The stochastic dominance (SD) is based on an axiomatic model of risk-averse preferences and there... more The stochastic dominance (SD) is based on an axiomatic model of risk-averse preferences and therefore, the SD-efficiency is an important property of selected portfolios. As defined with a continuum of criteria representing some measures of failure in achieving several targets , the SD does not provide us with a simple computational recipe. While limiting to a few selected target values one gets a typical multiple criteria optimization model approximating the corresponding SD approach. Although, it is rather difficult to justify a selection of a few target values, this difficulty can be overcome with the effective use of fuzzy target values. While focusing on the first degree SD and extending the target membership functions to some monotonic utility functions we get the multiple criteria model which preserves the consistency with both the first degree and the second degree SD. Further applying the reference point methodology to the multiple criteria model and taking advantages of fuzzy chance specifications we get the method that allows to model interactively the preferences by fuzzy specification of the desired distribution. The model itself guarantees that every generated solution is efficient according to the SD rules.
— The rapid growth of traffic induced by Internet services makes the simple over-provisioning of ... more — The rapid growth of traffic induced by Internet services makes the simple over-provisioning of resources not economical and hence imposes new requirements on the di-mensioning methods. Therefore, the problem of network design with the objective of minimizing the cost and at the same time solving the tradeoff between maximizing the service data flows and providing fair treatment of all demands becomes more and more important. In this context, the so-called Max-Min Fair (MMF) principle is widely considered to help finding reasonable bandwidth allocation schemes for competing demands. Roughly speaking, MMF assumes that the worst service performance is maximized, and then is the second worst performance, the third one, and so on, leading to a lexicographically maximized vector of sorted demand bandwidth allocations. It turns out that the MMF optimal solution cannot be approached in a standard way (i.e., as a mathematical programming problem) due to the necessity of lexicographic maximization of ordered quantities (bandwidth allocated to demands). Still, for convex models, it is possible to formulate effective sequential procedures for such lexicographic optimization. The purpose of the presented paper is threefold. First, it discusses resolution algorithms for a generic MMF problem related to telecommunications network design. Second, it gives a survey of network design instances of the generic formulation, and illustrates the efficiency of the general algorithms in these particular cases. Finally, the paper discusses extensions of the formulated problems into more practical (unfortunately non-convex) cases, where the general for convex MMF problems approach fails.
Badania operacyjne i systemowe 2004: Podejmowanie decyzji --- podstawy matematyczne i zastosowania, R.Kulikowski, J.Kacprzyk, R.Słowiński (red.),
Na przykładzie problemu wielotowarowego przepływu w sieci rozważane jest podejście umożliwiające ... more Na przykładzie problemu wielotowarowego przepływu w sieci rozważane jest podejście umożliwiające zastosowanie techniki generacji kolumn w wielokryterialnych zadaniach sprawiedliwego rozdziału zasobów (przepustowości), gdzie sprawiedliwość realizuje się przez stosowanie zasady MMF (Max-Min Fairness). Rozwiązanie zadania opiera się na maksyminimalizacji leksykograficznej i jest wyznaczane przez sekwencyjną optymalizację maksyminową z eliminacją blokujących funkcji oceny. Na każdym etapie odpowiednie zadanie maksyminimalizacji jest rozwiązywane z użyciem techniki generacji kolumn. Szczególna struktura zadania umożliwia wykorzystanie algorytmu znajdowania najkrótszej ścieżki w grafie do generowania nowych kolumn. Przedstawiono wyniki wstępnych testów numerycznych dla prezentowanego podejścia.
2012 15th International Telecommunications Network Strategy and Planning Symposium (NETWORKS), 2012
ABSTRACT Allocating bandwidth to maximize service flows with fair treatment of all the services i... more ABSTRACT Allocating bandwidth to maximize service flows with fair treatment of all the services is a key issue in network dimensioning. In such applications, the so-called Max-Min Fairness (MMF) solution concept is widely used. It is based on the worst service performance maximization with additional regularization by the lexicographic maximization of the second worst performance, the third one etc. The basic sequential procedure is applicable only for convex models, thus it allows to deal with basic design problems but fails if practical discrete restrictions commonly arriving in telecommunications network design are to be taken into account. We analyze alternative sequential approaches allowing to solve non-convex MMF network dimensioning problems. The directly defined sequential criteria can be introduced into the original model with some auxiliary variables and linear inequalities. The approaches guarantee the exact MMF solution for a complete set of criteria. However, they can be simplified by reducing the number of criteria thus generating effectively approximated MMF solutions.
The stochastic dominance (SD) is based on an axiomatic model of risk-averse preferences and there... more The stochastic dominance (SD) is based on an axiomatic model of risk-averse preferences and therefore, the SD-efficiency is an important property of selected portfolios. As defined with a continuum of criteria representing some measures of failure in achieving several targets , the SD does not provide us with a simple computational recipe. While limiting to a few selected target values one gets a typical multiple criteria optimization model approximating the corresponding SD approach. Although, it is rather difficult to justify a selection of a few target values, this difficulty can be overcome with the effective use of fuzzy target values. While focusing on the first degree SD and extending the target membership functions to some monotonic utility functions we get the multiple criteria model which preserves the consistency with both the first degree and the second degree SD. Further applying the reference point methodology to the multiple criteria model and taking advantages of fuzzy chance specifications we get the method that allows to model interactively the preferences by fuzzy specification of the desired distribution. The model itself guarantees that every generated solution is efficient according to the SD rules.
