# Probabilistic Linguistic Preference Relation-Based Decision Framework for Multi-Attribute Group Decision Making

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## Abstract

**:**

## 1. Introduction

- (1)
- Investigation of decision process using the preference relation proves to be effective than investigation using attribute driven methods [28]. The reason for this is evident from the ease of pair-wise comparison mechanism, which allows DMs to produce sensible preference information about each object with respect to a specific criterion. Also, the process of pair-wise comparison closely resembles with the practical decision process. Thus, motivated by the power of pair-wise comparison, we set our proposal in this context.
- (2)
- Since PLPR is a recent research topic, the challenge of automatic filling of missing values under PLPR context needs to be addressed. DMs often get confused between objects (alternatives) due to external pressure and lack of sufficient knowledge. This forces DMs to be ignorant and hesitant towards a certain pair of objects which eventually leads to missing values in the preference relation(s).
- (3)
- Checking and repairing the consistency of PLPRs in an automated fashion by using a systematic procedure is also an interesting challenge to be addressed. The consistency of preference relation is substantial aspect for rational and reasonable decision-making. Due to various external pressures, DMs often face difficulty in providing a consistent preference relation for evaluation and manual repairing of the preference relation is an ordeal and unreasonable. Though, Xie et al. [32] presented a method for consistency check and repair, they are complex and computationally intensive as they involve logarithmic function and iterative calculation of Eigen vectors.
- (4)
- Furthermore, extension of ranking methods under PLPR context is also an attractive challenge to be addressed for sensible prioritization of objects. DMs prefer systematic scientific procedure for selection of objects rather than random guess. Though, Xie et al. [32] extended AHP method, they converted the PLTS information into single value by using possibility degree measure which causes potential loss of information leading to unreasonable prioritization of objects.

- (1)
- In the first phase of the proposal, a new automated procedure for filling the missing values is presented.
- (2)
- Following this, a new systematic procedure is proposed for checking the consistency of PLPRs and inconsistent PLPRs are repaired automatically in an iterative manner. Unlike method discussed in [32], the proposed procedure uses simple and straightforward operational law(s) of PLEs.
- (3)
- Further, in the second phase of the proposal, a new extension to AHP method under PLPR context is presented for suitable selection of the object from the set of objects. Unlike method [32], the proposed extension for AHP retains the PLTS information throughout the formulation and mitigates information loss which allows reasonable prioritization of objects.
- (4)
- Finally, the practicality, strength, and weakness of the proposal are realized by using green supplier selection problem.

## 2. Preliminaries

**Definition**

**1**

**.**Consider a LTS $S$ defined by $\left\{{s}_{\alpha}|\alpha \in \left[-n,n\right]\right\}$ with $n$ being the limits of the term set and ${s}_{-n}$ and ${s}_{n}$ are the lower and upper bounds of the term set. The ${s}_{\alpha}$ is a linguistic term set with the following characteristics:

- (a)
- ${s}_{\alpha}$and${s}_{\beta}$are two linguistic term sets with${s}_{\alpha}>{s}_{\beta}$only if$\alpha >\beta $.
- (b)
- The negation of${s}_{\alpha}$is denoted by$neg\left({s}_{\alpha}\right)$and is given by$neg\left({s}_{\alpha}\right)={s}_{-\alpha}$. As a special case,$neg\left({s}_{0}\right)={s}_{0}$.

**Definition**

**2**

**.**Consider a LTS $S$ defined by $\left\{{s}_{\alpha}|\alpha \in \left[-n,n\right]\right\}$, then the PLTS is defined by:

^{th}linguistic term and ${p}^{t}$ is the associated occurring probability of the t

^{th}linguistic term.

**Note 1:**The concept of PLTS [23] is a generalization to LDA [22] that allows partial ignorance $\left({{\displaystyle \sum}}_{i}{p}_{i}\le 1\right)$ in preference elicitation and the concept of incomplete LDA [24] is similar to PLTS.

**Remark**

**1.**

**Definition**

**3**

**.**The PLPR is a square matrix of the form $R={\left({L}_{ij}^{t}\left({p}_{ij}^{t}\right)\right)}_{n\times n}$ with ${L}_{ii}^{t}=\left\{{s}_{0}\right\}$, ${p}_{ji}^{t}={p}_{ij}^{t}$ and ${L}_{ji}^{t}=neg\left({L}_{ij}^{t}\right)$.

**Definition**

**4**

**.**Consider two PLEs, ${L}_{1}\left(p\right)$ and ${L}_{2}\left(p\right)$ as defined before. Then,

**Remark**

**2.**

## 3. Proposed Decision Framework under Probabilistic Linguistic Preference Relation (PLPR) Context

#### 3.1. Proposed Architecture of PLPR Based Decision Framework

#### 3.2. Proposed Automatic Procedure for Filling Missing Values and Consistency Check and Repair for PLPRs

**Step 1:**Consider a PLPR $R={\left\{{L}_{ij}^{t}\left({p}_{ij}^{t}\right)\right\}}_{n\times n}$ which has PLEs. Identify the instance which is missing. If $j>i+1$, then the missing instance can be automatically estimated (follow steps below), else follow Equation (4).

