Elsevier

Fuzzy Sets and Systems

Volume 446, 5 October 2022, Pages 222-232
Fuzzy Sets and Systems

Aggregation with dependencies: Capacities and fuzzy integrals

https://doi.org/10.1016/j.fss.2021.03.009Get rights and content

Abstract

We outline recent trends in capacity-based aggregation in large universes. Capacities (fuzzy measures) model dependencies among the inputs, and aggregation by the discrete Choquet, Sugeno and other fuzzy integrals accounts for synergies and redundancies. For large number of inputs the exponential complexity of all interactions is a major obstacle. We exemplify the need for aggregation of a large number of dependent inputs on several applications and discuss the challenges and approaches to reducing the complexity of capacity-based aggregation. We also state which mathematical and computational tools are required for large scale capacity modelling.

Introduction

In this paper we outline the new trends in aggregation with inputs dependencies, and in particular we focus on capacities – normalised monotone set functions that model importances of not only individual inputs but also all their subsets, called coalitions. Capacities, also known as fuzzy measures and transferable utility (TU) cooperative games [11], [24], [38], [66], provide a powerful modelling tool to account for interactions within each coalition of inputs, and have been used for decades in Games theory and multiple criteria decision making (MCDM) [38], [40]. The aggregation is usually performed with the help of the Choquet integral [24], Sugeno integral [73] or other fuzzy integrals, see a review in [11].

The main challenge of this modelling approach is the high number of inputs coalitions (for n inputs it is 2n), and subsequently huge computational complexity, which involves not only the parameters but relations among them, such as monotonicity or convexity. Modern practical applications produce large universal sets, which imply the need to develop suitable representation, simplification, characterisation, eliciting, computation, and other tools. This paper reviews some of the recent developments in those areas with particular focus on large universes.

The paper is structured as follows. Section 2 outlines some of the practical aggregation problems which involve dependencies and motivate the use of capacities as a tool. Section 3 presents the capacities and fuzzy integrals. Section 4 presents the challenges when using capacities to model large scale problems and outlines some of the approaches to deal with those challenges. In Section 5 we summarise the needs for development of computational tools that would allow to successfully use those approaches. Section 6 presents the conclusions.

Section snippets

Why capacities?

The dependencies are everywhere. Aggregation in which only the individual inputs are weighted implicitly assume inputs independence, yet in reality a large number of practical problems involve inputs synergies and redundancies. Let us outline a few of such problem domains with specific emphasis on large universes.

The first area of application is the decision problems involving multiple criteria, multiple decision makers and multiple objectives. Take for example selection of a combat vehicle [62]

How they work

We consider the set of inputs N={1,2,…,n}, n⩾2, which can be either decision criteria, optimisation objectives, portfolio items, or any other aggregation variables. Let P(N) be its power set and let |S| denote the cardinality of a subset S⊆N. We only provide some basic definitions, the rest can be found in [6], [11], [38], [40], [81].

Definition 1

A capacity (fuzzy measure) on N is a set function μ:P(N)→[0,1] such that (i) μ(∅)=0, μ(N)=1; (ii) ∀A,B⊆N, A⊆B implies μ(A)⩽μ(B).

The value of μ(A) reflects the

Problems, challenges and approaches

While aggregation based on a given capacity seems to be straightforward, for large numbers of criteria n the exponential complexity of capacities starts to bite. There are 2n−2 parameters that need to be specified, and of the order of n2n monotonicity constraints to be ensured. Not only the storage and computation with capacities becomes problematic, but to a greater degree the interpretation, acquisition, and quantification of interactions is unrealistic in practice. For example even if

Tools required

We start with the capacity representations, because depending on the representation one can achieve significant savings in terms of storage and processing capacities of a particular type. The foremost current representations are the standard and the Möbius representations, related to each other by a system of linear equations. Various interaction indices also offer alternative representations, for example the Shapley interaction index [37], [41], the nonadditivity and nonmodularity indices [81]

Conclusions

We have presented examples of applications which require modelling of interactions of large numbers of inputs. The theory of capacities, also known as fuzzy measures and cooperative games, provides suitable mathematical formalism for modelling interactions, and fuzzy integrals accommodate specified interactions at the aggregation stage. There is high computational complexity when capturing all possible interactions, which renders general capacities unsuitable for large numbers of inputs.

In this

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

The publication was supported by the Australian Research Council Discovery Project DP210100227 and by the RUDN University Strategic Academic Leadership Program.

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