A decision-making framework under probabilistic linguistic term set for multi-criteria group decision-making problem

Abstract

This paper focuses on proposing a new decision framework under probabilistic linguistic term set (PLTS) for rational decision making. The PLTS concept is a generalization of hesitant fuzzy linguistic term set (HFLTS) which overcomes the limitation of HFLTS by associating occurring probability to each linguistic term. Initially, a new aggregation operator is presented for fusing decision makers’ (DMs) preferences. Following this, a new extension is put forward for statistical variance (SV) method under PLTS for criteria weight calculation and a new extension is presented for WASPAS (weighted arithmetic sum product assessment) method under PLTS context for ranking objects. The applicability of the proposed decision framework is demonstrated by using a numerical example and the strength and weakness of the proposal are investigated by comparison with other state-of-the-art methods.

Keywords

Decision making probabilistic linguistic term set statistical variance and WASPAS method

1. Introduction

Uncertainty and vagueness have become the part and parcel of decision-making process. Scholars widely present theories and concepts to mitigate such issues to some extent [1 –4]. The HFLTS by Rodriguez et al. [5] allowed DMs to enter multiple linguistic terms, but the occurring probability of each term was ignored. To circumvent the issue, Zhang et al. [6] came up with a concept called linguistic distribution assessment that associates probability values to each linguistic term. Later, Pang et al. [7] proposed PLTS (probabilistic linguistic term set) concept which is a generalization of [6] that allows incomplete linguistic distribution assessment. Recently, Wu et al. [8] proposed an interesting model called maximum support degree model that provides maximum support degree and guarantees accuracy of group opinions. Zhang et al. [9] came up with the concept of incomplete linguistic distribution assessment which is equivalent to PLTS and applied the same for consensus reaching. Motivated by the power of PLTS, scholars presented new theories and extensions which are presented below.

Bai et al. [10] proposed a new comparison method using area concept for ranking PLTS information. Gou et al. [11] presented new operational laws using transformation concept for PLTS information. Further, Zhai et al. [12] presented a new concept called probabilistic linguistic vector term set (PLVTS) which was used by recommender system for making a rational decision. Zhang and Xing [13] extended the popular VIKOR ranking method under PLTS context for green supplier selection. Zhang et al. [14] extended the idea of preference relation to PLTS environment and analyzed the consistency of the preferences using additive consistency measure. Lin et al. [15] proposed a new concept called probabilistic uncertain linguistic term set (PULTS) and applied the same for cloud service selection problem. Liao et al. [16] extended the linear programming to PLTS environment and used it for selecting suitable hospital.

From the review made above, it is clear that PLTS is an attractive concept for exploration and the research in PLTS has just taken pace. Some challenges that motivate the research focus are given below:

Based on the literature analysis (see Table 1) made above and from intuition, it is evident that a decision framework under PLTS which presents (i) an aggregation operator for fusing preferences, (ii) a criteria weight estimation method for systematic calculation of weight values and (iii) a ranking method for prioritizing objects all three concepts under one roof (that is PLTS context) is missing. Clearly, Table 1 supports the claim.

Scholars proposed some aggregation operators under PLTS context [7 , 20] that produce virtual sets. This affects the rationality of the decision process and causes a challenge to the decision-making process. To elaborate, when the linguistic term is aggregated, certain values are produced which are undefined in the LTS. These virtual values cannot be associated with an apt linguistic term and as a result, the rationality is affected during preference aggregation which is a crucial challenge to be addressed.

Further, criteria weights are provided directly [16] or calculated methodically [13] under PLTS context but, they do not properly reflect the hesitation of the DMs. This is also a key challenge to be addressed for rational decision-making.

Finally, selection of a suitable object from the set of objects and maintaining of backup plans for uncertain critical situations is also an interesting challenge which must be focused.

Table 1
Literature analysis of different PLTS based methods

Ref# PLTS based methods: Proposed vs. Others

Aggregation Weight Ranking Application

[7] yes, PLWA/G no TOPSIS Strategy project selection

[10] no no TOPSIS Hospital selection

[13] no yes, Defuzzification VIKOR Green supplier selection

[16] no no LINMAP Hospital selection

[11] no no TOPSIS Hospital selection

[17] no QFD ORESTE Panda car selection

[18] no correlation measure MULTIMOORA Karoke shared television brand selection

[19] yes, PLPA/G and WPLPA/G no no University professor assessment

[20] yes, PLAMM, PLAWMM, PLADMM, PLAD no no Project selection

Proposed yes, PLSWG PLTS-SV method PLTS-WASPAS Strategic project selection and green supplier selection

Ref#	PLTS based methods: Proposed vs. Others
[7]	yes, PLWA/G	no	TOPSIS	Strategy project selection
[10]	no	no	TOPSIS	Hospital selection
[13]	no	yes, Defuzzification	VIKOR	Green supplier selection
[16]	no	no	LINMAP	Hospital selection
[11]	no	no	TOPSIS	Hospital selection
[17]	no	QFD	ORESTE	Panda car selection
[18]	no	correlation measure	MULTIMOORA	Karoke shared television brand selection
[19]	yes, PLPA/G and WPLPA/G	no	no	University professor assessment
[20]	yes, PLAMM, PLAWMM, PLADMM, PLAD	no	no	Project selection
Proposed	yes, PLSWG	PLTS-SV method	PLTS-WASPAS	Strategic project selection and green supplier selection

By gaining strong motivation from these challenges, efforts are made to propose a new decision-making framework under PLTS context with new aggregation operator, criteria weight estimation method and ranking method. Some key contributions of this research are presented below:

A new aggregation operator called probabilistic linguistic simple weighted geometry (PLSWG) is proposed for fusing DMs’ preference information which can circumvent the challenge (2) by mitigating virtual set formation. Detailed explanation is given in Section 3.1.

