Multi-attribute Decision Method Based on Normal Random Variable in Economic Management Risk Control

Traditional economic evaluation theory assumes that the subject of investment decision-making has both ‘economic’ characteristics and ‘rationality’ in the course of action. Such people know all aspects of their environment. Even if this knowledge is not complete, at least it is pretty rich. Furthermore, such people have strong calculation skills, and they rely on these skills to determine which programme can reach the highest point on their preference scale [4]. However, this utterly rational assumption is difficult to realise in real life, and the rationality of the evaluation subject has limitations: first, the environment is complicated. In the form of impersonal exchange, people are faced with a complex and uncertain world. And the more transactions, the greater the uncertainty and the less complete the information. The second is that people’s ability to calculate and understand the environment is limited. Therefore, in this case, the project’s economic evaluation results will not be entirely rational.

According to the degree of completeness available and the possibility of corresponding decision failure, traditional project economic evaluation theory divides information into three states: deterministic information, risk information, and uncertainty information [5]. From the above definition of information, it can be seen that traditional project economic evaluation ignores information that cannot be reached or obtained by human rationality.

Traditional project economic evaluation methods tend to quantify the absolute number of fuzzy information in processing fuzzy information. The realisation of fuzzy linguistic values will make the information distorted or even lost. In addition, the evaluation subject may not be sure of a particular attribute value of the item and can only give an approximate range of the attribute value. Traditional economic evaluation theory rarely mentions the processing of this kind of fuzzy information.

In the traditional investment project evaluation, the good or bad of a project mainly determines the choice of the project from the level of economic income. As a result, it often leads to insufficient understanding of the role of investment projects, emphasising economic benefits and ignoring social and ecological environmental effects [6]. In this way, some projects, even though they are with good social and ecological environmental impacts, have been rejected because of poor economic returns.

The comparison of the size of standard random variables has probabilistic significance and can therefore be defined as:

Definition 1: We set ξ ∼ N(μ_A,σ_A²) and η ∼ N(μ_B,σ_B²) and they are independent of each other, then define the comparison result of the size of standard random variables ξ and η as (3) p{ξ>η}=Φ(μA−μBσA2+σB2) p\{\xi > \eta \} = \Phi \left({{{{\mu _A} - {\mu _B}} \over {\sqrt {\sigma {A^2} + \sigma {B^2}}}}} \right) (4) p{η>ξ}=Φ(μB−μAσA2+σB2) p\{\eta > \xi \} = \Phi \left({{{{\mu _B} - {\mu _A}} \over {\sqrt {\sigma {A^2} + \sigma {B^2}}}}} \right)

Where Φ(·) is the cumulative distribution function of the standard normal distribution: (5) Φ(x)=12π∫−∞xφ(t)dt,φ(x)=12πe−x2/2 \Phi (x) = {1 \over {\sqrt {2\pi}}}\int_{- \infty}^x \varphi (t)dt,\quad \varphi (x) = {1 \over {\sqrt {2\pi}}}{e^{- {x^2}/2}}

Obviously, according to the nature of the normal distribution: (6) P{ξ>η}=Φ(μA−μBσA2+σB2)=1−Φ(μB−μAσA2+σB2)=1−P{η>ξ} P\{\xi > \eta \} = \Phi \left({{{{\mu _A} - {\mu _B}} \over {\sqrt {\sigma {A^2} + \sigma {B^2}}}}} \right) = 1 - \Phi \left({{{{\mu _B} - {\mu _A}} \over {\sqrt {\sigma {A^2} + \sigma {B^2}}}}} \right) = 1 - P\{\eta > \xi \}

Namely: P{ξ > η} + P{η > ξ} = 1. It is not difficult to see from the above Definition 1:

(1) If μ_A > μ_B, there must be: (7) P{ξ>η}=Φ(μA−μBσA2+σB2)>12 P\{\xi > \eta \} = \Phi \left({{{{\mu _A} - {\mu _B}} \over {\sqrt {\sigma {A^2} + \sigma {B^2}}}}} \right) > {1 \over 2}

The mean of the normal distribution plays a leading role in the comparison of standard random variables. The following proves that the comparison probability of the size of the standard random variable given by Equation (7) is moderately transitive.

