Abstract
We calculate Zadeh’s max-min composition operators for two 3-dimensional triangular fuzzy numbers. We prove that if the 3-dimensional result is limited to 2 dimensions, it is the same as the 2-dimensional result, which is shown as a graph. Since a 3-dimensional graph cannot be drawn, the value of the membership function is expressed with color density. We cut a 3-dimensional triangular fuzzy number by a perpendicular plane passing a vertex, and consider the cut plane as a domain. The value of the membership function for each point on the cut plane is also expressed with color density. The graph expressing the value of the membership function, defined in the plane as a 3-dimensional graph using the -axis value instead of expressing with color density, is consistent with the results in the 2-dimensional case.
1. Introduction
Many results exist for Zadeh’s max-min composition operators. The results for triangular fuzzy numbers are well known [1–5]. We have generalized 1-dimensional triangular fuzzy number to 2-dimensional triangular fuzzy number and calculated Zadeh’s max-min composition operators for 2-dimensional fuzzy numbers [6]. In the 1-dimensional case, Zadeh’s max-min composition operator can be calculated using -cuts. By defining parametric operations between two region valued -cuts, we obtained parametric operations for two triangular fuzzy numbers defined on [7]. In the case of 3-dimensional fuzzy numbers, the -cuts are subsets of . By defining parametric operations between two ellipsoids including interior valued -cuts, we calculated Zadeh’s max-min composition operators for two 3-dimensional fuzzy numbers [8]. This will help facilitate further study of triangular fuzzy matrices [1, 3, 9].
In this paper, we prove that Zadeh’s max-min composition operators for two 3-dimensional triangular fuzzy numbers on constitutes the generalization of Zadeh’s max-min compositions for two 2-dimensional triangular fuzzy numbers on . In addition, by limiting the graph of the 3-dimensional result to the 2-dimensional case, we prove that the result expressed as a graph is consistent with the graph of the 2-dimensional result.
2. Preliminaries
We define -cut and -set of the fuzzy set on with the membership function .
Definition 1. (see [10]). An -cut of the fuzzy number is defined by if and . For , the set is said to be the -set of the fuzzy set , is the boundary of , and .
Following Zadeh, Dubois, and Prade, the extension principle is defined as follows.
Definition 2. (see [11–13]). The extended addition , extended subtraction , extended multiplication , and extended division are fuzzy sets with membership functions as follows. For all and ,
Definition 3. (see [7]). A fuzzy set with a membership function:where is called the 2-dimensional triangular fuzzy number and denoted by .
Let . Then, the -cut of a 2-dimensional triangular fuzzy number is an interior of an ellipse in an -plane, including the boundary
Theorem 1. (see [7]). Let be a continuous convex fuzzy number defined on , and be the -set of . Then, for all , there exist continuous functions and defined on such that
If is a continuous convex fuzzy number defined on , then the -cut is a closed convex subset in .
Definition 4. (see [7]). Let and be convex fuzzy numbers defined on andbe the -sets of and , respectively. For , we define that the parametric addition , parametric subtraction , parametric multiplication , and parametric division of two fuzzy numbers and are fuzzy numbers that have their -sets as follows:(1): .(2): where(3): .(4): whereFor and , and , where .
Theorem 2. (see [7]). Let and be two 2-dimensional triangular fuzzy numbers. Then, we have the following:(1).(2).(3), where(4), where
Therefore, and become 2-dimensional triangular fuzzy numbers, but and are not 2-dimensional triangular fuzzy numbers.
Theorem 3. (see [7]). Parametric operations on in Definition 4 are the generalization of Zadeh’s extension principles on .
3. Zadeh’s Max-Min Composition Operator for 3-Dimensional Triangular Fuzzy Numbers
We define parametric operations between two 3-dimensional triangular fuzzy numbers and calculate Zadeh’s max-min composition operators for 3-dimensional triangular fuzzy numbers.
Definition 5. A fuzzy set with a membership function:where is called the 3-dimensional triangular fuzzy number and denoted by .
Note that is a cone in , but we cannot know the shape of in . The -cut of a 3-dimensional triangular fuzzy number is the following set:
Definition 6. A 3-dimensional fuzzy number defined on is called a convex fuzzy number if, for all , the -cutsare convex subsets in .
