A note on constrained OWA aggregation

doi:10.1016/S0165-0114(03)00185-4

Fuzzy Sets and Systems

Volume 139, Issue 3, 1 November 2003, Pages 543-546

https://doi.org/10.1016/S0165-0114(03)00185-4 Get rights and content

Abstract

Yager considered the problem of maximizing an OWA aggregation of a group of variables that are interrelated and constrained by a collection of linear inequalities and he showed how this problem can be modelled as a mixed integer linear programming problem. In this short communication we show a simple algorithm for exact computation of optimal solutions to a constrained OWA aggregation problem with a single constraint on the sum of all decision variables.

References (1)

R.R. Yager
Constrained OWA aggregation
Fuzzy Sets and Systems
(1996)

Cited by (19)

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Besides the mean, the median is the most widely used single-valued descriptor of data sets. It is well-known that the orness level of the median operator is 1/2. In this paper we provide approximations of the median operator under a given level of orness. We find the exact optimal weighting vector for the 1-norm approximation problem in all conceivable cases and for 2-norm approximation up to nine aggregates. The analytical results show that if the orness level is not 1/2, then not only the middle, but the extreme values also matter.
Best approximation of OWA Olympic weights under predefined level of orness
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Citation Excerpt :
This problem is called a constrained OWA aggregation problem. Then, in [5] the authors provide the analytical solution in the case of a single constraint and all coefficients are equal. A new approach to solve this problem is given in [1] where, in addition, the author also proposes some problems with constraints over the argument of the OWA operator and the weights as well.
An ideal type of OWA aggregation operator is the one constructed on the so-called Olympic weights. The orness of this operator is 1/2. Therefore, sometimes we need a trade-off between the desire of having an OWA aggregation operator with weights as close as possible to the Olympic ones and the desire of having a predefined level of orness. In this paper, we choose these optimal weights by minimizing the Euclidean distance to the Olympic weights vector under the constraint of preserving a given level of orness. First, in the case $n = 4$ , we propose an iterative algorithm where the optimal solution is given for all possible values of the orness, values that are grouped along a partition of the unit interval. This iterative algorithm will inspire us to deduce the optimal weights in the general case. Consequently, we will obtain the analytical solution of the optimal weights as a function depending on the orness level.
Choquet integral optimisation with constraints and the buoyancy property for fuzzy measures
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This work concerns solving optimisation problems where the objective function is expressed as a Choquet integral. This objective generalises a linear objective function (with positive weights) and allows for interaction to be modeled between coalitions of the decision variables. We leverage results from optimising the ordered weighted averaging (OWA) operator and propose efficient solution approaches for the asymmetric objectives both for the simplest case of a single constraint and then for multiple comonotone constraints. To solve problems with a large number of variables, we rely on the so-called antibuoyancy property, previously applied to OWA weights, and which we extend to general fuzzy measures. This characterisation not only facilitates a restriction of the domain on which the solution lies but also allows us to relate the Choquet integral’s behavior in such cases to the Pigou-Dalton progressive transfers principle. We characterise the Choquet integrals consistent with the Pigou-Dalton principle. Theoretical results are supported by numerical experiments, which illustrate significant gains in performance. Our results offer opportunities for scalability to a much higher number of variables.
Constrained ordered weighted averaging aggregation with multiple comonotone constraints
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Citation Excerpt :
The solution therein involved mixed integer linear programming, which may not be computationally feasible in most real-world contexts. In [5], the authors developed an analytical expression of the optimal solution for maximizing the objective function in the case of a single constraint and equal coefficients, which was generalized in [6] to the case of arbitrary coefficients. The idea works in the case of minimization too, as it was proved in [7].
The constrained ordered weighted averaging (OWA) aggregation problem arises when we aim to maximize or minimize a convex combination of order statistics under linear inequality constraints that act on the variables with respect to their original sources. The standalone approach to optimizing the OWA under constraints is to consider all permutations of the inputs, which becomes quickly infeasible when there are more than a few variables, however in certain cases we can take advantage of the relationships amongst the constraints and the corresponding solution structures. For example, we can consider a land-use allocation satisfaction problem with an auxiliary aim of balancing land-types, whereby the response curves for each species are non-decreasing with respect to the land-types. This results in comonotone constraints, which allow us to drastically reduce the complexity of the problem.
In this paper, we show that if we have an arbitrary number of constraints that are comonotone (i.e., they share the same ordering permutation of the coefficients), then the optimal solution occurs for decreasing components of the solution. After investigating the form of the solution in some special cases and providing theoretical results that shed light on the form of the solution, we detail practical approaches to solving and give real-world examples.
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Citation Excerpt :
An effective and fast algorithm for the exact solution of the general problem addressed in [10] is still an open problem. In this contribution we extend the result of paper [2] as this time the coefficients in the constraint are variable. In this way, the case of a single constrained OWA aggregation problem is completely solved.
Weights determination of OWA operators by parametric identification
2008, Mathematics and Computers in Simulation
This contribution presents a new approach on weights determination in industrial decision making aided by OWA operators. Multi-criteria decision aid is a good way, for an industrialists, to determine his preferred compromise products, in the case of risk products or innovative products. The multi-criteria decision support chosen is the Ordered Weighted Average (OWA) operators, introduced by Yager [R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. Syst. Man Cybern. 18 (1988) 183–190]. The interest of this aggregation method is, beyond its simplicity of use, product evaluation according unique scale. Furthermore, the weights are not fixed by criterion but according to utility level. First, a learning sample is ranked by the decision-maker. Then, this ranked sample is used in order to determine the weights by parametric identification. For this, an hypothesis of equipartition of the scores of each sample is used. An industrial application, from a food production, illustrates this approach. The ranks obtained from several samples are compared.

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^☆: Partially supported by the Hungarian Research Funds OTKA T32412 and FKFP-0157/2000.

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A note on constrained OWA aggregation☆

Abstract

Fuzzy Sets and Systems