# Analysis and application of extension correlation and correspondence under uncertainty.

1. INTRODUCTION

Extenics is a new science subject (CAI Wen, Sep.1999), research on the extension possibility and extension laws of things to resolve contradictions problems. Its research object is the contradictions problems in real world and theoretical base is matter-element and extension set theory.

In classical mathematics, with the characteristic function to describe whether the elements in domains having a certain nature or not; in fuzzy mathematics, with the membership function to characterize it having the extent of a certain nature; in extenics, with the correlation function to portray it so. Correlation function indicates that when the value of matter-element is a point in solid axis, with matter-element to meet the requirements of the value range, it can quantitatively and objectively describe the elements having the nature or character at a certain extent and the process of quantitative change and qualitative change. (WANG Ze-min, Jun.1998) succeeded in promoting the concept of extension set to n-dimensional Euclidean space,and analyzed the nature and application of the n-variables correlation function. [3] studied the concept of correlation function space [OMEGA]([bar.X]) and discussed the properties of [OMEGA]([bar.X]), obtained the product theorem and combination theorem of the [[lambda](x). (HU, WANG, HE, June 2000) introduced concepts of extension set, interval distance and interval place distance on intervals and discusses their properties. The extension set based on correlation function has successfully applied to the forecast of container accumulation (WANG Chao, No.10, 2008), classifying of reservoir flow unit (LI, WANG, Dec.2007), cluster evaluation for ore rock cavability (DENG, ZHOU, Jan. 2008), product improvement design (HUANG, LIN, Jan.2008), risk investment decision (BAI, Jun.2008), knowledge acquisition (YANG, CAI, Aug.2008), data mining (CHEN, 2006), operational command (ZHANG Su, 2007), multiple fault diagnosis (JIN, CHEN, JIN, August 2006), etc.

Extension group decision-making, a problem of group decision-making described with matter-element extension set, studies extension associated and extension decision-making space, compatibility and conflict changes among decision-making systems by combining extension transformation with group decision optimization. Thus it produces the relevant policy strategies and realizes comparison and selection of objects in changing environment. Correlation function is an important component of extension group decision-making, through which studies the correspondence of extension associated. Then it can lead to the data interval analysis under uncertainty, by which can not only estimate the possible range of missing data based on available data, but also predict the efficient space of missing data under a certain constraint condition.

2. CORRELATION FUNCTION

Through the correlation function can quantitatively describe the area to which any one element belongs in the domain. For elements belonging to a same region distinguish from different levels by values size of the correlation function, as is of great significance to research the properties and changes of the elements under uncertainty.

2.1 Distance and position

Definition 1: if [absolute value of [X.sub.1]] = [absolute value of b - a] is the modulus of bounded interval [X.sub.1] = (a,b), then is the [absolute value of x] = 0 is the modulus of point x.

Nature 1: if the interval [X.sub.1] [contains or equal to] [X.sub.2], then [absolute value of [X.sub.1]] [greater than or equal to] [X.sub.2].

In particular, if the interval [X.sub.1] [contains] [X.sub.2], and at most only a common endpoint, then [absolute value of [X.sub.1]] > [absolute value of [X.sub.2]]

Definition 2 (CAI Wen, 1999): let the distance between the point x and the limited real range [X.sub.1] = (a,b) is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

Nature 2: for a given range of X = (a,b), then:

(1) when x [member of] X, [rho] = (x,X) < 0 is the necessary and sufficient conditions of x [not equal to] a,b;

(2) when x [member of] X, [rho] = (x,X) > 0 is the necessary and sufficient conditions of x [not equal to] a,b;

(3) [rho] = (x,X) = 0 is the necessary and sufficient conditions of x = a or x = b.

The distance is introduced to accurately describe the location relationship between points and intervals in the quantitative form. When the point is in the range, the classical mathematics assume that the distance between point and interval is zero, while in the extension set, by distance describes the position of point in the range. It realizes the quantitative description that from "category shall be the same" develops until a level of class also having extent differences.

In practical problems, in addition to the need to consider the position between the point and interval, but also has to consider the interval with the interval and consider the position between a point and two ranges.

Definition 3: if [X.sub.1] = (a,b), [X.sub.2] = (c,d), [X.sub.1] [subset] [X.sub.2], x is an any point in real domain, said that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

is the bit values of interval sets of x on [X.sub.1] and [X.sub.2], denoted by D(x, [X.sub.1], [X.sub.2]) .

When the [X.sub.1] and [X.sub.2] without common endpoints, there

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

2.2 Correlation function

Definition 4(CAI Wen, 1998): If [X.sub.1] = (a,b), [X.sub.2] = (c,d), [X.sub.1] [subset] [X.sub.2], [X.sub.1] and [X.sub.2] without common endpoints, so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

called k (x) is the elementary correlation function of x on [X.sub.1] and [X.sub.2] which is obtain the maximum value in the mid-point of [X.sub.1].

Correlation function to enable the calculation of correlation do not have to rely on subjective judgments or statistics, it can quantitatively and objectively describe the elements having the nature or character at a certain extent and the process of quantitative change and qualitative change. This allows correlation function out of bias caused by subjective judgments.

Nature 3: if [X.sub.1] = (a,b) and [X.sub.2] = (c,d) are two intervals in the real axis, [X.sub.2] [subset] [X.sub.1], and no common endpoints, then for any x, has a correlation function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

Nature 4: if [X.sub.1] = (a,b) and [X.sub.2] = (c,d) are two intervals in the real axis, [X.sub.2] [contains] [X.sub.1], and no common endpoints, then for any x to [X.sub.1] and [X.sub.2], correlation function k(x) has the following property:

(1) x [member of] [X.sub.1], and x [not equal to] a,b [left and right arrow] k(x) > 0;

(2) x = a or x = b [left and right arrow] k(x) = 0;

(3) x [??] [X.sub.1], x [member of] [X.sub.2], and x [not equal to] a,b,c,d [left and right arrow] -1 < k(x) < 0;

(4) x = c or x = d [left and right arrow] k(x) = -1;

(5) x [??] [X.sub.2], and x [not equal to] c,d [left and right arrow] k(x) < - 1;

(6) x = a + b/2, k(x) is the greatest.

Let [alpha] = (a + b)/2 is the midpoint of [X.sub.1], [beta] = (c + d)/2 is the midpoint of [X.sub.2], then according to the distance between point x and a limited real range can be:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

Correlation function can be determined based on the required value scope [X.sub.1] = (a,b) and the qualitative change interval [X.sub.2] = (c,d) of things on certain characteristics, according to set the four points, you can create correlation function, which has an important role for the quantitative research of extension transformation of extension set.

