A method of ranking interval numbers based on degrees for multiple attribute decision making

In order to deal with the difficulty of ranking interval numbers in the multiple attribute decision making process, interval numbers are expressed in the Rectangular Coordinate System. On the basis of this, two-dimensional relations of interval numbers are analyzed. For interval numbers, their advantage degree functions of the symmetry axis and the length are deduced after an information mining process, and then the advantage degree function of interval numbers is defined. Procedures of ranking interval numbers based on degrees for multiple attribute decision making are given. Finally, the feasibility and the effectiveness of this method are verified through an example. 6


Introduction
Multiple attribute decision making approach has been widely applied in areas such as economy, management and construction [1-3].In the actual decision making process, the evaluation values are always not real numbers but intervals, i.e., interval numbers, due to the fuzziness of human's mind and the uncertainty of the objective world [4,5].Using interval numbers to represent evaluation values for multiple attribute decision making is closer to the reality of uncertainty and more consistent with human's fuzzy mind than using real numbers [6,22].Especially, qualitative indexes are evaluated by linguistic information [26,27,29], interval numbers have good effect to represent them [28,30].
Interval number was first proposed by Dwyer [7] in 1951, the formal system establishment and value evidence of it was provided by Moore [8] and Moore Motivated by the aforementioned discussions, we focus on providing a simple method of ranking a group of interval numbers in multiple attribute decision making, which can also rank interval numbers with equal symmetry axis.The remainder of this paper is organized as follows.In Section 2, a brief account of current works on comparing interval numbers was given.In Section 3, the basic knowledge of interval numbers was introduced, and the method of expressing them in RCS was proposed.In Section 4, the method of ranking interval numbers was presented, especially the Advantage Degree Function of Interval Numbers and its effectiveness.In Section 5, an example to verify the effectiveness and advantages of the developed approach is given.Conclusions are drawn in Section 6, with recommendations on future studies.

Related works
Moore [15] proposed a method of comparing two interval numbers in 1979, but this method cannot compare them when they have overlap range.Ishibuchi and Tanaka [10] defined weak preference order relation of two interval numbers in linear programming in 1990, which made a significant improvement.The shortage of it is that the relation does not discuss "how much higher" when one interval is known to be higher than another [13,16].Kundu [17] claimed that the selection of least (or most) preferred item in two interval numbers can be made by using Left(A, B) (or Right(A, B)) in 1997, which based on the calculation of the limits of interval numbers, but it rank interval numbers with equal symmetry axis.On the basis of Ishibuchi et al. [10] and Kundu [17], Sengupta and Pal [13] defined an acceptability index in 2000, to measure "how much higher or smaller" of one interval number than another including interval numbers with equal symmetry axis.In addition, some scholars made other attempts to 109 calculate interval numbers.For example, Kůrka [18] 110 let an interval number system be given by an initial 111 interval cover of the extended real line and by a finite   139 Definition 4. [12,13] Let ã = [a L , a U ], then l + (ã) and 140 l − (ã) are defined as the symmetry axis and the length of 141 the interval number ã, respectively, i.e., l

The goal interval number (GIN)
Let {ã 1 , ã2 , . . ., ãn } be a group of interval num-153 bers, and suppose ãm is one of them.If there is a need to compare their lower limits, and the 161 one with the biggest lower limit is the goal interval number.The purpose of selecting the GIN is to determine a target before comparing a group of interval numbers.The method will reduce the time and burden of the comparison work, and make the comparison efficient.

RCS
In the decision making science, the upper and lower limits of interval numbers evaluation values are always positive real numbers.So, positive interval numbers and the situation of no degeneration are (1) The upper and lower limits of the interval number are expressed by y-axis and x-axis of the RCS, respectively.
(2) Suppose ã = [a L , a U ] being the GIN of a group of interval numbers, it is easy to obtain that the arbitrary interval number ã * of the group corresponding to the Point (a L * , a U * ) can be expressed in the triangle area which is bounded by lines y = a U , x = 0 and y = x.
Use four other lines to divide the triangle area into five smaller areas, the regularity and the significance of each area and each line are summarized in Table 2.
The propositions summarized from Fig. 1 and Table 2 189 are as follows: Proof: According to the distance formula of a point to a line, the distance from the Point (a L * , a U * ) to the Line Intersected, longer length, lower symmetry axis Contained, equal upper limit, longer length, lower symmetry axis 0 < a L * < a L Note: The part of the Line y = a U which is above Area is selected only as no interval number in the group has the equal upper limit and bigger lower limit compared to the GIN after the GIN is selected.

