## Aggregation Functions for Engineering Design Trade-offs (1997)

Citations: | 27 - 14 self |

### BibTeX

@MISC{Scott97aggregationfunctions,

author = {Michael J. Scott and Erik K. Antonsson},

title = {Aggregation Functions for Engineering Design Trade-offs},

year = {1997}

}

### Years of Citing Articles

### OpenURL

### Abstract

The choice of an aggregation function is a common problem in Multi Attribute Decision Making (MADM) systems. The Method of Imprecision (MoI) is a formal theory for the manipulation of preliminary design information that represents preferences among design alternatives with the mathematics of fuzzy sets. The MoI formulates the preliminary design problem as a MADM problem. To date, two aggregation functions have been developed for the MoI, one representing a compensating strategy and one a noncompensating strategy. Much of the prior fuzzy sets research on aggregation functions has been inappropriate for application to engineering design. In this paper, the selection of an aggregation function for MADM schemes is discussed within the context of the MoI. The general restrictions on designappropriate aggregation functions are outlined, and a family of functions, modeling a range of trade-off strategies, is presented. The results are illustrated with an example.

### Citations

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Citation Context ...I. Figure 4 shows Ps plotted for several positive values of s. As s !1, Ps ! max . If the designer preferences are taken to represent utilities, then the aggregation functions used by utility theory (=-=Keeney and Raiffa, 1993) can be similarly plotted. Th-=-e additive utility function (P((��1 ; !1 ); (��2 ; !2 )) = !1��1 + !2��2 ) corresponds to P1 in Figure 4. The "multi-linear" function, also shown in Figure 4 (P((��1 ; !1... |

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Citation Context ...their list, as do weighted sums. However, the general weighted means discussed in this paper do not. 4 Weighted Means Fuzzy set researchers have productively applied the study of functional equations =-=[1]-=- to explore t-norms and t-conorms. This section applies the same general approach to design-appropriate aggregation functions: an intuitively reasonable set of axioms is translated into a set of funct... |

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Citation Context ...o represent and manipulate imprecise design information. The MoI casts the design decision-making problem as a Multi Attribute Decision Making (MADM) or Multi Objective Decision Making (MODM) problem =-=[4, 24]-=-. The choice of an aggregation function is a crucial part of any MADM or MODM scheme. The MoI has attempted to approach this choice axiomatically. The axioms of the MoI restrict the choice of candidat... |

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Citation Context ...o represent and manipulate imprecise design information. The MoI casts the design decision-making problem as a Multi Attribute Decision Making (MADM) or Multi Objective Decision Making (MODM) problem =-=[4, 24]-=-. The choice of an aggregation function is a crucial part of any MADM or MODM scheme. The MoI has attempted to approach this choice axiomatically. The axioms of the MoI restrict the choice of candidat... |

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Citation Context ... extensions of classical intersection and union (Dubois and Prade, 1985). The operators of binary logic have been applied to fuzzy truth values to develop a theory for the aggregation of fuzzy goals (=-=Bellman and Zadeh, 1970). In this-=- system, a decision maker may have one of three attitudes towards goals: a "conjunctive attitude", in which goals must be simultaneously satisfied, modeled by the logical and and the interse... |

167 |
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Citation Context ... the MoI and to the methods listed above. The possible application of utility theory to engineering design has been considered previously, and shown to be problematic [12]. Matrix methods such as QFD =-=[6]-=- and Pugh charts [13] also support decision making by simple additive aggregation over several requirements. Aggregation operators are important in all MADM methods, from the most formal to the most c... |

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Citation Context ...ests, such as those conducted by other researchers [15] can help determine which aggregation functions best model 1 Both of these aggregations are analogous to Pareto-optimal solutions in game theory =-=[9, 18]-=-, and the product of powers is analogous to a Nash solution. However, neither correspondence is mathematically precise, since preferences are not equivalent to utilities.ScottandAntonsson,“Aggregatio... |

