# Cross-Docking Center Location Selection Based on Interval Multi-Granularity Multicriteria Group Decision-Making

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## Abstract

**:**

## 1. Introduction

- We propose a new interval multi-granular uncertainty language model in a dynamic heterogeneous environment;
- We propose a conversion function to standardize different granularity values;
- We introduce a consensus threshold. If the threshold is not reached within a limited number of times, a higher status expert will be added to change the opinion;
- We add or delete experts and cross-docking center alternatives anytime, anywhere;
- We apply IMG-MCGDM to the actual cross-terminal selection problem.

## 2. Research Backgrounds

**Definition**

**1**

**.**Suppose that language evaluation set ${S}_{[0,T-1]}^{T}=\{{s}_{0}^{T},{s}_{1}^{T},\dots ,{s}_{T-1}^{T}\}$, where T is the granularity of elements in language evaluation, and it is generally odd. This kind of evaluation set is called the discrete language evaluation set.

**Definition**

**2**

**.**Let $\tilde{s}=[{s}^{T-},{s}^{T+}]$, where ${s}^{T-},{s}^{T+}\in {S}_{[0,T-1]}^{T}$, ${s}^{T-}\le {s}^{T+}$, ${s}^{T-}$, and ${s}^{T+}$ are the lower and upper limits, respectively; we then call $\tilde{s}$ the uncertain linguistic variable.

**Definition**

**3**

**.**Let ${s}_{i}^{T}\in {S}_{[0,T-1]}^{T}$ be a language table in a continuous language evaluation set, through function:

**Definition**

**4**

**.**Let ${s}_{i}^{T}\in {S}_{[0,T-1]}^{T}$ be a language table in a continuous language evaluation set through the inverse function:

**Definition**

**5.**

## 3. A Novel IMG-MCGDM Method for Heterogeneous and Dynamic Contexts

#### 3.1. Define Parameters

#### 3.2. Preference Matrix

#### 3.3. Consistency of the Multi-Granularity Uncertain Language

#### 3.4. Aggregating the Results

- Aggregation of different criteria: Each expert gives a corresponding preference interval for different criteria. However, when evaluating different criteria, each expert will have a different emphasis on different criteria, that is each expert will give a different weight ${\lambda}_{{e}_{i}}=\{{\lambda}_{i1},{\lambda}_{i2},\dots ,{\lambda}_{il}\}$ to different criteria. How to allocate the weight of the criterion and how to use it after allocation depend on the problem to be solved. Generally speaking, experts give more weight to more important criteria and less weight to less important criteria. If the criteria are almost as important, they give the same weight.Therefore, the preference aggregation matrix of each expert is calculated as follows:$${\Gamma}_{{e}_{i}}={\lambda}_{i1}{P}^{1}+{\lambda}_{i2}{P}^{2}+\cdots +{\lambda}_{il}{P}^{l}$$
- Aggregation of expert opinions: After the aggregation of preferences, it is obvious that experts rank the results for the first time. However, then, we need to aggregate the opinions of experts. When aggregating expert opinions, there are some gaps in the social status and knowledge level of experts, so experts also have different weights $W=\{{w}_{1},{w}_{2},\dots ,{w}_{n}\},{w}_{i}\in [0,1],i=1,2\dots ,n,{\sum}_{i=1}^{n}=1$ when aggregating opinions. Generally, the higher the social status and professional level of experts, the higher the weight will be given, and vice versa.Therefore, we get the final decision result through two aggregations:$$\Gamma ={\Gamma}_{{e}_{1}}{w}_{1}+{\Gamma}_{{e}_{2}}{w}_{2}+\cdots +{\Gamma}_{{e}_{l}}{w}_{n}$$

#### 3.5. Decision Results

**Definition**

**6**

**.**A ULOWA operator of dimension n is a mapping that has an associated n vector $W={\{{w}_{1},{w}_{2},\dots ,{w}_{n}\}}^{T}$ such that ${w}_{i}\in [0,1],i=1,2\dots ,n,$ ${\sum}_{i=1}^{n}=1$. Furthermore:

**Definition**

**7**

**.**Let $a=[{s}_{a-}^{T},{s}_{a+}^{T}]$ and $b=[{s}_{b-}^{T},{s}_{b+}^{T}]$ be two uncertain language variables; if:

#### 3.6. Consensus Reached

- The consensus of experts is very low: First, identify those experts who do not agree with the majority of experts; then, in the next decision-making round, other experts can reach consensus by persuading these experts. If no consensus can be reached in the maximum number of decision rounds, some operators are needed.
- The consensus of the standard value is very low: If there is a low consensus for different criteria, the focus is on the discussion of alternatives focusing on the criteria that are likely to reach consensus.

#### 3.7. Modify Alternatives Criteria Values and Experts

- Consensus reached: If the experts have a high degree of consensus in the decision-making process, which is higher than the set consensus threshold, then we believe that consensus is reached; at this time, the experts’ decision is the final decision-making result and the best alternative.
- Reach the maximum number of discussions: Experts may not get high consensus once or several times. When we reach the maximum number of discussions, we think the whole decision-making process is over.

- Number of experts: During the decision-making process, new experts may be invited to join the discussion. It is possible that when discussing a certain point, specific experts are invited to solve it together, because specific experts have unique opinions and influence status in this regard. It is also possible that experts are attracted to the problem and decide to participate. It is also possible that when making decisions, individual experts will give up on the discussion and will not participate in the following discussion.
- Criteria value: During the discussion, experts may add some previously ignored criteria or remove some unnecessary criteria. The change of criteria also has a great impact on decision-making.

## 4. Application in Cross-Docking

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Deng, X.; Qu, S.
Cross-Docking Center Location Selection Based on Interval Multi-Granularity Multicriteria Group Decision-Making. *Symmetry* **2020**, *12*, 1564.
https://doi.org/10.3390/sym12091564

**AMA Style**

Deng X, Qu S.
Cross-Docking Center Location Selection Based on Interval Multi-Granularity Multicriteria Group Decision-Making. *Symmetry*. 2020; 12(9):1564.
https://doi.org/10.3390/sym12091564

**Chicago/Turabian Style**

Deng, Xuchen, and Shaojian Qu.
2020. "Cross-Docking Center Location Selection Based on Interval Multi-Granularity Multicriteria Group Decision-Making" *Symmetry* 12, no. 9: 1564.
https://doi.org/10.3390/sym12091564