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

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1 The Scientific World Journal Volume 03, Article ID 0085, 8 ages htt://dx.doi.org/0.55/03/0085 Research Article A Method to Dynamic Stochastic Multicriteria Decision Maing with Log-Normally Distributed Random Variables Xin-Fan Wang,, Jian-Qiang Wang, and Sheng-Yue Deng School of Science, Hunan University of Technology, Zhuzhou 4007, China School of Business, Central South University, Changsha 40083, China Corresondence should be addressed to Jian-Qiang Wang; jqwang@csu.edu.cn Received 4 August 03; Acceted August 03 Academic Editors: A. Amirteimoori and S. W. Chiu Coyright 03 Xin-Fan Wang et al. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original wor is roerly cited. We investigate the dynamic stochastic multicriteria decision maing (SMCDM) roblems, in which the criterion values tae the form of log-normally distributed random variables, and the argument information is collected from different eriods. We roose two new geometric aggregation oerators, such as the log-normal distribution weighted geometric (LNDWG) oerator and the dynamic log-normal distribution weighted geometric (DLNDWG) oerator, and develo a method for dynamic SMCDM with log-normally distributed random variables. This method uses the DLNDWG oerator and the LNDWG oerator to aggregate the log-normally distributed criterion values, utilizes the entroy model of Shannon to generate the time weight vector, and utilizes the exectation values and variances of log-normal distributions to ran the alternatives and select the best one. Finally, an examle is given to illustrate the feasibility and effectiveness of this develoed method.. Introduction In the socioeconomic activities, there are a large number of stochastic multicriteria decision maing (SMCDM) roblems in which the criterion values tae the form of random variables [ 3]. In SMCDM roblems, the normal distribution with well-nown bell-shaed curve is most often assumedtodescribetherandomvariationthatoccursin the criterion values, and each criterion value is commonly characterized and described by two values: the arithmetic mean and the standard deviation [, 4, 5]. However, many measurements of criterion values show a more or less sewed distribution. Particularly, sewed distributionsarecommonwhenmeanvaluesarelow,variances large, and values cannot be negative. Such sewed distributions often aroximately fit the log-normal distribution [6, 7]. The log-normal distribution is a continuous robability distribution of a random variable whose logarithm is normally distributed [6, 8]. It is similar to the normal distribution, but there are still several major differences between them: first, the normal distribution is symmetrical; thelog-normaldistributionissewedtotheleft.second, both forms of normal and log-normal variability are based on a variety of forces acting indeendent of one another, but a major difference is that the effects can be additive or multilicative, thus leading to normal or log-normal distributions, resectively. A variable might be distributed as log-normally if it can be thought of as the multilicative roduct of a large number of indeendent random variables each of which is ositive. Third, the sum of several indeendent normal distributed random variables itself is a normal distributed random variable. For log-normally distributed random variables, however, multilication is the relevant oeration for combining them in most alications; that is, the roduct of several indeendent log-normal random variables also follows a log-normal distribution. The lognormal distribution can model many instances, such as the loss of investment ris, the change in rice distribution of a stoc, and the failure rates in roduct tests [6, 9 ]. This is because the time series creates random variables. By taing 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 roblems in which the criterion values tae the form of log-normally distributed random variables.

