Some properties of Graded Mean Integration representation of L-R type fuzzy numbers.Abstract Recently, fuzzy fuzz·y adj. fuzz·i·er, fuzz·i·est 1. Covered with fuzz. 2. Of or resembling fuzz. 3. Not clear; indistinct: a fuzzy recollection of past events. 4. numbers are largely applied on data analysis, artificial intelligence, and decision making. Using extension principle as fuzzy arithmetic principle, the multiplication multiplication, fundamental operation in arithmetic and algebra. Multiplication by a whole number can be interpreted as successive addition. For example, a number N multiplied by 3 is N + N + N. of fuzzy numbers is a fuzzy number with very complicated membership function. For example, the multiplication of two fuzzy numbers, each with a trapezoidal membership function, is a fuzzy number with a two-sided parabolic par·a·bol·ic also par·a·bol·i·cal adj. 1. Of or similar to a parable. 2. Of or having the form of a parabola or paraboloid. drum-like Adj. 1. drum-like - shaped in a form resembling a drum drum-shaped formed - having or given a form or shape shape membership function. The multiplication of two fuzzy numbers, one with a trapezoidal membership function, the other with a two-sided parabolic drum-like shape membership function, becomes a fuzzy number with a two-sided cubic shape membership function. It gets more difficult to operate and to represent the fuzzy numbers when the original numbers become more complicated. One of the applications of the Fuzzy Multi-Criteria Decision Making is to evaluate alternatives and often uses the multiplication of at least two fuzzy numbers. The representation of the results is sometimes difficult to make. In this paper, we apply the mathematical software Mathematical software The collection of computer programs that can solve equations or perform mathematical manipulations. The developing of mathematical equations that describe a process is called mathematical modeling. Maple 7 to derive de·rive v. 1. To obtain or receive from a source. 2. To produce or obtain a chemical compound from another substance by chemical reaction. some general formulas of the representation of fuzzy numbers and the multiplication of fuzzy numbers under Extension Principle by using Graded Mean Integration Representation method. It is helpful to reduce the trouble and tediousness te·di·ous adj. 1. Tiresome by reason of length, slowness, or dullness; boring. See Synonyms at boring. 2. Obsolete Moving or progressing very slowly. of the massive operation of the original membership function and makes the application of Fuzzy Multi-Criteria Decision Making more convenient and acceptable. Keywords Keywords are the words that are used to reveal the internal structure of an author's reasoning. While they are used primarily for rhetoric, they are also used in a strictly grammatical sense for structural composition, reasoning, and comprehension. and Phrases: Extension Principle, Graded Mean Integration Representation, L-R L-R Left to Right L-R Lenoir-Rhyne College (Hickory, North Carolina) type fuzzy number. 1. Introduction Many fuzzy representation methods have been proposed in the literature. In 1970s, Jain Jain also Jai·na n. A believer or follower of Jainism. [Hindi jaina, from Sanskrit jaina-, relating to the saints, from jina [18] used the concept of maximizing max·i·mize tr.v. max·i·mized, max·i·miz·ing, max·i·miz·es 1. To increase or make as great as possible: set to defuzzify fuzzy numbers. In 1980s, Adamo (database) ADAMO - A data management system written at CERN, based on the Entity-Relationship model. [1] and Campos Campos (käm`p  s), city (1996 pop. 391,299), Rio de Janeiro state, SE Brazil, on the Paraíba River near its mouth. et al. [3]
discussed [alpha]-preference of fuzzy number for representing fuzzy
numbers. Yager Ya´gern. 1. (Mil.) In the German army, one belonging to a body of light infantry armed with rifles, resembling the chasseur of the French army. [27] discussed two indexes of fuzzy number that included center of gravity Gravity The gravitational attraction at the surface of a planet or other celestial body. The quantity g is often referred to simply as “gravity’’ or “the force of gravity’’ of Earth, both of which are incorrect. of fuzzy numbers and mean values. Chen [9] used total utility value which was based on Maximizing set and Minimizing set to defuzzify the fuzzy numbers. Lee et al. [23] proposed both the mean and dispersion dispersion, in chemistry dispersion, in chemistry, mixture in which fine particles of one substance are scattered throughout another substance. A dispersion is classed as a suspension, colloid, or solution. of alternatives, and give two groups of indexes based on the uniform and the proportional proportional values expressed as a proportion of the total number of values in a series. proportional dwarf the patient is a miniature without disproportionate reductions or enlargements of body parts. probability distributions Many probability distributions are so important in theory or applications that they have been given specific names. Discrete distributions With finite support
American writer and editor known especially for his caustic, polysyllabic wit. [2] introduced a method of the interval interval, in music, the difference in pitch between two tones. Intervals may be measured acoustically in terms of their vibration numbers. They are more generally named according to the number of steps they contain in the diatonic scale of the piano; e.g. of [alpha]-cuts of fuzzy numbers. Tseng et al. [26] designed an algorithm algorithm (ăl`gərĭth'əm) or algorism (–rĭz'əm) [for Al-Khowarizmi], a clearly defined procedure for obtaining the solution to a general type of problem, often numerical. used to represent fuzzy numbers. In 1990s, Kim Kim orphan wanders streets of India with lama. [Br. Lit.: Kim] See : Adventurousness et al. [22] suggested a method combined by maximizing possibility and minimizing possibility with an index of optimism Optimism See also Hope. Bontemps, Roger personification of cheery contentment. [Fr. Lit.: “Roger Bontemps” in Walsh Modern, 66] Candide beset by inconceivable misfortunes, hero indifferently shrugs them off. [Fr. k in [0, 1] for comparison of fuzzy numbers. McCahon et al. [25] proposed the proportion of fuzzy number ranking procedure measures the fuzzy number under comparison with the fuzzy ideas of max and min. Liou et al. [24] proposed a total integral value generated by the left and right integral value of fuzzy number for representing the fuzzy number. Chang Chang (chăng) or Yangtze (yăng`sē`, yäng`dzŭ`), Mandarin Chang Jiang, longest river of China and of Asia, c.3,880 mi (6,245 km) long, rising in the Tibetan highlands, SW Qinghai prov. and Lee's [4] emphasis is placed on the classification and comparison of some ranking methods. Heilpern Heilpern is a variation of the Jewish surname Heilprin and may refer to:
[18] presented the notions of the expected value Expected value The weighted average of a probability distribution. Also known as the mean value. based on the lower and upper expected values of fuzzy number. Chen [12] used the averages of four vertex values of trapezoidal fuzzy numbers to represent trapezoidal fuzzy numbers. Cheng [13] proposed a ranking function by a distance index based on the original point and the centroid centroid In geometry, the centre of mass of a two-dimensional figure or three-dimensional solid. Thus the centroid of a two-dimensional figure represents the point at which it could be balanced if it were cut out of, for example, sheet metal. point ([x.sub.0], [y.sub.0]) of fuzzy number. Delgado Delgado is a surname, and may refer to:
adj. 1. Involving an entire organ, as when an epileptic seizure involves all parts of the brain. 2. Not specifically adapted to a particular environment or function; not specialized. 3. fuzzy number. In this paper, we use Graded Mean Integration Representation to represent fuzzy numbers because of its simplicity Simplicity is the property, condition, or quality of being simple or un-combined. It often denotes beauty, purity or clarity. Simple things are usually easier to explain and understand than complicated ones. Simplicity can mean freedom from hardship, effort or confusion. and accuracy. The comparison of GMIRM with some described methods, please see [5,8]. In section 2, we describe Chen and Hsieh's GMIRM of L-R type fuzzy numbers, especially focus on trapezoidal membership function fuzzy numbers. By applying the mathematical software Maple 7, the representation of fuzzy numbers with kth order plane curve membership function is described in section 3, the representation of the multiplication of two fuzzy numbers is computed in section 4, and the representation of the multiplication of three fuzzy numbers with trapezoidal membership functions is derived de·rive v. de·rived, de·riv·ing, de·rives v.tr. 1. To obtain or receive from a source. 2. in section 5. Finally, we give a brief conclusion in section 6. 2. The Graded Mean Integration Representation of L-R type Fuzzy Numbers In general, a generalized L-R type fuzzy number A can be described as any fuzzy subset fuzzy subset - In fuzzy logic, a fuzzy subset F of a set S is defined by a "membership function" which gives the degree of membership of each element of S belonging to F. of the real line R whose membership function [[mu].sub.A] satisfies the following conditions. (1) [[mu].sub.A] is a continuous mapping from R to the closed interval closed interval n. A set of numbers consisting of all the numbers between a pair of given numbers and including the endpoints. closed interval [0, 1], (2) [[mu].sub.A](x)=0, -[infinity infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. ] < x [less than or equal to] c, (3) [[mu].sub.A](x)=L(x) is strictly increasing on [c, a], (4) [[mu].sub.A](x)=w, a [less than or equal to] x [less than or equal to] b, (5) [[mu].sub.A](x)=R(x) is strictly decreasing on [b, d], (6) [[mu].sub.A](x)=0, d [less than or equal to] x < [infinity], where 0<w[less than or equal to]1, and a, b, c, d are real numbers. This generalized L-R type fuzzy number is denoted as A = (c, a, b, d; w)[.sub.LR]. When w=1, we denote de·note tr.v. de·not·ed, de·not·ing, de·notes 1. To mark; indicate: a frown that denoted increasing impatience. 2. A = (c, a, b, d)[.sub.LR]. For example, when one says that something is "around 1000" with 90% of confidence, he can give w=0.9. Let [L.sup.-1] and [R.sup.-1] be the inverse functions inverse function Mathematical function that undoes the effect of another function. For example, the inverse function of the formula that converts Celsius temperature to Fahrenheit temperature is the formula that converts Fahrenheit to Celsius. of the functions L and R, respectively. The graded mean h-level value of generalized L-R type fuzzy number A=(c, a, b, d; w)[.sub.LR] is h[[L.sup.-1](h) + [R.sup.-1](h)]/2 as shown in Figure 1. Now, we define the representation of a generalized L-R type fuzzy number based on the integration value of graded mean h-levels as follow. Definition 1. Let A = (c, a, b, d; w)[.sub.LR] be a generalized L-R type fuzzy number, [L.sup.-1] and [R.sup.-1] be the inverse functions of the functions L and R respectively. Then the Graded Mean Integration Representation (GMIR GMIR GPO Marc Internet Resources GMIR Global Material Integration and Reporting ) of A is, P(A)= [[integral].sub.0.sup.w] h ([lambda] [L.sup.-1](h) + (1 - [lambda])[R.sup.-1](h))dh/ [[integral].sub.0.sup.w] h dh, (1) where h is between 0 and w, 0 < w [less than or equal to] 1, 0 [less than or equal to] [lambda] [less than or equal to] 1. We call P(A) as graded [lambda]-preference integration representation of fuzzy number A. Remark 1. When [lambda] = 1/2 in Definition 1, we call P(A) a GMIR of A. The value of [lambda] depends on the preference of the decision maker. Usually we choose [lambda] = 1/2, since it does not bias to left or right. Remark 2. When [L.sup.-1](h) or [R.sup.-1](h) does not exist, or [[integral].sub.0.sup.w] h[[[L.sup.-1](h) + [R.sup.-1] (h)]/2] dh cannot integrate, we can divide [0, w] into n equal intervals, and let P(A) = [[n.summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument) over (i=1)] w(i/n)([L.sup.-1](w [i/n])+[R.sup.-1](w [i/n])) / 2]/ [n.summation over (i=1)] w(i/n). Remark 3. From definition of generalized L-R type fuzzy number in Definition 1, we have -[infinity] < c [less than or equal to] a [less than or equal to] b [less than or equal to] d < [infinity]. 3. The GMIR of the Kth Order Plane Curve Fuzzy Numbers In this paper, generalized trapezoidal fuzzy number and generalized triangular fuzzy number are denoted as (c, a, b, d; w) and (c, a, d; w) respectively. Now, we discuss the general formulas for the representation of the kth order plane curve fuzzy numbers. It is possible to extend the trapezoidal membership function to the kth order plane curve membership function as, [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE re·pro·duce v. re·pro·duced, re·pro·duc·ing, re·pro·duc·es v.tr. 1. To produce a counterpart, image, or copy of. 2. Biology To generate (offspring) by sexual or asexual means. IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ]. (2) Where c, a, b, d, are positive real numbers, 0<w[less than or equal to]1 and k >0. For conveniences, we denote this type of fuzzy number as A=(c, a, b, d; w)[.sup.k]. Since, L(x) = w ([x - c]/[a - c])[.sup.k], c [less than or equal to] x [less than or equal to] a, R(x) = w ([x - d]/[b - d])[.sup.k], b [less than or equal to] x [less than or equal to] d, then, [L.sup.-1](h) = c + (a - c)(h/w)[.sup.1/k], 0 [less than or equal to] h [less than or equal to] w, [R.sup.