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Fuzzy network for decision support systems
Fuzzy Sets and Systems 58 (1993) 59-72 North-Holland
59
Fuzzy network for decision support systems Hiroshi Kawamura Department of Architecture, Faculty of Engineering, Kobe University, Kobe, Japan Received April 1991 Revised March 1992
Abstract: Fuzzy network composed of non-directed fuzzy systems with inputs and outputs exchangeable for each other is proposed as a decision support system. This fuzzy network is composed of fuzzy unknown variables and fuzzy known constraints. To resolve this fuzzy network practically and systematically, successive shadow method and boundary point method are proposed. Finally, simplified examples are shown to illustrate and verify these proposed methods.
Keywords: Fuzzy system; fuzzy network; decision making; optimization; fuzzy variable; fuzzy constraint; fuzzy relation; engineering design~
1. Introduction
In the engineering fields, design means optimization or decision making with respect to unknown variables under given constraints such as design formulas, design criteria, mathematical equations, and so on. However, there are inevitable uncertainties in such variables and constraints, so it is a very important subject in the engineering design to describe and resolve these problems of optimization and decision making. (Appendix A.) Zadeh [9, 10] introduced fuzzy sets and fuzzy systems, and Bellman and Zadeh [1] proposed maximizing decisions in fuzzy environments. These fundamental ideas are very useful for the engineering problems mentioned above. On the other hand, Serrano and Gossard [6] proposed an artificial intelligent method, i.e., constraint-objected networks for engineering designs including unknown variables and given constraints. In this paper, based on these ideas proposed by the above researchers, fuzzy network is proposed as a decision support system in order to resolve such fuzzy engineering design problems to obtain optimal design variables. The author [3] already proposed an idea of a fuzzy network composed of fuzzy systems whose 'state' can be assumed as conditioned fuzzy set [1], fuzzy confluence rule [2], and fuzzy identifier [4]. Generally, if fuzzy relations [9] or conditioned fuzzy sets are used as 'state', such fuzzy systems represent fuzzy relational equations proposed by Sanchez [5]. In this paper, however, non-directed fuzzy relations and/or crisp relations are used to represent engineering functions and/or equations. Therefore, reasoning directions can be arbitrarily selected in the fuzzy network proposed here. In OR, linear programming is well known as an optimization method. Furthermore, fuzzy logic has already been applied to this method by Tanaka, Okuda, and Asai [8] and by Zimmermann [12], and later by many researchers [13, 7]. In the real engineering world, however, constraints have possibilities to be described with any type of functions, e.g., linear, nonlinear, equal, unequal, deterministic, probabilistic, objective, subjective, continuous, discrete, theoretical, empirical functions, and so on. Correspondence to: Dr. Hiroshi Kawamura, Department of Architecture, Faculty of Engineering, Kobe University, Kobe, 657 Japan. 0165-0114/93/$06.00 (~) 1993--Elsevier Science Publishers B.V. All rights reserved
60
H. Kawamura / Fuzzy network for decision support systems
Fig. 1. Fuzzy network.
In this paper, non-directed fuzzy and crisp relations are employed as 'state' in fuzzy systems and as design constraints which are described with matrices so that any type of functions can be represented and calculations can be carried out by using the max-min operation (composition) [9] (Appendix B). It is a main purpose of this paper to investigate resolution methods for a more general engineering design tool, i.e., fuzzy network.
2. Preliminary definitions and assumptions 2.1. F u z z y network
Let R be a non-directed and n-ary fuzzy relation with fuzzy variables X 1 , . . . , Xn. When rn kinds of R, i.e., R1 . . . . , Rm are considered, a fuzzy network can be constituted as shown in Figure 1 which is analogous to multi-variable simultaneous equations. If the membership function/z of a fuzzy relation R is transferred to one of its supports and a fuzzy modification M whose support corresponds to the membership function ~ of the fuzzy relation R is introduced, then the fuzzy relation R is reduced to a (n + 1)-ary crisp relation Re. Such a fuzzy modification M belongs to Type-2 fuzzy sets [11] by means of which the evaluation grade in the crisp relation Rc can be adjusted according to the theory of mapping [9]. Besides fuzzy variables X~ . . . . . X,, a fuzzy relation R may include fuzzy known variables, fuzzy constants, fuzzy parameters and so on which are expressed by Y1. . . . , Yk. Adding M and Y to the fuzzy network shown in Figure 1, it is replaced by an expanded fuzzy network shown in Figure 2 in which R ~ , . . . , R~m are non-directed and (n + k + 1)-ary crisp relations. 2.2. Decision support system
At the beginning of the application of the expanded fuzzy network shown in Figure 2 to decision support systems, it is assumed that every fuzzy modification M, crisp relation Rc, and fuzzy variable (or constant) Y are given and that every fuzzy variable X is unknown. In decision making, crisp relations Re mean given constraints with respect to fuzzy variables X and Y, the latter of which is given. The degrees of standard or temporary evaluation included in the crisp relation Rc can be adjusted with its fuzzy modification M, e.g., 'not changed,' 'not,' 'nearly,' 'roughly'
Fig. 2. Expanded fuzzy network.
H. Kawamura / Fuzzy network for decision support systems
61
and so on (Appendix C). M and Y can be combined with Re by the m a x - m i n operation as follows: MioY1 . . . . .
YkoRci=Ri,
(1)
i=l,...,m.
Finally, the expanded fuzzy network shown in Figure 2 is reduced to the fuzzy network shown in Figure 1. Decision making is performed by means of the resolution of the fuzzy network with respect to X shown in Figure 1 and an optimal solution x* is given by x ( e X ) whose membership value is maximal. 3. Resolution m e t h o d s 3.1. Total intersection m e t h o d
The most direct and exact resolution method is to get the total intersection of R 1 , . . . , Rm shown in Figure 1, i.e., the fuzzy relation Rt in the total product space X1 x • • • x X, given by (2)
Rt = R~ D . • • D R m.
The optimal point or region is given by the combination of x * , . . . , x*, whose membership value in Rt is maximal. This method is fundamentally based on the maximizing decision ]1]. In the same way, based on the concept of a-level sets [11], the optimal combination of x* . . . . . x* is given by the intersection of a-level sets of every R with the maximal a at which such an intersection exists. The fundamental idea of this method was already proposed by Tanaka, Okuda and Asai [8]. In the application of the fuzzy network shown in Figure 1 to the real world problems, the numbers of X and its elements are so large that this resolution method requires a great number of memories in computers. For example, if the number of X is a and each X has the same number of elements b, then the total number of elements of R t becomes b". 3.2. Successive s h a d o w m e t h o d
In the real fuzzy networks, every R is not necessarily defined in the total product space X1 x • • • x An. If X1 belongs only to 81, 82 and R3, o n e can eliminate X1 according to the following three optional compositions of R: (1)
R1oR2=R12, xIEXI
(2)
(3)
xIEX]
82081 xIEX
RLOR3=R13,
= 821 , 1
R3°R1 : R3l, XI~X 1
82o83
: 823 .,
(3)
x}Ex I
R 3 ° R 2 : 832. XlE:X 1
Equation 3 implies that one can eliminate the same number of R as the number of X in the fuzzy network shown in Figure 1. After such successive compositions of R the fuzzy network is finally reduced to the following three types shown in Figure 3. Figure 3(a), (b) and (c) correspond to the cases that n = m + r, n = m and n = m - r, respectively, where r is a redundant number, and i, j are positive integers.
