Dynamic Stochastic Multiattribute Decision-Making That Considers Stochastic Variable Variance Characteristics under Time-Sequence Contingency Environments

This paper presents a dynamic stochastic decision-makingmethod that considers the characteristics of stochastic variable variances under time-sequence contingency environments for solving stochastic decision-making problems with information from different periods and of indeterminate attribute weights. First, time-sequence weights are obtained using the technique for order preference by similarity to ideal solution (TOPSIS), corresponding with the idea of “stressing the present rather than the past.” After determining the time degree and fully considering the characteristics of normally distributed stochastic variable variances, the attribute weight is determined based on vertical projection distance. Decision-making information is then assembled from two dimensions of time-sequence and attributes, based on the two categories of weighted arithmetic averaging operators of normally distributed stochastic variables, resulting in comprehensive dynamic decision-making from single solution dimensions and a priority sequence of solutions per the order relation criteria of normally distributed stochastic variables. Finally, the validity and practicability of the methods proposed in this paper are verified using an example numerical analysis.

Compared with dynamic stochastic decision-making, which comprehensively considers multiple time-sequences, stochastic multi-attribute decision-making under a single static time-sequence environment has received more widespread attention from scholars.These research findings mainly expand on information aggregation operators [1], stochastic dominance [2,3], stochastic multiattribute analysis [4][5][6], set pair connection number analysis [7], prospect stochastic dominance [8,9], bivariate expectation in decision-making [10], probability weighted means [3], and possibility degree interval-valued numbers [11,12], of attribute weights in an unknown state-further expanding the research boundary of stochastic multiattribute decisionmaking.It is difficult for most decision-making results to be comprehensive and rationally optimised based on single time-sequence nodes.Therefore, past and current decision-making information should also be considered when conducting stochastic decision-making.Appropriately determining the weight of different time-sequences is key to successful dynamic stochastic decision-making.Current 2 Mathematical Problems in Engineering scholars have mainly developed the following objective methods based on sample information to establish time-sequence weights: discrete time-sequences [12], normal distribution [15], time-sequence information entropy [16], difference geometric progression [26], and exponential distribution [27].Moreover, scholars such as Park et al. [17], Li et al. [24], Cao et al. [28], and Liu et al. [29] et al. have conducted weight allocation on different time-sequences by relying on expert experience and subjective preference.These methods provide a reference for solving for time-sequence weights in dynamic stochastic problems, but they derive weights based on an agent's preference in each period, which inevitably results in randomness in the time-sequence weight distribution.For instance, methods based on objective information often tend to emphasise time-sequences with more information in an historical period; at these times, sequence weight increases [12,15,16].However, methods that merely rely on the subjective perception of information at different times, owing to diversity in subjective perception, greatly impact the stability of decision-making results [17,24,28,29].In real life, people pay more attention to the most current information when making judgments of the present.Therefore, this paper focuses on allocating time-sequence weights in accordance with the closeness of historical information to the latest information.
Dynamic stochastic multiattribute decision-making problems possess a time dimension and an attribute dimension; therefore, determining attribute weights is a prerequisite for assembling the attribute information required for the final decision-making result.Relevant scholars have developed a variety of methods for successfully determining attribute weight.For instance, Chen and Li have obtained attribute weights [22] by solving the grey relation function of attribute information per the grey correlation model.Scholars, such as Yue [18], Cao et al. [19], and Zhang et al. [30] et al., have designed linear optimisation models of attribute weight using TOPSIS-based Euclidean distance.Xu and Wan [31] have calculated attribute weights based on an uncertaintyordered weighted averaging operator and an attribute information ordering method that relates to attribute weight size and attribute evaluation value size and order.This idea provides a basis for attribute weight calculation in normally distributed stochastic decision-making problems; however, due to its assumed conditions and the characteristic requirements of normally distributed stochastic variables, this method's applications are limited.For instance, grey system theory and TOPSIS are based on Euclidean distance.Since Euclidean distance does not consider the distance relationship between a decision-making solution and the positive and negative ideal solutions, the weight calculation can easily lead to a "reversed order" phenomenon, in which the attribute information ordering method assigns weights per the attribute information size criteria, but attribute information is the decision-making information after inverting the normally distributed stochastic variables into common interval numbers, and thus the attribute information of the original normally distributed stochastic variables is lost completely.Therefore, attribute weight information that considers normally distributed stochastic variable characteristics is expanded upon in this paper.
Based on these analyses, this study proposes a dynamic stochastic multiattribute decision-making method that considers random variable variance characteristics in timesequence contingencies with respect to time-sequence and attribute.The remainder of this paper is organised as follows: Section 2 reviews random decision-making and related algorithms in which attribute values are normally distributed stochastic variables.Section 3 provides a time-sequence weight calculation model using the ideal time-sequence solution by introducing the concept of time degree.The validity and applicability of the proposed dynamic stochastic decision-making method are examined in Section 4 using a numerical example.Conclusions and directions for future work are discussed in Section 5.