**Step 2:**When $j>i+1$, apply Equation (5) to automatically estimate the missing values.

**Note 2:**The result from Equation (5) is also a PLE and the values that go out of bounds when $\oplus $ operator is applied are transformed using Remark 2.

**Step 3:**Check the consistency of the matrix $R={\left(\left\{{L}_{ij}^{t}\left({p}_{ij}^{t}\right)\right\}\right)}_{n\times n}$ by using Equations (6) and (7).

**Step 4:**Calculate the distance between ${R}_{ij}$ and ${R}_{ij}^{z}$ by using Equation (8) to determine the consistency index ($CI$).

**Note 3:**The distance formula described in Equation (8) obeys the desirable distance properties viz., non-negative, non-degenerate, symmetric and transitive.

**Step 5:**The consistency values obtained from step 4 $\left(CI\left(R\right)\right)$ are compared with the standard consistency value $\left(\tilde{CI}\left(R\right)\right)$ (suggested as 0.05 by DMs). If $CI\left(R\right)\le \tilde{CI}\left(R\right)$ then, $R$ is acceptable; else $R$ is unacceptable and automatic repairing must be done by following the steps below.

**Step 6:**Repair the inconsistent PLPR automatically by using Equation (9).

**Step 7:**Repeat the steps 5 and 6 iteratively till a PLPR of acceptable consistency is obtained.

#### 3.3. Proposed Analytic Hierarchy Process (AHP) Method under PLPR Context

**Step 1:**Define the problem under multi-attributes decision-making context and determine the number of objects, attributes and DMs. Use PLEs as preference information.

**Step 2:**Suppose, m objects and n attributes are considered, n PLPRs of order $\left(m\times m\right)$ is formed. Following this, a PLPR of order $\left(n\times n\right)$ is formed for the attributes.

**Step 3:**Check the consistency of all PLPRs using the procedure presented in Section 3.2 and repair the inconsistent PLPR. Apply Equation (2) to the PLPR of order $\left(n\times n\right)$. This forms a weight vector for the attributes which is probabilistic linguistic in nature.

**Step 4:**Following step 3, we aggregate the PLEs from $\left(m\times m\right)$ matrices using Equation (2) to form a decision matrix with PLTS information of order $\left(m\times n\right)$ where m is the number of alternatives and n is the number of attributes.

**Step 5:**The attribute weights and decision matrix are taken from steps 3 and 4 respectively and Equation (2) is applied to obtain a vector of order $\left(m\times 1\right)$ for each of the $m$ alternatives.

**Step 6:**The vector obtained from step 6 contains PLTS information which is used for the final ranking by applying Equation (10).

## 4. Numerical Example

#### 4.1. Green Supplier Selection for Healthcare Center

_{2}emission and energy usage, healthcare must tune their thoughts towards green technologies and selection of equipment suppliers who follow green standards ISO 14000 and 14001 actively.

**Hygiene and safety**$\left({\mathit{C}}_{\mathbf{1}}\right)$**:**This attribute measures the amount of care given by the suppliers in adhering to the green technologies and standards.**Quality of equipment**$\left({\mathit{C}}_{\mathbf{2}}\right)$**:**The longevity and correctness of the product is determined from this attribute.**On-time delivery**$\left({\mathit{C}}_{\mathbf{3}}\right)$**:**Delivery of product at right time under critical scenario is determined from this attribute.**Cost of equipment**$\left({\mathit{C}}_{\mathbf{4}}\right)$**:**This attribute determines the total cost involved during the product life cycle.

**Step 1:**Construct four PLPRs of order $\left(4\times 4\right)$ with PLTS information. Each criterion is taken and the DMs form pairwise comparison matrices with each supplier over a specific criterion.

**Step 2:**Construct one PLPR matrix of order $\left(4\times 4\right)$ to determine the weights of the attributes. The Equation (2) is used to determine the weight of each criterion. The weight values are probabilistic linguistic in nature.

**Step 3:**Check the consistency of all PLPRs and repair those PLPRs that are inconsistent in nature. Follow the procedure from Section 3.2 for automatic repairing of inconsistent PLPR.

**Step 4:**Apply the proposed ranking method from Section 3.3 over the consistent PLPRs and obtain a suitable supplier for the process. The four PLPR matrices of order $\left(m\times m\right)$ and attributes weight matrix of order $\left(n\times 1\right)$ are aggregated using the $\oplus $ operator defined in Definition 4. The resultant matrix is given in Table 5.

**Step 5:**Compare the strength and weakness of the proposal with state of the art methods. Readers are encouraged to refer Section 5 for the same.

#### 4.2. Green Supplier Selection for Automobile Industry in India

**Step 1.**Form the PLPRs supplier wise for each criterion. This produces four matrices of order $4\times 4$ that correspond to one preference relation for each criterion.

**Step 2:**Fill the missing values by using the proposed procedure given in Section 3.2. The missing values are represented by “X” in Table 7 and these values are filled systematically using procedure proposed in Section 3.2 and the values are PLEs (refer Table 8).