(a) n=m+r
(b) n=m Fig. 3. Final fuzzy networks.
(c) n=m-r
H. Kawamura / Fuzzy network for decision support systems
62
Fig. 4. Hierarchical fuzzy network.
Optimal solutions are given as follows: In the case of Figure 3(a) n = m + r, because Rj is the fuzzy relation in the product space X / x • • • × X/+r, the combination of x * . . . . , X*+r whose membership value in Rj is maximal is the optimal solution. In the case of Figure 3(b) n = m, because Rj shows Xi itself, xi with maximal membership value in Rj is the optimal solution. In the case of Figure 3(c) n = m - r , because all Rj . . . . , Rj+r show Xi, x * with maximal membership value in R; is the optimal solution, and R; is the intersection given by R; = Rj n...
(4)
D Rj+ r.
As pointed out in [9], the max-rain operation followed by the elimination of some fuzzy sets produces a shadow of the total intersection of fuzzy relations projected on remaining fuzzy sets. The R's in Figure 3 are shadows projected on final remaining fuzzy sets. This is the reason why this method is named successive shadow method in this paper. This method becomes very practical and useful in a somewhat hierarchical fuzzy network where the number of R is very large, while each number of X connected to each R is not so large as shown in Figure 4. Exactly saying, Rj in Figure 3(a), Rj in Figure 3(b) and R; in (4) are the shadows of the total intersection of R 1 , . . . , Rm in Figure 1 projected on the fuzzy relation in the product space X i x • • • x Xi+~, the fuzzy set X / a n d the fuzzy set X~, respectively. If the membership function of the total intersection of R is of single-peak type, then these shadows also have the same type membership functions. In such a case, the optimal point x* and the optimal combination of x * , . . . , x* can be determined uniquely. However, if the membership function of the total intersection of R has plural and same extreme values, then optimal solutions are given by the regions of x*, and the truly optimal combinations of x * , . . . , x* are included partially in all the possible combinations of x * , . . . , x*. 3.3. B o u n d a r y p o i n t m e t h o d
'Boundary point method' is proposed here to find out a point on the boundaries of the region of truly optimal solutions obtained in the previous section. In this method, the following theorem is used: The maximal membership value of a shadow fuzzy set is the same as the one of the original total intersection of R. Let amax be the maximal membership value of the total intersection of R obtained by the above successive shadow method, and consider the spatial region ~ with OLrnax in the product space X1 x • • • x Xn. Furthermore, suppose a crisp relation R~ to be described with non-fuzzy singletons with respect to xi whose membership value is amax in the original R. Then the membership function ~ in every original R is replaced by the membership f u n c t i o n / ~ in R~ as follows: If
/-L(x'I. . . . . x'n) = a . . . .
then
/~(x'l . . . . , x'n) = 1.
If
/z(x[ . . . . . x " ) ~ a . . . .
then
/z~(Xl'. . . . . x ' ) = 0.
¢
x~ ~ Xi, i = 1. . . . , n,
(5)
x~'~ X'~ i = 1 . . . .
(6)
, n,
When amax does not exist in the original R, the least membership value more than O/max should be employed instead of OLmax in (5), (6).
H. Kawamura / Fuzzy network for decision support systems
(x-y-z o)
63
z_
Fig. 5. Assumedexpanded fuzzynetwork.
The sharpest boundary of g2~ is the intersectional point at which every R~ meets each other. This point is easily obtained by applying the successive shadow method to the rewritten R~ network in the case that n = m. If possible plural combinations of points are obtained, truly optimal points satisfying every Re must be selected by means of trial and error. The next sharpest boundary of ~2~ is the intersectional hyper-line on which rn - 1 kinds of R~ meet. This line is obtained as a set of boundary points by applying the successive shadow method to the network composed of m - 1 kinds of R~ and a kind of parameter Re in the case that n = m. This hyper-line exists in the range for the parameter R~ with non-fuzzy singletons corresponding to the membership values equal to or more than O~max in its original R. The intersectional hyper-plane and so on, can be given in the same way. In the case that n = m + r, the above process begins with the (r + 1)th sharpest boundary. In the case that n = m - r, the all above mentioned procedures are carried out regarding Rj . . . . . Rj+r, after which the intersection of each region is given as the optimal solution. This method is a complementary one of the successive shadow method presented in the previous section. By this method, outlines of optimal solutions can be given speedily, which can be considered to be very useful from practical and engineering points of view.
4. Examples 4.1.
Fuzzy
network
To illustrate and verify the methods proposed above, an expanded fuzzy network as shown in Figure 5 is supposed here, where X, Y, Z are unknown fuzzy variables, M is a fuzzy modification and C is a fuzzy constant, x - y - z - 0 , 2 y - z ~<0, 2 z - x - c ~ < 0 are fuzzy constraints assumed as linear algebraic equations for simplicity here and expressed in Tables 1, 2 and 3, where x, y and z are supposed to be positive integers. Assume M and C as follows: M:
(M means 'a little more gently')
~
0 0.1 0.5 0 (I.3 0.5
C:
0.9 0.7
1 1
(7)
(c is nearly equal to 2)
~_~0 0
1 0.2
2
3
1 0.2
4 0
(8)
Combining Table 2 with (7) and Table 3 with (8) by means of the max-min operation, the expanded
64
H. Kawamura / F u z z y network f o r decision support systems
Tablel. x- y- z - O x
y 0
0
1
2
3
4
1
2
z
0
1
2
3
0
1
2
3
0
1
2
3
p.
1
0.3
0
0
0.3
0
0
0
0
0
0
0
z
0
1
2
3
0
1
2
3
0
1
2
3
/z 0.3
1
0.3
0
1 0.3
0
0
0.3
0
0
0
z
0
1
2
3
0
1
2
3
0
1
2
3
p. 0
0.3
0.3
1
0.3
0
1 0.3
0
0
2
3
1 0.3
z
0
1
2
3
0
1
#
0
0
0.3
1
0
0.3
z
0
1
2
3
0
1
2
3
/z
0
0
0
0.3
0
0
0.3
1
1 0.3
0
1
2
3
0.3
1
0.3
0
0
1
2
3
0
0.3
1
0.3
fuzzy network shown in Figure 5 is reduced to the fuzzy network shown in Figure 6, where Rt is the same as Table 1, and R2 and R3 are given by Tables 4 and 5. 4.2. Total intersection method
According to (2), the total intersection of Rt of X, Y, and Z, is given by Table 6 which shows that there are three sets of optimal combinations of x*, y*, and z*, i.e., (0, 0, 0), (1, 0, 1), and (3, 1, 2). By using the a-level set method, the same optimal solutions must be obtained. 4.3. Successive shadow method
By (9), X is eliminated and the fuzzy network shown in Figure 6 is reduced to the one shown in Figure 7, where R13 is expressed by Table 7.
