DESIGN EVALUATION WITH MECHATRONICS INDEX USING THE DISCRETE CHOQUET

INTEGRAL

Moulianitis V. C. and Aspragathos N. A.

University of Patras

Mechanical Engineering and Aeronautics Department

26500, Patras, Greece

Tel: +302610997268, +302610997212 (FAX)

e-mail: moulian@mech.upatras.gr, asprag@mech.upatras.gr.

Abstract: In this paper the mechatronics index for design evaluation is modelled using the

discrete Choquet Integral where the interaction among the criteria is taken into account.

In use of fuzzy measures the complexity is shown and it is compensated by the richness

of them when interactive criteria are modelled. The weighted arithmetic mean and the

discrete Choquet Integral are compared for the calculation of the evaluation score in the

mechatronics design of grippers for handling fabrics in order to show its effectiveness in

the evaluation. Copyright © 2006 IFAC

Keywords: Mechatronics index, Discrete Choquet Integral, Fuzzy Measures, Interactive

Criteria, Grippers.

1. INTRODUCTION

According to Ullman (1992) the design procedure is

divided in six phases. It starts with the specification

development/planning phase where the designers

must understand the design problem and plan the

design process. In the second phase, which is called

the conceptual design, the main activities are the

concept generation and evaluation. In the product

design phase, the product is designed in detail and re-

evaluated. In the next phases of the design process,

the product is formed and the main design tasks are

devoted to its production planning and its support

until its retirement. Every evaluation of the product

may loop back the design phase causing delays or

even cancellation of the design process.

Ullman (1992) has analysed four methods that can be

used to evaluate different concepts. The evaluation

results depend on the experience of the designer

and/or the knowledge accumulated within the design

team. Decision-making is easier if quantitative

evaluation take place. A single score would rank

easier the alternative solutions and novice designer

would be more confident to obtain the right

solutions. In this case, the use of design indices

benefits the evaluation process (Moulianitis et al.,

2004). In addition, design indices contribute towards

systematic and automated mechatronic design

process concerning the mechatronics products and

systems as well as in the development of more

knowledge-based systems in the field of

mechatronics design.

Examples of indices presented in the literature, refer

to the producibility of a product, its

manufacturability and the easiness of its

maintenance. Wani and Gandhi (1999) developed a

maintainability index for mechanical systems.

Maintainability is an important aspect for the life

cycle of design and plays significant role during the

service period of the product. It facilitates the

performance of various maintenance activities, such

as inspection, repair, replacement and diagnosis that

must be performed as faster as possible and with

optimal resources. Yanoulakis et al. (1994) defined a

quantitative measure of manufacturability for

rotational parts. This index is based on the

geometrical characteristics of the part and the

characteristics of the cutting process. Byun (1996)

presented a producibility index vector, which

provides a quantified measure that takes into account

both quality and cost requirements for production. He

defined as producibility a measure of the desirability

of a product and its process design in terms of quality

and manufacturing cost. Shewchuk and Moodie

(1998) presented a frame and a classification scheme

for the definition and classification of the various

terms regarding flexibility of manufacturing systems.

This framework serves as a guide for developing new

flexibility terms, whereas the classification scheme

provides a mechanism for summarizing the important

aspects of a given term and any assumptions made

about it.

Moulianitis et. al (2001, 2002, 2004) introduced a

mechatronic index that includes three criteria

namely, intelligence, flexibility and complexity

which characterize most of the mechatronic products.

The attributes of every criterion is analysed and

formulated. The intelligence level of a system is

determined by its control functions and the structure

for information processing of mechatronic systems is

used to model intelligence. A technique to measure

the flexibility of manufacturing systems was used for

the estimation of the flexibility of a mechatronic

product. The various types of flexibility were

classified in three main categories, namely: product

flexibility, operation flexibility and capacity

flexibility. The complexity was modelled using

seven elements.

In addition, a model for concept evaluation using

design indices is presented by Moulianitis et. al

(2004). This model is based on the use of T-norms

and averaging operators and its main objective is to

automate the development of design indices for the

evaluation in conceptual design phase.

Aggregation operators that were used in the

development of indices for mulicriteria decision

making for non-interacting criteria have been

presented by Dubois and Prade (1985), Fodor and

Rubens (1994).

Designers may choose to use independent criteria in

order to simplify the decision-making problem (see

Fig. 1). The interaction among the criteria increases

the complexity of the problem but if it is not modeled

then some information about the problem is not used.

