This paper proposes a new scientific decision framework (SDF) under interval valued intuitionistic fuzzy (IVIF) environment for supplier selection (SS). The framework consists of two phases, where, in the first phase, criteria weights are estimated in a sensible manner using newly proposed IVIF based statistical variance (SV) method and, in the second phase, the suitable supplier is selected using ELECTRE (ELimination and Choice Expressing REality) ranking method under IVIF environment. This method involves three categories of outranking, namely, strong, moderate, and weak. Previous studies on ELECTRE ranking reveal that scholars have only used two categories of outranking, namely, strong and weak, in the formulation of IVIF based ELECTRE, which eventually aggravates fuzziness and vagueness in decision making process due to the potential loss of information. Motivated by this challenge, third outranking category, called moderate, is proposed, which considerably reduces the loss of information by improving checks to the concordance and discordance matrices. Thus, in this paper, IVIFELECTRE (IVIFE) method is presented and popular TOPSIS method is integrated with IVIFE for obtaining a linear ranking. Finally, the practicality of the proposed framework is demonstrated using SS example and the strength of proposed SDF is realized by comparing the framework with other similar methods.
Uncertainty and vagueness are an integral part of SS process [
Based on the discussion made above, we observe that IVIF based decision making methods are computationally attractive and effective for handling fuzziness and vagueness. In general, MCDM (multicriteria decision making) methods are classified into two groups, namely, utility based and outranking based methods. In utility based approach, alternatives are selected based on the utility function. On the other hand, in outranking based approach, alternatives are selected based on the preference function. Liao and Xu [
Roy [
To address such issues and to give a simple and straightforward approach for decision making, in this paper, we make efforts to extend the Wu and Chen [
The rest of the paper is organized as literature review in Section
In this section, literature review is conducted using twostage approach, where both application and method are concentrated. The application considered here is SS and the method is IVIF based MCDM methods. So far, in the field of survey and analysis, this twostage approach is considered to be effective and straightforward. Thus, our study also uses this approach for the review process. We apply filters to the year of publication from 2013 to 2017, owing to the extension of work done by [
Survey on supplier selection under IVIF environment from 2013 to 2016.
Ref #  Year  Aggregation  Fuzzy methods  Application(s)  Discussion 

[ 
2016  Yes  IVIF based ranking  Supplier selection  Linguistic terms were converted to IVIF terms and aggregation was performed to fuse different opinions into a single matrix. TOPSIS method was used for criteria weight estimation. Finally, ranking is done using linear programming model and SS example is used to demonstrate the practicality. 


[ 
2016  No  IVIF based ranking  Supplier selection  A new framework was proposed to select suitable business intelligent system supplier under IVIF domain. Criteria weights were estimated using ordered pairwise comparison method and weight of experts was determined using entropy method. A new algorithm was used for ranking suppliers and its practicality was tested using SS example. 


[ 
2016  No  IVIF based ranking  Green supplier selection  IVIF terms were used for evaluation. Delphi method was used for choosing criteria for SS. Ranking was done using Choquet integral and fuzzy measures. 


[ 
2016  Yes  IVIF based ranking  Supplier selection  IVIF terms were used for rating suppliers. AHP method was used for estimation of criteria weights and TOPSIS method was extended to IVIF domain for ranking suppliers. Preference values were aggregated using weighted arithmetic operator under interval domain. 


[ 
2015  Yes  IVIF based ranking  Supplier selection  The DMs gave their preferences in IVIF fashion. Criteria and DMs weights were calculated using cross entropy measures and ranking of suppliers was done using optimization mechanism. 


[ 
2015  No  IVIF based ranking  Green supplier selection  Heterogeneous set of terms were used for evaluation. IVIF based LINMAP approach was used for ranking of suppliers. Criteria weights were also determined using fuzzy programming model. 


[ 
2014  Yes  IVIF based ranking  Supplier selection  IVIF information was used for rating suppliers. ELECTRE 1 was extended to IVIF domain for ranking suppliers. Score and accuracy were used for estimating concordance and discordance matrix. Criteria weights were calculated using entropy method. 


[ 
2013  No  IVIF based ranking  Supplier selection  Heterogeneous values were used for evaluation of suppliers. LINMAP method was extended to IVIF domain and new distance measure was adopted to rank suppliers. 