2 The Scientific World Journal At resent, the SMCDM roblems, in which the criterion values tae the form of normally distributed random variables, have attracted lots of attentions from researchers [ 8]. But regarding the SMCDM roblems, in which the criterion values tae the form of log-normally distributed randomvariables,thereisstillfewrelatedresearch. Moreover, in some SMCDM situations, such as multieriods investment decision maing, medical dynamic diagnosis, ersonnel dynamic examination, military system efficiency dynamic evaluation, etc., the original decision information may be collected at different eriods (for convenience, wecallthisindofsmcdmroblemsthedynamicsmcdm roblems) [8]. Thus, accordingly, time should be taen into account, and it is an interesting and imortant research issue. In this aer, we shall focus on the dynamic SMCDM roblems, in which the criterion values tae the form of log-normally distributed random variables and the argument information is given at different eriods, and develo a method for dynamic SMCDM with log-normally distributed random variables. This method uses two new geometric aggregation oerators to aggregate the log-normally distributed criterion values, utilizes the entroy model of Shannon to generate the time weight vector, and utilizes the exectation values and variances of log-normal distributions to ran the alternatives and select the best one. To do so, this aer is organized as follows. Section introduces some oerational laws of log-normal distributions and resents a method for the comarison between two log-normal distributions. Section 3 rooses two new geometric aggregation oerators, such as the log-normal distribution weighted geometric (LNDWG) oerator and the dynamic log-normal distribution weighted geometric (DLNDWG) oerator. Section4 develos an aroach to solve the dynamic SMCDM roblems, in which the criterion values tae the form of log-normally distributed random variables, and the argument information is given at different eriods. Section 5 gives an illustrative examle. Finally, we conclude the aer in Section 6.. Preliminaries The normal distribution is a continuous robability distribution defined by the following robability density function []: f X (x) = πσ e (x μ) /σ, < x < +, () where μ is the exectation, σ>0isthe standard deviation, and σ is the variance. Generally, we use X N(μ,σ ) as a mathematical exression meaning that X is distributed normally with the exectation μ and variance σ. The log-normal distribution is a robability distribution of a random variable whose logarithm is normally distributed [6]; that is, if ln Y N(μ,σ ),theny has a log-normal distribution. The robability density function of the lognormal distribution has the following form: y μ) f Y (y) = /σ e (ln, y > 0. () yσ π If Y is distributed log-normally with arameters μ and σ, then we write Y log N(μ, σ ),andforconvenience,wecall β=log N(μ, σ ) alog-normaldistribution,andletθ be the set of all log-normal distributions. Definition (see [6]). Let β = log N(μ,σ ) and β = log N(μ,σ ) be two log-normal distributions, then () β β = log N(μ +μ,σ +σ ); () β a = log N(aμ,a σ ), a =0. It is easy to rove that all oerational results are still lognormal distributions, and by these two oerational laws, we have () β β =β β ; () (β β ) β 3 =β (β β 3 ); (3) (β β ) a =β a βa, a =0; (4) β a βa =βa +a, a,a =0. Furthermore, if log N(μ, σ ) is a log-normal distribution, then its exected value μ log and standard deviation σ log can be calculated by the following formulas [6]: μ log =e μ+(/)σ, σ log =e μ+(/)σ e σ. According to the relation between exectation and variance in statistics, in the following, we roose a method for the comarison between two log-normal distributions, which isbasedontheexectedvalueμ log and the standard deviation σ log. Definition. Let β = log N(μ,σ ) and β = log N(μ, σ ) be two log-normal distributions, then () if μ log (β ) < μ log (β ),thenβ is smaller than β, denoted by β <β ; () if μ log (β )=μ log (β ),then (3) (i) if σ log (β ) = σ log (β ),thenβ is equal to β, denoted by β =β ; (ii) if σ log (β )<σ log (β ),thenβ is bigger than β, denoted by β >β ; (iii) if σ log (β )>σ log (β ),thenβ is smaller than β, denoted by β <β. 3. The LNDWG and DLNDWG Oerators To aggregate the log-normally distributed criterion values, in what follows, based on Definition, wefirstrooseanew geometric aggregation oerator, which is called the LNDWG oerator.

