Dynamic grey target decision making method with grey numbers

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Journal of Intelligent & Fuzzy Systems 28 (2015) 2159–2168 DOI:10.3233/IFS-141497 IOS Press

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Dynamic grey target decision making method with grey numbers based on existing state and future development trend of alternatives Shuli Yana,∗ , Sifeng Liub , Jiefang Liub and Lifeng Wub a School

of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, PR China of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, PR China

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b College

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Abstract. The classic dynamic decision making methods mainly focus on aggregating the existing attribute values, not considering the development trend of alternatives through the whole period. Both the two aspects should be taken into account for reflecting the dynamic nature comprehensively. This paper is to present a dynamic grey target method from a new viewpoint, where the attribute values take the form of grey numbers. In the proposed method, an extended grey target technique based on approaching degree is applied twice, which is firstly applied in seeking the ranking of alternatives with existing attribute values in each period, and secondly applied in exploring the ranking of fluctuation of alternatives with respect to each attribute. Then the comprehensive approaching degree of alternative is to be aggregated concerning the two aspects with respect to all attributes through all the periods. The new concept of approaching degree in grey target method reflects the similarity and closeness of sequence. Both the information performance and the development trend of alternatives through the multi-periods are reflected. An evaluation of sub-companies in an enterprise and discussion are given to demonstrate the proposed method’s effectiveness and superiority.

1. Introduction

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Keywords: Dynamic decision making, grey numbers, grey target, development trend of alternatives

Multiple attribute decision making (MADM) is to obtain the best alternative from the set of feasible ones based on the information of each attribute. A great deal of researches about MADM methods have focused on single period, such as a certain week, a certain month, or a certain year. For example, Liu et al. [1] presented penalty-based continuous aggregation operators for the aggregation of multiple arguments. Liu et al. [2] proposed a method based on prospect theory about risk decision making problems with uncertain linguistic variables, where the probabilities are taken on ∗ Corresponding author. Shuli Yan, School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, PR China. Tel.: +8615090181269; Fax: +86 379 64231482; E-mail: [email protected].

interval numbers. Pei [3] proposed two attribute reduction algorithms to get the simplified attribute subsets, the core of the fuzzy MADM problem. Kuo et al. [4] and Wei [5–7] came up with MADM methods based on the grey relational analysis, and Li et al. [8] put forward an approach using grey system theory in solving supplier selection problem. However, in many practical cases, such as medical diagnosis, enterprise investment investigation, personnel dynamic evaluation, and equipment efficiency dynamic performance, etc., the required information frequently needs to be gathered through multi-periods, therefore, the corresponding decision making problems are usually dynamic, and the dynamic characteristic has been received more attention in MADM research fields [9–20], which is often called as dynamic multiple attribute decision making (DMADM). Xu [9] presented

1064-1246/15/$35.00 © 2015 – IOS Press and the authors. All rights reserved

S. Yan et al. / Dynamic grey target decision making method with grey numbers based on existing state

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their existing state and future development trend. Meanwhile, lots of DMADM methods only obtained the comprehensive evaluating values and the ranking of alternatives, the adjacency between each alternative and the ideal project is not reflected, so it is probable that although the alternative ranks the first, but it is not the desired, and the results may lead to a wrong decision. In view of this, we will present a novel method about DMADM problems with grey numbers from another point of view. Based on aggregating the attribute values of alternatives all the periods, we excavate fluctuation of alternatives with respect to each attribute between the adjacent periods, and meanwhile the development trend of alternatives in the whole period is researched. So the characteristics of the dynamic decision problem are fully considered. And we will provide a new and more appropriate method that not only gets the ranking of alternatives, but also excavates the adjacency between alternatives and the ideal project, which helps decision makers to make a comprehensive and correct decision. The grey target thought is a part of the grey system, which is initially presented by professor Deng [21], it has been widely used in MADM because of its simplicity and convenience [22], and has been applied in Saccadic tracking of mantis, engineering design project selection, and equipment condition monitoring [23–25]. In recent years, a method of uniform effect measures and intelligent grey target decision model was presented [26], a harden grey target model with interval number owning preference information was established [27], and a grey target model about three-parameter interval grey number was proposed [28]. In addition, Deng [29] defined the approaching degree of alternative in space of differences of information, which replaced the frequently-used target distance, and reflected more accurately similarity and closeness between alternatives and bull’s eye in grey target space. As a supplement to MADM, the grey target method is very important in solving decision making problem. Hence, this paper is to develop a new method of DMADM with grey number information, which considers both the attribute value aggregation of all periods and the fluctuation of attribute values from one period to the next period, and utilizes grey target method to calculate approaching degree of every alternative to the positive ideal target and negative ideal target, and approaching degree of fluctuation series to the positive and negative ideal fluctuation series, then aggregates the two approaching degrees to reflect information about the two aspects.

