In this paper, we propose a new method for choosing implications. Our method allows to compare two fuzzy implications. If the truth value of the antecedent and the truth value of the implication are given, by means of inverse fuzzy implications we can easily optimize the truth value of the implication consequent.
In 1975 Lotfi Zadeh introduced the theory of fuzzy logic [
The rest of this paper is organized as follows. In Sect. 2 the main research
problem is formulated. Sect. 3 presents the solution of the given research
problem. An example illustrating our approach is given in Sect. 4. Sect. 5 includes
remarks on directions for further research.
There is given a basic fuzzy implication z = I(x; y), where x, y belong to [
There are a few examples of basic fuzzy implications in Table 1. 3
Year
1923, [
The resulting inverse functions can be compared with each other so that it
is possible to order them. However, some of those functions are incomparable in
the whole domain. Nevertheless, by dividing the domain into separable areas (see
Figure 1), we have obtained the below given inequalities for any x; z belonging to
(0,1). The interval to which x and z belong to is open, since we did not manage
to deduce any inequalities on the edges of the domain. From the definition of a
fuzzy implication ([
InvILK(x; z) = z + x 1 InvIGD(x; z) = z InvIRC(x; z) = z+x 1
x InvIKD(x; z) = z InvIGG(x; z) = xz
1 InvIY G(x; z) = z x InvIF D(x; z) = z Values of those functions on this edge are equal to zero, hence, no inequality can be inferred from them. Likewise, on the edge x = 1 the inverse functions are InvILK , InvIGD, InvIRC , InvIKD, InvIGG, InvIY G, InvIF D. Values of these functions are equal z on this edge. For this reason it is not possible to infer any inequality. 4. For z < 1 x and z > x 1 xx
InvIGG < InvIY G 3. For z > 1 x and z < x 1 xx
InvILK < InvIRC < InvIY G < InvIGG < InvIGD = InvIKD = InvIF D 5. For z > 1 x and z x and z < 1+1x
InvILK < InvIRC < InvIGG < InvIY G < InvIKD 6. For z > x 1 xx and z < 1 x and z < 1+1x
InvILK < InvIRC < InvIGG < InvIY G < InvIKD = InvIGD = InvIF D 7. For z > 1+1x and z x
InvILK < InvIGG < InvIRC < InvIY G < InvIKD 8. For z > 1+1x and z < x
InvILK < InvIGG < InvIRC < InvIY G < InvIKD = InvIGD = InvIF D 9. For z = 1 x and x 2 (0; 12 )
InvILK = InvIRC < InvIGG < InvIY G 10. For z = 12 and x = 12
InvILK = InvIRC < InvIGG = InvIY G 11. For z = 1 x and x 2 ( 21 ; 1)
InvILK = InvIRC < InvIY G < InvIGG < InvIGD 12. For z = x 1 xx and x 2 (0; 21 )
InvIY G = InvIGG 13. For z = x 1 xx and x 2 ( 21 ; 1)
InvILK < InvIRC < InvIY G = InvIGG < InvIGD = InvIKD = InvIF D 14. For z = 1+1x and x 2 (0; 1)
InvILK < InvIRC = InvIGG < InvIY G < InvIKD
By means of these inequalities it is possible to find a fuzzy implication which has, for example, the greatest or the smallest truth value of a consequent y whereas the truth value of the antecedent x and the truth value of the implication z are given. First, it should be checked to which area of the domain the point (x; z) belongs. It is one of the areas 1 - 14. Then, according to the inequality in the given area, the function of the smallest or the greatest truth value of the consequent InvI(x; z) can be selected.