2.3 Correlation function standardization

In practice, matter-element of problems are often multi-dimensional, multi-attribute, multi-object, therefore, an object may have several correlation functions, based on the same premise of the domain and changing conditions, these correlation functions would have a comparable. According to the principle of judging logic operation of correlation functions can be achieved.

"OR" operator: [k.sub.1] v [k.sub.2] = max{[k.sub.1],[k.sub.2]},it is said that the maximum principle of a number of correlation functions of an object.

"AND" operator: [k.sub.1] ^ [k.sub.2] = min{[k.sub.1],[k.sub.2]},it is said that the minimum principle of a number of correlation functions of an object.

Then, based on the logic operation standard correlation function of an object as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

3. CORRESPONDENCE OF EXTENSION ASSOCIATED

Nature 5: if [X.sub.1] = (a, b) is the interval in the real axis, [X.sub.1] [contains] x, then for any x, there is correlation function:

K (x) = -[rho] (x, [X.sub.1])/[absolute value of [X.sub.1]](8)

Let [alpha] = (a + b)/2 is the mid-point of [X.sub.1], then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

Inference 1: When [X.sub.2] [contains] [X.sub.1] [contains] x, there must be a corresponding point [pi] = 2[alpha] - x, x and [pi] are symmetric on [alpha].

Proof:

(1) When x [greater than or equal to] [alpha], there must be [pi] < [alpha]; When x < [alpha], there must be [pi]> [alpha], [pi] = 2[alpha]- x. So, when [X.sub.2] [contains] [X.sub.1] [contains] x ,there must be a corresponding point [pi] = 2[alpha] - x.

(2) [absolute value of x] is the distance of x to [alpha], [absolute value of [pi] - a] is the distance of the corresponding point [pi] to [alpha]. Let [absolute value of x - [alpha]] = [absolute value of [pi] - [alpha]]

When x > [alpha], there must be a corresponding point [pi] < [alpha], then: x - [alpha] = [alpha] - [pi], [pi] = 2[alpha] - x substituted into the above formula will get:

x - [alpha] = [alpha]- (2[alpha] - x) [right arrow] x - [alpha] = x - [alpha],

Similarly, when x < [alpha], there must be a corresponding point [pi] = 2[alpha] - x, then [alpha] - x = [alpha] - x.

So, when [X.sub.2] [contains] [X.sub.1] [contains] x, there must be a corresponding point [pi] = 2[alpha] - x, x and [pi] are symmetric on [alpha].

Inference 2: When [X.sub.2] [contains] [X.sub.1] [contains] x , x = a or x = b , there is minimal correlation

K(x) = K(n) = 0 ; when x = [alpha], [pi] is the corresponding point of X, and has the greatest correlation K(x) = K([pi]) = 0.5 .

Proof:

(1) When [X.sub.2] [contains] [X.sub.1] [contains] x and x = a, obtain x < [alpha] , then:

K (a) = x - b/b - a = 0/b - a = 0

When [X.sub.2] [contains] [X.sub.1] [contains] x, there must be a corresponding point [pi] 2[alpha]- x,then [pi] 2[alpha]- x b.

Similarly, when x = b, x > [alpha], then K(b) = 0, [pi] = a.

So, when [X.sub.2] [contains] [X.sub.1] [contains] x, x = a or x = b , there is minimal correlation K(x) = K([pi]) = 0.

(2) Let an arbitrary point a < x < [alpha], according to the title need to prove that 0 < K(x) < 0.5 , then

Since b > x > a, get K (x) = x - x/ b - a > 0,

Based on K(x) = x - a / b - a < 0.5 [right arrow] x - a/b -a - 1/2 > 0,then 2x - (a + b)/2(b - a) = x - [alpha]/b - a.

Since x < [alpha], then x - [alpha] /b - a<0.

Similarly, when any point [alpha] < x < b, there exists 0 < K(x) < 0.5 .

When x = [alpha], to obtain [pi] = a under [pi] = 2[alpha]- x.

Therefore, when x = [alpha], [pi] is the corresponding point of x, and has the greatest correlation K(x) K([pi]) 0.5.

Nature 6: if [X.sub.1] = (a, b) and [X.sub.2] = (c, d) are two intervals in the real axis, [X.sub.2] [contains] [X.sub.1], and no common endpoints, then for any x , has a correlation function k(x), set [alpha] = (a + b)/2 is the mid-point of [X.sub.1], [beta] = (c + d)/2 is the mid-point of [X.sub.2], then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

As the distance between point x and the limited real range X is relative distance, then on the other side of X will inevitably be a corresponding point [pi], and there is [K.sub.X] (x) = [K.sub.X] ([pi]).

Inference 3: if [X.sub.1] = (a, b) and [X.sub.2] = (c, d) are two intervals in the real axis, [X.sub.2] [contains] [X.sub.1] ,and no common end points, then for any x, x [??] [X.sub.1], x [member of] [X.sub.2], has a correlation function K(x), there must exists an association corresponding point [pi] to meet [K.sub.X] (x) = [K.sub.X] (n).

(1) When [alpha] < x < [beta], there exists a corresponding point c < [pi] < a, [pi] = a(c - x) + c(b -x)/b + c -2x

(2) When [beta] < x < [alpha], there exists a corresponding point b < [pi] < d, [pi] = b(x - d) + d(x - a)/2x - (a + d)

(3) When x > [alpha], x > [beta] and a [less than or equal to] [beta], there exists a corresponding point c < [pi] < a, [pi] = a(x - d) + c(b - x)/b -d

(4) When x > [alpha], x > [beta], a > [beta] and b < b(d - c) + d(2a - (c + d))/2(a - c), there exists a corresponding point [beta] < [pi] < a, [pi] a(x - d) + d(x - b)/2x - (b - d); When x > [alpha], x > [beta], a > [beta] and b(d - c) + d (2a - (c + d))/2(a - c), there exists a corresponding point c < [pi] < [beta], [pi] c(x - b) + a(d - x)/d - b

(5) When x < [alpha], x < [beta] and b [greater than or equal to] [beta], there exists a corresponding point

b < [pi] < d, [pi] = b(c - x) + d(x - a)/c - a;

(6) When x < [alpha], x < [beta], b < [beta] and a(d - c) + c(c + d - 2b)/2(d - b), there exists a corresponding point b < [pi] < [beta], [pi] = c(a - x) + b(c - x)/c + a - 2x; When x < [alpha], x < [beta], b < [beta] and c < x < a(d - c) + c(c + d - 2b)/2(d - b), there exists a corresponding point [pi] = b(c - x) + d(x - a)/c - a

Proof (1):

According to the known conditions [K.sub.X] (x) = [K.sub.X] ([pi]), since [alpha] < x < [beta], c < [pi] < a, then x - b/b + c - 2 x = a - n/c - a

[pi] = a(c - x) + c(b - x)/b + c - 2 x

When [pi] > c , we can get a(c - x) + c(b - x)/b + c - 2x > c

Because a(c - x) + c(b - x) < 0 and b + c - 2x < 0 ,then a(c - x) + c(b - x) < c(b + c - 2x), (a - c)(c - x) < 0

Because (a - c) > 0 and (c - x) < 0 , then [pi] > c.

