Fuzzy Number Addition with the Application of Horizontal Membership Functions

The paper presents addition of fuzzy numbers realised with the application of the multidimensional RDM arithmetic and horizontal membership functions (MFs). Fuzzy arithmetic (FA) is a very difficult task because operations should be performed here on multidimensional information granules. Instead, a lot of FA methods use α-cuts in connection with 1-dimensional classical interval arithmetic that operates not on multidimensional granules but on 1-dimensional intervals. Such approach causes difficulties in calculations and is a reason for arithmetical paradoxes. The multidimensional approach allows for removing drawbacks and weaknesses of FA. It is possible thanks to the application of horizontal membership functions which considerably facilitate calculations because now uncertain values can be inserted directly into equations without using the extension principle. The paper shows how the addition operation can be realised on independent fuzzy numbers and on partly or fully dependent fuzzy numbers with taking into account the order relation and how to solve equations, which can be a difficult task for 1-dimensional FAs.


Introduction
Fuzzy arithmetic [1][2][3][4][5][6][7][8][9][10][11][12][13][14] is used in uncertainty theory [15][16][17][18][19][20], grey systems [21,22], granular computing [9,23], computing with words [23][24][25][26], decision-making [27,28], and other sciences and engineering branches [4,26,29,30]. Authors of first concepts of the fuzzy arithmetic (shortly FA) based on -(left-right) fuzzy numbers were Dubois and Prade [1]. With years, FA has been improved and its new versions have been introduced, for example, the popular -cuts' version [4,6,31], which in this paper will be called -cuts' FA. In general, all versions of FA can be divided [4] into elementary FA, standard FA, and advanced FA versions. Examples of advanced FA methods can be the generalized vertex method [32], constrained FA [7,8,21], algorithmic FA [9], transformation method and extended transformation method [4], and inverse FA [4]. It seems that mostly used FA versions are based on -cuts [2] and on Moore's interval arithmetic (shortly IA) [33][34][35]. In the FA literature, many examples of practical FA applications can be found, for example, [4,26,27,36,37]. Practical problems require effective FAs which would enable solving uncertain linear and nonlinear equation systems, differential equations, integral calculations, and so forth [38,39]. Therefore, many scientists investigate these problems and publish achieved results from this area . However, almost always they emphasise that the achieved level of FA is not satisfactory and that further investigations are necessary. Therefore, investigations on IA and FA have been continued nonstop. It is testified by new publications in journals, conferences, and new books. Though FA has achieved many application successes, it further on has many weak points. For example, Dymova and Sevastjanov report in their papers [27,61,62] the present FA has considerable difficulties in solving even simple equations with one unknown variable. There exist also difficulties in defining neutral and inverse elements for addition and multiplication.
The most popular version of FA is the arithmetic based on -cuts, where an -cut of a fuzzy set , denoted as , is defined by (1), where means domain of the set and means an element of this domain [4]. Consider = cut (̃) = { ∈ |̃( ) ≥ } .
(1) 2 The Scientific World Journal FA based on -cuts uses principles of the classical Moore's interval arithmetic [33,34] for realisation of elementary arithmetic operations such as addition, subtraction, multiplication, and division. Though this arithmetic has many applications, its possibilities are limited, because of some drawbacks, which are commented on also by Moore himself in his books. For example, if , , and are intervals, the distributive law, does not always hold. Additionally, an additive inverse of an interval is not defined. If = [ , ], then + (− ) = [ − , − ] ̸ = [0, 0]. As Dymowa shows in [27], there are great difficulties in solving even simple interval equations with one unknown. Let us assume that in a system there exists the dependence + = . This dependence can be presented in few equivalent forms: If only approximate values of and are known, ∈ [1,3] and ∈ [2,5], and value is not known, then, on the basis of (3), four interval equations (4)- (7) can be written: Solving (4), two equations, 1 + = 2 and 3 + = 5, are obtained. They give the solution [ , ] = [1,2]. Solving (5), two equations, 1 = 2 − and 3 = 5 − , are obtained. They give the solution [ , ] = [2, 1] being an inverse interval. Solving (6), the solution [−1, 4] is obtained and from (7) we get the solution [4, −1], which is also an inverse interval. Thus, various solutions, [1,2], [2,1], [−1, 4], and [4,1], are obtained depending on extension form of the original dependence. In the case of more complicated mathematical dependencies, the solution number can be considerably higher.
Another paradox of FA based on -cuts and Moore arithmetic (shortly -cuts' FA) [33][34][35] is not satisfying the cancellation law for multiplication. For example, equation = in the general case does not mean that = . It can be testified by an example shown below in which notation [1,2,3] means the triangle membership function (shortly MF) with support beginning, core position, and support end. Consider Because of this feature of -cuts' FA, transformations of formulas are not allowable. Why? It will be shown further on. Next important paradox of -cuts' FA is the observation that, during calculation of results of nonlinear formulas, for example, = − 2 , we obtain different, nonunique solutions depending on which form of the formula is used: Which solution is correct? The above phenomenon means that each transformation of an equation form, in the case of -cuts' FA, can change its solution and that solutions are not unique. Further on, it will be shown that in the case of the multidimensional RDM FA such paradoxes do not occur.
The above examples show that the classical IA, which is a basis of the -cut version of FA, is not ideal, though it can solve certain problems. Therefore, it can and should be further developed. Further on, a version of FA that is based on horizontal membership functions (MFs) and on -cuts will be presented. It also applies the multidimensional RDM arithmetic (M-RDM arithmetic) which has been elaborated by Andrzej Piegat