A new extension is proposed for SV method under PLTS for calculating criteria weights (relative importance). Kao [21] proved that SV method can better reflect DMs’ hesitation during the process of criteria weight calculation. Inspired by the work of Kao [21] and with a view of addressing challenge (3), this contribution is put forward. Detailed explanation is given in Section 3.2.

A new extension is proposed for WASPAS method under PLTS for ranking objects and selecting suitable object for the task. Mardani et al. [22] conducted an interesting analysis on WASPAS method and claimed that (a) WASPAS is a simple and straightforward method that provides efficient prioritization of objects; (b) also, objects are prioritized based on the generalized measure of fusing multiplicative and additive methods and (c) finally, WASPAS method is also used in many practical decision-making problems. Motivated by these claims, an extension to WASPAS under PLTS context is put forward. Detailed explanation is given in Section 3.3.

Further, the practicality of the proposed framework is demonstrated by using a numerical example. Detailed demonstration of the numerical example is presented in Section 4.

Finally, the strength and weakness of the proposed framework is realized under both theoretic and numeric perspectives [23] by comparison with other methods. The proposed framework enjoys the following superiorities over other state-of-the-art methods: (a) it is highly consistent which is evident from the correlation test; (b) it is highly stable which is evident from the sensitivity analysis test; (c) it is highly robust against rank reversal issue, which is evident from the adequacy test and (d) finally, the proposed framework produces broad and sensible rank value set that help DMs to make rational backup management at critical situations. Detailed analysis can be obtained from Section 5.

The rest of the paper is organized as Section 2 for understanding the basic of linguistic term set (LTS) and PLTS, Section 3 for discussing the proposed idea in detail, Section 4 for demonstrating the applicability of the proposed decision framework, Section 5 for comparing the strengths and weaknesses of the proposed decision framework with other methods and finally Section 6 for conclusion.

2. Preliminaries

Let us review some basics of LTS and PLTS concepts.

Definition 1. [24] Let S be a linguistic term set (LTS) defined as S = {s_k| ∀ k = 0, 1, …, n}, where n is a positive integer representing the number of terms in the set, s₀ and s_n are the lower and upper limits of the LTS and s_k has the following characteristics:

S_α and S_β are two linguistic terms, the relation s_α > s_β holds true if α > β.

Negation of s_α is given by neg (s_α) = s_β such that α + β = n.

Definition 2. [7] Let S be a LTS, then the PLTS is defined as L (p) = {L^f (p) ^f|L^f ∈ S, p^f ≥ 0, ∑_fp^f ≤ 1, f = 0, 1, …, # L (p)} where # L (p) is the number of instance, L^f is the linguistic term of the fth instance and p^f is the associated occurring probability of the fth instance.

Note 1: As mentioned in Section 1, PLTS is a generalization of [6] which allows incomplete linguistic distribution assessment. The concept of PLTS and incomplete linguistic distribution assessment presented in [9] are equivalent. When ∑_fp^f = 1, f = 0, 1, …, # L (p) in Definition 2, it is the concept discussed in [6].

Remark 1. For convenience, we represent the probabilistic linguistic element (PLE) as $α_{i} = {r_{i}^{f}, (p_{i}^{f})}$ where r^f is the subscript of the linguistic term L^f and p^f is the associated occurring probability value.

Definition 3. [7] Let S be a LTS, then the normalized PLTS is given by L (p^*) = {L^f (p^*) ^f|L^f ∈ S, p^*f ≥ 0, ∑_fp^*f = 1, f = 0, 1, …, # L (p^*)} and $p^{* f} = \frac{p^{f}}{\sum_{f = 1}^{# L (p)} p^{f}}$ .

Definition 4. [7] Let L (p), L₁ (p) and L₂ (p) are three PLTSs, then the operational laws are given by, $L_{1} (p) \oplus L_{2} (p) = \cup_{L_{1}^{f} \in L_{1} (p), L_{2}^{f} \in L_{2} (p)} (p_{1}^{f} L_{1}^{f} \oplus p_{2}^{f} L_{2}^{f})$ (1) $L_{1} (p) \otimes L_{2} (p) = \cup_{L_{1}^{f} \in L_{1} (p), L_{2}^{f} \in L_{2} (p)} (p_{1}^{f} L_{1}^{f} \otimes p_{2}^{f} L_{2}^{f})$ (2) $(L (p))^{λ} = \cup_{L^{f \in L (p)}}, (L^{f})^{λ p} f, 0 \leq λ \leq 1$ (3) $λ L (p) = \cup_{L^{f \in L (p)}} λ p^{f} L^{f}, 0 \leq λ \leq 1$ (4) $E (L (p)) = S_{\bar{k}}$ (5) $σ (L (p)) = \frac{\sqrt{\sum_{f = 1}^{# L (p)} (p^{f} (r^{f} - \bar{k})^{2}}}{\sum_{f = 1}^{# L (p)} p^{f}}$ (6) where $\bar{k} = \frac{\sum_{f = 1}^{# L (p)} r^{f} p^{f}}{\sum_{f = 1}^{# L (p)} p^{f}}$ with r^f as the subscript of the linguistic term L^f.

Note 2: Further, Gou et al. [11] proposed new operational laws under PLTS context by using transformation functions.

