# A New Interactive Decision Approach with Probabilistic Linguistic Data: An Application in the Academic Sector

Authors: K.S. Ravichandran; R. Krishankumar; Sibnath Deb; Dragan Pamucar; David Paul; Darko Božanić
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

As an innovative generalization to a linguistic term set, the probabilistic variant is gaining abundant attraction in the decision process. However, earlier studies with this variant for decision-making have not adequately explored hesitation in data articulation and interactive ranking. Driven by the claim, in this paper, a new integrated approach is put forward under the probabilistic linguistic context, which attempts to address the claims by presenting a regret/rejoice technique and an interactive WASPAS algorithm for determining the significance of factors and personalized ranking of alternatives. To test the usefulness of the approach, the online course prioritization problem based on empirical data is exemplified, and a comparison demonstrates the benefits and limitations of the proposed work.


---

## Body

## 1. Introduction

Decision-making under uncertainty is an interesting and complex problem in day-to-day life [1]. Zadeh [2] introduced the concept of linguistic decision-making that was further ameliorated by the work of Herrera et al. [3]. Rodriguez identified that the linguistic term set (LTS) could not accept more than one instance at a given point in time. To resolve the issue, hesitant fuzzy linguistic information (HFLI) [4] was put forward, which embedded the idea of hesitation and allowed more than one instance as preference information. Driven by this feature, many researchers used HFLI for decision-making [5]. Pang et al. [6] identified that though HFLI allowed multiple information, the confidence associated with each element is either ignored or assumed to be the same. A probabilistic linguistic term set (PLTS) was introduced to handle the issue.PLTS can accept more than one term and associate occurrence probability to each term. This setting gained a lot of attention in the decision-making context in both in theoretical (such as operational laws/aggregation functions [7–13]; ranking [14–19]; measures [20–22]) and the application perspectives [23–29]. Two popular reviews dealing with the applicability of PLTS in decision-making [30] and aggregation operators in the PLTS context [31] give a holistic view of the core importance of the structure for the decision process.Based on these two reviews, specific challenges can be identified, as follows:(i)
consideration of views from a large population that is heterogeneous is challenging to manage,(ii)
hesitation during opinion sharing is not adequately captured during weight estimation,(iii)
personalized prioritization based on personal choices on alternatives is lacking, along with the idea of capturing interaction among factors.These challenges motivated the present study, and the following contributions are henceforth made:(i)
PLTS is used to transform the Likert-scale rating from multiple participants to a holistic decision matrix for prioritization of alternatives,(ii)
the regret/rejoice approach is put forward to effectively capture hesitation under the PLTS context; weights of the rating personnel are considered during factor significance calculation,(iii)
a new interactive algorithm with the WASPAS (weighted aggregated sum product assessment) technique as the base formulation is developed to consider personal choices on alternatives and the nature of factors.These contributions add value to the PLTS-based decision-making and aid in rational selection. Further organization of the paper is as follows. First, literature relating to regret theory and WASPAS are presented in Section2, tracked by the methodology in Section 3, which covers the core aspect of the paper by detailing step-wise the methods for significance calculation and prioritization. An empirical case example is exemplified in Section 4 to clarify the usefulness of the developed interactive approach. Comparison with other models is also presented to showcase the efficacy and shortcomings of the work. Finally, concluding remarks with future research scope are provided in Section 5.


## 2. Literature Review

### 2.1. Regret Theory

Regret theory (RT) is a concept adapted from psychology into decision-making that deals with a clear understanding of the mindset of an expert upon choosing a candidate over another [32]. This idea resembles the human decision process and is a viable approach for determining the hesitation of experts during the decision process [33]. Some review articles [34–37] on RT are prepared by scholars that infer that the concept (i) is elegant and effective in decision-making; (ii) allows efficient capturing of experts' hesitation; and (iii) promotes methodical determination of weights of factors. Chen et al. [38] used the fuzzy context with axiomatic design and RT for logistic provider evaluation under an omnichannel environment. Wang et al. [39] presented a stock grading model with RT and compared the result with prospect theory in the context of fuzzy interval-neutrosophic probability. Qu et al. [40] gave an extended RT with group satisfaction measure under dual hesitant context for grading shared-bikes for an investment. Xia [41] developed the multiobjective model by adopting the RT idea under the hesitant fuzzy linguistic domain to solve firms' decision problems. Gong et al. [42] gave a cloud model with linguistic structure by extending the RT concept over dual expectation of stock evaluation. Wang et al. [43] developed RT-based TOPSIS (a technique for order of preference by similarity to ideal solution) with interval type-2 data in a three-way decision context. Liu et al. [44] assessed projects for venture capitalists under probabilistic hesitant setting by adopting RT and mathematical model formulated using entropy and water-filling strategy. Ren et al. [45] developed an extension to RT under an intuitionistic fuzzy environment by adopting Canberra distance for solving decision problems of supplier selection for assembly components. Gong et al. [46] performed a portfolio assessment with multiobjective programming by considering the RT formulation and the DEA (data envelopment analysis) approach. Liang et al. [47] solved decision problems in probabilistic interval hesitant fuzzy data by developing an integrated RT-gain/lost dominance approach.


