# A Method to Dynamic Stochastic Multicriteria Decision Making with Log-Normally Distributed Random Variables

Authors: Xin-Fan Wang; Jian-Qiang Wang; Sheng-Yue Deng
Journal: The Scientific World Journal
 (2013)

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## Abstract

We investigate the dynamic stochastic multicriteria decision making (SMCDM) problems, in which the criterion values take the form of log-normally distributed random variables, and the argument information is collected from different periods. We propose two new geometric aggregation operators, such as the log-normal distribution weighted geometric (LNDWG) operator and the dynamic log-normal distribution weighted geometric (DLNDWG) operator, and develop a method for dynamic SMCDM with log-normally distributed random variables. This method uses the DLNDWG operator and the LNDWG operator to aggregate the log-normally distributed criterion values, utilizes the entropy model of Shannon to generate the time weight vector, and utilizes the expectation values and variances of log-normal distributions to rank the alternatives and select the best one. Finally, an example is given to illustrate the feasibility and effectiveness of this developed method.


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## Body

## 1. Introduction

In the socioeconomic activities, there are a large number of stochastic multicriteria decision making (SMCDM) problems in which the criterion values take the form of random variables [1–13]. In SMCDM problems, the normal distribution with well-known bell-shaped curve is most often assumed to describe the random variation that occurs in the criterion values, and each criterion value is commonly characterized and described by two values: the arithmetic mean and the standard deviation [1, 14, 15].However, many measurements of criterion values show a more or less skewed distribution. Particularly, skewed distributions are common when mean values are low, variances large, and values cannot be negative. Such skewed distributions often approximately fit the log-normal distribution [16, 17]. The log-normal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed [16, 18]. It is similar to the normal distribution, but there are still several major differences between them: first, the normal distribution is symmetrical; the log-normal distribution is skewed to the left. Second, both forms of normal and log-normal variability are based on a variety of forces acting independent of one another, but a major difference is that the effects can be additive or multiplicative, thus leading to normal or log-normal distributions, respectively. A variable might be distributed as log-normally if it can be thought of as the multiplicative product of a large number of independent random variables each of which is positive. Third, the sum of several independent normal distributed random variables itself is a normal distributed random variable. For log-normally distributed random variables, however, multiplication is the relevant operation for combining them in most applications; that is, the product of several independent log-normal random variables also follows a log-normal distribution. The log-normal distribution can model many instances, such as the loss of investment risk, the change in price distribution of a stock, and the failure rates in product tests [16, 19–21]. This is because the time series creates random variables. By taking the natural log of each of the random variables, the resulting set of numbers shall be distributed log-normally. Thus, in real-life, there are many SMCDM problems in which the criterion values take the form of log-normally distributed random variables.At present, the SMCDM problems, in which the criterion values take the form of normally distributed random variables, have attracted lots of attentions from researchers [1–8]. But regarding the SMCDM problems, in which the criterion values take the form of log-normally distributed random variables, there is still few related research.Moreover, in some SMCDM situations, such as multi-periods investment decision making, medical dynamic diagnosis, personnel dynamic examination, military system efficiency dynamic evaluation, etc., the original decision information may be collected at different periods (for convenience, we call this kind of SMCDM problems the dynamic SMCDM problems) [8]. Thus, accordingly, time should be taken into account, and it is an interesting and important research issue.In this paper, we shall focus on the dynamic SMCDM problems, in which the criterion values take the form of log-normally distributed random variables and the argument information is given at different periods, and develop a method for dynamic SMCDM with log-normally distributed random variables. This method uses two new geometric aggregation operators to aggregate the log-normally distributed criterion values, utilizes the entropy model of Shannon to generate the time weight vector, and utilizes the expectation values and variances of log-normal distributions to rank the alternatives and select the best one.To do so, this paper is organized as follows. Section2 introduces some operational laws of log-normal distributions and presents a method for the comparison between two log-normal distributions. Section 3 proposes two new geometric aggregation operators, such as the log-normal distribution weighted geometric (LNDWG) operator and the dynamic log-normal distribution weighted geometric (DLNDWG) operator. Section 4 develops an approach to solve the dynamic SMCDM problems, in which the criterion values take the form of log-normally distributed random variables, and the argument information is given at different periods. Section 5 gives an illustrative example. Finally, we conclude the paper in Section 6.