— The rapid growth of traffic induced by Internet services makes the simple over-provisioning of ... more — The rapid growth of traffic induced by Internet services makes the simple over-provisioning of resources not economical and hence imposes new requirements on the di-mensioning methods. Therefore, the problem of network design with the objective of minimizing the cost and at the same time solving the tradeoff between maximizing the service data flows and providing fair treatment of all demands becomes more and more important. In this context, the so-called Max-Min Fair (MMF) principle is widely considered to help finding reasonable bandwidth allocation schemes for competing demands. Roughly speaking, MMF assumes that the worst service performance is maximized, and then is the second worst performance, the third one, and so on, leading to a lexicographically maximized vector of sorted demand bandwidth allocations. It turns out that the MMF optimal solution cannot be approached in a standard way (i.e., as a mathematical programming problem) due to the necessity of lexicographic maximization of ordered quantities (bandwidth allocated to demands). Still, for convex models, it is possible to formulate effective sequential procedures for such lexicographic optimization. The purpose of the presented paper is threefold. First, it discusses resolution algorithms for a generic MMF problem related to telecommunications network design. Second, it gives a survey of network design instances of the generic formulation, and illustrates the efficiency of the general algorithms in these particular cases. Finally, the paper discusses extensions of the formulated problems into more practical (unfortunately non-convex) cases, where the general for convex MMF problems approach fails.
Badania operacyjne i systemowe 2004: Podejmowanie decyzji --- podstawy matematyczne i zastosowania, R.Kulikowski, J.Kacprzyk, R.Słowiński (red.),
Na przykładzie problemu wielotowarowego przepływu w sieci rozważane jest podejście umożliwiające ... more Na przykładzie problemu wielotowarowego przepływu w sieci rozważane jest podejście umożliwiające zastosowanie techniki generacji kolumn w wielokryterialnych zadaniach sprawiedliwego rozdziału zasobów (przepustowości), gdzie sprawiedliwość realizuje się przez stosowanie zasady MMF (Max-Min Fairness). Rozwiązanie zadania opiera się na maksyminimalizacji leksykograficznej i jest wyznaczane przez sekwencyjną optymalizację maksyminową z eliminacją blokujących funkcji oceny. Na każdym etapie odpowiednie zadanie maksyminimalizacji jest rozwiązywane z użyciem techniki generacji kolumn. Szczególna struktura zadania umożliwia wykorzystanie algorytmu znajdowania najkrótszej ścieżki w grafie do generowania nowych kolumn. Przedstawiono wyniki wstępnych testów numerycznych dla prezentowanego podejścia.
2012 15th International Telecommunications Network Strategy and Planning Symposium (NETWORKS), 2012
ABSTRACT Allocating bandwidth to maximize service flows with fair treatment of all the services i... more ABSTRACT Allocating bandwidth to maximize service flows with fair treatment of all the services is a key issue in network dimensioning. In such applications, the so-called Max-Min Fairness (MMF) solution concept is widely used. It is based on the worst service performance maximization with additional regularization by the lexicographic maximization of the second worst performance, the third one etc. The basic sequential procedure is applicable only for convex models, thus it allows to deal with basic design problems but fails if practical discrete restrictions commonly arriving in telecommunications network design are to be taken into account. We analyze alternative sequential approaches allowing to solve non-convex MMF network dimensioning problems. The directly defined sequential criteria can be introduced into the original model with some auxiliary variables and linear inequalities. The approaches guarantee the exact MMF solution for a complete set of criteria. However, they can be simplified by reducing the number of criteria thus generating effectively approximated MMF solutions.
The Enhanced Index Tracking Problem (EITP) calls for the determination of an optimal portfolio of... more The Enhanced Index Tracking Problem (EITP) calls for the determination of an optimal portfolio of assets with the bi-objective of maximizing the excess return of the portfolio above a benchmark and, simultaneously, minimizing the tracking error. The EITP is capturing a growing attention among academics, both for its practical relevance and for the scientific challenges that its study, as a multi-objective problem, poses. Several optimization models have been proposed in the literature, where the tracking error is measured in terms of standard deviation or in linear form using, for instance, the mean absolute deviation. More recently, reward-risk optimization measures, like the Omega ratio, have been adopted for the EITP. On the other side, shortfall or quantile risk measures have nowadays gained an established popularity in a variety of financial applications. In this paper, we propose a class of bi-criteria optimization models for the EITP, where risk is measured using the Weighted multiple Conditional Value-at-Risk (WCVaR). The WCVaR is defined as a weighted combination of multiple CVaR measures, and thus allows a more detailed risk aversion modeling compared to the use of a single CVaR measure. The application of the WCVaR to the EITP is analyzed, both theoretically and empirically. Through extensive computational experiments, the performance of the optimal portfolios selected by means of the proposed optimization models is compared, both in-sample and, more importantly, out-of-sample, to the one of the portfolios obtained using another recent optimization model taken from the literature.
The Conditional Value-at-Risk (CVaR) has become a very popular concept to measure the risk of an ... more The Conditional Value-at-Risk (CVaR) has become a very popular concept to measure the risk of an investment. In fact, though originally proposed for adoption in a financial context, the concept has potential for a broad range of applicability. In this paper, we consider problems that are formulated as mixed integer linear programming (MILP) models and show that a discrete version of the CVaR, that we call Discrete CVaR (DCVaR), can be adopted. The DCVaR mediates between a conservative Min-imax/Maximin criterion and an aggressive minimum cost/maximum profit criterion, in all cases where uncertainty or variability matters. We show that the Discrete CVaR satisfies properties that make it an attractive measure. In particular, the models resulting from the adoption of the Discrete CVaR remain MILP models. To illustrate the relevance of the proposed model, we apply the DCVaR measure to several instances of the multidimensional knapsack problem and the p-median problem.
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