When [pi] < a, we can get a(c - x) + c(b - x)/b + c - 2 x < a

Because a(c - x) + c(b - x) < 0 and b + c - 2x < 0 ,then (b - x)(c - a) > 0

Because (b - x) < 0 and (c - a) < 0 , then [pi] < a.

Therefore, when [alpha] < x < [beta], there exists a corresponding point c < [pi] < a.

The same reason, we can prove that (2) and (3)

Proof (4):

Since x > [alpha] > [beta], so the corresponding point of X may be p < [pi] < a or c < [pi] < [beta].

When [beta] < [pi] < a, according to the known conditions [K.sub.X] (x) = [K.sub.X] ([pi]), then a - [pi]/2n- (a + d) = x - b/ b - d,

we can get x d([pi] - a) + b([pi] - d)/2[pi] - (a + d)[pi] = a(x - d) + d(x - b)/2x - (b + d),

if [beta] < [pi] < a, then b < x b(d - c) + d (2a - (c + d))/2(a-c).

When c < [pi] < [beta], then a - [pi]/c - a = x - b/b - d

we can get x = b(c -n) + d(n - a)/c - a, [pi] c(x - b) + a(d - x)/d - b

if c < [pi] < [beta], then b(d - c) + d (2a - (c + d))/2(a-c) < x < d

Therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

4. EXTENSION GROUP DECISION-MAKING INTERVAL BASED ON ASSOCIATED CORRESPONDENCE UNDER UNCERTAINTY

4.1 Correlation range based on the correspondence

Nature 7: if [X.sub.1] = (a, b) is an intervals in the real axis, [X.sub.1] [contains] x, for any x, has a correlation function k(x) , set [alpha] = (a + b)/2 is the mid-point of [X.sub.1], then the maximum correlation range k(x) of x on the [X.sub.1] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

Nature 8: if [X.sub.1] = (a, b) and [X.sub.2] = (c, d) are two intervals in the real axis, [X.sub.2] [contains] [X.sub.1], and no common endpoints, then for any x, has a correlation function K(x), set [alpha] = (a + b)/2 is the mid-point of [X.sub.1], [beta] = (c + d)/2 is the mid-point of [X.sub.2], then when x = [alpha], there are correlation interval as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

Then when x = p, there are correlation interval as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)

When x and interval endpoint a,b,c, d overlap , then correlation degree as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)

4.2 Extension group decision-making interval based on associated correspondence under missing data

Let [a.sub.i] = ([a.sub.1],[a.sub.2],...,[a.sub.n]) means n schemes, [c.sub.j] = {[c.sub.1],[c.sub.2],...,[c.sub.m]} means m decision-makers of [a.sub.i], the value of [a.sub.i] is [c.sub.j] ([a.sub.i]) = ([u.sub.i1],[u.sub.i2],...,[u.sub.im]). Then the composite matter-element of multi-dimensional group decision-making is [a.sub.i] = (N, [c.sub.j], [u.sub.ij]). i = 1,2, ... n, j = 1,2, ... m.

Let a = {a|[a.sub.i] = (N, [c.sub.j], [u.sub.ij]) [member of] R, [u.sub.ij] [member of] U} is the composite element set of group decision-making, [??] = {([u.sub.ij], y)|[u.sub.ij] [member of] U, y = k ([u.sub.ij])} is the extension set, then a matter-element extension set of group decision making in R is as follows:

[??] = {(a, y)|a = (N, [c.sub,j], [u.sub.ij]) [member of] R, y = K([a.sub.i]), y' = [T.sub.K]K([T.sub.u] ([a.sub.i]))}

Among them, [v.sub.pj] = [[a.sub.pj], [b.sub.pj]] is the joint field of the matter-element extension set, [V.sub.ij] = [[a.sub.ij], [b.sub.ij]] is the classical field of the matter-element extension set, [v.sub.ij] [member of] [v.sub.pj], g indicated that the number of classical field classification, (g = 1,2, ...) .

[k.sub.j] ([u.sub.ij]) means the association degree between value and interval of assessment, thus, the integrated association degree based on weights [X.sub.j] of decision-maker [c.sub.j] is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)

If you use [[??].sub.ij]- to represent the individual values of missing or wrong, on the basis of the known weights [X.sub.j] of decision-makers [c.sub.j], then the integrated relational degree as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (16)

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], then, Definition 5 can be obtained as follows:

Definition 5: [??] is the composite element set of the matter-element of group decision-making [a.sub.i] = (N, [c.sub.j], [u.sub.j]), if there are missing or uncertain values of [[??].sub.ij], when the arbitrary [[??].sub.ij] [member of] [v.sub.pj], there is [K.sub.i] ([a.sub.i]) < 0, then [d.sub.g] ([K.sub.i] ([a.sub.i])) = 0 , for any [u.sub.ij]-, the evaluation that program at does not belong to l has no affect; when the arbitrary [[??].sub.ij] [member of] [v.sub.pj], there is [K.sub.i] ([a.sub.i]) [greater than or equal to] 0, then [d.sub.g] ([K.sub.i] ([a.sub.i])) = l, for any [[??].sub.ij], the evaluation that program [a.sub.i] belongs to l has no affect.

Inference 4: If there is missing or uncertain value [[??].sub.ij] in extension group decision-making set [??], given the conditions [K.sub.i] ([a.sub.i]) < 0 , [d.sub.g] ([K.sub.i] ([a.sub.i])) = [empty set] , if [[??].sub.ij] = [[??].sub.ij] [member of] [v.sub.pj], to obtain [K.sub.i] ([a.sub.i]) [greater than or equal to] 0, [d.sub.g] ([K.sub.i] ([a.sub.i])) = l, then the point [u.sub.ij] as the boundary points to [[??].sub.ij] on l. Then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)

Set the classical field v{j is I, and [K.sub.i] ([a.sub.i]) > 0 , [d.sub.g] ([K.sub.i] ([a.sub.i])) = I, if [u.sub.ij] = [u.sub.ij] [member of] [v.sub.pj], when the classical field v{j is I +1, there is [K.sub.i] ([a.sub.i]) > 0 , [d.sub.g]+1 ([K.sub.i] ([a.sub.i])) = I + 1, then the point [u.sub.ij] is the separation points to [u.sub.ij] about l and l + 1. Then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)

Inference 5: if [[??].sub.j] is the boundary points to [[??].sub.ij] about l, there must be a corresponding points [[??].sub.j], makes [k.sub.i] ([[??].sub.ij]) = [k.sub.i] ([[??].sub.ij]), then [[[??].sub.ij], [[??].sub.j]] is the interval for [d.sub.g] ([K.sub.i] ([a.sub.i])) = l , [[a.sub.pj], [[??].sub.ij]] and [[[??].sub.ij], [b.sub.pj]] are the intervals for [d.sub.g] ([K.sub.i] ([a.sub.i])) = [empty set].

