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

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.


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][2][3][4][5][6][7][8][9][10][11][12][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 lognormal 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][20][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.

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The Scientific World Journal 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][2][3][4][5][6][7][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 multiperiods 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. Section 2 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.

Preliminaries
The normal distribution is a continuous probability distribution defined by the following probability density function [22]: where is the expectation, > 0 is the standard deviation, and 2 is the variance. Generally, we use ∼ ( , 2 ) as a mathematical expression meaning that is distributed normally with the expectation and variance 2 .
The log-normal distribution is a probability distribution of a random variable whose logarithm is normally distributed [16]; that is, if ln ∼ ( , 2 ), then has a log-normal distribution. The probability density function of the lognormal distribution has the following form: If is distributed log-normally with parameters and , then we write ∼ log − ( , 2 ), and for convenience, we call = log − ( , 2 ) a log-normal distribution, and let Θ be the set of all log-normal distributions.
Definition 1 (see [16]). Let 1 = log − ( 1 , 2 1 ) and 2 = log − ( 2 , 2 2 ) be two log-normal distributions, then It is easy to prove that all operational results are still lognormal distributions, and by these two operational laws, we have is a log-normal distribution, then its expected value log and standard deviation log can be calculated by the following formulas [16]: 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 value log and the standard deviation log .

The LNDWG and DLNDWG Operators
To aggregate the log-normally distributed criterion values, in what follows, based on Definition 1, we first propose a new geometric aggregation operator, which is called the LNDWG operator.
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.
(1) Idempotency: If all ( = 1, 2, . . . , ) are equal, that is, = for all , then (2) Boundary: 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 Definitions 1 and 3, we propose another new aggregation operator for aggregating the log-normally distributed criterion values given at different periods.

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The Scientific World Journal Definition 6. Let be a time variable and be a random variable, if ∼ log − ( ( ), ( ( )) 2 ) at the period , where ( ) and ( ( )) 2 is the expectation and the variance of at the period , respectively, then we call log − ( ( ), ( ( )) 2 ) the log-normal distribution of at the period , denoted by ( ) = log − ( ( ), ( ( )) 2 ). Similar to Definitions 1 and 3, we have the following. ( 2 )) 2 ) be two log-normal distributions at two different periods 1 , 2 , respectively; then their operational laws can be defined as follows: the dynamic log-normal distribution weighted geometric (DLNDWG) operator.
In (12) and (13), the time weight vector ( ) 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 ( ) = ( ( 1 ), ( 2 ), . . . , ( )) by using the entropy model of Shannon [29][30][31]. Consider that, on one hand, the real weights 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 values of different periods are random variables and we can utilize the time weight vector's entropy ( ( )) to describe the uncertainty of the time weight vector ( ) [30], which is defined as On the other hand, we can associate with a concept of the relative average age of the data [31], which is defined as where 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 = ∑ =1 ( − ) ( ), but only can be obtained by using approximate method. The relative average age 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 is close to 0, it indicates that the decision makers pay more attention to recent data; when is close to 1, it indicates that the decision makers pay more attention to distant data; when = 0.5, it indicates that the decision makers pay the same attention to every period, with no preference. Particularly, when = 1, then ( ) = (1, 0, . . . , 0) ; when = 0, then ( ) = (0, 0, . . . , 1) ; when = 0.5, then ( ) = (1/ , 1/ , . . . , 1/ ) . Thus, we can obtain the time weights by maximizing the time weight vector's entropy ( ( )) for a specified level of the relative average age and then find a set of weights that satisfies the following mathematical programming model for the ( ):

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. Based on the above decision information, in the following, we develop a practical procedure to rank the alternatives and select the most desirable one.
Note that standard deviation is relative to mean, so (20) is suitable for all ∈ .
Step 4. Utilize the LNDWG operator to aggregate the overall criterion values̃in the th column of the normalized decision matrix̃= (̃) × into the complex overall values̃of the alternatives 6 The Scientific World Journal   Step 5. Utilize (3) to calculate the expected values log (̃) and the standard deviations log (̃) of the complex overall values of the alternatives ( = 1, 2, . . . , ).

Illustrative Example
In this section, we use a practical dynamic SMCDM problem (adapted from [2]) to illustrate the application of the developed approach.
To get the best company, the following steps are involved.
Step 1. Suppose that the relative average age = 0.2 by taking advice from the decision makers; then we use (M-1) to construct the optimization model and obtain time weight vector ( ) = (0.0819, 0.2363, 0.6818) .

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 lognormally 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.