1 Introduction

The success of an organization is directly governed by their stakeholders (Fülöp 2001; O’Haire et al. 2011). Suppliers are one such important stakeholder, who determines the key success in an organization (Zhaoxia et al. 2013). In the process of SCM, supplier outsourcing plays an inevitable role and it is given the highest priority in the chain (Choi et al. 2007). The proper assessment and evaluation of suppliers bring enormous gain to the organization. They promote customer satisfaction, improve economic status, mitigate risk and boost the relationship between clients and service providers (Yücel and Güneri 2011). Such claims motivated researchers in the past decade to propose a huge set of methodologies for SO. This becomes evident from the discussion made in Sect. 2. (Öztayşi and Sürer 2014; Xu and Liao 2015) made a clear argument that evaluation of suppliers based on their preference rating involved a great deal of imprecision and vagueness due to the intervention of humans in the decision-making process. They claimed that human aspect and cognition involved implicit vagueness and ambiguity which needs proper treatment for better decisions.

To address such vagueness better, scholars adopted different variants of fuzzy concepts. One such variant is the hesitant fuzzy set (HFS), proposed by Torra and Narukawa (2009). This is by far the best variant of fuzzy that can model human cognition and thought process better. The claim is evident from the nature of HFS, which gets n different preferences for the same instance at different time intervals. These n instances denote n different choices made by the DM for a single case and this eventually helps DMs to represent fuzziness better. This claim by Torra and Narukawa (2009), inspired many scholars to work with HFS concepts in decision making. Some prominent contributions of HFS-based decision making are (Liao et al. 2014b; Liao and Xu 2014; Xia and Xu 2011; Xu and Xia 2011a; Yu et al. 2016a; Zhu et al. 2016). Thus, motivated by the power of HFS, in this paper, we adopt hesitant fuzzy information as input for supplier evaluation.

In the field of decision making, group decision making (GDM) has always experienced great appetite from research communities, as they closely resemble the real case decision-making scenarios and involves a good amount of fuzziness to be addressed. Researchers often aggregate different DMs viewpoint into a single entity and then apply decision methods for evaluation. This eases the process of decision making and builds confidence among the group by giving equal chance to everyone for rating alternatives and by considering all their preferences equally for evaluation. Some of the popular aggregation operators in the hesitant fuzzy environment are discussed in (Thakur et al. 2014; Xia and Xu 2011; Zhou and Li 2014). From the analysis, we infer that all these aggregation operators lacked the ability to properly reflect the hesitancy involved in providing apt relative importance values for each DM and/or had complex formulation procedures. To circumvent the issue, in this paper, we adopted SHFWG operator which is an extended version of hesitant fuzzy-weighted geometry (HFWG) (Xia and Xu 2011) operator with hesitant fuzzy weight values for DMs. By considering DMs’ relative importance (weights) in the form of hesitant fuzzy information, the major issue of proper representation and realization of DMs’ relative importance is mitigated and a better understanding of each DM is obtained. Motivated by this challenge, we make such proposal.

Followed by the concept of aggregation, the research community has also paid significant attention toward criteria weight estimation. Many scholars have adopted either manual entry process or weight estimation through entropy methods (Jin et al. 2014; Wood 2016; Xia and Xu 2012; Zhao et al. 2016). (Liu et al. 2016) rightly pointed out that, such methods often lead to imprecision and unrealistic weight values. To alleviate the issue, (Liu et al. 2016) came up with an idea of using statistical variance (SV) for weight estimation. They claimed that, SV method was simple and had the property of considering all the points in its estimation which prevents loss of information and yields sensible weight values. This helped the SV method to produce realistic and sensible weight values. Thus, motivated by the ease and power of SV method, we proposed a new weight estimation method called HFSV method, which is an extension of SV method under hesitant fuzzy environment.

Finally, in this proposal, we set our hands on the ranking method. Many scholars have widely exploited this field by proposing different ranking methods under different fuzzy environments. Motivated by the power of compromise solution in yielding a near optimal result and from the deep survey conducted by Mardani et al. (2016), we identified VIKOR ranking as a potential candidate for the process. This method has the property of categorizing criteria into benefit and cost zones, which is lacking in other ranking methods. Though TOPSIS method is a close counterpart to VIKOR, we chose VIKOR for the process as it had optimistic style of ranking and supplier outsourcing problem closely resembled to this style of ranking. Also, VIKOR method chose optimal points based on their closeness to positive ideal solution. For further investigation on these two compromise solution methods, readers are encouraged to refer (Opricovic and Tzeng 2004). Motivated by the power of HFS in representing fuzziness and vagueness, we extend the VIKOR ranking method under hesitant fuzzy environment by proposing new formulation for estimation function with three categories viz., cost, benefit and neutral under the realm of Euclid distance. Though, (Liao and Xu 2013) have already proposed HF-VIKOR and adopted Manhattan distance for its formulation, they have not considered three categories for estimation in their formulation. In this proposal, we adopt cost, benefit and neutral categories for evaluation of estimation functions under the realm of Euclid distance. The main advantage of using these three categories for evaluation is that, it gives the DMs a clear understanding on the nature of each criterion and enables rational and better judgments. To clarify the scenario, consider an example of car purchase, criteria like, fuel consumption, cost must be low and so they are placed in cost category. Similarly, criteria like speed, mileage, safety must be high and hence, these are placed in benefit zone. Criteria like color and style are considered to be benefit in normal cases. But, paying close attention to these criteria will give us an intuition that, these are neither benefit nor cost and hence, there is an urge need for a new zone for evaluation. This challenge motivated us to propose a new three-way hesitant fuzzy VIKOR (TWHFV) under the realm of Euclid distance. The main motivation for choosing Euclid distance is that, scholars (Charulatha et al. 2013; Sinwar and Kaushik 2014) argue that, Euclid distance has a better rate of convergence and hence are suitable for decision-making problems involving imprecision and vagueness. Also, Euclid measure has a better chance for handling vagueness as they consider varying (unequal or biased) deviations with different values for weights. Hence, TWHFV method is proposed for outsourcing suitable supplier for the task.

