that the convergence of the Dinkelbach-type procedure is a direct consequence of the properties of this cutting plane method. Finally, a class of generalized fractional programs is considered where the standard positivity assumption on the denominators of the ratios of the objective function has to be imposed explicitly. It is also shown that, when using a Dinkelbach-type approach for this class of programs, the constraints ensuring the positivity on the denominators can be dropped. http://repub.eur.nl/res/pub/1358/ 1995-01-01T00:00:00Z A maintenance activity carried out on a technical system often involves a system-dependent set-up cost that is the same for all maintenance activities carried out on that system. Grouping activities thus saves costs since execution of a group of activities requires only one set-up. By now, there are several multi-component maintenance models available in the literature, but most of them suffer from intractability when the number of components grows, unless a special structure is assumed. An approach that can handle many components was introduced in the literature by Goyal et al. However, this approach requires a specific deterioration structure for components. Moreover, the authors present an algorithm that is not optimal and there is no information of how good the obtained solutions are. In this paper we present an approach that solves the model of Goyal et al. to optimality. Furthermore, we extend the approach to deal with more general maintenance models like minimal repair and inspection that can be solved to optimality as well. Even block replacement can be incorporated, in which case our approach is a good heuristic. http://repub.eur.nl/res/pub/1359/ 1995-01-01T00:00:00Z During the last two decades, many heuristic procedures for the joint replenishment problem have appeared in the literature. The only available optimal solution procedure was based on an enumerative approach and was computationally prohibitive. In this paper we present an alternative optimal approach based on global optimisation theory. By applying Lipschitz optimisation one can find a solution with an arbitrarily small deviation from an optimal value. An efficient procedure is presented which uses a dynamic Lipschitz constant and generates a solution in little time. The running time of this procedure grows only linearly in the number of items. http://repub.eur.nl/res/pub/1361/ 1995-01-01T00:00:00Z In multi-echelon distribution systems it is usually assumed that demand is only satisfied from the lowest echelon. In this paper we will consider the case where demand can be satisfied from any level in the system. However, then the problem arises of how to allocate orders from customers to the different locations. A possible way of dealing with this problem consists of using a so-called break quantity rule. This easy implementable rule is to deliver every order with a size exceeding the break quantity from a higher echelon. The use of the break quantity rule now results in a reduction of the demand variability at the retailer and hence less safety stocks need to be held. The concept is studied for a two-echelon distribution system, consisting of one warehouse and one retailer, where the inventory at the retailer is controlled by an order up to level policy, and where at the warehouse there is enough inventory to satisfy all orders from the retailer and the customers. For this system an approximation for the long run average costs as a function of the break quantity is derived, and an algorithm is presented to determine the cost-optimal break quantity. Computational results indicate that the break quantity rule can lead to significant cost reductions. http://repub.eur.nl/res/pub/11742/ 1994-05-01T00:00:00Z In this paper we present two algorithms for a machine allocation problem occurring in manufacturing systems. For the two algorithms presented we prove worst-case performance ratios of 2 and 312, respectively. The machlne allocat~on problem we consider is a general convex resource allocation problem, which makes the algorithms applicable to a varlety of resource allocation problems. Numerical results are presented for two real-life manufacturing systems. http://repub.eur.nl/res/pub/11633/ 1994-01-01T00:00:00Z This paper proposes a deep cut version of the ellipsoid algorithm for solving a general class of continuous convex programming problems. In each step the algorithm does not require more computational effort to construct these deep cuts than its corresponding central cut version. Rules that prevent some of the numerical instabilities and theoretical drawbacks usually associated with the algorithm are also provided. Moreover, for a large class of convex programs a simple proof of its rate of convergence is given and the relation with previously known results is discussed. Finally some computational results of the deep and central cut version of the algorithm applied to a min—max stochastic queue location problem are reported. http://repub.eur.nl/res/pub/11700/ 1993-09-03T00:00:00Z The first algorithms for the on-line two-dimensional rectangle packing problem were introduced by Coppersmith and Raghavan. They showed that for a family of heuristics 13/4 is an upper bound for the asymptotic worst-case ratios. We have investigated the Next Fit and the First Fit variants of their method. We proved that the asymptotic worst-case ratio equals 13/4 for the Next Fit variant and that 49/16 is an upper bound of the asymptotic worst-case ratio for the First Fit variant. http://repub.eur.nl/res/pub/11669/ 1993-06-01T00:00:00Z In this paper we analyse the optimal claim behaviour of a risk sensitive policy holder having a vehicle damage insurance. It is proved that the optimal decision is of the form: to claim for damages only if its amount exceeds a certain limit. Moreover, we also derive the optimal stopping rule to terminate the insurance. Finally, some computational results are presented. http://repub.eur.nl/res/pub/11643/ 1993-05-01T00:00:00Z In this note we give a short and easy proof of the equivalence of Hakimi's one-median problem and the k-server-facility-loss median problem as discussed by Chiu and Larson in Computer and Operation Research. The proof makes only use of a stochastic monotonicity result for birth and death processes and the insensitivity of the M/G/k/k loss model. http://repub.eur.nl/res/pub/11696/ 1993-01-29T00:00:00Z In this note we present a simplified proof of a lower bound for on-line bin packing. This