In Fig. 2 the interaction among criteria is shown as a

weighing-machine with four scales representing the

four criteria. If a criterion value is changed then it

affects the other criteria. Mechatronics products are

intelligent, flexible but complex. A product that

presents high degree of intelligence and flexibility

presents high complexity too. This interaction among

those elements is missing in the previous

developments of the mechatronics index.

Interaction between criteria can be represented using

fuzzy integrals. According to Grabisch, (1996),

interaction among criteria is presented in terms of

synergy, redundancy or symmetry. However, the

richness of fuzzy integrals is paid by the complexity

of the model, since the number of coefficients,

named as fuzzy measures, involved in the fuzzy

integrals are increasing exponentially.

Criterion 1

Criterion 4 Criterion 3

Criterion 2

Score calculation

Criteria Space

Fig. 1 Score calculation with independent criteria.

Criterion 1

Criterion 4 Criterion 3

Criterion 2

Criteria Space

Score calculation

Fig. 2 Score calculation with interacting criteria.

In this paper, the discrete Choquet integral is used to

aggregate interacting criteria in order to compute the

evaluation score of a mechatronics index of design

alternatives. The second section presents the

mathematical background of fuzzy measures and

integrals. In the third section, the Mechatronics Index

is modeled using fuzzy measures in order to take into

account the interactions between its elements. The

evaluation of design alternatives in the mechatronics

design of grippers for handling non-rigid materials is

used as an illustrative example. Finally, concluding

remarks ends this paper.

2. MATHEMATICAL BACKGROUND

Assuming that mSSSS ,...,, 21

is a set of

alternatives solutions, among which the designer

must choose. In addition, consider a set of n criteria

nCCCC ,...,, 21 and an association among

every member of S to C (Marichal, 2000):

ni

n

iii xxxx ,...,, 21 , mi ,...,1

A fuzzy measure (or Choquet capacity) on C is a

monotonic set function:

1,02: Cu with 0u and 1Cu .

Fuzzy measures represent the importance of the set

of criteria C. It is not only the weight factors defined

in every criterion separately but, in addition, weight

factors are defined for all the combinations among

the criteria of the set C.

A suitable aggregation operator, which generalizes

the weighted arithmetic mean, is the discrete

Choquet integral (Grabisch, 1996):

n

j

jj

i

j

i

u CAuCAuxxC1

1: (1)

Where, CFu , CF denotes the set of all fuzzy

measures on C, indicates a permutation of C such

that nj xx ... , njj CCCA ,..., ,

1nCA .

2.1 Specification of fuzzy measures – Modelling

interaction.

In this section, the difficult problem of specifying the

fuzzy measures is addressed.

It is difficult to show the importance of each criterion

using fuzzy measures iCu for ni ,...,1 alone.

The global importance of each criterion is easier to

be determined using a normalised value. The

importance index or Shapley value of criterion iC

with respect to u is defined by:

iCCT

ii TuCTun

TTnCu

\ !

!!1:, (2)

where, T indicates the cardinal of T, and T is

formed by all combinations on iCC \ . These

values are normalised, having the property that

1, jCu and it is easier for the designer to

indicate the importance of each criterion.

The interaction between two criteria is sufficiently

found by the sign of

jiji CuCuCCu , . If the sign is

negative (positive), the criteria iC , jC presents a

positive (negative) correlation which expresses a

negative (positive) interaction. If iC , jC are

negatively correlated then if one of them has a good

value then the other usually presents a bad value and

vice versa. If iC , jC are positively correlated then

if one of them has a good value then the other

usually presents a good value. If

jiji CuCuCCu , the criteria are

independent. A proper modelling of interaction

should take into consideration all the combinations

on C. The average interaction index between two

criteria iC and jC with respect to the fuzzy

measure u is defined by:

ji CCCT

jijiji TuCTuCTuCCTun

TTnCCuI

,\

,!1

!!2:,,

(3)

Two criteria iC and jC are symmetric if they can

be exchanged without changing the aggregation

mode. Then ji CTuCTu ,

ji CCCT ,/ .

