Heinrich Rommelfanger

Fuzzy linear programming and applications

Abstract This paper presents a survey on methods for solving fuzzy linear programs. First LP models with soft constraints are discussed. Then LP problems in which coefficients of constraints and/or of the objective function may be fuzzy are outlined. Pivotal questions are the interpretation of the inequality relation in fuzzy constraints and the meaning of fuzzy objectives. In addition to the commonly applied extended addition, based on the min-operator and used for the aggregation of the left-hand sides of fuzzy constraints and fuzzy objectives, a more flexible procedure, based on Yagers parametrized t -norm T p , is presented. Finally practical applications of fuzzy linear programs are listed.

Linear programming with fuzzy objectives

Abstract This paper presents a new method for solving linear programming problems with fuzzy parameters in the objective function. To determine a compromise solution the authors do not reduce the set of infinitely many objective functions to a single ‘compromise objective function’ as done in classical deterministic or stochastic procedures. Here the problem is only reduced to a few extreme objective functions. The information contained in the membership functions can be used to any extent by a method called ‘α-level related pair formation’. Moreover, if different level sets are considered the stability of the optimal solution can be tested.

Interactive decision making in fuzzy linear optimization problems

Abstract This paper presents an interactive method for solving a (multi-criteria) linear program, where coefficients of the objective functions and/or of the constraints are known exactly but imprecisely. Assuming (fuzzy) aspiration levels for each of the goals and basing on a comparison of (flat) fuzzy numbers, the original fuzzy problem is transformed into a crisp satisfactory model. To determine a suitable compromise solution we will maximize the minimum of the surpluses over the aspiration levels. This induces a problem with a nonlinear objective function, which is equivalent to a linear program.

Network analysis and information flow in fuzzy environment

Abstract In this paper, we present a new method for determining starting dates and slack times in project network models with fuzzy intervals. Compared with the well-known fuzzy network techniques in literature, the new approach has the advantage, that all the definitions of classical network procedure may easily be extended. The calculation of earliest and lates dates, slack times and the recognition of critical paths is demonstrated by means of an illustrative numerical example. By that the process of getting additional information and the inclusion of acceleration costs is also discussed.

A general concept for solving linear multicriteria programming problems with crisp, fuzzy or stochastic values

For modelling imprecise data the literature offers two different methods: either the use of probability distributions or the use of fuzzy sets. In our opinion these two concepts should be used in parallel or in combination, dependent on the real situation. Moreover, in many economic problems, the well-known probabilistic or fuzzy solution procedures are not suitable because the stochastic variables do not have a simple classical distribution and the fuzzy values are not fuzzy intervals. For example, in case of investment problems the coefficients may often be described by means of more complex distributions or more general fuzzy sets. In this case we propose to distinguish several scenarios and to describe the parameters of the different scenarios by fuzzy intervals. For solving such stochastic linear programs with fuzzy parameters we propose a new method, which retains the original objective functions dependent on the different states of nature. It is based on the integrated chance constrained program introduced by Klein Haneveld [On integrated chance constraints, in: Gargnano (Ed.), Stochastic Programming, Springer, Berlin, 1986, pp. 194-209] and the interactive solution process FULPAL, see Rommelfanger [Fuzzy Decision Support-Systeme - Entscheiden bei Unscharfe, second ed., Springer, Berlin, Heidelberg, 1994; FULPAL: an interactive method for solving multiobjective fuzzy linear programming problems, in: R. Slowinski, J. Teghem (Eds.), Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty, Reidel Publishing Company, Dordrecht, 1990, pp. 279-299; FULPAL 2.0-an interactive algorithm for solving multicriteria fuzzy linear programs controlled by aspiration levels, in: D. Scheigert (Ed.), Methods of Multicriteria Decision Theory, Pfalzakademie Lamprecht, 1995, pp. 21-34; The advantages of fuzzy optimization models in practical use, Fuzzy Optim. Decision Making 3 (2004) 295-310] and Rommelfanger and Slowinski [Fuzzy linear programming with single or multiple objective functions, in: R. Slowinski (Ed.), Fuzzy Sets in Decision Analysis, Operations Research and Statistics, Kluwer Academic Publishers, Norwell, MA, 1998, pp. 179-213]. An extensive numerical example illustrates the efficiency and the generality of the proposed new method.

Fuzzy mathematical programming based on some new inequality relations

Abstract This paper deals with inequality relations in fuzzy mathematical programming problem (FMP) not necessarily linear. Moreover, fuzzy parameters may have nonlinear membership functions. A new approach for comparing fuzzy sets is proposed, which is more general than the well known proposals in the literature.

A single- and a multi-valued order on fuzzy numbers and its use in linear programming with fuzzy coefficients

Abstract This paper deals with inequality relations in Fuzzy Linear Programming problem (FLP). The coefficients are trapezoidal fuzzy numbers or even more simple: left- or right trapezoidal numbers. Two new approaches for comparing left and right hand sides of constraints in FLP problems are investigated. An example demonstrating the two approaches is supplied.

FULP-A PC-Supported Procedure for Solving Multicriteria Linear Programming Problems with Fuzzy Data

This paper describes a PC-program for solving (multiobjective) fuzzy linear programming models. It is based on the interactive solution process FULPAL and support the decisionmaker in modelling the membership functions of fuzzy data, specifying the aspiration levels, constructing and solving the surrogate crisp LP-models.

Rangordnungsverfahren für unscharfe Mengen

ZusammenfassungDiese Arbeit behandelt das Problem, eine Rangfolge unscharfer Mengen über der gleichen GrundmengeU ⊂ ℝ aufzustellen oder zumindest die „optimale“ unscharfe Menge zu bestimmen. Die wichtigsten in der Literatur vorgeschlagenen Rangordnungsverfahren werden kurz dargestellt und anschließend mit empirisch ermittelten Präferenzaussagen verglichen. Dabei zeigt es sich, daß in unproblematischen Fällen alle Methoden zu vernünftigen Präferenzordnungen führen, in kritischen Fällen aber nur das Niveau-Ebenen-Verfahren von Rommelfanger und das Chen-Kriterium zufriedenstellende Ergebnisse liefern.SummaryThis paper deals with the problem of ranking fuzzy sets of the same intervalU ⊂ ℝ. The most important methods suggested in literature are reviewed and compared with empirical preference-statements. In simple cases all the methods are good, but in difficult cases only the Chen-criterion and theα-level-method suggested by Rommelfanger behave reasonably well.

Entscheidungsmodelle mit Fuzzy-Nutzen

Eine wesentliche Ursache fur die geringe praktische Bedeutung statistischer Entscheidungsmodelle sind die ihnen zugrunde liegenden realitatsfremden Pramissen, insbesondere die hohen Anforderungen an den Informationsstand des Entscheidungstragers. Dessen zumeist nur vages Wissen uber die Menge aller in Betracht kommenden Alternativen A={ai}, i=1,2,..,m, die Menge der moglichen Umweltzustande S={sj}, j=1,2,..,n, die Konsequenzen, die sich aus der Entscheidung fur eine Handlungsalternative ai ergeben, wenn sich der Umweltzustand sj einstellt, und die zumeist als Nutzen u(ai,sj) ausgedruckt werden, reicht im allgemeinen nicht aus, ein Entscheidungsmodell der klassischen Form aufzustellen.