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  • 11
    Electronic Resource
    Electronic Resource
    Springer
    Journal of optimization theory and applications 87 (1995), S. 727-746 
    ISSN: 1573-2878
    Keywords: Multilevel programming ; rational reaction sets ; efficient solutions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The prime concern in this paper is the interface between efficient solutions and rational reaction sets in multilevel programming. It is shown that there need be no rational reaction solution which is also efficient, although in the bilevel case such a solution always exists, and in the generalK-level case such a solution exists if certain conditions hold. Some theoretical properties of an extension of known methods for generating efficient solutions are given, which are important algorithmically.
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  • 12
    Electronic Resource
    Electronic Resource
    Springer
    Journal of optimization theory and applications 65 (1990), S. 117-128 
    ISSN: 1573-2878
    Keywords: Multiple objectives ; least elements ; linear problems ; pre-Leontief matrices ; inverse notions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Least element theory is extended, for linear problems, to general multiple-objective problems, and the pre-Leontief matrix and left-hand matrix inverse notions are generalized.
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  • 13
    Electronic Resource
    Electronic Resource
    Springer
    Journal of optimization theory and applications 93 (1997), S. 183-197 
    ISSN: 1573-2878
    Keywords: Trilevel programming ; penalty functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper, we study a trilevel decision-making situation in which decisionmaker 1 selects an action, within a specified constraint set, then decisionmaker 2 selects an action within a constraint set determined by the action of decisionmaker 2, and finally decisionmaker 3 selects an action within a constraint set determined by the actions of decisionmakers 1 and 2. Each decisionmaker has an objective function to be optimized within the imposed constraint set. Bard (Ref. 1) presents a five-step algorithm for solving this problem. This paper presents an alternative approach to the key first step of that algorithm, which has some qualitative advantages over it.
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  • 14
    Electronic Resource
    Electronic Resource
    Springer
    Journal of global optimization 12 (1998), S. 435-448 
    ISSN: 1573-2916
    Keywords: Multiple objectives ; Vector optimization ; Constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we consider efficient sets of multiple objective problems, in which the feasible action set is the intersection of two other sets, and where one of these sets has a special structure, such as an assignment or transportation structure. The objective is to find the efficient set of the special structure set, and its intersection with the other set, and to examine how good an approximation this set is to the desired efficient set. The approximation set is called an ε-efficient solution set. Some theoretical partition results are given for a special constraint structure with upper bounds on the objective function levels. For the case of 0-efficient solution sets, and finite explicit sets, a computational cost analysis of two computational sequences is given. We also consider two other 0-efficient solution set cases. Then ε-efficiency is considered for linear problems. Finally, the approach is illustrated by a special multiple objective transportation problem.
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  • 15
    Electronic Resource
    Electronic Resource
    Springer
    OR spectrum 14 (1992), S. 79-83 
    ISSN: 1436-6304
    Keywords: Markov decision processes ; variance penalty ; parametric algorithms ; Markov-Entscheidungsprozesse ; Varianz-Strafkosten ; parametrische Verfahren
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung Für ein Markov-Entscheidungsmodell, in dem der Durchschnittsgewinn um einen Teil der mittleren Varianz vermindert wird, werden drei Typen von Lösungsverfahren entwickelt, und zwar parametrische lineare Programmierung, parametrische Lagrange-Optimierung und parametrische Politik-Iterations-Algorithmen.
    Notes: Summary This paper develops three computational approaches for solving a variance-penalised Markov decision process, viz. parametric linear programming, parametric Lagrangean programming, and a parametric policy space approach.
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  • 16
    Electronic Resource
    Electronic Resource
    Springer
    OR spectrum 10 (1988), S. 13-22 
    ISSN: 1436-6304
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung Für unendlich-stufige diskontierte Markovsche Entscheidungsprozesse werden Bedingungen angegeben, unter denen sogenannte “diskont-isotone” optimale Politiken existieren. Eine diskont-isotone Familie von optimalen Politiken liegt vor, wenn die Zustands- und Aktionenräume halbgeordnet sind und für eine Menge von Diskontierungsfaktoren je eine optimale Politik existiert, so daß in jedem Zustand die optimalen Aktionen isoton vom Diskontierungsfaktor abhängen. Es kann günstiger sein, zunächst Probleme mit kleinem Diskontierungsfaktor zu lösen und dann die Isotonie-Eigenschaften zur Lösung für größere Diskontierungsfaktoren heranzuziehen.
    Notes: Summary This paper considers infinite horizon discounted Markov decision processes and conditions under which discount-isotone optimal policies exist. Given partial orders over the state and action spaces, a set of discount-isotone optimal policies is a set of optimal policies, one for each discount factor in a given set, such that, for each state, the optimal actions are partially ordered in such a manner as to match the ordering of the discount factors. It is easier to solve problems with small discount factors and the induced partial ordering facilitates the solutions for higher discount factor levels.
