Wierzbicki A.P. (1980), The use of reference objective in Multiobjective Optimization, in: G. Fandel, T. Gal (eds.), Multiple Criteria Decision Making, Theory and Application, Springer, Berlin, 468-486.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....In the on line approach it is possible to use a single objective metaheuristic applied to a substitute problem whose optimal solution corresponds to a Pareto optimal solution of the original MOCO problem. The substitute problem may for example be optimization of a scalarizing function [179] or optimization of one of the objectives with constrained values of the remaining objectives (g constraints method [154] ch. 8.5) In the off line approach multiple objective metaheuristics could be used. The off line generation of approximately Pareto optimal solutions has several advantages ....

....The first degree permits specification of his her aspiration levels defining the aspiration point. The aspiration point is then projected onto the nondominated set. Similar degree of freedom is given in the Interactive Goal Programming [39] the Naive method [13] 15] the Reference Point method [179] and the Satisficing Trade Off method [125] These procedures belong to the class of learning oriented interactive procedures [172] Although they guarantee free search over the entire nondominated set, they do not give any substantial support to the DM in the decision phases. The second degree ....

Wierzbicki A.P. (1980), The use of reference objective in Multiobjective Optimization, in: G. Fandel, T. Gal (eds.), Multiple Criteria Decision Making, Theory and Application, Springer, Berlin, 468-486.

....that are the closest to some utopia (ideal) point. Di erent norms have been used to measure the distance between the solutions and the utopia point. In particular, the family of L p norms has been extensively studied by many researchers, including [Yu, 1973] Zeleny, 1973] Gearhart, 1979] [Wierzbicki, 1980], Steuer and Choo, 1983] Steuer, 1986] and many others. The l 1 norm and the augmented l 1 norm turned out to be very useful in generating nondominated solutions of general continuous or discrete multiple criteria programs and led to the well known weighted (augmented) Tchebyche scalarization ....

Wierzbicki, Andrzej P. (1980). The Use of Reference Objectives in Multiobjective Optimization. Lecture Notes in Economics and Mathematical Systems, 177:468-486.

....theoretical foundations of DEA and the Reference Point model of MOLP, we discuss 2 similarities and dissimilarities between DEA and MOLP formulations from a mathematical standpoint. We show that mathematically the CCR model by Charnes et al. 1978] and the reference point approach proposed by Wierzbicki [1980] to solving MOLP problems use quite similar formulations. Since the BCC model by Banker, Charnes and Cooper [1984] is a variation of the CCR model, its structural closeness to the reference point formulation follows from the results of our paper. The implications of this structural closeness are ....

....vectors. The set of all efficient solutions is called the efficient set (denoted E) and the set of all nondominated criterion vectors is called the nondominated set (denoted N) For weakly efficient solutions, we use E W and N W , respectively. For characterizing the nondominated set, Wierzbicki [1980] suggested the use of an achievement (scalarizing) function (ASF) Consider the following problem: min s(g, v, w, r) min max iP [ g i v i ) w i ] r i=1 p (g i v i ) 3.2) s.t. v = V= v = Cx o oo x X , where s is the ASF, w 0, w R p , is a vector of weights, r 0 is a ....

[Article contains additional citation context not shown here]

Wierzbicki, A. (1980), "The Use of Reference Objectives in Multiobjective Optimization", in G. Fandel and T. Gal (Eds.), Multiple Objective Decision Making, Theory and Application, Springer-Verlag, New York.

....methods of generating nondominated solutions, the ultimate principle is the same in all methods: a single objective optimization problem is solved to generate a new solution or solutions. The objective function of this single objective problem may be called a scalarizing function according to Wierzbicki [1980]. It typically has the original objectives and a set of parameters as its arguments. The form of the scalarizing function, as well as what parameters are used, depends on the assumptions made concerning the DM s preference structure and behavior. Two classes of parameters are widely used in ....

....use l Cx as a scalarizing function, limiting considerations to efficient extreme points (see, e.g. Zionts and Wallenius [1976] 3.2. A Chebyshev type Scalarizing Function Currently, most solution methods are based on the use of a so called Chebyshev type scalarizing function first proposed by Wierzbicki [1980]. The same function was also used by Steuer and Choo [1983] but in a somewhat different form. We will refer to this function by the term achievement (scalarizing) function. The achievement (scalarizing) function projects any given (feasible or infeasible) point g k onto the set of ....

[Article contains additional citation context not shown here]

Wierzbicki, A. (1980), "The Use of Reference Objectives in Multiobjective Optimization", in G. Fandel and T. Gal (Eds.), Multiple Objective Decision Making, Theory and Application, Springer-Verlag, New York.

