From the literature review presented in the next section alongwith the summarized view of characteristics of the extant PHFS models, following research challenges can be identified: •

Direct elicitation of probability alongwith membership grade is difficult from experts’ point of view.

The challenges identified by the authors are supported by the summarized view of characteristics of extant PHFS models provided in the next section. Besides, these challenges bring out the crucial research questions, which infer that there is an urge for a novel integrated framework under PHFS context for promoting rational decision-making. Motivated by the research challenges/questions pointed out above, in this article, authors put forward integrated decision approaches with PHFS data to rationally make decisions with less human intervention and subjectivity. To achieve the objective, following research contributions are put forward.

The presented challenges motivated the authors to make the following research contributions:

Before proceeding further, it is important to discuss the reason behind proposing such approaches as contributions in this article. As discussed above, determination of occurrence probability (confidence) for each membership grade is not easy and hence, a mathematical model is formulated to determine the confidence values methodically, which not only reduces the overhead, but also mitigates bias and subjectivity. Further, missing entries and non-availability of preferences are natural in practical decision problems owing to the pressure and hesitation. To address the issue, a case based approach is developed, which not only imputes missing values, but also holds the properties of PHFS during data imputation. Further, weight values of decision-makers and criteria are methodically decided by considering hesitation and attitudes of experts for rational determination. Also, interdependencies among experts are captured during preference aggregation alongwith consideration of methodically determined weights with generalized function that can aggregate preferences and represent other functions as special cases based on parameter values. Finally, a ranking approach is presented with the intent of considering criteria type and providing ranks close to human cognition.

Thus, Section 2 offers the essential idea required for framing the theoretical ground. Section 3 ensures the core contributions of the current work. Section 4 demonstrates the applicability of the decision model developed. A comparative investigation is carried out in Section 5 to address the pros and cons of the model. Finally, concluding remarks with future research directions are described in the last section.

This section offers the main idea of HFS and PHFS that forms the foundation for the research work.

The PHFS has the following advantages that motivated authors to consider the set in this study: (i) it can accept multiple preferences for a particular instance owing to the hesitant fuzzy nature, which is not possible in other fuzzy variants; (ii) further, each element can be associated with an occurrence probability, which can be viewed as the confidence associated with that particular element. Such features are missing in other fuzzy forms. It must be noted that interval variant of PHFS also has merits and authors plan its utilization for the future. Definition 3

Consider three PHFEs, h, h 1 and h 2 , as highlighted before. Some operations are as follows: (3) h 1 ⊕ h 2 = ⋃ a = 1 , 2 , … # h 1 , b = 1 , 2 , … , # h 2 { γ a + γ b − γ a γ b | p a p b } , (4) h 1 ⊗ h 2 = ⋃ a = 1 , 2 , … # h 1 , b = 1 , 2 , … , # h 2 { γ a γ b | p a p b } , (5) h c = ⋃ a = 1 , 2 , … # h { ( 1 − γ a ) | p a } , (6) h λ = ⋃ a = 1 , 2 , … # h { ( γ a ) λ | p a } λ ⩾ 0 , (7) λ h = ⋃ a = 1 , 2 , … # h { 1 − ( 1 − γ a ) λ | p a } λ ⩾ 0 . The operations discussed in equations (3)–(7) form the theoretical base of PHFS.

This section discusses the systematic procedure for imputation of missing values in preference matrices. Due to implicit confusion, pressure, and hesitation, experts are unable to provide complete preference matrices. Pieces of literature on PHFS reveal that previous approaches do not consider missing values, and to effectively handle the problem, the case-based methodology is introduced in this section. General imputation methods like random fill, binning, and arithmetic residue (Han et al., 2012) are not suitable for imputation of values in preference matrices due to the interdependencies among experts and criteria along with the varying nature of criteria, which must be considered during imputation. Besides, the proposed case-based approach considers the personal choice of experts on each alternative, which is ignored by the general/common methods.

To circumvent the challenges, the case-based approach is put forward.