3. Proposed decision framework

3.1. Proposed PLSWG operator

This section presents a new aggregation operator under the PLTS context for fusing DMs’ preference information. The operator is implemented in two folds viz., the fusion of linguistic terms and the fusion of associated occurring probability values. In the first fold, the linguistic terms are fused in such manner that there is no virtual term produced. Secondly, the associated occurring probability values are fused in a sensible manner by considering the relative importance of each DM. Motivated by these claims, the operator is presented.

Definition 5. Let L_i (p) be a PLTS given by $L_{i} (p) = {L_{i}^{f} (p_{i})^{f} | L_{i}^{f} \in S, p_{i}^{f} \geq 0, \sum_{f} p_{i}^{f} \leq 1, f = 0, 1, \dots, # L_{i} (p)}$ then the fusion of PLTS information is a mapping Xⁿ → X such that, $\begin{matrix} {PLSWG}_{λ} (L_{i}^{f}) = \\ {\begin{matrix} Scheme a if all linguistic terms are unique \\ Scheme b otherwise \end{matrix}} \end{matrix}$ (7) ${PLSWG}_{λ} (p_{i}^{f}) = 1 - \prod_{l = 1}^{# DM} (1 - p_{i}^{f})^{λ l}$ (8) where λ₁ is the relative importance of the lth DM with 0 ≤ λ₁ ≤ 1 and ∑_lλ_l = 1.

Scheme a: The mean of the subscript of the linguistic terms are taken and rounding-off principle is applied to obtain non-virtual linguistic terms.

Scheme b: The occurrence of each linguistic term is calculated and the term that appears the maximum number of times is chosen as the fused value.

The proposed aggregation operator enjoys some interesting properties which are presented below:

Property 1. Idempotent. If all the PLEs are equal, then PLSWG_λ (α₁, α₂, …, α_n) = α.

Property 2. Bounded. For any arbitrary value of λ, we have α^- ≤ PLSWG_λ (α₁, α₂, …, α_n) ≤ α⁺ where α^- = min(α_i) and α⁺ = max(α_i) for all i = 0, 1, …, n.

Property 3. Monotonic. Consider $α_{i}^{*} = {r_{i}^{* f}, (p_{i}^{* f})}$ to be a PLE which is of the form $α_{i}^{*} \leq α_{i}$ , then ${PLSWG}_{λ} (α_{i}^{*}) \leq {PLSWG}_{λ} (α_{i})$ for all i = 0, 1, …, n.

Theorem 1. The fusion of PLTS information using PLSWG operator produces PLTS information.

Proof. We can prove the above theorem by considering two cases viz., the first fold for fusion of linguistic terms and a second fold for the fusion of associated occurring probability. The first fold of fusion of linguistic terms produces a non-virtual set which is clearly evident from the formulation presented in Equation (8). Now, we have to prove the second fold which is done in the following manner:

It is clearly known from [25] that $\sum_{l} λ_{l} x_{i} \geq \prod_{l} (x_{i})^{λ_{l}}$ and hence, $0 \leq \prod_{l} (p_{i})^{λ_{l}} \leq \sum_{l} λ_{l} p_{i} \leq \sum_{l} λ_{l} = 1$ . By these two cases, it is evident that both the linguistic term and the associated occurring probability are following the constraints defined in Definition 2. Thus, the aggregation of PLTS information by using PLSWG operator produces PLTS information■

Example 1. Let S = {s_k| ∀ k = 0, 1, …, 6} be a LTS with {s₀ = none, s₁ = very low, s₂ = low, s₃ = normal, s₄ = high, s₅ = very high, s₆ = perfect}. Consider a snippet of IVPHFS information provided by two DMs for a particular scenario as: {s₃, (0.35) , s₁, (0.42)} and {s₃, (0.44) , s₄, (0.38)} with (λ₁, λ₂) = (0.4, 0.6). Now, these values are aggregated using PLSWG operator and the result is given by, {s₃, (0.41) , s₂ (0.40)}. It can be noted that while aggregating preference information, the linguistic terms can repeat among instances. The redundancy is eliminated by considering the unique occurrence of the linguistic term with a mean value as associated occurring probability. To clarify the scenario, if we consider the rating as: {s₃, (0.35) , s₂, (0.42)} and {s₃, (0.44) , s₄, (0.38)}, then the aggregated value obtained by PLSWG operator is, {s₃, (0.41) , s₃, (0.40)} = {s₃, (0.405)}.

3.2. Extended SV method under PLTS context

This section presents a systematic method for calculating the weights of the criteria. Previous studies on criteria weight estimation have either directly associated weights to each criterion or used weight estimation procedures for calculating the weights of the criteria. From the literature it is clear that direct assignment of weight values is uncertain and inaccurate and hence, popular methods like AHP (analytic hierarchy process) [26 –28], entropy-based method [29 –31], optimization model [32] etc. are used for weight calculation. Liu et al. [33] claimed that these methods produce unreasonable weight values and are complex in nature. Hence to address the challenge, statistical variance (SV) method was put forward. They claimed that the SV method: (i) produced sensible and rational weight values, (ii) was simple and straightforward and (iii) concentrated on all data points rather than checking the extreme points alone for determining the probability distribution [34].

Motivated by these attractive claims, in this paper, efforts are made to extend SV method to PHFS. The systematic procedure for calculating criteria weights using extended SV method is presented below:

Step 1. Form an evaluation matrix of order l × n where l is the number of DMs and n is the number of criteria. The PLTS information is used for rating the criteria.