Horizontal Membership Function
Fuzzy systems use vertical membership functions which were introduced by Zadeh [23]. They have the following form: = ( ), where is an independent variable and is a dependent one. Figure 1 shows an example of the trapezoidal MF and formula (10) gives its description: The Scientific World Journal Figure 2: Visualization of the horizontal approach to fuzzy membership functions.
Vertical MF realises the mapping → . The present fuzzy arithmetic is based on just such MFs. The idea of horizontal MFs has been elaborated by Andrzej Piegat. In this paper, an example of a trapezium MF will be presented but horizontal MFs can be used for all types of MFs. A function ( ) is unambiguous in the direction of the variable ( Figure 1) and ambiguous in the direction of . Therefore, it seems impossible to define a membership function in thedirection. The function from Figure 1 assigns two values of , ( ) and ( ), for one value of . However, let us introduce the RDM variable: ∈ [0, 1]. This variable has meaning of the relative-distance-measure and allows for determining of any point between two borders ( ) and ( ) of the function ( Figure 2). RDM variable takes a value of zero on the left border and a value of 1 on the right border of the function. Between the left border and the right border it takes fractional values. The idea of RDM variables was successfully used in the multidimensional IA [37,[67][68][69][70]. The multidimensional IA has shown that full and precise solutions of granular problems have form of multidimensional granules that cannot be explained and understood in terms of 1dimensional approaches.
Formula (11) defines the left border and the right border of the trapezium MF ( Figure 2). Consider RDM variable allows for a gradual transition of points between the left border and the right border. The interval ( ) in Figure 2 is defined by the following formula: It can be noted in formula (12) that = ( , ) is an unambiguous function existing in the 3D space, which can be seen in Figure 3. It should be also noticed that RDM variables introduce the continuous Cartesian coordinate system in interval and fuzzy arithmetic calculations. In Moore's arithmetic and in -cuts' FA based on it, only borders of intervals or -cuts, for example, [ , ]+[ , ] = [ + , + ], take part in calculations. Insides of intervals are not taken into account. This fact hinders solving of more complicated problems and results in many paradoxes observed in IA and FA calculations. It also deprives IA and FA of many mathematical properties which the conventional mathematics of crisp numbers has. Few of these properties will be presented further on.
Thanks to the fact that RDM variables introduce in interval calculations local and continuous Cartesian coordinate system, RDM FA possesses almost the same mathematical properties as arithmetic of crisp numbers. The subtraction result ( ) − ( ) determined by (15) is a fuzzy interval of nonzero span which means that, in -cuts' FA additive, inverse element does not exist. Consider In the case of RDM FA, the element ( , ) is determined by (16) and its negative element − ( , ) by (17). Consider It is easy to check that the subtraction result − is exactly equal to zero: Thus, in the case of RDM FA, there exists the inverse additive element − ( , ).

