In medical science, disease diagnosis is one of the difficult tasks for medical experts who are confronted with challenges in dealing with a lot of uncertain medical information. And different medical experts might express their own thought about the medical knowledge base which slightly differs from other medical experts. Thus, to solve the problems of uncertain data analysis and group decision making in disease diagnoses, we propose a new rough set model called dual hesitant fuzzy multigranulation rough set over two universes by combining the dual hesitant fuzzy set and multigranulation rough set theories. In the framework of our study, both the definition and some basic properties of the proposed model are presented. Finally, we give a general approach which is applied to a decision making problem in disease diagnoses, and the effectiveness of the approach is demonstrated by a numerical example. In real-life disease diagnoses, due to the inherent uncertainty of human's expression of preferences, and the management, storage, and extraction of various useful information available to physicians which is not always presented as crisp numbers, it is believed that fuzzy numbers own many advantages for dealing with medical information systems. Moreover, in order to seek a diagnosis for the considered patients, it is essential for physicians to take into account a number of symptoms at the same time; this process might take a long time to reach a final conclusion. What is worse, the situation of overlooking a few trivial symptoms may trigger wrong disease diagnosis. To solve this complex decision making problem, lots of efforts have been made based on combining the uncertain decision making methods with the traditional disease diagnosis study. Fuzzy set theory , proposex can be expressed as {0.7,0.8}. Ever since the establishment of the hesitant fuzzy set theory, many researchers have studied the HFS from various facets and obtained an increasing number of achievements. In the extensions of hesitant fuzzy set, Zhu et al. introduced the concept of dual hesitant fuzzy set (DHFS) in 2012 , which can be expressed as the following mathematical symbol:hF(x) is a set of some different finite values in , describing the possible membership degrees of the element x ∈ U to the set F. For convenience, hF(x) is called a hesitant fuzzy element. The set of all hesitant fuzzy elements is called HFEs.Let Zhu et al.  further U be the universe of discourse; then a dual hesitant fuzzy set D on U is defined ash(x) and g(x) are two sets of some different finite values in , describing the possible membership degrees and nonmembership degrees of the element x ∈ U to the set D, respectively, with the conditions 0 ≤ γ, η ≤ 1, 0 ≤ γ+ + η+ ≤ 1, where γ ∈ h(x), η ∈ g(x), γ+ ∈ h+(x) = ∪γ∈h(x)max{γ}, and η+ ∈ g+(x) = ∪η∈g(x)max{η} for all x ∈ U. For convenience, the pair d(x) = (h(x), g(x)) is called a dual hesitant fuzzy element. The set of all dual hesitant fuzzy elements is called DHFEs.Let U is the universe of discourse; then the set of all dual hesitant fuzzy sets on U is denoted by DHF(U).Suppose that U = {x1, x2} be a universe set; then a dual hesitant fuzzy set defined by D = {〈x1, {0.6,0.7}, {0.2,0.3}〉, 〈x2, {0.3,0.4}, {0.5,0.6}〉} is a dual hesitant fuzzy set.Let (1)D  is referred to as an empty dual hesitant fuzzy set , denoting the possible membership degrees and nonmembership degrees for all  ∈ U × V, respectively. With the conditions 0 ≤ γ, η ≤ 1, 0 ≤ γ+ + η+ ≤ 1, where γ ∈ hR, η ∈ gR, γ+ ∈ hR+ = maxγ∈hR{γ}, and η+ ∈ gR+ = maxη∈gR{η} for all  ∈ U × V. Moreover, the family of all dual hesitant fuzzy relations over U × V is denoted by DHFR(U × V).Let U, V be two nonempty