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Evaluation of design alternatives in mechanical engineering
Fuzzy Sets and Systems 47 (1992) 269-280 North-Holland
269
Evaluation of design.alternatives in mechanical engineering Ryszard Knosala Institute of Mechanics and Foundations of Machine Design, Silesian Technical University, Gliwice, 44-100 Poland
Witold Pedrycz Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 Received July 1987 Revised January 1989
Abstract: The paper gives a description of a structure dedicated to evaluation of design alternatives in mechanical engineering. A list of particular features of the system such as its ability to cope with fuzziness and a uniform way of treating various criteria involved in design processes is given and discussed in length. Special attention is focused on empirical engineering-oriented aspects of membership functions estimation with a particular emphasis on expressing achieved precision. The paper re-analyzes some existing methods of multiobjective decision-making. Some original contributions of the study to this area are focussed on developing new indices significantly facilitating interpretation of results of ranking (such as e.g. specificity of different alternatives as well as strength of dominance observed among them). An illustration of the performance of the system is completed with the aid of an evaluation of different types of joints found in hydraulic cylinders.
Keywords: Decision-making; designing; Saaty's priority method; mechanical engineering.
1. Introductory remarks. Problem formulation When dealing with any designing activity in mechanical engineering, one is usually faced with a complex single or multi-stage decisionmaking process. At different stages of this process a series of alternatives is evaluated with respect to a set of criteria. The essence of designing imposes that these criteria are usually contradictive and our ultimate goal is to select the alternative for which all are fulfilled to the highest possible degree. Furthermore a collection of criteria usually consists of elements
described in an ambiguous and imprecise way by making use of commonly accepted linguistic terms. Unfortunately there is still no common practice to have a well-developed computer system which enables any designer (here mechanical engineer) to guide the entire complex process of choice and ranking within the set of alternatives and handle the fuzzy character of individual criteria as well as partial evaluations considered in this process. This stands in sharp contrast in with situations observed in the field of parametric optimization of mechanical design nowadays forming a well-developed area of number crunching procedures, see for instance [7, 8, 9]. A major objective of the paper is to re-analyze some existing models of multiobjective decisionmaking (as those originally introduced by Bellman and Zadeh [1]) and incorporate some particular features which are of significant interest for mechanical engineers working in the field of designing mechanical constructions or evaluation of existing ones. Two characteristic properties developed within this approach are worth noticing: - user-friendliness of the proposed structure of the model that enables the user to work with fuzzy quantities, thus allowing him/her to use linguistic terms in describing his/her preferences. - an ability to cope with different types of criteria involving various sources of uncertainty (namely of fuzzy, probabilistic and deterministic nature). The paper is partitioned into two general parts. In the first we will expose basic ideas behind models of decision-making in a fuzzy environment. Obviously, our intent is not to introduce new methods but rather to refer to existing ones and enrich them by several significant application-driven extensions. Bearing this in mind, special attention will be paid to the
0165o0114/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved
270
R. Knosala, W. Pedrycz / Evaluation of design alternatives 1
2
3
l,
•
5
6
7
8
9
10
11
12
13
lt,
15
16
17
18
19
Fig. 1. Complete set of design alternativesstudied in this paper.
central and still open question of membership function estimation. As far as applications and final computer implementations are concerned this essential issue has not been properly addressed in existing solutions. The remaining part of the paper deals with the detailed solution of a problem that arose in design practice. The specific problem at hand concerns the choice of a suitable hydraulic cylinder that is to be used in a mining industry. To make our presentation more readable and more focussed, let us introduce the problem which is going to be studied. We deal with evaluation of 19 alternatives representing different types of joint of a piston and rod of the piston of the hydraulic cylinder. All of them are found in practice. An overall set of alternatives is summarized concisely in Figure 1. From a decision-making point of view, the list of objectives by means of which these types of joints will be evaluated can be split into two groups. By doing that one can make a clear distinction between requirements and criteria. The first group of objectives (called here requirements) consists of the following features
(properties): dismounting property of the joint, - flexibility in changes of velocity at boundary positions, - m i n i m a l length of the passive part of the piston, joint of piston with rod, of taking into account the difference between the diameter of the cylinder and the diameter of the rod smaller than twice the thickness of the seal. The set of criteria taken into account contains: - reliability of performance of the structure, easiness of manufacturing, the number of elements in the joint, - coaxiality of the cylinder and the rod. At a glance one can recognize that some of these features are of Boolean character. Nevertheless a significant number of them involves fuzzy sets as a means to represent uncertainty present there. Before proceeding with a detailed structure of the decision problem we will concentrate on estimation of all fuzzy sets. -
-
- p o s s i b i l i t y
-
R. Knosala, W. Pedrycz / Evaluation of design alternatives
2. Identification and determination of fuzzy sets in the decision problem As we have already noticed an overall evaluation of design alternatives contains two components: their partial evaluation and the importance of the criteria taken into account• First we will study the way of determination of fuzzy sets B~j representing partial (i.e. completed with respect to a single criterion) evaluation of the alternatives in the problem, i - - 1 , 2 . . . . . n, j = 1, 2 . . . . . m. Second, we deliberate on specification of fuzzy sets representing the importance of various criteria. 2. 1. Determination o f f u z z y sets" Bij
In a very concise setting, each alternative (i.e. construction) is evaluated in terms of a set of criteria. We can schematically summarize results coming from the evaluation of the i-th alternative as in Table 1. In the scheme of the decision-making procedure discussed here we consider all partial results, Bil, Bi2 . . . . . Bim and Wl, W2 . . . . . Wm as fuzzy sets defined in the interval [0, 1]. Formally we put them down as Bij, Wj:[O, 1 ] ~ [ 0 , 1]. Subsequently, at the second stage all partial information is aggregated into a final rating. For the i-th alternative it is specified accordingly,
be considered as convenient. Let us recall briefly its essence. Having a set of n objects of any specified character (alternatives) they have to be ordered in the sense of a certain prespecified property. Instead of making this ordering and assigning some values reflecting this order at 'one shot', just considering them altogether (which might not be feasible, especially for larger n), a pairwise comparison of objects is performed. The procedure is realized as follows: Each pair of different objects, say the k-th and the /-th one (assume that k ~ l ) is considered and ranked on a scale consisting of 7 + 2 levels. If the k-th object is completely preferred to the/-th one, the result of evaluation is equal to the highest value in the scale. If prefrence is not so evident then this fact is reflected by intermediate values of the scale. Of course this kind of assignment is a subjective one; however the method gives a certain mechanism forcing the decision-maker to produce consistent judgements across all alternatives. If the /-th object is preferred to the k-th one, the result of comparison is given as a number smaller than 1. Denoting by ukt the result of pairwise comparison (evaluation), we produce a matrix containing findings for all possible comparisons, U = [akl],
Ukk = 1
where U~ stands for a fuzzy set resulting from this aggregation expressed by the function f. Due to a variety of judgements concerning the expression of the suitability of different alternatives it is required to look for some well-suited methods that might be useful for this purpose. Here Saaty's priority method [12] can
1 ugl-----
B,m, W,, We . . . . .