Distributive Law of Multiplication.
In the -cuts' FA, similarly as in Moore's interval arithmetic, the distributive law holds only in the limited form: In the multidimensional, continuous RDM FA, the distributive law holds fully. Consider we have   Figure 4: Three different membership functions of "results" of the fuzzy formula = − 2 written in three equivalent with the application of -cuts' FA, three different results are obtained: Each of the results, 1 , 2 , and 3 , is different. For = 0, results mean spans (widths) of MF supports. They are equal to [−4, 2], [−2, 2], and [−4, 4]. The differences are considerable. Thus, the formula = − 2 cannot be calculated uniquely with -cuts' FA. If we use the multidimensional RDM FA, then interval is expressed by (24) and its square 2 by (25), where is generic variable of . One should notice that functions expressing and 2 are fully coupled (correlated); that is, values of and have to be identical both in and in 2 . Consider Formula 1 = − 2 can be determined by As can be seen, result for the second form 2 is the same as for 1 . Result for the third form 3 can be determined by The Scientific World Journal The result 3 is identical to 1 and 2 . As the analysed example shows the multidimensional RDM FA provides unique solutions for nonlinear formulas independently of their mathematical form, it cannot be said about -cuts' FA. To better realise this fact, particular solutions will be visualised in Figure 4.
One can easily see that all three solutions, 1 , 2 , and 3 , are incorrect. According to the solution 1 , it is possible that the difference = − 2 for ∈ [0, 2] could be equal to, for example, −4 or +2. However, these values cannot be obtained with any value of ∈ [0, 2]. According to the "result" 2 , possible values of = − 2 could be equal to +1 or +2. But, in fact, these values are impossible. According to the "result" 3 , possible values of are equal to −4 or +4. However, it is not true.
The full and precise 3D solution, in which formula (obtained with use of the RDM FA) is given by is shown in Figure 5.
If we are interested in the simplified, traditional MF of the full 3D result in the horizontal version, it can be easily determined analytically on the basis of formula (30) for particular membership levels ( -cuts). Consider where ( , ) is determined by formula (29). Extremes of ( , ) can be detected by usual, analytical function examination. However, one should remember that function extremes can lie not only on domain borders of variable ( = 0, = 1) but also inside of the domain (fractional values of ). Then, the extremes can be detected by identification of critical points of the derivative d ( , )/d = 0.
In the analysed problem, examination of the function ( , ) has shown that its left border ( ) and right border ( ) have mathematical form expressed by the following formula: Membership function representing the span (width) of the full result ( , ) of the fuzzy formula = − 2 is shown in Figure 6. Their solutions usually will be less uncertain than solutions obtained with not-multidimensional methods.

Addition of Two Independent Triangle Fuzzy Numbers
Triangle numbers (Figure 7) are used in practice very frequently. Two numbers "about 1.0" and "about 1.1" can represent weights of two stones, S1 and S2, that were weighed on the scales. The scales have shown 1 = 1.0 kg for stone 1 and 2 = 1. Horizontal MF of the stone S1 is given by (32) and horizontal MF of the stone S2 is given by (33). Consider If we want to calculate the sum of weights of both stones, then the calculation with horizontal MFs can be made similarly as in a classical arithmetic, without use of the extension principle of Zadeh [37]. It is necessary only when vertical MFs are used. According to this principle, the membership function of the addition result is determined by (34), where 1 and 2 are fuzzy numbers [4,10]. Consider If horizontal MFs are used, then the sum = 1 + 2 is created directly by adding 1 and 2 determined by (32) and (33). Consider Formula (35) shows that the sum = 1 + 2 is not 1dimensional. It is a function defined in the 4D space because = ( , 1 , 2 ). Thus, it cannot be visualised but it can be shown as the projection onto the 3D space:  Figure 8 shows only two border cuts for = 0 and = 1. Apart from the 3D projection shown in Figure 8, other projections of the full 4D granule are also possible. Figure 9 shows the projection onto the 3D space 1 × 2 × . In this figure, addition results = 1 + 2 are shown in a form of contour lines = 1 + 2 = const.
Each of the contour lines = const is the set of infinite number of tuples { 1 , 2 } satisfying the condition 1 + 2 = .
As can be seen in Figure 9 for 1 = 1.01 and 2 = 1.09. A cardinality of particular sets [69] can be calculated with the following formula: where is a contour line defined as The cardinality of particular sets is equal to an area of -cuts of the granule (Figure 9). It is easy to calculate. For example, the cardinality of the set 2.1 equals It is the greatest of all cardinalities (Max Abs Card( )). It can be shown that the cardinality of particular sets is a square function of the sum = 1 + 2 ( (39) and (40)). For < 1 + 2 , For ≥ 1 + 2 , The absolute cardinality of the sum = 1 + 2 can take various numeric values. Therefore, it should be normalised to interval [0, 1] for convenience ( (41) and (42)). The normalised cardinality Norm Card( ) can also be called the relative cardinality Rel Card( ). For For It should be noticed that formulas (41) and (42) concern fuzzy numbers with an identical support. In the considered example, 1 − 1 = 2 − 2 = 0.2, so they take the following form: The cardinality distribution of particular values of the sum = 1 + 2 for the considered example is shown in Figure 10. It should be emphasised once more that the distribution determined by formula (43) and shown in Figure 10 is not the addition result of two fuzzy numbers but only 2D information about the cardinality of particular sum values = 1 + 2 derived from the full 4D addition result given by (35). The representation provided by Zadeh's extension principle (34) can also be obtained on the basis of the exact solution (35). The full 4D solution is given below. Consider The minimal value of the sum (the left border) for various levels of ∈ [0, 1] is obtained for 1 = 2 = 0: The maximal sum value (the right border) is obtained for 1 = 2 = 1: The 2D representation of the addition result of two fuzzy numbers "about 1.0" and "about 1.1" obtained with formulae (45) and (46) is shown in Figure 11.
The 2D representation obtained according to Zadeh's extension principle (34) informs only about the maximal spread of the full solution granule on particular -levels ( Figure 9). Geometrically, the distribution from Figure 11 corresponds to the cross section going from the corner =