Case 1: Out of t preference matrices, an entry ( i , j ) is missing from a single preference matrix, which may be imputed in the following manner: (8) h i j = ( ∏ k k = 1 t a v ( γ i j l ) d w k k | ∏ k k = 1 t a v ( p i j l ) d w k k ) , where t a v are number of experts who provide their values to a particular ( i , j ) entry; and d w k k = 1 t a v and ∑ k k d w k k = 1.

Case 2: Out of t preference matrices, a particular entry ( i , j ) is missing in all t matrices, for which we adopt equation (9)) to impute the values: (9) h i j = ( ∏ i i = 1 m a v ( γ i j l ) d w i i | ∏ i i = 1 m a v ( p i j l ) d w i i ) , where m a v is the number of alternatives that have values; and d w i i is the weight of the i ith alternative that is in the unit interval and ∑ i i d w i i = 1.

Each expert provides his/her personal choice for each alternative in the form of PHFE. Except for the alternative that has the missing value, PHFEs of all other alternatives are considered. They are converted to a single value using ∑ l ( γ i j l . p i j l ), which are then normalized to obtain the weights in the unit interval and ∑ l ( γ i j l . p i j l ) = 1.

Case 3: In a preference matrix, a particular entry ( i , j ) is available, and others are missing. It can be imputed by repeating the available entry row-wise if the entire row is missing and/or column-wise if the entire column is missing.

Case 4: In all t preference matrices, a particular column is missing and can be imputed in the following manner. First, the characteristic of the criterion (column) is clarified. Provided that it is a non-cost type, then the mean of other non-cost type criteria (columns) in that row is calculated. If the entire row is missing, it is a special case of Case 2, and hence, it can be imputed by using equation (9).

This section offers a new probability calculation approach that utilizes the available information for formulating a mathematical model that could be solved to obtain the occurrence probability for each element. Previous studies on PHFS have shown that occurrence probabilities are directly given by the experts, causing difficulties and overheads. The study by Zhou and Xu (2017c) provides a method for the probability calculation in preference relations, which motivated the authors to develop an approach for the occurrence probability calculation in decision matrices.

Occurrence probability, as discussed earlier, is seen as a confidence value and determination of the probability supports the usage of PHFS, unlike the direct assignment of probability value by an expert during rating. Elicitation of elements is comfortable, while assigning a confidence value is ordeal for experts and hence, we propose a procedure to calculate. A mathematical model is developed with the help of HFEs and can be solved via the optimization toolbox of MATLAB^® to extract occurrence probability values for each element.

Model 1: Min Z = ∑ i = 1 m ∑ j = 1 n p i j k ( | γ i j k − γ i j k + | − | γ i j k − γ i j k − | ) . Subject to: p i j k ∈ [ 0 , 1 ] , γ i j k ∈ [ 0 , 1 ] , γ i j k + ∈ [ 0 , 1 ] , γ i j k − ∈ [ 0 , 1 ] , ∑ k p i j k ⩽ 1 , ∀ k = 1 , 2 , … , h .

In Model 1, γ i j k + = max j ∈ benefit ( γ i j k ) or min j ∈ cost ( γ i j k ) and γ i j k − = max j ∈ cost ( γ i j k ) or min j ∈ benefit ( γ i j k ). When Model 1 is solved using an optimization toolbox, the occurrence probability for each HFE is determined, and the PHFS property is held (as per Definition 2). We formulate the optimization problem using distance from ideal and anti-ideal solutions. It is a minimization problem where distance of datapoint to ideal solution must be minimum and distance of datapoint to anti-ideal solution must be maximum. Some typical advantages of the proposed mathematical model are: (i) it is easy and straightforward; and (ii) takes the nature of criteria into account for determining the occurrence probability.

This sub-section focuses on a new framework for deciding the weights of experts. As mentioned earlier, Koksalmis and Kabak (2019) pointed out the need for a methodical computation of decision makers’ weights to mitigate inaccuracies. Driven by such claim, in this section, a regret/rejoice factor-based weight calculation approach is introduced. The steps are depicted below.

Step 1: t decision matrices are obtained with order m × n, where m demonstrates the number of alternatives, and n demonstrates the number of factors. PHFS information is adopted as the input.