**Step 3:**Determine the consistency of each PLPR and repair the inconsistent PLPR iteratively using the proposed procedure given in Section 3.2. The $d\left({R}_{1},{R}_{1}^{z}\right)$ is 0.13 which is inconsistent and it is made consistent in two iterations with $d\left({R}_{1},{R}_{1}^{z}\right)$ as 0.013. Further, $d\left({R}_{2},{R}_{2}^{z}\right)$ is 0.091 which is inconsistent and it is made consistent in two iterations with $d\left({R}_{2},{R}_{2}^{z}\right)$ as 0.03. The $d\left({R}_{3},{R}_{3}^{z}\right)$ and $d\left({R}_{4},{R}_{4}^{z}\right)$ are 0.14 and 0.11 respectively which is inconsistent and it is made consistent with in a single iteration with $d\left({R}_{3},{R}_{3}^{z}\right)$ as 0.021 and 0.025 respectively.

**Step 4:**From step 3, we obtain consistent PLPRs which are used for prioritizing green suppliers and selection of a suitable green supplier for the automobile industry. The extended AHP under PLPR context (from Section 3.3) is used for prioritization of green suppliers.

**Step 5:**Compare the superiority and weakness of the proposed framework with other methods (refer Section 5 for details).

## 5. Comparative Analysis: PLPR Based Decision Framework vs. Others

- (1)
- (2)
- Though, method [27] presents a procedure for consistency check and repair, it is complex and computationally intensive as it involves Eigen vector calculation and uses logarithmic function. To circumvent the issue, the proposed framework presents a systematic procedure for consistency check and repairing inconsistent PLPRs. The procedure automatically repairs inconsistency in an iterative manner with less intervention from DMs. The proposed procedure is computationally feasible as it uses operational law(s) of PLTS.
- (3)
- Method [21] extends AHP under HFLTS context for ranking objects which loses potential probability information and hence, produces unreasonable ranking of objects. Further, method [27] extends AHP under PLTS context but, loses some information when transforming PLTS information to single values using possibility degree. To circumvent the issue, the proposed framework presents a method for ranking objects by extending the popular AHP under PLPR context. The preference information is retained throughout the formulation and hence, information loss is mitigated in an effective manner.
- (4)
- The practicality of the proposed framework is also realized by solving green supplier selection problem for a healthcare center.
- (5)
- Also, from the time complexity analysis, we can observe that proposed decision framework and method [27] has three crucial operations viz., (a) filling missing values, (b) check & repair of inconsistent and (c) ranking of objects with $m$ objects and $n$ attributes. Operation (a) takes $O\left({m}^{2}\right)$ time complexity, operation (b) takes $O\left({m}^{2}\right)$ time complexity and operation (c) takes $O\left({m}^{2}\left(n+1\right)\right)$. So, the complexity of the proposed decision framework is $O\left(3{m}^{2}+n{m}^{2}\right)\approx O\left({m}^{2}\right)$. In contrary, the complexity of [27] (by similar analysis) is $O({m}^{3}+{m}^{2}+n{m}^{2})\approx O\left({m}^{3}\right)$ which is evidently complex than the proposed decision framework.

- (1)
- It is computationally complex because of the idea of pair-wise comparison.
- (2)
- Also, the agility for judgment is slow (refer Table 7) because of the pair-wise comparison.

## 6. Conclusions

- (1)
- The proposed framework can be used as a “ready-to-use” framework for rational decision-making under uncertain situations.
- (2)
- Also, the consistency of the information is ensured by using a systematic procedure without loss of substantial preference information.
- (3)
- This framework can be used by the managers for proper planning of inventory and management of profit and risk the organization.
- (4)
- Further, customers can use this framework as a supplementary aid for making rational decisions.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Attributes | Supplier | ${\mathit{S}}_{1}$ | ${\mathit{S}}_{2}$ | ${\mathit{S}}_{3}$ | ${\mathit{S}}_{4}$ |
---|---|---|---|---|---|

${C}_{1}$ | ${S}_{1}$ | $\left\{\begin{array}{c}2,\left(0.3\right),\\ 1,\left(0.2\right),\\ -2,\left(0.42\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.33\right),\\ 1,\left(0.44\right),\\ -1,\left(0.2\right)\end{array}\right\}$ | $\left\{\begin{array}{c}1,\left(0.35\right),\\ -1,\left(0.25\right),\\ -2,\left(0.3\right)\end{array}\right\}$ | |

${S}_{2}$ | $\left\{\begin{array}{c}-1,\left(0.3\right),\\ 1,\left(0.25\right),\\ -2,\left(0.42\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.22\right),\\ 1,\left(0.4\right),\\ -2,\left(0.35\right)\end{array}\right\}$ | |||

${S}_{3}$ | $\left\{\begin{array}{c}2,\left(0.42\right),\\ -2,\left(0.25\right),\\ -1,\left(0.15\right)\end{array}\right\}$ | ||||