and finite universes of discourse and R ∈ DHFR(U × V); the pair  is called a dual hesitant fuzzy approximation space over two universes. For any A ∈ DHF(V), the lower and upper approximations of A with respect to , denoted by Let A with respect to , respectively. The pair A with respect to  and In what follows, based on the constructive approach to dual hesitant fuzzy rough set over two universes, we extend the dual hesitant fuzzy relation into the background of multigranulation rough set. Both the definitions and some basic properties of optimistic and pessimistic DHF multigranulation rough sets over two universes will be elaborated on.U, V be two nonempty and finite universes of discourse and Ri ∈ DHFR(U × V)   are m dual hesitant fuzzy relations over U × V; the pair  is called a dual hesitant fuzzy multigranulation approximation space over two universes. For any A ∈ DHF(V), the optimistic lower and upper approximations of A with respect to  are defined as follows:i=1m⋁y∈V{hRi∧gA(y)}, i=1m⋁y∈V{hRi∧hA(y)}, and i=1m⋀y∈V{gRi∨gA(y)}.Let A with respect to . If A optimistic-definable in ; otherwise, A is optimistic-undefinable in . It is noted that the optimistic DHF multigranulation rough set over two universes will reduce to a DHF rough set over two universes if m = 1.We call the pair U, V be two nonempty and finite universes of discourse and Ri ∈ DHFR(U × V)   are m dual hesitant fuzzy relations over U × V. For any A, A′ ∈ DHF(V), the optimistic DHF multigranulation rough set over two universes has the following properties:(1)(2)(3)(4)Let x ∈ U, we have x, ⋁i=1m⋀y∈V{gRi∨hA~(y)}, ⋀i=1m⋁y∈V{hRi∧gA~(y)}〉∣x ∈ U} =  {〈x, ⋁i=1m⋀y∈V{gRi∨gA(y)}, ⋀i=1m⋁y∈V{hRi∧hA(y)}〉∣x ∈ U} =  (1) For all A⊆A′, then by hAσ(k)(y) ≤ hA′σ(k)(y) and gAσ(k)(y) ≥ gA′σ(k)(y) for all y ∈ V. So it follows that {〈x, ⋁i=1m⋀y∈V{gRiσ(k)∨hAσ(k)(y)}, ⋀i=1m⋁y∈V{hRiσ(k)∧gAσ(k)(y)}〉∣x ∈ U}≤ {〈x, ⋁i=1m⋀y∈V{gRiσ(k)∨hA′σ(k)(y)}, ⋀i=1m⋁y∈V{hRiσ(k)∧gA′σ(k)(y)}〉∣x ∈ U}. Hence, for each x ∈ U, we have (2) Since x, ⋁i=1m⋀y∈V{gRi∨hA∩A′(y)}, ⋀i=1m⋁y∈V{hRi∧gA∩A′(y)}〉∣x ∈ U} =  {〈x, ⋁i=1m⋀y∈V{gRi∨(hA(y)∧hA′(y))}, ⋀i=1m⋁y∈V{hRi∧(gA(y)∧gA′(y))}〉∣x ∈ U} =  {〈x, ⋁i=1m⋀y∈V{∨hA(y))∧∧hA′(y))}, ⋀i=1m⋁y∈V{∧gA(y))∧∧gA′(y))}〉∣x ∈ U} =  {〈x, ⋁i=1m⋀y∈V{gRi∨hA(y)}, ⋀i=1m⋁y∈V{hRi∧gA(y)}〉∣x ∈ U}∧ {〈x, ⋁i=1m⋀y∈V{gRi∨hA′(y)}, ⋀i=1m⋁y∈V{hRi∧gA′(y)}〉∣x ∈ U} =  (3) Consider (4) From the discussions above, it is not difficult to prove that In the above theorem, (1) shows the complement of optimistic DHF multigranulation rough set over two universes; (2) shows the monotone of optimistic DHF multigranulation rough set over two universes with respect to the variety of dual hesitant fuzzy target; (3) and (4) show the multiplication and addition of optimistic DHF multigranulation rough set over two universes.U, V be two nonempty and finite universes of discourse and Ri, Ri′ ∈ DHFR(U × V)   are two dual hesitant fuzzy relations over U × V. If Ri⊆Ri′, for any A ∈ DHF(V), one has the following properties:(1)A ∈ DHF(V).(2)A ∈ DHF(V).Let Ri⊆Ri′, then by Definitions hRiσ(k) ≤ hRi′σ(k) and gRiσ(k) ≥ gRi′σ(k) for any  ∈ (U × V). So it follows that {〈x, ⋁i=1m⋀y∈V{gRiσ(k)∨hAσ(k)(y)}, ⋀i=1m⋁y∈V{hRiσ(k)∧gAσ(k)(y)}〉∣x ∈ U}≥ {〈x, ⋁i=1m⋀y∈V{gRi′σ(k)∨hAσ(k)(y)}, ⋀i=1m⋁y∈V{hRi′σ(k)∧gAσ(k)(y)}〉∣x ∈ U}. Hence, for each x ∈ U, we have Since U, V be two nonempty and finite universes of discourse and Ri ∈ DHFR(U × V)   are m dual hesitant fuzzy relations over U × V; the pair  is called a dual hesitant fuzzy multigranulation approximation space over two universes. For any A ∈ DHF(V), the pessimistic lower and upper approximations of A with respect to  are defined as follows:i=1m⋁y∈V{hRi∧gA(y)}, i=1m⋁y∈V{hRi∧hA(y)}, and i=1m⋀y∈V{gRi∨gA(y)}.Let A with respect