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the arithmetic series based weight operator of period, normal distribution based weight operator of period, and geometric series based weight operator of period. Ben´ıtez et al. [10] proposed a dynamic method using AHP to aggregate decision makers’ preference, and developed an algorithm to select new elements from the dynamic input based on achieving the consistency. Campanella and Ribeiro [11] put forward a versatile framework of DMADM, which depicts the exploratory process that alternatives and attributes are modified by the new input and the iterations may be carried out lots of times. Ustun and Demirtas [12] proposed a multi-objective mixed integer linear programming model aggregating the analytic network process which conquers weakness of AHP. Khalili-Damghani et al. [13] developed a method based on TOPSIS technique which transforms multi-objective decision making to bi-objective one. Chen and Fu [14] proposed a fuzzy dynamic programming iteration model, reflecting the transformation of decision situations. Besides, with the development of society, the complexity and uncertainty of evaluation about decision problems will be increasing. In this case, many researches focus on DMADM problems with intuitionistic fuzzy information [15], triangular intuitionistic fuzzy numbers [16], linguistic variables [17, 18], hybrid attribute concluding numerical values, interval numbers and linguistic variables [19]. In many practical cases, there is not enough available information to judge complicated situations, so it is often given approximate ranges, i.e., grey number. And recently the DMADM problems with grey numbers attracted researchers’ attention [20]. Above all, most of researches on DMADM problems focus on the attribute value aggregation among all periods, little research has considered the fluctuation of alternative’s attribute value from one period to the next period, and its dynamic characteristic has not been reflected adequately. For example, when the economic situation of a certain province is investigated, the GDP is not the sole evaluating index, and the economic growth level should also be considered. And if the sub-companies in an enterprise are evaluated in some periods, the leaders not only focus on each subcompany’s profits of all the periods, but also focus on the change trend of profits in those periods so as to obtain whether the sub-company is rising, stable, or in decline. If the research only concentrates on the profits and gives too little care to their development trend, the decision result may mislead the leaders to make wrong adjustments about sub-companies. Therefore, the subcompanies should be judged synthetically through both

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k · ⊗1 ∈ ka1L , ka1U

(3)

Here, k is a positive real number. The distance formula of grey numbers is based on the core and length of interval grey numbers, which is defined as follows. Definition 5. Let ⊗1 ∈ [a1L , a1U ] and ⊗2 ∈ [a2L , a2U ] be two grey numbers, then the distance between ⊗1 and ⊗2 is defined as below. 1 ˆ1−⊗ ˆ 2 + |l(⊗1 ) − l(⊗2 )| (4) d(⊗1 , ⊗2 ) = ⊗ 2 3. The grey target method for dynamic decision making considering attribute values of all the periods and development trend of alternatives

In the structure of dynamic decision making, suppose s1 , s2 , . . . , sm are alternatives, c1 , c2 , . . . , cn are attributes, and W = {ω1 , ω2 , . . . , ωn } is the attributes’ weight vector, where ωj ∈ [0, 1] (j = 1, 2, . . . , n) n takes the same value in different period, and ωj = 1.

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In this section, some basic definitions are introduced, which contain the concept of grey number, the rank rules, operations of grey numbers and distance function between grey numbers. Grey number represents insufficient information, and is essentially different from interval number, which is denoted by an interval, and the real value in the interval obeys uniform distribution.

(2)

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2. Preliminaries

⊗1 − ⊗2 ∈ a1L − a2U , a1U − a2L

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The following section is arranged as follows: Section 2 interprets the basic definition, distance formula and rank rules of grey numbers. Next, Section 3 presents grey target method for dynamic decision making considering attribute values of existing state and attributes’ fluctuation of alternative from one period to the next period. Then, Section 4 illustrates the proposed method by evaluation of sub-companies in an enterprise, gives comparison of the new grey target thought with Liang’s method, and makes sensitivity analysis for the priority parameter. Finally, Section 5 summarizes the findings.