Theorem 4. (see [8]). Let be a continuous convex fuzzy number defined on , and be the -set of . Then, for all , there exist continuous functions , and , such that
Definition 7. Let and be two continuous convex fuzzy numbers defined on andbe the -sets of and , respectively. For , we define that the parametric addition, parametric subtraction, parametric multiplication, and parametric division of two fuzzy numbers and are fuzzy numbers that have their -sets as follows:(1)Parametric addition, :(2)Parametric subtraction, :(3)Parametric multiplication, :(4)Parametric division, :For and , , and , where .
Theorem 5. (see [8]). Let and be two 3-dimensional triangular fuzzy numbers. Then, we have the following:(1).(2).(3), where(4), where
Therefore, and become 3-dimensional triangular fuzzy numbers, but and are not 3-dimensional triangular fuzzy numbers.
Theorem 6. Parametric operations on in Definition 7 are the generalization of parametric operations on in Definition 4, which are the generalization of Zadeh’s max-min composition operations on .
Proof. Consider two 3-dimensional triangular fuzzy numbers and . By Theorem 5,(1).(2).(3), where(4), whereThe intersections of these 3-dimensional triangular fuzzy numbers and are as follows:(1): note that If , Thus, the intersection is the 2-dimensional triangular fuzzy number .(2): note that If , Thus, the intersection is the 2-dimensional triangular fuzzy number .(3): if , . Thus, the intersection is a fuzzy number on with -cut, not -set, , where(4): if . Thus, the intersection is a fuzzy number on with -cut, not -set, , whereOn the contrary, the intersection of 3-dimensional triangular fuzzy number and is , and the intersection of 3-dimensional triangular fuzzy number and is . For two 2-dimensional triangular fuzzy numbers and , the following results for parametric operations on have been proven [7]:The proof is complete.
4. Conclusion
In this section, by limiting the graph of the 3-dimensional result to the 2-dimensional case, we prove that the result expressed as a graph is consistent with the graph of the 2-dimensional result.
For two 3-dimensional triangular fuzzy numbers and , the values of membership function are expressed using color density in Figures 1 and 2 , respectively. Zadeh’s max-min operations expanded to 3 dimensions are expressed in Figures 3–6 as graphs.
The graph of and for is Figures 5 and 6, respectively. In Figures 7 and 8 , the graph for and is the union of , respectively.
We cut the graph of , and by a perpendicular plane passing a vertex (Figures 9–12 ). The values of the membership function of each point on the cross section are expressed with color density.
We cut two 3-dimensional triangular fuzzy numbers and by a perpendicular plane passing a vertex and considered the cut plane as a domain. The value of the membership function for each point on the cut plane is expressed with color density. Figures 13 and 14 are the graphs expressing the value of the membership function defined in the plane as a 3-dimensional graph using the -axis value, instead of expressing with color density. In this way, restricting the three-dimensional results (Figures 3–6) to two dimensions, we express the value of the membership function in a three-dimensional graph without using color density (Figures 15–18 ).
(a)
(b)
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Consider the 2-dimensional triangular fuzzy numbers and represented by Figures 13 and 14, respectively. The result of Zadeh’s max-min operators expanded to 2 dimensions can be found in [7]. When the results are graphed, it becomes clear that the graphs are identical to graphs in Figures 15–18. The commands in Mathematica for ten figures on page 7 and 8 are provided in appendix. Figure 19 summarizes what has been mentioned. In conclusion, therefore, the 3-dimensional result is naturally expanded to the 3 dimensions while satisfying the 2-dimensional result as it is, and it can be applied in this form. This paper will further study the extended applications of triangular fuzzy numbers [14–18].