to . If A pessimistic-definable in ; otherwise, A is pessimistic-undefinable in . It is also noted that the pessimistic DHF multigranulation rough set over two universes will reduce to a DHF rough set over two universes if m = 1.We call the pair U, V be two nonempty and finite universes of discourse and Ri ∈ DHFR(U × V)   are m dual hesitant fuzzy relations over U × V. For any A, A′ ∈ DHF(V), the pessimistic DHF multigranulation rough set over two universes has the following properties:(1)(2)(3)(4)Let In the above theorem, (1) shows the complement of pessimistic DHF multigranulation rough set over two universes; (2) shows the monotone of pessimistic DHF multigranulation rough set over two universes with respect to the variety of dual hesitant fuzzy target; (3) and (4) show the multiplication and addition of pessimistic DHF multigranulation rough set over two universes.U, V be two nonempty and finite universes of discourse and Ri, Ri′ ∈ DHFR(U × V)   are two dual hesitant fuzzy relations over U × V. If Ri⊆Ri′, for any A ∈ DHF(V), one has the following properties:(1)A ∈ DHF(V).(2)A ∈ DHF(V).Let U, V be two nonempty and finite universes of discourse and Ri ∈ DHFR(U × V)   are m dual hesitant fuzzy relations over U × V. For any A ∈ DHF(V), the DHF multigranulation rough set over two universes has the following properties:(1)(2)Let x ∈ U, {〈x, ⋁i=1m⋀y∈V{gRiσ(k)∨hAσ(k)(y)}, ⋀i=1m⋁y∈V{hRiσ(k)∧gAσ(k)(y)}〉∣x ∈ U}≥ {〈x, ⋀i=1m⋀y∈V{gRiσ(k)∨hAσ(k)(y)}, ⋁i=1m⋁y∈V{hRiσ(k)∧gAσ(k)(y)}〉∣x ∈ U}. Hence, we have For any From In this section, we introduce a new approach to the decision making problem in medical diagnoses by utilizing the proposed model based on DHF multigranulation rough set over two universes. The main points of our model and decision making methods can be summarized as the following steps.U = {x1, x2,…, xj} is a set of diagnoses and V = {y1, y2,…, yk} is a set of symptoms. Let Ri ∈ DHFR(U × V)   be m dual hesitant fuzzy relations over U × V, which reflects the medical knowledge base with dual hesitant fuzzy elements data given by m experts. We also let A ∈ DHF(V) be the set of symptoms characteristic for the considered patients. Then, we obtain a dual hesitant fuzzy decision information system  in medical diagnoses.Suppose that A with respect to , respectively. That is, we obtain the set d1(x) ⊕ d2(x) =  ∪γd1(x)∈hd1(x),ηd1(x)∈gd1(x),γd2(x)∈hd2(x),ηd2(x)∈gd2(x){{γd1(x) + γd2(x) − γd1(x)γd2(x)}, {ηd1(x)ηd2(x)}}, we further obtain the set of In the following, we present an approach to the decision making for the above-mentioned problem by using DHF multigranulation rough set over two universes. At first, according to Definitions ented in , d1(x) loped in , we presT1, T2, and T3 indicate the decision making index sets which are composed of the subscripts of the largest dual hesitant fuzzy element in corresponding dual hesitant fuzzy sets T1, T2, and T3 could be obtained. Moreover, based on the risk decision making principle of classical operational research, we could present the practical meaning for the above three index sets according to their definitions. Since the optimistic multigranulation rough set is based on “seeking common ground while reserving differences” (SCRD) strategy, which implies that one reserves both common decisions and inconsistent decisions at the same time, thus, this opinion can be seen as a risk-seeking decision making strategy. While the pessimistic multigranulation rough set is based on “seeking common ground while eliminating differences” (SCED) strategy, this strategy indicates that one reserves common decisions while deleting inconsistent decisions. Hence, this opinion can be seen as a risk-averse decision making