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Definition 1. [30] Grey number is a real number that the lower limit and upper limit are clear, separately, but the position in the limits is unknown. It is expressed mathematically as ⊗ ∈ [aL , aU ] L (a < aU ), where aL and aU are the lower limit and upper limit, separately.

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Definition 2. [30] Let ⊗ ∈ [aL , aU ] (aL < aU ), assume that the grey number’s information of distribution is unknown. ˆ = 21 (aL + aU ) is the core If ⊗ is continuous, then ⊗ of the grey number ⊗. 2. If ⊗ is discrete, ai ∈ [aL , aU ] (i = 1, 2, . . . , n) ˆ = areall the possible values ⊗ might take on, then ⊗ n 1 a is the core of the grey number ⊗. i=1 i n We suppose grey numbers are continuous in this paper. Definition 3. Let ⊗1 ∈ [a1L , a1U ] and ⊗2 ∈ [a2L , a2U ] be ˆ 2 , then ⊗1 is bigger than ˆ1 >⊗ two grey numbers, if ⊗ ˆ1 =⊗ ˆ 2 , and a1L = a2L , ⊗2 , marked as ⊗1 > ⊗2 ; if ⊗ then ⊗1 is the same as ⊗2 , marked as ⊗1 = ⊗2 ; if ˆ 2 , and a1L > a2L , then ⊗1 is bigger than ⊗2 , ˆ1 =⊗ ⊗ marked as ⊗1 > ⊗2 . Definition 4. Let ⊗1 ∈ [a1L , a1U ] and ⊗2 ∈ [a2L , a2U ] be two grey numbers, the operations between ⊗1 and ⊗2 are given as follows. (1) ⊗1 + ⊗2 ∈ a1L + a2L , a1U + a2U

2161

j=1

There are l different periods t(t = 1, 2, . . . , l), and the weight vector of periods is τ = (τ(1), τ(2), . . . , τ(l)), l where τ(t) ∈ [0, 1] (t = 1, 2, . . . , l), and τ(t) = t=1

1. The attribute value of the alternative si about the attribute cj at the period t is denoted as aijt (⊗), which takes the form of grey number. The decision making t matrices are A(t) = aij (⊗) (t = 1, 2, . . . , l), m×n

and the standardized making matrices are decision denoted as B(t) = bijt (⊗) (t = 1, 2, . . . , l). m×n

In the following, the grey target method is extended to solve DMADM with grey number information. The process of dynamic information aggregation in most of researches lose too much information due to the fact that the development trend of alternatives is not considered, which signifies a lack of accuracy and entirety in the ultimate results. Therefore, in order to overcome this disadvantage, we have extended the grey target method to take the development trend of alternatives in the whole period into account. Grey relational analysis is a very active area of grey systems theory and has been widely used in solving MADM problems [31–33]. Its basic idea is to research


3.1. The grey target model based on existing attribute values of all the periods

i

j

i

j

bjt+ (⊗) =

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m×n

⎤

bijt (⊗)

(7)

min

bijt (⊗)

(8)

i=1,2,...,m

j = 1, 2, . . . , n, t = 1, 2, . . . , l. Step 3. Computing the approaching coefficient of every alternative from positive ideal target and negative ideal target at period t(t = 1, 2, . . . , l) according to the formulae, respectively. i

j

(9)

j

i=1

1 + l(bijt (⊗)) − l(bjt+ (⊗)) , 2 d(bijt (⊗), bjt− (⊗)) = bˆ ijt (⊗) − bˆ jt− (⊗) 1 + l(bijt (⊗)) − l(bjt− (⊗)) . 2 Step 4. Computing the approaching degree of every alternative from positive ideal target and negative ideal target at period t(t = 1, 2, . . . , l) according to the formulae, respectively.