IV,,),
Table 1. Evaluation of alternatives Importance of criterion
No. of criterion
Result of evaluation of i-th alternative
Wj W2
1 2
Bil Bi2
"Wm
m
"B,,o
k,/=1,2 .....
n•
According to [12] two interesting properties of the matrix are preserved,
(1)
U, =f(B~,, B,2 . . . . .
271
(reflexivity), (reciprocity).
Ulk
There exists an additional feature of Saaty's method that makes its use especially convenient. It relies on testing consistency of overall results. Recall that, roughly speaking, in ideal conditions, we may expect satisfaction of the following transitivity property: f f the k-th object is preferred to the l-th one and the l-th object is preferred to the s-th one, then the k-th object is preferred to the s-th one. For numerical quantities ukt, ut~ and Uks expressing pairwise preference, the above statement can be converted into at least approximate equality: uktuts ~ u~.
(2)
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R. Knosala, W. Pedrycz / Evaluation of design alternatives
The closer this equality is satisfied, the stronger consistency of the evaluations is observed. In the original presentation given by Saaty [11], the result of an overall comparison is given by a vector v = [vl, v2 . . . . . vn] where each coordinate (entry) visualizes the preference of a corresponding object. Furthermore the vector v is an eigenvector of U. Additionally for completely consistent data (viz. satisfying (2) for all the possible pairs of comparisons) the greatest eigenvalue of u is equal to the dimension of the matrix U, dim(U) = n. If the obtained value of eigenvalue is greater than n, it indicates a certain level of inconsistency occurring between groups of some comparisons. The higher the deviation from n, the higher the inconsistency. One among existing approaches in solving the above problem is based upon the following approximation task: min
~ ~
vk,vt k = l I=1 k,l=l,2,...,n
(
Ukt--~/
va = 1.
a~= min v~(t),
(5)
l~t<~K
fl = max vk(t),
(6)
l<~t<<_K
1 K
# = -~ ~ vk(t).
(7)
t=l
(3)
with respect to constraints (playing the role of normalization conditions)
~
They lead consequently to a triangular fuzzy number with piecewise linear membership functions characterized by three characteristic points (c~, #, fl) such as lower (ce) and upper (fl) bounds already mentioned and a modal (g) one. In the situation we deal with several, say K, decision-makers, each of whom can be asked to provide his/her own evaluation. Then the corresponding triangular fuzzy number (tr, #, fl) can be constructed on the basis of the vector v coming from them. Denote the values of v obtained from the t-th decision-maker by v(t), t = 1, 2 , . . . , K. For the k-th object we get the following evaluation:
(4)
k=l
In consequence it leads us to a system of linear equations to be solved with respect to v. Summarizing, Saaty's method enables us to check and control consistency of the partial evaluations done by the decision-maker. It can be imposed in the form of a threshold level: as soon as the lack of satisfaction of the equality (2) exceeds this threshold, the decision-maker should be asked to repeat his evaluation process to come up with more consistent outcomes. The value of the threshold depends on the range of the scale utilized in evaluation. For a 7-point scale a difference over 4 is not acceptable and the decision-maker should introduce new scores. It may happen that the decision-maker is hesitant about choosing a proper number from a scale to be assigned to a certain pair of alternatives. An evident relaxation will be to allow him to assign two numbers instead of one. This implies much more flexibility and in this way allows expression of pessimistic (lower bound) and optimistic (upper bound) scores.
All possible shapes of triangle-like membership functions of grades of preference as generated by empirical results are visualized in Figure 2. Only in the case of complete consensus of all the decision-makers (which is extremely rare in practice) we obtain a situation displayed in the fourth graph in Figure 2. Usually the input data is characterized by a significant scattering (viz. the entries of the pairwise comparison matrix) then significant differences between tr and fl are reported, and, of course the obtained fuzzy set becomes 'fuzzy' indeed, spreading over a certain interval of values of [0, 1]. A special caution has to be paid to cases in which the criterion put into account is of a deterministic or of a probabilistic nature. The deterministic one is usually captured by degenerated fuzzy sets, see again Figure 2. For a probabilistic criterion the situation is much more complicated. We may face some statistical data describing this criterion and in order to work with them in the framework of fuzzy sets, it is necessary to convert them into a unique format required by fuzzy set theory. Assume for the moment that the relevant statistical data can be represented by their probability (density) functions. Then following the idea given in [5] or [6] a relevant probability function can be converted into a membership function (or, strictly
R. Knosala, W. Pedrycz / Evaluation of design alternatives
10
10
iO
k
~0
IO
273
10
o
~0
~o
Fig. 2. Triangle-like membership functions.
speaking, a possibility distribution) which after normalization of the universe of discourse up to the unit interval can be uniformly treated as a fuzzy set describing the result of evaluation of the alternative. In the sequel we will study how linguistic weights can be associated with individual criteria. 2. 2. Determination of weights of criteria The criteria which will serve for evaluation of the alternatives are usually described in a linguistic fashion. For instance we can consider a certain criterion as an important one while another can be viewed as e.g. very important or less important. To have a numerical representation of these quantities, we will study two ways of their determination. The first one is almost the same as that used for pairwise
comparison of the alternatives. Now the objects being compared are formed by diffrent criteria. Another approach relies again on pairwise comparison of values situated in [0, 1] scale when they are associated with two criteria of different importance. An example of such two tables containing pairwise comparisons with the aid of which we can derive the membership function of a criterion which is more important or less important is visualized in Table 2. Notice that in general both the tables contain a sort of inverse entries, namely the numbers greater than 1 are situated on opposite sides of the main diagonal of the matrices. Then the obtained membership functions (viz. for models of more important and less important criteria) are given in discrete elements of [0, 1]. To obtain a continuous membership function, a leastsquares method of approximation can be utilized. A suitable function that fits well the
Table 2. Results of pairwise comparison used towards determination of membership functions of linguistic importance of criteria
less imporfonf 0 1 .2 3 ./* 5 6 7 .8 .9 1.0
\,
o"1 ~ X ~ X ~ I ~
o~,/.1) ~'.35 /.5 56 .66
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11
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6
6
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11 1 1
1 2 1 1
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I
1 1
1
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2
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3 X1/3~'2 1 1 1 1 1 2
6~/~s ~I//3~2
7 6 6 5 5 8 6 6 6 5
more irnportonf
0\0 I 2 .3 /* .5 .6 .7 .8 .91.0 1 12 3 /* /* 55/.. 6 6 6
I
1
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Ilo
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274
R. Knosala, W. Pedrycz / Evaluation of design alternatives
10
7
a5
101
30
2C
0.5
lc 1 I
09
0.2
04
0.6
0.8
1'0
0.0
I
0.2
I
I
0.4
I
i
0.6
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:
0,8
.