strategy. According to the above different decision making strategies, xi (i ∈ T1) is the optimistic diagnostic result for the considered patient, xi (i ∈ T2) is the pessimistic diagnostic result for the considered patient, and xi (i ∈ T3) is the weighted diagnostic result for the considered patient, where T3 is the weighted decision making index set of T1 and T2 with the weighted value 0.5. Based on the above definition, the decision rules can be presented as follows:(1)T1∩T2∩T3 ≠ ∅, then xi (i ∈ T1∩T2∩T3) is the determined diagnosis for the patient.If (2)T1∩T2∩T3 = ∅ and T1∩T2 ≠ ∅, then xi (i ∈ T1∩T2) is the determined diagnosis for the patient. Otherwise, if T1∩T2∩T3 = ∅ and T1∩T2 = ∅, then xi (i ∈ T3) is the determined diagnosis for the patient.If It is noted that T1, T2, and T3 which come from optimistic and pessimistic information fusion strategies based on medical expert's risk preference, the proposed decision rules could be regarded as a multifaceted diagnostic scheme through considering multiple situations. Moreover, by utilizing the multifaceted diagnostic scheme, medical experts could obtain more reasonable and accurate diagnostic results than other approaches. Hence, the decision rules provide medical experts with a more flexible access to determine the diagnostic results for the patients.In light of the above decision rules in medical diagnoses, by virtue of the decision making index sets In what follows, we present an algorithm for the medical diagnoses model based on DHF multigranulation rough set over two universes as follows.Require. The relation between the universes U and V is provided by an expert  and a set of symptoms characteristic for the considered patients A. Ensure. The determined diagnosis for the patient:(1)calculate (2)calculate (3)determine the score function values for the sets (4)T1, T2, T3, T1∩T2∩T3, and T1∩T2, and confirm the determined diagnosis for the patient.compute In this section, to illustrate the efficiency of the proposed algorithm, we use a medical diagnosis problem with DHFS information which was previously studied and modeled by Farhadinia . FarhadiU = {x1, x2, x3, x4, x5} be a set of diagnoses, where xi stands for viral fever, malaria, typhoid, stomach problem, and chest problem, respectively. A patient with the given values of symptoms is denoted by V = {y1, y2, y3, y4, y5}, where yi stands for temperature, headache, cough, stomach pain, and chest pain, respectively. The medical knowledge base with DHFS data is presented in Tables Let A with all the symptoms, which is represented by the following dual hesitant fuzzy set information:In medical diagnoses, assume that we take a sample from a patient A with respect to , respectively:Following the steps of Then, we further obtain x2 > x4 > x1 > x3 > x5. Therefore, it is not difficult to obtain T1∩T2∩T3 = {2} ≠ ∅, which means x2 is the determined diagnosis for the patient. From the arguments of the above results, we can find that the considered patient is suffering from malaria.In what follows, according to dj   be a collection of DHFEs and we let w = T be the weight vector of dj with the equal weight. Then we have the following aggregation operators:The dual hesitant fuzzy averaging (DHFA) operator:The dual hesitant fuzzy geometric (DHFG) operator:In the following, in order to validate the effectiveness of the proposed model based on DHF multigranulation rough set over two universes, a comparison analysis is conducted by utilizing the most commonly used aggregation operators for dual hesitant fuzzy information. As presented in , we let R1, R2, and R3 in Tables R for DHFA operator and DHFG operator, respectively. Then, within the background of dual hesitant fuzzy