Thus, to follow the context, the rest of the paper is organized as, related works in Sect. 2, followed by, basic knowledge in Sect. 3, proposed TSDMF framework in Sect. 4, illustrative example in Sect. 5, comparative analysis in Sect. 6 and conclusion in Sect. 7.

2 Related works

In this section, we conduct a survey to gain inference and to identify research lacuna. The procedure applied for surveying is a two-stage approach, where application and method are both concentrated. Here, the application is ‘supplier outsourcing’ and method is ‘hesitant fuzzy decision-making method’. We set these terms as the keywords and perform a search to identify potential papers for investigation. We identified some potential papers that are closely related to the keywords. Table 1 conducts an investigation on these 15 papers. All the fields of Table 1 are easy to follow. So, let us conduct the review process.

Table 1 Survey on supplier selection
Full size table

(Chai et al. 2013) conducted a thorough survey on multi-criteria decision-making methods and its application in supplier selection and evaluation. They conducted the survey from 2008 to 2012 and inferred that compromise solution viz., VIKOR and TOPSIS and hierarchical method viz., analytical hierarchy process (AHP) are good candidates for suitable selection of suppliers. (Gul et al. 2016) conducted a thorough investigation on VIKOR method and its extension in fuzzy concepts. From the analysis, we infer that the extension of VIKOR under HFS context is highly attractive and needs good exploration for taking sensible and rational decision. Motivated by this inference, efforts are made in this paper to extend VIKOR method under HFS context. (Simić et al. 2017) conducted a deep investigation with fifty years of articles in the field of fuzzy-based decision models for supplier selection. From the investigation, it can be observed that fuzzy-based decision methods are an effective tool for decision making and the complex and uncertain application of supplier selection is very well handled by fuzzy methods. Also, (Liu and Liao 2017) performed a detailed investigation on various fuzzy-based decision methods from 1970 to 2015 and inferred that hesitant-based decision methods are attractive and powerful for handling uncertainty and vagueness in the decision process. Motivated by the investigation made in Table 1 and from these four surveys, concrete research lacunas can be identified.

From Table 1, we make the following inferences:

  1. (1)

    Supplier selection/outsourcing is an attractive and hot area for research in SCM, which is earning high appetite from the research community.

  2. (2)

    This problem is better handled using MCDM approaches and viewing of SO problem from the lens of HFS-based MCDM methods has just started and it needs wide exploration.

  3. (3)

    Aggregation of preferences in such problem becomes a trivial aspect and hence, operators that make rational and logical aggregation are to be proposed.

  4. (4)

    Scholars have paid only little attention towards the integration of VIKOR ranking method under HFS for solving SO problem. On the other hand, the use of AHP, TOPSIS and QUALIFLEX under HFS for SO has been greatly exploited.

  5. (5)

    Criteria weight estimation for effective selection of supplier has been done either by expert advice or methodical way. Common methods include entropy measures which often yields unrealistic weights.

From these inferences, following research lacunas are identified:

  1. (1)

    Since the number of research articles under ‘hesitant fuzzy-based supplier selection’ is less, there is an urge need for a new decision-making framework under HFS for SO. Thus, we propose a new framework called TSDMF for SO.

  2. (2)

    Since aggregation is a trivial concept in SO and DMs want to have a sensible aggregation, we propose a new aggregation operator called, SHFWG, which considers DMs’ weights (relative importance) not as single term but as hesitant fuzzy values.

  3. (3)

    Since the criteria weight evaluation is often unrealistic and imprecise, we propose a new method for criteria weight estimation called HFSV, which is an extension to SV under HFS.

  4. (4)

    Finally, from the inference, we observe that, the use of VIKOR under HFS for SO is only rarely addressed and hence, we set our hands on this lacuna, by proposing a new ranking method called, TWHFV, which is an extension to VIKOR under HFS that formulates estimation function (group utility and individual regret) using three categories viz., cost, benefit and neutral under the realm of Euclid distance.

Thus, the proposal of such decision-making framework for supplier outsourcing will surely benefit the SCM community and the flexibility of the framework will also benefit research communities in making rational and critical decisions with regard to other selection problems.

3 Background knowledge

Definition 1 (Torra and Narukawa 2009)

Let A be a fixed set, then the hesitant fuzzy set on A is a function h, which when applied gives a subset of \( \left[ {0,1} \right] \). Mathematically, it is represented as,

$$ \bar{A} = \left\{ {a,h_{{\bar{A}}} \left( a \right)|a \in A} \right\} $$
(1)

where \( h_{{\bar{A}}} \left( a \right) \) is a subset of \( \left[ {0,1} \right] \) and it represents the possible membership degree for the element \( a \in A \).