## 2. Preliminaries

The normal distribution is a continuous probability distribution defined by the following probability density function [22]:
(1)fX(x)=12πσe-(x-μ)2/2σ2,-∞<x<+∞,
where μ is the expectation, σ>0 is the standard deviation, and σ2 is the variance. Generally, we use X~N(μ,σ2) as a mathematical expression meaning that X is distributed normally with the expectation μ and variance σ2.The log-normal distribution is a probability distribution of a random variable whose logarithm is normally distributed [16]; that is, if lnY~N(μ,σ2), then Y has a log-normal distribution. The probability density function of the log-normal distribution has the following form:
(2)fY(y)=1yσ2πe-(lny-μ)2/2σ2,y>0.IfY is distributed log-normally with parameters μ and σ, then we write Y~log-N(μ,σ2), and for convenience, we call β=log-N(μ,σ2) a log-normal distribution, and let Θ be the set of all log-normal distributions.Definition 1 (see [16]).
Letβ1=log-N(μ1,σ12) and β2=log-N(μ2,σ22) be two log-normal distributions, then(1)
β1⊗β2=log-N(μ1+μ2,σ12+σ22);(2)
β1a=log-N(aμ1,a2σ12), a≠0.It is easy to prove that all operational results are still log-normal distributions, and by these two operational laws, we have(1)
β1⊗β2=β2⊗β1;(2)
(β1⊗β2)⊗β3=β1⊗(β2⊗β3);(3)
(β1⊗β2)a=β1a⊗β2a, a≠0;(4)
β1a1⊗β1a2=β1a1+a2, a1,a2≠0.Furthermore, iflog-N(μ,σ2) is a log-normal distribution, then its expected value μlog and standard deviation σlog can be calculated by the following formulas [16]:
(3)μlog=eμ+(1/2)σ2,σlog=eμ+(1/2)σ2eσ2-1.According to the relation between expectation and variance in statistics, in the following, we propose a method for the comparison between two log-normal distributions, which is based on the expected valueμlog and the standard deviation σlog.Definition 2.
Letβ1=log-N(μ1,σ12) and β2=log-N(μ2,σ22) be two log-normal distributions, then(1)
ifμlog(β1)<μlog(β2), then β1 is smaller than β2, denoted by β1<β2;(2)
ifμlog(β1)=μlog(β2), then
(i)
ifσlog(β1)=σlog(β2), then β1 is equal to β2, denoted by β1=β2;(ii)
ifσlog(β1)<σlog(β2), then β1 is bigger thanβ2, denoted by β1>β2;(iii)
ifσlog(β1)>σlog(β2), then β1 is smaller than β2, denoted by β1<β2.