Step 2. Transform the PLTS information into single-valued entity using Equation (9). The order of the evaluation matrix remains unchanged after this transformation. ${sve}_{ij} = \sum_{f = 1}^{# instance} (r_{ij}^{f} \times p_{ij}^{f})$ (9) where sve_ij is the single-valued entity for the ith DM and jth criterion, $r_{ij}^{f}$ is the subscript of the linguistic term and $p_{ij}^{f}$ is the associated probability value of the linguistic term.

Step 3. Calculate the statistical variance (SV) for each criterion using Equation (10). The evaluation matrix obtained from step 2 is used for this purpose. ${var}_{j} = \frac{\sum_{i = 1}^{l} ({sve}_{ij} - \bar{{sve}_{ij}})^{2}}{l - 1}$ (10) where $\bar{{sve}_{ij}}$ is the mean value calculated for each criterion of the evaluation matrix obtained from step 2.

Step 4. The weights of the criteria are determined by normalizing the SV value obtained from step 3. The formula for normalization is given in Equation (11). $w_{j} = \frac{{var}_{j}}{\sum_{j} {var}_{j}}$ (11) where w_j is the weight of the jth criterion with 0 ≤ w_j ≤ 1 and ∑_jw_j = 1.

3.3. Extended WASPAS method

This section proposes a new extension to WASPAS method under PLTS context. The WASPAS method [35] is a compromise solution method that is based on the idea of weighted sum method (WSM) and weighted product method (WPM). The overall idea of WASPAS is closely related to the popular VIKOR ranking method [36]. The WASPAS method enjoys the following advantages:

It follows joint optimality concept for determining the final rank of the object.

It is simple and straightforward.

It determines the final ranking order based on the linear combination of WSM and WPM and also uses different strategy values to understand the effect on uncertainty in the preference information.

Motivated by these advantages, in this paper efforts are made to extend WASPAS for PLTS environment. The systematic procedure for the proposed ranking method is given below:

Step 1. Obtain the aggregated decision matrix and the criteria weight vector from Section 3.1 and 3.2 respectively.

Step 2. Determine the weighted sum value for each object using Equation (12). $Q_{i}^{1} = \sum_{j = 1}^{n} w_{j} (\sum_{f = 1}^{# L (p)} (r_{ij}^{f} + p_{ij}^{f}))$ (12) where $Q_{i}^{1}$ is the WSM which is estimated for the ith object, r_ij is the subscript of the linguistic term, p_ij is the associated probability value, w_j is the weight of the jth criteria, n is the number of criteria and # L (p) is the number of instances.

Step 3. Determine the weighted product value for each object using Equation (13). $Q_{i}^{2} = \prod_{j = 1}^{n} {(\sum_{f = 1}^{# L (p)} (r_{ij}^{f} \times p_{ij}^{f}))}^{w_{j}}$ (13) where $Q_{i}^{2}$ is the WPM which is estimated for the ith object.

Step 4. Determine the final ranking using Equation(14). The values from step 2 and step 3 are considered and the linear combination of WSM and WPM forms the final ranking order. $Q_{i}^{3} = v (\frac{Q_{i}^{1} - Q_{i}^{1 *}}{Q_{i}^{1} - Q_{i}^{1 *}}) + (1 - v) (\frac{Q_{i}^{2} - Q_{i}^{2 *}}{Q_{i}^{2} - Q_{i}^{2 *}})$ (14) where $Q_{i}^{1 *} = min (Q_{i}^{1})$ , $Q_{i}^{1 -} = max (Q_{i}^{1})$ , $Q_{i}^{2 *} = min (Q_{i}^{2})$ , $Q_{i}^{2 -} = max (Q_{i}^{2})$ and v is the strategy of DM which is of the form 0 ≤ v ≤ 1.

Step 5. The values from step 4 are assigned in ascending order and the object with minimum value is highly preferred.

4. Numerical example

4.1. Strategic project selection problem

This section presents the practical use of the proposed decision framework by demonstrating a strategic project planning example adapted from [7]. The board members decide three projects as the candidate for analysis and four criteria viz., finance C₁, customer satisfaction C₂, internal business process C₃ and learning and growth C₄ are chosen for evaluation using balanced scorecard method. Three experts are chosen for rating projects with respect to four criteria and PLTS information is used for the rating purpose.

The proposed decision framework consist of the following systematic procedure for analysis:

Step 1. Form three matrices of order 3 × 4 with each matrix using the PLTS information for rating projects over criteria. Table 2 depicts the preference information.

Table 2
Decision matrix with PLTS information

DMs	Projects	Evaluation criteria
DMs	Projects	C ₁	C ₂	C ₃	C ₄
e ₁	a ₁	{s₄, (0.2)}	{s₂, (0.3)}	{s₄, (0.1)}	{s₅, (0.2)}
	a ₂	{s₃, (0.3)}	{s₀, (0)}	{s₁, (0.1)}	{s₃, (0.2)}
	a ₃	{s₄, (0.5)}	{s₃, (0.1)}	{s₅, (0.2)}	{s₄, (0.1)}
e ₂	a ₁	{s₄, (0.2)}	{s₄, (0.3)}	{s₄, (0.2)}	{s₃, (0.2)}
	a ₂	{s₅, (0.1)}	{s₂, (0.3)}	{s₀, (0.1)}	{s₄, (0.2)}
	a ₃	{s₃, (0.3)}	{s₃, (0.1)}	{s₄, (0.2)}	{s₆, (0.1)}
e ₃	a ₁	{s₄, (0.1)}	{s₄, (0.5)}	{s₄, (0.1)}	{s₃, (0.2)}
	a ₂	{s₃, (0.1)}	{s₄, (0.6)}	{s₃, (0.1)}	{s₃, (0.2)}
	a ₃	{s₃, (0.1)}	{s₀, (0.1)}	{s₃, (0.1)}	{s₄, (0.1)}

Step 2. Aggregate these matrices to form a single matrix of order 3 × 4 by using the operator presented in section 3.1. The aggregated matrix also yields PLTS information which is depicted in Table 3.