Definition 2 (Xia and Xu 2011)

Some operational laws on hesitant fuzzy element (HFE) are as follows:Consider three HFEs \( h \), \( h_{1} \) and \( h_{2} \) with \( \lambda \) being a positive real number, then,

$$ h^{\lambda } = \cup_{\beta \in h} \left\{ {\beta^{\lambda } } \right\} $$
(2)
$$ \lambda h = \cup_{\beta \in h} \left\{ {1 - \left( {1 - \beta } \right)^{\lambda } } \right\} $$
(3)
$$ h_{1} \oplus h_{2} = \cup_{{\beta_{1} \in h_{1} ,\beta_{2} \in h_{2} }} \left\{ {\beta_{1} + \beta_{2} - \beta_{1} \beta_{2} } \right\} $$
(4)
$$ h_{1} \otimes h_{2} = \cup_{{\beta_{1} \in h_{1} ,\beta_{2} \in h_{2} }} \left\{ {\beta_{1} \beta_{2} } \right\} $$
(5)

Definition 3 (Xia and Xu 2011)

For a HFE \( h_{1} \), the score function \( s\left( {h_{1} } \right) \) is given by,

$$ s\left( {h_{1} } \right) = \frac{1}{{l_{{h_{1} }} }}\mathop \sum \limits_{{\beta \in h_{1} }} \beta $$
(6)

where \( l_{{h_{1} }} \) is the length of the HFE \( h_{1} \).

Definition 4 (Liao et al. 2014)

For a HFE \( h_{1} \), the variance function \( v\left( {h_{1} } \right) \) is given by,

$$ v\left( {h_{1} } \right) = \frac{1}{{l_{{h_{1} }} }}\sqrt {\mathop \sum \limits_{{\beta_{i,} \beta_{j} \in h_{1} }} \left( {\beta_{j} - \beta_{i} } \right)^{2} } $$
(7)

where \( \beta_{i} \), \( \beta_{j} \) are the different membership values for the same instance.

Remark 1

To rank two HFEs \( h_{1} \) and \( h_{2} \), we apply Eqs. (6, 7) as follows:

  • If \( s\left( {h_{1} } \right) < s\left( {h_{2} } \right) \) then, \( h_{1} < h_{2} \)

  • If \( s\left( {h_{1} } \right) = s\left( {h_{2} } \right) \) then, we follow,

    • If \( v\left( {h_{1} } \right) < v\left( {h_{2} } \right) \) then, \( h_{1} > h_{2} \)

    • If \( v\left( {h_{1} } \right) = v\left( {h_{2} } \right) \) then, \( h_{1} = h_{2} \)

Definition 5 (Xu and Xia 2011b)

The Euclid distance measure for two HFEs, \( h_{1} \) and \( h_{2} \) is given by,

$$ d\left( {h_{1} ,h_{2} } \right) = \frac{1}{l}\sqrt {\mathop \sum \limits_{i,j = 1}^{l} \left( {\beta_{i} - \beta_{j} } \right)^{2} } $$
(8)

where, \( \beta_{i} \in h_{1} \), \( \beta_{j} \in h_{2} \) and \( l \) is the length of the HFE. The length of the HFEs must be equal. If they are not equal, then, the procedure suggested by Liao et al. (2014a) is adopted for making them equal.

4 Proposed two-stage decision-making framework (TSDMF)

4.1 Architecture of TSDMF

In this section, Fig. 1 depicts the architecture of the proposed TSDMF. The architecture is a two tier framework used for decision making. We demonstrate the applicability of TSDMF using supplier outsourcing problem. The framework is also flexible enough to handle any type of selection problem. The architecture is self-contained and straightforward depiction of the working process and hence, we confine our discussion on the architecture and elaborate upon each step of the decision-making process in the upcoming sections.

Fig. 1
figure 1

Proposed TSDMF for supplier selection

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4.2 Proposed SHFWG operator

The aggregation is the process of combining DMs’ preferences in a manner such that, every DMs viewpoint is addressed equally. Many scholars have worked on the concept of aggregation of preferences and have given significant insights. Based on the understanding, scholars infer that, aggregation of preferences must preserve originality and must pay equal attention to all DMs’ viewpoint.

(Xia and Xu 2011) inspired by this idea, proposed HFWG operator which is an extension to simple intuitionistic fuzzy-weighted geometry (SIFWG) (Liao and Xu 2015) operator under HFS context. The HFWG operator has a unique property of yielding consistent fused decision matrix. They also proved that, this operator had the ability to yield complete consistent and acceptable consistent matrix based on the DMs’ initial preference matrices. They also claimed that, other operators like ordered weighted arithmetic, ordered weighted geometry, symmetric weighted geometry, hybrid arithmetic, hybrid geometry etc. lacked this property. Though HFWG operator enjoys such strength, it does suffer from the problem of proper representation of DMs’ relative importance values.

Motivated from the work of Xu (2014) and based on the investigation, we observe that, most of the operators in the hesitant fuzzy family use only single value to represent DMs’ weight, which causes some problems in the representation of fuzziness and vagueness. The use of single value to determine DMs’ relative importance causes fuzziness and imprecision in the decision-making process (Yue 2011). Motivated by this challenge, we make efforts to extend HFWG operator for better representation of DMs’ relative importance using hesitant fuzzy values. This idea of associating multiple membership degrees as relative importance (weight) to the DM reflects the hesitancy in the process better. Thus, the definition of SHFWG operator is given below:

Definition 6

The aggregation is a mapping defined by \( U^{n} \to U \) for aggregating decision matrices. The operator is given by,

$$ {\text{SHFWG}} = \mathop \prod \limits_{h = 1}^{m} \beta_{ij}^{{\lambda_{h}^{*} }} $$
(9)

where \( \lambda^{*} \) is the normalized relative importance for each DM with \( \mathop \sum \nolimits_{h = 1}^{m} \lambda_{h}^{*} = 1 \), m is the total number of DMs and \( \mu \) is the degree of preference rating.