## 3. The LNDWG and DLNDWG Operators

To aggregate the log-normally distributed criterion values, in what follows, based on Definition1, we first propose a new geometric aggregation operator, which is called the LNDWG operator.Definition 3.
Letβj=log-N(μj,σj2)  (j=1,2,…,n) be a collection of log-normal distributions, and let LNDWG Θn→Θ, if
(4)LNDWGw(β1,β2,…,βn)=β1w1⊗β2w2⊗⋯⊗βnwn,
then LNDWG is called the log-normal distribution weighted geometric operator of dimension n, where w=(w1,w2,…,wn)T is the weight vector of βj(j=1,2,…,n), with wj≥0 and ∑j=1nwj=1.Theorem 4.
Letβj=log-N(μj,σj2)  (j=1,2,…,n) be a collection of log-normal distributions, and let w=(w1,w2,…,wn)T be the weight vector of βj(j=1,2,…,n), with wj≥0 and ∑j=1nwj=1; then their aggregated result using the LNDWG operator is also a log-normal distribution, and
(5)LNDWGw(β1,β2,…,βn)=log-N(∑j=1nwjμj,∑j=1nwj2σj2).Proof.
Obviously, from Definition1, the aggregated value by using the LNDWG operator is also a log-normal distribution. In the following, we prove (5) by using mathematical induction on n.(1)
Forn=2, since
(6)β1w1=log-N(w1μ1,w12σ12),β2w2=log-N(w2μ2,w22σ22),then
(7)LNDWGw(β1,β2)=β1w1⊗β2w2=log-N(w1μ1+w2μ2,w12σ12+w22σ22)=log-N(∑j=12wjμj,∑j=12wj2σj2).(2)
If (5) holds for n=k, that is,
(8)LNDWGw(β1,β2,…,βk)=log-N(∑j=1kwjμj,∑j=1kwj2σj2).Then, when n=k+1, by Definition 1, we have
(9)LNDWGw(β1,β2,…,βk,βk+1)=log-N(∑j=1kwjμj,∑j=1kwj2σj2)⊗(log-N(μk+1,σk+12))wk+1=log-N(∑j=1kwjμj,∑j=1kwj2σj2)⊗log-N(wk+1μk+1,wk+12σk+12)=log-N(∑j=1kwjμj+wk+1μk+1,∑j=1kwj2σj2+wk+12σk+12)=log-N(∑j=1k+1wjμj,∑j=1k+1wj2σj2).
That is, (5) holds for n=k+1.
Thus, based on (1) and (2), (5) holds for all n∈N, which completes the proof of Theorem 4.The LNDWG operator is an extension of the well-known weighted geometric averaging (WGA) operator [23]. Similar to the WGA operator, the LNDWG operator has the following properties.Theorem 5 (properties of LNDWG).
Letβj=log-N(μj,σj2)(j=1,2,…,n) be a collection of log-normal distributions, and let w=(w1,w2,…,wn)T be the weight vector of βj(j=1,2,…,n), with wj∈[0,1] and ∑j=1nwj=1; then we have the following.(1)
Idempotency: If allβj(j=1,2,…,n) are equal, that is, βj=β for all j, then
(10)LNDWGw(β1,β2,…,βn)=β.(2)
Boundary:min(β1,β2,…,βn)≤LNDWGw(β1,β2,…,βn)≤max(β1,β2,…,βn).(3)
Monotonicity: Letβj=log-N(μj,σj2) and βj*=log-N(μj*,(σj*)2)(j=1,2,…,n) be two collections of log-normal distributions. If βj≤βj*, for all j, then
(11)LNDWGw(β1,β2,…,βn)≤LNDWGw(β1*,β2*,…,βn*).Consider that in many SMCDM problems, the original decision information is usually collected at different periods; then the aggregation operator and its associated weights should not keep constant. In the following, based on Definitions1 and 3, we propose another new aggregation operator for aggregating the log-normally distributed criterion values given at different periods.Definition 6.
Lett be a time variable and Y be a random variable, if Y~log-N(μ(t),(σ(t))2) at the period t, where μ(t) and (σ(t))2 is the expectation and the variance of Y at the period t, respectively, then we call log-N(μ(t),(σ(t))2) the log-normal distribution of Y  at the period t, denoted by β(t)=log-N(μ(t),(σ(t))2).
Similar to Definitions1 and 3, we have the following.Definition 7.
Letβ(t1)=log-N(μ(t1),(σ(t1))2) and β(t2)=log-N(μ(t2),(σ(t2))2) be two log-normal distributions at two different periods t1, t2, respectively; then their operational laws can be defined as follows:(1)
β(t1)⊗β(t2)=log-N(μ(t1)+μ(t2),(σ(t1))2+(σ(t2))2);(2)
(β(t1))a=log-N(aμ(t1),a2(σ(t1))2), a≠0.Definition 8.