Table 3

Aggregated PLTS information

DMs	Projects	Evaluation criteria
DMs	Projects	C ₁	C ₂	C ₃	C ₄
e ₁₂₃	a ₁	{s₄, (0.17)}	{s₄, (0.37)}	{s₄, (0.14)}	{s₃, (0.2)}
	a ₂	{s₃, (0.17)}	{s₂, (0.34)}	{s₁, (0.1)}	{s₃, (0.2)}
	a ₃	{s₃, (0.32)}	{s₃, (0.1)}	{s₄, (0.17)}	{s₄, (0.1)}

Step 3. Calculate the weight vector for the criteria using the method presented in Section 3.2. Initially, an evaluation matrix is obtained with order 3 × 4 with three experts and four criteria and the information used for rating is PLTS. Table 4 shows the DMs’ preference information for each criterion. The method discussed in section 3.2 is used for calculating the weights of the criteria. Using Equation (10) the variance is calculated and it is normalized by using Equation (11) to obtain weights of the criteria as w = (0.01, 0.10, 0.66, 0.23).

Table 4

Criteria weight calculation matrix

DMs	Evaluation criteria
DMs	C ₁	C ₂	C ₃	C ₄
e ₁	{s₃, (0.1)}	{s₄, (0.1)}	{s₃, (0.7)}	{s₅, (0.5)}
e ₂	{s₃, (0.3)}	{s₃, (0.6)}	{s₂, (0.2)}	{s₃, (0.1)}
e ₃	{s₃, (0.2)}	{s₄, (0.4)}	{s₀, (0.5)}	{s₄, (0.1)}

Step 4. Use the data from step 2 and step 3 to select a suitable project from the set of projects using the method proposed in section 3.3. The Table 5 depicts WSM and WPM values for the projects which are calculated using Equations (12, 13). The final ranking is calculated using Equation (14) and based on the sensitivity analysis of DMs’ strategy values (see Table 5) a suitable project is selected. The sensitivity analysis is performed to understand the effect of uncertainty on the proposed framework. From the analysis of Table 6, it can be observed that the final ranking order is given by a₁ > a₃ > a₂ and the suitable project for the strategic management is a₁. This infers that the proposed decision framework is stable even after adequate changes are made to the DMs’ strategy values and to the criteria weights.

Table 5

Evaluation parameters for WASPAS method

Projects	Biased weight(s)		Unbiased weight(s)
Projects	WSM	WPM	WSM	WPM
a ₁	0.661	0.629	0.829	0.764
a ₂	0.275	0.185	0.469	0.378
a ₃	0.585	0.561	0.584	0.529

Table 6

Sensitivity analysis: Criteria weights and strategy values

v values	Final rank value(s)		Ranking order
v values	Biased	Unbiased	Ranking order
0.1	1	1	a₁ > a₃ > a₂
	0	0
	0.842	0.384
0.2	1	1	a₁ > a₃ > a₂
	0	0
	0.838	0.377
0.3	1	1	a₁ > a₃ > a₂
	0	0
	0.834	0.370
0.4	1	1	a₁ > a₃ > a₂
	0	0
	0.829	0.362
0.5	1	1	a₁ > a₃ > a₂
	0	0
	0.825	0.355
0.6	1	1	a₁ > a₃ > a₂
	0	0
	0.821	0.348
0.7	1	1	a₁ > a₃ > a₂
	0	0
	0.816	0.340
0.8	1	1	a₁ > a₃ > a₂
	0	0
	0.812	0.334
0.9	1	1	a₁ > a₃ > a₂
	0	0
	0.807	0.327

Step 5. Finally, the strength and weakness of the proposed decision framework are realized by comparison with other state-of-the-art methods (refer section 5).

4.2. Green supplier selection problem

In this section, another example is demonstrated for understanding the practical ability of the proposed decision framework. The selection panel consisted of three DMs viz., technical officer e₁, financial officer e₂ and support manager e₃. The panel identified three potential green suppliers for supplying raw materials to a leading fashion brand company. The panel also identified four potential criteria viz., manufacturing ability C₁, purchase quality C₂, logistics C₃ and market C₄ for evaluating these three green suppliers. These criteria are adapted from [13] and PLTS information is used for rating the suppliers.

The procedure used for the evaluation of green supplier is given below:

Step 1. Obtain the decision matrix of order 3 × 4 from three DMs and use PLTS information for rating. Aggregate these matrices using PLSWG operator. Table 7 shows the DMs’ preference information and aggregated preference information.