Addition of Two Fuzzy Numbers with Taking into Account the Order Relation
Let us analyse an example of a fuzzy number addition similar to that in Section 3, but with small difference, with the additional knowledge that not only MFs of added numbers but also their order relation is known. Thus, there are two stones S1 and S2 which had been weighed on spring scales with maximal error equal to 0.1 kg. Therefore, weights 1 and 2 of stones are uncertain and can be expressed in the form of fuzzy numbers: 1 is [0.9, 1.0, 1.1] and 2 is [1.0, 1.1, 1.2]. MFs of uncertain weights are shown in Figure 7. After weighting the stones on the spring scales, their weights were compared on balance scales. The scales showed that the weight 2 , though uncertain, is greater than 1 . Therefore, the weights order 2 > 1 is known and can be taken into account in the weights' adding. The knowledge of the order relation changes (constraints) the domain of possible weight tuples { 1 , 2 } in comparison with the example from Section 3. Now less tuples will be feasible ( Figure 12).
In the considered example, uncertain weights 1 and determined by more constraints (formula (47)) as in the case without the weights' order relation. Consider Equations (47) determine the full and precise addition result = ( 1 , 2 , ) that has a form of the 4D information granule. This granule cannot be visualised. However, we can visualise its 3D projection onto the space 1 × 2 × that is shown in Figure 13.
A comparison of addition result granules obtained without ( Figure 8) and with ( Figure 13) taking into account the order relation 2 > 1 shows that it considerably changes the granules shape. Figure 14 shows other projection of the full 4D result granule onto the space 1 × 2 × . Values of the sum = 1 + 2 are shown in this figure in a form of contour lines of constant values.
The granule of the addition result from Figure 14 can be compared to the result granule of addition without the order relation 2 > 1 shown in Figure 9. The comparison shows that the relation 2 > 1 considerably decreases the domain of possible solutions. At the same time, we can observe decrease of cardinalities of particular values for For Norm Card ( ) Norm Card ( ) It should be noticed that the 2D representation obtained with use of Zadeh's extension principle (34) is still identical and has the form shown in Figure 16 independently whether the addition is realised with or without taking into account the order relation 2 > 1 .
The extension principle (34) does not "perceive" the domain loss (compare Figures 8 and 13) which is its important drawback. It provides even less information about the full 4D addition result than the 2D representation determined with formulas (50)- (56) and shown in Figure 15.