Inference 6: if [[??].sub.j] is the separation points to [[??].sub.ij] about l and l +1, there must be a corresponding points [[??].sub.ij], makes [k.sub.i] (uj) = [k.sub.i] ([[??].sub.ij]) = [k.sub.i] ([[??].sub.ij], then [[[??].sub.ij], [[??].sub.ij]] is the interval for [d.sub.g] ([K.sub.i]([a.sub.i]) = l + 1, [[a.sub.pj], [[??].sub.ij] and [[[??].sub.uij], [[??].sub.ij] are the intervals for [d.sub.g] ([K.sub.i] ([a.sub.i])) = l.

5. THE MODEL OF EXTENSION GROUP DECISION-MAKING BASED ON ASSOCIATED CORRESPONDENCE UNDER UNCERTAINTY

5.1 Purpose and ideas

The treatment method of missing data based on the extension correspondence is a kind of reverse thinking, with the correlation to predict the possible scope and choice for the missing data under the purpose and the certain constraint condition.. This method can not only take the advantages of point estimates of normal order, but also highlight the merits of interval estimation of reversed order.

The flow of missing data processing based on the extension correspondence mainly includes three parts: the first part is extension transformation, which applies the extension transformation to first evaluating the programs characterized by partial absence of value by setting the basic extension matter-element, extension set and judge set; the second part is the interval estimate, which is through correlation analysis to analyze demarcation point and class point of programs and estimate the potential scope of the missing data and the condition for missing data processing; the third part is a dynamic analysis, which is according to the effect of compensate for missing values, to observe possible changes of the program evaluation based on changing the evaluation value, the evaluation field and weights and adjust and improve missing value so as to enhance the estimated effect.

[FIGURE 1 OMITTED]

5.2 Steps

(1) To build composite matter-element of extension group decision-making [a.sub.i] = (N, [c.sub.j], [u.sub.ij]) and extension group decision-making set [??], set the weight [X.sub.j] of policy makers [c.sub.j];

(2) To input evaluation values [c.sub.j] ([a.sub.i]) of to policy makers [c.sub.j] about program [a.sub.i], building the matter-element matrix of extension group decision-making;

(3) To set joint field [v.sub.pj] = [[a.sub.pj], [b.sub.pj]] and the classical field [v.sub.ij] = [[a.sub.ij], [b.sub.ij]], and analysis whether the data is missing or exceed the festival field in [a.sub.i] = (N, [c.sub.j], [u.sub.ij]);

(4) To remove uncertainty values, calculated correlation degree [k.sub.j] ([u.sub.ij]) from each policy makers [c.sub.j] to program [a.sub.i] on [v.sub.pj] and [v.sub.ij];

(5) To calculate [K.sub.i] ([a.sub.i]) , by Definition 5, for any [[??].sub.ij], to determine that program [a.sub.i] whether belonging to l, if no effect then go to step (8).

(6) According to Inference 6, to determine whether there is a boundary point [[??].sub.ij], if there is a critical point, then calculate the corresponding point [[??].sub.ij], to determine [[[??].sub.ij], [[??].sub.ij]] is the interval for [d.sub.g] ([K.sub.i] ([a.sub.i])) = l, if there is no boundary point, then turned step (8);

(7) To determine whether there is a separation point [[??].sub.ij], if there is a separation point to make [d.sub.g] = l, then calculate the corresponding point [[??].sub.ij], to determine [[a.sub.pj], [[??].sub.ij]] and [[[??].sub.ij], [b.sub.pj]] are intervals for [d.sub.g] ([K.sub.i] ([a.sub.i])) = l, if there is no boundary point, then turned step (8); if there is a separation point to make [d.sub.g] = l + 1, then calculate the corresponding point [[??].sub.ij], to determine [[[??].sub.ij], [[??].sub.ij]] is the interval for [d.sub.g] ([K.sub.i] ([a.sub.i])) = l + 1, and turned step (8), or direct go to step (8);

(8) To estimate the range of [[??].sub.ij];

(9) If [[??].sub.ij] can meet the requirements of decision-making, go to step (10), otherwise go to step (1); if you do not need updating weights, go to step (2); if you do not need to update data, then go to step (3);

(10) End.

6. CONCLUSION

The present decision-making problems are often ones under uncertainty which are a kind of dynamic, complex and correlated decision-making process. Due to the complexity of the decision-making problem, the multi-hierarchy of decision-making process, the difference methods in data processing it will lead to the absence of decision-making data, error or ambiguity, further affecting the decision making. In terms of the features of missing data and the requirements of data processing in extension group decision-making, combined correlation function with extension transformation, the extension correspondence is applied to analyze the possible scope and range of decision-making data from the reverse perspective. It not only reduces the decision-making bias for subjective estimation, but also improves the results of data forecasting and analysis and provides a basis for recognition, judging and selection extension group decision-making under uncertainty.

ACKNOWLEDGMENT

The authors owe a lot to the funding from the National Natural Science Fund project (70971020) and the Colleges and Universities' Philosophy and Social Science Fund Project (09SJD790055) of Education Department of Jiangsu Province. And the authors of this paper would like to sincerely acknowledge the valuable suggestions of Xiaoqiang Zhong, the Associate Professor in Wuxi Institute of Commerce, which have immensely helped to enhance the quality of the paper.

* Received 23 January 2010; accepted 4 April 2010

REFERENCES

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ZHU Jia-jun (1)

ZHENG Jian-guo (2)

LI Jin-bing (3)

(1) School of Business & Management, Donghua University.Department of Accountancy & Finance, Wuxi Institute of Commerce, Shanghai, China. Email: zhujiajun@mail.dhu.edu.cn

(2) School of Business & Management, Donghua University, Shanghai, China. Email:zjg@dhu.edu.cn

(3) National School of Development, Peking University, Beijing, China. Email: lambo@mail.dhu.edu.cn

Extenics is a new science subject (CAI Wen, Sep.1999), research on the extension possibility and extension laws of things to resolve contradictions problems. Its research object is the contradictions problems in real world and theoretical base is matter-element and extension set theory.