-1](h) = d - (d - b)(h/w)[.sup.1/k], 0 [less than or equal to] h [less than or equal to] w. The graded [lambda]-preference integration representation of A is, P(A) = [[integral].sub.0.sup.w] h (L-1(h)+(1-)R-1(h)) dh / [[integral].sub.0.sup.w] h dh = [[integral].sub.0.sup.w] h ([[lambda](c + (a - c)(h/w)[.sup.1/k]) + (1-[lambda])(d - (d - b)(h/w)[.sup.1/k])]/2)dh / [[integral].sub.0.sup.w] hdh = [lambda] c+(1-[lambda])d +[[lambda] (a-c)+(1-[lambda])(b-d)](2k)/(2k+1). (3) Chen and Hsieh [7,8] is a special case of (3) with k=1. They have completed the result of property 1 as following. Property 1. Let A=(c, a, b, d; w)[.sup.1] be a generalized trapezoidal fuzzy number with 0<w [less than or equal to] 1, and c, a, b, d, are positive real numbers. Under Extension Principle of fuzzy numbers, the GMIR of A is, P(A)=[[lambda] (c+2a)+(1-[lambda])(2b+d)]/3. Remark 4. When [lambda] = 1/2, P(A) = [c + 2a + 2b + d]/6. Remark 5. The generalized triangular fuzzy number X=(c, a, d; w)[.sup.1] is a special case of generalized trapezoidal fuzzy number with b=a. Therefore, P(X) = [c + 4 a + d]/6. 3.1 The GMIR of the Case of (c, a, b, d; w)[.sup.2] Let k=2 in (3), We have Property 2. Let A=(c, a, b, d; w)[.sup.2] be a fuzzy number with 0<w[less than or equal to] 1, and c, a, b, d, are positive real numbers. Under Extension Principle of fuzzy numbers, the GMIR of A is, P(A)= [[lambda] (c+4a)+(1-[lambda])(4b+d)]/5. Remark 6. When [lambda] = 1/2, P(A) = [c + 4a + 4b + d]/10. Remark 7. The fuzzy number X=(c, a, d; w)[.sup.2] is a special case of fuzzy number A=(c, a, b, d; w)[.sup.2] with b=a. Therefore, P(X) = [c + 8a + d]/10. 3.2 The GMIR of the Case of (c, a, b, d; w)[.sup.1/2] Let k=1/2 in (3), we get Property 3. Let A=(c, a, b, d; w)[.sup.1/2] be a fuzzy number with 0<w[less than or equal to] 1, and c, a, b, d, are positive real numbers. Under Extension Principle of fuzzy numbers, the GMIR of A is, P (A)= [[lambda] (c+a)+(1-[lambda])(b+d)]/2. Remark 8. When [lambda] = 1/2, P(A) = [c + a + b + d]/4. Remark 9. The fuzzy number Y=(c, a, d; w)[.sup.1/2] is a special case of fuzzy number A=(c, a, b, d; w)[.sup.1/2] with b=a. Therefore, P(Y) = [c + 2a + d]/4. 3.3 Comparison of the Three Cases (k=1, 1/2 and 2) For [lambda] = 1/2, Suppose the representation value P(A) is a function of a, b, c, d and k. Then from (3), we can easy to get P(A)=P(c, a, b, d; k)=(c+d)/2+{k[a-c-(d-b)]}/(2k+1) (4) Now, we can use partial derivative partial derivative In differential calculus, the derivative of a function of several variables with respect to change in just one of its variables. Partial derivatives are useful in analyzing surfaces for maximum and minimum points and give rise to partial differential with respect to k to find the properties of function P(a, b, c, d; k). That is [partial derivative]P/[partial derivative]k = [a - c - (d - b)]/[(2k+1)[.sup.2]] (5) From (5), since (2k+1)[.sup.2] always positive for any real k, it shows that the monotonicity of P(A) does not depend on k, but depend on the length of the left span and the right span of the L-R type fuzzy number. It tells us that (a) When fuzzy number biased to right (see figure 3), then the left span (a-c) will bigger than the right span (d-b), or (a-c) > (d-b). Therefore, from formula (5), [partial derivative]P/[partial derivative]k > 0, [for all] k [not equal to] -1/2. Obviously, P(c, a, b, d; 2) > P(c, a, b, d; 1) > P(c, a, b, d; 1/2). These relation shows that the representation value of fuzzy number (c, a, b, d; 2) is the largest, and the representation value of fuzzy number (c, a, b, d; 1/2) is the smallest between these three type fuzzy numbers. (b) When fuzzy number biased to left (see figure (4), then the left span (a-c) will smaller than the right span (d-b), or (a-c)<(d-b). Therefore, from formula (5), [partial derivative]P/[partial derivative]k < 0, [for all] k [not equal to] -1/2. Obviously, P(c, a, b, d; 2) < P(c, a, b, d; 1) < P(c, a, b, d; 1/2). (c) When fuzzy number is symmetry symmetry, generally speaking, a balance or correspondence between various parts of an object; the term symmetry is used both in the arts and in the sciences. (see figure 2) that is (a-c)=(d-b). Therefore, from formula (5), [partial derivative]P/[partial derivative]k = 0, [for all] k [not equal to] -1/2.it shows that P(a, b, c, d; k) is constant with respect to k. Obviously, P(c, a, b, d; 2) = P(c, a, b, d; 1) = P(c, a, b, d; 1/2). 4. The GMIR of The Multiplication of Two Fuzzy Numbers 4.1 The GMIR of the Multiplication of Two Trapezoidal Fuzzy Numbers Suppose A = (c, a, b, d), B = (q, o, p, r) are two fuzzy numbers, each with a trapezoidal membership function, then the multiplication of A and B, X = A [cross product] B, is a fuzzy number with two-sided parabolic drum-like shape(shown in fig fig, name for members of the genus Ficus of the family Moraceae (mulberry family). This large genus contains some 800 species of widely varied tropical vines (some of which are epiphytic); shrubs; and trees, including the banyan, the peepul, or bo tree, and . 5), the membership function [mu] is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6) where c, a, b, d, o, p, q, r are all positive real numbers. We use X = (l, e, g, g', e', l') to denote the fuzzy number with a two-sided parabolic drum-like shape membership function, where l, g, g', and l' are four vertexes. We can rewrite re·write v. re·wrote , re·writ·ten , re·writ·ing, re·writes v.tr. 1. To write again, especially in a different or improved form; revise. 2. the membership function as followed. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Where, e = 1/(a - c)(o - q), e' = 1/(b - d)(p - r), f = 1/2[e((a - c)q + (o - q)c)], f' = -1/2[e'((b - d)r + (p - r)d)]. Then the graded [lambda]-preference integration representation of X is, P (X) = [[lambda] (3ao+co+aq+cq) + (1-[lambda])(3bp+dp+br+dr)]/6. Property 4. Let A = (c, a, b, d), B = (q, o, p, r) be two trapezoidal fuzzy numbers, and c, a, b, d, q, o, p, r be positive real numbers. Let X be the multiplication of A and B, under Extension Principle of fuzzy numbers, the GMIR of X is, P(X) = [[lambda](3ao+co+aq+cq) + (1-[lambda])(3bp+dp+br+dr)]/6. Remark 10. When [lambda] =1/2, P(X) = [3ao + aq + co + cq + 3bp + br + dp + dr]/12. Remark 11. The triangular fuzzy numbers A = (c, a, d) and B = (q, o, r) are special cases of trapezoidal fuzzy numbers with b=a, p=o. The GMIR of the multiplication of these two fuzzy numbers is, P (A [cross product] B) = [aq + co + cq + 6 ao + ar + do + dr]/12. 