### 2.2. WASPAS Method

The method's inception is from [48], which linearly combines the weighted sum and product measures. Driven by the simplicity of the method, many researchers used the technique for decision-making. Mardani et al. [49] prepared a detailed review on WASPAS showcasing the method's usefulness in the decision process. Mishra et al. [50] evaluated green suppliers in an industry by extending WASPAS with exponential divergence concept with hesitant fuzzy data. Tus and Adali [51] prepared the CRITIC-WASPAS approach as an integrated framework for software evaluation in firms. Pamucar et al. [52] gave a new extension to WASPAS under neutrosophic context for assessing advisors in the process of transporting hazardous goods. Krishankumar et al. [53] came up with a variance-WASPAS integrated method for green supplier selection in a linguistic environment. Bouchraki et al. [54] provided an integrated AHP-WASPAS with fuzzy numbers for assessing claims of customers concerning drinking water service in a firm. Ilbahar et al. developed a Pythagorean fuzzy WASPAS model for renewable energy selection by considering sustainable factors. Krishankumar et al. [55] ranked risk management strategies in the construction sector by proposing a combined framework with variance and WASPAS technique under a double hierarchy setting. Pamucar et al. [56] selected suitable transport modes for reaching airports in Istanbul by preparing a model with fuzzy numbers, a level-based weight assessment technique, and WASPAS. Simic et al. [57] ranked last-mile travel modes of goods by presenting WASPAS under picture fuzzy context. Ali et al. [58] came up with a new framework under an uncertain linguistic setting with arithmetic operations, fusion functions, entropy, and the WASPAS technique for selecting suppliers in a firm. Osintsev et al. [59] assessed compression methods for aerial images by gathering linguistic ratings and adopting neutrosophic WASPAS algorithm. Bozanic et al. [60] prepared a new extension of WASPAS and AHP approaches to ordered fuzzy values for rationally ranking improvement projects.


### 2.3. Insights from the Review

Based on the previously prepared previously review, it is clear that PLTS is a sophisticated preference structure that can associate confidence levels to terms and aid in heterogeneous data transformation into holistic data forms. Further, determining significance values by properly considering hesitation is crucial in rational decision-making. Finally, the idea of personalized ordering with appropriate consideration to the nature of criteria is key for effective decision-making. The insights are in line with the challenges discussed in the study, which are circumvented by the contributions made in this work.Figure1 presents the working model of the proposed interactive approach by utilizing PLD. As claimed in [6], PLD is a flexible preference style that can effectively model diverse opinions from heterogeneous candidates/participants. The procedure adopted for converting the rating information into PLD is explained in Section 3. Data on each online course is given by a diverse set of participants with a different count, background, expertise, demography, and so on. PLTS is a flexible structure adopted to transform the various data into a holistic data matrix to prioritize online professional courses. The data is collected empirically from participants of the short-term online certification course hosted by RGNIYD, an academic institution during the pandemic time. Officials who hosted the certification program acted as experts and offered their opinion on each factor considered for rating the courses. By using the regret/rejoice technique, the significance values of elements are calculated. Later, the data matrix and the significance vector are used by the interactive WASPAS algorithm for the prioritization of online courses given job opportunities for Indian youths. The officials collect personal choices on each course as a choice vector. The nature of factors is also being considered in this interactive WASPAS algorithm for rational prioritization of courses.Figure 1
Working model for online professional course selection with PLD.


## 2.1. Regret Theory

Regret theory (RT) is a concept adapted from psychology into decision-making that deals with a clear understanding of the mindset of an expert upon choosing a candidate over another [32]. This idea resembles the human decision process and is a viable approach for determining the hesitation of experts during the decision process [33]. Some review articles [34–37] on RT are prepared by scholars that infer that the concept (i) is elegant and effective in decision-making; (ii) allows efficient capturing of experts' hesitation; and (iii) promotes methodical determination of weights of factors. Chen et al. [38] used the fuzzy context with axiomatic design and RT for logistic provider evaluation under an omnichannel environment. Wang et al. [39] presented a stock grading model with RT and compared the result with prospect theory in the context of fuzzy interval-neutrosophic probability. Qu et al. [40] gave an extended RT with group satisfaction measure under dual hesitant context for grading shared-bikes for an investment. Xia [41] developed the multiobjective model by adopting the RT idea under the hesitant fuzzy linguistic domain to solve firms' decision problems. Gong et al. [42] gave a cloud model with linguistic structure by extending the RT concept over dual expectation of stock evaluation. Wang et al. [43] developed RT-based TOPSIS (a technique for order of preference by similarity to ideal solution) with interval type-2 data in a three-way decision context. Liu et al. [44] assessed projects for venture capitalists under probabilistic hesitant setting by adopting RT and mathematical model formulated using entropy and water-filling strategy. Ren et al. [45] developed an extension to RT under an intuitionistic fuzzy environment by adopting Canberra distance for solving decision problems of supplier selection for assembly components. Gong et al. [46] performed a portfolio assessment with multiobjective programming by considering the RT formulation and the DEA (data envelopment analysis) approach. Liang et al. [47] solved decision problems in probabilistic interval hesitant fuzzy data by developing an integrated RT-gain/lost dominance approach.


## 2.2. WASPAS Method

The method's inception is from [48], which linearly combines the weighted sum and product measures. Driven by the simplicity of the method, many researchers used the technique for decision-making. Mardani et al. [49] prepared a detailed review on WASPAS showcasing the method's usefulness in the decision process. Mishra et al. [50] evaluated green suppliers in an industry by extending WASPAS with exponential divergence concept with hesitant fuzzy data. Tus and Adali [51] prepared the CRITIC-WASPAS approach as an integrated framework for software evaluation in firms. Pamucar et al. [52] gave a new extension to WASPAS under neutrosophic context for assessing advisors in the process of transporting hazardous goods. Krishankumar et al. [53] came up with a variance-WASPAS integrated method for green supplier selection in a linguistic environment. Bouchraki et al. [54] provided an integrated AHP-WASPAS with fuzzy numbers for assessing claims of customers concerning drinking water service in a firm. Ilbahar et al. developed a Pythagorean fuzzy WASPAS model for renewable energy selection by considering sustainable factors. Krishankumar et al. [55] ranked risk management strategies in the construction sector by proposing a combined framework with variance and WASPAS technique under a double hierarchy setting. Pamucar et al. [56] selected suitable transport modes for reaching airports in Istanbul by preparing a model with fuzzy numbers, a level-based weight assessment technique, and WASPAS. Simic et al. [57] ranked last-mile travel modes of goods by presenting WASPAS under picture fuzzy context. Ali et al. [58] came up with a new framework under an uncertain linguistic setting with arithmetic operations, fusion functions, entropy, and the WASPAS technique for selecting suppliers in a firm. Osintsev et al. [59] assessed compression methods for aerial images by gathering linguistic ratings and adopting neutrosophic WASPAS algorithm. Bozanic et al. [60] prepared a new extension of WASPAS and AHP approaches to ordered fuzzy values for rationally ranking improvement projects.