Step 2: Transform the t matrices into single-valued matrices by applying equation (10): (10) s h i j = ∑ l = 1 # h ( γ i j l . p i j l ) , where # h represents no of instances in an IVPHFE.

Step 3: Calculate the utility value from the regret theory formulation using equation (11), which is adapted from Gong et al. (2019): (11) UT l = ∑ i = 1 m ∑ j = 1 n ( v f ( s h i j ) + RT ( v f ( s h i j ) ) − v f p o s ( s h i j ) ) , where v f ( . ) is the von Neuman Morgestern utility function with a power operation denoted as ( . ) a , and a ranges from 0 to 1. RT ( τ ) is the regret theory function that is given by 1 − e − ( ζ τ ) , where ζ ⩾ 0 is the risk aversion factor. v f p o s ( s h i j ) is maximum von Neuman utility for benefit criteria/attribute type and v f p o s ( s h i j ) is minimum von Neuman utility for cost criteria/attribute type. From Eq. (11), the regret/rejoice of selection over no selection is considered.

Step 4: Standardize the utility values from previous step to obtain the weight values of the experts, which subsequently forms a 1 × t vector. Equation (12) is employed to get the weight vector: (12) d w l = UT l ∑ l UT l , where d w l is the weight of the lth expert.

Herein, we introduce a novel framework for the criteria weight computation by presenting an attitude-based entropy measure. Generally, weights of criteria are decided either with partially known information or fully unknown information. If the weights of criteria are determined with partially known, decision-makers share their opinions of each criterion as inequality constraints, and in the latter context, such information is not existing. It should be noted that the former context adds an overhead to the expert and may not be possible in several practical situations.

To mitigate this issue, the latter context was further developed with popular approaches, such as AHP (Peng and Liu, 2017), BWM (Mi and Liao, 2019), and information/divergence measures (Mishra et al., 2020) for ratio analysis, among others. However, these methods do not consider the attitude of experts and are unable to capture the hesitation during preference elucidation. Driven by these challenges, herein, a new attitude-based entropy measure is introduced for factors weight determination under PHFS context. Steps for weight determination are given below:

Step 1: Form an expert opinion matrix ( t × n) with PHFEs as preference information for each criterion, where t represents the number of experts, and n represents the number of factors.

Step 2: Equation (7) must be applied to all PHFEs of the matrix to obtain a weighted opinion of experts. Attitude values of decision-makers that are gathered as weights from the previous sub-section are used as the scalar value, and a weighted matrix is obtained based on equation (7).

Step 3: Apply a deviation measure to all elements of the matrix (from Step 2) using equation (13): (13) D l j = | s h l j − s h j ‾ | , where s h l j = ∑ k = 1 # h ( γ l j k . p l j k ); and s h j ‾ is the mean for the jth factor.

Step 4: The information entropy measure is applied to form a vector of entropy values that is of order 1 × n. Equation (14) is applied for determining the entropy values: (14) E Y j = ∑ l ( − 1 n ( D l j ∑ l D l j ln ( D l j ∑ l D l j ) ) ) , where D l j ∑ l D l j is the normalized deviation, E Y j is the entropy value of the jth criterion or factor.

Step 5: Equation (15) is utilized to normalize the entropy values to form a weight vector of order 1 × n, which provides the weights of criteria: (15) c w j = E Y j ∑ j E Y j , where c w j is the weight of the jth factor.

It should be mentioned that the weight of each criterion is in the unit interval.

This section focuses on presenting a new aggregation operator under PHFS context for preference aggregation. Existing aggregation operators under PHFS (Jiang and Ma, 2018), Li and Wang (2018a) do not capture interdependencies among experts effectively, and to circumvent this issue, the Maclaurin operator is extended to PHFS for aggregation.

The Maclaurin symmetric mean (MSM) (Maclaurin, 1729) operator is a generalized operator that is capable of representing other arithmetic/geometric operators by parameter adjustments. The operator has the ability to capture interdependencies among decision-makers by adopting weight values and risk appetite values of experts during the formulation. Bearing in mind the literature survey above, it is clear that experts’ weights are not calculated systematically, which inspired inaccuracies, as argued by Koksalmis and Kabak (2019).