Let(tk)=log-N(μ(tk),(σ(tk))2)(k=1,2,…,p) be a collection of p log-normal distributions at p different periods tk(k=1,2,…,p), and let λ(t)=(λ(t1),λ(t2),…,λ(tp))T be the weight vector of the periods tk(k=1,2,…,p), with λ(tk)≥0 and ∑k=1pλ(tk)=1; then we call
(12)DLNDWGλ(t)(β(t1),β(t2),…,β(tp))=(β(t1))λ(t1)⊗(β(t2))λ(t2)⊗⋯⊗(β(tn))λ(tn)
the dynamic log-normal distribution weighted geometric (DLNDWG) operator.Theorem 9.
Let(tk)=log-N(μ(tk),(σ(tk))2)(k=1,2,…,p) be a collection of p log-normal distributions at p different periods tk(k=1,2,…,p), and let λ(t)=(λ(t1),λ(t2),…,λ(tp))T be the weight vector of the periods tk(k=1,2,…,p), with λ(tk)≥0 and ∑k=1pλ(tk)=1; then their aggregated result using the DLNDWG operator is also a log-normal distribution, and
(13)DLNDWGλ(t)(β(t1),β(t2),…,β(tp))=log-N(∑k=1pλ(tk)μ(tk),∑k=1p(λ(tk))2(σ(tk))2).In (12) and (13), the time weight vector λ(t) reflects the importance degree of different periods, which can be given by decision maker(s) or can be obtained by using one of the existing methods, including the arithmetic series based method [24], the geometric series based method [24], the BUM function based method [25], the normal distribution based method [25], the exponential distribution based method [26], the Poisson distribution based method [27], the binomial distribution based method [28], and the average age method [25]. In the following, we propose another method to generate the time weight vector λ(t)=(λ(t1),λ(t2),…,λ(tp))T by using the entropy model of Shannon [29–31]. Consider that, on one hand, the real weights of different periods are random variables and we can utilize the time weight vector’s entropy H(λ(t)) to describe the uncertainty of the time weight vector λ(t) [30], which is defined as
(14)H(λ(t))=-∑k=1pλ(tk)lnλ(tk).On the other hand, we can associate with a concept of the relative average age of the data [31], which is defined as
(15)τ=1p-1∑k=1p(p-k)λ(tk),
where τ indicates the relative average age of the data.The concept of relative average age is an extension of the average age concept [25, 31]. The average age of the data is defined by t-=∑k=1p(p-k)λ(tk), but t- only can be obtained by using approximate method. The relative average age τ reflects the degree paid attention to the data of different periods by the decision makers in the process of information aggregation and can be represented by using a 0.1–0.9 scale (Table 1). When τ is close to 0, it indicates that the decision makers pay more attention to recent data; when τ is close to 1, it indicates that the decision makers pay more attention to distant data; when τ=0.5, it indicates that the decision makers pay the same attention to every period, with no preference. Particularly, when τ=1, then λ(t)=(1,0,…,0)T; when τ=0, then λ(t)=(0,0,…,1)T; when τ=0.5, then λ(t)=(1/p,1/p,…,1/p)T.Table 1
0.1–0.9 scale for the relative average ageτ.
τ
Implication
0.1
Paying more attention to recent data
0.3
Paying much attention to recent data
0.5
Paying the same attention to every period
0.7
Paying much attention to distant data
0.9
Paying more attention to distant data
0.2, 0.4, 0.6, 0.8
Intermediate values between adjacent  scale valuesThus, we can obtain the time weights by maximizing the time weight vector’s entropyH(λ(t)) for a specified level of the relative average age τ and then find a set of weights that satisfies the following mathematical programming model for the λ(tk):
(M-1)Maximize:H(λ(t))=-∑k=1pλ(tk)lnλ(tk),Subjectto:τ=1p-1∑k=1p(p-k)λ(tk)∑k=1pλ(tk)=1λ(tk)≥0,k=1,2,…,p.