proof also covers the well-known result given by Liang in Inform. Process Lett. 10 (1980) 76–79. http://repub.eur.nl/res/pub/11664/ 1991-12-01T00:00:00Z In this note we derive the expected total discounted profit of an insurer due to a single policy holder within a third-party liability insurance. We consider both a policy holder claiming optimally and non-optimally. http://repub.eur.nl/res/pub/15939/ 1991-11-01T00:00:00Z In this paper approximation results for the M/G/s queueing model are used to derive an empirically verified shop type dependent non-homogeneous relation between labour volume and sales in retail trade. Moreover, we formulate the retailer's labour management as a formal minimization problem. http://repub.eur.nl/res/pub/11667/ 1991-07-01T00:00:00Z In this paper we analyse the optimal claim behaviour of a policy holder having a third-party liability insurance in which one is allowed to decide at the end of an insurance year which damages occurred during that year should be claimed. This analysis can only be carried out in detail in case the damages are negative exponentially distributed. Moreover, we present some computational results using an existing bonus—malus system and a horizon of 10 and 25 years and compare these results with similar computations for a corresponding third-party liability insurance in which the policy holder has to decide within a limited time period after the accident to claim or not to claim. http://repub.eur.nl/res/pub/11733/ 1991-06-01T00:00:00Z In the dual bin packing problem, the objective is to assign items of given size to the largest possible number of bins, subject to the constraint that the total size of the items assigned to any bin is at least equal to 1. We carry out a probabilistic analysis of this problem under the assumption that the items are drawn independently from the uniform distribution on [0, 1] and reveal the connection between this problem and the classical bin packing problem as well as to renewal theory. http://repub.eur.nl/res/pub/11663/ 1990-03-01T00:00:00Z It is proved that the optimal decision rule to claim or not to claim for damage is of the form: ‘to claim for damage only if its amount exceeds a certain limit’. Optimal critical claim sizes are derived, and a sensitivity analysis is given with respect to changes in (the parameters of) the distributions of the number of claims and the claim size. http://repub.eur.nl/res/pub/11738/ 1990-01-01T00:00:00Z http://repub.eur.nl/res/pub/11691/ 1987-09-01T00:00:00Z We present a new approximation algorithm for the two-dimensional bin-packing problem. The algorithm is based on two one-dimensional bin-packing algorithms. Since the algorithm is of next-fit type it can also be used for those cases where the output is required to be on-line (e. g. if we open an new bin we have no possibility to pack elements into the earlier opened bins). We give a tight bound for its worst-case and show that this bound is a parameter of the maximal sizes of the items to be packed. Moreover, we also present a probabilistic analysis of this algorithm. http://repub.eur.nl/res/pub/11690/ 1987-06-01T00:00:00Z Under mild conditions on the distribution functionF, we analyze the asymptotic behavior in expectation of the smallest order statistic, both for the case thatF is defined on (–, +) and for the case thatF is defined on (0, ). These results yield asymptotic estimates of the expected optiml value of the linear assignment problem under the assumption that the cost coefficients are independent random variables with distribution functionF. http://repub.eur.nl/res/pub/11645/ 1986-11-01T00:00:00Z A probabilistic analysis is presented of the Next Fit Decreasing bin packing heuristic, in which bins are opened to accomodate the items in order of decreasing size. http://repub.eur.nl/res/pub/11698/ 1986-06-01T00:00:00Z The LPT rule is a heuristic method to distribute jobs among identical machines so as to minimize the makespan of the resulting schedule. If the processing times of the jobs are assumed to be independent identically distributed random variables, then (under a mild condition on the distribution) the absolute error of this heuristic is known to converge to 0 almost surely. In this note we analyse the asymptotic behaviour of the absolute error and its first and higher moments to show that under quite general assumptions the speed of convergence is proportional to appropriate powers of (log log n)/n and 1/n. Thus, we simplify, strengthen and extend earlier results obtained for the uniform and exponential distribution. http://repub.eur.nl/res/pub/11692/ 1984-02-01T00:00:00Z In the hierarchical scheduling model to be considered, the decision at the aggregate level to acquire a number of identical machines has to be based on probabilistic information about the jobs that have to be scheduled on these machines at the detailed level. The objective is to minimize the sum of the acquisition costs and the expected average completion time of the jobs. In contrast to previous models of this type, the second part of this objective function corresponds to a well-solvable scheduling problem that can be solved to optimality by a simple priority rule. A heuristic method to solve the entire problem is described, for which strong asymptotic optimality results can be established. http://repub.eur.nl/res/pub/11689/ 1983-12-01T00:00:00Z In the distributive sorting method of Dobosiewicz, both the interval between the minimum and the median of the numbers to be sorted and the interval between the median and the maximum are partitioned inton/2 subintervals of equal length; the procedure is then applied recursively on each subinterval containing more than three numbers. We refine and extend previous analyses of this method, e.g., by establishing its asymptotic linear behaviour under various probabilistic assumptions. http://repub.eur.nl/res/pub/11647/ 1983-06-01T00:00:00Z Randomly generated polytopes are used frequently to test and compare algorithms for a variety of mathematical programming problems. These polytopes are constructed by generating linear inequality constraints with coefficients drawn independently from a distribution such as the uniform or the normal. It is noted that this class of 'random' polytopes has a special property: the angles between the hyperplanes, though dependent on the specific distribution used, tend to be equal when the dimension of the space increases. Obviously this structure of 'random' polytopes may bias test results.