Here, \( \lambda_{i} = \left\{ {x,\mu_{\lambda } \left( x \right)|x \in X} \right\} \). Now find the score for each of \( \lambda_{i} \) using Definition 3. Now normalize these score values using, \( \lambda_{i}^{*} = \frac{{\lambda_{i} }}{{\mathop \sum \nolimits_{i = 1}^{m} \lambda_{i} }} \), where m is the total number of DMs.

Remark 2

The length of every instance taken for aggregation must be equal. If they are unequal, follow the procedure given by Xu (2014) to make them equal.

The SHFWG operator satisfies the following properties:

Property 1 (Idempotent)

If the value of a particular instance is equal in all the decision matrices, then, the resultant value in the fused matrix remains the same.

$$ SHFWG\left( {\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{m} } \right) = \alpha $$

Property 2

(Boundedness) The aggregation using SHFWG operator yields values in the range,

$$ \alpha_{i}^{ - } \le SHFWG\left( {\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{m} } \right) \le \alpha_{i}^{ + } $$

where \( \alpha_{i}^{ - } \) is the minimum value of the instance and \( \alpha_{i}^{ + } \) is the maximum value of the instance.

Property 3 (Monotonic)

If \( \alpha^{i} < \alpha^{j} \) then,

$$ \textit{SHFWG}\left( {\alpha_{1}^{i} , \alpha_{2}^{i} , \ldots ,\alpha_{m}^{i} } \right) \le \textit{SHFWG}\left( {\alpha_{1}^{j} ,\alpha_{2}^{j} , \ldots ,\alpha_{m}^{j} } \right). $$

Theorem 1

The aggregation of hesitant fuzzy decision matrices using SHFWG operator yields a matrix which is hesitant fuzzy in nature.

Proof

To prove that, the aggregated matrix is also hesitant fuzzy in nature, we have to show that, the instances of the aggregated matrix is in the range \( \left[ {0,1} \right] \) and the order of the aggregated matrix is the same as the individual matrices. From the basic definition of aggregation, the second half of the proof is trivial and hence our aim is to show the instance of the aggregated matrix is in the range \( \left[ {0,1} \right] \). To follow the context;

Let us consider, \( \mu_{ij}^{*} = \mathop \prod \nolimits_{{{\text{h}} = 1}}^{\text{m}} \mu_{\text{ij}}^{{\uplambda_{\text{h}} }} \) as a particular instance of the aggregated matrix. We know that, \( \mathop \prod \nolimits_{h = 1}^{m} x_{i}^{{\lambda_{h} }} \le \mathop \sum \nolimits_{h = 1}^{m} \lambda_{h} x_{i} \forall x_{i} , i = 1,2, \ldots ,n \). Thus, applying this property, we get, \( \mathop \prod \nolimits_{h = 1}^{m} \mu_{ij}^{{\lambda_{h} }} \le \mathop \sum \nolimits_{h = 1}^{m} \lambda_{h} \mu_{ij} \forall \mu_{ij} \in \left[ {0,1} \right] \).

Now, we extend this idea to get \( 0 \le \mathop \prod \nolimits_{h = 1}^{m} \mu_{ij}^{{\lambda_{h} }} \le \mathop \sum \nolimits_{h = 1}^{m} \lambda_{h} \mu_{ij} \le \mathop \sum \nolimits_{h = 1}^{m} \lambda_{h} = 1 \forall \mu_{ij} \in \left[ {0,1} \right]{\text{and}} \lambda \in \left[ {0,1} \right] \). Thus, the instance of the aggregated matrix is in the range \( \left[ {0,1} \right] \).□

Example 1

Consider two DMs rating a particular alternative with respect to a particular criterion. The hesitant fuzzy elements \( h_{11}^{1} = \left( {0.3,0.4,0.5} \right) \) and \( h_{11}^{2} = \left( {0.33,0.48,0.62} \right) \). Let \( \lambda^{*} = \left( {\lambda_{1}^{*} , \lambda_{2}^{*} } \right) = \left( {0.5,0.5} \right) \). Calculate the aggregated instance \( h_{11}^{*} \). \( h_{11}^{*} = \Big\{ \left( {0.3^{0.5} \times 0.33^{0.5} } \right),\left( {0.4^{0.5} \times 0.48^{0.5} } \right),\left( 0.5^{0.5} \times 0.62^{0.5} \right) \Big\} = \left\{ {0.31,0.44,0.56} \right\} \). Thus, the idea of Theorem 1 can be realized.

4.3 Proposed HFSV method

The process of assigning weights to criteria is an essential operation in decision making. This weight value infers the relative importance of each criterion during evaluation. The assignment of weight is done using two popular methods. The first method is the manual entry of weight values and the second is the procedural way of estimating weight values. The former method is often difficult as the DMs are not fully aware of each of the criterion and its importance in the evaluation. Thus, the second method can be a solution for the issue. The popular methods for estimating weight values are entropy based (Zhang et al. 2014), optimization based (Chýna et al. 2013), AHP based (Saaty 1980) etc. But, all these methods produce weight values that are often unrealistic and irrational. To circumvent the issue, statisticians developed the idea of variance estimation for weight assignment.