....achievement scalarizing functions. These functions can be controlled either by varying weights (keeping aspiration levels fixed) or by varying the aspiration levels (keeping weights fixed) The same idea was originally proposed in a somewhat different form by Steuer and Choo [1983] and Wierzbicki [1980]. The Achievement Scalarizing Function is the main theoretical basis on which the Reference Direction Approach proposed by Korhonen and Laakso [1986a] also lies. The parameterizing of the achievement scalarizing function was one of the original ideas in the reference direction approach. When a ....

....Zionts and Wallenius [1983] proposed the use of the optimality conditions for extreme point solutions. That idea has been generalized for any nondominated solution in the reference direction approach. Thus the reference direction approach is a combination of the ideas of Geoffrion et al. 1972] Wierzbicki [1980], and Zionts and Wallenius [1976] flavored with some of our own ideas. The original reference direction approach has been further developed into many directions. First, Korhonen and Wallenius [1988] improved upon the original procedure by making it dynamic. The dynamic version was called Pareto ....

[Article contains additional citation context not shown here]

Wierzbicki, A. (1980). "The Use of Reference Objectives in Multiobjective Optimization", in G. Fandel and T. Gal (eds.), Multiple Criteria Decision Making Theory and Applications, Springer, New York.

....in [Sim58] another rationality framework, called bounded rationality or satisficing decision making. This framework has been extended further by many researchers (cf e.g. a summary given by Lewandowski and Wierzbicki in [LeW89b] One of the main directions in that field was set by Wierzbicki [Wie80] who formulated the principle of reference point optimization in multiobjective optimization and decision support. That principle has been extended by Wierzbicki (cf [Wie82, Wie84, Wie86] to principles of quasisatisficing decision making and has been extensively used both in research and in ....

....the method based on the aspiration level (sometimes referred to as reference point) concept. Following [OgL92] we will use for this method the acronym ARBDS (Aspiration Reservation Based Decision Support) The ARBDS method is an extension of the reference point method proposed by Wierzbicki in [Wie80] Several other extensions or similar approaches have been proposed and implemented (cf [LeW89a, LAP94, Nak94, SNT85, SeS88, Ste86] The differences between the approaches presented in those publications are discussed and a modular implementation of ARBDS approach is presented in more details in ....

A. Wierzbicki, The use of reference objectives in multiobjective optimization, in Multiple Criteria Decision Making, Theory and Applications, G. Fandel and T. Gal, eds., Lecture Notes in Economics and Mathematical Systems, vol. 177, Springer Verlag, Berlin, New York, 1980, pp. 468--486.

....methods of generating nondominated solutions, the ultimate principle is the same in all methods: a single objective optimization problem is solved to generate a new solution or solutions. The objective function of this single objective problem may be called a scalarizing function according to Wierzbicki [1980]. It typically has the original objectives and a set of parameters as its arguments. The form of the scalarizing function as well as what parameters are used depends on the assumptions made concerning the DM s preference structure and behavior. Two classes of parameters are widely used in multiple ....

....with weight vectors l k , limiting considerations to efficient extreme points (see, e.g. Zionts and Wallenius [1976] 3.2. A Chebyshev type Scalarizing Function Currently, the most solution methods are based on the use of a so called Chebyshevtype scalarizing function first proposed by Wierzbicki [1980]. We will refer to this function by the term achievement (scalarizing) function. The achievement (scalarizing) function projects any given (feasible or infeasible) point g k onto the set of nondominated solutions. Point g is called a reference point, and its components represent the desired ....

[Article contains additional citation context not shown here]

Wierzbicki, A. (1980), "The Use of Reference Objectives in Multiobjective Optimization", in G. Fandel and T. Gal (Eds.), Multiple Objective Decision Making, Theory and Application, Springer-Verlag, New York.

....are the closest to some utopia (ideal) point has been very successful. Different norms have been used to measure the distance between the solutions and the utopia point. In particular, the family of L p norms has been extensively studied by many researchers, including (Yu, 1973) Zeleny, 1973) (Wierzbicki, 1980), Steuer and Choo, 1983) Steuer, 1986) and many others. The l 1 norm and the augmented l 1 norm turned out to be very useful in generating nondominated solutions of general continuous or discrete multiple criteria programs and led to the well known weighted (augmented) Tchebycheff ....

Wierzbicki, A. P. (1980). The use of reference objectives in multiobjective optimization. Lecture Notes in Economics and Mathematical Systems, 177:468--486.

