Preference of an alternative over another alternative is a useful way to express the opinion of decision maker. In the process of group decision making, preference relations are used in preference modelling of the alternatives under given criteria. The probability is an important tool to deal with uncertainty; in many scenarios of decision making probabilities of different events affect the decision making process directly. In order to deal with this issue, in this paper, hesitant probabilistic fuzzy preference relation (HPFPR) is defined. Furthermore, consistency of HPFPR and consensus among decision makers are studied in the hesitant probabilistic fuzzy environment. In this respect, many novel algorithms are developed to achieve consistency of HPFPRs and reasonable consensus between decision makers and a final algorithm is proposed comprehending all other algorithms, presenting a complete decision support model for group decision making. Lastly, we present a case study with complete illustration of the proposed model and discussed the effects of probabilities on decision making validating the importance of the introduction of probability in hesitant fuzzy preference relation.

1. Introduction

Fuzzy set theory was initially introduced by Zadeh [1] in 1965 as an extension of the classical set theory. In classical set theory, an element either belongs to or does not belong to the set. In fuzzy set theory, the gradual assessment of elements of set is described by the membership function that is in . Fuzzy set theory can be used in which information is vague, incomplete, or imprecise and it is successfully used in decision making problems [2]. After the popularity of this extension in set theory, several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, interval-valued fuzzy sets, intuitionistic fuzzy sets, interval-valued intuitionistic fuzzy sets, trapezoidal-valued intuitionistic fuzzy sets, type-2 fuzzy sets, and fuzzy multisets. These extensions have been successfully used in several practical applications of real life problems and scientific problems. Applications of these extensions can be found in artificial intelligence, computer science, medicine, control engineering, decision theory, expert systems, logic, management science, operations research, pattern recognition, and robotics. Torra felt that there are some limitations or deficiencies in these extensions. He proposed another extension of fuzzy set theory which is named as hesitant fuzzy set (HFS) theory [3]. This extension permits the several possible membership degrees of an element in . HFS provides a much better description than the other extensions of fuzzy sets where the difficulty of establishing the membership degree and there is a specific set of possible values. Many studies on HFS have been conducted, such as extensions of HFS (see [49]).

Group decision making (GDM) is a procedure to find the best/optimal alternative from a set of alternatives under the basis of certain criteria and the alternatives are evaluated by the group of decision makers for the criteria [1014]. The opinion of DMs may also be in the form of preference of each pair of alternatives and provides a comparison of one alternative over another and this comparison is a preference relation; for some basics of preference relation see [15]. Preference relations have been developed and investigated in different modes, like multiplicative preference relations [16, 17], fuzzy preference relations [18], multiplicative fuzzy preference relation [10], incomplete fuzzy preference relation [11, 19], linguistic preference relations [20, 21], intuitionistic fuzzy preference relations [22, 23], and interval-valued hesitant preference relations [6]. Due to several external nonfeasible circumstances, like shortage of time, lack of knowledge, and available data resources, DMs provided their preferences opinion over the alternatives in the form of several possible numerical values. All the discussed preference relations do not handle such kind of situations. To overcome this problem, Zhu and Xu [24] introduced hesitant fuzzy preference relation (HFPR). Here the preference of an alternative over another alternative is a HFE; this HFE shows all possible preference values that denote the hesitant degree between two alternatives. HFPRs provide a better framework for the description of the DMs’ hesitation while providing their preferences among the alternatives [2528].

In preference relations based decision making problems, the concept of consistency plays an important role. It is the level or degree of satisfaction among the values in preference relation and these values are given by each DM [29]. Consensus is also an important and valuable concept in decision making problems based on preference relations. Consensus measure is used for the mutual understanding of DMs on the finally obtained alternatives [29]. Until now, several researchers of this area successfully made some progress to convert the preference relations as consistent and generate a certain level of consensus in decision making problems [3033]. Nowadays, hesitant probabilistic fuzzy sets received good attention in multicriteria decision making. So, it is important to discuss the consistency and consensus measure for these preference relations. In this paper, we target to develop a group decision making model for HPFPRs where the consistency of the model and consensus among the DMs are under consideration. Our proposed model is efficient and practical; furthermore, it is strictly based on theoretical foundations. Pang et al. introduce the idea of probabilistic linguistic term set [34]. They also develop the aggregation operator of this set and proposed the extended TOPSIS version to handle the multicriteria group decision making problems. By getting the motivation from the hesitant fuzzy set and probabilistic linguistic term set, Xu and Zhou proposed the concept of hesitant probabilistic fuzzy set [35]. They investigated several aggregation operators with properties for hesitant probabilistic fuzzy set. A novel algorithm was developed to handle the multicriteria group decision making problems for hesitant probabilistic fuzzy set [36]. This concept was further extended to the continuous form of hesitant probabilistic fuzzy set [37]. Distance measures were also discussed for the continuous form and it was also applied to automotive industry safety evaluation problem. In these no one discusses the preference relations of the hesitant probabilistic fuzzy set. It is worth defining the preference relations for this set and discussing the consistency and consensus measure of the decision model.