(9)
R1 o R3 = R13 x~g
Table 2. 2 y - z ~<0. (This constraint is described with a crisp relation which is modified with M) z
y 0
0
1
2
3
1
2
m 0
0.1
0.5
0.9
1
0
0.1
0.5
0.9
1
0
0.1
0.5
0.9
1
/~ 0
0
0
1
0
0
1
0
0
0
1
0
0
0
0
m 0
0.1
0.5
0.9
1
0
0.1
0.5
0.9
1
0
0.1
0.5
0.9
1
/z 0
0
0
0
1
0
0
1
0
0
1
0
0
0
0
m 0
0.1
0.5
0.9
1
0
0.1
0.5
0.9
1
0
0.1
0.5
0.9
1
tz 0
0
0
0
1
0
0
0
1
0
0
1
0
0
0
m 0
0.1
0.5
0.9
1
0
0.1
0.5
0.9
1
0
0.1
0.5
0.9
1
/x 0
0
0
0
1
0
0
0
0
1
0
0
1
0
0
H. Kawamura / Fuzzy network for decision support systems
Fig. 6. Reduced fuzzy network.
Table 3. 2z - x - c <~0 X
£
1
0
z
2
0
/x O.7 1
2
3
4
3
l
2
3
0
1
2
3
0
1
2
3
O.3
0
0
1
0.5
0
0
1
0.7
0.3
0
z
0
1
2
3
0
1
2
3
0
1
2
3
/x
1
0.5
0
0
1
0.7
0.2
0
1
1
0.5
0
z
0
1
2
3
0
1
2
3
0
1
2
3
1
O.7
O.3
0
1
1
0.5
0
1
1
0.7
0.3
z
0
1
2
3
0
1
2
3
0
1
2
3
/x
l
1
0.5
0
1
1
0.7
0.2
1
1
1
0.5
z
0
1
2
3
0
1
2
3
0
1
2
3
/z
1
1
0.7
0.3
1
l
1
0.5
1
1
1
0.7
Table 4. R 2 Y 0 z
1
0
/z 0.7
1
23
0
1
1
11
0.3
0.5
Table 5. R 3 X
Z
0 0 1 2 3 4
~ /x /x /z /.t
2
1
10.5 10.7 1 1 1 1 1 1
2
3
0.2 0 0.2 0 I).5 0.2 I).7 0.2 1 0.5
2 0.71
3
01
2
3
00
0.3
0.5
65
66
H. K a w a m u r a / F u z z y network f o r decision support systems Table 6. R t
x
y 0
0
z
1
0
/z 0.7 1
z
2
3
4
1
2
3
0
1
2
3
01
0.3
0.2
0
0.3
0
0
0
0
0
1
2
3
0
1
2
3
0.3 0.3
0
0
0
0
0
0
0
/z 0.3
2
1
2
3
0.7
0.2
0
z
0
1
2
3
tz
0
0.3
0.5
0.2
0
1
0.3 0.5
z
0
1
2
3
0
1
/z
0
0
0.3
0.2
0
0.3
z
0
1
2
3
0
1
iz
0
0
0
0.3
0
0
0
2
3
0
0
2
3
0
1
2
3
0.3
0
0
0
0
0
2
3
0
1
2
0.7 0.3 2
000.3
3
0.3 0.5
0
1
0
0
3 0
2
3
0.3 0.3
By (10), Y is eliminated and Z is given as follows: R13 ° R 2 = Z , yEY
z
Z:
(lO) 0 1 2 3 0.7 0.7 0.7 0.5"
(11)
By (12), Z is eliminated and Y is given as follows: R 1 3 o R 2 = Y, zEz
yt
Y:
(12) 0 1 2 0.7 0.7 0 . 3
(13)
By (14), Y is eliminated and the fuzzy network shown in Figure 6 is reduced to the one shown in
Fig. 7. Fuzzy network after eliminating X. Table 7. R13
Y 0 z
1
0
1
/x 1
0.7
2
3
0.5 0.3
2
0 1
2
3
1 1
0.7
0.5
01
2
3
1 1
1 0.3
H. Kawamura / Fuzzy network for decision support systems
67
Fig. 8. Fuzzy network after eliminating Y.
Figure 8. Rl oR2 = R12,
(14)
y~Y
where R12 is given by Table 8. By (15), Z is eliminated and X is given as follows: RlzoR3
:= X .
(15)
z~Z
xl 0 -/z~ 0.7
X:
1 0.7
2 0.5
3 0.7
4 0.5"
(16)
Equations (11), (13) and (16) are the shadows projected on Z, Y, X of the total intersection Rt shown in Table 6. Therefore, the optimal solutions are given with the range x*=0,1,3,
y*=0,1,
z*=0,1,2,
(17)
in which the real exact optimal combinations are included.
4.4. Boundary point method In the previous sections, 4.3, 4.4., it is shown that the maximal membership value of the optimal solutions, a . . . . is equal to 0.7. According to (5), (6), the crisp relations R~ 1, R~ 2, R , 3 with non-fuzzy singletons for the proposed boundary point method are assumed as follows: R,~,: amax is supposed to be 1.0, and R1 is reduced to the values shown in Table 9. R,,2: ~r~ax is 0.7, then Rz is reduced to the values shown in Table 10. R~3: amax is 0.7, then R3 is reduced to the values shown in Table 11. According to the same procedure as in the successive shadow method, the optimal solution on the sharpest boundary, i.e., the point, is given by (x*, y*, z*) = (3, 2, 1).
(18)
Secondly, let only R~ 3 be a parameter as specified in Table 12, which is derived from R3 by assuming that o~,ax -- 1.0 and non-fuzzy singletons are located on the membership value 1.0 adjacent to 0.5 in R3. According to the same procedure as in the successive shadow method, one of the optimal solutions
Table 8. R12 X
Z
0 0 1 2 3 4
/z /~ /z /x /z
1
0.7 0.3 0.3 1 0.3 0.5 0 0.3 0 0
2
3
0 0 0.3 0 1 0.3 0.7 1 0.3 1
H. Kawamura / Fuzzy network for decision support systems
68
Table 9. R~ x
y 0
0
1
2
3
4
1
2
z
0
123
0 1 2 3
0 1 2 3
#.
1
000
0 0 0 0
0 0 0 0
z
0
123
0 1 2 3
0
~
0
100
1 0 0 0
0 0 0 0
z
0
123
0
I
23
~
0
0
1
0
0
1
0
0
1
0
0
0
z
0
1
2
3
0
1
2
3
0
1
2
3
~
0
0
0
1
0
0
1
0
0
1
0
0
z ~
0 0
1 0
2 0
3 0
0 0
1 0
2 0
3 1
0 0
1 0
2 1
3 0
0 1 2 3
l
Table 10. R 2 Y 0 z
1
0
1
2
3
0
1
2
3
0
1
2
3
/,~1
0
0
0
0
0
1
0
0
0
0
0
Table 11. R 3 x
z 0 1 2 3
0 ~ 1 ~ 2 ~ 3~. 4~.