## 2.3. Insights from the Review

Based on the previously prepared previously review, it is clear that PLTS is a sophisticated preference structure that can associate confidence levels to terms and aid in heterogeneous data transformation into holistic data forms. Further, determining significance values by properly considering hesitation is crucial in rational decision-making. Finally, the idea of personalized ordering with appropriate consideration to the nature of criteria is key for effective decision-making. The insights are in line with the challenges discussed in the study, which are circumvented by the contributions made in this work.Figure1 presents the working model of the proposed interactive approach by utilizing PLD. As claimed in [6], PLD is a flexible preference style that can effectively model diverse opinions from heterogeneous candidates/participants. The procedure adopted for converting the rating information into PLD is explained in Section 3. Data on each online course is given by a diverse set of participants with a different count, background, expertise, demography, and so on. PLTS is a flexible structure adopted to transform the various data into a holistic data matrix to prioritize online professional courses. The data is collected empirically from participants of the short-term online certification course hosted by RGNIYD, an academic institution during the pandemic time. Officials who hosted the certification program acted as experts and offered their opinion on each factor considered for rating the courses. By using the regret/rejoice technique, the significance values of elements are calculated. Later, the data matrix and the significance vector are used by the interactive WASPAS algorithm for the prioritization of online courses given job opportunities for Indian youths. The officials collect personal choices on each course as a choice vector. The nature of factors is also being considered in this interactive WASPAS algorithm for rational prioritization of courses.Figure 1
Working model for online professional course selection with PLD.


## 3. Methodology

### 3.1. Preliminaries

The authors provide some basic concepts related to LTS, HFLTS, and PLTS.Definition 1.
Reference [3]. Let TS=sz|z=0,1,…,q be an LTS with s0 and sq as the initial and final objects with v being a positive integer. The features of S are as follows:
If(1)za>zbthensza>szb,negsza=szb,where (2)za+zb=q.Definition 2.
[4]. Let TS be as before. Then, an HFLTS is given by(3)DF=a,hDFa|a∈A,where(4)hDFa=ha=szk|k=1,2,…,#ha.Definition 3.
[6]. Let TS be as before. Then, PLTS is given by(5)Dp=Dkpk|Dk∈TS,0≤pk≤1,∑kpk≤1,k=1,2,…,#Dp,where Dkpk is the kth instance with pk being the occurrence probability that is associated with the term Dk and #Dp refers to the number of instances.
Note 1.di=szkipik is the probabilistic linguistic element (or) probabilistic linguistic data (PLE/PLD) and many such elements constitute the PLTS. Terms have the following semantics:(6)so=none,s1=extremely  low,s2=very  low,s2=low,s4=moderate,s5=high,s6=very  high,s7=extremely high..Definition 4.
Reference [21]. Two PLEs d1 and d2 considered. Then, the operations are given by(7)d1⊕d2=f−1fd1+fd2,d1⊕d2=f−1fd1×fd2,where f and f−1 are obtained from [21] (8)f:τ=z4q+0.5,and (9)f−1:sz=s2τ−1×2q.


### 3.2. Data Transformation

This section focuses on converting Likert scale ratings from a heterogeneous set of participants into holistic data for decision-making. PLTS is a suitable structure for supporting this conversion process. Occurrence probability values are associated as confidence values to the different rating terms that give an overview of all the participants and their rating for a particular instance.The present study considered four professional courses for short-term certification programs conducted online during 2020 (pandemic time). The participant count was diverse for each course, and a normal rating was insufficient to grasp the data correctly. Thus, the rating information is transformed to PLD. Proper consideration is given to the diverse participants pertaining to each course by doing the transformation. As a simple walkthrough example, suppose there are two courses, namely, course A and course B, with participants as three and five, respectively, for each course. Rating information is obtained as A(1) =s4, A(2) =  s3, and A(3) =  s4; B(1) =  s3, B(2) =  s3, B(3) =  s3, B(4) =  s2, and B(5) =  s5. It must be noted that A(1) to A(3) denotes Likert scale rating from three students on course A. Similarly, B(1) to B(5) denotes Likert scale rating from five students on course B. PLEs with respect to course A based on the rating information from three students are calculated as follows:(10)A=s423s313=s40.67s30.33,B=s335s215s515=s30.6s20.2s50.2.For course A, the factor2/3 for s4 is obtained since two students out of three rated course A as s4. On the other hand, for course B, a factor of 1/5 is obtained for s2 since one student rated course B as s2 out of five students. Similarly, other values can be determined. This mechanism is adopted to transform the input data matrix for ranking online professional courses within the perspective of job opportunities for Indian youths. It must be noted that the instances chosen for the study depend on the expert. For the present study, two cases in high probability are chosen.


