Random multi-attribute decision-making is a finite option selection problem related to multiple attributes, and the attribute values are random variables. Its application and supply chain risk management can transform interval decision numbers and fuzzy decision numbers into standardised decisions. Based on this research background, the article provides a primary method to determine the randomness of standard random variables based on expectations and variance through theoretical analysis. Second, the article determines the range of the total utility value of each supply chain selection plan based on the 3
Keywords
- normal random variables
- mixed multi-attribute decision-making
- entropy
- subjective weight
- objective weight
- supply chain risk management
MSC 2010
- 90B50
The complexity of decision-making issues leads to decision-making indicators, often including quantitative and qualitative indicators. The hybrid multi-attribute decision-making model can handle quantitative and qualitative indicators, which is more in line with actual decision-making situations. However, due to the complexity of the attributes and the bounded rationality of the decision-maker, it is difficult for the weight directly given by the decision maker's subjective judgement to be consistent with the actual situation [1]. The article presents a mathematical programming model that integrates decision-makers personal weight preference information and objective decision matrix information. At the same time, we propose a combined weight algorithm that can integrate all kinds of subjective weights and n−1 kinds of objective weights.
Suppose
We define
We assume that [
We assume that
The fuzzy maximum set is a fuzzy subset
In this way, the fuzzy number
In this way, the mixed decision matrix
The positive ideal solution is
After studying various methods of determining objective weights, some scholars have proposed mathematical optimisation models [4]. These models often use the following methods when solving objective weights. We transform the exact number decision matrix
Some scholars pointed out that the weight distribution mechanism as well as the meaning of model (8) is not precise, and it does not conform to the principle of entropy model weight distribution. Through case analysis, it is found that small changes in the decision matrix will lead to significant changes in weights, so the weight distribution mechanism of the model (8) is unreasonable. So, we proposed an entropy model to solve the objective weights [5]. The main methods are as follows:
To assign weights, the entropy model is guided by the following principle. If the evaluation value of each scheme under the
The weight distribution is not flexible. The entropy model defines
It is easy to cause too much weight difference. In actual decision-making, when an indicator is introduced into the evaluation system, it can generally be considered that it cannot exceed and equal to zero [6]. That is, the maximum weight cannot be >10 times the minimum weight.
The construction of a suitable mathematical model requires a deep understanding of the specific situation and rich mathematical experience of the issues involved in the decision-making problem. This isn’t easy. To judge the rationality of the objective weight model, we give Judgement Theorem 1.
The objective weight obtained by this model can reflect the information of the decision matrix. When the decision matrix changes, the degree of weight change should be consistent with the degree of change of the decision matrix.
Based on the model (8) and entropy model (10), we transform the exact number decision matrix
For the normalised matrix
Here are two examples to illustrate the difference between the entropy coefficient model (14), the model (8), and the entropy model (10):
Suppose there is a decision matrix
We use the entropy model (10) to calculate. If the evaluation value of each scheme under the jth attribute tends to be the same, that is,
3. We use the entropy coefficient model (15) for solving objective weight. According to formula (15), we can get:
In this way, we can set the system parameters
There is a decision matrix
Using model (8), we can get: w = (0.1384, 0.2232, 0.2783, 0.3601).
Using the entropy model (10), we can get: w = (0.4630, 0.3992, 0.1378, 0).
Using the entropy coefficient model (14), when the system parameter is
Using model (8), we can get w=(0.1821, 0.2937, 0.3662, 0.1579).
Using the entropy model (10), we can get w=(0.4630, 0.3992, 0.1378, 0).
Use the entropy coefficient model (14). When the system parameter is
When
Suppose that the decision-maker directly gives the emotional weight of the attribute as
If the scheme ranking is less sensitive to changes in the total weight, then the total weight is relatively reasonable.