⎤

⎢ yijLt yijUt ⎥ ⎥ ⎢ , m m ⎦ ⎣ yijUt yijLt i=1 i=1

(10)

i = 1, 2, . . . , m, j = 1, 2, . . . , n. Wherein, d(bijt (⊗), bjt+ (⊗)) = bˆ ijt (⊗) − bˆ jt+ (⊗)

(5)

For cost attributes, there is

bijt (⊗) = [bijLt , bijUt ] =

max

i=1,2,...,m

i j t− t d(bij (⊗), bj (⊗)) + 0.5 max max d(bijt (⊗), bjt− (⊗)) i j

⎢ aijLt aijUt ⎥ ⎥ bijt (⊗) = [bijLt , bijUt ] = ⎢ , m m ⎦ ⎣ Ut Lt aij aij

⎡

1≤i≤m

min min d(bijt (⊗), bjt− (⊗)) + 0.5 max max d(bijt (⊗), bjt− (⊗))

⎡

i=1

+ min aijUt .

d(bijt (⊗), bjt+ (⊗)) + 0.5 max max d(bijt (⊗), bjt+ (⊗))

dimension units, we give the normalization formulae of decision making matrices A(t) = aijt (⊗) (t = m×n 1, 2, . . . , l) into the matrices B(t) = bijt (⊗) (t = 1, 2, . . . , l) according to [34]. For benefit attributes, there is

1≤i≤m

− aijUt

min min d(bijt (⊗), bjt+ (⊗)) + 0.5 max max d(bijt (⊗), bjt+ (⊗))

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=

= max

aijUt

Step 2. Determining the positive ideal target sequence b+ = (b1t+ (⊗), b2t+ (⊗), . . . , bnt+ (⊗)) as the positive bull’s-eye and the negative ideal target sequence b− = (b1t− (⊗), b2t− (⊗), . . . , bnt− (⊗)) as the negative bull’s-eye at the period t(t = 1, 2, . . . , l).

i

γ(bijt (⊗), bjt− (⊗))

1≤i≤m

1≤i≤m

yijUt

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Step 1. In general, the attributes are classified as benefit and cost types in DMADM problems, in order to compare different type attributes directly and eliminate =

yijLt = max aijLt − aijLt + min aijLt ,

bjt− (⊗) =

The procedure of the proposed model is shown as follows.

γ(bijt (⊗), bjt+ (⊗))

Wherein,

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the degree of similarity and closeness of the geometric shapes of the available data sequences to determine whether their connections are closer or not [22, 29]. Professor Deng [29] put forward the standpoint that the grey target theory is the relational analysis theory dealing with mode sequences, and defined the grey relational degree between a certain mode (sequence of a certain alternative) and normal mode (bull’s eye) in space of difference of information as approaching degree. In this paper, we will provide two grey target models from two aspects about the attribute values among all the periods and development trend of alternatives.

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(6)

γ + (bit ) =

n

j=1

ωj · γ(bijt (⊗), bjt+ (⊗))

(11)


γ − (bit ) =

n

For cost attributes, there is

ωj · γ(bijt (⊗), bjt− (⊗))

(12)

j=1

Ut etij (⊗) = [eLt ij , eij ] = [

Step 5. Computing the approaching degree of each alternative from positive ideal target and negative ideal target all the evaluated periods. γ + (bi ) =

l

τ(t) · γ + (bit )

Step 6. Comprehensively, the better alternative should have bigger approaching degree between itself and positive bull’s-eye and simultaneously smaller approaching degree between itself and negative bull’s-eye. Then the relative approaching degree of every alternative from bull’s-eyes is computed using the following equation. γ + (bi ) γ + (bi ) + γ − (bi )

(15)

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Obviously, an alternative si is closer to the positive ideal target and simultaneously farther from negative ideal target as ui approaches 1. Consequently, according to the relative approaching degree ui , the ranking orders of all alternatives and the best one can be determined only by the existing attribute values all of periods.