## 4. A Procedure for Dynamic SMCDM with Log-Normally Distributed Random Variables

In this section, we consider a dynamic SMCDM problem where all criterion values take the form of log-normally distributed random variables collected at different periods. The following notations are used to depict the considered problems.(i)
A={A1,A2,…,Am}: a discrete set of  m feasible alternatives.(ii)
I={I1,I2,…,In}: a finite set of criteria. The criterion weight vector is w=(w1,w2,…,wn)T, with wj≥0 and ∑j=1nwj=1.(iii)
There arep different periods  tk(k=1,2,…,p) with tp being the most recent period and t1 being the most distant period.(iv)
R(tk)=(βij(tk))m×n(k=1,2,…,p): k log-normal distribution decision matrices at the periods tk(k=1,2,…,p), where βij(tk)=log-N(μij(tk),(σij(tk))2) are the criterion values of the alternatives Ai with respect to the criteria Ij at the periods tk(i=1,2,…,m, j=1,2,…,n, k=1,2,…,p).Based on the above decision information, in the following, we develop a practical procedure to rank the alternatives and select the most desirable one.Step 1.
Utilize the model(M-1) to generate the time weight vector λ(t)=(λ(t1),λ(t2),…,λ(tp))T.Step 2.
Utilize the DLNDWG operator:(16)βij=DLNDWGλ(t)(βij(t1),βij(t2),…,βij(tp)),
to aggregate all the log-normal distribution decision matrices R(tk)=(βij(tk))m×n(k=1,2,…,p) into an overall log-normal distribution decision matrix R=(βij)m×n=(log-N(μij,σij2))m×n, where
(17)βij=log-N(μij,σij2),μij=∑k=1pλ(tk)μij(tk),σij2=∑k=1p(λ(tk))2(σij(tk))2,i=1,2,…,m,j=1,2,…,n
and λ(t)=(λ(t1),λ(t2),…,λ(tp))T is the time weight vector, with λ(tk)≥0 and ∑k=1pλ(tk)=1.Step 3.
Normalize the decision matrixR=(βij)m×n. Let  Ib be the set of all benefit criteria, and let  Ic be the set of all cost criteria; then we can use the following formulas to transform the decision matrix R=(βij)m×n into the corresponding normalized decision matrix R~=(β~ij)m×n=(log-N(μ~ij,σ~ij2))m×n:
(18)μ~ij=μijmaxi{μij},Ij∈Ib,i=1,2,…,m,(19)μ~ij=mini{μij}μij,Ij∈Ic,i=1,2,…,m,(20)σ~ij=σijmaxi{μij},Ij∈I,i=1,2,…,m.
Note that standard deviation is relative to mean, so (20) is suitable for all Ij∈I.Step 4.
Utilize the LNDWG operator(21)β~i=LNDWGw(β~i1,β~i2,…,β~in),
to aggregate the overall criterion values β~ij in the ith column of the normalized decision matrix R~=(β~ij)m×n into the complex overall values β~i of the alternatives Ai(i=1,2,…,m), where
(22)β~i=log-N(μ~i,σ~i2),μ~i=∑j=1nwjμ~ijσ~i2=∑j=1nwj2σ~ij2,i=1,2,…,m.Step 5.
Utilize (3) to calculate the expected values μlog(β~i) and the standard deviations σlog(β~i) of the complex overall values β~i of the alternatives Ai(i=1,2,…,m).Step 6.
Use Definition2 to rank all the alternatives Ai(i=1,2,…,m) and then select the best one according to the values μlog(β~i) and σlog(β~i)(i=1,2,…,m).