Then, wanting to expand the set of tested fuzzy implications we have started
to study new fuzzy implications H1 H15 generated from the basic fuzzy
implications, which form an algebraic structure of lattice ([
{ 1 if x y H1(x; y) = max( xy ; 1 x + xy) if x > y
1 H1(x; y) = y
x 1 if x y if x > y x + xy if y < 1+xx
x 1+x H2(x;y) = {1 x + xy if x y
min(xy;1 x + xy) if x > y H11(x; y) = H12(x; y) = H12(x; y) = H13(x; y) = H13(x; y) = H14(x; y) = H15(x; y) = H15(x; y) = y 1x y 1 1 xy 1 1 In order to illustrate our approach, let us describe a simple example coming from the domain of train traffic control. We consider the following situation: a train B waits at a certain station for a train A to arrive in order to allow some passengers to change train A to train B. Now, a conflict arises when the train A is late. In this situation, the following alternatives can be taken into consideration: 1. Train B waits for train A to arrive. In this case, train B will depart with delay. 2. Train B departs in time. In this case, passengers disembarking train A have to wait for a later train. 3. Train B departs in time, and an additional train is employed for the train A passengers.
To make a decision, several inner conditions have to be taken into account such as the delay period, the number of passengers changing trains, etc. The discussion regarding an optimal solution to the problem of divergent aims such as: minimization of delays throughout the traffic network, warranty of connections for the customer satisfaction, efficient use of expensive resources, etc. is disregarded at this point. In order to describe the traffic conflict, we propose to consider the following three IF-THEN fuzzy rules: { r1: IF s2 OR s3 THEN s6 { r2: IF s1 AND s4 AND s6 THEN s7 { r3: IF s4 AND s5 THEN s8 where: { s1: ’Train B is the last train in this direction today’, { s2: ’The delay of train A is huge’, { s3: ’There is an urgent need for the track of train B’, { s4: ’Many passengers would like to change for train B’, { s5: ’The delay of train A is short’, { s6: ’(Let) train B depart according to schedule’, { s7: ’Employ an additional train C (in the same direction as train B)’, and { s8: ’Let train B wait for train A’.
In the further considerations we accept the following assumptions: 1. The logical operators OR, AND we interpret as max and min fuzzy operators, respectively. 2. The statements s1, s2, s3, s4 and s5 we assign the fuzzy values 0.6, 0.4, 0.7, 0.5 and 1, respectively. 3. The truth-values of rules r1; r2 and r3 are equal to 0.6, 0.7, 0.8, respectively. 4. The threshold values for three rules are equal to 0.1. 5. Each of rules r1, r2 and r3, firstly, we interpret as the Lukasiewicz implication.
Assessing the statements from s1 up to s5, we observe that the rule r1 and r3 can be fired (activated). Firing these rules according to the above assumptions allows computation of the support for the alternatives in question. In this way, the possible alternatives are ordered with regard to the preference they achieve from the knowledge base. This order forms the basis for further examinations and simulations and, ultimately, for the dispatching proposal. If one chooses a sequence of rules r1; r2 then they obtain the final value, corresponding to the statement s7, equal to 0. In the other case (i.e., for the rule r3 only), the final value, this time corresponding to the statement s8, equals 0.3. Secondly, if we interpret these three rules as the Reichenbach implications, and if we choose the same sequences of transitions as above we obtain the final values for the statements s7 and s8 equal to 0.3, 0.6, respectively. Thirdly, if we execute the similar simulation of approximate reasoning for three rules considered above and, if we interpret the rules as the Kleene-Dienes implications we obtain the final values for s7 and s8 equal to 0.7, 0.8, respectively. This example shows clearly that different interpretations for the rules may lead to quite different decision results. Nevertheless, they are conformable with the relationships between inverted fuzzy implication functions presented above (see, for example, items 3, 5-8, 13-14). Choosing a suitable interpretation for fuzzy implications we may try to obtain the optimal final values for decisions s7 and s8. The rest in this case certainly depends on the experience of the decision support system designer to a significant degree. 5
In this paper, we presented an approach to finding the fuzzy implication which has for example the largest or the smallest truth value of the consequent when the truth value of the antecedent and the truth value of the implication are given.
Our next problem to be solved is how to choose the fuzzy implication so that the truth value of the antecedent is optimized at the given truth value of implication’s conclusion and given value of truth of implication. Acknowledgment. This work was partially supported by the Center for Innovation and Transfer of Natural Sciences and Engineering Knowledge at the University of Rzesz´ow.