In classical mathematics, with the characteristic function to describe whether the elements in domains having a certain nature or not; in fuzzy mathematics, with the membership function to characterize it having the extent of a certain nature; in extenics, with the correlation function to portray it so. Correlation function indicates that when the value of matter-element is a point in solid axis, with matter-element to meet the requirements of the value range, it can quantitatively and objectively describe the elements having the nature or character at a certain extent and the process of quantitative change and qualitative change. (WANG Ze-min, Jun.1998) succeeded in promoting the concept of extension set to n-dimensional Euclidean space,and analyzed the nature and application of the n-variables correlation function. [3] studied the concept of correlation function space [OMEGA]([bar.X]) and discussed the properties of [OMEGA]([bar.X]), obtained the product theorem and combination theorem of the [[lambda](x). (HU, WANG, HE, June 2000) introduced concepts of extension set, interval distance and interval place distance on intervals and discusses their properties. The extension set based on correlation function has successfully applied to the forecast of container accumulation (WANG Chao, No.10, 2008), classifying of reservoir flow unit (LI, WANG, Dec.2007), cluster evaluation for ore rock cavability (DENG, ZHOU, Jan. 2008), product improvement design (HUANG, LIN, Jan.2008), risk investment decision (BAI, Jun.2008), knowledge acquisition (YANG, CAI, Aug.2008), data mining (CHEN, 2006), operational command (ZHANG Su, 2007), multiple fault diagnosis (JIN, CHEN, JIN, August 2006), etc.

Extension group decision-making, a problem of group decision-making described with matter-element extension set, studies extension associated and extension decision-making space, compatibility and conflict changes among decision-making systems by combining extension transformation with group decision optimization. Thus it produces the relevant policy strategies and realizes comparison and selection of objects in changing environment. Correlation function is an important component of extension group decision-making, through which studies the correspondence of extension associated. Then it can lead to the data interval analysis under uncertainty, by which can not only estimate the possible range of missing data based on available data, but also predict the efficient space of missing data under a certain constraint condition.

2. CORRELATION FUNCTION

Through the correlation function can quantitatively describe the area to which any one element belongs in the domain. For elements belonging to a same region distinguish from different levels by values size of the correlation function, as is of great significance to research the properties and changes of the elements under uncertainty.

2.1 Distance and position

Definition 1: if [absolute value of [X.sub.1]] = [absolute value of b - a] is the modulus of bounded interval [X.sub.1] = (a,b), then is the [absolute value of x] = 0 is the modulus of point x.

Nature 1: if the interval [X.sub.1] [contains or equal to] [X.sub.2], then [absolute value of [X.sub.1]] [greater than or equal to] [X.sub.2].

In particular, if the interval [X.sub.1] [contains] [X.sub.2], and at most only a common endpoint, then [absolute value of [X.sub.1]] > [absolute value of [X.sub.2]]

Definition 2 (CAI Wen, 1999): let the distance between the point x and the limited real range [X.sub.1] = (a,b) is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)

Nature 2: for a given range of X = (a,b), then:

(1) when x [member of] X, [rho] = (x,X) < 0 is the necessary and sufficient conditions of x [not equal to] a,b;

(2) when x [member of] X, [rho] = (x,X) > 0 is the necessary and sufficient conditions of x [not equal to] a,b;

(3) [rho] = (x,X) = 0 is the necessary and sufficient conditions of x = a or x = b.

The distance is introduced to accurately describe the location relationship between points and intervals in the quantitative form. When the point is in the range, the classical mathematics assume that the distance between point and interval is zero, while in the extension set, by distance describes the position of point in the range. It realizes the quantitative description that from "category shall be the same" develops until a level of class also having extent differences.

In practical problems, in addition to the need to consider the position between the point and interval, but also has to consider the interval with the interval and consider the position between a point and two ranges.

Definition 3: if [X.sub.1] = (a,b), [X.sub.2] = (c,d), [X.sub.1] [subset] [X.sub.2], x is an any point in real domain, said that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)

is the bit values of interval sets of x on [X.sub.1] and [X.sub.2], denoted by D(x, [X.sub.1], [X.sub.2]) .

When the [X.sub.1] and [X.sub.2] without common endpoints, there

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)

2.2 Correlation function

Definition 4(CAI Wen, 1998): If [X.sub.1] = (a,b), [X.sub.2] = (c,d), [X.sub.1] [subset] [X.sub.2], [X.sub.1] and [X.sub.2] without common endpoints, so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)

called k (x) is the elementary correlation function of x on [X.sub.1] and [X.sub.2] which is obtain the maximum value in the mid-point of [X.sub.1].

Correlation function to enable the calculation of correlation do not have to rely on subjective judgments or statistics, it can quantitatively and objectively describe the elements having the nature or character at a certain extent and the process of quantitative change and qualitative change. This allows correlation function out of bias caused by subjective judgments.

Nature 3: if [X.sub.1] = (a,b) and [X.sub.2] = (c,d) are two intervals in the real axis, [X.sub.2] [subset] [X.sub.1], and no common endpoints, then for any x, has a correlation function:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)

Nature 4: if [X.sub.1] = (a,b) and [X.sub.2] = (c,d) are two intervals in the real axis, [X.sub.2] [contains] [X.sub.1], and no common endpoints, then for any x to [X.sub.1] and [X.sub.2], correlation function k(x) has the following property:

(1) x [member of] [X.sub.1], and x [not equal to] a,b [left and right arrow] k(x) > 0;

(2) x = a or x = b [left and right arrow] k(x) = 0;

(3) x [??] [X.sub.1], x [member of] [X.sub.2], and x [not equal to] a,b,c,d [left and right arrow] -1 < k(x) < 0;

(4) x = c or x = d [left and right arrow] k(x) = -1;

(5) x [??] [X.sub.2], and x [not equal to] c,d [left and right arrow] k(x) < - 1;

(6) x = a + b/2, k(x) is the greatest.

Let [alpha] = (a + b)/2 is the midpoint of [X.sub.1], [beta] = (c + d)/2 is the midpoint of [X.sub.2], then according to the distance between point x and a limited real range can be:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

Correlation function can be determined based on the required value scope [X.sub.1] = (a,b) and the qualitative change interval [X.sub.2] = (c,d) of things on certain characteristics, according to set the four points, you can create correlation function, which has an important role for the quantitative research of extension transformation of extension set.

2.3 Correlation function standardization

In practice, matter-element of problems are often multi-dimensional, multi-attribute, multi-object, therefore, an object may have several correlation functions, based on the same premise of the domain and changing conditions, these correlation functions would have a comparable. According to the principle of judging logic operation of correlation functions can be achieved.