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3.2. The grey target model based on development trend of alternatives

Here, we concentrate on the changes of attribute values of alternatives about a certain attribute between the adjacent periods. If we adopt the above standardization formulae (5-6) in Section 3.1, the fluctuations of attribute values between different periods are possibly contrary to the original information. In order to reflect the comparability of attribute values between different periods, we give the standardization formulae from a new viewpoint. That is, it is based on the information in different periods, not different alternatives. For benefit attributes, there is Ut etij (⊗) = [eLt ij , eij ] = [

aijLt l t=1

aijUt

,

(17)

1≤t≤l

1≤t≤l

= max

1≤t≤l

aijUt

− aijUt

+ min aijUt . 1≤t≤l

k Let k,k+1 (⊗) = ek+1 ij ij (⊗) − eij (⊗) be the fluctuation of attribute value of the alternative si from the kth period to the (k + 1)th period with respect to attribute cj . (⊗) > 0, then the change from the kth period to If k,k+1 ij the (k + 1)th period is increasing, which indicates that the change is in the better direction, and the difference (⊗) < 0, then the change is to be expected. If k,k+1 ij from the kth period to the (k + 1)th period is decreasing, which indicates that the change is in the worse direction, and the difference is to be avoided. Moreover, whether the differences are positive or negative, the bigger the difference, the better the change, meanwhile, the smaller the difference, the worse the change. The fluctuation sequence of the alternative si between all the adjacent periods about the attribute cj is denoted 2,3 l−1,l (⊗)). as ij = (1,2 ij (⊗), ij (⊗), . . . , ij The procedure of the proposed model based on development trend of alternatives is listed as follows.

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ui =

t=1

]

yijLt

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t=1

l

yijLt = max aijLt − aijLt + min aijLt , yijUt

(14)

yijUt

yijUt

Wherein,

(13)

τ(t) · γ − (bit )

l

,

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γ − (bi ) =

yijLt t=1

t=1 l

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aijUt l t=1

aijLt

]

(16)

Step 1. Determining the positive ideal fluctuation 2,3 sequence of alternatives j+ = (1,2 j+ (⊗), j+ (⊗),

. . . , l−1,l j+ (⊗)) with respect to attribute cj as the positive bull’s eye for the development trend in all adjacent periods. The maximum difference of alternatives from the kth period to the (k + 1)th period indicates the best development trend, here k,k+1 j+ (⊗) =

max

i=1,2,...,m

k,k+1 (⊗) ij

(18)

Step 2. Determining the negative ideal fluctuation 2,3 sequence of alternatives j− = (1,2 j− (⊗), j− (⊗),

. . . , l−1,l j− (⊗)) with respect to attribute cj as the negative bull’s eye for the development trend in all adjacent periods. The minimum difference of alternatives from the kth period to the (k + 1)th period indicates the worst development trend, here

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k,k+1 j− (⊗) =

min

i=1,2,...,m

k,k+1 (⊗) ij

(19)

Step 3. Calculating the approaching coefficient of fluctuation sequence of alternative si from positive and negative ideal fluctuation sequences with respect to attribute cj from the kth period to the (k + 1)th period according to the following equations, respectively.

γ(k,k+1 (⊗), k,k+1 ij j− (⊗))

=

=

ωj uij

Obviously, an alternative si with best development trend is closer to the positive bull’s-eye, and farther from the negative bull’s eye for the development trend, simultaneously, ui will approaches 1.

k,k+1 (⊗), k,k+1 (⊗), k,k+1 min min d(k,k+1 ij j+ (⊗)) + 0.5 max max d(ij j+ (⊗)) i

i k k,k+1 k,k+1 d(ij (⊗), j+ (⊗)) + 0.5 max max d(k,k+1 (⊗), k,k+1 ij j+ (⊗)) i k k

k,k+1 (⊗), k,k+1 (⊗), k,k+1 min min d(k,k+1 ij j− (⊗)) + 0.5 max max d(ij j− (⊗)) i

(25)

j=1

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γ(k,k+1 (⊗), k,k+1 ij j+ (⊗))

ui =

n

i

k

k

k,k+1 d(k,k+1 (⊗), k,k+1 (⊗), k,k+1 ij j− (⊗)) + 0.5 max max d(ij j− (⊗)) i

(20)

(21)

k

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i = 1, 2, . . . , m, j = 1, 2, . . . , n, k = 1, 2, . . . , According to the relative approaching degree ui , l − 1. the ranking orders of all alternatives about development Wherein, 1 ˆ k,k+1 k,k+1 ˆ k,k+1 d(k,k+1 (⊗), k,k+1 (⊗) − (⊗)) − l(k,k+1 ij j+ (⊗)) = ij j+ (⊗) + l(ij j+ (⊗)) , 2 1 ˆ k,k+1 k,k+1 ˆ k,k+1 (⊗), k,k+1 (⊗) − (⊗)) − l(k,k+1 d(k,k+1 ij j− (⊗)) = ij j− (⊗) + l(ij j− (⊗)) . 2