To overcome the issue, in the present sub-section, a weight MSM operator is introduced to aggregate PHFEs by acquiring weights methodically from Section 3.3. We define the operator below and discuss key properties. Definition 4.

PHFEs are aggregated through the PH-WMSM operator ( β n → β) and is given by, (16) PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h 1 , h 2 , … , h t ) = ( ( 1 − ( ∏ l = 1 t ( 1 − ∏ l l = 1 v γ i j λ l l ) d w l ) ) ) 1 ∑ l l λ l l , ( ( 1 − ( ∏ l = 1 t ( 1 − ∏ l l = 1 v p i j λ l l ) d w l ) ) ) 1 ∑ l l λ l l , where v = ⌈ t 2 ⌉, λ 1 , λ 2 , … , λ v are risk appetite elements that could receive possible values from the set { 1 , 2 , … , t }; and d w l is the weight of the lth expert.

h i for all i = 1 , 2 , … , t be a set of PHFEs. If h i = h for all i = 1 , 2 , … , t, then PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h 1 , h 2 , … , h t ) = h. Proof.

PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h 1 , h 2 , … , h t ) = ( ( 1 − ( ∏ l = 1 t ( 1 − ∏ l l = 1 v γ i j λ l l ) d w l ) ) ) 1 ∑ l l λ l l , ( ( 1 − ( ∏ l = 1 t ( 1 − ∏ l l = 1 v p i j λ l l ) d w l ) ) ) 1 ∑ l l λ l l .

By expanding the terms: PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h 1 , h 2 , … , h t ) = ( ( 1 − ( ( 1 − ∏ l l = 1 v γ i j λ l l ) d w 1 + d w 2 + ⋯ + d w t ) ) ) 1 ∑ l l λ l l , ( ( 1 − ( ( 1 − ∏ l l = 1 v p i j λ l l ) d w 1 + d w 2 + ⋯ + d w t ) ) ) 1 ∑ l l λ l l . Since the sum of experts’ weights equal unity, we get: = ( ( 1 − ( ( 1 − ∏ l l = 1 v γ i j λ l l ) ) ) ) 1 ∑ l l λ l l , ( ( 1 − ( ( 1 − ∏ l l = 1 v p i j λ l l ) ) ) ) 1 ∑ l l λ l l = ( γ i j | p i j ) = h .  □

For all λ 1 , λ 2 , … , λ v ; h − ⩽ PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h 1 , h 2 , … , h t ) ⩽ h + , where h − = min ( ∑ k γ i j k . p i j k ); and h + = max ( ∑ k γ i j k . p i j k ) for all i = 1 , 2 , … , t. Proof.

Suppose that h be the aggregated PHFE. Based on monotonicity and idempotent properties. Then, PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h − , h − , … , h − ) ⩽ h and PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h + , h + , … , h + ) ⩾ h . Through combining the inequalities, one gets PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h − , h − , … , h − ) ⩽ h ⩽ PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h + , h + , … , h + ) . Thus, h − ⩽ PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h 1 , h 2 , … , h t ) ⩽ h + .  □

For any permutation h i ∗ ∗ , PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h 1 , h 2 , … , h t ) = PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h 1 ∗ ∗ , h 2 ∗ ∗ , … , h t ∗ ∗ ) . Proof.

PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h 1 ∗ ∗ , h 2 ∗ ∗ , … , h v ∗ ∗ ) = ( ( 1 − ( ∏ l = 1 t ( 1 − ∏ l l = 1 v γ i j ∗ ∗ λ l l ) d w l ) ) ) 1 ∑ l l λ l l , ( ( 1 − ( ∏ l = 1 t ( 1 − ∏ l l = 1 v p i j ∗ ∗ λ l l ) d w l ) ) ) 1 ∑ l l λ l l = ( ( 1 − ( ∏ l = 1 t ( 1 − ∏ l l = 1 v γ i j λ l l ) d w l ) ) ) 1 ∑ l l λ l l , ( ( 1 − ( ∏ l = 1 t ( 1 − ∏ l l = 1 v p i j λ l l ) d w l ) ) ) 1 ∑ l l λ l l = PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h 1 , h 2 , … , h t ) .