4.2 The GMIR of the Linear Combination of Two-Sided Parabolic Drum-Like Shape Membership Functionfuzzy Numbers Suppose [A.sub.i]= ([c.sub.i], [a.sub.i], [b.sub.i], [d.sub.i],) and [B.sub.i]= ([q.sub.i], [o.sub.i], [p.sub.i], [r.sub.i]) i = 1,2,..., n are n fuzzy numbers, each with a trapezoidal membership function, and 0 [less than or equal to] [c.sub.i], 0 [less than or equal to] [q.sub.i], i = 1,2,..., n, then the fuzzy linear function, X = [A.sub.1] [cross product] [B.sub.1] + [A.sub.2] [cross product] [B.sub.2] + ... + [A.sub.n] [cross product] [B.sub.n], is a fuzzy number with two-sided parabolic drum-like shape membership function, the membership function [mu] is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Where, e = 1 / [n.summation over (i=1)][([a.sub.i] - [c.sub.i])([o.sub.i] - [q.sub.i])], e' = 1 / [n.summation over (i=1)][([b.sub.i] - [d.sub.i])([p.sub.i] - [r.sub.i])], f = 1/2[e [n.summation over (i=1)](([a.sub.i] - [c.sub.i])[q.sub.i] + ([o.sub.i] - [q.sub.i])[c.sub.i])], f' = -1 / 2[e' [n.summation over (i=1)](([b.sub.i] - [d.sub.i])[r.sub.i] + ([p.sub.i] - [r.sub.i])[d.sub.i])]. Then the graded [lambda]-preference integration representation of X is, P (X) = [[lambda] (3 [n.summation over (i=1)][a.sub.i][o.sub.i] + [n.summation over (i=1)][a.sub.i][q.sub.i] + [n.summation over (i = 1)] [c.sub.i][o.sub.i] + [n.summation over (i = 1)] [c.sub.i][q.sub.i]) +(1-[lambda])(3 [n.summation over (i=1)][b.sub.i][p.sub.i] + [n.summation over (i=1)][b.sub.i][r.sub.i] + [n.summation over (i = 1)] [d.sub.i][p.sub.i] + [n.summation over (i = 1)] [d.sub.i][r.sub.i])]/6. Property 5. [A.sub.i]= ([c.sub.i], [a.sub.i], [b.sub.i], [d.sub.i],) and [B.sub.i]= ([q.sub.i], [o.sub.i], [p.sub.i], [r.sub.i]) i = 1, 2,..., n are n trapezoidal fuzzy numbers, and 0 [less than or equal to] [c.sub.i], 0 [less than or equal to] [q.sub.i]. Let X be the linear combination of the multiplication of A's and B's (X = [A.sub.1] [cross product] [B.sub.1] + [A.sub.2] [cross product] [B.sub.2]+ ... + [A.sub.n] [cross product] [B.sub.n]), under Extension Principle of fuzzy numbers, the GMIR of X is, P(X) = [[lambda] (3 [n.summation over (i=1)][a.sub.i][o.sub.i] + [n.summation over (i=1)][a.sub.i][q.sub.i] + [n.summation over (i = 1)] [c.sub.i][o.sub.i] + [n.summation over (i = 1)] [c.sub.i][q.sub.i]) +(1-[lambda])(3 [n.summation over (i=1)][b.sub.i][p.sub.i] + [n.summation over (i=1)][b.sub.i][r.sub.i] + [n.summation over (i = 1)] [d.sub.i][p.sub.i] + [n.summation over (i = 1)] [d.sub.i][r.sub.i])]/6. Remark 12. When [lambda] = 1/2, P(X) = [3 [n.summation over (i = 1)] [a.sub.i][o.sub.i] + [n.summation over (i = 1)] [a.sub.i][q.sub.i] + [n.summation over (i = 1)] [c.sub.i][o.sub.i] + [n.summation over (i = 1)] [c.sub.i][q.sub.i] + 3 [n.summation over (i = 1)] [b.sub.i][p.sub.i] + [n.summation over (i - 1)] [b.sub.i][r.sub.i] + [n.summation over (i = 1)] [d.sub.i][p.sub.i] + [n.summation over (i = 1)] [d.sub.i][r.sub.i]]/12. Remark 13. The triangular fuzzy numbers [A.sub.i]= ([c.sub.i], [a.sub.i], [d.sub.i],), [B.sub.i] = ([q.sub.i], [o.sub.i], [r.sub.i]), i = 1, 2 ..., n, are special cases of trapezoidal fuzzy numbers with [b.sub.i]=[a.sub.i], [p.sub.i]=[o.sub.i]. The GMIR of the fuzzy linear function, X= [A.sub.1] [cross product] [B.sub.1]+ [A.sub.2] [cross product] [B.sub.2]+ ... + [A.sub.n] [cross product] [B.sub.n], is, P(X) = [[n.summation over (i = 1)] [a.sub.i][q.sub.i] + [n.summation over (i = 1)][c.sub.i][o.sub.i] + [n.summation over (i = 1)] [c.sub.i][q.sub.i] + 6 [n.summation over (i = 1)] [a.sub.i][o.sub.i] + [n.summation over (i = 1)] [a.sub.i][r.sub.i] + [n.summation over (i = 1)] [d.sub.i][o.sub.i] + [n.summation over (i = 1)] [d.sub.i][r.sub.i]]/12. 4.3 The GMIR of the multiplication of two fuzzy numbers, one with a trapezoidal membership function and the other with a two-sided parabolic drum-like shape membership function Suppose A = (c, a, b, d) is a trapezoidal fuzzy number, B = (l, e, g, g', e', l') is a fuzzy number with two-sided parabolic drum-like shape, the membership function [[mu].sub.B] is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. where, f=[(g - l)e - 1]/2, f'=[1 - (g'-l')e']/2. The multiplication of A and B, X= A [cross product] B, is two-sided cubic shape (shown in fig. 6). The cubic equation an equation in which the highest power of the unknown quantity is a cube. See also: Cubic has at least one real root. The membership function shall be between 0 and 1, and there is one and only one real root that is between 0 and 1 in our cases. The membership function [mu] is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Where, [rho] = [2fa - 2fc + c]/[a - c], [beta] = [elc/[a - c]] - [[eta][rho]/3] + [[2[[rho].sup.3]]/27], [delta] = [eta] - [[[rho].sup.2]/3], [eta] = [ela - elc + 2fc]/(a - c), [rho]' = [2f'b - 2f'd + d]/[b - d], [beta]' = -[[eta]'[rho]/3]+[[2[[rho]'.sup.3]]/27], [delta]' = [eta]'-[[[rho]'.sup.2]/3], [eta]' = [e'l'b - e'l'd - 2f'd]/(b - d). Then, the graded [lambda]-preference integration representation of X is, P (X) = [[lambda] (5 l ec+5gec+5 l ea+ 15gea-3a-2c)/e+ (1-[lambda])(5 l'e'd+5g'e'd+5 l'e'b+15g'e'b-3b-2d)/e']/30. Property 6. A = (c, a, b, d) is a trapezoidal fuzzy number, B = (l, e, g, g', e', l') is a fuzzy number with two-sided parabolic drum-like shape. Let X be the multiplication of two fuzzy numbers, X = A [cross product] B, under Extension Principle of fuzzy numbers, t he GMIR of X is, P(X)= [[lambda] (5 l ec + 5gec + 5 l ea + 15gea-3a-2c)/e+ (1 - [lambda])(5 l'e'd + 5g'e'd+5 l'e'b+15g'e'd-3b-2d)/e']/30. Remark 14. When [lambda] = 1/2, P(X)= [(5lec + 5gec + 5lea + 15gea - 3a - 2c)/e + (5l'e'd + 5g'e'd + 5l'e'b + 15g'e'b - 3b - 2d)/e']/60. Remark 15. A = (c, a, d) is a triangular fuzzy number, B = (l, e, g, e', l') is a fuzzy number with two-sided parabolic drum-like shape, both are special cases of trapezoidal fuzzy numbers with b=a, g'=g. The GMIR of the multiplication of two fuzzy numbers is, P(A [cross product] B) = [(5lec + 5gec + 5lea + 15gea - 3a - 2c)/e + (5l'e'd + 5ge'd + 5l'e'a + 15ge'a - 3a - 2d)/e']/60. 4.4 The GMIR of the Linear Combination of Fuzzy Numbers with a Two-sided Cubic Shape Membership Function Suppose [A.sub.i]= ([c.sub.i], [a.sub.i], [b.sub.i], [d.sub.i],) i = 1,2,..., m are m fuzzy numbers, each with a trapezoidal membership function, [B.sub.i] = ([l.sub.i], [e.sub.i], [g.sub.i], [g'.sub.i], [e'.sub.i], [l'.sub.i]), i = 1,2,..., m are m fuzzy numbers with each a two-sided parabolic drum-like shape membership function, then the fuzzy linear function, X = [m.summation over (i=1)][A.sub.i] [cross product] [B.sub.i], is two-sided cubic shape, the membership function [mu] is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Where, [rho] = [[m.summation over (i = 1)] (2[f.sub.i][a.sub.i] - 2[f.sub.i][c.sub.i] + [c.sub.i])]/[[m.summation over (i = 1)] ([a.sub.i] - [c.sub.i])], [beta] = [[[m.summation over (i = 1)] [e.sub.i][l.sub.i][c.sub.i]]/[[m.summation over (i = 1)] ([a.sub.i] - [c.sub.i])]]-[[eta][rho]/3]+ [[2[[rho].sup.3]]/27], [delta] = [eta]-[[[rho].sup.2]/3], [eta] = [[m.summation over (i = 1)] ([e.sub.i][l.sub.i][a.sub.i] - [e.sub.i][l.sub.i][c.sub.i] + 2[f.sub.i][c.sub.i])]/[[m.summation over (i = 1)] ([a.sub.i] - [c.sub.i])], [rho]' = [[m.summation over (i = 1)] (2[f'.sub.i][b.sub.i] - 2[f'.sub.i][d.sub.i] + [d.sub.i])]/[[m.summation over (i = 1)] ([b.sub.i] - [d.sub.i])], [beta]' = [[[m.summation over (i = 1)] [e'.sub.i][l'.sub.i][d.sub.j]]/[[m.summation over (i = 1)] ([b.sub.i] - [d.sub.i])]]-[[eta]'[rho]'/3]+[[2[[rho]'.sup.3]]/27], [delta]' = [eta]'-[[[rho]'.sup.2]/3], [eta]' = [[m.summation