Background
Here, we offer a concise overview of the essentials of normally distributed numbers and stochastic variables and the operation rules thereof.
Definition 1 (see [32]).A stochastic (random) variable, usually written as , is a variable whose possible values are numerical outcomes of a random phenomenon.There are two types of stochastic variables: discrete and continuous.

Determination of Time-Sequence Weight.
In dynamic stochastic multiattribute decision-making problems, timesequence weight vector (  ) = (( 1 ), ( 2 ), . . ., (  ))  is a prerequisite for assembling time-sequence information and obtaining a decision-making result.Contrary to previous similar research, for example, Sun and Xu [12] and Tan and Chen [16], which are merely based on the absolute quantity of time-sequence information or completely rely on subjective arbitrary judgments, this paper obtains a timesequence weight vector using TOPSIS and by introducing time-degree criteria in accord with the findings of Yager [34] and Zhang and Zhu [35].
Per the definition of Euclidean distance, the distance between time-weight vectors  1 (  ) = ( 1 ( 1 ),  1 ( 2 ), . . .,  1 (  ))  and  2 (  ) = ( 2 ( 1 ),  2 ( 2 ), . . .,  2 (  ))  can be denoted by Thus, the distances of a time-weight vector (  ) = (( 1 ), ( 2 ), . . ., (  ))  from the positive and negative ideal time-weight vectors are, respectively, The closeness of time-weight vector (  ) to the ideal time-weight vector can be obtained by The greater the closeness degree  is, the more the attention is attached to current information.To maximise the closeness of a time-sequence weight vector, the following model is optimised based on nonlinear programming under the given time-degree conditions Mathematical Problems in Engineering solved via Lingo software, from which time-sequence weight vector (  ) = (( 1 ), ( 2 ), . . ., (  ))  is acquired.Upon determining the time-sequence weights, the decision-making information matrix  (  ) = (  (  ),   (  ) 2 ) × of different time-sequences is assembled with the DNDNWAA () operator from Formula (4) to form a comprehensive decision-making information matrix