,
_
10
Fig. 3. A collection of standard membership functions modelling linguistic terms of importance of criteria.
data points met in this problem is a twoparameterized family of the following form: /t(X) = 2 -- e a(1-x)k,
(8)
x • [0, 1], where a and k are adjustable parameters whose role is to minimize the sum of squared errors between observed and computed values of grades of membership. In comparison to the first method the second ones involves an additional assumption about the case when we are dealing with several decision-makers. All of them have to reach a preliminary consensus about the ordering of importance of the criteria. In other words all have to decide which criterion should be viewed as important, less important, very important, more important, etc. Afterwards it is possible to estimate their values of the membership functions. Instead of determining the membership functions of different linguistic values of importance, the decision-maker can simplify his/her strategy by choosing among a set of standard (to a certain extent) membership functions modelling grades of importance of criteria; see Figure 3. There exists a family of curves modelling the following linguistic notions and gradually shifting away from very important to quite important. The list of terms consists of frequently found items: 1. the most important 2. very important
3. more important 4. rather less important 5. less important 6. more or less important 7. important Also the two models (curves) for each of the above notions have been introduced. Therefore the user may have higher flexibility to find a suitable model of the notion considered. At this point we have arrived to a stage in which we have explained all the elements of the estimation of membership functions of all the basic elements of the decision-maker procedure. Now we will pay attention to the problem of aggregation of these partial evaluations referring to the variety of grades of importance of the criteria involved, and, subsequently, present a way of interpretation of the results of an overall evaluation.
3. Aggregation of partial evaluations and interpretation of ranking of derived results. Interaction and specificity Having at our disposal all the partial results, namely fuzzy sets expressing evaluation of the alternatives in light of the specified criteria as well as the weights of criteria themselves we are faced with a problem of overall evaluation. Recall that B i l , Bi2 . . . . , B i n and I411, W2. . . . . Wm denote the respective fuzzy sets derived in the previous section (index i denotes the i-th
R. Knosala, W. Pedrycz / Evaluation of design alternatives
alternative taken into account). An overall evaluation comprises all of them yielding a fuzzy set Z;:[0, 1]---~[0, 1] which in general is equal to Z; = F ( B , , , B n , . . . , B,m, W,, W2 . . . . .
Win)
(9)
with F standing for the aggregation function. The role of Z; is to capture the global preference of the i-th alternative. Since at least one partial result (either B o or IV/) is a fuzzy set, Z; becomes a fuzzy set as well. Due to the extension principle [13] the above expression reads as supremum followed by a sequence of min compositions: Z,(z) = sup[min(B;l(b,), Bi2(b2)
. . . . .
Bern(bin),
(10)
W,(w,) .....
with supremum taken subject to all constraints formed by F,
of fuzzy sets. This in turn requires an extra step not needed in deterministic pointwise rating results, i.e. ranking fuzzy sets with respect to the global preference of each alternative. In a general setting this process can be viewed as a set-to-point mapping g:Zi--->E(or ~+),
cf. [3, 11]. R ( ~ + ) stands for the line of real (real positive) numbers. The choice of a particular form of mapping is rather complicated and not evident. Moreover the results produced by different g's are incomparable; on this topic refer to survey paper [4]. Here we will adopt an algorithm of weighted mean (centre of gravity), cf. [3], in which a fuzzy set Z; is replaced by a single numerical quantity T; being its global (optimal) pointwise representative equal to 1
z = F ( b l , b2 . . . . .
b,,,, wl, w2 . . . . .
b/wj
wi
l
Win).
From a formal point of view (9) can be considered as an optimization problem with nonlinear constraints imposed by the function F. Of course, the proper choice of the aggregation function F may form a genuine applicational problem. In general it is difficult to decide in advance which form may fit a particular problem at hand. Neglecting the specificity of the decision problem from now on we will concentrate on a weighted scheme of combination of partial evaluations of alternatives with weights induced by the importance of individual criteria. Thus we have z =
275
(11)
j=l
for z, bi, % • [0, 1]. An assumption of linearity of the underlying function with respect to bj enables us to obtain a profound reduction of the amount of computations required to calculate fuzzy sets of ranking Zi. Then the algorithm discussed in [2, 6, 10] can be easily implemented. Its essence is to replace all fuzzy sets by a collection of their successive nested o:-cuts thus convertin~ the original problem of computation of Z/s into a series of tasks involving sets. As we have clearly underlined the results of the overall evaluation are represented in terms
The higher the value of T;, the more preferable the corresponding alternative, i.e. if T; > Ti then the i-th alternative is preferred to the ]-th one. Additionally the entire membership functions illustrate mutual connections between Z/s. This ability is lost to a significant extent when only pointwise representatives T; are chosen. Therefore it becomes necessary to preserve information by introducing measures of interactivity between pairs Z; and Zj. First we study the so-called interactivity index defined as flo =
fo' (Zi D Zj) dz f ) min(Zi(z), Z j ( z ) ) Oz.
(13)
It yields values close to 1 if Z, and Z; overlap significantly and tends to zero if Z, and Zj become almost disjoint. The possibility measure [14] 7;/= z ( Z i , Z i) = sup[min(Zi(z), Z~(z))] z
can form an alternative index highlighting existing interactivity. In comparison to rio, it aggregates over [0, 1] by taking supremum, thus strongly reflecting the strongest (highest) link among overalp occurring at individual values of
276
R. Knosala, W. Pedrycz / Evaluation of design alternatives
z. Let us also recall the well-known fact that sup-min composition may introduce too 'fuzzified' preference sets Z~, Z2 . . . . . Z,,. This may be especially cumbersome for interpretation including lower values of these membership functions. To avoid these difficulties it is worthwhile to analyze the impact of these fuzzy sets on the final ranking with respect to a certain threshold level. The value of this threshold depends on the precision of the partial evaluations (viz. Bq and Wj). For its determination introduce an index of specificity which for any fuzzy set A :[0, 1]---* [0, 1] provides a number Sp(A) = 1 -
A(x) dx/mes(supp(A))
(14)
where mes(supp(A)) denotes a measure of support of ,4. Notice several properties of
Sp(A): - i f A c_.4' then Sp(A)/> - f o r two singleton and A(x) = {~
and they have the same support, Sp(A'), special limit fuzzy sets, viz. a an interval, say
if x =Xo, otherwise,
and A(x)=
1 ifxE[x1,x2], 0 otherwise,
respectively, Sp(A) attains its extremal values; for the singleton it is equal to 1, while in the second case we obtain O. For triangular membership functions (tr, #, fl) we get Sp(A)= 1. For different shapes of fuzzy sets Bq and Wj standing in the partial evaluation, an overall specificity Sp is obtained by averaging partial evaluations: Sp = - Sp(Bq) + Sp(W~) . nm + m Li=lj= 1 j=l
(15)
Then the grades of preferability T~ as well as the values of interactivity flq and yq are determined by considering only the values of membership functions which exceed this threshold level Sp. 4. Analysis of results of ranking
Now the problem formulated in Section 1 will be solved in detail. Three experts were involved
in ranking alternatives namely (a) principal expert, (b) constructor, (c) manufacturer. All four criteria were modelled as fuzzy sets. Even if some of them could have been potentially determined as deterministic or probabilistic quantities (such as e.g., coaxiality of the cylinder and the rod) their exact specification was not feasible. Furthermore determination of the number of elements in the joint in some concrete cases can be a difficult task not free from ambiguity. A nine level qualitative scale used for completing pairwise comparisons corresponds to the following linguistic evaluations (they refer both to the levels of preference as well as mutual importance), 1. equal importance (preference 2. intermediate (between 1 and 3) 3. weak importance (preference) of one over another 4. intermediate (between 3 and 5) 5. essential or strong importance (preference) 6. intermediate (between 5 and 7) 7. demonstrated importance (preferences) 8. intermediate (between 7 and 9) 9. absolute importance (preference) Figure 4 summarizes a part of the matrix containing pairwise evaluations. The same figure illustrates membership functions describing the preference of the six first alternatives evaluated with respect to reliability of performance of the structure. It becomes obvious that the sixth alternative viewed in the light of this criterion is the best one while the first one was definitely conceived as the weakest one. At the next state the experts determined the importance of the criteria involved in the problem. All criteria are expressed in a linguistic form accordingly, -reliability of performance of the structure: the most important - easiness of manufacturing: important - t h e number of elements in the joint: less important - coaxiality of the cylinder and the rod: rather less important Aggregation of partial evaluation with respect to the above set of criteria, see Eq. (10)-(11) yields fuzzy sets of preference as shown in Figure 5.