3 The Scientific World Journal 3 Definition 3. Let β j = log N(μ j,σ j ) (j =,,...,n) be a collection of log-normal distributions, and let LNDWG Θ n Θ,if LNDWG w (β,β,...,β n )=β w β w β w n n, (4) then LNDWG is called the log-normal distribution weighted geometric oerator of dimension n, where w = (w,w,...,w n ) T is the weight vector of β j (,,...,n),withw j 0and n w j =. Theorem 4. Let β j = log N(μ j,σ j ) (j =,,...,n) be a collection of log-normal distributions, and let w = (w,w,...,w n ) T be the weight vector of β j (j =,,...,n), with w j 0 and n w j = ; then their aggregated result using the LNDWG oerator is also a log-normal distribution, and LNDWG w (β,β,...,β n )=log N ( n n w j σ j ). Proof. Obviously, from Definition, theaggregatedvalueby using the LNDWG oerator is also a log-normal distribution. In the following, we rove (5) by using mathematical induction on n. then () For n=,since (5) β w w μ,w σ ), (6) β w w μ,w σ ), LNDWG w (β,β )=β w β w w μ +w μ,w σ +w σ ) () If (5)holdsforn=,thatis, LNDWG w (β,β,...,β )=log N ( w j σ j ). (7) w j σ j ). (8) Then, when n=+,bydefinition,wehave LNDWG w (β,β,...,β,β + ) w j σ j ) (log N (μ +,σ + ))w + w j σ j ) log N (w + μ +,w + σ + ) + w j μ j +w + μ +, + w j σ j ). w j σ j +w + σ + ) That is, (5)holdsforn=+. Thus,basedon()and(),(5)holdsforalln N,which comletes the roof of Theorem 4. The LNDWG oerator is an extension of the well-nown weighted geometric averaging (WGA) oerator [3]. Similar tothewgaoerator,thelndwgoeratorhasthefollowing roerties. Theorem 5 (roerties of LNDWG). Let β j = log N(μ j, σ j ) (,,...,n)beacollection of log-normal distributions, and let w=(w,w,...,w n ) T be the weight vector of β j (j =,,...,n),withw j [0, ] and n w j =;thenwehavethe following. () Idemotency: If all β j (j =,,...,n)are equal, that is, β j =βfor all j,then (9) LNDWG w (β,β,...,β n ) =β. (0) () Boundary: min(β,β,...,β n ) LNDWG w (β,β,...,β n ) max(β,β,...,β n ). (3) Monotonicity: Let β j = log N(μ j,σ j ) and β j = log N(μ j,(σ j ) ) (j =,,...,n) be two collections of log-normal distributions. If β j β j,forallj,then LNDWG w (β,β,...,β n ) LNDWG w (β,β,...,β n ). () Consider that in many SMCDM roblems, the original decision information is usually collected at different eriods; then the aggregation oerator and its associated weights should not ee constant. In the following, based on Definitions and 3, we roose another new aggregation oerator for aggregating the log-normally distributed criterion values given at different eriods.

4 4 The Scientific World Journal Definition 6. Let t be a time variable and Y be a random variable, if Y log N(μ(t), (σ(t)) ) at the eriod t, where μ(t) and (σ(t)) is the exectation and the variance of Y at the eriod t, resectively, then we call log N(μ(t), (σ(t)) ) the log-normal distribution of Y at the eriod t, denoted by β(t) = log N(μ(t), (σ(t)) ). Similar to Definitions and 3,wehavethefollowing. Definition 7. Let β(t )=log N(μ(t ), (σ(t )) ) and β(t )= log N(μ(t ), (σ(t )) ) be two log-normal distributions at two different eriods t, t, resectively; then their oerational laws can be defined as follows: () β(t ) β(t ) = log N(μ(t )+μ(t ), (σ(t )) + (σ(t )) ); () (β(t )) a = log N(aμ(t ), a (σ(t )) ), a =0. Definition 8. Let β(t ) = log N(μ(t ), (σ(t )) )( =,,...,) be a collection of log-normal distributions at different eriods t ( =,,...,), and let λ(t) = (λ(t ), λ(t ),...,λ(t )) T be the weight vector of the eriods t ( =,,...,),withλ(t ) 0and λ(t )=;then we call DLNDWG λ(t) (β (t ),β(t ),...,β(t )) =(β(t )) λ(t ) (β(t )) λ(t ) (β(tn )) λ(t n) () the dynamic log-normal distribution weighted geometric (DLNDWG) oerator. Theorem 9. Let β(t ) = log N(μ(t ), (σ(t )) )( =,,...,) be a collection of log-normal distributions at different