### 3.3. Significance Calculation: Regret/Rejoice Technique

The factors considered for evaluating online courses are competing with one another and pose unbiased or heterogeneous importance. Kao [56] stated that this needs to be methodically determined to avoid subjectivity and inaccuracies from direct elicitation. Popularly, weights are determined either with partial information or unknown information. Programming models are used for partial context, and methods such as entropy [20], analytical methods [62–64], variance [53], and so on are used for unknown context. Though these methods estimate weights, hesitation during opinion sharing is not adequately captured, and the regret factor incurred by the personnel during the decision process is not properly realized. The regret/rejoice technique is put forward under the PLTS context to mitigate the issue. Regret theory [32] is an exciting concept that deals with the psychological aspect and demonstrates the behavior based on a particular choice made by the expert. The technique that evolves from the theory (i) is simple and elegant, (ii) effectively represents the hesitation of experts, and (iii) captures the regret factor incurred by an expert during the decision process. From the review, it is clear that the technique is suitable for the significance determination of factors.Inspired by the claims, steps for calculation of significance of factors are presented as follows:Step 1. Experts share their opinions on each factor as a PLE. g vectors of 1×y order are obtained.Step 2. Neumann utility function is applied to the score measures of PLEs by using (5) and (6). Neumann values are determined for each opinion value from the expert leading to g×y values.(11)Sdlj=∑kzljk×pljk,(12)UFj=∑l=1gvSdlj+RvSdlj−vSdlj˜,wherevSdlj=Sdljη is the von-Neumann function, RvSdlj=1−e−β×vSdlj is the regret function and(13)vSdlj˜=maxj∈BenefitvSdljor minj∈CostvSdlj.Here,η is the power value, and β is the risk aversion value. Both are considered as 0.50 in this study.Step 3. Normalize the values from the utility function by using (7) to get the significance vector for the factors. Values from Step 2 are considered as input to determine the significance values.(14)Sfj=UFj∑jUFj,where Sfj is in the range 0 to 1, with the sum being unity, referred to as the significance value.


### 3.4. Ranking Alternatives: WASPAS Method

The focus of the section is to present an approach for ranking alternatives based on the set of criteria. As discussed earlier, criteria are heterogeneous and pose biased significance. Therefore, the calculated significance vector from the previous section is utilized here, along with the preference data.Based on the review previously made, it is clear that WASPAS [48] (i) is simple and elegant; (ii) popularly used in decision-making, and (iii) works using sum/product functions as base formulation. It can be noted that the WASPAS method does not consider the nature of criteria and cannot accept the personal choices of experts during the ranking process. Motivated by the issue, in this section, an interactive extension of WASPAS is put forward, and the stepwise formulation is given as follows.Step 1. Based on the participants rating, a holistic PLTS data matrix is formed of x×y order by adopting the procedure given in Section 3.2. Here x denotes the number of professional courses (alternatives), and y represents the number of factors.Step 2. Apply transformation functions given in (15) to effectively accommodate the nature of factors (criteria) in the rank estimation.(15)dij=dij=dijforj∈benefit,dij=dijcforj∈cost,wheredijc=sq−zijk1−pijk is the complement of dij. Ideally, the probability values are normalized to retain the property of a PLE. Here q is as denoted in Definition 1.Step 3. Determine each alternative's weighted sum and product values based on the significance vector and holistic data matrix. Equations (9) and (10) are applied for this purpose.(16)AFi=∑j=1ySfj×∑kpoi×zijk×pijk∑jzijk×pijk,(17)PFi=∏j=1y∑kpoi×zijk×pijk∑jzijk×pijkSfj,wherepoi is the personal opinion on alternative i in the unit interval with ∑ipoi as unity, poi is a vector of 1×x order, Sfj is the significance of factor j, zijk is the subscript of the linguistic term in the kth instance for ith professional course rated based on jth factor, pijk is the occurrence probability value in the kth instance for ith professional course rated based on jth factor, and AFi and PFi are the weighted sum and weighted product values associated with each alternative.Equations (9) and (10) yield two vectors each of order 1×x, and Sfj is the significance value of the factor j calculated by applying (14). poi is another parameter that denotes the personal opinion on an alternative i given by the experts.It must be noted that when instances of the PLEs are not equal, the procedure mentioned in Definition 6 of [6] is adopted, which makes the instances in the PLEs equal.Step 4. Calculate the net rank TRi of each alternative i by adopting the idea of a linear combination of weighted sum and weighted product determined from (9) and (10). Equation (18) is used for the calculation that yields a rank vector of order 1×x.(18)TRi=θ×AFi+1−θ×PFi,where θ is the strategy measure between 0 and 1.It must be noted that (18) yields a vector of 1×x order that contains the rank values of each professional certification course that is considered in the case example. By increasing the strategy values stepwise from 0.1 to 0.9, nine vectors of 1×x can be obtained from (18). Arrange the values obtained from (18) in the descending order for forming the ordering of alternatives.


## 3.1. Preliminaries

The authors provide some basic concepts related to LTS, HFLTS, and PLTS.Definition 1.
Reference [3]. Let TS=sz|z=0,1,…,q be an LTS with s0 and sq as the initial and final objects with v being a positive integer. The features of S are as follows:
If(1)za>zbthensza>szb,negsza=szb,where (2)za+zb=q.Definition 2.
[4]. Let TS be as before. Then, an HFLTS is given by(3)DF=a,hDFa|a∈A,where(4)hDFa=ha=szk|k=1,2,…,#ha.Definition 3.
[6]. Let TS be as before. Then, PLTS is given by(5)Dp=Dkpk|Dk∈TS,0≤pk≤1,∑kpk≤1,k=1,2,…,#Dp,where Dkpk is the kth instance with pk being the occurrence probability that is associated with the term Dk and #Dp refers to the number of instances.
Note 1.di=szkipik is the probabilistic linguistic element (or) probabilistic linguistic data (PLE/PLD) and many such elements constitute the PLTS. Terms have the following semantics:(6)so=none,s1=extremely  low,s2=very  low,s2=low,s4=moderate,s5=high,s6=very  high,s7=extremely high..Definition 4.
Reference [21]. Two PLEs d1 and d2 considered. Then, the operations are given by(7)d1⊕d2=f−1fd1+fd2,d1⊕d2=f−1fd1×fd2,where f and f−1 are obtained from [21] (8)f:τ=z4q+0.5,and (9)f−1:sz=s2τ−1×2q.