"OR" operator: [k.sub.1] v [k.sub.2] = max{[k.sub.1],[k.sub.2]},it is said that the maximum principle of a number of correlation functions of an object.

"AND" operator: [k.sub.1] ^ [k.sub.2] = min{[k.sub.1],[k.sub.2]},it is said that the minimum principle of a number of correlation functions of an object.

Then, based on the logic operation standard correlation function of an object as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

3. CORRESPONDENCE OF EXTENSION ASSOCIATED

Nature 5: if [X.sub.1] = (a, b) is the interval in the real axis, [X.sub.1] [contains] x, then for any x, there is correlation function:

K (x) = -[rho] (x, [X.sub.1])/[absolute value of [X.sub.1]](8)

Let [alpha] = (a + b)/2 is the mid-point of [X.sub.1], then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

Inference 1: When [X.sub.2] [contains] [X.sub.1] [contains] x, there must be a corresponding point [pi] = 2[alpha] - x, x and [pi] are symmetric on [alpha].

Proof:

(1) When x [greater than or equal to] [alpha], there must be [pi] < [alpha]; When x < [alpha], there must be [pi]> [alpha], [pi] = 2[alpha]- x. So, when [X.sub.2] [contains] [X.sub.1] [contains] x ,there must be a corresponding point [pi] = 2[alpha] - x.

(2) [absolute value of x] is the distance of x to [alpha], [absolute value of [pi] - a] is the distance of the corresponding point [pi] to [alpha]. Let [absolute value of x - [alpha]] = [absolute value of [pi] - [alpha]]

When x > [alpha], there must be a corresponding point [pi] < [alpha], then: x - [alpha] = [alpha] - [pi], [pi] = 2[alpha] - x substituted into the above formula will get:

x - [alpha] = [alpha]- (2[alpha] - x) [right arrow] x - [alpha] = x - [alpha],

Similarly, when x < [alpha], there must be a corresponding point [pi] = 2[alpha] - x, then [alpha] - x = [alpha] - x.

So, when [X.sub.2] [contains] [X.sub.1] [contains] x, there must be a corresponding point [pi] = 2[alpha] - x, x and [pi] are symmetric on [alpha].

Inference 2: When [X.sub.2] [contains] [X.sub.1] [contains] x , x = a or x = b , there is minimal correlation

K(x) = K(n) = 0 ; when x = [alpha], [pi] is the corresponding point of X, and has the greatest correlation K(x) = K([pi]) = 0.5 .

Proof:

(1) When [X.sub.2] [contains] [X.sub.1] [contains] x and x = a, obtain x < [alpha] , then:

K (a) = x - b/b - a = 0/b - a = 0

When [X.sub.2] [contains] [X.sub.1] [contains] x, there must be a corresponding point [pi] 2[alpha]- x,then [pi] 2[alpha]- x b.

Similarly, when x = b, x > [alpha], then K(b) = 0, [pi] = a.

So, when [X.sub.2] [contains] [X.sub.1] [contains] x, x = a or x = b , there is minimal correlation K(x) = K([pi]) = 0.

(2) Let an arbitrary point a < x < [alpha], according to the title need to prove that 0 < K(x) < 0.5 , then

Since b > x > a, get K (x) = x - x/ b - a > 0,

Based on K(x) = x - a / b - a < 0.5 [right arrow] x - a/b -a - 1/2 > 0,then 2x - (a + b)/2(b - a) = x - [alpha]/b - a.

Since x < [alpha], then x - [alpha] /b - a<0.

Similarly, when any point [alpha] < x < b, there exists 0 < K(x) < 0.5 .

When x = [alpha], to obtain [pi] = a under [pi] = 2[alpha]- x.

Therefore, when x = [alpha], [pi] is the corresponding point of x, and has the greatest correlation K(x) K([pi]) 0.5.

Nature 6: if [X.sub.1] = (a, b) and [X.sub.2] = (c, d) are two intervals in the real axis, [X.sub.2] [contains] [X.sub.1], and no common endpoints, then for any x , has a correlation function k(x), set [alpha] = (a + b)/2 is the mid-point of [X.sub.1], [beta] = (c + d)/2 is the mid-point of [X.sub.2], then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)

As the distance between point x and the limited real range X is relative distance, then on the other side of X will inevitably be a corresponding point [pi], and there is [K.sub.X] (x) = [K.sub.X] ([pi]).

Inference 3: if [X.sub.1] = (a, b) and [X.sub.2] = (c, d) are two intervals in the real axis, [X.sub.2] [contains] [X.sub.1] ,and no common end points, then for any x, x [??] [X.sub.1], x [member of] [X.sub.2], has a correlation function K(x), there must exists an association corresponding point [pi] to meet [K.sub.X] (x) = [K.sub.X] (n).

(1) When [alpha] < x < [beta], there exists a corresponding point c < [pi] < a, [pi] = a(c - x) + c(b -x)/b + c -2x

(2) When [beta] < x < [alpha], there exists a corresponding point b < [pi] < d, [pi] = b(x - d) + d(x - a)/2x - (a + d)

(3) When x > [alpha], x > [beta] and a [less than or equal to] [beta], there exists a corresponding point c < [pi] < a, [pi] = a(x - d) + c(b - x)/b -d

(4) When x > [alpha], x > [beta], a > [beta] and b < b(d - c) + d(2a - (c + d))/2(a - c), there exists a corresponding point [beta] < [pi] < a, [pi] a(x - d) + d(x - b)/2x - (b - d); When x > [alpha], x > [beta], a > [beta] and b(d - c) + d (2a - (c + d))/2(a - c), there exists a corresponding point c < [pi] < [beta], [pi] c(x - b) + a(d - x)/d - b

(5) When x < [alpha], x < [beta] and b [greater than or equal to] [beta], there exists a corresponding point

b < [pi] < d, [pi] = b(c - x) + d(x - a)/c - a;

(6) When x < [alpha], x < [beta], b < [beta] and a(d - c) + c(c + d - 2b)/2(d - b), there exists a corresponding point b < [pi] < [beta], [pi] = c(a - x) + b(c - x)/c + a - 2x; When x < [alpha], x < [beta], b < [beta] and c < x < a(d - c) + c(c + d - 2b)/2(d - b), there exists a corresponding point [pi] = b(c - x) + d(x - a)/c - a

Proof (1):

According to the known conditions [K.sub.X] (x) = [K.sub.X] ([pi]), since [alpha] < x < [beta], c < [pi] < a, then x - b/b + c - 2 x = a - n/c - a

[pi] = a(c - x) + c(b - x)/b + c - 2 x

When [pi] > c , we can get a(c - x) + c(b - x)/b + c - 2x > c

Because a(c - x) + c(b - x) < 0 and b + c - 2x < 0 ,then a(c - x) + c(b - x) < c(b + c - 2x), (a - c)(c - x) < 0

Because (a - c) > 0 and (c - x) < 0 , then [pi] > c.

