J.B.G. Frenk (Hans)
http://repub.eur.nl/ppl/32/
List of Publicationsenhttp://repub.eur.nl/logo.jpg
http://repub.eur.nl/
RePub, Erasmus University RepositoryEnd-of-life inventory decisions for consumer electronics service parts
http://repub.eur.nl/pub/37785/
Sat, 01 Sep 2012 00:00:01 GMT<div>M. Pourakbar</div><div>J.B.G. Frenk</div><div>R. Dekker</div>
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. End-of-Life Inventory Decisions for Consumer Electronics Service Parts
http://repub.eur.nl/pub/18332/
Tue, 02 Mar 2010 00:00:01 GMT<div>M. Pourakbar</div><div>J.B.G. Frenk</div><div>R. Dekker</div>
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.On the Economic Order Quantity Model With Transportation Costs
http://repub.eur.nl/pub/16675/
Wed, 09 Sep 2009 00:00:01 GMT<div>S.I. Birbil</div><div>K. Bulbul</div><div>J.B.G. Frenk</div><div>H.M. Mulder</div>
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.Modelling and optimizing imperfect maintenance of coatings on steel structures
http://repub.eur.nl/pub/18266/
Fri, 01 May 2009 00:00:01 GMT<div>R.P. Nicolai</div><div>J.B.G. Frenk</div><div>R. Dekker</div>
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.The Role of Robust Optimization in Single-Leg Airline Revenue Management
http://repub.eur.nl/pub/16323/
Thu, 01 Jan 2009 00:00:01 GMT<div>S.I. Birbil</div><div>J.B.G. Frenk</div><div>J.A.S. Gromicho</div><div>J. Zhang</div>
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.Risk measures and their applications in asset management
http://repub.eur.nl/pub/13050/
Thu, 21 Aug 2008 00:00:01 GMT<div>S.I. Birbil</div><div>J.B.G. Frenk</div><div>B. Kaynar</div><div>N. N. Nilay</div>
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.Lagrangian Duality and Cone Convexlike Functions
http://repub.eur.nl/pub/19268/
Wed, 01 Aug 2007 00:00:01 GMT<div>J.B.G. Frenk</div><div>G. Kassay</div>
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.Modelling and Optimizing Imperfect Maintenance of Coatings on Steel Structures
http://repub.eur.nl/pub/10455/
Tue, 03 Jul 2007 00:00:01 GMT<div>R.P. Nicolai</div><div>J.B.G. Frenk</div><div>R. Dekker</div>
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.Modelling and optimizing imperfect maintenance of coatings on steel structures
http://repub.eur.nl/pub/10437/
Mon, 02 Jul 2007 00:00:01 GMT<div>R.P. Nicolai</div><div>J.B.G. Frenk</div><div>R. Dekker</div>
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.An elementary proof of the Fritz-John and Karush–Kuhn–Tucker conditions in nonlinear programming
http://repub.eur.nl/pub/19255/
Sun, 01 Jul 2007 00:00:01 GMT<div>S.I. Birbil</div><div>J.B.G. Frenk</div><div>G.J. Still</div>
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.Approximating the Randomized Hitting Time Distribution of a Non-stationary Gamma Process
http://repub.eur.nl/pub/10149/
Thu, 24 May 2007 00:00:01 GMT<div>J.B.G. Frenk</div><div>R.P. Nicolai</div>
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.Application of a General Risk Management Model to Portfolio Optimization Problems with Elliptical Distributed Returns for Risk Neutral and Risk Averse Decision Makers
http://repub.eur.nl/pub/10151/
Thu, 24 May 2007 00:00:01 GMT<div>B. Kaynar</div><div>S.I. Birbil</div><div>J.B.G. Frenk</div>
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.A deterministic inventory/production model with general inventory cost rate function and piecewise linear concave production costs
http://repub.eur.nl/pub/19256/
Wed, 16 May 2007 00:00:01 GMT<div>Z.P. Bayindir</div><div>S.I. Birbil</div><div>J.B.G. Frenk</div>
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.Approximating the randomized hitting time distribution of a non-stationary gamma process
http://repub.eur.nl/pub/10095/
Thu, 10 May 2007 00:00:01 GMT<div>J.B.G. Frenk</div><div>R.P. Nicolai</div>
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.Application of a general risk management model to portfolio optimization problems with elliptical distributed returns for risk neutral and risk averse decision makers.
http://repub.eur.nl/pub/9412/
Wed, 28 Mar 2007 00:00:01 GMT<div>B. Kaynar</div><div>S.I. Birbil</div><div>J.B.G. Frenk</div>
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.On Linear Programming Duality and Necessary and Sufficient Conditions in Minimax Theory
http://repub.eur.nl/pub/19261/
Thu, 01 Mar 2007 00:00:01 GMT<div>J.B.G. Frenk</div><div>P. Kas</div><div>G. Kassay</div>
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.An Integrated Approach to Single-Leg Airline Revenue Management: The Role of Robust Optimization
http://repub.eur.nl/pub/7808/
Sat, 10 Jun 2006 00:00:01 GMT<div>S.I. Birbil</div><div>J.B.G. Frenk</div><div>J.A.S. Gromicho</div><div>S. Zhang</div>
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.On Noncooperative Games, Minimax Theorems and Equilibrium Problems
http://repub.eur.nl/pub/7809/
Sat, 10 Jun 2006 00:00:01 GMT<div>J.B.G. Frenk</div><div>G. Kassay</div>
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.An integrated approach to single-leg airline revenue management: The role of robust optimization
http://repub.eur.nl/pub/7755/
Thu, 11 May 2006 00:00:01 GMT<div>S.I. Birbil</div><div>J.B.G. Frenk</div><div>J.A.S. Gromicho</div><div>S. Zhang</div>
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.On noncooperative games, minimax theorems and equilibrium problems
http://repub.eur.nl/pub/7756/
Thu, 11 May 2006 00:00:01 GMT<div>J.B.G. Frenk</div><div>G. Kassay</div>
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.