The advantage degree function of interval numbers
The Interval Numbers Advantage Degree Function which based on the principles of Limit and Piecewise function is summarized based on the following information: -The symmetry axes and lengths of interval numbers.
-The variation rules of the distances between the points (which correspond to interval numbers) and the lines(which correspond to the Interval Equallength Function and the Interval Equal-symmetry-axis Function) in RCS.
When the relation of two interval numbers is not deviated, the Advantage Degree Function of Interval Numbers Symmetry Axis S 1 and the Advantage Degree Function of Interval Numbers Length S 2 are as follows. (1) (2) Function S 1 and S 2 should be continuous functions in the function range (0,1).

Proof:
(1) Prove the continuity of the functions first.
It is easy to know that Function S 1 is a monotone and linear function for the independent variable l + (ã * ) when l + (ã * ) / = l + (ã), so S 1 is continuous when its independent variable locates in two piecewise ranges.To prove the continuity of S 1 , the only thing needs to do is to prove S 1 is continuous when l + (ã * ) = l + (ã).The continuity of Function S 2 can be proved in the same way.
So there is a need to prove the left limit and right limit of the piecewise functions are both equal to the function value when l + (ã * ) = l + (ã).i.e., S 1 and S 2 are therefore both continuous functions.
Function S 1 and S 2 are both monotone continuous 297 functions.So, their function ranges are both (0,1).

298
Then the Advantage Degree Function of Interval Numbers can be defined as follows.
Consider the interval number is a set of possible interval numbers are compared in the condition that one's lower limit is not bigger than the one's upper limit, their symmetry axes are compared first, then compare their lengths if they have an equal symmetry axis.

Procedures of the method
(i) Use the method which is introduced in Section 3.2, select the GIN ã from a group of interval numbers.Try to rank all of the interval numbers.
Use the method of ranking interval numbers based on degrees which is introduced in Section 4.2.
(i) Compare the upper limits of the interval numbers, . ., a U 7 ) = 30, then the GIN is ã1 .

348
P(ã 6 ã1 ) > 0.5, then ã6 ã1 .Add this rank to the 349 one that is made in the last step, then the five interval 350 numbers will be ranked as ã3 ã6 ã1 ã5 ã2 .

356
Stressed is that, ã1 and ã6 are equal-symmetry-axis 357 interval numbers.It is obvious that this method is more 358 advanced than Nakahara's method [6,11].The ranking 359 method is in accordance with people's prospective and 360 decision making habit, the interval number which has 361 bigger length has more risk, showing effectiveness on 362 some level.

An example of applying the method for
Through adopting the method to rank the compre-374 hensive interval-number-values, the procedures are as (iii) Repeat Procedure (i) and (ii) for Ã2 and Ã4 .
The final rank of all the interval numbers is Ã3 Ã5 Ã1 Ã4 Ã2 , i.e., the rank of the five methods is X 3 X 5 X 1 X 4 X 2 , which is the same with the one of Liu's [21] but with less calculation burden.This proposed method, which is applicable, will create good profit for multiple attribute decision making approach which uses interval numbers to represent evaluation values.

Conclusion
In order to develop a method for resolving the multiple attribute decision making problem, interval numbers can be expressed in RCS.On the basis of this, interval numbers are expressed in different areas of the RCS according to their different properties after information mining.This approach seems to clarify relations of interval numbers.] is the dimensionalized value of it.The values of each evaluation index are made dimensionless (0, 1) through Equation (4).The dimensionalized interval-number-values of evaluation indexes show in Table 5. Suppose [A L i , A U i ] is the comprehensive values of Method X i , and w j is the weight of Evaluation Index G j .Through Equation (5), calculate the comprehensive interval-number-values of each method, and the results show in Table 6.
show that the method of ranking interval numbers based 404 on degrees is feasible, simple and effective.In addition, 405 as a practical method, the equal-symmetry-axis interval 406 numbers can be ranked by using it.

407
There are still further works that we will perform.
The main contributions of this work can be summarized below.(i) Relations of interval numbers are expressed in the Rectangular Coordinate System (RCS) firstly instead of on the Number Axis.The twodimensional relations of interval numbers in RCS are analyzed.It may provide a new perspective of processing interval numbers.(ii) On the basis of this, Advantage Degree Function of Interval Numbers was proposed for ranking interval numbers simply and feasibly based on degrees, especially for a group of them.(iii) Interval numbers with equal symmetry axis can be ranked easily by using this method.