## 3.2. Data Transformation

This section focuses on converting Likert scale ratings from a heterogeneous set of participants into holistic data for decision-making. PLTS is a suitable structure for supporting this conversion process. Occurrence probability values are associated as confidence values to the different rating terms that give an overview of all the participants and their rating for a particular instance.The present study considered four professional courses for short-term certification programs conducted online during 2020 (pandemic time). The participant count was diverse for each course, and a normal rating was insufficient to grasp the data correctly. Thus, the rating information is transformed to PLD. Proper consideration is given to the diverse participants pertaining to each course by doing the transformation. As a simple walkthrough example, suppose there are two courses, namely, course A and course B, with participants as three and five, respectively, for each course. Rating information is obtained as A(1) =s4, A(2) =  s3, and A(3) =  s4; B(1) =  s3, B(2) =  s3, B(3) =  s3, B(4) =  s2, and B(5) =  s5. It must be noted that A(1) to A(3) denotes Likert scale rating from three students on course A. Similarly, B(1) to B(5) denotes Likert scale rating from five students on course B. PLEs with respect to course A based on the rating information from three students are calculated as follows:(10)A=s423s313=s40.67s30.33,B=s335s215s515=s30.6s20.2s50.2.For course A, the factor2/3 for s4 is obtained since two students out of three rated course A as s4. On the other hand, for course B, a factor of 1/5 is obtained for s2 since one student rated course B as s2 out of five students. Similarly, other values can be determined. This mechanism is adopted to transform the input data matrix for ranking online professional courses within the perspective of job opportunities for Indian youths. It must be noted that the instances chosen for the study depend on the expert. For the present study, two cases in high probability are chosen.


## 3.3. Significance Calculation: Regret/Rejoice Technique

The factors considered for evaluating online courses are competing with one another and pose unbiased or heterogeneous importance. Kao [56] stated that this needs to be methodically determined to avoid subjectivity and inaccuracies from direct elicitation. Popularly, weights are determined either with partial information or unknown information. Programming models are used for partial context, and methods such as entropy [20], analytical methods [62–64], variance [53], and so on are used for unknown context. Though these methods estimate weights, hesitation during opinion sharing is not adequately captured, and the regret factor incurred by the personnel during the decision process is not properly realized. The regret/rejoice technique is put forward under the PLTS context to mitigate the issue. Regret theory [32] is an exciting concept that deals with the psychological aspect and demonstrates the behavior based on a particular choice made by the expert. The technique that evolves from the theory (i) is simple and elegant, (ii) effectively represents the hesitation of experts, and (iii) captures the regret factor incurred by an expert during the decision process. From the review, it is clear that the technique is suitable for the significance determination of factors.Inspired by the claims, steps for calculation of significance of factors are presented as follows:Step 1. Experts share their opinions on each factor as a PLE. g vectors of 1×y order are obtained.Step 2. Neumann utility function is applied to the score measures of PLEs by using (5) and (6). Neumann values are determined for each opinion value from the expert leading to g×y values.(11)Sdlj=∑kzljk×pljk,(12)UFj=∑l=1gvSdlj+RvSdlj−vSdlj˜,wherevSdlj=Sdljη is the von-Neumann function, RvSdlj=1−e−β×vSdlj is the regret function and(13)vSdlj˜=maxj∈BenefitvSdljor minj∈CostvSdlj.Here,η is the power value, and β is the risk aversion value. Both are considered as 0.50 in this study.Step 3. Normalize the values from the utility function by using (7) to get the significance vector for the factors. Values from Step 2 are considered as input to determine the significance values.(14)Sfj=UFj∑jUFj,where Sfj is in the range 0 to 1, with the sum being unity, referred to as the significance value.


## 3.4. Ranking Alternatives: WASPAS Method

The focus of the section is to present an approach for ranking alternatives based on the set of criteria. As discussed earlier, criteria are heterogeneous and pose biased significance. Therefore, the calculated significance vector from the previous section is utilized here, along with the preference data.Based on the review previously made, it is clear that WASPAS [48] (i) is simple and elegant; (ii) popularly used in decision-making, and (iii) works using sum/product functions as base formulation. It can be noted that the WASPAS method does not consider the nature of criteria and cannot accept the personal choices of experts during the ranking process. Motivated by the issue, in this section, an interactive extension of WASPAS is put forward, and the stepwise formulation is given as follows.Step 1. Based on the participants rating, a holistic PLTS data matrix is formed of x×y order by adopting the procedure given in Section 3.2. Here x denotes the number of professional courses (alternatives), and y represents the number of factors.Step 2. Apply transformation functions given in (15) to effectively accommodate the nature of factors (criteria) in the rank estimation.(15)dij=dij=dijforj∈benefit,dij=dijcforj∈cost,wheredijc=sq−zijk1−pijk is the complement of dij. Ideally, the probability values are normalized to retain the property of a PLE. Here q is as denoted in Definition 1.Step 3. Determine each alternative's weighted sum and product values based on the significance vector and holistic data matrix. Equations (9) and (10) are applied for this purpose.(16)AFi=∑j=1ySfj×∑kpoi×zijk×pijk∑jzijk×pijk,(17)PFi=∏j=1y∑kpoi×zijk×pijk∑jzijk×pijkSfj,wherepoi is the personal opinion on alternative i in the unit interval with ∑ipoi as unity, poi is a vector of 1×x order, Sfj is the significance of factor j, zijk is the subscript of the linguistic term in the kth instance for ith professional course rated based on jth factor, pijk is the occurrence probability value in the kth instance for ith professional course rated based on jth factor, and AFi and PFi are the weighted sum and weighted product values associated with each alternative.Equations (9) and (10) yield two vectors each of order 1×x, and Sfj is the significance value of the factor j calculated by applying (14). poi is another parameter that denotes the personal opinion on an alternative i given by the experts.It must be noted that when instances of the PLEs are not equal, the procedure mentioned in Definition 6 of [6] is adopted, which makes the instances in the PLEs equal.Step 4. Calculate the net rank TRi of each alternative i by adopting the idea of a linear combination of weighted sum and weighted product determined from (9) and (10). Equation (18) is used for the calculation that yields a rank vector of order 1×x.(18)TRi=θ×AFi+1−θ×PFi,where θ is the strategy measure between 0 and 1.It must be noted that (18) yields a vector of 1×x order that contains the rank values of each professional certification course that is considered in the case example. By increasing the strategy values stepwise from 0.1 to 0.9, nine vectors of 1×x can be obtained from (18). Arrange the values obtained from (18) in the descending order for forming the ordering of alternatives.