The distance from the first plan to the positive ideal plan is:
The distance from the
The relative closeness of the
Arrange the pros and cons of the schemes in descending order of
Use the weighted trade-off coefficient
A company's production line needs to choose robots among the four submitted models. Now four suppliers are providing four solutions:
The six attribute values of the four robots
2 | 2.5 | [55,56] | [94,114] | Normal (0.4, 0.5, 0.6) | Very high (0.85, 0.9, 0.95, 1) | |
2.5 | 2.7 | [30,40] | [84,104] | Low (0.2, 0.3, 0.4) | Normal (0.3, 0.4, 0.6, 0.7) | |
1.8 | 2.4 | [50,60] | [100,120] | High (0.6, 0.7, 0.8) | High (0.5, 0.6, 0.8, 0.9) | |
2.2 | 2.6 | [35,45] | [90,110] | Normal (0.4, 0.5, 0.6) | Normal (0.3, 0.4, 0.6, 0.7) |
Subjective weight
The positive ideal solution is
Use model (14). If we set the system parameter
The distance from each plan to the positive ideal plan is
The distance from each plan to the negative ideal plan is
The relative closeness of each scheme to the positive ideal scheme is
So, the sorting result:
Sensitivity analysis. Sensitivity analysis observes the influence of the trade-off coefficient
0 | 0.6726 | 0.0944 | 0.7366 | 0.3451 |
0.2 | 0.6705 | 0.0994 | 0.7358 | 0.3472 |
0.4 | 0.6683 | 0.1053 | 0.735 | 0.3496 |
0.6 | 0.666 | 0.1118 | 0.734 | 0.3522 |
0.8 | 0.6637 | 0.1191 | 0.7329 | 0.355 |
1 | 0.6613 | 0.1296 | 0.7316 | 0.358 |
When the system parameter is
2. Using model (8), we can get the objective weight W=(0.0777,0.5411,0.0392,0.3091,0.0157,0.0171). The sensitivity analysis is shown in Figure 3.
From Figure 3, the weight is more sensitive to the change of the program ranking, and it is difficult for decision-makers to choose. The main reason is that the objective weight is not very reasonable, and the evaluation value of each scheme under the second attribute is the most consistent. This causes the weight of the second attribute to be too large and it exceeds 50%. It is 35 times the minimum weight. In Figures 1 and 2, the pros and cons of the scheme are more pronounced, so it is easier for decision-makers to judge.
This article studied the mixed multi-attribute decision-making problem with quantitative and qualitative indicators and converts interval and fuzzy numbers into exact numbers to obtain a standardised judgement matrix. This method can resolve some of the issues involved with the mixed decision-making problem and simplify the calculation with the undefined index being the non-linear fuzzy number. We have established an entropy coefficient model for solving the objective weights of attributes. This model has a certain degree of flexibility, and the obtained weights are relatively reasonable.
The six attribute values of the four robots
2 | 2.5 | [55,56] | [94,114] | Normal (0.4, 0.5, 0.6) | Very high (0.85, 0.9, 0.95, 1) | |
2.5 | 2.7 | [30,40] | [84,104] | Low (0.2, 0.3, 0.4) | Normal (0.3, 0.4, 0.6, 0.7) | |
1.8 | 2.4 | [50,60] | [100,120] | High (0.6, 0.7, 0.8) | High (0.5, 0.6, 0.8, 0.9) | |
2.2 | 2.6 | [35,45] | [90,110] | Normal (0.4, 0.5, 0.6) | Normal (0.3, 0.4, 0.6, 0.7) |
β influence on the closeness of each plan
0 | 0.6726 | 0.0944 | 0.7366 | 0.3451 |
0.2 | 0.6705 | 0.0994 | 0.7358 | 0.3472 |
0.4 | 0.6683 | 0.1053 | 0.735 | 0.3496 |
0.6 | 0.666 | 0.1118 | 0.734 | 0.3522 |
0.8 | 0.6637 | 0.1191 | 0.7329 | 0.355 |
1 | 0.6613 | 0.1296 | 0.7316 | 0.358 |
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