The main advantage of the variance method is that, (1) it is simple and rational; (2), unlike other statistical methods, the variance measure considers every data point for determining the distribution without ignoring any potential information. This preserves the viewpoint of all DMs and keeps the estimation sensible. Motivated by the strength of SV method, we make efforts to extend SV method to HFS context.

The procedure for HFSV method is given below:

Step 1: Construct a criteria rating matrix which involves different DMs rating each criterion using hesitant fuzzy values.

Step 2: Rearrange the matrix in some order and then calculate the variance using Definition 4 for each instance of the matrix obtained from Step 1.

Step 3: Calculate statistical variance between instances obtained from Step 2 using Eq. (10).

$$ \textit{HFSV}_{l} = \frac{1}{n}\mathop \sum \nolimits_{k = 1}^{n} \left( {v_{kl} - \bar{v}_{kl} } \right)^{2} $$
(10)

where, \( \bar{v}_{kl} \) is the mean value of each criterion (matrix taken from Step 2), \( l \) refers to the \( l^{th} \) criterion and n is the total number of DMs.

Step 4: Normalize the values obtained by Step 3 to estimate the actual weight values of each criterion. This is mathematically given by Eq. (11).

$$ \omega_{l}^{*} = \frac{{\textit{HFSV}_{l} }}{{\mathop \sum \nolimits_{l = 1}^{m} \textit{HFSV}_{l} }} $$
(11)

where m is the total number of criteria and \( \omega_{l}^{*} \) is the normalized weight value.

4.4 Proposed TWHFV method

The VIKOR method is a compromise ranking method which has the ability to categorize criteria as cost and benefit. The base idea for VIKOR is inspired from \( L_{\text{p}} \)-metric. The VIKOR method finds optimal points that are close to the positive ideal solution and provides ranking in an optimistic fashion. Since the application of supplier outsourcing in SCM closely resembles to the optimistic style of ranking, we gain motivation and use VIKOR ranking method in our framework. The optimistic style infers that selection is based on the domination of benefit criteria. Thus, in SO problem, suppliers with good service and quality are preferred more. Details on the procedure for VIKOR can be found in (Opricovic and Tzeng 2004).

Inspired by the strength of VIKOR, (Liao and Xu 2013) extended the method for HFS with Manhattan distance. With the view of tackling the weakness of traditional VIKOR ranking method, in this paper, we made rational and effective modification to this method by proposing TWHFV method which uses three categories (benefit, cost and neutral) for evaluation of estimation function viz., group utility (S) and individual regret (R) functions under the realm of Euclid distance rather than Manhattan distance. The motivation for this modification to HF-VIKOR (Liao and Xu 2013) is gained from intuition and statistical theories, which claims Euclid distance to be more effective in handling fuzziness than Manhattan distance (Charulatha et al. 2013; Xu and Xia 2011b). Also, our motivation is strengthened by understanding the concept of \( L_{\text{p}} \)-metric, which is the base for VIKOR method. (Liao and Xu 2013) made an argument that, the greater the value of p, the estimation moves from minimizing the sum of regrets to minimizing maximum regret. Thus, they stated that, for p = 1, deviations are equally weighted and for p = 2, the deviations are unequal and they are weighted based on the magnitude. Manhattan distance is analogous to p = 1 condition and hence, cannot offer rational inference. So, we make efforts to move to a distance measure which is analogous to p = 2, i.e., Euclid distance. Thus, the Euclid distance can produce sensible and rational inference by handling fuzziness effectively.

Based on these claims, we gained motivation to extend VIKOR method under hesitant fuzzy environment with three categories for evaluation of S and R under the realm of Euclid distance. The procedure for TWHFV is given below:

Step 1: Calculate positive ideal solution (PIS) and negative ideal solution (NIS) using Eqs. (12 and 13).

$$ \begin{aligned} {\text{PIS}}\left( {h^{ *} } \right)& = { \hbox{max} }_{\text{benefit}} \left( {h_{ij} } \right) {\text{or min}}_{\text{cost}} \left( {h_{ij} } \right) {\text{or }}\\ &\quad \left( { \hbox{max} }_{\text{nuetral}} \left( {h_{ij} } \right) \oplus { \hbox{min} }_{\text{neutral}} \left( {h_{ij} } \right) \right)\end{aligned} $$
(12)
$$ \begin{aligned} {\text{NIS}}\left( {h^{ - } } \right)& = { \hbox{min} }_{\text{benefit}} \left( {h_{ij} } \right) {\text{or max}}_{\text{cost}} \left( {h_{ij} } \right) {\text{or }} \\ &\quad \left( { \hbox{max} }_{\text{nuetral}} \left( {h_{ij} } \right) \otimes { \hbox{min} }_{\text{neutral}} \left( {h_{ij} } \right) \right)\end{aligned} $$
(13)

where \( \oplus \) and \( \otimes \) are operators from Definition 2, \( h_{ij} \) is that instance of the decision matrix whose score value is either maximum or minimum depending on the ideal solution considered. When score values are equal, consider variance value. The lesser the variance value, the higher the rating.