Table 7
Decision matrix with PLTS information from different DMs

DMs	Supplier(s)	Evaluation criteria
C ₁	C ₂	C ₃	C ₄
e ₁	a ₁	${\begin{matrix} s_{3}, (0.4) \\ s_{4}, (0.2) \end{matrix}}$	${\begin{matrix} s_{4}, (0.2) \\ s_{0}, (0.3) \end{matrix}}$	${\begin{matrix} s_{2}, (0.5) \\ s_{1}, (0.4) \end{matrix}}$	${\begin{matrix} s_{5}, (0.6) \\ s_{0}, (0.3) \end{matrix}}$
	a ₂	${\begin{matrix} s_{2}, (0.2) \\ s_{1}, (0.4) \end{matrix}}$	${\begin{matrix} s_{3}, (0.3) \\ s_{2}, (0.1) \end{matrix}}$	${\begin{matrix} s_{4}, (0.1) \\ s_{3}, (0.7) \end{matrix}}$	${\begin{matrix} s_{4}, (0.1) \\ s_{3}, (0.2) \end{matrix}}$
	a ₃	${\begin{matrix} s_{1}, (0.5) \\ s_{0}, (0.1) \end{matrix}}$	${\begin{matrix} s_{4}, (0.1) \\ s_{2}, (0.2) \end{matrix}}$	${\begin{matrix} s_{3}, (0.1) \\ s_{3}, (0.6) \end{matrix}}$	${\begin{matrix} s_{4}, (0.1) \\ s_{4}, (0.2) \end{matrix}}$
e ₂	a ₁	${\begin{matrix} s_{2}, (0.8) \\ s_{3}, (0.1) \end{matrix}}$	${\begin{matrix} s_{4}, (0.1) \\ s_{2}, (0.2) \end{matrix}}$	${\begin{matrix} s_{3}, (0.2) \\ s_{3}, (0.1) \end{matrix}}$	${\begin{matrix} s_{0}, (0.1) \\ s_{1}, (0.2) \end{matrix}}$
	a ₂	${\begin{matrix} s_{4}, (0.2) \\ s_{1}, (0.4) \end{matrix}}$	${\begin{matrix} s_{1}, (0.7) \\ s_{4}, (0.1) \end{matrix}}$	${\begin{matrix} s_{3}, (0.2) \\ s_{3}, (0.1) \end{matrix}}$	${\begin{matrix} s_{2}, (0.3) \\ s_{3}, (0.6) \end{matrix}}$
	a ₃	${\begin{matrix} s_{1}, (0.1) \\ s_{4}, (0.2) \end{matrix}}$	${\begin{matrix} s_{2}, (0) \\ s_{3}, (0.1) \end{matrix}}$	${\begin{matrix} s_{3}, (0.4) \\ s_{5}, (0.1) \end{matrix}}$	${\begin{matrix} s_{4}, (0.6) \\ s_{1}, (0.1) \end{matrix}}$
e ₃	a ₁	${\begin{matrix} s_{0}, (0.2) \\ s_{1}, (0.3) \end{matrix}}$	${\begin{matrix} s_{3}, (0.6) \\ s_{2}, (0.1) \end{matrix}}$	${\begin{matrix} s_{0}, (0.7) \\ s_{1}, (0.2) \end{matrix}}$	${\begin{matrix} s_{1}, (0.1) \\ s_{1}, (0.1) \end{matrix}}$
	a ₂	${\begin{matrix} s_{1}, (0.1) \\ s_{1}, (0.4) \end{matrix}}$	${\begin{matrix} s_{2}, (0.2) \\ s_{2}, (0.3) \end{matrix}}$	${\begin{matrix} s_{1}, (0.4) \\ s_{2}, (0.3) \end{matrix}}$	${\begin{matrix} s_{4}, (0.9) \\ s_{0}, (0) \end{matrix}}$
	a ₃	${\begin{matrix} s_{2}, (0.5) \\ s_{2}, (0.2) \end{matrix}}$	${\begin{matrix} s_{1}, (0.5) \\ s_{4}, (0.1) \end{matrix}}$	${\begin{matrix} s_{3}, (0.2 \\ s_{2}, (0.1) \end{matrix}}$	${\begin{matrix} s_{2}, (0) \\ s_{3}, (0.6) \end{matrix}}$
e ₁₂₃	a ₁	${\begin{matrix} s_{1}, (0.0.58) \\ s_{2}, (0.19) \end{matrix}}$	${\begin{matrix} s_{4}, (0.32) \\ s_{2}, (0.20) \end{matrix}}$	${\begin{matrix} s_{1}, (0.48) \\ s_{1}, (0.23) \end{matrix}}$	${\begin{matrix} s_{2}, (0.29) \\ s_{1}, (0.20) \end{matrix}}$
	a ₂	${\begin{matrix} s_{2}, (0.17) \\ s_{1}, (0.4) \end{matrix}}$	${\begin{matrix} s_{2}, (0.48) \\ s_{2}, (0.17) \end{matrix}}$	${\begin{matrix} s_{2}, (0.24) \\ s_{2}, (0.4) \end{matrix}}$	${\begin{matrix} s_{4}, (0.58) \\ s_{3}, (0.35) \end{matrix}}$
	a ₃	${\begin{matrix} s_{1}, (0.37) \\ s_{2}, (0.17) \end{matrix}}$	${\begin{matrix} s_{2}, (0.21) \\ s_{3}, (0.13) \end{matrix}}$	${\begin{matrix} s_{3}, (0.26) \\ s_{3}, (0.29) \end{matrix}}$	${\begin{matrix} s_{4}, (0.33) \\ s_{2}, (0.32) \end{matrix}}$

Step 2. Determine the weights of the criteria using an evaluation matrix of order 3 × 4 with PLTS information by applying extended SV method discussed in section 3.2. Table 8 depicts the evaluation matrix for criteria weight calculation.

