An Information Axiom based decision making approach under hybrid uncertain environments

Abstract

This paper aims to develop a decision making approach based on the Information Axiom under hybrid uncertain environments. In these environments fuzziness and randomness co-exist. To model the hybrid uncertainties, the design range and system range of an evaluation criterion are expressed as a fuzzy variable and a random variable respectively. Calculating the Information Contents for these criteria would involve integrals whose bounds are fuzzy variables. To calculate these integrals, an expected Information Content Model (eICM) is proposed based upon the credibility theory. To reflect the designer’s risk appetite accurately, the confidence levels are introduced and a credibility Information Content Model (cICM) based on the chance constrains programming model is constructed. Two algorithms based on fuzzy simulation methods and Genetic Algorithm are developed for these two models. Finally, a case study is illustrated to highlight the effectiveness and accuracy of the proposed approach.

Introduction

Axiomatic design developed by Suh [37] was motivated by the absence of scientific design principles [7]. According to axiomatic design, a design work could be regarded as a zigzag mapping process between the Functional Requirements (FRs) and the Design Parameters (DPs). Two fundamental design principles, the Independence Axiom and the Information Axiom should be satisfied in this mapping. The Independence Axiom maintains the independence of FRs and enhances the controllability of a design. The Information Axiom is used to minimise the Information Content of a design and reduce the design complexity. The Information Content is defined as a measure of complexity, and it is related to the probability of conceived solutions meeting FRs. The objective of this paper is to extend the Information Axiom under hybrid uncertain environments characterized by randomness and fuzziness co-existing. Randomness is about an occurrence (or non-occurrence) of a certain event (typically described by some set) [29]. Fuzziness is not about an occurrence but about a degree of membership to a certain concept [29].

Randomness and fuzziness sometimes co-exist in actual decision making problems. Take the evaluation criterion “timeliness” in a service selecting problem for example. On the one hand, “timeliness” is a random variable obeying some pdf (probability distribution function) for the reason that each execution of this service happens at different time and different places by different staffs, and the pdf is denoted as f(FR). On the other hand, “timeliness” is usually described in imprecise linguistic terms such as “very slow”, “slow”, “medium”, “fast”, “very fast”, which could be modelled by the fuzzy set theory [28], [39], [42], [43]. We refer to this kind of criterion as a hybrid uncertain criterion [24]. Two difficult issues will be encountered under hybrid uncertain environments. The first one is how to calculate the integrals whose bounds are fuzzy variables. The integrals are used to calculate the probability of a design solutions meeting FRs (successful probability) which is related to the Information Content. For a hybrid uncertain criterion, the integral bounds are fuzzy variables. For example, the success probability of “timeliness” is ${\int }_{''“''\text{medium}''”''}^{''“''\text{very fast}''”''}f(\mathit{FR})\mathit{dFR}$, and the integral bounds are two fuzzy variables “medium” and “very fast”. How to calculate this kind of integrals characterised by fuzzy bounds is seldom mentioned in existing researches. The second problem is how to reflect the risk appetite of decision makers in the Information Content. Due to the multi-value characteristic of fuzzy variables, the success probability of a hybrid uncertain criterion is also multi-value. Take the success probability of “timeliness” ${\int }_{''“''\text{medium}''”''}^{''“''\text{very fast}''”''}f(\mathit{FR})\mathit{dFR}$ for example again. The up and down fuzzy bound “very fast” and “medium” are both multi-value. Every combination of the up and down bound’s value determines a success probability of “timeliness”. In order to calculate the Information Content, an integrate value should be obtained from the multiple values, which is influenced by decision makers’ risk appetite. The bigger the integrated value takes, the more risky the decision makers’ risk appetite is. Up to now, little research deals with these problems. The existing Information Axiom approaches, considering randomness or fuzziness alone, could not describe the reality perfectly, and the evaluation results could not express the opinions of decision makers accurately.

An extension of the Information Axiom under hybrid uncertain environments is presented in this paper. The remaining part is organized as follows. Section 2 reviews the extensive literatures about the Information Axiom and the credibility theory. In Section 3 some important concepts are represented. In Section 4 two Information Content models are developed. Section 5 demonstrates the application of the proposed evaluation approach by a case. Section 6 compares the proposed approach with the traditional Information Axiom and concludes this paper.