Step 2: Calculate group utility (S) and individual regret (R) using (14) and (15).

$$\begin{aligned} & S_{i} = \mathop \sum \nolimits_{{k \in {\text{benefit}}}} \omega_{k} \left( {\frac{{d\left( {h_{k}^{*} ,h_{ik} } \right)}}{{d\left( {h_{k}^{*} ,h_{k}^{ - } } \right)}}} \right)+ \mathop \sum \nolimits_{{k \in {\text{cost}}}} \omega_{k} \left( {\frac{{d\left( {h_{ik} ,h_{k}^{*} } \right)}}{{d\left( {h_{k}^{*} ,h_{k}^{ - } } \right)}}} \right)\\ &\qquad + \mathop \sum \nolimits_{{k \in {\text{neutral}}}} \omega_{k} \left( {\frac{{d\left( {h_{ik} ,h_{k}^{ - } } \right) + d\left( {h_{k}^{*} ,h_{ik} } \right)}}{{d\left( {h_{k}^{*} ,h_{k}^{ - } } \right)}}} \right)\end{aligned} $$
(14)
$$\begin{aligned} R_{i} &= { \hbox{max} }_{{k \in {\text{benefit}},\;{\text{cost}},\;{\text{nuetral}}}} \left( \omega_{k} \left( {\frac{{d\left( {h_{k}^{*} ,h_{ik} } \right)}}{{d\left( {h_{k}^{*} ,h_{k}^{ - } } \right)}}} \right),\right.\\&\left.\omega_{k} \left( {\frac{{d\left( {h_{ik} ,h_{k}^{*} } \right)}}{{d\left( {h_{k}^{*} ,h_{k}^{ - } } \right)}}} \right),\omega_{k} \left( {\frac{{d\left( {h_{ik} ,h_{k}^{ - } } \right) + d\left( {h_{k}^{*} ,h_{ik} } \right)}}{{d\left( {h_{k}^{*} ,h_{k}^{ - } } \right)}}} \right) \right)\end{aligned} $$
(15)

Step 3: Calculate the merit function (Q) using (16).

$$ Q_{i} = v\left( {\frac{{S_{i} - S^{*} }}{{S^{ - } - S^{*} }}} \right) + \left( {1 - v} \right)\left( {\frac{{R_{i} - R^{*} }}{{R^{ - } - R^{*} }}} \right) $$
(16)

where v is the strategy adopted by the DM which is in the range 0 to 1, \( S^{*} = \hbox{min} \left( {S_{i} } \right) \), \( R^{*} = \hbox{min} \left( {R_{i} } \right) \), \( S^{ - } = \hbox{max} \left( {S_{i} } \right) \) and \( R^{ - } = \hbox{max} \left( {R_{i} } \right) \).

Step 4: Determine the final ranking order by arranging merit functions in the ascending order. The alternative that has the minimum Q value is ranked first and then the process follows.

Before demonstrating the practicality of the proposed framework, it is worth to understand certain details of the method. The discussion is as follows:

  1. (1)

    The process of aggregation and weight estimation are self-contained and straightforward. So for brevity, we confine our discussion on these two concepts and move on with the ranking of alternatives.

  2. (2)

    In the step (1) of TWHFV, the PIS and NIS are calculated for each of the criterion based on its respective category. We introduce a new category called neutral for enhancing choice representation by DM. Since the idea of benefit and cost are similar to that of HF-VIKOR, we see neutral category. Suppose, two criteria are neutral, we find the maximum and minimum of these two criteria and perform operations from Definition 2. The result of these operations yields a set of value which is hesitant fuzzy in nature.

  3. (3)

    In step 2 of TWHFV, the values of S and R are estimated using Euclid distance. For the neutral part, both the distances of an instance from the PIS as well as the NIS are taken in consideration. This ensures better rationality of the decision process from the DMs’ point of view.

  4. (4)

    The Step 3 of TWHFV is used to calculate the merit function (Q) which is similar to the HF-VIKOR method. Here, the strategy value (v) is explored for different types of strategy (v = 0.1 to 0.9) to verify the robustness of the method.

5 An illustrative example

In this section let us consider the supplier outsourcing example in SCM. An automobile industry needs to purchase auto parts for a specific job. The industry identified six suppliers and five criteria for evaluating those suppliers. The industry decides to have three DMs viz., senior technical officer \( \left( {D_{1} } \right) \), chief finance officer \( \left( {D_{2} } \right) \) and senior executive \( \left( {D_{3} } \right) \). Based on the initial screening, four potential suppliers were selected and they were evaluated based on the five competing criteria (see Appendix). The DMs adopted hesitant fuzzy information for rating suppliers. Let us now review the procedure involved in decision-making process.

Step 1: Construct the decision matrix of order \( \left( {4 \times 5} \right) \) for all three DMs with hesitant fuzzy information as the source for rating alternatives.

Step 2: Rearrange the decision matrices in some order and then aggregate these matrices into a single decision matrix of order \( \left( {4 \times 5} \right) \) using the aggregation operator defined in Sect. 4.2. Table 2 shows the matrix from both step 1 and step 2.

Table 2 Decision matrix
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The relative importance of each DM is expressed in the form of hesitant fuzzy values, given by, \( \lambda_{1} = \left( {0.3,0.32,0.35} \right) \), \( \lambda_{2} = \left( {0.35,0.25,0.38} \right) \), \( \lambda_{3} = \left( {0.36,0.4,0.30} \right) \) and these values are converted to single valued weights based on the procedure given in Definition 6. The values are given by \( \lambda_{1} = 0.2941 \), \( \lambda_{2} = 0.3390 \) and \( \lambda_{3} = 0.3667 \). Finally, these values are used by SHFWG operator for computing the aggregated matrix.