γ(ij+ ) =

l−1

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Step 4. Computing the approaching degree of fluctuation sequence of alternative si from positive and negative ideal fluctuation sequences with respect to attribute cj all the evaluated periods according to the following equations, respectively. τ(k ∼ k + 1)γ(k,k+1 (⊗), k,k+1 ij j+ (⊗))

k=1

τ(k ∼ k

k=1

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γ(ij− ) =

l−1

(22)

+ 1)γ(k,k+1 (⊗), k,k+1 ij j− (⊗))

(23) Here, τ(k ∼ k + 1) (k = 1, 2, . . . , l − 1) is the weight of the period from k to k + 1. Step 5. Computing the relative approaching degree of fluctuation sequence of alternative si from bull’s eyes with respect to attribute cj all the evaluated periods using the following equation. uij =

γ(ij+ ) γ(ij− ) + γ(ij+ )

(24)

Step 6. Computing the relative approaching degree of each fluctuation sequence of alternative si from bull’seye all the evaluated periods by the following equation

trend and the one with best development trend can be determined. Finally, in order to combine the existing attribute values of all the periods with development trend of attribute values of alternatives, we define the comprehensive approaching degree between alternative si and bull’s eye as zi = λui + (1 − λ)ui

(26)

Where λ is the priority parameter, and 0 < λ < 1. If the decision maker pays more attention to exiting attribute values of all the periods, the parameter can be assumed as 0.5 < λ < 1; if the decision maker pays more attention to development trend of alternatives, the parameter can be assumed as 0 < λ < 0.5; and if the decision maker deems the two aspects are fair, the parameter λ is assumed as 0.5. Ranking all the alternatives is on the basis of the values of comprehensive approaching degree zi (i = 1, 2, . . . , m). 4. Illustrative example For comparison, we adopt an example from Liang [34]. The core enterprise wants to evaluate its four sub-companies and select the best one to strengthen


Table 2 The approaching degrees, relative approaching degrees of fluctuation sequences in all adjacent periods with respect to every attribute

s2

s3

s4

degree

c1

c2

c3

c4

γ(1j+ ) γ(1j− ) u1j γ(2j+ ) γ(2j− ) u2j γ(3j+ ) γ(3j− ) u3j γ(4j+ ) γ(4j− ) u4j

0.8750 0.5944 0.5955 0.7752 0.7406 0.5114 0.6387 0.9252 0.4084 0.6492 0.7334 0.4696

0.9398 0.6185 0.6031 0.7922 0.7920 0.5001 0.7299 0.8368 0.4659 0.7175 0.8132 0.4687

0.6242 0.8134 0.4342 0.7032 0.8462 0.4539 0.8124 0.6153 0.5690 0.8294 0.6615 0.5563

0.8470 0.7596 0.5272 0.7820 0.6902 0.5312 0.7851 0.6728 0.5385 0.7058 0.6995 0.5022

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s1

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adjacent periods with respect to every attribute by using formulae (16–25), and results are given in Table 2. Table 2 shows that, with respect to attribute c1 and c2 , the relative approaching degrees of sub-company s1 about development trend is the biggest, the one of subcompany s2 is the second biggest, then, the next is s4 , and s3 owns the smallest relative approaching degree about development trend. With respect to attribute c3 , the relative approaching degree of sub-company s3 about development trend is the biggest, the one of subcompany s4 is the second biggest, then, the next is s2 , and s1 owns the smallest relative approaching degree about development trend. With respect to attribute c4 , the relative approaching degrees of sub-company s3 about development trend is the biggest, the one of subcompany s2 is the second biggest, then, the next is s1 , and s4 owns the smallest relative approaching degree about development trend.