0 0 0 0 0
0 1 0 0 0
0 0 0 1 0
0 0 0 0 0
Table 12. R,~ 3 z
z 0 1 2 3
0 1 2 3 4
2
/.,,,, p,, /.,, P,o~ p,,
1 0 0 0 0
0 0 1 0 0
0 0 0 0 1
0 0 0 0 0
23
H. Kawamura / Fuzzy network for decision support systems
69
Table 13. R,,2 Y
0 z 0 1 2 3 g~o 1 0 0
1
2
0 1 2 3 0 0 0 1
0 1 2 3 0 0 0 0
on a next sharpest boundary, i.e., a line, is given by (x*, y*, z*) = (0, 0, 0).
(19)
Thirdly, let only R~ 2 be a parameter as given in Table 13, which is derived from R2 by assuming that amax = 1.0 and non-fuzzy singletons are located on the membership value 1.0 adjacent to 0.7 in R2. According to the same procedure as in the successive shadow method, one of the optimal solutions on another next sharpest boundary, i.e., a line, is given by (x*, y*, z*) = (1, 0, 1).
(20)
Furthermore replacements of R 2 and R3 by R~, and R ~ no longer bring optimal solutions. In this case study, as obtained by the total intersection method, (18), (19) and (20) are all the optimal solutions.
5. Summary and conclusions In this paper, fuzzy network is proposed for engineering design in which optimal variables are determined under fuzzy constraints. This fuzzy network is composed of non-directed fuzzy systems with unknown fuzzy variables described with fuzzy sets corresponding to inputs and outputs and known fuzzy constraints described with fuzzy relations corresponding to states. Furthermore, by introducing fuzzy modifications, fuzzy relations can be reduced to crisp relations and membership values can be modified easily. Besides the most direct and exact methods, i.e., total intersection method and/or a-level set method, the following practical and systematic methods are proposed: (1) Successive shadow method and (2) Boundary point method. The former (1) is suitable for the fuzzy network (the hierarchical fuzzy network shown in Figure 4) in which each number of fuzzy variables connected to each fuzzy constraint is small relative to the number of fuzzy constraints. The latter (2) is a complementary method of the former one by which optimal solutions are obtained not as a point but as a region. These two methods are valid for any types of constraints, as far as they can be described with matrices. For the purpose of obtaining all the regions of optimal solutions in under- or over-constrained fuzzy networks where the number of fuzzy constraints differs from the one of fuzzy variables, the application of the boundary point method may become so complex that more detailed investigations of such fuzzy networks will be needed in the future.
Acknowledgement The author is grateful to Prof. James T.P. Yao in Texas A&M University for his encouraging to submit this paper. The author is in debt to Research Associate Mr. Akinori Tani at the Kobe University for his help in preparing this paper.
H. Kawamura / Fuzzy network for decision support systems
70
Appendix A. Consider a stub column with sectional area A subjected to axial load P. The safety factor of this column s is given by (A.1)
s = P/A~ra,
where o a is a given allowable stress of the material of this column. Suppose that a design critiera is given by s ~
(A.2)
where Sc is a given critical stress. In this case, P, A, O'a, s and Sc are variables and (A.1) and (A.2) are constraints. So a design problem is described with a network such as shown in Figure A.1.
Fig. A.1. Stub column design network.
For example, if all the constraints and variables except A and s are known in Figure A.1, the minimal A and the maximal s can be obtained by solving this network.
Appendix B. It is well known that the extension principle [11] is applicable to any type of functions. In the same sense, this principle can be replaced by the max-min operation [9] between fuzzy sets and a crisp relation described by a matrix. For example, consider an equation y =
(B.1)
where )7 and ~ are expressed by fuzzy sets Y and X. If (B.1) is expressed by a crisp relation described by a matrix, e.g., R as given in Table B.1, then )7 is given by Y = X o R. Table B1 x
y 2
4
1 1 0 0 2 0 1 0 3 0 0 1
6
(B.2)
71
14. Kawamura / Fuzzy network for decision support systems
F u r t h e r m o r e , if Eq. B.1 is fuzzified as follows: (B.3)
y - 26~, then R in T a b l e B1 can be r e p l a c e d by a fuzzy relation.
Appendix C. The effects of the proposed fuzzy modification M are illustrated as follows: Suppose that standard membership values m' are given by broken lines as shown in Figure C.1. If the fuzzy modification M has membership values rn such as shown in the left hand side figures, the membership values m' of X are transformed into m (solid lines) as shown in the right hand side figures. Figures C.1 (a), (b), (c) and (d) represent 'not changed,' 'not,' 'nearly' and 'roughly,' respectively. m
m
~-~
m' I
0
(a)
~
0
o
I ×
I x
(b) "not"
"not changed"
,q m' I
0
m
m
/ / ! \',
0 O0
I x
(e) "nearly"
m" I
0
o
I x
(d) "roughly"
Fig. C.1. Effects of fuzzy modification M.
References [1] R.E. Bellman and L.A. Zadeh, Decision-making in a fuzzy environment, Manag. Sci. 17 (1970) B-141-B-164. [2] H. Kawamura, A. Tani, M. Kawamura, S. Matsumoto and M. Yamada, A general formulation of the confluence rule of fuzzy goal and constraint and its non-numerical maximization, Proc. of 3rd Fuzzy System Symposium, Tokyo, Japan (1987) 71-76 (in Japanese). [3] H. Kawamura and J.T.P. Yao, Applications of fuzzy systems based on conditioned fuzzy sets to structural engineering, J. Struct. Eng. 36B (1990) 51-56 (in Japanese). [4] H. Kawamura and A. Tani, Multi-variable fuzzy identifier, Proc. of 6th Fuzzy System Symposium, Tokyo, Japan (1990) 179-182 (in Japanese). [5] E. Sanchez, Resolution of composite fuzzy relation equations, Inf. Contr. 30 (1976) 38-48. [6] D. Serrano and D. Gossard, Constraint management in MCAE, in: J.S. Gero (Ed.) Artificial Intelligence in Engineering: Design (Elsevier, Boston, 1988) 217-240.
72
H. Kawamura / Fuzzy network for decision support systems
[7] R. Slowinski and J. Teghem (Eds.) Stochastic versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty (Kluwer Academic Publishers, Dordrecht, 1990). [8] H. Tanaka, T. Okuda and K. Asai, Fuzzy mathematical programming, Trans. Soc. Instr. Contr. Eng. 9 (1973) 607-613 (in Japanese). [9] L.A. Zadeh, Fuzzy sets, Inf. Contr. 8 (1965) 338-353. [10] L.A. Zadeh, Toward a theory of fuzzy systems, in: R.E. Kalman and N. Declaris (Eds.) Aspects of Network and System Theory (Holt, Rinehart and Winston, New York, 1971) 469-490. [11] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Part 1, Inf. Sci. 8 (1975) 199-249. [12] H.-J. Zimmermann, Description and optimization of fuzzy systems, Intern. J. Gen. Syst. 2 (1976) 209-215. [13] H.-J. Zimmermann, L.A. Zadeh and B.R. Gaines (Eds.) Fuzzy Sets and Decision Analysis (North-Holland, Amsterdam, 1984).