112
system of nonnegative Möbius transformations; Xu 113 [19] used normal distribution based method to assume 114 the probability density function of interval numbers 115 before measuring advantage possibility degree.Wei et 116 al. [24] define operations on hesitant fuzzy linguistic 117 term sets (HFLTSs) and give possibility degree formu-118 las for comparing HFLTSs.Dong et al. [25] propose a 119 consistency-driven automatic methodology to set inter-120 val numerical scales of 2-tuple linguistic term sets in the 121 decision making problems with linguistic preference 122 relations.
The basic definitions of interval numbers 125 Definition 1.[13, 20]  Let ã = [a L , a U ] = {a|a L ≤ 126 a ≤ a U ,a L , a U ∈ R} be an interval number, where 127 a L and a U are the upper and lower limits of ã on the 128 real line R, respectively.Especially, if a L = a U , then ã 129 degenerates into a real number (where ã is also called 130 degenerate interval number).131 Definition 2. [20] Let ã = [a L , a U ] and b = [b L , b U ], 132 then ã = b, if a L = b L and a U = b U .133 Definition 3. [13, 14, 20] Let ã = [a L , a U ] and 134 b = [b L , b U ], let "⊕" and "⊗" be the arithmetic 135 operations on the set of interval numbers, then ã ⊕ 136

137
where a L , b L > 0, ã and b are positive interval 138 numbers.

190Proposition 1 .
The length of the arbitrary interval num-191 ber in the triangle area ã * = [a L * , a U * ] is √ 2 times of 192 the distance (which is named as d * ) from the corre-193 sponded Point (a L * , a U * ) of the interval number to the 194 Line y = x, i.e., d * = l − (ã * )/ √ 2.

275
Referring toFig.1, the farthest points to both sides of 276 the lines of the Interval Equal-length Function y = x + 277 l − (ã) and the Interval Equal-symmetry-axis Function 278 y = −x + 2l + (ã) of the GIN ã in Area , , and 279 (plus the boundaries) are (0, a U ), (a i , a i ), (0, a L ) 280 and (a U , a U ). Especially, (a i , a i ) is the arbitrary point 281 on the Line y = x(a L < x < a U ).The symmetry axis 282 l + (ã * ) and the length l − (ã * ) of the interval numbers 283 which correspond to farthest points are extremums.
299 values, from the Advantage Degree Function of Inter-300 val Numbers, it can be found that when two interval 301 numbers are compared, the one with an upper symme-302 try axis is superior to the other because it has a bigger 303 average value.If two interval numbers have an equal 304 symmetry axis, then the one with a shorter length is superior to the other because it has more concentrative value around the symmetry axis.So when two

(
ii) Analyze the relation of the arbitrary interval number (in the group) ã * and ã.Use the Advantage Degree Function of Interval Numbers which is given in Section 4.1.If a U * ≤ a L , then P(ã * ã) = 0.If a U * > a L , use Equation (1) to calculate the advantage degrees of symmetry axes of the interval numbers to the GIN, and rank the interval numbers according to the advantage degrees.If two or more interval numbers have equal symmetry axis, then use Equation (2) to calculate the advantage degrees of lengths of the interval numbers to the GIN, and rank them according to the results as a complementary to the initial ranking.(iii) If two or more P(ã * ã) = 0, repeat Procedure (i) and (ii) for these interval numbers until all interval numbers of the group are ranked.The detailed procedure of ranking interval numbers is described in Fig. 2. Example.Let ã1 , ã2 , ã3 , ã4 , ã5 , ã6 , and ã7 be a group of interval numbers as such ã1 = [10, 30], ã2 = [6, 18], ã3 = [24, 30], ã4 = [4, 10], ã5 = [12, 20], ã6 = [15, 25] and ã7 = [2, 8]

408
For example, in this work, the length of interval num-409 bers is the second attribute of ranking work.Indexes 410 can be given to make comprehensive consideration of 411 symmetry and length.In addition, new two-dimensional 412 relations of interval numbers in RCS could be mined, 413 which can show different attitudes in the ranking work.the National Science & Technology Pillar Program 417 during the Twelfth Five-year Plan Period of China 418 (No. 2011BAB05B03).The authors of this article thank 419 the Key Projects in the National Science & Technology 420 Pillar Program during the Twelfth Five-year Plan Period 421 of China.

Table 2
The regularity and significance of interval numbers in each area or on each line G 1 , G 2 , G 3 and G 4 are quantitative indexes, G 1 is an income index, and the others are cost indexes.G 5 , G 6 , G 7 and G 8 are qualitative indexes, and all of them are income indexes.Suppose interval number [a L ij , a U ij ] is the value of Evaluation Index G j of Method X i , and [a L ij , a U ij