The score of the aggregated matrix is evaluated using (6) and if the score values are found to be equal between the alternatives, then variance is estimated for such alternatives using Eq. (7). Table 3 shows the score and variance value of each alternative with respect to each criterion.

Table 3 Score and variance evaluation
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Step 3: Construct a weight evaluation matrix for assigning weights to each of the criterion. Table 4 shows this matrix which is of order \( \left( {3 \times 5} \right) \). The rating is in the form of hesitant fuzzy information. Apply HFSV method from Sect. 4.3, to estimate the weight values for each of the criterion.

Table 4 Rating of criteria
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Using the score values from Table 5, the weight values are estimated using HFSV method and are given by \( \omega \approx \left( {0.2, 0.2,0.4,0.1,0.1} \right) \).

Table 5 Score evaluation of criteria for weight calculation
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Step 4: Determine the ranking order for all four suppliers using step 2, step 3 and TWHFV ranking method.

Using Eqs. (12, 13), PIS and NIS are calculated and it is given by.

$$ \begin{aligned} h^{*} & = \left( {\begin{array}{*{20}c} {\left( {0.45,0.52,0.58,0.61} \right),\left( {0.43,0.48,0.53,0.57} \right),\left( {0.58,0.63,0.66,0.69} \right),} \\ {\left( {0.79,0.82,0.85,0.88} \right),\left( {0.66,0.71,0.75,0.78} \right)} \\ \end{array} } \right)\quad {\text{and}} \\ h^{ - } & = \left( {\begin{array}{*{20}c} {\left( {0.46,0.52,0.56,0.59} \right),\left( {0.51,0.56,0.60,0.65} \right),\left( {0.60,0.63,0.65,0.68} \right),} \\ {\left( {0.30,0.33,0.38,0.43} \right),\left( {0.59,0.66,0.70,0.73} \right)} \\ \end{array} } \right) \\ \end{aligned} $$

Table 6 shows the S, R and Q values for different alternatives which are calculated using Eqs. (14–16) for a strategy value of \( v = 0.5 \).

Table 6 Evaluation of VIKOR parameters
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Step 5: Finally, sensitivity analysis is performed for TWHFV ranking method to validate its robustness of the method using Table 7. Also, in the next section, we perform a comparative investigation to verify other strengths of HFV ranking method with another ranking method.

Table 7 Sensitivity analysis for proposed TWHFV method
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6 Comparative study on proposed TWHFV and HF-VIKOR ranking methods

In this section, we demonstrate the power of TWHFV ranking method (belongs to TSDMF) by performing a comparative investigation with its close counterpart, HF-VIKOR method (belongs to Liao and Xu 2013 framework). We analyze these two methods via sensitivity analysis and realize the robustness of these two methods. Further, some strengths of TSDMF are also discussed by investigating their features with another framework. With a view to maintain homogeneity, we compare our proposed decision-making framework with the decision-making framework of Liao and Xu (2013). Table 8 shows the sensitivity analysis for HF-VIKOR method and Table 9 shows the comparative analysis of features for proposed TSDMF and (Liao and Xu 2013) framework.

Table 8 Sensitivity analysis of HF-VIKOR (Liao and Xu 2013) method
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Table 9 Ranking order of various ranking method(s)
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We further understand the strength of the proposal by comparison with other methods under HFS context. With the view of maintaining homogeneity in the process of comparison, methods viz., HF-VIKOR (Liao and Xu 2013), HF-TOPSIS (Xu and Zhang 2013), HF-ELECTRE (Chen and Xu 2015) and HF-PROMETHEE (Mahmoudi et al. 2016) are taken for analysis with the proposed method.

From Table 9, it is clear that the proposed decision framework produces a unique ranking order with \( A_{3} \) as the compromise solution. Moreover to understand the consistency of the proposed method, Spearman correlation (Spearman 1904) is applied over different ranking order. From Fig. 2, it can be inferred that the proposed decision framework is consistent with other methods.

Fig. 2
figure 2

Spearman correlation of different method(s)

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Based on the investigation of Tables 7, 8, 9, 10, we make the following inferences about the novelty of the proposed decision framework:

Table 10 Investigation of features: TSDMF versus Liao and Xu (2013)
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  1. (1)

    Tables 7 and 8 show the sensitivity analysis for two ranking methods viz., proposed TWHFV method and HF-VIKOR method. Nine unique values from 0.1 to 0.9 are considered for the evaluation and the final ranking order is given by

  2. (2)

    \( {\text{A}}_{3} \succ {\text{A}}_{1} { \succcurlyeq }{\text{A}}_{2} \succ {\text{A}}_{4} \) (same ranking order with TWHFV and HF-VIKOR methods). From the sensitivity analysis, we observe that proposed TWHFV method shows a change in ranking order only at v = 0.9, while HF-VIKOR showed a change of order at v = 0.1. This signifies that proposed TWHFV method and HF-VIKOR method are less sensitive to vagueness and uncertainty. Also, the competition among suppliers \( A_{1} \) and \( A_{2} \) is clearly

  3. (3)

    The main advantage of proposed TSDMF is that it allows only little intervention from the DM and has an automated setup for evaluation of alternatives. Such automation reduces the DMs’ effort in the process of evaluation and reduces imprecision and vagueness to a certain extent.

  4. (4)

    Another interesting feature of TSDMF is that it has a ranking method (TWHFV) which follows much sensible and rational procedure for ranking, which helps DMs to make clarified judgments. This method offers better flexibility to DMs by proposing three categories for classification of criteria, which drives proper and effective understanding of the nature of each criterion and thereby alleviating the issue of imprecision and vagueness in the classification of criteria.