${S}_{4}$ | |||||

${C}_{2}$ | ${S}_{1}$ | $\left\{\begin{array}{c}2,\left(0.28\right),\\ 1,\left(0.35\right),\\ -1,\left(0.3\right)\end{array}\right\}$ | $\left\{\begin{array}{c}0,\left(0.22\right),\\ -2,\left(0.45\right),\\ -1,\left(0.15\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.18\right),\\ 1,\left(0.35\right),\\ -1,\left(0.25\right)\end{array}\right\}$ | |

${S}_{2}$ | $\left\{\begin{array}{c}-2,\left(0.4\right),\\ 0,\left(0.33\right),\\ 1,\left(0.25\right)\end{array}\right\}$ | X | |||

${S}_{3}$ | $\left\{\begin{array}{c}1,\left(0.25\right),\\ -2,\left(0.33\right),\\ -1,\left(0.25\right)\end{array}\right\}$ | ||||

${S}_{4}$ | |||||

${C}_{3}$ | ${S}_{1}$ | $\left\{\begin{array}{c}1,\left(0.45\right),\\ 0,\left(0.3\right),\\ -1,\left(0.11\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.25\right),\\ 1,\left(0.37\right),\\ -1,\left(0.33\right)\end{array}\right\}$ | X | |

${S}_{2}$ | $\left\{\begin{array}{c}2,\left(0.28\right),\\ -1,\left(0.35\right),\\ 0,\left(0.3\right)\end{array}\right\}$ | $\left\{\begin{array}{c}1,\left(0.35\right),\\ -1,\left(0.25\right),\\ -2,\left(0.3\right)\end{array}\right\}$ | |||

${S}_{3}$ | $\left\{\begin{array}{c}-2,\left(0.33\right),\\ 1,\left(0.4\right),\\ 2,\left(0.22\right)\end{array}\right\}$ | ||||

${S}_{4}$ | |||||

${C}_{4}$ | ${S}_{1}$ | $\left\{\begin{array}{c}-1,\left(0.3\right),\\ 1,\left(0.42\right),\\ -2,\left(0.25\right)\end{array}\right\}$ | $\left\{\begin{array}{c}-2,\left(0.33\right),\\ 1,\left(0.4\right),\\ 2,\left(0.22\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.33\right),\\ 0,\left(0.35\right),\\ -1,\left(0.2\right)\end{array}\right\}$ | |

${S}_{2}$ | $\left\{\begin{array}{c}-2,\left(0.3\right),\\ 2,\left(0.24\right),\\ 1,\left(0.4\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.3\right),\\ -1,\left(0.35\right),\\ 1,\left(0.25\right)\end{array}\right\}$ | |||

${S}_{3}$ | $\left\{\begin{array}{c}1,\left(0.4\right),\\ 0,\left(0.22\right),\\ -1,\left(0.15\right)\end{array}\right\}$ | ||||

${S}_{4}$ |

Attributes | Supplier | ${\mathit{S}}_{1}$ | ${\mathit{S}}_{2}$ | ${\mathit{S}}_{3}$ | ${\mathit{S}}_{4}$ |
---|---|---|---|---|---|

${C}_{1}$ | ${S}_{1}$ | $\left\{\begin{array}{c}2,\left(0.3\right),\\ 1,\left(0.2\right),\\ -2,\left(0.42\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.33\right),\\ 1,\left(0.44\right),\\ -1,\left(0.2\right)\end{array}\right\}$ | $\left\{\begin{array}{c}1,\left(0.35\right),\\ -1,\left(0.25\right),\\ -2,\left(0.3\right)\end{array}\right\}$ | |

${S}_{2}$ | $\left\{\begin{array}{c}-1,\left(0.3\right),\\ 1,\left(0.25\right),\\ -2,\left(0.42\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.22\right),\\ 1,\left(0.4\right),\\ -2,\left(0.35\right)\end{array}\right\}$ | |||

${S}_{3}$ | $\left\{\begin{array}{c}2,\left(0.42\right),\\ -2,\left(0.25\right),\\ -1,\left(0.15\right)\end{array}\right\}$ | ||||

${S}_{4}$ | |||||

${C}_{2}$ | ${S}_{1}$ | $\left\{\begin{array}{c}2,\left(0.28\right),\\ 1,\left(0.35\right),\\ -1,\left(0.3\right)\end{array}\right\}$ | $\left\{\begin{array}{c}0,\left(0.22\right),\\ -2,\left(0.45\right),\\ -1,\left(0.15\right)\end{array}\right\}$ | $\left\{\begin{array}{c}1,\left(0.19\right),\\ -2,\left(0.22\right),\\ -1,\left(0.19\right)\end{array}\right\}$ | |

${S}_{2}$ | $\left\{\begin{array}{c}-2,\left(0.4\right),\\ 0,\left(0.33\right),\\ 1,\left(0.25\right)\end{array}\right\}$ | $\left\{\begin{array}{c}-2,\left(0.4\right),\\ 0,\left(0.33\right),\\ 1,\left(0.25\right)\end{array}\right\}$ | |||

${S}_{3}$ | $\left\{\begin{array}{c}1,\left(0.25\right),\\ -2,\left(0.33\right),\\ -1,\left(0.25\right)\end{array}\right\}$ | ||||