(2) If μ_A = μ_B, there must be: (18) P{ξ>η}=Φ(0)=1/2,P{ξ>η}=P{η>ξ}=1/2 P\{\xi > \eta \} = \Phi (0) = 1/2,\quad P\{\xi > \eta \} = P\{\eta > \xi \} = 1/2

It can be seen that this definition cannot distinguish when the mean is equal. To this end, it is necessary to compare the variance of the normal distribution further. In the case of equal means, the smaller the variance, the better the solution. From this, the basic principle of the ordering of MADM schemes with standard random variables can be expressed as follows: First, the schemes are sorted according to the mean value of the total evaluation value of each scheme. The larger the mean, the better the scheme. Then, when the mean values are equal, compare the variances. Again, the smaller the variance, the better the solution. When the variances are also equal, it is determined that the two schemes are in the same order.

This paper proposes the ordering steps of MADM schemes with standard random variables as follows:

Step 1: We set the attribute weight vector as ω = (ω₁,ω₂, … , ω_n)_T;

The choice of multi-attribute economic evaluation methods for investment projects depends on the environment in which the investment project system is located (the combination of project information environment, time and space environment, and subject environment).

Complete project information and no need to investigate dynamic benefits Project information are complete means that the project meets the following conditions simultaneously: First, the project attribute index value can be determined [10]. Quantitative indicators of the project can calculate the quantitative value, and qualitative indicators can be given based on a certain level of superiority and inferiority. They can give a clear quantitative score based on this. Second, the attribute weight value of an investment project can be determined. The attribute weights of investment projects can be determined in advance or determined by the subjective weighting method or objective weighting method.

We can choose the corresponding multi-attribute economic evaluation method according to the different combinations of investment project environments. Then, when the investment project information is complete, and there is no need to examine the dynamic benefits of the project, according to the corresponding relationship between the investment project information environment and the investment project subject environment. At this time, we adopt the accurate individual number evaluation method to obtain good investment projects.

When the investment project information is complete, and the dynamic benefits of the investment project need to be examined, the economic evaluation of the investment project starts from two dimensions [13]. One is the individual accurate number evaluation, which evaluates the comprehensive benefits of the project at the current point in time. The second is time series evaluation, which evaluates the total benefit and benefits increment during the inspection period.

When the investment project information is not complete, we do not need to examine the dynamic benefits of the project. The group fuzzy economic evaluation method can be carried out by first carrying out the individual fuzzy evaluation. Then integrate the individual fuzzy evaluation results to achieve.

When the investment project information is incomplete, and the dynamic benefits of the project need to be examined, we need to adopt the group fuzzy economic evaluation method. This needs to eliminate as much as possible the influence of individual decision-making preferences while adopting time-series economic evaluation methods to evaluate the dynamic benefits of the project over time.

According to the multi-attribute economic evaluation flowchart shown in Figure 1, we first judge whether the project information is confirmed or not. Then, in a deterministic environment, a multi-attribute economic evaluation is carried out at the individual level. Individual fuzzy evaluation is carried out in an uncertain environment first, and then the group fuzzy evaluation ranking result is obtained by weighted integration.

Fig. 1

Suppose there are four schemes, and the evaluation indicators involved in each scheme have three attributes. Then the decision matrix we give is (20) R=(rij)4×3=[{0.49,0.023}{0.52,0.005}{0.73,0.003}{0.45,0.047}{0.61,0.010}{0.69,0.090}{0.66,0.053}{0.68,0.020}{0.50,0.033}{0.64,0.055}{0.70,0.065}{0.70,0.010}] R = {({r_{ij}})_{4 \times 3}} = \left[ {\matrix{{\{0.49,0.023\}} & {\{0.52,0.005\}} & {\{0.73,0.003\}} \cr {\{0.45,0.047\}} & {\{0.61,0.010\}} & {\{0.69,0.090\}} \cr {\{0.66,0.053\}} & {\{0.68,0.020\}} & {\{0.50,0.033\}} \cr {\{0.64,0.055\}} & {\{0.70,0.065\}} & {\{0.70,0.010\}} \cr}} \right]

Among them, r_ij = {μ_ij,σ_ij²}. If the weight vector of these three attributes is (21) ω=(0.258,0.373,0.369)T \omega = {(0.258,0.373,0.369)^T}

Then it is not difficult to calculate the comprehensive evaluation value of these four options (22) G1∼N(0.5898,0.0026),G2∼N(0.5982,0.0168) {G_1} \sim N(0.5898,0.0026),\quad {G_2} \sim N(0.5982,0.0168) (23) G3∼N(0.6084,0.0108),G4∼N(0.6845,0.0141) {G_3} \sim N(0.6084,0.0108),\quad {G_4} \sim N(0.6845,0.0141)

According to the scheme ranking method given in Section “The sorting method of the scheme”, A₄ ≻ A₃ ≻ A₂ ≻ A₁. Scheme 4 is the optimal scheme. The ranking results are the same as the actual results, but the method is more superficial.