59
Fuzzy network for decision support systems Hiroshi Kawamura Department of Architecture, Faculty of Engineering, Kobe University, Kobe, Japan Received April 1991 Revised March 1992
Abstract: Fuzzy network composed of non-directed fuzzy systems with inputs and outputs exchangeable for each other is proposed as a decision support system. This fuzzy network is composed of fuzzy unknown variables and fuzzy known constraints. To resolve this fuzzy network practically and systematically, successive shadow method and boundary point method are proposed. Finally, simplified examples are shown to illustrate and verify these proposed methods.
Keywords: Fuzzy system; fuzzy network; decision making; optimization; fuzzy variable; fuzzy constraint; fuzzy relation; engineering design~
1. Introduction
In the engineering fields, design means optimization or decision making with respect to unknown variables under given constraints such as design formulas, design criteria, mathematical equations, and so on. However, there are inevitable uncertainties in such variables and constraints, so it is a very important subject in the engineering design to describe and resolve these problems of optimization and decision making. (Appendix A.) Zadeh [9, 10] introduced fuzzy sets and fuzzy systems, and Bellman and Zadeh [1] proposed maximizing decisions in fuzzy environments. These fundamental ideas are very useful for the engineering problems mentioned above. On the other hand, Serrano and Gossard [6] proposed an artificial intelligent method, i.e., constraint-objected networks for engineering designs including unknown variables and given constraints. In this paper, based on these ideas proposed by the above researchers, fuzzy network is proposed as a decision support system in order to resolve such fuzzy engineering design problems to obtain optimal design variables. The author [3] already proposed an idea of a fuzzy network composed of fuzzy systems whose 'state' can be assumed as conditioned fuzzy set [1], fuzzy confluence rule [2], and fuzzy identifier [4]. Generally, if fuzzy relations [9] or conditioned fuzzy sets are used as 'state', such fuzzy systems represent fuzzy relational equations proposed by Sanchez [5]. In this paper, however, non-directed fuzzy relations and/or crisp relations are used to represent engineering functions and/or equations. Therefore, reasoning directions can be arbitrarily selected in the fuzzy network proposed here. In OR, linear programming is well known as an optimization method. Furthermore, fuzzy logic has already been applied to this method by Tanaka, Okuda, and Asai [8] and by Zimmermann [12], and later by many researchers [13, 7]. In the real engineering world, however, constraints have possibilities to be described with any type of functions, e.g., linear, nonlinear, equal, unequal, deterministic, probabilistic, objective, subjective, continuous, discrete, theoretical, empirical functions, and so on. Correspondence to: Dr. Hiroshi Kawamura, Department of Architecture, Faculty of Engineering, Kobe University, Kobe, 657 Japan. 0165-0114/93/$06.00 (~) 1993--Elsevier Science Publishers B.V. All rights reserved
60
H. Kawamura / Fuzzy network for decision support systems
Fig. 1. Fuzzy network.
In this paper, non-directed fuzzy and crisp relations are employed as 'state' in fuzzy systems and as design constraints which are described with matrices so that any type of functions can be represented and calculations can be carried out by using the max-min operation (composition) [9] (Appendix B). It is a main purpose of this paper to investigate resolution methods for a more general engineering design tool, i.e., fuzzy network.
2. Preliminary definitions and assumptions 2.1. F u z z y network
Let R be a non-directed and n-ary fuzzy relation with fuzzy variables X 1 , . . . , Xn. When rn kinds of R, i.e., R1 . . . . , Rm are considered, a fuzzy network can be constituted as shown in Figure 1 which is analogous to multi-variable simultaneous equations. If the membership function/z of a fuzzy relation R is transferred to one of its supports and a fuzzy modification M whose support corresponds to the membership function ~ of the fuzzy relation R is introduced, then the fuzzy relation R is reduced to a (n + 1)-ary crisp relation Re. Such a fuzzy modification M belongs to Type-2 fuzzy sets [11] by means of which the evaluation grade in the crisp relation Rc can be adjusted according to the theory of mapping [9]. Besides fuzzy variables X~ . . . . . X,, a fuzzy relation R may include fuzzy known variables, fuzzy constants, fuzzy parameters and so on which are expressed by Y1. . . . , Yk. Adding M and Y to the fuzzy network shown in Figure 1, it is replaced by an expanded fuzzy network shown in Figure 2 in which R ~ , . . . , R~m are non-directed and (n + k + 1)-ary crisp relations. 2.2. Decision support system
At the beginning of the application of the expanded fuzzy network shown in Figure 2 to decision support systems, it is assumed that every fuzzy modification M, crisp relation Rc, and fuzzy variable (or constant) Y are given and that every fuzzy variable X is unknown. In decision making, crisp relations Re mean given constraints with respect to fuzzy variables X and Y, the latter of which is given. The degrees of standard or temporary evaluation included in the crisp relation Rc can be adjusted with its fuzzy modification M, e.g., 'not changed,' 'not,' 'nearly,' 'roughly'
Fig. 2. Expanded fuzzy network.
H. Kawamura / Fuzzy network for decision support systems
61
and so on (Appendix C). M and Y can be combined with Re by the m a x - m i n operation as follows: MioY1 . . . . .
YkoRci=Ri,
(1)
i=l,...,m.
Finally, the expanded fuzzy network shown in Figure 2 is reduced to the fuzzy network shown in Figure 1. Decision making is performed by means of the resolution of the fuzzy network with respect to X shown in Figure 1 and an optimal solution x* is given by x ( e X ) whose membership value is maximal. 3. Resolution m e t h o d s 3.1. Total intersection m e t h o d
The most direct and exact resolution method is to get the total intersection of R 1 , . . . , Rm shown in Figure 1, i.e., the fuzzy relation Rt in the total product space X1 x • • • x X, given by (2)
Rt = R~ D . • • D R m.
The optimal point or region is given by the combination of x * , . . . , x*, whose membership value in Rt is maximal. This method is fundamentally based on the maximizing decision ]1]. In the same way, based on the concept of a-level sets [11], the optimal combination of x* . . . . . x* is given by the intersection of a-level sets of every R with the maximal a at which such an intersection exists. The fundamental idea of this method was already proposed by Tanaka, Okuda and Asai [8]. In the application of the fuzzy network shown in Figure 1 to the real world problems, the numbers of X and its elements are so large that this resolution method requires a great number of memories in computers. For example, if the number of X is a and each X has the same number of elements b, then the total number of elements of R t becomes b". 3.2. Successive s h a d o w m e t h o d
In the real fuzzy networks, every R is not necessarily defined in the total product space X1 x • • • x An. If X1 belongs only to 81, 82 and R3, o n e can eliminate X1 according to the following three optional compositions of R: (1)
R1oR2=R12, xIEXI
(2)
(3)
xIEX]
82081 xIEX
RLOR3=R13,
= 821 , 1
R3°R1 : R3l, XI~X 1
82o83
: 823 .,
(3)
x}Ex I
R 3 ° R 2 : 832. XlE:X 1
Equation 3 implies that one can eliminate the same number of R as the number of X in the fuzzy network shown in Figure 1. After such successive compositions of R the fuzzy network is finally reduced to the following three types shown in Figure 3. Figure 3(a), (b) and (c) correspond to the cases that n = m + r, n = m and n = m - r, respectively, where r is a redundant number, and i, j are positive integers.