${S}_{4}$ | |||||

${C}_{3}$ | ${S}_{1}$ | $\left\{\begin{array}{c}1,\left(0.45\right),\\ 0,\left(0.3\right),\\ -1,\left(0.11\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.25\right),\\ 1,\left(0.37\right),\\ -1,\left(0.33\right)\end{array}\right\}$ | $\left\{\begin{array}{c}0,\left(0.06\right),\\ 0,\left(0.08\right),\\ 1,\left(0.05\right)\end{array}\right\}$ | |

${S}_{2}$ | $\left\{\begin{array}{c}2,\left(0.28\right),\\ -1,\left(0.35\right),\\ 0,\left(0.3\right)\end{array}\right\}$ | ||||

${S}_{3}$ | $\left\{\begin{array}{c}-2,\left(0.33\right),\\ 1,\left(0.4\right),\\ 2,\left(0.22\right)\end{array}\right\}$ | ||||

${S}_{4}$ | |||||

${C}_{4}$ | ${S}_{1}$ | $\left\{\begin{array}{c}-1,\left(0.3\right),\\ 1,\left(0.42\right),\\ -2,\left(0.25\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.33\right),\\ 0,\left(0.35\right),\\ -1,\left(0.2\right)\end{array}\right\}$ | ||

${S}_{2}$ | $\left\{\begin{array}{c}-2,\left(0.3\right),\\ 2,\left(0.24\right),\\ 1,\left(0.4\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.3\right),\\ -1,\left(0.35\right),\\ 1,\left(0.25\right)\end{array}\right\}$ | |||

${S}_{3}$ | $\left\{\begin{array}{c}1,\left(0.4\right),\\ 0,\left(0.22\right),\\ -1,\left(0.15\right)\end{array}\right\}$ | ||||

${S}_{4}$ |

Attributes | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ |
---|---|---|---|---|

${C}_{1}$ | $\left\{\begin{array}{c}2,\left(0.4\right),\\ 1,\left(0.32\right),\\ -1,\left(0.25\right)\end{array}\right\}$ | $\left\{\begin{array}{c}-1,\left(0.25\right),\\ 0,\left(0.3\right),\\ 1,\left(0.35\right)\end{array}\right\}$ | $\left\{\begin{array}{c}-1,\left(0.15\right),\\ 1,\left(0.25\right),\\ 2,\left(0.4\right)\end{array}\right\}$ | |

${C}_{2}$ | $\left\{\begin{array}{c}1,\left(0.35\right),\\ -1,\left(0.12\right),\\ 0,\left(0.25\right)\end{array}\right\}$ | $\left\{\begin{array}{c}0,\left(0.35\right),\\ -2,\left(0.32\right),\\ 2,\left(0.25\right)\end{array}\right\}$ | ||

${C}_{3}$ | $\left\{\begin{array}{c}-1,\left(0.18\right),\\ 1,\left(0.4\right),\\ 2,\left(0.33\right)\end{array}\right\}$ | |||

${C}_{4}$ |

$\mathit{C}\mathit{I}\left(\mathit{R}\right)$ | Value(s) |
---|---|

$d\left({R}_{1},{R}_{1}^{z}\right)$ | 0.0675 |

$d\left({R}_{2},{R}_{2}^{z}\right)$ | 0.1267 |

$d\left({R}_{3},{R}_{3}^{z}\right)$ | 0.0717 |

$d\left({R}_{4},{R}_{4}^{z}\right)$ | 0.0453 |

Supplier vs. Attributes | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ |
---|---|---|---|---|

${S}_{1}$ | $\left\{\begin{array}{c}2,\left(0.55\right),\\ 2,\left(0.55\right),\\ 2,\left(0.57\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.56\right),\\ 2,\left(0.57\right),\\ 2,\left(0.53\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.53\right),\\ 1,\left(0.54\right),\\ 2,\left(0.52\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.58\right),\\ 2,\left(0.62\right),\\ 2,\left(0.55\right)\end{array}\right\}$ |

${S}_{2}$ | $\left\{\begin{array}{c}2,\left(0.49\right),\\ 2,\left(0.51\right),\\ 2,\left(0.53\right)\end{array}\right\}$ | $\left\{\begin{array}{c}1,\left(0.54\right),\\ 0,\left(0.51\right),\\ 2,\left(0.49\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.51\right),\\ 1,\left(0.51\right),\\ 2,\left(0.51\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.53\right),\\ 2,\left(0.57\right),\\ 2,\left(0.53\right)\end{array}\right\}$ |

${S}_{3}$ | $\left\{\begin{array}{c}2,\left(0.55\right),\\ 1,\left(0.55\right),\\ 2,\left(0.56\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.57\right),\\ 2,\left(0.57\right),\\ 2,\left(0.53\right)\end{array}\right\}$ | $\left\{\begin{array}{c}1,\left(0.56\right),\\ 2,\left(0.58\right),\\ 2,\left(0.58\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.59\right),\\ 2,\left(0.60\right),\\ 0,\left(0.56\right)\end{array}\right\}$ |