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investment. Four main attributes, including output value c1 , technical innovation c2 , administrative expenses c3 , and the rate of production and marketing c4 are taken into consideration during the sub-company evaluation process. The four sub-companies with respect to the above attributes are to be evaluated using grey numbers in the years 2005–2010. The data are given in Liang [34]. For comparing conveniently, we also suppose the attributes’ weight vector is W = (0.2218, 0.3012, 0.1971, 0.2799) in each year which is identical with that in [34]. Here, we suppose the periods’ weights under two aspects are equal, and the exiting attribute values of all the periods and development trend of alternatives are equally important, so, the priority parameter is assumed as 0.5. By using formulae (7–15), we determine the approaching degrees of alternatives from positive and negative ideal target, the relative degrees of alternatives in the years 2005–2010, and results are given in Table 1. Table 1 shows that the relative approaching degrees of sub-company s2 is the biggest of the sub- companies from year 2005 to year 2009, and only in year 2010, it is smaller than the relative approaching degrees of sub-company s1 . The sub-company s3 owns the second biggest relative approaching degrees from year 2005 to year 2007, but from year 2008 to year 2010, the sub-company s1 owns the second biggest relative approaching degrees, and both of them exceed 0.5. The relative approaching degrees of sub-company s3 are totally bigger than the ones of sub-company s4 from year 2005 to year 2010. Then, we calculate the approaching degrees of fluctuation sequences of alternatives from positive and negative ideal fluctuation sequences, the relative approaching degrees of fluctuation sequences in all

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Table 1 The approaching degrees, relative degrees of alternatives s1

s2

s3

s4

γ + (b1t ) γ − (b1t ) u1 γ + (b2t ) γ − (b2t ) u2 γ + (b3t ) γ − (b3t ) u3 γ + (b4t ) γ − (b4t ) u4

2005

2006

2007

2008

2009

2010

0.7312 0.9032 0.4474 0.8169 0.7626 0.5172 0.7896 0.8057 0.4949 0.7509 0.8047 0.4828

0.7455 0.8927 0.4552 0.8444 0.7250 0.5381 0.7999 0.8210 0.4935 0.7190 0.8481 0.4588

0.7287 0.8080 0.4742 0.7976 0.6866 0.5374 0.7548 0.7622 0.4976 0.6782 0.8139 0.4545

0.7307 0.7024 0.5099 0.7802 0.6236 0.5558 0.6745 0.7326 0.4794 0.6089 0.7331 0.4537

0.7443 0.5955 0.5555 0.7590 0.5711 0.5706 0.6287 0.7283 0.4633 0.5458 0.7802 0.4116

0.7855 0.6013 0.5664 0.6944 0.5804 0.5447 0.6251 0.7137 0.4669 0.5636 0.7591 0.4261

4.1. Comparison The relative approaching degrees of each alternative from bull’s-eyes concerning existing attribute values of all the periods and development trend of alternatives, the comprehensive approaching degree of alternatives and ranking of alternative are computed according to formulae (15, 25, 26), and results are given in Table 3. Table 3 shows that the sub-company s2 is the best one with the best existing attribute values, while, the subcompany s1 is the best one with the best development trend. The rankings of alternatives are different based on the existing attribute value state and development trend of alternatives. The sub-company s1 is the best one under both the two aspects, and the comprehensive ranking of alternatives is the same as Liang [34], which

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S. Yan et al. / Dynamic grey target decision making method with grey numbers based on existing state Table 3 The relative approaching degrees and ranking of alternatives Existing information Development trend Both information

The relative degree of sub-company

Ranking

u1 = 0.5014, u2 = 0.5440, u3 = 0.4826, u4 = 0.4479 u1 = 0.5469, u2 = 0.5022, u3 = 0.4938, u4 = 0.4956 z1 = 0.5241, z2 = 0.5231, z3 = 0.4882, z4 = 0.4717

s 2 s1 s3 s4 s 1 s 2 s 4 s3 s 1 s2 s3 s4

weights based on the principle of laying more stress on the present than on the past. We calculate the weights of periods using the geometric series based method in [9] as follows: τ(t1 ) = 0.0481, τ(t2 ) = 0.0722, τ(t3 ) = 0.1083, τ(t4 ) = 0.1624, τ(t5 ) = 0.2436, τ(t6 ) = 0.3654. And the weights of two adjacent periods by the same methods in [9] are as follows: τ(t1 ∼ t2 ) = 0.0758, τ(t2 ∼ t3 ) = 0.1137, τ(t3 ∼ t4 ) = 0.1706, τ(t4 ∼ t5 ) = 0.2559, τ(t5 ∼ t6 ) = 0.3839. The relative approaching degrees of each alternative from bull’seyes of existing attribute values of all the periods and fluctuation sequence of alternatives, the comprehensive approaching degree of alternatives and ranking of each alternative are calculated by using formulae (15, 25, 26) and results are given in Table 4. Comparing Tables 3 and 4, we know that the relative degrees of sub-companies are different, and the rankings about development trend are different, which is caused by the importance of different periods is paid different attention by decision makers.