over (i = 1)] ([e'.sub.i][l'.sub.i][b.sub.i] - [e'.sub.i][l'.sub.i][d.sub.i] + 2[f'.sub.i][f.sub.i])]/[[m.summation over (i = 1)] ([b.sub.i] - [d.sub.i])]. Then, the graded [lambda]-preference integration representation of X is, P (X) = [[lambda] [m.summation over (i = 1)] [[5[l.sub.i][e.sub.i][c.sub.i] + 5[g.sub.i][e.sub.i][c.sub.i] + 5[l.sub.i][e.sub.i][a.sub.i] + 15[g.sub.i][e.sub.i][a.sub.i] - 3[a.sub.i] - 2[c.sub.i]]/[e.sub.i]] +(1 - [lambda]) [m.summation over (i = 1)] [[5[l'.sub.i][e'.sub.i][d.sub.i] + 5[g'.sub.i][e'.sub.i][d.sub.i] + 5[l'.sub.i][e'.sub.i][b.sub.i] + 15[g'.sub.i][e'.sub.i][b.sub.i] - 3[b.sub.i] - 2[d.sub.i]]/[e.sub.i]]]/30. Property 7. [A.sub.i]= ([c.sub.i], [a.sub.i], [b.sub.i], [d.sub.i],), i =1,2,..., m are m trapezoidal fuzzy numbers, [B.sub.i] = ([l.sub.i], [e.sub.i], [g.sub.i],[g'.sub.i], [e'.sub.i], [l'.sub.i]) i = 1,2,..., m are m fuzzy numbers with two-sided parabolic drum-like shape. Let X be the linear combination of the multiplication of two fuzzy numbers (X = [m.summation over (i = 1)] [A.sub.i] [cross product] [B.sub.i]), under Extension Principle of fuzzy numbers, the GMIR of X is, P(X)= [[lambda] [m.summation over (i = 1)] [[5[l.sub.i][e.sub.i][c.sub.i] + 5[g.sub.i][e.sub.i][c.sub.i] + 5[l.sub.i][e.sub.i][a.sub.i] + 15[g.sub.i][e.sub.i][a.sub.i] - 3[a.sub.i] - 2[c.sub.i]]/[e.sub.i]] + (1 - [lambda]) [m.summatioin over (i = 1)] [[5[l'.sub.i][e'sub.i][d.sub.i] + 5[g'.sub.i][e'.sub.i][d.sub.i] + 5[l'.sub.i][e'.sub.i][b.sub.i] + 15[g'.sub.i][e'.sub.i][b.sub.i] - 3[b.sub.i] - 2[d.sub.i]]/[e.sub.i]]]/30. Remark 16. When [lambda] = 1/2, P(X) = [[m.summation over (i = 1)] [[5[l.sub.i][e.sub.i][c.sub.i] + 5[g.sub.i][e.sub.i][c.sub.i] + 5[l.sub.i][e.sub.i][a.sub.i] + 15[g.sub.i][e.sub.i][a.sub.i] - 3[a.sub.i] - 2[c.sub.i]]/[e.sub.i]] + [m.summation over (i = 1)] [[5[l'.sub.i][e'.sub.i][d.sub.i] + 5[g'.sub.i][e'.sub.i][d.sub.i] + 5[l'.sub.i][e'.sub.i][b.sub.i] + 15[g'.sub.i][e'.sub.i][b.sub.i] - 3[b.sub.i] - 2[d.sub.i]]/[e.sub.i]]]/60. Remark 17. A = ([c.sub.i], [a.sub.i], [d.sub.i]) i = 1,2,..., m are m triangular fuzzy numbers, B = ([l.sub.i], [e.sub.i], [g.sub.i], [e.sub.i]', [l.sub.i]') i = 1,2,..., m are m fuzzy numbers with two-sided parabolic drum-like shape membership functions, are special cases of trapezoidal fuzzy numbers with [b.sub.i]=[a.sub.i], [g.sub.i]'=[g.sub.i]. The GMIR of the linear combination of the multiplication of these fuzzy numbers, X = [m.summation over (i = 1)] [A.sub.i] [cross product] [B.sub.i], is, P(X)=[[m.summation over (i = 1)] [[5[l.sub.i][e.sub.i][c.sub.i] + 5[g.sub.i][e.sub.i][c.sub.i] + 5[l.sub.i][e.sub.i][a.sub.i] + 15[g.sub.i][e.sub.i][a.sub.i] - 3[a.sub.i] - 2[c.sub.i]]/[e.sub.i]] + [m.summation over (i = 1)] [[5[l'.sub.i][e'.sub.i][d.sub.i] 5[g.sub.i][e'.sub.i][d.sub.i] + 5[l'.sub.i][e'.sub.i][a.sub.i] + 15[g.sub.i][e'.sub.i][a.sub.i] - 3[a.sub.i] - 2[d.sub.i]]/[e.sub.i]]]/60. 5. The GMIR of the Multiplication of Three Fuzzy Numbers 5.1 The GMIR of the multiplication of three trapezoidal fuzzy numbers Suppose A = (c, a, b, d), B = (q, o, p, r) and C = (u, s, t, v) are three fuzzy numbers, each with a trapezoidal membership. The multiplication of A, B and C, X = A [cross product] B [cross product] C, is two-sided cubic shape, too. The membership function [mu] is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Where, [rho] = [c/[a - c]]+[q/[o - q]]+[u/[s - u]], [beta] = [cqu/[(a - c)(o - q)(s - u)]]-[[eta][rho]/3]+ [[2[[rho].sup.3]]/27], [delta] = [eta]-[[[rho].sup.2]/3], [eta] = [cq(s - u) + cu(o - q) + qu(a - c)]/[(a - c)(o - q)(s - u)], [rho]' = [d/[d - b]]+[r/[r - p]]+[v/[v - t]], [beta]' = [drv/[(d - b)(r - p)(v - t)]]-[[eta]'[rho]'/3]+[[2[[rho]'.sup.3]]/27], [delta]' = [eta]'-[[[rho]'.sup.2]/3], [eta]' = [dr(v - t) + dv(r - p) + rv(d - b)]/[(d - b)(r - p)(v - t)]. Then, the graded [lambda]-preference integration representation of X is, P (X) = [[lambda] (3aou + 12aos + 2cqs + 3aqs + 2aqu + 3cqu + 3cos + 2cou) +(1 - [lambda])(3brt + 3drv + 12pbt + 2brv + 3dpt + 2dpv + 3pbv + 2drt)]/30. Property 8. A = (c, a, b, d), B = (q, o, p, r), and C= (u, s, t, v) are three trapezoidal fuzzy numbers. Let X be the multiplication of three fuzzy numbers (X = A [cross product] B [cross product] C), under Extension Principle of fuzzy numbers, the GMIR of X is, P (X) = [[lambda] (3aou + 12aos + 2cqs + 3aqs + 2aqu + 3cqu + 3cos + 2cou) +(1 - [lambda])(3brt + 3drv + 12pbt + 2brv + 3dpt + 2dpv + 3pbv + 2drt)]/30. Remark 18. When [lambda] = 1/2, P(X) = [3aou + 12aos + 2cqs + 3aqs + 2aqu + 3cqu + 3cos+ 2cou + 3brt + 3drv + 12pbt + 2brv + 3dpt + 2dpv + 3pbv + 2drt]/60. Remark 19. The triangular fuzzy numbers A = (c, a, d), B = (q, o, r) and C= (u, s, v) are special cases of trapezoidal fuzzy numbers with b=a, p=o, t=s. The GMIR of the multiplication of these three fuzzy numbers is, P(A [cross product] B [cross product] C) = [3aou + 24aos + 2cqs + 3aqs + 2aqu + 3cqu + 3cos+ 2cou + 3ars + 3drv + 2arv + 3dos + 2dov + 3aov + 2drs]/60. 5.2 The GMIR of the linear combination of the multiplications of three trapezoidal fuzzy numbers Suppose [A.sub.ij]= ([c.sub.ij], [a.sub.ij], [b.sub.ij], [d.sub.ij],) i=1,2,..., m, j = 1,2,..., [n.sub.i] are [n.sub.i] fuzzy numbers, each with a trapezoidal membership function [[mu].sub.A], [B.sub.ij]= ([q.sub.ij], [o.sub.ij], [p.sub.ij], [r.sub.ij]) i=1,2,..., m, j =1,2,..., [n.sub.i] are [n.sub.i] fuzzy numbers, each with a trapezoidal membership function [[mu].sub.B], and Ci= ([u.sub.i], [s.sub.i], [t.sub.i], [v.sub.i]) i = 1,2,..., m are m fuzzy numbers, each with a trapezoidal membership function [[mu].sub.C], and 0 [less than or equal to] [c.sub.ij], 0 [less than or equal to] [q.sub.ij], 0 [less than or equal to] [u.sub.i], j = 1,2,..., [n.sub.i], i = 1,2,..., m, then the fuzzy linear function, X= [m.summation over (i=1)][C.sub.i] [[n.sub.i].summation over (j=1)] [A.sub.ij] [cross product] [B.sub.ij], is two-sided cubic shape. The membership function [mu] is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Where, [beta] = [[[m.summation over (i=1)][u.sub.i] [[n.sub.i].summation over (j=1)] [c.sub.ij][q.sub.ij]]/[[m.summation over (i=1)]([s.sub.i] - [u.sub.i]) [[n.sub.i].summation over (j=1)]([q.sub.ij]([a.sub.ij] - [c.sub.ij]) + [c.sub.ij]([o.sub.ij] - [q.sub.ij]))]]-[[eta][rho]/3]+ [[2[[rho].sup.3]]/27], [delta] = [eta]-[[[rho].sup.2]/3], [rho] = [[m.summation over (i = 1)][u.sub.i] [[n.sub.i].summation over (j = 1)] ([a.sub.ij] - [c.sub.ij])([o.sub.ij] - [q.sub.ij]) + [m.summation over (i = 1)] ([s.sub.i] - [u.sub.i])[[n.sub.i].summation over (j = 1)] ([q.sub.ij]([a.sub.ij] - [c.sub.ij]) + [c.sub.ij]([o.sub.ij] - [q.sub.ij]))]/[[m.summation over (i=1)]([s.sub.i] - [u.sub.i]) [[n.sub.i].summation over (j=1)] ([a.sub.ij] - [c.sub.ij])([o.sub.ij] - [q.sub.ij])], [eta] = [[m.summation over (i = 1)][u.sub.i][[n.sub.i].summation over (j = 1)] [c.sub.ij][o.sub.ij] + [m.summation over (i = 1)] [u.sub.i] [[n.sub.i].summation over (j = 1)] [a.sub.ij][q.sub.ij] + [m.summation over (i = 1)] [s.sub.i] [[n.sub.i].summation over (j = 1)] [c.sub.ij][q.sub.ij] - 3[m.summation over (i = 1)] [u.sub.i] [[n.sub.i].summation over (j = 1)] [c.sub.ij][q.sub.ij]]/[[m.summation over (i=1)]([s.sub.i] - [u.sub.i]) [[n.sub.i].summation over (j=1)] ([a.sub.ij] - [c.sub.ij])([o.sub.ij] - [q.sub.ij])], [beta]' = [[[m.summation over (i=1)][v.