□

If there exist a set of PHFEs h i ∗ for all i = 1 , 2 , … , t, such that h i ⩽ h i ∗ for all i = 1 , 2 , … , e, then PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h 1 , h 2 , … , h t ) ⩽ PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h 1 ∗ , h 2 ∗ , … , h t ∗ ) . Proof.

Let γ i j ∗ = ( ( 1 − ( ∏ l = 1 t ( 1 − ∏ l l = 1 v γ i j ∗ λ l l ) d w l ) ) ) 1 ∑ l l λ l l and p i j ∗ = ( ( 1 − ( ∏ l = 1 t ( 1 − ∏ l l = 1 v p i j ∗ λ l l ) d w l ) ) ) 1 ∑ l l λ l l . Also, PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h 1 , h 2 , … , h t ) = h and PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h 1 ∗ , h 2 ∗ , … , h t ∗ ) = h ∗ . Based on the score and deviation functions adapted from Xu and Zhou (2016), we can infer that s ( h i ) ⩽ s ( h i ∗ ), and if s ( h i ) = s ( h i ∗ ), then d ( h i ) ⩾ d ( h i ∗ ) for all i = 1 , 2 , … , e as h i ⩽ h i ∗ . Now, s ( h ) ⩽ s ( h ∗ ) and when s ( h ) = s ( h ∗ ), d ( h ) ⩾ d ( h ∗ ). Thereby, h ⩽ h ∗ , and so PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h 1 , h 2 , … , h t ) ⩽ PH-WMSM ( v , λ 1 , λ 2 , … , λ v ) ( h 1 ∗ , h 2 ∗ , … , h t ∗ ).  □

This sub-section introduces a fresh extension to the EDAS technique under the PHFS environment for rational ranking of alternatives. EDAS (Keshavarz Ghorabaee et al., 2015) is an attractive ranking method that follows the distance measure between possible choices and average values. Because of the simplicity and flexibility of the technique, numerous researchers adopt EDAS for MCDM. For instance, Keshavarz Ghorabaee et al. (2016) gave a fuzzy extension to EDAS for supplier selection. Kahraman et al. (2017) extended EDAS to IFS and applied the disposal technique selection method for solid wastes. Feng et al. (2018) proposed a new model by extending EDAS to hesitant fuzzy linguistic information and focused on the project selection problem. Peng and Liu (2017) developed a new model via a neutrosophic soft EDAS methodology using a novel similarity measure to select an optimal investment for software projects. Ecer (2018) developed an integrated framework with fuzzy AHP and EDAS for third party logistic provider selection. Karaşan and Kahraman (2018) made a suitable selection of sustainable goals for the United Nations by proposing the EDAS approach with interval-valued neutrosophic fuzzy information. Mi and Liao (2019) proposed an integrated approach under HFS by using BWM and EDAS methods for choosing insurance products in commercial sectors. Zhang et al. (2019) developed a framework with picture 2-tuple linguistic information by extending the geometric operator and EDAS to evaluate green suppliers. Li et al. (2019b) suggested a decision model with a power operator and EDAS for linguistic neutrosophic information using the property management company selection model. Recently, Mishra et al. (2020) employed the IFS-based EDAS approach for HCWT assessment. Liang et al. (2020) also presented an IFS-based EDAS for the proper selection of energy-efficient projects for green building construction. Aldalou and Perçin (2020) prepared a fuzzy decision model by integrating an entropy measure and EDAS for assessing financial indicators of food and drink firms in Istanbul. Yanmaz et al. (2020) introduced a new approach with interval-valued Pythagorean EDAS for car selection. Further, Ecer et al. performed the intuitionistic fuzzy EDAS model for evaluating cryptocurrencies, Lei et al. (2022) introduced the PDHL-EDAS approach, Mishra et al. (2022) developed a Fermatean fuzzy EDAS methodology, Batool et al. (2022) offered a Pythagorean probabilistic hesitant fuzzy EDAS approach, Hashemkhani Zolfani et al. (2021) utilized the EDAS method for international market selection, and Menekse and Camgoz Akdag (2023) introduced interval-valued spherical fuzzy EDAS. Torkayesh et al. (2023) prepared a detailed review on EDAS approach and discussed the applicability and possible extesions in the decision-making field. Motivated by the flexibility of EDAS, in this sub-section, a new extension of EDAS with PHFEs and the steps for ranking are presented as follows.