R. Knosala, W. Pedrycz / Evaluation of design alternatives
Expert no.1
Expert no.2
AI A2 A3 A4A5 A6..
Jl I ~ ¼ ½ X ~
277
Expert no. 3
AI A2 A3 N.,.A5 A6.
AI A2 A3 A4 A5 A6..
AII X~¼½ A23
1 I
2 51
A3 3 I I 2 5 1 A3 912[11ZI411}}A3 8 2 1 L, ~ ZX A4 7 / l / x / l J z / ¼ l l A. s ¼ X 1 1 Y~ IA,'I ~ ½ 1 AS IS I~IXI£111XII A5 S X X 1 1 X ~IXX~ 1 X A6L~~[~_ ~ A6 9 : 3 2 5 S I ~i A 6 3 I I 3 5 1
Alternative 1 10
Atternofive 2 ,ol
I0
1286 0653
10
0
Z632
3859
li0 = 6.500 7908
34~,8
Alternative L,.
Alternative S
Atfernofive 6
i
1o / i ~ 3 4 4 0
I
I0 1320 1646
Alternative 3
~ 1714 080~, 1646
Z652
8552 10
Fig. 4. The matrix of pairwise comparisons and derived fuzzysets of preference.
In addition to this purely graphical representation, which in our case might be sometimes difficult to interpret (especially for several selected alternatives), one can support it by indices flu and Yu" Both describe the strength of links (relationships) between alternatives. They are again shown in Figure 5 in the form of flu/~'u" Noticeable is the fact that relationships between 'vertical' alternatives are much more weaker than 'horizontal' ones, which values are close to 1, Some slight changes are visible when the index of precision Sp is included, see again Figure 5. For comparative purposes Figure 6 contains results of ranking performed by another independent group of constructors. Their results of ranking are very similar in comparison to our generated series, in particular
for the best and the weakest alternatives. Furthermore the algorithmic approach developed here induces a powerful interpretation framework. It enables us to reveal the level of preference of some selected alternatives. This holds, for instance, for the following list: -alternative 10: fulfills the requirement of dismounting property of the joint, -alternative 3: satisfies the requirement of minimal length of the passive part of the piston, - alternative 13: provides flexibility in changes of velocity at boundary positions.
5.
Conclusions
In this study methodology of
we have investigated the solving of multi-objective
278
R. Knosala, W. Pedrycz I Evaluation of design alternatives ,o~-
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Fig. 5. Ordering of design alternatives.
decision-making problems in mechanical engineering. Fuzzy sets when used to formulate a problem are able to cope with ambiguity directly resulting from a series of criteria taken into account as well as their uncertain character. In addition to that, fuzzy sets representing final outcomes (viz. ranking results) contribute a lot towards understanding the nature of all alternatives and reveal the form of relationships
between them as far as the dominance of some of them is concerned. Two new indices associated with pointwise global representation allow us to clarify a chain-like structure with respect to the dominance relationships. In a detailed analysis of a discussed example the produced results and their format were found convincing for the users. Definitely, the list of feasible methods for
R. Knosala, W. Pedrycz / Evaluation of design alternatives
279
~ooo
\, / /
\ \,
0__~6
013
012
0 75
0 7t*
0 7~
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o~, o.~/ o . ~ / \,oo ~ ,oo,':-~
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Fig. 6. Results of r a n k i n g by a n o t h e r g r o u p of c o n s t r u c t o r s .
ranking and the number of aggregating functions becomes an extensive one. Therefore it is evident that for any problem at hand some of them should be applied and the derived results be studied with respect to the generated ranking structures. The methods devised in this paper are directly designed to explore all essential components of dominance structures occurring between discussed alternatives. By no means the pursued research is complete; still a lot of room is left for
generalized versions of ranking methods. This in turn may call for methods of cross-ranking analysis.
Acknowledgement Support from the Natural Sciences and Engineering Research Council of Canada is greatly appreciated.
280
R. Knosala, W. Pedrycz / Evaluation of design alternatives
References [1] R.E. Bellman and L.A. Zadeh, Decision-making in a fuzzy environment, Management Sci. 17 (1970) 141164. [2] S.M. Baas and H. Kwakernaak, Rating and ranking of multi-aspect alternatives using fuzzy sets, Automatica 13 (1977) 47-58. [3] J.F. Baldwin and N.C.F. Guild, Comparison of fuzzy sets on the same decision space, Fuzzy Sets and Systems 2 (1979) 213-233. [4] G. Bortolan and R. Degani, A review of some methods for ranking fuzzy subsets, Fuzzy Sets and Systems 15 (1985) 1-19. [5] M.R. Civanlar and H.J. Trussel, Constructing membership functions using statistical data, Fuzzy Sets and Systems 18 (1986) 1-13. [6] D. Dubois and H. Prade, Unfair coins and necessity measures: toward a possibilistic interpretation of histograms, Fuzzy Sets and Systems 10 (1983) 15-20. [7] G.D. Erkrath, Die sicherheitstechnische Bewertung yon Mensch-Maschine-Systemen, VDI-Z 113, Reihe 1
(VDI-Verlag, Diisseldorf, 1984). [8] H. Goldbecker, Die betriebswirtschaftliche bewertung yon CAD-Systemen in Rahmen des lnvestitionsentscheidungsprozesses, VID-Z 38, Reihe 2 (VDIVerlag, Diisseldorf, 1979). [9] R. Hechler, Bewertung von Alternativen fiir Beleuchtungsanlagne mit Hiife der Nutzwertanalyse, Diss. TU Braunschweig (1976). [10] H. Kwakernaak, An algorithm for rating multipleaspect alternatives using fuzzy sets, Automatica 15 (1979) 615-616. [11] W. Pedrycz, Ranking multiple aspect alternativesfuzzy relational equations approach, Automatica 22 (1986) 251-253. [12] T.L. Saaty, Exploring the interfaces between hierarchies, multiple objectives and fuzzy sets, Fuzzy Sets and Systems 1 (1978) 57-68. [13] L.A. Zadeh, The Concept of a Linguistic Variable and its Application to Approximate Reasoning (Elsevier, New York, 1973). [14] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978) 3-28.