When [pi] < a, we can get a(c - x) + c(b - x)/b + c - 2 x < a

Because a(c - x) + c(b - x) < 0 and b + c - 2x < 0 ,then (b - x)(c - a) > 0

Because (b - x) < 0 and (c - a) < 0 , then [pi] < a.

Therefore, when [alpha] < x < [beta], there exists a corresponding point c < [pi] < a.

The same reason, we can prove that (2) and (3)

Proof (4):

Since x > [alpha] > [beta], so the corresponding point of X may be p < [pi] < a or c < [pi] < [beta].

When [beta] < [pi] < a, according to the known conditions [K.sub.X] (x) = [K.sub.X] ([pi]), then a - [pi]/2n- (a + d) = x - b/ b - d,

we can get x d([pi] - a) + b([pi] - d)/2[pi] - (a + d)[pi] = a(x - d) + d(x - b)/2x - (b + d),

if [beta] < [pi] < a, then b < x b(d - c) + d (2a - (c + d))/2(a-c).

When c < [pi] < [beta], then a - [pi]/c - a = x - b/b - d

we can get x = b(c -n) + d(n - a)/c - a, [pi] c(x - b) + a(d - x)/d - b

if c < [pi] < [beta], then b(d - c) + d (2a - (c + d))/2(a-c) < x < d

Therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

4. EXTENSION GROUP DECISION-MAKING INTERVAL BASED ON ASSOCIATED CORRESPONDENCE UNDER UNCERTAINTY

4.1 Correlation range based on the correspondence

Nature 7: if [X.sub.1] = (a, b) is an intervals in the real axis, [X.sub.1] [contains] x, for any x, has a correlation function k(x) , set [alpha] = (a + b)/2 is the mid-point of [X.sub.1], then the maximum correlation range k(x) of x on the [X.sub.1] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)

Nature 8: if [X.sub.1] = (a, b) and [X.sub.2] = (c, d) are two intervals in the real axis, [X.sub.2] [contains] [X.sub.1], and no common endpoints, then for any x, has a correlation function K(x), set [alpha] = (a + b)/2 is the mid-point of [X.sub.1], [beta] = (c + d)/2 is the mid-point of [X.sub.2], then when x = [alpha], there are correlation interval as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

Then when x = p, there are correlation interval as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)

When x and interval endpoint a,b,c, d overlap , then correlation degree as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)

4.2 Extension group decision-making interval based on associated correspondence under missing data

Let [a.sub.i] = ([a.sub.1],[a.sub.2],...,[a.sub.n]) means n schemes, [c.sub.j] = {[c.sub.1],[c.sub.2],...,[c.sub.m]} means m decision-makers of [a.sub.i], the value of [a.sub.i] is [c.sub.j] ([a.sub.i]) = ([u.sub.i1],[u.sub.i2],...,[u.sub.im]). Then the composite matter-element of multi-dimensional group decision-making is [a.sub.i] = (N, [c.sub.j], [u.sub.ij]). i = 1,2, ... n, j = 1,2, ... m.

Let a = {a|[a.sub.i] = (N, [c.sub.j], [u.sub.ij]) [member of] R, [u.sub.ij] [member of] U} is the composite element set of group decision-making, [??] = {([u.sub.ij], y)|[u.sub.ij] [member of] U, y = k ([u.sub.ij])} is the extension set, then a matter-element extension set of group decision making in R is as follows:

[??] = {(a, y)|a = (N, [c.sub,j], [u.sub.ij]) [member of] R, y = K([a.sub.i]), y' = [T.sub.K]K([T.sub.u] ([a.sub.i]))}

Among them, [v.sub.pj] = [[a.sub.pj], [b.sub.pj]] is the joint field of the matter-element extension set, [V.sub.ij] = [[a.sub.ij], [b.sub.ij]] is the classical field of the matter-element extension set, [v.sub.ij] [member of] [v.sub.pj], g indicated that the number of classical field classification, (g = 1,2, ...) .

[k.sub.j] ([u.sub.ij]) means the association degree between value and interval of assessment, thus, the integrated association degree based on weights [X.sub.j] of decision-maker [c.sub.j] is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)

If you use [[??].sub.ij]- to represent the individual values of missing or wrong, on the basis of the known weights [X.sub.j] of decision-makers [c.sub.j], then the integrated relational degree as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (16)

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], then, Definition 5 can be obtained as follows:

Definition 5: [??] is the composite element set of the matter-element of group decision-making [a.sub.i] = (N, [c.sub.j], [u.sub.j]), if there are missing or uncertain values of [[??].sub.ij], when the arbitrary [[??].sub.ij] [member of] [v.sub.pj], there is [K.sub.i] ([a.sub.i]) < 0, then [d.sub.g] ([K.sub.i] ([a.sub.i])) = 0 , for any [u.sub.ij]-, the evaluation that program at does not belong to l has no affect; when the arbitrary [[??].sub.ij] [member of] [v.sub.pj], there is [K.sub.i] ([a.sub.i]) [greater than or equal to] 0, then [d.sub.g] ([K.sub.i] ([a.sub.i])) = l, for any [[??].sub.ij], the evaluation that program [a.sub.i] belongs to l has no affect.

Inference 4: If there is missing or uncertain value [[??].sub.ij] in extension group decision-making set [??], given the conditions [K.sub.i] ([a.sub.i]) < 0 , [d.sub.g] ([K.sub.i] ([a.sub.i])) = [empty set] , if [[??].sub.ij] = [[??].sub.ij] [member of] [v.sub.pj], to obtain [K.sub.i] ([a.sub.i]) [greater than or equal to] 0, [d.sub.g] ([K.sub.i] ([a.sub.i])) = l, then the point [u.sub.ij] as the boundary points to [[??].sub.ij] on l. Then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)

Set the classical field v{j is I, and [K.sub.i] ([a.sub.i]) > 0 , [d.sub.g] ([K.sub.i] ([a.sub.i])) = I, if [u.sub.ij] = [u.sub.ij] [member of] [v.sub.pj], when the classical field v{j is I +1, there is [K.sub.i] ([a.sub.i]) > 0 , [d.sub.g]+1 ([K.sub.i] ([a.sub.i])) = I + 1, then the point [u.sub.ij] is the separation points to [u.sub.ij] about l and l + 1. Then:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)

Inference 5: if [[??].sub.j] is the boundary points to [[??].sub.ij] about l, there must be a corresponding points [[??].sub.j], makes [k.sub.i] ([[??].sub.ij]) = [k.sub.i] ([[??].sub.ij]), then [[[??].sub.ij], [[??].sub.j]] is the interval for [d.sub.g] ([K.sub.i] ([a.sub.i])) = l , [[a.sub.pj], [[??].sub.ij]] and [[[??].sub.ij], [b.sub.pj]] are the intervals for [d.sub.g] ([K.sub.i] ([a.sub.i])) = [empty set].