${S}_{4}$ | $\left\{\begin{array}{c}2,\left(0.49\right),\\ 2,\left(0.51\right),\\ 2,\left(0.52\right)\end{array}\right\}$ | $\left\{\begin{array}{c}1,\left(0.53\right),\\ 2,\left(0.51\right),\\ 2,\left(0.49\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.47\right),\\ 2,\left(0.48\right),\\ 2,\left(0.49\right)\end{array}\right\}$ | $\left\{\begin{array}{c}1,\left(0.53\right),\\ 2,\left(0.57\right),\\ 2,\left(0.51\right)\end{array}\right\}$ |

Supplier(s) | PLTS Information | Ranking Value(s) |
---|---|---|

${S}_{1}$ | $\left\{\begin{array}{c}2,\left(0.56\right),\\ 1,\left(0.57\right),\\ 2,\left(0.54\right)\end{array}\right\}$ | 2.7899 |

${S}_{2}$ | $\left\{\begin{array}{c}1,\left(0.52\right),\\ 0,\left(0.52\right),\\ 2,\left(0.52\right)\end{array}\right\}$ | 1.5612 |

${S}_{3}$ | $\left\{\begin{array}{c}1,\left(0.57\right),\\ 1,\left(0.58\right),\\ 0,\left(0.55\right)\end{array}\right\}$ | 1.1554 |

${S}_{4}$ | $\left\{\begin{array}{c}1,\left(0.51\right),\\ 2,\left(0.52\right),\\ 2,\left(0.5\right)\end{array}\right\}$ | 2.5677 |

Attributes | Supplier | ${\mathit{S}}_{1}$ | ${\mathit{S}}_{2}$ | ${\mathit{S}}_{3}$ | ${\mathit{S}}_{4}$ |
---|---|---|---|---|---|

${C}_{1}$ | ${S}_{1}$ | $\left\{\begin{array}{c}3,\left(0.35\right),\\ 2,\left(0.22\right),\\ -2,\left(0.40\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.35\right),\\ 1,\left(0.44\right),\\ -2,\left(0.25\right)\end{array}\right\}$ | $\left\{\begin{array}{c}3,\left(0.35\right),\\ -2,\left(0.25\right),\\ 2,\left(0.3\right)\end{array}\right\}$ | |

${S}_{2}$ | $\left\{\begin{array}{c}1,\left(0.35\right),\\ 2,\left(0.3\right),\\ -2,\left(0.40\right)\end{array}\right\}$ | $\left\{\begin{array}{c}3,\left(0.25\right),\\ -1,\left(0.3\right),\\ 2,\left(0.45\right)\end{array}\right\}$ | |||

${S}_{3}$ | $\left\{\begin{array}{c}-2,\left(0.45\right),\\ 2,\left(0.30\right),\\ -3,\left(0.25\right)\end{array}\right\}$ | ||||

${S}_{4}$ | |||||

${C}_{2}$ | ${S}_{1}$ | $\left\{\begin{array}{c}-2,\left(0.38\right),\\ 1,\left(0.44\right),\\ -1,\left(0.30\right)\end{array}\right\}$ | $\left\{\begin{array}{c}0,\left(0.25\right),\\ 2,\left(0.45\right),\\ -1,\left(0.45\right)\end{array}\right\}$ | $\left\{\begin{array}{c}3,\left(0.18\right),\\ -2,\left(0.35\right),\\ -1,\left(0.25\right)\end{array}\right\}$ | |

${S}_{2}$ | $\left\{\begin{array}{c}-2,\left(0.45\right),\\ 0,\left(0.27\right),\\ 1,\left(0.35\right)\end{array}\right\}$ | X | |||

${S}_{3}$ | $\left\{\begin{array}{c}1,\left(0.25\right),\\ 2,\left(0.35\right),\\ -1,\left(0.35\right)\end{array}\right\}$ | ||||

${S}_{4}$ | |||||

${C}_{3}$ | ${S}_{1}$ | $\left\{\begin{array}{c}1,\left(0.45\right),\\ 0,\left(0.35\right),\\ -2,\left(0.18\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.25\right),\\ 1,\left(0.44\right),\\ -2,\left(0.35\right)\end{array}\right\}$ | X | |

${S}_{2}$ | $\left\{\begin{array}{c}2,\left(0.42\right),\\ 1,\left(0.35\right),\\ 0,\left(0.35\right)\end{array}\right\}$ | $\left\{\begin{array}{c}1,\left(0.35\right),\\ -1,\left(0.45\right),\\ -2,\left(0.50\right)\end{array}\right\}$ | |||

${S}_{3}$ | $\left\{\begin{array}{c}-2,\left(0.35\right),\\ 1,\left(0.42\right),\\ 0,\left(0.32\right)\end{array}\right\}$ | ||||