Table 8

Criteria weight calculation matrix

DMs	Evaluation criteria
DMs	C ₁	C ₂	C ₃	C ₄
e ₁	${\begin{matrix} s_{2}, (0.1) \\ s_{3}, (0.4) \end{matrix}}$	${\begin{matrix} s_{5}, (0.3) \\ s_{1}, (0.4) \end{matrix}}$	${\begin{matrix} s_{4}, (0.2) \\ s_{3}, (0.1) \end{matrix}}$	${\begin{matrix} s_{2}, (0.2) \\ s_{4}, (0) \end{matrix}}$
e ₂	${\begin{matrix} s_{4}, (0.2) \\ s_{3}, (0.1) \end{matrix}}$	${\begin{matrix} s_{5}, (0.4) \\ s_{5}, (0.5) \end{matrix}}$	${\begin{matrix} s_{4}, (0.2) \\ s_{3}, (0.3) \end{matrix}}$	${\begin{matrix} s_{1}, (0.2) \\ s_{2}, (0.2) \end{matrix}}$
e ₃	${\begin{matrix} s_{2}, (0.8) \\ s_{1}, (0) \end{matrix}}$	${\begin{matrix} s_{3}, (0.9) \\ s_{4}, (0) \end{matrix}}$	${\begin{matrix} s_{4}, (0.7) \\ s_{3}, (0.2) \end{matrix}}$	${\begin{matrix} s_{4}, (0.8) \\ s_{5}, (0.1) \end{matrix}}$

Step 3. The aggregated decision matrix obtained from step 1 and the criteria weight vector obtained from step 2 are used by the extended WASPAS method (given in section 3.3) for selecting a suitable green supplier for the fashion company. Table 9 calculates presents the WASPAS parameter values for each supplier.

Table 9

Parameter calculation for WASPAS method

Supplier(s)	Calculated value(s)
Supplier(s)	WSM	WPM
a ₁	0.986	0.927
a ₂	2.417	2.163
a ₃	1.614	1.525

Step 4. The strength and weakness of the proposed decision framework are realized by comparison with other methods.

From Table 10 (sensitivity analysis) it can be observed that the proposed decision framework produces a ranking order as: α₂ > α₃ > α₁ with α₂ as the suitable green supplier for the fashion company. From the table we can also infer that the proposed decision framework is stable even after adequate changes are made to the criteria and to the DMs’ strategy values.

Table 10

Sensitivity analysis: Weights and strategy values

v values	Final rank value(s)		Ranking order
v values	Biased	Unbiased	Ranking order
0.1	0	0	a₂ > a₃ > a₁
	1	1
	0.479	0.430
0.2	0	0	a₂ > a₃ > a₁
	1	1
	0.475	0.426
0.3	0	0	a₂ > a₃ > a₁
	1	1
	0.470	0.422
0.4	0	0	a₂ > a₃ > a₁
	1	1
	0.466	0.417
0.5	0	0	a₂ > a₃ > a₁
	1	1
	0.461	0.413
0.6	0	0	a₂ > a₃ > a₁
	1	1
	0.457	0.409
0.7	0	0	a₂ > a₃ > a₁
	1	1
	0.452	0.405
0.8	0	0	a₂ > a₃ > a₁
	1	1
	0.448	0.401
0.9	0	0	a₂ > a₃ > a₁
	1	1
	0.443	0.397

5. Comparative analysis: Proposed framework vs. Others

In this section, the strength and weakness of the proposed decision framework are investigated from both theoretic as well as numeric perspectives. The theoretic factors are chosen based on previous literature and intuition, while the numeric factors are adapted from [23]. Table 10 shows the ranking order obtained from different ranking methods. To retain the homogeneity in the comparison process, the methods taken for analysis are: PLTS based TOPSIS (technique for order of preference by similarity to ideal solution) [7], PLTS based aggregation [7] and PLTS based VIKOR (vlsekriterijumska optimizacija kompromisno resenje) [13].

From Table 11 and Fig. 1 it can be observed that the proposed decision framework is highly consistent with other state-of-the-art methods. Two cases viz., strategic project selection [7] and green supplier selection [13] are demonstrated to understand the consistency of the proposed framework. Table 12 investigates the theoretic and numeric features of different methods.

Table 11
Ranking order from different methods

Method(s)	Project selection [7]			Supplier selection [13]			Ranking order
Method(s)	a ₁	a ₂	a ₃	a ₁	a ₂	a ₃	Ranking order
Proposed	1	3	2	3	1	2	a₁ > a₃ > a₂
							a₂ > a₃ > a₁
PL-TOPSIS [7]	1	3	2	3	1	2	a₁ > a₃ > a₂
							a₂ > a₃ > a₁
PL-aggregate [7]	1	3	2	2	1	3	a₂ > a₁ > a₃
PL-VIKOR [13]	1	3	2	3	1	2	a₁ > a₃ > a₂
							a₂ > a₃ > a₁

Fig. 1

Spearman corrplot: Proposed vs. Others. (a) Example 1, (b) Example 2.

Table 12

Investigation of different factors: Proposed vs. Others

Context(s)	Proposed	PL-TOPSIS [7]	PL-aggregate [7]	PL-VIKOR [13]
Input	PLTS information
Aggregate	PLSWG operator	PLWA and PLWG operators		no
Criteria weights	Extended SV method	Directly given by DMs		Defuzzification method
Ranking method	Extended WASPAS method	PL-TOPSIS	PL-aggregate	PL-VIKOR
Ranking style	Linear combination of WSM and WPM.	Relative index	Aggregation operator	Merit function which is a linear combination of group utility and individual regret.
Rank value set	Broad and sensible values are produced.	Negative and unreasonable values are produced.		Broad and sensible values are produced.
Adequacy test	Ranking order remains unchanged even after adequate changes are made to the alternatives.	Ranking order changes when alternatives and criteria are changed adequately.	Ranking order remains unchanged when adequate changes are made to the alternatives.	Ranking order remains unchanged when adequate changes are made to the criteria.
Backup management	Proper and reasonable backup management is possible.	Backup management is improper as negative values are produced in the rank value set.		Proper and reasonable backup management is possible.
Scalability	Follows the Saaty’s principle [37].