Determination of Attribute
Weight.Attribute weight determination in normally distributed stochastic decisionmaking seldom stems from the characteristics of the attribute itself, and, from the characteristics of normally distributed stochastic variables, the superiority or inferiority of a variable attribute is closely related to its variance size-the smaller the variance, the better the attribute [36].With the previous analysis, the variance characteristics of normally distributed stochastic variables can be used to determine attribute weights.Hence, this study obtains attribute weight by establishing an attribute-weight-solving optimised model based on stochastic variable variance characteristics guided by the shortest vertical projection distance.Vertical projection distance refers to the distance between the planes that take the connections crossing the positive and negative ideal solution points as a normal vector [37].Vertical projection distance can prevent a reversed order solution, which may occur in traditional Euclidean distance methods, and allows the characteristics of stochastic variable variances to be highlighted when calculating attribute weight in stochastic decision-making problems.Vertical projection distance has been elaborated upon in the literature [37]; in this study, it mainly provides an optimised model for determining attribute weight.
Like the comprehensive decision-making information matrix   = (  ,   2 ) × , assembled based on time-seq-uences, owing to the difference in data dimensionality of different attributes, nondimensionalisation is carried out for comprehensive decision-making information, to avoid the influence of dimensionality differences on the decisionmaking result.Considering the normal distribution number   = (  ,   ) to which each attribute corresponds, the nondimensionalisation formulas for benefit-oriented and cost-oriented indexes and for index variance are, respectively [11,12], A normal distribution number after nondimensionalisation is denoted by x = (μ  , σ ) and the decision-making information matrix is denoted by D = (μ  , σ 2 ) × .
Attribute weights are determined after conducting nondimensionalisation of the comprehensive decision-making information using vertical projection distance.The specific steps of this calculation are as follows.
Per the characteristics of stochastic variable variances,  +  = min 1≤≤ {σ  2 } can be obtained; then To determine the negative ideal variance of matrix  as  − = ( −  |  = 1, 2, . . ., ), the ideal variance is (0, 0, . . ., 0), where Set , where 1 ≤  ≤ .Vertical projection distance   of each variance and the ideal variance, because the distance between the positive and negative ideal variances is constant for each solution, may be determined by The variances in normally distributed stochastic variables   can be thought of as the degree of closeness between the variance and ideal variance of all attributes in different solutions-smaller values of   indicate that the variance and ideal variance of attributes in a solution are closer.To achieve more optimised stochastic variables, variances should be as small as possible.Therefore, these problems can be inverted to the following optimised ones: Attribute weight  = ( 1 ,  2 , . . .,   )  can be solved via Lingo software.

Decision-Making
Based on Order Relation.Upon obtaining attribute weight, comprehensive decision-making information is assembled after nondimensionalisation from attribute dimensions, per the NDNWAA  operator in Formula (3), forming a decision-making information matrix D  = (μ  , σ 2 ) ×1 constituted by a single solution dimension.Because of the characteristics of expectation and variance attributes in normally distributed stochastic variables, the greater expectation μ is, the smaller σ 2 is and the greater corresponding stochastic variable   is [36].Normally, σ 2 > 0; if σ 2 = 0, then the normally distributed stochastic variable is degraded into a real number.Therefore, the order relation criteria between any two arbitrary normally distributed stochastic variables  1 ∼ (μ 1 , σ1 2 ) and  2 ∼ (μ 2 , σ2 2 ) can be defined as follows: The order relation criteria of normally distributed stochastic variables state that the superiority-inferiority degree value (  =  1 ,  2 , . . .,   ) of related normally distributed stochastic variables can be obtained to determine the priority sequence of the solution; that is, the greater   is, the more optimised corresponding th solution is.

Summary of the New Dynamic Stochastic Decision-Making
Method.In brief, the calculation steps for dynamic stochastic multiattribute decision-making that considers stochastic variable characteristics under time-sequence contingencies are as follows.
Step 2. Assemble  moments of dynamic stochastic decisionmaking information with the time-sequence weights and Formula (4) to form a comprehensive decision-making information matrix   = (  ,   2 ) × comprising attributes and solutions.
Step 4. Further assemble decision-making information matrix D from attributes, utilising attribute weights and Formula (3) to form a comprehensive decision-making information matrix D  = (μ  , σ 2 ) ×1 that is only constituted of a single solution dimension.
Step 5. Finally determine the order (  =  1 ,  2 , . . .,   ) of each alternative solution using the order relation criteria of normally distributed stochastic variables and further determine the priority sequence of alternative solutions based on this order.