${S}_{4}$ | |||||

${C}_{4}$ | ${S}_{1}$ | $\left\{\begin{array}{c}-1,\left(0.3\right),\\ 1,\left(0.42\right),\\ -2,\left(0.25\right)\end{array}\right\}$ | $\left\{\begin{array}{c}-2,\left(0.35\right),\\ -1,\left(0.42\right),\\ 2,\left(0.25\right)\end{array}\right\}$ | $\left\{\begin{array}{c}3,\left(0.43\right),\\ 0,\left(0.35\right),\\ 1,\left(0.25\right)\end{array}\right\}$ | |

${S}_{2}$ | $\left\{\begin{array}{c}2,\left(0.35\right),\\ -2,\left(0.25\right),\\ 1,\left(0.45\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.4\right),\\ -1,\left(0.35\right),\\ 3,\left(0.25\right)\end{array}\right\}$ | |||

${S}_{3}$ | $\left\{\begin{array}{c}2,\left(0.45\right),\\ 0,\left(0.35\right),\\ -1,\left(0.25\right)\end{array}\right\}$ | ||||

${S}_{4}$ |

Attributes | Supplier | ${\mathit{S}}_{1}$ | ${\mathit{S}}_{2}$ | ${\mathit{S}}_{3}$ | ${\mathit{S}}_{4}$ |
---|---|---|---|---|---|

${C}_{1}$ | ${S}_{1}$ | $\left\{\begin{array}{c}3,\left(0.35\right),\\ 2,\left(0.22\right),\\ -2,\left(0.40\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.35\right),\\ 1,\left(0.44\right),\\ -2,\left(0.25\right)\end{array}\right\}$ | $\left\{\begin{array}{c}3,\left(0.35\right),\\ -2,\left(0.25\right),\\ 2,\left(0.3\right)\end{array}\right\}$ | |

${S}_{2}$ | $\left\{\begin{array}{c}1,\left(0.35\right),\\ 2,\left(0.3\right),\\ -2,\left(0.40\right)\end{array}\right\}$ | $\left\{\begin{array}{c}3,\left(0.25\right),\\ -1,\left(0.3\right),\\ 2,\left(0.45\right)\end{array}\right\}$ | |||

${S}_{3}$ | $\left\{\begin{array}{c}-2,\left(0.45\right),\\ 2,\left(0.30\right),\\ -3,\left(0.25\right)\end{array}\right\}$ | ||||

${S}_{4}$ | |||||

${C}_{2}$ | ${S}_{1}$ | $\left\{\begin{array}{c}-2,\left(0.38\right),\\ 1,\left(0.44\right),\\ -1,\left(0.30\right)\end{array}\right\}$ | $\left\{\begin{array}{c}0,\left(0.25\right),\\ 2,\left(0.45\right),\\ -1,\left(0.45\right)\end{array}\right\}$ | $\left\{\begin{array}{c}3,\left(0.18\right),\\ -2,\left(0.35\right),\\ -1,\left(0.25\right)\end{array}\right\}$ | |

${S}_{2}$ | $\left\{\begin{array}{c}-2,\left(0.45\right),\\ 0,\left(0.27\right),\\ 1,\left(0.35\right)\end{array}\right\}$ | $\left\{\begin{array}{c}1,\left(0.18\right),\\ 1,\left(0.23\right),\\ -1,\left(0.23\right)\end{array}\right\}$ | |||

${S}_{3}$ | $\left\{\begin{array}{c}1,\left(0.25\right),\\ 2,\left(0.35\right),\\ -1,\left(0.35\right)\end{array}\right\}$ | ||||

${S}_{4}$ | |||||

${C}_{3}$ | ${S}_{1}$ | $\left\{\begin{array}{c}1,\left(0.45\right),\\ 0,\left(0.35\right),\\ -2,\left(0.18\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.25\right),\\ 1,\left(0.44\right),\\ -2,\left(0.35\right)\end{array}\right\}$ | $\left\{\begin{array}{c}0,\left(0.11\right),\\ 2,\left(0.09\right),\\ 0,\left(0.08\right)\end{array}\right\}$ | |

${S}_{2}$ | $\left\{\begin{array}{c}2,\left(0.42\right),\\ 1,\left(0.35\right),\\ 0,\left(0.35\right)\end{array}\right\}$ | $\left\{\begin{array}{c}1,\left(0.35\right),\\ -1,\left(0.45\right),\\ -2,\left(0.50\right)\end{array}\right\}$ | |||

${S}_{3}$ | $\left\{\begin{array}{c}-2,\left(0.35\right),\\ 1,\left(0.42\right),\\ 0,\left(0.32\right)\end{array}\right\}$ | ||||

${S}_{4}$ | |||||

${C}_{4}$ | ${S}_{1}$ | $\left\{\begin{array}{c}-2,\left(0.35\right),\\ -1,\left(0.42\right),\\ 2,\left(0.25\right)\end{array}\right\}$ | $\left\{\begin{array}{c}3,\left(0.43\right),\\ 0,\left(0.35\right),\\ 1,\left(0.25\right)\end{array}\right\}$ | ||

${S}_{2}$ | $\left\{\begin{array}{c}2,\left(0.35\right),\\ -2,\left(0.25\right),\\ 1,\left(0.45\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.4\right),\\ -1,\left(0.35\right),\\ 3,\left(0.25\right)\end{array}\right\}$ | |||