rough set over two universes introduced in x2 > x4 > x1 > x3 > x5, which is consistent with the ranking results of DHF multigranulation rough sets over two universes. Thus, the diagnostic result also shows the considered patient is suffering from malaria. In light of the above comparison analysis, though the diagnostic outcomes for the two types of information fusion strategies are indistinguishable. It is noted that the information fusion strategies for DHFA and DHFG operators are onefold. By utilizing the optimistic and pessimistic DHF multigranulation rough sets over two universes, the proposed decision rules provide a multifaceted diagnostic scheme for medical experts, which enable them to obtain more reasonable and accurate diagnostic results than DHFA and DHFG operators.Through utilizing the above two aggregation operators, we can aggregate the DHF relation From the above analysis, the DHF multigranulation rough set over two-universe model takes full advantage of dual hesitant fuzzy set and multigranulation rough set in medical diagnoses. On one hand, compared with other generalizations of fuzzy sets, the dual hesitant fuzzy set takes into account much more information given by medical experts. That is, the nonmembership hesitancy function enables medical experts to express his or her opinions from the viewpoint of whether a patient is not suffering from a certain disease, and the hesitant information enables medical experts to hesitate among several numerical numbers when evaluating whether a patient is suffering from a certain disease or not. Thus, the dual hesitant fuzzy set provides medical experts with a more exemplary and flexible access to convey their understandings about the medical knowledge base. On the other hand, the method of multigranulation rough set is an ideal information fusion strategy which could synthesize each medical expert's view to form a final conclusion by providing optimistic and pessimistic information fusion strategies. In light of the above, the superiorities of DHF multigranulation rough set over two-universe model could decline the uncertainty to a great extent and enhance the accuracy and reliability of medical diagnoses effectively.In this paper, we have proposed a new rough set model through combining multigranulation rough set and the dual hesitant fuzzy set, called a DHF multigranulation rough set over two-universe model. In this framework, the definition and some properties of optimistic and pessimistic DHF multigranulation rough sets over two universes have been studied. Finally, we have established a general approach to the decision making problem in medical diagnoses. The outcomes of the example show that the approach proposed in this paper could deal with group decision making problems effectively. Furthermore, comparing to those theoretical results in the existing literature, the main contribution of the proposed decision making model consists in taking into account three decision making index sets based on optimistic and pessimistic information fusion strategies. By virtue of the decision making index sets, the proposed decision making model provides a multifaceted diagnostic scheme for medical experts. And with the aid of multifaceted diagnostic scheme, it is convenient for medical experts to obtain more reasonable and accurate diagnostic outcomes than other methods.This study develops a framework of DHF multigranulation rough set over two universes, in which there are still many interesting issues to be explored. In the future, we can discuss various uncertainty measures and attribute reduction approaches. It is also desirable to further apply our proposed model to other practical applications.