[ 
2013  Yes  Dynamic fuzzy set (DFS) based ranking  Supplier selection  The preferences were made in DFS fashion. The proposal discusses few weight models and ranking was done using a newly proposed DFS function. DFS was compared with IVIF over SS example and the power of DFS was realized. 


[ 
2013  Yes  IVIF based ranking  Supplier selection  A new decision framework was proposed under IVIF domain for SS. The input to the framework was assessment in the form of IFVs and ranking was done using aggregated index. 


[ 
2013  No  IVIF based ranking  Supplier selection  Heterogeneous information was used to rate suppliers. The LINMAP method was extended over IVIF and distance measure was used to rank suppliers. Criteria weights were calculated using fuzzy optimization model. 


[ 
2013  No  IVIF based ranking  Supplier selection  The DMs rate the suppliers using IVIF values. The criteria weights were determined using newly proposed entropy measures. Ranking was also done using score and accuracy methods. 


[ 
2013  No  IVIF based ranking  Supplier selection  The DMs adopt IVIF information to rate suppliers. The criteria were weighed using cotangent function based entropy measure. Ranking of suppliers was done using score and accuracy function and suitable supplier was selected for the job. 


[ 
2013  Yes  IVIF based ranking  Supplier selection  IVIF information was used by the DM to gauge suppliers. A new aggregation method was proposed under IVIF environment for fusing preferences and final ranking was done using this method. Power of the proposed model was validated using sensitivity analysis and results infer better practicality and robustness of the model. 


[ 
2013  No  IVIF based ranking  Supplier selection  IVIF values were used for rating. Criteria weights were estimated using AHP method and VIKOR was extended to IVIF for ranking suppliers. 
From Table
Supplier selection is an attractive area for research in the field of SCM which has gained welcoming interest from the researchers under the MCDM perspective.
The use of IVIF based MCDM methods for SS problem has just started and exploration in this field of study is becoming essential. Moreover, researchers have started realizing the strength of IVIF over IFS and hence, the use of IVIF based MCDM concepts to SS problem becomes an interesting and challenging task.
Many scholars have extended the LINMAP approach under IVIF environment for solving SS problem and the extension of ELECTRE method to SS problem has just started.
Estimation of criteria weights is another attractive area for research, where many scholars have focused their attention. These scholars have dominantly extended entropy measures and optimization models for calculating criteria weights, but Liu et al. [
Based on these inferences, the following research lacunas can be identified:
From inference (
Inference (
Inference (
The claim from inference (
Consider a fixed set
For the ease of understanding Atanassov [
Consider a fixed set
The IFS obeys certain operational laws given in (
The interval numbers also obey some operational laws which are given by (
In this section, we demonstrate the working procedure of proposed SDF. The flow chart idea is adopted for depicting the working procedure. Figure
Workflow of proposed SDF.
The estimation of weights or relative importance for each criterion is an attractive area for research. Many scholars have worked towards this field of study and proposed new methods for weight estimation. Broadly, the idea of criteria weight estimation can be classified into two groups, namely, manual weight estimation and methodical weight estimation. In manual weight estimation, DMs give weights to the criteria directly without following any procedure for estimation. On the other hand, methodical type of weight estimation is a systematic procedure used for estimation of criteria weights. Popular among them are entropy based [
The steps involved in IVIF based SV method is discussed below.
Construct a criteria weight evaluation matrix of order
Calculate the mean of each instance and convert IVIF values to IFVs. Calculate the accuracy for each instance and convert IFV to a single value term.
Calculate the mean of each criterion (values taken from Step
Normalize these variance values using (
The IVIFE method is a novel approach for solving MCDM problems, which integrates IFS and interval numbers into ELECTRE ranking scheme. We also combine classical TOPSIS approach with this method to obtain final ranking. The procedure for IVIFE is given below.
Construct a judgment matrix as in (
For each of the
Form the concordance and discordance matrix (CM and DM) using the conditions given in Table
The three categories of concordance and discordance, namely, strong (
Condition for concordance and discordance.
Constraints  Concordance  Discordance  

Strong  Lower 


Upper 

 


Moderate  Lower 


Upper 

 