269
Evaluation of design.alternatives in mechanical engineering Ryszard Knosala Institute of Mechanics and Foundations of Machine Design, Silesian Technical University, Gliwice, 44-100 Poland
Witold Pedrycz Department of Electrical and Computer Engineering, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2 Received July 1987 Revised January 1989
Abstract: The paper gives a description of a structure dedicated to evaluation of design alternatives in mechanical engineering. A list of particular features of the system such as its ability to cope with fuzziness and a uniform way of treating various criteria involved in design processes is given and discussed in length. Special attention is focused on empirical engineering-oriented aspects of membership functions estimation with a particular emphasis on expressing achieved precision. The paper re-analyzes some existing methods of multiobjective decision-making. Some original contributions of the study to this area are focussed on developing new indices significantly facilitating interpretation of results of ranking (such as e.g. specificity of different alternatives as well as strength of dominance observed among them). An illustration of the performance of the system is completed with the aid of an evaluation of different types of joints found in hydraulic cylinders.
Keywords: Decision-making; designing; Saaty's priority method; mechanical engineering.
1. Introductory remarks. Problem formulation When dealing with any designing activity in mechanical engineering, one is usually faced with a complex single or multi-stage decisionmaking process. At different stages of this process a series of alternatives is evaluated with respect to a set of criteria. The essence of designing imposes that these criteria are usually contradictive and our ultimate goal is to select the alternative for which all are fulfilled to the highest possible degree. Furthermore a collection of criteria usually consists of elements
described in an ambiguous and imprecise way by making use of commonly accepted linguistic terms. Unfortunately there is still no common practice to have a well-developed computer system which enables any designer (here mechanical engineer) to guide the entire complex process of choice and ranking within the set of alternatives and handle the fuzzy character of individual criteria as well as partial evaluations considered in this process. This stands in sharp contrast in with situations observed in the field of parametric optimization of mechanical design nowadays forming a well-developed area of number crunching procedures, see for instance [7, 8, 9]. A major objective of the paper is to re-analyze some existing models of multiobjective decisionmaking (as those originally introduced by Bellman and Zadeh [1]) and incorporate some particular features which are of significant interest for mechanical engineers working in the field of designing mechanical constructions or evaluation of existing ones. Two characteristic properties developed within this approach are worth noticing: - user-friendliness of the proposed structure of the model that enables the user to work with fuzzy quantities, thus allowing him/her to use linguistic terms in describing his/her preferences. - an ability to cope with different types of criteria involving various sources of uncertainty (namely of fuzzy, probabilistic and deterministic nature). The paper is partitioned into two general parts. In the first we will expose basic ideas behind models of decision-making in a fuzzy environment. Obviously, our intent is not to introduce new methods but rather to refer to existing ones and enrich them by several significant application-driven extensions. Bearing this in mind, special attention will be paid to the
0165o0114/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved
270
R. Knosala, W. Pedrycz / Evaluation of design alternatives 1
2
3
l,
•
5
6
7
8
9
10
11
12
13
lt,
15
16
17
18
19
Fig. 1. Complete set of design alternativesstudied in this paper.
central and still open question of membership function estimation. As far as applications and final computer implementations are concerned this essential issue has not been properly addressed in existing solutions. The remaining part of the paper deals with the detailed solution of a problem that arose in design practice. The specific problem at hand concerns the choice of a suitable hydraulic cylinder that is to be used in a mining industry. To make our presentation more readable and more focussed, let us introduce the problem which is going to be studied. We deal with evaluation of 19 alternatives representing different types of joint of a piston and rod of the piston of the hydraulic cylinder. All of them are found in practice. An overall set of alternatives is summarized concisely in Figure 1. From a decision-making point of view, the list of objectives by means of which these types of joints will be evaluated can be split into two groups. By doing that one can make a clear distinction between requirements and criteria. The first group of objectives (called here requirements) consists of the following features
(properties): dismounting property of the joint, - flexibility in changes of velocity at boundary positions, - m i n i m a l length of the passive part of the piston, joint of piston with rod, of taking into account the difference between the diameter of the cylinder and the diameter of the rod smaller than twice the thickness of the seal. The set of criteria taken into account contains: - reliability of performance of the structure, easiness of manufacturing, the number of elements in the joint, - coaxiality of the cylinder and the rod. At a glance one can recognize that some of these features are of Boolean character. Nevertheless a significant number of them involves fuzzy sets as a means to represent uncertainty present there. Before proceeding with a detailed structure of the decision problem we will concentrate on estimation of all fuzzy sets. -
-
- p o s s i b i l i t y
-
R. Knosala, W. Pedrycz / Evaluation of design alternatives
2. Identification and determination of fuzzy sets in the decision problem As we have already noticed an overall evaluation of design alternatives contains two components: their partial evaluation and the importance of the criteria taken into account• First we will study the way of determination of fuzzy sets B~j representing partial (i.e. completed with respect to a single criterion) evaluation of the alternatives in the problem, i - - 1 , 2 . . . . . n, j = 1, 2 . . . . . m. Second, we deliberate on specification of fuzzy sets representing the importance of various criteria. 2. 1. Determination o f f u z z y sets" Bij
In a very concise setting, each alternative (i.e. construction) is evaluated in terms of a set of criteria. We can schematically summarize results coming from the evaluation of the i-th alternative as in Table 1. In the scheme of the decision-making procedure discussed here we consider all partial results, Bil, Bi2 . . . . . Bim and Wl, W2 . . . . . Wm as fuzzy sets defined in the interval [0, 1]. Formally we put them down as Bij, Wj:[O, 1 ] ~ [ 0 , 1]. Subsequently, at the second stage all partial information is aggregated into a final rating. For the i-th alternative it is specified accordingly,
be considered as convenient. Let us recall briefly its essence. Having a set of n objects of any specified character (alternatives) they have to be ordered in the sense of a certain prespecified property. Instead of making this ordering and assigning some values reflecting this order at 'one shot', just considering them altogether (which might not be feasible, especially for larger n), a pairwise comparison of objects is performed. The procedure is realized as follows: Each pair of different objects, say the k-th and the /-th one (assume that k ~ l ) is considered and ranked on a scale consisting of 7 + 2 levels. If the k-th object is completely preferred to the/-th one, the result of evaluation is equal to the highest value in the scale. If prefrence is not so evident then this fact is reflected by intermediate values of the scale. Of course this kind of assignment is a subjective one; however the method gives a certain mechanism forcing the decision-maker to produce consistent judgements across all alternatives. If the /-th object is preferred to the k-th one, the result of comparison is given as a number smaller than 1. Denoting by ukt the result of pairwise comparison (evaluation), we produce a matrix containing findings for all possible comparisons, U = [akl],
Ukk = 1
where U~ stands for a fuzzy set resulting from this aggregation expressed by the function f. Due to a variety of judgements concerning the expression of the suitability of different alternatives it is required to look for some well-suited methods that might be useful for this purpose. Here Saaty's priority method [12] can
1 ugl-----
B,m, W,, We . . . . .