....formulated [S4] another rationality framework, called bounded rationality or satisficing decision making. This framework has been extented further by many researchers (cf e.g. a summary given by Lewandowski and Wierzbicki in [L4] One of the main directions in that field was set by Wierzbicki [W3], who formulated the principle of reference point optimization in multiobjective optimization and decision support. That principle has been extended by Wierzbicki (cf [W4, W5, W6] to principles of quasisatisficing decision making and has been extensively used both in research and in applications ....

....part of the criteria space. There are many methods for the scanning and analysis of solutions on the Pareto surface. The most natural method which best corresponds to a real life DMP seems to be the method based on the aspiration level (sometimes referred to as reference point) concept (cf [L4, W1, W3]) 7 There are some technical difficulties for the determination of the nadir point, if there are more than two criteria and if nadir point is defined as the point composed of the worst (over a set of all Paretoefficient solutions) values for each criterion. Therefore for practical applications ....

Wierzbicki, A.P., The use of reference objectives in multiobjective optimization. In G. Fandel and T. Gal, eds., Multiple Criteria Decision Making, Theory and Applications, Springer Verlag, Heidelberg 1980.

....no direction of improvement is found. A critical point in the procedure is the efficiency of the projection operation. This efficiency of our algorithm is considered theoretically and numerically. The projection is made by parametrizing an achievement scalarizing function originally proposed by Wierzbicki (1980) to project any single point onto the nondominated set. Keywords: Multiple Criteria, Discrete, Evaluation, Reference Point, Reference Direction. Acknowledgments The research was supported, in part, by grants from the Academy of Finland, the Foundation for Economic Education, and the Foundation ....

....achievement scalarizing functions. These functions can be controlled either by varying weights (keeping aspiration levels fixed) or by varying the aspiration levels (keeping weights fixed) The same idea was originally proposed in a somewhat different form by Steuer and Choo [1983] and Wierzbicki [1980]. The Achievement Scalarizing Function is also the main theoretical basis in the Reference Direction Approach proposed by Korhonen and Laakso [1986] Instead of projecting one point at a time onto the nondominated frontier like in the original reference point approach by Wierzbicki [1980] ....

[Article contains additional citation context not shown here]

Wierzbicki, A. (1980), "The use of reference objectives in multiobjective optimization", in: G. Fandel and T. Gal (eds.), Multiple Criteria Decision Making, Theory and Application, Springer, New York.

.... refer to the efficient solutions in the criterion space p m (set T, where T = y, x) y = Yl, x = Xl, l L ) A possible and currently popular way to perform the search for solutions on the efficient frontier of a MOLP problem is to use the achievement (scalarizing) function suggested by Wierzbicki [1980]. Following Wierzbicki, we call the method the Reference Point Method. To characterize the efficient set of problem (2.1) we may use the following formulation: min s(g, u, w,d ) min max i P [ a i (g u i ) w i ] d i P a i (g i u i ) 2.2) s.t. u = y, x) T, where s is the ASF, w ....

.... (MOLP) are useful in analyzing the efficiency by using Data Envelopment Analysis (DEA) To characterize the efficient frontier of a MOLP problem, a widely used technique is to transform the problem into a single objective problem by using an achievement scalarizing function as proposed by Wierzbicki [1980]. This transformation leads to a so called reference point model in which the search on the efficient frontier is controlled by varying the aspiration levels of the values of the objective functions. For each given aspiration level point, the minimization of the achievement scalarizing function ....

Wierzbicki, A. (1980), "The Use of Reference Objectives in Multiobjective Optimization", in G. Fandel and T. Gal (Eds.), Multiple Objective Decision Making, Theory and Application, Springer-Verlag, New York.

No context found.

A. Wierzbicki, "The Use of Reference Objectives in Multiobjective Optimization," in Multiple Criteria Decision Making, Theory and Application, G. Fandel and T. Gal, (Eds.), Springer-Verlag, Berlin, 1980, 468-486.

No context found.

Wierzbicki A.P. (1980), The use of reference objective in Multiobjective Optimization. In: Fandel G. and Gal T. (eds.) Hy#vfyr 8...v#r...vh 9rpv+v' Hhxvt# Uur'...'hq6ffyvph#v', Springer-Verlag, Berlin, 468-486.

No context found.

Wierzbicki, A., "The Use of Reference Objectives in Multiobjective Optimization," in Multiple Criteria Decision Making, Theory and Application, G. Fandel and T. Gal, (Eds.), Springer-Verlag, Berlin, 1980, pp. 468-486.

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