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][r.sub.ij]]/[[m.summation over (i=1)]([t.sub.i] - [v.sub.i]) [[n.sub.i].summation over (j=1)] ([b.sub.ij] - [d.sub.ij])([p.sub.ij] - [r.sub.ij])]]-[[eta]'[rho]'/3]+[[[2[rho]'.sup.3]]/27], [delta]'=[eta]'-[[[rho]'.sup.2]/3], [rho]' = [[m.summation over (i = 1)] [v.sub.i] [[n.sub.i].summation over (j = 1)] ([b.sub.ij] - [d.sub.ij])([p.sub.ij] - [r.sub.ij]) + [m.summation over (i = 1)] ([t.sub.i] - [v.sub.i]) [[n.sub.i].summation over (j = 1)] ([r.sub.ij]([b.sub.ij] - [d.sub.ij]) + [d.sub.ij] ([p.sub.ij] - [r.sub.ij]))]/[[m.summation over (i=1)]([t.sub.i] - [v.sub.i]) [[n.sub.i].summation over (j=1)] ([b.sub.ij] - [d.sub.ij])([p.sub.ij] - [r.sub.ij])], [eta]' = [[m.summation over (i=1)] [v.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][p.sub.ij] + [m.summation over (i=1)] [v.sub.i] [[n.sub.i].summation over (j=1)] [b.sub.ij][r.sub.ij] + [m.summation over (i=1)] [t.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][r.sub.ij] - 3[m.summation over (i=1)] [v.sub.i][[n.sub.i].summation over (j=1)] [d.sub.ij][r.sub.ij]]/[[m.summation over (i=1)]([t.sub.i] - [v.sub.i])[[n.sub.i].summation over (j=1)]([b.sub.ij] - [d.sub.ij])([p.sub.ij] - [r.sub.ij])]. Then, the graded [lambda]-preference integration representation of Y is, [P.sub.[lambda]](X) = [[lambda](3 [m.summation over (i=1)][u.sub.i] [[n.sub.i].summation over (j=1)][a.sub.ij][o.sub.ij] + 2[m.summation over (i=1)][u.sub.i][[n.sub.i].summation over (j=1)][a.sub.ij][q.sub.ij] + 2 [m.summation over (i=1)][u.sub.i] [[n.sub.i].summation over (j=1)] [c.sub.ij][o.sub.ij] + 3 [m.summation over (i=1)][u.sub.i] [[n.sub.i].summation over (j=1)] [c.sub.ij][q.sub.ij] + 3 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [a.sub.ij][q.sub.ij] + 3 [m.summation over (i=1)] [s.sub.i] [[n.sub.i].summation over (j=1)] [c.sub.ij][o.sub.ij] +2 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [c.sub.ij][q.sub.ij] + 12 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [a.sub.ij][o.sub.ij]) + (1-[lambda])(3 [m.summation over (i=1)][v.sub.i] [[n.sub.i].summation over (j=1)] [b.sub.ij][p.sub.ij] +3 [m.summation over (i=1)][v.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][r.sub.ij] +2 [m.summation over (i=1)][v.sub.i] [[n.sub.i].summation over (j=1)] [b.sub.ij] [r.sub.ij] +2 [m.summation over (i=1)][v.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij] [p.sub.ij] + 3 [m.summation over (i=1)][t.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][p.sub.ij] + 2 [m.summation over (i=1)][t.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][r.sub.ij] + 12 [m.summation over (i=1)] [t.sub.i] [[n.sub.i].summation over (j=1)] [b.sub.ij][p.sub.ij] +3 [m.summation over (i=1)][t.sub.i] [[n.sub.i].summation over (j=1)] [b.sub.ij][r.sub.ij])/30. Property 9. [A.sub.ij] = ([c.sub.ij], [a.sub.ij], [b.sub.ij], [d.sub.ij],) and [B.sub.ij] = ([q.sub.ij], [o.sub.ij], [p.sub.ij], [r.sub.ij]) i=1,2,..., m, j =1,2,..., [n.sub.i] are [n.sub.i] trapezoidal fuzzy numbers, and Ci= ([u.sub.i], [s.sub.i], [t.sub.i], [v.sub.i]) i =1,2,..., m are m trapezoidal fuzzy numbers, and 0[less than or equal to][c.sub.ij], 0[less than or equal to][q.sub.ij], 0[less than or equal to][u.sub.i], i =1,2,..., m, j =1,2,..., [n.sub.i], then the fuzzy linear function (X = [m.summation over (i=1)][C.sub.i] [[n.sub.i].summation over (j=1)] [A.sub.ij] [cross product] [B.sub.ij]) is two-sided cubic shape. Under Extension Principle of fuzzy numbers, the GMIR of X is, P(X) = [[lambda] (3 [m.summation over (i=1)][u.sub.i] [[n.sub.i].summation over (j=1)] [a.sub.ij][o.sub.ij] + 2 [m.summation over (i=1)][u.sub.i] [[n.sub.i].summation over (j=1)] [a.sub.ij][q.sub.ij] +2 [m.summation over (i=1)][u.sub.i] [[n.sub.i].summation over (j=1)] [c.sub.ij][o.sub.ij] +3 [m.summation over (i=1)][u.sub.i] [[n.sub.i].summation over (j=1)] [c.sub.ij][q.sub.ij] +3 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [a.sub.ij][q.sub.ij] +3 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [c.sub.ij][o.sub.ij] +2 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [c.sub.ij][q.sub.ij] +12 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [a.sub.ij][o.sub.ij]) + (1-[lambda])(3 [m.summation over (i=1)][v.sub.i] [[n.sub.i].summation over (j=1)] [b.sub.ij][p.sub.ij] +3 [m.summation over (i=1)][v.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][r.sub.ij] +2 [m.summation over (i=1)][v.sub.i] [[n.sub.i].summation over (j=1)] [b.sub.ij][r.sub.ij] +2 [m.summation over (i=1)][v.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][p.sub.ij] +3 [m.summation over (i=1)][t.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][p.sub.ij] +2 [m.summation over (i=1)][t.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][r.sub.ij] +12 [m.summation over (i=1)][t.sub.i] [[n.sub.i].summation over (j=1)] [b.sub.ij][p.sub.ij] +3 [m.summation over (i=1)][t.sub.i] [[n.sub.i].summation over (j=1)] [b.sub.ij][r.sub.ij])/30. Remark 20. When [lambda] =1/2, P(X)=(3 [m.summation over (i=1)][u.sub.i] [[n.sub.i].summation over (j=1)] [a.sub.ij][o.sub.ij] +2 [m.summation over (i=1)][u.sub.i] [[n.sub.i].summation over (j=1)] [a.sub.ij][q.sub.ij] +2 [m.summation over (i=1)][u.sub.i] [[n.sub.i].summation over (j=1)] [c.sub.ij][o.sub.ij] +3 [m.summation over (i=1)][u.sub.i] [[n.sub.i].summation over (j=1)] [c.sub.ij][q.sub.ij] +3 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [a.sub.ij][q.sub.ij] +3 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [c.sub.ij][o.sub.ij] + 2 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [c.sub.ij][q.sub.ij] +12 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [a.sub.ij][o.sub.ij] +3 [m.summation over (i=1)][v.sub.i] [[n.sub.i].summation over (j=1)] [b.sub.ij][p.sub.ij] +3 [m.summation over (i=1)][v.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][r.sub.ij] +2 [m.summation over (i=1)][v.sub.i] [[n.sub.i].summation over (j=1)] [b.sub.ij][r.sub.ij] +2 [m.summation over (i=1)][v.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][p.sub.ij] +3 [m.summation over (i=1)][t.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][p.sub.ij] +2 [m.summation over (i=1)][t.