To accomplish these goals, this paper is structured in the following way. In Section 2, some preliminary concepts are discussed to understand our proposal. Section 3 is devoted to design the basic structure of hesitant probabilistic fuzzy preference relations. A distance measured between the preference relations is developed, based on it, consistency measure of preference relations is derived. Also, two novel algorithms are designed to achieve acceptable consistency. In Section 4, consensus measure is defined for the group decision making and a novel algorithm is presented for reaching acceptable consensus among decision makers. Section 5 is dedicated to presenting a complete group decision making model dealing with both issues of consistency and consensus. In Section 6, numerical analysis of the developed model through a case study is performed to understand the importance of our proposal. Section 7 is dedicated for comparison between proposed model and existing ones. Section 8 ends the paper with some concluding remarks.

2. Preliminaries

Definition 1 (see [39]). For , a fixed set, a preference relation is a subset of , which satisfies the following two:(1)(Completeness) for all and for all , either or ; that is, or .(2)(Transitivity) for all , for all , and for all if and , then ; that is, and

Definition 2 (see [1, 40]). For , a fixed set, a HFPR is expressed by a matrix , where is a HFE, giving all the possible preference degrees of the alternative over . Also satisfy the following conditions for all : where is the smallest value in and also elements of are arranged in increasing order.

The notion of the hesitant fuzzy set given by Torra [3] is well known and has been successfully used to model vagueness of real life. It allows decision makers to give multiple membership values, but has the deficiency to deal with probabilities of preference degrees. To make hesitant fuzzy sets more compatible with real life, Xu and Zhou [35] defined hesitant probabilistic fuzzy element (HPFE) and hesitant probabilistic fuzzy set (HPFS).

Definition 3 (see [35]). Consider a fixed set The HPFS on is defined as a mathematical symbol:where is HPFE comprising the elements of the form , expressing the hesitant fuzzy information with probabilities to the set , where is the number of elements in , is the respective hesitant probability for , and

Score function, deviation function, and comparison laws are given to compare different HPFEs.

Definition 4. For a HPFE where , is called the score function of , where is the number of possible elements in

Definition 5. For a HPFE where , is called the deviation function of , where is the score function of and is the number of possible elements in

The score and deviation functions are similar to the expectation and variance of the random variable, respectively, and, thus, the comparison laws for two HPFEs and can be presented as follows:If , then ,If and , then ,If and , then ,If and , then

3. Hesitant Probabilistic Fuzzy Preference Relation

In order to build a complete model for group decision making first some operations are defined for HPFEs of the same length. Let , , and be HPFEs with Thenwhere and are elements of and , respectively.

For simplicity will be written as throughout this paper.

To allow decision makers to provide the preferences in hesitant probabilistic environment, we define hesitant probabilistic fuzzy preference relation (HPFPR).

Definition 6 (HPFPR). Let be the set of alternatives. The HPFPR is a matrix , where is the HPFE expressing the possible preference degrees of the alternative over with probabilities and with satisfying the following conditions:where and are the elements in and , respectively.

Remark 7. The above definition is very much alike to the definition of probabilistic hesitant fuzzy preference relation proposed by Zhou and Xu [41, Definition 4]. They take But if one adopted the technique of -normalization, that is, making the length of the same by adding elements to HPFEs of shorter length, it implicates numerous errors and difficulties when dealing with consistency of HPFPRs and consensus among decision makers based on -normalization. So fixing the length of diagonal elements to one does not match with the condition So by allowing variation, the probabilities of diagonal HPFEs helps us to maintain the spirit of HPFE in discussing consistency and consensus in the context of -normalization. The diversity in probabilities of diagonal preference degrees does not cause any harm; the net impact remains the same as the sum of all probabilities is .

Often the length of HPFEs has been different, but to apply the above defined operations (3), (4), and (7), the length is needed to be equal for all HPFEs. Some elements will be added to HPFE who has less elements, but the information it provides that should not be changed. Now, the definition of normalized hesitant probabilistic fuzzy preference relation (NHPFPR) is proposed.

Definition 8 (normalized HPFPR). A HPFPR is called NHPFPR if the length of all is the same for all .

Let be a HPFE. For preference degrees Zhu et al. [28] define a way to add elements in HFE; for an optimized parameter the preference degree that will be added to is , where is the largest and is the smallest among The decision maker can choose the value of according to his risk preferences. The added element will be and for and , respectively, which demonstrate the optimistic and pessimistic approach of decision maker proposed by Xu and Xia [42]. In hesitant probabilistic fuzzy environment, some way is needed to assign probability to the added preference degree such that the information of HPFPR is not changed. There are many ways to do it; one option is to assign to added preference degree , but for the extreme cases pessimistic approach, that is, , and optimistic approach, that is, , the added element in HPFE is and , respectively, where is the required length of HPFEs and and are the probabilities of and , respectively.

For a given HPFPR , we normalize it as follows. Let and . For optimized parameter ,For ,For , for where

Now is NHPFPR; next we deal with consistency.