A criterion iC presents a veto (pass) effect if a bad

(good) score of it leads to a global bad (good) score

whatever the degree of satisfaction of the other

criteria. Then the score is computed by:

iui CCCCS (veto effect) (4)

iui CCCCS (pass effect) (5)

3. MODELLING THE MECHATRONICS INDEX

Buur (1990) presented the main characteristics that

define the set of products and systems found in the

literature as mechatronic. He concluded that the

mechatronic products are intelligent, flexible and

complex. An approach for the estimation of these

criteria is presented by Moulianitis et al. (2001,

2004) and Moulianitis and Aspragathos (2002). In

the following, the Mechatronics index is briefly

presented.

The mechatronics index is composed by three

criteria:

CMFMIMC ,, (6)

where,

:IM the intelligence measurement

:FM the flexibility measurement

:CM the complexity measurement

Fuzzy measures for every criterion and pairs of

criteria must be addressed. In this case six (6) fuzzy

measures must be addressed. The designer must

address three fuzzy measures for the importance of

each criterion and three for the interaction between

them. The fuzzy measures for the interaction between

criteria should be addressed in order to satisfy

specific criteria. In this case, the alternatives that

present high intelligence, high flexibility and low

complexity are favoured:

IMFMuFMuIMu

FMCMuCMuFMu

IMCMuCMuIMu

,

,

,

(7)

Table 1 Sets for fuzzy measures.

j 1

jx jCA

1jCA

1 11

1 IMxx CCMFMIMCCC },,{,, 321 },{, 32 CMFMCC

2 11

2 FMxx },{, 32 CMFMCC }{3 CMC

3 11

3 CMxx }{3 CMC

According to (7), there is synergy between

intelligence and flexibility and redundancy between

intelligence and complexity, flexibility and

complexity. There is no symmetry among the

criteria.

The evaluation score is determined by the discrete

Choquet integral presented in section 2:

3

1

1:j

jj

i

j

i CAuCAuxxScore (8)

For example, assuming that the values for the

alternative solution 1S associated with a

mechatronics profile1x such as

1111

3

1

2

1

1

1 ,,,, CMFMIM xxxxxxx where

111

CMFMIM xxx . The sets for the calculation of

the score according to formula (8) are shown in

Table 1.

4. EVALUATION IN THE CONCEPTUAL

DESIGN

In the following paragraph, evaluation of design

alternatives in the conceptual design phase will be

presented. The mechatronics index will be formed

using the Choque Integral and will be compared to

the weighted arithmetic mean. The evaluation of

design alternatives in the mechatronics design of

grippers for handling non-rigid materials is used as

an illustrative example.

A gripper is a key mechatronic component of a

robotics workcell. It is the mean of interaction

between the robot and the workpiece. The more

intelligent and flexible the grippers are, the more

flexible and efficient the workcell or the

manufacturing system is (Heilala et al. 1992). In

addition, correct design of a gripper can reduce the

cost of a workcell. When designing such devices the

synergy of different technical disciplines is needed.

The knowledge regarding the mechatronics design of

grippers for handling non-rigid materials has been

presented in other papers by Moulianitis et al.

(1999). In these papers the evaluation of design

alternatives was based on T-norms and averaging

operators. In this example the Mechatronics Index

based on fuzzy measures is compared to one, which

is formed using the weighted arithmetic mean.

(a)

(b)

Fig. 3 (a) Separation of a single ply of fabric from a

stack. (b). An air jet gripper (Zoumponos and

Aspragathos, 2004).

Assuming that the handling task to be accomplished

by the gripper is the separation of a single ply from a

stack (see Fig. 3.a), the material characteristics

constrain the number of feasible solutions to five (see

Table 2). In Fig. 3.b the Air-Jet operating principle is

shown.

Table 2. Alternative solutions

Alternative Solution Operating Principle

1 Vacuum

2 Air-Jet

3 Pin

4 Freezing

5 Clamp

Table 3 Values of the criteria of the mechatronics

index.

Crit. 1 2 3 4 5

IM 0.01 0.5 0.01 0.01 0.5

FM 0.173 0.3367 0.8333 0.01 0.6667

CM 0.7471 0.6429 0.8114 0.83 0.4

Assuming that the criteria to be used in order to

select the best alternative are the intelligence, the

flexibility and the complexity of every alternative.

The values of the criteria of the mechatronics index

for every concept are shown in Table 3. The first row

of Table 3 indicates the values of Intelligent

measurement of each concept, and the second and

third the measurement of Flexibility and Complexity

respectively.

The weighted arithmetic mean is defined as:

i

CMC

i

FMF

i

IMI

i xwxwxwxScore (9)

Where, i

CM

i

FM

i

IM

i xxxx ,, is the associated

profile of the alternative solution iS for the set of

criteria and 1 CFI www .