(a) n=m+r
(b) n=m Fig. 3. Final fuzzy networks.
(c) n=m-r
H. Kawamura / Fuzzy network for decision support systems
62
Fig. 4. Hierarchical fuzzy network.
Optimal solutions are given as follows: In the case of Figure 3(a) n = m + r, because Rj is the fuzzy relation in the product space X / x • • • × X/+r, the combination of x * . . . . , X*+r whose membership value in Rj is maximal is the optimal solution. In the case of Figure 3(b) n = m, because Rj shows Xi itself, xi with maximal membership value in Rj is the optimal solution. In the case of Figure 3(c) n = m - r , because all Rj . . . . , Rj+r show Xi, x * with maximal membership value in R; is the optimal solution, and R; is the intersection given by R; = Rj n...
(4)
D Rj+ r.
As pointed out in [9], the max-rain operation followed by the elimination of some fuzzy sets produces a shadow of the total intersection of fuzzy relations projected on remaining fuzzy sets. The R's in Figure 3 are shadows projected on final remaining fuzzy sets. This is the reason why this method is named successive shadow method in this paper. This method becomes very practical and useful in a somewhat hierarchical fuzzy network where the number of R is very large, while each number of X connected to each R is not so large as shown in Figure 4. Exactly saying, Rj in Figure 3(a), Rj in Figure 3(b) and R; in (4) are the shadows of the total intersection of R 1 , . . . , Rm in Figure 1 projected on the fuzzy relation in the product space X i x • • • x Xi+~, the fuzzy set X / a n d the fuzzy set X~, respectively. If the membership function of the total intersection of R is of single-peak type, then these shadows also have the same type membership functions. In such a case, the optimal point x* and the optimal combination of x * , . . . , x* can be determined uniquely. However, if the membership function of the total intersection of R has plural and same extreme values, then optimal solutions are given by the regions of x*, and the truly optimal combinations of x * , . . . , x* are included partially in all the possible combinations of x * , . . . , x*. 3.3. B o u n d a r y p o i n t m e t h o d
'Boundary point method' is proposed here to find out a point on the boundaries of the region of truly optimal solutions obtained in the previous section. In this method, the following theorem is used: The maximal membership value of a shadow fuzzy set is the same as the one of the original total intersection of R. Let amax be the maximal membership value of the total intersection of R obtained by the above successive shadow method, and consider the spatial region ~ with OLrnax in the product space X1 x • • • x Xn. Furthermore, suppose a crisp relation R~ to be described with non-fuzzy singletons with respect to xi whose membership value is amax in the original R. Then the membership function ~ in every original R is replaced by the membership f u n c t i o n / ~ in R~ as follows: If
/-L(x'I. . . . . x'n) = a . . . .
then
/~(x'l . . . . , x'n) = 1.
If
/z(x[ . . . . . x " ) ~ a . . . .
then
/z~(Xl'. . . . . x ' ) = 0.
¢
x~ ~ Xi, i = 1. . . . , n,
(5)
x~'~ X'~ i = 1 . . . .
(6)
, n,
When amax does not exist in the original R, the least membership value more than O/max should be employed instead of OLmax in (5), (6).
H. Kawamura / Fuzzy network for decision support systems
(x-y-z o)
63
z_
Fig. 5. Assumedexpanded fuzzynetwork.
The sharpest boundary of g2~ is the intersectional point at which every R~ meets each other. This point is easily obtained by applying the successive shadow method to the rewritten R~ network in the case that n = m. If possible plural combinations of points are obtained, truly optimal points satisfying every Re must be selected by means of trial and error. The next sharpest boundary of ~2~ is the intersectional hyper-line on which rn - 1 kinds of R~ meet. This line is obtained as a set of boundary points by applying the successive shadow method to the network composed of m - 1 kinds of R~ and a kind of parameter Re in the case that n = m. This hyper-line exists in the range for the parameter R~ with non-fuzzy singletons corresponding to the membership values equal to or more than O~max in its original R. The intersectional hyper-plane and so on, can be given in the same way. In the case that n = m + r, the above process begins with the (r + 1)th sharpest boundary. In the case that n = m - r, the all above mentioned procedures are carried out regarding Rj . . . . . Rj+r, after which the intersection of each region is given as the optimal solution. This method is a complementary one of the successive shadow method presented in the previous section. By this method, outlines of optimal solutions can be given speedily, which can be considered to be very useful from practical and engineering points of view.
4. Examples 4.1.
Fuzzy
network
To illustrate and verify the methods proposed above, an expanded fuzzy network as shown in Figure 5 is supposed here, where X, Y, Z are unknown fuzzy variables, M is a fuzzy modification and C is a fuzzy constant, x - y - z - 0 , 2 y - z ~<0, 2 z - x - c ~ < 0 are fuzzy constraints assumed as linear algebraic equations for simplicity here and expressed in Tables 1, 2 and 3, where x, y and z are supposed to be positive integers. Assume M and C as follows: M:
(M means 'a little more gently')
~
0 0.1 0.5 0 (I.3 0.5
C:
0.9 0.7
1 1
(7)
(c is nearly equal to 2)
~_~0 0
1 0.2
2
3
1 0.2
4 0
(8)
Combining Table 2 with (7) and Table 3 with (8) by means of the max-min operation, the expanded
64
H. Kawamura / F u z z y network f o r decision support systems
Tablel. x- y- z - O x
y 0
0
1
2
3
4
1
2
z
0
1
2
3
0
1
2
3
0
1
2
3
p.
1
0.3
0
0
0.3
0
0
0
0
0
0
0
z
0
1
2
3
0
1
2
3
0
1
2
3
/z 0.3
1
0.3
0
1 0.3
0
0
0.3
0
0
0
z
0
1
2
3
0
1
2
3
0
1
2
3
p. 0
0.3
0.3
1
0.3
0
1 0.3
0
0
2
3
1 0.3
z
0
1
2
3
0
1
#
0
0
0.3
1
0
0.3
z
0
1
2
3
0
1
2
3
/z
0
0
0
0.3
0
0
0.3
1
1 0.3
0
1
2
3
0.3
1
0.3
0
0
1
2
3
0
0.3
1
0.3
fuzzy network shown in Figure 5 is reduced to the fuzzy network shown in Figure 6, where Rt is the same as Table 1, and R2 and R3 are given by Tables 4 and 5. 4.2. Total intersection method
According to (2), the total intersection of Rt of X, Y, and Z, is given by Table 6 which shows that there are three sets of optimal combinations of x*, y*, and z*, i.e., (0, 0, 0), (1, 0, 1), and (3, 1, 2). By using the a-level set method, the same optimal solutions must be obtained. 4.3. Successive shadow method
By (9), X is eliminated and the fuzzy network shown in Figure 6 is reduced to the one shown in Figure 7, where R13 is expressed by Table 7.