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][r.sub.ij] +12 [m.summation over (i=1)][t.sub.i] [[n.sub.i].summation over (j=1)] [b.sub.ij][p.sub.ij] +3 [m.summation over (i=1)][t.sub.i] [[n.sub.i].summation over (j=1)] [b.sub.ij][r.sub.ij])/60. Remark 21. [A.sub.ij]=([c.sub.ij], [a.sub.ij], [d.sub.ij],) and [B.sub.ij]=([q.sub.ij], [o.sub.ij], [r.sub.ij]) i=1,2,..., m, j=1,2,..., [n.sub.i] are [n.sub.i] fuzzy numbers, and Ci= ([u.sub.i], [s.sub.i], [v.sub.i]) i=1,2,..., m, are m fuzzy numbers, each with a triangular membership function, all are special cases of trapezoidal fuzzy numbers with [b.sub.ij]=[a.sub.ij], [p.sub.ij]=[o.sub.ij], [t.sub.i],=[s.sub.i]. The GMIR of the fuzzy linear function, X = [m.summation over (i=1)][C.sub.i] [[n.sub.i].summation over (j=1)] [A.sub.ij] [cross product] [B.sub.ij], is, P(X) = (3 [m.summation over (i=1)][u.sub.i] [[n.sub.i].summation over (j=1)] [a.sub.ij][o.sub.ij] +2 [m.summation over (i=1)][u.sub.i] [[n.sub.i].summation over (j=1)] [a.sub.ij][q.sub.ij] +2 [m.summation over (i=1)][u.sub.i] [[n.sub.i].summation over (j=1)] [c.sub.ij][o.sub.ij] +3 [m.summation over (i=1)][u.sub.i] [[n.sub.i].summation over (j=1)] [c.sub.ij][q.sub.ij] +3 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [a.sub.ij][q.sub.ij] +3 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [c.sub.ij][o.sub.ij] +2 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [c.sub.ij][q.sub.ij] +24 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [a.sub.ij][o.sub.ij] +3 [m.summation over (i=1)][v.sub.i] [[n.sub.i].summation over (j=1)] [a.sub.ij][o.sub.ij] +3 [m.summation over (i=1)][v.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][r.sub.ij] +2 [m.summation over (i=1)][v.sub.i] [[n.sub.i].summation over (j=1)] [a.sub.ij][r.sub.ij] + 2 [m.summation over (i=1)][v.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][o.sub.ij] +3 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][o.sub.ij] +2 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [d.sub.ij][r.sub.ij] +3 [m.summation over (i=1)][s.sub.i] [[n.sub.i].summation over (j=1)] [a.sub.ij][r.sub.ij])/60. 6. Conclusion In this paper, we derive some properties of the representation of fuzzy numbers by using the GMIR method under fuzzy arithmetical operations with extension principle. These properties can help us to simplify the calculation of representation of kth order plane curve fuzzy numbers, the multiplication of two or three fuzzy numbers and the linear combination of the multiplication of fuzzy numbers. When using these formulas, we need not to go through the operations of the membership function to get the representation of fuzzy numbers. What we need here are the vertexes of the original membership functions of the fuzzy numbers only. The Fuzzy Multi-Criteria Decision Making problems, our study can be very helpful. We can use these results in evaluation problems such as the two level hierarchies evaluating problems. For example, in case that the alternatives have m goals and each has [n.sub.i] criteria criteria (krītēr´ē  ),n. . We can let the fuzzy numbers Ci, i =1,2,..., m, be the weights of the goals, fuzzy numbers [B.sub.ij], i =1,2,..., m, j =1,2,..., [n.sub.i], be the characteristic values of the [n.sub.i] criteria, and fuzzy numbers [A.sub.ij], i = 1,2,..., m, j = 1,2,..., [n.sub.i], be the weights of the criteria. Once we decide the vertexes of the original membership functions of these fuzzy numbers, we'll we'll Contraction of we will. we'll we will or we shall we'll will ~shall get the representation of each alternative from the formula mentioned in section 5.2. [FIGURE 1 OMITTED] [FIGURE 2 OMITTED] [FIGURE 3 OMITTED] [FIGURE 4 OMITTED] [FIGURE 5 OMITTED] [FIGURE 6 OMITTED] References [1] J. M. Adamo, Fuzzy decision trees, Fuzzy Sets and Systems Fuzzy sets and systems A fuzzy set is a generalized set to which objects can belong with various degrees (grades) of memberships over the interval [0,1]. Fuzzy systems are processes that are too complex to be modeled by using conventional mathematical methods. 4 (1980), 207-219. [2] J. J. Buckley, A fast method of ranking alternatives using fuzzy numbers, Fuzzy sets and Systems, 30 (1989) 337-338. [3] L. Campos and J. L. Verdegay, Linear programming problems and ranking of fuzzy numbers, Fuzzy sets and Systems 32 (1989) 1-11. [4] P. T. Chang and E. S. Lee, Fuzzy arithmetics and comparison of fuzzy numbers, Fuzzy optimization optimization Field of applied mathematics whose principles and methods are used to solve quantitative problems in disciplines including physics, biology, engineering, and economics. : Resent re·sent tr.v. re·sent·ed, re·sent·ing, re·sents To feel indignantly aggrieved at. [French ressentir, to be angry, from Old French resentir, advances (M. Delgado, J. Kacprzyk, J.-L. Verdegay, M. A. Vila Vila or Port-Vila Seaport, capital, and largest town (pop., 2003 est.: 33,987) of Vanuatu, southwestern South Pacific Ocean. Although French in appearance, the town has a multinational population including British, French, and Vietnamese. ), Physica-Verlag, Heidelberg Heidelberg (hī`dəlbĕrkh), city (1994 pop. 139,430), Baden-Württemberg, SW Germany, picturesquely situated on the Neckar River. Manufactures include machinery, precision instruments, leather goods, and tobacco and wood products. , Germany Germany (jûr`mənē), Ger. Deutschland, officially Federal Republic of Germany, republic (2005 est. pop. 82,431,000), 137,699 sq mi (356,733 sq km). , 69-81. [5] S. H. Chen and C. H. Hsieh, Graded Mean Integration Representation of Generalized Fuzzy Number, Journal of the Chinese Chinese, subfamily of the Sino-Tibetan family of languages (see Sino-Tibetan languages), which is also sometimes grouped with the Tai, or Thai, languages in a Sinitic subfamily of the Sino-Tibetan language stock. Fuzzy System Association 5(2) (1999) 1-7. [6] S. H. Chen and C. H. Hsieh, Ranking generalized fuzzy number with graded mean integration representation, Proceeding of Eighth International Fuzzy Systems Association World Congress (IFSA'99 Taiwan Taiwan (tī`wän`), Portuguese Formosa, officially Republic of China, island nation (2005 est. pop. 22,894,000), 13,885 sq mi (35,961 sq km), in the Pacific Ocean, separated from the mainland of S China by the 100-mi-wide (161-km) Taiwan ), Taiwan, Republic of China (2) (1999), 551-555. [7] S. H. Chen and C. H. Hsieh, Graded mean representation of generalized fuzzy numbers, Proceeding of Sixth Conference on Fuzzy Theory and its Applications (CD ROM CD ROM Compact Disk Read Only Memory ), Chinese Fuzzy Systems Association, Taiwan, Republic of China, filename file·name also file name n. A name given to a computer file to distinguish it from other files, often containing an extension that classifies it by type. 031.wdl(1998), 1-6. [8] S. H. Chen and C. H. Hsieh, Representations, Ranking, Distance, And Similarity Similarity is some degree of symmetry in either analogy and resemblance between two or more concepts or objects. The notion of similarity rests either on exact or approximate repetitions of patterns in the compared items. Of L-R Type Fuzzy Number and Application, Australia Australia (ôstrāl`yə), smallest continent, between the Indian and Pacific oceans. With the island state of Tasmania to the south, the continent makes up