Some genuine strengths of the proposed decision framework are given below:

The proposed decision framework uses PLTS information for rating objects. The PLTS information allows DMs to associate occurring probability value with each linguistic term. The PLTS information also allows partial ignorance.

The framework presents an aggregation operator which aggregates PLTS information in a much sensible manner without forming any virtual sets.

The proposed framework also presents a method for calculating weights of the criteria which produces rational and reasonable weight values by getting the preferences for each criterion from different DMs. The extended SV method under PLTS context properly reflects the hesitation of DMs during preference elicitation.

Finally, the proposed framework presents a ranking method for selecting a suitable object from the set of objects. The method is simple and straightforward and resembles closely to the powerful VIKOR method. Compared to the VIKOR method, WASPAS method under PLTS environment is easy to implement as it involves less computation overhead.

The proposed framework is highly consistent (Spearman correlation) with its close counterpart and it can be clearly observed from Fig. 1.

The proposed framework is also highly stable even after adequate changes are made to the criteria weights and DMs’ strategy values which is evident from the sensitivity analysis test. The framework is also highly robust from rank reversal issue even after adequate changes are made to the objects. The idea of adequacy test is adapted from [23] and different test cases are formed by repeating object and criterion. The test is intended to see if there exists rank reversal issue when such adequate changes are made. Clearly, the proposed framework remains robust when objects are repeated to form new test cases.

The proposed decision framework produces broad and sensible rank value set for effective and rational backup management under uncertain situations. To realize the strength of broad rank value set, an investigation using simulated datasets is adopted. In this study, 500 decision matrices of order 3 × 4 are taken and PLTS information is used as preference information. Proposed decision framework is fed with these 500 decision matrices and the standard deviation is calculated for each rank value set. These values are compared with the standard deviation of other rank value sets obtained from different ranking methods (viz., [7, 13]). From Fig. 2, it can be observed that the proposed decision framework produces much sensible and broad rank value set which helps DMs to make rational backup plans under uncertain scenarios. The PL-VIKOR [13] method also produces broad rank value set but from Fig. 2 it is clear that the proposed decision framework outperforms PL-VIKOR [13] method. Also, the implementation overhead of PL-VIKOR method [13] is more than the proposed decision framework and the problem of divide by zero which arises when the positive ideal and negative ideal solutions are same does not occur in the proposed framework. Further, the methods discussed in [7] sometimes produce negative rank value set which is transformed to a positive value by using $| x | = \sqrt{(x)^{2}}$ operator and the ranking order is formed by arranging these values in ascending order. Standard deviation is calculated for these values for analysis.

Fig. 2

Analysis of rank value set by standard deviation: Proposed vs. Others.

Some weaknesses of the proposed framework are:

In the practical sense, the DMs need some training with the PLTS environment for sharing their rating over the objects.

Also, manually associating occurring probability value with each linguistic term is difficult and prone to inaccuracies.

6. Conclusion

In this paper, efforts are made to propose a new decision framework which presents a new aggregation operator for fusing DMs’ PLTS information. Further, the SV method is extended under PLTS environment for determining the weights of the criteria. WASPAS method is also extended under PLTS environment for selecting a suitable object from the set of objects. The practical use of the proposed framework is demonstrated by using two numerical examples viz., strategic project selection problem and green supplier selection problem. Some distinct features of the proposed decision framework are:

The proposed framework presents systematic scientific methods for selecting suitable object form the set of objects under group decision making context with PLTS information used for rating objects and criteria.

The proposed framework reduces DMs’ intervention to a certain extent which mitigates uncertainty and vagueness in the decision-making process. The DMs are used only for presenting their opinions and the systematic methods are applied for arriving at inferences in the decision-making process.

Such decision frameworks help managers to a greater extent by providing a ready-made tool for analysis and rational decision making. With this framework, the managers can easily choose the suitable alternative and effectively plan for backup management during critical uncertain scenarios.

Further, the framework is highly flexible and can be used for solving any selection problem which adopts PLTS information for rating objects and criteria.

As a part of the future work, we plan to address the weaknesses of the proposed decision framework. Also, the concept of PLTS can be extended for consensus reaching which is motivated by [38, 39]. Further, motivated by the work of Li et al. [40 –42] plans are made to propose new decision framework under personalized linguistic semantics. Motivated by [43, 44], plans are made for strategic weight manipulation of criteria under PLTS context based on DMs’ desire to change weight values. The proposed framework can be integrated with machine learning methods for solving critical decision-making problems in the field of medical sciences, management etc. Also, new fuzzy sets can be proposed for effectively managing uncertainty and vagueness.

Footnotes

Acknowledgments

Authors thank funding agencies University Grants Commission (UGC), India and Department of Science and Technology (DST), India for their financial support from grant no. F./2015-17/RGNF-2015-17-TAM-83 and SR/FST/ETI-349/2013. Authors also thank the editor and the anonymous reviewer(s) for their constructive comments which improved the quality of the paper.

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A decision-making framework under probabilistic linguistic term set for multi-criteria group decision-making problem

Abstract

Keywords

1. Introduction

3.1. Proposed PLSWG operator

4.1. Strategic project selection problem

Table 2 Decision matrix with PLTS information

Table 7 Decision matrix with PLTS information from different DMs

Table 11 Ranking order from different methods

Footnotes

Acknowledgments

References

Table 2
Decision matrix with PLTS information

Table 7
Decision matrix with PLTS information from different DMs

Table 11
Ranking order from different methods