Step 1: Aggregated matrix of order m × n is gathered from the previous sub-section as input for the PHFS-based EDAS method. Also, the criteria weight vector of order 1 × n obtained from the previous sub-section is used as the input.

Step 2: Determine the weighted aggregated matrix by using equation (17) that is of order m × n: (17) h w i j = ( 1 − ( 1 − γ i j k ) c w j | 1 − ( 1 − p i j k ) c w j ) , where c w j is the weight of the jth criterion obtained from the previous section.

Step 3: Determine the average value of preferences for each alternative over different criteria using equation (18): (18) h w i ‾ = ( ∑ j = 1 n γ i j k ( a ) n | ∑ j = 1 n p i j k ( a ) n ) , where h w i ‾ is the average PHFE for the ith alternative.

It must be noted that h w i j is a PHFE in the form ( γ i j k ( a ) | p i j k ( a ) ).

Step 4: Determine the positive and negative distances from the average ( PDA , NDA ) for each alternative that yields a vector of order 1 × m: (19) PDA i = d ( h w i j , h w i ‾ ) , (20) NDA i = d ( h w i ‾ , h w i j ) , where d ( a 1 , a 2 ) is the distance between two PHFEs, a 1 and a 2.

Although equations (19) and (20) seem similar, they vary in terms of the nature of criteria. That is, in equation (19), PDA i is determined by taking the complement of the preferences in the cost type, and in equation (20), N D A i is calculated by taking the complement of the preferences in the benefit type. By this way, the ranking method effectively considers the nature of criteria in their formulation. Both equations (19) and (20) form a vector of order 1 × m that is used in the next step for ranking alternatives. Equation (5) presents the complement operation. d ( a 1 , a 2 ) = ∑ k = 1 # h ( ( γ i j k . p i j k ) a 1 − ( γ i j k . p i j k ) a 2 ) 2 . Here, a 1 and a 2 are any two PHFEs.

Step 5: Estimate the rank values of each alternative with Eq. (21), in which values from Step 4 are utilized to form the rank values: (21) R V i = ( PDA i − min i ( PDA i ) max i ( PDA i ) − min i ( PDA i ) ) + ( NDA i − min i ( NDA i ) max i ( NDA i ) − min i ( NDA i ) ) , where R V i is the rank value of the ith alternative; min i ( ∗ ) is the minimum operator; and max i ( ∗ ) is the maximum operator.

Arrange the rank values in descending order to find the ranking/prioritization order of the alternatives.

Before presenting the case study and to clearly recognise the practicality of the introduced methodology, it is essential to explain the working mechanism of the proposed decision model with PHFS information. Fig. 1 provides the working model that begins with collecting HFEs from experts for each alternative over each criterion. The team of experts are decided by the top officials. These experts finalize the alternatives and criteria for the process of decision-making. The model begins with the imputation of HFEs, then occurrence probability values are calculated. Sections 3.1 and 3.2 describe achieving the task. Weights of experts and criteria are determined methodically in order to mitigate human bias and inaccuracies in decision-making. Sections 3.3 and 3.4 are used for this purpose. Based on the decision matrices and experts’ weights, aggregation is performed to obtain a single aggregated matrix (refer to Section 3.5). Finally, the alternatives are ranked using the aggregated matrix and criteria weight vector. The detailed procedure for ranking is provided in Section 3.6.

The two stage model is desirable because extant decision models do not consider the preprocessing module in the decision process, which is addressed in this study. Imputing missing data and determination of probability is provided as preprocessing module that improves the input to the decision approach where weights of experts and criteria are determined along with data fusion and ranking. Many studies in decision-making considers data to be complete, which may not be practical in real life due to diverse reasons and as a result, a preprocessing procedure is required.