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Citation Context ...both trade-off strategies to industrial applications is ongoing (Law and Antonsson, 1994). A class of aggregation functions on fuzzy sets called t-norms and t-conorms has been studied in some detail (=-=Dubois and Prade, 1985-=-; Roychowdhury and Wang, 1994) by researchers of fuzzy sets. Unfortunately, these functions do not satisfy the axioms of the MoI, as will be explained below. Much less research has been devoted to mix... |

60 |
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Citation Context ...ty is often used as a basis for the choice of an aggregation function, however inappropriate that may be. The development of the MoI has been to appeal to intuitive notions of rational human behavior =-=[16]-=-, and to formalize this rationality in a set of axioms that the aggregation functions must follow. The axioms of the MoI (see Table 1) [10] are a formal description of restrictions on any aggregation ... |

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Citation Context ... [19]. A rigorous treatment of design decision-making is necessary for the construction of computer tools to aid the designer in that decision-making. One such tool is the Method of Imprecision (MoI) =-=[20]-=-, a formal method for preliminary design decision-making that uses the mathematics of fuzzy sets to represent and manipulate imprecise design information. The MoI casts the design decision-making prob... |

49 |
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Citation Context ...methods listed above. The possible application of utility theory to engineering design has been considered previously, and shown to be problematic [12]. Matrix methods such as QFD [6] and Pugh charts =-=[13]-=- also support decision making by simple additive aggregation over several requirements. Aggregation operators are important in all MADM methods, from the most formal to the most casual. The arithmetic... |

36 |
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Citation Context ....g., [14]). MADM schemes have applied a wide range of t-norms and other operators to decision problems. While averaging operators, which fall between min and max, have been acknowledged for some time =-=[5, 22, 23]-=-, comparatively little study has been devoted to these connectives. Averaging operators are not appropriate for binary logic, but they are well suited to engineering design decisions: indeed, the axio... |

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(Show Context)
Citation Context ... too small to be distinguished. A designer does not actually specify a continuous preference function on the interval [0, 1], but rather gives the values for each variable on several different α-cuts =-=[2]-=-. For example, for a particular design variable, the designer may specify which values correspond to µ =0,µ=0.25, µ =0.5,µ=0.75, andµ=1. The discontinuous manner in which preferences are specified pro... |

26 |
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Citation Context ...esign decisions are among the most important in engineering. It is when the details of a design are still unknown, when the design is imprecise, that the most costly and important decisions are made (=-=Whitney, 1988-=-). Despite the importance of preliminary design, there are a limited number of formal tools available to the engineer for representing and manipulating imprecise design information. One such tool is t... |

20 |
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Citation Context ... axioms will be investigated below. Three of the axioms are particularly important to the discussion here. The annihilation axiom is crucial for design, as others have argued (Biegel and Pecht, 1991; =-=Vincent, 1983-=-). It states that if the preference for any one attribute of the design sinks to zero (unaceptable) then the overall preference for the design is zero. For example, given a fixed material, the tensile... |

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Citation Context ...e lowest-ranked attribute. Idempotency and monotonicity together lead to the requirement that min ≤P≤max. The annihilation axiom is specific to engineering design, and others have argued its validity =-=[3, 12, 17]-=-. It states that if the preference for any one attribute of the design sinks to zero (unacceptable) then the overall preference for the design is zero. For example, given a fixed material, the tensile... |

16 | Trade-Off Strategies in Engineering Design
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(Show Context)
Citation Context ...ng uncertainty in engineering design employing the mathematics of fuzzy sets. It has been shown previously that the MoI can be used to combine design information using different trade-off strategies (=-=Otto and Antonsson, 1991-=-). Specifically, the designer can combine preferences in a compensating or non-compensating manner, using one of two aggregation functions. The MoI can handle importance weightings using these two tra... |

16 |
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Citation Context ...unctions used be justifiable models for decision-making behavior. The choice of an aggregation function may be justified in several ways. Empirical tests, such as those conducted by other researchers =-=[15]-=- can help determine which aggregation functions best model 1 Both of these aggregations are analogous to Pareto-optimal solutions in game theory [9, 18], and the product of powers is analogous to a Na... |