(9)
R1 o R3 = R13 x~g
Table 2. 2 y - z ~<0. (This constraint is described with a crisp relation which is modified with M) z
y 0
0
1
2
3
1
2
m 0
0.1
0.5
0.9
1
0
0.1
0.5
0.9
1
0
0.1
0.5
0.9
1
/~ 0
0
0
1
0
0
1
0
0
0
1
0
0
0
0
m 0
0.1
0.5
0.9
1
0
0.1
0.5
0.9
1
0
0.1
0.5
0.9
1
/z 0
0
0
0
1
0
0
1
0
0
1
0
0
0
0
m 0
0.1
0.5
0.9
1
0
0.1
0.5
0.9
1
0
0.1
0.5
0.9
1
tz 0
0
0
0
1
0
0
0
1
0
0
1
0
0
0
m 0
0.1
0.5
0.9
1
0
0.1
0.5
0.9
1
0
0.1
0.5
0.9
1
/x 0
0
0
0
1
0
0
0
0
1
0
0
1
0
0
H. Kawamura / Fuzzy network for decision support systems
Fig. 6. Reduced fuzzy network.
Table 3. 2z - x - c <~0 X
£
1
0
z
2
0
/x O.7 1
2
3
4
3
l
2
3
0
1
2
3
0
1
2
3
O.3
0
0
1
0.5
0
0
1
0.7
0.3
0
z
0
1
2
3
0
1
2
3
0
1
2
3
/x
1
0.5
0
0
1
0.7
0.2
0
1
1
0.5
0
z
0
1
2
3
0
1
2
3
0
1
2
3
1
O.7
O.3
0
1
1
0.5
0
1
1
0.7
0.3
z
0
1
2
3
0
1
2
3
0
1
2
3
/x
l
1
0.5
0
1
1
0.7
0.2
1
1
1
0.5
z
0
1
2
3
0
1
2
3
0
1
2
3
/z
1
1
0.7
0.3
1
l
1
0.5
1
1
1
0.7
Table 4. R 2 Y 0 z
1
0
/z 0.7
1
23
0
1
1
11
0.3
0.5
Table 5. R 3 X
Z
0 0 1 2 3 4
~ /x /x /z /.t
2
1
10.5 10.7 1 1 1 1 1 1
2
3
0.2 0 0.2 0 I).5 0.2 I).7 0.2 1 0.5
2 0.71
3
01
2
3
00
0.3
0.5
65
66
H. K a w a m u r a / F u z z y network f o r decision support systems Table 6. R t
x
y 0
0
z
1
0
/z 0.7 1
z
2
3
4
1
2
3
0
1
2
3
01
0.3
0.2
0
0.3
0
0
0
0
0
1
2
3
0
1
2
3
0.3 0.3
0
0
0
0
0
0
0
/z 0.3
2
1
2
3
0.7
0.2
0
z
0
1
2
3
tz
0
0.3
0.5
0.2
0
1
0.3 0.5
z
0
1
2
3
0
1
/z
0
0
0.3
0.2
0
0.3
z
0
1
2
3
0
1
iz
0
0
0
0.3
0
0
0
2
3
0
0
2
3
0
1
2
3
0.3
0
0
0
0
0
2
3
0
1
2
0.7 0.3 2
000.3
3
0.3 0.5
0
1
0
0
3 0
2
3
0.3 0.3
By (10), Y is eliminated and Z is given as follows: R13 ° R 2 = Z , yEY
z
Z:
(lO) 0 1 2 3 0.7 0.7 0.7 0.5"
(11)
By (12), Z is eliminated and Y is given as follows: R 1 3 o R 2 = Y, zEz
yt
Y:
(12) 0 1 2 0.7 0.7 0 . 3
(13)
By (14), Y is eliminated and the fuzzy network shown in Figure 6 is reduced to the one shown in
Fig. 7. Fuzzy network after eliminating X. Table 7. R13
Y 0 z
1
0
1
/x 1
0.7
2
3
0.5 0.3
2
0 1
2
3
1 1
0.7
0.5
01
2
3
1 1
1 0.3
H. Kawamura / Fuzzy network for decision support systems
67
Fig. 8. Fuzzy network after eliminating Y.
Figure 8. Rl oR2 = R12,
(14)
y~Y
where R12 is given by Table 8. By (15), Z is eliminated and X is given as follows: RlzoR3
:= X .
(15)
z~Z
xl 0 -/z~ 0.7
X:
1 0.7
2 0.5
3 0.7
4 0.5"
(16)
Equations (11), (13) and (16) are the shadows projected on Z, Y, X of the total intersection Rt shown in Table 6. Therefore, the optimal solutions are given with the range x*=0,1,3,
y*=0,1,
z*=0,1,2,
(17)
in which the real exact optimal combinations are included.
4.4. Boundary point method In the previous sections, 4.3, 4.4., it is shown that the maximal membership value of the optimal solutions, a . . . . is equal to 0.7. According to (5), (6), the crisp relations R~ 1, R~ 2, R , 3 with non-fuzzy singletons for the proposed boundary point method are assumed as follows: R,~,: amax is supposed to be 1.0, and R1 is reduced to the values shown in Table 9. R,,2: ~r~ax is 0.7, then Rz is reduced to the values shown in Table 10. R~3: amax is 0.7, then R3 is reduced to the values shown in Table 11. According to the same procedure as in the successive shadow method, the optimal solution on the sharpest boundary, i.e., the point, is given by (x*, y*, z*) = (3, 2, 1).
(18)
Secondly, let only R~ 3 be a parameter as specified in Table 12, which is derived from R3 by assuming that o~,ax -- 1.0 and non-fuzzy singletons are located on the membership value 1.0 adjacent to 0.5 in R3. According to the same procedure as in the successive shadow method, one of the optimal solutions
Table 8. R12 X
Z
0 0 1 2 3 4
/z /~ /z /x /z
1
0.7 0.3 0.3 1 0.3 0.5 0 0.3 0 0
2
3
0 0 0.3 0 1 0.3 0.7 1 0.3 1
H. Kawamura / Fuzzy network for decision support systems
68
Table 9. R~ x
y 0
0
1
2
3
4
1
2
z
0
123
0 1 2 3
0 1 2 3
#.
1
000
0 0 0 0
0 0 0 0
z
0
123
0 1 2 3
0
~
0
100
1 0 0 0
0 0 0 0
z
0
123
0
I
23
~
0
0
1
0
0
1
0
0
1
0
0
0
z
0
1
2
3
0
1
2
3
0
1
2
3
~
0
0
0
1
0
0
1
0
0
1
0
0
z ~
0 0
1 0
2 0
3 0
0 0
1 0
2 0
3 1
0 0
1 0
2 1
3 0
0 1 2 3
l
Table 10. R 2 Y 0 z
1
0
1
2
3
0
1
2
3
0
1
2
3
/,~1
0
0
0
0
0
1
0
0
0
0
0
Table 11. R 3 x
z 0 1 2 3
0 ~ 1 ~ 2 ~ 3~. 4~.