IV,,),
Table 1. Evaluation of alternatives Importance of criterion
No. of criterion
Result of evaluation of i-th alternative
Wj W2
1 2
Bil Bi2
"Wm
m
"B,,o
k,/=1,2 .....
n•
According to [12] two interesting properties of the matrix are preserved,
(1)
U, =f(B~,, B,2 . . . . .
271
(reflexivity), (reciprocity).
Ulk
There exists an additional feature of Saaty's method that makes its use especially convenient. It relies on testing consistency of overall results. Recall that, roughly speaking, in ideal conditions, we may expect satisfaction of the following transitivity property: f f the k-th object is preferred to the l-th one and the l-th object is preferred to the s-th one, then the k-th object is preferred to the s-th one. For numerical quantities ukt, ut~ and Uks expressing pairwise preference, the above statement can be converted into at least approximate equality: uktuts ~ u~.
(2)
272
R. Knosala, W. Pedrycz / Evaluation of design alternatives
The closer this equality is satisfied, the stronger consistency of the evaluations is observed. In the original presentation given by Saaty [11], the result of an overall comparison is given by a vector v = [vl, v2 . . . . . vn] where each coordinate (entry) visualizes the preference of a corresponding object. Furthermore the vector v is an eigenvector of U. Additionally for completely consistent data (viz. satisfying (2) for all the possible pairs of comparisons) the greatest eigenvalue of u is equal to the dimension of the matrix U, dim(U) = n. If the obtained value of eigenvalue is greater than n, it indicates a certain level of inconsistency occurring between groups of some comparisons. The higher the deviation from n, the higher the inconsistency. One among existing approaches in solving the above problem is based upon the following approximation task: min
~ ~
vk,vt k = l I=1 k,l=l,2,...,n
(
Ukt--~/
va = 1.
a~= min v~(t),
(5)
l~t<~K
fl = max vk(t),
(6)
l<~t<<_K
1 K
# = -~ ~ vk(t).
(7)
t=l
(3)
with respect to constraints (playing the role of normalization conditions)
~
They lead consequently to a triangular fuzzy number with piecewise linear membership functions characterized by three characteristic points (c~, #, fl) such as lower (ce) and upper (fl) bounds already mentioned and a modal (g) one. In the situation we deal with several, say K, decision-makers, each of whom can be asked to provide his/her own evaluation. Then the corresponding triangular fuzzy number (tr, #, fl) can be constructed on the basis of the vector v coming from them. Denote the values of v obtained from the t-th decision-maker by v(t), t = 1, 2 , . . . , K. For the k-th object we get the following evaluation:
(4)
k=l
In consequence it leads us to a system of linear equations to be solved with respect to v. Summarizing, Saaty's method enables us to check and control consistency of the partial evaluations done by the decision-maker. It can be imposed in the form of a threshold level: as soon as the lack of satisfaction of the equality (2) exceeds this threshold, the decision-maker should be asked to repeat his evaluation process to come up with more consistent outcomes. The value of the threshold depends on the range of the scale utilized in evaluation. For a 7-point scale a difference over 4 is not acceptable and the decision-maker should introduce new scores. It may happen that the decision-maker is hesitant about choosing a proper number from a scale to be assigned to a certain pair of alternatives. An evident relaxation will be to allow him to assign two numbers instead of one. This implies much more flexibility and in this way allows expression of pessimistic (lower bound) and optimistic (upper bound) scores.
All possible shapes of triangle-like membership functions of grades of preference as generated by empirical results are visualized in Figure 2. Only in the case of complete consensus of all the decision-makers (which is extremely rare in practice) we obtain a situation displayed in the fourth graph in Figure 2. Usually the input data is characterized by a significant scattering (viz. the entries of the pairwise comparison matrix) then significant differences between tr and fl are reported, and, of course the obtained fuzzy set becomes 'fuzzy' indeed, spreading over a certain interval of values of [0, 1]. A special caution has to be paid to cases in which the criterion put into account is of a deterministic or of a probabilistic nature. The deterministic one is usually captured by degenerated fuzzy sets, see again Figure 2. For a probabilistic criterion the situation is much more complicated. We may face some statistical data describing this criterion and in order to work with them in the framework of fuzzy sets, it is necessary to convert them into a unique format required by fuzzy set theory. Assume for the moment that the relevant statistical data can be represented by their probability (density) functions. Then following the idea given in [5] or [6] a relevant probability function can be converted into a membership function (or, strictly
R. Knosala, W. Pedrycz / Evaluation of design alternatives
10
10
iO
k
~0
IO
273
10
o
~0
~o
Fig. 2. Triangle-like membership functions.