14 | A Formal Representational Theory for Engineering Design
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(Show Context)
Citation Context ...empted to approach this choice axiomatically. The axioms of the MoI restrict the choice of candidate aggregation functions so as to reflect the natural structure of the design decision-making process =-=[10, 11]-=-. Within the axiomatic framework of the MoI, a designer can combine preliminary design information using different trade-off strategies. In particular, two aggregation functions have been used for mul... |

13 | Implementing the Method of Imprecision: An Engineering Design Example
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(Show Context)
Citation Context ...son, 1995a; Otto and Antonsson, 1991), and different attributes can be combined hierarchically in hybrid strategies. Research applying both trade-off strategies to industrial applications is ongoing (=-=Law and Antonsson, 1994-=-). A class of aggregation functions on fuzzy sets called t-norms and t-conorms has been studied in some detail (Dubois and Prade, 1985; Roychowdhury and Wang, 1994) by researchers of fuzzy sets. Unfor... |

9 |
Design trade-offs made easy
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Citation Context ...tions that satisfy these axioms will be investigated below. Three of the axioms are particularly important to the discussion here. The annihilation axiom is crucial for design, as others have argued (=-=Biegel and Pecht, 1991-=-; Vincent, 1983). It states that if the preference for any one attribute of the design sinks to zero (unaceptable) then the overall preference for the design is zero. For example, given a fixed materi... |

7 | Hierarchical Imprecise Design with Weights
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(Show Context)
Citation Context ...e designer can combine preferences in a compensating or non-compensating manner, using one of two aggregation functions. The MoI can handle importance weightings using these two trade-off strategies (=-=Law and Antonsson, 1995-=-a; Otto and Antonsson, 1991), and different attributes can be combined hierarchically in hybrid strategies. Research applying both trade-off strategies to industrial applications is ongoing (Law and A... |

5 |
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Citation Context ....g., [14]). MADM schemes have applied a wide range of t-norms and other operators to decision problems. While averaging operators, which fall between min and max, have been acknowledged for some time =-=[5, 22, 23]-=-, comparatively little study has been devoted to these connectives. Averaging operators are not appropriate for binary logic, but they are well suited to engineering design decisions: indeed, the axio... |

5 | Imprecision in Engineering Design." ASME Journal of Mechanical Design 117 - Antonsson, Otto - 1995 |

5 | H.Decisions with Multiple Objectives: Preferences and Value Tradeoffs - KEENEY, RAIFFA - 1993 |

3 |
Comparing Fuzzy and Probability Calculus for Representing Imprecision in Preliminary Engineering Design
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Citation Context ...in. As this uncertainty is not probabilistic, but will be resolved by further refinement and specification later in the design process, it is appropriately modeled using the mathematics of fuzzy sets =-=[20, 21]-=-. The Method of Imprecision is a formal system for the representation and manipulation of such imprecise design data. When a design is finalized, standard analysis tools (e.g., finite element analysis... |

2 |
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Citation Context ...s to industrial applications is ongoing (Law and Antonsson, 1994). A class of aggregation functions on fuzzy sets called t-norms and t-conorms has been studied in some detail (Dubois and Prade, 1985; =-=Roychowdhury and Wang, 1994-=-) by researchers of fuzzy sets. Unfortunately, these functions do not satisfy the axioms of the MoI, as will be explained below. Much less research has been devoted to mixed connectives, those functio... |

1 |
Imprecision in Engineering Design," ASME Journal of Mechanical Design Invited paper, to appear
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(Show Context)
Citation Context ...ce are too small to be distinguished. A designer does not actually specify a continous preference function on the interval [0; 1]; she gives the values for each variable on several different ff-cuts (=-=Antonsson and Otto, 1995). For example, -=-for a particular design variable, Figure 5. The product of powers, with various weights the designer may specify which values correspond to �� = 0, �� = 0:25, �� = 0:5, �� = 0:75, and ... |