## 5. Illustrative Example

In this section, we use a practical dynamic SMCDM problem (adapted from [2]) to illustrate the application of the developed approach.An investment company wants to invest a total amount of money in the best option. There are five possible companiesAi(i=1,2,…,5) to be invested: (1)A1 is an arms company; (2)A2 is a computer company; (3)A3 is a food company; (4)A4 is an auto company; and (5)A5 is a TV company. The criteria considered here in selection of the five possible companies are the following: (1)  I1 is cost; (2)I2 is net present value; and (3)I3 is loss, whose weight vector w=(0.3100,0.3600,0.3300)T. The investment company evaluates the performance of these companies Ai(i=1,2,…,5) in 2009–2011 according to the criteria Ij(j=1,2,3) and constructs the decision matrices R(tk)=(βij(tk))m×n  (k=1,2,3, here, t1 denotes “2009”, t2 denotes “2010”, t3 denotes “2011”) as listed in Tables 2, 3, and 4 (unit: ten thousands RMB). In the decision matrices R(tk)=(βij(tk))m×n, all the criterion values are expressed in log-normal distributions βij(tk)=log-N(μij(tk),(σij(tk))2), where μij(tk) and (σij(tk))2 can be estimated by using statistic methods (i=1,2,…,5, j=1,2,3, k=1,2,3).Table 2
Decision matrixR(t1).
I
1
I
2
I
3
A
1
log-N(385,9.22)
log-N(259,7.92)
log-N(139,5.62)
A
2
log-N(392,10.12)
log-N(266,8.52)
log-N(136,6.12)
A
3
log-N(358,8.92)
log-N(253,6.82)
log-N(130,6.82)
A
4
log-N(468,10.92)
log-N(317,7.52)
log-N(166,7.22)
A
5
log-N(451,9.62)
log-N(303,6.92)
log-N(159,7.52)Table 3
Decision matrixR(t2).
I
1
I
2
I
3
A
1
log-N(371,9.62)
log-N(251,7.62)
log-N(134,5.52)
A
2
log-N(385,10.22)
log-N(269,9.32)
log-N(138,6.12)
A
3
log-N(359,9.32)
log-N(253,8.62)
log-N(135,6.52)
A
4
log-N(463,10.92)
log-N(319,9.12)
log-N(169,7.52)
A
5
log-N(455,9.72)
log-N(319,8.92)
log-N(155,8.62)Table 4
Decision matrixR(t3).
I
1
I
2
I
3
A
1
log-N(369,9.22)
log-N(255,7.92)
log-N(131,5.72)
A
2
log-N(391,9.82)
log-N(269,9.22)
log-N(136,6.12)
A
3
log-N(351,10.62)
log-N(257,8.62)
log-N(133,6.72)
A
4
log-N(467,11.12)
log-N(316,9.32)
log-N(168,7.12)
A
5
log-N(469,11.72)
log-N(306,8.82)
log-N(158,7.62)To get the best company, the following steps are involved.Step 1.
Suppose that the relative average ageτ=0.2 by taking advice from the decision makers; then we use (M-1) to construct the optimization model and obtain time weight vector λ(t)=(0.0819,0.2363,0.6818)T.Step 2.
Utilize (16) to aggregate all the log-normal distribution decision matrices R(tk)=(βij(tk))5×3 (k=1,2,3) into the overall log-normal distribution decision matrix R=(βij)5×3 (Table 5).Table 5
Overall decision matrixR.
I
1
I
2
I
3
A
1
log-N(370.7830,6.71262)
log-N(254.3824,5.71452)
log-N(132.3641,4.12342)
A
2
log-N(389.6641,7.15112)
log-N(268.7543,6.68272)
log-N(136.4726,4.42992)
A
3
log-N(353.4637,7.58892)
log-N(255.7272,6.23062)
log-N(133.2269,4.85142)
A
4
log-N(466.1367,8.04402)
log-N(316.7908,6.72362)
log-N(168.0725,5.18862)
A
5
log-N(464.2176,8.33702)
log-N(308.8262,6.38282)
log-N(157.3730,5.59972)Step 3.
Utilize (18), (19), and (20) to normalize the decision matrix R=(βij)5×3 into the corresponding decision matrix R~=(β~ij)5×3 (Table 6). Note that the criterion I2 is benefit criterion, and the criteria I1 and I3 are cost criteria.Table 6
Normalized decision matrixR~.
I
1
I
2
I
3
A
1
log-N(0.9533,0.01442)
log-N(0.8030,0.01802)
log-N(1.0000,0.02452)
A
2
log-N(0.9071,0.01532)
log-N(0.8484,0.02112)
log-N(0.9699,0.02642)
A
3
log-N(1.0000,0.01632)
log-N(0.8072,0.01972)
log-N(0.9935,0.02892)
A
4
log-N(0.7583,0.01732)
log-N(1.0000,0.02122)
log-N(0.7875,0.03092)
A
5
log-N(0.7614,0.01792)
log-N(0.9749,0.02012)
log-N(0.8411,0.03332)Step 4.
Utilize (21) to aggregate the overall criterion values β~ij in the ith column of the normalized decision matrix R~=(β~ij)m×n and derive the complex overall values β~i of the alternatives Ai(i=1,2,…,5):
(23)β~1=log-N(0.9146,0.01132),β~2=log-N(0.9067,0.01252),β~3=log-N(0.9285,0.01292),β~4=log-N(0.8550,0.01382),β~5=log-N(0.8645,0.01432).Step 5.
Use (3) to calculate the expected values μlog(β~i)(i=1,2,…,5):
(24)μlog(β~1)=2.4959,μlog(β~2)=2.4763,μlog(β~3)=2.5309,μlog(β~4)=2.3516,μlog(β~5)=2.3741.
Thus,
(25)μlog(β~3)>μlog(β~1)>μlog(β~2)>μlog(β~5)>μlog(β~4).Step 6.
Use Definition2 to rank all the alternatives Ai(i=1,2,…,5): A3≻A1≻A2≻A5≻A4. Therefore, the best alternative (company) is A3.


## 6.  Conclusions

In this paper, we have proposed two new geometric aggregation operators, such as the LNDWG operator and the DLNDWG operator. Both operators can be used to aggregate the log-normally distributed random variables, can avoid losing the original decision information, and thus ensure the veracity and rationality of the aggregated results. But weights represent different aspects in both the LNDWG operator and the DLNDWG operator. The weights of the LNDWG operator only reflect the importance degrees of the given log-normal distributions themselves, whereas the weights of the DLNDWG operator only reflect the importance degrees of different periods. Thus, the LNDWG operator is a time independent operator, and because of taking time into account in the aggregation process, the DLNDWG operator is a time-dependent operator. The weights associated with the DLNDWG operator can be given by decision maker(s) or can be obtained by using the existing methods, but we have proposed another method by using the entropy model of Shannon. We have also developed an approach to dynamic SMCDM, in which the criterion values take the form of log-normally distributed random variables, and the argument information is given at different periods. This method has been detailedly illustrated with a practical example. This paper enriches and develops aggregation operator theory and SMCDM theory, and it can be widely applied in medical dynamic diagnosis, personnel dynamic examination, military system efficiency dynamic evaluation, and other related decision making fields.



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*Source: 202085-2013-09-26.xml*