Weak  Lower 


Upper 


Calculate the concordance payoff matrix (CPM). This is a square matrix of order
Equation (
Calculate the discordance payoff matrix (DPM). This is also a square matrix of order
Equations (
The distance measure is calculated using
Calculate the dominance concordance (DC) and dominance discordance (DD) matrix using
Identify the largest value from DC matrix and name it as
Now, form the rank matrix
Determine the optimal ranking order using TOPSIS method. The rank estimate
This section demonstrates the working of SDF with the popular SS example in the context of auto parts manufacturing agents in an automobile factory. Here, we consider 8 suppliers for the initial analysis and based on the prescreening process, 3 suppliers were removed and 5 potential suppliers were chosen for further investigation. These potential suppliers were judged based on 6 criteria. The criteria used for evaluation are cost, product delivery time, service satisfaction, quality of end product, risk, and supplier profile. These are functional criteria inspired from [
Step
Just as an example, we will consider single entry
Table
IVIF values for relative importance of criteria.
Criteria weights 




























IFVs for relative importance of criteria.
Criteria weights 




























Steps
As an example, let us consider the entry
Steps
We obtain the preference order using Steps
In this section, we compare our proposed IVIFE method with other stateoftheart methods in the same IVIF environment for maintaining homogeneity. We consider VIKOR, PROMETHEE, and ELECTRE ranking methods under IVIF environment as potential candidates for comparison with IVIFE. Table
Single value for relative importance of criteria.
Criteria weights 








0.4  0.56  0.55  0.45  0.37  0.52 

0.4  0.48  0.46  0.48  0.43  0.42 

0.5  0.45  0.48  0.48  0.40  0.46 
From Table
From Table
Supplier
Unlike the method described in [
Unlike method [
In case of proposed IVIFE method, the final ranking values that are used for constructing the preference order set are broader in nature. This helps DMs to clearly form rank value set and to easily make decisions under uncertainty by providing well distinguishable values for clear evaluation of suitable supplier for the process. Such rank value set can also help DMs to make backup suppliers ready either for other processes or for the same process. On the other hand, the method [
IVIFE is
IVIFE [
Based on the values shown above, we can affirm that proposed IVIFE method yields a reasonable and rational preference order than [
The proposed IVIFE method is moderately consistent (inference is gained from Spearman rank correlation; see Figure
As we observe, these advantages are realized from the theoretical lens, but one inference (see point (
Comparison of proposed IVIFE with other IVIF methods.
Method(s)  Alternatives  Preference order  





 
IVIFE (proposed)  4  1  5  2  3 

IVIFV [ 
1  4  2  5  3 

IVIFP [ 
4  1  5  2  3 

IVIFE [ 
5  2  4  3  1 

Spearman correlation for different ranking methods.
Though the proposed IVIFE method enjoys such attractive advantages, it does suffer from some disadvantages as well. Some disadvantages of IVIFE method are as follows:
The extra constraint check causes an additional overhead in manipulation and thereby increases the complexity of estimation.
Also, linear ranking by TOPSIS method is done with single valued entity which causes some loss of information. These disadvantages can be addressed in the future work.
In the previous section, we investigated the strength of IVIFE method from the theoretic perspective (see Table
These 5 test cases are given as input to IVIFE method [
On the other hand, when these 5 test cases are given as input to proposed IVIFE method, the ranking order remains unaffected and the stability of the method is also ensured.
The crux of this investigation is that though adequate changes are made to the alternatives, unlike method [
These 6 test cases are given as input to the method discussed in [
On the other hand, when these 6 test cases are given as input to the proposed IVIFE method, we observe that the rank order changes for criteria 3, 4, and 5. The normal rank order is
The crux of this investigation is that proposed IVIFE method suffers from rank reversal issue when adequate changes are made to the criteria. This can be considered as a weakness of the proposed method and can be addressed in the future.
The method discussed in [
On the other hand, proposed IVIFE method uses three zones for evaluation of concordance and discordance matrix and hence the judgment needed for consensus is given by
Thus, for the example demonstrated above, method discussed in [
Clearly from the discussion made above, proposed IVIFE method is slow in arriving at a particular consensus compared to method [
Characteristics of different IVIF based ranking methods.
Characteristics  IVIFE  IVIFV [ 
IVIFP [ 
IVIFE [ 

Type of input  Interval values  Interval values  Interval values are initially used and these are converted to IFS using mean operator  Interval values 