speaking, a possibility distribution) which after normalization of the universe of discourse up to the unit interval can be uniformly treated as a fuzzy set describing the result of evaluation of the alternative. In the sequel we will study how linguistic weights can be associated with individual criteria. 2. 2. Determination of weights of criteria The criteria which will serve for evaluation of the alternatives are usually described in a linguistic fashion. For instance we can consider a certain criterion as an important one while another can be viewed as e.g. very important or less important. To have a numerical representation of these quantities, we will study two ways of their determination. The first one is almost the same as that used for pairwise
comparison of the alternatives. Now the objects being compared are formed by diffrent criteria. Another approach relies again on pairwise comparison of values situated in [0, 1] scale when they are associated with two criteria of different importance. An example of such two tables containing pairwise comparisons with the aid of which we can derive the membership function of a criterion which is more important or less important is visualized in Table 2. Notice that in general both the tables contain a sort of inverse entries, namely the numbers greater than 1 are situated on opposite sides of the main diagonal of the matrices. Then the obtained membership functions (viz. for models of more important and less important criteria) are given in discrete elements of [0, 1]. To obtain a continuous membership function, a leastsquares method of approximation can be utilized. A suitable function that fits well the
Table 2. Results of pairwise comparison used towards determination of membership functions of linguistic importance of criteria
less imporfonf 0 1 .2 3 ./* 5 6 7 .8 .9 1.0
\,
o"1 ~ X ~ X ~ I ~
o~,/.1) ~'.35 /.5 56 .66
& 31 5/*3 5 /*3 6
¼1~XX~X~ 3 1 Z X£~X
5/*
~
11
11
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1 2
1 11
~-.3~ 11 - 4 ~ 1
5~//33!~ 71~X
5 /*
3 /*
5
6
6
z 3 /* 4 s 5
11 1 1
1 2 1 1
3 /* 4 5 z 3 L, 4
I
1 1
1
1
2
3 3
3 X1/3~'2 1 1 1 1 1 2
6~/~s ~I//3~2
7 6 6 5 5 8 6 6 6 5
more irnportonf
0\0 I 2 .3 /* .5 .6 .7 .8 .91.0 1 12 3 /* /* 55/.. 6 6 6
I
1
1
1
1 2
3 1 1 1~
i
1 1 19i6.0' 6 6 6 5 5 / , , 2 1 21 1 6 616 6 5 5 / * 3
Ilo
l, ll 1 o8 1 1o711
9¼~P~XX%~l
s
1 1
I
274
R. Knosala, W. Pedrycz / Evaluation of design alternatives
10
7
a5
101
30
2C
0.5
lc 1 I
09
0.2
04
0.6
0.8
1'0
0.0
I
0.2
I
I
0.4
I
i
0.6
:
:
0,8
.
,
_
10
Fig. 3. A collection of standard membership functions modelling linguistic terms of importance of criteria.
data points met in this problem is a twoparameterized family of the following form: /t(X) = 2 -- e a(1-x)k,
(8)
x • [0, 1], where a and k are adjustable parameters whose role is to minimize the sum of squared errors between observed and computed values of grades of membership. In comparison to the first method the second ones involves an additional assumption about the case when we are dealing with several decision-makers. All of them have to reach a preliminary consensus about the ordering of importance of the criteria. In other words all have to decide which criterion should be viewed as important, less important, very important, more important, etc. Afterwards it is possible to estimate their values of the membership functions. Instead of determining the membership functions of different linguistic values of importance, the decision-maker can simplify his/her strategy by choosing among a set of standard (to a certain extent) membership functions modelling grades of importance of criteria; see Figure 3. There exists a family of curves modelling the following linguistic notions and gradually shifting away from very important to quite important. The list of terms consists of frequently found items: 1. the most important 2. very important
3. more important 4. rather less important 5. less important 6. more or less important 7. important Also the two models (curves) for each of the above notions have been introduced. Therefore the user may have higher flexibility to find a suitable model of the notion considered. At this point we have arrived to a stage in which we have explained all the elements of the estimation of membership functions of all the basic elements of the decision-maker procedure. Now we will pay attention to the problem of aggregation of these partial evaluations referring to the variety of grades of importance of the criteria involved, and, subsequently, present a way of interpretation of the results of an overall evaluation.
3. Aggregation of partial evaluations and interpretation of ranking of derived results. Interaction and specificity Having at our disposal all the partial results, namely fuzzy sets expressing evaluation of the alternatives in light of the specified criteria as well as the weights of criteria themselves we are faced with a problem of overall evaluation. Recall that B i l , Bi2 . . . . , B i n and I411, W2. . . . . Wm denote the respective fuzzy sets derived in the previous section (index i denotes the i-th
R. Knosala, W. Pedrycz / Evaluation of design alternatives
alternative taken into account). An overall evaluation comprises all of them yielding a fuzzy set Z;:[0, 1]---~[0, 1] which in general is equal to Z; = F ( B , , , B n , . . . , B,m, W,, W2 . . . . .
Win)
(9)
with F standing for the aggregation function. The role of Z; is to capture the global preference of the i-th alternative. Since at least one partial result (either B o or IV/) is a fuzzy set, Z; becomes a fuzzy set as well. Due to the extension principle [13] the above expression reads as supremum followed by a sequence of min compositions: Z,(z) = sup[min(B;l(b,), Bi2(b2)
. . . . .
Bern(bin),
(10)
W,(w,) .....
with supremum taken subject to all constraints formed by F,
of fuzzy sets. This in turn requires an extra step not needed in deterministic pointwise rating results, i.e. ranking fuzzy sets with respect to the global preference of each alternative. In a general setting this process can be viewed as a set-to-point mapping g:Zi--->E(or ~+),
cf. [3, 11]. R ( ~ + ) stands for the line of real (real positive) numbers. The choice of a particular form of mapping is rather complicated and not evident. Moreover the results produced by different g's are incomparable; on this topic refer to survey paper [4]. Here we will adopt an algorithm of weighted mean (centre of gravity), cf. [3], in which a fuzzy set Z; is replaced by a single numerical quantity T; being its global (optimal) pointwise representative equal to 1
z = F ( b l , b2 . . . . .
b,,,, wl, w2 . . . . .
b/wj
wi
l
Win).
From a formal point of view (9) can be considered as an optimization problem with nonlinear constraints imposed by the function F. Of course, the proper choice of the aggregation function F may form a genuine applicational problem. In general it is difficult to decide in advance which form may fit a particular problem at hand. Neglecting the specificity of the decision problem from now on we will concentrate on a weighted scheme of combination of partial evaluations of alternatives with weights induced by the importance of individual criteria. Thus we have z =
275
(11)
j=l
for z, bi, % • [0, 1]. An assumption of linearity of the underlying function with respect to bj enables us to obtain a profound reduction of the amount of computations required to calculate fuzzy sets of ranking Zi. Then the algorithm discussed in [2, 6, 10] can be easily implemented. Its essence is to replace all fuzzy sets by a collection of their successive nested o:-cuts thus convertin~ the original problem of computation of Z/s into a series of tasks involving sets. As we have clearly underlined the results of the overall evaluation are represented in terms
The higher the value of T;, the more preferable the corresponding alternative, i.e. if T; > Ti then the i-th alternative is preferred to the ]-th one. Additionally the entire membership functions illustrate mutual connections between Z/s. This ability is lost to a significant extent when only pointwise representatives T; are chosen. Therefore it becomes necessary to preserve information by introducing measures of interactivity between pairs Z; and Zj. First we study the so-called interactivity index defined as flo =
fo' (Zi D Zj) dz f ) min(Zi(z), Z j ( z ) ) Oz.