eriods t ( =,,...,),andletλ(t) = (λ(t ), λ(t ),...,λ(t )) T be the weight vector of the eriods t ( =,,...,),withλ(t ) 0and λ(t )=;then their aggregated result using the DLNDWG oerator is also a log-normal distribution, and DLNDWG λ(t) (β (t ),β(t ),...,β(t )) λ(t )μ(t ), (λ (t )) (σ (t )) ). (3) In () and(3), the time weight vector λ(t) reflects the imortance degree of different eriods, which can be given by decision maer(s) or can be obtained by using one of the existing methods, including the arithmetic series based method [4], the geometric series based method [4], the BUM function based method [5], the normal distribution based method [5], the exonential distribution based method [6], the Poisson distribution based method [7], the binomial distribution based method [8], and the average age method [5]. In the following, we roose another method to generate the time weight vector λ(t) = (λ(t ), λ(t ),...,λ(t )) T by using the entroy model of Shannon [9 3]. Consider that, on one hand, the real weights Table : scale for the relative average age τ. τ Imlication 0. Paying more attention to recent data 0.3 Paying much attention to recent data 0.5 Paying the same attention to every eriod 0.7 Paying much attention to distant data 0.9 Paying more attention to distant data 0., 0.4, 0.6, 0.8 Intermediate values between adjacent scale values of different eriods are random variables and we can utilize the time weight vector s entroy H(λ(t)) to describe the uncertainty of the time weight vector λ(t) [30], which is defined as H (λ (t)) = λ(t ) ln λ(t ). (4) On the other hand, we can associate with a concet of the relative average age of the data [3], which is defined as τ= ( ) λ (t ), (5) where τ indicates the relative average age of the data. The concet of relative average age is an extension of the average age concet [5, 3]. The average age of the data is defined by t= ( )λ(t ),butt only can be obtained by usingaroximatemethod.therelativeaverageageτ reflects the degree aid attention to the data of different eriods by the decision maers in the rocess of information aggregation and can be reresented by using a scale (Table ). When τ is close to 0, it indicates that the decision maers ay more attention to recent data; when τ iscloseto,itindicates that the decision maers ay more attention to distant data; when τ = 0.5, it indicates that the decision maers ay the same attention to every eriod, with no reference. Particularly, when τ =,thenλ(t) = (,0,...,0) T ;when τ=0,thenλ(t) = (0,0,...,) T ;whenτ = 0.5, thenλ(t) = (/,/,...,/) T. Thus, we can obtain the time weights by maximizing the time weight vector s entroy H(λ(t)) for a secified level of the relative average age τ andthenfindasetofweightsthat satisfies the following mathematical rogramming model for the λ(t ): Maximize : H(λ (t)) = λ(t ) ln λ(t ), Subject to : τ = ( ) λ (t ) λ(t ) 0, λ(t )=,,...,. (M-)

5 The Scientific World Journal 5 Table : Decision matrix R(t ). I I I 3 A log-n (385, 9. ) log-n (59, 7.9 ) log-n (39, 5.6 ) A log-n (39, 0. ) log-n (66, 8.5 ) log-n (36, 6. ) A 3 log-n (358, 8.9 ) log-n (53, 6.8 ) log-n (30, 6.8 ) A 4 log-n (468, 0.9 ) log-n (37, 7.5 ) log-n (66, 7. ) A 5 log-n (45, 9.6 ) log-n (303, 6.9 ) log-n (59, 7.5 ) Table 3: Decision matrix R(t ). I I I 3 A log-n (37, 9.6 ) log-n (5, 7.6 ) log-n (34, 5.5 ) A log-n (385, 0. ) log-n (69, 9.3 ) log-n (38, 6. ) A 3 log-n (359, 9.3 ) log-n (53, 8.6 ) log-n (35, 6.5 ) A 4 log-n (463, 0.9 ) log-n (39, 9. ) log-n (69, 7.5 ) A 5 log-n (455, 9.7 ) log-n (39, 8.9 ) log-n (55, 8.6 ) Table 4: Decision matrix R(t 3 ). I I I 3 A log-n (369, 9. ) log-n (55, 7.9 ) log-n (3, 5.7 ) A log-n (39, 9.8 ) log-n (69, 9. ) log-n (36, 6. ) A 3 log-n (35, 0.6 ) log-n (57, 8.6 ) log-n (33, 6.7 ) A 4 log-n (467,. ) log-n (36, 9.3 ) log-n (68, 7. ) A 5 log-n (469,.7 ) log-n (306, 8.8 ) log-n (58, 7.6 ) 4. A Procedure for Dynamic SMCDM with Log-Normally Distributed Random Variables In this section, we consider a dynamic SMCDM roblem where all criterion values tae the form of log-normally distributed random variables collected at different eriods. The following notations are used to deict the considered roblems. (i) A = {A,A,...,A m }: a discrete set of m feasible alternatives. (ii) I={I,I,...,I n }: a finite set of criteria. The criterion weight vector is w=(w,w,...,w n ) T,withw j 0 and n w j =. (iii) There are different eriods t ( =,,...,)with t being the most recent eriod and t being the most distant eriod. (iv) R(t )=(β ij (t )) m n ( =,,...,): log-normal distribution decision matrices at the eriods t ( =,,...,),whereβ ij (t )=log N(μ ij (t ), (σ ij (t )) ) are the criterion values of the alternatives A i with resect to the criteria I j at the eriods t (i =,,...,m,,,...,n,,,...,). Basedontheabovedecisioninformation,inthefollowing, we develo a ractical rocedure to ran the alternatives and select the most desirable one. Ste. Utilize the model (M-) to generate the time weight vector λ(t) = (λ(t ), λ(t ),...,λ(t )) T. Ste. Utilize the DLNDWG oerator: β ij = DLNDWG λ(t) (β ij (t ),β ij (t ),...,β ij (t )), (6) to aggregate all the log-normal distribution decision matrices R(t ) = (β ij (t )) m n ( =,,...,) into an overall log-normal distribution decision matrix R = (β ij ) m n = (log N(μ ij,σ ij )) m n,where β ij μ ij,σ ij ), μ ij = σ ij = (λ (t )) (σ ij (t )), i=,,...,m,,,...,n λ(t )μ ij (t ), (7) and λ(t) = (λ(t ), λ(t ),...,λ(t )) T is the time weight vector, with λ(t ) 0and λ(t )=. Ste 3. Normalize the decision matrix R = (β ij ) m n.let I b be the set of all benefit criteria, and let I c be the set of all cost criteria; then we can use the following formulas to transform the decision matrix R = (β ij ) m n into the corresonding normalized decision matrix R = ( β ij ) m n = (log N( μ ij, σ ij )) m n: μ ij μ ij = max i {μ ij }, I j I b, i=,,...,m, (8) μ ij = min i {μ ij } μ ij, I j I c, i=,,...,m, (9) σ ij σ ij = max i {μ ij }, I j I, i=,,...,m. (0) Note that standard deviation is relative to mean, so (0) is suitable for all I j I. Ste 4. Utilize the LNDWG oerator β i = LNDWG w ( β i, β i,..., 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 comlex overall values β i of the alternatives A i (i =,,...,m),where β i μ i, σ i ), μ i = σ i = n n w j μ ij w j σ ij, i=,,...,m. ()

6 6 The Scientific World Journal Table 5: Overall decision matrix R. I I I 3 A log-n ( , 6.76 ) log-n (54.384, ) log-n (3.364, 4.34 ) A log-n ( , 7.5 ) log-n ( , ) log-n (36.476, ) A 3 log-n ( , ) log-n (55.77, ) log-n (33.69, ) A 4 log-n ( , ) log-n ( , ) log-n (68.075, ) A 5 log-n (464.76, ) log-n (308.86, ) log-n ( , ) Table 6: Normalized decision matrix R. I I I 3 A log-n (0.9533, ) log-n (0.8030, ) log-n (.0000, ) A log-n (0.907, ) log-n (0.8484, 0.0 ) log-n (0.9699, ) A 3 log-n (.0000, ) log-n (0.807, ) log-n (0.9935, ) A 4 log-n (0.7583, ) log-n (.0000, 0.0 ) log-n (0.7875, ) A 5 log-n (0.764, ) log-n (0.9749, 0.00 ) log-n (0.84, ) Ste 5. Utilize (3) to calculate the exected values μ log ( β i ) and the standard deviations σ log ( β i ) of the comlex overall values β i of the alternatives A i (i=,,...,m). Ste 6. Use Definition to ran all the alternatives A i (i =,,...,m) andthenselectthebestoneaccordingtothe values μ log ( β i ) and σ log ( β i ) (i=,,...,m). 