${S}_{3}$ | $\left\{\begin{array}{c}2,\left(0.45\right),\\ 0,\left(0.35\right),\\ -1,\left(0.25\right)\end{array}\right\}$ | ||||

${S}_{4}$ |

Supplier vs. Attributes | ${\mathit{C}}_{1}$ | ${\mathit{C}}_{2}$ | ${\mathit{C}}_{3}$ | ${\mathit{C}}_{4}$ |
---|---|---|---|---|

${S}_{1}$ | $\left\{\begin{array}{c}2,\left(0.57\right),\\ 2,\left(0.56\right),\\ 2,\left(0.59\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.58\right),\\ 2,\left(0.58\right),\\ 2,\left(0.57\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.55\right),\\ 2,\left(0.55\right),\\ 2,\left(0.55\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.61\right),\\ 2,\left(0.64\right),\\ 2,\left(0.60\right)\end{array}\right\}$ |

${S}_{2}$ | $\left\{\begin{array}{c}2,\left(0.50\right),\\ 1,\left(0.51\right),\\ 2,\left(0.55\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.55\right),\\ 1,\left(0.52\right),\\ 2,\left(0.52\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.52\right),\\ 1,\left(0.52\right),\\ 2,\left(0.55\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.54\right),\\ 2,\left(0.58\right),\\ 2,\left(0.60\right)\end{array}\right\}$ |

${S}_{3}$ | $\left\{\begin{array}{c}2,\left(0.56\right),\\ 1,\left(0.57\right),\\ 2,\left(0.58\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.60\right),\\ 2,\left(0.58\right),\\ 2,\left(0.59\right)\end{array}\right\}$ | $\left\{\begin{array}{c}1,\left(0.57\right),\\ 1,\left(0.59\right),\\ 2,\left(0.61\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.61\right),\\ 2,\left(0.63\right),\\ 0,\left(0.61\right)\end{array}\right\}$ |

${S}_{4}$ | $\left\{\begin{array}{c}2,\left(0.51\right),\\ 2,\left(0.51\right),\\ 2,\left(0.54\right)\end{array}\right\}$ | $\left\{\begin{array}{c}1,\left(0.53\right),\\ 0,\left(0.52\right),\\ 2,\left(0.51\right)\end{array}\right\}$ | $\left\{\begin{array}{c}2,\left(0.49\right),\\ 1,\left(0.50\right),\\ 2,\left(0.52\right)\end{array}\right\}$ | $\left\{\begin{array}{c}1,\left(0.55\right),\\ 2,\left(0.59\right),\\ 2,\left(0.56\right)\end{array}\right\}$ |

Context(s) | Method(s) | ||
---|---|---|---|

Proposed | Xie et al. [32] | Tuysuz and Simsek [21] | |

Input | PLTS information | PLTS information | HFLTS information |

Aggregation | Ring sum operator | PLWG | no |

Weight calculation | Ring sum operator | Eigen vectors | no |

Fuzziness | yes | yes | yes |

Occurring probability | yes | yes | no |

Total preorder | yes | yes | yes |

Missing value(s) | Filled automatically using a systematic procedure | no | no |

Consistency | Check & repair using systematic procedure | Check & repair using expectation measure for geometric consistency index | no |

Adequacy test | Causes rank reversal issue when adequate changes are made to objects and attributes. | ||

Scalability | Obeys Saaty’s principle [37] | ||

Agility | $\left(n\left(n-1\right)/2\right)+n\left(m\left(m-1\right)/2\right)$ where n is the number of attributes and m is the number of objects. | ||

Ranking principle | Pair-wise comparison and Equation (10) are used for ranking objects. | Pair-wise comparison and possibility degree measure are used for ranking objects. | Pair-wise comparison and pessimistic & optimistic preference evaluation. |

Information loss | Mitigated to a great extent by retaining the PLTS information throughout the decision process | Some information is lost when PLTS information is converted into a single value using possibility degree | Crucial occurring probability value is missing in HFLTS context |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Krishankumar, R.; Ravichandran, K.S.; Ahmed, M.I.; Kar, S.; Tyagi, S.K.
Probabilistic Linguistic Preference Relation-Based Decision Framework for Multi-Attribute Group Decision Making. *Symmetry* **2019**, *11*, 2.
https://doi.org/10.3390/sym11010002

**AMA Style**

Krishankumar R, Ravichandran KS, Ahmed MI, Kar S, Tyagi SK.
Probabilistic Linguistic Preference Relation-Based Decision Framework for Multi-Attribute Group Decision Making. *Symmetry*. 2019; 11(1):2.
https://doi.org/10.3390/sym11010002

**Chicago/Turabian Style**

Krishankumar, R., K. S. Ravichandran, M. Ifjaz Ahmed, Samarjit Kar, and Sanjay K. Tyagi.
2019. "Probabilistic Linguistic Preference Relation-Based Decision Framework for Multi-Attribute Group Decision Making" *Symmetry* 11, no. 1: 2.
https://doi.org/10.3390/sym11010002