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indicates that the proposed method is effective and reasonable. Besides, this paper analyses the information in decision matrix from two aspects about time series and cross-section data. The degree of superiority of every sub-company in all the periods is showed distinctly through the grey target method according to crosssection data, and meanwhile, the development trend of every sub-company with respect to every attribute is showed distinctly through the grey target method based on time series. These results can be provided to help the leader to adjust the sub-companies according to each specific aspect. Although the grey target methods are also used in [34], there are essential differences between two methods. The approaching degree in grey target method investigate similarity, closeness and integrity between the evaluated sequence and bull’s eye, but the off-target distance in [34] only reflects the closeness between the evaluated sequence and bull’s eye. And the literature [34] doesn’t reflect the development trend of alternatives. So the proposed method in this paper is more suitable for DMADM. 4.2. Discussion

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4.2.1. Comparison with the method using different weights of periods The weights of periods in the example are assumed as equal for observing the fair development trend of decision information in every period. But in other literatures the periods are often assumed as different

4.2.2. Sensitivity analysis for the priority parameter The priority parameter λ is very important in Equation (26). Normally, the different values of λ can lead to different values of zi . The aim of sensitivity analysis is to see if the different values of λ effect on the ranking of alternatives or not. To realize it, we show the change of rankings of four sub-companies as Table 5.

Table 4 The relative approaching degrees and ranking of alternatives under different weights of periods The relative degrees of sub-companies

Ranking

Information emerged u1 = 0.5308, u2 = 0.5502, u3 = 0.4747, u4 = 0.4352 s 2 s1 s3 s4 Development trend u1 = 0.5398, u2 = 0.4975, u3 = 0.4995, u4 = 0.5025 s1 s4 s3 s2 Both information z1 = 0.5353, z2 = 0.5239, z3 = 0.4871, z4 = 0.4689 s 1 s2 s3 s4 Table 5 Overall relative approaching degrees and rankings of alternatives with respect to λ Relative approaching degrees of alternatives λ = 0.2 λ = 0.4 λ = 0.6 λ = 0.8

z1 z1 z1 z1

= 0.5378, = 0.5287, = 0.5196, = 0.5105,

z2 z2 z2 z2

= 0.5105, = 0.5189, = 0.5273, = 0.5356,

z3 z3 z3 z3

= 0.4916, = 0.4893, = 0.4871, = 0.4848,

z4 z4 z4 z4

= 0.4860 = 0.4765 = 0.4670 = 0.4574

Ranking s1 s1 s2 s2

s2 s2 s1 s1

s3 s3 s3 s3

s4 s4 s4 s4


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5. Conclusion

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In this paper, we studied the dynamic decision making problem with grey numbers, and proposed the grey target method not only aggregating the attribute values all of the periods but also considering the development trend of alternatives with respect to each attribute in the whole period. The approaching degree of alternative about the two aspects reflects the alternative’s existing state and future developments, and the new grey target method using the principle of grey relational analysis can reflect closeness, similarity and integrity of connection between evaluated sequences and the ideal one, which is more suitable for decision making than off-target distance. Moreover, the different periods can be assumed as different weights according to decision maker’s attention, and the parameter λ in Equation (26) is also assumed as different value according to the different attention paid by decision maker to the existing information and development trend of alternatives, which describes influences of different psychological behavior on decision results.

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Acknowledgments

The authors are very grateful to the editor and the anonymous referees for their insightful and constructive comments on this paper. This research was supported by a Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme (Grant No. FP7-PIIF-GA-2013-629051), the National Natural Science Foundation of China(91324003, 71401051, 71171112, 71171113 and 71271226), the joint research project of both the NSFC(71111130211) and the RS of UK, the National Special Major Project for Large Aircraft(2009ZX1102), the major project and key project of Social Science Foundation of the China (10zd&014, 12AZD102).

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