5. Illustrative Examle In this section, we use a ractical dynamic SMCDM roblem (adated from []) to illustrate the alication of the develoed aroach. An investment comany wants to invest a total amount of money in the best otion. There are five ossible comanies A i (i=,,...,5)to be invested: () A is an arms comany; () A is a comuter comany; (3) A 3 is a food comany; (4) A 4 is an auto comany; and (5) A 5 is a TV comany. The criteria considered here in selection of the five ossible comanies are the following: () I is cost; () I is net resent value; and (3) I 3 is loss, whose weight vector w = (0.300, , ) T. The investment comany evaluates the erformance of these comanies A i (i =,,...,5) in according to the criteria I j (j =,,3)and constructs the decision matrices R(t )=(β ij (t )) m n (,,3, here, t denotes 009, t denotes 00, t 3 denotes 0 ) as listed in Tables, 3, and4 (unit: ten thousands RMB). In the decision matrices R(t ) = (β ij (t )) m n, all the criterion values are exressed in log-normal distributions β ij (t ) = log N(μ ij (t ), (σ ij (t )) ),whereμ ij (t ) and (σ ij (t )) can be estimated by using statistic methods (i =,,...,5, j =,, 3,,,3). To get the best comany, the following stes are involved. Ste. Suose that the relative average age τ = 0. by taing advice from the decision maers; then we use (M-) to construct the otimization model and obtain time weight vector λ(t) = (0.089, 0.363, 0.688) T. Ste. Utilize (6) to aggregate all the log-normal distribution decision matrices R(t ) = (β ij (t )) 5 3 ( =,,3) into the overall log-normal distribution decision matrix R= (β ij ) 5 3 (Table 5). Ste 3. Utilize (8), (9), and (0) to normalize the decision matrix R=(β ij ) 5 3 into the corresonding decision matrix R = ( β ij ) 5 3 (Table 6). Note that the criterion I is benefit criterion, and the criteria I and I 3 are cost criteria. Ste 4. Utilize () 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 comlex overall values β i of the alternatives A i (i=,,...,5): β 0.946, 0.03 ), β , 0.05 ), β , 0.09 ), β , ), β , ). (3) Ste 5. Use (3) to calculate the exected values μ log ( β i )(i=,,...,5): Thus, μ log ( β ) =.4959, μ log ( β ) =.4763, μ log ( β 3 ) =.5309, μ log ( β 4 ) =.356, μ log ( β 5 ) =.374. (4) μ log ( β 3 )>μ log ( β )>μ log ( β )>μ log ( β 5 )>μ log ( β 4 ). (5)

7 The Scientific World Journal 7 Ste 6. Use Definition to ran all the alternatives A i (i =,,...,5): A 3 A A A 5 A 4. Therefore, the best alternative (comany) is A Conclusions In this aer, we have roosed two new geometric aggregation oerators, such as the LNDWG oerator and the DLNDWG oerator. Both oerators 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 reresent different asects in both the LNDWG oerator and the DLNDWG oerator. The weights of the LNDWG oerator only reflect the imortance degrees of the given log-normal distributions themselves, whereas the weights of the DLNDWG oerator only reflect the imortance degrees of different eriods. Thus, the LNDWG oerator is a time indeendent oerator, and because of taing time into account in the aggregation rocess, the DLNDWG oerator is a time-deendent oerator. The weights associated with thedlndwgoeratorcanbegivenbydecisionmaer(s) or can be obtained by using the existing methods, but we have roosed another method by using the entroy model of Shannon. We have also develoed an aroach to dynamic SMCDM, in which the criterion values tae the form of lognormally distributed random variables, and the argument information is given at different eriods. This method has been detailedly illustrated with a ractical examle. This aer enriches and develos aggregation oerator theory and SMCDM theory, and it can be widely alied in medical dynamic diagnosis, ersonnel dynamic examination, military system efficiency dynamic evaluation, and other related decision maing fields. Acnowledgments The authors are very grateful to the anonymous reviewers and the editors for their constructive comments and suggestions. 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