The fuzzy measures for intelligence, flexibility and

complexity and their permutations are shown in

Table 4. Intelligence and flexibility in this example

are considered to be more important than the

complexity. The values of fuzzy measures for the

pairs of criteria are specified according formula (7).

Intelligence (or flexibility) and complexity are

negatively correlated because usually high values in

intelligence (or flexibility) are simultaneous with low

values in complexity:

IMCMuCMuIMu ,9.03.045.0 .

In contrary, flexibility and intelligence are in a sense

redundant. Usually, high values in flexibility are

simultaneous with high values in intelligence:

FMIMuFMuIMu ,5.045.045.0 .

The weights for the formula (9) are the normalised

fuzzy measures taken from Table 4, e.g.

CMuFMuIMuIMuwI . The

scores for each method are shown in Table 5.

Table 4 Fuzzy measures of criteria and their

permutations.

Set of Criteria Fuzzy Measure

0

IM 0.45

FM 0.45

CM 0.3

FMIM , 0.5

CMIM , 0.9

CMFM , 0.9

CMFMIM ,, 1

Choquet integral (CI) is more sensitive in the change

of values. It favours the alternatives that has high

complexity and diminishes the alternatives that have

low one. This is obvious in alternative solution 5

where the weighted arithmetic mean (WAM) is

higher than the CI because the complexity is low

while the flexibility and the intelligence are high. In

contrary, alternative solution 2 while it has almost

the same score given by CI than alternative solution

5 it has lower score using WAM than the alternative

solution 5. In addition solution 3 is found to be better

than solution 5 using CI while it is the opposite using

WAM. In this case the two scores are close using

WAM, while CI presents a higher difference between

these scores.

Table 5 Scores for weighted arithmetic mean and

Choquet Integral for each alternative.

Alternative

solution

Choquet

Integral

Weighted

Arithmetic

Mean

1 0.329 0.255

2 0.526 0.474

3 0.741 0.519

4 0.256 0.215

5 0.525 0.537

In Fig. 4-Fig. 6 the sensitivity in change of values of

CI against WAM is shown. Eleven profiles of criteria

values in each figure are used to compare the two

methods. In Fig. 4, intelligence and flexibility are

kept constant while complexity varies among 0 and 1

with constant step. The slope of CI is higher than

WAN’ s one, indicating that small changes in

complexity produce more readable results. In Fig. 5

and Fig. 6 the complexity kept constant with low and

high values respectively. When, complexity is low

(see Fig. 5) CI diminishes the score, while when

complexity is high (see Fig. 6) CI favours the score.

In Fig. 5 and Fig. 6 WAM seams to be more sensitive

in the changes but it fails to favour the score.

Using CI, designers can favour those alternatives that

present specific profiles for the criteria. In the case of

mechatronics index design alternatives that presents

high intelligence and flexibility with simultaneous

low complexity should be favoured. Complexity of

CI is the trade off for this method. For more than

three criteria fuzzy measures specification becomes

very complex.

5. CONCLUSIONS

In this paper, the Mechatronics Index is modelled

using Choquet Integral. The Mechatronics Index in

this form takes into account the interactions among

its criteria and its score has a compensatory effect.

Mechatronics index using fuzzy measures is more

sensitive than the weighted arithmetic mean as it is

shown in the illustrative example.

In contrary the model based on the Choquet Integral

is more complex than the weighted arithmetic mean

and needs three additional measures to be addressed.

In the case of the mechatronics index, is not difficult

to specify those values, but when the cardinality of

the criteria set is higher the complexity of the index

increases exponentially.

ACKNOWLEDGEMENTS

This research is a part of work for the project funded

by the Hellenic Ministry of National Education and

Religious Affairs in the “Operational Programme for

Education and Initial Vocational Training” of the

action “Pythagoras II”.

University of Patras is partner of the EU-funded FP6

Innovative Production Machines and Systems

(I*PROMS) Network of Excellence.

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Fig. 4 Choquet integral against weighted arithmetic

mean value with constant intelligence, constant

flexibility and variable complexity.

Fig. 5 Choquet integral against weighted arithmetic

mean value with constant intelligence, constant

low valued complexity and variable flexibility.

Fig. 6 Choquet integral against weighted arithmetic

mean value with constant intelligence, constant

high valued complexity and variable flexibility.