## 4. Case Example: Online Professional Course Selection

This section attempts to exemplify the applicability of the research model. For this, a case example with empirical data from participants is adopted to select suitable online courses for Indian youths focused on job creation for the youth population with an IT background. Four online courses, namely, data science, machine learning, cyber security, and cloud computing were conducted as short-term certifications in RGNIYD, an academic institution. Youths with fair IT background attended the program, and it accounted for 69, 164, 64, and 103, respectively. Out of these, participants who volunteered for data collection were 47, 110, 44, and 83, respectively. It can be seen that the population size is heterogeneous for each online course. To better formulate the data, PLTS is adopted. During the pandemic, these courses were hosted online for the betterment of youths. Resource personnel from the institute of national importance were invited to deliver lectures and hands-on training to the participants.To further understand the efficacy of these courses in terms of job creation for youths, we invited volunteers to participate in a semistructured questionnaire created using Google forms and circulated online for data collection. The study aimed to rank online courses as per the perception of youth. Factors utilized in the study for rating online courses are the usefulness of the course, job creation from the course, resource personnel content knowledge, expected prerequisite/preparation of the course, stress due to pandemic, and connectivity issues. Based on the literature [65–67] and intuition, these factors are finalized for the study. The last three factors are cost type, and the other factors are benefit type.For the sake of implementation, authors refer to online courses (cyber security, machine learning, data science, and cloud computing) asCC1, CC2, CC3, and CC4; factors as FC1, FC2, FC3, FC4, FC5, and FC6; course organizers as CO1, CO2, and CO3. The last three factors are of cost type, while the rest are benefit type. Steps for ranking the online courses are given as follows.Step 1. Consider rating data from each participant on the four online courses based on the six factors. Likert-scale rating is adopted by participants transformed to PLE by adopting the procedure described in Section 3.2.Table1 gives the PLEs as a data matrix for the participants' four courses rated on six linguistically (Likert-scale). The transformation procedure presented in Section 3.2 is used for constructing PLEs from the linguistic data. This is crucial because participants for each course are heterogeneous in terms of count, demography, and so on. The authors considered the top two linguistic terms based on the associated occurrence probability values to build the decision matrix.Step 2. Officials/organizers of the course (online) provide their rating on the factors that helps in determining the significance of the elements (Table 2). The procedure developed in Section 3.3 is used for this purpose.The procedure put forward in Section3.3 is applied to determine the utility values of factors by considering regret/rejoice factors and von-Neumann values (as shown in Figure 2). vSdlj˜ is determined as 2.025, 2.023, 2.098, 1.643, 1.265, and 1.732. Equations (6) and (7) yield the significance values of factors as 0.097, 0.230, 0.090, 0.203.0.170, and 0.210, respectively.Step 3. With the help of data from Step 1 and vector from Step 2, online courses are ranked by adopting the algorithm proposed in Section 3.4. From the data transformation procedure, the heterogeneous participants' data of each course is holistically transformed into a data matrix of order 4×6. Significance value is a vector of order 1×6.Personal Opinion.poi is considered for each course as 0.25, 0.20, 0.35, and 0.20, respectively. From Table 3, the parameter values associated with the improved WASPAS for each online professional course are obtained. The TRi values indicate the ordering of courses as CC3≻CC1≻CC4≽CC2, which infers that the data science course is considered most suited for job opportunities for Indian youths, followed by cyber security, machine learning, and cloud computing. In particular, machine learning and cloud computing are equally preferred by Indian youths in terms of job opportunities in organizations.Step 4. Conduct sensitivity analysis with significance and strategy values of factors and experts by altering values systematically by shift operations.Table 1
Linguistic data transformed to PLE for decision-making.
FactorsProfessional online coursesCC1CC2CC3CC4FC1s50.5s40.44s40.46s60.36s50.5s20.5s40.44s30.56FC2s50.45s30.4s40.66s50.3s60.52s30.4s20.35s30.47FC3s30.6s50.4s60.55s40.4s60.46s40.52s50.54s60.43FC4s50.6s20.33s40.44s30.33s30.63s40.33s50.44s30.33FC5s20.25s30.27s30.6s40.4s50.55s60.33s40.62s20.28FC6s60.5s40.5s60.35s50.6s30.42s20.25s30.52s40.44Table 2
Opinions for determining factors' significance.
FactorsOfficials/organizersCO1CO2CO3FC1s30.45s40.44s40.4s50.5s30.6s50.3FC2s50.5s40.4s50.6s30.25s40.45s50.4FC3s40.4s30.5s40.35s60.5s40.4s30.6FC4s20.55s40.4s50.35s60.4s40.45s20.55FC5s30.3s20.35s50.4s40.45s40.5s50.5FC6s30.6s40.3s50.5s40.45s50.6s20.4Table 3
Rank values associated with each online professional course.
Online coursesParameters of interactive WASPAS algorithmAFiPFiTRiCC10.0665230.0624330.064478CC20.0481260.045090.046608CC30.0901130.0831390.086626CC40.0477290.0456340.046891Figure 2
(a) von-Neumann values and (b) regret/rejoice values.
(a)(b)Sensitivity measure is investigated in both inter/intracontext by varying the significance of factors through shift operations and periodically increasing step size of strategy values. In the intercontext, the effect of new sets of significance vectors on rank values is determined, and in the intracontext, the impact of strategy values on rank values is determined. Figures3(a) to 3(f) show the effect of both the values (alteration of weights (inter) and alteration of strategy values (intra) on the ordering of the online professional courses. Six bar graphs are depicted for six weight sets (obtained by shift operation of significance values), and within each graph, strategy values are altered from 0.1 to 0.9. Rank values of each course is plotted as the bar. It can be seen that the Indian youths highly prefer data science in terms of job opportunities in organizations. Courses such as machine learning and cloud computing are equally considered in their respective ranking regarding job opportunities for the youth population. The empirical case study conducted by RGNIYD, an academic institute, serves as a pilot study in effectively understanding the importance of online training (teaching/learning) during pandemic situations and the courses that fetch job opportunities to Indian youths based on their data. The inter/intrasensitivity analysis shows that the proposed framework is robust even after adequate alterations are incorporated.Figure 3
Sensitivity measure of factors' significance: (a) to (f) is set 1 to set 6 (X axis 1 to 9 indicate strategy values from 0.1 to 0.9, resp., with step size 0.1).
(a)(b)(c)(d)(e)(f)