Appendix
(i)(: A^3 :) DensityPlot3D[1 − Sqrt[(x − 3)^2/6 + (y − 5)^2/8 + (z − 7)^2/4],{x, y, z} in Ellipsoid[{3, 5, 7}, {Sqrt[6], Sqrt[8], 2}], PlotPoints->100, ColorFunction->“SunsetColors”, OpacityFunction->0.05, BoxRatios->{Sqrt[6], Sqrt[8], 2}, PlotLegends->Automatic](ii)(: (A + B)^3 :) DensityPlot3D[1 − Sqrt[(x − 9)^2/10+(y − 13)^2/14+(z − 12)^2/7],{x, y, z}in Ellipsoid[{9, 13, 12}, {Sqrt[10], Sqrt[14], Sqrt[7]}], PlotPoints->100, ColorFunction->“SunsetColors”, OpacityFunction->0.05, BoxRatios->{Sqrt[10], Sqrt[14], Sqrt[7]}, PlotLegends- > Automatic](iii)(: (AB)1/2 :) ParametricPlot3D[{18 + 24Cos[s] + 6(Cos[s])^2, 40 + 47 Sin[s]Cos[t] + 12 (Sin[s])^2(Cos[t])^2, 35 + 41/2 Sin[s]Sin[t] + 3 (Sin[s])^2(Sin[t])^2}, {s, 0, 2Pi}, {t, −Pi/2, Pi/2}, Axes->True](iv)(: (A/B)^3 :) g[a_]\coloneq ParametricPlot3D[{(3 + 6(1 − a)Cos[s])/(6–4(1 − a)Cos[s]), (5 + 8(1 − a) Sin[s]Cos[t])/(8 − 6(1 − a) Sin[s]Cos[t]),(7 + 4(1 − a) Sin[s]Sin[t])/(5 − 3(1 − a) Sin[s]Sin[t]) }, {s, 0, 2Pi}, {t, −Pi/2, Pi/2},PlotStyle->Directive[RGBColor[0.2,0.5 + a/2,0.5 + a/2],Opacity[0.3]], BoxRatios->{1, 1, 1}]; tg = Table[g[i],{i, 0, 1.0, 0.01}]; Show[tg](v)(: A/2 :) reg1 = ImplicitRegion[0{\textless}=(x − 3)^2/6 + (y − 5)^2/8 + (z − 7)^2/4{\textless} = 1 && z{\textless} = 7, {x, y, z}]; DensityPlot3D[1 − Sqrt[(x − 3)^2/6 + ((y − 5)^2/8 + ((z − 7)^2/4], {x, y, z}in reg1, PlotPoints->100, ColorFunction->“SunsetColors”, Opacity Function->1, BoxRatios->{Sqrt[4], Sqrt[5], Sqrt[6]/2}, PlotLegends->Automatic] (vi)(: (A + B)/2 :) reg1 = ImplicitRegion[0{\textless}=(x − 9)^2/10 + (y − 13)^2/14 + (z − 12)^2/7{\textless} = 1&&z{\textless} = 12, {x, y, z}]; DensityPlot3D[1 − Sqrt[(x − 9)^2/10 + (y − 13)^2/14 + (z − 12)^2/7], {x, y, z}in reg1, PlotPoints->100, ColorFunction->“SunsetColors”, OpacityFunction->1, BoxRatios->{Sqrt[10], Sqrt[14], Sqrt[7]}, PlotLegends->Automatic](vii)(: A^2 :) Plot3D[1 − Sqrt[(x − 3)^2/6 + (y − 5)^2/8], {x, y} in Ellipsoid[{3, 5}, {Sqrt[6], Sqrt[8]}], PlotPoints->50, ColorFunction->“SunsetColors”, BoxRatios->{Sqrt[6], Sqrt[8], 1}, PlotLegends->Automatic](viii)(: (A + B)^2 :) Plot3D[1 − Sqrt[(x − 9)^2/10 + (y − 13)^2/14], {x, y} in Ellipsoid[{9, 13}, {Sqrt[10], Sqrt[14]}], PlotPoints->50, ColorFunction->“SunsetColors”, BoxRatios->{Sqrt[6], Sqrt[8], 1}, PlotLegends- > Automatic](ix)(: (AB)^2 :) g[a_]\coloneq ParametricPlot3D[{18 + 48(1 − a)Cos[s]+24(1 − a)^2(Cos[s])^2, 40 + 94(1 − a)Sin[s]+48(1 − a)^2(Sin[s])^2, a}, {s, 0, 2Pi}, PlotStyle->Directive[RGBColor[0.3, 0.5 + a, 0.5 + a], Opacity[0.3]], BoxRatios->{1, 1, 1}]; tg = Table[g[i],{i, 0, 1.0, 0.001}]; Show[tg](x)(: (A/B)^2 :) g[a_]\coloneq ParametricPlot3D[{(3 + 3Cos[s])/(6 − 2Cos[s]), (5 + 4 Sin[s])/(8 − 3/2 Sin[s]), a},{s, 0, 2Pi}, PlotStyle->Directive[RGBColor[0.3, 0.5 + a, 0.5 + a], Opacity[0.3]], BoxRatios->{1, 1, 1}]; tg = Table[g[i], {i, 0, 1.0, 0.001}]; Show[tg]Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This research was supported by the 2021 scientific promotion program funded by Jeju National University.