  5. (5)

    Also, the proposed TWHFV method yields a broader rank value set compared to the HF-VIKOR method, which helps DMs to make sensible and rational decisions. Though, the rank value set appears close for both the methods, estimation of standard deviation clarifies the fact that, proposed TWHFV method is broader than HF-VIKOR method. The standard deviation values of TWHFV method and HF-VIKOR (Liao and Xu 2013) method for each set of v values (0.1–0.9) is given by:

    $$ \begin{aligned} &{\text{TWHFV}}{:}\left( 0.4546,0.4560,0.4574,0.4590,0.4606,\right.\\&\left.\quad 0.4623,0.4640,0.4660,0.4678 \right) \hfill \\ &{\text{HFVIKOR}}{:}\left( 0.4677,0.4654,4633,0.4613,0.4590,\right.\\&\left.\quad 0.4574,0.4556,0.4538,0.4522 \right) \hfill \\ \end{aligned} $$

    From the values shown above, an instance of v = 0.5 is taken which clearly shows that the proposed TWHFV method (0.4606 at v = 0.5) produces broader rank value set than HF-VIKOR (0.4590 at v = 0.5) method (Liao and Xu 2013). The value 0.5 is taken specifically for analysis as this is the ideal strategy value adopted by DM for making rational decisions.

  6. (6)

    The Spearman correlation is applied to different ranking order obtained from different methods to analyze the consistency of the proposed framework. From Fig. 2, it can be observed that the proposed framework is highly consistent with other methods.

  7. (7)

    Further, a simulation study is conducted to understand the strength of the proposed framework. The study initially forms 300 decision matrices of order (4 × 5) with four objects and five criteria. In these datasets, three criteria are considered as benefit, one as cost and one as neutral. The constraints discussed in Definition 1 are followed for the creation of datasets and four instances of membership degrees are considered in each dataset. Using these datasets as input, the ranking method is adopted and standard deviation is calculated for the rank value set at v = 0.5. These standard deviation values are depicted in Fig. 3.

    Fig. 3
    figure 3

    Estimation of broadness of rank value set(s): Proposed versus Others

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    From the analysis, we infer that the proposed TSDMF has broader and sensible rank value set than its close counterpart (Liao and Xu 2013) method. Thus, the proposed framework can be used for proper and rational backup management under uncertain situations.

  8. (8)

    Though the proposed framework enjoys such attractive strengths, it does suffer from some weakness. They are: (a) the framework needs trained DMs for its implementation and (b) the framework is computationally complex as it uses hesitant fuzzy values for evaluation. These weaknesses will be addressed in the future.

7 Conclusion

In this paper, a new two-stage automated scientific decision-making framework called TSDMF is proposed for solving supplier outsourcing problem. The framework uses hesitant fuzzy values as input and consists of aggregation stage, followed by weight estimation and ranking stage. The DMs’ preferences are aggregated using newly proposed SHFWG operator, which considers DMs’ weight as hesitant fuzzy values for sensible evaluation. Following this, the criteria weights are estimated using newly proposed HFSV method, which is an extension to SV method under HFS. This method yields a more reasonable and sensible weight value compared to other state of the art methods. Finally, the ranking is done using newly proposed TWHFV method, which is an extension to VIKOR method under HFS. The TWHFV method uses three categories viz., cost, benefit and neutral for clear classification of each criterion and Euclid distance in its formulation which is claimed to have better robustness than its close counterpart. Some significant contributions of proposed TSDMF are pointed out:

  1. (1)

    The proposed TSDMF is the first scientific framework under hesitant fuzzy environment to follow the combination of SHFWG operator for aggregation, HFSV method for weight estimation and TWHFV (new extension of VIKOR) which is a compromise ranking method for selecting a suitable supplier for the task.

  2. (2)

    This framework complements the work done by Liao and Xu (2013), by taking full advantage of the hesitant fuzzy environment. Unlike the framework (Liao and Xu 2013), the TSDMF offers better scope for handling imprecision and vagueness by allowing only little intervention from humans. The framework by Liao and Xu (2013), uses hesitant fuzzy property only for ranking, while, criteria weights are directly given by DMs and there is no aggregation concept in this framework.

  3. (3)

    Though, ranking by Liao and Xu (2013) uses HFS property, the difficulty in proper understanding of each criterion still remained unresolved. To better circumvent the issue and to help DMs in better understanding the nature of each criterion, we proposed a three-way classification setup involving cost, benefit and neutral zones. Also, the framework by Liao and Xu (2013), formulates the parameters S and R only for benefit zone and with the view of improving the formulation, proposed TWHFV method formulates S and R for all three classification zones viz., cost, benefit and neutral.

  4. (4)

    Finally, the strength of the proposed TSDMF is realized using sensitivity analysis (Sect. 6) and inferences clarify the fact that, proposed TSDMF is robust and less sensitive to imprecision and vagueness. Also, we observe that, proposed TSDMF provides much broader rank value set which helps DMs to make rational decisions with ease and efficacy.

As a part of future work, we address the weakness of the framework and also make efforts to extend the framework to other decision-making applications like healthcare management, resource management etc. Also, we make efforts to propose new automated scientific decision-making frameworks under different fuzzy sets like soft sets, neutrosophic sets, shadow sets, m-polar fuzzy set etc.