### 4.1. Comparison Study

The authors attempt to showcase the efficacy of the proposed work by comparing the model with a close counterpart method [48]. From the sensitivity graph shown in Figure 3, it is clear that the proposed work is highly robust even after alterations to factor significance, and strategy values are done adequately. To further realize the efficacy of the work, the model in [53] is compared with the proposed work. Table 4 gives the summarized view of the theoretical benefits of the proposed work over [53]. Besides, authors also compare the proposed work with extant models, namely, [13, 17] and [23], which actively use PLTS in their framework for attaining rational decisions. Finally, data from the case example are provided to these models, and the courses are ranked, and they are shown in Table 5.Table 4
Summary of characteristics: proposed and other models.
ContextProposed work[53]DataPLDPLDWeights of factorsCalculated methodicallyCalculated methodicallyRegret attitude characterizationDone, regret/rejoice factorNot doneNature of factorsConsidered, both during weight assessment and rankingNot consideredPersonal choices of expertsConsidered during rankingNot consideredTable 5
Order of professional courses obtained from PLTS models: proposed versus others.
CourseProposedReference [48] (0.2)[13][17][23]CC123233CC241111CC312322CC434444Table5 and Figure 4 show that the proposed model produces a unique ranking order of professional courses. It can be intuitively inferred owing to the proposed model's ability to consider the nature of factors during the ranking of courses (alternatives) and capture experts' personal opinions during ranking, which provides a sense of personalization and is lacking in existing models. Apart from this, specific theoretical merits of the framework are listed below. Based on the briefing in Table 4, it is inferred that the proposed work is novel and innovative. To further detail the claim, specific points are presented as follows:(i)
PLD is a sophisticated structure that allows for the elegant transformation of rating data from multiple heterogeneous candidates/users. Furthermore, the structure ensures data without loss of generality.(ii)
Driven by the inference from Kao [61], weights are rationally determined by considering risk attitudes and the nature of factors, which is lacking in the close counterpart approach.(iii)
Unlike the framework [53], the ranking algorithm in the proposed work considers the nature of factors and personal choices to provide an interactive, personalized variant of WASPAS with PLD.Figure 4
Test of uniqueness in ordering.O The nature of factors, which is a potential parameter in the decision process, is considered both in weight assessment and ranking, lacking in [53].O Also, the new formulation allows experts/agents to share their personal opinion on each alternative option (as a vector) that acts as potential information in influencing the ordering of options.Statistically, the comparison is further extended to realize the superiority of the proposed work. For this purpose, 300 matrices are generated that are used in the simulation experiment. These matrices are of the same dimension as the data in the case example. They are given as input to both the proposed models [53]. Rank vectors are estimated via the algorithm provided in each model. It can be seen that each algorithm obtains 300 vectors of 1×4. Statistical variance is calculated for each vector, so 300 values are obtained, which are plotted in Figure 5. The graph shows that the proposed model can better discriminate alternatives (online courses here) by producing broader vectors than its close counterpart. The graph shows that the proposed model has about six times better discrimination than the counterpart approach. Besides the test for uniqueness of the proposed model put forward in Figure 4, the uniqueness measure is also determined for the 300 orders yielded by the proposed model from the simulation experiment. Spearman rank correlation is applied for the rank values produced by the proposed and counterpart approach. It can be inferred that due to consideration of personal choices, the proposed model produces an order that is unique compared to its close counterpart with an average uniqueness score of 0.7807 for the 300 simulated matrices (Figure 6).Figure 5
Broadness measure for realizing discrimination factor.Figure 6
Consistency measure through Spearman correlation.In a nutshell, the proposed model is analyzed from both the theoretical and statistical perspective to effectively understand the superiority of the model that is put forth in the present study. It is clear that (i) the model adds value to the PLTS-based decision-making by putting forward a novel framework, whose usefulness is demonstrated by using real case example of professional online course evaluation by collecting empirical data from RGNIYD, an academic institution in India; (ii) later, the test for uniqueness (Figures4 and 6) shows that the proposed model produces an unique order of alternatives (online courses), which is intuitively backed by the novelty in the formulation of the model that allows consideration of personal choices; and (iii) finally, the test for discriminative power (Figure 5) also reveals that the proposed model produces broader and sensible rank values that aid in better discrimination of alternatives (online courses) for rational decision-making.