the Commonwealth of Australia, a federal parliamentary state (2005 est. pop. Journal of Intelligent Processing System 6(4) (2000), 217-229. [9] S. H. Chen and C. H. Hsieh, A New Method of Representing Generalized Fuzzy Number, Tamsui Oxford Journal of Management Sciences 13-14 (1998), 133-143. [10] S. H. Chen, Ranking fuzzy numbers with maximizing set and minimizing set, Fuzzy sets and Systems 17 (1985), 113-129. [11] S. H. Chen and Fuzzy linear combination of fuzzy linear function under extension principle and second function principle, Tamsui Oxford Journal of Management Science 1 (1985), 11-31. [12] S. M. Chen, A new method for tool steel material selection under fuzzy environment, Fuzzy Sets and Systems 92 (1997), 265-274. [13] C. H. Cheng, A new approach for ranking fuzzy numbers by distance method, Fuzzy Sets and Systems 95(1998) 307-317. [14] M. Delgado and M. A. Vila, W. Voxman, on a canonical The standard or authoritative method. The term comes from "canon," which is the law or rules of the church. See canonical name and canonical synthesis. canonical - (Historically, "according to religious law") 1. [15] D. Dubois People Dubois (also spelled DuBois or Du Bois) is the name of several people:
1. (programming) int - A common name for the integer data type. In C for example, it means a (signed) integer of the computer's native word length. . J. Systems Science 9 (1978), 613-626. [16] D. Dubois and H. Prade, Fuzzy Sets Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets have been introduced by Lotfi A. Zadeh (1965) as an extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent and Systems-Theory and Applications, Academic press, New York New York, state, United States New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of , 1980. [17] P. Fortemps and M. Roubens, Ranking and defuzzification Defuzzification is the process of producing a quantifiable result in fuzzy logic. Typically, a fuzzy system will have a number of rules that transform a number of variables into a "fuzzy" result, that is, the result is described in terms of membership in fuzzy sets. methods based on area compensation, Fuzzy Sets and Systems 82 (1996), 310-330. [18] S. Heilpern, Representation and application of fuzzy numbers, Fuzzy Sets and Systems 91 (1997)259-268. [19] R. Jain, Decision-making decision-making, n the process of coming to a conclusion or making a judgment. decision-making, evidence-based, n a type of informal decision-making that combines clinical expertise, patient concerns, and evidence gathered from in the presence of fuzzy variables, IEEE (Institute of Electrical and Electronics Engineers, New York, www.ieee.org) A membership organization that includes engineers, scientists and students in electronics and allied fields. Trans., Systems Man and Cybern. 6 (1976), 698-703. [20] R. Jain, A procedure for mutli-aspect decision making using fuzzy sets, Int. J. Systems Science 8 (1977), 1-7. [21] A. Kaufmann and M. M. Gupta Gupta (g p`tə), Indian dynasty, A.D. c.320–c.550, whose empire at its height encompassed much of N India. Ancient Indian culture reached a high point during this period. , Introduction to Fuzzy Arithmetic
Theory and Applications, Van Nostrand Reinhold Reinhold is a surname and given name, and may refer to:As a surname:
[22] K. Kim and K. S. Park, Ranking fuzzy numbers with index of optimism, Fuzzy Sets and Systems, 35 (1990), 143-150. [23] E. S. Lee and R. J. Li, Comparison of fuzzy numbers based on the probability probability, in mathematics, assignment of a number as a measure of the "chance" that a given event will occur. There are certain important restrictions on such a probability measure. measure of fuzzy events, Computer and Mathematics with Applications 15 (1988), 887-896. [24] T. S. Liou and M. J. J. Wang (Wang Laboratories, Inc., Lowell, MA) A computer services and network integration company. Wang was one of the major early contributors to the computing industry from its founder's invention that made core memory possible, to leadership in desktop calculators and word processors. , Ranking fuzzy numbers with integral values, Fuzzy Sets and Systems 50 (1992), 247-255. [25] C. S. McCahon and E. S. Lee, Comparing fuzzy numbers: the proportion of the optimum method, Int. J. Approximatie Reasoning 4 (1990), 159-181. [26] T. Y. Tseng and C. M. Klein Klein , Melanie 1882-1960. Austrian-born British psychoanalyst who first introduced play therapy and was the first to use psychoanalysis to treat young children. , New algorithm for the ranking procedure in fuzzy decision-making, IEEE Trans. Stst. Man. Cybern 19 (1989), 1289-1296. [27] R. R. Yager, A procedure for ordering fuzzy subsets of the unit interval For the data transmission signaling interval, see . In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. , Information Science 24 (1981), 143-161. [28] L. A. Zadeh, Similarity relations and fuzzy orderings, Information Sciences 3 (1971), 177-200. [29] L. A. Zadeh, Fuzzy set, Information and Control 8 (1965), 338-353. Shan Shan Shan Tai Any member of a Southeast Asian people who live primarily in eastern and northwestern Myanmar (Burma) and also in Yunnan province, China. The Shan are the largest minority group in Myanmar, numbering more than four million. Huo Chen Department of Information Management, Ching Yun University History The Presidents Organization A president (校長) heads the University. Each college (院) is headed by a dean (院長), and each department (系) by a chairman (系主任). , Jung-Li, 320, Taiwan Shiu Tung Wang Department of Shipping and Transportation Management, National Taiwan Ocean University National Taiwan Ocean University (NTOU 國立臺灣海洋大學) is a national university in Keelung, Taiwan. History The predecessor of NTOU was a junior college for the study of maritime science and technology, founded in 1953. , Keelung Keelung: see Chilung, Taiwan. , 202, Taiwan and Shu Shu In Egyptian religion, the god of the air and supporter of the sky, created by the god Atum. Shu and his sister Tefnut (goddess of moisture) were the first couple of the group of nine gods called the Ennead of Heliopolis. Of their union were born Geb and Nut. Man Chang * Department of Shipping Business Management, China College of Marine Technology and Commerce, Taipei Taipei (tībā`), city (1995 est. pop. 2,632,863), N Taiwan, capital of Taiwan and provisional capital of the Republic of China. Taiwan's largest city, it is the administrative, cultural, and industrial center of the island. , 111, Taiwan Received May 4, 2005, Accepted October October: see month. 18, 2005. * Corresponding author. Tel.:+886 2 28102292 ext5500; fax:+886 2 28106688 E-mail:smchang@mail.ccmtc.edu See .edu. (networking) edu - ("education") The top-level domain for educational establishments in the USA (and some other countries). E.g. "mit.edu". The UK equivalent is "ac.uk". .tw Postal Postal can refer to:
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