(13)
It yields values close to 1 if Z, and Z; overlap significantly and tends to zero if Z, and Zj become almost disjoint. The possibility measure [14] 7;/= z ( Z i , Z i) = sup[min(Zi(z), Z~(z))] z
can form an alternative index highlighting existing interactivity. In comparison to rio, it aggregates over [0, 1] by taking supremum, thus strongly reflecting the strongest (highest) link among overalp occurring at individual values of
276
R. Knosala, W. Pedrycz / Evaluation of design alternatives
z. Let us also recall the well-known fact that sup-min composition may introduce too 'fuzzified' preference sets Z~, Z2 . . . . . Z,,. This may be especially cumbersome for interpretation including lower values of these membership functions. To avoid these difficulties it is worthwhile to analyze the impact of these fuzzy sets on the final ranking with respect to a certain threshold level. The value of this threshold depends on the precision of the partial evaluations (viz. Bq and Wj). For its determination introduce an index of specificity which for any fuzzy set A :[0, 1]---* [0, 1] provides a number Sp(A) = 1 -
A(x) dx/mes(supp(A))
(14)
where mes(supp(A)) denotes a measure of support of ,4. Notice several properties of
Sp(A): - i f A c_.4' then Sp(A)/> - f o r two singleton and A(x) = {~
and they have the same support, Sp(A'), special limit fuzzy sets, viz. a an interval, say
if x =Xo, otherwise,
and A(x)=
1 ifxE[x1,x2], 0 otherwise,
respectively, Sp(A) attains its extremal values; for the singleton it is equal to 1, while in the second case we obtain O. For triangular membership functions (tr, #, fl) we get Sp(A)= 1. For different shapes of fuzzy sets Bq and Wj standing in the partial evaluation, an overall specificity Sp is obtained by averaging partial evaluations: Sp = - Sp(Bq) + Sp(W~) . nm + m Li=lj= 1 j=l
(15)
Then the grades of preferability T~ as well as the values of interactivity flq and yq are determined by considering only the values of membership functions which exceed this threshold level Sp. 4. Analysis of results of ranking
Now the problem formulated in Section 1 will be solved in detail. Three experts were involved
in ranking alternatives namely (a) principal expert, (b) constructor, (c) manufacturer. All four criteria were modelled as fuzzy sets. Even if some of them could have been potentially determined as deterministic or probabilistic quantities (such as e.g., coaxiality of the cylinder and the rod) their exact specification was not feasible. Furthermore determination of the number of elements in the joint in some concrete cases can be a difficult task not free from ambiguity. A nine level qualitative scale used for completing pairwise comparisons corresponds to the following linguistic evaluations (they refer both to the levels of preference as well as mutual importance), 1. equal importance (preference 2. intermediate (between 1 and 3) 3. weak importance (preference) of one over another 4. intermediate (between 3 and 5) 5. essential or strong importance (preference) 6. intermediate (between 5 and 7) 7. demonstrated importance (preferences) 8. intermediate (between 7 and 9) 9. absolute importance (preference) Figure 4 summarizes a part of the matrix containing pairwise evaluations. The same figure illustrates membership functions describing the preference of the six first alternatives evaluated with respect to reliability of performance of the structure. It becomes obvious that the sixth alternative viewed in the light of this criterion is the best one while the first one was definitely conceived as the weakest one. At the next state the experts determined the importance of the criteria involved in the problem. All criteria are expressed in a linguistic form accordingly, -reliability of performance of the structure: the most important - easiness of manufacturing: important - t h e number of elements in the joint: less important - coaxiality of the cylinder and the rod: rather less important Aggregation of partial evaluation with respect to the above set of criteria, see Eq. (10)-(11) yields fuzzy sets of preference as shown in Figure 5.
R. Knosala, W. Pedrycz / Evaluation of design alternatives
Expert no.1
Expert no.2
AI A2 A3 A4A5 A6..
Jl I ~ ¼ ½ X ~
277
Expert no. 3
AI A2 A3 N.,.A5 A6.
AI A2 A3 A4 A5 A6..
AII X~¼½ A23
1 I
2 51
A3 3 I I 2 5 1 A3 912[11ZI411}}A3 8 2 1 L, ~ ZX A4 7 / l / x / l J z / ¼ l l A. s ¼ X 1 1 Y~ IA,'I ~ ½ 1 AS IS I~IXI£111XII A5 S X X 1 1 X ~IXX~ 1 X A6L~~[~_ ~ A6 9 : 3 2 5 S I ~i A 6 3 I I 3 5 1
Alternative 1 10
Atternofive 2 ,ol
I0
1286 0653
10
0
Z632
3859
li0 = 6.500 7908
34~,8
Alternative L,.
Alternative S
Atfernofive 6
i
1o / i ~ 3 4 4 0
I
I0 1320 1646
Alternative 3
~ 1714 080~, 1646
Z652
8552 10
Fig. 4. The matrix of pairwise comparisons and derived fuzzysets of preference.
In addition to this purely graphical representation, which in our case might be sometimes difficult to interpret (especially for several selected alternatives), one can support it by indices flu and Yu" Both describe the strength of links (relationships) between alternatives. They are again shown in Figure 5 in the form of flu/~'u" Noticeable is the fact that relationships between 'vertical' alternatives are much more weaker than 'horizontal' ones, which values are close to 1, Some slight changes are visible when the index of precision Sp is included, see again Figure 5. For comparative purposes Figure 6 contains results of ranking performed by another independent group of constructors. Their results of ranking are very similar in comparison to our generated series, in particular
for the best and the weakest alternatives. Furthermore the algorithmic approach developed here induces a powerful interpretation framework. It enables us to reveal the level of preference of some selected alternatives. This holds, for instance, for the following list: -alternative 10: fulfills the requirement of dismounting property of the joint, -alternative 3: satisfies the requirement of minimal length of the passive part of the piston, - alternative 13: provides flexibility in changes of velocity at boundary positions.
5.
Conclusions
In this study methodology of
we have investigated the solving of multi-objective
278
R. Knosala, W. Pedrycz I Evaluation of design alternatives ,o~-
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0
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.
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os6
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Fig. 5. Ordering of design alternatives.
decision-making problems in mechanical engineering. Fuzzy sets when used to formulate a problem are able to cope with ambiguity directly resulting from a series of criteria taken into account as well as their uncertain character. In addition to that, fuzzy sets representing final outcomes (viz. ranking results) contribute a lot towards understanding the nature of all alternatives and reveal the form of relationships
between them as far as the dominance of some of them is concerned. Two new indices associated with pointwise global representation allow us to clarify a chain-like structure with respect to the dominance relationships. In a detailed analysis of a discussed example the produced results and their format were found convincing for the users. Definitely, the list of feasible methods for
R. Knosala, W. Pedrycz / Evaluation of design alternatives
279
~ooo
\, / /
\ \,
0__~6
013
012
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0 7t*
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ranking and the number of aggregating functions becomes an extensive one. Therefore it is evident that for any problem at hand some of them should be applied and the derived results be studied with respect to the generated ranking structures. The methods devised in this paper are directly designed to explore all essential components of dominance structures occurring between discussed alternatives. By no means the pursued research is complete; still a lot of room is left for
generalized versions of ranking methods. This in turn may call for methods of cross-ranking analysis.
Acknowledgement Support from the Natural Sciences and Engineering Research Council of Canada is greatly appreciated.
280
R. Knosala, W. Pedrycz / Evaluation of design alternatives
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