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  • 17
    Electronic Resource
    Electronic Resource
    Springer
    OR spectrum 15 (1994), S. 225-230 
    ISSN: 1436-6304
    Keywords: Markov decision processes ; ɛ-optimal ; variance ; Markov'sche Entscheidungsprozesse ; ɛ-Optimalität ; Gewinnvarianz
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung In dieser Arbeit geben wir drei Algorithmen zur Lösung eines stochastischen dynamischen Problems an. Dabei wird ein optimaler Ausgleich zwischen erwartetem Gewinn und Gewinnvarianz pro Zeiteinheit gesucht. Die Algorithmen liefern für beliebigesɛ〉0 jeweilsɛ-optimale Lösungen.
    Notes: Abstract In this paper we present three algorithms for solving a problem in which it is required to get an optimal compromise between the average expected reward per unit time and the variance of the reward per unit time. The algorithms lead to anɛ-optimal solution, whereɛ〉0 is arbitrary.
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  • 18
    Electronic Resource
    Electronic Resource
    Springer
    OR spectrum 4 (1982), S. 41-45 
    ISSN: 1436-6304
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung Wir betrachten einen Markovschen Entscheidungsprozeß vom random walk Typ. Der ZustandsraumI sei eine Teilmenge des IRm, wobeii εI ganzzahlige Komponenten habe. Die MengeK der zulässigen Aktionen ini εI sei unabhängig voni εI. Sei die natürliche Ordnung aufI und ′ sei eine Quasiordnung aufK. Die Erträge {r i k }seienbedingt konvex, darüberhinaus seien weitere Voraussetzungen über diese Erträge und die Übergangswahrscheinlichkeiten in Bezug auf die Ordnungen und ′ erfüllt. Eine Politik δ heißt negativ isoton, falls ausi i′ folgtδi⊁′δ(i′) (d. h.δ(i) ′δ(i)′ oderδ(i)′ ′δ(i)′). Wir zeigen, daß unter gewissen Voraussetzungen einenegativ isotone optimale Politik existiert: Auch diskutieren wir einige Folgerungen für die Numerik, insbesondere hinsichtlich Howards Politikiteration.
    Notes: Summary This paper considers a random walk type Markov decision process in which the state spaceI is an integer subset of IR m , and the action spaceK is independent ofi εI. The natural order , overI, and a quasi order, ′, overK, is assumed, together with aconditional convexity assumption on the returns {r i k }, and certain other assumptions about these rewards and the transition probabilities in relationship to the orders and ′.A negatively isotone policy is one for whichi i′→δ(i)⊁′)δ(i′) (i.e.δ(i) ′δ(i)′ orδ(i′) ′δi)). It is shown that, under specified conditions, a negatively isotone optimal policy exists. Some consideration is given to computational implications in particular relationship to Howard's policy space method.
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  • 19
    Electronic Resource
    Electronic Resource
    Springer
    OR spectrum 11 (1989), S. 143-149 
    ISSN: 1436-6304
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung Hastings [1] benutzt Reparaturgrenzen für Ersetzungsprobleme: wenn zu einem Zeitpunkt die Reparatur eines Fahrzeugs (z. B.) hinreichend teuer ist, dann ist es ökonomisch sinnvoll, das Fahrzeug durch ein neues zu ersetzen. Numerische Resultate zeigten, daß diese Reparaturgrenzen monoton fallend sind mit zunehmendem Alter. Außerdem wurde angenommen, daß die Fahrzeuge in einem gewissen Alter unabhängig von den Raparaturkosten ersetzt werden. Die vorliegende Arbeit befaßt sich mit der theoretischen Untersuchung dieser beiden Fragen.
    Notes: Summary Hastings [1] introduced the concept of repair limits for replacement problems, the basic idea being that if, at a given time, the repair of a vehicle, for example, is sufficiently expensive, then it will be economical to replace the vehicle with a new one. Numerical results also demonstrated that repair limits decreased with age. In addition, it was assumed that, at a certain age, vehicles should be replaced whatever the repair cost. This paper addresses theoretical issues surrounding these questions.
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  • 20
    Electronic Resource
    Electronic Resource
    Springer
    OR spectrum 9 (1987), S. 13-22 
    ISSN: 1436-6304
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Description / Table of Contents: Zusammenfassung Es werden diskontierte Markovsche Entscheidungsprozesse behandelt mit endlichem Zustandsraum I, wobei die Mengen der zulässigen Entscheidungen K(i, t) vom Zustandi∈ I und vom Zeitpunktt abhängen können. Es werden verschiedene Zielfunktionen betrachtet, die jeweils als Funktion des diskontierten Gesamtgewinns (nicht dessen Erwartungswerts) formuliert werden. Um optimale oder fast-optimale Politiken zu erhalten ohne die gesamte Vorgeschichte zu registrieren, wird der Zustandsraum um die akkumulierten diskontierten Auszahlungen erweitert. Eine Auswahl solcher Probleme wird exemplarisch diskutiert einschließlich einiger Aspekte der numerischen Behandlung.
    Notes: Summary This paper deals with discounted Markov decision processes, Markov with respect to a finite statespaceI, where for eachi∈I, and each decision epocht, there is a finite action space K(i, t). The paper is concerned with problems which are formulated in terms of the discounted rewards in several ways. In order to ensure that optimal, or near optimal, policies are obtained, the state spaceI is extended to augmented state-spaces A(n), or A(∞), including the accumulated discounted rewards. Specimen problems are formulated and some computational aspects examined.
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