0 0 0 0 0
0 1 0 0 0
0 0 0 1 0
0 0 0 0 0
Table 12. R,~ 3 z
z 0 1 2 3
0 1 2 3 4
2
/.,,,, p,, /.,, P,o~ p,,
1 0 0 0 0
0 0 1 0 0
0 0 0 0 1
0 0 0 0 0
23
H. Kawamura / Fuzzy network for decision support systems
69
Table 13. R,,2 Y
0 z 0 1 2 3 g~o 1 0 0
1
2
0 1 2 3 0 0 0 1
0 1 2 3 0 0 0 0
on a next sharpest boundary, i.e., a line, is given by (x*, y*, z*) = (0, 0, 0).
(19)
Thirdly, let only R~ 2 be a parameter as given in Table 13, which is derived from R2 by assuming that amax = 1.0 and non-fuzzy singletons are located on the membership value 1.0 adjacent to 0.7 in R2. According to the same procedure as in the successive shadow method, one of the optimal solutions on another next sharpest boundary, i.e., a line, is given by (x*, y*, z*) = (1, 0, 1).
(20)
Furthermore replacements of R 2 and R3 by R~, and R ~ no longer bring optimal solutions. In this case study, as obtained by the total intersection method, (18), (19) and (20) are all the optimal solutions.
5. Summary and conclusions In this paper, fuzzy network is proposed for engineering design in which optimal variables are determined under fuzzy constraints. This fuzzy network is composed of non-directed fuzzy systems with unknown fuzzy variables described with fuzzy sets corresponding to inputs and outputs and known fuzzy constraints described with fuzzy relations corresponding to states. Furthermore, by introducing fuzzy modifications, fuzzy relations can be reduced to crisp relations and membership values can be modified easily. Besides the most direct and exact methods, i.e., total intersection method and/or a-level set method, the following practical and systematic methods are proposed: (1) Successive shadow method and (2) Boundary point method. The former (1) is suitable for the fuzzy network (the hierarchical fuzzy network shown in Figure 4) in which each number of fuzzy variables connected to each fuzzy constraint is small relative to the number of fuzzy constraints. The latter (2) is a complementary method of the former one by which optimal solutions are obtained not as a point but as a region. These two methods are valid for any types of constraints, as far as they can be described with matrices. For the purpose of obtaining all the regions of optimal solutions in under- or over-constrained fuzzy networks where the number of fuzzy constraints differs from the one of fuzzy variables, the application of the boundary point method may become so complex that more detailed investigations of such fuzzy networks will be needed in the future.
Acknowledgement The author is grateful to Prof. James T.P. Yao in Texas A&M University for his encouraging to submit this paper. The author is in debt to Research Associate Mr. Akinori Tani at the Kobe University for his help in preparing this paper.
H. Kawamura / Fuzzy network for decision support systems
70
Appendix A. Consider a stub column with sectional area A subjected to axial load P. The safety factor of this column s is given by (A.1)
s = P/A~ra,
where o a is a given allowable stress of the material of this column. Suppose that a design critiera is given by s ~
(A.2)
where Sc is a given critical stress. In this case, P, A, O'a, s and Sc are variables and (A.1) and (A.2) are constraints. So a design problem is described with a network such as shown in Figure A.1.
Fig. A.1. Stub column design network.
For example, if all the constraints and variables except A and s are known in Figure A.1, the minimal A and the maximal s can be obtained by solving this network.
Appendix B. It is well known that the extension principle [11] is applicable to any type of functions. In the same sense, this principle can be replaced by the max-min operation [9] between fuzzy sets and a crisp relation described by a matrix. For example, consider an equation y =
(B.1)
where )7 and ~ are expressed by fuzzy sets Y and X. If (B.1) is expressed by a crisp relation described by a matrix, e.g., R as given in Table B.1, then )7 is given by Y = X o R. Table B1 x
y 2
4
1 1 0 0 2 0 1 0 3 0 0 1
6
(B.2)
71
14. Kawamura / Fuzzy network for decision support systems
F u r t h e r m o r e , if Eq. B.1 is fuzzified as follows: (B.3)
y - 26~, then R in T a b l e B1 can be r e p l a c e d by a fuzzy relation.
Appendix C. The effects of the proposed fuzzy modification M are illustrated as follows: Suppose that standard membership values m' are given by broken lines as shown in Figure C.1. If the fuzzy modification M has membership values rn such as shown in the left hand side figures, the membership values m' of X are transformed into m (solid lines) as shown in the right hand side figures. Figures C.1 (a), (b), (c) and (d) represent 'not changed,' 'not,' 'nearly' and 'roughly,' respectively. m
m
~-~
m' I
0
(a)
~
0
o
I ×
I x
(b) "not"
"not changed"
,q m' I
0
m
m
/ / ! \',
0 O0
I x
(e) "nearly"
m" I
0
o
I x
(d) "roughly"
Fig. C.1. Effects of fuzzy modification M.
References [1] R.E. Bellman and L.A. Zadeh, Decision-making in a fuzzy environment, Manag. Sci. 17 (1970) B-141-B-164. [2] H. Kawamura, A. Tani, M. Kawamura, S. Matsumoto and M. Yamada, A general formulation of the confluence rule of fuzzy goal and constraint and its non-numerical maximization, Proc. of 3rd Fuzzy System Symposium, Tokyo, Japan (1987) 71-76 (in Japanese). [3] H. Kawamura and J.T.P. Yao, Applications of fuzzy systems based on conditioned fuzzy sets to structural engineering, J. Struct. Eng. 36B (1990) 51-56 (in Japanese). [4] H. Kawamura and A. Tani, Multi-variable fuzzy identifier, Proc. of 6th Fuzzy System Symposium, Tokyo, Japan (1990) 179-182 (in Japanese). [5] E. Sanchez, Resolution of composite fuzzy relation equations, Inf. Contr. 30 (1976) 38-48. [6] D. Serrano and D. Gossard, Constraint management in MCAE, in: J.S. Gero (Ed.) Artificial Intelligence in Engineering: Design (Elsevier, Boston, 1988) 217-240.
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[7] R. Slowinski and J. Teghem (Eds.) Stochastic versus Fuzzy Approaches to Multiobjective Mathematical Programming under Uncertainty (Kluwer Academic Publishers, Dordrecht, 1990). [8] H. Tanaka, T. Okuda and K. Asai, Fuzzy mathematical programming, Trans. Soc. Instr. Contr. Eng. 9 (1973) 607-613 (in Japanese). [9] L.A. Zadeh, Fuzzy sets, Inf. Contr. 8 (1965) 338-353. [10] L.A. Zadeh, Toward a theory of fuzzy systems, in: R.E. Kalman and N. Declaris (Eds.) Aspects of Network and System Theory (Holt, Rinehart and Winston, New York, 1971) 469-490. [11] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, Part 1, Inf. Sci. 8 (1975) 199-249. [12] H.-J. Zimmermann, Description and optimization of fuzzy systems, Intern. J. Gen. Syst. 2 (1976) 209-215. [13] H.-J. Zimmermann, L.A. Zadeh and B.R. Gaines (Eds.) Fuzzy Sets and Decision Analysis (North-Holland, Amsterdam, 1984).