Inference 6: if [[??].sub.j] is the separation points to [[??].sub.ij] about l and l +1, there must be a corresponding points [[??].sub.ij], makes [k.sub.i] (uj) = [k.sub.i] ([[??].sub.ij]) = [k.sub.i] ([[??].sub.ij], then [[[??].sub.ij], [[??].sub.ij]] is the interval for [d.sub.g] ([K.sub.i]([a.sub.i]) = l + 1, [[a.sub.pj], [[??].sub.ij] and [[[??].sub.uij], [[??].sub.ij] are the intervals for [d.sub.g] ([K.sub.i] ([a.sub.i])) = l.

5. THE MODEL OF EXTENSION GROUP DECISION-MAKING BASED ON ASSOCIATED CORRESPONDENCE UNDER UNCERTAINTY

5.1 Purpose and ideas

The treatment method of missing data based on the extension correspondence is a kind of reverse thinking, with the correlation to predict the possible scope and choice for the missing data under the purpose and the certain constraint condition.. This method can not only take the advantages of point estimates of normal order, but also highlight the merits of interval estimation of reversed order.

The flow of missing data processing based on the extension correspondence mainly includes three parts: the first part is extension transformation, which applies the extension transformation to first evaluating the programs characterized by partial absence of value by setting the basic extension matter-element, extension set and judge set; the second part is the interval estimate, which is through correlation analysis to analyze demarcation point and class point of programs and estimate the potential scope of the missing data and the condition for missing data processing; the third part is a dynamic analysis, which is according to the effect of compensate for missing values, to observe possible changes of the program evaluation based on changing the evaluation value, the evaluation field and weights and adjust and improve missing value so as to enhance the estimated effect.

[FIGURE 1 OMITTED]

5.2 Steps

(1) To build composite matter-element of extension group decision-making [a.sub.i] = (N, [c.sub.j], [u.sub.ij]) and extension group decision-making set [??], set the weight [X.sub.j] of policy makers [c.sub.j];

(2) To input evaluation values [c.sub.j] ([a.sub.i]) of to policy makers [c.sub.j] about program [a.sub.i], building the matter-element matrix of extension group decision-making;

(3) To set joint field [v.sub.pj] = [[a.sub.pj], [b.sub.pj]] and the classical field [v.sub.ij] = [[a.sub.ij], [b.sub.ij]], and analysis whether the data is missing or exceed the festival field in [a.sub.i] = (N, [c.sub.j], [u.sub.ij]);

(4) To remove uncertainty values, calculated correlation degree [k.sub.j] ([u.sub.ij]) from each policy makers [c.sub.j] to program [a.sub.i] on [v.sub.pj] and [v.sub.ij];

(5) To calculate [K.sub.i] ([a.sub.i]) , by Definition 5, for any [[??].sub.ij], to determine that program [a.sub.i] whether belonging to l, if no effect then go to step (8).

(6) According to Inference 6, to determine whether there is a boundary point [[??].sub.ij], if there is a critical point, then calculate the corresponding point [[??].sub.ij], to determine [[[??].sub.ij], [[??].sub.ij]] is the interval for [d.sub.g] ([K.sub.i] ([a.sub.i])) = l, if there is no boundary point, then turned step (8);

(7) To determine whether there is a separation point [[??].sub.ij], if there is a separation point to make [d.sub.g] = l, then calculate the corresponding point [[??].sub.ij], to determine [[a.sub.pj], [[??].sub.ij]] and [[[??].sub.ij], [b.sub.pj]] are intervals for [d.sub.g] ([K.sub.i] ([a.sub.i])) = l, if there is no boundary point, then turned step (8); if there is a separation point to make [d.sub.g] = l + 1, then calculate the corresponding point [[??].sub.ij], to determine [[[??].sub.ij], [[??].sub.ij]] is the interval for [d.sub.g] ([K.sub.i] ([a.sub.i])) = l + 1, and turned step (8), or direct go to step (8);

(8) To estimate the range of [[??].sub.ij];

(9) If [[??].sub.ij] can meet the requirements of decision-making, go to step (10), otherwise go to step (1); if you do not need updating weights, go to step (2); if you do not need to update data, then go to step (3);

(10) End.

6. CONCLUSION

The present decision-making problems are often ones under uncertainty which are a kind of dynamic, complex and correlated decision-making process. Due to the complexity of the decision-making problem, the multi-hierarchy of decision-making process, the difference methods in data processing it will lead to the absence of decision-making data, error or ambiguity, further affecting the decision making. In terms of the features of missing data and the requirements of data processing in extension group decision-making, combined correlation function with extension transformation, the extension correspondence is applied to analyze the possible scope and range of decision-making data from the reverse perspective. It not only reduces the decision-making bias for subjective estimation, but also improves the results of data forecasting and analysis and provides a basis for recognition, judging and selection extension group decision-making under uncertainty.

ACKNOWLEDGMENT

The authors owe a lot to the funding from the National Natural Science Fund project (70971020) and the Colleges and Universities' Philosophy and Social Science Fund Project (09SJD790055) of Education Department of Jiangsu Province. And the authors of this paper would like to sincerely acknowledge the valuable suggestions of Xiaoqiang Zhong, the Associate Professor in Wuxi Institute of Commerce, which have immensely helped to enhance the quality of the paper.

* Received 23 January 2010; accepted 4 April 2010

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ZHU Jia-jun (1)

ZHENG Jian-guo (2)

LI Jin-bing (3)

(1) School of Business & Management, Donghua University.Department of Accountancy & Finance, Wuxi Institute of Commerce, Shanghai, China. Email: zhujiajun@mail.dhu.edu.cn

(2) School of Business & Management, Donghua University, Shanghai, China. Email:zjg@dhu.edu.cn

(3) National School of Development, Peking University, Beijing, China. Email: lambo@mail.dhu.edu.cn

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Author: | Zhu, Jia-jun; Zheng, Jian-guo; Li, Jin-bing |
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Publication: | Advances in Natural Science |

Article Type: | Report |

Geographic Code: | 9CHIN |

Date: | Jun 30, 2010 |

Words: | 5001 |

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