## 4.1. Comparison Study

The authors attempt to showcase the efficacy of the proposed work by comparing the model with a close counterpart method [48]. From the sensitivity graph shown in Figure 3, it is clear that the proposed work is highly robust even after alterations to factor significance, and strategy values are done adequately. To further realize the efficacy of the work, the model in [53] is compared with the proposed work. Table 4 gives the summarized view of the theoretical benefits of the proposed work over [53]. Besides, authors also compare the proposed work with extant models, namely, [13, 17] and [23], which actively use PLTS in their framework for attaining rational decisions. Finally, data from the case example are provided to these models, and the courses are ranked, and they are shown in Table 5.Table 4
Summary of characteristics: proposed and other models.
ContextProposed work[53]DataPLDPLDWeights of factorsCalculated methodicallyCalculated methodicallyRegret attitude characterizationDone, regret/rejoice factorNot doneNature of factorsConsidered, both during weight assessment and rankingNot consideredPersonal choices of expertsConsidered during rankingNot consideredTable 5
Order of professional courses obtained from PLTS models: proposed versus others.
CourseProposedReference [48] (0.2)[13][17][23]CC123233CC241111CC312322CC434444Table5 and Figure 4 show that the proposed model produces a unique ranking order of professional courses. It can be intuitively inferred owing to the proposed model's ability to consider the nature of factors during the ranking of courses (alternatives) and capture experts' personal opinions during ranking, which provides a sense of personalization and is lacking in existing models. Apart from this, specific theoretical merits of the framework are listed below. Based on the briefing in Table 4, it is inferred that the proposed work is novel and innovative. To further detail the claim, specific points are presented as follows:(i)
PLD is a sophisticated structure that allows for the elegant transformation of rating data from multiple heterogeneous candidates/users. Furthermore, the structure ensures data without loss of generality.(ii)
Driven by the inference from Kao [61], weights are rationally determined by considering risk attitudes and the nature of factors, which is lacking in the close counterpart approach.(iii)
Unlike the framework [53], the ranking algorithm in the proposed work considers the nature of factors and personal choices to provide an interactive, personalized variant of WASPAS with PLD.Figure 4
Test of uniqueness in ordering.O The nature of factors, which is a potential parameter in the decision process, is considered both in weight assessment and ranking, lacking in [53].O Also, the new formulation allows experts/agents to share their personal opinion on each alternative option (as a vector) that acts as potential information in influencing the ordering of options.Statistically, the comparison is further extended to realize the superiority of the proposed work. For this purpose, 300 matrices are generated that are used in the simulation experiment. These matrices are of the same dimension as the data in the case example. They are given as input to both the proposed models [53]. Rank vectors are estimated via the algorithm provided in each model. It can be seen that each algorithm obtains 300 vectors of 1×4. Statistical variance is calculated for each vector, so 300 values are obtained, which are plotted in Figure 5. The graph shows that the proposed model can better discriminate alternatives (online courses here) by producing broader vectors than its close counterpart. The graph shows that the proposed model has about six times better discrimination than the counterpart approach. Besides the test for uniqueness of the proposed model put forward in Figure 4, the uniqueness measure is also determined for the 300 orders yielded by the proposed model from the simulation experiment. Spearman rank correlation is applied for the rank values produced by the proposed and counterpart approach. It can be inferred that due to consideration of personal choices, the proposed model produces an order that is unique compared to its close counterpart with an average uniqueness score of 0.7807 for the 300 simulated matrices (Figure 6).Figure 5
Broadness measure for realizing discrimination factor.Figure 6
Consistency measure through Spearman correlation.In a nutshell, the proposed model is analyzed from both the theoretical and statistical perspective to effectively understand the superiority of the model that is put forth in the present study. It is clear that (i) the model adds value to the PLTS-based decision-making by putting forward a novel framework, whose usefulness is demonstrated by using real case example of professional online course evaluation by collecting empirical data from RGNIYD, an academic institution in India; (ii) later, the test for uniqueness (Figures4 and 6) shows that the proposed model produces an unique order of alternatives (online courses), which is intuitively backed by the novelty in the formulation of the model that allows consideration of personal choices; and (iii) finally, the test for discriminative power (Figure 5) also reveals that the proposed model produces broader and sensible rank values that aid in better discrimination of alternatives (online courses) for rational decision-making.


## 5. Conclusion

The present study offers an integrated technique put into a framework for decision-making with PLD. The framework adds value to the research in the field of PLTS. The method in the framework calculates weights/significance of factors rationally by considering the hesitation attitudes of experts/agents. Furthermore, an extension to the WASPAS approach is provided with PLD that enables the method to consider the nature of factors during the ordering and personal choices of alternative options from experts/agents. This personal choice as a vector offers a sense of the personalized ordering of options. The framework's benefits can be theoretically and statistically verified by an empirical case study of professional online course selection during the pandemic time. Comprehensive inter-/intrasensitivity analysis and comparison (with close counterpart) reveal the framework's superiority in robustness and acceptable discrimination level.Some shortcomings of the study are (i) the importance of experts/agents is not methodically derived, and (ii) consideration of top two linguistic instances with its associated single confidence value to the terms (Likert scales) may cause some information loss in the practical sense. On the other hand, a few implications from the managerial perspectives are (i) the framework can be readily adapted for other decision problems in academics and other fields; (ii) transforming rating data from heterogeneous participants into a holistic data matrix by using the PLTS concept is a flexible way for data representation; (iii) though some loss exists in the data, it may be addressed by extending the framework to complete data zone; (iv) the framework gives educational policymakers to effectively plan courses for youth population so that they gain the state-of-the-art skill and knowledge to become ready of industry; (v) finally, some training with the model is expected to aid policymakers in the decision process.In the future, authors plan to resolve the previously mentioned shortcomings by presenting algorithms for expert weight assessment and considering complete data zone for decision-making. Further, plans are made to propose an integrated approach in the fuzzy variants, such as orthopair sets [68–70] and interval variants of linguistic forms, such as PLTS [6] and double hierarchy variants [71]. Finally, machine learning concepts can be embedded with decision approaches for solving large-scale decision problems in academic and other contexts.



---

*Source: 9299246-2022-04-21.xml*