Addition of Two Fully Dependent Fuzzy Numbers
Correlations between variables have no significance in the case of an addition of two crisp numbers. However, in the case of intervals, fuzzy intervals, and fuzzy numbers, mutual dependencies are of great importance. Let us consider another example. Small John did not know exactly how much money he had in his money box after a month of saving. He evaluated that the sum was about $10, at least $7, and no more than $13. This evaluation can be expressed by the fuzzy number [46,49,51]. His father promised to double the saved sum after 1 month. He opened the box, checked the sum, and added exactly the same sum to the box. However, father did not inform John of the sum he added. How much money does small John have in the box now?
Let us denote by 1 the uncertain sum of money which small John had in the money box at the beginning: 1 is 12 The Scientific World Journal "about 10" [46,49,51]. The horizontal MF of the set "about 10" can be determined on the base of formula (32): Though the precise value of the sum 1 is not known for us and for John, his father knew it and added the same sum 2 to John's money box. Because 1 = 2 , then from (58) we can conclude that 1 = 2 . Consider It means that both RDM variables and also variables 1 and 2 are mutually fully coupled (correlated). The resulting sum = 1 + 2 can be expressed by the following formula: and it is visualised in Figure 17.
As Figure 17 shows, the addition result "about 20" is in this case the flat and not 3D information granule as in the case of the addition of fully independent fuzzy numbers; see Figure 9. As before, the flat 2D representation in the form of the horizontal MF = ( ) can be determined on the basis of exact solution (59). This function can be characterized with left and right border functions ( ) and ( ). Both functions can be determined from the full result: The left, minimal function border is obtained for minimal values of RDM variables 1 = 2 = 0. Consider The right border of the 2D representation, the border of maximal values, is obtained for maximal values of RDM variables 2D representation of the full 4D addition result (59) is shown in Figure 18.

Solving Additive Fuzzy Equation: + =
As it was mentioned in Section 1, solving equations with one unknown makes great difficulties even for the interval arithmetic which is considerably simpler than the fuzzy arithmetic. It will be shown in this section that solving fuzzy equations with use of horizontal MFs is possible and not difficult. Let us consider the last example.
Corn harvested from field 1 was weighed on the scales that have maximal error equal to 1 ton. The scales indicated 5 tons. Thus, the true weight of the corn can be expressed in a form of the fuzzy number = [4,5,6]. Corn harvested from field 2 was not weighed on the field. Therefore, its true weight was not known even approximately. Both crops were brought to a warehouse. There, they were mixed and weighed together on 3 scales that also have the maximal error equal to 1 ton. The scales indicated 15 tons. Thus, the sum of crops can be expressed in a form of the fuzzy number = [14,15,16]. How large is the weight of the crop from field 2? The knowledge of and is important, because farmers 1 and 2 should be paid fairly for their delivery.
This problem cannot be solved uniquely with -cut method based on the traditional interval arithmetic. However, it can be solved with an application of horizontal MFs (63) which can be constructed on the basis of the general formula (12). Consider It can be easily noticed that formula (64) is fourdimensional: = ( , , ). It cannot be expressed exactly in the 2D space, which is suggested by the extension principle and various versions of the fuzzy arithmetic. However, it can be visualised in a form of the 3D projection on subspaces, for example, on the subspace × × (variable is equivalent to variable because of the transformation = 4 + 2 ) ( Figure 19). Figure 19 shows the support of the membership function of solution (64) being the -cut of this function on the level = 0. It can be noticed that this support does not have a shape of a rectangle but of a parallelogram, which is shown additionally in Figure 20. Figure 20 shows that the solution of the fuzzy equation + = is not 1-dimensional one and therefore it is not possible to express it in a form of the interval [ , ] which is usually done by these versions of fuzzy arithmetics that are based on the classical IA. Moore's IA provides us with two possible solutions: (65) and (66). The first result (65) One can easily convince himself that this result is not the precise solution of the considered equation. It is only information about the greatest span [8,12] of the precise solution (64), which has a form of a parallelogram ( Figure 20). As can be seen in Figure 20 A simple analysis of this equation gives the conclusion that the minimal value occurs for = 1 and = 0. This

Conclusions
The paper presents the new concept of horizontal MFs which were used for the addition as the exemplary operation on fuzzy numbers. The application of such functions in combination with the RDM arithmetic enables multidimensional approach to operations on fuzzy numbers and thereby it allows for removing some drawbacks and weaknesses of the fuzzy arithmetic. The use of horizontal MFs considerably facilitates calculations because now uncertain values can be inserted directly into equations without using the extension principle.
Additionally, the RDM arithmetic enables taking into account correlations occurring between arguments of mathematical operations which is not possible using onedimensional approach offered by classic FA methods.