Total preorder  Yes  Yes  Yes  Yes 


Criteria weights  Given by DM  Given by DM  Given by DM  Given by DM 


Concept adopted  Pareto dominance  Compromise solution  Preference function  Pareto dominance 


Algorithm feature  Concordance and discordance measure  Ideal solution  Preference and indifference measure  Concordance and discordance measure 


Metric adopted  Three categories of outranking relation 

Partial and total outranking  Two categories of outranking relation 


Ranking procedure  Concordance and discordance matrices for both lower and upper bounds are constructed under all three categories. Finally, TOPSIS is used for linear ranking  Group utility, individual regret, and merit function are determined for both lower and upper bounds and finally linear ranking is obtained using ranking measures  Score measure is used along with 
Score measure is used for the construction of concordance and discordance matrices. Single valued matrices are processed to obtain linear ranking using TOPSIS method 


Ranking category  Outranking  Utility  Outranking  Outranking 


Fuzziness handled  Better because the estimation is done in three levels and interval values are retained till the final stage of ranking  Normal  Normal  Weak because the estimation is done only in two levels and interval values are initially converted into single valued terms using score measure 


Alternative chosen 




In this section, we clearly bring out the power of proposed IVIFE method by conducting an experiment using simulation process. The experiment clarifies the need for an additional outranking category, that is, moderate. We generate 300 decision matrices with IVIF information under 3 categories, namely, (a)
Form 300 decision matrices with IVIF information under 3 categories mentioned above. For brevity, we consider all these 100 matrices to be complete and they all obey the condition given in Definition
These decision matrices are given as input to the proposed ranking method and different ranking order is observed for each matrix. We use unbiased weights for each criterion and weights of concordance and discordance in each outranking category are taken from Section
In order to maintain homogeneity and closeness in comparison, we give these decision matrices as input to IVIFE [
From the ranking order obtained for each matrix by using different ranking methods as in Steps
To better realize the need for an additional outranking category moderate in the proposal, we plot the deviation vectors of each method as shown in Figure
Analysis of standard deviation of rank values for different ranking methods.
Based on the analysis conducted in Section
In this study, we have demonstrated a new computational framework for cloud vendor selection by proposing IVIF, which is an extension to ELECTRE under IVIF environment with three categories of outranking relation for the construction of concordance and discordance matrices. This new scheme also combines TOPSIS ranking method for achieving linear preference order. The proposed IVIFE method represents vagueness better by adopting IVIF concept. The method tackles the issue of information loss better by preserving the interval concept during concordance and discordance estimation. The proposed IVIFE method also handles information loss and vagueness better by constructing concordance and discordance matrices for all three outranking categories under both lower and upper bounds separately. The introduction of the third category (moderate) mitigates the problem of vagueness and information loss within the data by bringing out additional constraint checks for concordance and discordance matrices. This proposal also helps DMs to make rational decisions at uncertain and critical times. An illustrative numerical example is also demonstrated to verify the practicality of the proposed IVIFE scheme.
Some key contributions of the proposed SDF are given below:
The proposed SDF is the first framework for supplier selection under IVIF environment which uses a combination of SV method for criteria weight estimation and ELECTRE method for ranking suppliers.
The SDF complements the work done in [
The power of SDF is realized from both theoretic and numeric perspective and to the best of our knowledge, this is the first time such an analysis is carried out over a decision framework under IVIF environment.
From the analysis, we infer that both proposed IVIFE and method [
Along with the future works planned above, we also have ideas for developing some new fuzzy concepts which might be integrated along with ELECTRE for solving MCDM problems. We also have plans to extend the existing fuzzy concepts over some new outranking methods. Also, the proposed framework, SDF, can be extended to other applications like medical for nurse selection, equipment selection, and so forth, management for manager selection, personnel selection, and so forth, and manufacturing for material selection and so on.
The authors declare that they have no conflicts of interest.
The authors would like to thank the funding agencies, University Grants Commission, for their financial support for the research from Rajiv Gandhi National Fellowship of Grant no. F./201517/RGNF201517TAM83. The authors would like to thank the Department of Science and Technology, India, for their financial support for providing cloud environment through FIST programme (Grant no. SR/FST/ETI349/2013). They also thank the SASTRA University for providing an excellent infrastructure for performing their research work.