Example 9. Let Then and are HPFPRs. Now by taking optimized parameters , for and , respectively, the following NHPFPR are obtained.

3.1. Consistency Measure of Hesitant Probabilistic Fuzzy Preference Relation

In order to obtain valuable decision from preference relations they should be consistent in a sense that, let us say, is preferable to and is preferable to then must be preferable to Several authors have pursued consistency issues for preference relations [1, 21, 32, 33, 43].

Additive consistency for fuzzy preference degrees is well known; actually Tanino [44] defined the additive consistency for fuzzy preference relation based on moderate stochastic transitivity, well known in the probabilistic choice theory [45, page 27]. Furthermore, many kinds of transitivity are proposed and studied for probabilities in comparing the preferences in the choice theory.

Considering HPFPR by (10), (11) calculate NHPFPR The weak stochastic transitivity for probability means

This will provide a platform to define consistency for HPFPR.

Definition 10 (consistency). For a given HPFPR, and its NHFPR with optimized parameter Iffor all , then is called consistent HPFPR with optimized parameter

But many times preference relations are not consistent and, for meaningful decision making, some level of consistency is required in the least if it is not fully consistent. For preference degrees, take the summation of (16) for all therefore,Thus, (19) is satisfied by a consistent HPFPR, if not putone can check that the preference degrees obtained from the above equation are consistent. For probabilities, matter is not that simple; some mechanism is needed to make consistent probabilities with all the restrictions of HPFPR like and Let and Then defineto keep account for all and keeping in mind we modify (21) asHence and and if and for all then surely . But, it is possible that and are not true for some that will lead to a situation where and . Now if another convex combination is calculated by (22) then obtained probability will increase. These observations lead to the following novel algorithm producing a sequence of HPFPRs convergent to fully consistent HPFPR.

Algorithm 11 (consistent HPFPR calculator).
Input. HPFPR and optimized parameter .
Output. NHPFPR , consistent HPFPR , and number of iterations
Step 1. Compute NHPFPR by (10) or (11). Let and be defined asStep 2. If the following condition is true, then go to Step 4; otherwise, go to Step 3. Step 3. is defined asput Go to Step 2.
Step 4. Output NHPFPR , consistent HPFPR , and number of iterations
Step 5. End.

Proposition 12. Let be a HPFPR with its NHFPR with optimized parameter Then output of Algorithm 11 is consistent HPFPR.

Proof. Definition 10, (23) and (24) directly imply this proposition.

This result also gives the following theorem.

Proposition 13. Consider a HPFPR , its NHFPR , and consistent HPFPR with optimized parameter Then, is consistent if and only if

The above algorithm is quite efficient; to see this fact, we generate 1000 random HPFPRs with different values of , and apply Algorithm 11 to find their consistent HPFPRs. Table 1 shows the average value of the number of iterations in Algorithm 11.

Example 14. Take and their optimized parameter the same as in Example 9. Then by Algorithm 11, the following consistent HPFPRs and are obtained.

Remark 15. To see the consistency of HPFPR geometrically, three area graphs of fuzzy preference degrees , probabilities of preference degrees , and score values are made. The procedure to make these graphs is explained for , as follows.
Three matrices , , and of order of fuzzy preference values, probability values, and score values are made from HPFPR , The area graphs are made of the above matrices by using Matlab drawing tool bar. Figures 1, 2, and 3 and Figures 4, 5, and 6 show the comparison of area graphs for fuzzy preference degrees, probability values, and score values between , and , , respectively. The areas are more smooth for consistent HPFPRs and

Once the consistent HPFPR is found, we are in a position to make HPFPR acceptably consistent by defining consistency measure. In order to define consistency measure, first distance between two HPFEs is defined. Let and with Then

Based on the above equation the distance between two HPFPRs is defined as follows.

Definition 16 (distance). Consider two HPFPRs and and their NHPFPRs and The distance is defined as

It is clear that the following properties are satisfied for :(1);(2);(3) if and only if

While the decision maker provides its preference in form of HPFPR, it should be noted that it can be used for decision making with good results only if it has sufficient consistency.

Definition 17 (consistency index). For a given HPFPR , its NHPFPR and consistent HPFPR with optimized parameter obtained from Algorithm 11. The consistency index of is defined to be the distance between and , denoted as

It is clear that if and only if is consistent. Ideally, the decision maker should provide consistent HPFPR so that it can be used for meaningful decision making. However, some margin of error should be provided to decision maker relative to the practical problems.

Definition 18 (acceptably consistent HPFPR). Consider a HPFPR For a given tolerance value HPFPR is said to be acceptably consistent HPFPR if

If a HPFPR is not even acceptably consistent, then the decision maker should revisit and modify it. A novel algorithm is proposed to make a HPFPR acceptably consistent.

Algorithm 19 (acceptably consistent HPFPR calculator).
Input. The HPFPR , the consistency tolerance value , and the parameter .
Output. The acceptably consistent HPFPR , consistency index