Illustrative Cases of Study
There is a major emerging technical project seeking technical partners.Five enterprises, denoted by   = { 1 ,  2 ,  3 ,  4 ,  5 }, are options for potential partners or solutions.The attribute parameters for evaluating each enterprise refer to the stochastic variables subject to stochastic normal distribution: the R&D cycle of the enterprise's emerging technology ( 1 ), the market share of its emerging technology products ( 2 ), the Mathematical Problems in Engineering life of its emerging technology products ( 3 ), the conversion rate of its emerging technology achievements ( 4 ), and the attribute set   = { 1 ,  2 ,  3 ,  4 }, corresponding to attribute weights  = ( 1 ,  2 ,  3 ,  4 ).In addition, the investor of the emerging technical project has investigated related attribute information in four historical periods for the previously mentioned enterprises and synthesised historical attribute information for reference.Suppose that the time-sequence set of different historical periods is   = { 1 ,  2 ,  3 ,  4 }.
To obtain decision-making information for each evaluation attribute that obeys or approximately obeys the law of normal distribution, we first collect sample data from completed emerging technical project developments of each enterprise as the basic data sample {   (  )}.Basic data were provided by each candidate enterprise for attribute set Then, the collected sample data were transformed; thus the expectation   (  ) and variance   (  ) 2 of each attribute index can be calculated by the sample data in accordance with Formulas (1) and (2).Finally, the basic data samples were converted into the original evaluation attribute information data shown in Tables 1-4.
The four historical time-sequences were assembled as in Step 2 and synthesised into comprehensive decision-making attribute information shown in Table 5.
Nondimensionalisation was conducted on the comprehensive original evaluation attributes in Table 5 with the procedure in Step 3, as shown in Table 6.
Attribute weights were obtained as in Step 3:  = ( 1 ,  2 ,  3 ,  4 ) = (0.421, 0.103, 0.177, 0.299) .(18) Comprehensive decision-making information was assembled from attribute information, based on Step 4, forming comprehensive normal distribution numbers constituted by target single dimensions: The priority sequence of the five alternative enterprises,  5 >  2 >  3 >  1 >  4 , was obtained by arranging each enterprise's value from largest to smallest and the optimal solution was determined to be  5 .
Further, the decision-making solution sequence was also calculated using the possibility degree method described in the literature [11,12].
Obviously, the result based on the possibility degree method described in the literature [11,12] was consistent with the result of Section 4, indicating the rationality of the method proposed in this study.Compared with the possibility degree method, the method proposed herein requires fewer nonlinear conversions during calculation and model construction, and its calculation is simple and easy to implement.In addition, compared with similar normally distributed stochastic decision-making methods, this study considers the characteristics of normally distributed stochastic variable variances while determining attribute weight; thus, the final ordering result of the method proposed in this paper is more comprehensive.

Conclusions
This paper describes a dynamic stochastic decision-making method that fully considers the characteristics of normally distributed stochastic variable variances with time-sequence Decision-making information is then assembled from the attribute and time-sequence weights via related operators, to obtain the stochastic normally distributed comprehensive decision-making matrix constituted by target single dimensions, finally providing the priority sequence of alternative solutions using order relation criteria.The primary contributions of this paper are as follows: (1) In time-sequence weight calculations, this method fully considers decision-makers' preference to "stress the present rather than the past," which more accurately models real-world decision-making processes.
(2) The characteristics of normally distributed stochastic variable variances are highlighted in attribute-weight calculations and solution priority sequence ordering, making the final ordering result more comprehensive and reasonable.
(3) The method has a clear theoretical backing, requires only simple calculation, and is easy to implement.It provides a new, broadly applicable way of thinking of dynamic stochastic decision-making problems with time-sequence contingencies.
Our future study will extend the proposed method for multiattribute decision-making problems with intuitionistic fuzzy numbers and two-dimensional linguistic information.

Additional Points
Highlights.Stochastic decision-making is based on variance and time-sequence characteristics.A time-sequence weight calculation in line with TOPSIS and time degree is introduced.A new vertical projection distance-based attributeweight model is proposed.The final decision-making result in accordance with the order relation criteria is given.

( 21 )
Then, the possibility matrices of the two compared schemes are calculated:

Table 1 :
Original evaluation attribute information at  1 .

Table 2 :
Original evaluation attribute information at  2 .

Table 3 :
Original evaluation attribute information at  3 .

Table 4 :
Original evaluation attribute information at  4 .